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arXiv:physics/0001001v1 [physics.flu-dyn] 1 Jan 2000Under consideration for publication in J. Fluid Mech. 1 Capillary-gravity wave transport over spatially random drift By GUILLAUME BAL∗and TOM CHOU† ∗Department of Mathematics, University of Chicago, Chicago , IL 60637 †Department of Mathematics, Stanford University, Stanford , CA 94305 (Received 2 February 2008) We derive transport equations for the propagation of water w ave action in the pres- ence of a static, spatially random surface drift. Using the W igner distribution W(x,k,t) to represent the envelope of the wave amplitude at position xcontained in waves with wavevector k, we describe surface wave transport over static flows consis ting of two length scales; one varying smoothly on the wavelength scale, the ot her varying on a scale com- parable to the wavelength. The spatially rapidly varying bu t weak surface flows augment the characteristic equations with scattering terms that ar e explicit functions of the cor- relations of the random surface currents. These scattering terms depend parametrically on the magnitudes and directions of the smoothly varying dri ft and are shown to give rise to a Doppler coupled scattering mechanism. The Doppler interaction in the presence of slowly varying drift modifies the scattering processes an d provides a mechanism for coupling long wavelengths with short wavelengths. Conserv ation of wave action (CWA), typically derived for slowly varying drift, is extended to s ystems with rapidly varying flow. At yet larger propagation distances, we derive from the transport equations, an equation for wave energy diffusion. The associated diffusion constant is also expressed in terms of the surface flow correlations. Our results provid e a formal set of equations to analyse transport of surface wave action, intensity, ene rgy, and wave scattering as a function of the slowly varying drifts and the correlation fu nctions of the random, highly oscillatory surface flows. 1. Introduction Water wave dynamics are altered by interactions with spatia lly varying surface flows. The surface flows modify the free surface boundary condition s that determine the dis- persion for propagating water waves. The effect of smoothly v arying (compared to the wavelength) currents have been analysed using ray theory (P eregrine (1976), Jonsson (1990)) and the principle of conservation of wave action (CW A) (cf. Longuet-Higgins & Stewart (1961), Mei (1979), White (1999), Whitham (1974) a nd references within). These studies have largely focussed on long ocean gravity wa ves propagating over even larger scale spatially varying drifts. Water waves can also scatter from regions of un- derlying vorticity regions smaller than the wavelength Fab rikant & Raevsky (1994) and Cerda & Lund (1993). Boundary conditions that vary on capill ary length scales, as well as wave interactions with structures comparable to or small er than the wavelength can also lead to wave scattering (Chou, Lucas & Stone (1995), Gou , Messiter & Schultz (1993)), attenuation (Chou & Nelson (1994), Lee et al. (1993)), and Bragg reflections (Chou (1998), Naciri & Mei (1988)). Nonetheless, water wave propagation over random2 Bal & Chou static underlying currents that vary on both large and small length scales, and their interactions, have received relatively less attention. In this paper, we will only consider static irrotational cur rents, but derive the transport equations for surface waves in the presence of underlying flo ws that vary on bothlong and short (on the order of the wavelength) length scales. Rat her than computing wave scattering from specific static flow configurations (Gerber ( 1993), Trulson & Mei (1993), Fabrikant & Raevsky (1994)), we take a statistical approach by considering ensemble averages over realisations of the static randomness. Differ ent statistical approaches have been applied to wave propagation over a random depth (Elter & Molyneux (1972)), third sound localization in superfluid Helium films (Kleinert (199 0)), and wave diffusion in the presence of turbulent flows (Howe (1973), Rayevskiy (1983), Fannjiang & Ryzhik (1999)). In the next section we derive the linearised capillary-grav ity wave equations to low- est order in the irrotational surface flow. The fluid mechanic al boundary conditions are reduced to two partial differential equations that couple th e surface height to velocity potential at the free surface. We treat only the “high freque ncy” limit (Ryzhik, Papani- colaou, & Keller (1996)) where wavelengths are much smaller than wave propagation dis- tances under consideration. In Section 3, we introduce the W igner distribution W(x,k,t) which represents the wave energy density and allows us to tre at surface currents that vary simultaneously on two separated length scales. The dyn amical equations developed in section 2 are then written in terms of an evolution equatio n forW. Upon expanding Win powers of wavelength/propagation distance, we obtain tr ansport equations. In Section 4, we present our main mathematical result, equat ion (4.1), an equation describing the transport of surface wave action. Appendix A gives details of some of the derivation. The transport equation includes advection by the slowly varying drift, plus scattering terms that are functions of the correlation s of the rapidly varying drift, representing water wave scattering. Upon simultaneously t reating both smoothly varying and rapidly varying flows using a two-scale expansion, we find that scattering from rapidly varying flows depends parametrically on the smoothly varyin g flows. In the Results and Discussion, we discuss the regimes of validity, consider sp ecific forms for the correlation functions, and detail the conditions for doppler coupling. CWA is extended to include rapidly varying drift provided that the correlations of the drift satisfy certain constraints. We also physically motivate the reason for considering two s cales for the underlying drift. In the limit of still larger propagation distances, a fter multiple wave scattering, wave propagation leaves the transport regime and becomes di ffusive (Sheng (1995)). A diffusion equation for water wave energy is also given, with a n outline of its derivation given in Appendix B. 2. Surface wave equations Assume an underlying flow V(x,z)≡(U1(x,z),U2(x,z),Uz(x,z))≡(U(x,z),Uz(x,z)), where the 1,2 components denote the two-dimensional in-pla ne directions. This static flow may be generated by external, time independent sources s uch as wind or inter- nal flows beneath the water surface. The surface deformation due to V(x,z) is denoted ¯η(x) where x≡(x,y) is the two-dimensional in-plane position vector. An addit ional variation in height due to the velocity v(x,z) associated with surface waves is denoted η(x,t). When all flows are irrotational, we can define their associa ted velocity poten- tialsV(x,z)≡(∇x+ˆz∂z)Φ(x,z) andv(x,z,t)≡(∇x+ˆz∂z)ϕ(x,z,t). Incompressibility requires ∆ϕ(x,z,t) +∂2 zϕ(x,z,t) = ∆Φ( x,z) +∂2 zΦ(x,z) = 0, (2.1)Water wave transport 3 +ε−y |y|~1/k x1x2 O(1)X -+y O(L)x x Figure 1. The relevant scales in water wave transport. Initially, the system size, observation point, and length scale of the slowly varying drift is O(L), with surface wave wavelength and scale of the random surface current of O(1). Upon rescaling, the system size becomes O(1), while the wavelength and random flow variations are O(ε). where ∆ = ∇2 xis the two-dimensional Laplacian. The kinematic condition (Whitham (1974)) applied at z= ¯η(x) +η(x,t)≡ζ(x,t) is ∂tη(x,t) +U(x,ζ)· ∇xζ(x,t) =Uz(x,z=ζ) +∂zϕ(x,z=ζ,t). (2.2) Upon expanding (2.2) to linear order in ηandϕabout the static free surface, the right hand side becomes Uz(x,ζ) +∂zϕ(x,ζ,t) =Uz(x,¯η) +η(x,t)∂zUz(x,¯η) +∂zϕ(x,¯η,t) +O(η2).(2.3) At the static surface ¯ η,U(x,¯η)·∇x¯η(x) =Uz(x,¯η). Now assume that the underlying flow is weak enough such that Uz(x,z≈0) and ¯ηare both small. A rigid surface approximation is appropriate for small Froude numbers U2/c2 φ∼ |∇x¯η|2∼Uz(x,0)/|U(x,0)| ≪1 (cφ is the surface wave phase velocity) when the free surface bou ndary conditions can be approximately evaluated at z= 0 (Fabrikant & Raevsky (1994)). Although we have assumedUz(x,z≈0) =∂zΦ(x,z≈0)≈0 and a vanishing static surface deformation ¯η(x)≈0,∇x·U(x,0) =−∂zUz(x,0)∝ne}ationslash= 0. Combining the above approximations with the dynamic bounda ry conditions (derived from balance of normal surface stresses at z= 0 (Whitham (1974))), we have the pair of coupled equations ∂tη(x,t) +∇x·(U(x,z= 0)η(x,t)) = lim z→0−∂zϕ(x,z,t) lim z→0−[ρ∂tϕ(x,z,t) +ρU(x,z)· ∇xϕ(x,z,t)] =σ∆η(x,t)−ρgη(x,t)(2.4) whereσandgare the air-water surface tension and gravitational accele ration, respec- tively. Although it is straightforward to expand to higher o rders in ¯η(x) andη(x,t), or to include underlying vorticity, we will limit our study to equ ations (2.4) in order to make the development of the transport equations more transparen t. The typical system size, or distance of wave propagation sho wn in Fig. 1 is of O(L) with4 Bal & Chou L≫1. Wavelengths however, are of O(1). To implement our high frequency (Ryzhik, Papanicolaou, & Keller (1996)) asymptotic analyses, we res cale the system such that all distances are measured in units of L≡ε−1. We eventually take the limit ε→0 as an approximation for small, finite ε. Surface velocities, potentials, and height displacement s are now functions of the new variables x→x/ε,z→z/εandt→t/ε. We shall further nondimensionalise all distances in terms of the capillary l engthℓc=/radicalbig σ/gρ. Time, velocity potentials, and velocities are dimensionalised i n units of/radicalbig ℓc/g,/radicalbig gℓ3c, and√gℓc respectively, e.g.for water,U= 1 corresponds to a surface drift velocity of ∼16.3cm/s. SinceUz(x,z≈0)≈0, we define the flow at the surface by U(x,z= 0)≡U(x) +√εδU(x/ε). (2.5) In these rescaled coordinates, U(x) denotes surface flows varying on length scales of O(1) much greater than a typical wavelength, while δU(x/ε) varies over lengths of O(ε) comparable to a typical wavelength. The amplitude of the slowly varying flow U(x) is O(ε0), while that of the rapidly varying flow δU(x/ε), is assumed to be of O(√ε). A more detailed discussion of the physical motivation for conside ring the√εscaling is deferred to the Results and Discussion. After rescaling, the boundar y conditions (2.4) evaluated atz= 0 become ∂tη(x,t) +∇x·/bracketleftbig/parenleftbig U(x) +√εδU(x/ε)/parenrightbig η(x)/bracketrightbig = lim z→0−∂zϕ(x,0) ∂tϕ(x,t) +U(x)· ∇xϕ(x,t) +√εδU(x/ε)· ∇xϕ(x,t) =ε∆η(x,t)−ε−1η(x,t). (2.6) Although drift that varies slowly along one wavelength can b e treated with characteristics and WKB theory, random flows varying on the wavelength scale r equire a statistical approach. Without loss of generality, we choose δUto have zero mean and an isotropic two-point correlation function ∝an}b∇acketle{tδUi(x)δUj(x′)∝an}b∇acket∇i}ht ≡Rij(|x−x′|), where (i,j) = (1,2) and ∝an}b∇acketle{t...∝an}b∇acket∇i}htdenotes an ensemble average over realisations of δU(x). We now define the spatial Fourier decompositions for the dyna mical wave variables ϕ(x,−h/lessorequalslantz/lessorequalslantζ,t) =/integraldisplay qϕ(q,t)e−iq·xcoshq(h+z) coshqh, η(x,t) =/integraldisplay qη(q,t)e−iq·x, (2.7) the static surface flows U(x) =/integraldisplay qU(q)e−iq·x, δU/parenleftigx ε/parenrightig =/integraldisplay qδU(q)e−iq·x/ε, (2.8) and the correlations Rij(x) =/integraldisplay qRij(q)e−iq·x, (2.9) where q= (q1,q2) is an in-plane two dimensional wavevector, q≡ |q|=/radicalbig q2 1+q2 2, and/integraltext q≡(2π)−2/integraltext dq1dq2. The Fourier integrals for ηexclude q= 0 due to the incompress- ibility constraint/integraltext xη(x,t) = 0, while the q= 0 mode for ϕgives an irrelevant constant shift to the velocity potential. Note that ϕin (2.7) manifestly satisfies (2.1). SubstitutingWater wave transport 5 (2.8) into the boundary conditions (2.4), we obtain, ∂tη(k,t)−i/integraldisplay qη(k−q)U(q)·k−i√ε/integraldisplay qη(k−q/ε)δU(q)·k=ϕ(k,t)ktanhεkh ∂tϕ(k,t)−i/integraldisplay qU(q)·(k−q)ϕ(k−q)−i√ε/integraldisplay qδU(q)·(k−q/ε)ϕ(k−q/ε) =−(εk2+ε−1)η(k). (2.10) where theδU(q) are correlated according to ∝an}b∇acketle{tδUi(p)δUj(q)∝an}b∇acket∇i}ht=Rij(|p|)δ(p+q). (2.11) Since the correlation Rij(x) is symmetric in i↔j, and depends only upon the magnitude |x|,Rij(|p−q|) is real. In the case where δU= 0 and U(x)≡U0is strictly uniform, equations (2.10) can be simplified by assuming a e−iωtdependence for all dynamical variables. Uniform drift yields the familiar capillary-gravity wave dispersion rel ation ω(k) =/radicalbig (k3+k)tanhkh+U0·k≡Ω(k) +U0·k. (2.12) However, for what follows, we wish to derive transport equat ions for surface waves (action, energy, intensity) in the presence of a spatially varying dr ift containing two length scales: U=U(x) +√εδU(x/ε). 3. The Wigner distribution and asymptotic analyses The intensity of the dynamical wave variables can be represe nted by the product of two Green functions evaluated at positions x±εy/2. The difference in their evaluation points, εy, resolves the waves of wavevector |k| ∼2π/(εy). Elter & Molyneux (1972) used this representation to study shallow water wave propagation ove r a random bottom. However, for the arbitrary depth surface wave problem, where the Gree n function is not simple, and where two length scales are treated, it is convenient to u se the Fourier representation of the Wigner distribution (Wigner (1932), G´ erard et al. (1 997), Ryzhik, Papanicolaou, & Keller (1996)). Defineψ= (ψ1,ψ2)≡(η(x),ϕ(x,z= 0)) and the Wigner distribution: Wij(x,k,t)≡(2π)−2/integraldisplay eik·yψi/parenleftig x−εy 2,t/parenrightig ψ∗ j/parenleftig x+εy 2,t/parenrightig dy (3.1) where xis a central field point from which we consider two neighbouri ng points x±εy 2, and their intervening wave field. Fourier transforming the xvariable using the definition (2.7) we find, Wij(p,k,t) = (2πε)−2ψi/parenleftbiggp 2−k ε,t/parenrightbigg ψ∗ j/parenleftbigg −p 2−k ε,t/parenrightbigg . (3.2) The total wave energy, comprising gravitational, kinetic, and surface tension contribu-6 Bal & Chou tions is E=1 2/integraldisplay x/bracketleftbig |∇xη(x)|2+|η(x)|2/bracketrightbig +1 2/integraldisplay x/integraldisplay0 −hdz|U(x,z) +ˆ zUz(x,z) +v(x,z)|2 −1 2/integraldisplay x/integraldisplay0 −hdz|U(x,z) +ˆ zUz(x,z)|2 =1 2/integraldisplay k(k2+ 1)|η(k)|2+ktanhkh|ϕ(k,z= 0)|2.(3.3) The energy above has been expanded to an order in η(x,t) andϕ(x,z,t) consistent with the approximations used to derive (2.4). In arriving at the l ast equality in (3.3), we have integrated by parts, used the Fourier decompositions (2.7) and imposed an impenetrable bottom condition at z=−h. The wave energy density carried by wavevector kis (G´ erard et al. (1997)) E(k,t) =1 2Tr [A(k)W(k,t)], (3.4) whereA11(k) =k2+1,A22(k) =ktanhkh,A 12=A21= 0. Thus, the Wigner distribution epitomises the local surface wave energy density. In the presence of slowly varying drift, we identify W(x,k,t) as the localWigner distribution at position xrepresenting waves of wavevector k. The time evolution of its Fourier transform W(p,k,t), can be derived by considering time evolution of the vector fieldψimplied by the boundary conditions (2.4): ˙ψj(k,t) +iLjℓ(k)ψℓ(k,t) =i/integraldisplay qU(q)·(k−qδj2)ψj(k−q,t) +i√ε/integraltext qδU(q)·(k−qδj2/ε)ψj(k−q/ε,t),(3.5) where the operator L(k) is defined by L(k) = 0i|k|tanhε|k|h −i(εk2+ε−1) 0 . (3.6) We have redefined the physical wavenumber to be k/εso thatk∼O(1). Upon using (3.5) and the definition (3.2), (see Appendix A) ˙Wij(p,k,t) =iWiℓ(p,k,t)L† ℓj/parenleftbiggk ε+p 2/parenrightbigg −iLiℓ/parenleftbiggk ε−p 2/parenrightbigg Wℓj(p,k,t) +i/integraldisplay qU(q)·/parenleftbigg −k ε+p 2−qδi2/parenrightbigg Wij(p−q,k+εq/2,t) −i/integraldisplay qU(q)·/parenleftbigg −k ε−p 2+qδj2/parenrightbigg Wij(p−q,k−εq/2,t) +i√ε/integraldisplay qδU(q)·/parenleftbigg −k ε+p 2−q εδi2/parenrightbigg Wij(p−q/ε,k+q/2,t) −i√ε/integraldisplay qδU(q)·/parenleftbigg −k ε−p 2+q εδj2/parenrightbigg Wij(p−q/ε,k−q/2,t),(3.7) where only the index ℓ= 1,2 has been summed over. If we now assume that W(x,k,t) can be expanded in functions that vary independently at the t wo relevant length scales,Water wave transport 7 functions of the field p(dual to x) can be replaced by functions of a slow variation in p and a fast oscillation ξ/ε;p→p+ξ/ε. This amounts to the Fourier equivalent of a two-scale expans ion in which xis replaced byxandy=x/ε(Ryzhik, Papanicolaou, & Keller (1996)). The two new indepe ndent wavevectors pandξare both of O(1). Expanding the Wigner distribution in powers of√εand using p→p+ξ/ε, W(p,k,t)→W0(p,ξ,k,t) +√εW1/2(p,ξ,k,t) +εW1(p,ξ,k,t) +O(ε3/2),(3.8) we expand each quantity appearing in (3.7) in powers of√εand equate like powers. Upon expanding the off-diagonal operator L(−k/ε+p/2) =ε−1L0(k)+L1(k,p)+O(ε), where L0(k) = 0iktanhkh −i(k2+ 1) 0 ,L1(k,p)≡ 0ip·kf(k) ip·k 0 (3.9) and f(k)≡ −hk+ sinhkhcoshkh 2kcosh2kh. (3.10) 3.1.Orderε−1terms The terms of O(ε−1) in (3.7) are W0(p,ξ,k,t)L† 0(k+)−L0(k−)W0(p,ξ,k,t) = 0,k±≡k±ξ 2(3.11) To solve (3.11), we use the eigenvalues and normalised eigen vectors for L0and its complex adjoint L† 0, τΩ(k)−iγ,bτ= iτ/radicalbig α(k)/2 1/radicalbig 2α(k) ;τΩ(k) +iγ,cτ= iτ/radicalbig 2α(k) /radicalbig α(k)/2 ,(3.12) whereα(k)≡Ω(k) k2+ 1,τ=±1, andiγ→0 is a small imaginary term. A W0(p,ξ,k,t) that manifestly satisfies (3.12) can be constructed by expan ding in the basis of 2 ×2 matrices composed from the eigenvectors: W0(p,ξ,k,t) =/summationdisplay τ,τ′=±aττ′(p,k,t)bτ(k−)b† τ′(k+).(3.13) Right[left] multiplying (3.11) (using (3.13)) by the eigen vectors of the adjoint prob- lem,cτ(k−)/bracketleftbig c† τ(k+)/bracketrightbig , we find that a+−=a−+= 0, anda−−(x,k,t)≡a−(x,k,t) = a++(x,−k,t)≡a+(x,−k,t). Furthermore, a+,a−∝ne}ationslash= 0 only ifξ= 0. Thus W0has the form W0(p,ξ,k,t) =W0(p,k,t)δ(ξ). (3.14) From the definition of W0, we see that the (1,1) component of W0is the local envelop of the ensemble averaged wave intensity |η(x,k,t)|2≃a+(x,k,t)α(k). Similarly, from the energy (Eq. (3.4)), we see immediately that the local ensemb le averaged energy density ∝an}b∇acketle{tE(x,k,t)∝an}b∇acket∇i}ht=A11(k)α(k)∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}ht+A22(k)∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}ht = Ω(k)∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}ht.(3.15)8 Bal & Chou Therefore, since the starting dynamical equations are line ar, we can identify ∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}htas the ensemble averaged local wave action associated with wav es of wavevector k(Henyey et al. (1988)). The wave action ∝an}b∇acketle{ta(x,k,t)∝an}b∇acket∇i}ht, rather than the energy density ∝an}b∇acketle{tE(x,k,t)∝an}b∇acket∇i}ht is the conserved quantity (Longuet-Higgins & Stewart (1961 ), Mei (1979), Whitham (1974)). The physical origin of γarises from causality, but can also be explicitly derived fr om considerations of an infinitesimally small viscous dissipa tion (Chou, Lucas & Stone (1995)). Although we have assumed γ→0, for our model to be valid, the viscosity need only be small enough such that surface waves are not atte nuated before they have a chance to multiply scatter and enter the transport or diffus ion regimes under consid- eration. This constraint can be quantified by noting that in t he frequency domain, wave dissipation is given by γ= 2νk2(Landau (1985)) where νis the kinematic viscosity and cg(k)≡ |∇kΩ(k)| (3.16) is the group velocity. The corresponding decay length k−1 d∼cg(k)/(νk2) must be greater than the relevant wave propagation distance. Therefore, we require ε2cg(k/ε) νk2≫(1,ε−1) (3.17) for (transport, diffusion) theories to be valid. The inequal ity (3.17) gives an upper bound for the viscosity νk2≪(εcg(k/ε),ε2cg(k/ε)) (3.18) which is most easily satisfied in the shallow water wave regim e for transport. Otherwise we must at least require ν <o(√ε). The upper bounds for ν(and hence γ) given above provide one criterion for the validity of transport theory. 3.2.Orderε−1/2terms Collecting terms in (3.7) of order ε−1/2, we obtain W1/2(p,ξ,k,t)L† 0(k+)−L0(k−)W1/2(p,ξ,k,t) +/integraldisplay qU(q)·ξW1/2(p−q,ξ−q,k,t) −/integraldisplay qδU(q)·k−W0(p,ξ−q,k+q/2,t) +/integraldisplay qδU(q)·k+W0(p,ξ−q,k−q/2,t) −/integraldisplay qδU(q)·q[W0(p,ξ−q,k+q/2,t)S+SW0(p,ξ−q,k−q/2,t)] = 0 (3.19) where S=/bracketleftbigg 0 0 0 1/bracketrightbigg . Similarly decomposing W1/2in the basis matrices composed of bτ(k−)b† τ′(k+) (as in 3.13), substituting W0(p,0,k,t)δ(ξ) from (3.13 into (3.19), and inverse Fourier trans- forming in the slow variable p, we obtain W1/2(x,k,ξ,t) =/summationdisplay τ,τ′=±δU(ξ)·Γτ,τ′(x,ξ,k,t)bτ(k−)b† τ′(k+) τ′Ω(k+)−τΩ(k−) +U(x)·ξ+ 2iγ, (3.20)Water wave transport 9 where Γτ,τ′(x,ξ,k,t)≡k−aτ′(x,k+,t)c† τ(k−)bτ′(k+)−k+aτ(x,k−,t)b† τ(k−)cτ′(k+) +ξ 2/summationdisplay µ=±/bracketleftbig aµ(x,k+,t)c† τ(k−)bµ(k+) +aµ(x,k−,t)b† µ(k−)cτ′(k+)/bracketrightbig . (3.21) 3.3.Orderε0terms The terms of order ε0in (3.7) read ˙W0(p,k,t) =iW0(p,k,t)L† 1(−p)−iL1(p)W0(p,k,t)−i/integraldisplay qk·U(q)q· ∇kW0(p−q,ξ,k,t) +i/integraldisplay qU(q)·pW0(p−q,ξ,k,t)−i/integraldisplay qU(q)·q[SW0(p−q,ξ,k,t) +W0(p−q,ξ,k,t)S] +i/integraldisplay qδU(q)·k+W1/2(p,ξ−q,k−q/2,t)−i/integraldisplay qδU(q)·k−W1/2(p,ξ−q,k+q/2,t) −/integraldisplay qδU(q)·q/bracketleftbig SW1/2(p,ξ−q,k+q/2,t) +W1/2(p,ξ−q,k−q/2,t)S/bracketrightbig +iW1L† 0−iL0W1+/integraldisplay qU(q)·ξW1(p−q,ξ,k,t). (3.22) To obtain an equation for the statistical ensemble average ∝an}b∇acketle{ta+(x,k,t)∝an}b∇acket∇i}ht, we multiply (3.22) by c† +(k) on the left and by c+(k) on the right and substitute W1/2from equation (3.20). We obtain a closed equation for a(x,k,t)≡ ∝an}b∇acketle{ta+(x,k,t)∝an}b∇acket∇i}ht(we henceforth suppress the∝an}b∇acketle{t...∝an}b∇acket∇i}htnotation for a(x,k,t) andE(x,k,t)) by truncating terms containing W1. Clearly, from (3.12), c† +(k)(iW1L† 0−iL0W1)c+(k) = 0. Furthermore, we assume ∝an}b∇acketle{tξW1(p− q,ξ,k,t)∝an}b∇acket∇i}ht ≈0 which follows from ergodicity of dynamical systems, and ha s been used in the propagation of waves in random media (see Ryzhik, Papani colaou, & Keller (1996), Bal et al. (1999)). The transport equations resulting from t his truncation are rigorously justified in the scalar case (Spohn (1977), Erd¨ os & Yau (1998 )). 4. The surface wave transport equation The main mathematical result of this paper, an evolution equ ation for the ensemble averaged wave action a(x,k,t) follows from equation (3.22) above (cf. Appendix A) and reads, ˙a(x,k,t) +∇kω(x,k)· ∇xa(x,k,t)− ∇xω(x,k)· ∇ka(x,k,t) =−Σ(k)a(x,k,t) +/integraldisplay qσ(q,k)a(x,q,t),(4.1) where ω(x,k) =/radicalbig (k3+k)tanhkh+U(x)·k≡Ω(k) +U(x)·k. (4.2) The left hand side in (4.1) corresponds to wave action propag ation in the absence of random fluctuations. It is equivalent to the equations obtai ned by the ray theory, or a WKB expansion (see section 5.1). The two terms on the right h and side of (4.1) represent refraction, or “scattering” of wave action out of and into waves with wavevector10 Bal & Chou krespectively. In deriving (4.1) we have inverse Fourier tra nsformed back to the slow field point variable x, and used the relation ( α(k)−f(k)α−1(k))k≡ ∇kΩ(k). To obtain (4.1), we assumed Rij(q)qi=Rij(q)qj= 0, which would always be valid for divergence-free flows in two dimensions. Although the perturbation δUis not divergence-free in general, ∇·δU(x,z= 0) = −∂zδUz(x,0)∝ne}ationslash= 0, using symmetry considerations, we will show in section 5.2 that Rij(q)qi=Rij(q)qj= 0. Explicitly, the scattering rates are Σ(k)≡2π/integraldisplay qqiRij(q−k)kj/summationdisplay τ=±b† +(k)cτ(q)b† τ(q)c+(k)δ(τω(x,τq)−ω(x,k)) σ(q,k)≡2π/summationdisplay τ=±τqiRij(τq−k)kj|b† τ(τq)c+(k)|2δ(τω(x,q)−ω(x,k)) (4.3) where b† +(k)cτ(q)b† τ(q)c+(k) =(τα(k) +α(q))(τα(q) +α(k)) 4α(k)α(q) |b† τ(k)cτ′(q)|2=(τα(q) +α(k))2 4α(k)α(q).(4.4) Physically, Σ( k) is a decay rate arising from scattering of action out of wave vector k. The kernel σ(q,k) represents scattering of action from wavevector qintoaction with wavevector k. Note that the slowly varying drift U(x) enters parametrically in the scat- tering viaω(x,k) in theδ−function supports. The arguments ω(x,k) in theδ−functions mean that we can consider the transport of waves of each fixed f requencyω0≡ω(x,k) independently. The typical distance travelled by a wave before it is signific antly redirected is defined by the mean free path ℓmfp=cg(k) Σ(k)∼O(1). (4.5) The mean free path described here carries a different interpr etation from that considered in weakly nonlinear, or multiple scattering theories (Zakh arov, L’vov & Falkovich (1992)) where one treats a low density of scatterers. Rather than str ong, rare scatterings over every distance ℓmfp∼O(1), we have considered constant, but weak interaction with an extended, random flow field. Although here, each scatterin g isO(ε) and weak, over a distance of O(1), approximately ε−1interactions arise, ultimately producing ℓmfp∼ O(1). 5. Results and Discussion We have derived transport equations for water wave propagat ion interacting with static, random surface flows containing two explicit length scales. We have further as- sumed that the amplitude of δUscales asεβwithβ= 1/2: The random flows are correspondingly weakened as the high frequency limit is tak en. Since scattering strength is proportional to the power spectrum of the random flows and i s quadratic in δU, the mean free path can be estimated heuristically by ℓmfp∼cg(k)/Σ(k)ε1−2β. Forβ >1/2, the scattering is too weak and the mean free path diverges. In this limit, waves are nearly freely propagating and can be described by the slowly varyin g flows alone, or WKB the- ory. Ifβ <1/2,ℓmfp→0 and the scattering becomes so frequent that over a propagat ion distance of O(1), the large number of scatterings lead to diffusive (cf. Se ction 5.4) be-Water wave transport 11 haviour (Sheng (1995)). Therefore, only random flows that ha ve the scaling β= 1/2 contribute to the wave transport regime. We also note that β > 0 precludes any wave localisation phenomena. In a two- dimensional random environment, the localisation length o ver which wave diffusion is inhibited is approximately (Sheng (1995)) ℓloc∼ℓmfpexp/parenleftig ε−1kℓmfp/parenrightig ∼ε1−2βexp/parenleftbig ε−2β/parenrightbig . (5.1) As long as the random potential is scaled weaker ( β >0),ℓloc→ ∞, and strong lo- calisation will not take hold. In the following subsections , we systematically discuss the salient features of water wave transport contained in Eq. (4 .1) and derive wave diffusion for propagation distances /greaterorsimilarO(1). 5.1.Slowly varying drift: U(x)∝ne}ationslash= 0,δU= 0 First consider the case where surface flows vary only on scale s much larger than the longest wavelength 2 π/kconsidered, i.e.,δU= 0. The left-hand side in (4.1) represents wave action transport over slowly varying drift and may desc ribe short wavelength modes propagating over flows generated by underlying long ocean wa ves. We first demonstrate that the nonscattering terms of the tran sport equation (4.1) is equivalent to the results obtained by ray theory (WKB expans ion) and conservation of wave action (CWA) (Longuet-Higgins & Stewart (1961), Mei (1 979), Peregrine (1976), White (1999), Whitham (1974)). Assume the WKB expansion (Ke ller (1958), Bender & Orszag (1978)) ηε=Aη(x,t)eiS(x,t)/εandϕε=Aϕ(x,t)eiS(x,t)/ε, (5.2) with smoothly varying AηandAϕ. Upon using the above ansatz in (3.1) and setting ε→0, we have a(x,k,t) =|A|2(x,t)δ(k− ∇xS(x,t)) where |A|2= 2α(k)|Aϕ|2= 2α−1(k)|Aη|2. Substitution of this expression for a(x,k,t) into (4.1), we obtain the fol- lowing possible equations for S(x,t) and|A|2(x,t) ∂tS+ω(x,∇xS) = 0, (5.3) ∂t|A|2(x,t) +∇x·/parenleftig |A|2∇kω(x,∇xS)/parenrightig = 0. (5.4) The first equation is the eikonal equation, while the second e quation is the wave action amplitude equation. Recalling that |Aη|2=α(k)|A|2/2, we obtain the following transport equation for the height amplitude: ∂t/parenleftig|Aη|2 α(∇xS)/parenrightig +∇x·/parenleftig|Aη|2 α(∇xS)∇kω(x,∇xS)/parenrightig = 0. (5.5) Equation (5.5) is the same as Eq. (8) of White (1999), except t hat his ¯Ω is replaced here withαdue to our inclusion of surface tension. Wave action conservation can be understood by noting that d dta(X(t),K(t),t) = 0, (5.6) where the characteristics ( X(t),K(t)) satisfy the Hamilton equations dX(t) dt=∇kω(X(t),K(t)), anddK(t) dt=−∇xω(X(t),K(t)). (5.7) The solutions to the ordinary differential equations (5.7) a re the characteristic curves used to solve (5.3) and (5.4) (Courant & Hilbert (1962)).12 Bal & Chou 5.2.Correlation functions and conservation laws We now consider the case where δU∝ne}ationslash= 0. The scattering rates defined by (4.3) depend upon the precise form of the random flow correlation Rij. There are actually six additional terms in (4.3) in the calculation of σand Σ, which vanish because 2/summationdisplay j=1Rij(q)qj= 0 for i= 1,2. (5.8) We prove relation (5.8) provided that δUz(k,kz) andδUz(k,−kz) have the same proba- bility distribution. Thus, ∝an}b∇acketle{tδUi(p,pz)δUz(k,kz)∝an}b∇acket∇i}ht=∝an}b∇acketle{tδUi(p,pz)δUz(k,−kz)∝an}b∇acket∇i}ht (5.9) This symmetry condition is reasonable, and is compatible wi th the divergence-free con- dition forδUin three dimensions. We show that Hypothesis (5.9) implies ( 5.8) by first using incompressibility/summationtext2 j=1δUj(k,kz)kj+δUz(k,kz)kz= 0: 2/summationdisplay j=1δ(p+k)Rij(k)kj=2/summationdisplay j=1∝an}b∇acketle{tδUi(p,0)δUj(k,0)kj∝an}b∇acket∇i}ht =2/summationdisplay j=1∝an}b∇acketle{tδUi(p,0)/integraldisplay kzδUj(k,kz)kj∝an}b∇acket∇i}ht =−/integraldisplay∞ −∞∝an}b∇acketle{tδUi(p,0)δUz(k,kz)∝an}b∇acket∇i}htkzdkz = 0, where the last equality follows from (5.9). Thus, (5.8) is ve rified, and (4.3) derived. The formRij(|q|)qi=Rij(|q|)qj= 0, requires the correlation function to be transverse: Rij(|q|) =R(q)/bracketleftbigg δij−qiqj q2/bracketrightbigg , (5.10) whereR(q) is a scalar function of q. The correlation kernels in the scattering integrals can now be written as qiRij(|τq−k|)kj=R(|τq−k|)/bracketleftbigg q·k−q·(τq−k)k·(τq−k) |τq−k|2/bracketrightbigg =τR(|τq−k|) |τq−k|2q2k2sin2θ(5.11) whereθdenotes the angle between qandk. The scattering must also satisfy the support of theδ-functions; for U(x) = 0 only |q|=|k|satisfy the the δ−function constraints. In the presence of slowly varying drift, the evolution of a(x,|k| ∝ne}ationslash=|q|) can “doppler” couple toa(x,q,t). It is straightforward to show from the explicit expressions (4.3) that Σ(k) =/integraldisplay qσ(k,q). (5.12) This relation indicates that the scattering operator on the right hand side of (4.1) is conservative: Integrating (4.1) over the whole phase space yields d dt/integraldisplay x/integraldisplay ka(x,k,t) = 0. (5.13)Water wave transport 13 Figure 2. (a). Contour plot of ω(q). Each grayscale corresponds to a different constant value ofω(q) =ω(k)≡ω0. (b). The band of qthat satisfies 0 .625< ω0<0.6625. Wavevectors qand kthat lie in this band can couple a(x,k, t) toa(x,q, t) via wave scattering. Equation (5.13) is the generalization of CWA to include scat tering of action from rapidly varying random flows δU(x/ε). Although a(x,k,t) is conserved, the total water wave energy E(x,k,t) = Ω( k)a(x,k,t) will not be conserved. For example, if U(x) is small enough such that the δ−function in the σ(q,k) integral is triggered only when τ= +1, d dtE=d dt/integraldisplay x/integraldisplay k[ω(x,k)−k·U(x)]a(x,k,t) =−d dt/integraldisplay x/integraldisplay kk·U(x)a(x,k,t)∝ne}ationslash= 0.(5.14) This nonconservation results from the energy that must be su pplied in order to sustain the stationary underlying flow. For small U(x), the quantity ω(x,k)a(x,k,t) is conserved. In that case, the evolution of ω(x,k)a(x,k,t) obeys an equation identical to (4.1). When there is doppler coupling with τ=−1, an additional term arises and ω(x,k)a(x,k,t) is no longer conserved under scattering. 5.3.Doppler coupled scattering In addition to the correlation functions, the wave action sc attering terms involving Σ( k) and integrals over σ(q,k) depend also on the support of the δ−function. Consider action contained in water waves of fixed wavevector k. When U(x) = 0, only τ= +1 terms contribute to the the integration over qas long as |q|=|k|. In this case, we can define the angle q·k=k2cosθand reduce the cross-sections to single angular integrals o ver qiRij(|q−k|)qj=R/parenleftbigg/vextendsingle/vextendsingle2ksinθ 2/vextendsingle/vextendsingle/parenrightbiggk2 4sin2θ sin2θ 2, τ= +1. (5.15) In this case ( U(x) = 0), assuming R(|q|) is monotonically decreasing, the most important contribution to the scattering occurs when qandkare collinear. When U(x)∝ne}ationslash= 0, andτ= +1, the sets of qwhich satisfy Ω( q) +U(x)·q= Ω(k) + U(x)·k≡ω0trace out closed ellipse-like curves and are shown in the con tour plots ofω(q) in Figure 2(a). The parameters used are U(x)·k1=−0.5k1andh=∞(the −k1,−q1directions are defined by the direction of U(x)). Each grayscale corresponds to a curve defined by fixed ω(k) =ω0. All wavevectors q in each contour contribute to the integration in the express ions for Σ( k) andω(q,k). Thus, slowly varying drift can induce an indirect doppler co upling between waves with14 Bal & Chou different wavenumbers, with the most drastic coupling occur ring at the two far ends of a particular oval curve. For example, in Figure 2(b), the dar k band denotes qsuch that ω(q) =ω0when 0.625< ω0<0.6625. The wavevectors q≈(−0.3,0) and q≈(0.8,0) are two of many that contribute to the scattering terms. Ther efore, the evolution of a(x,k≈(−0.3,0),t) also depends on a(x,q≈(0.8,0),t) via the second term on the right side of (4.1). Provided U(x) is sufficiently large, the τ=−1 terms can also contribute to scatter- ing. The dissipative scattering rate Σ( k)a(x,k,t) will change quantitatively since addi- tional q’s will contribute to Σ( k). However, this decay process depends only on kand is not coupled to a(x,|q| ∝ne}ationslash=|k|,t). Wavevectors qthat satisfy the δ−function in the σ(q,k)a(x,q,t) term will, as when τ= +1, lead to indirect doppler coupling. This oc- curs whenω(q) =−ω0and, as we shall see, allows doppler coupling of waves with mo re widely varying wavelengths than compared to the τ= +1 case. Observe that if τ=−1 terms arise, the drift frame energy a(x,k,t)ω(x,k) is no longer conserved. Figure 3(a) plotsω(q1,q2= 0) forU(x) = 1<√ 2,U(x) =√ 2, andU(x) = 1.6>√ 2. Sinceω0 andω(q) are identical functions, −ω0can take on values below the upper dotted line (ω0/lessorsimilar0.22 forU= 1.6). Therefore, coupling for τ=−1 andq2= 0 occurs for values of −ω0between the dotted lines. Note that depending upon the value ofω0, coupling can occur at two or four different points q= (q1,0). Figure 3(b) shows a contour plot of |ω(q)| as a function of ( q1,q2). A level set lying between the dotted lines in ( a) will slice out two bands; one band corresponds to all values of kthat couple to qlying in the associated second band. The two bands determined by the interval 0 .414<−ω0<0.468 are shown in Fig. 3(c). For any klying in the inner band of Fig. 3(c), all qlying in the outer band will contribute to doppler coupling for τ=−1, and vice versa . As−ω0is increased, the inner(outer) band decreases(increases) in size, with the c entral band vanishing when −ω0 approaches the upper dotted line in ( a) where the τ=−1 coupling evaporates. If −ω0 is decreased, the two bands merge, then disappear as −ω0reaches the lower limit. Fig. 3(d) is an expanded view of the two bands for small 0 .0756<−ω0<0.1368. Note that a small island of qorkappears for very small wavevectors. The water wave scatteri ng represented by σ(q,k) can therefore couple very long wavelength modes with very s hort wavelength modes (the two larger bands to the right in Fig. 3( d)). However, the strength of this coupling is still determined by the magnitude of qiRij(|q−k|)kj, which may be small for |q−k|large. The depth dependence of doppler coupling will be relevant wh enhq,hk /lessorsimilar1 whereq andkare the magnitudes of the wavevectors of two doppler-couple d waves. For τ= +1, finite depth reduces the ellipticity of the coupling bands, r esulting in weaker doppler effects. Since the water wave phase velocity decreases with h, a finite depth will also reduce the critical U(x) required for τ=−1 doppler coupling. For small U(x), it is clear that theδ−functions associated with the τ=−1 terms inσ(q,k) are first triggered when theqandkare antiparallel, U·k=−k|U|,U·q= +q|U|. Figure 4(a) shows the phase velocity for various depths h. In order for τ=−1 to contribute to scattering, U/greaterorequalslantcφ(k;h). ForU≈1.6, this condition holds in the h=∞case for 0.5/lessorsimilark/lessorsimilar2 (the dashed region of cφ(k,∞)). Recall that our starting equations (2.4) are valid only in the small Froude number limit. However, for water waves propagating over infinite depth, τ=−1 coupling requires U >U min= min k{cφ(k)}, withcφ(kmin)≃ 22cm/s. Therefore, in such “supersonic” cases, where τ=−1 is relevant, our treatment is accurate only at wavevectors k∗such thatU≪cφ(k∗;h),e.g.,the thick solid portion ofcφ(k;∞). ForU/greaterorsimilarUmin, theτ=−1 term can couple wavevectors q≈0≪kmin withk≈2−3≫kmin. The rich τ=−1 doppler coupling displayed in Figures 3 isWater wave transport 15 Figure 3. Conditions for doppler coupling when τ=−1. (a). Plot of ω(q1, q2= 0;h=∞) for U= 1,U=√ 2, and U= 1.6. Only for U >√ 2 does ω(q1, q2= 0;h=∞)<0. (b). Contour plot of |ω(q)|. Each grayscale corresponds to a different constant value of ω(q) =ω(k)≡ −ω0. (c). The bands of qsatisfying 0 .414<−ω0<0.468. (d). An expanded view of the coupling bands for 0 .0756<−ω0<0.1368. Note that wavenumbers of very small modulus can couple with wavenumbers of significantly larger modulus. particular to water waves with a dispersion relation ω(q) that behaves as q3/2,U·q, or q1/2depending on the wavelength. Doppler coupling in water wave propagation is very different from that arising in acoustic wave propagation in a n incompressible, randomly flowing fluid (Howe (1973), Fannjiang & Ryzhik (1999), Vedant ham & Hunter (1997)) whereω(q) =cs|q|. An additional doppler coupling analogous to the τ=−1 coupling for water waves arises only for supersonic random flows when U(x)/greaterorequalslantcs, independent of q. In such instances, compressibility effects must also be con sidered. Figure 4(b) plots the minimum drift velocity Umin(h) whereτ=−1 doppler coupling first occurs at any wavevector. The wavevector at which coupl ing first occurs is also shown by the dashed curve. For shallow water, h≪√ 3,Umin(h)∝√ hand very long wavelengths couple first (small k(Umin)). For depths h >√ 3 (∼3cm for water), the minimum drift required quickly increases to U∗(∞) =√ 2, while the initial coupling occurs at increasing wavevectors until at infinite depth, wh ere the first wavevector to doppler couple approaches k→1 (in water, this corresponds to wavelengths of ∼6.3cm). The conditions for τ=−1 doppler coupling outlined in Figures 2 and 3 apply to both16 Bal & Chou 0 1 2 3 4 5 k0123 phase velocity cφ(k;h) h=∞ h=2 h=1 h=0.5 0 1 2 3 4 h00.511.5Umin for τ=−1 coupling k(Umin) Umin1.021/2 (a).(b). Figure 4. U > c φ(k) is required for τ=−1 coupling. (a). The phase velocity cφ(k) for various depths h. The velocity shown by the solid horizontal line U≈1.6> cφ(k;h=∞) for 0 .5/lessorsimilark/lessorsimilar2. (b). The minimum Umin(h) required for existence of τ=−1 coupling at any wavevector k, and the wavevector k(Umin) at which this first happens. Σ(k) andσ(q,k), with the proviso that qandkare parallel for Σ( k) and antiparallel forσ(q,k). However, even when U <U minsuch that only τ= +1 applies, the set of q corresponding to a constant value of ω(k) =ω0, traces out a noncircular curve. There is doppler coupling between wavenumbers q∝ne}ationslash=kas long asU∝ne}ationslash= 0. 5.4.Surface wave diffusion We now consider the radiative transfer equation (4.1) over p ropagation distances long compared to the mean free path ℓmfp. Imposing an additional rescaling and measuring all distances in terms of the mean free path, we introduce anothe r scalingǫ−1, proportional to the number of mean free paths travelled. Since β= 1/2, transport of wave action prevails when O(ε)<|x| ∼O(1), while diffusion holds when O(ǫ−1)∼ |x|<ℓloc. Since waves of each frequency satisfy (4.1) independently, we consider the diffusion of waves of constant frequency ω0. To derive the diffusion equation, we assume for simplicity thatUis constant and small such that ω0+ω(x,q)∝ne}ationslash= 0 (theτ=−1 terms are never triggered by the δ−functions). Expanding all quantities in the transport equa tion (4.1) in powers of ǫ, we find ˙a0+¯U· ∇xa0− ∇x·D· ∇xa0= 0. (5.16) The derivation of this equation is given in Appendix B. The di ffusion tensor Dis given in (B 12) and is a function of the power spectrum Rij. The effective drift ¯Uis given by (B 7): ¯U=/integraltext k∇kω(k)δ(k·U+ Ω(k)−ω0)/integraltext kδ(k·U+ Ω(k)−ω0). (5.17) Up to a change of basis, we can assume that U=Ue1, whereU >0. Then the set of points k·U+ Ω(k)−ω0= 0 is symmetric with respect to the x1−axis and ¯Uis parallel toU. Also notice that the total energy given in (3.15) is asympto tically conserved in theWater wave transport 17 diffusive regime. Indeed, the total energy variations are gi ven by (5.14). Assuming that all water waves have frequency ω0, we have in the diffusive regime d dtE=−d dt/integraldisplay x/integraldisplay kk·U(x)a(x,k,t) ≈ −/parenleftig/integraldisplay x˙a0(x,t)/parenrightig/integraldisplay kk·Uδ(ω0−ω(k)), sinceUis constant. Recasting the diffusion equation as ˙ a0=−∇x·(¯Ua0+D· ∇xa0), we deduce that/integraldisplay x˙a0(x,t) = 0, which conserves the total energy E. Now consider the simplified case U≡0,h=∞and Ω ∞(k) =√ k3+k. Since U= ¯U= 0, (5.15) holds and we have for all k, /integraldisplay qqiRij(|q−k|)qjq=0. (5.18) We deduce that the corrector χin (B 8) is given by χ(k) =−∇kΩ∞(k) Σ(k)=−|∇kΩ∞(k)| Σ(k)ˆk=−cg Σ(k)ˆk, where k=kˆk. The isotropic diffusion tensor Dis thus given by D=1 Σ(k)Vω0/integraldisplay q|∇qΩ∞(q)|2ˆqˆqTδ(Ω∞(q)−ω0) =c2 g(k) 2Σ(k)I, (5.19) whereIis the 2 ×2 identity matrix. Thus, the diffusion equation for a0(x,t) assumes the standard form (Sheng (1995)) ˙a0−c2 g(k) 2Σ(k)∆a0= 0. (5.20) 6. Summary and Conclusions In this paper, we have used the Wigner distribution to derive the transport equations for water wave propagation over a spatially random drift com posed of a slowly varying partU(x), and a rapidly varying part√εδU(x/ε). The slowly varying part determines the characteristics on which the waves propagate. We recove r the standard result obtained from WKB theory: conservation of wave action. Provided Rij(q)qj= 0, we extend CWA to include wave scattering from correlations Rijof the rapidly varying random flow. Evolution equations for the nonconserved wave intensity an d energy density can be read- ily obtained from (4.1). Moreover, conservation of drift fr ame energy a(x,k,t)ω(x,k) requires small U <U minand absence of τ=−1 contributions to scattering. Explicit expressions for the scattering rates Σ( k) andσ(q,k) are given in Eqs. (4.3). For fixedω(k), we find the set of qsuch that the δ−functions in (4.3) are supported. This set of qindicates the wavevectors of the background surface flow tha t can me- diate doppler coupling of the water waves. Although widely v arying wavenumbers can doppler couple, supported by the δ−function constraints, particularly for τ=−1, the correlation Rij(|q−k|) also decreases for large |q−k|. For long times, multiple weak scat- tering nonetheless exchanges action among disparate waven umbers within the transport18 Bal & Chou regime. Our collective results, including water wave actio n diffusion, provide a model for describing linear ocean wave propagation over random flows o f different length scales. The scattering terms in (4.1) also provide a means to correla te sea surface wave spectra to statistics Rijof finer scale random flows. Although many situations arise where the underlying flow is r otational (White (1999)), the irrotational approximation used simplifies the treatme nt and allows a relatively simple derivation of the transport and diffusive regimes of water wa ve propagation. The recent extension by White (1999) of CWA to include rotational flows a lso suggests that an explicit consideration of velocity and pressure can be used to generalise the present study to include rotational random flows. Other feasible ext ensions include the analysis of a time varying random flow, as well as separating the underl ying flows into static and wave dynamic components. The authors thank A. Balk, M. Moscoso, G. Papanicolaou, L. Ry zhik, and I. Smol- yarenko for helpful comments and discussion. GB was support ed by AFOSR grant 49620- 98-1-0211 and NSF grant DMS-9709320. TC was supported by NSF grant DMS-9804780. Appendix A. Derivation of the transport equation Some of the steps in the derivation of (4.1) are outlined here . By taking the time derivative of Wijin (3.2) and using the definition (3.5) for ˙ψ, we obtain (2πε)2˙Wij(p,k,t) = (2πε)2iWiℓ(p,k)L∗ ℓj/parenleftbiggk ε−p 2/parenrightbigg −(2πε)2iLiℓ/parenleftbiggk ε+p 2/parenrightbigg Wℓj(p,k) +i/integraldisplay qU(q)·/parenleftbigg −k ε+p 2−qδi2/parenrightbigg ψi/parenleftbigg −k ε+p 2−q/parenrightbigg ψ∗ j/parenleftbigg −k ε−p 2/parenrightbigg −i/integraldisplay qU∗(q)·/parenleftbigg −k ε−p 2−qδj2/parenrightbigg ψi/parenleftbigg −k ε+p 2/parenrightbigg ψ∗ j/parenleftbigg −k ε−p 2−q/parenrightbigg +i√ε/integraldisplay qδU(q)·/parenleftbigg −k ε+p 2−q εδi2/parenrightbigg ψi/parenleftbigg −k ε+p 2−q ε/parenrightbigg ψ∗ j/parenleftbigg −k ε−p 2/parenrightbigg −i√ε/integraldisplay qδU∗(q)·/parenleftbigg −k ε−p 2−q εδj2/parenrightbigg ψi/parenleftbigg −k ε+p 2/parenrightbigg ψ∗ j/parenleftbigg −k ε−p 2−q ε/parenrightbigg (A 1) To rewrite the above expression as a function of Wijonly, we relabel appropriately, e.g., −k ε−p 2=−k′ ε−p′ 2 −k ε+p 2−q=−k′ ε+p′ 2(A 2) for the third term on the right hand side of (A1). Similarly re labelling for all relevant terms yields the integral equation (3.7). TheO(ε−1/2) terms of (3.7) determine W1/2. Decomposing W1/2(p,ξ,k)≡/summationdisplay τ,τ′=±a(1/2) τ,τ′(p,ξ,k)bτ(k−)b† τ′(k+) (A 3) and substituting into (3.19) we find the coefficients a(1/2) τ,τ′, where in this case a(1/2) +−,a(1/2) −+∝ne}ationslash=Water wave transport 19 0. Due to the nonlocal nature of the third term on the right of ( 3.19), we must first inverse Fourier transform the slow wavevector variable back to x. To extract the O(ε0) terms from (3.7) we need to expand Lto orderε0, theL1term. Similarly, the terms W(p−q,ξ,k±εq/2) must be expanded: W(p−q,ξ,k±εq/2) =W(p−q,ξ,k)±ε 2q· ∇kW(p−q,ξ,k) +O(ε2).(A 4) Theεq·∇kW(p−q,ξ,k) terms combine with the −ε−1U(q)·k−+ε−1U(q)·k+terms from the third and fourth terms in (3.7) to give the third term on the right of Eq. (3.22). TheδU-dependent, order ε0terms (the sixth, seventh, and eighth terms on the right side of (3.22)) come from collecting ±√εδU(q)·/parenleftbigg −k ε±ξ 2ε/parenrightbigg√εW1/2(p,ξ−q,k±q/2) (A 5) from the last two terms in (3.7). The ensemble averaged time e volution of the Wigner amplitudeaσ(x,k) can be succinctly written in the form: ˙a+(x,k,t)− ∇xω(x,k)· ∇ka+(x,k,t) +∇kω(x,k)· ∇xa+(x,k,t) = Σ+,µ(k)aµ(x,k,t) +/integraldisplay qσ+,µ(q,k)aµ(x,k,t). (A 6) Using the form for W0found from (3.11) in (3.19) to find W1/2, we substitute into (3.22) to find (4.1), the transport equation for one of the dia gonal intensities of the Wigner distribution. We have explicitly used eigenbasis orthonor mality b† τ(k)·cτ′(k) =δτ,τ′and the fact that a−(x,k,t) =a+(x,−k,t). Appendix B. Derivation of the diffusion equation The derivation of diffusion of water wave action is outlined b elow and follows the established mathematical treatment of Larsen & Keller (197 4) and Dautray & Lions (1993). For simplicity we assume that the flow Uis constant and small enough so that for a considered range of frequencies, the relation ω(q) +ω(k) = 0 is never satisfied for any kandq∝ne}ationslash=0. The diffusion approximation is valid after long times and la rge distances of propagation X(see Fig. 1) such that the wave has multiply scattered and its dynamics are determined by a random walk. We therefore resca le time and space as ˜t=t ǫ2, ˜x=x ǫ. (B 1) The small parameter ǫin this further rescaling represents the transport mean fre e path ℓmfpand not the wavelength as in the initial rescaling used to der ive the transport equation. We drop the tilde symbol for convenience and rewri te the transport equation in the new variables: ˙aǫ(x,k,t)+1 ǫ∇kω(k)·∇xaǫ(x.k,t) =1 ǫ2/integraldisplay qQ(q,k)(aǫ(x,q,t)−aǫ(x,k,t))δ(ω(q)−ω(k)), (B 2) with obvious notation for Q(q,k). Since the frequency is fixed, the equation is posed forksatisfyingω(k) =ω0. The transport equation assumes the form (B 2) because the scattering operator is conservative. Since U∝ne}ationslash= 0, wave action is transported by the flow, and diffusion takes place on top of advection. Therefore, we i ntroduce the main drift ¯U,20 Bal & Chou which will be computed explicitly later, and define the drift -free unknown ˜ aǫ(x,k,t) as ˜aǫ(x,k,t) =aǫ(x+¯U ǫt,k,t). (B 3) It is easy to check that ˜ aǫsatisfies the same transport equation as aǫwhere the drift term∇kωhas been replaced by ∇kω−¯U. We now derive the limit of ˜ aǫasǫ→0. Consider the classical asymptotic expansion ˜aǫ= ˜a0+ǫ˜a1+ǫ2˜a2+.... (B 4) Upon substitution into (B 2) and equating like powers of ǫ, we obtain at order ǫ−2, for fixed frequency ω0, /integraldisplay qQ(q,k)(˜a0(x,q,t)−˜a0(x,k,t))δ(ω(q)−ω0) = 0. (B 5) It follows from the Krein-Rutman theory (Dautray & Lions (19 93)) that ˜a0is independent ofq. At orderǫ−1, we obtain (∇kω(k)−¯U)· ∇x˜a0=/integraldisplay qQ(q,k)(˜a1(x,q,t)−˜a1(x,k,t))δ(ω(q)−ω0). (B 6) The compatibility condition for this equation to admit a sol ution requires both sides to vanish upon integration over δ(ω(k)−ω0)dk. Therefore, ¯Usatisfies ¯U=1 Vω0/integraldisplay k∇kω(k)δ(ω(k)−ω0), where Vω0=/integraldisplay kδ(ω(k)−ω0).(B 7) Once the constraint is satisfied, we deduce from Krein-Rutma n theory the existence of a vector-valued mean zero corrector χsolving (∇kω(k)−¯U) =/integraldisplay qQ(q,k)(χ(q)−χ(k))δ(ω(q)−ω0)≡ Lχ. (B 8) There is no general analytic expression for χ, which must in practice be solved numer- ically. This is typical of problems where the domain of integ ration in qdoes not have enough symmetries (cf. Allaire & Bal (1999), Bal (1999)). We now have ˜a1(x,k,t) =χ(k)· ∇x˜a0(x,t). (B 9) It remains to consider O(ǫ0) in the asymptotic expansion. This yields ˙˜a0+ (∇kω(k)−¯U)· ∇x˜a1=L(˜a2). (B 10) The compatibility condition, obtained by integrating both sides over k, yields the wave action diffusion equation ˙˜a0− ∇x·D· ∇x˜a0= 0, (B 11) where the diffusion tensor is given by D=−1 Vω0/integraldisplay k(∇kω(k)−¯U)χT(k) =−1 Vω0/integraldisplay kL(χ)χT(k). (B 12) The second form shows that Dis positive definite since Lis a nonpositive operator. The formal asymptotic expansion can be justified rigorously usi ng the techniques in Dautray & Lions (1993). As ǫ→0, we obtain that the error between ˜ aǫand ˜a0is at most of orderǫ. Therefore, we have that aǫconverges to a0satisfying the following drift-diffusion equation ˙a0+¯U ǫ· ∇xa0− ∇x·D· ∇xa0= 0, (B 13)Water wave transport 21 with suitable initial conditions. Equation (B 13) is the coo rdinate-scaled version of (5.16). REFERENCES Allaire G. & Bal, G. 1999 Homogenization of the criticality spectral equation i n neutron transport M2AN Math. Model. Numer. Anal. 33721-746. Bal, G. 1999 First-order Corrector for the Homogenization of the Cr iticality Eigenvalue Problem in the Even Parity Formulation of the Neutron Transport SIAM J. Math. Anal. 301208- 1240. Bal, G., Fannjiang, A., Papanicolaou, G. & Ryzhik, L. 1999 Radiative Transport in a Periodic Structure J. Stat. Phys. 95479-494. Bender, C. M., & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. (McGraw-Hill, New York). Cerda, E. & Lund, F. 1993 Interaction of surface waves with vorticity in shallow water Phys. Rev. Lett. 703896-3899. Chou, T. 1998 Band structure of surface flexural-gravity waves along periodic interfaces J. Fluid Mech. 369333-350. Chou, T., Lucas, S. K., & Stone, H. A. 1995 Capillary Wave Scattering from a Surfactant Domain Phys. Fluids 71872-1885. Chou, T., & Nelson, D. R. 1994 Surface Wave Scattering from Nonuniform Interfaces J. Chem. Phys. 1019022-9031. Courant, R. & Hilbert, D. ,Methods of Mathematical Physics , Vol. II. Wiley Interscience. 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G´erard, P., Markowich, P. A., Mauser, N.J., & Poupaud, F. 1997 Homogenization limits and Wigner transforms Comm. Pure Appl. Math. 50, 323-380. Henyey, F. S., Creamer, D. B., Dysthe, K. B., Schult, R. L., & Wrig ht, J. A. 1988 The energy and action of small waves riding on large waves J. Fluid Mech. 189, 443-462. Howe, M. 1973 Multiple scattering of sound by turbulence and other in homogeneities J. Sound and Vibration 27, 455-476. Jonsson, I.G. 1990 Wave-current interactions Ocean Engineering Science: The Sea (ed. B. Le Mehaute & D. Hanes), 65-120, Wiley. Keller, J. B. 1958 Surface Waves on water of non-uniform depth J. Fluid Mech. 4, 607-614. Kleinert, P. 1990 A field-theoretical treatment of Third-Sound localiza tionPhys. Stat. Sol. B 158, 539-550. Lee, K. Y., Chou, T., Chung, D. S. & Mazur, E. 1993 Direct Measurement of the Spatial Damping of Capillary Waves at Liquid-Vapor Interfaces J. Phys. Chem. 97, 12876-12878. Landau 1985 Fluid Mechanics. Pergamon. Larsen, E. W. & Keller, J. 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arXiv:physics/0001002v1 [physics.gen-ph] 3 Jan 2000We can know more in double-slit experiment Gao Shan Institute of Quantum Mechanics 11-10, NO.10 Building, YueTan XiJie DongLi, XiCheng Distri ct Beijing 100045, P.R.China E-mail: gaoshan.iqm@263.net We show that we can know more than the orthodox view does, as on e example, we make a new analysis about double-slit experiment, and demonstrate th at we can measure the objective state of the particles passing through the two slits while not dest roying the interference pattern, the measurement method is to use protective measurement. Double-slit experiment has been widely discussed, and near ly all textbooks about quantum mechanics demonstrated the weirdness of quantum world using it as one example, as Fey nman said, it contains all mysteries of quantum mechanics, but have we disclosed these mysteries and unders tood the weirdness in double-slit experiment? as we think, the answer is definitely No. When discussing double-slit experiment, the most notoriou s question is which slit the particle passes through in each experiment, it is just this problem that touches our sor e spots in understanding quantum mechanics, according to the widely-accepted orthodox view, this question is actu ally meaningless, let’s see how it gets this bizarre answer, it assumes that only an measurement can give an answer to the abo ve question, then detectors need to be put near both slits to measure which slit the particle passes through, but when this is done the interference pattern will disappear, thus the orthodox view asserts that the above question is mea ningless since we can not measure which slit the particle passes through while not destroying the interference patte rn. In fact, the above question is indeed meaningless, and at it h appens the orthodox answer is right, but its reason is by no means right, the genuine reason is that if the particl e passes through only one slit in each experiment, the interference pattern will not be formed at all∗, thus it is obviously wrong to ask which slit the particle pas ses through in each experiment, it does not pass through a single slit at a ll! On the other hand, we can still ask the following meaningful q uestion, namely how the particle passes through the two slits to form the interference pattern? now as to this que stion, the deadly flaw of the orthodox view is clearly unveiled, what is its answer? as we know, its answer will be th ere does not exist any objective motion picture of the particle, the question is still meaningless, but how can it g et this conclusion? it can’t! and no one can. Since we have known that the particle does not pass through a s ingle slit in each experiment, the direct position measurement near both slits is obviously useless for finding the objective motion state of the particle passing through the two slits, and it will also destroy the objective motion s tate of the particle, then the operational basis of the orthodox view disappears, it also ruins, thus the orthodox d emonstrations can’t compel us to reject the objective motion picture of the particle†, it only requires that the motion picture of classical conti nuous motion should be rejected, this is undoubtedly right, since the motion of mic roscopic particle will be not classical continuous motion a t all, it will be one kind of completely different motion. Once the objective motion picture of the particle can’t be es sentially rejected, we can first have a look at it using the logical microscope, since the particle does not pass thr ough a single slit in each experiment, it must pass through both slits during passing through the two slits, it has no oth er choices! this kind of bizarre motion is not impossible since it will take a period of time for the particle to pass thr ough the slits, no matter how short this time interval is, so far as it is not zero, the particle can pass through both sli ts during this finite time interval, what it must do is just discontinuously move, nobody can prevent it from moving in s uch a way! in fact, as we have demonstrated [4], this is just the natural motion of particle. ∗Here we assume the only existence of particle, thus Bohm’s hi dden-variable theory [3] is not considered. †Why we can’t detect which slit the particle passes through wh en not destroying the interference pattern is not because th ere does not exist any objective motion picture of the particle, but because the particle does not pass through a single slit a t all. 1On the other hand, in order to find and confirm the objective mot ion picture of the particle passing through the two slits, which will be very different from classical contin uous motion, we still need a new kind of measurement, which will be very different from the position measurement, f ortunately it has been found several year ago [1,2], its name is protective measurement, since we know the state of th e particle beforehand in double-slit experiment, we can protectively measure the objective motion state of the p article when it passes through the two slits, while the state of the particle will not be destroyed after such protec tive measurement, and the interference pattern will not be destroyed either, thus by use of this kind of measurement w e can find the objective motion picture of the particle passing through the two slits while not destroying the inter ference pattern, and the measurement results will reveal that the particle indeed passes through both slits as we see u sing the logical microscope. Now, the above analysis has strictly demonstrated that we ca n know more than the orthodox view does in double- slit experiment, namely we know that the particle passes thr ough both slits to form the interference pattern, while the orthodox view never knows this. [1] Y.Aharonov, L.Vaidman, Phys.Lett.A 178, 38 (1993) [2] Y.Aharonov, J.Anandan, and L.Vaidman, Phys.Rev.A. 47, 4616 (1993) [3] D.Bohm, Phys.Rev. 85,166-193. (1952) [4] Gao Shan, quant-ph/. 2

arXiv:physics/0001003v1 [physics.chem-ph] 2 Jan 2000Tentative statistical interpretation of non-equilibrium entropy Lajos Di´ osi∗ Research Institute for Particle and Nuclear Physics H-1525 Budapest 114, POB 49, Hungary January 1, 2000 Abstract We suggest a certain statistical interpretation for the ent ropy pro- duced in driven thermodynamic processes. The exponential f unction of the irreversible entropy re-weights the probability of t he standard Ornstein-Uhlenbeck-type thermodynamic fluctuations. ∗E-mail: diosi@rmki.kfki.hu 0In 1910 Einstein [1], paraphrasing Boltzmann’s lapidary fo rmula S= logW, expressed the probability distribution of thermodynamic variables x through the entropy function S(x): W(x)∼eS(x). (1) This equation describes thermodynamic fluctuations in Gaus sian approxima- tion properly. Going beyond the stationary features, the ti me-dependence of fluctuations xtcan be characterized by a certain probability functional W[x] over complete paths {xt;t∈(−∞,∞)}. I suggest that, in driven thermody- namic processes, this probability is related to the irrever sible entropy Sirr[x]. Symbolically, we can write the following relationship: W[x]∼WOU[x−¯x]eSirr[x], (2) where ¯ xtis the ‘driving’ value of parameter xtandWOU[z] turns out to correspond to fluctuations ztof Ornstein-Uhlenbeck type. This relationship offers Sirra certain statistical interpretation, somehow resembling Einstein’s suggestion (1) for the equilibrium entropy S(x). In this short note, Ein- stein’s approach to the thermodynamic fluctuations is outli ned and standard equations of time-dependent fluctuations are invoked from i rreversible ther- modynamics. Then I give a precise form to the relationship (2 ) for driven thermodynamic processes. The equilibrium conditions for isolated composite thermod ynamic sys- tems derive from the maximum entropy principle: S(x) =max , (3) where S(x) is the total entropy of the system in function of certain fre e thermodynamic parameters x[2]. If the function S(x) is maximum at x= ¯x then ¯xis the equilibrium state. For example, xmay be the temperature Tof a small (yet macroscopic) subsystem in the large isolated system of temperature T= ¯x. Then, the function S(x) must be the total entropy of the isolated system, depending on the variation of the subsyste m’s temperature around its equilibrium value. The equilibrium value ¯ x[as well as S(x) itself] may vary with the deliberate alteration of the initial condi tions. Surely, in our example the temperature Tof the whole isolated system can always be controlled at will. For later convenience, especially in treating driven 1thermodynamic processes, we may prefer the explicit detail ed notation S(x|¯x) forS(x). Though S(¯x)−S(x) might qualify the lack of equilibrium, nearby values x≈¯xhave no interpretation in phenomenological thermodynamic s. They only have it in the broader context of statistical physi cs. In finite thermodynamic systems there are fluctuations around the equ ilibrium state ¯xand their probability follows Eq. (1): W(x|¯x)dx=NeS(x|¯x)−S(¯x|¯x)dx . (4) Assume, for simplicity, that there is a single free variable x. The Taylor expansion of the entropy function yields Gaussian fluctuati ons: W(x|¯x) =1√ 2πσ2exp/parenleftBig −1 2σ2(x−¯x)2/parenrightBig , (5) where 1 σ2=−S′′(¯x)≡ −∂2S(x|¯x) ∂x2/vextendsingle/vextendsingle/vextendsingle x=¯x. (6) In our concrete example σ2=T2/Cwhere Cis the specific heat of the subsystem. We are going to regard the time-dependence of the parameter xtfluctu- ating around ¯ x, according to the standard irreversible thermodynamics [2 ]. The time-dependent fluctuation zt≡xt−¯xis an Ornstein-Uhlenbeck (OU) stochastic process [3] of zero mean /an}bracketle{tzt/an}bracketri}ht ≡0 and of correlation /an}bracketle{tztzt′/an}bracketri}htOU=σ2e−λ|t−t′|. (7) The relaxation rate λof fluctuations is related to the corresponding Onsager kinetic constant γbyλ=γ/σ2. It can be shown that the probability dis- tribution of xt=zt+ ¯xat any fixed time tis the Gaussian distribution (5) as it must be. For the probability of the complete fluctuation pathzt, the zero mean and correlation (7) are equivalent with the follow ing functional measure: WOU[z]Dz= exp/parenleftBig −1 2γ/integraldisplay (˙z2 t+λ2z2 t)dt/parenrightBig Dz , (8) where a possible constant of normalization has been absorbe d into the func- tional measure Dz. In order to construct and justify a relationship like (2) one needs to proceed to driven thermodynamic processes. In fact, we assu me that we 2are varying the parameter ¯ xwith small but finite velocity. Formally, the parameter ¯ xbecomes time-dependent. For simplicity’s sake we assume th at the coefficients σ, γdo not depend on ¯ xor, at least, that we can ignore their variation throughout the driven range of ¯ xt. We define the irreversible entropy production during the driven process as follows: Sirr[x|¯x] =1 σ2/integraldisplay (¯xt−xt)dxt. (9) In our concrete example dSirr= (C/T2)(T−T)dT≈dQ(T−1−T−1) which is indeed the entropy produced randomly by the heat transfer dQfrom the surrounding to the subsystem. By partial integration, Eq. ( 9) leads to an alternative form: Sirr[x|¯x] =1 σ2/integraldisplay (¯xt−xt)d¯xt+1 σ2(x−∞−¯x−∞)2−1 σ2(x∞−¯x∞)2.(10) In relevant driven processes the entropy production is macr oscopic, i.e., Sirr≫1 inkB-units, hence it is dominated by the integral term above. I exploit this fact to replace expression (9) by Sirr[x|¯x] =1 σ2/integraldisplay (¯xt−xt)d¯xt (11) which vanishes for constant ¯ x[4]. In the sense of the guess (2), I suggest the following form for the probability distribution of the driv en path: W[x|¯x] =N[¯x]WOU[x−¯x]eSirr[x|¯x]. (12) The non-trivial normalizing pre-factor is a consequence of ¯x’s time-depen- dence and will be derived below. Since the above distributio n is a Gaussian functional and Sirr[x|¯x] is a linear functional (11) of x, we can easily calculate the expectation value of the irreversible entropy: Sirr[¯x]≡ /an}bracketle{tSirr[x|¯x]/an}bracketri}ht=1 2σ2/integraldisplay /integraldisplay ˙¯xt˙¯xt′e−λ|t−t′|dtdt′. (13) In case of moderate accelerations ¨¯x≪λ˙¯x, this expression reduces to the standard irreversible entropy γ−1/integraltext˙¯x2 tdtof the phenomenological theory of driven processes [6]. Coming back to the normalizing factor in Eq. (12), we 3can relate it to the mean entropy production (13): N[¯x] =exp(−Sirr[¯x]). Hence, the ultimate form of Eq. (12) will be: W[x|¯x] =WOU[x−¯x]eSirr[x|¯x]−Sirr[¯x]. (14) This result gives the precise meaning to our symbolic relati onship (2). If the entropy production Sirrwere negligible then the thermodynamic fluctu- ations xt−¯xtwould follow the OU statistics (7) like in case of a steady sta te ¯xt=const. Even in slow irreversibly driven processes Sirrmay grow essen- tial and exp[Sirr] will re-weight the probability of OU fluctuations. The true stochastic expectation value of an arbitrary functional F[x] can be expressed by the OU expectation values of the re-weighted functional: /an}bracketle{tF[x]/an}bracketri}ht=/angbracketleftBig F[x]eSirr[x|¯x]−Sirr[¯x]/angbracketrightBig OU. (15) I can verify the plausibility of Eq. (14) for the special case of small ac- celerations. Let us insert Eqs. (8,11) and also Eq. (13) whil e ignore ¨¯xin comparison with λ˙¯x. We obtain: W[x|¯x] =WOU[x−¯x+λ−1˙¯x]. (16) Obviously, the fluctuations of the driven system are governe d by the OU process zt(7) in the equilibrium case when ˙¯x≡0. In driven process, when ˙¯x/ne}ationslash= 0, there is only a simple change: The OU fluctuations happen a round the retarded value ¯ xt−τ˙¯x≈¯xt−τof the driven parameter. The lag τis equal to the thermodynamic relaxation time 1 /λ. Consequently, the driven random path takes the following form: xt= ¯xt−τ+zt, (17) where ztis the equilibrium OU process (7). This result implies, in pa rticular, the equation /an}bracketle{txt/an}bracketri}ht= ¯xt−τwhich is just the retardation effect well-known in the thermodynamic theory of slightly irreversible driven proc esses. For example, in case of an irreversible heating process the subsystem’s a verage temperature will always be retarded by τ˙Twith respect to the controlling temperature T[6]. Finally, let us summarize the basic features of Einstein’s f ormula (1) and of the present proposal (2). They characterize via pretty co mpact formu- lae the lack of equilibrium in steady and driven states, resp ectively. They 4do it in terms of thermodynamic entropies while they refer to a statistical context lying outside both reversible and irreversible the rmodynamics. Both formulae are only valid in the lowest non-trivial order and t heir correctness in higher orders is questionable [7]. Contrary to their limite d validity, they can no doubt give an insight into the role of thermodynamic entro py in statistical fluctuations around both equilibrium or non-equilibrium st ates. Acknowledgments. I thank Bjarne Andresen, Karl Heinz Hoffmann, At- tila R´ acz, and Stan Sieniutycz for useful remarks regardin g the problem in general. This work enjoyed the support of the EC Inco-Copern icus program Carnet 2 . References [1] A. Einstein, Ann.Phys.(Leipzig) 33, 1275 (1910). [2] L.D. Landau and E.M. Lifshitz, Statistical Physics (Clarendon, Oxford, 1982). [3] G.E.Uhlenbeck and L.S.Ornstein, Phys.Rev. 36, 823 (1930). [4] I am puzzled by the fact that one could retain the original definition (9) forSirrif one used the well-known Onsager–Machlup functional [5]: WOM[z]Dz= exp/parenleftBig −1 2γ/integraldisplay ( ˙zt+λzt)2dt/parenrightBig Dz , instead of (8). With this replacement all results would rema in, including the central ones (14,15). Unfortunately the OM functional, unlike the OU functional, can not be considered a resonable equilibriu m distribu- tion since, e.g., it orders the same probability to the class of unbounded fluctuations {zt+cexp(−λt);c∈(−∞,∞)}. [5] L. Onsager and S. Machlup, Phys.Rev. 91, 1505 (1953); 91, 1512 (1953); R.L. Stratonovitch, Sel.Transl.Math.Stat.Prob. 10, 273 (1971); R. Gra- ham, Z.Phys. B26, 281 (1976). [6] L.Di´ osi, Katalin Kulacsy, B.Luk´ acs and A.R´ acz, J.Ch em.Phys. 105, 11220 (1996). 5[7] Einstein’s ansatz fails obviously beyond the Gaussian a pproximation. Our present proposal is first of all limited to small velociti es˙¯x. In fact, the fluctuations of the thermodynamic parameters are govern ed by the phenomenological Langevin equation (see, e.g., in [2]): ˙xt=−λ(xt−¯x) +/radicalBig 2γ wt which can be generalized for time-dependent ¯ xt. To lowest order in ˙¯x the result (17) comes out. In higher orders the Langevin equa tion gives different results from the present proposal. 6

arXiv:physics/0001004v1 [physics.acc-ph] 3 Jan 2000SINGLE–PASS LASER POLARIZATION OF ULTRARELATIVISTIC POSITRONS. Alexander Potylitsyn Tomsk Polytechnic University, pr. Lenina 2A, Tomsk, 634050, Russia e-mail: pap@phtd.tpu.edu.ru Abstract The new method for producing of the polarized relativistic p ositrons is suggested. A beam of unpolarized positrons accelerated up to a few GeV ca n be polarized during a head-on collision with an intense circularly polar ized lazer wave. After a multiple Compton backscattering process the initial posit rons may lose a substantial part of its energy and, as consequence, may acquire the signi ficant longitudinal polarization. The simple formulas for the final positron ene rgy and polarization degree depended on the laser flash parameters have been obtai ned. The comparison of efficiences for the suggested technique and known ones is ca rried out. Some advantages of the new method were shown. The experiments with polarized electron-positron beams in future linear colliders will furnish a means for studying a number of intriguing physical problems [1]. While the problem of generation and acceleration of longitudinally p olarized electron beams seems to be solved [2], the approach for production of polarized po sitron beams with the required parameters has not been finally defined yet. In [3-7] methods w ere proposed for the generation of longitudinally polarized positrons during e+e−– pair production by circularly polarized photons with the energy ω∼101MeV, which are, in their turn, generated by either passing electrons with the energy ∼102GeV through a helical undulator [3], or through Compton backscattering of circularly polarized laser photons on a beam of electrons with the energy ∼5 GeV [4,5], or through bremsstrahlung of longitudinally polarized ∼50 MeV electrons [6,7]. To achieve the needed intensity of a p ositron source (Ne+,pol∼1010particles/bunch) it is suggested to use an undulator of the l ength L >100 m [8], or to increase the laser power [9], or to use a high-curr ent accelerator of polarized electrons [10]. The present paper considers an alternative way to approach t his problem. A beam of unpolarized positrons from a conventional source b eing cooled in a damping ring and preliminary accelerated to an energy E 0can be polarized during a head-on- collision with a high-intensity circularly polarized lase r wave. It is well known that during Compton backscattering of circu larly polarized laser pho- tons on unpolarized positrons (electrons) with the energy E 0∼100 GeV the scattered photon takes up to 90% of the initial positron energy while th e recoil positron aquires ∼100% longitudinal polarization [11,12]. At E 0≤10 GeV, however, the positron loses too little of its energy during single Compton backscatteri ng (a few percent), and the lon- gitudinal polarization of the recoil positron is, therefor e, of the same order of magnitude. Current advances of laser physics make it possible to obtain parameters of laser flash such 1that the positron successively interacts with N ≫1 identical circularly polarized photons. It is apparent that in this case the positron can lose a substa ntial fraction of its energy (comparable with E 0). To evaluate the resulting polarization of the recoil posi tron, let us consider multiple Compton backscattering in greater detai l. Let us carry out calculations in a positron rest frame (PRF) a nd in a laboratory frame (LF). Following [13], let us write the Compton scattering cr oss section in PRF where spin correlations of three particles will be viewed– initia l photon, and initial and recoil positrons (upon summation over the scattered photon polari zations): dσ dΩ= 2r2 0/parenleftBigk k0/parenrightBig2/braceleftBig Φ0+ Φ2(Pc,/vectorξ0) + Φ 2(Pc,/vectorξ) + Φ 2(/vectorξ0,/vectorξ) + Φ 3(Pc,/vectorξ0,/vectorξ)/bracerightBig (1) Here r 0is the electron classical radius; k 0, k are the initial and scattered photon energy; Pc=±1 is the circular polarization of the initial photon; and /vectorξ0,/vectorξare the spin vectors of the initial and final positrons. Functions Φ 0,Φ2,Φ3were obtained in paper [13]. In (1) and further in the paper use is made of the system of unit s ¯ h = m e= c = 1, unless otherwise indicated. Since the scattered photons are not detected, the cross sect ion (1) has to be integrated over the photon outgoing angles. Due to azymuthal symmetry, it will depend on the average longitudinal polarization components ξ0l, ξlsolely. On this basis we will keep only these components which remain the same in LF. For positrons with γ0≤104(γ0is the Lorentz-factor of the initial positron), the laser photon energy in PRF ( ω0∼1 eV in LF ) will satisfy the relation k0= 2γ0ω0≪1 (2) Using (2) let us write the expression for the scattered photo n energy in PRF: k =k0 1 + k 0(1−cosθ)≈k0[1−k0(1−cosθ)] (3) Hereθis the polar angle of the scattered photon in PRF. Leaving the terms not higher than k2 0, let us write in explicit form the Φ ifunctions derived in [13] for electrons : Φ0=1 8/bracketleftBig 1 + cos2θ+ k2 0(1−cosθ2)/bracketrightBig , Φ2(Pc, ξ0l) =−1 8Pcξ0lk0cosθ , (4) Φ2(Pc, ξl) =−1 8Pcξl(1−cosθ)/bracketleftBig 2k0cosθ−k2 0(cosθ−cos2θ+ sin2θ)/bracketrightBig , Φ2(ξ0l, ξl) =1 8ξ0lξl[1 + cos2θ−k2 0cosθsin2θ], Φ3(Pc, ξ0l, ξl) = 0. 2Upon routine integration we obtain: σ=πr2 0 2/braceleftBig8 3(1−2k0) +4 3Pcξ0lk0(1−2k0) +ξl/bracketleftBig8 3ξ0l(1−2k0) +4 3Pck0/bracketrightBig/bracerightBig (5) It is obvious that in averaging with respect to the initial pa rticles spin and taking the summation with respect to two spin states of the recoil posit ron, instead of (5) we get Klein-Nishina’s cross section for k 0≪1 [11]: σ=8 3πr2 0(1−2k0) (6) From (5) follows that longitudinal polarization of a recoil positron (electron) is deter- mined by both its initial polarization and the circular pola rization of a photon (later the longitudinal polarization indices lwill be omitted): ξ=ξ0∓k0 2Pc 1∓k0 2Pcξ0(7) The upper (lower) sing refers to a positron (electron). If the initial positron is unpolarized ( ξ0= 0), then upon a single interaction with a laser photon the recoil positron becames polarized : |ξ(1)|=|−k0 2Pc| ≪1. (8) In order to consider the next scattering act, let us calculat e the average longitudinal momentum <k/bardbl>along the initial photon direction and the average energy <k>of the scattered photon in PRF using the same approximation as b efore: <k/bardbl>=/integraltextk cosθ/parenleftBigk k0/parenrightBig2Φ0dΩ /integraltext/parenleftBigk k0/parenrightBig2Φ0dΩ=6 5k2 0, (9) <k>=/integraltextk/parenleftBig k k0/parenrightBig2Φ0dΩ /integraltext/parenleftBigk k0/parenrightBig2Φ0dΩ= k0(1−k0). Thus, upon the first event of interaction, the photon in LF aqu ires, on average, the energy < ωsc>=γ0(<k>−β0<k/bardbl>)≈γ0<k>=γ0k0. (10) In (10) β0= 1−γ−2 0 2is the velocity of PRF with respect to LF. 3It is apparent that the recoil positron loses its energy (10) and hence γ(1)=γ0−< ωsc>=γ0(1−k0) =γ0(1−2γ0ω0). (11) In PRF before the second interaction the initial photon, in v iew of (11), will have a lower energy k(1)= 2γ(1)ω0= 2γ0ω0(1−2γ0ω0) = k 0(1−k0), (12) and the recoil positron will have a polarization: ξ(2)=ξ(1)−k(1) 2Pc 1−k(1) 2Pcξ(1). (13) Substituting its value from (8) for ξ(1), we obtain: ξ(2)=−Pck0 2+k(1) 2 1−Pck0 2k(1) 2,|ξ(2)|>|ξ(1)| (14) It follows from (14) that as a result of multiple Compton back scattering the longitudinal polarization of positrons builds up, while their energy dec reases in LF (so-called laser cooling, see [14, 15]). Let us write expressions relating the polarization and ener gy for two subsequent acts of scattering: γ(i+1)=γ(i)(1−2ω0γ(i)), (15) ξ(i+1)=ξ(i)−γ(i)ω0Pc 1−γ(i)ω0PCξ(i). (16) From these we can obtain the equations for the finite differenc es: ∆γ(i)=γ(i+1)−γ(i)= 2ω0γ2 (i), (17) ∆ξ(i)=ξ(i+1)−ξ(i)≈ −ω0Pcγ(i)(1−ξ2 (i)). (18) When N ≫1, instead of (17) and (18) we can arrive at differential equat ions, whose solution with proper initial conditions will yield γ(N)=γ0 1 + 2γ0ω0N(19) ξ(N)=γ0ω0N 1 +γ0ω0N. (20) When deriving the above relation, there was taken the left ci rcular polarization P c=−1 for the sake of simplicity. 4Equations (19) and (20) describe the positron characterist ics after Ncollisions with circularly polarized laser photons. The number of collisio ns N is controlled by the lumi- nosity of the process L: N =Nscat N+e= N 0L = N 08 3πr2 0 2π(σ2 e++σ2 ph). (21) In (21) N 0= A/ω0is the number of photons per laser flash, A is the laser energy, and σph, σe+, are the laser focus and positron bunch radii. We can expect t hat after cooling in the damping ring σe+≪σph. In this case, substituting (21) into (19) and (20), we obtai n the following simple formulas for positron’s characterist ics: γ(N)=γ0 1 + 2µ, (22) ξ(N)=µ 1 +µ, (23) which depend on the dimensionless parameter µsolely µ=γ0ω0N =4 3A mc2γ0/parenleftBigr0 σph/parenrightBig2. (24) It follows from (24) that the µparameter depends linearly on the laser flash energy and the initial positron energy, but it is inversely proportion al to the laser focus area and does not depend on the interaction time (duration flash). Having w ritten (22) as: γ0 γ(N)= 1 + 2 µ , (25) we will compare the result with the estimate by V. Telnov [15] obtained in a classical approximation. Substituting into (24) the estimate used in [15]σ2 ph=λ0le 8π(λ0is the laser photon wavelength and l eis the positron bunch length), we get : γ0 γ(N)= 1 +64 3A mc2γ0πr2 0 λ0le. (26) The resulting expression is closed to a similar one in [15] bu t the second term in (26) is by a factor of πsmaller. This dicrepancy is connected with rough calculati on of the luminosity (constant area of the laser focus) used in (21). By way of illustration let us consider an example (see [15]): γ0= 104,A = 5 J , λ0= 500 nm ,le= 0.2 mm, σ2 ph=λ0le/8π . (27) In this case µ= 1.6 and, therefore, γ(N)≈0.3γ0;ξl≈60%. 5Thus, when a positron bunch interacts with a laser flash of the given parameters, all the positrons acquire longitudinal polarization of about 60%. The change in the polarization sign is obtained by inverting the sign of the circular polari zation of laser radiation. It should be noted that with a proper selection of the sign of c ircular polarization, the process of laser cooling would give rise to a longitudina l polarization increase of the electrons rather than to depolarization of electrons beam ( as in the case of unpolarized laser radiation considered in [15]). Note that, generally speaking, the laser parameters (27) co rrespond to the so-called ”strong” field, when the contribution from non-linear Compt on scattering [4] would be considerably high. Non-linear processes, i.e., simultaneous scattering of a f ew laser photons on the moving positron, are characterized by an increase in the effective p ositron mass in PRF, which, in its turn, leads to a decrease in the Lorentz-factor and the energy transferred to the positron through scattering. It is to be expected that the µparameter (24) for a fixed value of the laser flash energy A will be sufficiently lower for a non-linear case as compared to the linear one, and hence a lower attainable polarization (23). In order to reach a linear mode of the Compton scattering proc ess, one has to stretch the laser flash (the length of its interaction with the positr on bunch) (see, for instance, [16]). In conclusion, let us estimate the energy A +,polnecessary to obtain one polarized positron with the energy E +>101MeV and the longitudinal polarization ξl>0,5 i.e., the parameters acceptable for consequent acceleration. i) According to the estimates [8] an electron with the energy E−∼200 GeV passing through a helical undulator of the length L ∼150 m can generate a number of circu- larly polarized photons needed to obtain one polarized posi tron to be later accelerated (conversion efficiency η= N e+,pol/Ne−≈1). Hence, A +,pol∼E−/η= 200 GeV. ii) The author of the paper [9] considered a scheme for produc tion of ∼Ne+= 109 polarized positrons when the laser radiation of the total en ergy A Σ∼20 J is scattered on an electron bunch with E −= 5 GeV and N e−= 1010e−/ bunch. Thus A+,pol≈Ne−E−+ AΣ Ne+≈170 GeV . iii) In [6] the author estimated the conversion efficiency for longitudinally polarized electrons with the energy E −= 50 MeV: η≈10−3. Therefore A +,pol∼E−/η= 50 GeV. iv) For the method suggested in the present paper, evaluatio n of A +,polcan be made for parameters of the unpolarized positron source used in SL AC [17]. The conversion efficiency for the electron energy E −= 33 GeV equals: η≈1. 6Therefore, for a bunch with N e+= 1010and the positron energy E 0= 5 GeV interacting with the laser flash (A = 5 J) we have: A+,pol=E− η+ E0+A Ne+= 33 GeV + 5 GeV + 3 GeV ∼40 GeV . Thus, the scheme proposed here seems to be most energy effecti ve. The author is grateful to V. Telnov and J. Clendenin for stimu lating discussions. References 1. P.M. Zerwas. Preprint DESY 94-001, 1994. 2. J.E. Clendenin, R. Alley, J. Frish, T. Kotseroglou, G.Mul hollan, D.Schultz, H.Tang, J.Turner and A.D. Yeremian. The SLAC Polarized Electron Sou rce. AIP Conf. Proceedings, N. 421, pp. 250-259, 1997. 3. V.E. Balakin, A.A.Mikhailichenko. Preprint INP 79-85, N ovosibirsk, 1979. 4. Yung Su Tsai. Phys.Rev.D, v.48 (1993), pp.96-115. 5. T. Okugi, Y. Kurihara, M. Chiba, A. Endo, R. Hamatsu, T. Hir ose, T. Kumita, T. Omori, Y. Takeuchi, M. Yoshioka. Jap.J.Appl.Phys. v. 35(19 96), pp. 3677-3680. 6. A.P. Potylitsyn. Nucl. Instr. and Meth. v. A398 (1997), pp .395-398. 7. E.G. Bessonov, A.A. Mikhailichenko. Proc. of V European P article Accelerator Conference, 1996, pp. 1516-1518. 8. K. Flottmann. Preprint DESY 93-161, 1993. 9. T. Omori. KEK- Proceedings 99-12, 1999, pp. 161-179. 10. T. Kotseroglou, V. Bharadwaj, J.E. Clendenin, S. Ecklun d, J. Frisch, P. Krejcik, A.V. Kulikov, J. Liu, T. Maruyama, K.K. Millage, G. Mulholla n, W.R. Nelson, D.C. Schultz, J.C. Sheppard, J. Turner, K. Van Bibber, K. Flo ttmann, Y. Namito. Particle Accelerator Conference (PAC’99). Proceedings (t o be published). 11. H. Tolhoek. Rev. of Mod. Phys. v. 28 (1956), pp. 277-298. 12. G.I.Kotkin, S.I. Polityko, V.G. Serbo. Physics of Atomi c Nuclei, v. 59 (1996), pp. 2229-2234. 13. F.W.Lipps, H.A.Tolhoek. Physika, v. 20 (1954), pp. 395- 405. 14. P.Sprangle, E.Esarey. Phys. Fluids. v.B4 (1992), pp. 22 41-2248. 15. V. Telnov. Phys. Rev. Lett. v.78 (1997), pp.4757-4760. 716. I.V. Pogorelsky, I. Ben-Zvi, T. Hirose. BNL Report No.65 907, October, 1998. 17. S. Ecklund. SLAC-R-502 (1997), pp. 63-98. 8

arXiv:physics/0001005v1 [physics.class-ph] 3 Jan 2000An asymptotic form of the reciprocity theorem with applications in x-ray scattering Ariel Caticha Department of Physics, University at Albany-SUNY, Albany, NY 12222, USA. ariel@cnsvax.albany.edu Abstract The emission of electromagnetic waves from a source within o r near a non-trivial medium (with or without boundaries, crystall ine or amor- phous, with inhomogeneities, absorption and so on) is somet imes studied using the reciprocity principle. This is a variation of the m ethod of Green’s functions. If one is only interested in the asymptotic radia tion fields the generality of these methods may actually be a shortcoming: o btaining expressions valid for the uninteresting near fields is not ju st a wasted ef- fort but may be prohibitively difficult. In this work we obtain a modified form the reciprocity principle which gives the asymptotic r adiation field directly. The method may be used to obtain the radiation from a pre- scribed source, and also to study scattering problems. To il lustrate the power of the method we study a few pedagogical examples and th en, as a more challenging application we tackle two related problem s. We calcu- late the specular reflection of x rays by a rough surface and by a smoothly graded surface taking polarization effects into account. In conventional treatments of reflection x rays are treated as scalar waves, p olarization effects are neglected. This is a good approximation at grazin g incidence but becomes increasingly questionable for soft x rays and UV at higher incidence angles. PACs: 61.10.Dp, 61.10.Kw, 03.50.De 1 Introduction The principle of reciprocity can be traced to Helmholtz in th e field of acoustics. It states that everything else being equal the amplitude of a wave at a point Adue to a source at point Bis equal to the amplitude at Bdue to a source atA. With its extension to electromagnetic waves by Lorentz [1] and later to quantum mechanical amplitudes [2], the applicability to al l sorts of fields was made manifest. Nowadays the principle is regarded as a symme try of Green’s functions when the source point and the field point are revers ed. This symmetry is actually quite general. As shown in [3] the conditions of t ime-reversal invari- 1ance and hermiticity of the Hamiltonian are sufficient to guar antee reciprocity, but they are not necessary; in fact, reciprocity holds even i n the presence of complex absorbing potentials. In the case of electromagnet ic waves the only requirement is that the material medium be linear and descri bed by symmetric permittivity and permeability tensors [4][5]. This exclud es plasmas and ferrite media in the presence of magnetic fields. In the field of x-ray optics the principle was used by von Laue [ 6] to explain the diffraction patterns generated by sources within the cry stal, the so-called Kossel lines [7]. More recently there has been a widespread r ecognition that these interference patterns contain information not just a bout intensities but also about phases and can be thought of as holographic record s from which real space images of the location of the internal sources can be re constructed. Thus, under the modern name of ‘x-ray holography’ there has been a c onsiderable revival of interest in this subject [8]. However, powerful as it is, the usual formulation of the reci procity principle suffers from a rather serious drawback: it refers to the excha nge of source and fieldpoints . As a consequence, a careful application of the principle re quires one to consider the emission of spherical waves which in crystal line media or even in the mere presence of plane boundaries, can be surprisingl y difficult (recall e.g.studying the radiation by an antenna in the vicinity of the co nducting surface of the Earth [9] or of layered media [10]). Furthermo re, one is typically interested in the asymptotic radiation fields so the relevan t exchange should involve a source point herewith a field point at infinity . These technical difficulties have not deterred the users of re ciprocity from using the principle to make valuable predictions, but a high price has been paid. The required asymptotic limits are usually taken verbally and no accounts are given of where and how spherical waves are replaced by plane w aves. Such sleights of hand, because skillfully performed, have not le ad to wrong results, but intensities are predicted only up to undetermined propo rtionality factors and this excludes applications to classes of problems where absolute intensities are needed. Moreover one is left with the uneasy feeling that the validity of the predictions is justified mostly on the purely pragmatic g rounds that for the problem at hand they seem to work which, again, limits applic ations to problems that are already familiar. The main goal of this paper (section 2) is to obtain a modified f orm of the reciprocity theorem that gives the asymptotic radiation fie lds directly and that accommodates plane waves and both point and extended source s in a natu- ral way. Remarkably the resulting expressions, which inclu de all the relevant proportionality factors and yield absolute, not just relat ive intensities, are very simple. For many problems the Asymptotic Reciprocity Theorem (ART) obtained here represents an improvement not only over the usual form o f the reciprocity theorem but also over the method of Green’s functions. Compu ting the Green’s function requires solving a boundary value problems for sph erical waves in the presence of plane boundaries and/or periodic media; this ma y well be an in- tractably difficult problem. Furthermore, a considerable eff ort is wasted by first 2obtaining both near fields and far fields and then discarding t he uninteresting near fields. The ART is a shortcut that discards the near fields before, rather than after they are computed. To illustrate the power of the method we consider several app lications. The first three (section 3) are brief pedagogical examples of inc reasing complexity. First the ART is used to calculate the fields radiated by an arb itrary prescribed source in vacuum; next as an application to scattering probl ems we reproduce the kinematical theory of diffraction by crystals. The third example, the radi- ation by a current located near a plane dielectric boundary, is straightforward when the ART is used but not if other methods are used. One must emphasize that what is new in these examples are not the results, but the method; the first two are standard textbook material, a special case of the thi rd is treated in [9]. As a more involved application of the ART, in section 4 we comb ine ideas from the three previous examples to study two other related scatt ering problems, the specular reflection of polarized x rays by a rough surface and by a continuously graded surface. The technique of the grazing-incidence reflection of x-rays has received con- siderable attention [11]-[17] from both the theoretical an d the experimental sides as a means to obtain structural information about surfaces. The effect of sur- face roughness on the reflection is taken into account by mult iplying the Fresnel reflectivity of an ideal sharp and planar surface by a “static Debye-Waller” fac- tor. The problem is to calculate this corrective factor. The calculation has been carried out in several different approximations. The Ra yleigh or Born ap- proximation [11] is satisfactory for rough surfaces with lo ng lateral correlation lengths but for x-rays the situations of interest generally involve short lateral correlation lengths. Here other approximations such as the distorted-wave Born approximation [13][14] and the Nevot-Croce approximation [15] are used. For variations and interpolations between these two methods se e [16], and for a gen- eralization to surfaces with non-Gaussian roughness and to graded interfaces of arbitrary profile see [17]. In these treatments ([14] is an ex ception) the x rays are treated as scalar waves. One expects this approximation to hold at graz- ing incidence but at higher incidence angles ( e.g., for soft x rays) its validity becomes increasingly questionable. Using a modified first Bo rn approximation Dietrich and Haase [14] took the vector character of the x ray s into account but they point out that the validity of their approximation is no t in general easy to assess and they restrict themselves to studying special int erface profiles. In section 4 we study this problem using a different approxima tion; we use the ART to develop approximations of the Nevot-Croce type [1 7]. There is, of course, a trivial polarization dependence that is already d escribed by Fresnel formulas for the reflectivity of the ideal flat step surface. T he question we address here is whether the “static Debye-Waller” factor sh ows any additional dependence on polarization. The final result is remarkably s imple: the “static Debye-Waller” factor for the specularly reflected vector wa ves is the same for both polarizations and coincides with that for scalar waves . Finally, some brief concluding remarks are collected in section 5. 3Figure 1: (a) In the usual form of the reciprocity theorem the surface Sencloses the medium, and all sources. (b) For the asymptotic form of th e reciprocity theorem the connecting field /vectorEcis a radiation field, its source lies outside the surface S=S++S′. 2 The reciprocity theorem and its asymptotic form We wish to calculate the asymptotic radiation fields /vectorEand/vectorHgenerated by a prescribed current /vectorJ(t,/vector r) located near or within a linear medium, Di=εijEjand Bi=µijHj. We will assume that the tensors εij(/vector r) and µij(/vector r) are symmetric, but otherwise the situation remains quite general, the medium may have an i rregular shape, or be inhomogeneous, crystalline or amorphous, absorbing, dispersive, etc. As in the usual deduction of the reciprocity theorem (see e.g., [4]), we con- sider a second set of fields /vectorEcand/vectorHc, which we will call the “connecting fields”, generated by a source /vectorJc(see fig.1a). For simplicity we will also assume that all fields and sources are monochromatic /vectorE=/vectorE(/vector r)e−iωt,/vectorJ=/vectorJ(/vector r)e−iωt, etc. For linear media this is not a restriction. From Maxwell’s equations ∇ ×/vectorE=iK/vectorB and ∇ ×/vectorH=−iK/vectorD+4π c/vectorJ, (1) where K≡ω/c, one easily obtains the following identity ∇ ·/parenleftBig /vectorE×/vectorHc−/vectorEc×/vectorH/parenrightBig =4π c/parenleftBig /vectorEc·/vectorJ−/vectorE·/vectorJc/parenrightBig , 4which, on integrating over a large volume Vbounded by the surface S, can be rewritten as /integraldisplay S/parenleftBig /vectorE×/vectorHc−/vectorEc×/vectorH/parenrightBig ·d/vector s=4π c/integraldisplay V/parenleftBig /vectorEc·/vectorJ−/vectorE·/vectorJc/parenrightBig dv. (2) This expression simplifies if one deals with point sources. F or example, consider oscillating point dipoles /vector poe−iωtand/vector pce−iωt, located at /vector roand/vector rcrespectively. The current density /vectorJis given by /vectorJ=−iω/vector poδ(/vector r−/vector ro)e−iωtand/vectorJcis given by an analogous expression. Further simplification is achieve d if one assumes that the surface Sis so remote that the surface integral is negligibly small [1 8], then /vectorEc(/vector ro)·/vector po=/vectorE(/vector rc)·/vector pc. (3) This is the usual form of the reciprocity theorem; it says tha t if we know /vectorEcat the location of /vector powe can calculate /vectorEat the location of /vector pc. This elegant result takes us a long way toward a final answer for /vectorE, but the remaining problem of calculating /vectorEc, that is, the calculation of how the spherical wave generate d by /vector pcis scattered by the medium, can still be too difficult. A more useful version of the theorem can be obtained once one r ealizes that the connecting field is merely a tool that codifies inform ation about the influence of the non-trivial medium. Above, the field /vectorEchas been introduced by first specifying a source /vectorJc, but clearly this is an unnecessary additional complication. In fact, since the most convenient /vectorJcis that which results in the simplest /vectorEcit is best to focus attention directly on the field rather than its source. Thus we move /vectorJcoutside the surface S, to infinity (see fig.1b) so that throughout the volume Vthe connecting field /vectorEcis a pure radiation field. Furthermore, let the surface Sitself be so distant that on Sitself both /vectorEand /vectorEcarevacuum radiation fields. Then /integraldisplay S/parenleftBig /vectorE×(∇ ×/vectorEc)−/vectorEc×(∇ ×/vectorE)/parenrightBig ·d/vector s=4πiK c/integraldisplay V/vectorEc·/vectorJ dv. (4) At this point it is not yet clear that this form of the reciproc ity theorem is simpler than eq. (3) but one remarkable feature can already b e seen: eq. (4) relates the field /vectorEat a distant surface Sto its source /vectorJwithin a nontrivial medium without having to calculate /vectorEin the vicinity of /vectorJ. The “connection” between the distant radiation field /vectorEand its source /vectorJis achieved through the much simpler (i.e., hopefully calculable) “connecting” fie ld/vectorEc. To bring eq. (4) into a form that is manifestly simpler than (3 ) the surface Sis chosen as a cube with edges of length L→ ∞. In fig.1b the upper face, defined by a constant zcoordinate, z=z+, has been singled out as S+, the remaining seven faces are denoted S′. On the upper face S+we write the field /vectorEas a superposition of outgoing plane waves of wave vector /vectorksatisfying /vectork·/vectork=ω2/c2=K2andkz>0, /vectorE(/vector r) =/integraldisplay kz>0d3k (2π)32πδ(k−K)/vectorE(/vectork)ei/vectork·/vector r, (5) 5where, in a self-explanatory notation, /vectork·/vector r=/vectork⊥·/vector r⊥+kzz+. It is here, by the very act of writing /vectorEin this form, that the asymptotic limit of discarding near fields is being taken. For z << z +additional terms describing the near fields should be included. The integral over dkzis most easily done using δ(k−K) =K kz/bracketleftbigg δ/parenleftbigg kz−/radicalBig K2−k2 ⊥/parenrightbigg −δ/parenleftbigg kz+/radicalBig K2−k2 ⊥/parenrightbigg/bracketrightbigg .(6) The result is /vectorE(/vector r) =/integraldisplayd2k⊥ (2π)2K kz/vectorE(/vectork)ei/vectork·/vector r, (7) where kz= +/radicalbig K2−k2 ⊥. The choice of the connecting field /vectorEcis dictated purely by convenience. A particularly good choice for /vectorEcis the superposition of an incoming plane wave of unit amplitude (we use ˆ to denote vectors of unit length) a nd wave vector /vectorkc, with /vectorkc·/vectorkc=K2andkcz<0, plus all the waves scattered by the medium, /vectorEc(/vector r) = ˆecei/vectorkc·/vector r+/vectorE′ c(/vector r). (8) OnS+, the scattered field /vectorE′ cis a superposition of outgoing plane waves and is given also in a form analogous to eq. (7), /vectorE′ c(/vector r) =/integraldisplayd2k′ ⊥ (2π)2K k′ z/vectorE′ c(/vectork′)ei/vectork′·/vector r, (9) with/vectork′·/vector r=/vectork′ ⊥·/vector r⊥+k′ zz+andk′ z= +/radicalBig K2−k′2 ⊥. Now we are ready to calculate the surface integral on the left hand side of eq. (4). Substituting (8) into (4) the integral over S+separates into two terms, one due to the incoming plane wave ˆ ecei/vectorkc·/vector r, and the other due to the scattered waves /vectorE′ c(/vector r). The first term is I1=/integraldisplay S+dxdy ˆez·/parenleftBig /vectorE×(i/vectorkc׈ec)−ˆec×(∇ ×/vectorE)/parenrightBig ei/vectorkc·/vector r, (10) and substituting (7) its evaluation is straightforward. Th e integral over dxdy yields (2 π)2δ(/vectork⊥+/vectorkc⊥). Since k2=k2 c=K2,kcz<0 and kz>0 this implies that the plane waves superposed in (7) yield a vanishing cont ribution except when /vectork=−/vectorkc. Thus, I1=−2iKˆec·/vectorE(−/vectorkc) (11) The contribution of the scattered waves /vectorE′ c(/vector r), I2=/integraldisplay S+dxdy ˆez·/parenleftBig /vectorE×(∇ ×/vectorE′ c)−/vectorE′ c×(∇ ×/vectorE)/parenrightBig , (12) 6is calculated in a similar way. Substitute (7) and (9) and int egrate over dxdy to obtain a delta function. This eliminates all Fourier comp onents except those with/vectork⊥=−/vectork′ ⊥. Since k2=k′2=K2, and both kz, k′ z>0 this implies kz= k′ z>0. Thus, I2=/integraldisplayd2k′ ⊥ (2π)2K2 k′2 zˆez·/bracketleftBig /vectorE(/vectork)×/parenleftBig i/vectork′×/vectorE′ c(/vectork′)/parenrightBig −/vectorE′ c(/vectork′)×/parenleftBig i/vectork×/vectorE(/vectork)/parenrightBig/bracketrightBig ,(13) where /vectork=−/vectork′+ 2k′ zˆez. Further manipulation using /vectork′·/vectorE′ c(/vectork′) =/vectork·/vectorE(/vectork) = 0 gives ˆez·[···] = 0, so that I2= 0. (14) According to eq. (11) and (14) the only contributions to the s urface integral over the distant plane S+come from products of outgoing with incoming waves. Products of two outgoing waves yield vanishing contributio ns. This result ap- plies also to the remaining seven faces of the cube S. Since on each of these faces there are only outgoing waves we find that the integral o verS′makes no contribution to the left hand side of (4). Incidentally, thi s argument completes our previously unfinished deduction of the usual form of the r eciprocity theo- rem, eq. (3): if both sources /vectorJand/vectorJcare internal to the surface Sthe surface integral in eq. (2) vanishes because it only involves produc ts of outgoing waves. Substituting (11) into (4) leads us to the main result of this paper, the asymptotic reciprocity theorem, /hatwidee·/vectorE(/vectork) =−2π c/integraldisplay V/vectorEc·/vectorJ dv. (15) In words: The field /vectorE(/vectork)radiated in a direction /vectorkwith a certain polarization /hatwideeis −2π/ctimes the “component” of the source /vectorJ(/vector r)“along” a connecting field /vectorEc(/vector r)with incoming wave vector /vectorkc=−/vectorkand polarization /hatwideec=/hatwidee. Typically one is interested in the intensity radiated into a solid angle dΩ; since the amplitude /vectorE(/vectork) that appears in (7) and (15) is not quite the Fourier transform of /vectorE(/vector r) it may be useful to derive an explicit expression for dW/d Ω. The total power radiated through the plane S+is given by the flux of the time- averaged Poynting vector,c 8πRe [/vectorE×/vectorB∗], W=/integraldisplay d2x⊥c 8πRe [/vectorE×/vectorB∗]·ˆez=/integraldisplay dΩdW dΩ(16) Using (7) and d2k⊥=k⊥dk⊥dφ=KkzdΩ (where φis the usual azimuthal angle about the zaxis) we get W=c 8π/integraldisplayd2k⊥ (2π)2K kz/vectorE(/vectork)·/vectorE∗(/vectork), (17) so that dW dΩ=c 8π/parenleftbiggK 2π/parenrightbigg2 /vectorE(/vectork)·/vectorE∗(/vectork). (18) In the next section we offer a few illustrative examples of the ART in action. 73 Some simple examples The ART, eq. (15), holds for an arbitrary linear medium. In pa rticular it holds if the medium is vacuum. Our first trivial example is the radiati on by a prescribed current in vacuum. Next, to show that the ART can be used to stu dy scattering problems we deal with another equally trivial example, the k inematical theory of diffraction by crystals. The third example, the radiation by currents located near a dielectric boundary, is also straightforward. What i s remarkable here is the ease with which the results are obtained compared to conv entional methods [9][10]. 3.1 Radiation in vacuum In this case the connecting field is just an incoming plane wav e,/vectorEc(/vector r) = ˆecei/vectorkc·/vector r. The ART, eq. (15), gives the radiated field with polarization ˆe= ˆecas /hatwidee·/vectorE(/vectork) =−2π c/integraldisplay V/vectorJ(/vector r)·ˆe e−i/vectork·/vector rdv=−2π cˆe·/vectorJ(/vectork), (19) so that /vectorE(/vectork) =2π cˆk×(ˆk×/vectorJ(/vectork)). (20) The radiated power, eq. (18), is dW dΩ=K2 8πc/vextendsingle/vextendsingle/vextendsingleˆk×(ˆk×/vectorJ(/vectork))/vextendsingle/vextendsingle/vextendsingle2 , (21) as expected. (For radiation by a point dipole just substitut e/vectorJ(/vectork) =−icK/vector p.) 3.2 Bragg diffraction Consider a crystal described by its dielectric susceptibil ityχ(/vector r) [19] which for x rays is quite small (typically about 10−5or less). An incident plane wave /vectorEoei/vectorko·/vector rinduces a current /vectorJ(/vector r) =−iω/vectorP(/vector r) =−iω 4πχ(/vector r)/vectorEoei/vectorko·/vector r, (22) which radiates. The connecting field needed to calculate thi s radiation is a simple incoming plane wave, /vectorEc(/vector r) = ˆecei/vectorkc·/vector r, and the ART, eq. (15), gives the radiated field as /vectorE(/vectork) =−iω 2cˆk×(ˆk×/vectorEo)χ(/vectork−/vectorko). (23) The scattered field is proportional to the Fourier transform of the susceptibility of the medium; for a periodic medium this is Bragg diffraction . 83.3 Radiation in the vicinity of a reflecting surface Consider a current /vectorJin(/vector r) located within a uniform medium with dielectric sus- ceptibility χ0occupying the region z <0 (see fig. 2). To calculate the radiation in the direction /vectorkwith polarization ˆ ewe choose as connecting field an incom- ing plane wave with wave vector /vectorkc=−/vectorkand unit amplitude ˆ ec= ˆeplus the corresponding reflected and transmitted waves, /vectorEc(/vector r) =/braceleftBigg ˆecei/vectorkc·/vector r+/vector εcrei/vectorkcr·/vector r,forz >0 /vector εctei/vectorkct·/vector r, forz <0(24) The various wave vectors are given by /vectorkc=−Kcosθˆex−qˆez=−/vectork , (25) /vectorkcr=−Kcosθˆex+qˆez, (26) /vectorkct=−Kcosθˆex−¯qˆez, (27) where K=ω/c, and the normal components qand ¯qare given by q=Ksinθand ¯ q=K/parenleftbig sin2θ+χ0/parenrightbig1/2. (28) The amplitudes /vector εcrand/vector εctof the reflected and transmitted waves are given by the Fresnel expressions /vector εcr=rsˆecrwhere rs=q−¯q q+ ¯qˆecr·ˆect ˆec·ˆect, (29) and /vector εct=tsˆectwhere ts=2q q+ ¯q1 ˆec·ˆect. (30) (ˆecrand ˆectare unit vectors describing the polarization of the specula r reflected and transmitted waves.) Then for a source /vectorJin(/vector r) located within the medium, the ART, eq.(15), gives the radiated field as /hatwidee·/vectorE(/vectork) =−2π c/integraldisplay z<0/vectorJin(/vector r)·/vector εctei/vectorkct·/vector rdv . (31) On the other hand, had the source /vectorJout(/vector r) been located outside the dielectric medium ( z >0) the corresponding radiated field would be /hatwidee·/vectorE(/vectork) =−2π c/integraldisplay z>0/vectorJout(/vector r)·/parenleftBig ˆecei/vectorkc·/vector r+/vector εcrei/vectorkcr·/vector r/parenrightBig dv . (32) For an oscillating dipole on the zaxis, /vectorJ(/vector r) =−iω/vector pδ(/vector r−zpˆez), eq.(31) and (32) give /hatwidee·/vectorE(/vectork) =/braceleftbigg 2πiK/parenleftbig /vector p·ˆee−iqzp+/vector p·ˆecrrseiqzp/parenrightbig ifzp>0 2πiK /vector p ·ˆecttse−i¯qzp ifzp<0.(33) 9Figure 2: The connecting field for radiation in the presence o f a reflecting medium includes reflected and transmitted waves. Here the so urce/vectorJinis shown within the medium ( z <0). The power radiated with polarization ˆ e, eq.(18), is dW dΩ=/bracketleftBigc 8πK4(ˆe·/vector p)2/bracketrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 +ˆecr·/vector p ˆe·/vector prse2iqzp/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 forzp>0, (34) and dW dΩ=/bracketleftBigc 8πK4(ˆe·/vector p)2/bracketrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆect·/vector p ˆe·/vector ptse−i¯qzp/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 forzp<0. (35) In these two expressions we can recognize the first factor (in square brackets) as the power radiated by a dipole in vacuum. The second factor accounts for the presence of the dielectric medium. 4 Specular reflection of polarized x rays In this section ideas from the three previous examples are co mbined to study two similar and considerably more involved scattering prob lems, the specular reflection of polarized x rays by a rough surface and by graded interfaces. We show that within approximations of the Nevot-Croce type gra ding and roughness affect the specular reflectivity in a manner that is independe nt of the polarization of the incident radiation. 104.1 Reflection by rough surfaces The dielectric susceptibility χ(/vector r) that describes the rough surface from which we wish to scatter x rays is given by χ(x, y, z ) =/braceleftbigg 0 for z > ζ(x, y) χ0forz < ζ(x, y)(36) where the height ζ(x, y), is a Gaussian random variable with zero mean, /angb∇acketleftζ/angb∇acket∇ight= 0, and variance/angbracketleftbig ζ2/angbracketrightbig =σ2(see fig. 3). To apply the ART it is convenient to rewrite χ(/vector r) as χ(/vector r) =χs(/vector r) +δχ(/vector r), (37) where χs(/vector r) represents a medium with an ideally flat surface at z0, χs(/vector r) =/braceleftbigg0 for z > z 0 χ0forz < z 0(38) andδχ(/vector r) represents the roughness. Let ˆe0ei/vectork0·/vector rbe the incident field. The total scattered field /vector ε(/vector r) includes the wave/vector εs(/vector r) specularly reflected by the step χs(/vector r) plus waves δ/vector ε(/vector r) scattered by δχ(/vector r) /vector ε(/vector r) =/vector εs(/vector r) +δ/vector ε(/vector r). (39) The first term on the right is /vector εs(/vector r) = ˆerrse−2iqz0ei/vectorkr·/vector r, (40) where rs=q−¯q q+ ¯qˆer·ˆet ˆe0·ˆet(41) (ˆerand ˆetare unit vectors describing the polarization of the specula r reflected and transmitted waves). The second contribution in eq.(39) , the field δ/vector ε(/vector r) includes a specular component plus diffusely scattered and e vanescent waves, δ/vector ε(/vector r) =δεrˆerei/vectorkr·/vector r+δ/vector εd(/vector r). (42) Using eq.(7) this may be written as δ/vector ε(/vector r) =/integraldisplayd2k′ ⊥ (2π)2K k′ zδ/vector ε(/vectork′)ei/vectork′·/vector r, (43) where δ/vector ε(/vectork′) =k′ z Kδεrˆer(2π)2δ(/vectork′ ⊥−/vectork0⊥) +δ/vector εd(/vectork′). (44) To calculate δ/vector ε(/vector r) we can proceed exactly as in the previous section (3.3): δ/vector ε(/vector r) is the field radiated by a current δ/vectorJ(/vector r) in the presence of the medium χs(/vector r). The current δ/vectorJ(/vector r) =−iω 4πδχ(/vector r)/vectorE(/vector r), (45) 11Figure 3: The problem of scattering by a rough surface can be t ackled using the ART by adding a fictitious overlayer δχ. originates in the polarization of the roughness δχ(/vector r) by the total electric field /vectorE(/vector r) due to the incident and all scattered waves, including thos e generated by the roughness itself. Thus, the challenge here is that the fie ld/vectorE(/vector r) is itself unknown; an approximation for it must be obtained as part of o ur solution. We can exploit the arbitrariness in the separation of χ(/vector r) into χs(/vector r) plus δχ(/vector r) to suggest a self-consistent approximation for /vectorE. Suppose we choose z0 positive and considerably larger than the roughness σ(see fig. 3). Then δχ(/vector r) represents a fictitious overlayer that extends well into the vacuum; the sign of δχ(/vector r) is opposite to that of χs(/vector r) and in the vicinity of z0they completely cancel out. The field δ/vector ε(/vectork) in a direction /vectorkwith polarization ˆ eis given by eq.(31) /hatwidee·δ/vector ε(/vectork) =iK 2/integraldisplay dv δχ(/vector r)/vectorE(/vector r)·/vector εctei/vectorkct·/vector r, (46) where the connecting field is precisely as in eqs.(24)-(30) e xcept for phase shifts due to the reflecting surface being at z0, /vector εcr=e−2iqz0rsˆecrand /vector εct=ei(¯q−q)z0tsˆect. (47) The reason behind the somewhat surprising choice for z0will now become clear: slightly above z0, in vacuum, the exact field is /vectorE(/vector r) = ˆe0ei/vectork0·/vector r+/vector ε(/vector r) = ˆe0ei/vectork0·/vector r+/vector εs(/vector r) +δ/vector ε(/vector r), (48) but slightly below z0and, in fact, over all of the extension occupied by δχ(/vector r), we are also in vacuum ( δχ(/vector r) and χs(/vector r) cancel each other) and therefore /vectorE(/vector r) is 12given by the same expression (48). The last term δ/vector ε(/vector r), given by (42), includes some weak diffusely scattered and evanescent waves δ/vector εd(/vector r). Our approximation consists of neglecting them. Therefore, /vectorE(/vector r)≈ˆe0ei/vectork0·/vector r+rˆerei/vectorkr·/vector r, (49) where the specular reflections by χs(/vector r) and δχ(/vector r) have been combined into the single, and still unknown, reflection coefficient r, r=rse−2iqz0+δεr. (50) Substituting into eq.(46) yields /hatwidee·δ/vector ε(/vectork) =−iKχ0 2/integraldisplay dxdy/integraldisplayz0 ζ(x,y)dz/bracketleftBig ˆe0ei/vectork0·/vector r+rˆerei/vectorkr·/vector r/bracketrightBig ·/vector εctei/vectorkct·/vector r.(51) From now on we focus our attention on the specularly reflected component; let ˆe= ˆec= ˆer, ˆecr= ˆe0,/vectork=/vectorkr=−/vectorkc. Substituting eq.(44) into the left hand side (l.h.s.), using (2 π)2δ(k⊥−k0⊥) = (2 π)2δ(0) =/integraltext dxdy we get l.h.s.=q Kδεr(2π)2δ(0) =q K/parenleftbig r−rse−2iqz0/parenrightbig/integraldisplay dxdy. (52) This shows that the unknown reflection coefficient rwe want to calculate appears in both the left and the right hand sides of (51), as part of the radiated field and also as part of the field that induces the source; eq.(51) perm its a self-consistent calculation of r. The integral over dzin the right hand side of eq.(51) is elementary and the remaining integral over dxdy is performed using the identity /integraltext dxdy e−iQζ(x,y) /integraltext dxdy=/angb∇acketlefte−iQζ/angb∇acket∇ight=e−Q2σ2/2, (53) where ζis a Gaussian random variable with zero mean, /angb∇acketleftζ/angb∇acket∇ight= 0, and variance/angbracketleftbig ζ2/angbracketrightbig =σ2. The right hand side ( r.h.s.) of eq.(51) becomes r.h.s. =Kχ0 2/parenleftbigg/integraldisplay dxdy/parenrightbigg /braceleftbiggˆe0·/vector εct q+ ¯q/bracketleftBig e−i(q+¯q)z0−e−(q+¯q)2σ2/2/bracketrightBig −rˆer·/vector εct q−¯q/bracketleftBig ei(q−¯q)z0−e−(q−¯q)2σ2/2/bracketrightBig/bracerightbigg , (54) which can be further rewritten by substituting /vector εctas given by eq.(47), and using ¯q2−q2=K2χ0, and ˆe0·ˆect ˆe·ˆect=ˆecr·ˆect ˆe·ˆect=ˆer·ˆet ˆe0·ˆetandˆer·ˆect ˆe·ˆect= 1. (55) 13Finally, equating eq.(52) to (54) yields a self-consistent approximation to r, r=q−¯q q+ ¯qˆer·ˆet ˆe0·ˆete−2q¯qσ2=rse−2q¯qσ2. (56) This coincides exactly with the Nevot-Croce result for the p olarization ˆ e0= ˆet= ˆerfor which the ratio ˆ e0·ˆet/ˆer·ˆetis unity, and provides the correct generalization to all polarizations. According to this approximation the s pecular reflection coefficient rhas no polarization dependence beyond that already implici t in the reflection coefficient rsfor the ideal flat step surface; the “static Debye-Waller” factor exp(−2q¯qσ2) is polarization independent. Notice that any possible dependence on the arbitrary choice ofz0has can- celled out. 4.2 Reflection by smoothly graded surfaces The problem of scattering by a smoothly graded interface is s imilar and some- what simpler. Here the susceptibility χ(z) depends only on the normal coor- dinate zand not on the transverse coordinates xandy. This implies that the tangential component of momentum is conserved in the scatte ring; there are no diffuse waves, there is only specular scattering. As before, it is convenient to separate χ(z) into χ(z) =χs(z) +δχ(z), (57) where χs(z) represents an ideally flat surface at z0, χs(z) =/braceleftbigg 0 for z > z 0 χ0forz < z 0(58) andδχ(z) is an overlayer (see fig.4) describing the smooth transitio n from bulk to vacuum. Let ˆe0ei/vectork0·/vector rbe the incident field. The total scattered field /vector ε(/vector r), eq.(39), /vector ε(/vector r) =/vector εs(/vector r) +δ/vector ε(/vector r). (59) includes the wave /vector εs(/vector r) reflected by the step χs(z), eq.(40), plus waves δ/vector ε(/vector r) scattered by the overlayer δχ(z). While diffusely scattered waves are not present inδ/vector ε(/vector r), faint evanescent waves could be; these are weak near field e ffects and we neglect them. Thus δ/vector ε(/vector r) =δεrˆerei/vectorkr·/vector r, (60) and the Fourier expansion, eq.(43), and transform δ/vector ε(/vectork), eq.(44), remain other- wise unchanged. Once again, δ/vector ε(/vector r) is radiated by a current δ/vectorJ(/vector r) =−iω 4πδχ(z)/vectorE(/vector r), (61) where the field /vectorE(/vector r) includes the incident and the unknown reflected waves; /vectorE(/vector r) must be self-consistently obtained as part of the solution . Then the ART, 14Figure 4: The problem of reflection by a graded surface can be t ackled using the ART by adding a fictitious overlayer δχ. The hatched region shows the transition region from δχ= 0 to δχ=−χ0. in the form of eq.(31), gives the field δ/vector ε(/vectork) in a direction /vectorkwith polarization ˆ e as /hatwidee·δ/vector ε(/vectork) =iK 2/integraldisplay dv δχ(z)/vectorE(/vector r)·/vector εctei/vectorkct·/vector r, (62) with the same connecting field given back in eq.(47). The approximation we use for /vectorE(/vector r) is the same as in last section. The arbitrariness of z0can be exploited by choosing it large enough that the overlay er extends well into the vacuum. Near z0the overlayer and the sharp step χs(/vector r) cancel each other out; slightly above z0, in vacuum, the field is /vectorE(/vector r) = ˆe0ei/vectork0·/vector r+rˆerei/vectorkr·/vector r, (63) where ris the unknown reflection coefficient we want to calculate, r=rse−2iqz0+δεr. (64) Slightly below z0and over most of the extension occupied by δχ(z) we are also in vacuum (provided the bulk to vacuum transition is not too g radual) and we approximate /vectorE(/vector r) by the same expression, eq.(63). Substituting into eq.(62 ) yields an equation for r, q K/parenleftbig r−rse−2iqz0/parenrightbig (2π)2δ(k⊥−k0⊥) = 15=iK 2/integraldisplayz0 −∞dz δχ(z)/integraldisplay dxdy/bracketleftBig ˆe0ei/vectork0·/vector r+rˆerei/vectorkr·/vector r/bracketrightBig ·/vector εctei/vectorkct·/vector r.(65) The integral over dxdy yields a delta function, (2 π)2δ(k⊥−k0⊥), and we can substitute ˆ e= ˆec= ˆer, ˆecr= ˆe0,/vectork=/vectorkr=−/vectorkc. The integral over zis conveniently expressed as /integraldisplayz0 −∞dz δχ(z)e−iQz=χ0 iQ[e−iQz0+χ′(Q) χ0], (66) where χ′(Q) is the Fourier transform of dχ(z)/dz, χ′(Q) =/integraldisplay+∞ −∞dzdχ(z) dze−iQz(67) Eq.(66) is proved by integrating the left hand side by parts, using δχ(z0)≈ −χ0, anddδχ(z)/dz=dχ(z)/dz. Using ¯ q2−q2=K2χ0and the identities in eq.(55) the final result is r=rsχ′(¯q+q) χ′(¯q−q). (68) Notice that any possible dependence on the arbitrary choice ofz0has cancelled out. This coincides exactly with the scalar wave result [17] and p rovides the correct generalization to all polarizations. Within these approximations the specular reflection coefficient rhas no polarization dependence beyond that already implicit in the reflection coefficient rsfor the ideal flat step surface; the “static Debye-Waller” factor is polarization independent . To conclude we mention some illustrative examples: (a) The error-function profile χ(z) =χ0√ 2πσ2/integraldisplayz −∞dxexp−/parenleftbiggx2 2σ2/parenrightbigg , (69) gives χ′(¯q+q) χ′(¯q−q)=e−2q¯qσ2, (70) the same factor obtained in the previous section for a Gaussi an rough surface. This is as expected, the error function is the averaged profil e for the Gaussian rough surface. (b) The Epstein (or Fermi distribution) profile [17] χ(z) =χ0 1 +e−z/σ, (71) gives χ′(¯q+q) χ′(¯q−q)=¯q+q ¯q−qsinh[πσ(q−¯q)] sinh[πσ(q+ ¯q)]. (72) 16(c) The triangular profile χ(z) =  χ0 forz <−σ/2 χ0(1−2z/σ) for |z|< σ/2 0 for z > σ/ 2, (73) gives χ′(¯q+q) χ′(¯q−q)=q−¯q q+ ¯qsin[(q+ ¯q)σ/2] sin[(q−¯q)σ/2]. (74) The reliability of these approximations was studied in [17] in the case of scalar waves. There is no reason to expect any difference from the conclusions reached there: the “static Debye-Waller” in eq.(68) provid es a remarkably good approximation for the intensities reflected by interfaces o f arbitrary grading profile even for transition regions that are quite wide ( σas large as several nanometers). The phase of the reflected waves is however more sensitive; eq.(68) provides a good approximation for more abrupt transitions ( σof the order of 1 nmor less). 5 Conclusion The main result of this work, eq.(15), is an asymptotic form o f the reciprocity theorem which can be used as the basis for a practical method f or calcula- tions. The theorem states that the field radiated in the prese nce of a nontriv- ial medium, in a certain direction and with a given polarizat ion, is a suitable ‘component’ of the radiating source. This ‘component’ is to be extracted by introducing an auxiliary ‘connecting’ field which contains the necessary infor- mation about the medium. The practical advantage of the meth od lies in the simplifications achieved by systematically avoiding unnec essary calculations; it thereby allows one to tackle problems of increasing complex ity. In forthcoming papers we will further explore the applicati on of the ART to the study of the dynamical diffraction of radiation generate d by sources within a crystal, the so-called Kossel lines. Even this well explor ed topic has not been exhausted. Of particular interest are situations where the Bragg angle lies close toπ/2 and the Kossel cones degenerate into single beams [20], and situations where the source location is revealed by the oscillatory ‘Pe ndell¨ osung’ structure of the diffraction pattern [21]. Other applications will inc lude a new approach to thermal diffuse scattering under conditions of dynamical diffraction [22]. Acknowledgments . I am indebted to P. Zambianchi and E. Sutter for valuable discussions. References [1] H. A. Lorentz, Proc. Amsterdam Acad., 8, 401 (1905). [2] See e.g., J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (Dover, 1979); E. Merzbacher, Quantum Mechanics (Wiley, 1998). 17[3] D. E. Bilhorn, L. L. Foldy, R. M. Thaler and W. Tobocman, J. Math. Phys. 5, 435 (1964). [4] L. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Contin- uous Media (Butterworth-Heinemann, 1984). [5] J. A. Kong, Electromagnetic Wave Theory (Wiley, New York 1990). [6] M. v. Laue, Ann. Physik 23, 705 (1935). [7] W. Kossel, V. Loeck and H. Voges, Zeit. f. Physik, 94, 139 (1935). [8] M.Tegze and G. Feigel, Europhys. Lett. 16, 41 (1991), Nature (London) 380, 49 (1996); T. Gog et al., Phys. Rev. B 51, 6761 (1995), Phys. Rev. Lett.76, 3132 (1996); P. M. Len et al., Phys. Rev. B 55, R3323 (1997). [9] J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York 1941). [10] W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York 1990); J. R. Wait, Electromagnetic Waves in Stratified Media (IEEE-Oxford, New York 1996). [11] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, MA 1987); J. Lekner, The- ory of Reflection of Electromagnetic and Particle Waves (Martinez Nijof, Dordrecht, Holland). [12] D. G. Stearns, J. Appl. Phys. 65, 491 (1989) and 71, 4286 (1992); A. V. Vinogradov et al., Sov. Phys. JETP 62, 1225 (1985) and 67, 1631 (1988); W. Weber and B. Lengeler, Phys. Rev. B 46, 7953 (1992); J. C. Kimball and D. Bittel, J. Appl. Phys. 74, 887 (1993). [13] S. K. Sinha, et al., Phys. Rev. B 38, 2297 (1988); V. Holy et al., Phys. Rev.B 47, 15896 (1993). [14] S. Dietrich and A. Haase, Phys. Rep. 260, 1-138 (1995). [15] L. Nevot and P. Croce, Revue Phys. Appl. 15, 761 (1980). [16] R. Pynn, Phys. Rev. B 45, 602 (1992); D. K. G. de Boer, Phys. Rev. B 49, 5817 (1994). [17] A. Caticha, Phys. Rev. B 52, 9214 (1995). [18] This is a nontrivial statement; its justification is giv en in the paragraph following eq. 14. [19] To avoid factors of 4 πelsewhere it is usual in x-ray optics to define the susceptibility by 4 πP=χE. [20] P. Zambianchi and A. Caticha, “Dynamical diffraction of x rays generated within the crystal: the case of θB≈π/2.” 18[21] E. Sutter and A. Caticha, “Dynamical diffraction of x ray s generated within the crystal: the Laue case.” [22] P. Zambianchi and A. Caticha, “Thermal diffuse dynamica l diffraction of x rays.” 19JcEcEc E E J JS'S ε(r) ε(r) (b) (a)S+kcr S+ kctkc z θ x Jin

arXiv:physics/0001006v1 [physics.bio-ph] 4 Jan 2000Temporal correlations and neural spike train entropy Simon R. Schultz1and Stefano Panzeri2 1Howard Hughes Medical Institute and Center for Neural Scien ce, New York University, 4 Washington Place, New York, NY 10003, U.S.A. 2Department of Psychology, Ridley Building, University of N ewcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, U.K. Sampling considerations limit the experimental condition s under which information theoretic analyses of neurophysiological data yield reliable result s. We develop a procedure for computing the full temporal entropy and information of ensembles of neura l spike trains, which performs reliably for extremely limited samples of data. This approach also yield s insight upon the role of correlations between spikes in temporal coding mechanisms. The method is applied to recordings from the monkey visual cortex, yielding 1.5 and 0.5 bits per spike for simple and complex cells respectively. PACS numbers: 87.19.Nn,87.19.La,89.70.+c,07.05.Kf Cells in the central nervous system communicate by means of stereotypical electrical pulses called action po- tentials, or spikes [1]. The information content of neural spike trains is fully described by the sequence of times of spike emission. In principle, the pattern of spike times provides a large capacity for conveying information be- yond that due to the code commonly assumed by phys- iologists, the number of spikes fired [2]. Reliable quan- tification of this spike timing information is made diffi- cult by undersampling problems that can increase expo- nentially with the precision of spike observation. While advances have been made in experimental preparations where extensive sampling may be undertaken [3–6], our understanding of the temporal information properties of nerve cells from less accessible preparations such as the mammalian cerebral cortex is limited. This Letter presents an analytical expression which al- lows the ensemble spike train entropy to be computed from limited data samples, and relates the entropy and information to the instantaneous probability of spike oc- currence and the temporal correlations between spikes. This is achieved by power series expansion of the entropy in the time window of observation [7], keeping terms of up to second order, and subtraction of an analytical es- timate of the bias due to finite sampling. Comparison is made with other procedures such as the ‘brute force’ approach [4,9]; the analytical expression is found to give substantially better performance for data sizes of the or- der typically obtained from mammalian neurophysiology experiments, as well as providing insight into potential coding mechanisms. Consider a time period of duration T, associated with a dynamic or static sensory stimulus, during which the activity of Ccells is observed. The neuronal population response to the stimulus is described by the collection of spike arrival times {ta i},ta ibeing the time of the i-th spike emitted by the a-th neuron. The spike time is ob- served with finite precision ∆ t, and this bin width is used to digitise the spike train. The total entropy of the spike train ensemble is H({ta i}) =−/summationdisplay ta iP({ta i})log2P({ta i}), (1)where the sum over ta iis over all possible spike times within Tand over all possible total spike counts from the population of cells. This entropy quantifies the to- tal variability of the spike train. Each different stimu- lus history (time course of characteristics within T) is denoted as s. The noise entropy, which quantifies the variability to repeated presentations of the same stimu- lus, is Hnoise=/an}bracketle{tH({ta i}|s)/an}bracketri}hts, where the angular brackets indicate the average over different stimuli,/summationtext s∈SP(s). The mutual information that the responses convey about which stimulus history invoked the spike train is the dif- ference between these two quantities. These entropies may be expanded as a Taylor series in the time window of measurement, H=THt+T2 2Htt+O(T3). (2) This becomes essentially an expansion in the total num- ber of spikes emitted; the only responses which contribute to order kare those with up to kspikes emitted in total. The conditional firing probabilities can be written P(ta i|tb j;s)≡ra(ta i;s) ∆t/bracketleftbig 1 +γab(ta i, tb j;s)/bracketrightbig +O(∆t2),(3) and are assumed to scale proportionally to ∆ tin the short timescale limit. Only the first order conditional firing probability affects the entropy to second order in the time window. In the above, ra(t;s) is the time- dependent instantaneous firing rate and is measurable from the data. The bar indicates the average over mul- tiple trials in which the same stimulus history was pre- sented. The scaled correlation function γabis measured as [8,10]: γab(ta i, tb j;s) =ra(ta i;s)rb(tb j;s) ra(ta i;s)rb(tb j;s)−1, a/ne}ationslash=borta i/ne}ationslash=tb j γaa(ta i, ta i;s) =−1. (4) Denoting the no spikes event as 0and the joint occur- rence of a spike from cell aat time ta 1and a spike from cell bat time tb 2asta 1tb 2, the conditional response probabilities are:P(0|s) = 1 −C/summationdisplay a=1/summationdisplay ta 1ra(ta 1;s)∆t+1 2/summationdisplay ab/summationdisplay ta 1/summationdisplay tb 2ra(ta 1;s)rb(tb 2;s)/bracketleftbig 1 +γab(ta 1, tb 2;s)/bracketrightbig ∆t2 P(ta 1|s) =ra(ta 1;s)∆t−ra(ta 1;s)C/summationdisplay b=1/summationdisplay tb 2rb(tb 2;s)/bracketleftbig 1 +γab(ta 1, tb 2;s)/bracketrightbig ∆t2a= 1,· · ·, C P(ta 1tb 2|s) =1 2ra(ta 1;s)rb(tb 2;s)/bracketleftbig 1 +γab(ta 1, tb 2;s)/bracketrightbig ∆t2a= 1,· · ·, C, b = 1,· · ·, C. (5) where/summationtext abindicates the sum over both aandbfrom 1 to C. The unconditional response probabilities are simply p({ta i}) =/an}bracketle{tp({ta i}|s)/an}bracketri}hts. Inserting p({ta i}) into Eq. 1 and keeping only terms up to and including O(T) yields for the first order total entropy THt=1 ln 2/summationdisplay a/summationdisplay ta 1/an}bracketle{tra(ta 1;s)∆t/an}bracketri}hts−/summationdisplay a/summationdisplay ta 1/an}bracketle{tra(ta 1;s)∆t/an}bracketri}htslog2/an}bracketle{tra(ta 1;s)∆t/an}bracketri}hts. (6) Similarly, inserting p({ta i}|s) yields an expression for the first order noise entropy THnoise twhich is identical, except that there is a single stimulus average /an}bracketle{t·/an}bracketri}htsaround the entire second term. Continuing the expansion, th e additional terms up to second order are: T2 2Htt=1 2 ln2/summationdisplay ab/summationdisplay ta 1/summationdisplay tb 2/braceleftbig/angbracketleftbig ra(ta 1;s)rb(tb 2;s)/bracketleftbig 1 +γab(ta 1, tb 2;s)/bracketrightbig/angbracketrightbig s− /an}bracketle{tra(ta 1;s)/an}bracketri}hts/angbracketleftbig rb(tb 2;s)/angbracketrightbig s/bracerightbig ∆t2 +/summationdisplay ab/summationdisplay ta t/summationdisplay tb 2/angbracketleftbig ra(ta 1;s)rb(tb 2;s)/bracketleftbig 1 +γab(ta 1, tb 2;s)/bracketrightbig ∆t2/angbracketrightbig slog2/an}bracketle{tra(ta 1;s)/an}bracketri}hts/radicalBig/angbracketleftbig ra(ta 1;s)rb(tb 2;s)/bracketleftbig 1 +γab(ta 1, tb 2;s)/bracketrightbig/angbracketrightbig s(7) T2 2Hnoise tt=1 2 ln2/summationdisplay ab/summationdisplay ta 1/summationdisplay tb 2/angbracketleftbig ra(ta 1;s)rb(tb 2;s)γab(ta 1, tb 2;s)/angbracketrightbig s∆t2 +/summationdisplay ab/summationdisplay ta t/summationdisplay tb 2/angbracketleftBigg ra(ta 1;s)rb(tb 2;s)/bracketleftbig 1 +γab(ta 1, tb 2;s)/bracketrightbig ∆t2log2ra(ta 1;s)/radicalBig ra(ta 1;s)rb(tb 2;s)/bracketleftbig 1 +γab(ta 1, tb 2;s)/bracketrightbig/angbracketrightBigg s.(8) It is easily verified that the difference between the total and noise entropies gives the expression for the mutual information detailed in [10]. It has recently been found that correlations, even if independent of the stimulus identity, can increase the information present in a neural population [11,8]. This effect is governed by the similarity of tuning of the cells across the stimuli, and applies both to correlations be- tween neurons and between different spikes emitted by the same neuron [10]. The equations derived above ex- plain how this is realised in terms of entropy. The sec- ond order total entropy can be rewritten in a form which shows that it depends only upon the grand mean firing rates across stimuli, and upon the correlation coefficient of the whole spike train, Γ( ·) (defined across alltrials rather than for fixed stimulus as with Eqn. 4): T2 2Htt=T2 2 ln 2/summationdisplay ab/summationdisplay ta 1/summationdisplay tb 2/an}bracketle{tra(ta 1;s)/an}bracketri}hts/angbracketleftbig rb(tb 2;s)/angbracketrightbig s(9) ×/braceleftbig Γab(ta i, tb j)−[1 + Γ ab(ta i, tb j)] ln[1 + Γ ab(ta i, tb j)]/bracerightbig . It follows that the second order entropy is maximal when Γ(·) = 0, and non-zero correlations in the spike trains (indicating statistical dependence) always decrease thetotal response entropy. However, statistical dependence in the full spike train recorded across all stimuli does not necessarily imply neuronal interaction [12]. If the signal correlation νab(ta i, tb j) =<ra(ta i;s)rb(tb j;s)>s <ra(ta i;s)>s<rb(tb j;s)>s−1 (10) is negative, then positive γ(s)’s reduce the overall sta- tistical dependency and thus increase the entropy of the whole spike train. The entropy increase is maximum for theγvalue which leads to exactly Γ = 0. The effect is opposite when signal correlation is positive. In com- parison, the noise or conditional entropy is always de- creased by γ/ne}ationslash= 0 - at fixed stimulus the only statistical dependencies in the spike train are those measured by γ. The increase/decrease of the population information depending upon the signs of the signal and noise corre- lation is thus not due to a change in the behaviour of the noise entropy, but to the increase or decrease of the total entropy. Statistical dependence always decreases entropy, but neuronal (or spike time) interaction may in- crease entropy itself, by eliminating or reducing the sta- tistical dependencies introduced by other covariations. I tis intriguing to speculate a specific role for synaptic in- teractions in compensating for the statistical dependency introduced by necessary covariations such as firing rate and slow systemic covariations. The rate and correlation functions must be estimated from a limited number of experimental trials, which leads to a bias in each of the entropy components. This bias can be estimated by the method derived in [13]: Hbias=−R 2Nln 2Hnoise bias=−1 2Nln 2/summationdisplay s∈SRs(11) where Ris the number of relevant (non-zero) response bins. For the first order entropy, it is the number of non- zero bins of the dynamic rate function; for the second order entropy, it is the number of relevant bins in the space of pairs of spike firing times. For the noise en- tropy, the conditional response space is used, and for the frequency entropy, the response space is that of the full temporal words. There is some subtlety as to how the number of ‘rele- vant’ response bins should be counted. If zero occupancy count is observed, it is ambiguous whether that indicates true zero occupancy probability or local undersampling. Naive counting of the bins based on raw occupancy prob- abilities results in underestimation of the bias (in fact providing a lower bound upon it) and thus underesti- mation of the entropy. An alternative strategy is to use Bayes’ theorem to reestimate the expectation value of the number of occupied bins, as described in [13]. These procedures were compared by estimating the entropy of a time-dependent simulated Poisson process for different data sizes. The series estimator using the Bayesian bias estimate is the only one which gives accept- able performance in the range of 10-20 trials per stimulus, although there is little to choose between it and the naive series estimator above about 50 trials. The uncorrected frequency (‘brute force’) estimator is inadequate. Also shown in Fig. 1 is the Ma lower bound upon the entropy [14], which performs comparatively poorly. The Ma bound has been proposed as a useful bound which is relatively insensitive to sampling problems [6]. The Ma bound is tight only when the probability distribution of words at fixed spike count is close to uniform; this is not the case in general. To understand the behaviour of the Ma bound for short time windows, we calculated se- ries terms. The Ma entropy already differs from the true entropy at first order: THMa t=1 ln 2/summationdisplay a/summationdisplay ta 1/an}bracketle{tra(ta 1;s)∆t/an}bracketri}hts −/summationdisplay a;ta 1/an}bracketle{tra(ta 1;s)∆t/an}bracketri}htslog2/summationtext a;ta 1/an}bracketle{tra(ta 1;s)/an}bracketri}ht2 s∆t /summationtext a;ta 1/an}bracketle{tra(ta 1;s)/an}bracketri}hts(12) The first order approximation coincides with the true en- tropy rate if and only if there are no variations of rate across time and cells. If there were higher frequency ratevariations, or more cells with different response profiles, the Ma bound would be worse than depicted. 10010110210310400.511.522.5 Trials per stimulusEntropy (bits)< < total entropy (Bayes counting) noise entropy (Bayes counting) total entropy (naive counting) frequency entropy (naive counting) frequency entropy (uncorrected) Ma bound 1001011021031040.10.20.30.40.50.6 coding efficiency FIG. 1. Estimates of the entropy of an inhomogeneous Poisson process with mean rate r(t) = 50 sin(2 π50t) spikes/sec. in response to one stimulus, and zero spikes/se c. for a second equiprobable stimulus. 30 ms time windows of data are used for all curves, with a bin width of 3 ms. The ar- rowhead at the upper right corner indicates the true entropy , calculated analytically. Inset : the effect of entropy and in- formation bias on the estimated coding efficiency. Compared are the bias-corrected series entropy (our best estimate; s olid line) and the uncorrected frequency entropy (dot-dashed li ne). The mutual information suffers less from sampling problems than do the individual entropies, since to some extent the biases of the total and noise entropies elim- inate. This is particularly true for a small number of stimuli; the effect will diminish as the number of stim- uli increases, and the bias behaviour of the information can be expected to become worse (see Eqn. 11). A re- lated quantity often used to characterise neural coding, the coding efficiency [15,5] (defined as the mutual infor- mation divided by the total entropy), does not have this elimination property, and in fact compounds the effects of both total and noise entropy biases. This is shown in the inset of Fig. 1, which shows the coding efficiency versus the number of trials of data per stimulus for both the bias-corrected series estimator (solid) and the raw frequency (‘brute-force’ ) estimator (dot-dashed). One might be cautioned against use of the brute-force ap- proach for calculating the information efficiency. To demonstrate their applicability, we applied these techniques to data recorded from the primary visual cor- tex (V1) of anaesthetised macaque monkeys [17]. Theone empirical assumption made in this analysis (Eq. 3 - that the probability of observing a spike at time tigiven that one has been observed at time tjscales with ∆ t) was examined by computing the average conditional spiking probability as the bin width is decreased (i.e. spikes are observed with higher precision). This assumption might be expected to break down if there were spikes synchro- nised with near-infinite precision. For all cells examined, the assumption was valid, as is shown in the inset of Fig. 2. 2 3 4 5 6 7 8 9 1000.511.522.533.544.5 bits in 30 ms Precision ∆t (ms)Entropies Informationcomplex cells simple cells 1 1010−310−210−1100 Precision ∆ t (ms)Probability 00.10.20.3 Coding efficiencyEfficiencies FIG. 2. Temporal entropy and information in spike trains recorded from 7 simple and 14 complex cells in the monkey primary visual cortex. For clarity population mean and stan - dard error are shown. Black symbols at the far right indicate complex cell spike count entropy (diamond) and information (circle). White symbols similarly for simple cells. Coding ef- ficiencies are also shown (lines without error bars; right ax es). Inset : The conditional spiking probability scales with ∆ tas the binwidth becomes small. Each symbol type represents a different cell (black, a complex cell and white, a simple cell ). Fig. 2 shows entropy estimates for two classes of V1 cells. The entropy of the spiking process continued to rise as the observation precision was increased, up to a resolution of 2 ms. For 30 ms time windows and 2 ms bin width, the information rate for the complex cells was 9±1 (s.e.m.) bits/sec., or 0.5 bits per spike. For the simple cells it was 11 ±2 bits/sec. or 1.5 bits per spike. The coding efficiencies of up to 19 and 31% (maximal in the spike count limit) for this type of stimulation were substantially below the >50% efficiencies that have been reported for insect visual neurons [5,6,16]. As neuroscience enters a quantitative phase of develop- ment, information theoretic techniques are being found useful for the analysis of data from physiological experi- ments. Sampling considerations have however prevented their application to many interesting experimental prepa-rations. The methods developed here broaden the scope of the study of neuronal information properties consider- ably. In particular, they make possible the reliable anal- ysis of recordings from both the anaesthetised and awake mammalian cerebral cortex. SRS is supported by the HHMI, and SP by the Well- come Trust. [1] N. Wedenskii, Bull. de l’Acad. de St. Petersbourg XXVIII , 290 (1883); E. D. Adrian, J. Physiol. (Lond.) 61, 49 (1926). [2] D. MacKay and W. S. McCulloch, Bull. Math. Biophys. 14, 127 (1952). [3] F. Theunissen et al., J. Neurophys. 75, 1345, 1996. [4] R. R. de Ruyter van Steveninck et al., Science 275, 1805 (1997); [5] F. Rieke, D. Warland, R. de Ruyter van Steveninck, and W. Bialek, Spikes: exploring the neural code (MIT Press, Cambridge, MA, USA, 1997). [6] S. Strong et al., Physical Review Letters 80, 197 (1998). [7] A number of previous studies have reported first order expansions of the information: W. E. Skaggs, B. L. Mc- Naughton, K. Gothard, and E. Markus, in Advances in Neural Information Processing Systems , eds. S. Hanson, J. Cowan, and C. Giles (Morgan Kaufmann, San Mateo, 1993), Vol. 5, pp. 1030–1037; S. Panzeri et al.,Network 7, 365 (1996).; N. Brenner et al.physics/9902067 . Sec- ond order series expansion of the spike count information from an ensemble of cells was performed in [8]. An alter- native cluster expansion method has also been used by M. de Weese, Network 7, 325 (1996). [8] S. Panzeri et al.Proc. R. Soc. Lond. B 266, 1001 (1999). [9] G. T. Buracas et al., Neuron 20, 959 (1998). [10] S. Panzeri and S. R. Schultz, physics/9908027 . [11] L.F. Abbott and P. Dayan, Neur. Comp. 11, 91-101 (1999); M. W. Oram et al., Trends in Neurosci. 21, 259- 265 (1998). [12] C. F. Brody, Neur. Comp. 11, 1536 (1999). [13] S. Panzeri and A. Treves, Network 7, 87 (1996). [14] S.-K. Ma, J. Stat. Phys. 26, 221 (1981). [15] F. Rieke, D. Warland and W. Bialek, Europhys. Lett. 22, 151 (1993). [16] A. Dimitrov and J. P. Miller, Neurocomputing, in press. [17] The data used was from procedures to extract the ori- entation tuning of V1 cells. The stimuli were sinusoidal gratings of 16 different orientations placed over the re- ceptive field of the cell. Each cycle of the moving grating was considered to be an experimental trial; only cells with ≥96 cycles available were selected from the database, in order to study timing precisions as high as 2 ms. J. R. Cavanaugh, W. Bair and J. A. Movshon, Society for Neu- roscience Abstracts, 24, 1875 (1998); J. R. Cavanaugh, W. Bair and J. A. Movshon, Society for Neuroscience Ab- stracts, 25, 1048 (1999). See J. A. Movshon and W. T. Newsome, J. Neurosci. 16, 7733, 1996 for experimental methods from this laboratory. We thank J. Cavanaugh, W. Bair and J.A. Movshon for making their data avail- able to us.

arXiv:physics/0001007v1 [physics.ins-det] 4 Jan 2000The preliminary results of fast neutron flux measurements in the DULB laboratory at Baksan J.N.Abdurashitova, V.N.Gavrina, A.V.Kalikhova, A.A.Klimenkoa,b, S.B.Osetrova,b, A.A.Shikhina, A.A.Smolnikova,b, S.I.Vasilieva,b, V.E.Yantza, O.S.Zaborskayaa. aInstitute for Nuclear Research, 117312 Moscow, Russia bJoint Institute for Nuclear Research, 141980 Dubna, Russia (4 Jan 2000) One of the main sources of a background in underground physic s experiments (such as the investi- gation of solar neutrino flux, neutrino oscillations, neutr inoless double beta decay, and the search for annual and daily Cold Dark Matter particle flux modulation) a re fast neutrons originating from the surrounding rocks. The measurements of fast neutron flux in t he new DULB Laboratory situated at a depth of 4900 m w.e. in the Baksan Neutrino Observatory ha ve been performed. The relative neutron shielding properties of several commonly availabl e natural materials were investigated too. The preliminary results obtained with a high-sensitive fas t neutron spectrometer at the level of sensitivity of about 10−7neutron cm−2s−1are presented and discussed. PACS numbers: 06.90.+v, 29.30.Hs INTRODUCTION It is well known that one of the main sources of a back- ground in underground physics experiments (such as the investigation of solar neutrino flux, neutrino oscillation s, neutrinoless double beta decay, and the search for annual and daily Cold Dark Matter particle flux modulation) are fast neutrons originating from the surrounding rocks. The sources of the fast neutrons are ( α, n) reactions on the light elements contained in the rock (C, O, F, Na, Mg, Al, Si). Neutrons from spontaneous fission of238U take an additional contribution in a total fast neutron flux of about 15-20%. Several research groups have investi- gated the neutron background at different underground laboratories [1–3]. Some of them used6Li-dopped liquid scintillator technique [3], and others used in addition a Pulse Shape Discrimination technique [1]. The measurements of fast neutron flux in the Deep Un- derground Low Background Laboratory of Baksan Neu- trino Observatory (DULB BNO) have been performed with using of a special, high-sensitive fast neutron spec- trometer [4]. This laboratory is located under Mt. Andyrchy (Northen Caucasus Mountains, Russia) in a tunnel that penetrates 4.5 km into the mountain, at a depth of 4900 meters of water equivalent. The results of such measurements lead to a conclu- sion that a neutron background places a severe limitation on the sensitivity of current and planned experiments. Owing this fact, the development of new cost-effective, high-strength radiation shielding against neutrons be- comes a very important task for modern non-accelerator physics experiments. For such purposes the relative neu- tron shielding properties of several commonly available natural materials were investigated too. Specially, these materials are planned for use in the construction of large- volume underground facilities which will be covered with suitable shielding materials and are situated in the DULB Laboratory at Baksan.NEUTRON DETECTOR The spectrometer was constructed to measure low background neutron fluxes at the level up to 10−7cm−2s−1in the presence of intensive gamma-ray background. The detector consists of 30 l liquid organic scintilla- tor viewed by photomultipliers with 19 neutron coun- ters (3Heproportional counters) uniformly distributed through the scintillator volume (see [4] for detail). The spectrometer schematic view and the principle of opera- tion are shown in Fig. 1. Fast neutrons with En>1 MeV entering the liquid scintillator (LS) are moderated down to thermal energy, producing a LS signal. Then they diffuse through the de- tector volume to be captured in3Hecounters or on pro- tons in the scintillator. The LS signal starts the recording system. After triggering the system waits a signal from any of the helium counters for a specific time. This time window corresponds to the delay time between correlated events in the scintillator and in the helium counters. This is one of specific features of the detector. The signal from the LS is ’marked’ as a coincident with a neutron capture in the3Hecounters only in the case if a single counter is triggered during the waiting period. An amplitude of the ’marked’ LS signal corresponds to an initial neutron energy. This method allows us to suppress the natural γ-ray background considerably. The described event discriminating procedure allows us to measure extremely low neutron fluxes at the level up to 10−7cm−2s−1reliably even if the LS counting rate is as large as several hundred per second. The dead time of the detector is equal to the delay time (variable value, but generally about 120 µs) plus about 400 µs, which is needed to analyze a LS event whether it corresponds to neutron or not. The detection efficiency depends in a complicated manner on the response function of the de- tector. As a rough estimation, we use the value of the efficiency, which is equal to 0 .04±0.02 in the energy range 1FIG. 1. The neutron spectrometer schematic view and a principle of operation. (1) is a PMT, (2) is a liquid scintill ator and (3) is a3Hecounter. from 1 to 15 MeV. This is based on preliminary measure- ments performed with a Pu-Be source. Owing this fact, an absolute values of the neutron fluxes can be estimated with an uncertainty of 50% on the basis of available cal- ibration data. The delay time is a specific feature of the detector and depends on the detector design. The acqui- sition system allows us to measure the delay time for the neutron events directly. Such measurements were carried out using a Pu−Besource with a time window selected to be equal to 300 µs. A typical delay time distribution is shown in Fig. 2 A fitting procedure leads to a time constant of T1/2∼55µs. According to this result it is sufficient to select the time window to be equal 120 µsfor an actual measurement. MEASUREMENTS A. The geometry It has been mentioned that we have no yet precise information about the detection efficiency, that is why one can calculate absolute value of neutron fluxes with only 50% certainty. However, it is possible to measure the relative neutron absorption abilities of various shiel ds FIG. 2. Delay time distribution for Pu-Be neutron source TABLE I. Concentration of U, Th and K in the rock sam- ples. Sample238U, g/g232Th, g/g40K, g/g Quartzite (1 .1±0.1) 10−7(4.3±0.1) 10−7(1.9±0.03) 10−7 Serpentine (2 .2±0.5) 10−8(2.0±0.9) 10−8<1.2 10−8 Surrounding (1 .6±0.3) 10−6(4.0±0.1) 10−6(1.6±0.1) 10−6 mine rock with high precision. This information will be very use- ful for development of new low background experiments and searching for cost-effective neutron absoption shields . Such measurements were carried out in the DULB BNO with using the described neutron spectrometer. This new laboratory, consisted of 8 separate counting facilities, i s located under Mt. Andyrchy in a tunnel, which pen- etrates 4.3 kminto the mountain, at a depth of 4900 m w.e. Quartzite and serpentine were selected as materials to be tested because of comparatively low concentrations of uranium- and thorium-bearing compounds contained in these rocks. For instant, the measured concentrations of uranium and thorium for rock serpentine are about 10−8g/g in comparison with 10−6g/g for the surround- ing rock. As for potassium (40K) contained in serpen- tine, it has been found less than 10−8g/g in comparison with 10−6g/g for the surrounding rock. Measurements of gamma-activity of different rock samples have been per- formed with using a well-type NaI gamma spectrometer with level of sensitivity of about 10−9g/g, operated in one of the underground low counting facilities at BNO [5]. The measured Th, U, and K concentrations in different rock samples are given in Table I. Four series of measurements were performed with the neutron spectrometer surrounded by different radiation 2FIG. 3. Schematic view of cross-section of the DULB ex- perimental site and the detector inside the neutron shield. All sizes in millimeters. shields. In the first series the spectrometer was sur- rounded with a lead shield 4 cmthickness (to reduce the natural gamma-ray counting rate), and measurements of the natural neutron background radition field existing in the open experimental site were performed. In the second and third series the spectrometer was surrounded with shields of quartzite and serpentine, respectively. The rock shields consisted of broken pieces of various sizes, ranging from 1 cmto 15 cm, with an effective shield thickness of 35 cmin all directions. The mean relaxation length of fast neutrons in these shields is about 15 cm(25 g/cm2for quartzite and 21 g/cm2for serpentine). In the fourth series we measured the internal background of the detector using a neutron-absorbing shield consisted of 40- cm thick section of polyethylene containing an admixture of boron and water about 30 cmthick. Schematic view of one of the investigated neutron shield and cross-section of the DULB experimental site are shown in Fig.3. B. Calibration A60Co γ-source has been used to calibrate the LS- channel. The energy of the middle of the Compton edge was assumed to be equal to 1 MeV in the electron en- ergy scale, which corresponds to ∼3MeV in the neutron energy scale (see Fig.4a). A Pu-Be source was used to calibrate the NC-channel of3Hecounters. The spectrum produced by the Pu-Be source in the3Hecounters has a specific shape due to a wall effect which distorts the counter event spectrum (see Fig. 4b). In spite of this distortion, the range of energies observed for true neu- tron events is less narrow compared to the broad back- FIG. 4. Calibration spectra. (a) liquid scintillator irra- diated with60Co; (b)3He counters irradiated with Pu-Be source. ground spectrum produced by internal alphas. Using of the only events from the neutron window coincident with LS signals makes it possible to suppress the internal back- ground of the detector. C. Conditions of measurements Main conditions for all series of measurements, such as measuring times, LS- and NC-counting rates are given in the Table II. The typical exposure time for each series was a few weeks. The γ-ray background in the open experimen- tal site is high enough that leads to γ-counting rate in the LS-channel of about 700 s−1. Due to this fact, fol- lowing values of dead time were determined for different series: 12% of the total exposure time for measurements with the lead shield, 4 .3% for quartzite series, 2 .7% for serpentine series, and 1 .5% for measurements with the polyethylene/water shield. To calculate the true neutron counting rates a proper dead time correction has been performed. 3TABLE II. Conditions of measurements. Value No shield Quartzite Serpentine Water+ (5 cm lead) Polyethylene The measuring time, h 400 290 950 605 Dead Time, % 12 4.3 2.7 1.5 Total LS-rate, s−1202 83 62 21 Total NC-rate, h−1123±0.6 103 ±0.6 92 ±0.3 95 ±0.4 NC-rate in neutron window, h−158±0.4 37 ±0.4 27 ±0.2 27 ±0.2 Random coincidences rate, h−11.41±0.005 0.38 ±0.002 0.19 ±0.001 0.07 ±0.001 Rnneutron counting rate, h−129.6±0.5 9.6 ±0.5 -0.2 ±0.3 — DATA TREATMENT AND RESULTS Contamination of222Rngas inside the experimental site can make a considerable contribution (up to 20%) to the background γ-counting rate, which can influence results of the performed measurements because222Rn activity can vary significantly for a period of a measure- ment. To suppress the count rate variation effect we used a special procedure for treatment of experimental data. It consists of the following steps. Two types of data files are stored as a result of a mea- surement. One of them contains the information about neutron energy losses ( LS-signal amplitudes),3Hecoun- ters signal amplitudes, and delay time for each ’neutron’ candidate event. Data accumulation was stopped every half-hour and overall numbers of NC-counts, LS-counts, LS-counts above 1 MeV, and elapsed time were saved in a file. Total background γ- spectra for every half-hour run were measured simultaneously and saved in a sec- ond file to make it possible to take into account a time variation of the background γ-counting rate. We consider three contributions into the experimen- tally measured counting rate Rmeas: the random coinci- dence rate Rrnd, the internal detector background count- ing rate Rbkg, and the ’neutron’ counting rate Rn, so that Rn=Rmeas−Rrnd−Rbkg (1) We have made obvious assumption that the total back- ground γ-spectrum and the random coincidence spec- trum have the same shapes. To obtain random coin- cidence spectrum for further subtraction procedure the total background γ-spectrum has been normalized with a factor corresponding to the calculated random coinci- dence rate. The maximal evaluation for the random co- incidence rate, if the LS- and NC- events are absolutely independent, can be calculated by the following way: Rrand=rγrw n∆t, (2) where rγis the γ-rate, rw nis the3He-counters count- ing rate in the determined neutron energy window, ∆ t is the time window. In the case of the performed mea- surements ( RLS≫RHe), this evaluation is very close to the real counting rate of random coincidences. Due to a variation in time of the222Rnactivity, the current value of ri γdepends on time too. Owing to this fact,we applied the described subtraction procedure to each half-hour run with corresponding current value of Ri rnd, and then summarized resulting neutron spectra in a total serial spectrum. The accumulated LS-spectra of all coin- cidented events ( Rmeas) and the recalculated spectra of random coincidence ( Rrnd) for the no-shield, quartzite, and serpentine series are presented in Fig. 5. An internal detector background spectrum Rbkghas been accumulated inside the neutron-absorbing shield consisting of polyethylene and water. Obtained count- ing rate of the internal background correlated (neutron- type, but non-neutron) events was measured as 27 counts per hour, which in terms of a neutron flux corresponds to (8.1±0.5) 10−7s−1cm−2. The residual LS-spectra (Rmeas−Rrnd) in comparison with the internal back- ground LS-spectrum ( Rbkg) are presented in Fig. 6. Performing the total subtraction procedure in accord- ing with the equation (1) we obtain values of the neutron counting rate Rnfor the no-shield, quartzite, and serpen- tine series. Taking into account the detection efficiency uncertainty ( ε= 0.04±0.02) the obtained values of fast neutron fluxes (above 700 keVof neutron energy) are presented here in a following way: a(3.5±1.1) 10−7s−1cm−2for the no-shield measure- ment, a(2.9±1.1) 10−7s−1cm−2for quartzite shield, a(0.6±0.7) 10−7s−1cm−2for serpentine shield, were a = ( ε+ ∆ε)/ε. One can see that the resulting neutron flux measured when the serpentine shield was in place were found to be at about the minimum level of sensitivity of the spec- trometer. It means that a neutron background inside the serpentine shield is consisted with a neuron flux less than 0.7 10−7s−1cm−2. It indicates that serpentine is indeed clear from uranium and thorium, and is, therefore, the most likely candidate for use as a cost-effective neutron shield component material for large-scale low background experiments. A delay time distribution analysis was performed to understand the origin of a high level of the internal de- tector background. DELAY TIME DISTRIBUTIONS Decays of Bi and Po radioactive isotopes, such as 4FIG. 5. The accumulated LS-spectra of all coincidented events ( Rmeas, solid) and the recalculated spectra of random coincidences ( Rrnd, dashed) 214Bi(e,˜ν)164µs→214Po(α)→..., (3) which can take place in the helium counter walls, have been considered as main possible sources of the signifi- cant internal background. To imitate an actual neutron event beta decay of214Bican fire the liquid scintillator, followed by a delayed capture α- signal from Po decay in helium counters. The delay time distribution of the neutron-type coincidented events obtained for the series in the water shield is shown in Fig. 7. Fitting procedure leads to the time constant T1/2= 164 µs. It means that, as it was supposed, the origin of the internal background of our detector is mostly due to con- tamination of214Biin the3He-counter walls. The delay time distributions for other series of measurements are shown in Fig. 8. The following fitting function was used to analize these distributions ( tis expressed in µs): A+Ne−t ln2/55+Be−t ln2/164, (4) where A is a constant, N is an amplitude corresponding to neutrons and B corresponds to internal background. FIG. 6. Residual LS-spectra ( Rmeas-Rrnd, solid) in com- parison with the internal background LS-spectrum ( Rbkg, dashed). The ratio N/B, which was obtained in this manner, de- creases from measurements in the lead shield to the mea- surements in the serpentine shield. CONCLUSIONS The main results of the measurements can be summa- rized as follows. (I). The preliminary results obtained from the fast neutron spectrum accumulated in the open experimen- tal site of the DULB Laboratory at Baksan is consisted with a neutron flux (for neutrons with energy above 700 keV) estimated as values from 5 .3×10−7cm−2s−1to 1.8×10−7cm−2s−1depending on the present uncertainty in determination of the detection efficiency. (II). The neutron spectrometer sensitivity in a shielded experimental site is estimated as 0 .5×10−7cm−2s−1for a measuring time of about 1000 h. (III). It is shown that the main source of the detec- tion sensitivity limitation, rather then random coinci- dences, is the internal background of the spectrometer, 5FIG. 7. Delay time distribution for the coincident events measured in the water shield. which is mostly due to the presence of α-particle emitters (214Bi−214Podecays) in the3He-counters walls. (IV). The achieved neutron background inside the ser- pentine shield is consisted with a neutron flux less than 0.7 10−7s−1cm−2. It indicates that serpentine is one of the more likely candidate for use as a cost-effective neu- tron shield component material for large-scale low back- ground experiments. We have obtained the presented results using the sim- ple event discrimination procedure and did not use pulse shape discrimination yet. Nevertheless, it takes us a pos- sibility to measure extremely low neutron fluxes up to 10−7cm−2s−1even when external γ-counting rate is more than 200 s−1. ACKNOWLEDGMENTS We are grateful to I.I.Pyanzin for the management in proving of reserves and quarrying of the domestic ultra basic rock samples. We thank P.S.Wildenhain for care- ful reading of this article and his critical remarks. We acknowledge the support of the Russian Foundation of Basic Research. This research was made possible in part by the grants of RFBR No. 98–02 16962 and No. 98-02- 17973. FIG. 8. Delay time distributions for the coincident events measured in the series with no shield, quartzite and serpen- tine. [1] V. Chazal, B. Chambon, M. De Jesus et al. Astroparticle Physics, v.9, n.2 (1998) pp. 163-172 [2] F. Arneodo, F. Cavanna, S. Parlatti et al. INFN/AE- 97/52 (1997) [3] R.Alexan et al. NIM A274 203(1989). [4] J.N. Abdurashitov, V.N. Gavrin, G.D. Efimov, A.V. Ka- likhov, A.A. Shikhin and V.E. Yants, “Instrum. and Exp. Tech.” , Vol. 40, No 6, 1997, pp. 741–752. [5] A.A.Klimenko, A.A.Pomansky, A.A.Smolnikov, NIM B17 445(1986). 6

arXiv:physics/0001008v1 [physics.atom-ph] 5 Jan 2000A semi–classical over–barrier model for charge exchange be tween highly charged ions and one–optical electron atoms Fabio Sattin∗ Consorzio RFX, Corso Stati Uniti 4, 35127 Padova, ITALY Absolute total cross sections for electron capture between slow, highly charged ions and alkali targets have been recently measured. It is found that these c ross sections follow a scaling law with the projectile charge which is different from the one pre viously proposed basing on a classical over–barrier model (OBM) and verified using rare gases and mo lecules as targets. In this paper we develop a ”semi–classical” (i.e. including some quantal features) OBM attempting to recover experimental results. The method is then applied to ion–hyd rogen collisions and compared with the result of a sophisticated quantum-mechanical calculat ion. In both cases the present method is found to underestimate by a factor two the correct result but , where comparison can be made, it is superior to other OBMs. A qualitative explanation for the di screpancies is also given. PACS numbers: 34.70+e, 34.10.+x I. INTRODUCTION The electron capture processes in collisions of slow, highl y charged ions with neutral atoms and molecules are of great importance not only in basic atomic ph ysics but also in applied fields such as fusion plasmas and astrophysics. In the past years a number of measurements have been carried o n the collisions between highly charged ions and rare gases [1] or molecules [2], in which one or several electrons were transferred from the neutral target to a charged projectile: A+q+B→A(q−j)++Bj+. (1) Their results-together with those from a number of other lab oratories-yielded a curve which can be fitted within a single scaling law (a linear relationship) when plotting cross section σversus projectile charge q: it is almost independent of the projectile species and of th e impact velocity v(at least in the low–speed range v <1 au). When one extends experiments to different target species, the same linear relation holds between σandq/I2 t, with Itthe ionization potential of the target [3,4]. It is found that this scaling law could to be predicted, in the limit of very high projectile charge, by a modification of an extended classical over–barrier model ( ECBM), allowing for multiple electron capture, proposed by Niehaus [5]. Quite recently a confirmat ion of this scaling has come from a sophisticated quantum–mechanical calculation [6]. Similar experiments were carried on more recently for colli sions between ions and alkali atoms [7]. The results show that the linear trend is roughly satisfi ed, but the slope of the straight line is grossly overestimated by the ECBM: in Fig. 1 we show some data points (stars with error bars) together with the analytical curve from the ECBM (dashed cur ve) which, for one–electron atoms, is written [3,4] σ= 2.6×103q/I2 t[10−20m2] (2) (Itin eV). It should be noticed that experimental data are inste ad well fitted by the results of a Classical Trajectory MonteCarlo (CTMC) code [7]. The ECBM of ref. [3] works in a simplified one-dimensional geo metry where the only physically meaningful spatial dimension is along the internuclear axi s. It does not take into account the fact that the electrons move in a three-dimensional space. This m eans that only a fraction of the electrons actually can fulfil the conditions dictated by the model. For rare gases and molecules, which have a large number of active electrons, this can be not a trouble (i .e., there are nearly always one or more ∗E-mail: sattin@igi.pd.cnr.it 1electrons which can participate to the collision). For alka li atoms with only one active electron, on the other hand, an overestimate of the capture probability b y OBM’s could be foreseen. With present–days supercomputers there are relatively few difficulties in computing cross sec- tions from numerical integration of the time-dependent Sch r¨ odinger equation (e.g. refer to ref. [6]). Notwithstanding this, simple models are still valuable sin ce they allow to get analytical estimates which are easy to adapt to particular cases, and give physica l insight on the features of the problem. For this reason new models are being still developed [8,9]. In this paper we present a modified OBM which allows to get a bet ter agreement with the experi- mental data of ref. [7]. II. THE MODEL We start from the same approach as Ostrovsky [8] (see also [10 ]): berthe electron vector relative to the neutral atom ( T) andRthe internuclear vector between Tand the projectile P(see Fig. 2 for a picture of the geometry: it is an adaptation of Figure 1 from ref. [8]). Let us consider the plane containing the electron, PandT, and use cylindrical polar coordinates ( ρ,z, φ ) to describe the position of the electron within this plane. We can choose the angle φ= 0 and the zdirection along the internuclear axis. We will assume that the target atom ca n be described as an hydrogenlike atom, which is not a bad approximation when dealing with alka li atoms. The total energy of the electron is E=p2 2+U=p2 2−Zt/radicalbig ρ2+z2−Zp/radicalbig ρ2+ (R−z)2. (3) ZpandZtare the charge of the projectile and the effective charge of th e target seen by the electron, respectively, and we are using atomic units. We can also approximate Eas E(R) =−En−Zp R. (4) Enis given by the quantum–mechanical value: En=Z2 t/(2n2). This expression is asimptotically correct as R→ ∞. On the plane (e, P,T) we can draw a section of the equipotential surface U(z, ρ,R ) =En−Zp R. (5) This represents the limit of the region classically allowed to the electron. When R→ ∞ this region is decomposed into two disconnected circles centered around e ach of the two nuclei. Initial conditions determine which of the two regions actually the electron liv es in. AsRdiminishes there can be eventually a time where the two regio ns become connected. It is easy to solve eq. (5) for Rby imposing that ρm= 0 and that there must be an unique solution for z with 0 < z < R : Rm=Zt+/radicalbig ZtZp En. (6) In the spirit of OBMs it is the opening of the equipotential cu rve between PandTwhich leads to a leakage of electrons from one nucleus to another, and the refore to charge exchange. Along the internuclear axis the potential Uhas a maximum at z=z0=R√Zt/radicalbig Zp+√ Zt. (7) Whether the electron crosses this potential barrier depend s upon its initial conditions. These are chosen from a statistical ensemble, which we will leave unsp ecified for the moment. Let NΩbe the fraction of trajectories which lead to electron loss at the t imetandW(t) the probability for the electron to be still bound to the target, always at time t. The fraction of losses in the interval t, t+dt is given by dW(t) =−NΩdt TemW(t), (8) 2withTemthe period of the electron motion along its orbit. A simple in tegration yields the leakage probability Pl= 1−exp/parenleftbigg −1 Tem/integraldisplay+∞ −∞NΩdt/parenrightbigg . (9) In order to actually integrate Eq. (9) we need to know the coll ision trajectory; an unperturbed straight line with bimpact parameter is assumed: R=/radicalbig b2+ (vt)2. (10) At this point it is necessary to give an explicit expression f orNΩ. The electron is supposed to be in the ground state ( n= 1, l=m= 0). Tembecomes therefore [11] Tem= 2π/Z3 t. (11) Ref. [8] adopts a geometrical reasoning: the classical elec tron trajectories, with zero angular momen- tum, are ellipses squeezed onto the target nucleus. The only trajectories which are allowed to escape are those whose aphelia are directed towards the opening wit hin the angle ±θm. The integration over this angle yields an analytical expression for NΩ(Eq. 17 of ref. [8]). In Fig. 1 we show the results obtained using Ostrovsky’s model ( dotted curve–eq ns. 8,17 of ref. [8])1. Notice that from direct inspection of the analytical formula, one sees that t he scaling law is not exactly satisfied, at least at small values of the parameter q/I2 t, and this is clearly visible in the plot. The result is almost equal to the scaling (2). The present approach is based on the electron position instead than on electron direction . The recipe used here is (I) to neglect the dependence from the ang le: all electrons have the same proba- bility of escaping, regardless of their initial phase. Inst ead, (II) the lost electrons are precisely those which, when picked up from the statistical ensemble, are fou nd farther from nucleus Tthan the distance z0: NΩ=/integraldisplay∞ z0f(r)dr , (12) withf(r) the electron distribution function. There is not a unique choice for f(r): the (phase-space) microcanonical distribution ˜f(r,p)∝δ/parenleftbigg En+p2 2−Zt r/parenrightbigg (13) (δis the Dirac delta) has been often used in literature since th e works [12] as it is known that, when integrated over spatial coordinates, it reproduces th e correct quantum–mechanical momentum distribution function for the case of the electron in the gro und state [13] (more recently the same formalism has been extended to Rydberg atoms [14]). After in tegration over momentum variables one gets instead a spatial distribution function [15] fmc(r) =Zt(2Zt)3/2 πr2/radicalbigg 1 r−Zt 2, r < 2/Zt (14) and zero elsewhere (The lowerscript ”mc” is to emphasize tha t it is obtained from the microcanonical distribution). However, this choice was found to give poor r esults. It could be expected on the basis of the fact that (14) does not extend beyond r= 2/Ztand misses therefore all the large impact– parameter collisions. In the spirit of the present approach , it should be instead important to have an accurate representation of the spatial distribution. We us e therefore for f(r) the quantum mechanical formula for an electron in the ground state: f1s(r) = 4Z3 tr2exp(−2Ztr) (15) which, when substituted in (12), gives 1Beware of a small difference in notation between the present p aper and [8]: here we use an effective charge for the target, Zt=√2En, while [8] uses an effective quantum number nt= 1/√2Enwith the effective charge of the target set to 1. 3NΩ=/bracketleftbig 1 + 2r0Zt+ 2(r0Zt)2/bracketrightbig exp(−2r0Zt). (16) Since the choice for f(r) does not derive from any classical consideration, we call t his method a “semi–classical” OBM. Notice that, in principle, one could go further and compute f(r) from a very accurate wavefunction, fruit of quantum mechanical computations (see [16]), but th is is beyond the purpose of the present paper (it could be worthy mentioning a number of other attemp ts of building stationary distributions f(r), mainly in connections with CTMC studies, see [17–19]). Thef(r) of Eq. (15) does not reproduce the correct momentum distrib ution, nor the correct energy distribution (which could be obtained only by using eq. (13) . However, it is shown in [15] that this choice gives an energy distribution for the electrons, f(E), peaked around the correct value En, and < E > =En, where < . . . > is the average over f(E). Some important remarks are to be done here. First of all, a que stion to be answered is: why use an unperturbed distribution, when the correct one should be se nsitively modified by the approaching of the projectile. The answer is, obviously, that this choice a llows to perform calculations analitically. We are doing here a sort of classical counterpart of a quantum –mechanical Born calculation: there, too, the matrix elements are computed as scalar products ove r unperturbed states, regardless of any perturbation induced by the projectile. In the followin g, however, some considerations about possible improvements over this simple approximation will be done. A second question regards the meaning of the factor dt/T emin eq. (8): in Ostrovsky’s paper this is the fraction of electrons which enter the loss zone du ring the time interval dtand is valid under the hypothesis of a uniform distribution of initial ph ases of the electrons. In our case this this assumption ceases to be valid: electrons actually spen d different fractions of their time at different radial distances from T, depending on their energy. We will do a (hopefully not too se vere) assumption by assuming that, on the average, the expression (8) still holds. III. RESULTS A. Iodine - Cesium This study has been prompted by the ion-atom experiments of [ 7]: first of all, therefore, we apply the above model to the process of electron capture Iq++ Cs→I(q−1)++ Cs+(17) withq= 6÷30. Impact energy is 1 .5×qkeV [7]. The ionization potential of Cesium is It= 3.9 eV. Solid line in Fig. 1 is the result of the present model: the ratio between the OBM result and the true one is passed from a factor near three to something le ss than 50% (which means that, in absolute value, the disagreement is reduced). We will attempt here a possible explanation of this underest imate: we have used in Eq. (15) an unperturbed wavefunction, while the electron distribut ion function must be modified by the approaching of the projectile. Since the electron is attrac ted by P, the distribution function is likely to be more weighted at larger r’s: in a quantum mechanical treatment, the perturbation of t he electron due to the projectile would translate in an excitat ion to higher states. We will find in the following subsection some results in favour of this hypothe sis. B. Bare ions - Hydrogen As second test, we have computed cross section for captures H + O8+→H++ O7+(18) and H + He2+→H++ He+(19) and compared it with similar calculations done using the mol ecular approach by Harel et al[20]. The results are summarized in fig. 3. There is a sharp discrepa ncy in the behaviour for v→0, where the present model predicts an increasing cross sectio n. At very low speed it is the concept 4itself of atomic distribution function which becomes quest ionable, and molecular aspects become important. Besides, quantum effects such as the discretenes s of the energy levels also play a major role and are completely missed by this approach. In the highe r velocity part, the present model underestimates the more accurate value by a factor 2 for proc ess (18), but the error is much less, just 25 %, for process (19). These two ions have been chosen ad hoc : they correspond to values of the ratio Zt/Zp= 1/8 and 1/2 respectively. In the (I, Cs) test this ratio ranged f rom≈1/12 to ≈1/60 depending upon the projectile charge. This means that in t he former case the perturbation of the projectile on the electron distribution function is c omparable to the (I, Cs) case, while in the latter it is much less. We expect the electron distribution f unction to be more and more perturbed asZt/Zp→0. IV. SUMMARY AND CONCLUSIONS We have developed in this paper a very simple OBM for charge ex change. It exploits some features of the quantum mechanical version of the problem, thus differ ing from similar models which are solely classical. The agreement with experiment is better than pre vious calculations where a comparison could be made. It is far from excellent, but reasons for the (p artial) failure have been suggested. As it stands, the model is well suited for one-optical-elect ron atoms (since it uses hydrogen–like wavefunctions), therefore we do expect that other classica l OBM’s can still work better in the many-electrons targets studied in previous experiments. Some improvements are likely to be added to the present model : a possible line of investigation could be coupling the present method with a very simplified ca lculation of the evolution of the wavefunction, using quantum mechanics. From this one shoul d not compute the fas coming from a single state, but as a linear combination including also ex cited wavefunctions (the relative weights in the combination should be given by the quantum mechanical calculation). Work in this direction is currently underway. ACKNOWLEDGMENTS It is a pleasure to thank the staff at National Institute for Fu sion Science (Nagoya), and in particular Prof. H. Tawara and Dr. K. Hosaka for providing th e data of ref. [7] and for useful discussions about the subject. The referees through their s uggestions and criticism have made the manuscript readable. 5[1] Hiroyuki A et al1997 Fus Eng Design 34-35 785 [2] Hosaka K et al1997 Phys Scr T73273 [3] Kimura M et al1995 J Phys B: At Mol Opt Phys 28L643 [4] Hosaka K et al1997 Fus Eng Design 34-35 781 [5] Niehaus A 1986 J Phys B: At Mol Phys 192925 [6] Nagano R, Yabana K, Tazawa T and Abe Y 1999 J Phys B: At Mol Op t Phys 32L65 [7] Hosaka K et al Electron capture cross sections of low energy highly ch arged ions in collisions with alkali atoms , poster presented at the International Seminar on Atomic Processes in Plasmas held in Toki (Japan, 1999). To be published as a report of the National Institute for Fusion Science (NIFS) [8] Ostrovsky V N 1995 J Phys B: At Mol Opt Phys 283901 [9] Ivanovski G, Janev R K, and Solov’ev E A 1995 J Phys B: At Mol Opt Phys 284799 [10] Ryufuku H, Sasaki K and Watanabe T 1980 Phys Rev A 21745 [11] Landau L D and Lifshitz E M 1977 Quantum Mechanics (Oxford, Pergamon) Eq. (48.5) [12] Abrines R and Percival I C 1966 Proc Phys Soc 88861 [13] Sattin F and Bolzonella T 1998 Phys Scr 5853 [14] Samengo I 1998 Phys Rev A 582767 [15] Cohen J S 1985 J Phys B: At Mol Phys 181759 [16] Ema J et al1999 At Data Nucl Data Tables 7257 [17] Eichenauer D, Gr¨ un N and Scheid W 1981 J Phys B: At Mol Phy s143929 [18] Hardie D J W and Olson R E 1983 J Phys B: At Mol Phys 161983 [19] Montemajor V J and Schiwietz G 1989 J Phys B: At Mol Opt Phy s222555 [20] Harel C, Jouin H and Pons B 1998 At Data Nucl Data Tables 68279 6FIGURE CAPTIONS FIG. 1. Comparison between experimental data and predictio n from models for electron capture cross section of process (17). Stars, experiment with 20% error bar; dashed line, sca ling law from Niehaus (Eq. 2); dotted line, Ostrovsky’s scal ing law; solid line, scaling law from present model. σis in 10−20m2,Itin eV. FIG. 2. Geometry of the scattering. PandTare the projectile and target nucleus respectively. The env eloping curve shows a section of the equipotential surface U=E, i.e., it is the border of the region classically accessible to the electron. Ris the internuclear distance. The parameter ρmis the radius of the opening which joins the potential wells, θmthe opening angle fromT;z0is the position of the potential’s saddle point. 7FIG. 3. Capture cross section versus impact velocity. Upper, H–O8+collisions; lower, H–He2+collisions. Diamonds, data from ref. 20; solid line, present model. 8

arXiv:physics/0001009v1 [physics.atom-ph] 5 Jan 2000An Exact Approach to the Oscillator Radiation Process in an Arbitrarily Large Cavity N.P. Andion(b), A.P.C. Malbouisson(a)and A. Mattos Neto(b) (a)Centro Brasileiro de Pesquisas F´ ısicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro CEP 22290-180-RJ, Brazil. E-mail: adolfo@lafex.cbpf.br (b)Instituto de Fisica - Universidade Federal da Bahia Campus Universitario de Ondina, 40210-340-BA Salvador Bra zil E-mail: andion@ufba.br,arthur@fis.ufba.br Abstract Starting from a solution of the problem of a mechanical oscil lator coupled to a scalar field inside a reflecting sphere of radius R, we study the behaviour of the system in free space as the limi t of an arbitrarily large radius in the confined solution. From a m athematical point of view we show that this way of facing the problem is not equivalent to consi der the system a priori embedded in infinite space. In particular, the matrix elements of the t ransformation turning the system to principal axis, do not tend to distributions in the limit of a n arbitrarily large sphere as it shouldbe the case if the two procedures were mathematically equiva lent. Also, we introduce ”dressed” coordinates which allow an exact description of the oscilla tor radiation process for any value of the coupling, strong or weak. In the case of weak coupling, we recover from our exact expressions the well known decay formulas from perturbation theory. 11 Introduction Since a long time ago the experimental and theoretical inves tigations on the polarization of atoms by optical pumping and the possibility of detecting changes in their polarization states has al- lowed the observation of resonant effects associated to the c oupling of these atoms with strong radiofrequency fields [1]. As remarked in [2], the theoretic al understanding of these effects using perturbative methods requires the calculation of very high -order terms in perturbation theory, what makes the standard Feynman diagrams technique practic ally unreliable in those cases. The trials of treating non-perturbativelly such kind of system s consisting of an atom coupled to the electromagnetic field, have lead to the idea of ”dressed atom ”, introduced in refs [3] and [4]. This approach consists in quantizing the electromagnetic field a nd analyzing the whole system consist- ing of the atom coupled to the electromagnetic field. Along th e years since then, this concept has been extensively used to investigate several situation s involving the interaction of atoms and electromagnetic fields. For instance, atoms embedded in a st rong radiofrequency field background in refs. [5] and [6], atoms in intense resonant laser beans in ref. [7] or the study of photon correla- tions and quantum jumps. In this last situation, as showed in refs. [8], [9] and [10], the statistical properties of the random sequence of outcoming pulses can be analyzed by a broadband photode- tector and the dressed atom approach provides a convenient t heoretical framework to perform this analysis. Besides the idea of dressed atom in itself, another aspect th at desserves attention is the non- linear character of the problem involved in realistic situa tions, which implies, as noted above, in 2very hard mathematical problems to be dealt with. An way to ci rcunvect these mathematical difficulties, is to assume that under certain conditions the c oupled atom-electromagnetic field system may be approximated by the system composed of an harmo nic oscillator coupled linearly to the field trough some effective coupling constant g. In this sense, in a slightly different context, recently a sig nificative number of works has been spared to the study of cavity QED, in particular to the th eoretical investigation of higher- generation Schrodinger cat-states in high-Q cavities, as h as been done for instance in [11]. Linear approximations of this type have been applied along the last years in quantum optics to study decoherence, by assuming a linear coupling between a cavity harmonic mode and a thermal bath of oscillators at zero temperature, as it has been done in [12 ] and [13]. To investigate decoherence of higher generation Schrodinger cat-states the cavity fiel d reduced matrix for these states could be calculated either by evaluating the normal-ordering cha racteristic function, or by solving the evolution equation for the field-resevoir state using the no rmal mode expansion, generalizing the analysis of [12] and [13]. In this paper we adopt a general physicist’s point of view, we do not intend to describe the specific features of a particular physical situation, inste ad we analyse a simplified linear version of the atom-field system and we try to extract the more detaile d information we can from this model. We take a linear simplified model in order to try to have a clearer understanding of what we believe is one of the essential points, namely, the need of non-perturbative analytical treatments to coupled systems, which is the basic problem underlying th e idea of dressed atom. Of course, 3such an approach to a realistic non-linear system is an extre melly hard task and here we make what we think is a good agreement between physical reality an d mathematical reliability, with the hope that in future work our approach could be transposed to m ore realistic situations. We consider a non relativistic system composed of a harmonic oscillator coupled linearly to a scalar field in ordinary Euclidean 3-dimensional space. We start from an analysis of the same system confined in a reflecting sphere of radius R, and we assume that the free space solution to the radiating oscillator should be obtained taking a radius arbitrarily large in the R-dependent quantities. The limit of an arbitrarily large radius in the m athematics of the confined system is taken as a good description of the ordinary situation of th e radiating oscillator in free space. We will see that this is not equivalent to the alternative con tinuous formulation in terms of distributions, which is the case when we consider a priori the system in unlimited space. The limiting procedure adopted here allows to avoid the inheren t ambiguities present in the continuous formulation. From a physical point of view we give a non-pert urbative treatment to the oscillator radiation introducing some coordinates that allow to divid e the coupled system into two parts, the ”dressed” oscillator and the field, what makes unecessary to work directly with the concepts of ”bare” oscillator, field and interaction to study the radiat ion process. These are the main reasons why we study a simplified linear system instead of a more reali stic model, to make evident some subtleties of the mathematics involved in the limiting proc ess of taking a cavity arbitrarily large, and also to exhibit an exact solution valid for weak as well as for strong coupling. These aspects would be masked in the perturbative approach used to study no n-linear couplings. 4

arXiv:physics/0001010v1 [physics.atom-ph] 5 Jan 2000Spectrum of atomic radiation at sudden perturbation Victor I. Matveev Heat Physics Department of Uzbek Academy of Sciences, 28 Katartal St., 700135 Tashkent, Uzbekistan March 20, 2013 Abstract A general expression for the spectrum of photons emitted by a tom at sudden perturbation is obtained. Some concrete examples of application of the obtained result are considered. The conclusion about th e coherence of radiation of the atomic electrons under the such influences i s made. PACS numbers: 32.30.* 0It is known many examples when the excitation or ionization o f atoms occurs as result of the action of sudden perturbations. First of all th ese are atomic excitation or ionization in the nuclear reactions [1,2]. For example in β-decay of nucleus, when the fastβ-electron’s escape is perceived by atomic electrons as a sud den changing of nuclear charge or in neutron impact with nucleus, when the su dden of momentum transfer to the nucleus occurs etc. The sudden approximation [3] can be used for consideration m ultielectron transi- tion in complex atoms, when transition occurring in interna l shells, are perceived by relatively slow electrons of external shells as instantane ous (see [4,5]). As a result of action of sudden perturbation can be considered inelastic p rocesses in the collisions of fast multicharged ions with atoms [6 - 12] and in the collis ions of charged parti- cles with highly-excited atoms [13]. After action of sudden perturbation, the excited atom can relax with radiation of photons belonging to known s pectrum of isolated atom. However, if sudden perturbation causes the change of v elocities of atomic electrons, atom can radiate during the action of perturbati on. Classical analogue of such a problem is the [14] radiation of a free electron unde r the sudden changing of velocity. Thus, it is necessary to state a general problem on the spectrum of photons emitted by atom during the time of action of sudden pe rturbation, i.e. - on the spectrum of photons emitted simultaneously by all atomi c electrons as a result of action of perturbation. In many practically important cases perturbation is not suffi ciently small to use a perturbation theory. However the situations when the time o f action of perturbation is considerably less than the characteristic atomic time th at enables one to solve the problem without restricting the value of perturbation [9,1 5-17]. In this paper we derive a general expression for the spectrum of photons emitted by the atom under sudden perturbation and apply this result t o some concrete processes. Consider ”collision” type sudden perturbation [3], when th e perturbation V(t)≡ V(ra,t) , where ra- coordinates of atomic electrons, acts only during the time τ, which is much smaller than the characteristic period of unpe rturbed atom, describing by Hamiltonian H0. To be definite we will assume that V(t) is not equal zero near t= 0 only. Then in the exact solution of Schr¨ odinger equation ( atomic units are used throughout in this paper) i∂ψ ∂t= (H0+V(t))ψ one can neglect by evolution of ψ(during the time τ) caused by unperturbed Hamil- tonianH0. Therefore the transition amplitude of atom from the initia l stateϕ0to 1a final state ϕn, as a result of actions of sudden perturbation V(t), has the form [3]: a0n=/angbracketleftϕn|exp(−i+∞/integraldisplay −∞V(t)dt)|ϕ0/angbracketright, (1) whereϕ0andϕnbelong to the full set of orthonormalized eigenfunctions of the unperturbated Hamiltonian H0, i.e.H0ϕn=ǫnϕn. Thus in the sudden perturbation approximation the evolutio n of the initial state has the form ψ0(t) =exp(−it/integraldisplay −∞V(t′)dt′)ϕ0, (2) whereψ0(t) satisfies the equation i∂ψ0(t) ∂t=V(t)ψ0(t), (3) ψ0(t)→ϕ0undert→ −∞ . Let’s introduce full and orthonormal set of functions Φn(t) =exp(i+∞/integraldisplay tV(t′)dt′)ϕn, (4) obeying eq. (3), and Φ n(t)→ϕnt→+∞. Obviously the amplitude (1) can be rewritten as a0n=/angbracketleftΦn(t)|ψ0(t)/angbracketright. Therefore the radiation amplitude can be calculated in the fi rst order of per- turbation theory (as a corrections to the states (2) and (4)) over the interaction of atomic electrons with electromagnetic field [18,19]. W=−/summationdisplay a,k,σ/parenleftbigg2π ω/parenrightbigg1 2ukσ(a+ kσe−ikra+akσe−ikra)ˆ pa, wherea+ kσandakσare the creation and annihilation operators of the photon wi th a frequencyω, momentum kand polarization σ,(σ= 1,2),ukσare the unit vectors of polarization, raare the coordinates of atomic electrons ( a= 1,..,Z a), hereZa is the number of atomic electrons, ˆ paare the momentum operators of atomic elec- trons. Then in the dipole approximation the amplitude of emi ssion of photon with simultaneous transition of atom from the state ϕ0to a stateϕnhas the form b0n(ω) =i/parenleftbigg2π ω/parenrightbigg1 2ukσ+∞/integraldisplay −∞dteiωt/angbracketleftΦn(t)|/summationdisplay aˆ pa|ψ0(t)/angbracketright. 2Integrating this expression by parts over the time and omitt ing the terms vanishing (t→ ±∞ ) in turning off the interaction with electromagnetic field we have b0n(ω) =i/parenleftbigg2π ω/parenrightbigg1 2ukσ+∞/integraldisplay −∞dteiωt ω× ×/angbracketleftϕn|/summationdisplay a∂V(t) ∂raexp(−i+∞/integraldisplay −∞V(t′)dt′)|ϕ0/angbracketright. (5) Summing |b0n(ω)|2over polarization and integrating over the photon’s emissi on angles and summing, after this, over all final states of the at omϕn, we find the total radiation spectrum dW dω=2 3π1 c3ω/angbracketleftϕ0|/summationdisplay a∂˜V∗(ω) ∂ra/summationdisplay b∂˜V(ω) ∂rb|ϕ0/angbracketright, (6) where c = 137 .u. is the speed of light, ˜V(ω) =+∞/integraldisplay −∞V(t)eiωtdt. (7) Thus we have obtained the radiation spectrum of atom during t he time of sudden perturbation V(t). As an application we consider the radiation spectrum of atom in the sudden transmission of momentum pto the atomic electrons when V(t) has the form V(t) =f(t)/summationdisplay ara,p=+∞/integraldisplay −∞dtf(t), (8) f(t) is the perturbing force which not depends on raand interacts during a time τ that is considerable less than the characteristic periods o f the unperturbed atom. The total radiation spectrum (6) in this case has the form dW dω=2 3π1 c3ω|˜f(ω)|2·Z2 a, (9) where˜f(ω),is the Fourier transform of the functions f(t), defined according to (7), Zais the number of atomic electrons. In this case the spectrum c oincides (after producting to ω) with the radiation spectrum of the classical particle with mass equal to electron’s one and with charge Za, moving in the field of homogeneous forcesf(t). This gives us the information about the value of the spectr um (9). Since f(t)/negationslash= 0 just during the time τ, and the spectrum (9) is proportional to |˜f(ω)|2, only the photons belonging to continuum with characteristi c frequencies ω≤1/τ can be emitted by atom. 3Analogously one can consider the radiation of atom in the ”sw itching” type sudden perturbation (we use the classification of sudden per turbations introduced in [3]). Formula (5) allows one to obtain the spectrum of photons in th e transition of atom from the state ϕ0to a stateϕnunder the influence of perturbation (8): dw0n dω=2 3π1 c3ω|˜f(ω)|2Z2 a| /angbracketleftϕn|exp(−ip/summationdisplay ara)|ϕ0/angbracketright |2. (10) HeredW/dω =/summationtext ndw0n/dω, where/summationtext nmeans summing over the complete set of atomic states. Formula (10) allows one to express the relati ve contribution of tran- sitions with excitation to an arbitrary state ϕnto the total spectrum (9) dw0n/dω dW/dω=| /angbracketleftϕn|exp(−ip/summationdisplay ara)|ϕ0/angbracketright |2. via the well known [2] inelastic formfactors /angbracketleftϕn|exp(−ip/summationtext ara)|ϕ0/angbracketright. In the most simple case of transferring to atomic electrons t he momentum p, when in (8) f(t) =p·δ(t),whereδ(t) is the Dirac δ-function, then ˜f(ω) =p and spectrum (9) coincides, after producting to ω, with the radiation spectrum of classical particle [11] with charge Za, which takes (suddenly) a velocity p. As an another example we give the radiation spectrum in the in fluence of mo- mentum having the Gausian form. f(t) =f0exp(−α2t2)cos(ω0t), respectively ˜V(ω) =√π 2αf0/summationdisplay ara/braceleftBigg exp/bracketleftBigg −(ω−ω0)2 4α2/bracketrightBigg +exp/bracketleftBigg −(ω+ω0)2 4α2/bracketrightBigg/bracerightBigg . Therefore the radiation spectrum has the form dW dΩ=f2 0 6Ωc3α2/braceleftBig exp/bracketleftBig −(Ω + Ω 0)2/bracketrightBig +exp/bracketleftBig −(Ω−Ω0)2/bracketrightBig/bracerightBig Z2 a, where for the sake of convenience the frequencies Ω = ω/(2α) and Ω 0=ω0/(2α) are introduced. One should note an important generality of radiation at sudd en perturbation, namely, the radiation intensity for the multielectron atom s is proportional to the square of the number of atomic electrons.(see [12]) This fac t allows one to conclude on the coherence of radiation of atomic electrons under such type influences. References 1. A.B. Migdal, Qualitative Methods in Quantum Theory (Moscow: Nauka, 1975) 42. L.D. Landau and E.M. Lifshitz, Quantum Menchanics (Moscow: Nauka, 1989) 3. A.M. Dykhne, G.L. Yudin, Usp. Fiz. Nauk, 125, 377 (1978). [Sov.Phys. Usp. 21, 549 (1978)]. 4. T. Aberg, in ” Photoionization and Other Probes of Many Electron Interact ions” (F. Wuillemier, ed. Plenum, New York, 1976, p. 49). 5. V.I. Matveev, E.S. Parilis, Usp. Fiz. Nauk, 138, 573 (1982). [Sov. Phys. Usp. 1982,25, 881 (1982)]. 6. J. Eichler, Phys.Rev.A. 15, 1856(1977). 7. G.L. Yudin, Zh.Eksp.Teor. Fiz. 1981, 80, 1026 (1981). 8. J.H. McGuire, Advances in Atomic, Molecular and Optical P hysics, 29, 217 (1992). 9. V.I. Matveev, Phys.Part. Nuclei, 26, 329 (1995). 10. P.K. Khabibullaev, V.I. Matveev, D.U. Matrasulov, J. Ph ys. B,31, L607 (1998). 11. V.I.Matveev, Kh.Yu.Rakhimov, D.U.Matrasulov. J.Phys . B,32, 3849 (1999). 12. V.I. Matveev, J. Phys. B, 24, p. 3589 (1991). 13. I.C. Percival, in ” Atoms in Astrophysics ”, (Edited by P.G. Burke, W.B. Eissner, D.G. Hammer and I.C. Percival Plenum Press, New York and Lond on, 1983, p. 87-113.) 14. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fiel d (Moscow: Nauka, 1988). 15. M.Ya. Amusia, The Bremsstrahlung , (Moscow: Energoatomizdat, 1990.) 16. A.J. Baltz, Phys. Rev. A, 52, 4970 (1995). 17. A.J. Baltz, Phys. Rev. Lett. 78, p.1231 (1997). 18. The sudden perturbation V(t) accounted in the functions Φ n(t)ψ0(t) without limitation of value V(t). 19. V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Quantum Electrodynamics (Moscow: Nauka, 1989) 5

arXiv:physics/0001011v1 [physics.chem-ph] 5 Jan 2000The Hydration Number of Li+in Liquid Water∗ Susan B. Rempe, Lawrence R. Pratt, Gerhard Hummer, Joel D. Kr ess, Richard L. Martin, and Antonio Redondo Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA (February 9, 2008) Abstract A theoretical treatment based upon the quasi-chemical theo ry of solutions predicts the most probable number of water neighbors in the i nner shell of a Li+ion in liquid water to be four. The instability of a six water m olecule inner sphere complex relative to four-coordinated structu res is confirmed by an ‘ab initio’ molecular dynamics calculation. A classical Monte Carlo simu- lation equilibrated 26 water molecules with a rigid six-coo rdinated Li(H 2O)6+ complex with periodic boundary conditions in aqueous solut ion. With that initial configuration for the molecular dynamics, the six-c oordinated structure relaxed into four-coordinated arrangements within 112 fs a nd stabilized. This conclusion differs from prior interpretations of neutron an d X-ray scattering results on aqueous solutions. ∗LA-UR-99-3360. 1The hydration of ions in water is not only fundamental to phys ical chemistry but also relevant to the current issue of selectivity of biological i on channels. In the context of potassium channels [1,2,3], for example, the free energies for replacement of inner shell water ligands with peptide carbonyls donated by proteins of the channel structure seem decisive to the selectivity of the channel, specifically for preference of K+over Na+. Studies to elucidate the thermodynamic features of such inner shell exchange reactions require prior knowledge of the ion hydration structures and energetics. Unfortunately, our understanding of the inner hydration sh ell structure of ions in wa- ter is not as clear as it might be [4]. The simplest and most fav orable case to pursue is the Li+solute. Neutron scattering measurements on LiCl solutions in liquid water have led to a firm conclusion that the Li+ion has six near-neighbor water molecule partners [4,5,6,7,8,9,10]. That result, however, has not been entir ely uniform across studies of similar aqueous solutions [11,12] containing Li+ions. X-ray scattering results have been interpreted similarly [13] to indicate a hydration number of six, again w ith some nonuniformity [14]. In contrast, some spectroscopic studies have suggested tetra hedral coordination of the Li+ion in water [15] and an array of physical chemical inferences le nd some support to that con- clusion [16]. On the theoretical side, electronic structur e calculations on the Li+ion with six water molecules predict a slightly, but distinctly, low er energy for a structure with four inner shell and two outer shell water molecules than for stru ctures with six water molecules in the innermost shell [17,18]; results such as those seem to be universally supported by other electronic structure efforts [19,20]. Simulations ha ve produced a range of results in- cluding both four and six inner shell water neighbors with co nsiderable statistical dispersion [21,22,23,24,25,26,27,28,29,30,31,32]. It is well recog nized, of course, that simulations are typically not designed to provide a sole determination of su ch properties, though they do shed light on the issues determining the hydration number of ions in water. The theoretical scheme used here to address these problems f or the Li+(aq) ion is based upon the quasi-chemical organization of solution theory, w hich is naturally suited to these problems [33,34,35,36]. The first step is the study of the rea ctions Li++nH2O⇀↽Li(H2O)n+(1) that combine nwater molecule ligands with the Li+ion in a geometrically defined inner sphere under ideal gas conditions. At a subsequent step an ap proximate, physical description of the aqueous environment surrounding these complexes is i ncluded [33,34,35,36]. The geometric definition of an inner sphere region enforces a phy sical balance in this method. The goal of this approach is to treat inner sphere ligands exp licitly, in molecular detail, but at the same time to achieve a description of outer sphere h ydration thermodynamics that is consistent from one complex to another. If minimum en ergy complex geometries were to shift different numbers of ligands to outer sphere reg ions, that would unbalance the thermodynamic description of the hydration of the inner sph ere materials. For example, in the quantitative implementation of the quasi-chemical app roach we specifically do not use the Li[(H 2O)4][(H2O)2]+complex cited above, with two water molecules outside the in ner sphere, even though this structure helpfully clarifies the p hysical issue. Gas-phase thermochemical data required for the equilibria in Eq. (1) were obtained by electronic structure calculations using the Gaussian98 programs with the B3LYP hybrid density functional theory approximation [37]. All structu res were fully optimized with a basis 2including polarization functions on Li+(6-31G*) and both polarization and diffuse functions (6-31++G**) on the oxygen and hydrogen centers. At the optim um geometry and with the same basis set, harmonic vibrational frequencies of the clu sters were calculated and atomic charges determined using the ChelpG capability in Gaussian 98. Partition functions were then calculated, thus providing a determination of the free energy changes of the equilibria in Eq. (1) due to atomic motions internal to the clusters with in the harmonic approximation. Interactions of these complexes with the external aqueous e nvironment [34] were treated with a dielectric model following the previous study of the h ydrolysis of the ferric ion [35]. Classic electrostatic interactions based upon the ChelpG p artial atomic charges were the only solution-complex interactions treated; in particular, re pulsive force (overlap) interactions were neglected based on the expectation that they make a seco ndary contribution to the thermodynamic properties considered here. The external bo undary of the volume enclosed by spheres centered on all atoms defined the solute molecular surface. The sphere radii were those determined empirically by Stefanovich and Truong [38 ], except R Li+=2.0˚A for the lithium ion. Because the lithium ion is well buried by the inn er shell waters, slight variations of the lithium radius were found to be unimportant. The value RLi+=2.0˚A was identified as slightly larger than the nearest Li-O distances and signi ficantly smaller than the Li-O distances (3.5 – 4.0 ˚A) for second shell pairs. Results of the calculations are summarized in Fig. 1. Geomet ry optimization of each of then-coordinated clusters confirms that the inner shell structu res used in these calculations are not necessarily the lowest energy structures for a given number of water neighbors. Although a tetrahedral cluster of inner shell water molecul es is the lowest energy structure for Li(H 2O)4+, a cluster with five inner shell water molecules is slightly h igher in energy than a cluster with one outer shell and four inner shell water molecules. Similarly, the lowest energy cluster with six water molecules contains fou r inner shell water molecules arranged tetrahedrally and two outer shell water molecules . Fig. 1 shows that the n=4 inner sphere cluster has the lowest free energy for a dilut e (p=1 atm) ideal gas phase. Adjustment of the concentration o f water molecules to the value ρW= 1 g/cm3, to match the normal density of liquid water, changes the mos t favored cluster to the one with n=6 inner shell water molecules. Outer sphere interactions d escribed by the dielectric model progressively destabilize the larger clusters, as they should since larger numbers of water molecules are being treated explicitly as m embers of the inner shell. As a consequence of including the outer sphere contributions, the final position of minimum free energy is returned to the n=4 structure, with the n=3 complex predicted to be next most populous in liquid water at T=298.15 K and p=1 atm. The me an hydration number predicted by this calculation is n=4.0. The current quasi-chemical prediction for the absolute hyd ration free energy of the Li+ ion under these conditions is -128 kcal/mol, not including a ny repulsive force (packing) con- tributions. An extreme increase of R Li+to 2.65 ˚A raises this value to about -126 kcal/mol, showing that the theoretical results are insensitive to the ion radius, as remarked above. Experimental values are -113 kcal/mol [39], -118 kcal/mol [ 40], and -125 kcal/mol [41], converted to this standard state. This dispersion of experi mental values for the absolute hydration free energy of the Li+(aq) ion is accurately mirrored in the dispersion of refer- ence values adopted for the absolute hydration free energy o f the H+(aq) ion. Inclusion of repulsive force contributions would reduce the present c alculated value slightly. Further- 3more, Li+(aq) is believed to have a strongly structured second hydrat ion shell [23], which is treated only approximately in this calculation. Neverthel ess, this level of agreement between calculation and experiment is satisfactory. We additionally emphasize that the Li(H 2O)n+complexes are treated in the harmonic approximation, although fully quantum mechanically. The l ow-nclusters might have more entropy than is being accounted for by the harmonic approxim ation. If this were the case, then low- nclusters would be more populous than currently represented . This would likely raise the theoretical value also. To further test the n=4 prediction, ‘ab initio’ molecular dynamics calculation s were car- ried out utilizing the VASP program [42]. Two checks establi shed the consistency for these problems between the electronic structure calculations de scribed above and the energet- ics involved in the molecular dynamics calculations. First , the electron density functional alternative implemented in VASP [43] was checked by compari ng the electronic structure results obtained with the B3LYP hybrid electron density fun ctional and the PW91 general- ized gradient approximation exchange-correlation functi onal, using the Gaussian98 program and the same basis sets. As expected, satisfactory agreemen t was observed in the binding energies for sequential addition of a water molecule to the L i(H2O)n+clusters. Then the issues of pseudo-potentials and basis set were checked by op timizing cluster geometries with the VASP program and comparing to the results obtained for th e same problems with Gaus- sian98. Again agreement was observed. For example, both pro cedures predicted the same lowest energy six-coordinated structure, the characteris tic Li[(H 2O)4][(H2O)2]+cluster, with nearly identical geometries. To initiate the ‘ab initio’ molecular dynamics calculation , the optimum n=6 inner sphere structure, rigidly constrained, was first equilibrated wit h 26 water molecules under conven- tional Monte Carlo liquid simulation conditions for liquid water, including periodic boundary conditions. This system of one Li+ion and 32 water molecules was then used as an initial configuration for the molecular dynamics calculation. As sh own in Fig. 2, the initial n=6 structure relaxed to stable n=4 alternatives within 112 fs. The results of longer molecul ar dynamics calculations will be reported later. The ‘ab initio’ molecular dynamics and the quasi-chemical t heory of liquids exploit dif- ferent approximations and produce the same conclusion here . This agreement supports the prediction that Li+(aq) has four inner shell water ligands at infinite dilution i n liquid water under normal conditions. This prediction differs from inter pretations of neutron and X-ray scattering data on aqueous solutions. The conditions studied by these calculations and those targ eted in the neutron scattering work do not match perfectly, particularly with regard to Li+concentration. Nevertheless, the theoretical methods are straightforward and physical, and , moreover, the distinct methods used here conform in their prediction of hydration number. T herefore, it will be of great importance for future work to fully resolve the differences b etween calculations and scattering experiments for these problems. This work was supported by the US Department of Energy under c ontract W-7405-ENG- 36 and the LDRD program at Los Alamos. References and Notes [1] Doyle, D. A.; Cabral, J. M.; Pfuetzner, R. A.; Kuo, A. L.; G ulbis, J. M.; Cohen, S. L.; 4Chait, B. T.; MacKinnon, R. Science 1998, 280, 69–77. [2] Guidoni, L.; Torre, V.; Carloni, P. Biochem. 1999, 38, 8599–8604. [3] Laio, A.; Torre, V. Biophys. J. 1999, 76, 129–148. [4] Friedman, H. L. Chem. Scr. 1985, 25, 42–48. [5] Newsome, J. R.; Neilson, G. W.; Enderby, J. E. J. Phys. C: Solid St. Phys. 1980, 13, L923–L926. [6] Enderby, J. E.; Neilson, G. W. Rep. Prog. Phys. 1981, 44, 593–653. [7] Hunt, J. P.; Friedman, H. L. Prog. Inorg. Chem. 1983, 30, 359–387. [8] Ichikawa, K.; Kameda, Y.; Matsumoto, T.; Masawa, M. J. Phys. C: Solid State Phys. 1984, 17, L725–L729. [9] van der Maarel, J. R. C.; Powell, D. H.; Jawahier, A. K.; Le yte-Zuiderweg, L. H.; Neilson, G. W.; Bellissent-Funel, M. C. J. Chem. Phys. 1989, 90, 6709–6715. [10] Howell, I.; Neilson, G. W. J. Phys.: Condens. Matter 1996, 8, 4455–4463. [11] Cartailler, T.; Kunz, W.; Turq, P.; Bellissent-Funel, M. C. J. Phys.: Condens. Matter 1991, 3, 9511–9520. [12] Yamagami, M.; Yamaguichi, T.; Wakita, H.; Misawa, M.; J. Chem. Phys. 1994, 100, 3122–3126. [13] Radnai, T.; P´ alink´ as, G.; Szasz, G. I.; Heinzinger, K .Z. Naturforsch. A 1981, 36, 1076–1082. [14] Narten, A. H.; Vaslow, F.; Levy, H. A. J. Chem. Phys. 1973, 85, 5017–5023. [15] Michaellian, K. H.; Moskovits, M. Nature 1978, 273, 135 – 136. [16] Ohtaki, H.; Radnai, T.; Chem. Rev. 1993, 93, 1157–1204. [17] Feller, D.; Glendening, E. D.; Kendall, R. A.; Peterson , K. A.; J. Chem. Phys. 1994, 100, 4981–4997. [18] Feller, D.; Glendening, E. D.; Woon, D. E.; Feyereisen, M. W.; J. Chem. Phys. 1995, 103, 3526–3542. [19] Bishof, G.; Silbernagel, A.; Hermansson, K.; Probst, M .Int. J. Quant. Chem. 1997, 65, 803–816. [20] Tongraar, A.; Liedl, K. R.; Rode, B. M. Chem. Phys. Letts. 1998, 286, 56–64. [21] Heinzinger, K.; P´ alink´ as, G. The Chemical Physics of Solvation Elsevier, Amsterdam, 1985; pp. 313. 5[22] Mezei, M.; Beveridge, D. L. J. Chem. Phys. 1981, 74, 6902–6910. [23] Impey, R. W.; Madden, P. A.; McDonald, I. R. J. Phys. Chem. 1983, 87, 5071–5083. [24] Chandrasekhar, J.; Spellmeyer, D. C.; Jorgensen, W. L. J. Am. Chem. Soc. 1984, 106, 903–910. [25] Bounds, D. G. Mol. Phys. 1985, 54, 1335–1355. [26] Zhu, S. B.; Robinson, G. W. Z. Naturforsch. A 1991, 46, 221–228. [27] Romero, C. J. Chim. Phys. 1991, 88, 765–777. [28] Heinzinger, K. in Water-Biomolecule Interactions eds. M. U. Palma, M. B. Palma- Vittorelli, and F. Patak: SIF, Bologna, 1993. p. 23. [29] Lee, S. H.; Rasaiah, J. C. J. Chem. Phys. 1994, 101, 6964–6974. [30] Toth, G. J. Chem. Phys. 1996, 105, 5518–5524. [31] Obst, S; Bradaczek, H. J. Phys. Chem. 1996, 100, 15677–15687. [32] Koneshan, S.; Rasaiah, J. C.; Lynden-Bell, R. M.; Lee, S . H.J. Phys. Chem. B 1998, 102, 4193–4204. [33] Pratt, L. R.; LaViolette, R. A. Mol. Phys. 1998, 94, 909. [34] Hummer, G.; Pratt, L. R.; Garc´ ıa, A. E. J. Phys. Chem. A 1998, 102, 7885–7895. [35] Martin, R. L.; Hay, P. J.; Pratt, L. R. J. Phys. Chem. A 1998, 102, 3565–3573. [36] Pratt, L. R.; Rempe, S. B. Quasi-Chemical Theory and Implicit Solvent Models for Simulations 1999, LA-UR-99-3125. [37] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria , M. A. Robb, J. R. Cheese- man, V. G. Zakrzewski, J. A. Montgomery, R. E. Stratmann, J. C . Burant, S. Dap- prich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O . Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Ada mo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokum a, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V . Ortiz, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R . L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, C. Gonza lez, M. Challa- combe, P. M. W. Gill, B. G. Johnson, W. Chen, M. W. Wong, M. Head -Gordon, E. S. Replogle, and J. A. Pople. Gaussian 98 (Revision A.2) . Gaussian, Inc., Pittsburgh PA, 1998. [38] Stefanovich, E. V.; Truong, T. N. Chem. Phys. Lett. 1995, 244, 65–74. [39] Marcus, Y. Biophys. Chem. 1994, 51, 111–127. [40] Conway, B. E. J. Soln. Chem. 1978, 7, 721–770. 6[41] Friedman, H. L., and Krishnan, C. V., in Water A Comprehensive Treatise Vol. 3, edited by F. Franks: Plenum Press, New York, 1973, p. 1. [42] Kresse, G.; Hafner, J. Phys. Rev. B 1993, 41, 558. [43] Perdew, J.; Burke, K.; Wang, Y. Phys. Rev. B 1996, 54, 16533. [44] Kresse, G.; Hafner, J. J. Phys.: Condens. Mat. 1994, 6, 8245. [45] Sprik, M.; Hutter, J.; Parrinello, M. J. Chem. Phys. 1996, 105, 1142. 7FIGURES -140-120-100-80-60-40-200 1 2 3 4 5 6Energy (kcal/mol) nΔG(0) -RT ln xn + ΔµLi+ Figure 1. Free energies for Li+ion hydration in liquid water as a function of the number of inner shell water neighbors at T=298.15 K. The results mar ked ∆G(0)(open circles) are the free energies predicted for the reaction Li++nH2O = Li(H 2O)n+under standard ideal conditions, including p = 1 atm. The minimum value is at n=4. The next lower curve (squares) incorporates the replacement free energy - nRT ln(RT ρW/1 atm) that adjusts the concentration of water molecules to the normal concentr ation of liquid water, ρW= 1 g/cm3so that RT ρW= 1354 atm [35]. The minimum value is at n=6. The topmost graph (diamonds) plots µ∗ Li(H2O)n+−nµ∗ H2O, the external-cluster contributions obtained from the standard dielectric model [34,35]. The bottommost results (solid circles) are the final, net values. The label provides the quasi-chemical expression o f these net values [33,36] with xn the fraction of lithium ions having ninner shell water neighbors and ∆ µLi+the interaction part of the chemical potential of the lithium ions. This grap h indicates that the n=4 inner sphere structure is most probable in liquid water under norm al conditions. 8Figure 2. Structures from molecular dynamics calculations based upo n a gradient- corrected electron density functional description of the i nteratomic forces. The ions were represented by ultrasoft pseudopotentials [44] and a kinet ic energy cutoff of 396 eV, which was found satisfactory in related calculations [45], limit ed the plane wave expansions . The top panel is the configuration used as an initial condition. A hexa-coordinate inner sphere structure, rigidly constrained, was equilibrated with 26 a dditional water molecules by Monte Carlo calculations using classical model force fields and as suming a partial molar volume of zero. The bottom panel is the structure produced 112 fs later . The bonds identify water oxygen atoms within 2.65 ˚A of the Li+ion. The hydrogen, lithium, and oxygen atoms are shown as open, black, and gray circles, respectively. 9

arXiv:physics/0001012v1 [physics.gen-ph] 6 Jan 2000What quantum mechanics describes is discontinuous motion Gao Shan Institute of Quantum Mechanics 11-10, NO.10 Building, YueTan XiJie DongLi, XiCheng Distri ct Beijing 100045, P.R.China E-mail: gaoshan.iqm@263.net We show that the natural motion of particles in continuous sp ace-time (CSTM) is not classi- cal continuous motion (CCM), but one kind of essentially dis continuous motion, the wave func- tion in quantum mechanics is the very mathematical complex d escribing this kind of motion, and Schr¨ odinger equation is just its simplest nonrelativisti c motion equation, we call such motion quan- tum discontinuous motion or quantum motion; furthermore, w e show that, when considering gravity the space-time will be essentially discrete, and the motion in discrete space-time (DSTM) will nat- urally result in the collapse process of the wave function, t his finally brings about the appearance of classical continuous motion (CCM) in macroscopic world. I. INTRODUCTION The analysis about motion has never ceased since the old Gree ce times, but from Zeno Paradox to Einstein’s relativity [6], only CCM is discussed, and its uniqueness is taken for granted ever since, but as to whether or not CCM is the only possible and objective motion, whether CCM is the real motion or apparent motion, no one has given a definite answer up to now, in fact, people have been indulging in the study of the motion law, but omitted the study of the motion itself. On the other hand, we have entered into the microscopic world for nearly one century, but our understanding about it is still in confusion, the orthodox view [5] renunci ates CCM in microscopic world, but permits no existence of objective motion mode for the microscopic particles, whi le the opponents [3,4,7] still recourse to CCM to lessen the pain of losing realism, no other objective motion modes have been presented for the microscopic particles till now, thus the above problem is more urgent than ever. In this paper, we will mainly address the above problem, afte r given a deep logical and physical analysis about motion, we demonstrate that the natural motion in continuou s space-time or CSTM is not CCM, but one kind of essentially discontinuous motion, we call it quantum disco ntinuous motion or quantum motion, since we show that the wave function in quantum mechanics is the very mathemati cal complex describing it, and Schr¨ odinger equation of the wave function is also its simplest nonrelativistic mo tion equation; Furthermore, since the combination of quantum mechanics and general relativity will result in the discreteness of space-time, namely the real space-time will be essentially discrete, we further study the motion in discrete space-time, or DSTM, and demonstrate that it will naturally result in the collapse process of the wave fun ction, and finally bring about the appearance of CCM in macroscopic world. The plan of this paper is as follows: In Sect. 2 we first give a ge neral analysis about CSTM, the motion state of particle is physically defined, its general form and descr iption are also given based on the mathematical analysis in the Appendix. In Sect. 3 we work out the simplest evolution law of CSTM, which turns out to be Schr¨ odinger equation in quantum mechanics. In Sect. 4 we give a strict phy sical definition of CSTM, and further discuss the constant ¯hinvolved in its law. In Sect. 5 we point out that space-time is essentially discrete due to the ubiquitous existence of gravity, and give a simple demonstration. In Se ct. 6 we further give a general analysis about DSTM, and the general form of motion state in such space-time is given. In Sect. 7 the evolution law of DSTM is worked out, and we demonstrate that it will naturally result in the collapse process of the wave function. In Sect. 8 we further show that CCM and its evolution law can be consistently derived fr om the evolution law of DSTM. At last, conclusions are given. II. GENERAL ANALYSIS ABOUT MOTION IN CONTINUOUS SPACE-TIME (CSTM) In this section, we will give a deep logical and physical anal ysis about CSTM. 1A. The motion state of particle First, we should define the motion state of particle, there ar e two alternatives, one is the instant state of particle, the other is the infinitesimal interval state of particle, it has been generally accepted that the motion state of particl e should be the infinitesimal interval state of particle, not t he instant state of particle, while people usually omit thei r essential difference, here we will present some of them. (1).The instant state of particle contains only one point in space, its potential in mathematics is zero, while the infinitesimal interval state of particle contains infinite i nnumerable points in space, its potential in mathematics is ζ1. (2).The instant state of particle contains no motion, but on ly the existence of particle, while the infinitesimal interv al state of particle may contain abundant motion elements, sin ce it contains infinite innumerable points in space. (3).The instant state of particle possesses no physical mea ning, since we can not access it through physical mea- surement, while the infinitesimal interval state of particl e possesses real physical meaning, since we can measure it by means of the following infinite process:∆ t→dt. (4).We can only find and confirm the law for the infinitesimal in terval state of particle, while as to the instant state of particle, even if its law exists, we can not find it, let alon g confirm it. In fact, in physics there exist only the description quantit ies defined during infinitesimal time interval, this fact can be seen from the familiar differential quantities such as dt a nd dx, whereas the quantities defined at instants come only from mathematics, people always mix up these two kinds of qua ntities, this is a huge obstacle for the development of physics. Thus we can only discuss the motion state and releva nt quantities defined during infinitesimal time interval, as well as their differential laws, if we study the point set co rresponding to real physical motion. For simplicity, in the following we say the motion state of pa rticle at one instant, but it still denotes the infinitesimal interval state of particle, not the instant state of particl e. B. The general form of the motion state of particle Secondly, we will give the general form of the motion state of particle, according to the analysis about point set(see Appendix), the natural assumption in logic is that the motio n state of particle in infinitesimal time interval is a general dense point set in space, since we have no a priori rea son to assume a special form, its proper description is the measure density ρ(x,t) and measure density fluid j(x,t). Certainly, at some instant t the motion state of particle may assume some kind of special form, such as the continuous point set described by dx or ρ(x,t) =δ(x−x(t)), but whether or not this kind of special form can exist for other instants should be determined by the motion law, no t our prejudices. III. THE EVOLUTION OF MOTION IN CONTINUOUS SPACE-TIME (CSTM ) In the following, we will give the main clues for finding the po ssible evolution equations of CSTM, and show that Schr¨ odinger equation in quantum mechanics is just its simp lest nonrelativistic evolution equations. Here we mainly analyze one-dimension motion, but the results can be easily extended to three-dimension situation. A. The first motion principle First, we should find the first motion principle similar to the first Newton principle, this means that we need to find the simplest solution of the motion equation, in which we can get the invariant quantity during free motion, it is evident that the simplest solution of the motion equation is : ∂ρ(x,t) ∂t= 0 (1) ∂j(x,t) ∂x= 0 (2) ∂j(x,t) ∂t= 0 (3) 2∂ρ(x,t) ∂x= 0 (4) using the relation j(x,t)=ρ(x,t)·vwe can further get the solution, namely ρ(x,t)=1,j(x,t)=v=p/m, wheremis the mass of the particle, pis defined as the momentum of particle. Now, we get the first mot ion principle, namely during the free motion of particle, the momentum of the particle is i nvariant, but it can be easily seen that, contrary to classical continuous motion, for the free particle with one constant momentum, its position will not be limited in the infinitesimal space dx, but spread throughout the whole spac e with the same position measure density. Similar to the quantity position, the natural assumption in logic is also that the momentum (motion) state of particle in infinitesimal time interval is still a general de nse point set in momentum space, thus we can also define the momentum measure density f(p,t), and the momentum measure fluid density J(p,t), their meanings are similar to those of position. B. Two kinds of description bases Now, we have two description quantities, one is position, th e other is momentum, and position descriptions ρ(x,t) andj(x,t) provide a complete local description of the motion state, w e may call it local description basis, similarly momentum descriptions f(p,t) andJ(p,t) provide a complete nonlocal description of the motion stat e, since for the particle with any constant momentum, its position will spre ad throughout the whole space with the same position measure density, we may call it nonlocal description basis. Furthermore, at any instant the motion state of particle is u nique, thus there should exist a one-to-one relation between these two kinds of description bases, namely there s hould exist a one-to-one relation between position de- scription (ρ,j) and momentum description ( f,J), and this relation is irrelevant to the concrete motion sta te, in the following we will mainly discuss how to find this one-to-one r elation, and our analysis will also show that this relation essentially determines the evolution of motion. C. One-to-one relation First, it is evident that there exists no direct one-to-one r elation between the measure density functions ρ(x,t) and f(p,t), since even for the above simplest situation, we have ρ(x,t) = 1 and f(p,t) =δ2(p−p0)∗, and there is no one-to-one relation between them. Then in order to obtain the one-to-one relation, we have to cr eate new properties on the basis of the above position description ( ρ,j) and momentum description ( f,J), this needs a little mathematical trick, here we only give t he main clues and the detailed mathematical demonstrations are omi tted, first, we disregard the time variable tand lett= 0, as to the above free evolution state with one momentum, we hav e (ρ,j) = (1,p0/m) and (f,J) = (δ2(p−p0),0), thus we need to create a new position state function ψ(x,0) using 1 and p0/m, a new momentum state function ϕ(p,0) usingδ2(p−p0) and 0, and find the one-to-one relation between these two sta te functions, this means there exists a one-to-one transformation between the state functions ψ(x,0) andϕ(p,0), we generally write it as follows: ψ(x,0) =/integraldisplay+∞ −∞ϕ(p,0)T(p,x)dp (5) whereT(p,x) is the transformation function and generally continuous a nd finite for finite pandx, since the function ϕ(p,0) will contain some form of the basic element δ2(p−p0), normally we may expand it as ϕ(p,0) =/summationtext∞ i=1aiδi(p−p0), while the function ψ(x,0) will contain the momentum p0, and be generally continuous and finite for finite x, then it is evident that the function ϕ(p,0) can only contain the term δ(p−p0), because the other terms will result in infiniteness. On the other hand, since the result ϕ(p,0) =δ(p−p0) implies that there exists the simple relation f(p,0)=ϕ(p,0)∗ϕ(p,0)†, and owing to the equality between the position description and momentum description, we also have the similar relation ρ(x,0)=ψ(x,0)∗ψ(x,0), thus we may let ψ(x,0) =eiG(p0,x)and haveT(p,x) =eiG(p,x), then ∗This result can be directly obtained when considering the ge neral normalization relation/integraltext Ωρ(x,t)dx=/integraltext Ωf(p, t)dp. †Evidently, another simple relation f(p,0)= ϕ(p,0)2permit no existence of one-to-one relation 3considering the symmetry between the properties position a nd momentum‡, we have the general extension G(p,x) =/summationtext∞ i=1bi(px)i, furthermore, this kind of symmetry also results in the symm etry between the transformation T(p,x) and its reverse transformation T−1(p,x), whereT−1(p,x) satisfies the relation ϕ(p,0) =/integraltext+∞ −∞ψ(x,0)T−1(p,x)dp, thus we can only have the term pxin the function G(p,x), and the resulting symmetry relation between these two transformations will be T−1(p,x) =T∗(p,x) =e−ipx, we letb1= 1/¯h, where ¯his a constant quantity(for simplicity we let ¯h= 1 in the following discussions), then we get the basic one-t o-one relation, it is ψ(x,0) =/integraltext+∞ −∞ϕ(p,0)eipxdp, whereψ(x,0) =e−ip0xandϕ(p,0) =δ(p−p0), it mainly results from the essential symmetry involved in CSTM itself. In order to further find how the time variable tis included in the functions ψ(x,t) andϕ(p,t), we may consider the superposition of two single momentum states, namely ψ(x,t) =1√ 2[eip1x−ic1(t)+eip2x−ic2(t)], then the position measure density is ρ(x,t) = [1 + cos( △c(t)− △px)]/2, where △c(t) =c2(t)−c1(t) and△p=p2−p1, now we let △p→0, then we have ρ(x,t)→1 and △c(t)→0, especially using the measure conservation relation we ca n get dc(t)/dt=dp·p/m, namelydc(t) =d(p2/m)·tordc(t) =dE·t, whereE=p2/m, is defined as the energy of the particle in the nonrelativistic domain, thus as to any si ngle momentum state we have the time-included formula ψ(x,t) =eipx−iEt. In fact, there may exist other complex forms for the state fun ctionsψ(x,t) andϕ(p,t), for example, they are not the above simple number functions but multidimensional vector functions such as ψ(x,t) = (ψ1(x,t),ψ2(x,t),...,ψ n(x,t)) andϕ(p,t) = (ϕ1(p,t),ϕ2(p,t),...,ϕ n(p,t)), but the above one-to-one relation still exists for every component function, and these vector functions still satisfy the above modulo sq uare relations, namely ρ(x,t) =/summationtextn i=1ψi(x,t)∗ψi(x,t) and f(p,t) =/summationtextn i=1ϕi(p,t)∗ϕi(p,t), these complex forms will correspond to the particles with more complex structure, say, involving more inner properties of the particle such as charge and spin etc. At last, since the one-to-one relation between the position description and momentum description is irrelevant to the concrete motion state, the above one-to-one relation for th e free motion state with one momentum should hold true for any motion state, and the states satisfying the one-to-one r elation will be the possible motion states. Furthermore, it is evident that this one-to-one relation will directly result in the famous Heisenberg uncertainty relation △x·△p≥¯h/2. D. The evolution law of motion in continuous space-time (CST M) Now, we will work out the evolution law of CSTM. First, as to the free motion state with one momentum, namely t he single momentum state ψ(x,t) =eipx−iEt, using the above definition of energy E=p2 mand including the constant quantity ¯ hwe can easily find its nonrelativistic evolution law, which is i¯h∂ψ(x,t) ∂t=−¯h2 2m·∂2ψ(x,t) ∂x2(6) then owing to the linearity of this equation, this evolution equation also applies to the linear superposition of the single momentum states, namely all possible free notion sta tes, or we can say, it is the free evolution law of CSTM. Secondly, we will consider the evolution law of CTSM under ou tside potential, when the potential U(x,t) is a constantU, the evolution equation will be i¯h∂ψ(x,t) ∂t=−¯h2 2m·∂2ψ(x,t) ∂x2+U·ψ(x,t) (7) then when the potential U(x,t) is related to x and t, the above form will still hole true, nam ely i¯h∂ψ(x,t) ∂t=−¯h2 2m·∂2ψ(x,t) ∂x2+U(x,t)·ψ(x,t) (8) for three-dimension situation the equation will be i¯h∂ψ(x,t) ∂t=−¯h2 2m· ∇2ψ(x,t) +U(x,t)·ψ(x,t) (9) ‡This symmetry essentially stems from the equivalence betwe en these two kinds of descriptions, the direct implication i s for ρ(x,0) =δ2(x−x0) we also have f(p,0) = 1. 4this is just the Schr¨ odinger equation in quantum mechanics§. At last, the above analysis also shows that the state functio nψ(x,t) provides a complete description of CSTM, since CSTM is completely described by the measure density ρ(x,t) and measure fluid density j(x,t), and according to the above evolution equation the state function ψ(x,t) can be expressed by these two functions, namely ψ(x,t) = ρ1/2·eiS(x,t)/¯h, whereS(x,t) =m/integraltextx −∞j(x′,t)/ρ(x′,t)dx′+C(t)∗∗, and these two functions can also be expressed by the state function, namely ρ(x,t) =|ψ(x,t)|2andj(x,t) = [ψ∗∂ψ/∂t −ψ∂ψ∗/∂t]/2i, then there exists a one-to-one relation between ( ρ(x,t),j(x,t)) andψ(x,t) when omitting the absolute phase, thus the state function ψ(x,t) also provides a complete description of CSTM. On the other hand, we can see that the absolute phase of the wav e functionψ(x,t), which may depend on time, is useless for describing CSTM, since according to the above an alysis it disappears in the measure density ρ(x,t) and measure fluid density j(x,t), which completely describe CSTM, thus it is natural that th e absolute phase of the wave function possesses no physical meaning. IV. FURTHER DISCUSSIONS ABOUT MOTION IN CONTINUOUS SPACE-T IME (CSTM) A. The definition of motion in continuous space-time (CSTM) Now we can give the physical definition of CSTM in three-dimen sion space, the definition for other abstract spaces or many-particle situation can be easily extended. (1). The motion of particle in space is described by dense poi nt set in four-dimension space and time. (2). The motion state of particle in space is described by the position measure density ρ(x,t) and position measure fluid density j(x,t) of the corresponding dense point set. (3). The evolution of motion corresponds to the evolution of the dense point set, and the simplest evolution equation is Schr¨ odinger equation in quantum mechanics. Compared with classical continuous motion, we may call CSTM quantum discontinuous motion, or quantum motion, the commonness of these two kinds of motion is that they are bo th the motion of particle, namely the moving object exists only in one position in space at one instant, their diff erence lies in the moving behavior, namely the behavior of the particle during infinitesimal time interval [t,t+dt] , for classical motion, the particle is limited in a certain l ocal space interval [V,V+dV], while for quantum motion, the part icle moves throughout the whole space with a certain position measure density ρ(x,t). In fact, all physical states of CSTM are defined during infinit esimal time interval in the meaning of measure, not at one instant, for example, the single momentum state ψp(x,t) =eipx−iEt, especially even the single position state ψ(x,t) =δ(x−x0) is still defined during infinitesimal time interval. B. Some discussions about the constant ¯h First, from the above analysis about CSTM, we can understand why the constant ¯ hwith dimension J·sshould appear in the evolution equation of CSTM, or Schr¨ odinger eq uation, the existence of ¯ hessentially results from the equivalence between the nonlocal momentum description and local position description of CSTM, it is this equivalence that results in the one-to-one relation between these two ki nds of descriptions, which requires the existence of a certa in constant ¯hwith dimension J·sto cancel out the dimension of the physical quantities pxandEtin the relation, at the same time, the existence of ¯ halso indicates some kind of balance between the dispersion o f the position distribution and that of momentum distribution limited by the one-to-one relation ( there is no such limitation for CCM and ¯ h= 0 ), or we can say, the existence of ¯ hessentially indicates some kind of balance between the nonl ocality and locality of motion in continuous space-time. §In fact, Schr¨ odinger equation is also the simplest evoluti on equation of dense point set, this fact manifests the ident ity of physical reality and mathematical reality to some extent. ∗∗When in three-dimension space, the formula for S(x,y,z,t) w ill be S(x,y, z, t ) =m/integraltextx −∞jx(x′, y,z, t )/ρ(x′, y, z, t )dx′+C(t) = m/integraltexty −∞jy(x, y′, z, t)/ρ(x,y′, z, t)dy′+C(t) =m/integraltextz −∞jz(x, y,z′, t)/ρ(x,y, z′, t)dz′+C(t), since in general there exists the relation ∇ × {j(x,y,z,t)/ρ(x,y,z,t)}=0. 5Secondly, even though the appearance of ¯ hin the evolution equation of CSTM is inevitable, its value ca n not be determined by CSTM itself, we only know that ¯ hpossesses a finite nonzero value, certainly, just like the ot her physical constants such as c and G, its value can be determined by the ex perience, but there may exist some deeper reasons for the special value of ¯ hin our universe, although motion can not determine this valu e alone, the solution may have to resort to other subtle realities in this world, for exampl e, gravity (G), space-time(c), or even the existence of our mankind. V. THE REAL SPACE-TIME IS DISCRETE The combination of quantum mechanics and general relativit y strongly implied space-time is essentially discrete, and the minimum space-time unit will be Planck size TpandLp, thus owing to the ubiquitous existence of gravity, the real space-time will be essentially discrete with the mi nimum size TpandLp. Here we will give a simple operational demonstration about t he discreteness of space-time, consider a measurement of the length between points A and B, at point A place a clock wi th massmand sizeato register time, at point B place a reflection mirror, when t=0 a photon signal is sent fr om A to B, at point B it is reflected by the mirror and returns to point A, then the clock registers the return ti me, for classical situation the measured length will be L=1 2ct, but when considering quantum mechanics and general relati vity, the existence of the clock introduces two kinds of uncertainties to the measured length, the uncertai nty resulting from quantum mechanics is: δLQM≥(¯hL mc)1/2, the uncertainty resulting from general relativity is: δLGR∼=1 22Gm c2lna+L a≥Gm c2, then the total uncertainties is: δL= δLQM+δLGR≥(L·Lp2)1/3, whereLp= (G¯h c3)1/2, is Planck length, thus we conclude that the minimum measura ble length is Planck length Lp, in a similar way, we can also work out the minimum measurable time, it is just Planck timeTp. VI. GENERAL ANALYSIS ABOUT MOTION IN DISCRETE SPACE-TIME (D STM) In the discrete space-time, there exist absolute minimum si zesTpandLp, namely the minimum distinguishable size of time and position of the particle is respectively TpandLp, thus in physics the existence of the particle is no longer in one position at one instant as in the continuous space-tim e, but limited in a space interval Lpduring a finite time intervalTp, it can be seen that this state corresponds to the instant sta te of particle in continuous space-time, we define it as the instant state of particle in discrete space-t ime, this state evidently contains no motion, but only the existence of particle. Furthermore, during the finite time interval Tpthe particle can only be limited in a space interval Lp, since if it can move throughout at least two different local regions with separation size larger than Lpduring the time interval Tp, then there essentially exists a smaller distinguishable fi nite time interval than Tp, which evidently contradicts the fact thatTpis the minimum time unit, thus the discreteness of space-tim e essentially results in the existence of local position state of the particle, in which the particle stays i n a local region with size Lpfor a time interval Tp, we may call such general local position state Planck cell state. Similar to the analysis of the motion state in continuous spa ce-time, in discrete space-time, the natural assumption in logic is that the motion state of particle in finite time int erval, which is much longer than Tpbut still small enough, is a general discrete point set or cell set in space, since we h ave no a priori reason to assume a special form, its proper description is still the measure density ρ(x,t) and measure density fluid j(x,t), but in the meaning of time average. Certainly, we can also get the motion state of particle in dis crete space-time from that in continuous space-time, since in continuous space-time the particle, which instant state is the particle being in one position at one instant, moves throughout the whole space during infinitesimal time i nterval, while in discrete space-time the instant state of particle turns to be the particle being in a space interval Lpduring a finite time interval Tp, the motion state of particle in discrete space-time will naturally be that duri ng a finite time interval much larger than Tpthe particle moves throughout the whole space with the position measure d ensityρ(x,t) in the meaning of time average. Now the visual physical picture of DSTM will be that during a fi nite time interval Tpthe particle will stay in a local region with size Lp, then it will still stay there or ”jump” to another local regi on, which may be very far from the original region, while during a time interval much larger th anTpthe particle will move throughout the whole space with a certain average position measure density ρ(x,t). As we can see, on the one hand, the particle undergoing DSTM st ays in a local region during infinitesimal time interval, this may generate the display of CCM, on the other h and, during a finite time interval much larger than Tp the particle will continually jumps from one local region to another local region, and move throughout the whole space with a certain position measure density ρ(x,t), this may generate the display of CSTM, then DSTM is evident ly some 6kind of unification of CCM and CSTM, in fact, owing to the ubiqu itous existence of quantum and gravity, space-time is essentially discrete, and DSTM will be the only real motio n in Nature, while CSTM and CCM are the apparent motion modes, they are only two ideal approximators of DSTM i n microscopic and macroscopic world, thus DSTM is just the lost reality unifying quantum motion (CSTM) and cla ssical motion (CCM), and it will undoubtedly provide an uniform realistic picture for microscopic world and macr oscopic world, the following analysis will confirm this conclusion more convincingly. VII. THE EVOLUTION OF MOTION IN DISCRETE SPACE-TIME (DSTM) A. A general discussion Since CSTM is some kind of time average of DSTM, the evolution of DSTM will follow the evolution law of CSTM in the meaning of time average, on the other hand, the particl e undergoing DSTM does stay in a local region for a finite nonzero time interval, and jump from this local region to another local region stochastically, thus the position measure density ρ(x,t) of the particle will be essentially changed in a stochastic way due to the finite nonzero stay time in different stochastic region††, and the corresponding wave function will be also stochasti cally changed, then this kind of stochastic jump inevitably introduce the stochasti c element to the evolution, so the evolution law of DSTM will be the combination of the deterministic linear evoluti on and stochastic nonlinear evolution, in the following we will work out this law. B. Two rules At first, since CSTM is some kind of average of DSTM during a fini te time interval much larger than Tp, thus the position measure density ρ(x,t) of the particle undergoing CSTM will be also the average of t he position distribution of the particle undergoing DSTM during this time interval, the n it is natural that the position of the particle undergoing DSTM will satisfy the position measure density ρ(x,t), namely for DSTM the stochastic stay position of the partic le satisfies the distribution P(x,t) =|ψ(x,t)|2(10) this is the first useful rule for finding the evolution law of DS TM. Secondly, according to the definition of the position measur e densityρ(x,t), the finite nonzero stay time of the particle in a local region evidently implies that the positi on measure density ρ(x,t) in that region will be increased after this finite nonzero stay time interval, and the increas e will be larger when the stay time is longer. We consider the general situation that the particle undergoing DSTM sta ys in a local region Lpfor a time interval T, in the first rank approximation the increase of the position measure den sityρ(x,t) in this region can be written as follows after normalization: ρ(x,t+T) =1 A(T)(ρ(x,t) +T/T m) (11) whereA(T) is the normalization factor, Tmis a certain time size to be determined, which may be relevant to the concrete motion state of the particle, this will be the secon d useful rule. We first work out the normalization factor A(t), considering the following two conditions:(1)when T= 0,ρ(x,t+T) = ρ(x,t), andA(0) = 1;(2)when T→ ∞,ρ(x,t+T)→1, andA(∞)→T/T m, we can get A(T) = 1 +T/T m, then the above formula will be: ρ(x,t+T) =ρ(x,t) +T/T m 1 +T/T m(12) or it can be written as follows: ††For CSTM, the stay time of the particle in any position is zero , so its position measure density ρ(x,t) is not influenced by the stochastic jump. 7∆ρ(x,t) =T Tm+T(1−ρ) (13) in general, when T > T p, namely when the particle undergoing DSTM stays in a local re gionLpfor a time interval longer than Tp, we can divide the whole time interval Tinto many Planck cell Tp, and the above formula is still valid for every cell, thus in the following discussions we let T=Tpfor simplicity. In order to further find the formula of Tm, we need to study the limitation on the jump of the particle un dergoing DSTM, since the item Tmjust denotes this kind of limitation, this can be seen from th e following two extreme situations:(1)when Tm→ ∞, we have ∆ ρ→0, this denotes that the position measure density will be not influenced by the jump, and the particle can jump freely;(2) Tm→0, we have ρ→1, this denotes that the position measure density will turn to be one in the region where the particle st ays, and the position measure density in other regions will turn to be zero, so the particle can not jump at all. In fac t, from the physical analysis about the jump we can see that the limitation results from the principle of energy conservation, according to which during a finite nonzero time interval ∆ tthe possible change of energy ∆ Ejresulting from jump will be limited by the uncertainty relat ion ∆Ej≈¯h/∆t, now we consider two situations, first, if the total differenc e of energy ∆ Ebetween the original stay region and other regions satisfies the condition ∆ E≫∆Ej, then the particle can hardly jump from its original region to other regions, namely after the stay time ∆ tthe position measure density ρ(x,t) in the original region will be greatly increased, especially when ∆ E→ ∞, we haveρ(x,t)→1, andTm→0‡‡; Secondly, if the total difference of energy ∆Ebetween the original stay region and other regions satisfies the condition ∆ E≪∆Ej, then the particle can jump more easily from its original region to other region s, namely after the stay time ∆ tthe position measure densityρ(x,t) will be only changed slightly, especially when ∆ E→0, we have ∆ ρ(x,t)→0, andTm→ ∞. Then we can see that Tmis inversely proportional to ∆ E, considering the dimension requirement their relation wil l be Tm= ¯h/k∆E, wherekis a dimensionless constant. Now, the change of the position measure density after stay ti meTpcan be formulated in a more complete way: ρ(x,t+T) =ρ(x,t) +k∆E/E p 1 +k∆E/E p(14) or it can be written as follows: ∆ρ(x,t) =∆E kEp+ ∆E(1−ρ) (15) whereEp= ¯h/Tpis Planck energy, thus we get the second useful rule for findin g the evolution law of DSTM. C. The evolution law of motion in discrete space-time (DSTM) Now, according to the above two rules, we can give the evoluti on equation of DSTM. For simplicity but lose no generality, we consider a one-dim ension initial wave function ψ(x,0), according to the above analysis, the concrete evolution equation of DSTM wil l be essentially one kind of revised stochastic evolution equation based on Schr¨ odinger equation, here we assume the form of stochastic differential equation ( SDE ), it can be written as follows: dψ(x,t) =1 i¯hHQψ(x,t)dt+1 2[δxxN ρ(x,t)−1]∆E(xN,xN) kEp+ ∆E(xN,xN)ψ(x,t)dt (16) where the first term in right side represents the evolution el ement resulting from CTSM, the average behavior of DSTM,HQis the corresponding Hamiltonian, the second term in right s ide represents the evolution element resulting from the stochastic jump resulting from DSTM itself, δxxNis the discrete δ-function,kis a dimensionless constant, ρ(x,t) =|ψ(x,t)|2, is the position measure density, ∆ E(xN,xN) is the total difference of energy of the particle between the cell containing xNand all other cells xN,xNis a stochastic position variable, whose distribution is P(xN,t) =ρ(xN,t). In physics, this stochastic differential equation is essent ially a discrete evolution equation, all the quantities are defined relative to the Planck cells TpandLp, and the equation should be also solved in a discrete way. ‡‡In fact, in this situation the wave function has collapsed in to this local region in order to satisfy the requirement of en ergy conservation, and this also indicates that in order to satis fy the principle of energy conservation DSTM will naturally result in the collapse of the wave function. 8D. Some further discussions Now we will give some physical analyses about the above evolu tion equation of DSTM, first, the linear item in the equation will result in the spreading process of the wave fun ction as for the evolution of CSTM, while the nonlinear stochastic item in the equation will result in the localizin g process of the particle or collapse process of the wave function, this can also be seen qualitatively, since accord ing to the nonlinear stochastic item, in the region where the position measure density is larger the stay time of the parti cle will be longer, moreover, the longer stay time of the particle in one region will further increase the position me asure density in that region much more, thus this process is evidently one kind of positive feedback process, the partic le will finally stay in a local region, and the wave function of particle will also collapse to that region, so the evolution of DSTM will be some kind of combination of the spreading process and localizing process. Secondly, the strength of the spreading process and localiz ing process is mainly determined by the energy difference between different branches of the wave function, if the energ y difference is so small, then the evolution of DSTM will be mainly dominated by the spreading process, or we can say, t he display of DSTM will be more like that of quantum motion (CSTM), this is just what happens in microscopic worl d; while if the energy difference is so large, then the evolution of DSTM will be mainly dominated by the localizing process, or we can say, the display of DSTM will be more like that of classical motion (CCM), this is just what ha ppens in macroscopic world, and the boundary of these two worlds can also be estimated, the following example indi cates that the energy difference in the boundary may assume ∆E∼=/radicalbig ¯hEp≈7Mev, the corresponding collapse time will be in the level of seco nds. Thirdly, if the particle finally stay in a local region during the evolution of DSTM, the localizing probability of the particle, or the collapse probability the wave function in o ne local region is just the initial position measure density of the particle in that region, namely the probability satisfie s the Born rule in quantum mechanics, since the stochastic evolution of DSTM satisfies the Martingale condition, this c an be seen from the following fact, namely during every jump the position measure density ρsatisfies the equation P(ρ) =ρP(ρ+∆E kEp+∆E(1−ρ)) + (1 −ρ)P(ρ−∆E kEp+∆Eρ) [12], where P(ρ) is the probability of ρturning into one in one local region, namely the probability of the particle localizing in a local region, moreover, the solution of this equation is P(ρ) =ρ, this just means that the localizing probability of the particle in one region is just the initial position measure density of the particle in that region. Fourthly, the collapse process resulting from the evolutio n of DSTM has no tails, since the evolution is essentially discrete, the wave function is just the description of the mo tion of particle, and its existence is only in the meaning of time average, while the particle, the real object, always exists in one local position, thus in the last stage of the collapse process, when the particle stays in one of the branc hes long enough it will de facto collapse into that branch owing to the limitation of energy conservation, and the wave function, the apparent ”object”, will also completely disappears in other branches§§. At last, the existence of DSTM will help to tackle the well-kn own time problem involved in formulating a complete theory of quantum gravity [13], since as to DSTM, the local po sition state of particle will be the only proper state, and the only real physical existence, during a finite time int ervalTpthe particle can only be limited in a local space intervalLp, namely there does not exist any essential superposition of different positions at all, the superposition of the wave function is only in the meaning of time average, th us the essential inconsistency of the superposition of different space-time in the theory of quantum gravity, which results from the existence of the essential superposition of the wave function, will naturally disappear, and the real physical picture based on DSTM will be that at any instant ( during a finite time interval Tp) the structure of space-time determined by the existence of the particle ( in a local space interval Lp) is definite or ”classical”, while during a finite time interv al much larger than Tpbut still small enough it will be stochastically disturbed by the stoc hastic jump of the particle undergoing DSTM, this kind of stochastic disturbance will be the real quantum nature of the space-time and matter. E. One simple example In this section, as one example we will analyze the DSTM evolu tion of a simple two-state system, and quantifica- tionally show that the evolution of DSTM will indeed result i n the collapse process of the wave function. §§If the wave function is taken as some kind of essential existe nce, and its evolution is essentially continuous, then the t ails problem will be inevitable. 9We suppose the initial wave function of the particle is ψ(x,0) =α(0)1/2ψ1(x)+β(0)1/2ψ2(x), which is a superposition of two static states with different energy levels E1andE2, these two static states are located in separate regions R1 andR2with the same size. Since the energy of the particle inside the region of each sta tic state is the same, we can consider the spreading space of both static states as a whole local region, and only study t he stochastic jump between these two regions resulting from the evolution of DSTM, namely we directly consider the d ifference of the energy ∆ E=E2−E1between these two states, through some mathematical calculations we can w ork out the density matrix of the two-state system, it is: ρ11(t) =α(0) (17) ρ22(t) =β(0) (18) ρ12(t) = [1−1 2(∆E kEp+ ∆E)2]t/Tp/radicalbig α(0)β(0)≈(1−(∆E)2 2k2¯hEpt)/radicalbig α(0)β(0) (19) ρ21(t) = [1−1 2(∆E kEp+ ∆E)2]t/Tp/radicalbig α(0)β(0)≈(1−(∆E)2 2k2¯hEpt)/radicalbig α(0)β(0) (20) It is evident that these results confirm the above qualitativ e analysis definitely, namely, the evolution of DSTM indeed results in the collapse of the wave function describi ng DSTM, and the distribution of the collapse results satisfies the Born rule in quantum mechanics, besides, we als o get the concrete collapse time for two-state system, it isτc≈2k2¯hEp (∆E)2∗∗∗. VIII. THE APPEARANCE OF CLASSICAL MOTION IN MACROSCOPIC WOR LD The above analysis has indicated that, when the energy differ ence between different branches of the wave function is large enough, say for the macroscopic situation†††, the linear spreading of the wave function will be greatly su ppressed, and the evolution of the wave function will be dominated by th e localizing process, in fact, the motion state of the particle will be only local position state in appearance, an d the evolution of this state will be only still or continuous ly move in space, this is just the display of CCM in macroscopic w orld. Furthermore, we will show that the evolution law of CCM can al so be derived, in fact, some people have strictly given the demonstration based on revised quantum dynamics [ 9,10], here we simply use the Enrenfest theorem, namely d<x> dt=< p > andd<p> dt=<−∂U ∂x>, as we have demonstrated, for macroscopic object its wave fu nction will no longer spread, thus the average items in the theorem will rep resent the effective description quantities for the classic al motion of the macroscopic object, and the classical motion l aw is also naturally derived in such a way, the result is dx dt=p, the definition of the momentum, anddp dt=−∂U ∂x, the motion equation. IX. CONCLUSIONS In this paper, we strictly demonstrate the logical inevitab ility of the existent form and evolution law of CSTM, the existence of discrete space-time in Nature and resultin g real existence of DSTM and its evolution law, this not only explains the appearance of classical motion in macrosc opic world, as well as quantum motion in microscopic world consistently and objectively, but also presents a cle ar logical connection between quantum motion and classical motion, and unveils the unified realistic picture of microsc opic and macroscopic world. ∗∗∗This result has also been obtained by Hughston [11] and Fivel [8] from different point of views, and discussed by Adler etc [1,2]. †††The largeness of the energy difference for macroscopic objec t results mainly from the environmental influences such as thermal energy fluctuations. 10Acknowledgments Thanks for helpful discussions with X.Y.Huang ( Peking Univ ersity ), A.Jadczyk ( University of Wroclaw ), P.Pearle ( Hamilton College ), F.Selleri ( University di Bari ), Y.Shi ( University of Cambridge ), A.Shimony, A.Suarez ( Center for Quantum Philosophy ), L.A.Wu ( Institute Of Physics, Aca demia Sinica ), Dr S.X.Yu ( Institute Of Theoretical Physics, Academia Sinica ), H.D.Zeh. Appendix:Mathematical Analysis About Motion In Continuou s Space-time First, we will give three general presuppositions about the relation between physical motion and mathematical point set, they are basic conceptions and correspondence ru les before we discuss the physical motion of particles in continuous space-time. (1). Time and space in which the particle moves are both conti nuous point set. (2). The moving particle is represented by one point in time a nd space. (3). The motion of particle is represented by the point set in time and space. The first presupposition defines the continuity of space-tim e, the second one defines the existent form of particle in time and space, the last one relates the physical motion of pa rticle with the mathematical point set. For simplicity but lose no generality, in the following we wi ll mainly analyze the point set in two-dimension space- time, which corresponds to one-dimension motion in continu ous space-time. A. Point set and its law—a general discussion As we know, the point set theory has been deeply studied since the beginning of this century, nowadays we can grasp it more easily, according to this theory, we know that t he general point set is dense point set, whose basic property is the measure of the point set, while the continuou s point set is one kind of special dense point set, its basic property is the length of the point set. Naturally, as to the point set in two-dimension space-time, the general situation is the dense point set in this two-dimension space-time, while the continuous curve is on e kind of extremely special dense point set, surely it is a wonder that so many points bind together to form one continu ous curve by order, in fact, the probability for its natural formation is zero. Now, we will generally analyze the law of the point set, as we k now, the law about the points in point set, which can be called point law, is the most familiar law, and it is wid ely taken as the only rational law, for example, as to the continuous curve in two-dimension space-time there may exi st a certain expressible analytical formula for the points in this special point set‡‡‡, but as to the general dense point set in two-dimension space -time the point law possesses no mathematical meaning, since the dense point set is discon tinuous everywhere, even if the difference of time is very small, or infinitesimal, the difference of space can be very la rge, then infinitesimal error in time will result in finite error in space, thus even if the point law exists we can not for mulate it in mathematics, and owing to finite error in time determination and calculation, we can not prove it eith er, let alone predict the evolution of the point set using it, in one word, there does not exist point law for dense point set in mathematics. B. Deep analysis about dense point set Now, we will further study the differential description of po int set in detail. First, in order to find the differential description of the spe cial dense point set—continuous curve, we may measure the rise or fall size of the space ∆ xcorresponding to any finite time interval ∆ tnear each instant tj, then at any instanttjwe can get the approximate information about the continuous curve through the quantities ∆ tand ∆xat that instant, and when the time interval ∆ tturns smaller, we will get more accurate information about t he curve. In theory, we can get the complete information through this i nfinite process, that is to say, we can build up the ‡‡‡People cherish this kind of point laws owing to their infrequ ent existence, but perhaps Nature detests and rejects them, since the probability of creating them is zero. 11basic description quantities for the special dense point se t—continuous curve, which are the differential quantities d t and dx, then given the initial condition the relation betwee n dt and dx at all instants will completely describe the continuous curve. Then, we will analyze the differential description of the gen eral dense point set, as to this kind of point set, we still need to study the concrete situation of the point set corresp onding to finite time interval near every instant. Now, when time is during the interval ∆ tnear instant tj, the points in space are no longer limited in the local space i nterval ∆x, they distribute throughout the whole space instead, so we s hould study this new point set, which is also dense point set, for simplicity but lose no generality, we conside r finite space such as x ∈[0,1], first, we may divide the whole space in small equal interval ∆ x, the dividing points are denoted as xi, then we can define and calculate the measure of the local dense point set in the space interval ∆ xnear eachxi, which can be written as M∆x,∆t(xi,tj), since the measure sum of all local dense point sets in time interval ∆ tjust equals to the length of the continuous time interval ∆t, we have: /summationdisplay iM∆x,∆t(xi,tj)= ∆t (21) On the other hand, since the measure M∆x,∆t(xi,tj) will also turn to be zero when the intervals ∆ xand ∆tturn to be zero, it is not an useful quantity, and we have to create a new quantity on the basis of this measure. Through further analysis, we find that a new quantity ρ∆x,∆t(xi,tj) =M∆x,∆t(xi,tj)/(∆x·∆t), which can be called average measure density, will be an useful one, it generally does not turn to be zero when ∆ xand ∆tturn to be zero, especially if the limit lim∆x→0,∆t→0ρ∆x,∆t(xi,tj) exists, it will no longer relate to the observation sizes ∆ xand ∆t, so it can accurately describe the whole dense point set, as well as all local dense point sets near every instant, now we let: ρ(x,t) =lim∆x→0,∆t→0ρ∆x,∆t(x,t) (22) then we can get: /integraldisplay Ωρ(x,t)dx= 1 (23) this is just the normalization formula, where ρ(x,t) is called position measure density, Ω denotes the whol e integral space. Now, we will analyze the new quantity ρ(x,t) in detail, first, the position measure density ρ(x,t) is not a point quantity, it is defined during infinitesimal interval, this f act is very important, since it means that if the measure densityρ(x,t) exists, then it will be continuous relative to both t and x, t hat is to say, contrary to the position function x(t), there does not exist the discontinuous situa tion for the measure density function ρ(x,t), furthermore, this fact also results in that the continuous function ρ(x,t) is the last useful quantity for describing the dense po int set; Secondly, the essential meaning of the position measur e densityρ(x,t) lies in that it represents the dense degree of the points in the point set in two-dimension space and time , and the points are denser where the position measure densityρ(x,t) is larger. C. The evolution of dense point set Now, we will further discuss the evolution law for dense poin t set. Just like the continuous position function x(t), although the continuous position measure density functi onρ(x,t) completely describes the dense point set, it is one kind of st atic description about the point set, and it can not be used for prediction itself, so in order to predict the evolution o f the dense point set we must create some kind of quantity describing its change, enlightened by the theory of fluid mec hanics we can define the fluid density for the position measure density ρ(x,t) as follows: ∂ρ(x,t) ∂t+∂j(x,t) ∂x= 0 (24) we call this new quantity j(x,t) position measure fluid density, this equation measure cons ervation equation, it is evident that this quantity just describes the change of the m easure density of dense point set, thus the general evolution equations of dense point set can be written as in th e following: ∂ρ(x,t) ∂t+∂j(x,t) ∂x= 0 (25) 12∂j(x,t) ∂t+...= 0 (26) [1] S.L.Adler, L.P.Horwitz, e-print quant-ph/9904048, (1 999) [2] S.L.Adler, L.P.Horwitz, e-print quant-ph/9909026, (1 999) [3] J. S. Bell,Speakable and unspeakable in quantum mechani cs(Cambridge University Press, Cambridge 1987) [4] D.Bohm, Phys.Rev. 85,166-193. (1952) [5] N.Bohr, Nature( London ). 121, 580-590 (1927) [6] Albert Einstein, Ann.Phys.( Leipzig ) 17, 132 (1905) [7] H.Everett. Rev.Mod.Phys. 29, 454-462.(1957) [8] D.I.Fivel, e-print quant-ph/9710042, (1997) [9] G.C.Ghiradi, A.Rimini, and T.Weber, Phys. Rev. D. 34470-491. (1986) [10] G.C.Ghiradi, A.Rimini, and T.Weber, Phys. Rev. D. 421057-1064. (1990) [11] L.P.Hughston, Proc.Roy.Soc.Lond.A, 452,953 (1996) [12] P.Pearle, Combining stochastic dynamical state-vect or reduction with spontaneous localization. Phys. Rev. A 39,2277- 2289. (1989) [13] R.Penrose, On gravity’s role in quantum state reductio n.Gen. Rel. and Grav. ,28,581-600. (1996) 13

arXiv:physics/0001013v1 [physics.gen-ph] 6 Jan 2000Everyone can understand quantum mechanics Gao Shan Institute of Quantum Mechanics 11-10, NO.10 Building, YueTan XiJie DongLi, XiCheng Distri ct Beijing 100045, P.R.China E-mail: gaoshan.iqm@263.net We show that everyone can understand quantum mechanics, onl y if he rejects the following prejudice, namely classical continuous motion (CCM) is the only possible and objective motion of particles. I think I can safely say that nobody today understands quantu m mechanics. ——Feynman (1965) When people talk about motion, they only refer to CCM, its uni queness is taken for granted absolutely but unconsciously, people never dream of another different moti on in Nature, but to our surprise, as to whether or not CCM is the only possible and objective motion, and whether CC M is the real motion or apparent motion, no one has given a definite answer up to now. In classical mechanics, CCM is undoubtedly the leading acto r, while in quantum mechanics, CCM is rejected by the orthodox interpretation from stem to stern, but why did peop le never guess what quantum mechanics describes is just another different motion from CCM? as we think, this is the mos t direct and natural idea, since classical mechanics describes CCM, then correspondingly quantum mechanics wil l describe another kind of motion. The only stumbling block is just the huge prejudice rooted in the mind of people, it is that classical continuous motion (CCM) is the only possible and objective motion of par ticles, now let’s see it more clearly through looking back to the history. Bohr [2] and his enthusiastic supporters held this prejudic e strong, they insisted that Copenhagen interpretation is the only possible interpretation of quantum mechanics, s ince CCM can no longer account for the phenomena in quantum mechanics, we must essentially discard it, the only possible and objective motion, then it is evident that quantum mechanics provides no objective description of Nat ure at all, but only our knowledge about Nature. Einstein [3] held this prejudice stronger, he believed that if the objective picture of classical continuous motion contradicts with quantum mechanics, the wrong side can only be quantum mechanics, not classical continuous motion, since in any case we can not lose the reality, while classical continuous motion is the only reality of Nature, thus he became the strongest opponent of Copenhagen interpretatio n, but his acerbic comments did not help him so much, he failed in persuading Bohr, as well as his contemporary. Bohm [1] also held this prejudice, his cleverness lies in tha t he provided a compromise hidden-variable picture between those of Bohr and Einstein, but neither one was satis fied with his way, and he himself was also tortured by the dualistic monster he created. Everett [4] still held this prejudice, even though he presen ted a crazy many worlds interpretation for quantum mechanics, his interpretation is still in the framework of C CM, only for every branch of the expensive many worlds, and no supporters would like to attempt quantum suicide to co nvince themselves the many worlds interpretation is right, let alone convince anyone else. More and more followers have been trying to understand quant um mechanics, but they still held this prejudice firmly and unconsciously, they are doomed to fail, this is the ir destiny due to the prejudice. Then why cling to it till death like a miser? unloosen it! plea se reject it! and don’t walk along this wrong way any more, it only leads to the blind alley, the impasse, no way out there. In our previous paper [5], from the clear logical and physica l analyses about motion, we have shown that the natural motion in continuous space-time is not CCM, but one kind of es sentially discontinuous motion, and Schr¨ odinger equation in quantum mechanics is just its simplest nonrelat ivistic motion equation; while in the real discrete space- time, the natural motion is also discontinuous, and it will r esult in the collapse process of the wave function, this brings about the appearance of CCM in macroscopic world, thu s CCM is by no means the real motion in Nature, let alone be the only possible and objective motion, it is just on e kind of ideal apparent motion in the macroscopic world where we live, while the real motion is essentially disconti nuous. 1Once we reject the apparent CCM, and find the real motion in Nat ure, understanding quantum mechanics is just an easy task, we can safely say that everybody can understand qu antum mechanics easily from now on, nobody will be plagued by its weirdness any more, since quantum mechanics i s just the theory describing the real motion in Nature, even though the real motion is more complex than CCM, it also h as a clear picture just like CCM, its weirdness results only from its particular existence and evolution, in fact, f rom a logical point of view, its existence and evolution are more natural than those of CCM, only because we are unfamilia r with it, it looks very bizarre for us. Concretely speaking, the wave function ψ(x,t) in quantum mechanics is an indirect mathematical complex t o describe the state of the real motion of particle, the direct description quantities are ρ(x,t) andj(x,t), their relation isψ(x,t) =ρ1/2·eiS(x,t)/¯h, whereS(x,t) =m/integraltextx −∞j(x′,t)/ρ(x′,t)dx′+C(t), the apparent wave-like form of ψ(x,t) results essentially from the discontinuity of the real moti on, not from any objective existence of wave or field. The evolution of the real motion includes two parts, one is th e linear evolution part, it results in the interference pattern, which is usually the display of classical wave, but the pattern is undoubtedly formed by a large number of particles undergoing the real motion; the other is the nonli near stochastic evolution part, it results in the collapse process of the wave function, during measurement this proce ss happens very soon, and the wave function of the particle collapses into a local region, this brings about th e appearance of single event in measurement, this process is stochastic and indeterministic due to the essential discon tinuity and randomicity of the real motion itself. Certainly, one point needs to be stressed, even though the wa ve function does provide a complete description of the state of the real motion, present quantum theory does not pro vide a complete description of the evolution of the real motion, and needs to be revised to include the stochastic evo lution part. Now we may also understand why people haven’t understood qua ntum mechanics yet after they found it more than seventy years ago, the reason is very simple, because people always discuss and picturize it in the framework of CCM, they can only see the sky of CCM, some of them would ruthlessly reject the reality in the quantum world rather than give another possible motion a glance, the others would never ever give up CCM, this is indeed the sorriness of science, but the most heart-struck is that people are always very complacent about their own choices, and care little about the ideas of others, all these will be fundamentally ch anged from now on. [1] D.Bohm, Phys.Rev. 85,166-193. (1952) [2] N.Bohr, Nature. 121, 580-590 (1927) [3] A.Einstein, B.Podolsky, and N.Rosen, Physical Review. 47, 777-780 (1935) [4] H.Everett. Rev.Mod.Phys. 29, 454-462.(1957) [5] Gao Shan, physics/0001012. 2

arXiv:physics/0001014v1 [physics.chem-ph] 6 Jan 2000PHOTO-INDUCED INTERMOLECULAR CHARGE TRANSFER IN PORPHYRIN COMPLEXES Michael Schreiber, Dmitry Kilin, and Ulrich Kleinekath¨ of er Institut f¨ ur Physik, Technische Universit¨ at, D-09107 Ch emnitz, Germany Optical excitation of the sequential supermolecule H2P−ZnP−Qinduces an electron transfer from the free-base porphyrin ( H2P) to the quinone ( Q) via the zinc porphyrin ( ZnP). This process is modeled by equations of motion for the reduced density matrix which are solved numer ically and approximately analytically. These two solutions agree ver y well in a great region of parameter space. It is shown that for the majority o f solvents the electron transfer occurs with the superexchange mechanism . I. INTRODUCTION The investigation of photoinduced charge transfer is impor tant both for the de- scription of natural photosynthesis [] and for the creation of artificial photoenergy- converting devices []. For experimental realizations of su ch artificial devices porphyrin complexes are good candidates []. Of major interest are thos e complexes with an ad- ditional bridging block between donor and acceptor []. Electron transfer reactions can occur through different mec hanism []: sequential transfer (ST) or superexchange (SE). Changing a building bl ock of the complex [] or changing the environment [] can modify which mechanism is most significant. To clarify which mechanism is present one sequentially vari es the energetics of the complex []. This is done by radical substituting the porphyr in complexes [] or by changing the polarity of the solvent []. Also the geometry an d size of a bridging block can be varied and in this way the length of the subsystem throu gh which the electron has to be transfered []. SE [] occurs due to coherent mixing of the levels [] and plays a role for any detuning of the energy levels []. The transfer rate in this channel dec reases exponentially with increasing length of the bridge []. When incoherent effects s uch as dissipation and dephasing dominate [], the transfer is mainly sequential [] , i. e., the levels are occupied mainly in sequential order []. An increase in the bridge leng th induces only a small reduction in the transfer rate []. In the case of coherent SE the dynamics is mainly Hamiltonian and can be de- scribed on the basis of the Schr¨ odinger equation. The physi cally important results can be obtained by perturbation theory [], most successfull y by the Marcus theory []. In case of ST the environmental influence has to be taken in to account. The more natural description of the relaxation process is based on the density matrix (DM) formalism []. The master equation that governs the DM ev olution as well as the appropriate relaxation coefficients can be derived from s uch basic information as system-environment coupling strength and spectral densit y of the environment []. The main physics of the system can be described by a DM equatio n which accounts for relaxation effects phenomenologically []. The master eq uation is analytically solv- able only for the simplest models []. Most investigations ar e based on the numerical solution of this equation []. However, an estimations can be obtained within the steady-state approximation []. Here we perform numerical a s well as approximate analytical calculations. II. MODEL We investigate the photoinduced electron transfer in super molecules that consist of sequentially connected molecular blocks, namely donor, bridge, and acceptor. The donor (D) is not able to transfer its charge directly to the ac ceptor (A) because ofD B AD B AEnergyD B A D B A D B A D B A*+ - * + - + -Energy+ Charge TransferSuperexchangeSequentialtransferExcitationFIG. 1. Schematic view of the energy lev- els in the H2P−ZnP− Qcomplex taken into account in calculation. The three states in the boxes define the charge separation which can happen either by ST or by SE. their spatial separation. D and A can exchange their charges only through B (Fig. 1). In the present investigation the supermolecule consists of free-base porphyrin( H2P) as donor, zinc substituted porphyrin( ZnP) as bridge, and benzoquinone as accep- tor []. In each of those molecular blocks we consider only two molecular orbitals, the LUMO and the HOMO. Each of those orbitals can be occupied b y an elec- tron ( |1/an}bracketri}ht) or not ( |0/an}bracketri}ht). This model allows us to describe the neutral nonexcited molecule |1/an}bracketri}htHOMO|0/an}bracketri}htLUMO and the following three states of the molecule: neutral excited molecule |0/an}bracketri}htHOMO|1/an}bracketri}htLUMO, positive ion |0/an}bracketri}htHOMO|0/an}bracketri}htLUMO ,and negative ion |1/an}bracketri}htHOMO|1/an}bracketri}htLUMO. Below Roman indices indicate molecular orbitals ( m= 0 - HOMO, m= 1 - LUMO), while Greek indices indicate molecular blocks ( µ= 1 - donor, µ= 2 - bridge, µ= 3 - acceptor). Each of the electronic states has its own vibr ational substructure. However the time of vibrational relaxation [ ] is two orders of magni- tude faster than the characteristic time of the electron tra nsfer []. Because of this we assume that only the vibrational ground states play a domi nant role in electron transfer. One can describe the occupation of an orbital by an electron w ith the appropriate creation operator c+ µm=|1/an}bracketri}htµm/an}bracketle{t0|µmas well as its annihilation cµm=|0/an}bracketri}htµm/an}bracketle{t1|µm. Then ˆ nµ=/summationtext mc+ µmcµmgives the number of electrons in the molecular block µ. For the description of charge transfer and other dynamical p rocesses in the system we introduce the Hamiltonian ˆH=ˆHS+ˆHB+ˆHSB, (1) where HScharacterizes the supermolecule, HBthe dissipative bath, and HSBthe interaction between the two. HS, however, includes the static influence of the envi- ronment, namely of the solvent dipoles, which gives rise to a reduction of the energy levels, ˆHS=/summationdisplay µmEµmˆnµm+3 ǫs+ 2(ˆEel+ˆEion) +ˆV , . (2) The energies Eµmare calculated in the independent particle approximation [ ].ǫs denotes the static dielectric constant of the solvent. ˆEel=/summationtext µ(ˆnµ−1)e2/(4ǫ0rµ) describes the energy to create an isolated ion. This term dep ends on the characteristic radius rµof the molecular blocks. ˆEion=/summationtext µ/summationtext ν(ˆnµ−1)(ˆnν−1)e2/(4πǫ0rµν) includes the interaction between the already created ions. It depend s on the distance between the molecular blocks rµν. The last contribution to the system Hamiltonian is the hopping term ˆV=/summationtext µνvµν(ˆV+ µν+ˆV− µν)((ˆnµ−1)2+ (ˆnν−1)2),which includes the coherent hopping between each pair of LUMO ˆV− µν=c+ ν1cµ1,ˆV+= (ˆV−)+as well as the corresponding intensities vµν. The matrix elements of this operator give nonzero contribution only if one of the states has a charge separatio n. Because there is no direct connection between donor and acceptor we assume v13= 0.As usual the bath is given by harmonic oscillators with creat ion and anhilation operators a+ λandaλ. The system bath interaction comprises both irradiative an d radiative transitions. For t≪1−10nsthe latter one can be neglected . The irradiative contribution corresponds to energy transfer t o the solvent and spreading of energy over vibrational modes of the supermolecule ˆHSB=/summationdisplay λ/summationdisplay µνKλ,µνvµν(a+ λ+aλ)(ˆV+ µν+ˆV− µν), (3) where Kλ,µνreflects the interaction strength between bath mode λand quantum transition between LUMO levels of molecules µandν. Initially we use the whole density matrix of system and bath f or the description of the dynamics. After applying the Markov and rotating wave approximations and tracing out the bath modes [] we obtain the equation of motion for the reduced density matrix (RDM) ˙σ=−i/¯h[ˆHS, σ] +/summationdisplay µνΓµν{(n(ωµ1ν1) + 1)([ ˆV− µνσ,ˆV+ µν] + [ˆV− µν, σˆV+ µν]) +n(ωµ1ν1)([ˆV+ µνσ,ˆV− µν] + [ˆV+ µν, σˆV− µν])}, (4) where the dissipation intensity Γ µν=πK2 µνρ(ωµ1ν1)v2 µνdepends on the coupling Kµν of the transition µ1;ν1 and on the bath mode of the same frequency. Γ µνdepends also on the density ρof bath modes at the transition frequency ωµ1ν1and on the corresponding coherent coupling vµνbetween the system states. n(ω) denotes Bose- Einstein distribution. For simplicity we introduce a superindex i={µm}, the intensities of the dissi- pative transitions dij= Γ ijn(−ωij) between each pair of states, as well as the corre- sponding dephasing intensities γij= 1/2/summationtext k(dik+dkj). Taking these simplifications into account one gets ˙σii=−i/¯h/summationdisplay j(Vijσji−σijVji)−/summationdisplay idijσii+/summationdisplay jdjiσjj, (5) ˙σij= (−iωij−γij)σij−i/¯hVij(σjj−σii). (6) The simplification is that we do not calculate the system para meters, rather we extract them from experimental data. III. EXTRACTION OF SYSTEM PARAMETERS The porphyrin absorption spectra [] consist of high frequen cy Soret bands and low frequency Qbands. In case of ZnP theQband has two subbands, Q(0,0) and Q(1,0). In the free-base porphyrin H2Pthe reduction of symmetry induces a splitting of each subband into two, namely Qx(0,0),Qy(0,0) and Qx(1,0),Qy(1,0). So the emission spectra of ZnPandH2Pconsist of two and four bands, respectively. Each of the abovementioned spectra can be represented as a sum of L orentzians with good precision. It is important to note that the spectra of porphy rin complexes contain all bands of the isolated porphyrins without essential changes . We use the lowest band of each spectrum. The corresponding frequencies and widths are shown in table 1. On the basis of the experimental spectra we determine ED∗BA= 1.82eVand EDB∗A= 2.03eV(inCH2Cl2). The authors of Ref. give the energies of two other levels, ED+B−A= 2.44eVandED+BA−= 1.42eV. This allows to calculate EDB+A−= 1.21eV. The hopping intensity v23=v= 2.2meV is calculated in Ref. . On the other hand Rempel et al. [] estimate the electron coupling of the initially excited and charged bridge states v12=V= 65meV. We take the intensity of the intermolecular conversions Γ 21, Γ23in range 1 −10×1011s−1[].Table 1: Low-energy bands of the porphyrin spectra for CH2Cl2as solvent. Absorption Emission Frequency, eV Width, eV Frequency, eV Width, eV H2P νx 00= 1.91 γx 00= 0.06 νx 01= 1.73 γx 01= 0.05 ZnP ν 00= 2.13 γ00= 0.07 ν01= 1.92 γ01= 0.05 The main parameter which controls the electron transfer in a triad is the relative energy of the state D+B−A. This state has a strong coupling to the solvent that changes the energy of the state. The values of the energy ED+B−Acalculated in the present model are shown in table 2 for some solvents. In ta ble 2ǫsdenotes the static dielectric permittivity, ǫ∞the optic dielectric permittivity, MTHF 2-methil- tetrahydrofuran, and CYCLO denotes cyclohexane. The calcu lated value ED+B−A= 2.86eVdeviates 15% from the data of Ref. . IV. RESULTS The time evolution of charge transfer within the supermolec ule is described by Eqs. (5) and (6). At initial time only the donor state is occup ied. The calcula- tions were performed with two methods, direct numerical int egration and analytic approximation. For the numerical simulation the eigenvalues and -vectors o f the system are cal- culated and with these the time evolution of the system is kno wn. The simulation of the system dynamics with the parameters determined in the pr evious section shows exponential growth of the acceptor population. Such a behav ior can be accurately fitted to the formula P3(t) =P3(∞)[1−exp (−kETt)],where kET≃5×109s−1and P3(∞)≃0.95 for CH2Cl2as solvent. The population of the bridge state does not ex- ceed 0 .005. This shows that the SE mechanism dominates over the ST fo r the chosen set of parameters. In this case the system dynamics can be des cribed by two values: the acceptor population at infinite time P3(∞) and the reaction rate kETthat we deduce from the dynamics via the following formula kET=P3(∞)/{/integraltext∞ 0[1−P3(t)]dt}. The analytical approach is valid for the kinetic limit t≫1/γij. In Laplace-space we can replace 1 /(iωij+γij+s) by 1/(iωij+γij), where sdenotes the Laplace variable. This allows to simplify Eqs. (5) and (6) and we define a new rela xation operator (Lσ)new ii=−/summationtext igijσii+/summationtext jgjiσjj.In this expression the transition coefficients gij contain both, dissipative and coherent contributions gij=dij+vijvjiγij/[¯h2(ω2 ij+γ2 ij)]. (7) Assuming the bridge population to be zero allows us to find the dynamics of the acceptor state in the form P3(t) =P3(∞)[1−exp (−kETt)],where the final population P3(∞) and the reaction rate kETare expressed in terms of the coefficients gij kET=g23+g23(g12−g32) g21+g23, P 3(∞) =g12g23 g21+g23(kET)−1. (8) Table 2: Energy of the charged bridge state and transfer rate s in different solvents. Solvent 75% CH2Cl2 +25% CH3CNCH2Cl2 MTHF CYCLO ǫs 15.75 9 .08 6 .24 2 .02 ǫ∞ 2.00 2 .01 2 .03 2 .03 ED+B−A,eV 1.89 2 .86 3 .18 5 .30 kET,s−13.98×10115.01×1097.94×1083.80×10810 11 12 13 14 lg(V)891011lg(kLT) 10 11 12 13 14 lg(v)10 11 12 13 14 lg(Γ)101112131415 lg(γ) 10 11 12 13 140.00.20.40.60.81.0P3(∞) 10 11 12 13 14 10 11 12 13 14 10 11 12 13 14numerical analytical FIG. 2. The dependence of the reaction rate (upper row) and fin al population of the acceptor state (lower row) on the parameters V=v12,v=v23, Γ = Γ 21,γ= Γ23. Solid lines correspond to the numerical solution and dashed lines to the analytical solution. The circles show the realistic parameter values for CH2Cl2as solvent. −0.5 0.0 0.5 1.091011121314lg(kLT) −0.5 0.0 0.5 1.0 E, eV0.00.40.8P3(∞)numerical analyticalFIG. 3. Dependence of reaction rate (upper picture) and final accep- tor population (lower pic- ture) on the energy of the bridge state E= ED+B−A. Solid lines cor- respond to the numeri- cal solution and dashed lines to the analytical so- lution. V. DISCUSSION The following question will now be discussed: How does the me chanism and speed of the reaction depend on a deviation of the parameters from t he determined values? Namely which parameters have to be changed in order to change not only the reaction rate quantitatively, but the dominant mechanism of reactio n and the qualitative behavior of dynamics at all. To answer these questions we cal culate the system dynamics while varying one parameter at a time and keeping th e other parameters unchanged. The dependencies of transfer rate kETand final population P3(∞) on coherent couplings V=v12,v=v23and dissipation intensities Γ = Γ 21,γ= Γ23are shown in Fig. 2. In particular, the decrease of the coherent coupling Vinduces a quadratic decrease of the reaction rate kETuntil saturation V∼1010ps−1. Then kETreaches its lower bound and does not depend on Vanymore. This corresponds to the crossover of the reaction mechanism from SE mechanism to ST. But, due to the bi g energy differencebetween donor and bridge state the efficiency of this ST is extr emely low, i. e., P3;0. The considered variation of the coherent coupling can be exp erimentally performed by exchanging building blocks in the supermolecule. The most crucial change in the reaction dynamics can be induc ed by changing the energies of the system levels. As discussed above this ca n be done by altering the solvent. Most important is the relative energy of the bridge state|D+B−A/an}bracketri}ht. The results of the corresponding calculations are presented in Fig. 3. For high energies of the bridge state ED+B−A≫ED∗BAthe numerical and analytical results do not differ from each other. The reaction occurs with the SE mechanism th at coincides with the conclusion of Ref. . This is the case for the most of solven ts (see table 2). The smooth decrease of energy induces an increase of the reactio n rate up to the maximal value near 1 ps−1. While the bridge energy approaches the energy of the donor st ate the ST mecha- nism starts to contribute to the process. As can be seen in tab le 2 this regime can be reached by the use strong polar solvents. The analytical sol ution does not coincide with the numerical one anymore because the used approximati ons are no more valid in this region. In the case ED+B−A< E D+BA−one cannot approximate the dynamics of the acceptor population in the form P3∼[1−exp (−kETt)]. A high value of the bridge energy ensures the transition of the whole populatio n to the acceptor state |D+BA−/an}bracketri}ht. In the intermediate case, when the bridge state has the same energy as the acceptor state, the final population spreads itself over the se two states P3(∞) = 0.5. At even lower bridge energies the population gets trapped at the bridge state. We performed calculations for the electron transfer in the s upermolecular complex H2P−ZnP−Qwithin the RDM formalism. The resulting analytical and nume rical reaction rates are in good agreement with each other and in qu alitative correspon- dence with experimental data []. The SE mechanism of electro n transfer dominates over the sequential one. The qualitative character of the tr ansfer reaction is stable with respect to a small variation of the system parameter. Th e crossover between the reaction mechanisms can be forced by lowering the bridge sta te energy to the energy of the donor state. REFERENCES 1. D. G. Johnson et al., J. Am. Chem. Soc. ,115, 5692, (1993). 2. M. R. Wasielewski, Chem. Rev. ,92, 345, (1992). 3. U. Rempel et al., Chem. Phys. Lett. ,245, 253, (1995). 4. J. Zaleski, C. Chang, and D. Nocera, J. Phys. Chem. ,97, 13206, (1993). 5. E. Zenkevich et al., J. Lumin. ,76-77 , 354, (1998). 6. S. S. Scourtis and S. Mukamel, Chem. Phys. ,197, 367, (1995). 7. W. Davis et al., J. Phys. Chem. ,101, 6158, (1997). 8. M. Schreiber, C. Fuchs, and R. Scholz, J. Lumin. ,76-77 , 482, (1998). 9. R. A. Markus and N. Sutin, Biochim. Biophys. Acta ,811, 256, (1985). 10. H. M. McConnel, J. Chem. Phys. ,35, 508, (1961). 11. V. Mujica, M. Kemp, and M. A. Ratner, J. Chem. Phys. ,101, 6856, (1994). 12. O. K¨ uhn, V. May, and M. Schreiber, J. Chem. Phys. ,101, 10404, (1994). 13. D. Kilin and M. Schreiber, J. Lumin. ,76-77 , 433, (1998). 14. R. Loudon, The Quantum Theory of Light , Clarendon Press, Oxford, (1973). 15. V. May and M. Schreiber, Phys. Rev. A ,45, 2868, (1992). 16. O. K¨ uhn, Th. Renger, and V. May, Chem. Phys. ,101, 99, (1996). 17. M. Schreiber and D. Kilin, in: Proc. 2nd Int. Conf. Excitonic Processes in Condensed Matter , Editor M. Schreiber, p. 331, (1996). 18. D. A. Weitz et al., J. Chem. Phys. ,78, 5324, (1983). 19. D. Kilin, U. Kleinekath¨ ofer, and M. Schreiber (in prepa ration). 20. K. Wynne et al., J. Am. Chem. Soc. ,117, 3749, (1995).

arXiv:physics/0001015v1 [physics.atom-ph] 6 Jan 2000Using atomic interference to probe atom-surface interacti on Roberta Marani, Laurent Cognet, Veronique Savalli, Nathal ie Westbrook, Christoph I. Westbrook, Alain Aspect Laboratoire Charles Fabry de l’Institut d’Optique, Unit´ e Mixte du CNRS n◦8501, BP 147, 91403 Orsay CEDEX, France (February 2, 2008) We show that atomic interference in the reflection from two su itably polarized evanescent waves is sensitive to retardation effects in the atom–surface intera ction for specific experimental parameters. We study the limit of short and long atomic de Broglie wavelen gth. The former case is analyzed in the semiclassical approximation (Landau-Zener model). The latter represents a quantum regime and is analyzed by solving numerically the associated coupl ed Schr¨ odinger equations. We consider a specific experimental scheme and show the results for rubid ium (short wavelength) and the much lighter meta-stable helium atom (long wavelength). The mer its of each case are then discussed. I. INTRODUCTION The interaction between a ground–state atom and a dielectri c or conducting wall has been investigated theoretically ( [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]) and experi mentally ( [11], [12], [13], [14]). Theoretical studies hav e been performed on different levels, from a simple model of a dipole –dipole interaction of the atom and its mirror image, to the full QED relativistic quantum treatment. Interesting i n particular are the long–range Casimir interactions [6] th at were recently observed in cavity QED experiments [13], [14] . When the atom–wall distance zis not small compared to the wavelength of the dominant atomic transitions, the z−3law associated with the instantaneous electrostatic interaction is no longer valid. The full QED treatment leads to the famous long distance z−4law. Recent experimental developments enable precise manipula tion of cold atoms by lasers, see e.g. Ref. [15]. Small and well defined velocities of the atoms can be achieved using adv anced cooling and launching techniques and a detuned laser field can be used to create controlled and adjustable po tentials for the atoms. Under these conditions, atoms can be used to explore the atom–surface potential, for examp le using evanescent–wave mirrors. Classical reflection from such an atomic mirror was used to measure the van der Waal s force between a dielectric surface and an atom in its ground state [11]. This experiment though, could not f ully discriminate between the electrostatic and the QED expressions. Segev et al. [16] considered a similar experiment in the quantum regime ( atoms undergoing above– barrier, classically forbidden reflection). Unlike classi cal reflection, which can only be used to identify thresholds and to measure the height of the potential barrier, quantum prob abilities are determined by the complete potential curve, and are sensitive to the short- and long–range behavior of th e potential. It was found that, for velocities of the order of the recoil speed, the quantum reflection probabilities ar e indeed sensitive to the long–range (Casimir) interaction . In this work we study how the form of the atom–surface interac tion can be observed using atomic interference of the type reported in [17]. We consider atoms with multiple gr ound state sub-levels, which feel different potentials in the evanescent radiation field. These potentials can be pr obed by using stimulated Raman transitions within the evanescent wave. These transitions exhibit interference e ffects whose phases depend on the atomic trajectories and on the entire potential, as in quantum reflection. An important aspect of the effects we discuss here is that they occur for higher incident velocities than those considered in [16 ] and may therefore be experimentally easily accessible. Furthermore, even for small velocities (near the recoil vel ocity) the effects are more dramatic than those shown by quantum reflection. As discussed in Refs. [18] and [17], the experiment is analog ous to an atom interferometer whose size is of the order of a fraction of the evanescent wave decay length, κ−1. We will focus on three experimental cases: in the first, the atomic de Broglie wavelength λdB≪κ−1. This leads to interference fringes analogous to those in an y other interferometer. It corresponds to the experiment describe d in Ref. [17] in which rubidium (Rb) atoms are dropped from a height of about 2 cm. Second, we will repeat the situati on of the first case, but with Rb replaced by meta-stable helium (He*). In this case we have λdB/2π≃a few κ−1, which allows for only a few interference fringes. This allo ws us to illustrate how the signal changes when only the atomic s pecies is changed. Finally, we will examine the case where λdB/2π≃κ−1, taking He* atoms at the recoil velocity as our example. No fr inges are present, but a strong dependence on the nature of the atom-wall potential is demon strated. To simplify the discussion, we work with a two-level model, between the initial atomic state and the ad jacent one in the ground state manifold, thus neglecting the populations of the other ground state sub-levels as well as the excited levels. For the atom-wall interaction we will use the published values for each atom. 1In the short λdBlimit, the motion of the atom can be treated semi-classicall y. We thus calculate the transition probability between the two atomic levels according to the L andau-Zener model for adiabatic transitions. In the other cases, the study of the atomic motion requires a fully quantu m mechanical treatment. We study these cases by solving the associated coupled Schr¨ odinger equations numericall y. In Sec.II we describe the physical system and the theory for a tomic interference. In Sec.III the results are presented. Discussion and conclusions are given in Sec.IV. II. MODEL FOR INTERFERENCE OF MULTIPLE GROUND STATE SUB-LEV ELS We will focus on an experimental setup along the lines of Fig. 1. Here, a strong laser beam with frequency ω and transverse magnetic (TM) polarization (i.e. magnetic fi eld perpendicular to the plane of incidence) creates an evanescent wave, which is nearly σ−circularly polarized with respect to the y-axis. A second we ak laser beam with frequency ω−∆ωand transverse electric (TE) polarization (i.e. electric fi eld perpendicular to the plane of incidence) creates another evanescent wave with πpolarization along the y-axis. Atoms normally incident on the evanescent wave move in an effe ctive optical potential ˆVlight(which in general depends on the internal state of the atom) and an attractive a tom–wall interaction ˆVwall. The total potential is given by ˆU(z) =ˆVlight(z) +ˆVwall(z). (1) A. Optical potential Let us consider an atom in the evanescent field and assume for s implicity that the σ−polarized wave is strong compared to the πwave. The σ−component lifts the magnetic (Zeeman) degeneracy of the ato mic ground state (the quantization axis is in the ˆ ydirection), so that each magnetic sub-level feels a differen t optical potential. To first order, the πpolarized wave produces a coupling between the atomic sub-l evels via a Raman transition: for example, starting from the sub-level mi, the atom absorbs a σ−polarized photon and emits a stimulated photon with πpolarization. The atom thus ends up in the mi−1 sub-state with its total energy increased by ¯ h∆ω. This transition is resonant when ¯ h∆ωis equal to the energy difference between the magnetic sub-le vels. If the two evanescent waves are counter-propagating, in the reference frame moving wit h the optical grating, the situation corresponds to grazing incidence diffraction [17]. For a review of the theoretical u nderstanding of atomic diffraction and interference from evanescent waves, see [19]. In the limit of low saturation and a detuning δlarge compared to both the frequency difference ∆ ωand the natural linewidth, the excited state manifold may be elimin ated adiabatically, and, for a ground state of total angular momentum (including nuclear spin) Jg, the atomic wave function is described by the 2 Jg+ 1 Zeeman components |m >, m =−Jg, ...,+Jg. An atom at a distance zfrom the surface of the mirror is subject to an optical potent ial ˆVlight(z) whose matrix elements are of the form [we suppose that the fr equency difference ∆ ωand Zeeman shift are negligible compared to the detuning δand the hyperfine structure of the excited level] < m|ˆVlight(z)|m′>=d2 ¯hδ/summationdisplay q,q′,me,JeE∗ q(z)Eq′(z)(Jgm; 1q|Jeme)(Jeme|Jgm′; 1q′), (2) where a product of Clebsch–Gordon–coefficients appears on th e rhs, the electric field polarization is expanded in the usual spherical basis with coefficients Eq, (q=−1,0,+1) and dis the reduced dipole moment. The optical potential couples different Zeeman sub-levels if the field is not in a pur e polarization state with respect to this basis, as in the setup considered in this work. The optical potential due to t he two evanescent waves can then be written as < m|ˆVlight(z)|m′>=d2 ¯hδCmm′exp(−2κz), (3) where the Cmm′coefficients are given by the Clebsch-Gordon coefficients time s the field amplitudes at the surface z= 0 (see Eq. 2). The inverse decay length κisκ=k[(nsinθ)2−1)]1/2, where k=ω/cis the free field wave vector, assumed to be the same for each laser, nis the refraction index of the prism and θis the angle of incidence of the lasers with the surface. In our calculations we will always u sen= 1.87 and θ= 53◦. The wave-vector kis different for each atom. 2B. Atom–wall potential The simplest model for the interaction of a ground state atom and a wall of dielectric constant ǫconsiders the interaction between a dipole dand its mirror image and yields the Lennard–Jones potential , VLJ wall(z) =−ǫ−1 ǫ+ 1/parenleftBigg < d2 /bardbl>+2< d2 ⊥> 64πǫ0/parenrightBigg 1 z3≡ −ǫ−1 ǫ+ 1C(3) z3, (4) where < d2 /bardbl>and< d2 ⊥>are the expectation values of the squared dipole parallel an d perpendicular to the surface [1], [20]. This expression for the potential is approximate ly valid for ǫindependent of frequency and kzmuch smaller than unity. If we take into account retardation effects, the Casimir-Pol der potential is obtained, where the finite propagation time between the dipole and its image results in a different po wer-law behavior for large z[2], lim z→∞VCP wall(z)∝z−4. (5) In the complete QED theory the interaction potential betwee n an atom of polarizability α(ω) at a distance zfrom a dielectric wall can be written as [5]: VQED wall(z) =−¯h 8π2c3/integraldisplay∞ 0dωω3α(ω)×/parenleftbigg/integraldisplay1 0dp+/integraldisplayi∞ 0dp/parenrightbigg H(p, ǫ)exp(−2ipωz/c ) (6) with H(p) =/radicalbig ǫ−1 +p2−p/radicalbig ǫ−1 +p2+p+ (1−2p2)/radicalbig ǫ−1 +p2−ǫp/radicalbig ǫ−1 +p2+ǫp. (7) The numerical values of the constant C(3)for Rb and He* atoms used in this paper are given in Table I, as w ell as the numerical values of some other important parameters. Since the atom is in a ground state with l= 0, the value of C(3)is the same for any magnetic or hyperfine sub-level. The detai led interaction between a ground state Rb or meta-stable He atom and a dielectric surface were recentl y calculated. For the van der Waals potential we have used data from [21] for Rb and from [22] for He*. In the former w ork an interpolation formula for the van der Waals potential is given as VQED wall(z) =VLJ wall(z)0.987[(1 + 1 .098z)−1−0.00493 z(1 + 0 .00987 z3−0.00064 z4)−1] where zis expressed in units of the laser wavelength λ/(2π). This formula approximates the numerical calculation wit h a 0.6% accuracy between 0 and 10 λ/(2π). C. Population transfer In this work we will consider only population transfer from t he incident sub-level ito the final one f(mf=mi−1), that is only two levels. This is a good approximation if the co upling is weak enough for the population of the other levels to be negligible. An example of the total interaction potential from Eq. 1 is shown in Fig.2. The Uffpotential curve has been shifted vertically by −¯h∆ω, corresponding to the kinetic energy change. Then the two (a diabatic) potential curves cross at the point of resonance. The coupli ng turns the exact crossing into an avoided crossing. An incoming wave function in the michannel is split in two parts that are subsequently reflected from their respective repulsive potentials and recombined after the second passa ge at the crossing. Thus, the evanescent wave realizes a “Michelson interferometer” with a single beam splitter and two mirrors. When the atomic λdBis short, the avoided crossing can be treated by means of the s emi-classical Landau–Zener model for non–adiabatic transitions [23]. Assuming that th e atom moves through the crossing with a constant velocity vc(fixed by energy conservation), the Landau–Zener formula al lows one to compute the probability amplitude for the two atomic levels after the crossing. The transition probab ilitywiffrom the initial sub-level ito the final sub-level f is given by wi,f= 4TLZRLZcos2(δφ), (8) where RLZ= 1−TLZand the transmission coefficient is TLZ= exp(−πΛ), with Λ = 2 |< i|ˆU|f >|2/(¯h2κ∆ωvc). The phase difference δφis given by the difference in the phase shifts between the cros sing and the turning points in the semi-classical approximation, plus a correction ter m [24]: 3δφ=1 ¯h/bracketleftBigg/integraldisplayzc zf,rdzpf(z)−/integraldisplayzc zi,rdzpi(z)/bracketrightBigg +π 4+Λ 2log/parenleftbiggΛ 2e/parenrightbigg + arg/parenleftbigg Γ/parenleftbigg 1−iΛ 2/parenrightbigg/parenrightbigg , (9) where zcis the position of the crossing point, zn,rthe classical return point for an atom in the n-th level, pn(z) its momentum and Γ the Gamma function. Changing the frequency difference ∆ ω/(2π) causes a change in the length of one of the interferometer ar ms, thus a change in the phase difference δφbetween the two paths. As a consequence we expect the transit ion probability to show oscillations in ∆ ω(St¨ uckelberg oscillations). We will see in the next sectio n that δφis very sensitive to the exact shape of the potential. The amplitude of the oscillations al so depends on ∆ ωboth explicitly and implicitly through vc(the crossing point moves with changing ∆ ω). The Landau–Zener model is a good approximation only when the atom speed is approximately constant during the crossing. In particular it is not valid when the classical re turn point and the crossing are close to each other or when the de Broglie wavelength of the atom is of the order of the wid th of the interaction region. In order to explore this long-wavelength regime, we have to forgo the semi-classica l Landau-Zener model and solve numerically the coupled Schr¨ odinger equations for the system. Since atoms that cro ss the potential barrier stick to the dielectric surface, th e appropriate boundary conditions at the surface are those fo r a running wave propagating downward ( z→ −∞ ), while forz→ ∞ the solution is a superposition of downward (incident) and u pward (reflected) waves. We have integrated the system of Schr¨ odinger equations using the renormalize d Numerov method [25]. To avoid the singularity at the surface we have modified the potential to be a large negative c onstant near the interface and verified that the transition probability does not change by varying the value of the const ant. III. RESULTS We will present our calculations in two parts. First we discu ss the short wavelength regime, and point out various experimental strategies to observe retardation effects. Th en we discuss two cases in which the de Broglie wavelength is not small compared to the evanescent wave decay length (th e ”long wavelength regime”), in which a numerical integration of the Schr¨ odinger equation is necessary. A. Short wavelength regime Fig. 3 shows a calculation of the population transfer wifas a function of the frequency difference ∆ ωfor the Lennard–Jones (LJ) and QED model of the van der Waals interac tion. The value of the light potential is the same for the two curves (i.e. the laser intensity is the same), and the incident momentum is 115¯ hk, which corresponds to Rb atoms dropped from a height of 2.3 cm. We use the Landau-Zen er approach to calculate the transfer probability (see Eq. 8). As in [18] and [17], we observe St¨ uckelberg osci llations in the transfer probability. These oscillations can be understood as the variations in the accumulated phase difference between the two different paths taken by the atoms after the level crossing shown in Fig.2. The de Broglie wavelength λdBis such that several fringes appear as the position of the level crossing is moved through its possi ble range. The last oscillation at the higher frequencies is where the crossing point and the classical return point ar e very close to each other and the Landau-Zener model breaks down. We have set the probability to zero beyond this l imit. In reality, the transition probability falls roughly exponentially with frequency, as one finds solving the Schr¨ odinger equations numerically (see next subsection). In general, we find that the dephasing between the two curves ( with and without retardation effects) is greatest when the atoms are incident at an energy close to the top of the potential barrier. Note that the height of the barrier is greater when retardation effects are included, since thes e reduce the strength of the atom-surface interaction. The effect of retardation is roughly to shift these fringes by hal f a fringe to the left. The major cause of this shift is the increase in the height of the total potential which is gre ater for the ilevel than the flevel. A 10% increase in the value of the light-shift potential would exhibit nearly the same shift. Therefore, since it is not possible to turn retardation on and off, it would be necessary to measure absol utely the light-shift potential to better than 10% in order to distinguish a retardation effect. Experimentally t his is rather difficult. Instead of attempting to measure the absolute light shift, one could rather measure the absol ute height of the potential by observing the threshold of reflection as in [11], and using the known kinetic energy of th e atoms to get an absolute calibration of the height. Let us assume then that the barrier height, instead of the light i ntensity, is known. In Fig.4a we show the result with the same parameters as in Fig.3 except for the light intensity in the LJ model, which has been changed so as to have the same barrier height as the QED model. In this case the shift is much smaller, about 1/5 of a fringe. We have verified 4that even taking into account an experimental uncertainty o f a few % in the height of the corresponding potentials, the two models are still clearly resolved. This approach seems feasible, but a third method of observin g the effects of retardation is possible if one uses more of the information available in the oscillation curve. Fig. 4b shows the same curves as Fig. 4a but with the QED curve numerically shifted so as to coincide with the LJ curve at its maximum. One sees that the period of these oscillations is not the same. It decreases with decreasing d etuning, faster for the full QED potential, so that there is a difference in the spacing of the minima in the population tra nsfer. Thus with fringes with sufficiently high signal to noise, one can distinguish retardation while leaving the absolute barrier height as a free parameter in a fit to the data. It seems to us that a viable experimental method is to us e a combination of the second two approaches. Careful measurements of the barrier height can be used to cross check a fit to the St¨ uckelberg oscillations with the barrier height as a free parameter. The incident energy, momentum and barrier height used in Fig s. 3 and 4a were arbitrarily chosen to correspond to the experiment in [17], but it would be interesting to know ho w Figs. 3 and 4a would change, especially for different incident momenta (de Broglie wavelengths). We studied this question by repeating the calculation of Fig. 4a, for different incident momenta, while always keeping the barrie r height 10% above the incident kinetic energy. We find that, roughly speaking, the number of oscillations increas es as the incident momentum. This is because the number of fringes in the interferogram increases with decreasing w avelength. The accumulated phase difference between the oscillations in the LJ and QED models, over the correspondin g frequency range, also increases approximately linearly with incident momentum. Thus if we consider the fringe shift divided by the fringe period as a figure of merit, the sensitivity of the experiment to retardation effects increa ses with increasing incident momentum. B. Long wavelength regime We now examine the large de Broglie wavelength limit, where t he semi-classical model breaks down since the atomic wavelength is not small compared to the interferometer size . We first consider the case of He* atoms dropped from a height of a few cm as for Rb. We show the results in Fig. 5. For a n initial distance of 2.3 cm from the mirror, the incident momentum of He* is 7 .4¯hk, much lower than for Rb. Again we will chose the intensities o f the strong σ−wave so as to have the same barrier height for the two potentia ls, about 10% above the incident kinetic energy. In this regime, to observe interference, the detuning betwe en the evanescent waves must be of only a few MHz, in fact, beyond about 10MHz the crossing point is closer to the s urface than the return point. One only sees one or two St¨ uckelberg oscillations since the momentum involved is small and the atoms do not accumulate enough phase difference to show more oscillations. The two potentials giv e similar results (Fig.5), the main difference being in the shape and height of the big peak. We have also looked at even lower incident momenta. Near the r ecoil limit, one can expect interference only for detunings of less than one MHz and the accumulated phase diffe rence is too small in this range to show any St¨ uckelberg oscillations. Both for He* and Rb one obtains a big peak in the transition probability and no St¨ uckelberg oscillations, as expected. The difference between the van der Waals potenti al and the full QED potential still shows up in the different shape and height of the peaks (Fig.6). Here the widt h of the curve is delimited by the frequency for which the classical return point and the crossing point coincide. Finally we note that the qualitative features of the short an d long λdBlimit are the same for Rb and He*, e.g. He* with an incident momentum of about 100¯ hkgives the same type of oscillation pattern as Rb. IV. CONCLUSIONS In summary, we have proposed an experiment to probe van der Wa als like surface interactions by exploiting interference mechanisms for well defined Zeeman sub-levels of atoms moving in two evanescent waves. Retardation can be resolved using atoms incident at speeds which are easi ly obtained in free fall over a few centimeters. The controlling parameter is here the detuning between the two e vanescent waves. One then measures the fraction of the atoms which have undergone a change of magnetic sub-level as a function of this detuning. For the situation in which the atomic de Broglie wavelength i s sufficiently small the experiment resembles typical interferometry experiments. The theoretical description is semi-classical, employing well defined atomic trajector ies, while experimentally, one seeks a particular (non sinusoid al) fringe pattern as a signature of retardation. This shoul d be possible with an improved version of the experiment of Ref . [17]. Another approach is to investigate the interaction of atoms whose de Broglie wavelength is not small compared to the length scale of the interferometer. In this regime mos t of the information is to be found in the shape of the 5population transfer curve, since there are very few or zero i nterference fringes. Note though, that we have assumed throughout that the incident atoms are mono-energetic. Thi s means that the velocity spread of the incident atoms must be small compared to the atomic recoil. Nevertheless th is may be worth the effort, because the predicted effect of retardation is quite dramatic. For a quantitative comparison however, the presence of all t he sub-levels (which give rise to the multiple crossing of the dressed potentials at the same distance zcfrom the surface) and, possibly, losses from spontaneous em ission have to be taken into account, but the results are not qualita tively different. V. ACKNOWLEDGMENTS R. M. acknowledges support from the Training and Mobility of Researchers (European Union, Marie–Curie Fellow- ship contract n. ERBFMBIC983271) and would like to thank Pau l Julienne and Olivier Dulieu for their help with the numerical code, and J. Babb for providing the data for the atom-wall interaction for meta-stable helium. This work was also supported by the R´ egion Ile de France. [1] L. E. Lennard-Jones, Trans. Faraday Soc. 28, 333 (1932). [2] H. B. G. Casimir and D. Polder, Phys. Rev. 73, 360 (1948). [3] I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevski i, Adv. Phys. 10, 165 (1961). [4] L. Spruch and Y. Tikochinsky, Phys. Rev. A 48, 4213 (1993). [5] Y. Tikochinsky and L. Spruch, Phys. Rev. A 48, 4223 (1993). [6] L. Spruch, Science 272, 1452 (1996). [7] M. Fichet, F. Schuller, D. Bloch, and M. Ducloy, Phys. Rev . A51, 1553 (1995), and references therein. [8] J. M. Wylie and J. E. Sipe, Phys. Rev. A 30, 1185 (1984). [9] J. M. Wylie and J. E. Sipe, Phys. Rev. A 32, 2030 (1985). [10] C. Mavroyannis, Mol. Phys. 6, 593 (1963). [11] A. Landragin et al., Phys. Rev. Lett. 77, 1464 (1996). [12] M. Kasevich et al., inAtomic Physics 12 , Vol. 233 of AIP Conf. Proc. No. 233 , edited by J. C. Zorn and R. R. Lewis (AIP, New York, 1991), p. 47. [13] C. I. Sandoghdar, V. Sukenik, E. A. Hinds, and S. Haroche , Phys. Rev. Lett. 68, 3432 (1992). [14] C. I. Sukenik et al., Phys. Rev. Lett. 70, 560 (1993). [15] S. Chu, C. N. Cohen-Tannoudji, and W. D. Phillips, Rev. M od. Phys. 70, 685 (1998). [16] B. Segev, R. Cˆ ot´ e, and M. G. Raizen, Phys. Rev. A 56, R3350 (1997). [17] L. Cognet et al., Phys. Rev. Lett. 81, 5044 (1998). [18] C. Henkel, K. Molmer, R. Kaiser, and C. I. Westbrook, Phy s. Rev. A 56, R9 (1997). [19] C. Henkel et al., Appl. Phys. B 69, 277 (1999). [20] R. Cˆ ot´ e, B. Segev, and M. G. Raizen, Phys. Rev. A 58, 3999 (1998). [21] A. Landragin, Ph.D. thesis, Universit´ e de Paris–Sud, 1997. [22] Z.-C. Yan and J. F. Babb, Phys. Rev. A 58, 1247 (1998). [23] C. Zener, Proc. Roy. Soc., Ser. A 137, 696 (1932). [24] A. P. Kazantsev, G. A. Ryabenko, G. Surdutovich, and V. Y akovlev, Phys. Rep. 129, 75 (1985). [25] B. R. Johnson, J. Chem. Phys. 67, 4086 (1977). 6Parameter Rb He* λ 780×10−9m 1083×10−9m Γ 2π5.9×106s−12π1.6×106s−1 Er 6.4×10−4¯hΓ 2.6×10−2¯hΓ C(3)0.113¯hΓ/λ30.125¯hΓ/λ3 state i 5S1/2, F= 2, mF= 2 23S1, J= 1, mJ= 1 state f 5S1/2, F= 2, mF= 1 23S1, J= 1, mJ= 0 Cii |E−1(0)|21/3 |E−1(0)|21/6 Cff |E−1(0)|21/2 |E−1(0)|21/2 Cif E−1(0)∗E0(0)√ 2/3 E−1(0)∗E0(0)1/3 TABLE I. Values of parameters used in the text for Rb and He*: l aser wavelength λ, atomic transition width Γ, recoil energy Er, van der Waals coefficient C(3)(see Eq.4). Definition of the initial and final atomic state ( iandf). Optical potential coefficients Cij(see Eq.3). We assume here that the σ−wave ( E−1) is much stronger than the πwave ( E0). The dielectric constant of the wall is ǫ= 3.49 and κ= 1.52k. 7Atoms before the bounce σ-, π ωL ωL−ΔωAtoms after the bounceProbe beam z xy TM TE FIG. 1. Typical experimental setup. The atoms are released f rom a MOT above the prism. Two slightly detuned laser beams of polarizations TM (and frequency ω/(2π)) and TE (and frequency ( ω−∆ω)/(2π)) form evanescent waves on the surface of the prism of polarizations σ−andπrespectively ( yis the quantization axis). The reflected atoms are detected b y a detection beam. 0 0.5 1 1.5 2 2.5−400−2000200400600800 z/λEnergy/Er Uii Uff−Δω h/2πincident energy state i state f FIG. 2. Typical potential curves experienced by the atoms du ring reflection. The potential for state fis shifted by the Raman energy −¯h∆ω. The atoms approach the potential barrier in state i, pass through the curve crossing twice and can end up on either state iorf. For each final situation, two paths are possible (between th e crossing and the turning points) and can interfere producing fringes as a function of the locatio n of the crossing. 800.10.20.30.40.50.60.70.80.91 0 510 15 20 25 30Population transfer Frequency difference (MHz)LJ QED FIG. 3. Population transfer from state ito state fafter reflection ( wiffrom Eq. 8) vs frequency difference ∆ ω/(2π) for rubidium atoms, released 2.3 cm above the mirror, (i.e. with incident momentum 115¯ hk). The solid line is for the QED model and the dashed line for the Lennard–Jones (LJ) model. The cou pling coefficients defined in Eq. 3 are |Cif|= 859 ERb rand |Cii|= 3.8×104ERb r . 900.10.20.30.40.50.60.70.80.91 0 510 15 20 25 30Population transferLJ QED 00.10.20.30.40.50.60.70.80.91Population transfer Frequency difference (MHz)LJ QED FIG. 4. Upper figure: population transfer wif(see Eq. 8) vs frequency difference ∆ ω/(2π) for rubidium atoms released 2.3 cm above the mirror (i.e. with incident momentum 115¯ hkRb). The solid line is for the QED model and the dashed line for th e Leenard–Jones (LJ) model. The height of the potential barri er is 1 .48×104ERb r, i.e. the intensity of the strong laser beam has been adjusted to give the same barrier height for both models . The light–shift coefficients for the QED model are the same as for Fig. 3, while for the LJ model |Cii|= 4.11×104ERb rand|Cif|= 890 ERb r. Lower figure: same parameters as in the previous figure with the LJ curve artificially shifted 0.79 MHz to the le ft in order to show the changes in fringe spacing. 1000.020.040.060.080.10.120.140.160.180.2 02468101214Population transfer Frequency difference (MHz)LJ QED FIG. 5. Population transfer wif(see Eq. 8) vs frequency difference ∆ ω/(2π) for meta-stable helium atoms released 2.3 cm above the mirror (i.e. with incident momentum 7 .4¯hkHe). The solid line is for the QED model and the dashed line for th e Lennard–Jones (LJ) model. The height of the potential barri er is 61 EHe r, i.e. the intensity of the strong laser beam has been adjusted to give the same barrier height for both models ( 10% above the incident kinetic energy). The light–shift coeffici ents for the QED model are |Cii|= 252 EHe r|Cif|= 20.8EHe rwhile for the LJ model |Cii|= 303 EHe rand|Cif|= 22.7EHe r. 00.010.020.030.040.050.060.07 00.20.40.60.811.21.41.6Population transfer Frequency difference (MHz)LJ QED FIG. 6. Population transfer wifvs frequency difference ∆ ω/(2π) for meta-stable helium atom with incident momentum ¯hkHe. The solid line is for the QED model and the dashed line for the Lennard–Jones (LJ) model. The height of the potential barrier is 1 .28EHe r, i.e. the intensity of the strong laser beam has been adjuste d to give the same barrier height for both models. The light–shift coefficients for the QED model are |Cii|= 0.8EHe r|Cif|= 0.16EHe rwhile for the LJ model |Cii|= 1.47EHe rand |Cif|= 0.29EHe r. 11

arXiv:physics/0001016v1 [physics.plasm-ph] 7 Jan 2000New tests for a singularity of ideal MHD Robert M. Kerr1and Axel Brandenburg2 1NCAR, Boulder, CO 80307-3000;2Mathematics, University of Newcastle, NE1 7RU, UK Analysis using new calculations with 3 times the resolu- tion of the earlier linked magnetic flux tubes confirms the transition from singular to saturated growth rate reported by Grauer and Marliani [2] for the incompressible cases is con- firmed. However, all of the secondary tests point to a transi- tion back to stronger growth rate at a different location at la te times. Similar problems in ideal hydrodynamics are discuss ed, pointing out that initial negative results eventually led t o bet- ter initial conditions that did show evidence for a singular ity of Euler. Whether singular or near-singular growth in ideal MHD is eventually shown, this study could have bearing on fast magnetic reconnection, high energy particle producti on and coronal heating. The issue currently leading to conflicting conclusions about ideal 3D, incompressible MHD is similar [1,2] to what led to conflicting results on whether there is a sin- gularity of the 3D incompressible Euler. With numeri- cal simulations, it was first concluded that uniform mesh calculations with symmetric initial conditions such as 3D Taylor-Green were not yet singular [3]. Next, a prelimi- nary spectral calculation [4] found weak evidence in favor a singularity in a series of Navier-Stokes simulations at increasing Reynolds numbers, but larger adaptive mesh or refined mesh calculations did not support this result [5,6]. Eventually, numerical evidence in favor of a sin- gularity of Euler was obtained using several independent tests applied to highly resolved, refined mesh calculations of the evolution of two anti-parallel vortex tubes [7]. To date, these calculations have met every analytic test for whether there could be a singularity of Euler. Several other calculations have also claimed numerical evidence for a singularity of Euler [8–10]. While in all of these cases the evidence is plausible, with the perturbed cylindrical shear flow [10] using the BKM /bardblω/bardbl∞test [11], for none has the entire battery of tests used for the anti- parallel case been applied. We have recently repeated one of the orthogonal cases [8] and have applied the BKM test successfully. In all cases using the BKM test, |ω/bardbl∞≈ A/(Tc−t) with A≈19. To be able to make a convincing case for the existence of a singularity in higher dimensional partial differential equations, great care must be taken with initial condi- tions, demonstrating numerical convergence, and com- parisons to all known analytic or empirical tests. On the other hand, if no singularity is suspected, some quan- tity that clearly saturates should be demonstrated, such as the strain causing vorticity growth [5]. It is an even more delicate matter to claim that someone else’s calcu-lations or conclusions are incorrect. If it is a matter of suspecting there is inadequate resolution, one must at- tempt to reproduce the suspicious calculations as nearly as possible and show where inadequate resolution begins to corrupt the calculations and how improved resolution changes the results. An example of how a detailed search for numerical errors should be conducted can be found in the exten- sive conference proceeding [12] that appeared prior to the publication of the major results supporting the ex- istence of a singularity of Euler for anti-parallel vortex tubes [7]. The primary difference with earlier work was in the initial conditions. It was found that compact pro- files [13] were an improvement, but only if used in con- junction with a high wavenumber filter. Otherwise, the initial unfiltered energy spectrum of the bent anti-paralle l vortex tubes went as k−2. Oscillations in the spectrum at high wavenumber in unfiltered initial conditions for linked magnetic flux tubes are shown in Figure 1, show- ing that the initial MHD spectrum is steep enough that eventually these oscillations are not important. FIG. 1. Filtered and unfiltered initial and final spectrum . The unfiltered spectrum is initialized on a 3843mesh. The purpose of this letter is to address the claim that a new adaptive mesh refinement (AMR) calculation by Grauer and Marliani [2] supercedes our uniform mesh cal- culations [1] and that eventually there is a transition to exponential growth. Note that this claim was made with- out any evidence for whether their numerical method was converged. In all of our earlier calculations, once the cal- culations become underresolved, we also saw transitions to exponential growth. Not knowing exactly the initial condition used by the new AMR calculations [2], where and how much grid re- 1finement was used, and the short notice we have been given to reply has proven a challenge. Fortunately, we were in the process of new 6483calculations in a smaller domain of 4 .33, yielding effectively 3 times the local res- olution of our earlier work [1] in a (2 π)3domain on a 3843mesh. The case with an initial flux tube diameter ofd= 0.65, so that the tubes slightly overlap, appears to be closer to their initial condition and so will be the focus of this letter. The importance of our other initial condi- tion, with d= 0.5, and no initial overlap of the tubes, is that it is less influenced by an initial current sheet that forms near the origin and is claimed to be the source of the saturation of the nonlinear terms. This was used for the compressible calculations. FIG. 2. Replot of /bardblJ/bardbl∞,/bardblω/bardbl∞, and PΩJfor the new in- compressible calculations on 4 .33domain with initial condi- tiond= 0.65 in semi-log coordinates. All plots are from the 6483calculation except one 3843plot of 1 //bardblJ/bardbl. Exponential and inverse linear fits are shown for t= 1.75 to 1.98. Each works equally well for /bardblJ/bardbl∞, inverse linear is better for /bardblω/bardbl∞, and exponential is better for PΩJ. Multiplying /bardblJ/bardbl∞,/bardblω/bardbl∞, andPΩJby (Tc−t) in the inset emphasizes that /bardblJ/bardbl∞and /bardblω/bardbl∞might be showing consistent singular behavior. The large growth of PΩJis discussed. Using semi-log coordinates, Figure 2 plots the growth of/bardblω/bardbl∞and/bardblJ/bardbl∞for our new high resolution incom- pressible calculation and Figure 3 plots /bardblJ/bardbl∞for a new compressible calculation. By taking the last time all rel- evant quantities on the 3843and 6483grids were con- verged, /bardblJ/bardblbeing the worst, then by assuming that the smallest scales are decreasing linearly towards a possible singular time, an estimate of the last time the 6483cal- culation was valid was made. To test exponential versus inverse linear growth, fits were taken between T= 1.72 and 1.87, then extrapolated to large T. The large figureshows that either an exponential or a singular 1 /(Tc−t) form could fit the data, while the inset shows that taking an estimated singular time of Tc= 2.15 and multiplying by (Tc−T) that at least /bardblJ/bardbl∞and/bardblω/bardbl∞have consistent singular behavior over this time span. The strong growth ofPΩJ=/integraltext dV(ωieijωj−ωidijJj−2εijkJidjℓeℓk), which is the production of/integraltext dV(ω2+J2), is discussed below. The 3843curve for 1 //bardblJ/bardbldemonstrates that lack of res- olution tends to exaggerate exponential growth. For the compressible calculations it can be seen that there also is an exponential regime that changes into a regime with 1//bardblJ/bardbl∞∼(Tc−t). FIG. 3. Semi-logarithmic plot of /bardblJ/bardbl∞for a compressible 2403calculation in a domain of size 4 (dotted line: filtered, and solid line: unfiltered initial conditions) together wit h fits to exponential growth and blow-up behavior, respectively. The latter are better fits at later times. Using the new incompressible calculations and apply- ing the entire battery of tests, based upon Figure 2 we would agree that for the incompressible case there is a transition as reported [2] and signs of saturation at this stage are shown below. Whether the transition is to ex- ponential for all times as claimed [2], or whether there is a still later transition to different singular behavior, wil l be the focus of this letter. We will look more closely at the structure of the current sheet we all agree exists [1,2] for signs of saturation. The case against a singularity in early calculations of Euler [5,14,15] was the appearance of vortex sheets, and through analogies with the current in 2D ideal MHD, a suggestion that this leads to a depletion of nonlinearity. The fluid flow most relevant to the linked flux rings is 3D Taylor-Green, due to the initial symmetries [3]. For both TG and linked flux tubes, two sets of anti-parallel vortex pairs form that are skewed with respect to each other and are colliding. In TG, just after the anti-parallel vortex tubes form there is a period of slightly singular development. This is suppressed once the pairs collide with each other, and then vortex sheets dominate for a period. The vortex sheets are very thin, but go across the domain, so fine localized resolution might not be an 2advantage at this stage. At late phases in TG, the ends of the colliding pairs begin to interact with each other, so that at 4 corners locally orthogonal vortices begin to form. Due to resolution limitations, an Euler calculation of Taylor-Green has not been continued long enough to determine whether, during this phase, singular behavior might develop. We would draw a similar conclusion for all of MHD cases studied to date [2,16,17], that there might not be enough local resolution to draw any final conclusions even if AMR is applied. While Taylor-Green has not been continued far enough to rule out singularities, the final arrangement of vortex structures led first to studies of interacting orthogonal vortices [8], and then anti-parallel vortices (see referen ces in [7]). Both of these initial conditions now appear to de- velop singular behavior. An important piece of evidence for a singularity of Euler was that near the point of a possible singularity, the structure could not be described simply as a vortex sheet. Therefore, there is a precedent to earlier work suggesting sheets, suppression of nonlin- earity, and no singularities to later work showing fully three-dimensional structure and singular behavior. The initial singular growth of /bardblJ/bardbl∞and/bardblω/bardbl∞for the linked flux rings, then the transition to a saturated growth rate, might be due to the same skewed, anti- parallel vortex pair interaction as in Taylor-Green. Even if this is all that is happening, the strong initial vortic- ity production and shorter dynamical timescale (order of a few Alfv´ en times) than earlier magnetic reconnection simulations with anti-parallel flux tubes [17] is a signif- icant success of these simulations. It might be that the vortices that have been generated are strong enough to develop their own Euler singularity. However, the in- teresting physics is how the magnetic field and current interact with the vorticity. Do they suppress the ten- dency of the vorticity to become singular, or augment that tendency? FIG. 4. Positions of /bardblJ/bardbl∞and/bardblω/bardbl∞ford= 0.65 in a 4 .33 domain. One sign for saturation of the linked flux ring inter-action would be if the strongest current remains at the origin in this sheet. Figure 4 plots the positions of /bardblJ/bardbl∞ and/bardblω/bardbl∞from the origin as a function of time. Dur- ing the period where exponential growth is claimed [2], /bardblJ/bardbl∞is at the origin, which would support the claims of saturation. However, this situation does not persist. By analogy to the movement of the L∞norms of the components of the stress tensor ui,jin Euler, we expect that the positions of /bardblJ/bardbl∞and/bardblω/bardbl∞should approach each other and an extrapolated singular point in ideal MHD. Figure 4 supports the prediction that the positions of/bardblJ/bardbl∞and/bardblω/bardbl∞should approach each other but so far not in a convincingly linear fashion. This is addressed next. We have similar trends for the positions of /bardblJ/bardbl∞ and/bardblω/bardbl∞in the compressible calculations. FIG. 5. For t= 1.97 on the inner 1623grid points, the cur- rent sheet is shown with arrows of− →Joverlaid in dark. The current through the ( x/y=z) plane containing /bardblJ/bardbl∞is in lower right. Contours of |J|4are shown to emphasize where /bardblJ/bardbl∞is located. Dark lines are− →Band light lines are− →ω that originated in the vicinity of /bardblJ/bardbl∞. The vortex lines are predominantly those in the double vortex rings that were ori g- inally generated by the Lorenz force, then became responsib le for spreading out the current sheet. Where the− →Blines cross in the upper left and lower right corners are around the lo- cations of /bardblJ/bardbl∞, which due to symmetries are different views of the same structure. Near /bardblJ/bardbl∞,− →Bnearly overlies and is parallel to− →ωand both− →Band− →ωare nearly orthogonal to their partners across the current sheet, where− →Band− →ωare anti-parallel. Taken from the d= 0.65 calculation in a 4 .33 domain on a 6483mesh. Figure 5 gives an overall view of the current, vorticity and magnetic field around the inner current sheet. The vortex pattern has developed out of the four initial vor- tices, two sets of orthogonal, anti-parallel pairs that are responsible for the initial compression and stretching of the current sheet. By this late time, the ends of the those vortices have begun to interact as new sets of orthogonal vortex pairs. The lower right inset in Figure 5 is a 2D x/(y=z) slice through this domain that goes through 3/bardblJ/bardbl∞att= 1.97 to show that while /bardblJ/bardbl∞is large at the origin (0 ,0,0),/bardblJ/bardbl∞is larger where it is being squeezed between the new orthogonal vortices. Along one of the new vortices− →Bis parallel to and overlying− →ωand on the orthogonal partner they are anti-parallel and overlying. The location of /bardblω/bardbl∞is not in the vortex lines shown, but is on the outer edges of the current sheet. Therefore, the exact position of /bardblω/bardbl∞in Figure 4 is an artifact of the initial development and does not accurately reflect the position of− →ωmost directly involved in amplifying /bardblJ/bardbl∞, which is probably why the positions of /bardblJ/bardbl∞and /bardblω/bardbl∞are not approaching each other faster. The con- tinuing effects of the initial current sheet is probably also behind the strong exponential growth of PΩJin Figure 2, stronger even than the the possible singular growth of /bardblJ/bardbl∞and/bardblω/bardbl∞in the inset. More detailed analysis in progress should show that near the position of /bardblJ/bardbl∞, the growth of PΩJand the position of /bardblω/bardbl∞are more con- sistent with our expectations for singular growth and has already shown that some of the components of PΩJhave consistent singular growth. As noted, for Euler all available calculations find |ω/bardbl∞≈A/(Tc−t) with A≈19.Arepresents how much smaller the strain along /bardblω/bardbl∞is than /bardblω/bardbl∞. Here, A≈4, indicating stronger growth in /bardblω/bardbl∞for ideal MHD than Euler. Another Euler result was that the asymptotic energy spectrum as the possible singularity was approached was k−3, whereas purely sheet-like struc- tures in vorticity should yield k−4spectrum. k−3indi- cates a more complicated 3D structure than sheets. In Figure 1 the late time spectra are again k−3. The next the initial condition we will investigate will be magnetic flux and vortex tubes that nearly overlay each other and are orthogonal to their partners. Our new cal- culations of orthogonal vortex tubes for Euler show that they start becoming singular as filaments are pulled off of the original tubes and these filaments become anti- parallel, suggesting that the fundamental singular inter- action in Euler is between anti-parallel vortices. Whether the next step for ideal MHD is to become anti-parallel or something else can only be determined by new cal- culations. AMR might be useful, but great care must be taken with the placement of the inner domains and a large mesh will still be necessary. The complicated struc- tures in the domain in Figure 5 are not fully contained in this innermost 1623mesh points and the innermost domain should go out to the order of 3003points. There are examples of how to use AMR when there are strong shears on the boundaries of sharp structures [18]. This uncertainty of where to place the mesh is why we believe in using uniform mesh calculations as an unbiased first look at the problem. These final results are hardly robust and their useful- ness is primarily to suggest a new more localized initial condition and to show that none of the calculations to date is the full story. For Jandωto show singular be- havior as long as they have has been surprising. Recall that for Euler, velocity, vorticity and strain are all man-ifestations of the same vector field, but for ideal MHD there are two independent vector fields even though the only analytic result in 3D is a condition on the combi- nation,/integraltextdV[/bardblω/bardbl∞(t) +/bardblJ/bardbl∞(t)]dt→ ∞ [19]. Even- tually, one piece of evidence for singular growth must be a demonstration of strong coupling between the cur- rent and vorticity so that they are acting as one vector field. It could be that our strong growth is due to the strongly helical initial conditions and there are no singu- larities. This would still be physically interesting since helical conditions could be set up by footpoint motion in the corona. Could the magnetic and electric fields blow up too? There are signs this might be developing around the fi- nal position of /bardblJ/bardbl∞, in which case there might exist a mechanism for the direct acceleration of high energy par- ticles. This has been considered on larger scale [20], but to our knowledge a mechanism for small-scale production of super-Dreicer electric fields has not been proposed be- fore. A singular rise in electric fields could explain the sharp rise times in X-ray production in solar coronal mea- surements [21], which could be a consequence of particle acceleration coming from reconnection. This would also have implications for the heating of the solar corona by nanoflares [22] and the production of cosmic rays. This work has been supported in part by an EPSRC visiting grant GR/M46136. NCAR is support by the National Science Foundation. [1] R.M. Kerr and A. Brandenburg, Phys. Rev Lett. 83, 1155 (1999). [2] R. Grauer and C. Marliani, submitted to PRL (1999). [3] M.E. Brachet, D.I. Meiron, S.A. Orszag, B.G. Nickel, R.H. Morf, and U. Frisch, J. Fluid Mech. 130, 411 (1983). [4] R. M. Kerr and F. Hussain, Physica D 37, 474 (1989). [5] A. Pumir and E. D. Siggia, Phys. Fluids A 2220 (1990). [6] M. J. Shelley, D. I. Meiron, and S. A. Orszag, J. Fluid Mech.246613 (1993). [7] R. M. Kerr, Phys. Fluids A 5, 1725 (1993). [8] O.N Boratav, R.B. Pelz, N.J. Zabusky, Phys. Fluids A 4, 581 (1992). [9] O.N. Boratav and R.B. Pelz, Phys. Fluids 6, 2757 (1994). [10] R. Grauer, C. Marliani, K. Germaschewski, Phys. Rev Lett80, 4177 (1998). [11] J. T. Beale, T. Kato, and A. Majda, Comm. Math. Phys. 94, 61 (1984). [12] R. M. Kerr, In Topological aspects of the dynamics of flu- ids and plasmas (Proceedings of the NATO-ARW Work- shop at the Institute for Theoretical Physics, University of California at Santa Barbara), H. K Moffatt, G. M. Za- slavsky, M. Tabor, and P. Comte, Eds. Kluwer Academic Publishers, Dordrecht, The Netherlands 309 (1992). [13] M.V. Melander, F. Hussain, Phys. Fluids A 1633 (1989). [14] A. Pumir and R. M. Kerr, Phys. Rev. Let., 58, 1636 4(1987). [15] M.E. Brachet, M. Meneguzzi, A. Vincent, H. Politano, and P.L. Sulem, Phys. Fluids A 4, 2845 (1992). Sulem, Brachet, etc. [16] H. Politano, A. Pouquet, and P.L. Sulem, Phys. Plasmas 2, 2931 (1995). [17] Y. Ono, M. Yamada, T. Akao, T. Tajima, and R. Mat- sumoto, Phys. Rev. Lett. 76, 3328 (1996). [18] W. W. Grabowski and T.L. Clark. J. Atmos. Sci. 50, 555 (1993). [19] R. E. Caflisch, I. Klapper, and G. Steele, Comm. Math. Phys.184, 443 (1997). [20] J.A. Miller, P.J. Cargill, A.G. Emslie, G.D. Holman, B. R. Dennis, T.N. LaRosa, R.M. Winglee, S.G. Benka and S. Tsuneta, J. Geo. Res. 102, 14631 (1997) [21] A.L. Kiplinger, B.R. Dennis, K.J. Frost, L.E. Orwig, As - trophys. J. Lett. 287, L105 (1984). [22] E.N. Parker, Astrophys. J. 244, 644 (1981). 5

arXiv:physics/0001017v1 [physics.atom-ph] 7 Jan 2000Inelastic semiclassical Coulomb scattering Gerd van de Sand †and Jan M Rost ‡ †– Theoretical Quantum Dynamics – Fakult¨ at f¨ ur Physik, Universit¨ at Freiburg, Hermann–He rder–Str. 3, D–79104 Freiburg, Germany ‡Max-Planck-Institute for Physics of Complex Systems, N¨ ot hnitzer Str. 38, D-01187 Dresden, Germany Abstract. We present a semiclassical S-matrix study of inelastic coll inear electron-hydrogen scattering. A simple way to extract all n ecessary information from the deflection function alone without having to compute the stability matrix is described. This includes the determination of the releva nt Maslov indices. Results of singlet and triplet cross sections for excitatio n and ionization are reported. The different levels of approximation – classical , semiclassical, and uniform semiclassical – are compared among each other and to the full quantum result. PACS numbers: 34.80D, 03.65.Sq, 34.10+x 1. Introduction Semiclassical scattering theory was formulated almost 40 y ears ago for potential scattering in terms of WKB-phaseshifts [1]. Ten years later , a multidimensional formulation appeared, derived from the Feynman path integr al [2]. Based on a similar derivation Miller developed at about the same time his ’clas sical S-matrix’ which extended Pechukas’ multidimensional semiclassical S-mat rix for potential scattering to inelastic scattering [3, 4, 5]. These semiclassical conc epts have been mostly applied to molecular problems, and in a parallel development by Bali an and Bloch [6] to condensed matter problems, i.e. to short range interaction s. Only recently, scattering involving long range (Coulomb) f orces has been studied using semiclassical S-matrix techniques, in parti cular potential scattering [7], ionization of atoms near the threshold [8, 9] and chaotic sca ttering below the ionization threshold [10]. The latter problem has also been studied pur ely classically [11] and semiclassically within a periodic orbit approach [12]. While there is a substantial body of work on classical collis ions with Coulomb forces using the Classical Trajectory Monte Carlo Method (C TMC) almost no semiclassical studies exist. This fact together with the re markable success of CTMC methods have motivated our semiclassical investigati on of inelastic Coulomb scattering. To carry out an explorative study in the full (12 ) dimensional phase space of three interacting particles is prohibitively expensive . Instead, we restrict ourselves tocollinear scattering, i.e. all three particles are located on a line wi th the nucleus in between the two electrons. This collision configuration w as proven to contain the essential physics for ionization near the threshold [8, 13, 14] and it fits well into the context of classical mechanics since the collinear phase sp ace is the consequence of aInelastic semiclassical Coulomb scattering 2 stable partial fixed point at the interelectronic angle θ12= 180◦[14]. Moreover, it is exactly the setting of Miller’s approach for molecular reac tive scattering. For the theoretical development of scattering concepts ano ther Hamiltonian of only two degrees of freedom has been established in the liter ature, the s-wave model [15]. Formally, this model Hamiltonian is obtained by avera ging the angular degrees of freedom and retaining only the zeroth order of the respect ive multipole expansions. The resulting electron-electron interaction is limited to the line r1=r2, where the ri are the electron-nucleus distances, and the potential is no t differentiable along the line r1=r2. This is not very important for the quantum mechanical treat ment, however, it affects the classical mechanics drastically. Indeed, it h as been found that the s-wave Hamiltonian leads to a threshold law for ionization very diff erent from the one resulting from the collinear and the full Hamiltonian (which both lead to the same threshold law) [16]. Since it is desirable for a comparison of semiclas sical with quantum results that the underlying classical mechanics does not lead to qua litative different physics we have chosen to work with the collinear Hamiltonian. For th is collisional system we will obtain and compare the classical, the quantum and the pr imitive and uniformized semiclassical result. For the semiclassical calculations the collinear Hamiltonian was amended by the so called Langer correction, introduced by La nger [17] to overcome inconsistencies with the semiclassical quantization in sp herical (or more generally non- cartesian) coordinates. As a side product of this study we give a rule how to obtain the c orrect Maslov indices for a two-dimensional collision system directly fr om the deflection function without the stability matrix. This does not only make the sem iclassical calculation much more transparent it also considerably reduces the nume rical effort since one can avoid to compute the stability matrix and nevertheless o ne obtains the full semiclassical result. The plan of the paper is as follows: in section 2 we introduce t he Hamiltonian and the basic semiclassical formulation of the S-matrix in term s of classical trajectories. We will discuss a typical S-matrix S(E) at fixed total energy Eand illustrate a simple way to determine the relevant (relative) Maslov phases. In s ection 3 semiclassical excitation and ionization probabilities are compared to qu antum results for singlet and triplet symmetry. The spin averaged probabilities are also compared to the classical results. In section 4 we will go one step further and uniformi ze the semiclassical S- matrix, the corresponding scattering probabilities will b e presented. We conclude the paper with section 5 where we try to assess how useful semicla ssical scattering theory is for Coulomb potentials. 2. Collinear electron-atom scattering 2.1. The Hamiltonian and the scattering probability The collinear two-electron Hamiltonian with a proton as a nu cleus reads (atomic units are used throughout the paper) h=p2 1 2+p2 2 2−1 r1−1 r2−1 r1+r2. (1) The Langer-corrected Hamiltonian reads H=h+1 8r2 1+1 8r2 2. (2)Inelastic semiclassical Coulomb scattering 3 For collinear collisions we have only one ’observable’ afte r the collision, namely the state with quantum number n, to which the target electron was excited through the collision. If its initial quantum number before the collisi on was n′, we may write the probability at total energy Eas Pn,n′(E) =|/angbracketleftn|S|n′/angbracketright|2(3) with the S-matrix S= lim t→∞ t′→−∞eiHfte−iH(t−t′)e−iHit′. (4) Generally, we use the prime to distinguish initial from final state variables. The Hamiltonians HiandHfrepresent the scattering system before and after the interaction and do not need to be identical (e.g. in the case o f a rearrangement collision). The initial energy of the projectile electron i s given by ǫ′=E−˜ǫ′(5) where ˜ ǫ′is the energy of the bound electron and Ethe total energy of the system. In the same way the final energy of the free electron is fixed. Ho wever, apart from excitation, ionization can also occur for E >0 in which case |n/angbracketrightis simply replaced by by a free momentum state |p/angbracketright. This is possible since the complicated asymptotics of three free charged particles in the continuum is contained i n the S-matrix. 2.2. The semiclassical expression for the S-matrix Semiclassically, the S-matrix may be expressed as Sn,n′(E) =/summationdisplay j/radicalBig P(j) n,n′(E)eiΦj−iπ 2νj, (6) where the sum is over all classical trajectories jwhich connect the initial state n′and the final ’state’ nwith a respective probability of P(j) n,n′(E). The classical probability P(j) n,n′(E) is given by P(j) n,n′(E) =P(j) ǫ,ǫ′(E)∂ǫ ∂n=1 N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ǫ(R′) ∂R′ j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1 ∂ǫ ∂n, (7) see [9] where also an expression for the normalization const antNis given. Note, that due to the relation (5) derivatives of ǫand ˜ǫwith respect to norR′differ only by a sign. From now on we denote the coordinates of the initially f ree electron by capital letters and those of the initially bound electron by small le tters. If the projectile is bound after the collision we will call this an ’exchange proc ess’, otherwise we speak of ’excitation’ (the initially bound electron remains bound) or ionization (both electrons have positive energies). The deflection function ǫ(R′) has to be calculated numerically, as described in the next section. The phase Φ jis the collisional action [18] given by Φj(P, n;P′, n′) =−/integraldisplay dt/parenleftBig q˙n+R˙P/parenrightBig (8) with the angle variable q. The Maslov index νjcounts the number of caustics along each trajectory. ’State’ refers in the present contex t to integrable motion for asymptotic times t→ ±∞ , characterized by constant actions, J′= 2π¯h(n′+ 1/2). The (free) projectile represents trivially integrable mot ion and can be characterized byInelastic semiclassical Coulomb scattering 4 5101520r2 [a.u.] 0 5 10 15 r1 [a.u.]051015r2 [a.u.] 5 10 15 r1 [a.u.]5 10 15 20 r1 [a.u.](a) (b) (c) (d) (e) (f) Figure 1. Scattering trajectories at a total energy of E= 0.125 a.u. with initial conditions marked in figure 2. The labels (a-f) refer to repre sentative trajectories with initial values R′shown in figure 2. The left column corresponds to classical exchange n′= 1→n= 1, the middle column represents ionization events and the right column shows elastic back-scattering with n′= 1→n= 1. its momentum P′. In our case, each particle has only one degree of freedom. He nce, instead of the action J′we may use the energy ˜ ǫ′for a unique determination of the initial bound state. In the next sections we describe how we c alculated the deflection function, the collisional action and the Maslov index. 2.2.1. Scattering trajectories and the deflection function The crucial object for the determination of (semi-)classical scattering probabilit ies is the deflection function ǫ(R′) where ǫis the final energy of the projectile electron as a function of its initial position R0+R′. Each trajectory is started with the bound electron at an arb itrary but fixed phase space point on the degenerate Kepler ellipse w ith energy ˜ ǫ′=−1/2 a.u.. The initial position of the projectile electron is cha nged according to R′, but always at asymptotic distances (we take R0= 1000 a.u.), and its momentum is fixed by energy conservation to P′= [2(E−˜ǫ′)]1/2. The trajectories are propagated as a function of time with a symplectic integrator [19] and ǫ=ǫ(t→ ∞) is in practice evaluated at a time twhen dln|ǫ|/dt < δ (9) where δdetermines the desired accuracy of the result. Typical traj ectories are shown in figure 1, their initial conditions are marked in the deflect ion function of figure 2. In the present (and generic) case of a two-body potential tha t is bounded from below the deflection function must have maxima and minima acc ording to the largest and smallest energy exchange possible limited by the minimu m of the two-body potential. The deflection function can only be monotonic if t he two-body potential is unbounded from below as in the case of the pure (homogeneous) Coulomb potentialInelastic semiclassical Coulomb scattering 5 0 1 2 3 4 5 6 7 ∆R’ [a.u.]−1.00.01.0ε [a.u.] (a)(b)(c)(d) (e) (f) Figure 2. The deflection function at an energy of E= 0.125 a.u. and for an initial state as described in the text. The energy interval e nclosed by dashed lines marks ionizing initial conditions and separates the exchan ge region ( ǫ <0) from the excitation region ( ǫ > E ), where ǫis the energy of the projectile after the collision. without Langer correction (compare, e.g., figure 1 of [8]). T his qualitative difference implies another important consequence: For higher total en ergies Ethe deflection function is pushed upwards. Although energetically allowe d, for E > 1 a.u. the exchange-branch vanishes as can be seen from figure 3. As we wi ll see later this has a significant effect on semiclassical excitation and ioni zation probabilities. 2.2.2. The form of the collisional action The collisional action Φ jalong the trajectory jin (6) has some special properties which result from the form of the S-matrix (4). The asymptotically constant states are repre sented by a constant action Jor quantum number nand a constant momentum Pfor bound and free degrees of freedom respectively. Hence, in the asymptotic integrable situation with ˙ n=˙P= 0 before and after the collision no action Φ jis accumulated and the collisional action has a well defined value irrespectively of the actual propagatio n time in the asymptotic regions. This is evident from (8) which is, however, not suit able for a numerical realization of the collision. The scattering process is muc h easier followed in coordinate space, and more specifically for our collinear case, in radia l coordinates. In the following, we will describe how to extract the action accord ing to (8) from such a calculation in radial coordinates (position rand momentum pfor the target electron, RandPfor the projectile electron). The discussion refers to exci tation processes to keep the notation simple but the result (13) holds also for the other cases. TheInelastic semiclassical Coulomb scattering 6 0 5 10 15 ∆R’ [a.u.]1.02.03.0ε [a.u.] Figure 3. The deflection function at an energy of E= 2 a.u. and for an initial state as described in the text. The dashed line separates ion izing initial conditions from excitation events. collisional action Φ can be expressed through the action in c oordinate space ˜Φ by [3] Φ(P, n;P′, n′) =˜Φ(P, r;P′, r′) +F2(r′, n′)−F2(r, n), (10) where ˜Φ(P, r;P′, r′) = lim t→∞ t′→−∞t/integraldisplay t′dτ/bracketleftBig −R˙P+p˙r/bracketrightBig (11) is the action in coordinate space and F2is the generator for the classical canonical transformation from the phase space variables ( r, p) to (q, n) given by F2(r, n) = sgn( p)r/integraldisplay ri(2m[ǫ(n)−v(x)])1 2dx. (12) Here, ridenotes an inner turning point of an electron with energy ǫ(n) in the potential v(x). Clearly, F2will not contribute if the trajectory starts end ends at a tur ning point of the bound electron. Partial integration of (11) transfor ms to momentum space and yields a simple expression for the collisional action in ter ms of spatial coordinates: Φ(P, n;P′, n′) = lim ti→∞ t′ i→−∞−ti/integraldisplay t′ idτ/bracketleftBig R˙P+r˙p/bracketrightBig . (13) Note, that t′ iandtirefer to times where the bound electron is at an inner turning point and the generator F2vanishes. Phases determined according to (13) may still differ for the same path depending on its time of termination. However, the difference can only amount to integer multiples of the (quantized !) act ion J=/contintegraldisplay p dr= 2π/parenleftbigg n+1 2/parenrightbigg (14) of the bound electron with ǫ <0. Multiples of 2 πfor each revolution do not change the value of the S-matrix and the factor2π 2is compensated by the Maslov index. InInelastic semiclassical Coulomb scattering 7 the case of an ionizing trajectory the action must be correct ed for the logarithmic phase accumulated in Coulomb potentials [18]. Summarizing this analysis, we fix the (in principle arbitrar y) starting point of the trajectory to be an inner turning point ( r′ i|p′= 0,˙p′>0) which completes the initial condition for the propagation of trajectories desc ribed in section 2.2.1. In order to obtain the correct collisional action (8) in the for m (13) we also terminate a trajectory at an inner turning point riafter the collision such that Φ is a continuous function of the initial position R′. Although this is not necessary for the primitive semiclassical scattering probability which is only sensit ive to phase differences up to multiples of Jas mentioned above, the absolute phase difference is needed f or a uniformized semiclassical expression to be discussed late r. 2.3. Maslov indices 2.3.1. Numerical procedure In position space the determination of the Maslov index is rather simple for an ordinary Hamiltonian with kinetic en ergy as in (2). According to Morse’s theorem the Maslov index is equal to the number of c onjugate points along the trajectory. A conjugate point in coordinate space is defi ned by ( fdegrees of freedom, ( qi, pi) a pair of conjugate variables) det(Mqp) =det ∂(q1, . . ., q f) ∂/parenleftBig p′ 1, . . ., p′ f/parenrightBig = 0. (15) The matrix Mqpis the upper right part of the stability or monodromy matrix w hich is defined by /parenleftbiggδ/vector q(t) δ/vector p(t)/parenrightbigg ≡M(t)/parenleftbiggδ/vector q(0) δ/vector p(0)/parenrightbigg . (16) In general, the Maslov index νjin (6) must be computed in the same representation as the action. In our case this is the momentum representatio n in (13). However, the Maslov index in momentum space is not simply the number of conjugate points in momentum space where det(Mpq) = 0. Morse’s theorem relies on the fact that in position space the mass tensor Bij=∂2H/∂p i∂pjis positive definite. This is not necessarily true for Dij=∂2H/∂q i∂qjwhich is the equivalent of the mass tensor in momentum space. How to obtain the correct Maslov index from t he number of zeros of det(Mpq) = 0 is described in [20], a review about the Maslov index and i ts geometrical interpretation is given in [21]. 2.3.2. Phenomenological approach for two degrees of freedo mFor two degrees of freedom, one can extract the scattering probability direct ly from the deflection function without having to compute the stability matrix and its determinant explicitly [8]. In view of this simplification it would be desirable to de termine the Maslov indices also directly from the deflection function avoiding the complicated procedure described in the previous section. This is indeed possible s ince one needs only the correct difference of Maslov indices for a semiclassical scattering amplitude . A little thought reveals that trajectories starting from br anches in the deflection function of figure 2 separated by an extremum differ by one conj ugate point. This implies that their respective Maslov indices differ by ∆ ν= 1. For this reason it is convenient to divide the deflection function in different bra nches, separated by anInelastic semiclassical Coulomb scattering 8 extremum. Trajectories of one branch have the same Maslov in dex. Since there are two extrema we need only two Maslov indices, ν1= 1 and ν2= 2. The relevance of just two values of Maslov indices (1 ,2) can be traced to the fact that almost all conjugate points are trivial in the sense that they belong to turning points of bound two-body motion. We can assign the larger index ν2= 2 to the trajectories which have passed one more conjugate point than the others. As it is almost evident from their topology, these are the trajectories with dǫ/dR′>0 shown in the upper row of figure 1. (They also have a larger collisional action Φ j). The two non-trivial conjugate points for these trajectories compared to the single conjugate point for orb its with initial conditions corresponding to dǫ/dR′<0 can be understood looking at the ionizing trajectories (b) and (e) of each branch in figure 1. Trajectories from both bran ches have in common the turning point for the projectile electron ( P= 0). For trajectories of the lower row all other turning points belong to complete two-body revolu tions of a bound electron and may be regarded as trivial conjugate points. However, fo r the trajectories from the upper row there is one additional turning point (see, e.g ., figure 1(b)) which cannot be absorbed by a complete two-body revolution. It is the sour ce for the additional Maslov phase. We finally remark that dǫ/dR′>0 is equivalent to dn/d¯q <0 of [25] leading to the same result as our considerations illustrated above. 3. Semiclassical scattering probabilities Taking into account the Pauli principle for the indistingui shable electrons leads to different excitation probabilities for singlet and triplet , P+ ǫ(E) =|Sǫ,ǫ′(E) +SE−ǫ,ǫ′(E)|2 P− ǫ(E) =|Sǫ,ǫ′(E)−SE−ǫ,ǫ′(E)|2, (17) where the probabilities are symmetrized a posteriori (see [ 24]). Here, Sǫ,ǫ′denotes the S-matrix for the excitation branch, calculated accordi ng to (6), while SE−ǫ,ǫ′ represents the exchange processes, at a fixed energy ǫ <0, respectively. Ionization probabilities are obtained by integrating the d ifferential probabilities over the relevant energy range which is due to the symmetriza tion (17) reduced to E/2: P± ion(E) =E/2/integraldisplay 0P± ǫ(E)dǫ . (18) 3.1. Ionization and excitation for singlet and triplet symm etry We begin with the ionization probabilities since they illus trate most clearly the effect of the vanishing exchange branch for higher energies a s illustrated in figure 3. The semiclassical result for the Langer Hamiltonian (2) sho ws the effect of the vanishing exchange branch in the deflection function figure 3 which leads to merging P±probabilities at a finite energy E, in clear discrepancy to the quantum result, see figure 4. Moreover, the extrema in the deflection function lead to the sharp structures below E= 1 a.u.. The same is true for the excitation probabilities wh ere a discontinuity appears below E= 1 a.u. (figure 5). Note also that due to theInelastic semiclassical Coulomb scattering 9 0.1 1.0 10.0 E [a.u.]0.0010.0100.100Pion(E)P P+ − Figure 4. Ionization probabilities for singlet and triplet accordin g to (18) with the Hamiltonian (2) (solid line) compared to quantum mechan ical calculations (dotted line). 0.0100.1001.000P+ ε (E) 1 10 E [a.u.]0.0010.0100.1001.000P− ε (E)n=1 n=2 n=3n=1 n=2 n=3(a) (b) Figure 5. Semiclassical excitation probabilities ( n= 1,2,3) according to (17) for singlet (part a) and triplet (part b) in the LSA (solid line) c ompared to quantum mechanical calculations (dotted line). violated unitarity in the semiclassical approximation pro babilities can become larger than unity, as it is the case for the n= 1 channel. Singlet and triplet excitation probabilities represent th e most differential scattering information for the present collisional system . Hence, the strongest deviations of the semiclassical results from the quantum va lues can be expected. Most experiments do not resolve the spin states and measure a spin -averaged signal. In ourInelastic semiclassical Coulomb scattering 10 0.0 0.2 0.4 0.6 0.8 1.0 E [a.u.]0.0100.0200.0300.0400.0500.060Pion (E) Figure 6. Spin averaged quantum results for ionization (dotted line) compared to averaged semiclassical probabilities (solid line) from (19) and classical probabilities (dashed line) from (20). model this can be simulated by averaging the singlet and trip let probabilities to Pǫ(E) =1 2(P+ ǫ(E) +P− ǫ(E)). (19) The averaged semiclassical probabilities may also be compa red to the classical result which is simply given by PCL ǫ(E) =/summationdisplay j(P(j) ǫ,ǫ′(E) +P(j) ǫ,E−ǫ′(E)) (20) withP(j) ǫ,ǫ′(E) from (7). Figure 6 shows averaged ionization probabilities. They are very similar to each other, and indeed, the classical result is not much worse tha n the semiclassical result. In figure 7 we present the averaged excitation probabilities . Again, on can see the discontinuity resulting from the extrema in the deflection f unction. As for ionization, the spin averaged semiclassical probabilities (figure 7b) a re rather similar to the classical ones (figure 7a), in particular the discontinuity is of the same magnitude as in the classical case and considerably more localized in e nergy than in the non- averaged quantities of figure 5. Clearly, the discontinuities are an artefact of the semicla ssical approximation. More precisely, they are a result of the finite depth of the two -body potential in the Langer corrected Hamiltonian (2). Around the extrema of the deflection function the condition of isolated stationary points, necessary to a pply the stationary phase approximation which leads to (6), is not fulfilled. Rather, o ne has to formulate a uniform approximation which can handle the coalescence of t wo stationary phase points. 4. Uniformized scattering probabilities We follow an approach by Chester et. al. [23]. The explicit expression for the uniform S-matrix goes back to Connor and Marcus [22] who obtained for two coalescingInelastic semiclassical Coulomb scattering 11 0.010.101.00Pε (E) 1 10 E [a.u.]0.010.101.00Pε (E)n=1 n=2 n=3n=1 n=2 n=3(a) (b) Figure 7. Spin averaged quantum results (dotted line) for excitation (n= 1,2,3) compared to classical probabilities (solid line, part a) fr om (20) and averaged semiclassical probabilities (solid line, part b) from (19) . trajectories 1 and 2 Sn,n′(E) = Bi+(−z)/radicalBig P(1) n,n′(E)eiΦ1+iπ 4+ Bi−(−z)/radicalBig P(2) n,n′(E)eiΦ2−iπ 4(21) where Bi±(−z) =√π/bracketleftBig z1 4Ai (−z)∓iz−1 4Ai′(−z)/bracketrightBig e±i/parenleftBig 2 3z3 2−π 4/parenrightBig (22) The argument z=/bracketleftbig3 4(Φ2−Φ1)/bracketrightbig2 3of the Airy function Ai(z) contains the absolute phase difference. We assume that Φ 2>Φ1which implies for the difference of the Maslov indices that ν2−ν1= 1 (compare (6) with (21) and (23)). Since the absolute phase difference enters (21) it is important to ensure that th e action is a continuous function of R′avoiding jumps of multiples of 2 π, as already mentioned in section 2.2.2. For large phase differences (6) is recovered since lim z→∞Bi±(−z) = 1. (23) Our uniformized S-matrix has been calculated by applying (2 1) to the two branches for exchange and excitation separately and adding or subtracting the results according to a singlet or triplet probability. In the corres ponding probabilities of figure 8 the discontinuities of the non-uniform results ar e indeed smoothed in comparison with figure 5. However, the overall agreement wit h the quantum probabilities is worse than in the non-uniform approximati on. A possible explanation could lie in the construction of the uniform approximation. It works with an integral representation of the S-matrix, where the oscillating phas e (the action) is mapped onto a cubic polynomial. As a result, the uniformization wor ks best, if the deflectionInelastic semiclassical Coulomb scattering 12 0.0100.1001.000P+ ε (E) 1 10 E [a.u.]0.0010.0100.1001.000P− ε (E)(a) (b)n=1 n=2 n=3 n=1 n=2 n=3 Figure 8. Uniformized semiclassical excitation probabilities ( n= 1,2,3) according to (21) (solid line) for singlet (part a) and tripl et (part b) compared to quantum mechanical calculations (dotted line). function can be described as a quadratic function around the extremum. Looking at figure 2 one sees that this is true only in a very small neighbor hood of the extrema because the deflection function is strongly asymmetric arou nd these points. We also applied a uniform approximation derived by Miller [4] which gave almost identical results. Finally, for the sake of completeness, the spin averaged uni form probabilities are shown in figure 9. As can be seen, the discontinuities have vanished almost completely. However, the general agreement with quantum me chanics is worse than for the standard semiclassical calculations, similarly as for the symmetrized probabilities. 5. Conclusion In this paper we have described inelastic Coulomb scatterin g with a semiclassical S- matrix. To handle the problem for this explorative study we h ave restricted the phase space to the collinear arrangement of the two electrons redu cing the degrees of freedom to one radial coordinate for each electron. In appreciation of the spherical geometry we have applied the so called Langer correction to obtain the co rrect angular momentum quantization. Thereby, a lower bound to the two-body potent ial is introduced which generates a generic situation for bound state dynamics sinc e the (singular) Coulomb potential is replaced by a potential bounded from below. The finite depth of the two- body potential leads to singularities in the semiclassical scattering matrix (rainbow effect) which call for a uniformization. Hence, we have carried out and compared among each other clas sical (where applicable), semiclassical, and uniformized semiclassic al calculations for the singlet, triplet and spin-averaged ionization and excitation proba bilities. Two general trendsInelastic semiclassical Coulomb scattering 13 1 10 E [a.u.]0.010.101.00Pε (E)n=1 n=2 n=3 Figure 9. Spin averaged uniformized excitation probabilities ( n= 1,2,3, solid line) compared to quantum results (dotted line). may be summarized: Firstly, the simple semiclassical proba bilities are overall in better agreement with the quantum results for the singlet/t riplet observables than the uniformized results. The latter are only superior close to t he singularities. Secondly, for the (experimentally most relevant) spin-averaged prob abilities the classical (non- symmetrizable) result is almost as good as the semiclassica l one compared to the exact quantum probability. This holds for excitation as wel l as for ionization. Hence, we conclude from our explorative study that a full semiclass ical treatment for spin- averaged observables is probably not worthwhile since it do es not produce better results than the much simpler classical approach. Clearly, this conclusion has to be taken with some caution since we have only explored a colline ar, low dimensional phase space. Acknowledgments We would like to thank A. Isele for providing us with the quant um results for the collinear scattering reported here. This work has been supp orted by the DFG within the Gerhard Hess-Programm. References [1] Ford K W and Wheeler J A 1959 Ann. Phys. 7259 [2] Pechukas P 1969 Phys. Rev. 181166 [3] Miller W H 1970 J. Chem. Phys. 531949 [4] Miller W H 1970 J. Chem. Phys. 533578 Miller W H 1970 Chem. Phys. Lett. 7431 [5] Miller W H 1974 Adv. Chem. Phys. 2569 Miller W H 1975 Adv. Chem. Phys. 3077 [6] Balian R and Bloch C 1974 Ann. Phys. 85514 [7] Rost J M and Heller E J 1994 J. Phys. B: At. Mol. Opt. Phys. 271387 [8] Rost J M 1994 Phys. Rev. Lett. 721998 [9] Rost J M 1995 J. Phys. B: At. Mol. Opt. Phys. 283003 [10] Rost J M and Wintgen D 1996 Europhys. Lett. 3519 [11] Gu Y and Yuan J M 1993 Phys. Rev. A 47R2442Inelastic semiclassical Coulomb scattering 14 [12] Ezra G S, Richter K, Tanner G and Wintgen D 1991 J. Phys. B: At. Mol. Opt. Phys. 24L413 Wintgen D, Richter K and Tanner G 1992 CHAOS 219 Tanner G and Wintgen D 1995 Phys. Rev. Lett. 752928 [13] Wannier G H 1953 Phys. Rev. 90817 [14] Rost J M 1998 Phys. Rep. 297291 [15] Handke G, Draeger M, Ihra W and Friedrich H 1993 Phys. Rev. A 483699 [16] Friedrich H, Ihra W and Meerwald P 1999 Aust. J. Phys. 52323 [17] Langer R 1937 Phys. Rev. 51669 [18] Child M S 1974 Molecular Collision Theory (London: Academic Press) [19] Yoshida H 1990 Phys. Lett. A 150262 [20] Levit S, M¨ ohring K, Smilansky U and Dreyfus T 1978 Ann. Phys. 114223 [21] Littlejohn R G 1992 J. Stat. Phys. 687 [22] Connor J N L and Marcus R A 1971 J. Chem. Phys. 555636 [23] Chester C, Friedman B and Ursell F 1957 Proc. Camb. phil. Soc. 53555 [24] Joachain C J 1975 Quantum Collision Theory (Amsterdam: North Holland) [25] Marcus R 1972 Chem. Phys. 574903

arXiv:physics/0001018v1 [physics.optics] 9 Jan 2000Hierarchy of time scales and quasitrapping in theN-atom micromaser Georgii Miroshnichenko†, Andrei Rybin‡, Ilia Vadeiko†‡, and Jussi Timonen‡ †Fine Mechanics and Optics Institute Sablinskaya 14, St. Petersburg, Russia ‡University of Jyv¨ askyl¨ a, Department of Physics PO Box 35, Jyv¨ askyl¨ a, Finland Abstract We study the dynamics of the reduced density matrix(RDM) of the field in the micromaser. The resonator is pumped by N-atomic clusters of two-level atoms. At each given instant there is o nly one cluster in the cavity. We find the conditions of the independe nt evo- lution of the matrix elements of RDM belonging to a (sub)diag onal of the RDM, i.e. conditions of the diagonal invariance for th e case of pumping by N-atomic clusters. We analyze the spectrum of the evolution operator of the RDM and discover the existence of t he qu- asitrapped states of the field mode. These states exist for a w ide range of number of atoms in the cluster as well as for a broad ra nge of relaxation rates. We discuss the hierarchy of dynamical p rocesses in the micromaser and find an important property of the field st ates corresponding to the quasi-equilibrium: these states are c lose to either Fock states or to a superposition of the Fock states. A possib ility to tune the distribution function of photon numbers is discuss ed. PACS number 42.50 communicating author: Andrei.Rybin@phys.jyu.fi 0Recent developments in the cavity electrodynamics [1] gave rise to the cre- ation of a real physical device - micromaser which operates o n highly excited Rydberg atoms pumped through a high-Q resonator [2]. Existi ng literature mostly focuses on the ideal (basic) model which is the so-cal led one-atom micromaser [3, 4, 5, 6]. This device is assumed to operate in s uch a way that no more than one atom excited with the probability 1 can b e found in the cavity at each given instant of time. The basic model is justified by the following assumptions: the average velocity of injecti onR, and the time of interaction τ(in which a cluster passes through the resonator) are small. The rate of relaxation γof the field is low, which means a high-Q resonator. The coupling constant gof the field mode interacting with internal degrees of freedom of the atom is sufficiently large. Trajectories are assumed to be quasiclassical. In more exact terms these assumptions can b e recapitulated as Rτ≪1, τγ≪1, gτ≥1 (1) The micromaser operating on a periodic sequence of N-atomic clusters which are created by laser pulses in the gas of unexcited atoms is in troduced in [7]. It is assumed that the size of the cluster is much less than the wavelength of the microwave radiation. Effects of finite cluster size [8] are comparable in magnitude with effects of inhomogeneous field at the edges o f resonator. The latter observation was reported in Ref. [9]. In this Lett er we study the one-cluster extension of the basic model. This means tha t we assume a point-like structure of N-atomic cluster (i.e. the finite size effects are not taken into account) as well as the fulfillment of the condi tions Eq.(1). This formulation generalizes greatly the basic model while leaves intact the simplifying assumption that process of interaction of the c luster and the field (within the time interval τ) and the process of the field relaxation to the thermodynamic equilibrium (time interval T∼1/R) are separated in time. This latter assumption allows to factorize the evolution op erator of the RDM (see Eq. (3) below) and greatly simplifies the analysis of the properties of evolution operator and the dynamics of RDM. One-cluster model of the micromaser assumes that the N-particle Tavis- Cummings Hamiltonian [10] H=H0+V=ω/parenleftbigg a†a+S3+N 2/parenrightbigg + 2g/parenleftBig a†S−+aS+/parenrightBig (2) 1is applicable. Here ωis the frequency of the quantum transition which is in exact resonance with the field mode. The collective spin va riables S3, S± are the generators of the su(2) algebra, while a†, aare the creation and annihilation operator of the field mode, ¯ h= 1. The operation of the micromaser for each cluster is divided i nto two time intervals: the interaction time τand the relaxation time T. This means that the vector of the main diagonal ρ(l)of RDM satisfies the following equation ρ(l+1)=S(N)ρ(l)=Q(Nex)Spat/parenleftBig e−iHτρat⊗ρ(l)eiHτ/parenrightBig =Q(Nex)W(τ)ρ(l). (3) This difference equation connects the main diagonals of RDM t aken at the instants when the l-th and ( l+ 1)-th clusters enter the cavity. This allows to understand the number of passing clusters las a discrete ”time variable”. HereNex=R/γ, andQ(Nex) is the evolution operator of RDM at the relax- ation stage, i.e. in the empty resonator [5]. The operator W(τ) describes the evolution of RDM at the stage of interaction of ( l+1)-th cluster with the field, ρatis the density matrix of N-atomic cluster before it enters the resonator. The operation Spatmeans the trace with respect to atomic variables. In our work we consider clusters of fully excited atoms, whil e the field is ini- tially prepared in the state of the thermal equilibrium with the mean number of photons nb= 0.1. In our forthcoming publication we will rigorously show that if there is an additive with respect to atoms and field int egral of mo- tion [H, H 0] = 0 and for unpolarized initial state of the cluster, then th e dynamics of RDM is diagonally invariant . This important property of the evolution operator W(τ) means that each (sub)diagonal of RDM in the Fock basis evolves independently of other elements of RD matrix. In this Letter we concentrate on the dynamics of the main diagonal of RDM, i. e. on the number of photons probability distribution function. In th e space of vectors with components ρ(l) n, the evolution operators Q(Nex) can be represented as the following matrix Q(Nex) =/parenleftbigg 1 +L Nex/parenrightbigg−1 , (4) where the operator Lin the matrix form reads 2Lnm= [−(nb+ 1)nb(n+ 1)]δnm−(nb+ 1)(n+ 1)δn+1,m+nbnδn−1,m.(5) The matrix of the evolution operator W(τ) in the Fock basis is low-triangular. In the present Letter we analyze this matrix by numerical met hods. The property of diagonal invariance simplifies greatly the anal ysis of RDM dy- namics. The vector of the main diagonal of RDM satisfies the following difference equation ρ(l+1) n−ρ(l) n=J(l+1) n−J(l) n. (6) HereJ(l)is the vector of the probability flux for the l-th passage. The components of this vector are J(l) n=−n−1/summationdisplay n′=0/summationdisplay n′′=nS(N)n′n′′ρ(l) n′′+/summationdisplay n′=nn−1/summationdisplay n′′=0S(N)n′n′′ρ(l) n′′ (7) This vector determines the rate of change (after one passage ) of the sum of probabilities of the photon numbers in the interval of Fock n umbers between n=n0andn=n1. This rate is equal to the difference of fluxes through the chosen boundary values, viz n1/summationdisplay n=n0/parenleftBig ρ(l+1) n−ρ(l) n/parenrightBig =J(l) n0−J(l) n1+1 (8) The dependencies of the eigenvalues Wnon the number of photons are given in Figure 1 for the number of atoms in the cluster N=1,5,10. Th e interaction time is chosen as gτ= 1.355 . The eigenvalues Wnare positive and do not exceed 1. Their mutual positions are defined by the parameter τand the number of atoms N. For the one-atom micromaser the so-called trapped states are known. These are the Fock states for the number of p hotons n corresponding to the eigenvalue Wn= 1 of the matrix W(τ) . This number of photons fulfills the trapping condition √ n+ 1 =πχ gτ(9) where χis an integer number. The trapped states do not decay in the ab sence of relaxation, and thus determine the dynamics of ρ(l)for large l. The recent 3experimental realization of the trapped states was reporte d in [11]. The Figure 1 shows that in the multi-atomic case there are no trap ped states. There are however a few eigenvalues which are close to 1. The c orresponding eigenvectors in the space of the number of photons are locali zed around the numbers nfor which Wn≈1. Such long-living vectors is natural to call quasitrapped states . The Figure 2 shows the spectrum S(N) in ascending order. It is interesting to notice that the eigenvalues of the evolution operator ten d to group around zero when the number of atoms in the cluster increases. In the hierarchy of dynamical processes in the micromaser the small eigenvalue s are responsi- ble for the rapid phase of the dynamics (with respect to the di screte time l ). The quasitrapped states corresponding to the eigenvalue s in the interval [0.9,1) are in turn responsible for the slow phase of dynamics. Pro babilities of the states with corresponding photon numbers at certain s tages of the field formation can be rather high. In Figure 2 we compare the spect rum of the evolution operator S(N) for the cases with ( Nex= 20) and without relax- ation. The Figure 2 shows in particular that for bigger Nthe spectrum of S(N) is more stable towards the influence of relaxation. The rela tion Eq.(3) describes the transition of the diagonal elements of RDM to a stationary state. This transition process is determined by the pumping of the cavity field by passing clusters as well as by the relaxation of the fie ld. The re- cent literature discusses mostly [3, 5] the properties of th e stationary state, which can be achieved when a large number of clusters has gone through the cavity. This case corresponds to the asymptotic limit l→ ∞ . In this work we concentrate on the properties of the transition process w hich, due to the existence of the quasitrapped states, are very interesting . The field rather rapidly ”forgets” its initial state of the thermal equilibr ium. The dynamics of the population of the Fock states shows instead the format ion of long- living (with respect to the ”time” l) quasi-equilibrium distributions. This is illustrated by the properties of spectrum of the evolution o perator given in Figures 1,2. The existence of the eigenvalues close to 1 indi cates consider- able probabilities of the Fock states with photon numbers in the vicinities of the maxima. The small eigenvalues correspond to the sharp de pletion of the corresponding Fock states. The Figure 1 shows that in the cho sen interval of Fock numbers, 0 ≤n≤60, and for gτ= 1.355 there are three domains capturing considerable probabilities. These domains, whi ch are natural to call the domains of quasitrapping are localized in the vicinities of the Fock 4numbers n= 5,18,40 and contain almost all the probabilities. This means that they are getting populated at different ”moments” of ”ti me”lin relays: the next domain cannot get populated until the previous one i s depleted. This relay of populations is illustrated in Figures 3,4,5. T he Figure 3 show forN= 10 how the sums of probabilities of the Fock states change wi thl in the second (14 ≤n≤24) and the third (39 ≤n≤49) domains of qu- asitrapping. The rates of probability change are calculate d through Eq. (8) i.e. as the flux differences through the boundaries of the chos en domains. The Figure 3 allows to identify the following stages of the l-dynamics: a period of accumulation of the probability which correspond s to the positive values of the probability rate as well as an extended in time ( l) period of the negative probability rates. The lasting nature of the la tter period indi- cates that the life-time of the quasitrapped states is consi derable. The rate of decay of the second quasitrapped state ( n≈18) is approximately the rate of accumulation in the third state ( n≈40). This means that through a passage of a cluster the probability is almost fully relayed from the second quasitrapped state to the third. Since the dynamics of the de cay of the sec- ond quasitrapped state is slow, so is the dynamics of the accu mulation in the third state. In Figures 4 and 5 are given dependencies on lof total popula- tions curves of the domains of quasitrapping. It is evident f rom Figures 4 and 5 that the sum of populations of two subsequent domains of qua sitrapping is close to 1. This again manifests the full accumulation of the probabilities in the domains of quasitrapping as well as the relay of probabil ities indicated above. The Figure 6 shows the distributions of the diagonal e lements of the RDM taken at the l-moments of maximal probabilities of the Fock states in the corresponding domains of quasitrapping. As follows f rom Figure 6 it is possible to govern the vector of photon number distributi on. This can be achieved by the variation of the number of atoms passed throu gh the res- onator. It is possible in particular to create states close t o the Fock states localized at certain photon numbers. We can also report that a domain of the localization changes smoothly in accord with variation s of parameters τ, NandNex. The dependence of the stationary field on these parameters f or N= 1 was discussed in Refs. [3, 5, 6]. The possibility to engine er quantum states is actively studied in the recent literature [12]. 5Conclusions and discussion The main result of our work is the discovered possibility to p urposefully create in the cavity quasistable states close to Fock states . We analyzed the dynamics of the micromaser pumped by N-atomic clusters [7]. Our approach generalizes the basic model of the one atom microma ser [3] and can be experimentally realized. We assumed the point-like n ature of N- atomic clusters. This assumption can easily be realized in p ractice when clusters are created in a gas flow by focused laser pulses in th e light range. In this case the width of the beam is of order of few microns whi le the size of the cavity can be of order of few millimeters. In our wo rk we have pointed out the conditions when the time evolution of a (sub) diagonal of the reduced density matrix is independent of the other elements of the density matrix. We have investigated the properties of the spectrum of the evolution operator (see Figures 1,2) and discussed their connection t o the properties of the RDM dynamics. We have discussed the hierarchy of the ti me scales of the micromaser dynamics and have shown that the sectors of th e spectrum around zero are responsible for rapid processes while the se ctors close to 1 correspond to quasi-equilibrium. For the first time in the ex isting literature we have introduced an important notion of the quasitrapped states . The Figure 6 shows that these states are close to the Fock states. The domains in the Fock space corresponding to quasitrapping are rather narrow, their locations change smoothly with variations of the number of a toms in the cluster. This means that the overall picture of the dynamics is stable with respect to small variations of the number of atoms in a cluste r. In our future work we plan to investigate this phenomenon in a greater deta il as well as to study how the properties of the quasitrapped states depend o n the choice of the initial density matrix of the N-atomic cluster. References [1] S. Haroche, D. Kleppner, Phys. Today, 42(1), 24(1989). [2] D. Meschede, H. Walther, G. Muller, Phys. Rev. Lett. 54, 551(1985); G. Rempe, M. Scully, H. Walther, Physica Scripta, 34, 5 (1991); G.M. Brune, J. Raimond, P. Goy, L. Davidovich, S. Haroche, Phys. R ev. Lett. 59, 1899 (1987). 6[3] P. Filipowicz, L. Javanainen, P. Meystre, Phys. Rev. A, 34, 3077 (1986). [4] P. Meystre, M. Sargent III. Elements of Quantum Optica, S pringer- Verlag, Berlin, 1990. [5] P. Elmfors, B. Lautrup, B. Shagerstam, Phys. Rev. 54, 5171 (1996). [6] P. Meystre, G. Rempe, H. Walther, Opt. Lett. 13, 1078 (1988). [7] G. D’Ariano, N. Sterpi, A. Zucchetti, Phys. Rev. Lett. 74, 900 (1995). [8] M. Orszag, R. Ramirez, J. Retamal, C. Saavedra, Phys. Rev . A49, 2933 (1994); L. Ladron, M. Orszag, R. Ramirez, Phys. Rev. A 55, 2471 (1997); M. Kolobov, F. Haake, Phys. Rev. A 55, 3033 (1997). [9] C. Yang, K. An, Phys. Rev. A 55, 4492 (1997); F. Gasagrande, A. Lulli, S. Ulrega, Phys. Rev. A 60, 1582 (1999). [10] M. Tavis, E. Cummings, Phys. Rev. 170, 379 (1968); M. Sculle, G. Meyer, H. Walther, Phys. Rev. Lett. 76, 4144 (1996); A. Rybin, G. Kastelewicz, J. Timonen, N. Bogoliubov, J. Phys. A: Math. An d Gen. 31, 4705 (1998). [11] M. Weidinger, B.T.H. Varcoe, R. Heerlein, and H. Walthe r, Phys. Rev. Lett.82, 3795 (1999). Phys. Rev. Lett. 82, 3795 (1999). [12] K. Vogel, V. Akulin, and W. Schleich, Phys. Rev. Lett. 71, 1816 (1993); Shi-Biao Zeng, Guang-Gan Guo, Phys. Lett. A, 244, 512 (1998); A. Kozhekin, G. Kurizki, and B. Sherman, Phys. Rev. A, 54, 3535 (1996). 7Figure captions Figure 1. The spectrum of W(0, τ) forN= 1,5,10 and gτ= 1.355 . Figure 2. The spectrum of the evolution operator S(N) in ascending order forN= 1,15,Nex= 20,∞, andgτ= 1.355 . Figure 3. The rates of change of integral probabilities of th e Fock states in the second 14 ≤n≤24 and the third 39 ≤n≤49 quasitrapping domains forN= 10 . Figure 4. Integral probabilities of the Fock states in the se cond 14 ≤n≤24 and the third 39 ≤n≤49 quasitrapping domains for N= 1 . Figure 5. Integral probabilities of the Fock states in the se cond 14 ≤n≤24 and the third 39 ≤n≤49 quasitrapping domains for N= 10 . Figure 6. The photon number distributions of the diagonal el ements of RDM at the l-moments of maximal probabilities of the Fock states in the s econd 14≤n≤24 and the third 39 ≤n≤49 domains of quasitrapping. 80 10 20 30 40 50 600,00,20,40,60,81,0 nFIGURE 1.Spectrum N=1 N=5 N=100 10 20 30 40 50 600,00,20,40,60,81,0 FIGURE 2.Spectrum of S(N) N=1, Nex=∞ N=1, Nex=20 N=15, Nex=∞ N=15, Nex=200 1000 2000 3000-0,020,000,020,040,060,080,100,12 FIGURE 3.Dynamics rate 14<n<24 39<n<490 1000 2000 30000,00,20,40,60,81,0 FIGURE 4.Integral probability 14<n<24 39<n<490 100 200 3000,00,20,40,60,8 FIGURE 5.Integral probability 14<n<24 39<n<49048121620242832364044485256600,00,10,20,30,40,5 na) N=1, /G22/G22 =350 FIGURE 6.nPn 048121620240,00,10,20,30,40,5 b) NPn 048121620242832364044485256600,00,10,20,30,40,5 c) N=10, /G22/G22 =25Pn 048121620240,00,10,20,30,40,5 d) Pn

arXiv:physics/0001019 9 Jan 2000F.V.Tkachov 2000-Jan-09 17:21 Page 1 of 4 APPROACHING THE PARAMETER ESTIMATION QUALITY OF MAXIMUM LIKELIHOOD VIA GENERALIZED MOMENTS Fyodor V. Tkachov Institute for Nuclear Research of Russian Academy of Sciences Moscow 117312 Russia A simple criterion is presented for a practical construction of generalized moments that allow one to ap- proach the theoretical Rao-Cramer limit for parameter estimation while avoiding the complexity of the maximum likelihood method in the cases of complicated probability distributions and/or very large event samples. INTRODUCTION. The purpose of this note is to describe a result that was discovered in a rather special context of the theory of so-called jet finding algorithms [1] but seems to be basic enough to belong to the core statistical wisdom of pa- rameter estimation. Namely, I would like to present a simple formula (Eq.(20)) that connects the method of generalized moments with the maximum likelihood method by explicitly describing devia- tions from the Rao-Cramer limit on precision of parameter es- timation with a given event sample; see e.g. [2], [3]. The formula leads to practical prescriptions (the method of quasi-optimal momentsa; see after Eq.(24)) that offer a practi- cal alternative to the maximum likelihood method in precision measurement problems when the use of the maximum likeli- hood method is impractical due to complexity of theoretical expressions for the probability distribution or a large size of the sample of events. Although closely related to the well-known results and mathematical techniques, the prescription is new to the extent that I’ve seen no trace in the literature of its being known to physicists despite its immediate relevance to precision meas- urements. THE PROBLEM. One deals with a random variable P whose instances (specific values) are called events. Their probability density is denoted as π(P). It is assumed to depend on a parameter M which has to be estimated from an experi- mental sample of events {Pi}i. The standard method of generalized moments consists in choosing a function f(P) defined on events (the generalized moment), and then finding M by fitting its theoretical average value, f f =zd()PPPπ(), (1) against the correspond ing experimental value: f Nfii exp=∑1()P. (2) The problem is to find f which would allow one to extract M with the highest precision from the event sample. a In the quantum-theoretic context of [1] generalized moments are natu- rally interpreted as quantum observables, so the method was called the method of quasi-optimal observables.OPTIMAL MOMENTS. In the context of precision meas- urements one can assume the magnitude of errors to be small. Then fluctuations in the values of M are related to fluctuations in the values of f as δ δMf Mf =∂ ∂FHGIKJ−1 . (3) The derivative is applied only to the probability distribution: ∂ ∂=∂ ∂zf Mf MdPPP()()π. (4) This is because M is unknown, so even though the solution, fopt, will depend on M, any such dependence is coincidental and therefore “frozen” in this calculation. For small fluctuations δfN f =−12/Var, where Var () f ffff = − ≡ −zd()PPPπb g22 2. (5) In terms of variances, Eq .(3) becomes: Var[] Var. Mff Mf =∂ ∂FHGIKJ−2 (6) The problem is to minimized this by a suitable choice of f. A necessary condition for a minimum can be written in terms of functional derivatives:b δ δfMf()Var[]P=0. (7) Substitute Eq .(6) into (7) and use the following relations: b An interesting mathematical exercise of casting the following reasoning (the functional derivatives, etc.) into a rigorous form is left to interested mathematical parties. A premature emphasis on rigor would have obscured the simple analogy with the study of minima of ordinary functions via the usual Taylor expa nsion. For practical purposes it is sufficient to remember that the range of va- lidity of the prescriptions we obtain is practically the same as for the maximum likelihood method. Note that the derivation in terms of func- tional derivatives can be related to the proofs of the Rao-Cramer inequality in terms of Hilbert statistics, etc.; cf. e.g. [4].F.V.Tkachov 2000-Jan-09 17:21 Page 2 of 4 δ δπδ δπ δ δπfffff ff MM()(),()()(), ()().PPPPP PP= = ∂ ∂=∂ ∂22 (8) After some simple algebra one obtains: ff M()ln()PP= +∂ ∂constπ, (9) where the constant is independent of P. The constant plays no role since f is defined by this reasoning only up to a constant factor. Noticing that d d PPPPP z z∂ ∂=∂ ∂≡∂ ∂= πππ ()ln()() M M M10, (10) we arrive at the following general family of solutions: fC MC ()ln(), PP=∂ ∂+1 2π(11) where Ci are independent of P but may depend on M. For convenience of formal investigation we will usually deal with the following member of the family (11): f Mopt()ln(). PP=∂ ∂π(12) Then Eq.(10) is essentially the same as fopt=0. (13) A SIMPLE EXAMPLE. Consider the familiar Breit-Wigner shape. Let P be random real numbers distributed a ccording to π() ()P P∝ − +1 22M Γ(14) in some fixed interval around P=M. Suppose M is unknown. Then the optimal m oment is fMM MM,()ln()() ().optP PP P=∂ ∂= −− − +π2 22Γ(15) (Remember that P-independent additive and multiplicative constants can be dropped in such expressions; see Eq .(11).) It is interesting to observe how fM,opt emphasizes contribu- tions of the slopes of the bump — exactly where the magnitude of π(P) is most sensitive to variations of M — and taking con- tributions from the two slopes with a different sign maximizes the signal. At the same time the expression (15) suppresses contributions from the middle part of the bump (14) that gen- erates mostly noise as far as M is concerned. CONNECTION WITH MAXIMUM LIKELIHOOD. Eq.(12) can be regarded as a translation of the method of maximum likelihood (which is known to yield the theoretically best esti- mate for M; cf. the Rao-Cramer inequality [2], [3]) into the language of generalized moments.c Indeed, the maximum like- lihood method prescribes to estimate M by the value which maximizes the likelihood fun ction, ii∑ln()πP, (16) c Rather surprisingly, none of a dozen or so textbooks and monographs on mathematical statistics that I checked (including a comprehensive practical guide [2] and a comprehensive mathematical treatment [3]) explicitly for- mulated the prescription in terms of the method of moments /G03 although equivalent formulas do occur e.g. in simple examples of specific estimates for the parameters of standard distributions; cf. [4].where summation runs over all events from the sample. The necessary condition for the maximum of (16) is ∂ ∂=∂ ∂∝ = ∑ ∑M Mfiiiiln()ln()ππPP optexp0.(17) This agrees with (12) thanks to (13). DEVIATIONS FROM fopt. Next we are going to consider how small deviations from fopt affect the precision of extracted M. Consider (6) as a functional of f, VarM[f]. Assume ϕ is a function of events such that ϕ2<∞. We are going to evalu- ate the functional Taylor expansion of VarM[fopt+ϕ] with re- spect to ϕ through quadratic terms: Var[]Var[] Var[] ()()Mf Mf Mf ffffopt opt opt()() dd+ = +L NMO QP=+ zϕ δ δ δϕ ϕ1 22 PQPQPQ K(18) The term which is linear in ϕ does not occur because fopt satis- fies (7). To evaluate the quadratic term in (18), it is sufficient to use functional derivatives and relations such as (8) and δ δδ δ ϕ ϕff()()(,),(,)()().PQPQ PQPPQ = =zd (19) A straightforward calculation yields our main technical result: Var[] Mf f ff fopt opt2opt2opt2opt+ = + × − × +ϕ ϕ ϕ1 1 322{ } K(20) where ϕ ϕ ϕ= − . Non-negativity of the factor in curly braces follows from the standard Schwartz inequality.d The first term on the r.h.s. of (20), fopt2−1, is the absolute minimum for the variance of M as established by the Rao- Cramer inequality [2], [3]. The latter is valid for all ϕ and therefore is somewhat stronger than the result (20) which we have obtained only for sufficiently small ϕ. However, Eq.(20) gives a simple explicit description of the deviation from opti- mality and so makes possible the practical prescriptions pre- sented below after Eq .(24). It is convenient to talk about informativeness If of a gener- alized moment f with respect to the parameter M, defined by I Mff=−Var[]. b g1(21) The informativeness of fopt is Ifoptopt2=, (22) which corresponds to the Rao-Cramer limit. And the expansion (20) explicitly describes the deviations from the limit. Informativeness is closely related to Fischer’s information [2], [3] which, however, is an attribute of data whereas infor- mativeness is a property of the moment. d Note that the Schwartz inequality figures in standard rigorous proofs of the Rao-Cramer theorem.F.V.Tkachov 2000-Jan-09 17:21 Page 3 of 4 THE METHOD OF QUASI-OPTIMAL MOMENTS. The fact that the solution (12) is the point of a quadratic minimum means that any moment fquasi which is close to (12) would be practically as good as the optimal solution (we will call such moments quasi-optimal). A quantitative measure of closeness is given by comparing the O(1) and O(ϕ2) terms on the r.h.s. of (20): f f fopt2opt opt2ϕ ϕ2 2 21−<<, (23) where ϕ= − −fffquasiquasiopt. The subtracted term in the numerator of (23) is non- negative, so dropping it results in a sufficient condition for Eq.(23). Furthermore, foptϕ would tend to be suppressed anyway whenever fquasi oscillates around fopt. Assuming with- out loss of generality that fquasi=0, we obtain the following convenient sufficient criterion: ff f quasiopt opt2− <<2. (24) Taking into account this and Eq.(20) and denoting the usual σ for M for the optimal and quasi-optimal cases as σopt and σquasi, respectively, one obtains: σ σquasi optquasiopt opt2≈+ ×− 11 22ff f. (25) Now the method of quasi-optimal moments is as follows: (i) construct a generalized moment fquasi using (12) as a guide so that fquasi were close to fopt in the integral sense of Eq.(24); (ii) find M by fitting fquasi against fquasiexp; (iii) estimate the error for M via (6); (iv) fquasi may depend on M to find which one can optionally use an iterative procedure starting from some value M0 close to the true one. For practical construction of quasi-optimal moments fquasi it is useful to reformulate (24) in terms of integrands. The ex- plicit form for (24) is d dquasi opt opt2PPPP PPP z z− << π π()()() ()(). f f f2 (26) As a rule of thumb, one would aim to minimize the bracketed expression on the l.h.s. of (26): f f f quasi opt opt2()() (). PP P − <<2(27) This should hold for “most” P, i.e. taking into account the magnitude of π(P): the inequality (27) may be relaxed in the regions which yield small contributions to the integral on the l.h.s. of (26). THE EXAMPLE (14). Suppose the exact probability distri- bution differs from (14) by, say, a mild but complicated de- pendence of Γ on P (as seen e.g. from some sort of perturbative calculations of theoretical corrections — a situation typical of high-energy physics problems [5]). Then the r.h.s. of (15) with a constant Γ would correspond to a generalized moment which is only quasi-optimal but deviations from optimality may bepractically negligible (depending on the “mildness” of the P- dependence). So one could still use the moment given by the simplest formula (15) without significant loss of informative- ness. Alternatively, one could replace the analytical shape (15) by cruder piecewise constant or, better, piecewise linear approxi- mations that would imitate the expression (15): PMπ()P fopt()P fquasi()P (a) (b) (c)(d)(28) In either case, the effect of non-optimality can be easily es- timated via Eq.(25): the piecewise linear shape (d) deviates from optimality in the sense of (25) by a few per cent (in infi- nite domains, the slowly decreasing tails of the probability distribution may spoil this conclusion somewhat so one may wish to extend fquasi by additional linear pieces as well as in- sert flat linear pieces at the sharp peaks). DISCUSSION. Eq.(27) allows one to talk about non- optimality of moments (i.e. their lower informativeness com- pared with fopt) in terms of sources of non-optimality, i.e. the deviations of fquasi(P) from fopt(P) which give sizeable contri- butions to the l.h.s. of (24). The simplest example is when fopt is a continuous smoothly varying function whereas fquasi is a piecewise constant approximation (see (28), figure (c)). Then fquasi would usually deviate most from fopt near the discontinui- ties which, therefore, are naturally identified as sources of non- optimality. Then a natural way to improve fquasi is by “regulating” discontinuities via continuous (e.g. linear) inter- polations. Intuitively, one could think about sources of non-optimality as “leaks” through which information about M is lost, and the improvement of fquasi would then correspond to patching up those leaks. It is practically sufficient to take Eq.(12) at some value M=M0 close to the true one (which is unknown anyway). This is usually possible in the case of precision measurements. One could also perform an iterative procedure for M starting from M0, then replacing M0 with the value newly found, etc. — a procedure closely related to the optimization in the maximum likelihood method. If π(P) is given by a perturbation theory with increasingly complex but decreasingly important contributions, it is possible to use an approximate shape for the r.h.s. of (12) such as given by a few terms of a perturbative expansion in which the de- pendence on the parameter manifests itself. Theoretical up- dates of the complete π(P) need not be always reflected in the quasi-optimal moments. If the dimensionality of the space of events is not large then it may be possible to construct a suitable fquasi in a brute force fashion, i.e. build an interpolation formula for π(P) for two or more values of M near the value of interest, and perform the differentiation in M numerically. Also, one can use different expressions for fquasi: e.g. per- form a few first iterations with a simple shape for faster calcu- lations and then switch to a more sophisticated interpolation formula for best precision.F.V.Tkachov 2000-Jan-09 17:21 Page 4 of 4 SEVERAL PARAMETERS. With several parameters to be extracted from data there are the usual ambiguities due to reparametrizations but one can always define a moment per pa- rameter according to (12). Then the informativeness (21) is a matrix (as is Fischer’s information). Since the covariance matrix of (quasi-)optimal moments is known (or can be computed from data), the mapping of the cor- responding error ellipsoids for different confidence levels from the space of moments into the space of parameters is straight- forward. OPTIMAL MOMENTS AND THE LEAST SQUARES METHOD. The popular χ2 method makes a fit with a number of non-optimal moments (bins of a histogram). The histogram- ming implies a loss of information but the method is universal, verifies the probability distribution as a whole, and is imple- mented in standard software routines. On the other hand, the choice of fquasi requires a problem-specific effort but then the loss of information can in principle be made negligible by suit- able adjus tments of fquasi. The balance is, as usual, between the quality of custom so- lutions and the readiness of universal ones. However, once quasi-optimal moments are found, the quality of maximum likelihood method seems to become available at a lower com- putational cost. The two methods are best regarded as complementary: One could first employ the χ2 method to verify the shape of the probability distribution and obtain the value of M0 to be used as a starting point in the method of quasi-optimal moments in order to obtain the best final est imate for M. An additional advantage of the method of quasi-optimal moments may be that some of the more sophisticated theoreti- cal formalisms yield predictions for probability densities in the form of singular (and therefore not necessarily positive-definite everywhere) generalized functions (cf. the systematic gauge- invariant quantum-field-theoretic perturbation theory with un- stable particles outlined in [6]). In such cases theoretical pre- dictions for generalized moments (quasi-optimal or not) may exceed in quality predictions for probability densities, so that the use of the χ2 method would be somewhat disfavored com- pared with the method of quasi-optimal moments for the high- est-precision measurements of unknown p arameters. Note that the data processing for the LEP1 experiments [5] has been performed in several iterations over several years and it would have been entirely possible to design, say, five quasi- optimal moments for the five parameters measured at the Z resonance back in the ‘80s and to use them ever since. CONCLUSIONS. It is clear that the method of quasi- optimal moments may be a useful addition to the data- processing arsenal e.g. in situations encountered in precision measurement problems in high-energy particle physics (cf. [5]) where one deals with O(106) events and very complicated probability distributions obtained via quantum-field-theoretic perturbation theory so that the optimization involved in the maximum likelihood method is unfeasible. It also does not seem impossible to design universal software routines for a numerical construction of fquasi in the form of dynamically gen- erated interpolation fo rmulas. Lastly, the usefulness of the concept of quasi-optimal mo- ments is not limited to purely numerical situations: It also proved to be useful in a theoretical context of [1] as a guiding principle for studying an important class of data processing al- gorithms (the so-called jet finding alg orithms).ACKNOWLEDGMENTS. I thank Dima Bardin for a help with clarifying the bibliographic status of the concept of opti- mal moments. This work was supported in part by the Russian Foundation for B asic Research under grant 99-02-18365. References [1]F.V.Tkachov, A theory of jet definition [hep-ph/9901444, revised January, 2000]. [2]W.T.Eadie et al., Statistical methods in experimental physics. North-Holland, 1971. [3]A.A.Borovkov, Mathematical statistics. Parameter estimation and tests of hypotheses. NAUKA: Moscow, 1984 (in Russian). [4]Yu.P.Pyt’ev and I.A .Shyshmarev, A course of the theory of pro b- ability and mathematical statistics for physicists. Moscow State Univ.: Moscow, 1983 (in Russian). [5]J.Ellis and R .Peccei (eds.), Physics at LEP , CERN: Geneva, 1986. [6]F.V.Tkachov, in: Proc. of The V St .Petersburg School on The o- retical Physics, 8–12 February, 1999, Gatchina , eds. Ya.I.Azimov et al. [hep-ph/9802307].

arXiv:physics/0001020v1 [physics.gen-ph] 10 Jan 2000Why Occam’s Razor Russell K. Standish High Performance Computing Support Unit University of New South Wales Sydney, 2052 Australia R.Standish@unsw.edu.au http://parallel.hpc.unsw.edu.au/rks Abstract In this paper, I show why in an ensemble theory of the universe , we should be inhabiting one of the elements of that ensemble with least information content that satisfies the anthropic principle. This explai ns the effectiveness of aesthetic principles such as Occam’s razor in predicting us efulness of scientific theories. I also show, with a couple of reasonable assumptio ns about the phenomenon of consciousness, that quantum mechanics is the most general linear theory satisfying the anthropic principle. 03.65.Bz,01.70.+w Typeset using REVT EX 1I. INTRODUCTION Wigner [1] once remarked on “the unreasonable effectiveness of mathematics”, encap- sulating in one phrase the mystery of why the scientific enter prise is so successful. There is an aesthetic principle at large, whereby scientific theor ies are chosen according to their beauty, or simplicity. These then must be tested by experime nt — the surprising thing is that the aesthetic quality of a theory is often a good predict or of that theory’s explanatory and predictive power. This situation is summed up by William de Ockham “Entities should not be multiplied unnecessarily” known as Ockam’s Razor. We start our search into an explanation of this mystery with t heanthropic principle [2]. This is normally cast into either a weak form (that physical r eality must be consistent with our existence as conscious, self-aware entities) or a stron g form (that physical reality is the way it is because of our existence as conscious, self-aware entities). The an thropic principle is remarkable in that it generates significant constraints o n the form of the universe [2,3]. The two main explanations for this are the Divine Creator explanation (the universe was created deliberately by God to have properties sufficient to s upport intelligent life), or the Ensemble explanation [3] (that there is a set, or ensemble, of different universes, differing in details such as physical parameters, constants and even law s, however, we are only aware of such universes that are consistent with our existence). I n the Ensemble explanation, the strong and weak formulations of the anthropic principle are equivalent. Tegmark introduces an ensemble theory based on the idea that every self-consistent mathematical structure be accorded the ontological status ofphysical existence . He then goes on to categorize mathematical structures that have been dis covered thus far (by humans), and argues that this set should be largely universal, in that all self-aware entities should be able to uncover at least the most basic of these mathematic al structures, and that it is unlikely we have overlooked any equally basic mathematical structures. An alternative ensemble approach is that of Schmidhuber’s [ 4] — the “Great Program- mer”. This states that all possible programs of a universal t uring machine have physical existence. Some of these programs will contain self-aware s ubstructures — these are the programs deemed interesting by the anthropic principle. No te that there is no need for the UTM to actually exist, nor is there any need to specify whi ch UTM is to be used — a program that is meaningful on UTM 1can be executed on UTM 2by prepending it with another program that describes UTM 1in terms of UTM 2’s instructions, then executing the individual program. Since the set of all programs (infinite l ength bitstrings) is isomorphic to the set of whole numbers N, an enumeration of Nis sufficient to generate the ensemble that contains our universe. The information content of this complete set is precisely zero, as no bits are specified. This has been called the “zero inform ation principle”. In this paper, we adopt the Schmidhuber ensemble as containi ng all possible descrip- tions of all possible universes, whilst remaining agnostic on the issue of whether this is all there is.1Each self-consistent mathematical structure (member of th e Tegmark ensemble) is completely described by a finite set of symbols, and a count able set of axioms encoded in 1For example, this ensemble does not include uncomputable nu mbers — but should these peculiar mathematical beasts be accorded physical existence? 2those symbols, and a set of rules (logic) describing how one m athematical statement may be converted into another.2These axioms may be encoded as a bitstring, and the rules enco ded as a program of a UTM that enumerates all possible theorems de rived from the axioms, so each member of the Tegmark ensemble may be mapped onto a Schmi dhuber one.3. The Tegmark ensemble must be contained within the Schmidhuber o ne. An alternative connection between the two ensembles is that the Schmidhuber ensemble is a self-consistent mathematical structure, and is theref ore an element of the Tegmark one. However, all this implies is that one element of the ense mble may in fact generate the complete ensemble again, a point made by Schmidhuber in that the “Great Programmer” exists many times, over and over in a recursive manner within his ensemble. This is now clearly true also of the Tegmark ensemble. II. UNIVERSAL PRIOR The natural measure induced on the ensemble of bitstrings is the uniform one, i.e. no bitstring is favoured over any other. This leads to a problem in that longer strings are far more numerous than shorter strings, so we would conclude tha t we should expect to see an infinitely complex universe. However, we should recognise that under a UTM, some strings e ncode for identical programs as other strings, so one should equivalence class t he strings. In particular, finite strings (ones in which the bits after some bit number nare “don’t care” bits) are in fact equivalence classes of all infinite length strings that shar e the firstnbits in common. These strings correspond to halting programs under the UTM. One ca n see that the size of the equivalence class drops off exponentially with the amount of information encoded by the string. Under a UTM, the amount of information is not necessa rily equal to the length of the string, as some of the bits may be redundant. The sum PU(s) =/summationdisplay p:Ucomputessfrompand halts2|p|, (1) where |p|means the length of p, gives the size of the equivalence class of all halting progr ams generating the same output sunder the UTM U. This measure distribution is known as auniversal prior , or alternatively a Solomonoff-Levin distribution [5]. We a ssume the self- sampling assumption [6,7], essentially that we expect to find ourselves in one of t he universes with greatest measure, subject to the constraints of the ant hropic principle. This implies we should find ourselves in one of the simplest possible univers es capable of supporting self-aware substructures (SASes). This is the origin of physical law — w hy we live in a mathematical, 2Strictly speaking, these systems are called recursively en umerable formal systems, and are only a subset of the totality of mathematics, however this seem in keeping with the spirit of Tegmark’s suggestion 3In the case of an infinite number of axioms, the theorems must b e enumerated using a dovetailer algorithm. 3as opposed to a magical universe. This is why aesthetic princ iples, and Ockam’s razor in particular are so successful at predicting good scientific t heories. This might also be called the “minimum information principle”. There is the issue of what UTM Ushould be chosen. Schmidhuber sweeps this issue under the carpet stating that the universal priors differ onl y by a constant factor due to the compiler theorem, along the lines of PV(s)≥PUVPU(s) wherePUVis the universal prior of the compiler that interprets U’s instruction set in terms ofV’s. The inequality is there because there are possibly nativ eV-code programs that computesas well. Inverting the symmetric relationship yields: PUVPU(s)≤PV(s)≤(PVU)−1P(U)(s) The trouble with this argument, is that it allows for the poss ibility that: PV(s1)≪PV(s2),but P U(s1)≫PU(s2) So our expectation of whether we’re in universe s1ors2depends on whether we chose Vor Ufor the interpreting UTM. There may well be some way of resolving this problem that lead s to an absolute measure over all bitstrings. However, it turns out that an absolute m easure is not required to explain features we observe. A SAS is an information processing enti ty, and may well be capable of universal computation (certainly homo sapiens seems capable of universal computation). Therefore, the only interpreter (UTM) that is relevant to th e measure that determines which universe a SAS appears in is the SAS itself. We should expect t o find ourselves in a universe with one of the simplest underlying structures, according t o our own information processing abilities. This does not preclude the fact that other more co mplex universes (by our own perspective) may be the simplest such universe according to the self-aware inhabitants of that universe. This is the bootstrap principle writ large. III. THE WHITE RABBIT PARADOX A criticism levelled at ensemble theories is to consider uni verses indistinguishable from our own, except for the appearance of something that breaks t he laws of physics temporarily, e.g. a white rabbit is observed to fly around the room at specifi c time and place.4There are two possible explanations for this: 1. that there is some previously unknown law of physics that c aused this rather remark- able phenomenon to happen. However, it would have to be an ext remely complex law, and thus belong to a rather unlikely universe. 4This problem was first discussed in Marchal [8], where it is ca lled the Universal Dovetailer Paradox . Marchal used the term “White Rabbit” in [9], presumably in a literary reference to Lewis Carrol. 42. that there is some “glitch” or “bug” in the program governi ng the universe, that allows some of the “don’t care” bits to be interpreted. Since there a re many more ways a program can fail, than be correct, surely then, the “White Ra bbit” universes should outnumber the lawlike ones. Consider more carefully the latter scenario. In most of the u niverses where the “don’t care” bits are interpreted, the “don’t care” bits will be dev oid of information, and appear as random noise to the self-aware entity, and thus the univer se is indistinguisable from a law-like one. Only when the “don’t care” bits form a pattern r ecognisable by the self-aware entity, will a breakdown of physical laws be observed (such a s seeing a flying white rabbit). Such patterns, of course, will be sparse in the space of all su ch “don’t care” bitstrings, and so the vast majority of the pathological universes would be i ndistinguishable from the law abiding universe they approximate. Another viewpoint on this explanation is to realise that SAS es are themselves finite entities, with finite discriminatory powers and memory. The refore, the SAS imposes an interpretation filter on the data perceived from the univers e it inhabits, imposing order (or compressibility) in its interpretation of the universe , even if no such order exists. Even though incompressible strings vastly outnumber compressi ble ones, the “interpretation filter” of the SAS maps these incompressible strings onto compressi ble ones. This implies a large number of “don’t care bits” in any description of a universe, with correspondingly larger numbers of “don’t care bits” for simpler descriptions, givi ng rise to the universal prior. Any white rabbit universe must therefore take on the appeara nce of being a consequence of complicated physical law, (i.e. case 1 above) which must be r are accoding to the universal prior. Marchal’s universal dovetailer paradox is expressed somew hat differently to the preceding description of the white rabbit paradox. He assumes that all SASes are neither more nor less than universal turing machines, and concious experiences a re implemented as computations. This is a form of strong AI he calls COMP . The set of all possible computational continua- tions can be generated by the dovetailing algorithm. Since a ll such continuations exist, and bizarre experiences (eg the white rabbit) by far dominate th e numbers of continuations, the paradox is why we experience order in the world. This specification of the problem does not admit an obvious me asure on which to decide which experiences are more likely than others. However, eac h such computational continua- tion can be identified with a string from the Schmidhuber ense mble, so the universal prior is defined over the set of such experiences, and the above argume nts about the general white rabbit problem also apply to the universal dovetailer parad ox. IV. QUANTUM MECHANICS In this section, I ask the question of what is the most general (i.e. minimum information content) description of an ensemble containing self-aware substructures. Firstly, it seems that time is critical for consciousness — i.e. in order for th ere to be a “flow of consciousness”. Denote the state of an ensemble by ψ. This induces an evolution equation ∂ψ ∂t=H(ψ,t) (2) 5Now conscious observers induce a partitioning for each obse rvableA:ψ−→ {ψa,µa}, where aindexes the allowable range of “classical” observable valu es corresponding to A, andµa is the measure associated with ψa(/summationtext aµa= 15). Theψawill also, in turn, be solutions to equation (2). If we further assume that the states ψare elements of a vector space, and that the evolution equation (2) is linear, we may write ψ=/summationtext aµaψa, and the observable operators can be written compactly as a linear operator/summationtext aψaψ† a, whereψ†is the dual of ψ. Measure is clearly given by µa=ψ† aψ ψ† aψa. Furthermore, since scaling does not change the physical st ate represented by ψ, we can assert without loss of generality that∂ ∂t(ψ†ψ) = 0, implying that H=iH, whereHis Hermitian, and ψis a vector in a Hilbert space. The most general Hilbert space is one over the field of complex numbers. In shor t, by means of 3 assumptions, 2 of which appear to be irreducible properties of consciousn ess, and the third being that of linearity, Quantum Mechanics is derived from the anthropic principle applied to an ensemble. Returning then, to the issue of linearity. This is not an obvi ous requirement for anthropic universes, so must have an explanation. Weinberg [10,11] ex perimented with a possible non- linear generalisation of quantum mechanics, however found great difficulty in producing a theory that satisfied causality. This is probably due to the n onlinear terms mixing up the partitioning {ψa,µa}over time. It is usually supposed that causality [3], at leas t to a certain level of approximation, is a requirment for a self-aware sub structure to exist. V. ACKNOWLEDGEMENTS I would like to thank the following people from the “Everythi ng” email discussion list for many varied and illuminating discussions on this and relate d topics: Wei Dai, Hal Finney, Gilles Henri, James Higgo, George Levy, Alastair Malcolm, C hristopher Maloney, Jaques Mallah, Bruno Marchal and J¨ urgen Schmidhuber. 5Here, as elsewhere, we use Σ to denote sum or integral respect ively as ais discrete of continuous 6REFERENCES [1] E. P. Wigner, Symmetries and Reflections (MIT Press, Cambridge, 1967). [2] J. D. Barrow and F. J. Tipler, The Anthropic Cosmological Principle (Clarendon, Ox- ford, 1986). [3] M. Tegmark, Annals of Physics 270, 1 (1998). [4] J. Schmidhuber, in Foundations of Computer Science: Potential-Theory-Cogni tion, Vol. 1337 of Lecture Notes in Computer Science , edited by C. Freska, M. Jantzen, and R. Valk (Springer, Berlin, 1997), pp. 201–208. [5] M. Li and P. Vit´ anyi, An Introduction to Kolmogorov Complexity and its Applicati ons, 2nd ed. (Springer, New York, 1997). [6] J. Leslie, The End of the World (Routledge, London, 1996). [7] B. Carter, Phil. Trans. Roy. Soc. Lond. A310 , 347 (1983). [8] B. Marchal, in Proceedings of WOCFAI ’91 , edited by M. de Glas and D. Gabbay (Angkor, Paris, 1991), pp. 335–345. [9] B. Marchal, Technical Report No. TR/IRIDIA/95, Brussel s University (unpublished). [10] S. Weinberg, Annals of Physics 194, 336 (1989). [11] S. Weinberg, Dreams of a Final Theory (Pantheon, New York, 1992). 7

arXiv:physics/0001021v1 [physics.gen-ph] 10 Jan 2000Evolution in the Multiverse Russell K. Standish High Performance Computing Support Unit University of New South Wales Sydney, 2052 Australia R.Standish@unsw.edu.au http://parallel.hpc.unsw.edu.au February 29, 2008 Abstract In the Many Worlds Interpretation of quantum mechanics, the range of possible worlds (or histories) provides variation, and t he Anthropic Principle is a selective principle analogous to natural sel ection. When looked on in this way, the “process” by which the laws and cons tants of physics is determined not too different from the process that gave rise to our current biodiversity, i.e. Darwinian evolution. This h as implications for the fields of SETI and Artificial Life, which are based on a p hilosophy of the inevitability of life. 1 Introduction TheMany Worlds Interpretation (MWI)[4] of Quantum Mechanics has become increasingly favoured in recent years over its rivals, with a recent straw poll of eminent physicists[14, pp170–1] showing more than 50% su pport it. David Deutsch[5] provides a convincing argument in favour of MWI, and the multi- verse in the title is due to him. Tegmark[13] has somewhat waggishl y suggested that a Principle of Plenitude (alternatively All Universes Hypothesis — AUH), coupled with the Anthropic Principle1[3] could be the ultimate theory of ev- erything (TOE). Tegmark’s Plenitude consists of all mathematically consistent logical systems, the principle of plenitude according each of these systems phys- ical existence, however by the anthropic principle, we shou ld only expect to find ourselves in a system capable of supporting self-aware substructures (SAS), 1In the all universes hypothesis, the anthropic principle ac ts to select those universes that are “interesting”, i.e. capable of supporting self aware co nsciousness. In this ensemble picture, the distinction between the weak and strong forms of the anth ropic principle is meaningless. 1i.e. conciousness. Alternative Plenitudes have been sugge sted, for example Schmidh¨ uber’s all possible programs for a universal turin g machine. I have ar- gued elsewhere[12], that the quantum mechanical subset of t he Plenitude, i.e. the multiverse, by far dominates the set of systems supporti ng SASes. In this paper, we accept the MWI as a working hypothesis, and c onsider what the implications are for evolutionary systems. An evol utionary system consists of a means of producing variation, and a means of sel ecting amongst those variations (natural selection). Now variations are p roduced by chance, in the MWI picture, this corresponds to a branching of histor ies, whereby a particular entity’s offspring will have different forms in di fferent histories. The measure of each variant is related to the proportions in whic h the variants are formed, and the measure of each variant evolves in time throu gh a strictly deterministic application of Schr¨ odinger’s equation. What, then, determines which organisms we see today, given t hat a priori, any possible history, and hence any mix of organisms may corr espond to our own? Is natural selection completely meaningless? The first principle we need to apply is the anthropic principl e, i.e. only those histories leading to complex, self-aware substructu res will be selected. We also need to apply the self sampling assumption , namely that we expect to find ourselves in an anthropic principle consistent history tha t is nearly maximal in its measure. This, then, gives an alternative interpretati on of natural selection as being the process that differentiates the measure attribu ted to each variant. 2 Complexity Growth in Evolution As I argued elsewhere[12], lawful universes with simple ini tial states by far dominate the set consistent with the AP. This implies that th e AP fixes the end point of our evolutionary history (existence of complex, se lf-aware organisms), and the SSA fixes the beginning (evolutionary history is most likely started with the simplest organisms). We should expect to see an increase in complexity through time in a system governed by these two principles. What about systems not governed by the anthropic principle? Examples include extra terrestrial life (within our own universe, if it exists) and artificial life systems. Proponents of SETI (the Search for Extra-Terrestrial Intelligence ) believe in an inevitability of the evolution of intelligent life, given the laws of physics. The anthropic principle does indeed ensure that th e laws of physics are compatible with the evolution of intelligence, but does not mandate that this should be likely (excepting, obviously in our own case). Han son[6] has studied a model of evolution based on easy and hard steps to make predic tions about what the distribution of such steps should be within the fossil re cord. He finds that the fossil record is consistent with there being 4–5 hard steps i n getting to intelligent life on Earth. By hard steps, he means steps who’s expected du ration greatly exeeds the present age of the universe. This would imply that intelligent life is 2fairly unique within our own universe, to the chagrin of the S ETI proponents. Of course, it is also true that a single example of extra terre strial intelligence would be an important counterexample to these arguments bas ed on the AP and SSA, so SETI is by itself not a fruitless exercise. It is pr obable, however, that extraterrestrial life, whilst ubiquitous, will be no m ore complex than the most primitive prokaryote, as suggested by the recent Marti an meteorite finds. Likewise, for artificial life, it would seem plausible that a serious of easy and hard steps are required to climb the complexity ladder. Alre ady, the first such hard transition (the creation of replicators from the prime val soup) has been observed[8, 7], but equivalents of other transitions (eg tr ansition to sexual re- production, prokaryote to eukaryote or multicellularity) have not been observed to date. Ray is leading a major experiment designed to probe t he transition to multicellularity[9, 10] — success in this experiment wil l provide remarkable constraints on just how finely tuned the physics and chemistr y needs to be in order for the system to pass through a hard transistion. Adami[1, 2] and co-workers examined the Avida alife system for evidence of complexity growth during evolution. They did find this, alth ough this is largely seen as the artificial organisms learning how to solve arithm etic problems that have been imposed artificially on the system. An analogous st udy by myself[11] of Tierra showed no such increase in complexity over time — if anything the trend was to greater simplicity. This work is still in progre ss. 3 Evolutionary Physics? Returning back to the picture of the “All Universes Hypothes is”, we can see that our current universe is made up from contingency and nec essity. The ne- cessity comes from the requirements of the anthropic princi ple, however when a particular aspect of the universe is not constrained by the AP, its value must be decided by chance (according to the SSA) the first time it is “measured” by self-aware beings (this measurement may well be indirect — properties of the microscopic or cosmic worlds will need to be consistent w ith our everyday observations at the macroscopic level, so may well be determ ined prior to the first direct measurements). Evolution is also described as a mixture of con- tingency and necessity. When understood in terms of the AP su pplying the necessary, and the SSA supplying the rationale for resolvin g chance, the con- nection between the selection of phyical laws and the select ion of organisms in evolution is made clear. It is as though the laws of physics an d chemistry have themselves evolved. Perhaps applying evolutionary princi ples to the underlying physico-chemical laws of an alife system will result in an al ife system that can pass through these hard transitions. References 3[1] Chris Adami. Introduction to Artificial Life . Springer, 1998. [2] Chris Adami. Physical complexity of symbolic sequences .Physica D , 1999. To appear. [3] J. D. Barrow and F. J. Tipler. The Anthropic Cosmological Principle . Clarendon, Oxford, 1986. [4] Bryce de Witt and R. Neill Graham. The Many Worlds Interpretation of Quantum Mechanics . Princeton UP, 1973. [5] David Deutsh. The Fabric of Reality . Penguin, 1997. [6] Robin Hanson. Must early life be easy? the rythm of ma- jor evolutionary transitions. Origins of Life , 2000. submitted; http://hanson.gmu.edu/hardstep.pdf. [7] A. N. Pargellis. The evolution of self-replicating comp uter organisms. Phys- ica D, 98:111, 1996. [8] A. N. Pargellis. The spontaneous generation of digital l ife.Physica D , 91:86, 1996. [9] Tom Ray. A proposal to create two biodiversity reserves: One digi- tal and one organic. See ftp://tierra.slhs.udel.edu/tier ra/doc/reserves.tex, http://www.hip.atr.co.jp/˜ray/pubs/reserves/reserve s.html. Also see New Scientist, vol 150, no 2034, pp32–35. [10] Tom Ray and Joseph Hart. Evolution of differentiated mul ti-threaded digi- tal organisms. In Chris Adami, Richard Belew, Hiroaki Kitan o, and Charles Taylor, editors, Artificial Life VI , pages 295–304, Cambridge, Mass., 1998. MIT Press. [11] Russell K. Standish. Some techniques for the measureme nt of complexity in Tierra. In Advances in Artificial Life , volume 1674 of Lecture Notes in Computer Science , page 104, Berlin, 1999. Springer. [12] Russell K. Standish. Why Occam’s razor? Physics Review , 2000. submit- ted. [13] Max Tegmark. Is ”the theory of everything” merely the ul timate ensemble theory. Annals of Physics , 270:1–51, 1998. [14] Frank J. Tipler. The Physics of Immortality . Doubleday, 1994. 4

arXiv:physics/0001022v1 [physics.chem-ph] 10 Jan 2000Quantum Monte Carlo Methods in Statistical Mechanics Vilen Melik-Alaverdian and M.P. Nightingale Department of Physics, University of Rhode Island, Kingsto n, Rhode Island 02881, USA (January 15, 2014) Abstract This paper deals with the optimization of trial states for th e computation of dominant eigenvalues of operators and very large matrice s. In addition to preliminary results for the energy spectrum of van der Waa ls clusters, we review results of the application of this method to the com putation of relaxation times of independent relaxation modes at the Isi ng critical point in two dimensions. I. INTRODUCTION The computation of eigenvalues and eigenstates of operator s and large matrices is a ubiquitous problem. In this paper we review recent applicat ions of the Quantum Monte Carlo methods that we have developed for this purpose. The re ader is referred to other papers for introductory or more technical discussions of ea rlier work. [1–4] II. MATHEMATICAL PRELIMINARIES For an operator G, the power method can be used to compute the dominant eigenst ate and eigenvalue, |ψ0∝angb∇acket∇ightandλ0. This well-know procedure can be summarized as follows: 1. Choose a generic initial state |u(0)∝angb∇acket∇ightof the appropriate symmetry. 2. Iterate: |u(t+1)∝angb∇acket∇ight=1 ct+1G|u(t)∝angb∇acket∇ight, (1) wherectputs|u(t)∝angb∇acket∇ightin standard form. For projection time t→ ∞ the following is almost always true: 1. Eigenstate: |u(t+1)∝angb∇acket∇ight → |ψ0∝angb∇acket∇ight (2) 12. Eigenvalue: ct→λ0 (3) To see this, expand the initial state in normalized eigensta tes |u(0)∝angb∇acket∇ight=/summationdisplay kw(0) k|ψk∝angb∇acket∇ight (4) with spectral weights w(0) k. Then |u(t)∝angb∇acket∇ighthas spectral weights w(t) k∼/parenleftiggλk λ0/parenrightiggt . (5) This method can be implemented by means of a Monte Carlo metho d and, unlike varia- tional Monte Carlo, it has the advantage of producing unbias ed results for large projection timest. The disadvantage is, however, that at the same time the stat istical noise increases exponentially, unless Gis a Markov (stochastic) matrix, or can be explicitly transf ormed to one. The statistical errors grow with the extent to which Gfails to conserve probility, and to alleviate this problem, approximate dominant eigenstat es can be used. In the case of Markov matrices, computation of the dominant e igenvalue is of no interest, since it is equals unity, but sampling the corresponding eig enstate has numerous applications. A. Subspace iteration Given a set of basis states, one can construct trial states as linear combinations to obtain approximate excited or, more generally, sub-domina nt states and the corresponding eigenvalues. These are computed by solving a linear variati onal problem. In a Monte Carlo context, the Metropolis method can be used to evaluate the required matrix elements. Subsequently, a variation of the power method can again be us ed to remove systematically the variational bias. [5–7] Again, the price to be paid for re duction of the variational bias is increased statistical noise, a problem that can be mitiga ted by the use of optimized trial states. The linear variational problem to be solved for the computat ion of excited states is the following one. Given nbasis functions |ui∝angb∇acket∇ight, find then×nmatrix of coefficients D(j) isuch that |˜ψj∝angb∇acket∇ight=n/summationdisplay i=1D(j) i|ui∝angb∇acket∇ight (6) are the “best” variational approximations for the nlowest eigenstates |ψi∝angb∇acket∇ightof some Hamilto- nianH. In this problem we shall, at least initially, use the langua ge of quantum mechanical systems, where one has to distinguish the Hamiltonian from t he imaginary-time evolution operatorG= exp( −τH). In the statistical mechanical application discussed bel ow, we shall encounter only the equivalent of the latter, which is t he Markov matrix governing the stochastic dynamics. In the expressions to be derived below , the substitution HGp→Gp+1 2will produce the expressions required for the statistical m echanical application, at least if we assume that the non-symmetric Markov matrix that appear i n that context has been symmetrized, which can always be accomplished if detailed b alance is satisfied. Given these basis states, one seeks the “best” solution to th e linear variational problem in Eq. (6) in the sense that for all ithe Rayleigh quotient ∝angb∇acketleft˜ψi|H|˜ψi∝angb∇acket∇ight/∝angb∇acketleft˜ψi|˜ψi∝angb∇acket∇ightis stationary with respect to variation of the coefficients of the matrix D. The solution is that the matrix of coefficients D(j) ihas to satisfy the following generalized eigenvalue equati on n/summationdisplay i=1HkiD(j) i=˜Ejn/summationdisplay i= 1NkiD(j) i, (7) where Hki=∝angb∇acketleftuk|H|ui∝angb∇acket∇ight,andNki=∝angb∇acketleftuk|ui∝angb∇acket∇ight. (8) We note a number of important properties of this scheme. Firs tly, the basis states |ui∝angb∇acket∇ightin general are not orthonormal. Secondly, it is clear that any n onsingular linear combination of the basis vectors will produce precisely the same results, o btained from the correspondingly transformed version of Eq. (7). The final comment is that the v ariational eigenvalues bound the exact eigenvalues from above, i.e.,˜Ei≥Ei, where we assume E1≤E2≤.... One recovers exact eigenvalues Eiand the corresponding eigenstates, if the |ui∝angb∇acket∇ightspan the same space as the exact eigenstates, or in other words, have no adm ixtures of more than nstates. The required matrix elements can be computed using the stand ard variational Monte Carlo method. The power method can subsequently be used to re duce the variational bias. Formally, one simply defines new basis states |u(p) i∝angb∇acket∇ight=Gp|ui∝angb∇acket∇ight (9) and substitutes these new basis states for the original ones . In quantum mechanical appli- cations, where G= exp( −τH), the corresponding matrices H(p) ki=∝angb∇acketleftu(p) k|H|u(p) i∝angb∇acket∇ight (10) and N(p) ki=∝angb∇acketleftu(p) k|ui(p)∝angb∇acket∇ight (11) can be computed by pure-diffusion Monte Carlo. [8] We note tha t, Monte Carlo yields these matrix elements up to an irrelevant overall normalization c onstant. As an explicit example illustrating the nature of the Monte C arlo time-averages that one has to evaluate in this approach, we write down the expres sion forN(p) ijas used for the computation of eigenvalues of the Markov matrix relevant to the problem of critical slowing down, discussed in detail in the next section. One estimates this matrix as N(p) ij∝/summationdisplay tui(St) ψB(St)uj(St+p) ψB(St+p), (12) where the Stare configurations forming a time series that is designed to s ample the distri- bution of a system in thermodynamic equilibrium, i.e.,the Boltzmann distribution ψ2 B. It 3turns out that in this particular case, this distribution, t he dominant eigenstate, has suffi- cient overlap with the magnitude of the sub-dominant states so that one can compute all matrix elements N(p) ijsimultaneously without introducing a separate guiding fun ction [5]. The expression given in Eq. (12) yields the u/ψB-auto-correlation function at lag p. The expression for H(p) ijis similar, and represents a cross-correlation function in volving the configurational eigenvalues of the Markov matrix in the vari ous basis states. Compared to the expressions one usually encounters in applications t o quantum mechanical problems, Eq. (12) takes a particularly simple form in which products o f fluctuating weights are absent, because one is dealing with a probability conserving evolut ion operator from the outset in this particular problem. III. UNIVERSAL AMPLITUDE RATIOS IN CRITICAL DYNAMICS Before continuing our general discussion, we temporarily c hange the topic to introduce stochastic dynamics of critical systems. What make such sys tems interesting, is that one can distinguish universality classes in which behavior doe s not depend on many of the microscopic details. For static critical phenomena, it is k nown that universality classes can be identified by dimensionality, symmetry of the order param eter, and the range of the interactions. For dynamical phenomena, there are addition al features such as whether or not the dynamics is local or subject to conservation laws. On approach of a critical point, the correlation length ξdiverges. The dynamical ex- ponentzgoverns the corresponding divergence of the correlation ti meτby means of the relationτ∝ξz. Since the critical exponent zis one of the universal quantities, it has been used to identify universality classes. Unfortunately ,zdoes not vary by much from one universality class to another, and this poses a serious comp utational problem in terms of the accuracy required to obtain significant differences. One of the outcomes of the work reviewed here is that there are other quantities within comp utational reach, namely univer- sal amplitude ratios. [4] These ratios may serve as addition al, and possibly more sensitive identifiers of universality classes. We shall consider vari ous systems belonging to a single universality class, and we assume that the representatives of the class are parameterized by κ. If a thermodynamic system is perturbed out of equilibrium, d ifferent thermodynamic quantities relax back at a different rates. More generally, t here are infinitely many indepen- dent relaxation modes for a system in the thermodynamic limi t. The Monte Carlo methods reviewed here have been used to compute relaxation times of I sing models on square L×L lattices at the critical point. [4] Let us denote by τκi(L) the relaxation time of mode iof a system of linear dimension L. As indeed scaling theory suggests, it turns out that the rel axation time has the following factorization property τκi(L)≈mκAiLz, (13) wheremκis anon-universal metric factor, which differs for different representatives o f the same universality class as indicated; Aiis auniversal amplitude which depends on the mode i; andzis the universal dynamical exponent introduced above. 4Formulated as a computational problem, one has the followin g. Suppose S= (s1,...,s L2), withsi=±1, is a spin configuration and ρt(S) is the probability of finding S at timet. The probability distribution evolves in time according to ρt+1(S) =/summationdisplay S′P(S|S′)ρt(S′). (14) The detailed structure of the Markov matrix Pis of no immediate importance for the current discussion. All that matters is that it satisfies det ailed balance, has the Boltzmann distribution ψ2 Bas its stationary state. Also, Pis a single-spin flip matrix, i.e.P(S|S′) vanishes if SandS′differ by more than a single spin. The desired relaxation time of mode iis given by τi(L) =−L−2/lnλi(L), (15) whereλiis an eigenvalue of Markov matrix P. We obtained the previous expression by assuming a single-spin flip Markov matrix, so that the L2in the denominator produces a relaxation time measured in units of sweeps, i.e.flips per spin. IV. TRIAL STATE OPTIMIZATION To verify Eq. (13), it is important to obtain estimates that a re exact within the range of the estimated error. For this purpose we use a set of optimize d variational basis functions, to which we subsequently apply the projection procedure des cribed in Section 2 to remove the variational bias. As mentioned, the Monte Carlo projection increases the stat istical noise, and the solution to this problem is to improve the variational basis function s. We shall now discuss how this is done and we consider the problem using the language of the S chr¨ odinger equation. We first consider the ground state and review how one can optim ize a many-, say 50- parameter trial function ψT(R). [9] The local energy E(R) is defined by HψT(R)≡ E(R)ψT(R). (16) The variance of the local energy is given by χ2=∝angb∇acketleft(H −E)2∝angb∇acket∇ight=/integraldisplay |ψT(R)|2[E(R)−E]2dR//integraldisplay |ψT(R)|2dR. (17) A property that we shall exploit later is that χ2= 0 for any eigenstate, not just the ground state. The following sums up the Monte Carlo optimization procedur e for a single state: 1. SampleR1,...,R sfromψT2a typical sample size has s≈3,000. 2. Approximate the integrals in Eq. (17) by Monte Carlo sums. 3. Minimize χ2as follows, while keeping this sample fixed. For each member of the sample R1,...,R s: 54. Compute ψT(R1),...,ψ T(Rs). 5. Compute HψT(R1),...,HψT(Rs). 6. Find Efrom least-squares fit of HψT(Rσ) =EψT(Rσ), σ= 1,...,s. (18) 7. Minimize the sum of squared residues of Eq. 18.1 This procedure can be generalized immediately to a setof basis functions, as required to implement Eq. (6). The only new ingredient is a guiding fun ctionψ2 gthat has sufficient overlap with all basis states used in the computation. For th is purpose one can conveniently use the groundstate raised to some appropriate power less th an unity. This yields the following algorithm to optimize basis state s forndominant eigenvalues: 1. SampleR1,...,R sfromψ2 g. 2. Compute the arrays  u(1)(R1) u(2)(R1) ... ,..., u(1)(Rs) u(2)(Rs) ... . (19) 3. Compute the arrays  Hu(1)(R1) Hu(2)(R1) ... ,..., Hu(1)(Rs) Hu(2)(Rs) ... . (20) 4. Find the matrix elements Eijfrom the appropriately weighted least-squares fit to Hu(i)(Rσ) =n/summationdisplay j= 1Eiju(j)(Rσ), σ= 1,...,s. (21) 5. Vary the parameters to optimize the fit, as explained below . In case of a perfect fit, the eigenvalues of the truncated Hami ltonian matrix E= (Eij)n i,j=1are the required eigenvalues, but in real life one has to opti mize the parameters of the basis functions, which can be done as follows: 1. Divide the sample in blocks and compute one Hamiltonian ma trixEper block. 1Once the parameters are changed from the values they had in st ep 1, one should use an appro- priately weighted sum of squared residues. [9] 62. Minimize the variance of the E-spectra over the blocks. The variance vanishes if the basis functions u(i)arelinear combinations ofneigenstates of H. This gives rise to a computational problem, viz., the variance is near-invariant under linear transformation of the u(i). This approximate “gauge invariance” gives rise to near-si ngular, non-linear optimization problem. This can be avoided by sim ultaneously minimizing the variance of both the spectrum of the “local” Hamiltonian mat rixEand the local energy E of the individual basis functions. Finally, the variational bias of the eigenvalue estimates o btained with the optimized basis states is reduced by using Monte Carlo to make the substituti on discussed previously |u(i)∝angb∇acket∇ight →e−Hτ|u(i)∝angb∇acket∇ight. (22) For this purpose, one has to use the short-time approximatio n of exp( −Hτ). [5] To apply the preceding scheme to the problem of critical dynamics, al l one has to do is to make use of the fact the analog of the quantum mechanical evolution is the symmetrized Markov ˆP of stochastic dynamics, which is defined as ˆP(S|S′) =1 ψB(S)P(S|S′)ψB(S′), (23) in terms of which we have the correspondence e−Hτ→ˆPt. (24) V. XE TRIMER: A TEST CASE As an example that illustrates the accuracy one can obtain by means of the optimization schemes discussed above, we present results for a Xe trimer i nteracting via a Lennard-Jones potential. To be precise, we write the Hamiltonian of this sy stem in reduced units as H=−1 2m∇2+/summationdisplay i<j(r−6 ij−2)r−6 ij, (25) where therijdenote the dimensionless interparticle distances. We define Xe to correspond tom−1= 7.8508×10−5, which probably to four significant figures [10] agrees with L eitner et al.. [11] Table I shows results for variational energies of the lowest five completely symmetric states of a Lennard-Jones Xe trimer. The results are compare d with results obtained by the discrete variable representation truncation-diagonaliz ation method. [11] The basis functions used in this computation are of the same general form used in e arlier work with an additional polynomial prefactor for excited states. [12,13] Clearly, we obtain consistently lower reduced energies, wh ich we attribute to lack of convergence of the results of Leitner et al. [14] 7TABLES k Ek σ Leitner et al. 0 -2.845 241 50 1×10−8-2.844 1 -2.724 955 8 1×10−7-2.723 2 -2.675 065 1×10−6-2.664 3 -2.608 612 2×10−6-2.604 4 -2.592 223 3×10−6-2.580 TABLE I. Variational reduced energies compared with estima tes of Leitner et al. VI. CRITICAL POINT DYNAMICS: RESULTS Next we briefly address the issue of the choice of trial functi ons for the eigenstates of symmetrized Markov matrix ˆP. We write u(S) =f(S)×ψB(S). (26) For the modes we considered, f(S) was chosen to be a rotationally and translationally invariant polynomial in long-wavelength Fourier componen ts ofS, the lowest-order one of which is simply the magnetization. Corresponding to the ord er parameter and energy-like modes, we considered polynomials either odd or even under th e transformation S→ −S. We briefly discuss some of the results that illustrate the val idity of Eq. (13). Figure 1. shows plots of the effective amplitudes for the three dominan t odd, and two dominant even modes of three different Ising models on L×Llattices. Of the three Ising models we studied, the first one, the NN model, had nearest-neighbor co uplings only. The other two also had next-nearest-neighbor couplings. In one of them, t he equivalent neighbor or EQN model, both couplings were of equal ferromagnetic strength s. In the third or NEQ model, the nearest-neighbor coupling was chosen ferromagnetic an d of twice the magnitude of the antiferromagnetic next-nearest-neighbor coupling. 8FIGURES 0.010.1110 4 6 8 10 12 14 16 18 20ALi L✸✸✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ +++++ + + + + + + + + + + + ✷✷✷✷✷✷✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷××××××××× × × × × × × × △△△△△△△△△△ △ △ △ △ △ △⋆⋆⋆ ⋆⋆ ⋆ ⋆⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ❜❜❜❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜❜ ❝❝❝❝❝❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❡❡❡❡❡❡❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡rrrrrrr r r r r r r r r rsssss s s s s s s s s ss s ✉✉✉✉✉✉✉ ✉ ✉ ✉ ✉✉✉✉✉✉ ✸✸✸✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ +++++ + + + + + + + + ++ +✷✷✷✷✷ ✷ ✷ ✷ ✷ ✷✷✷ ✷ ✷ ✷ ✷ FIG. 1. Universality of relaxation-time amplitudes, shown in a plot of the effective, size-dependent amplitudes ALion a logarithmic scale. To separate data points for the three mod- els, the NEQ data were displaced to the left and the EQN data to the right. The data collapse predicted by Eq. (13) was produced by fitting the metric facto rs of the NN and NEQ models. Amplitudes of odd and even states alternate in magnitude. To obtain estimates of the amplitudes of the relaxation mode s, we fit the computed correlation times to expressions of the form τi(L)≈Lznc/summationdisplay k= 0αkiL−2k. (27) In our computation of the non-universal metric factors, thi s quantity was set equal to unity by definition for the EQN model. Table II shows the metri c factors computed for each mode separately as the ratio of the computed amplitudes. In a greement with the scaling prediction in Eq. (13), the computed metric factors depend o nly on the model but not on the mode. TABLE II. Non-universal metric factors mκ, as defined in Eq. (13), computed for the NN and NEQ models. The modes indicated by o1, o2, and o3 are odd under spin inversion; the remaining two, e2 and e3, are even. NEQ NN o1 2.389(1) 1.5569 (5) e2 2.394(2) 1.5569 (5) o2 2.393(2) 1.5567 (6) e3 2.391(2) 1.554 (2) o3 2.385(4) 1.554 (2) 9Finally we mention that the spectral gaps of the Markov matri x vary over a considerable range 1−λi(L)≈L−(d+z)≈L−4.17, (28) i.e.from approximately 3 ×10−3forL= 4 to 3 ×10−6forL= 21. For details of the numerical analysis based on Eq. (27) we refer the interested reader to Ref. [4]. Suffice it to mention that the value obtained for the universal dynamic cr itical exponent zfeatured in Eq. (13) is z= 2.167±0.002 which is indistinguishable from 13/6. ACKNOWLEDGMENTS This work was supported by the (US) National Science Foundat ion through Grants DMR-9725080 and CHE-9625498. It is a pleasure to acknowledg e helpful e-mail exchanges with David Leitner. 10REFERENCES [1] M.P. Nightingale and C.J. Umrigar, Monte Carlo Eigenvalue Methods in Quantum Me- chanics and Statistical Mechanics, in Advances in Chemical Physics, Vol. 105, Monte Carlo Methods in Chemistry, edited by David M. Ferguson, J. I lja Siepmann, and Don- ald G. Truhlar, series editors I. Prigogine and Stuart A. Ric e, Chapter 4 (John Wiley and Sons, New York 1999). [2] M.P. Nightingale and C.J. Umrigar (eds.), Quantum Monte Carlo methods in Physics and Chemistry, NATO Science Series, Series C: Mathematical and Physical Sc iences - Vol. 525 (Kluwer Academic Publishers, Dordrecht, 1998). [3] M. P. Nightingale and H.W.J. Bl¨ ote, Phys. Rev. Lett. 76, 4548 (1996). [4] M. P. Nightingale and H.W.J. Bl¨ ote, Phys. Rev. Lett. 80, 1007 (1998). Also see http://xxx.lanl.gov/abs/cond-mat/9708063. [5] D.M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988). [6] B. Bernu, D.M. Ceperley, and W.A. Lester, Jr., J. Chem. Phys. 93, 552 (1990). [7] W.R. Brown, W.A. Glauser, and W.A. Lester, Jr., J. Chem. Phys. 103, 9721 (1995). [8] M. Caffarel and P. Claverie, J. Chem. Phys. 88, 1088 (1988); ibid.p. 1100. Also see S. Baroni and S. Moroni in Ref. [2]. [9] C.J. Umrigar, K.G. Wilson, and J.W. Wilkins, Phys. Rev. Lett. 60, 1719 (1988); Computer Simulation Studies in Condensed Matter Physics , edited by D.P. Landau, K.K. Mon, and H.-B. Sch¨ uttler, Springer Proceedings in Phy sics Vol. 33 (Springer- Verlag, Berlin, 1988), p.185. [10] This number was obtained by using mass and Lennard-Jone s parameters of Ref. [11], as given in Table I and the fundamental physical constants of E.R. Cohen and B. N. Taylor, in the supplement to the August 1999 issue of Physics Today, BG5. [11] D.M. Leitner, J.D. Doll, and R.M. Whitnell, J. Chem. Phys. 94, 6644 (1991). [12] Andrei Mushinski and M. P. Nightingale, J. Chem. Phys. 101, 8831 (1994). [13] M. Meierovich, A. Mushinski, and M.P. Nightingale, J. Chem. Phys. 105, 6498 (1996). [14] Indeed recent, improved computations by Leitner (priv ate communication) produce lower energies than the results quoted in Table II of Ref. [11 ]. 11

arXiv:physics/0001023v1 [physics.ins-det] 11 Jan 2000Measurement of mechanical vibrations excited in aluminium resonators by 0.6 GeV electrons. G.D. van Albadaa, E. Cocciab, V. Fafonec, H. van der Graafd, G. Heijboerd, J.W. van Holtend, W.J. Kasdorpd, J.B. van der Laand, L. Lapik´ asd, G. Mazzitellic, G.J.L. Noorend, C.W.J. Noteboomd, J.E.J. Oberskid, G. Pallottinoe, H.Z. Peekd, F. Rongac, A. Schimmeld, T.G.B.W. Sluijkd, P. Stemand, J. Venemad, P.K.A. de Witt Hubertsd. a) Dept. of Computer Science, Univ. van Amsterdam;, The Neth erlands b) Dept. of Physics, Univ. of Rome, ”Tor Vergata”, and INFN, I taly; c) Lab. Nazionale di Frascati, INFN, Italy; d) NIKHEF, P.O.B. 41882, 1009 DB Amsterdam, The Netherlands ; e) Dept. of Physics, Univ. of Rome ”La Sapienza”, and INFN, It aly email: J.Oberski@nikhef.nl February 2, 2008 NIKHEF 99-036 To be published by Review of Scientific Instruments, May 2000 Abstract We present measurements of mechanical vibrations induced b y 0.6 GeV elec- trons impinging on cylindrical and spherical aluminium res onators. To monitor the amplitude of the resonator’s vibrational modes we used piez oelectric ceramic sen- sors, calibrated by standard accelerometers. Calculation s using the thermo-acoustic conversion model, agree well with the experimental data, as demonstrated by the specific variation of the excitation strengths with the abso rbed energy, and with the traversing particles’ track positions. For the first longit udinal mode of the cylindri- cal resonator we measured a conversion factor of 7.4 ±1.4 nm/J, confirming the model value of 10 nm/J. Also, for the spherical resonator, we found the model values for the L=2 and L=1 mode amplitudes to be consistent with our measurement. We thus have confirmed the applicability of the model, and we not e that calculations based on the model have shown that next generation resonant m ass gravitational wave detectors can only be expected to reach their intended u ltra high sensitivity if they will be shielded by an appreciable amount of rock, where a veto detector can reduce the background of remaining impinging cosmic rays eff ectively. 04.80.Nn 07.07.Df 07.64.+z 07.77.Ka 29.40.Gx 29.40.Wk 43. 20.Ks 43.35.Ud 95.55.Ym 96.40.Vw 96.40.z1 Introduction A key issue for a Resonant Mass Gravitational Wave Detector [ 1] of improved sen- sitivity with respect to the existing detectors, is the back ground due to impinging cosmic ray particles [2, 3]. The energy deposited in the dete ctor’s mass along a particle’s track may excite the very vibrational modes that are to signal the pass- ing of a gravitational wave. Computer simulations of such eff ects are based on the thermo-acoustic conversion model and earlier measurement s of resonant effects in Beron et al. [4] and Grassi Strini et al. [5]. According to the model, the energy de- posited by a traversing particle heats the material locally around the particle track, which leads to mechanical tension and thereby excites acous tic vibrational modes [6]. At a strain sensitivity of the order of 10−21envisaged for a next generation gravitational-wave detector, computer simulations [3, 7] show that operation of the instrument at the surface of the earth would be prohibited by the effect of the cos- mic ray background. Since the applicability of the thermo-a coustic conversion model would thus yield an important constraint on the operating co nditions of resonant mass gravitational wave detectors, Grassi-Strini, Strini and Tagliaferri [5] measured the mechanical vibrations in a bar resonator bombarded by 0. 02 GeV protons and 5*10−4GeV electrons. We extended that experiment by measuring the excitation patterns in more detail for a bar and a sphere excited by 0.6 Ge V electrons. Even though we cannot think of a reason why the model, if applicabl e to the bar, would not hold for a sphere, we did turn to measuring with a sphere al so. We exposed [8] two aluminium 50ST alloy cylindrical bars and an aluminium alloy sphere, each equipped with piezoelectric ceramic sensors, to a beam of ≈0.6 GeV electrons used in single bunch mode with a pulse width of up to ≈2µs, and ad- justable intensity of 109to 1010electrons. We recorded the signals from the piezo sensors, and Fourier analysed their time series. Before and after the beam run we calibrated the sensor response of one of the bars for its first longitudinal vibrational mode at ≈13 kHz to calibrated accelerometers. 2 Experiment setup and method In the experiment we used three different setups in various ru ns, as summarised in table I: two bars and a sphere. With the un-calibrated bar BU w e explored the feasibility of the measurement. Also, bar BU proved useful t o indirectly determine the relative excitation amplitudes of higher longitudinal vibrational modes, see sec. 4.1. With bar BC calibrated at its first longitudinal vibrati onal mode, we measured directly its excitation amplitude in the beam. Finally, wit h the sphere we further explored the applicability of the model.Table I. Characteristics of our setup. Setup code name: BC BU SU Resonator type: bar bar sphere Diameter: 0.035 m 0.035 m 0.150 m Length: 0.2 m 0.2 m - Suspension: plastic string plastic string brass rod 0.15 m*0.002 m Piezo sensors: 1 2 2 Piezo hammer: 0 1 1 Capacitor driver 1 0 0 Direct calibration yes no no Beam energy 0.76 GeV 0.62 GeV 0.35 GeV Beam peak current 3 mA 18 mA 19 mA Electrons per burst ≈109≈5·1010≈5·1010 Mean absorbed energy per electron 0.02 GeV 0.02 GeV 0.1 GeV Typical absorbed energy per burst 0.01 J 0.6 J 3.0 J 2.1 Electron beam We used the Amsterdam linear electron accelerator MEA [9, 10 ] delivering an elec- tron beam with a pulse-width of up to 2 µs in its hand-triggered, single bunch mode. The amount of charge per beam pulse was varied, recorde d by a calibrated digital oscilloscope, photographed and analysed off line to determine the number of impinging electrons per burst. 2.2 Suspension and positioning In both setups BC and BU, see fig. 1, the cylindrical aluminium bar was horizontally suspended in the middle, as indicated in the figure, with a pla stic string. The bar’s Figure 1: Calibrated bar setup BC. The beam traverses the bar ’s front side perpendicular to the drawing plane. cylinder axis was positioned at 900to the beam direction. The bar’s suspension string was connected to a horizontally movable gliding cons truction, enabling us tohandle the resonator by remote control, and let the impingin g electron beam hit it at different horizontal positions. The aluminium sphere SU, se e fig. 2, was suspended from its centre by a brass rod. Either the bar’s gliding const ruction, or the sphere’s suspension bar, was attached to an aluminium tripod mounted inside a vacuum chamber [9], which was evacuated to about 10−5mbar. By remote control, we rotated the tripod and moved it vertically to either let the b eam pass the resonator completely, or let it traverse the resonator. We let the beam traverse the sphere at different heights and different incident angles with respe ct to the piezo sensors positions on the sphere. We mark the beam heights as E (Equato r) and A (Africa) Figure 2: The spherical resonator setup, SU. at 0.022 m below the equator. The E beam passed horizontally t hrough the sphere’s origin, remaining in the same vertical plane for the A beam. 2.3 Sensors and signal processing In setup BC we used a single piezo sensor of ≈15∗3∗1 mm3and glued it over it full length at 0.01 m off the centre on the top of the bar. Bar BC w as equipped with a capacitor plate of 0.03 m diameter at a distance of ≈0.004 m from one of its end faces. In setup BU one piezo sensor of ≈3∗6∗0.3 mm3was fixed on one end face of the bar. A similar sensor of about the same dimensions was fixed in the s ame manner, ori- ented parallel to the cylinder’s long axis at a position 35 mm away from the end-face, In the third setup, SU, see fig. 2, two piezo sensors of ≈3∗6∗0.3 mm3were glued to the sphere’s surface. One was situated at the equator, wit h respect to the vertical rotation axis, the other one at a relative displacement of 450west longitude, and at 450north latitude. For the setup in use, each sensor was connected to a charge amp lifier of ≈2∗1010V/C gain. The signals were sent through a Krohn-Hite 3202R low-p ass 100 kHz pre-filter,to a R9211C Advantest spectrum analyser with internal 2 MHz p re-sampling and 125 kHz digital low-pass filtering. The oscillation signals were recorded for 64 ms periods at a 4 µs sample rate. The beam pulse could be used as a delayed trigge r to the Advantest. Using the memory option of the Advantest, the piezo signals were recorded from 0.3 ms onward before the arrival of the trigger. The data were stored on disk and were Fourier analysed off line. 2.4 Checks and stability The data were taken at an ambient temperature of ≈230C. By exciting the res- onator with the piezo-hammer we checked roughly its overall performance. As to be discussed in section 3, setup BC was calibrated before and after the beam run. The instrument’s stability was checked several times durin g the run by an electric driving signal on its capacitor endplate. 3 Calibration of bar BC’s piezo ceramic sensor A standard accelerometer mounted on the bar damped the vibra tions too strongly to confidently measure their excitations in the electron bea m. Therefore the re- sponse of the piezoelectric ceramic together with its ampli fier was first calibrated against two 2.4 gramme Bruel&Kjaer 4375 accelerometers glu ed, one at a time, to bar BC’s end face and connected to a 2635 charge amplifier. The resonator was excited through air by a nearby positioned loud-speaker dri ven from the Advantest digitally tunable sine-wave generator. The output signals from both the piezoelec- tric ceramic amplifier and the accelerometer amplifier were f ed into the Advantest. Stored time series were read out by an Apple Mac 8100 AV, runni ng Lab-View for on-line Fourier analysis, peak selection, amplitude and de cay time determination. We took nine calibration runs varying the charge amplifier’s sensitivity setting, and dismounting and remounting either of the two accelerometer s to the bar. For the lowest longitudinal vibrational mode we calculated the rat io of the Fourier peak signal amplitudes, R, from the piezoelectric ceramic and accelerometer. With the calibrated bar BC positioned in the electron beam li ne we checked the stability of the piezoelectric ceramic’s response intermi ttently with the beam runs by exciting the bar through its capacitor plate at one end fac e, electrically driving it at and around half the bar’s resonance frequency. We found th e response to remain stable within a few percent. After the beam runs we took additional calibration values in air with a newly ac- quired Bruel&Kjaer 0.5 g 4374S subminiature accelerometer and a Nexus 2692 AOS4 charge amplifier. In fig. 3, typical frequency responses are s hown when driving the bar by a loud-speaker signal. The upper part gives the Fourie r peak amplitude of the bar’s 13 kHz resonance as measured with the accelerome ter. The lower part gives the corresponding amplitude for the signal from the pi ezoelectric ceramic. TheFigure 3: Response at the lowest longitudinal acous- tic frequency of bar BC’s piezo (lower) and accelerom- eter (upper) Fourier amplitudes by constant amplitude loud-speaker driving. Left: on-resonance, f=12950 Hz. Right: slightly off-resonance, f=12850 Hz. Figure 4: Response of bar BC’s piezo (lower) and ac- celerometer (upper) Fourier amplitudes by electrostatic capacitor plate driving. Left: on-resonance, f=6481.7 Hz. Right: slightly off-resonance, f=6480.0 Hz. The piezo peak of constant amplitude at 6.5 kHz arising from crosstalk is absent in the accelerometer, while the acous- tic resonance is clearly seen at 13 kHz in both. right hand side of the picture shows the amplitudes to be smal ler, as expected when driving the bar slightly off resonance. We calculate the deca y time, τ, of the k-th mode amplitude Ak(t) =Ak(0)·e−t/τto be τ= 0.4 s for this setup, equipped with the relatively light accelerometer. Figure 4 shows the corresponding two signals when driving th e bar by the capacitor plate at 6.5 kHz, that is at half the bar’s resonance frequenc y. Here, the direct elec- tric response of the piezoelectric ceramic’s signal to the d riving sine-wave is present, clearly without a mechanic signal, as would have shown up in t he accelerometer. The direct signal at 6.5 kHz remains constant. On the other ha nd, the bar’s me- chanical signals on and off its resonance frequency around 13 kHz show the expected amplitude change again, thereby demonstrating that around the bar’s resonance, the piezoelectric ceramic does only respond to the mechanical s ignal, not to the electric driving signal. See also the caption of fig. 4. We calculated the average value of R0=VFourier piezo /VFourier accel. and the error over all 29 measurements, finding for the calibration factor at f=13 k Hz, β=R0S(2πf)2= (2.2±0.3) V/nm (1) where S= 0.1 V/ms−2is the amplifier setting of the accelerometer.4 Beam experiments Sensor signals way above the noise level were observed for ev ery beam pulse hitting the sphere or the bar. We ascertained that: a) the signals aro se from mechanical vibrations in the resonator, and b) they were directly initi ated by the effect of the beam on the resonator, and not arising from an indirect effect of the beam on the piezo sensors. Our assertion is based on a combination of tes t results observed for both the bars and the sphere, as now to be discussed. Figure 5: The 2 µs electron bunch (middle trace) and the piezo sensors signals on the sphere (upper and lower traces). Figure 6: A typical Fourier spectrum of bar BC as excited by the electron beam. Data were analysed over 0.016 s duration, from 0.008 s onwards after the beam passed. Identified vibrational modes are indicated. First, when the beam passed underneath the resonator withou t hitting it, we observed no sensor signal above the noise. Second, as shown i n fig. 5, the sensors’ delayed responses after the impact of the beam agreed with th e sound velocity. Here the beam was hitting the sphere at a position 5 mm above the sph ere’s south pole. The middle trace shows the beam pulse of ≈2µs duration. The two other traces show both piezo sensors to respond with a transient signal ri ght from the start time of the beam’s arrival and to begin oscillating after some del ay, depending on their distance from the beam. The distance of the equatorial senso r to the beam hitting the sphere at the south pole was 0.11 m, corresponding to ≈22µs travel time for a sound velocity of ≈5∗103m/s. The signal is indeed seen in the lowest trace starting to oscillate at that delay time. The upper trace sho ws the signal from the second sensor situated on the northern hemisphere at 0.14 m f rom the traversing beam, correspondingly starting to oscillate with a delay of ≈28µs after the impact of the beam. Third, after dismounting the piezo-hammer from the resonator, we observed that the sensor signals did not change, which showe d that the activation is not caused by the beam inducing a triggering of the piezo-h ammer. Fourth, tosimulate the electric effect of the beam pulse on the sensors, we coupled a direct current of 60 mA and 2.5 µs duration from a wave packet generator to the bar. Apart from the direct response of the piezo-sensor during th e input driving wave, no oscillatory signal was detected above the noise level. Fi nally, we measured the dependence of the amplitudes in several vibrational modes o n the hit position of the beam, as will be described in the following sections. We foun d the amplitudes to follow the patterns as calculated with the thermo-acoustic conversion model. 4.1 Results for the bar In fig. 6 a typical Fourier spectrum of bar BC is shown up to 55 kH z. The arrows point to identified vibrational modes [11]. From a fit of Kandf0of the longitudinal frequencies fL=L·f0·(1−L2K) [12] of the modes for L=1,..,4, we find f0= 12933, K= 0.0022, where f0is related to the sound velocity by vs= 2l∗f0= 5173 m/s for our bar length of l=0.2 m. For the Poisson-ratio σ= 2l√K/(πr),r being the cylinder radius of the bar, from our fit we get σ= 0.338. The values agree well with the handbook [13] quoting σ=0.33 and vs=5000 m/s for aluminium. The root mean square error of the fit is 35 Hz, in correspondence wi th the 30 Hz frequency resolution used in the Fourier analysis. Other peaks corres pond to torsional and transverse modes [11, 12]. Figure 7: Correlation between the Fourier amplitude of the 12.6 kHz vibrational mode and the beam charge. Data points (*) and straight line fit. Figure 8: The measured, unnormalised Fourier ampli- tudes (+) and model calculations (–) as a function of the beam hit position along the cylinder axis for the four lowest longitudinal modes of bar BC. The Fourier amplitudes Akoff(t) =/summationtextAkeiωktof the modes depend linearly (as shown for the 13 kHz, L=1 mode in fig. 7) on the integrated charge in the beam pulse for a fixed beam position, and therefore also linearly o n the energy deposited by the beam, which ranged in these runs from 0.06-0.8 J. The sp read in the ratiosof the amplitudes to the beam charge, shows the Fourier ampli tudes to reproduce within ±10%. The agreement of the model to within 10% with the measured dat a is shown in fig. 8. The figure shows the measured Fourier amplitudes of bar BC at the piezo sensor and the calculations following Grassi Strini et al. [ 5, 14] as a function of the hit position along the cylinder axis for the first four longit udinal modes. For each mode the average model value was scaled to the average measur ed value. The best fit was found with a shift of the hit positions along the bar, by an overall offset ofx0=-0.0075 m, which corresponds to the crude way we aligned the bar with the beam line. 4.1.1 Lowest bar mode excitation amplitude For the 13 kHz, L=1 mode we determine the absolute amplitude for a comparison with the model calculation of ref. [5, 14]. Firstly, we use th e amplitude function B0(x), see eq. 9 of ref. [5, 14], by rewriting it in the form: B0(x) = 2·κ0·∆E/π×cos(πx/l)sin(πη/(2l))/πη/(2l) (2) with κ0=α·l/(cv·M) =α/(cv·ρO). (3) In this expressions xis the hit position along the the cylinder axis, lthe bar length, ηthe beam diameter, αthe thermal linear expansion coefficient, ρthe density, c vthe specific heat, Othe cylindrical surface area of the bar, and ∆ Ethe energy absorbed by the bar. From B0(x) we derive the functional form for the measured values of Wsensas Wsens(x) =B0(x) ∆EβDdE dQ(4) where dE/dQ is the beam energy absorbed by the bar per unit of impinging be am charge, βthe calibration factor as discussed in section 3, and D the de cay factor e−t/τ, since eq. 2 applies at excitation time and we have to correct the amplitude at measuring time for the mode’s decay, corresponding to its Q- factor. Therefore, Wsens(x) =κexp2/π×cos(πx/l)sin(πη/(2l))/πη/(2l), (5) with κexp=βDκ 0dE dQ(6) From fitting eq. 5 to the measured values Wsens(x) given in table II with κexpas the free variable, we find our presently measured value for κexp 0=κexp·/(dE/dQ ·D·β) which we compare to the model value in eq. 3. Secondly, the dec ay time was measured by recording the sensor signals after a trigger del ayed by up to 1.6 s at afixed beam hit position. An exponential fit A(t) =A0∗e−t/τto the mode amplitude gives τ= (0.36±0.01) s for the L=1 mode. This corresponds to a Q-value of ≈15000, a value consistent with the room temperature measure ment of aluminium as in ref. [15], and indicating a negligible influence of the s uspension and piezoelectric ceramic sensor for this mode. From the measured value of τand a mean delay time from the start of the beam pulse of 0.016 s, we calculate the de cay factor to be D= 0.95. Table II. Excitation values W sens, equalling the ratio of the measured Fourier amplitude and the measured beam pulse charge at each of the in dicated hit positions on the bar for the 13 kHz, L=1 mode. hit position xcm 0 1 2 3 4 5 6 7 8 9 10 WsensV/nC 0.185 0.216 0.167 0.180 0.225 0.152 0.152 0.157 0.112 0.089 0.057 Thirdly, as indicated, we use the data for W sensin the second row of table II to fit the variable κexpin eq. 5, where now x is the hit position as given in row 1, l=0.2 m , and η= 0.002 m. The value found in the fit is κexp= (0.300±0.025) V/nC. Fourthly, from a Monte Carlo simulation at the beam ene rgy of 570 MeV used for these runs, we calculate the mean absorbed energy an d the mean energy spread which results from the fluctuating energy losses of th e passing electrons and the energies of the secondaries escaping from the bar, as ∆E e=(19±2) MeV. The electron beam pulse thus deposits dE/dQ = (0.019±0.002) J/nC in the bar. Using the measured calibration value at f=12986 Hz as given in eq. 1 ,β=(2.2 ±0.3) V/nm, we arrive at κexp 0= (7.4±1.4) nm/J . (7) Finally, we calculate the model value of κ0from the material constants as being κ0=10 nm/J, neglecting the much smaller error as arising from s ome uncertainty in the parameters. We conclude that κexp 0/κ0= (0.74±0.14), a result that is consistent with the validity of the model of ref. [5, 14]. The measured maximum excitation amplitude at beam position x=0, see fig. 8 for the 13 kHz, L=1 longitudinal mode thus corresponds to (0.13 ±0.02) nm. 4.1.2 Higher bar mode excitation amplitudes Having determined the correspondence between the model cal culation and the ex- periment’s result for the first longitudinal vibrational mo de amplitude, we return to some of the higher vibrational modes. To compare the modes we need to take the sensor position on the bar into account. We rewrite the displ acement amplitude of eq. 5 from ref. [5] as a function of hit position xhand sensor position xsas: Φodd−L= (2κ/Lπ)sin(Lπx s/l)cos(Lπx h/l), Φeven−L= (2κ/Lπ)cos(Lπx s/l)sin(Lπx h/l), (8)where lis the bar length. We dropped the beam width correction term w hich would lead to a less than 0.1% correction even for L=4. We approximate the sensor response by the local strain along bar BC’s cylinder axis, th at is to the dΦ/dx sof eq. 8, arriving at a sensor response, S L: Sodd−L=BLcos(Lπx h/l), Bodd−L= (2ǫκ/l)cos(Lπx s/l), Seven−L=BLsin(Lπx h/l), Beven−L= (2ǫκ/l)sin(Lπx s/l), (9) where ǫis a sensor response parameter. The xsdependent term did not enter into the calculation of κexp 0in the previous section, since the calibration was done at th e same sensor position as the beam measurement. However, for a comparison between the modes, the dependence on the sensor position xshas to be taken into account. Since the variables are strongly correlated, we, first, fitte d for each mode the term BLin the xhdependent part of eq. 9 to the measured value of Wsensfor the mode, shifting the origin of xhby 0.0075 m, as mentioned before. The results are given in the first row of table III. Table III. Bar BC modes comparison. The piezoelectric ceram ic sensor responds to the bar’s strain. description symbol 13 kHz, L=1 25.6 kHz, L=2 38 kHz, L=3 50 kHz, L=4 amplitude Bmeas L 0.12±0.01 0.021±0.002 0.033±0.03 0.052±0.005 decay correction D 1.04±0.001 1.17±0.02 1.49±0.06 1.14±0.01 Bmeas L×D Bexp L 0.12±0.01 0.025±0.003 0.049±0.005 0.059±0.006 sensor position factor PL 1.02 2.61 1.20 1.41 ǫκ2/l=Bexp L×PL(arb.u. ) κ′0.12±0.01 0.065±0.007 0.059±0.006 0.083±0.008 Second, we corrected the amplitudes Bmeas Lfor the mode decay with a factor D, given in row 2, and corresponding to the times τ1= 0.36s, τ2= 0.10s, τ3= 0.04s, τ4= 0.12s, which leads to the values of Bexp Lin row 3. Finally, we multiplied with the factor Podd−L= 1/cos(Lπx s/l),Peven−L= 1/sin(Lπx s/l), where the bar length is l=0.2 m. Since the sensor extends from 0.005 through 0.020 m fr om the center of the bar, we use the mean sensor position xs= 0.0125 m. The resulting values of κ′= 2ǫκ/l, shown in the last row, should be independent of L. For L=2,3,4 they are rather closely scattered around a mean value of κ′= 0.07 which is, however, at about half the L=1 value. This discrepancy might have originated from some resonances of the sensor itself, and we suspect the strong pe ak at 23 kHz, shown in figure 6, to be an indication of such resonances playing a role . Since the amplitudes of the higher modes for bar BC do not comp ly with our ex- pectations we turn, as a further check, to our un-calibrated measurements with bar BU. It had been equipped with a piezoelectric sensor at one en d face where the longitudinal modes have maximum amplitude. The sensor had b een mounted flatly with about half of its surface glued to the bar, and respondin g to the bar’s sur- face acceleration, not its strain as at bar BC. We extract the κLvalues from our measurement analogously as for bar BC, following again the m odel calculations of Grassi Strini et al. [5], using the L=1 mode as the reference. The results are given in table IV.Table IV. Bar BU modes comparison. The piezoelectric cerami c sensor responds to the bar’s acceleration. The value of κmeas 1for the L=1, 13 kHz mode is used as the reference for the higher modes. description symbol 13 kHz, L=1 25.6 kHz, L=2 38 kHz, L=3 50 kHz, L=4 relative amplitude Bmeas1 1.15 11.5 3.6 relative decay correction D 1 3±1 0.7±0.2 1.3±0.9 (ωL=1/ωL)2Ω 1 0.26 0.12 0.07 Bmeas×D×Ω κmeas L/κmeas 1 1 0.8±0.3 0.9±0.4 0.3±0.3 After applying the decay correction factor Dand the frequency normalisation factor Ω, the results should be independent of L. The L=4 value is significantly low, which, again, might be due to some interfering resonanc e. The L=2 and L=3 values, however, do not significantly deviate from the L=1 value, thus confirming the model calculations for these higher modes too. 4.2 Results for the sphere Our measurements on the sphere consisted of a) hitting the sp here with the beam at one of two heights in the vertically oriented plane throug h its suspension: at the equator (E) and at 0.022 m southward (A); b) rotating the sphe re with its two fixed sensors over 1800around the suspension axis at each beam height, and measurin g several times back and forth by steps of 300to diminish the influence of temperature and beam fluctuations, ending up on a 100angular lattice. The Fourier amplitude Figure 9: The Fourier amplitude spectrum of the sphere SU averaged over all measured angles for sensor- 1, at beam height position E. The modes and frequencies as calculated, are indicated. Figure 10: Decay of sphere’s vibrational modes as mea- sured with delayed data taking of the spectrum analyser. The curve shows the fit of A(t) =A0∗e−t/τ. Upper: 17.6 kHz, L=2,τ=1.1 s. Middle: 24 kHz, L=1,τ=0.4 s. Lower: 37 kHz, L=0,τ=0.1s. spectrum of sensor-1, averaged over the angular positions, is shown in fig. 9. The lowest spheroidal mode is most relevant for a spherical resonant mass gravitationalwave detector, and we therefore focus on a few spheroidal mod es. As expected, the lowest spheroidal L=2 mode is seen at 17.6 kHz, the lowest spheroidal L=1 mode at 24 kHz, and the lowest spheroidal L=0 mode at 37 kHz. Some other peaks are also indicated in the figure, though not the toroidal mode s, which we neglect completely. It should be noted that while the L/negationslash=0 amplitudes oscillate over the angles, the L=0 amplitude does not, leading to a relative enhancement of t he latter in the angle-averaged fig. 9. The Fourier amplitudes, again, showed a linear dependence o n the deposited energy. To determine the decay times at f=17.6, 24 kHz and 37 kHz, see fig. 10, we took data with up to 4 s delay in the spectrum analyser, and found τ≈1 s, 0.4 s and 0.1 s respectively. The angular distributions for the amplitude of the 37 kHz, L=0 mode at the two Figure 11: 37 kHz Fourier-modulus angular distribu- tion. The beam hits the sphere at upper: E, lower: A. vertical beam positions E and A are shown in figures 11. The L=0 amplitude is independent of the angle, and since the amplitude is constan t to within 20% we infer from the Fourier-modulus’ deviation from flatness a 20 % variation of the beam intensity from one shot to another. During our measurement with the sphere we were unable to use t he beam pulse as a trigger, implying that the start time of data acquisition w ith respect to the beam pulse is unknown. In our further analysis we will therefore u se only the Fourier modulus, and not analyse the phases. The absolute scale of the 37 kHz Fourier amplitude turned out to be ≈5 times larger than the model value for sensor-2 and ≈50 times for sensor-1. We assume this discrepancy to be based on some interference effects, po ssibly with a sensor resonance and a suspension bar mode, and we do not further ana lyse the L=0mode. To disentangle the angular distributions in general, we felt, would squeeze the results of our simple measurement too much, for a couple o f reasons. First, the Fourier amplitude ALfor any multi-pole order Lin the sphere’s case is actually a sum of M-submodes. Though they would be degenerate for an ideal sphe re, in practice some M-modes might or might not turn out to be split beyond the frequency-resolution of ∆ f=30 Hz. Second, both sensors s1, s2should be taken to have unknown sensitivities, esj, in three orthogonal directions, with phase factors +1 or -1 for their orientation. Third, though each mode would start to be excited within the same sub-nanosecond time interval of the beam cro ssing, the building up of each mode’s resonance vibration may lead to a specific phas et0 ML,bkdepending on the mode’s spatial relation to the beam path. We now show that the calculated angular distributions have t he signature of the L- character of the measurement. Therefore, we write the Fouri er modulus at different impinging beam positions bkas a function of the angle φas: AL,sj,bk(φ) =|FL,sj,bkǫsj·+ML/summationdisplay −MLsL,M L,bkuL,M L,sj,bk(φ)eωLt0 ML,bk|, (10) where FL,sj,bkis a frequency response function for each sensor, which may d epend also on the beam position. This normalisation factor is expe cted to be of order 1, and is kept fixed at 1 for the L=2 distributions. It is used as a free parameter for the L=1 distributions to compensate for the rather inaccurate kn owledge of a) the sensor positions on the sphere’s surface, b) the beam tra ck location and c) the electrons and photons shower development along the track, s ince the exact excitation strengths of the modes are quite sensitive to such data. As th e first step in the fitting procedure we separately calculated the sL,M L,bkuL,M L,sj,bk, where sL,M L,bkis the mode’s strength from the beam excitation, as detailed in the append ix. We inserted the calculated sL,M L,bkuL,M L,sj,bkin a hierarchical fitting model, to simultaneously fit [16] the relevant parameters of eq. 10 to the 17.6 kHz, L=2 Fourier modulus AL,sj,bkfor both sensors s1ands2at both beam positions E and A. This fit led to a reduced χ2=1.3 at 59 degrees of freedom. Next, with fixed values for the s ensor efficiencies ǫsjso established, we fitted the relevant parameters for the 24 k Hz,L=1 Fourier peaks, including the L=1 sensor response factors F. At all stages the |t0 ML,bk|of the phases were kept within the bounds of the period of mode- L. With an uncertainty in the beam charge and in the Fourier peak amplitudes of ≈20% each, the error amounts to ≈30%, and we took a minimum absolute error of 2 ×10−5for sensor s1 and 1×10−5for sensor s2. In total we have 152 data points, while the total number of fitted parameters is 27, including a relative normalising factor for the mean beam current at beam position A with respect to the mean current at beam position E. We found for the total fit a reduced χ2=1.6 at 125 degrees of freedom. The L=1 response factors remain within 1.1 and 0.2. The results of th e fits to the 17.6 kHz and 24 kHz are given in figures 12 and 13 and table V. Note the diff erent vertical scales used for sensor-1 and sensor-2 in both pictures. Some of the parameters givenFigure 12: Data points (+) and fit results (—) for the sphere’s 24 kHz, L=1 mode. Left column: sensor-1, right colum: sensor-2. Upper row: beam position E, lower rows: beam position A. The x-axes give the angle of sensor-1. Note that the y-scales are different for the two sensors. Figure 13: Data points (+) and fit results (—) for the sphere’s 17.6 kHz, L=2 mode. Left column: sensor-1, right colum: sensor-2. Upper row: beam position E, lower rows: beam position A. The x-axes give the angle of sensor-1. Note that the y-scales are different for the two sensors. in the table are strongly correlated. Table V. Results of a hierarchical fit to the data of the sphere at the 19 mea- suring angles of both sensors s1ands2of the Fourier modulus at 17.6 kHz and 24 kHz at beam positions E and A. sensor efficiency fit result error ∗10−4V/ms−2∗10−4V/ms−2 ǫs1−r +0.10 0.02 ǫs1−θ -0.5 0.2 ǫs1−φ +0.4 0.1 ǫs2−r -0.5 0.2 ǫs2−θ -0.5 0.2 ǫs2−φ +4.0 0.3 intensity (beamA beamE) 0.7 0.1 response factor FL=1,s1,beamA 0.3 0.1 FL=1,s2,beamA 1.1 0.1 FL=1,s1,beamE 0.2 0.1 FL=1,s2,beamE 1.1 0.2 We conclude that the measured Fourier amplitude angular dis tributions are con- sistent with the model value for the L=2 and L=1 mode signature. 4.2.1 The sphere’s absolute displacement Finally, to estimate the order of magnitude of the sphere’s a bsolute displacement, we have to take an intermediate step by first normalising bar B U to the calibrated results for bar BC and then use bar BU as a calibration for the s phere. The sensors used on bar BU consisting of the same sensor material and havi ng been cut roughly to the same size, we assume to be identical to the ones used on s phere SU. The amplifiers used are identical. The bar BU sensors, however, d iffer strongly from those of bar BC.We arrive at an indirectly calibrated value for ǫBU,2= (3.4∗10−4±30%) V/ms−2of sensor-2 on bar BU. The value of ǫBU,1for sensor-1 is about ten times smaller. Then, for the sphere, the fitted value of ǫsjgiven in table V, shows the largest value of sensor-2, ǫφ−SU= (4∗10−4±10%) V/ms−2, to lead to a ratio ≈(1.2±0.4) with ǫBU. Again, the values for sensor-1 are about ten times smaller. The error of ≈33% is the propagated statistical error only. The result seems r easonable. So, the model calculation and our sphere measurement results are of the sa me order of magnitude on an absolute scale too. From the maximum Fourier modulus, Vmax= (0.003±0.001) V, of the 17.6 kHz, L=2 sphere mode measured in sensor sensor-2 as given in fig. 13, and the ab- sorbed energy of 3.1 J, we find the maximum sphere’s displacem ent to correspond to (0.2±0.1)nm/J. 5 Discussion Having confirmed the thermo-acoustic conversion model in th e present experiment, we discuss some points about extrapolating these results to the actual operation of a resonant gravitational wave detector. Firstly, in our exp eriment many incident particles deposited their energy in the resonator, in contr ast to a single muon hit- ting an actual detector. However, from this difference it see ms unlikely to reach different conclusions, especially since in the process of de positing energy along its track, the muon will generate lots of secondary particles to o. Also, we measured at room temperature while actual detectors would have to opera te in the millikelvin range. An aluminium resonator, for instance, at such a tempe rature, would be su- perconducting, and it is as yet unclear how the decoupling of the electron gas from the lattice would affect the process of acoustic excitation. Therefore, we hold it of particular importance for the prosp ected shielding of a next generation resonant mass gravitational wave detector that an existing millikelvin detector like the Nautilus [17, 18], would succeed in measur ing the impinging cos- mic rays in correlation with the resonator mode. Such a resul t, as a test for the further applicability of the thermo-acoustic conversion m odel at operating temper- ature, would come closest to the real situation envisaged fo r the new detectors. Apart from such temperature effects, the applicability of th e thermal acoustic con- version model [4, 5, 14] is confirmed by the data and therefore cosmic rays should be expected to seriously disrupt, as calculated by the model, t he possibility of detecting gravitational waves. It is beyond the scope of this article t o go into any detail [19]. We want to point, however, to earlier calculations [3, 7, 20] which, having used the model, clearly show, firstly, that a next generation spheric al resonant mass gravita- tional wave detector of ultra high sensitivity will be signi ficantly excited by cosmic rays. Secondly, the high impact rate of cosmic rays will proh ibit gravitational wave detection at the earth’s surface, with the required sensiti vity.Finally, shielding the instrument by an appreciable layer o f rock as available in, for instance, the Gran Sasso laboratory, would suppress the cos mic ray background by a factor of ≥106. Even then a vetoing system will be necessary and, at the radi cal reduction of the background rate so established, it may inde ed work effectively. Acknowledgements We thank A. Henneman for the computer code to calculate a sphe re’s vibra- tional modes, R. Rumphorst for his knowledgeable estimate o f sensor sensitivity, J. Boersma for digging out the formal orthogonality proof of a sphere’s eigen- modes, and the members of the former GRAIL team for expressin g their interest in this study, especially P.W. van Amersfoort, J. Flokstra, G. Frossati, H. Rogalla, A.T.M. de Waele. This work is part of the research programme o f the National In- stitute for Nuclear Physics and High-Energy Physics (NIKHE F) which is financially supported through the Foundation for Fundamental Research on Matter (FOM), by the Dutch Organisation for Science Research (NWO). References [1] P. Astone, G.V. Pallottino, M. Bassan, E. Coccia, Y. Mine nkov, I. Modena, A. Moleti, M.A. Papa, G. Pizzella, P. Bonifazi, R. 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Mazzitelli, Assessi ng the effects of cosmic rays on a resonant-mass gravitational wave detector, NIKHE F-97/3 [8] G.D. van Albada, H. van der Graaf, G. Heijboer, J.W. van Ho lten, W.J. Kas- dorp, J.B. van der Laan, L. Lapik´ as, G.J.L. Nooren, C.W.J. N oteboom, J.E.J. Oberski, H.Z. Peek, A. Schimmel, T.G.B.W. Sluijk, J. Venema, P.K.A. de Witt Huberts, p. 402 in: E. Coccia, G. Pizzella, G. V eneziano (eds.), Proc. 2nd Amaldi conf. on Gravitational Waves, CERN 1997, Wo rld Scientific 1999. [9] C. de Vries, C.W. de Jager, L. Lapik´ as, G. Luijckx, R. Maa s, H. de Vries, P.K.A. de Witt Huberts, Nucl.Instr. and Meth. 223(1984)1 [10] In the beginning of 1999 the Amsterdam MEA electron acce lerator, and AmPS stretcher ring ended their operations permanently, due to s topped funding. Sev- eral parts are being dispersed over labs in Europe, Russia an d the USA. [11] J.F. de Ronde, G.D. van Albada and P.M.A. Sloot in High Pe rformance Com- puting and Networking ’97, Lecture Notes in Computer Scienc e, pag. 200, Springer, 1997. J.F. de Ronde, G.D. van Albada and P.M.A. Sloot Computers in P hysics, 11(5):484–497, Sept/Oct 1997. J.F. de Ronde. Mapping in High Performance Computing, PhD. t hesis Univ. of Amsterdam, The Netherlands, 1997. J. de Rue. On the normal modes of freely vibrating elastic obj ects of various shapes, thesis, Univ. of Amsterdam, the Netherlands, 1996. [12] H. Kolsky, Stress waves in solids, Dover 1961, A.E.H. Love, Mathematical theory of elasticity, Dover 1944 , D. Bancroft, Phys.Rev. 59(1941)588, R.Q. Gram, D.H. Douglas, J.A. Tyson, Rev.Sci.Instr. 44,7(1 973)857 [13] Metals Handbook, vol. 6, 9th edition 1983, American Soc iety of Metals, Metal park, OH.[14] D. Bernard, A. de Rujula, B. Lautrup, Nucl.Phys. B242(1 984)93. [15] W. Duffy, J.Appl.Phys. 68(1990)5601. [16] MINUIT fitting tool, CERN, http://wwwinfo.cern.ch/as d/cernlib/minuit [17] E. Coccia, A. Marini, G. Mazzitelli, G. Modestino, F. Ri cci, F. Ronga, L. Votano, Nucl.Instr. and Meth. A355(1995)624 [18] P. Astone, M. Bassan, P. Bonifazi, P. Carelli, E. Coccia , V. Fafone, A. Marini, G. Mazzitelli, S.M. Merkowitz, Y. Minenkov, I. Modena, G. Mo destino, A. Mo- leti, G.V. Pallottino, M.A. Papa, G. Pizzella, F. Ronga, M. S pinetti, R. Terenzi, M. Visco, L. Votano, LNF-95/35, Nucl.Phys.B(Proc.Suppl.) 70(1999)461. [19] The Dutch subsidising agencies NWO/FOM have decided ag ainst further pur- suing the GRAIL project. They favoured more conventional, o n-going work, above the funding of our team’s proposal to research, develo p and open up a new field in The Netherlands with the GRAIL resonant sphere Gr avitational Wave detector, even though an evaluation committee of inter national experts gave GRAIL an almost embarrassingly positive judgement. We therefore see, sadly, no opportunity for a follow up to the current paper. [20] G.Pizzella, ”Do cosmic rays perturb the operation of a l arge resonant spherical detector of gravitational waves?”, LNF-99/001(R) [21] D.E. Groom, Passage of particles through matter, in Rev.Part.Phys C3,1-4(1998)148 [22] GEANT simulation tool, CERN, http://wwwinfo.cern.ch/asd/geant/index.html [23] Electron Gamma Shower development, OMEGA project, http://www-madrad.radiology.wisc.edu/omega/www/omeg aintro00.html [24] We thank Richard Wigmans for running the case. [25] At an even smaller frequency than the first quadrupole mo de of the full sphere, another quadrupole mode arises when the sphere has a spheric al hole, actually being a thick spherical shell. We did not consider the latter in our study, how- ever, since it has its maximum amplitude at the inner surface and a minimum amplitude at the outer surface. [26] M.E. Gurtin, The Linear Theory of Elasticity, sec. E.VI ., The free vibration problem, p. 261 in: C. Truesdell (ed.), Mechanics of Solids, Handbuch der Physik VIa/2, Springer 1972, proves for an ideal sphere, even if par tially clamped, the orthogonality of its modes. [27] A.A. Henneman, J.W. van Holten, J. Oberski, Excitation s of a wave-dominated spherical resonant-mass detector, NIKHEF-96-006Appendix: Sphere excitation model calculation Our calculation of the ( L, M)-mode excitation strengths is based on the source term of eq. 5.10/11 of ref. [14], s= Σ/(ρV)∗/integraltextdz∇⊥·u, with Σ = γdE/dz . Here, γis the Grueneisen constant, ρVde sphere’s mass, and dE/dz the absorbed energy per unit track length. The Fourier amplitudes, measuring the second time derivative of the mode amplitudes, are directly proportional to s, and the mode amplitudes follow froms/ω2, as in eq. 5.18 of [14]. However, the amount of energy absorbe d per unit length in our case depends on the particle’s position along t he track. We therefore re- included the Σ-term under the source term’s integral by lett ingdE(z)/dzrepresent the electromagnetic cascade development of ref. [21] as an a pproximation to the amount of energy absorbed per unit track length by the sphere at position z along the beam track, sL,M L=κ/integraldisplay L∇⊥·uL,M L(z)dE(z) dzdz, (11) where zis being measured from the beam’s entrance point into the sph ere. With Eabs being the total amount of energy absorbed by the sphere from t he electron bunch, we write dE(z)/dz=Eabs∗d(E(z)/Eabs)/dzand use the polynomial expansion d(E(z)/Eabs)/dz=/summationtext3 i=0cizi. Then/integraltextzmaxzmindzd(E(z)/Eabs)/dz=1. For the polynomial, measuring zin meter, we acquired the values c0=0.8332 m−1,c1=226 m−2,c2=- 1832 m−3,c3=4909 m−4from a fit to the form given in ref. [21], with less than a percent deviation for our case of 0 ≤z≤0.15 m. The value for the energy absorbed by the sphere from a single electron, Ee abs=123 MeV, we got from both our Monte Carlo simulation using GEANT [22], and from EGS4 [23, 24]. At the measured 25 nC beam pulse charge this corresponds to a total Eabs= 3.1 J absorbed by the sphere. Then the value of κ Eabs=γEabs/M=1.00 m2/s2, for our case of M=ρV=4.95 kg and γ=1.6. Our sphere has a suspension hole which leads to a slight shift in the frequencies and the spatial distribution of the modes, w ith respect to those of a sphere without a hole [25]. We approximated, however, our sp here’s modes by the ideal hole-free sphere’s eigenmode solutions u(z) [26], using the available computer code as established in ref. [27], and renormalising to/integraltextu·udV=V, as used in ref. [14] from eq. 5.6 onward. The source term sL,M Lwas calculated for each mode ( L, M) by numerically integrating eq. 11. We checked the surface te rm in the numerical procedure to be negligible, as assumed in the partial integr ation leading to the form ofsused by ref. [14]. Each u(φ) in eq. 10 is the eigenmode solution, calculated for each sensor on the φ-grid of the measured data, and each term sL,M L,bkis the excitation factor sL,M Lat the specific beam position bk.

arXiv:physics/0001024v1 [physics.atom-ph] 11 Jan 2000Scattering of positronium by H, He, Ne, and Ar P. K. Biswas and Sadhan K. Adhikari Instituto de F´ ısica Te´ orica, Universidade Estadual Paul ista, 01.405-900 S˜ ao Paulo, S˜ ao Paulo, Brazil February 2, 2008 Abstract The low-energy scattering of ortho positronium (Ps) by H, He , Ne, and Ar atoms has been investigated in the coupled-channel framework by usin g a recently proposed time- reversal-symmetric nonlocal electron-exchange model pot ential with a single parameter C. For H and He we use a three-Ps-state coupled-channel model a nd for Ar and Ne we use a static-exchange model. The sensitivity of the results is studied with respect to the parameter C. Present low-energy cross sections for He, Ne and Ar are in go od agreement with experiment. 1Low-energy collision of ortho positronium (Ps) atom with ne utral gas atoms and molecules is of interest in both physics and chemistry. Recently, ther e have been precise measurements of low-energy ortho-Ps scattering by H 2, N2, He, Ne, Ar, C 4H10, and C 5H12[1, 2, 3]. Due to internal charge and mass symmetry, Ps atom yields zero ela stic and even-parity transition potentials in the direct channel. Ps scattering is dominate d mainly by exchange correlation at low energies. If Nbasis states are included for both Ps and target in a coupled- channel formalism, the number of channels grow as N2. This complicates the tractability of the Ps scattering process in a coupled channel scheme compared to t he electron scattering. The dominance of the short-range interaction in Ps scattering c auses serious trouble towards the convergence of any coupled-channel formalism with a trunca ted basis [4, 5]. The use of 22 coupled Ps pseudostates for Ps-H system in the R-matrix appr oach has indicated convergence difficulties [5, 6] in Ps-H binding and resonance energies. To find a solution to the nonconvergence problem, we have sugg ested a nonlocal electron- exchange model potential [4, 5] with a single parameter Cand demonstrated its effectiveness by performing quantum coupled-channel calculations using the ab initio framework of close coupling approximation. Two versions of this potential wer e suggested: one is time-reversal- symmetric and the other nonsymmetric. The nonsymmetric pot ential has been applied for the study of Ps scattering by H [7], He [4] and H 2[8] using a three-Ps-state coupled-channel model. For He and H 2these studies yielded total cross sections in good agreemen t with experiments [1, 2, 3] in addition to producing the correct pick-off quench ing rate for Ps-He at low energies [9]. Higher Ps-excitation and ionization cross sections we re also calculated in these cases to reproduce the total cross sections at medium and high energi es. Previous theoretical studies [10, 11] on Ps-He produced results in strong disagreement [4 ] with experiment [1, 2, 3]. In a subsequent application of this model potential to Ps sca ttering by H [5], it was found that the symmetric form leads to by far superior result than t he nonsymmetric form. The symmetric form was able to reproduce accurate variational r esults very precisely for singlet Ps-H binding and resonance energies; the nonsymmetric form failed to yield such results. In view of this we reinvestigate the problem of low-energy Ps scattering by He [4] using the symmetric exchange potential employing the three-Ps-stat e model used before. We study how the low-energy cross sections for Ps-H and Ps-He change with the variation of the parameter Cof the potential. Then we also apply this exchange potential to the study of low-energy Ps scattering by Ne and Ar using a static-exchange model. The pr esent calculation accounts for the measured low-energy cross sections [1, 2, 3] satisfacto rily for He, Ne, and Ar. The total wave function of the Ps-target system is expanded i n terms of the Ps eigenstates 2as Ψ±(r1, ...,rN;rN+1,x) =/summationdisplay ν/bracketleftbigg Fν(ρN+1)χν(tN+1)φ0(r1, ...,rj, ...,rN) +N/summationdisplay j=1(−1)SN+1,jFν(ρj)χν(tj)φ0(r1, ...,rN+1, ...,rN)/bracketrightbigg , (1) where antisymmetrization with respect to Ps- and target-el ectron coordinates has been made. Hereρi= (x+ri)/2,ti= (x−ri), where ri, i= 1, ..., N denote the target electron coordinates, andrN+1andxare the electron and positron coordinates of Ps; φ0andχνdenote the target and Ps wave functions, and Fνis the continuum orbital of Ps with respect to the target. SN+1,jis the total spin of the Ps electron ( N+ 1) and target electron ( j) undergoing exchange and can have values of 1 or 0. The spin of the positron is conser ved in this process and the exchange profile of the Ps-target system is analogous to t he corresponding electron-target system. Projecting the resultant Schr¨ odinger equation on the Ps eigenstates and averaging over spin states, the resulting momentum-space Lippmann-S chwinger scattering equation for a particular total electronic spin state Scan, in general, be written as [11, 12] fS ν′ν(k′,k) =BS ν′ν(k′,k) −1 2π2/summationdisplay ν′′/integraldisplay dk′′BS ν′ν′′(k′,k′′)fS ν′′ν(k′′,k) k2 ν′′/4−k′′2/4 + i0, (2) where fS ν′νis the scattering amplitude, and Bν′νis the corresponding Born potential, νandν′ denote initial and final Ps states, kν′′=/radicalBig 2m(E −ǫ′′)/¯h2is the on-shell relative momentum of Ps in the channel ν′′withǫ′′the total binding energy of the intermediate Ps and target st ates, andEthe total energy of the Ps-target system and mthe reduced mass. Here BS ν′ν(k′,k) =BD ν′ν(k′,k) +N/summationdisplay j=1(−1)SN+1,jBEj ν′ν(k′,k); (3) where BDis the direct Born potential and BEjis the model potential for exchange between the Ps electron (denoted by N+1) with the jth target electron. For Ps-H scattering both S≡S2,1 = 0, 1 will contribute [12]; the corresponding potentials an d amplitudes are usually denoted by B± ν′νandf± ν′ν, respectively [5]. For Ps scattering from He, Ne, and Ar etc. there will be only one scattering equation (2) corresponding to total electro nic spin S= 1/2. For these targets with doubly occupied spatial orbitals, in the sum over jin Eq. (3) only half of the occupied target electrons in each sub-shell will contribute when the target is frozen to its ground state [13]. Consequently, only SN+1,j= 1 (f−andB−) will contribute to target-elastic scattering [11] for targets with doubly occupied spatial orbitals. 3For Ps-H scattering, the differential cross section is given bydσ/dΩ = [ |f+|2+ 3|f−|2]/4. For target-elastic Ps scattering by He, Ne, and Ar, the differ ential cross section is given by dσ/dΩ =|f−|2. In all Ps-scattering the direct potential, BD, is zero for elastic and all even- parity-state transitions of Ps. Thus the nonorthogonal exc hange kernel alone dominates the solution of Eq. (2) and this dominance is possibly responsib le for convergence difficulties to conventional approaches based on Eq. (2). The present exchange model was derived using Slater-type or bital for the H-atom, so that a generalization from a H-target to a complex target represen ted by a Hatree-Fock wave function becomes straight-forward. For Ps scattering from a H orbita l, the model exchange potential between the Ps-electron ( r2) and the orbital electron ( r1) was derived from the 1 /r12term and is given by [4, 5]: BE µ′ν′µν=4(−1)l+l′ < D >/integraldisplay φ∗ µ′(r2) exp(iQ.r2)φµ(r2)dr2 ×/integraldisplay χ∗ ν′(t2) exp(iQ.t2/2)χν(t2)dt2, (4) where landl′are angular momenta of the initial ( χν) and final ( χν′) Ps states, φµandφµ′are initial and final H states, and Q=ki−kf. Here kiandkfare initial and final Ps momenta, respectively. In Eq. (4) the symmetric form of the averaged q uantity < D > is [4, 5] < D > =k2 i+k2 f 8+C2/bracketleftBiggα2 µ+α2 µ′ 2+β2 ν+β2 ν′ 2/bracketrightBigg , (5) where α2 µ′/2 and β2 νare the binding energies of the final target and initial Ps sta tes, respectively, andCis the only parameter of the potential. Normally, the parame terCis taken to be unity [4, 5] which leads to reasonably good numerical results. How ever, it can be varied slightly from unity to obtain a precise fit of a low-energy scattering obser vable (experimental or variational), as have been done in some applications of model potentials [1 4, 15]. A variation of Cfrom unity leads to a variation of the binding energy parameters ( α2, β2etc.) used as average values for square of momentum [4] in the expression for /angbracketleftD/angbracketrightof Eq. (5). This, in turn, tunes the strength of the exchange potential (4) at low energies. At hi gh energies this model potential is insensitive to this parametrization and leads to the well -known Born-Oppenheimer form of exchange [16]. We have turned this flexibility to good advant age by obtaining precise agreement with low-energy results of Ps scattering by H, He, Ne, and Ar, as we shall see in the following. For a complex target the space part of the HF wave function [17 ] is given by Ψ( r1,r2, ...,rj,- ...,rN) =A[φ1(r1)φ2(r2)...φj(rj)...φN(rN)],where Ais the antisymmetrization operator. The position vectors of the electrons are ri, i= 1,2, ..., N andφj’s have the form: φj(rj) = 4/summationtext κaκjφκj(rj). The orbital φκj(rj) is a Slater-type orbital. Considering proper antisymmetr iza- tion with respect to Ps and target electrons, the final model e xchange potential obtained from (4) is given by [4] BEj ν′ν=/summationdisplay κκ′4aκjaκ′j(−1)l+l′ < D κκ′> ×/integraldisplay φ∗ κ′j(rj) exp(iQ.rj)φκj(rj)drj ×/integraldisplay χ∗ ν‘(tj) exp(iQ.tj/2)χν(tj)dtj. (6) with < D κκ′>= (k2 i+k2 f)/8 +C2[(α2 κj+α2 κ′j)/2 + (β2 ν+β2 ν‘)/2], where ακjis the energy parameter corresponding to the orbital φκj(rj) [17]. We use exact wave functions for H and Ps, HF wave functions for He, Ne, and Ar [17]. After a partial-wave projection, the one-dimensional scatterin g equations are solved by the method of matrix inversion. First we study the effect of the variation of the parameter Cof the exchange potential in the Ps-H system using a three-Ps-state model with Ps(1s,2 s,2p) states [7]. We start our discussion with the singlet S-wave resonance. In Fig. 1 we pl ot the S-wave phase shift at different energies which illustrate the resonance pattern o btained with different values of C. The resonance position shifts monotonically towards lower energies with decreasing of Cfrom unity. We have shown it in steps where the value of Cis varied from unity to 0.785. The resonance position matches with the accurate prediction of 4.01 eV [18] for C= 0.785. In Fig. 2, we plot kcotδversus k2for the corresponding low-energy S-wave phase shifts δ. Figure 2 demonstrates how the improvement in the resonance position simultaneously improves the Ps-H binding energy. For C= 1.0 the resonance and binding energies are 4.715 eV and 0.165 eV , respectively; for C= 0.9 the corresponding energies are 4.470 eV and 0.445 eV, respe ctively. AtC= 0.785, while the resonance position is correctly fitted to 4.01 eV (Fig. 1), we obtain an approximate binding energy of 1.02 eV from a linear extrap olation as in Fig. 2, and 0.99 eV with more precise fitting considering second order correc tions, compared to the accurate prediction of 1.0598 −1.067 eV [19]. This behavior of the low-energy phase shifts, which yields simultaneously the Ps-H resonance and binding energies, in dicates that the use of the model potential (4) in a coupled-channel scheme can lead to a good d escription of Ps-H scattering. We exhibit in Figs. 3 (a), (b), and (c) the present elastic cro ss sections, for Ps scattering by He, (three-Ps-state [4]) Ne and Ar (static-exchange), re spectively, for C= 1, 0.85, and 0.785. The value C= 0.785 yielded the good agreement in the case of Ps-H. For the clo sed- shell atoms, a typical Cclose to 0.85 works well for the 3-Ps-state model in Ps-He and for 5the static-exchange model in Ps-Ar and Ps-Ne. Although the p resent cross sections differ from other theoretical [10, 11] and experimental [20] works at lo w energies for Ps-He (See, Fig. 6 of Ref. [4]), they agree well with the recent measurements of Sk alsey et al. [3] and unpublished work of G. Peach as quoted in Ref. [2]. Table I: Low-energy S-wave phase shifts in radians for Ps-He , Ps-Ne, and Ps-Ar for different kin au. The entries for k= 0 correspond to the scattering lengths in units of a0, incident positronium energy E= 6.8k2eV. kAr(SE) Ne(SE) He(SE) He(3st) 0.0 1.65 1.41 1.03 0.90 0.1−0.164 −0.141 −0.103 −0.088 0.2−0.319 −0.277 −0.202 −0.172 0.3−0.457 −0.404 −0.294 −0.249 0.4−0.572 −0.518 −0.375 −0.315 0.5−0.656 −0.615 −0.444 −0.368 0.6−0.706 −0.694 −0.500 −0.408 0.7−0.720 −0.754 −0.541 −0.433 0.8−0.699 −0.792 −0.569 −0.445 Table II: Low-energy P-wave phase shifts in radians for Ps-H e, Ps-Ne, and Ps-Ar for different kin au. The incident positronium energy E= 6.8k2eV. The numbers in parenthesis denote powers of ten. k Ar(SE) Ne(SE) He(SE) He(3st) 0.1−2.71(−3)−1.63(−3)−8.19(−4)−6.24(−4) 0.2−1.94(−2)−1.20(−2)−6.11(−3)−4.63(−3) 0.3−5.51(−2)−3.55(−2)−1.85(−2)−1.38(−2) 0.4−1.06(−1)−7.17(−2)−3.81(−2)−2.80(−2) 0.5−1.65(−1)−1.17(−1)−6.33(−2)−4.53(−2) 0.6−2.23(−1)−1.66(−1)−9.15(−2)−6.26(−2) 0.7−2.75(−1)−2.15(−1)−1.20(−1)−7.63(−2) 0.8−3.17(−1)−2.60(−1)−1.47(−1)−7.89(−2) In Table I we present the S-wave phase shifts for Ps-Ar (stati c exchange model denoted SE), Ps-Ne (SE), and Ps-He (SE and three-Ps-state models) sc attering for C= 0.85. In Table 6II we present the same for the P wave. The magnitude of the scat tering lengths, low-energy cross sections and phase shifts (well below Ps-excitation t hreshold) increase monotonically as we move from He to Ne and from Ne to Ar. As the effective potentia l for elastic scattering in these cases is repulsive in nature, this indicates an incr ease in repulsion from He to Ne and from Ne to Ar. We find from Figs. 3 that the energy-dependences of the elasti c cross sections are similar for all the closed-shell atoms studied here. The cross secti on has a monotonic slow decrease with increasing energy. This trend is consistently found in all previous theoretical calculations in Ps-He. Also, this is expected as the underlying effective p otential for elastic scattering is repulsive in nature. In conclusion, we have reinvestigated the problem of low-en ergy elastic Ps scattering by H, and He (three-Ps-state) using a symmetric nonlocal elect ron-exchange potential with a parameter C. We further apply this potential to Ps scattering by Ne, and A r (static-exchange). Although the value C= 1 was originally suggested, a slightly lower value C≈0.8 leads to good agreement with accurate experiment [3] and accurate ca lculations [18, 19] in the present cases. Although, a non-symmetric form of the model potentia l provides a fairly good account of the cross section [7], we have found that the symmetric for m is able to provide a more precise description of scattering. The Ref. [4] we have demonstrate d the effectiveness of the present exchange potential in electron-impact scattering. Simpli city of the present exchange potential and the reliability of the present results calculated with i t from a two- (Ps-H) to a 19-electron (Ps-Ar) system reveals the effectiveness of the exchange mod el and warrants further study with it. We thank Prof. B. H. Bransden for his helpful and encouraging comments. The work is supported in part by the CNPq and FAPESP of Brazil. References [1] N. Zafar, G. Laricchia, M. Charlton, and A. Garner, Phys. Rev. Lett. 76 (1996) 1595. [2] A. J. Garner, G. Laricchia, and A. Ozen, J. Phys. B 29 (1996 ) 5961. [3] M. Skalsey, J. J. Engbrecht, R. K. Bithell, R. S. Vallery, and D. W. Gidley, Phys. Rev. Lett. 80 (1998) 3727. [4] P. K. Biswas and S. K. Adhikari, Phys. Rev. A 59 (1999) 363. [5] S. K. Adhikari and P. K. Biswas, Phys. Rev. A 59 (1999) 2058 . 7[6] C. P. Campbell, M. T. McAlinden, F. G. R. S. MacDonald, and H. R. J. Walters, Phys. Rev. Lett. 80 (1998) 5097. [7] P. K. Biswas and S. K. Adhikari, J. Phys. B 31 (1998) 3147; 3 1 (1998) 5403. [8] P. K. Biswas and S. K. Adhikari, J. Phys. B 31 (1998) L737. [9] S. K. Adhikari, P. K. Biswas, and R. A. Sultanov, Phys. Rev . A 59 (1999) 4828. [10] M. I. Barker, B. H. Bransden, J. Phys. B 1 (1968) 1109; 2 (1 969) 730. [11] N. K. Sarkar and A. S. Ghosh, J. Phys. B 30 (1997) 4591. [12] P. K. Sinha, P. Chaudhury, and A. S. Ghosh, J. Phys. B 30 (1 997) 4643. [13] M. E. Riley and D. G. Truhlar, J. Chem. Phys. 63 (1975) 218 2; see, especially, its Eq. (1) and related discussion below it. [14] M. A. Morrison, A. N. Feldt, and D. Austin, Phys. Rev. A 29 (1984) 2518. [15] T. L. Gibson and M. A. Morrison, J. Phys. B 15 (1982) L221. [16] J. R. Oppenheimer, Phys. Rev. 32 (1928) 361. [17] E. Clementi and C. Roetti, At. Data and Nucl. Data Tables 14 (1974) 177. [18] Y. K. Ho, Phys. Rev. A 17 (1978) 1675. [19] A. M. Frolov and V. H. Smith, Jr., Phys. Rev. A 55 (1997) 26 62 and references therein. [20] Y. Nagashima, T. Hyodo, K. Fujiwara, and A. Ichimura, J. Phys. B 31 (1998) 329 and private communication. Figure Caption 1. S-wave singlet Ps-H phase shifts in radian showing the var iation of the resonance position with the variation of Cin (5) using present three-Ps-state model. 2.kcotδandikversus k2plot showing the change in Ps-H binding energy with the varia tion inCas in figure 1 (Energy = 6 .8k2eV). The crossing of the kcotδandikcurves give the energy of the bound state. 3. Cross section of Ps scattering by (a) He, (three-Ps-state model) (b) Ne, and (c) Ar (static-exchange model) for different C:C= 1 (full line), C= 0.85 (dashed-dotted) line, C= 0.785 (dashed line), experiment (box, Ref. [3]). 83.0 3.5 4.0 4.5 5.0 Energy (eV)1.02.03.04.0S-wave Singlet Phase Shift (rad)Figure 1 C=1.0C=0.9C=0.8C=0.785-0.20 -0.10 0.00 0.10 0.20 k2 (units of a0-2)-0.5-0.4-0.3-0.2-0.10.00.10.2ik (units of a0-1)Figure 2 C=1.0 C=0.9 C=0.785 -0.5-0.4-0.3-0.2-0.10.00.10.2 k cot (units of a0-1) C=0.80 1 2 3 4 5 Energy (eV)01234Cross Section (10 -16 cm2)HeFigure 3 0.785 1.0(a) 0.850 4 Energy (eV)0246810Cross Section (10 -16 cm2) Ar(c) (b)Figure 3 1.00.785 0 4 Energy (eV)0246810 Ne 1.00.785 0.850.85

arXiv:physics/0001025v1 [physics.data-an] 12 Jan 2000Forecast and event control: On what is and what cannot be possible Karl Svozil Institut f¨ ur Theoretische Physik, Technische Universit¨ at Wien Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria e-mail: svozil@tuwien.ac.at http://tph.tuwien.ac.at/ /tildewidesvozil/publ/consist. {htm,ps,tex } Abstract Consequences of the basic and most evident consistency requ irement— that measured events cannot happen and not happen at the same time—are shortly reviewed. Particular emphasis is given to event forecast and event control. As a consequence, particular, v ery general bounds on the forecast and control of events within the known laws of physics are derived. These bounds are of a global, statistic al nature and need not affect singular events or groups of events. Principle of self-consistency An irreducible, atomic physical phenomenon manifests itse lf as a click of some detector. There can either be a click or there can be no cl ick. This yes-no scheme is experimental physics in-a-nutshell (at le ast according to a theoretician). From this type of elementary observation, a ll of our physical evidence is accumulated. Irreversibly observed events of p hysical reality (in the context in which they can be defined [1, 2, 3]) are subject t o the primary condition of consistency orself-consistency . Any particular irreversibly observed event can either happ en or cannot happen, but it must not both happen and not happen. Indeed, so trivial seems the requirement of consistency for the set of physically recorded events that David Hilbert polemicised against “another 1author” with the following words [4], “...for me, the opinio n that the [[phys- ical]] facts and events themselves can be contradictory is a good example of thoughtlessness.” Just as in mathematics, inconsistency, i.e., the coexisten ce of truth and falseness of propositions, is a fatal property of any physic al theory. Nev- ertheless, in a certain very precise sense, quantum mechani cs incorporates inconsistencies in a very subtle way which assures overall c onsistency. For instance, a particle wave function or quantum state is said t o “pass” a double slit through both slits, which is classically impossible. ( Such considerations may, however, be considered as mere trickery quanum talk, de void of any operational meaning.) Yet, neither a particle wave functio n nor quantum states are directly associable with any sort of irreversibl e observed event of physical reality. And just as in mathematics it can be argued that too strong cap acities of event forecast and event control renders the system overa ll inconsistent. Strong forecasting Let us consider forecasting the future first. Even if physica l phenomena occur deterministically and can be accounted for (”computed”) on a higher level of abstraction, from within the system such a complete descr iption may not be of much practical, operational use. Indeed, suppose there exists free will. Suppose further tha t an agent could predict allfuture events, without exceptions. We shall call this the strong form of forecasting. In this case, the agent could freely decide to counteract in such a way as to invalidate that prediction. He nce, in order to avoid inconsistencies and paradoxes, either free will has t o be abandoned or it has to be accepted that complete prediction is impossible . Another possibility would be to consider strong forms of for ecasting which are, however, not utilized to alter the system. Effectively, this results in the abandonment of free will, amounting to an extrinsic, detach ed viewpoint. After all, what is knowledge and what is it good for if it canno t be applied and made to use? It should be mentioned that the above argument is of an ancien t type. It has been formalized recently in set theory, formal logic and recursive function theory, where it is called “diagonalization method.” 2Strong event control A very similar argument holds for event control and the produ ction of “mira- cles” [5]. Suppose there exists free will. Suppose further t hat an agent could entirely control the future. We shall call this the strong form of event control. Then this observer could freely decide to invalidate the law s of physics. In order to avoid a paradox, either free will or some physical la ws would have to be abandoned, or it has to be accepted that complete event c ontrol is impossible. Stated differently, forecast and event control should be pos sible only if this capacity cannot be associated with any paradox or contradic tion. Thus the requirement of consistency of the phenomena seems t o impose rather stringent conditions on forecasting and event contr ol. Similar ideas have already been discussed in the context of time paradoxes in relativity theory (cf. [6] and [7, p. 272], “The only solutions to the laws of physics that can occur locally . . .are those which are globally self-consistent” ). Weak forcast and event control There is, however, a possibility that the forecast and contr ol of future events isconceivable for singular events within the statistical bounds. Such occur- rences may be “singular miracles” which are well accountabl e within classical physics. They will be called weak forms of forecasting and event control. It may be argued that, in order to obey overall consistency, s uch a frame- work should not be extendable to any forms of strong forecast or event con- trol, because, as has been argued before, this could either v iolate global consistency criteria or would make necessary a revision of t he known laws of physics. It may be argued that weak forms of forecasting and event cont rol amount to nothing else than the impossibility of any forms of forecasting and event control at all. This, however, needs not to be the case. The laws of statistic s impose rather lax constraints and do not exclude local, singular, i mprobable events. For example, a binary sequence such as 11111111111111111111111111111111 3is just as probable as the sequences 11100101110101000111000011010101 01010101010101010101010101010101 and its occurrence in a test is equally likely, although its s tatistical property and the “meaning” an observer could ascribe to it is rather ou tstanding. Just as it is perfectly all right to consider the statement “T his statement is true” to be true, it may thus be perfectly reasonable to spe culate that certain events are forecasted and controlled within the dom ain of statistical laws. But in order to be within the statistical laws, any such method needs not to be guaranteed to work all the time. To put it pointedly: it may be perfectly reasonable to become rich, say, by singular forecasts of the stock and future values or in hor se races, but such an ability must necessarily be irreproducible and secr etive. At least to such an extend that no guarantee of an overall strategy can be derived from it. The associated weak forms of forecasting and event control a re thus be- yond any global statistical significance. Their importance and meaning seem to lie mainly on a very subjective level of singular events. T his comes close to one aspect of what Jung imagined as the principle of “Synch ronicity” [8]. Against the odds This final paragraphs review a couple of experiments which su ggest them- selves in the context of weak forecast and event control. All are based on the observation that an agent forcasts or controls correctly fu ture events such as, say, the tossing of a fair coin. In the first run of the experiment, no consequence is derived f rom the agent’s capacity despite the mere recording of the data. The second run of the experiment is like the first run, but the meaning of the forecasts or controlled events are different. They are ta ken as outcomes of, say gambling, against other individuals (i) with or (ii) without similar capacities, or against (iii) an anonymous “mechanic” agent such as a casino or a stock exchange. As a variant of this experiment, the partners or adversaries of the agent are informed about the agent’s intentions. 4In the third run of experiments, the experimenter attempts t o counteract the agent’s capacity. Let us assume the experimenter has tot al control over the event. If the agent predicts or attempts to bring about to happen a certain future event, the experimenter causes the event not to happen and so on. It might be interesting to record just how much the agent’s ca pacity is changed by the setup. Such a correlation might be defined from a dichotomic observable e(A, i) =/braceleftBigg +1 correct guess −1 incorrect guess where istands for the i’th experiment and Astands for the agent A. A correlation function can then be defined as usual by the avera ge over N experiments; i.e., C(A) =1 NN/summationdisplay i=1e(A, i). From the first to the second type of experiment it should becom e more and more unlikely that the agent operates correctly, since h is performance is leveled against other agents with more or less the same capac ities. The third type of experiment should produce a total uncorrelation. References [1] Daniel B. Greenberger and A. YaSin. “Haunted” measureme nts in quan- tum theory. Foundation of Physics , 19(6):679–704, 1989. [2] Thomas J. Herzog, Paul G. Kwiat, Harald Weinfurter, and A nton Zeilinger. Complementarity and the quantum eraser. Physical Review Letters , 75(17):3034–3037, 1995. [3] Karl Svozil. Quantum interfaces. forthcoming, 2000. [4] David Hilbert. ¨Uber das Unendliche. Mathematische Annalen , 95:161– 190, 1926. [5] Philip Frank. Das Kausalgesetz und seine Grenzen . Springer, Vienna, 1932. 5[6] John Friedman, Michael S. Morris, Igor D. Novikov, Ferna ndo Echever- ria, Gunnar Klinkhammer, Kip S. Thorne, and Ulvi Yurtsever. Cauchy problem in spacetimes with closed timelight curves. Physical Review , D42(6):1915–1930, 1990. [7] Paul J. Nahin. Time Travel (Second edition) . AIP Press and Springer, New York, 1998. [8] Carl Gustav Jung. Synchronizit¨ at als ein Prinzip akaus aler Zusam- menh¨ ange. In Carl Gustav Jung and Wolfgang Pauli, editors, Natur- erkl¨ arung und Psyche . Rascher, Z¨ urich, 1952. 6

arXiv:physics/0001027 13 Jan 2000 1Newtonian Relativity, Gravity, and Cosmology Joseph L. McCauley+ Department of Physics University of Oslo Box 1048, Blindern N-0314 Oslo and Institute for Energy Technology Box 40 N-2007 Kjeller jmccauley@uh.edu Abstract Well-known to specialists but little-known to the wider audience is that Newtonian gravity can be understood as geodesic motion in space-time, where time is absolute and space is Euclidean. Newtonian cosmology formulated by Heckmann agrees implicitly with Cartan's formulation but does not unfold the underlying geometric picture in space-time. I present the transformation theory of Newtonian mechanics and gravity developed by Cartan and Heckmann, and show via coordinate transformations that Heckmann's Newtonian cosmological model has a center, so that the cosmological principle cannot hold globally. It is possible to confuse relativity principles with a position of relativism. Mach's principle has much to do with the latter and little to do with the former. Both general relativity and nonlinear dynamics inform us that most coordinate systems are defined locally by differential equations that are2globally nonintegrable: global extensions of local coordinates usually do not exist. I explain how defining inertial frames locally by free fall short circuits Mach's philosophic objections to Newtonian dynamics. + permanent address: Physics Department University of Houston Houston, Texas 77204 USA31. Newtonian dust In Newtonian theory we can consider a pressureless dust obeying the coupled quasilinear equations of hydrodynamics dv dt = ∂v ∂t + v⋅∇v = - 1 ρ ∇Φ ∂ρ ∂t + ∇⋅ρv = 0 ∇2Φ = 4πρ. (1) In a cosmological model each dust particle represents a galaxy. The connection with Newtonian particle dynamics is provided by the method of solution of (1), the method of characteristics. Equations (1) are quasilinear partial differential equations whose characteristic curves [1,2] are generated locally by the differential equations dt 1 = dxkvk = dvk - ∂Φ ∂xk . (4) These are simply the equations of Newtonian mechanics: if we think of N dust particles then each dust particle obeys Newton's law in the gravitational field defined by the other N-1 dust particles. We can rewrite the characteristic equations in the form4xk = vk vk = - ∂Φ ∂xk (4b) so that we might attempt to study the nonlinear dynamics of galaxy formation and evolution in the 3N dimensional phase space of the x's and v's, or as Newton's second law d2xi dt2 + ∂Φ ∂xi = 0 (5) in 3-space. We can think of streamlines in a 6-dimensional (x,v) phase space (phase flow picture) if and only if the solutions x and v are finite for all real finite times, in which case all singularities of power series solutions of (5) are confined to the complex time plane. We will see below that this condition is violated by spontaneous singularities in Newtonian cosmology. In this case the language of jet space and caustics [3] provides the right approach to the analysis of the nonlinear dynamics of galaxy motions. Newtonian cosmology is still of interest for at least two main reasons. First, it reproduces the same equation for the Hubble expansion as the Friedmann model in general relativity. Second, and more fundamentally, Cartan's formulation and interpretation of Newtonian dynamics provides the best take-off point for general relativity. The problem of defining average solutions in theoretical cosmology is unsolved in general relativity although much headway has been made in Newtonian cosmology. It is hoped that perturbation theory in Cartan's formulation of Newtonian theory may shed some light on how to handle the problem in general relativity [4]. 52. The cosmologic principle The Hubble expansion is inferred nonuniquely from redshift data. The galaxy distribution is obtained from the analysis of data derived from redshift data. The analogy of galaxies in an expanding universe with points on an expanding balloon is well-known. In the Hubble's law interpretation of redshifts the matter distribution of the observable part of the universe is implicitly presumed to be isotropic and homogeneous, so that galaxies appear to recede from one another with a steady radial velocity field [5] v = H(t)r (6) where H(t) is the very slowly-varying Hubble 'constant' and is given approximately by H(t) ≈h-1100km/s.Mpc, with h≈.5 to .6 in our present epoch. Since H(t) varies with time the presumed equivalent observers are accelerated relative to one another. The main point to bear in mind is that this simplest form (6) of the Hubble law implicitly presumes the cosmological principle, so that the usual method of data analysis in cosmology would have to be revised or abandoned if the cosmological principle could be shown to be false. For example, one does not yet know how to generalize (6) to handle the case of a hierarchical universe, or any other nonhomogeneous universe. The cosmologic principle is sometimes stated in the following form: All observers are equivalent; there are no preferred observers: there is no preferred vantage point from which to observe the gross motions of matter in the universe. This statement is not precise. I will emphasize in part 5 below6that the standard definition of equivalent observers simultaneously defines those observers as preferred : from their globally accelerated (but locally inertial) reference frames, and from no others, the Hubble expansion would look the same. Another way to express the content of the cosmological principle is to assume that the distribution of matter in the universe is globally (more or less) homogeneous and isotropic. Equivalently, one can use the assumption that the density is spatially constant at fixed times, with only small deviations from uniformity. The cosmological principle appears superficially to resemble a principle of relativity, but that principle is neither the basis for the general theory of relativity nor is it demanded by that theory. The cosmological principle is not required by any known law of physics. Some writers allude to 'the Copernican Principle' but that principle is merely another statement of belief in the cosmological principle (Copernicus certainly did not adhere to the 'principle' attributed to him, because his own model universe had a center). I defer the discussion of the observational basis for or against the cosmological principle until a second paper, and discuss here only the attempt to realize the cosmological principle within an infinite Newtonian universe. Newtonian cosmologies require flat space (a flat space is one where Cartesian coordinates exist globally because the Riemann curvature vanishes everywhere). Flat spaces (and Newtonian mechanics) can be realized in two ways: (i) as an infinite unbounded space, or (ii) as a finite unbounded space in the form of a flat 3-torus. The latter is equivalent to solving (1) with periodic7boundary conditions, a common procedure in numerical simulations of the N-body problem. I review next one of the main building blocks of Newtonian (and Einsteinean) cosmology: the principle of equivalence. The discussion leads us into the transformation theory of Newtonian mechanics, and to Cartan's formulation of Newtonian theory, which reflects the correct geometric nature of gravity: Cartan's formulation agrees with the infinite light-speed limit of general relativity, whereas the standard textbook interpretation of Newtonian gravity as a scalar potential giving rise to a covariant force does not. In order to arrive well-informed at the goal it's necessary to start at the beginning. 3. Free fall and the principle of equivalence Consider N particles of fixed mass mi interacting only via gravity in what follows. For any one of the N bodies the law of inertia states that dvi dt = 0 (7) whenever no net force acts on the body, and in a gravitational field with potential Φ Newton's second law d2xi dt2 + ∂Φ ∂xi = 0 (5) generalizes the law of inertia. I assume in all that follows that the potential Φ is independent of the mass m i of the body described by (5).8Standard treatments of Newtonian mechanics assume, without explanation, that the gravitational potential transforms like a scalar (Newton did not tell us how gravity should transform under arbitrary coordinate changes). The gravitational force is then forced to transform like a covariant vector. This assumption is not only unnecessary, it also does not agree with the infinite light-speed limit of general relativity. When it is used then the transformation theory of Newtonian mechanics does not emphasize the following fundamental fact about gravity: a reference frame in free-fall is locally inertial. In a freely falling frame a ball tossed into a vacuum obeys the law of inertia locally (not globally), is force-free and has a trajectory in the freely falling frame that is a straight line with constant speed (by local I mean the consideration of a small enough spatial region that the gradients of the gravitational field can be neglected for a short but finite time, whereas global refers to arbitrarily large spatial regions and time intervals). The experiment could be performed by an astronaut near a satellite falling freely about about the earth, or by a robot made to jump off a cliff on Mars. Treating the gravitational force as a covariant vector emphasizes that gravity cannot be transformed away globally (tidal forces cannot be transformed away, e.g.). We want to use instead a transformation theory that explicitly makes use of the fact that the gravitational force can be made to vanish locally. The second related fact, emphasized by Einstein, is the equivalence principle. If we transform from a local inertial frame (meaning here any freely falling frame, or any frame connected by a Galilean transformation to a freely falling frame) to a linearly accelerated one,9x'i = xi - bi(t) v'i = vi - bi(t) a'i = ai - bi(t), (8) where db i/dt is constant for Galilean transformations, then Newton's second law becomes d2xi dt2 + bi + ∂Φ ∂xi = 0 . (5b) When d2bi/dt2 is constant then this is the same as if the accelerated frame were replaced by a frame that is stationary in a gravitational field with local field strength gi = d2bi/dt2, corresponding to a planet whose surface is transverse to the xi-axis. By giving up the fiction that the gravitational force transforms globally like a covariant vector we shall see that both of these fundamental empirical facts about how gravity behaves locally can be adequately emphasized within the transformation theory of Newtonian dynamics. 4. Newtonian gravity transforming as a scalar potential Given Newton's second law, if one asks for the force that produces an ellipse with the point of attraction at one focus, then the result is the inverse square law of attraction. Because this method of derivation of gravity is due to Newton, he surely knew that his description of gravity as an inverse square law force is accurate only to the extent that the orbits of planets can be accurately described as (nonprecessing) ellipses [6]. Note that no assumption was made here about how gravity behaves under arbitrary coordinate10transformations. I next review the usual (and wrong) treatment of the transformation theory of Newtonian dynamics, where gravity is supposed to transform as a scalar potential. Assume that Newtonian gravity can be described under arbitrary coordinate transformations as a scalar potential Φ. This means that under differentiable coordinate transformations x'k = fk(x,t), x i = gi(x',t) (9) the gravitational potential transforms like Φ'(x',t) = Φ(x,t) (10) which seems obvious from a superficial standpoint1. The gravitational force F = - ∇Φ must then transform like a covariant vector, because ∂Φ' ∂x'k = ∂xi ∂x'k ∂Φ ∂xi. (11) It follows that if F = - ∇Φ = 0 globally in one frame then F' = - ∇'Φ' = 0 globally in all (global) frames. This global perspective is misleading because it masks the all-important fact that the gravitational force vanishes in local inertial frames that are defined by free fall. Why does this matter? 1 A scalar that is, in addition, invariant would transform like Φ(x',t) = Φ(x,t).11There are no known global inertial frames (stars and galaxies aren't fixed, but accelerate). All known inertial frames are local, and the only local inertial frames that we know in nature are those provided by free fall. By local, I mean simply that the gravitational field is approximated over a small but finite region by a constant, so that the derivatives of the field are ignored (tidal effects are global and are therefore ignored). The earth, in free fall about the sun, is a good local inertial frame over small spatial regions and over times that are short compared with the rotation period of the earth about it's axis. Were this not true then Galileo could not have discovered the local laws of free fall and inertia from his borderline medieval perspective of Archimedean empiricism combined with neo-Platonic argumentation. Galileo understood the law of inertia as a local, not global principle, but for the wrong reason [7]: he did not deviate from Plato and Aristotle in regarding uniform circular motion as 'natural', and as requiring no mechanical explanation (Aristotle and Galileo also required no explanation for gravitational free-fall and radial coalescence of mass via gravity). Galileo largely regarded constant speed linear motion (force-free motion, which we identify as the law of inertia) as only a local tangential description of a global orbit that he believed (in agreement with Plato and Aristotle) should be circular, at uniform speed. Galileo described parabolic trajectories of canon balls but did not realize that gravity also makes the planets accelerate in approximately circular orbits about the sun. Newton was the first to make the connection between the trajectories of apples and the orbit of the moon, and beyond, although Huyghens preceded Newton in using the law of inertia (for tangential motions) combined with the second law (for radial motions), before Newton, to prove that an inverse square force of attraction is necessary for uniform circular motion. Galileo's local dynamics was new and anti-12Aristotelian, but was inconsistent with his global view of astronomy, which remained Copernican-Platonic in spite of Kepler's revolutionary advances in the very same era. It was his literal belief in the Copernican system that motivated Galileo's argument and experiments supporting the law of inertia in order to explain to the Aristotelians why, as the earth moves, we are not aware of it. The modern viewpoint on Newtonian dynamics, informed by general relativity and nonlinear dynamics, abandons the search for global inertial frames. The modern viewpoint emphasizes that freely falling frames are locally inertial. Hence, not all inertial frames can be connected by Galilean transformations. For example, frames in free fall on opposite sides of the earth are both locally inertial, but because both are accelerated toward (or away from) each other they cannot be connected via a Galilean transformation. By abandoning the fiction of global inertial frames and adopting instead the local viewpoint we will not need Newton's global idea of absolute space. The latter requires the unnecessary (and unphysical, because unverifiable) assumption that Cartesian axes can be extended all the way to infinity, an idea that more or less goes back to Descartes, who arrived at the law of inertia independently of Galileo and regarded it as a global law of nature (Galileo's universe was Copernican-Platonic spherical and finite, Descarte's was infinite and unbounded). The noninertial effects identified by Newton as acceleration relative to absolute space are present if we transform from a local frame in free fall to a locally accelerated frame, but the acceleration is relative to a frame that is in free fall relative to the mass distribution that generates the local gravitational field. I will also discuss Mach's criticism of the law of inertia and Newton's second law in part 6 below.13Continuing with the traditional treatment of gravity as a scalar potential, Newton's second law in an inertial frame is given by d2xi dt2 + ∂Φ ∂xi = 0 , (5) and is covariant under Galilean transformations (set d2bi/dt2 = 0 in (8)). The law of inertia in an inertial frame is given by dvi dt = 0 (7) and is not merely covariant but is also invariant under Galilean transformations . This invariance principle (Galilean relativity) describes mathematically the fact that no mechanical experiment can be performed that detects motion at constant velocity relative to a local inertial frame. The mathematical basis for this is that solutions of (7) in two separate inertial frames connected by a Galilean transformation, but using the same initial conditions, are identical . Galilean invariance is the basis for the identity of outcome of identically prepared experiments in two separate inertial frames [6]. I should warn the reader that many older books and articles on special and general relativity (and also some newer ones) have (mis-)used the word "invariant" where instead they should have used the word "scalar". For example, statements like "ds2 = gµνdqµdqν is invariant" must be replaced by "ds2 is a scalar" under coordinate transformations. The distinction between14scalars and invariants is made clear by Hamermesh [7]. Havas [9] discusses Newtonian gravity transforming both as a covariant vector and as an affine connection, but does not distinguish the word "scalar" from the word "invariant". He also states that "... the fundamental equations of Newtonian mechanics are invariant under the Galilei group, those of the special theory of relativity under the transformations of the Lorentz group, and those of the general theory of relativity under all coordinate transformations ("principle of general covariance")", also confusing covariance of vector equations with invariance of differential equations and their solutions. Covariance of vector equations does not define the physics, which is defined by a principle of relativistic invariance: the law of inertia is covariant with respect to Galilean transformations, but the Galilean relativity principle is reflected by the invariance of the law of inertia (7) and its solutions under Galilean transformations. Newton's second law is covariant, but not invariant, under the same group of transformations. Covariance does not carry the weight of a principle that can be imposed externally because it can always be achieved for any equation of motion [10]. Rewriting a law of motion that is correct in a restricted class of frames of reference in covariant fashion does not change the underlying invariance principle that defines the physics [6,11]. As an example, I will show, following Heckmann, how Newton's second law can be made covariant under transformations that include linearly accelerated frames (covariance with respect to transformations to and among rotating frames is also possible [11]). The invariance principle is still Galilean invariance. Cartan [6] showed how Newton's second law can be written in covariant form with respect to15transformations to arbitrarily accelerated frames in Newtonian space-time by treating the gravitational force as a "nonintegrable connection". This beautiful geometric picture does not change the fact that Galilean relativity is still the basic invariance principle of classical mechanics. In parts 7 and 8 we see how Cartan brought to light that the only essential difference between Newtonian and Einsteinean mechanics is the replacement of local Galilean invariance by local Lorentz invariance. 5. Newtonian gravity transforms like a gauge potential I start by abandoning the unnecessary assumption that gravity behaves like a scalar potential under arbitrary coordinate changes. Instead, under a transformation from an inertial frame to a linearly accelerated frame, x'i = xi - bi(t) v'i = vi - bi(t) a'i = ai - bi(t) (8) if we allow the gravitational potential to transform like a gauge potential, Φ'(x') = Φ(x) + x ibi (12) then Newton's second law is covariant with respect to transformations to and among linearly accelerated frames [5], d2x'i dt2 + ∂Φ' ∂x'i = 0 . (5c)16The covariance of Newton's second law under transformations to accelerated frames does not mean that accelerated frames are equivalent for the purpose of doing experiments in physics [6]. Experiments prepared and performed identically in two differently linearly accelerated frames will necessarily yield different numbers, because the solutions of Newton's laws in these frames are not invariant under the transformation (8) whenever d2b/dt2≠ 02. Whenever d2b/dt2=0 then we retrieve the Galilean transformation and the equivalence in outcome of identically prepared experiments, which is the basis for the human ability to discover laws of nature in the first place [12]. This viewpoint disagrees with Einstein's claims about the physical importance of covariance and the use of arbitrary reference frames for the expression of the laws of physics. The modern viewpoint (emphasized earlier by Fock and Wigner) can be found in the texts by Weinberg [13] and by Misner, Thorne, and Wheeler [14], for example. If Cartan had thought only about motion in space, rather than in space-time, then he might have arrived at the point of view discussed in the next section. 6. Newtonian cosmology (Alta Via Heckmann) The cosmological principle, interpreted globally, would require a matter density that is everywhere constant, with only small fluctuations from its average value. It is well known (see Rindler [15]), but is often ignored in elementary texts and monographs, that a uniform density is impossible in an infinite Newtonian universe. In other words, the cosmological principle can't be realized globally in an infinite Newtonian universe, although the same17assumption is allowed on a flat 3-torus, where Newtonian mechanics also holds [16]. The first assertion that uniform densities of infinite extent are impossible in Newtonian mechanics is due to Neumann [17]. Milne [18] and McCrea thought that they had discovered a local approach that avoids the global infinity pointed out by Neumann [17,18], but later were proven to have been wrong on that point [19]. Heckmann [5] assumed without checking carefully enough that Milne and McCrea indeed had found a way to avoid the fact that the gravitational force can't be defined globally inside a uniform mass distribution that is infinite in extent. This is not the main point of interest here: Heckmann independently discovered a key result of Cartan's geometric formulation of Newton's laws of motion and gravity, namely, that the principle of equivalence can be built explicitly into Newtonian theory if one assumes that the gravitational potential transforms like a gauge potential under transformations to linearly accelerated frames [5]. I begin with the hydrodynamics description of a pressureless Newtonian dust, dv dt = ∂v ∂t + v⋅∇v = - 1 ρ ∇Φ ∂ρ ∂t + ∇⋅ρv = 0 ∇2Φ = 4πρ, (1) whose characteristic curves, generated by Newton's second law18dt 1 = dxkvk = dvk - ∂Φ ∂xk , (4) and are the trajectories of the dust particles. Solving the quasi-linear partial differential equations (1) is equivalent to solving the nonlinear differential equations (4) with specified initial conditions (and with some prescribed initial density distribution ρ(x,0) at t=0 that determines Φ(x,0)) Under transformations x'i = xi - bi(t) v'i = vi - bi(t) a'i = ai - bi(t) (8) the Abelian gauge transformation rule Φ'(x') = Φ(x) + x ibi (12) yields covariance of Newton's second law, d2x'i dt2 + ∂Φ' ∂x'i = 0 . (5c) Note that bk=constant describes translations, bk=Vkt with Vk = constant describes restricted Galilean transformations, and the frame is linearly accelerated if d2bk/dt2≠0.19A cosmological model is defined by a preferred class of relatively moving coordinate systems. In the preferred frames hypothetical equivalent observers with identical equipment and using equivalent techniques are supposed to be able to observe "the same coarse features of the universe". Clearly, such an assumption will require some yet-to-be specified degree of uniformity of the matter distribution. In Heckmann's model1 one defines preferred reference frames as those for which the dust velocity field is invariant (the analog in general relativity is a maximally-symmetric space, which can be defined via invariance of the metric and mass-energy tensor [13]). When x'i = xi - bi(t) (8b) we have vi(x,t) = v' i(x',t) + bi(t) (8c) which represents the Newtonian law of combination of velocities. Require in addition that v'i(x',t) = v i(x',t) (13) which means that the velocity field v must be invariant for the preferred class of frames that will be defined by using (13) to determine the functions bi(t) in (8b). From (8b), (8c), and (13) we obtain20vi(x,t) = v i(x - b,t) + bi (14) so that ∂vi(x,t) ∂xk = ∂vi(x - b,t) ∂xk , (15) which means that this derivative is independent of x, so that vi(x,t) = a ik(t)xk + ai(t). (16) I leave it as an exercise for the reader to show that a i=0 is required, yielding vi(x,t) = a ik(t)xk (17) which is a generally anisotropic Hubble law of expansion or contraction of the matter distribution. To go further we need the equation of continuity, which we can more conveniently impose in the particle mechanics form [6] dxi(t) = J(t) dxio∏ i=13 ∏ i=13 . (18) Here, xi(t) is the position of a dust particle at time t, xio = xi(0) is the initial condition at t = 0, and J(t) is the Jacobian of the one parameter transformation from the variables x io to the variables x i(t). It is well-known that21J = ∇⋅vJ (19) where v is the velocity field of the dust particles in the six dimensional phase space. Combining this with the Hubble law (17) yields J J = ∇⋅v = Tr a(t) (20) where a(t) is the 3x3 matrix in the Hubble law (17). Hydrodynamicists might say that we are working in the Lagrangian rather than Eulerian picture, but we are simply studying the characteristic equations (4) by standard methods of nonlinear dynamics. The Lagrangian picture of hydrodynamics amounts to integrating Newton's equations (5) backward in time while using the fact that the initial conditions xio are trivially conserved along characteristics. If the solutions of (5c) do not have spontaneous singularities (singularities at real, finite times) then we have a flow in a six dimensional (x,v) phase space, where the trajectories can be thought of as streamlines, and the transformation from the xio to the xi(t) can be regarded as a one-parameter coordinate transformation with Jacobian J(t). If the initial conditions and dynamics permit the definition of a once- differentiable matter density ρ then dρ dt = ρ + v⋅∇ρ = - ρ∇⋅v = -ρ J J (21) along dust particle trajectories so that22 ρ = ρo/J(t) (22) along a characteristic curve, where ρo is independent of t. In chaotic nonlinear dynamics initial conditions may not permit the definition of a smooth pointwise density but one can always work with coarsegrained pictures where, at least initially, densities are piecewise constant. The gravitational potential obeys - ∂Φ ∂xi = vi + vk∂vi ∂xk = - aikxk - aikaklxl (23) and with a generally anisotropic Hubble law v i=aikxk we obtain ∂2Φ ∂xi∂xk = ∂2Φ ∂xk∂xi , (24) This condition guarantees that Φ exists and yields daik/dt=da ki/dt, which means that a is a symmetric matrix, aik(t) = aki(t). (25) Therefore, the gravitational potential corresponding to the anisotropic Hubble motion (17) is given by Φ(x,t) = - 1 2 (aik(t) + ail(t)akl(t))xixl. (26)23It is easy to show that the potential is invariant under the transformations (8): Φ(x',t) = Φ'(x',t). We can now see the impossibility of a static Newtonian universe, where no evolution could occur. With ∇⋅v = Tr a(t) = - 1 ρ dρ dt (27) and ∇2Φ = - Tr a(t) - a(t)ik2∑ i,k = 4πGρ (28) we find that d dt 1 ρ dρ dt = a(t)ik2∑ i,k + 4πGρ ≥ 0 . (29) Therefore dρ/dt = 0 is impossible and ρ(t) cannot be a constant. Is is still possible that the density ρ is spatially constant for fixed times (is the cosmological principle realizable in this model)? I defer the analysis until the end of this section, assuming temporarily without proof that a uniform and isotropic Newtonian universe is possible at any given time t. Consider next only isotropic Hubble motions. Starting with vi(x,t) = a ik(t)xk (17)24and ∇⋅v = Tr a(t) = - J J (30) and then imposing the isotropy condition aik(t) = δik J 3J , (31) we find that can write vi(x,t) = J 3J xi (32) where it follows that the Hubble 'constant' is given by H(t) = (dJ/dt)/3J. Since the dust particle trajectories are generated by xi = J 3J xi (33) and one integration yields the time evolution rule xi(t) = (J(t))1/3 xio. (34) Writing dR/dt = (dJ/dt)/3J yields Hubble's law in the form vi(x,t) = R R xi (33b) so that the dust particle trajectories can be written as25xi(t) = R(t) x io. (34b) The time evolution of the dust particles is just a rescaling of the initial conditions x io where the scale factor is R(t) = (J(t))1/3 , and J(t) is the Jacobian of the time evolution transformation. We can also write ρ(t) = ρo R(t)3. (35) Next, I want to solve for the parameters bi(t) in order to see what the preferred class of equivalent reference frames looks like. Combining the Newtonian law of addition of velocities (8) with the generalized Hubble motion (17) and the invariance condition (13) that defines our cosmology we obtain vi(x',t) = a ik(t)xk - bi(t) =a ik(t)(x'k + bk(t)) - bi(t) = aik(t)x'k . (36) This yields the differential equations bi = aikbk (37) that define the preferred frames. In the isotropic model, a ik = δik (dJ/dt)3J = δik (dR/dt)R, we have26bi = R R bi (38) which yields bi(t) = cR(t) (39) with c a constant. Therefore the preferred frames are defined by the one parameter coordinate transformations (c is the parameter) x'k = xk - cR(t) , (40) where R(t) = (J(t))1/3. Translational invariance would require that we find an additive constant when we solve (the Friedmann equation) for R(t). Globally seen, the preferred frames are not inertial frames (we will see later that the form of R(t) rules out global Galilean invariance). However, because each frame is in free fall in the gravitational field of the other N-1 dust particles, these frames are locally inertial. We now obtain the equation of motion for the expanding universe. Combining - Tr a(t) = a(t)ik2∑ i,k + 4πGρ (41) with the local isotropy assumption a ik = δik(dR/dt)/R with ρ = ρo/R3 we find -3d dt R 3R = (R R)2 + 4πGρo R3 (42)27which we can rewrite as Newton's second law for free fall in the field of a point singularity R = - 4πGρo R2 (43) at R = 0 (since we are doing classical mechanics, this result should give us a hint that this universe has a center). Newton's equation of motion can be integrated once to yield the Friedmann equation [5] R2 2 - 4πGρo R = ε = constant (44) Note that R is only a scale factor so that one should not interpret ρo as the mass of the universe. With ε = 0 we have expansion with dR/dt=0 at R= ∞. ε>0 yields expansion with finite expansion rate at infinity, and when ε<0 there is expansion to a finite value Rm = 4πGM/(-ε) followed by collapse to R=0 in finite time (spontaneous singularity of equation (43)). The cosmological principle appears to hold locally: the density is spatially uniform except near the boundary of the mass distribution, which is ignored in this analysis (boundary conditions on Φ at large xi were never discussed), and except near the point of gravitational collapse (the singularity of the mass distribution). I will explain below why the mass distribution is necessarily finite in extent, why the uniform density must be cut off after a finite distance, and where the finite-sized boundary of the mass distribution was swept under the rug in Heckmann's treatment. Locally, inside the mass28distribution and far from the edges of the finite universe (imbedded in an infinite Euclidean space), the density is spatially uniform, the universe is locally homogeneous and isotropic, so that the cosmological principle holds locally (Milne, McCrea, and Heckmann had believed that the cosmological principle would hold globally in this model). A Newtonian universe where the cosmological principle holds globally cannot have a singularity in time for the density because a spontaneous singularity of (43) defines the center of the universe. I stress that this finite time singularity of Newton's law is not the same geometrically/qualitatively as the collapse of the entire space-time manifold in general relativity, although the Friedmann equation (43) describes both cases quantitatively. Where is the Neumann-Seeliger infinity hidden in the previous analysis if we would assume, as did Heckmann (and later Bondi [20], who repeated the simpler Milne-McCrea analysis), that we can take ρ(t) to be spatially constant all the way to infinity? I will show first that the universe defined by our model has a center. A hypothetical observer who's near neither the edge nor the center of the mass distribution won't know in advance where the center is located before the collapse occurs, unless he's carried out calculations that go beyond the scope of these lectures. In other words, a preferred observer sees the contraction that occurs for ε < 0 but, because of invariance of the velocity field, all observers see the same contraction so that none of these observers can say in advance where the collapse is going to take place. This center differs both qualitatively29and quantitatively from the anthropic center of the universe that Plato and Aristotle imagined, and also from Copernicus' neo-Platonic godly center. To see why there is a center consider the case where ε<0: 1. Gravitational collapse occurs in finite time (R(t) vanishes in finite time). 2. xi(t) - x j(t) = R(t)(x io - xjo) vanishes as the scale factor R vanishes; all particle displacements collapse as R vanishes. 3. x i(t) = R(t)x io vanishes as R vanishes, so that each and every dust particle approaches the origin of its coordinate system as the collapse proceeds. 4. x' k = xk - R(t) approaches xk as R vanishes, so that all coordinate origins coincide at the time of gravitational collapse. The entire mass distribution disappears into a spontaneous singularity of the nonlinear differential equation (43) in finite time. In other words, this model of the universe has a center. Furthermore, the uniform mass distribution cannot be infinite in extent and must have a boundary. Where was the violation of this condition hidden in the analysis? By introducing the idea of the density and then assuming that ρ(t)J(t) = ρo is constant we implicitly restricted our analysis to local internal regions far from any boundary, where the mass density (if it exists) may look nearly uniform. If we would assume that the same condition could hold all the way to infinity then we would not be able to define either Φ(x,0) or the gravitational force. In other words, the Neumann-Seeliger infinity was hidden in the initial condition ρo. Milne, McCrea and Heckmann did not face the infinity (that infinity informs us that a uniform mass distribution cannot be spatially infinite) because they ignored the boundary conditions on the30potential Φ (Milne had offered his model as an example of how pure mathematics can be used to short-circuit the need for physics [17]). In a global analysis the velocity field and potential cannot be invariant under the transformations (8) that define the preferred observers locally. The expansion/contraction does not look the same for observers near the edge as for observers far from the edges. An observer with a global viewpoint (a space traveler outside the mass distribution, e.g.) could know where the collapse is going to occur, while an observer well within the interior (who hasn't done the necessary calculations) would be ignorant of the fact that the universe has a center. The size of the universe can be taken as proportional to the scale factor R(t). For a spherical universe the center of the universe lies at the sphere's center. The cosmological principle doesn't hold globally but holds locally well within the interior, at times that are not too near to the collapse time. 7. Relativism or relativistic invariance? "An influence of the local inertial frame on the stars is not acceptable, and hence it must be assumed that the local inertial frame is determined by some average of the motion of the distant astronomical objects. This statement is known as Mach's principle." H. Bondi, in Cosmology [ 20] It is difficult to discuss Mach's principle. There is still no convincing realization of Mach's Principle. The number of different statements of Mach's principle may be on the order of magnitude of the number of different writers31on the subject. Interesting attempts to realize Mach's principle in the context of classical mechanics can be found in references [21,22,23,24,25]. Mach himself offered no principle but criticized both the law of inertia and Newton's second law. Mach [26] wrote from a standpoint of relativism in philosophy and physics (Descartes and Huyghens were earlier advocates of relativism2). Roughly speaking, Mach asserted that only relative positions, velocities, etc. between masses should enter into laws of motion, and that mechanics should not be local: the entire universe should be considered in defining inertia. This is a holistic point of view. I will argue below that local laws of motion are enough, that holism and relativism are unnecessary. Mach's call for relativism is not the same as a principle of relativistic invariance (Galilean or Lorentzian), and (upon closer inspection) amounts to a criticism of principles of relativistic invariance because those principles are based on translational invariance and invariance with respect to transformations to uniformly moving frames relative to a local inertial frame that replaces Newton's idea of a fixed, global inertial frame. Mach criticized the law of inertia [26] (which is essentially the same as criticizing Galilean, and also Lorentzian, invariance) as not having been derived from an 2 Barbour [28]argues that Huyghen's belief in Cartesian mechanism prevented his getting credit for both Newton's second law and the law of gravity. Newton, in contrast, hated Cartesian relativism and was driven to dispute it [29] . Newton understood the difference between (Galilean) relativity and relativism. Descartes adopted a position of relativism in an attempt to avoid the Index and the inquisition. By asserting that the difference between taking the sun to be fixed or the earth to be fixed (choice of coordinate system, in our language) is merely a matter of convenience, he hoped to avoid being charged with disputing the notion that the earth may stand still. Following Kepler, (whom Galileo completely ignored) Newton showed that this is not merely a matter of convenience, that the sun constitutes a better approximation to an inertial frame than does the earth for the calculation of planetary motions. One can calculate planetary motions from the earth's frame if one wishes, but the calculations (obtained via transformation from an inertial frame) will be much more complicated [30].32equation of relative motion of interacting bodies. Einstein was stimulated by the criticism of Newtonian and Galilean ideas found in Mach's historico- philosophical book [26], as were many other physicists, psychologists, and philosophers [29], but Einstein did not succeed in incorporating Mach's demand for relativism into physics, or even in clarifying what we should understand as Mach's principle. I will explain next how Mach's relativismic (as opposed to relativistic) criticism of Galileo and Newton can be short- circuited. It is not an accident, or a mere misfortune of old age (as some authors have lamented), that Mach did not see his ideas of relativism reflected in Einstein's work on either special relativity or gravity. Mach wanted to define forces and relative motions first, and then derive the law of inertia from a reformulated second law of Newton (Einstein did the opposite--see part 8). He was misled by the pre-Einsteinean assumption that inertial frames are globally possible. We now expect that they are not, and we can reinterpret Newtonian physics with the benefit of both Einstein's contributions and insight provided by modern nonlinear dynamics. This is done, following Cartan, in the next section. The following point of view eliminates Machian-style objections to the appearance of inertial terms interpreted as accelerations relative to absolute space, where absolute space is usually regarded as some undefined collection of global inertial frames, or as one global frame of reference fixed in the (only approximately) "fixed stars". In place of this global picture we can instead adopt the following position, which relies upon and emphasizes the typical nonintegrability of local laws of motion.33A neutral body subject to no other net force is always locally in free fall in the net gravitational field of the rest of the matter in the universe. A body in free fall defines a local inertial frame. There need be no other inertial frames in the universe. Noninertial effects then occur relative to a local frame that is accelerated relative to a local inertial frame, and therefore occur indirectly relative to the masses that determine the local gravitational field over laboratory or observational times (the earth falls freely about the sun, the moon falls freely about the earth in the field of both the earth and sun, etc.). A local frame is approximately inertial only for a finite time. In that frame the law of inertia holds, dvi dt = 0, (7) where we can [6] and should think of the Cartesian axes x i, xi(t) = vit + xio, (45) as generated by the motion of three free tracer-particles moving rectilinearly with constant speeds vi. In other words, the law of inertia (7) generates the local inertial frame, whose axes are defined by (45). Mathematically, (7) is globally integrable in Newtonian mechanics but we ask next whether the applicability of this formal mathematical condition might be preempted by the physics of tracer particles. Given finite velocities vi, the spatial extent of the Cartesian axes (45) would be limited by the finite time t max over which the local frame is approximately inertial, which is determined by the neglected gradients of the local gravitational field. Also, for any finite velocity v the differential equation (7) that generates the axes (45) locally will fail to hold at34large times t >> tmax because of the (neglected) field gradients, so that (7) would have to be replaced globally by (5). Even in classical mechanics there is no way to generate infinitely-long straight lines via particle motions when it is taken into account that matter is distributed throughout the universe. However, in classical mechanics the speeds vi can be as large as you like so that, in principle, the axes of the inertial frame can be imagined to extend spatially as far as you like (corresponding mathematically to the fact that (7) is globally integrable in a flat space) in spite of the fact that the frame is approximately inertial only over a finite time tmax. Summarizing, in classical mechanics a local inertial frame (45), with axes fixed in a freely falling body, can only be imagined to extend in all three directions to infinity by using the unphysical artifice of infinite tracer-particle velocities, but still that local frame is approximately inertial only for a finite time. Special relativity limits all speeds to no more than the speed of light. Therefore, local inertial frames can be realized in a gedanken experiment by tracer particles (including photons) only for finite times and are also finite in extent. Cartesian inertial frames extending to infinity cannot be constructed in a gedanken experiment that takes into account both the speed of light and the nonemptiness of the universe (gravity can be eliminated locally, but not globally in cosmology). As Havas [19] has noted, general relativity has two distinct mathematical limits where global inertial frames are allowed mathematically (the resulting three-space is Euclidean, (7) is globally integrable, and so (45) holds mathematically, if not physically, for -∞ ≤ t ≤ ∞): (1) neglect gravity but keep Lorentz invariance (special relativity in an empty35universe3), or (2) let the speed of light go to infinity but keep gravity (local Galilean invariance with Newtonian gravity). "... - the independence of the laws of nature from the state of motion in which it is observed, so long as it is uniform - is not obvious to the unpreoccupied mind. One of its consequences is that the laws of nature determine not the velocity but the acceleration of a body..." E. P. Wigner [12] In contrast, Mach refused to regard the law of inertia as an independent law of motion. He regarded it instead as already defined by the statement that acceleration vanishes when net force vanishes. This is a superficial viewpoint, but a viewpoint that one can easily adopt by default through failing adequately to appreciate that Galilean invariance is the foundation of Newton's second law [6,12]. "... [the Mach Principle] ... is the tendency to derive the meaning ... from the whole ... ." Otto Neurath [29] Mach's vague, holistic idea of obtaining the law of motion of interacting bodies first and then deriving the law of inertia from it puts the cart before the horse. However philosophically appealing and invulnerable Mach's criticism might have seemed earlier, we can now see it's weaknesses: Wigner [12] has pointed out the necessity of local invariance principles (like Galilean relativity) as the basis for our ability to discover of laws of nature in the first 3 One can solve the Kepler problem in special relativity [6], but that is not the point here.36place. Galileo's two local laws (the law of inertia and the principle of equivalence) were necessary before the second law could be formulated by either Huyghens (for uniform circular motion only) or Newton (most generally). Mach's objections to Newtonian mechanics were fueled by the mistaken (and historically-understandable) assumption that inertial frames, infinite in extent and defined by "the fixed stars", instead of local frames defined by freely falling bodies, could exist for arbitrarily long times. Summarizing, physics is described locally by differential equations. The differential equations of mechanics are universally applicable, locally, so long as we can find approximate inertial frames. Inertial frames are fixed in locally freely-falling bodies (gravitational interactions are always present in the real universe), and are therefore approximately inertial for limited times only. Whether or not global laws can be deduced from the local laws of motion is a question of integrability. From different geometric perspectives both modern nonlinear dynamics and differential geometry inform us that most systems of differential equations are nonintegrable in one sense or another, which means that the calculation of correct predictions for very long times, and over very long distances cannot be taken for granted. The realization that correct global results are very, very hard to determine leads one to the viewpoint that cosmology, in the end, may not produce much more than locally correct results. I have argued elsewhere [31] that an analog of the law of inertia would be necessary for economics, sociology, and psychology before there could be any hope to retrieve those fields from the mathematical lawlessness that is largely their present content. The existence of such an analog is unlikely because37people, unlike billiard balls and planets, can either cause or avoid collisions simply by changing their minds either systematically or arbitrarily. The ability to change one's mind is a prime example of mathematical lawlessness: a dynamical system (like the Newtonian three body problem) cannot change it's orbits arbitrarily, and cannot learn from the past. The absence of fixed mathematical law in brain-driven "motions" is the reason why artificial "law" is legislated (the motions of a deterministic or probabilistic dynamical system cannot be legislated). To argue that brain-driven motions may be more like a neural network that learns, than like a Newtonian dynamical system, is the same as admitting that brain-driven behavior is effectively mathematically lawless (such a system's dynamics must change unpredictably as it learns unexpectedly). Mach's thinking fed directly into and reinforced philosophical relativism [28], whose extreme wing (found mainly in literary criticism [32], cultural studies, anthropology, and sociology) now attacks physics as just another arbitrary activity like sociology, where there are no universal laws and where a "text" is presumed without proof to have no more meaning than a collection of abstract symbols on a printed page [33]. Interpretations of "texts" are regarded merely as arbitrary "representations" in the imprecise and undefined jargon of postmodernism. "... we have to consider science as a human enterprise by which man tries to adapt himself to the external world. Then a "pragmatic" criterion means ... the introduction of psychological and sociological considerations into every science, even into physics and chemistry. ....the sociology of science, the38consideration of science as a human enterprise, has to be connected in a very tight way with every consideration which one may call logical or semantical." Philipp Frank [29] Relativismic socio-literary criticism may describe contemporary literary theory and the confusion within the socio-economic sciences, but it fails to shed light on scientific fields like physics, chemistry, and genetics that are grounded in empirically-established local invariance principles, and in empirically-verified universally-applicable local laws of motion of interacting bodies that are based on those invariance principles. 8. Newtonian gravity is a nonintegrable connection in space-time Cartan [4,6,14,34], with hindsight informed by general relativity, noticed that we can rewrite Newton's second law for motion of a body of mass m in a net gravitational potential Φ d2xi dt2+∂Φ ∂xi=0 (5) in the following way: d2xk dλ2 + ∂2Φ ∂xk2 (dt dλ)2= 0, d2t dλ2 = 0 (46) The time t is linear in λ and is is affine. We can always choose t=λ. Go next to space-time coordinates xµ = (t,x1,x2,x3). Phoenician letters denote spatial39components (1,2,3) of coordinates, vectors, tensors, and connections. Note that we can write xµ + Γνλµxνxλ = 0 (47) where Γooi = ∂Φ ∂xi, Γbca = 0 . (48) Newtonian space-time is not flat because the Riemann curvature tensor is given by Rojoi = ∂2Φ ∂xj∂xi, Rbcda = 0 (49) Three-space is flat with Cartesian coordinates xi, and time is absolute (simultaneous spatially-separated events are allowed), but the four coordinates xµ are not globally Cartesian on four-dimensional space-time manifold because of curvature in the three (t,xi) sub-manifolds. Newtonian gravity is therefore not a covariant vector, but is instead a nonintegrable connection Γ in space-time. Kepler's orbits, as described by Newton, are therefore geodesics in curved space-time. Note that Heckmann's treatment of the gravitational potential Φ as a gauge potential, Φ(x',t) = Φ(x,t) + xk(d2bk/dt2), under transformations x'k = xk - bk(t) from inertial to linearly accelerated frames, and correspondingly the principle of equivalence, are automatically built into Cartan's interpretation of Newton's laws (equation (5) holds whether the frame is locally inertial, as is the earth in it's motion40about the sun, or is linearly accelerated, like a train leaving sentralstasjon in Oslo). Continuing, we find that the Ricci tensor is given by Roo = ∇2Φ = 4πρ, Rab = 0. (50) Newtonian space-time is not a metric space: a nondegenerate metric g cannot be defined that is consistent with the covariant derivative [14] (Heckmann may not have been aware of this restriction). This formulation can be extended to include transformations to rotating frames as well [6,14]. In other words, Newtons' second law is covariant with respect to transformations to arbitrarily accelerated frames in space-time, where the local relativity principle is Galilean invariance. General covariance plays no physical role here or in general relativity. Arbitrarily accelerated frames are certainly not equivalent for the performance of identically- prepared mechanical experiments. Local inertial frames are clearly preferred, for otherwise there is no identity of outcomes for experiments with identical preparation (solutions of (7) in two different frames connected by a Galilean transformation, but using identical initial conditions, are identical). A recent paper [35] on knot theory implicitly presumes Cartan's interpretation of Newtonian mechanics/gravity (geodesics in space-time due to the equivalence principle), but without reference or explanation.41Had Einstein been able to sidestep Mach and global inertial frames ("absolute space") he would, in principle, have been able to arrive more directly at the formulation of general relativity. Or, as Thomas Buchert [36] has put it, "If Lagrange (could) have had a beer with Newton, then they could have derived the Einstein-Cartan theory" (on a nice Biergarten tablecloth, perhaps). By starting with Cartan's description of Newtonian mechanics, and then replacing local Galilean invariance by Lorentz invariance, one arrives at the doorstep of general relativity, which is not a new or global relativity principle at all but is instead a locally Lorentz invariant description of gravity based upon the principle of equivalence. Cartan showed how to build the principle of equivalence into Newtonian mechanics explicitly. The equivalence principle was built into Newtonian mechanics independently and later by the cosmologist Heckmann. Heckmann seems to have been unaware of Cartan's work, which shows that Newtonian motion in a gravitational field can be understood as geodesic motion in space-time, where space is Euclidean and time is absolute. There are now attempts to use Cartan's interpretation of Newtonian dynamics as the framework for formulating and doing averaging and perturbation theory. 9. Einstein's theory of gravity To arrive at general relativity we could simply start with Cartan's description of Newtonian mechanics and replace local Galilean invariance by local Lorentz invariance [37]. Instead, I will follow an alternative path that uses only local Lorentz invariance, the local law of inertia (7), and the principle of equivalence (see also [13] or [14]). In other words, we start locally, making no42global assumptions. Global results, if they exist at all, must follow from global integrability of the resulting local law of motion. The law of inertia in any local Lorentz frame is given by d2xµ dτ2 = 0 (7b) where τ is the proper time. The law of inertia is globally integrable to yield Cartesian coordinates xµ(τ) = vµτ + xµo (51) (the vµ are constants) if and only if the Riemann curvature tensor vanishes everywhere (although no one need be concerned mathematically about distances greater than the speed of light times the age of the universe). Global integrability of (7b), with no points of the manifold excluded, is the formal mathematical (but not necessarily physical) condition to extend local Cartesian axes defined by (51) all the way to infinity (the space-time manifold is then globally flat). This is equivalent to the assumption that the Lorentz metric ηµν is globally diagonal and constant, and so is given by η = (-1,1,1,1). Instead of assuming that space-time is globally flat, begin by transforming the locally Lorentz invariant law of inertia (7b) to any other local coordinate system qµ = fµ(x), xν = hν(q),430 = d2xµ dτ2 = d2qν dτ2 ∂hµ ∂qν + ∂2hµ ∂qα∂qν qαqν . (52) So far, we have only assumed that the space-time manifold is locally Lorentzian (therefore locally flat). Using ∂fµ ∂xν ∂hν ∂qκ = δµκ (53) we obtain the law of inertia d2qµ dτ2 + Γαβµ qαqβ = 0 (50b) in any other coordinate system. The affine connections are given by Γαβµ = ∂hµ ∂xκ ∂fκ ∂qα∂qβ (54) and include all noninertial effects like linear and angular accelerations relative to freely falling frames, and also (according to Cartan [34] and Einstein [38]) global gravitational effects as well. The principle of equivalence was used implicitly because we cannot distinguish gravity locally from a linear acceleration.44The q's are called generalized coordinates. Synonyms for generalized coordinates are holonomic coordinates, or just coordinates [6,14]4. Generally, the coordinates qµ also do not exist globally (the differential equations that generate them are only locally integrable) unless they reflect the symmetry of the manifold, like spherical coordinates in the Schwarzschild solution of general relativity. Again, the condition for global parallelism, represented by the extension of (51) to τ = ±∞, is that the Riemann tensor Rηγβα = ∂Γβηα ∂qα - ∂Γηγα ∂qβ + ΓτγαΓτγα - ΓτβαΓγητ (55) vanishes everywhere. This represents the essential physics of the transition from special relativity and Newtonian mechanics to general relativity. The next step would be to generalize the Cartan-Newton equation (50) for the gravitational potential to include a locally-conserved mass-energy tensor on the right-hand side. Gravity is here described by a nonintegrable connection Γ in a locally Lorentzian space-time, and therefore is determined by the ten gravitational potentials g µν (in the absence of rotation, and using holonomic coordinates, Γ is symmetric). Starting with a Cartesian frame xµ in free fall and the local Lorentz metric η = (−1,1,1,1) , where ds2 = ηµνdxµdxν, the transformation to any other local coordinate system yields ds2 = ηµνdxµdxν = gαβdqαdqβ (56) 4 A crude dictionary relating turn-of-the-century phrases to modern terminolgy in differential geometry reads in part as follows: 'holonomic coordinates' [6] represent a 'coordinate basis' [14] while 'nonintegrable velocities' [6] correspond to a 'noncoordinate basis' [14].45where the metric is given at least locally by gαβ = ηµν ∂hµ ∂qα ∂hν ∂qβ. (57) The principle of equivalence combined with local Lorentz invariance, not general covariance, is the basis for general relativity, which is not a global relativity principle at all (and is also not a theory based on relativism in Mach's sense) but is a geometric theory of gravity. The principle of equivalence as the physical basis for the geometric theory of gravity was first emphasized by Einstein [38]. In another paper [39] I will discuss the observed distribution of the galaxies and ask whether there is any evidence to support either the cosmological principle or a hierarchical universe, e.g, a fractal universe. References 1. Sneddon, I., Elements of Partial Differential Equations , McGraw-Hill, New York (1957). 2. Duff, G.F.D., Partial Differential Equations , U. Toronto Pr., Toronto (1962). 3. Arnol'd, V. I., Geometric Theory of Ordinary Differential Equations , Chapter 1, Springer-Verlag, Berlin (1983). 4. Rüede, C., and Straumann, N., On Newton-Cartan Cosmology , Helv. Phys. Act. (1996). 5. Heckmann, O., Theorien der Kosmologie , Fortschritte der Astronomy, Band 2, Springer-Verlag, Berlin (1942).466. McCauley, J.L., Classical Mechanics, flows, transformations, integrability, and chaos , Cambridge, Cambridge (1997). 7. Galileo, Dialogue Concerning Two Chief World Systems , tr. by Stillman Drake, 2nd ed., U. Cal. Pr., Berkeley (1967). 8. Hamermesh, M. Group Theory , Addison Wesley, Reading (1962). 9. Havas, P., Rev. Mod, Phys. 36, 938 (1964). 10. Kretschmann, E., Ann. der Physik 53, 575 (1917). 11. McCauley, J.L., American Journal of Physics 45, 94 (1977). 12. Wigner, E.P., Symmetries and Reflections , Indiana, Bloomington (1967). 13. Weinberg, S., Gravitation and Cosmology , Wiley, New York (1972). 14. Misner, C.W., Thorn, K.S., and Wheeler, J.A., Gravitation , Freeman, San Francisco (1973).15. Rindler, W., Essential Relativity, 2nd ed., Springer-Verlag, (1977). 16. Buchert, T., and Ehlers, J., Averaging inhomogeneous Newtonian cosmologies, Astron. & Astrophys. (1996). 17. North. J.D., The Measure of the Universe , Clarendon Pr., Oxford (1965). 18. Milne, E.A., Quart. J. Math. 5, 64 (1934). 19. Layzer, D., Ap. J. 59, 268 (1954). 20. Bondi, H., Cosmology , Cambridge, Cambridge (1961). 21. Zanstra, H., Phys. Rev. 23, 528 (1924). 22. Schrödinger, E., Ann. Phys. 79, 325 (1925). 23. Barbour, J. B., and Bertotti, B., Nuovo Cim. B38, 1 (1977). 24. Barbour, J. B., and Bertotti, B., Proc. Roy. Soc. London A382 , 295 (1982). 25. Lynden-Bell, D., and Katz, J., Phys. Rev. D52, 7322 (1995). 26. Mach, E., The Science of Mechanics , Tr. from German, Open Court, LaSalle, Ill. (1942).4727. Barbour, J.E., Absolute or Relative Motion? , Vol. 1, Cambridge, Cambridge (1989). 28. Westfall, R.S., Never at Rest, A Biography of Isaac Newton , Cambridge, Cambridge (1980). 29. Holton, G., Science and Anti-Science , Harvard, Cambridge, Mass. (1993). 30. Hoyle, F., Nicolaus Copernicus , Heinemann, London (1973). 31. McCauley, J.L., Discrete Dynamics in Nature and Society 1, 17 (1997); Physica A237 , 387 (1997). 32. Horgan, J., The End of Science , Broadway Books, New York (1997). 33. Gross, P.R., and Levitt, N., Higher Superstition , Johns Hopkins, Baltimore (1994). 34. Cartan, E., On manifolds with an affine connection and the general theory of relativity , tr. from French, Biblios, Naples/Atlantic Highlands, N.J. (1986). 35. Moore, C., Phys. Rev. Lett. 70, 3675 (1993). 36. Buchert, T., preprint (1997). 37. Ehlers, J., in Relativity, Astrophysics, and Cosmology , ed. W. Israel, D. Reidel, Dordrecht (1972). 38. Einstein, A., The Meaning of Relativity, Princeton, Princeton (1953). 39. McCauley, J. L., The Galaxy Distribution: homogeneous, fractal, or neither?, preprint (1998). Acknowledgement These notes reflect in part lectures presented at the University of Oslo in the spring of 1997 during my sabbatical, during which time I was a guest of both the Institute for Theoretical Physics at the University and the Physics Department of The Institute for Energy Technology. I am grateful to Jan Frøyland and Arne Skjeltorp for guestfriendship, and to Finn Ravndal and48Øyvind Grøn for persistently stimulating questions and criticism during my lectures. Earlier in my sabbatical, from September 1996, through February 1997, I was guest professor in Lehrstuhl Wagner at Ludwig Maximillian's Universität in München, where Herbert Wagner and Thomas Buchert introduced me to Heckmann's papers, which are not available in English. Part 7 can be seen more or less as an edited and extended version of an informal translation of the Newtonian part of Heckmann's Theorie der Kosmologie . I am also grateful to Julian Barbour for friendly correspondence and criticism of my discussion of Mach's principle and relativism. This work was supported financially by The Research Council of Norway, the Institute for Energy Technology (at Kjeller, Norway), and the Departments of Physics at the Universities of Oslo (UIO), München (LMU) and Houston (UH).

arXiv:physics/0001028v1 [physics.comp-ph] 13 Jan 2000Protein Folding Simulations in a Deformed Energy Landscape Ulrich H.E. Hansmann1 Department of Physics Michigan Technological University Houghton, MI 49931-1295 ABSTRACT A modified version of stochastic tunneling, a recently intro duced global optimiza- tion technique, is introduced as a new generalized-ensembl e technique and tested for a benchmark peptide, Met-enkephalin. It is demonstrated tha t the new technique allows to evaluate folding properties and especially the glass tempe ratureTgof this peptide. Key words: Generalized-Ensemble Simulations, Protein Folding, Stoc hastic Tunneling. 1e-mail: hansmann@mtu.eduNumerical simulations of biological molecules can be extre mely difficult when the molecule is described by “realistic” energy functions wher e interactions between all atoms are taken into account. For a large class of molecules, in par ticular for peptides or proteins, the various competing interactions lead to frustration and a rough energy landscape. At low temperatures canonical simulations will get trapped in one of the multitude of local minima separated by high energy barriers and physical quant ities cannot be calculated ac- curately. One way to overcome this difficulty in protein simul ations is by utilizing so-called generalized ensembles [1], which are based on a non-Boltzmann probability distrib ution. Multicanonical sampling [2] and simulated tempering [3] ar e prominent examples of such an approach. Application of these techniques to the protein folding problem was first addressed in Ref. [4] and their usefulness for simulation of biological molecules and other complex systems [4]-[8] has become increasingly recognize d. However, generalized-ensemble methods are not without pro blems. In contrast to canonical simulations the weight factors are not a priori known. Hence, for a computer experiment one needs estimators of the weights, and the prob lem of finding good estima- tors is often limiting the use of generalized-ensemble tech niques. Here we describe and test a new generalized ensemble where determination of the w eights is by construction of the ensemble simple and straightforward. Our method is base d on a recently introduced global optimization technique, stochastic tunneling [9]. Canonical simulations of proteins at low temperature are ha mpered by the roughness of the potential energy surface: local minima are separated by high energy barriers. To enhance sampling we propose to weight conformations not wit h the Boltzmann factor wB(E) = exp( −E/k BT), but with a weight wf(E) = exp(f(E)/kBT). (1) Here,Tis a low temperature, kBthe Boltzmann constant, and f(E) is a non-linear transformation of the potential energy onto the interval [0 ,−1] chosen such that the relative location of all minima is preserved. The physical i dea behind such an approach is to allow the system at a given low temperature Tto “tunnel ” through energy barriers of arbitrary height, while the low energy region is still well r esolved. A transformation with 2the above characteristics can be realized by f1(E) =−e−(E−E0)/nF. (2) Here,E0is an estimate for the ground state and nFthe number of degrees of freedom of the system. Eq. 2 is a special choice of the transformation recen tly introduced under the name “stochastic tunneling” [9] to the corresponding problem of global minimization in complex potential energy landscapes. One can easily find further exa mples for transformations with the above stated properties, for instance, f2(E) =−(1 + (E−E0)/nF)−1. (3) We will restrict our investigation to these two transformat ions without claiming that they are an optimal choice. A simulation in the above ensemble, defined by the weight of Eq . 1 with a suitable chosen non-linear transformation f(E), will sample a broad range of energies. Hence, application of re-weighting technique [10] allows to calcu late the expectation value of any physical quantity Oover a large range of temperatures Tby <O>T=/integraldisplay dEO(E)Pf(E)w−1 f(E)e−E/k BT /integraldisplay dE P f(E)w−1 f(E)e−E/k BT. (4) In this point our method is similar to other generalized-ens emble techniques such as the multicanonical sampling [2], however, our method differs fr om them in that the weights are explicitly given by Eq. 1. One only needs to find an estimator f or the ground-state energy E0in the transforming functions f1(E) orf2(E) (see Eqs. 2 and 3) which in earlier work [11, 12] was found to be much easier than the determination of weights for multicanonical algorithm [2] or simulated tempering [3]. The new simulation technique was tested for Met-enkephalin , one of the simplest pep- tides, which has become a often used model to examine new algo rithms. Met-enkephalin has the amino-acid sequence Tyr-Gly-Gly-Phe-Met. The pote ntial energy function Etot that was used is given by the sum of the electrostatic term Ees, 12-6 Lennard-Jones term EvdW, and hydrogen-bond term Ehbfor all pairs of atoms in the peptide together with 3the torsion term Etorsfor all torsion angles: Etot=Ees+EvdW+Ehb+Etors, (5) Ees=/summationdisplay (i,j)332qiqj ǫrij, (6) EvdW=/summationdisplay (i,j)/parenleftBiggAij r12 ij−Bij r6 ij/parenrightBigg , (7) Ehb=/summationdisplay (i,j)/parenleftBiggCij r12 ij−Dij r10 ij/parenrightBigg , (8) Etors=/summationdisplay lUl(1±cos(nlχl)), (9) whererijis the distance between the atoms iandj, andχlis thel-th torsion angle. The parameters ( qi,Aij,Bij,Cij,Dij,Ulandnl) for the energy function were adopted from ECEPP/2.[13] The computer code SMC2was used. The simulations were started from completely random initial conformations (Hot Start) and one Monte Carl o sweep updates every tor- sion angle of the peptide once. The peptide bond angles ωwere fixed to their common value 180◦, which left 19 torsion angles ( φ, ψ, andχ) as independent degrees of freedom (i.e.,nF= 19). The interaction of the peptide with the solvent was neg lected in the simu- lations and the dielectric constant ǫset equal to 2. In short preliminary runs it was found thatT= 8 K was the optimal temperatures for simulations relying on the transformation f1(E) (Eq. 2), and T= 6Kfor simulations relying on the second chosen transformatio n f2(E) (Eq. 3). The free parameter E0was set in Eq. 2 or (3) to E0=−10.72 kcal/mol, the ground state energy as known from previous work. In addit ion, simulations were also performed where E0was dynamically updated in the course of the simulation and s et to the lowest ever encountered energy. In these runs the (known ) ground state was found in less than 5000 MC sweeps. Hence, determination of the weig hts is easier than in other generalized-ensemble techniques since in earlier work[4] it was found that at least 40,000 sweeps were needed to calculate multicanonical weights. We remark that a Monte Carlo sweep in both algorithm takes approximately the same amount of CPU time. All thermodynamic quantities were then calculated from a si ngle production run of 1,000,000 MC sweeps which followed 10,000 sweeps for therma lization. At the end of 2The program SMC was written by Dr. Frank Eisenmenger (eisenm enger@rz.hu-berlin.de) 4every sweep we stored the energies of the conformation and th e radius of gyration R=1 N2 atomsNatoms/summationdisplay i,j(/vector ri−/vector rj)2(10) for further analyses. In order to demonstrate the dynamical behavior of the algori thm the “time series” and histograms of potential energy are shown for both choices of the transforming functions f1(E) (Fig. 1) and f2(E) (Fig. 2). Both choices of the non-linear transformation wi th which the energy landscape was deformed in the simulations l ead to qualitatively the same picture. In Fig. 1a and Fig. 2a, respectively, one can se e that the whole energy range betweenE <−10 kcal/mol (the ground state region) and E≈20 kcal/mol (high-energy, coil states) is sampled. However, unlike in the multicanoni cal algorithm the energies are not sampled uniformly and low-energy states appear with hig her frequency than high energy states. However, as one can see from the logarithmic s cale of Fig. 1b and 2b where the histograms are displayed for these simulations, high-e nergy states are only suppressed by three orders of magnitude and their probability is still l arge enough to allow crossing of energy barriers. Hence large parts of the configuration spac e are sampled by our method and it is justified to calculate from these simulations therm odynamic quantities by means of re-weighting, see Eq. 4. Here, the average radius of gyration <R> , which is is a measure for the compactness of protein configurations and defined in Eq. 10, was calculate for various temperatures. In Fig. 3 the results for the new ensemble, using the defining n on-linear transformations f1(E) orf2(E), are compared with the ones of a multicanonical run with equ al number of Monte Carlo sweeps. As one can see, the values of < R > (T) agree for all three simulations over the whole temperature range. Hence, it is o bvious that simulations in the new ensemble are indeed well able to calculate thermodyn amic averages over a wide temperature range. After having established the new techniques as a possible al ternative to other generalized- ensemble techniques such as multicanonical sampling or sim ulated tempering, its useful- ness shall be further demonstrated by calculating the free e nergy of Met-enkephalin as a 5function of R: G(R) =−kBTlogP(R) (11) where P(R) =Pf(R)∗w−1 f(E(R))e−E(R)/kBT. (12) Here, a normalization is chosen where the minimal value of Gmin(R) = 0. The chosen temperature was T= 230K, which was found in earlier work [7] as the folding temp erature Tfof Met-enkephalin. The results, which rely on the transform ationf1(E) of the energy landscape given by Eq. 2 are displayed in Fig. 4. At this tempe rature one observes clearly a “funnel” towards low values of Rwhich correspond to compact structures. Such a funnel-like landscape was already observed in Ref. [8] for Met-enkephalin, utilizing a different set of order parameters, and is predicted by the lan dscape theory of folding [14]. The essence of the funnel landscape idea is competition betw een the tendency towards the folded state and trapping due to ruggedness of the landsc ape. One way to measure this competition is by the ratio [15]: Q=E−E0/radicalBig E2−¯E2, (13) where the bar denotes averaging over compact configurations . The landscape theory asserts that good folding protein sequences are characteri zed by large values of Q[15]. Using the results of our simulations and defining a compact st ructure as one where R(i)≤ 23˚A, we findE−E0= 13.96(3) Kcal/mol, E2−¯E2= 0.49(2), from which we estimate for the above ratio Q= 20.0(5). This value indicates that Met-enkephalin is good fold er and is consistent with earlier work [7] where we evaluated an alternative characterization of folding properties. Thirumalai and collaborators [16] h ave conjectured that the kinetic accessibility of the native conformation can be classified b y the parameter σ=Tθ−Tf Tθ, (14) i.e., the smaller σis, the more easily a protein can fold. Here Tfis the folding temperature andTθthe collapse temperature. With values for Tθ= 295 K and Tf= 230 K, as measured in Ref. [7], one has for Met-enkephalin σ≈0.2, indicating again that the peptide has good folding properties. 6Yet another characterization of folding properties relies on knowledge of the glass tem- peratureTgand is closely related to Eq. 13. As the number of available st ates gets reduced with the decrease of temperature, the possibility of local t rapping increases substantially. Glassy behavior appears when the residence time in some loca l traps becomes of the order of the folding event. Folding dynamics is now non-exponenti al since different traps have different escape times [17]. For temperatures above the glas s transition temperature Tg, the folding dynamics is exponential and a configurational di ffusion coefficient average the effects of the short lived traps [18]. It is expected that for a good folder the glass transi- tion temperature, Tg, where glass behavior sets in, has to be significantly lower t han the folding temperature Tf, i.e. a good folder can be characterized by the relation [19] Tf Tg>1. (15) I present here for the first time a numerical estimate of this g lass transition temperature for the peptide Met-enkephalin. The calculation of the estimat e is based on the approximation [19] Tg=/radicaltp/radicalvertex/radicalvertex/radicalbtE2−¯E2 2kBS0, (16) where the bar indicates again averaging over compact struct ures andS0is the entropy of these states estimated by the relation S0=logw(i) w(i)−log ˜z−C (17) Here, ˜z=/summationtext compactw(i) andCchosen such that the entropy of the ground state becomes zero. The results of the simulation in the new ensemble define d by the transformation f1(E), leads to a value of s0= 2.3(7). Together with the above quoted value for E2− ¯E2= 0.49(2) (in (Kcal/mol)2) one therefore finds as an estimate for the glass transition temperature Tg= 160(30) K . (18) Since it was found in earlier work [7] that Tf= 230(30), it is obvious that the ratio Tf/Tg>1 and again one finds that Met-enkephalin has good folding pro perties. Hence, we see that there is a strong correlation between all three fo lding criteria. 7Let me summarize my results. I have proposed to utilize a rece ntly introduced global optimization technique, stochastic tunneling, in such a wa y that it allows calculation of thermodynamic quantities. The new generalized-ensembl e technique was tested for a benchmark peptide, Met-enkephalin. It was demonstrated th at the new technique allows to evaluate the folding properties of this peptide and an est imate for the glass transition temperature Tgin that system was presented. Currently I am evaluating the e fficiency of the new method in simulations of larger molecules. Acknowledgements : This article was written in part when I was visitor at the Inst itute of Physics, Academia Sinica, Taipei, Taiwan. I like to thank the Institute and spe cially C.K. Hu, head of the Laboratory for Statistical and Computational Physics, for the kind hospitality extended to me. Financial support from a Research Excellence Fund of t he State of Michigan is gratefully acknowledged. References [1] U.H.E. Hansmann and Y. Okamoto, In Annual Reviews in Computational Physics VI. Edited by Stauffer D. Singapore: World Scientific; 1999, 129 -157. [2] B.A. Berg and T. Neuhaus, Phys. Lett. B 267, 249 (1991); Phys. Rev. Lett. 68, 9 (1992). [3] A.P. Lyubartsev, A.A.Martinovski, S.V. Shevkunov, and P.N. Vorontsov- Velyaminov, J. Chem. Phys. 96, 1776 (1992); E. Marinari and G. Parisi, Euro- phys. Lett. 19, 451 (1992). [4] U.H.E. Hansmann and Y. Okamoto, J. Comp. Chem. 14, 1333 (1993). [5] Y. Okamoto and U.H.E. Hansmann, J. Phys. Chem. 99, 2236 (1995). [6] F. Eisenmenger and U.H.E. Hansmann, J. Phys. Chem. B 101, 3304 (1997). 8[7] U.H.E. Hansmann, M. Masuya, and Y. Okamoto, Proc. Natl. Acad. Sci. U.S.A. 94, 10652 (1997). [8] U.H.E. Hansmann, Y. Okamoto and J.N. Onuchic, Proteins 34(1999) 472. [9] W. Wenzel and K. Hamacher, Phys. Rev. Let. 823003 (1999). [10] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988); Phys. Rev. Lett.63, 1658(E) (1989), and references given in the erratum. [11] U.H.E. Hansmann, Physica A 242250 (1997). [12] U.H.E. Hansmann and Y. Okamoto, Phy. Rev. E 56, 2228 (1997). [13] M.J. Sippl, G. N´ emethy, and H.A. Scheraga, J. Phys. Chem. 88, 6231 (1984), and references therein. [14] K.A. Dill and H.S. Chan, Nature Structural Biology 4, 10 (1997). [15] R.A. Goldstein, Z.A. Luthey-Schulten and P.G. Wolynes ,Proc. Natl. Acad. Sci. U.S.A. 894918 (1992). [16] D.K. Klimov and D. Thirumalai, Phys. Rev. Lett. 76, 4070 (1996). [17] N.D. Socci, J.N. Onuchic and P.G. Wolynes, Proteins 32136 (1998). [18] N.D. Socci, J.N. Onuchic, and P.G. Wolynes, J. Chem. Phys. 104, 5860 (1996). [19] J.D. Bryngelson, J.N. Onuchic, N.D. Socci, and P.G. Wol ynes,Proteins 21, 167 (1995). 9FIGURE CAPTIONS: 1. “Time series”(a) of potential energy Eof Met-enkephalin for a simulation in a gener- alized ensemble defined by the transformation f1(E) of Eq. 2 and the corresponding histogram (b) of potential energy. 2. “Time series”(a) of potential energy Eof Met-enkephalin for a simulation in a generalized ensemble defined by the transformation f2(E) of Eq. 3 (a) and the corresponding histogram (b) of potential energy. 3. Average radius of gyration < R > (in˚A2) as a function of temperature (in K). The results of a multicanonical simulation of 1,000,000 MC s weeps were compare with simulations of equal statistics in the new ensemble uti lizing either the no-linear transformation f1(E) orf2(E). 4. Free energy G(R) as a function of the radius of gyration RforT= 230 K. The results rely on a generalized-ensemble simulation based on the tran sformation f1(E) of the energy landscape s defined in Eq. 2. 10-10-505101520 0200000 400000 600000 800000 1e+06E [kcal/mol] t [MC-sweeps]f_1(E)0.00010.0010.010.1 -10-505101520P(E) E [kcal/mol]f_1(E)-10-5051015202530 0200000 400000 600000 800000 1e+06E [kcal/mol] t [MC-sweeps]f_2(E)0.00010.0010.010.1 -10-505101520f_2(E)2224262830323436 01002003004005006007008009001000<R> [Angstroem^2] T [K]MuCa f_1 f_2-20246810 1820222426283032343638G(R) RT=230 K

arXiv:physics/0001029 13 Jan 2000 1The new science of complexity Joseph L. McCauley Physics Department University of Houston Houston, Texas 77204 jmccauley@uh.edu key words: complexity simulations laws of nature invariance principles socio-economic sciences Abstract Deterministic chaos, and even maximum computational complexity, have been discovered within Newtonian dynamics. Encouraged by comparisons of the economy with the weather, a Newtonian system, economists assume that prices and price changes can also obey abstract mathematical laws of motion. Meanwhile, sociologists and other postmodernists advertise that physics and chemistry have outgrown their former limitations, that chaos and complexity provide new holistic paradigms for science, and that the boundaries between the hard and the soft sciences, once impenetrable, have disappeared along with the Berlin Wall. Three hundred years after the deaths of Galileo, Descartes, and Kepler, and the birth of Newton, reductionism would appear to be on the decline, with holistic approaches to science on the upswing. We therefore examine the evidence that dynamical laws of motion may be discovered from empirical studies of chaotic or complex phenomena, and also review the foundations of reductionism.2Socio-economic fields and "system theory" I define "system theory" to include mathematical models written in terms of systems of deterministic and stochastic ordinary and partial differential equations, iterated maps, and deterministic and stochastic automata. The idea is to include every possible kind of dynamical modelling. In attempts to describe socio-economic phenomena from the standpoint of system theory it is Platonically assumed that the probability distributions describing prices and price changes, or other social factors, are determined by an objective mathematical law that governs how the economic system evolves [1]. This assumption is not only sufficient but is also necessary if the idea of mathematical law in economics is to make any sense. In physics and chemistry the ideas of entropy, thermodynamics, and nonequilibrium statistical mechanics are grounded in universally-valid microscopic dynamics. Without the underlying dynamics of particles, fluids and solid or plastic bodies there would be no dynamical origin for macroscopic probability distributions. By a mathematical law of nature I mean a law of motion, a mathematical law of time-evolution. Galileo and Kepler discovered the simplest special cases. Their local laws were generalized by Newton to become three universally- valid laws of motion, along with a universal law of gravity. Newton's laws are "universal" in the following sense: they can be verified, often with very high decimal precision, regardless of where and when on earth (or on the moon or in an artificial satellite) careful, controlled experiments, or careful observations, are performed. It is the main purpose of this paper to stress the3implications of the fact that no comparable result has ever been found in the socio-economic fields. "Laws" of economics, "laws" of human behavior, and the Darwin-Wallace "laws" of fitness, competition, selection and adaptation are sometimes mentioned in the same context as laws of motion of inanimate matter (physics and chemistry), although since the time of Galileo the word "law" in the first three cases does not have the same import as in the case of physico- chemical phenomena. Confusion over what constitutes a law of nature is ancient: Aristotle invented a purely qualitative, holistic approach to the description of nature. Not recognizing any distinction between the different uses of the idea of natural law, he lumped together as "motion" the rolling of a ball, the education of a boy, and the growth of an acorn [2]. Ibn-Rushd realized that Aristotle's philosophy is consistent with a purely mechanistic picture of the universe. The growing influence of the mechanistic interpretation of nature in western Europe set Tomasso d'Aquino into motion in the thirteenth century. Aristotle did not use mathematics, but mechanism and mathematics go hand in hand. Is human nature, in some still-unknown mathematical sense, also mechanistic? In the first chapter of his text on elementary economics [3], Samuelson tries to convince both the reader and himself that the difference between the socio- economic fields and the laws of physics is blurry, so that economics can be treated as if it would also be a science subject to mathematical law. Samuelson claims that physics is not necessarily as lawful as it appears, that the laws of physics depend subjectively on one's point of view. His argument is based on a nonscientific example of ambiguity from the visual perception of art (figure41), and is genetically related through academic mutation and evolution to a viewpoint that has been advanced by the postmodernist and deconstructionist movement in art, literature, philosophy, psychology, and sociology. The latter argue that a text has no more meaning than the symbols on a printed page, that there is no universal truth, and therefore no universal laws of nature, and that Platonic-Ptolemeic astronomy and Aristotelian physics are still just as valid as fields of scientific study as are physics and astronomy since Galileo and Kepler (who revived the spirit of Archimedes). Samuelson notes that physics relies on controlled experiments, and adds that in the socio-economic fields it is generally impossible to perform controlled experiments. This is not an excuse for bad science: controlled experiments are also impossible in astronomy where mathematical laws of nature have been verified with high decimal precision. See also Feynman [4] for criticism of the lack of isolation of cause and effect in the psycho-social fields. Platonists in mathematics [5] form another category, believing that mathematical laws exist objectively and govern everything that happens. Physics is neither Aristotelian (qualitative and "holistic") nor Platonic (relying upon wishful thinking, because the "expected" mathematical laws are not grounded in careful, repeatable empiricism). The divorce of the study of nature from Platonic and Aristotelian notions was initiated by Galileo and Descartes [6], but that divorce was not complete: with Galileo's empirical discoveries of two local laws of nature, the law of inertia and the local law of gravity, physics became a precise mathematico-empirical science. Biology, excepting the study of heredity since Mendel and excepting5biochemistry and biophysics since the advent of quantum mechanics, has continued through the age of Darwin and beyond as a largely descriptive science in the tradition of Aristotle, with reliance upon vague, mathematically-undefined notions like "competition, natural selection and adaptation". I will explain why economic and other social phenomena lie beyond the bounds of understanding from the standpoint of dynamical modelling that attempts to describe the time-evolution of systems, even if the goal is merely to extract the crudest features like coarsegrained statistics. I will give reasons why mathematical laws of economics do not exist in any empirical or computationally-effective [7] sense. In order to make my argument precise, I first review some little-known and poorly-understood facts about deterministic dynamical systems that include Newton's laws of motion for particles and rigid bodies, and also nondiffusive chemically-reacting systems. What does "nonintegrable" mean? We expect that any system of ordinary differential equations generating critical (orbitally-metastable), chaotic (orbitally-unstable), or complex dynamics must be both nonlinear and nonintegrable. Most of us think that we can agree on the meaning of "nonlinear". Before asking "What is complexity?" we first define what "nonintegrable" means [8,9]. The ambiguity inherent in both serious and superficial attempts to distinguish "integrability" from "nonintegrability" was expressed poetically by Poincaré, who stated that a dynamical system is generally neither integrable6nor nonintegrable, but is more or less integrable [10]. For most scientists the explanation of various roots to chaos (via period doubling, e.g.) has tended to submerge rather than clarify the question how to distinguish those two ideas, but without eliminating many misconceptions. Modern mathematicians have managed to give some precise definitions of nonintegrability [11] that are hard to translate into simpler mathematics. Here, I try to describe what "nonintegrability" means geometrically and analytically. For the sake of precision I frame my discussion in the context of flows in phase space, dx dt = V(x), (1) where phase space is a flat inner product space so that the n axes labeled by (x1,...,x n) can be regarded as Cartesian [12], and V(x) is an n-component time- independent velocity field. Newtonian dynamical systems can always be rewritten in this form whether or not the variables xi defining the system in physical three dimensional space are Cartesian (for example, it is allowed have x1 = θ and x2 = dθ/dt, where θ is an angular variable). Flows that preserve the Cartesian volume element d Ω = dx1...dx n are defined by ∇⋅V = 0 (conservative flows) while driven dissipative-flows correspond to ∇⋅V ≠ 0, where ∇denotes the Cartesian gradient in n dimensions. For a velocity field whose components satisfy the condition V1 + ... + Vn = 0, then the global conservation law x1 + ... + xn = C follows. This abstract case includes chemically-reacting systems with concentration x i for species i.7For a flow and for any initial condition xo the solution xi(t) = U(t)x io has no finite time singularities [13] because singularities of trajectories of flows are confined to the complex time plane: the time evolution operator U(t) exists for all real finite times t and defines a one-parameter transformation group with inverse U-1(t) = U(-t), so that one can in principle integrate backward in time, xoi = U(-t)x i(t), as well as forward. In other words, even driven- dissipative flows are perfectly time-reversible. Many researchers use floating point arithmetic in numerical integrations of chaotic systems but uncontrollable errors are introduced into numerical integrations by the use of floating point arithmetic, and those errors violate time reversibility in the simplest of cases. Even for a nonchaotic driven- dissipative flow floating-point errors will prevent accurate numerical solutions either forward or backward in time after only a relatively short time [12] . The simplest example is given by the one dimensional flow dy/dt = y, all of whose streamlines have the positive Liapunov exponent λ = 1 forward in time, and the negative Liapunov exponent λ = - 1 backward in time. Consequently, the simple linear equation dy/dt = y cannot be integrated forward in time accurately numerically, for moderately-long times, if floating point arithmetic is used. Chaotic unimodal maps z n = f(zn-1) like the logistic map f(x) = Dx(1-x) have a multi-valued inverse zn-1 = f-1(zn) and therefore are not uniquely time- reversible. Contrary to superficial appearances based upon an unwarranted extrapolation of a numerical calculation, time reversal is not violated by the Lorenz model8 d x1 d t=σ(x2±x1) d x2 d t=ρx1±x2±x1x3 d x3 d t= ±βx3+x1x2 (1b) in the chaotic regime. The well-known numerically-suggested one dimensional cusp map (figure 2) z n = f(zn-1) that represents maxima of a time series [14] of x3(t) at discrete times to, t1, ..., t n, ... , where tn-tn-1 denotes the time lag between successive maxima zn-1 = x3(tn-1) and zn = x3(tn), can not have a double-valued inverse z n-1 = f-1(zn): backward integration zn-1 = U(tn-1 -tn)zn is unique for a flow, and the Lorenz model satisfies the boundedness condition for a flow [14] . Therefore, Lorenz's one dimensional cusp map z n = f(z n-1) is not continuous and may even be infinitely fragmented and nondifferentiable in order that the inverse map f-1 doesn't have two branches. Note that the Lorenz model may describe a chemically-reacting system if β = 0 and σ = ρ = 1, in which case the flow is driven-dissipative but is not chaotic (the flow is orbitally-stable, with no positive Liapunov exponent in forward integration). Surprise has been expressed that it was found possible to describe a certain chaotic flow by a formula in the form of an infinite series [8], but "nonintegrable" does not mean not solvable: any flow, even a critical, chaotic or complex one, has a unique, well-defined solution if the velocity field V(x) satisfies a Lipshitz condition (a Lipshitz condition requires the definition of a metric in phase space), or is at least once continuously differentiable, with9respect to the n variables xi. If, in addition, the velocity field is analytic in those variables then the power series xi(t) = xio + t(Lx i)o + t2(L2xi)o/2 + .... , (2) where L = V⋅∇, has a nonvanishing radius of convergence, so that the solution of (1) can in principle be described by the power series (2) combined with analytic continuation for all finite times [15] . It is well known that this is not a practical prescription for the calculation of trajectories at long times. The point is that a large category of deterministic chaotic and complex flows are precisely determined over any desired number of finite time intervals by analytic formulae . The Lorenz model (1b) provides an example. Analyticity is impossible for the case of truly "random" motion (like α-particle decays), where the specification of an initial condition does not determine a trajectory at all (as in Feynman's path integral), or for Langevin descriptions of diffusive motion, where almost all trajectories are also continuous and almost everywhere nondifferentiable (as in Wiener's functional integral). According to Jacobi and Lie, a completely integrable dynamical system has n-1 global time-independent first integrals (conservation laws) Gi(x1,...,x n) = Ci satisfying the linear partial differential equation dGi dt = V⋅∇Gi = Vk∂Gi ∂xk = 0 (3) along any streamline of the flow. In addition, these conservation laws must (in principle, but not necessarily via explicit construction) determine n-1 "isolating integrals" of the form x k = g k(xn,C1,...,C n-1) for k = 1,...,n-1. When all10of this holds then the global flow is a time-translation for all finite times t in the Lie coordinate system yi = Gi(x1,...,xn) = Ci, i = 1,...,n-1 yn = F(x1,...,xn) = t + D (4) defined by the n-1 conservation laws, and the system is called completely integrable. The solution reduces in principle to n independent integrations, and the flow is confined to a two-dimensional manifold that may be either flat or curved and is determined by the intersection of the n-1 global conservation laws. For the special case of a canonical Hamiltonian flow with f degrees of freedom, f commuting conservation laws confine the flow to a constant speed translation on an f dimensional flat manifold. The nth transformation function F(x1,...,x n) is defined by integrating dt = dxn/Vn(x1,...,x n) = dx/v n(xn,C1,...,C n-1) to yield t + D = f(xn,C1,...,C n-1). One then uses the n-1 conservation laws to eliminate the constants Ci in favor of the n-1 variables x i in f to obtain the function F. Whether one can carry out all or any of this constructively, in practice, is geometrically irrelevant: in the description (4) of the flow all effects of interactions have been eliminated globally via a coordinate transformation. The transformation (4) "parallelizes" (or "rectifies" [13]) the flow: the streamlines of (1) in the y- coordinate system are parallel to a single axis yn for all times, and the time evolution operator is a uniform time-translation U(t) = etd/dyn. Eisenhart asserted formally, without proof, that all systems of differential equations (1) are described by a single time translation operator [16], but this is possible globally (meaning for all finite times) only in the completely integrable case.11Although time-dependent first integrals are stressed in discussions of integrable cases of driven-dissipative flows like the Lorenz model [8], there is generally no essential difference between (3) and the case of n time-dependent first integrals G' i(x1,...,x n,t) = C' i satisfying dGi dt = V⋅∇Gi + ∂Gi ∂t = 0 . (3b) Relying on the implicit function theorem, one conservation law G'n(x1,...,x n,t) = C'n can be used to determine a function t = F'(x 1,...,x n,C'n), whose substitution into the other n-1 time-dependent conservation laws yields n-1 time-independent ones satisfying (3). The n initial conditions xio = U(-t)x i(t) of (1) satisfy (3b) and therefore qualify as time-dependent conservation laws, but initial conditions of (1) are generally only trivial local time-dependent conservation laws: dynamically seen, there is no qualitative difference between backward and forward integration in time. Nontrivial global conservation laws are provided by the initial conditions y io, for i = 1, 2, ... , n-1, of a completely integrable flow in the Lie coordinate system (4), where the streamlines are parallel for all finite times: dy i/dt = 0, i = 1,...,n-1, and dy n/dt = 1. Algebraic or at least analytic conservation laws [8] have generally been assumed to be necessary in order to obtain complete integrability. For example, Euler's description of a torque-free rigid body [12]12 d L1 d t=aL2L3 d L2 d t= ±bL1L3 d L d t=cL1L2, (5) with positive constants a , b , and c satisfying a - b + c = 0, defines a phase flow in three dimensions that is confined to a two dimensional sphere that follows from angular momentum conservation L12 + L22 + L32 = L2. Here, we have completely integrable motion that technically violates the naive expectation that each term in (4) should be given by a single function: for each period τ of the motion, the transformation function F has four distinct branches due to the turning points of the three Cartesian components Li of angular momenta on the sphere. In general, any "isolating integral" gk describing bounded motion must be multivalued at a turning point. Note also that the Lorenz model defines a certain linearly damped, driven symmetric top: to see this, set a = 0 and b = c = 1 in (5), and ignore all linear terms in (1b). The few mathematicians who have discussed conservation laws in the literature usually have assumed that first integrals must be analytic or at least continuous [13] (however, see also ref. [11] where nonanalytic functions as first integrals are also mentioned). This is an arbitrary restriction that is not always necessary in order to generate the transformation (4) over all finite times: a two-dimensional flow in phase space, including a driven-dissipative flow, is generally integrable via a conservation law but that conservation law is typically singular. The conservation law is simply the function G(x 1,x2) = C that describes the two-dimensional phase portrait, and is singular at sources and sinks like attractors and repellers (equilibria and limit cycles provide13examples of attractors and repellers in driven-dissipative planar flows) [12] . For the damped simple harmonic oscillator, for example, the conservation law has been constructed analytically [17] and is logarithmically singular at the sink. The planar flow where dr/dt = r and dθ/dt = 0 in cylindrical coordinates (r,θ) describes radial flow out of a source at r = 0. The conservation law is simply θ, which is constant along every streamline and is undefined at r = 0. This integrable flow is parallelizeable for all finite times t simply by excluding one point, the source at r = 0 (infinite time would be required to leave or reach an equilibrium point, but the infinite time limit is completely unphysical). "Nonintegrable" flows do not occur in the phase plane. What can we say about "nonintegrability" about in three or more dimensions? In differential equations [13] and differential geometry [18] there is also an idea of local integrability: one can parallelize an arbitrary vector field V about any "noncritical point", meaning about any point x o where the field V(x) does not vanish. The size ε(xo) of the region where this parallelization holds is finite and depends nonuniversally on the n gradients of the vector field. This means that we can "rectify" even chaotic and complex flows over a finite time, starting from any nonequilibrium point xo. By analytic continuation [11,19] , this local parallelization of the flow yields n-1 nontrival "local" conservation laws yi = Gi(x) = Ci that hold out to the first singularity of any one of the n-1 functions Gi, in agreement with the demands of the theory of first order linear partial differential equations (the linear partial differential equation (3) always has n-1 functionally independent solutions, but the solutions may be singular [17] ).14Contemplate the trajectory of a "nonintegrable" flow that passes through any nonequilibrium point xo, and let t = 0 when x = xo. Let t(x o) then denote the time required for the trajectory to reach the first singularity of one of the conservation laws Gk. Such a singularity must exist, otherwise the flow would be confined for all finite times ("globally") to a single, smooth two- dimensional manifold. The global existence of a two-dimensional manifold can be prevented, for example, by singularities that make the n-1 conservation laws G i multivalued in an extension of phase space to complex variables [11] . Generally, as with solutions of (1) defined locally by the series expansion (2), the n-1 local conservation laws Gi will be defined locally by infinite series with radii of convergence determined by singularities that lie in the complex extension of phase space. The formulae (4) then hold for a finite time 0 ≤t<t(x o) that is determined by the distance from xo to the nearest complex singularity. Let x 1(xo) denote the point in phase space where that singularity causes the series defining Gi to diverge. Following Arnol'd's [13] statement of the "basic theorem of ordinary differential equations", we observe that the streamline of a flow (1) passing through x o can not be affected by the singularity at x1(xo) in the following superficial sense (consistent with the fact that the singularities of the functions Gi are either branch cuts or phase singularities): we can again parallelize the flow about the singular point x1(xo) and can again describe the streamline for another finite time t(xo)≤t<t(x 1) by another set of parallelized flow equations of the form (4), where t(x 1) is the time required to reach the next singularity x2(xo) of any one of the n-1 conservation laws Gi, starting from the second initial condition x1(xo). Reparallelizing the flow about any one of these singularities is somewhat like resetting the calendar when crossing the international15dateline, except that a nonintegrable flow is generally not confined globally to a two dimensional analytic manifold. We have reasoned that a "nonintegrable" flow is piecewise integrable: different sets of formulae of the form (4) hold in principle for consecutive finite time intervals 0≤t(xo)<t(x 1), t(x1)≤t<t(x 2), ... t(x n-1)≤t<t(x n), .... , giving geometric meaning to Poincaré's dictum [10] that a dynamical system is generally neither integrable nor nonintegrable but is more or less integrable. Nonintegrable flows are describable over arbitrarily-many consecutive time intervals by the simple formulae of the form (4) except at countably many singular points x1(xo), x2(xo), ... , where the n-1 initial conditions yio and the integration constant D must be reset. The relevance for Takens's embedding theorem is discussed in [9]. Deterministic chaos as simple dynamics We have often read over the last twenty years that deterministic chaos can explain complex phenomena, but without having had a definition of "complex". This was the point of view in the era when computers were used to try to study chaoic motions via numerical integrations without error control, based upon floating point forward integrations of chaotic dynamical equations (or by forward iterations of chaotic maps). We have since learned that uncontrolled numerical integrations can be avoided, and correspondingly that chaotic dynamics can be understood from a certain topologic point of view as relatively simple dynamics. This "new" approach (roughly ten years old) is the consequence of analytic studies of chaotic16systems using controlled approximations via a purely digital method called "symbolic dynamics". Symbol sequences are equivalent to digit strings in some base of arithmetic. Since we are going to talk about digit strings it is both wise and useful to begin with the idea of a computable number [20,21]. The reason for this is simple: "algorithmically random" numbers and sequences "exist" in the mathematical continuum but require infinite time and infinite precision for their definition, and therefore have no application to either experiment or computation. By a computable number, we mean either a rational number or an algorithm that generates a digit expansion for an irrational number in some base of arithmetic, like the usual grade school algorithm for the square-root operation in base ten (the same algorithm also works in any other integer base). If we use computable numbers as control parameters and initial conditions, then the chaotic dynamical systems typically studied in physics and chemistry are computable, e.g. via (2) combined with analytic continuation. The Lorenz model (1b) provides one example. Systems of chemical kinetic equations provide other examples. Seen from the perspective of computability, the local solution (2) of a dynamical system (1) that is digitized completely in some base of arithmetic defines an "artificial automaton", an abstract model of a computer. The digitized initial condition constitutes the program for the automaton. In a chaotic dynamical system the part of the program that directs the trajectory17into the distant future is encoded as the end-string εN+1... of digits in an initial condition x o = . ε1.ε2...εN... . For example,the binary tent map x n = f(x n-1), f(x) = 2x, x< 1/2 2(1 - x), x > 1/2 , (6) can be rewritten and studied naturally in binary arithmetic by writing xn = .ε1(n)ε2(n)...εn(n)..., with εi(n) = 0 or 1. The map (2) is then represented by the simple automaton [21] εi(n) = εi+1(n-1), ε1(n-1) = 0 1 - εi+1(n-1), ε1(n-1) = 1. (6b) For every possible binary-encoded "computer program" x o = .ε1(0)ε2(0)...εN(0)... this automaton performs only a trivial computation: either it reads a bit in the program, or else flips the bit and reads it, then moves one bit to the right and repeats the operation. The logistic map at the period doubling critical point [22], in contrast, is capable of performing simple arithmetic. Unlike the binary tent map in binary arithmetic, most dynamical systems do not admit a "natural" base of arithmetic. The logistic map f(x) = Dx(1-x) with D arbitrary and the Lorenz model are examples. The series solutions of these dynamical systems can still be rewritten as automata in any integer base of arithmetic, albeit in relatively cumbersome fashion. However, there is a systematic generalization of solution of the binary Bernoulli shift map xn = 2xn-1 mod 1 via binary arithmetic that sometimes works: symbolic dynamics. The symbolic dynamics of a chaotic dynamical system can be defined, and solved digitally at least in principle, if the map has a generating18partition [23]. For the binary tent map (6) the generating partition, in generation n, consists of the 2n intervals l(n) = 2-n that are obtained by backward iteration of the entire unit interval by the map (a chaotic one dimensional map contracts intervals in backward iteration). Each interval in the generating partition can be labeled by an n-bit binary (L,R) address (L and R are defined in figure 3) called a symbol sequence, as is shown in figure 4. The symbol sequence tells us the itinerary of the map, for n forward iterations, for any initial condition that is covered by the interval l(n)(ε1ε2...εn) labeled by the n-bit address ε1ε2...εn, where εi = L or R [21]. Excepting pathological cases where the contraction rate in backward iteration is too slow, an infinite length symbol sequence corresponds uniquely to an infinitely-precise initial condition. Given a symbol sequence, coarsegrained statistics for any number Nn of bins in the generating partition (Nn = 2n for the binary tent map) can be obtained merely by reading the sequence while sliding an N-bit window one bit at a time to the right, as is indicated in figure 5. Clearly, orbital statistics depend on initial conditions, and it is very easy to construct algorithms for initial conditions whose orbital statistics do not mimic the uniform invariant density of the binary tent map (e. g., xo = .101001000100001... qualifies and follows from an obvious algorithm). I have explained elsewhere why "random" initial conditions may be a bad assumption for a dynamical system far from thermal equilibrium [9,21]. Because the binary tent map generates all possible infinite-length binary sequences (almost all of which are not computable via any possible algorithm [20]), we can use that map to generate any histogram that can be constructed in finitely-many steps, merely by a correct choice of initial conditions [21]. Many19different initial conditions will allow the dynamical system to generate the same coarsegrained statistics because the precise ordering of L's and R's in a symbol sequence doesn't matter in determining the histograms. Liapunov exponents depend strongly on initial conditions, a fact that is not brought out by concentration on excessively simple models like the symmetric tent map, or numerical attempts to extract "the largest Liapunov exponent" of a chaotic dynamical system like the Lorenz model. Chaotic dynamical systems like the Lorenz model or the logistic map generally generate an entire spectrum of Liapunov exponents (and therefore also a spectrum of largest Liapunov exponents). The easiest way to understand this is to solve for the generating partition and Liapunov exponents of the asymmetric tent map [21], where only simple algebra is needed. We define a class of initial conditions to consist of all initial conditions that yield the same Liapunov exponent λ. Correspondingly, we can say that a class of symbol sequences defines a single Liapunov exponent. The Boltzmann entropy per iteration s(λ) of all symbol sequences with the same Liapunov exponent λ defines the fractal dimension D(λ) = s(λ)/λ of that class of initial conditions [12,21], so that a chaotic dynamical system generally generates spectra of both Liapunov exponents and fractal dimensions. Both critical [22] and chaotic [23] dynamical systems may generate a natural partitioning of phase space, the generating partition, but not every nonintegrable dynamical system defines a generating partition. If a deterministic dynamical system has a generating partition then the symbolic dynamics can in principle be solved and the long-time behavior can be20understood qualitatively, without the need to compute specific trajectories algorithmically from the algorithmic construction of a specific computable initial condition. For example, one need only determine the possible symbol sequences and then read them with a sliding n-bit window in order to generate the statistics in the form of a hierarchy of histograms (figure 5). In other words, a high degree of "computational compressibility" holds even if the dynamical system is critical or chaotic. Every chaotic dynamical system generates infinitely-many different classes of statistical distributions for infinitely-many different classes of initial conditions, and at most one of those distributions is differentiable (unlike the case of equilibrium statistical mechanics, there is no empirical evidence to suggest that nature far from equilibrium evolves from unknown initial conditions to generate differentiable distributions [9]). The generating partition, if it exists, uniquely forms the support of every possible statistical distribution and also characterizes the particular dynamical system (the intervals l(n) = 2-n characterize the binary tent map and the binary Bernoulli shift). For a system with a generating partition, topologic universality classes can be defined that permit one to study the simplest system in the universality class [24]. The infinity of statistical distributions is topologically invariant and therefore can not be used to discern or characterize a particular dynamical system within a universality class [21]. For maps of the unit interval, both the symmetric and asymmetric logistic maps peaking at or above unity belong to the trivial universality class of the binary tent map [21] (where all possible binary sequences are allowed). The topologic universality class is described by figure 4, and is defined by the21complete binary tree. Dynamical systems that generate complete ternary trees or incomplete binary trees, e.g., define other universality classes. The two dimensional Henon map belongs to the universality class of chaotic logistic maps of the unit interval peaking beneath unity. The simplest model in this topologic universality class is the symmetric tent map with slope magnitude between 1 and 2, and the class is defined by a certain incomplete binary tree [24]. In these systems the long-time behavior can be understood qualitatively and statistically in advance , so that the future holds no surprises : the generating partition and symbol sequences can be used to describe the motion at long times, to within any desired degree of precision l(n), and multifractal scaling laws (via the D(λ) spectrum) show how finer-grained pictures of trajectories are related to coarser-grained ones. In other words, universality and scaling imply relatively simple dynamics in spite of the fact that the word "complex" has often been used to describe deterministic chaos. Complex dynamics Scale invariance based upon criticality has been suggested as an approach to "complex space-time phenomena" based upon the largely unfulfilled expectation of finding universal scaling laws, generated dynamically by many interacting degrees of freedom and yielding critical states independent of parameter-tuning [25,26] , that are ubiquitous in nature. This is equivalent to expecting that nature is mathematically relatively simple.22From the standpoint of computable functions and computable numbers we can generally think of a deterministic dynamical system as a computer with the initial condition as the program [21]. Thinking of dynamics from this point of view, it has been discovered that there is a far greater and far more interesting degree of complicated behavior in nonlinear dynamics than either criticality or deterministic chaos: systems of billiard balls combined with mirrors [27,27b] , and even two-dimensional maps [28], can exhibit universal computational capacity via formal equivalence to a Turing machine. A system of nine first order quasi-linear partial differential equations has been offered as a computationally-universal system [29] . A quasi-linear first order partial differential equation in n variables can be replaced by a linear one in n+1 variables. Maximum computational complexity is apparently possible in systems of linear first order partial differential equations . Such systems are nondiffusive but can describe damped-driven dynamics and wave propagation. For a dynamical system with universal computational capability a classification into topologic universality classes is impossible [28] . Given an algorithm for the computation of an initial condition to as many digits as computer time allows, nothing can be said in advance about the future either statistically or otherwise except to compute the dynamics with controlled precision for that initial condition, iteration by iteration, to see what falls out: there is no computational compressibility that allows us to summarize the system's long-time behavior, either statistically or otherwise. In contrast with the case where topologic universality classes exist there is no tree-like organization of a hierarchy of periodic orbits, stable, marginally stable, or unstable, that allows us to understand the fine-grained behavior of an orbit23from the coarse-grained behavior via scaling laws, or to look into the very distant future for arbitrary (so-called "random") initial conditions via symbolic dynamics. There can be no scaling laws that hold independently of a very careful choice of classes of initial conditions. We do not know whethe either fluid turbulence or Newton's three-body problem fall into this category. Some degrees of complexity are defined precisely in computer science [30] but these definitions, based soley on computability theory, have not satisfied physicists [ 31,31b,32] . According to von Neumann [33] a system is complex when it is easier to build than to describe mathematically. Under this qualitative definition the Henon map is not complex but a living cell is. In earlier attempts to model biologic evolution [34,35] information was incorrectly identified as complexity. The stated idea was to find an algorithm that generates information, but this is too easy: the square root algorithm and the logistic map f(x) = 4x(1-x) generate information at the rate of one bit per iteration from rational binary initial conditions. There is no correct model of a dynamic theory of the evolution of biologic complexity, neither over short time intervals (cell to embryo to adult) nor over very long time intervals (inorganic matter to organic matter to metabolizing cells and beyond). There is no physico-chemical model of the time-development of different degrees of complexity in nonlinear dynamics. No one knows if universal computational capability is necessary for biologic evolution, although DNA molecules in solution apparently are able to compute [36], but not error-free like a Turing machine or other deterministic dynamical system.24Moore has speculated that computational universality should be possible in a certain kind of conservative three degree of freedom Newtonian potential flow [28] , but so far no one has constructed an analytic example of the required potential energy. We do not yet know the minimum number of degrees of freedom necessary for universal computational capability in a driven-dissipative flow (a digital computer is a very high degree of freedom damped-driven dynamical system via electric circuit theory). Diffusive motion is time-irreversible (U-1(t) doesn't exist for diffusive motion), but arguments have been made that some diffusive dynamical systems may have an asymptotic limit that is reached asymptotically-fast, where the motion is non-diffusive and is even time reversible on a finite dimensional attractor [37,38,39] , and is therefore generated on the attractor by a finite dimensional deterministic dynamical system (1). However, if a diffusive dynamical system (the Navier-Stokes equations, e.g.) can be shown to be computationally- universal then it will be impossible to discover a single attractor that would permit the derivation of scaling laws for eddy cascades in open flows , or in other flows, independently of specific classes of boundary and initial conditions. With a computationally-universal (and therefore computable) dynamical system (1), given a specific computable initial condition xo, both that initial condition and the dynamics can in principle be encoded as the digit string for another computable initial condition yo. If the computable trajectory y(t) = U(t)y o could be digitally decoded, then we could learn the trajectory x(t) = U(t)x o for the first initial condition (self-replication without copying errors). This maximum degree of computational complexity may be possible25in low dimensional nonintegrable conservative Newtonian dynamics . Some features of nonintegrable quantum systems with a chaotic classical limit (the helium atom, e.g.) have been studied using uncontrolled approximations based on the low order unstable periodic orbits of a chaotic dynamical system [40], but we have no hint what might be the behavior of a low dimensional quantum mechanical system with a computationally-complex Newtonian limit. Interacting DNA molecules obey the laws of quantum mechanics but the biologically-interesting case can not be reduced to a few degrees of freedom. Can new laws of nature emerge from studies of complicated motions? [42] The empirical discovery of mathematical laws of nature arose from the study of the simplest possible dynamical systems: classical mechanics via Galilean trajectories of apples and Keplerian orbits of two bodies (the sun and one planet) interacting via gravity, and quantum mechanics via the hydrogen atom. Is there any reason to expect that simplicity can be short-circuited in favor of complexity in the attempt to discover new mathematical laws of nature? Some researchers expect this to be possible, but without saying how [41]. Consider first an example from fluid dynamics where an attempt has been made to extract a simple law of motion from a complicated time series. Fluid turbulence provides examples of complicated motions in both space and time in a Newtonian dynamical system of very high dimension. We know how to formulate fluid mechanical time evolution according to Newton's laws of motion, the Navier-Stokes equations, but infinitely many interacting degrees of freedom represented by second order coupled nonlinear26partial differential equations are the stumbling block in our attempt to understand fluid turbulence mathematically. We do not understand coupled nonlinear partial differential equations of either the first or second order well enough to be able to derive any of the important features of fluid turbulence in either the finite or infinite Reynold's number limit from the Navier- Stokes equations in a systematic way that starts with the laws of energy and momentum transport and makes controlled, systematic approximations. Can eddy-cascades in turbulent open flows [43] be understood by trying to build simpler mathematical models than the Navier-Stokes equations? So far, this goal remains nothing but an unfulfilled hope. Setting our sights much lower, is it possible to derive a mathematical law in the form of an iterated map that describes only the transition to turbulence, near criticality? We have noted above that the binary tent map can generate all possible histograms that can be constructed simply by varying classes of initial conditions. Statistics that are generated by an unknown dynamical system are therefore inadequate to infer the dynamical law that generates the observed statistical behavior [21] . That is why, in any effort to derive a simplified dynamical system that describes either turbulence or the transition to turbulence, one cannot rely upon statistics alone. Instead, it is necessary to extract the generating partition of the dynamical system from the empirical data, if there is a generating partition . Consider a low dimensional dynamical system that is described by an unknown iterated map defined finitely by a generating partition. With infinite precision and infinite time, it would be possible in principle to pin27down the map's universality class and also the map, from a chaotic time series by the empirical construction of the generating partition. With finite precision and finite time one must always resort to some guesswork after a few steps in the hierarchy of unstable periodic orbits, which are arranged naturally onto a tree of some order and degree of incompleteness [24]. In practice one can discover at most only a small section of the tree and its degree of pruning, so if one is to narrow down the practical choices to a few topologic universality classes of maps the observational data must be extremely precise. Given the most accurate existing data on a fluid dynamical system near a critical point, the unique extraction of the universality class of an iterated map from a chaotic time series has yet to be accomplished without physically-significant ambiguity [44], demonstrating how difficult is the empirical problem that one faces in any attempt to extract an unknown law of motion from the analysis of complicated empirical data. The method of topologic universality classes [21,23,24] is the only known way to study the long time behavior of a chaotic dynamical system systematically, meaning without the introduction of uncontrolled and uncontrollable errors. For truly complex dynamical systems, therefore, our analysis suggests that the extraction of laws of motion from empirical data is a hopeless task. This conclusion does not provide encouragement for experimental mathematicians who want to discover socio-economic or biologic laws of chaotic dynamics from raw statistics or the analysis of time series [45]. The alternative, to imagine that one could "guess" laws of nature without adequate empirical evidence or corresponding symmetry principles, would be to ignore the lessons of Archimedes and Galileo and revert to Platonism.28Einstein apparently became Platonic later in life, but Platonism was not Einstein's guiding light (or light-shade) during his generalization of Newton's theory of gravity, because that generalization is based upon a local invariance principle: no experiment can be performed to detect any difference between a linearly-accelerated frame of reference and the effect of a local gravitational field. This local symmetry principle was not accounted for by Newtonian theory, and motivated Einstein to discover a new set of gravitational field equations [46]. Is socio-economic behavior (mathematically-)lawful? Is it reasonable, even in principle, to expect that mathematical laws of socio- economic or other mathematical laws of human behavior exist in any humanly-discernable form? Is it possible abstractly to reduce some aspects of human behavior to a set of universal formulae, or even to a finite set of more or less invariant rules? Many economists and system theorists [32,47,48], and even some sociologists [49,50], assume that this is possible. By disregarding Galileo's historic and fruitful severing of the abstract study of inanimate motion from imprecise Aristotelian ideas of "motion" like youths alearning, acorns asprouting [2], and markets emerging, many mathematical economists have attempted to describe the irregularities of individual and collective human nature as if the price movements of a commodity, which are determined by human decisions and man-made political and economic rules, would define mathematical variables and abstract universal equations of motion analogous to ballistics and astronomy (deterministic models), or analogous to a drunken professor (stochastic models).29Mathematical economists often speak of the economy [48], which is determined by human behavior and man-made rules (and also in part by the weather, geology, and other limiting physical factors), as if the economy could be studied mathematically as an abstract dynamical system like the weather. In the latter case the equations of motion are known but cannot be solved approximately over large space-time regions by using floating point arithmetic on a computer without the introduction of uncontrollable errors. However, for specified boundary and initial conditions the weather is determined by the mathematical equivalent of many brainless interacting bodies that can not use intelligience to choose whether or not to obey the deterministic differential equations whose rigid mathematical rule they are condemned forever to follow. Chaos and complexity do not install either randomness, freedom of choice, or arbitrariness in the solutions of deterministic dynamical equations [21]. The absence of arbitrariness, or freedom of choice, is part of the key to understanding why mathematics works in physics but not in the socio-economic fields. Comparing the weather with socio-economic behavior is not a scientifically-sound theoretical analogy. Contrary to certain expectations [51] and to recent extraordinary claims [52] , there is no evidence to suggest that abstract dynamical systems theory can be used either to explain or understand socio-economic behavior. Billiard balls and gravitating bodies have no choice but to follow mathematical trajectories that are laid out deterministically, beyond the possibility of human convention, invention, or intervention , by Newton's laws of motion. The law of probability of a Brownian particle also evolves deterministically30according to the diffusion equation beyond the possibility of human convention, invention, or intervention. In stark contrast, a brain that directs the movements of a body continually makes willful and arbitrary decisions at arbitrary times that cause it to deviate from and eventually contradict any mathematical trajectory (deterministic models) or evolving set of probabilities (stochastic models) assigned to it in advance. Given a hypothetical set of probabilities for a decision at one instant, there is no algorithm that tells us how to compute the probabilities correctly for later times, excepting at best the trivial case of curve-fitting at very short times, and then only if nothing changes significantly. Socio-economic statistics can not be known in advance of their occurrence because, to begin with, there are no known socio-economic laws of motion that are correct. Economists stress that they study open systems, whereas physics concentrates on closed systems. This claim misses the point completely. We can describe and understand tornadoes and hurricanes mathematically because the equations of thermo-hydrodynamics apply, in spite of the fact that the earth's atmosphere is an open dynamical system. We can not understand the collapse of the Soviet Union or the financial crisis in Mexico on the basis of any known set of dynamics equations in spite of the fact that the world economy forms a closed financial system. Mathematical-lawlessness reigns supreme in the socio-economic fields, where nothing of any social or economic significance is left even approximately invariant by socio-economic evolution, including the "value" of the Mark. This is the reason that artificial law ("law") must be used by governments and central banks in the attempt to regulate human behavior,31both individually and collectively. Socio-economically, everything that is significant changes completely unregulated in the absence of police-enforced artificial law (the Roman method) or strong community traditions (the tribal method). In the socio-economic fields there are no fundamental constants because nothing is left invariant by the time evolution. That nothing is left invariant is the same as saying that the system is not describable by mathematics: dynamical systems, even discrete ones [52b], have local conservation laws. Deterministic dynamical systems obey n-1 local conservation laws that prevent any external constraint from being imposed on the system. You can not "legislate" a change in the dynamics of a system that obeys a deterministic law of motion. The division of observable phenomena into machine-like and not-machine- like behavior was made by Descartes [53]. In the Cartesian picture animals are supposed to behave more like machines, like robots that respond mechanically to stimuli. People, in contrast with robots, can reason and make decisions freely, or at least arbitrarily. Even the most illiterate or most stupid people can speak, can invent sentences creatively, and can behave unpredictably in other ways as well. The most intelligent dog, cat, or cow cannot invent intellectual complexity that is equivalent to a human language or a capitalist economy. We should ask: why should any part of nature behave mathematically, simulating an automaton? Why does the mathematics of dynamical systems theory accurately describe the motions studied in physics, but not the32"motions" (in Aristotle's sense) studied in economics, political science, psychology, and sociology? This question leads to Wigner's discussion of the "unreasonable effectiveness of mathematics" in describing the inanimate aspects of nature that physics traditionally studies, and that fields disconnected from physics have tried unsuccessfully to imitate merely by postulating laws of motion that do not pass the test of reproducibility of measurements. It is necessary to realize that, in spite of Newton's scholastic style of presentation of his laws, which many mechanics text books unfortunately mimic, physics is neither postulatory nor axiomatic. Physics since Galileo is grounded in a deep interplay of empiricism and mathematical abstraction, and the reason that this mathematical interplay is at all possible is due to certain invariance principles (physics would be impossible in the absence of certain fundamental constants of nature; those constants reflect certain invariance principles). Reductionism, invariance principles, and laws of nature Reductionism is the arbitrary division of nature into laws of motion and initial conditions, plus "the environment". We must always be able to neglect "the environment" to zeroth order, because if nothing can be isolated then a law of motion can never be discovered. For example air resistance had to be negligible in order that Galileo could discover the law of inertia and the local law of gravity.33The empirical discovery of mathematical laws of motion that correctly describe nature is impossible in the absence of empirically-significant invariance principles, but there are no laws of nature that can tell us the initial conditions. Following Wigner, laws of motion themselves obey laws called invariance principles, while initial conditions are completely lawless [54]. Why must mathematical laws of motion that describe nature obey invariance principles? "It is not necessary to look deeper into the situation to realize that laws of nature could not exist without principles of invariance. This is explained in many texts of elementary physics even though only few of the readers of these texts have the maturity necessary to appreciate these explanations. If the correlations between events changed from day to day, and would be different for different points of space, it would be impossible to discover them. Thus the invariances of the laws of nature with respect to displacements in space and time are almost necessary prerequisites that it be possible to discover, or even catalogue, the correlations between events which are the laws of nature. E. P. Wigner in Symmetries and Reflections [54] Nearly every elementary physics text shows that the experiments that Wigner had in mind are the parabolic trajectories of apples and blocks sliding down inclined planes, the two physical systems originally studied by Galileo in his empirical discovery of the local versions of Newton's first two laws of motion. Those discoveries would have been impossible in the absence of four fundamental invariance principles.34Without translational and rotational invariance in space and translational invariance in time (at least locally, on earth and within our solar system), simple mathematical laws of motion like the Keplerian planetary orbits and the Galilean trajectories of apples could not have been discovered in the first place. The experiments that are needed to discover the law of inertia are precisely reproducible because absolute position and absolute time are irrelevant as initial conditions, which is the same as saying that space is homogeneous and isotropic (space is locally Euclidean) and that the flow of time is uniform. The translational invariance of the law of inertia dp/dt=0 means that the law of inertia can be verified regardless of where, in a tangent plane on earth, you perform the required experiment. The law of Galilean invariance is inherent in the law of inertia. Socio-economic phenomena are not invariant in any empirically-discernable sense. Socio-economic time-development and the corresponding statistics depend upon absolute position and absolute time, which is the same (for all practical purposes) as admitting that socio-economic "motions" are not reducible to a well-defined dynamical system. Dynamical laws of motion are postulated in economics, but the laws of physics are not mere postulates: mathematical laws of time-development come second, invariance principles come first. The law of inertia had to be discovered first (Galileo/Descartes) before Newton could write down his second order differential equation that generalizes Galileo's two local empirical laws, the law of inertia and the local law of gravity. Described from the standpoint of invariance and symmetry, the law of inertia is the foundation of all of physics: from it and Galileo's local law of gravity follow35two of Newton's three laws of motion and his law of gravity as a generalization, when Kepler's first law and the action-reaction principle are used [12]. It is superficial and misleading to imagine that the law of inertia can be "derived" from Newton's second law merely by setting the net force equal to zero. If absolute time and absolute position were relevant initial conditions then neither the law of inertia nor the local law of gravity would hold: identically prepared experiments would yield entirely different outcomes in different places and at different times. In this case there could have been no regularities discovered by Galileo, and no generalizations to universal laws of classical mechanics could have been proposed by Newton. Physics would, in that case, have remained Aristotelian and consequently would have evolved like economics, sociology, psychology, and political science: the study of a lot of special cases with no universal time-evolution laws that permit the prediction, or at least understanding, of phenomena over more than the short time intervals where curve-fitting sometimes "works", and with no qualitative understanding whatsoever of the phenomena underlying the observed "motions" and their corresponding statistics. System theorists commonly assume that the economy operates like a dynamical system, the equivalent of an automaton that is too simple to simulate any kind of creative behavior, including the violation of politically- enforced laws as occurred during the collapse of the government of the former Soviet Union and the peasant rebellion in Chiapas. This is a strange assumption. Without human brains and human agreements based upon language, "laws" of economic behavior certainly could not exist. Dogs, cows36and even peasants generally don't invent money-economies. In contrast, the available geological and astronomical evidence indicates that Newton's laws of motion held locally in our corner of the universe long before human languages emerged on earth. Wigner considers that we can not rule out that "holistic" laws of nature (beyond general relativity, for example) might exist, but if so then we have no way to discover them. Reductionism can not explain everything mathematically, but reductionism is required in order to explain the phenomena that can be understood mathematically from the human perspective. Maybe an "oracle" would be required in order to discern the workings of a holistic law of motion. Summarizing, universal laws that are determined by regularities of nature differ markedly from human-created systems of merely conventional behavior. The latter consist of learned, agreed-on, and communally- or politically-enforced behavior, which can always be violated by willful or at least clever people. People and groups who violate artificial law are sometimes called either "progressive" or "outlaw", depending on which social group does the labeling. In Wigner's language, all socio-economic initial conditions matter because of the lack of invariance, so that it is impossible to discover any underlying correlations that could be identified as mathematical laws of socio-economic "progress" (note that the idea of progress is also a "motion" only in the Aristotelian rather than in the Galilean sense).37Darwinism and neo-Darwinism [551] "From a physicist's viewpoint, though, biology, history, and economics can be viewed as dynamical systems." P. Bak and M. Paczuski in Complexity, Contingency, and Criticality [52] "Reductionism" (a better word is "science") is criticized by "holists" for not taking us far enough in our understanding of the world (see the introduction to ref. [32] and also [56]; see also any attempt by the so-called postmodernists to discuss science [57]). Some holists hope to be able to mathematize Darwinism in order to go beyond physics and chemistry (see discussions of "complex adaptable systems" [32]), but so far they have not been able use their invented dynamics models to predict or explain anything that occurs in nature. Physics and astronomy, since the divorce from Platonic mathematics and Aristotlelian "holism" in the seventeenth century, have a completely different history (or "evolution") than "political economy" and most of biology. "Emergence, selection, and adaptation" are buzz words used by Darwin-oriented holists (see ref. [58] for an alternative form of "holism"), while postmodernists like to toss around the notion of "a new paradigm for science". According to the postmodernists, "chaos" (which is merely a part of classical mechanics or chemical kinetics) is an example of "a new paradigm". "Paradigms" are very important for philosophers who have not understood science at the level of Galilean kinematics, and who can not distinguish science from pseudo-seience. Paradigms and "metaphors" are also important 1 Kelly's book "Out of control" is a bible of "paradigms" of postmodernist "holistic" thought.38for people who know that a particular model doesn't represent what the researcher purports to study, but wants to claim that it does anyway. The Aristotelian dream of a holistic approach to physics, biology, economics, history, and other phenomena was revived by Bertanffly in 1968 [59] under the heading of system theory. System theory proposes to use mathematics to describe the time evolution of "the whole", like a living organism or a money-economy, but generally in the absence of adequate information about the local correlations of the connected links that determine the behavior of the whole. I call attempts to quantify the Aristotelian style of thought "reductionist holism", or "holistic reductionism" because any mathematization whatsoever is an attempt at reductionism [42]. Quantification necessarily ignores all nonquantifiable qualities, and there are plenty of qualitative and quantitative considerations to ignore if we want to restrict our considerations to a definite mathematical model. Some physicists tend to believe that physics, which is successful reductionism (often with several-to-high decimal accuracy in agreement between theory and reproducible observations), provides the basis for understanding everything in nature, but only in principle [60]. There is no effective way to "reduce" the study of DNA to the study of quarks but this is not a failure of reductionism: both quarks and DNA are accounted for by quantum mechanics at vastly different length scales. In order to adhere to the illusion that reductionism might also be able account for biological and societal phenomena beyond DNA in principle , physicists must leave out of39consideration everything that hasn't been accounted for by physics, which includes many practical problems that ordinary people face in everyday life. When sociologists [49,50] (who, unlike physicists, claim to interest themselves in the doings of ordinary people) try to follow suit but merely postulate or talk about dynamics "paradigms" in the absence of empirically-established invariance principles, then they reduce their considerations of society to groundless mathematical models, to artificial simulations of life that have nothing to do with any important quality of life. Every computer simulation of a society or an economy is merely the creation of an abstract artificial and brainless society or an artificial and brainless economy. Mathematical simulations cannot adequately describe real societies and real economies although, through adequate politico-financial enforcement, which is truly a form of selection, we can be constrained to simulate some economist's simulation of society and economics. A money economy represents a selection based upon material resources and human needs, desires, and illusions. The idealized free market system described by Adam Smith's "invisible hand" represents a vague notion of autonomy, or self-regulation, inspired in part by Calvinism and in part by Watt's flywheel governor, but is in no scientific sense a "natural" selection. Darwin's ideas of "natural selection, fitness, and adaptation" may appear to make sense in both sports and the socio-economic context of daily life but they are not scientifically-defined mathematical terms. That they remind us of the description of an organized market economy is not accidental: Darwin was strongly influenced during the cruise of the Beagle by his second reading of Malthus [61], who was both a protestant preacher and a worldly40philosopher. Terms like "selection" and "adaptation" are reminiscent of Adam Smith's vague "invisible hand" rather than of scientifically well- defined processes like the dissociation and recombination of DNA molecules described by quantum mechanics or chemical kinetics. In an attempt to model the origin of life, chemical kinetic equations have been used to try either to discover or to invent Darwinism at the molecular level [35] but the use of that terminology seems either superfluous or forced: a deterministic system of ordinary differential equations, whether chemical kinetic or not, can be described by the relatively precise, standard terminology of dynamical systems theory (stability, attractors, etc.). A stochastic system of chemical kinetic equations can be described by purely dynamic terminology combined with additional terms like "most probable distribution" and "fluctuations". There is far less reason to believe that Darwin's socio- economic terminology applies at the macromolecular level than there was, before 1925, to believe that the language of the Bohr model correctly described the motions of electrons relative to nuclei in hydrogen and helium atoms. There are two main sources of Darwin's vague notion of "natural selection". The social-Darwinist origin of the phrase is Malthus's socio-economic doctrine, which derives from Calvinism [61] and can be traced through the late medieval revival of puritanism by Luther, Calvin, and Zwingli back to the neo-Platonist St. Augustine [6], who bequeathed to the west the notion of selection called "predestination". In "predestination" humans are divided completely arbitrarily into "the elect" and "the damned" (according to Luther, man is only an ass ridden by both God and the devil, with no choice whatsoever as to his ultimate fate [62]). Here, "selection" is not a41mathematical idea that describes the time-evolution of a dynamical system. The second and only scientific motivation for Darwin's vague idea of "natural selection" came from plant and animal breeding, which he mislabeled as "artificial selection." Plant and animal breeding constitute the only true case of selection because they proceed via manipulating certain initial conditions in order to try to achieve a desired result. Darwinists, true to their Aristotelian heritage, are condemned to argue endlessly to try to find out what their terminology means because that terminology is, from a scientific standpoint (empiric or theoretic), completely undefined . The scientific foundation of organic evolution was established in Darwin's time by Mendel, who chose to become an Augustinian monk out of financial necessity [39] and was trained more in mathematics and physics than in biology. In contrast with Luther and Calvin, Mendel was not Augustinian in education and outlook: he was even a lecturer in experimental physics for a while, and approached the problem of heredity via isolation of cause and effect in the spirit of a physicist (or a good auto mechanic2 ). Darwin and his contemporaries, in contrast, accepted a holistic (or "integrated") picture of heredity that made the understanding of genetics impossible [64]. It was only after Mendel's reductionist discovery that some biologists began to dislodge themselves from the teleological notion of organic evolution as progress toward a goal predetermined by a selector (or 2 Personally, I would not entrust my auto to a self-proclaimed "holist" for trouble-shooting prior to necessary repairs. See also Ginsburg [63b] for a nonmathematical alternative to holism in the social sciences.42read by an "oracle" capable of "infinite knowledge" of both future and past). By ignoring "the whole" in favor of the most important parts inferred from performing simple, controlled experiments, Mendel found the key that divorced the study of heredity from unsystematic tinkering and socio- economic doctrine and changed it into a precise mathematical science [64b]. Today, Darwinist concepts play a part in genetics research that is comparable to the role played by "waves" in high energy physics. "Wave-particle duality", rather than the Dirac-Feynman interpretation of quantum mechanics, is still taught in physics and chemistry courses, but you may scour the literature to no avail in an attempt to find reference to this cumbersome and unnecessary philosophic principle in particle physics research papers. Human history is narrative. This includes the statistics of socio-economic phenomena, which constitute only one very small part of the entire narrative, a quantitative part. There is no reason to expect that the uncontrolled approximations of system theory modelling can tell us as much, quantitatively or qualitatively, about social or individual behavioral phenomena as we can learn from experience and by reading history and novels (see [65] for an uncontrolled approximation to the description of some of the consequences of the unrestricted mechanization). The reason why it is illusory to expect to discover objective laws of human history, including the history ("time-evolution") of socio-economic development, was explained prosaically in 1952: "There can be no 'pure history'---history-in-itself, recorded from nobody's point of view, for nobody's sake. The most objective history conceivable is still a selection and an interpretation, necessarily governed by some special43interests and based on some particular beliefs. It can be more nearly objective if those interests and beliefs are explicit, out in the open, where they can be freely examined and criticized. Historians can more nearly approach the detachment of the physicist when they realize that the historical 'reality' is symbolic, not physical, and that they are giving as well as finding meanings. The important meanings of history are not simply there, lined up, waiting to be discovered." Herbert J. Muller, in The Uses of the Past [65] One dimensional life "The nineteenth century, in western Europe and North America, saw the beginning of a process, today being completed by corporate capitalism, by which every tradition which has previously mediated between man and nature was broken." John Berger, in About Looking [66] John Berger, in a very beautiful essay introducing the latest edition of Pig Earth [67], emphasizes what he calls the peasants' view of "circular time" in contrast with the abstract idea of linear time used in Newtonian mechanics. A related viewpoint was developed earlier by the Spengler [68], who was one is three historians who attempted to construct evidence for a grand scheme according to which human "history" evolves. Following the anti-Newtonian Goethe, Spengler imagined human societies as "organisms" moving toward a "destiny". "Destiny" represents a vague idea of organic determinism that Goethe assumed to be in conflict with44mechanistic time-evolution that proceeds via local cause and effect. "Destiny" was imagined to be impossible to describe via mathematical ideas, via Newtonian-style mechanism. In trying to make a distinction between global "destiny" and local cause and effect Spengler was not aware of the idea of attractors in dynamical systems theory, whereby time evolution mimics "destiny" but proceeds purely mechanically according local cause and effect. The Lie-Klein idea of invariance of geometry under coordinate transformations, the forerunner of Nöther's theorem on symmetry, invariance, and conservation laws in physics, may have inspired Spengler's attempt to compare entirely different cultures, widely separated in time and space, as they evolved toward "destinies" that he identified as fully-developed civilizations. Spengler characterized western (European/North American) "civilization" in the following way: the entire countryside is dominated, Roman-style, by a few extremely overpopulated cities called megalopolises. Traditional cultures, derived from man's historic experience of wresting survival directly from nature, have been replaced by the abstract driving force of late civilization, the spirit of money-making. Spengler identified the transition from early Greek culture to late Roman civilization as an earlier example of the nearly "universal" evolution from local tribal culture to money-driven civilization. In modern and postmodern civilization, in a single uncontrolled approximation, all traditions and ideas that interfere with "progress" defined as large-scale and efficient economic development are rejected as unrealistic or irrelevant in the face of a one-dimensional quantitative position whose units may be dollars or marks. The dialogue paraphrased below can be found45on pg. 16 the book Complexity, Metaphors, Models, and Reality [32] about complex adaptable systems in biology, economics, and other fields. A, A', and A'', who are paraphrased, are theoretical physicists. A: Why try to define measures of complexity? A measure of complexity is just a number and that doesn't tell you anything about the system. Assume that there's a particular state that you want to create, a slightly better state of the economy, for example. Suppose that you want to know how complicated that problem is to solve on a computer, and that you're able to characterize complexity. One of the proposals of A' for defining the complexity of a problem is 'what's the minimum amount of money you'd need in order to solve it?' A'': The cost is proportional to computer time. A: Then maybe the unit of complexity should be "money". If you're able to formalize the difficulty of solving the problem of making the economy slightly better, and you find out that you can measure its complexity in terms of dollars or yen, then that kind of measure would be extremely useful. The prediction of a computable chaotic trajectory is limited, decimal by decimal or bit by bit, by computation time, but there are also integrable many body problems that are not complex but also require large amounts of marks or dollars. A'' also asserts in the same book Complexity (pg. 11) that low dimensional chaos is "not complex in a true sense: ... the number bits required for specification of where you are is highly limited." In part, this assertion is false: note that the binary specification of a single state xn in the46logistic map f(x) = 4x(1-x) requires precisely N(n) = 2n(No - 2) + 2 bits, where N o is the number of bits in any simple initial condition xo = .ε1...εNo000... . If the string representing x n is arbitrarily truncated to m ≤N(n) bits, then after on the order of m iterations the first bit (and all other bits) in x n', where n' ≈ n + m, is completely wrong [7]. Multiplication of two finite binary strings of arbitrary length cannot be carried out on any fixed-state machine [5], and if multiplication is done incorrectly at any stage then after only a few more iterations the bits in x n cannot be known even to one-bit accuracy . I expect that the complexity of a dynamical system, like fractal dimensions and Liapunov exponents, can not be described by a single number. "Paradise...was..the invention of a relatively leisured class. ... Work is the condition for equality. ... bourgeois and Marxist ideals of equality presume a world of plenty, they demand equal rights before a cornucopia ... to be constructed by science and the advancement of knowledge. ... The peasant ideal of equality recognizes a world of scarcity ... mutual fraternal aid in struggling against this scarcity and a just sharing of what the world produces. Closely connected with the peasant's recognition, as a survivor, of scarcity is his recognition of man's relative ignorance. He may admire knowledge and the fruits of knowledge but he never supposes that the advance of knowledge reduces the extent of the unknown. ... Nothing in his experience encourages him to believe in final causes ... . The unknown can only be eliminated within the limits of a laboratory experiment. Those limits seem to him to be naive. John Berger, in Pig Earth [67] Danksagung47Dieser Aufsatz basiert teilweise auf meinem Eröffnungsvortrag beim Winter seminar März 1996 auf dem Zeinisjoch. Mein Dank gilt Professor Dr. Peter Plath, der mich zu diesem Vortrag eingeladen hat und auch zu dieser schriftlichen Form ermunterte, sowie Familie Lorenz für ihre liebenswürdige Bewirtung auf dem Zeinisjoch. Obgleich der größte Teil des Seminars auf Deutsch abgehalten wurde, entschied ich mich doch, meinen Vortrag auf Englisch zu halten. Mein Deutsch hätte doch nicht ausgereicht, komplizierten Gedankengänge präzise darzustellen wiederzugeben. References 1. J. L. 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Neishtadt (1993) Mathematical Aspects of Classical and Celestial Mechanics, in Dynamical Systems III , ed. by V.I. Arnold (Springer-Verlag , Heidelberg). 12. J.L. McCauley (forthcoming in 1996) Classical Mechanics: flows, transformations, integrability, and chaos (Cambridge Univ. Pr., Cambridge). 13. V.I. Arnol'd (1981) Ordinary Differential Equations (M.I.T. Press, Cambridge, Mass.). 14. E. Lorenz (1963) J. Atm. Sc. 20 130. 15. H. Poincaré (1993) New Methods of Celestial Mechanics (AIP, Woodbury, NY). 16. L. P. Eisenhart (1961) Continuous Groups of Transformations (Dover, New York). 17. S.A. Burns and J.I. Palmore (1989) Physica D37 , 83. 18. P.J. Olver (1993) Applications of Lie Groups to Differential Equations (Springer-Verlag, New York) 30. 19. J. Palmore (1995) private conversation. 20. A. Turing (1937) Proc. Lon. Math. Soc. (2) 42 , 230. 21. J.L. 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Johnson (1979) Computers and Intr actability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco). 31. C. Bennett (1986) Found. Phys. 16 585. 31b. S. Lloyd and H. Pagels (1988) Annals of Phys. 188 , 186. 32. G. A. Cowan, D. Pines, and D. Meltzer, editiors (1994) Complexity, Metaphors, Models, and Reality (Addison-Wesley, Reading). 33. J. von Neumann (1966) Theory of Self-Reproducing Automata (Univ. of Ill., Urbana) pp. 47, 48. 34. M. Eigen und R. Winkler-Oswatitsch (1987) Stufen zum Leben, (R. Piper, München). 35. B. -O. Küppers (1985) The Molecular theory of Evolution (Springer, Heidelberg). 36. R.J. Lipton (1995) Science 268 542. 37. P. Constantin, C. Foias, O. P. Manley, and R. Temam (1985) J. Fluid Mech. 150 427. 38. C. Foias, G. R. Sell, and R. Temam (1988) J. Diff. Eqns. 73 309.5039. P. Constantin, C. Foias, B. Nicolaenko, and R. Teman (1980) Integral and Inertial Manifolds for dissipative Partial Differential Equations (Springer- Verlag, New York). 40. H. Friedrich (Apr. 1992) Physics World 32-36. 41. J. L. Casti (1995) Complexity 1 , 12. 42. J. L. McCauley (1996) Simulations, complexity, and laws of nature (refereed twice and accepted by editor John Casti for the Santa Fe Institute journal "Complexity", on 2/9/96 and then much later rejected on 5/4/96 because of "hostile" and "unprintable" reactions by four "commentators"). 43. J. L. McCauley (1991) in Spontaneous Formation of Space-Time Structures and Criticality , ed. D. Sherrington and T. Riste (Kluwer, Dordrecht). 44. A. Chhabra, R. V. Jensen, and K. R. Sreenivasan (1989) Phys. Rev. A 40 , 4593. 45. E. E. Peters (1991) Chaos and Order in the Capital Markets (Wiley, New York). 46. A. Einstein (1953) The Meaning of Relativity , Princeton Univ. Pr., Princeton. 47. J. L. Casti and A. Karlqvist (1991) Beyond Belief: Randomness, Prediction, and Explanation in Science (CRC Press, Boca Raton). 48. P. W. Anderson, K. J. Arrow, and D. Pines, eds. (1988) The Economy as an Evolving Complex System (Addison-Wesley, Redwood City). 49. R. L. Ackoff (1974) Redisigning the Future (Wiley-Interscience, New York). 50. R. Collins (1988) Theoretical Sociology (Harcourt, Brace, Janovich, San Diego).5151. D. Ruelle (July, 1994) Physics Today, pp. 24-30. 52. P. Bak and M. Paczuski (1995) Complexity, Contingency, and Criticality , subm. to Proc. Nat. Acad. Sci.. 52b. J. Palmore (1995) Chaos, Solitons, and Fractals 5 1397. 53. N. Chomsky (1968) Language and Mind , Harcourt Brace Janovich (New York). 54. E. P. Wigner (1967) Symmetries and Reflections (Univ. Indiana Pr., Bloomington). 55. K. Kelly (1994) Out of Control (Addison-Wesley, Reading) ch. 22. 56. H. Morowitz (1995) Complexity 1 4. 57. J.- F. Lyotard (1993) The Postmodern Condition: A Report on Knowledge , trans. from fr. by G. Bennington and B. Massumi (Univ. of Minn., Minneapolis). 58. P. Bak (1994) Self-Organized Criticality: A Holistic View of Nature in ref. 59. L. von Bertalanffy (1968) General Systems Theory (G. Braziller, New York). 60. S. Weinberg (10/5/1995) Reductionism Redux , New York Review of Books, pp. 39-42. 61. R. M. Young (1985) Darwin's Metaphor (Cambridge Univ. Pr., Cambridge). 62. H. A. Oberman (1982) Mensch zwischen Gott und Teufel (Severin und Seidler Ver., Berlin). 63. R. Olby (1985) Origins of Mendelism , 2nd Ed. (Univ. of Chicago Pr., Chicago). 63b. C. Ginsburg (1989) Clues, Myths, and the Historical Method transl. from Italian (Johns Hopkins Pr., Baltimore) pp. 96-125. 64. P. J. Bowler (1989) The Mendelian Revolution (Johns Hopkins, Baltimore). 64b. The Human Genome Project (1992) Los Alamos Science 20 . 65. K. Vonnegut (1952) Player Piano (Avon Books, New York).5266. H. J. Muller (1952) The Uses of the Past (Oxford Univ. Pr., New York). 66. J. Berger (1991) About Looking (Vintage Pr., New York). 67. J. Berger (1992) Pig Earth (Vintage Pr., New York). 68. O. Spengler (1922) Untergang des Abendlandes (C. H. Bech'sche Verlagsbuchhandlung, München).53Figure Captions 1. Samuelson's question: Is it a bird or an antelope? Answer: neither, it's a continuous line between two points plus a closed curve that, unlike both birds and antelopes, is topologically equivalent to a straight line plus a circle (from Samuelson [3]). 2. Successive maxima zn of a numerically-computed time series x3(t) for the Lorenz model are plotted against each other (from McCauley [21]). The drawing of a single continuous curve through all of these points would violate the time-reversibility. 3. Assignment of the symbols L and R for a unimodal map (from McCauley [21]). 4. The complete binary tree defines the topologic universality class of the binary tent map, and all unimodal maps of the unit interval that peak at or above unity and contract intervals exponentially fast in backward iteration (from McCauley [21]). 5. An n-bit sliding window (shown for n = 1, 2, and 3) is used to read a section of a binary symbol sequence in order to discover the corresponding orbital statistics, described by histograms with 2n bins (from McCauley [21]).

arXiv:physics/0001030v1 [physics.ed-ph] 17 Jan 2000Simple quantum systems in the momentum rep- resentation H. N. N´ u˜ nez-Y´ epez † Departamento de F´ ısica, Universidad Aut´ onoma Metropoli tana-Iztapalapa, Apar- tado Postal 55-534, Iztapalapa 09340 D.F. M´ exico, E. Guillaum´ ın-Espa˜ na, R. P. Mart´ ınez-y-Romero, ‡A. L. Salas-Brito ¶ Laboratorio de Sistemas Din´ amicos, Departamento de Cienc ias B´ asicas, Univer- sidad Aut´ onoma Metropolitana-Azcapotzalco, Apartado Po stal 21-726, Coyoacan 04000 D. F. M´ exico Abstract The momentum representation is seldom used in quantum mecha nics courses. Some students are thence surprised by the change in viewpoin t when, in doing advanced work, they have to use the momentum rather than the c oordinate repre- sentation. In this work, we give an introduction to quantum m echanics in momen- tum space, where the Schr¨ odinger equation becomes an integ ral equation. To this end we discuss standard problems, namely, the free particle , the quantum motion under a constant potential, a particle interacting with a po tential step, and the motion of a particle under a harmonic potential. What is not s o standard is that they are all conceived from momentum space and hence they, wi th the exception of the free particle, are not equivalent to the coordinate sp ace ones with the same names. All the problems are solved within the momentum repre sentation making no reference to the systems they correspond to in the coordin ate representation. †E-mail: nyhn@xanum.uam.mx ‡On leave from Fac. de Ciencias, UNAM. E-mail: rodolfo@dirac .fciencias.unam.mx ¶E-mail: asb@hp9000a1.uam.mx or asb@data.net.mx 11. Introduction In quantum mechanics the spatial position, ˆ x, and the linear momentum, ˆ p, operators play very symmetrical roles, as it must be obvious from the fundamental commutation relation [ˆp,ˆx] = ˆpˆx−ˆxˆp=−i¯h (1) where, apart from a minus sign, the roles of ˆ xand ˆpare the same. Notice that a hat over the symbols has been used to identify operators. Ho wever, this funda- mental symmetry is not apparent to many students of quantum m echanics because an excessive emphasis is put on the coordinate representati on in lectures and in textbooks. Some students are even lead to think of the coordi nate space wave functionψ(x) as more fundamental, in a certain way, than its momentum spa ce counterpart φ(p); for, even in one of those rare cases where the Schr¨ odinger equa- tion is solved in momentum space, as is often the case with the linear potential x(Constantinescu and Magyari 1978), many students feel that the quantum solu- tion is somewhat not complete until the coordinate space wav e function has been found. This is a pity since a great deal of physical consequen ces are understood better and many physical effects are more readily evaluated i n the momentum rather than in the coordinate representation; as an example just think of scat- tering processes and form factors of every kind (Taylor 1972 , Ch 3; Frauenfelder and Henley 1974; Bransden and Joachaim 1983; Griffiths 1987). To give another interesting example, let us remember the one-dimensional h ydrogen atom, an ap- parently simple system whose properties were finally unders tood, after thirty years of controversy (Loudon 1959, Elliot and Loudon 1960, Haines and Roberts 1969, Andrews 1976, Imbo and Sukhatme 1985, Boya et al. 1988, N´ u˜ nez-Y´ epez et al. 1988, 1989, Mart´ ınez-y-Romero et al.1989a,b,c) only after an analysis was carried out in the momentum representation (N´ u˜ nez-Y´ epez et al. 1987, 1988, Davtyan et al.1987). But, besides particular ocurrences, the advantages of an ea rly introduction to the momentum representation in quantum mechanics are man yfold: a) to em- phasize the basic symmetry between the representations, b) to introduce from the beginning and in a rather natural manner, distributions—pi npointing that the eigenfunctions are best regarded as generalized rather tha n ordinary functions—, 2non-local operators and integral equations, c) to help clar ify the different nature of operators in both representations, for example, in the mome ntum representation a free particle (vanishing potential in any representation )cannot be considered as equivalent to a particle acted by a constant (in momentum spa ce) potential, since this last system admits a bound state. According to us the pro blems discussed in this work make clear, using rather simple examples, the dist inct advantages and perhaps some of the disadvantages of working in the momentum representation. 2. Quantum mechanics in momentum space. For calculating the basic properties and the stationary sta tes of quantum mechanics systems, the fundamental equation is the time-in dependent Scrh¨ odinger equation which, in the coordinate representation, can be wr itten as the differential eigenvalue equation −¯h2 2md2ψ(x) dx2+U(x)ψ(x) =Eψ(x). (2) This can be obtained using, in the classical Hamiltonian H=p2/2m+U(x), the operator correspondence ˆ p→ −i¯hd/dx , ˆx→x[operators complying with (1)]. It is equally possible the use of the alternative operator corr espondence ˆ p→pand ˆx→i¯hd/dp [also complying with (1)] that can be shown to lead —though no t as straightforwardly as in the previous case— to the integral S chr¨ odinger equation in the momentum representation p2 2mφ(p) +/integraldisplay dp′U(p−p′)φ(p′) =Eφ(p), (3) whereU(q) is the Fourier transform of the potential energy function i n the coor- dinate representation: U(q) =1√ 2π¯h/integraldisplay+∞ −∞exp(−ipx/¯h)U(x)dx. (4) As it is obvious from (3), in this representation the potenti al energy becomes an integral operator (hence, usually non-local) in momentum s pace. Equations (2) and (3) are, in fact, Fourier transforms of each other; there fore the relationship between the coordinate and the momentum space wave function s is 3φ(p) =1√ 2π¯h/integraldisplay+∞ −∞exp(−ipx/¯h)ψ(x)dx, and ψ(x) =1√ 2π¯h/integraldisplay+∞ −∞exp(+ipx/¯h)φ(p)dp.(5) Both functions, ψ(x) andφ(p), characterize completely and independently the state of the system in question; although they differ slightl y in interpretation: whereasψ(x) is the probability amplitude that the a measurement of posi tion gives a value in the interval [ x, x+dx],φ(p) is the probability amplitude that a measurement of momentum gives a value in the interval [ p,p+dp]. In spite of their complete equivalence, the momentum repres entation could throw light in certain features that may remain hidden in the coordinate represen- tation; very good examples of this are the SO(4) symmetry of t he hydrogen atom first uncovered by Fock (1935) using his masterly treatment o f the problem in the momentum representation; or the treatment of resonant Gamo w states in the mo- mentum representation where they were found to be, contrary to what happens in the coordinate representation, square integrable solut ions to a homogeneous Lippmann-Schwinger equation (Hern´ andez and Mondrag´ on 1 984). In this work we calculate the bound energy eigenstates and co rresponding quantum levels of the simplest one-dimensional potential p roblems in the momen- tum representation. We choose to present them in order of inc reasing complexity, as it is usually done in basic quantum mechanics: 1) The free particle with U(p) = 0. 2) Particle in a constant potential: U(p) =−U0(U0>0). 3) Particle interacting with the potential step U(p) =  0, ifp≤0, iα(αa positive constant) ,ifp>0;(6) please notice that as we assume αto be a real number, the ifactor is necessary to assure the Hermiticity of the potential energy operator. 4) Motion in the harmonic potential U(p) =−f0cos(ap), (7) 4wheref0>0 andaare real numbers. As we intend to illustrate in this contribution, in many inst ances the eigen- functions are easier to calculate in momentum space than in t he coordinate space representation. We have to recognize though that the moment um space eigen- states are best understood as generalized functions or dist ributions —to which the Riemann interpretation of integrals does not apply; this is explicitly illustrated by examples A, B, and D below. The energy eigenvalues are calcul ated, in most cases discussed here (A, B, and D), as consistence conditions on th e eigenfunctions, and in the remaining one, C, from the univaluedness of the eigenf unctions. 3. The examples. We want to point out that albeit we are addressing the same typ e of systems that are used to introduce quantum mechanics, here we employ the same notion of simplicity but with problems posed in momentum space (mak ing them very different from the coordinate space ones). Please be aware th at we use atomic units wherever it is convenient in the rest of the paper: ¯ h=e=m= 1. A. The free particle. In the case of the free particle, as in the coordinate represe ntation,U(p) = 0 everywhere, so the Schr¨ odinger equation (3) is simply /parenleftbiggp2 2−E/parenrightbigg φ(p) = 0; (8) this deceptively simple equation has as its basic solutions φpE(p) =Aδ(p−pE) (9) wherepEis a solution of p2= 2EandAis a constant. This is so since, according to (8), the wave function vanishes excepting when the energy takes its “on shell” valueE=p2/2; furthermore as φ(p) cannot vanish everywhere, equation (9) follows. The energy eigenfunctions (9) are also simultaneo usly eigenstates of the linear momentum, ˆpφpE(p) =pδ(p−pE) =pEφpE, (10) and form a generalized basis — i.e.formed by Dirac improper vectors— for the states of a free particle with well defined energy and linear m omentum (B¨ ohm 1979, 5Sakurai 1985); for such a free particle the most general stat ionary momentum- space solution is then Φ(p) =A+δ(p+|pE|) +A−δ(p− |pE|) (11) where theA±are complex normalization constants; this solution repres ent a par- ticle traveling to the right with momentum |pE|and to the left with momentum −|pE|. The basic solutions (9) can be “orthonormalized” accordin g to (Sakurai 1985) /integraldisplay+∞ −∞φ∗ pEφp′ E(p)(p)dp=δ(pE−p′ E) (12) which requires A= 1 in (9). The possible energy values are constrained only by the classical dispersion relation E=p2 E/2mhence they form a continuum and the eigenstates cannot be bound. It is to be noted that for describing the eigenstates of a free particle, quantum mechanics uses generalized functions for which the probabi lity densities |φpE(p)|2 are not well defined! What it is well defined is their action on a ny square integrable function, hence on any physical state; therefore the eigens tates have to be regarded as linear functionals acting on L2(R), the set of all square integrable functions. The only physically meaningful way of dealing with free part icles requires thus the use of wave packets as follows Φ(p) =/integraldisplay+∞ −∞F(p′)δ(p−p′)dp′ =F(p),(13) whereF(p) is any square integrable function of p. According to their properties then, improper vectors, like those in (9), though very usefu l for formal manipula- tions can never strictly represent physically realizable s tates (Taylor 1972, section 1a). B. Motion under a constant potential Substitution of the constant value −U0<0 into (3), gives us /parenleftbiggp2 2−E/parenrightbigg φ(p) =−U0/integraldisplay+∞ −∞φ(p′)dp′; (14) 6to solve (14), let us define the number ˇ ϕas ˇϕ≡/integraldisplay+∞ −∞φ(p′)dp′; (15) with this definition, the momentum representation Schr¨ odi nger equation (14) re- duces to a purely algebraic equation for φ(p), /parenleftbiggp2 2−E/parenrightbigg φ(p) =−U0ˇϕ; (16) let us now define k2 0=−2E >0, then the eigenfunctions are easily seen to be φ(p) =−2U0ˇϕ p2+k2 0. (17) To determine the energy eigenvalues we integrate both sides of (17) to get ˇϕ=−2πU0ˇϕ k0orE=−2π2U0; (18) the system has a single energy eigenstate with the energy eig envalue given in (18). The associated normalized eigenfunction is then φ(p) =/radicalbigg 2 πk0k2 0 p2+k2 0. (19) It is important to emphasize what we have shown: a constant po tential in momentum space admits a bound state. Obviously then in this r epresentation we have not the freedom of changing the origin of the potential e nergy by adding a constant. In momentum space the potential energy is undetermined not u p to a constant value but up to a Dirac-delta function potential ; that is, if you take an arbitrary potential U(p) in momentum space, the physics of the problem is not changed when you consider instead the modified potential U′(p) =U(p) +γδ(p) withγan arbitrary constant, whereas the change U′′(p) =U(p)+γ/negationslash=U′(p)does indeed change the physics. The reader can prove by herself th is elementary fact. This discussion is going forward apparently with no trouble ; we have to ac- knowledge though that for getting to the condition (18), we q uickly passed over a very important point, the integral of the right hand side of (17) does not exist in the ordinary Riemann sense. To obtain our result you need t o do it instead 7in the distribution sense, regarding the momentum space fun ctionφ(p) as a lin- ear functional acting upon square integrable functions, as corresponds to possible state functions of a quantum system. Such idea is also behind the usefulness of the delta functions as generalized basis for the free partic le states in example A. To particularize to the present situation, this amounts to m ake ˆϕconvergent (hence meaningful) when acting on any state function (Richt myer 1978, B¨ ohm 1979). A direct way of accomplishing this is, as usually done in theoretical physics (Taylor 1972, Frauenfelder and Henley 1974, Griffiths 1987), to get the mentioned integral come into existence in a principal value sense (Mat hews and Walker 1970). To this end first multiply the right hand side of (17) times an e xp(−iǫp) complex factor, then perform the integral using contour integratio n in the complex plane and,at the very end , take the limit ǫ→0. With such provisos considered, it is not difficult getting the result (18). However, this means that th e functions involved in our discussion have to be considered as linear functionals o r generalized functions, as can be done—perhaps it would be better to say: should be done —for every wave function of a quantum system (Messiah 1976, B¨ ohm 1979) ; forgetting this fact can produce erroneous results as it is exemplified by the case discussed in (N´ u˜ nez-Y´ epez and Salas-Brito 1987). It is to be noted that the free particle potential acts as a con fining potential in momentum space; it allows, for each—out of a nonnegative c ontinuum—energy value, just two choices for the momentum: |pE|and−|pE|; such extreme restriction is also reflected in the wave functions, they are Dirac delta f unctions which peak at the just quoted values of p. On the other hand, the constant potential, which does not restrict the possible values of the momentum in the s evere way of the zero potential, is not as confining in momentum space and allo ws a single energy eigenvalue whose associated eigenstate requires a very wid e range of momenta [given in (19)] to exist. At this point we invite the reader to try to solve the problem of a particle inside an infinite potential box— in momentum space . This is a simple and nice exercise to test the intuition on the diffe rences between the momentum and the coordinate representation; it is not difficu lt to conclude that, in this case, the eigenfunctions are also Dirac delta functi ons with a lightly but subtly modified relation linking energy and momentum. C. Motion in a potential step 8In this case U(p) is given in (6). Using such potential, the Schr¨ odinger equ a- tion becomes a simple Volterra integral equation /parenleftbiggp2 2−E/parenrightbigg φ(p) +iα/integraldisplayp −∞φ(p′)dp′. (20) To solve this equation, we derive both members and, using k2 0≡ −2E, we obtain a very simple differential equation dφ(p) dp= 2p−iα p2+k2 0φ(p), (21) whose solution is φk0(p) =A p2+k2 0/bracketleftbiggk0−ip k0+ip/bracketrightbiggα/k0 (22) withAan integration constant. The energy eigenvalues follow, not from a consistency condi tion as in the last example, B, but from the requirement that the eigenfunction s be single valued. This is only possible if α/k0takes nonnegative integer values (Churchill 1960), i.e. ifk0=α/n,n= 1,2,..., the value n= 0 is not allowed for φ(p) would vanish identically in that case. Thus, the system has an infinite num ber of bound energy eigenstates with energies given by En=−α2 2n2, n= 1,2,...; (23) the normalization of the eigenfunctions requires that A= (2α3/n3π)1/2in equation (22). A very important property of the eigenfunctions is /integraldisplay+∞ −∞φ(p)dp= 0, (24) this is required to guarantee the Hermiticity of the Hamilto nian operator of the problem (Andrews 1976, N´ u˜ nez-Y´ epez et al.1987, Salas-Brito 1990). We pinpoint that the potential step in momentum space is particularly in teresting because it is closely related to the study of the momentum space behaviour of electrons interact- ing with the surface of liquid helium, with the properties of the an hydrogen atom in superstrong magnetic fields, and with certain supersymme tric problems (Cole 9and Cohen 1969, Imbo and Sukhatme 1985, N´ u˜ nez-Y´ epez et al. 1987, Mart´ ınez-y- Romero et al. 1989c, Salas-Brito 1990). D. Motion under a harmonic potential Let us, as our final example, study the motion of a particle und er the har- monic potential U(p) =−f0cos(ap), whereaandf0>0 are real constants. The Schr¨ odinger equation is then p2 2φ(p)−f0/integraldisplay+∞ −∞cos[a(p−p′)]φ(p′)dp′. (25) By changing pfor−pin (25) we can show that the Hamiltonian commutes with the parity operator, thus its eigenfunctions can be chosen a s even or odd functions, i.e.as parity eigenstates. For solving (25), let us define k2≡ −2Eand, using the identity 2 cos x= exp(ix) + exp( −ix), we easily obtain the eigenfunctions as φ(p) =f0 p2+k2/bracketleftbig ˇϕ+e+iap+ ˇϕ−e−iap/bracketrightbig , (26) where the numbers ˇ ϕ±are defined by ˇϕ±≡/integraldisplay+∞ −∞e±iap′φ(p′)dp′. (27) As in the constant potential (example B), the energy eigenva lues follow from using the definitions (27) back in the eigenfunctions (26)—please remember that we require the functions (26) to be regarded in the distributio n sense for doing the integrals (27). This gives us the following two conditions ˇϕ+=f0π k[ˇϕ++ ˇϕ−exp(−2ak)], ˇϕ−=f0π k[ˇϕ−+ ˇϕ+exp(−2ak)].(28) From (28), it follows that ˇ ϕ+=±ˇϕ−and, as anticipated, the eigenfunctions are even or odd, namely φ+(p) =A+ p2+k2cos(ap), φ−(p) =A− p2+k2sin(ap);(29) 10which correspond to the complete set of eigenfunctions of th e problem. From (28) we also get the equations determining the energy ei genvalues k f0π−1 =±e−2ak. (30) As can be seen in Figure 1, in general equations (30) admits tw o solutions, let us call themk+(for the even state) and k−(for the odd state). Therefore the system has a maximum of two eigenvalues E+=−k2 +/2 andE−=−k2 −/2; the ground state is always even and the excited (odd) state exist only if f0≤fcrit 0= 1/2aπ≃ 0.1592. The analysis is easily done using the graph shown in Figure 1, where we plot together −αk+ 1,αk−1 and exp( −2ak) [usingα≡1/(f0π) = 1] against k, for illustrating the roots of (30). In the plot, we have used the valuesa= 1, f0= 1/π≃0.3183, corresponding to the roots k−= 0.7968 (the leftmost root) and k+≃1.109 (the rightmost root); thus the energy eigenvalues are E+≃ −0.6148 (the ground state) and E−≃ −0.3175 (the excited state). The criterion for the existence of the excited state and the value for fcrit 0follows from Figure 1, by noting that such critical value stems from the equality of the slope s of the two curves meeting at the point (0 ,1) in the plot. Notice also that the results previously obtained for the constant potential (example B) can be recov ered as a limiting case of the harmonic potential if we let a→0. 4. Concluding remarks We have discussed four instructive one-dimensional exampl es in quantum me- chanics from the point of view of momentum space. Purportedl y we have not made any reference to the problems they represent in the coordina te representation. We expect to contribute with our approach to the development of physical insight for problems posed in the momentum representation and, furt hermore, to help students to understand the different features of operators, as opposed to classical variables, in different representations. We also expect to m ade clear that some- times it is better to treat a problem from the momentum space p oint of view since the solution can be simplified. The point at hand is the simple form in which the momentum space eigenfunctions are obtained in the probl ems discussed here; though these have to be regarded as distributions for obtain ing the associated energy eigenvalues. 11With goals as the mentioned in mind and to point out other adva ntages of the momentum representation, in a formal set of lectures and dep ending on the level of the students, it may be also convenient to discuss more com plex problems: as scattering and dispersion relations (Taylor 1972), or the s tudy of resonant states as solutions of a Lippmann-Schwinger equation in momentum spa ce (Hern´ andez and Mondrag´ on 1984), or the 3D hydrogen atom, whose solution us ing Fock’s method is nicely exposed in (Bransden and Joachaim 1983). Just in the case you are wondering and have not found the time f or doing the transformations yourself, let us say that the problems, save the free particle, we have posed and solved in this paper are known, in the coordi nate representa- tion, as 1) the attractive delta potential (Example B), 2) qu antum motion under the (quasi-Coulomb) potential 1 /x(Example C) and, finally, the problem of two equal (intensity: A=−πf0/√ 2), symmetrically placed, attractive delta function potentials, which are displaced by 2 afrom one another (Example D). Acknowledgements This paper was partially supported by PAPIIT-UNAM (grant IN –122498). We want to thank Q Chiornaya, M Sieriy, K Hryoltiy, C Srida, M M ati, Ch Cori, F Cucho, S Mahui, R Sammi, and F C Bonito for their encour agement. ALSB also wants to thank the attendants of his UAM-A lectures on quantum mechanics (F´ ısica Moderna, 99-O term), especially Arturo Vel´ azquez-Estrada and El´ ıas Serv´ ın-Hern´ andez, whose participation was relev ant for testing the ideas contained in this work. 12References Andrews M 1976 Am. J. Phys. 441064 B¨ ohm A 1979 Quantum Mechanics (New York: Springer) Boya J, Kmiecik M, and B¨ ohm A 1988 Phys. Rev. A 373567 Bransden B H and Joachaim C J 1983 Physics of Atoms and Molecules (Lon- don: Longman) Ch 2 Cole M W and Cohen M H 1969 Phys. Rev. Lett. 231238 Churchill R V 1960 Complex Variables and Applications (New York: McGraw- Hill) pp 59–60 Constantinescu F and Magyari E 1978 Problems in Quantum Mechanics (Lon- don: Pergamon) Ch V problem 118 Davtyan L S, Pogosian G S, Sissakian A N and Ter-Antonyan V M 19 87J. Phys. A: Math. Gen. 202765 Elliot R J and Loudon R 1960 J. Phys. Chem. Solids 15196 Fock V A 1935 Z. 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Phys. 8189 N´ u˜ nez-Y´ epez H N, Vargas C A and Salas-Brito A L 1988 J. Phys. A: Math. Gen.21L651 N´ u˜ nez-Y´ epez H N, Vargas C A and Salas-Brito A L 1989 Phys. Rev. A 39 4306 Richtmyer R D 1978 Principles of Advanced Mathematical Physics Vol I (New York: Springer) Ch 2 Sakurai J J 1985 Modern Quantum Mechanics (Reading: Addison-Wesley) Salas-Brito A L 1990 ´Atomo de Hidr´ ogeno en un Campo Magn´ etico Infinito: Un Modelo con Regla de Superselecci´ on , Tesis Doctoral, Facultad de Ciencias, Universidad Nacional Aut´ onoma de M´ exico (in Spanish) Taylor J R 1972 Scattering Theory (New York: Wiley) 14Figure Caption Figure 1 The figure illustrates the solution to equations (30) determ ining the energy eigen- values under the harmonic potential (7). We here plot exp( −2ak),αk−1 and −αk+ 1 against kall in the same graph. Just for illustration purposes, we hav e used the specific values a= 1,f0= 1/απ≃0.3183, (we have defined α= 1/f0π and usedα= 1). The critical value of f0, giving birth to the excited state, is fcrit 0= (2aπ)−1as can be obtained from the equality of the slopes of the two curves meeting at the point (0 ,1) in the graph. In the situation exemplified by this figure, fcrit 0≃0.1592 and the roots of equations (30) are k+≃1.109 and k−≃0.7968. 150 0.5 1 1.5 2 k-1-0.500.51f

arXiv:physics/0001031v1 [physics.data-an] 15 Jan 2000Application of Conditioning to the Gaussian-with-Boundar y Problem in the Unified Approach to Confidence Intervals Robert D. Cousins† Department of Physics and Astronomy, University of Califor nia, Los Angeles, CA 90095 (January 14, 2000) Abstract Roe and Woodroofe (RW) have suggested that certain conditio nal probabil- ities be incorporated into the “unified approach” for constr ucting confidence intervals, previously described by Feldman and Cousins (FC ). RW illustrated this conditioning technique using one of the two prototype p roblems in the FC paper, that of Poisson processes with background. The mai n effect was on the upper curve in the confidence belt. In this paper, we att empt to apply this style of conditioning to the other prototype problem, t hat of Gaussian errors with a bounded physical region. We find that the lower c urve on the confidence belt is also moved significantly, in an undesirabl e manner. PACS numbers: 06.20.Dk, 14.60.Pq Typeset using REVT EX 1I. INTRODUCTION Roe and Woodroofe [1] have made an interesting suggestion fo r modifying the “uni- fied approach” to classical confidence intervals which Feldm an and I advocated in Ref. [2]. They invoke the use of “conditioning”, namely replacing fre quentist coverage probabilities with conditional probabilities, still calculated in a freq uentist manner, but conditioned on knowledge gained from the result of the particular experime nt at hand. Roe and Woodroofe (RW) illustrate their suggestion using on e of the two prototype prob- lems, that of Poisson processes with background. Suppose, f or example that an experiment observes 3 events (signal plus background). Then the experi menters know that, in that particular experiment, there were 3 or fewer background eve nts. RW therefore calculate the frequentist coverage using an ensemble of experiments with 3 or fewer background events, rather than the larger unrestricted ensemble which we used. Thus, the RW ensemble changes from experiment to experiment. Conditioning on an equality has a long history in classical statistics. (Ref. [1] contains key references.) However, c onditioning on an inequality, as RW do when the number of events is greater than zero, is perhaps l ess well founded, and it is interesting to explore the consequences. In this paper, we attempt to apply RW-like conditioning to th e other prototype problem, that of Gaussian errors with a bounded physical region. The r esult is similar to the Poisson problem analyzed by RW, but difficulties which were apparentl y masked by the discrete nature of the Poisson problem now arise. In particular, the l ower endpoints of confidence intervals are moved significantly in an undesirable directi on. II. THE UNIFIED APPROACH TO THE GAUSSIAN-WITH-BOUNDARY PROBLEM As in Ref. [2], we consider an observable xwhich is the measured value of parameter µ in an experiment with a Gaussian resolution function with kn own fixed rms deviation σ, set here to unity. I.e., P(x|µ) =1√ 2πexp(−(x−µ)2/2). (2.1) We consider the interesting case where only non-negative va lues for µare physically allowed (for example, if µis a mass). The confidence-belt construction in Ref. [2] proceeded as fo llows. For a particular x, we let µbestbe the physically allowed value of µfor which P(x|µ) is maximum. Then µbest= max(0 , x), and P(x|µbest) =/braceleftBigg 1/√ 2π, x ≥0 exp(−x2/2)/√ 2π, x < 0.(2.2) We then compute the likelihood ratio R, R(x) =P(x|µ) P(x|µbest)=/braceleftBigg exp(−(x−µ)2/2), x≥0 exp(xµ−µ2/2), x < 0.(2.3) 2During our Neyman construction of confidence intervals, Rdetermines the order in which values of xare added to the acceptance region at a particular value of µ. In practice, this means that for a given value of µ, one finds the interval [ x1, x2] such that R(x1) =R(x2) and /integraldisplayx2 x1P(x|µ)dx=α, (2.4) where αis the confidence level (C.L.). We solve for x1andx2numerically to the desired precision, for each µin a fine grid. With the acceptance regions all constructed, w e then read off the confidence intervals [ µ1, µ2] as in Ref. [2]. III. INVOKING CONDITIONING IN THE GAUSSIAN-WITH-BOUNDARY PROBLEM In order to formulate the conditioning, we find it helpful to t hink of the measured value xas being the sum of two parts, the true mean µtand the random “noise” which we call ε: x=µt+ε. (3.1) We are considering the case where it is known on physical grou nds that µt≥0. Thus, if an experimenter obtains the value x0in an particular experiment, then he or she knows that, in that particular experiment , ε≤x0. (3.2) For example, if the experimenter measures µand obtains x0=−2, then the experimenter knows that ε≤ −2 in that particular experiment. This information is analog ous to the information in the Poisson problem above in which one knows t hat in the particular experi- ment, the number of background events is 3 or fewer. We thus us e it the manner analogous to that of RW: our particular experimenter will consider the ensemble of experiments with ε≤x0when constructing the confidence belt relevant to his or her e xperiment. We let P(x|µ, ε≤x0) be the (normalized) conditional probability for obtainin gx, given thatε≤x0. In notation similar to that of RW, this can be denoted as qx0µ(x): qx0 µ(x)≡P(x|µ, ε≤x0) =/braceleftBigg2√ 2πexp(−(x−µ)2/2)/(erf(x0/√ 2) + 1) , x≤µ+x0 0, x > µ +x0.(3.3) Given x0, at each xwe find µbest, that value of µwhich maximizes P(x|µ, ε≤x0): µbest=  x, x 0≥0 and x≥0 x−x0, x0<0 and x≥x0 0, otherwise(3.4) In the notation of Ref. [1], P(x|µbest, ε≤x0) is then max µ′qx0 µ′(x) =2√ 2π(erf(x0/√ 2) + 1)×  1, x 0≥0 and x≥0 exp(−x2 0)/2, x0<0 and x≥x0 exp(−x2)/2,otherwise(3.5) 3Then the ratio Rof Eqn. 2.3 is replaced by /tildewideRx0(µ, x) =qx0µ(x) max µ′qx0 µ′(x), (3.6) which vanishes if x > µ +x0, and otherwise is given by /tildewideRx0(µ, x) =  exp(−(x−µ)2/2), x 0≥0 and x≥0 exp((−(x−µ)2+x2 0)/2), x0<0 and x≥x0 exp(xµ−µ2/2), otherwise(3.7) Figures 1 through 3 show graphs of qx0µ(x), max µ′qx0 µ′(x), and/tildewideRx0(µ, x), for three values of µ, for each of three values of x0. We let /tildewidecx0(µ) be the value of cfor which /integraldisplay x:/tildewideRx0(µ,x)<cqx0 µ(x)dx=α. (3.8) The modified confidence interval consists of those µfor which /tildewideRx0(µ, x0)≥/tildewidecx0(µ). (3.9) Note that this entire construction depends on the value of x0obtained by the particular experiment. An experiment obtaining a different value of x0will have a different function in Eqn. 3.3, and hence a different confidence belt constructio n. Figure 4 shows examples of such constructions for six values of x0. The vertical axis gives the endpoints of the confi- dence intervals. Each different confidence belt constructio n is used only for an experiment obtaining the value x0which was used to construct the belt. The interval [ µ1, µ2] atx=x0 is read off for that experiment; the rest of that plot is not use d. Finally, we can form the graph shown in Fig. 5 by taking the mod ified confidence interval for each x0, and plotting them all on one plot. These are tabulated in Tab le I, which includes for comparison the unconditioned intervals from Table X of R ef. [2]. Fig. 6 shows the modified intervals plotted together with the unified intervals of Ref. [2]. The modified upper curve is shifted upward for negative x, which results in a less stringent upper limit when εis known to be negative; this feature is considered desirabl e by some. The lower curve, however, is also shifted upward: for all x0>0, the interval is two-sided. We find this to be a highly undesirable side-effect. It is interesting to consider what happens if one applies Fig . 5 to an unconditioned ensemble. The result can be seen by drawing a horizontal line at any µin Fig. 5 and inte- grating P(x|µ) (Eqn.2.1) along that line between the belts. For small µ, there is significant undercoverage, while for µnear 1.0, there is significant overcoverage. The undercover age was surprising, since the conditioned intervals always cov er within the relevant subset of the ensemble. However, conditioning on an inequality means tha t these subsets are not disjoint. The undesirable raising of the lower curve is present in the P oisson case, as can be seen in Figure 1 of Ref. [1]. However, there the discreteness of th e Poisson problem apparently prevents the curve from being shifted so dramatically, and t he two-sided intervals do not extend to such low values of the measured n. 4IV. CONCLUSION In this paper, we apply conditioning in the style Roe and Wood roofe to the Gaussian- with-boundary problem. We find that the transition from one- sided intervals to two-sided intervals undesirably moves to the origin. This reflects a ge neral feature of confidence interval construction: when moving one of the two curves, the other cu rve moves also. In the Poisson- with-background problem, the undesirable movement was not large, but in the Gaussian- with-boundary problem, the effect is quite substantial. ACKNOWLEDGMENTS I thank Gary Feldman, Byron Roe, and Michael Woodroofe for co mments on the paper. This work was supported by the U.S. Department of Energy. 5REFERENCES †cousins@physics.ucla.edu [1] B.P. Roe and M.B. Woodroofe, Phys. Rev. D60053009 (1999). [2] G.J. Feldman and R.D. Cousins, Phys. Rev. D573873 (1998). 6FIGURES 00.511.5 -4 -2 0 2 4q(x) xx0=-1.0 µ= 0.01 00.250.50.75 -4 -2 0 2 4q(x) xx0= 0.0 µ= 0.01 00.20.4 -4 -2 0 2 4q(x) xx0= 1.0 µ= 0.01 00.511.5 -4 -2 0 2 4max q(x) xx0=-1.0 µ= 0.01 00.250.50.75 -4 -2 0 2 4max q(x) xx0= 0.0 µ= 0.01 00.20.4 -4 -2 0 2 4max q(x) xx0= 1.0 µ= 0.01 00.51 -4 -2 0 2 4R(µ,x) xx0=-1.0 µ= 0.01 00.51 -4 -2 0 2 4R(µ,x) xx0= 0.0 µ= 0.01 00.51 -4 -2 0 2 4R(µ,x) xx0= 1.0 µ= 0.01 FIG. 1. Graphs of qx0µ(x) (top row), max µ′qx0 µ′(x) (middle row), and/tildewideRx0(µ,x) (bottom row), forµ= 0.01. The columns are for x0=−1, 0, and 1. Each graph in the bottom row is the quotient of the two graphs above it. 700.511.5 -4 -2 0 2 4q(x) xx0=-1.0 µ= 0.50 00.250.50.75 -4 -2 0 2 4q(x) xx0= 0.0 µ= 0.50 00.20.4 -4 -2 0 2 4q(x) xx0= 1.0 µ= 0.50 00.511.5 -4 -2 0 2 4max q(x) xx0=-1.0 µ= 0.50 00.250.50.75 -4 -2 0 2 4max q(x) xx0= 0.0 µ= 0.50 00.20.4 -4 -2 0 2 4max q(x) xx0= 1.0 µ= 0.50 00.51 -4 -2 0 2 4R(µ,x) xx0=-1.0 µ= 0.50 00.51 -4 -2 0 2 4R(µ,x) xx0= 0.0 µ= 0.50 00.51 -4 -2 0 2 4R(µ,x) xx0= 1.0 µ= 0.50 FIG. 2. Graphs of qx0µ(x) (top row), max µ′qx0 µ′(x) (middle row), and/tildewideRx0(µ,x) (bottom row), forµ= 0.5. The columns are for x0=−1, 0, and 1. Each graph in the bottom row is the quotient of the two graphs above it. 800.511.5 -4 -2 0 2 4q(x) xx0=-1.0 µ= 2.50 00.250.50.75 -4 -2 0 2 4q(x) xx0= 0.0 µ= 2.50 00.20.4 -4 -2 0 2 4q(x) xx0= 1.0 µ= 2.50 00.511.5 -4 -2 0 2 4max q(x) xx0=-1.0 µ= 2.50 00.250.50.75 -4 -2 0 2 4max q(x) xx0= 0.0 µ= 2.50 00.20.4 -4 -2 0 2 4max q(x) xx0= 1.0 µ= 2.50 00.51 -4 -2 0 2 4R(µ,x) xx0=-1.0 µ= 2.50 00.51 -4 -2 0 2 4R(µ,x) xx0= 0.0 µ= 2.50 00.51 -4 -2 0 2 4R(µ,x) xx0= 1.0 µ= 2.50 FIG. 3. Graphs of qx0µ(x) (top row), max µ′qx0 µ′(x) (middle row), and/tildewideRx0(µ,x) (bottom row), forµ= 2.5. The columns are for x0=−1, 0. and 1. Each graph in the bottom row is the quotient of the two graphs above it. 90123456 -2 -1 0 1 2 3 4 Measured Mean xMean µ 0123456 -2 -1 0 1 2 3 4x0 = -1.0 0123456 -2 -1 0 1 2 3 4 Measured Mean xMean µ 0123456 -2 -1 0 1 2 3 4x0 = -0.5 0123456 -2 -1 0 1 2 3 4 Measured Mean xMean µ 0123456 -2 -1 0 1 2 3 4x0 = 0.0 0123456 -2 -1 0 1 2 3 4 Measured Mean xMean µ 0123456 -2 -1 0 1 2 3 4x0 = 0.5 0123456 -2 -1 0 1 2 3 4 Measured Mean xMean µ 0123456 -2 -1 0 1 2 3 4x0 = 1.0 0123456 -2 -1 0 1 2 3 4 Measured Mean xMean µ 0123456 -2 -1 0 1 2 3 4x0 = 3.0 FIG. 4. Conditional confidence belts for the six sample value s ofx0indicated. Each plot is used only to find the [ µ1,µ2] interval at xequal to the x0used to construct it; that interval is transferred to Fig.5. 100123456 -2 -1 0 1 2 3 4 Measured Mean xMean µ 0123456 -2 -1 0 1 2 3 4 FIG. 5. Plot of RW-inspired 90% conditional confidence inter vals for mean of a Gaussian, constrained to be non-negative, described in the text. 110123456 -2 -1 0 1 2 3 4 Measured Mean xMean µ 0123456 -2 -1 0 1 2 3 4 FIG. 6. Plot of RW-inspired 90% conditional confidence inter vals (solid curves) , superimposed on the unconditioned intervals of Ref. [2] (dotted curves). 12TABLES TABLE I. 90% C.L. confidence intervals for the mean µof a Gaussian, constrained to be non-negative, as a function of the measured mean x0, for the RW conditioning method, and for the unified approach of Feldman and Cousins. All numbers are i n units of σ. The conditioned numbers may be inaccurate at the level of ±0.01 due to the computational grid used. x0 conditioned unconditioned -3.0 ( 0.00, 0.63) 0.00, 0.26 -2.9 ( 0.00, 0.66) 0.00, 0.27 -2.8 ( 0.00, 0.68) 0.00, 0.28 -2.7 ( 0.00, 0.68) 0.00, 0.29 -2.6 ( 0.00, 0.70) 0.00, 0.30 -2.5 ( 0.00, 0.73) 0.00, 0.32 -2.4 ( 0.00, 0.75) 0.00, 0.33 -2.3 ( 0.00, 0.77) 0.00, 0.34 -2.2 ( 0.00, 0.78) 0.00, 0.36 -2.1 ( 0.00, 0.80) 0.00, 0.38 -2.0 ( 0.00, 0.84) 0.00, 0.40 -1.9 ( 0.00, 0.86) 0.00, 0.43 -1.8 ( 0.00, 0.89) 0.00, 0.45 -1.7 ( 0.00, 0.92) 0.00, 0.48 -1.6 ( 0.00, 0.94) 0.00, 0.52 -1.5 ( 0.00, 0.97) 0.00, 0.56 -1.4 ( 0.00, 1.01) 0.00, 0.60 -1.3 ( 0.00, 1.04) 0.00, 0.64 -1.2 ( 0.00, 1.07) 0.00, 0.70 -1.1 ( 0.00, 1.11) 0.00, 0.75 -1.0 ( 0.00, 1.15) 0.00, 0.81 -0.9 ( 0.00, 1.19) 0.00, 0.88 -0.8 ( 0.00, 1.23) 0.00, 0.95 -0.7 ( 0.00, 1.27) 0.00, 1.02 -0.6 ( 0.00, 1.32) 0.00, 1.10 -0.5 ( 0.00, 1.37) 0.00, 1.18 -0.4 ( 0.00, 1.42) 0.00, 1.27 -0.3 ( 0.00, 1.47) 0.00, 1.36 -0.2 ( 0.00, 1.53) 0.00, 1.45 -0.1 ( 0.00, 1.58) 0.00, 1.55 0.0 ( 0.00, 1.65) 0.00, 1.64 0.1 ( 0.00, 1.71) 0.00, 1.74 0.2 ( 0.01, 1.77) 0.00, 1.84 0.3 ( 0.02, 1.84) 0.00, 1.94 0.4 ( 0.04, 1.91) 0.00, 2.04 0.5 ( 0.06, 1.98) 0.00, 2.14 0.6 ( 0.08, 2.06) 0.00, 2.24 0.7 ( 0.11, 2.13) 0.00, 2.34 130.8 ( 0.13, 2.21) 0.00, 2.44 0.9 ( 0.16, 2.29) 0.00, 2.54 1.0 ( 0.19, 2.38) 0.00, 2.64 1.1 ( 0.22, 2.46) 0.00, 2.74 1.2 ( 0.26, 2.55) 0.00, 2.84 1.3 ( 0.29, 2.64) 0.02, 2.94 1.4 ( 0.33, 2.76) 0.12, 3.04 1.5 ( 0.38, 2.90) 0.22, 3.14 1.6 ( 0.42, 3.04) 0.31, 3.24 1.7 ( 0.47, 3.18) 0.38, 3.34 1.8 ( 0.52, 3.30) 0.45, 3.44 1.9 ( 0.57, 3.43) 0.51, 3.54 2.0 ( 0.63, 3.55) 0.58, 3.64 2.1 ( 0.69, 3.67) 0.65, 3.74 2.2 ( 0.75, 3.79) 0.72, 3.84 2.3 ( 0.82, 3.90) 0.79, 3.94 2.4 ( 0.89, 4.01) 0.87, 4.04 2.5 ( 0.96, 4.12) 0.95, 4.14 2.6 ( 1.04, 4.22) 1.02, 4.24 2.7 ( 1.12, 4.33) 1.11, 4.34 2.8 ( 1.20, 4.43) 1.19, 4.44 2.9 ( 1.28, 4.54) 1.28, 4.54 3.0 ( 1.37, 4.64) 1.37, 4.64 3.1 ( 1.47, 4.74) 1.46, 4.74 14

arXiv:physics/0001032v1 [physics.atm-clus] 15 Jan 2000Decay channels and appearance sizes of doubly anionic gold a nd silver clusters Constantine Yannouleas and Uzi Landman School of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332-0430 (January 2000) Second electron affinities of Au Nand Ag Nclusters and the dissociation energies for fission of the Au2− Nand Ag2− N dianions are calculated using the finite-temperature shell - correction method and allowing for triaxial deformations. Di- anionic clusters with N > 2 are found to be energetically stable against fission, leaving electron autodetachment as the dominant decay process. The second electron affinities exhib it pronounced shell effects in excellent agreement with measur ed abundance spectra for Au2− N(N <30), with appearance sizes n2− a(Au)= 12 and n2− a(Ag)= 24. Pacs Numbers: 36.40.Wa, 36.40.Qv, 36.40.Cg Unlike the case of multiply charged cationic species, the production and observation of gas-phase doubly an- ionic aggregates had remained for many years a chal- lenging experimental goal. However, with the availabil- ity of large carbon clusters (which can easily accomo- date the repulsion between the two excess electrons) this state of affairs changed, including observation of doubly negative fullerenes,1C2− 60, and fullerene derivatives,2as well as a recent measurement of the photoelectron spec- trum of the citric acid dianion.3Moreover, such obser- vations are not limited to carbon based aggregates and organic molecules, with a first observation of doubly an- ionic metal clusters (specifically gold clusters) reported4,5 most recently. A few theoretical studies of multiply charged anionic fullerenes6,7and alkali-metal (sodium) clusters8have also appeared, but overall the field of mul- tiply anionic aggregates remains at an embryonic stage. In this paper, we investigate the stability and decay channels of Au2− Nand Ag2− Nat finite temperature, and determine their appearance sizes n2− a(clusters with N < n2− aare energetically unstable). Two decay channels of doubly anionic clusters need to be considered: (i) binary fission, M2− N→M− P+M− N−P, (1) which has a well known analog in the case of doubly cationic clusters,9–11and (ii) electron autodetachment via emission through a Coulombic barrier,8 M2− N→M− N+e , (2) with an analogy to proton and alpha decay in atomic nuclei.12,13The theoretical approach we use is a finite-temperature semi-empirical shell-correction method (SCM), which incorporates triaxial shapes and which has been previously used successfully to describe the properties of neutral and cationic metal clusters.14Our main conclusion is that, unlike the case of doubly cationic metal clusters,9,11fission of Au2− Nand Ag2− Nis not a dominant process, and that the appearance sizes of these doubly anionic clusters are determined by elec- tron autodetachment. Our results for the second electron affinities exhibit pronounced electronic shell effects and are in excellent agreement with most recent experimen- tal data5for Au2− Nwithn2− a= 12. For Ag2− N, we predict n2− a= 24. The finite-temperature multiple electron affinities of a cluster of Natoms of valence v(we take v= 1 for Au and Ag) are defined as AZ(N, β) =F(β, vN, vN +Z−1)−F(β, vN, vN +Z), (3) where Fis the free energy, β= 1/kBT, and Z≥1 is the number of excess electrons in the cluster (e.g., the first and second affinities correspond to Z= 1 and Z= 2, respectively). To determine the free energy, we use the shell correction method. In the SCM, Fis sepa- rated into a smooth liquid-drop-model (LDM) part /tildewideFLDM (varying monotonically with N), and a Strutinsky-type shell-correction term ∆ Fsp=Fsp−/tildewideFsp, where Fspis the canonical (fixed Nat a given T) free energy of the valence electrons, treated as independent single particles mov- ing in an effective mean-field potential (approximated by a modified Nilsson hamiltonian pertaining to triax- ial cluster shapes), and /tildewideFspis the Strutinsky-averaged free energy. The smooth /tildewideFLDMcontains volume, sur- face, and curvature contributions, whose coefficients are determined as described in Ref. 14, with experimental values and temperature dependencies. In addition to the finite-temperature contribution due to the electronic en- tropy, the entropic contribution from thermal shape fluc- tuations is evaluated via a Boltzmann averaging.14(a) We note here that the smooth contribution /tildewideAZ(N, β) to the full multiple electron affinities AZ(N, β) can be approximated8by the LDM expression /tildewideAZ=/tildewideA1−(Z−1)e2 R(N) +δ0=W−(Z−1 +γ)e2 R(N) +δ0,(4) where R(N) =rsN1/3is the radius of the positive back- ground ( rsis the Wigner-Seitz radius which depends weakly on Tdue to volume dilation), γ= 5/8,δ0is an electron spillout parameter, and the work function Wis assumed to be temperature independent [we take W(Au)= 5 .31 eV and W(Ag)= 4 .26 eV]. 101 5 10 15 20 25 3001 5 15 20 25 3001 5 15 20 25 30 1E-4 1E-30.010.11 NA (eV)2(a) (b)REL. ABUND.12 19Au Au FIG. 1. (a) Measured [see figure 3(a) in Ref. 5] average relative abundances of Au2− Nclusters (i.e., the ratio of the number of the observed dianions over the sum of the numbers of corresponding singly-anionic precursors and dianions) as a function of cluster size. Note the logarithmic ordinate sca ling. (b) Calculated second electron affinity ( A2in eV) for gold clusters at T= 300 K plotted versus N. Results from SCM calculations are connected by a solid line, and LDM results [see Eq. (4) with Z= 2] are depicted by the dashed line. A2>0 corresponds to stable dianionic Au2− Nclusters; note the appearance size n2− a= 12. Energies in units of eV. In a recent experiment,5singly anionic gold clusters Au− N(N≤28) were stored in a Penning trap, size se- lected, and transformed into dianions, Au2− N, through ir- radiation by an electron beam. The measured5relative intensity ratios of the dianions to their monoanionic pre- cursors are reproduced in Fig. 1(a); they exhibit size- evolutionary patterns (arising from electronic shell ef- fects) reminiscent of those found earlier in the mass abun- dance spectra, ionization potentials and first electron affinities of alkali- and coinage-metal clusters.15Since the stability of the dianions relative to their monoanionic pre - cursors depends on the second electron affinity A2, it may be expected that A2and the relative signal intensity of the Au2− Nclusters will exhibit correlated patterns as a function of size. Here we note that stable dianions must haveA2>0, whereas those with A2<0 are unstable and decay via process (ii), i.e., via electron emission through a Coulombic barrier8(see below). In Fig. 1(b), we display the SCM theoretical results16,17for the second electron affinity of Au Nclus- ters in the size range 10 ≤N≤30. These results cor- relate remarkably well with the measured relative abun- dance spectrum [see Fig. 1(a)]. Note in particular: (i) the observed and predicted appearance size n2− a(Au)= 12; (ii) the relative instability of Au2− 13[portrayed by its ab- sence in Fig. 1(a) and the negative A2value in Fig. 1(b)0246 0 10 20 300246 0 10 20 30 3579 0 5 10 15 203579 0 5 10 15 203579 0 5 10 15 20 NP(eV) N,1 N,P(eV)(a) (b)N=7N=14N=21 AuAu FIG. 2. (a) Fission dissociation energies (∆ N,Pin eV) for binary fission Au2− N→Au− P+Au− N−P, calculated at T= 300 K with the SCM for parent dianionic clusters with N= 7, 14 and 21, and plotted versus P. Note that in all cases the most favorable fission channel corresponds to P= 1. (b) SCM fis- sion dissociation energies, ∆ N,1, atT= 300 K for the most favorable channel, plotted versus cluster size. Exothermi c fis- sion (∆ N,1<0) is found only for the smallest cluster. associated with the closing of a spheroidal electronic sub- shell (containing 14 electrons) in the singly anionic Au− 13 parent cluster, see Ref. 14(b)]; (iii) the pronounced lower stability of Au2− 19relative to its neighboring cluster sizes [associated with the closing of a major electronic shell (containing 20 electrons) in the Au− 19parent cluster]; (iv) the overall similarity between the trends in Fig. 1(a) and Fig. 1(b)(that is, even-odd alternations for N≤19 with a sole discrepancy at N= 15, and the monotonic behav- ior for N≥19). Underlying the pattern shown in Fig. 1(b) are electronic shell effects [compare in Fig. 1(b) the shell-corrected results indicated by the solid line with th e LDM curve] combined with energy-lowering shape defor- mations of the clusters (which are akin to Jahn-Teller distortions and are associated with the lifting of spectral degeneracies for open-shell cluster sizes). To explore the energetic stability of the Au2− Nclus- ters against binary fission [see Eq. (1)], we show in Fig. 2(a) SCM results, at selected cluster sizes ( N= 7,14 and 21), for the fission dissociation energies ∆ N,P= F(Au− P)+F(Au− N−P)−F(Au2− N), with the total free en- ergies of the parent dianion and the singly-charged fis- sion products calculated at T= 300 K. For all Au2− N parent clusters, the energetically favorable channel (low - est ∆ N,P) corresponds to P= 1 (i.e., one of the fission 2-101 -101 -101 -1135 0 10 20 30 40-1135 0 10 20 30 40(eV) N,1(a) (b)A (eV)2 AgAg 1924 N FIG. 3. SCM second electron affinities [ A2in (a)] and fis- sion dissociation energies [∆ N,1in (b)] for the most favorable channel ( P= 1) for Ag2− Nclusters at T= 300 K, plotted ver- sus cluster size. In (a) LDM results [see Eq. (4) with Z= 2] are depicted by the dashed line. Note the appearance size n2− a= 24. Energies in units of eV. products is the closed-shell Au−anion). The influence of shell effects on the fission dissociation energies is evident particularly in cases where the fission channel involves closed-shell magic products (see P= 7, and equivalently P= 14, for N= 21, and the pronounced effect at P= 7 forN= 14 where both fission products are magic). The fission results summarized in Fig. 2(b) for the most fa- vorable channel ( P= 1) illustrate that exothermic fis- sion (that is ∆ N,P<0) is predicted to occur only for the smallest size ( N= 2). This, together with the existence of a fission barrier, leads us to conclude that the decay of Au2− Nclusters is dominated by the electron autode- tachment process (which is operative when A2<0 and involves tunneling through a Coulomb barrier8), rather than by fission. Finally, we show in Fig. 3 SCM results for the sec- ond electron affinity [ A2in Fig. 3(a)] and the fission dis- sociation energies [∆ N,Pin Fig. 3(b)] corresponding to the most favorable channel ( P= 1) for silver dianionic clusters Ag2− N. Again binary fission is seen to be en- dothermic except for N= 2, and the appearance size for Ag2− N(i.e., the smallest size with A2>0) is predicted to ben2− a(Ag)= 24. The shift of the appearance size to a larger value than that found for gold dianionic clusters [that is n2− a(Au)= 12, see above] can be traced to the smaller work function of silver, as can be seen from the LDM curves calculated through the use of Eq. (4) with Z= 2.18 This research was supported by a grant from the U.S.Department of Energy (Grant No. FG05-86ER45234). 1R.L. Hettich, R.N. Compton, and R.H. Ritchie, Phys. Rev. Lett.67, 1242 (1991). 2C. Jin et al., Phys. Rev. Lett. 73, 2821 (1994). 3X-B. Wang, C-F. Ding, and L-S Wang, Phys. Rev. Lett. 81, 3351 (1998). 4A. Herlert et al., Physica Scripta T80, 200 (1999). 5L. Schweikhard, A. Herlert, S. Kr¨ uckeberg, and M. Vogel, Philos. Mag. 79, 1343 (1999). 6C. Yannouleas and U. Landman, Chem. Phys. Lett. 217, 175 (1994). 7M.R. Pederson and A.A. Quong, Phys. Rev. B 46, 13 584 (1992). 8C. Yannouleas and U. Landman, Phys. Rev. B 48, 8376 (1993); Chem. Phys. Lett. 210, 437 (1993). 9C. Br´ echignac et al., Comments At. Mol. Phys. 31, 361 (1995). 10U. N¨ aher et al., Phys. Rep. 285, 245 (1997). 11C. Yannouleas, U. Landman, and R.N. Barnett, in Metal Clusters , edited by W. Ekardt (Wiley, New York, 1999), p. 145. 12S.˚Aberg, P.B. Semmes, and W. Nazarewicz, Phys. Rev. C 58, 3011 (1998). 13M.A. Preston and R.K. Bhaduri, Structure of the Nucleus (Addison-Wesley, London, 1975). 14C. Yannouleas and U. Landman, (a) Phys. Rev. Lett. 78, 1424 (1997); (b) Phys. Rev. B 51, 1902 (1995). 15See W.A. de Heer, Rev. Mod. Phys. 65, 611 (1993); see also Refs. 11 and 14, and references therein. 16For the rather small Au Nand Ag Nclusters discussed here (N≤30 and N≤40, respectively), our results at T= 0 K andT= 300 K differ only slightly. Since, however, the ex- periments are carried out at finite temperatures, we present here the T= 300 K results for the sake of completeness. For cases where the SCM reveals significant thermal effects portrayed by reduction and smearing out of electronic shell effects (e.g., for higher temperatures and/or larger rs’s as in the case of Na Nand K Nclusters), see Ref. 14(a). 17TheT= 0 parameters entering in the SCM calculation [for definitions of these, see Ref. 14(b)] are: (i) In the case of AuZ− Nclusters, U0=−0.045,rs= 3.01 a.u., t= 0.37 a.u.,δ0= 1.31 a.u., δ2= 0, W= 5.31 eV, αv=−8.06 eV,αs= 2.52 eV, and αc= 1.04 eV; (ii) In the case of AgZ− Nclusters, U0=−0.045,rs= 3.01 a.u., t= 0.47 a.u.,δ0= 1.31 a.u., δ2= 0, W= 4.26 eV, αv=−8.06 eV,αs= 2.05 eV, and αc= 0.86 eV. Experimental val- ues were used for the liquid-drop parameters for the sur- face tension, αs, and for the work function, W.αvand δ0were specified through a fit to extended-Thomas-Fermi total energy calculations for spherical clusters in conjun c- tion with a stabilized-jellium -LDA energy functional [see J.P. Perdew, H.Q. Tran, and E.D. Smith, Phys. Rev. B 42, 11 627 (1990)]. The curvature coefficient, αc, was spec- ified by assuming that its ratio over the stabilized-jellium 3value is the same as the ratio between the experimental and stabilized-jellium values for the surface tension. The need to use experimental values for αs,αc, and Warises from the fact that the stabilized-jellium-LDA omits con- tributions from the d-electrons and thus does not provide accurate values for the surface tension and the work func- tion of gold and silver. The mass of the delocalized valence s-electrons was taken equal to the free-electron mass. The experimental temperature dependence of the surface ten- sion and the coefficient of linear thermal expansion were taken from standard tables as elaborated in Ref. 14(a). 18The use of a finite value [see Ref. 14(b)] for the spillout parameter δ0in the case of negatively charged clusters is crucial for obtaining accurate results. Taking δ0= 0 (as was done in Ref. 5) yields a higher value for the appearance sizes. 4

arXiv:physics/0001033v1 [physics.soc-ph] 16 Jan 2000Dream of a Christmas lecture Alejandro Rivero∗ February 2, 2008 Abstract We recall the origins of differential calculus from a modern p erspective. This lecture should be a victory song, but the pain makes it to sound more as a oath for vendetta, coming from Syracuse two milenia befo re. A visitor in England, if he is bored enough, could notice that our old 20 pound notes are decorated with a portrait of Faraday imparti ng the first series of ”Christmas lectures for young people”, which began time a go, back in the XIXth century, at his suggestion. Today they have become tra ditional activity in the Royal Institution. This year the generic theme of the lectures was quantum theor y and the limits implied by it. The BBC uses to broadcast the full sessi ons during the holidays, and I decided to enjoy an evening seeing the record ing. This day, the third of the series, is dedicated to the time scale of quantum phenomena. The main hall is to be occupied, of course, by the children who hav e come to enjoy the experimental session, and the BBC director, a senior wel l trained to control this audience, keeps the attention explaining how the volun teers are expected to enter and exit the scene. While he proceeds to the customary n otice, that ” all the demonstrations here are done under controlled conditio ns and you should not try to repeat them at home ”, I dream of a zoom over a first bowl with some of the bank notes, and the teacher starting the lecture. He wears the white coat and in a rapid gesture drops a match in t he bowl, and the pieces of money take fire. The camera goes from the flame s to the speaker, who starts: Money. Man made, artificial, unnatural. Real and Untrue. And then a slide of a stock market chart: But take a look to this graph: Why does it move with the same equ ations that a grain of pollen? Why does it oscillate as randomly as a quant um mechanic system? . Indeed. It is already a popular topic that the equations used for the deriva- tive market are related to the heat equation, and there is som e research run- ning in this address. But the point resonating in my head was a protest, formulated[20] a couple of months before by Mr. W.T. Shaw, a r esearcher of financial agency, Nomura: ”Money analysts get volatility and other parameters from th e measured mar- ket data, and this is done by using the inverse function theor em. If a function has a derivative non zero in a point, then it is invertible at t his point. But, if ∗Email:rivero@wigner.unizar.es 1we are working out discrete calculus, if we are getting discr ete data from the market, how can we claim that the derivative is non zero? Shou ld we say that our derivative is almost non zero ? What control do we have over the inversion process?”. Most meditations in this sense drive oneself to understand t he hidings under the concept of stability of a numerical integration process . But consider just this: discrete, almost zero, almost nonze ro calculus!. It is a romantic concept by itself. Infinitesimals were at the core o f the greatest priority dispute in Mathematics. On one side, at Cambridge, the secon d Lucasian chair, Newton. On the other side, at the political service of the ele ctor of Mainz, the mathematical philosophy of Leibniz. And coming from the dar k antiquity, old problems: How do you get a straight line from a circle? How do you underst and the area of any figure? What is speed? Is the mathematical cont inuum composed of indivisible ”individua”, mathematical ”atoms without e xtension”? Really all the thinking of calculus is pushed by two paradoxe s. That one of the volume and that one of the speed. The first one comes, it is said, from Democritus. Cut a cone wit h a plane parallel and indefinitely near to the basis. Is the circle on t he plane smaller or equal than the basis? Other version makes the infinite more explicit. Simply cut th e cone parallel to the basis. The circle in the smaller cone should be equal to the one in the top of the trunk. But this happens for every cut, lets say you make infinite cuts, always the circles will be equal. How is that different of a cyl inder? You can say, well, that the shape, the area, decreases between the cuts, n o in the cuts. Ok, good point. But take a slice bounded by two cuts. As we keep cut ting we make the slice smaller, indefinitely thinner, until the distinct ion between to remove a slice and to make a cut is impossible. How can this distinctio n be kept? And we need to kept it in a mathematically rigorous way, if possib le. The second paradox is a more popular one, coming from the medi tations of Zeno. In more than one sense, it is dual to the previous one. Ta ke time instead of height and position instead of circular area. How can an ar row to have a speed? How can an arrow to change position if it is resting at e very instant? In other version, it is say that it can not move where it is fixed , and it can not move where it is not yet. Or, as Garcia-Calvo, a linguist a nd translator of Greek, formulated once: ”One does not kiss while he lives, on e does not live when he kisses”. Seriously taken, the paradox throws strong doubts about the concept of instantaneous speed. Or perhaps about the whole conception of what ”instan- taneous” is. While Democritus asked how indefinite parts of s pace could add up to a volume, Zeno wonders how a movement can be decomposed to r un across indefinite parts of time. A Wicked interplay. It is interesting to notice that physicists modernly do not l ike to speak of classical mechanics as a limit ¯ h→0, but as a cancellation of the trajectories that differ from the classical one. Perhaps this is more accep table. Anyway, the paradoxes were closed in false by Aristotle with some deep th oughts about the infinity. Old mathematics was recasted for practical uses an d, at the end, lost. But in the late mid ages, some manuscripts were translated ag ain. A man no far from my homeland, in the Ebro river in Spain, took over a Arabic book to be versed into Latin. It was the Elements , that book all you still ”suffer” in the first courses of math in the primary school, do you remem ber? Circles, 2angles, triangles, and all that. And, if your teacher is good enough, the art of mathematical skepticism and proof comes with it. Of course the main interesting thing in the mid ages is a new ar t, Algebra. But that is a even longer history. To us, our interest is that w ith the comeback of geometry, old questions were again to be formulated. If continuous becomes, in the limit, without extension, then is such limit divisible? And if it is indivisible, atomic, how can it be? That automatically brings up other deterred theory to compa re with. That one which postulates Nature as composed with indivisible at oms but, having somehow extension, or at least some vacuum between them. Such speculation had begun to be resuscitated in the start of the XVII, with Galileo Galilei himself using atomic theory to justify heat , colours, smell. His disciple Vincenzo Viviani will write, time-after, that the n, with the polemic of the book titled ”Saggiatore”, the eternal prosecution of Galileo actions and discourses began. Mathematics was needing also such atomic objects, and in fac t the first infinitesimal elements were named just that, atoms, before t he modern name was accepted. (By the way, Copernico in “De revolutionibus” explains how t he atomic model, with its different scales of magnitude, inspires the a stronomical world: the distance of the earth to the center of the stars sphere is s aid to be negligible by inspiration from the negligibility of atomic scale. It is very funny that some centuries later someone proposed the ”planetary” model of a toms.) Back to the lecture. Or to the dream. Now the laboratory has ac tivated a sort of TV projector bringing images from the past. Italy. Viviani. He made a good effort to recover Archimedes and other classical geometers. So it is not strange that the would-to-be first luc asian, Isaac Bar- row, become involved when coming to Florence. And Barrow und erstood how differentiation and integration are dual operations. Noises... Perhaps Barrow learn of it during his Mediterranean voyage Noises of swords and pirates sound here in the TV scene, and Ba rrow himself enters in the lecture room. He is still blooding from the encounter with the pirates. Gre ets the speaker, cleans himself, and smiles to the children in the first row: ”We become involved in a stupid war. Europe went to war about s acraments, you know, the mystery of eucharistic miracle and all that nic eties. And there we were, with individia, indivisilia, atoms... things that ru le out difference between substance and accidents. You can not make a bread into a divin e body if it is only atoms, they say.” Indeed, someone filed a denounce against Galileo claiming th at is theory was against the dogma of transustantiation[17]. Touchy matter , good for protestant faith but not for the dogmas from Trent concilium. For a moment he raises the head, staring to us, in the upper cir cle. Then he goes back to the young public: ”Yes, there was war. Protestants, Catholics, Anglicans. Dogmas and soldiers across Europe. Bad time to re ject Aristotle, worse even to bring again Democritus. With Democritus comes Lucretius, with Lucretius comes Epicurus. Politically inconvenient, you k now. Do the answers pay the risk?” 3He goes away. He went away to Constantinople, perhaps to read the only extant copy of the Archimedean law. Perhaps he found the lost Method. Perhaps he lost other books when his ship was burned in Venice. Yes, Bourbaki says (according [1]) that Barrow was the first o ne proofing the duality between derivatives and integration. At least, with his discrete ”almost zero” differential triangle, doubting about the ris ks of jumping to the limit, was closer to our modern [3] view. Three or four years a go Majid, then still in Cambridge, claimed its resurrection in the non comm utative calculus f(x)dx=dxf(x−λ). Even the formulation of fermions in the lattice, accordin g Luscher, depends on this relationship to proof the cancella tion of anomalies. Also we would note that his calculus was ”renormalized” to a fi nite scale, as instead of considering directly ∆ f/∆x, he first scaled this relationship to a finite triangle with side f(x). The freedom to choose either the triangle on f(x) or the one in f(x+λ) was lost when people start to neglect this finite scale. Really, this is mathematical orthodoxy. Consider a series s in(1/n) and an- other one 1 /n, both going to zero. The quotient, then, seems to go to an indefinite 0 /0, but if you scale all the series to a common denominator, cal l itSa(n)/a, you will find that Sa(n) goes to aasnincreases. Wilson in the seventies made the same trick for statistical field theory (o r for quantum field theory), which was at that moment crowded of problematic infi nities. There is also a infinite there in the Barrow idea, but it is a ver y trivial one. Just the relation between the vertical of the finite triangle and the horizontal of the small one,f(x) ∆x. It goes to infinity, but this divergence can be cured by subtracting another infinite quantity,f(x+λ) ∆x, so that the limit is finite1. Barrow died in sanctity. But in his library [13] there was no l ess that three copies of Lucretius ”De Rerum Natura”, a romam poem about ato mistic Nature, already critiquized in the antiquity because in supports th e Epicurean doctrine: that gods, if they exist, are not worried about the human affai rs, so we must build our moral values from ourselves and our relationships with our friends and society. In the lecture room, the slides fly one ager other. Back in the X VII, with heat, smell, colour, and other accidents, black storms blow in the air. It has been proposed that the sacred eucaristic mystery was in agreemen t with Aristotle, as it could be said that the substance of wine and bread was subst ituted by the substance of the Christ, while the accidents remained. Go te ll to the Luterans. First August 1632. The Compa˜ nia de Jesus forbids the teachi ng of the atomic doctrine. 22nd June 1633, Galileo recants . “Of all the days that was the one / An age of reason could have begun” [2] In the ”Saggiatore”, Galileo had begun to think of physical m ovement of atoms as the origin of the heat. It would take three centuries for Einstein to get the Brownian key. But even that was already disentangled of p ure mathematics, so it took some other half century for to discover the same equ ations again, now for the stock market products. The history has not finished. It sounds not to surprise that Dimakis has related discrete c alculus to the Ito calculus, the basic stochastic in the heat equation, the play of money, that Black and Scholes rediscovered. In some sense it is as if the p hysical world described by mathematics were dependent on mathematics onl y, as it it were 1This example was provided by Alain in Vietri at the request of the public, but it was not to be related to the hoft algebra of trees, as far as I can see 4the unique answer to organise things in a localized position . Dark clouds will block our view. Barrow survived to his ship and crossed Ger- many and come home to teach Newton. But Newton himself missed something greater when, for sake of simplicity, the limit to zero was ta ken. In this limit, he can claim the validity of series expansion to solve any diff erential equation, so it is a very reasonable assumption. Yes, but it had been more interesting to control the series expansion even without such limit. Leibniz come to the same methods and the jump to the limit is to be the standard. Mathematical atoms, scales and discrete calculu s will hide its inter- play with the infinitesimal ones for some centuries. Only two years ago Mainz, in voice of Dirk Kreimer, got again the clue to generalized Ta ylor series. The wood was found to be composed of trees. Vietri is a small village in the Tyrrenian sea, near Salerno, looking at the bay of Amalfi. Good fishing and intense limoncello liquor. About the 20th of Mars, 1998, there Alain come, to explain the way Kreimer had f ound a Hopf algebra structure governing perturbative renormalizatio n. The algebra of trees was not only related to Connes Moscovici algebra, but also wi th the old one proposed by Cayley to control the Taylor series of the vector field differential equation. And to close the circle, Runge-Kutta numerical integration algorithms can be classified with a hoft algebra of trees. Today it can be said[5] that the generic solution to a differential equation is not just the function, but also some in- formation codified in the Butcher group . Which can be related to the physics monster of this century, the renormalization group we have m entioned before. Can we control the inversion of the Taylor series using trees ? Then these doubts about the inverse function theorem in stock markets c ould be sorted out. Will us be able to expand in more than one variable? Still igno rabimus. Worse, it is progressively clear that this kind of pre-Newto nian calculus are a natural receptacle for quantum mechanics. Even the stock ma rket Ito equations are sometimes honoured as ”Feynman-Kac-Ito” formula, so ma rking its link with the quantum world. The difference comes from the format o f the time variable in both worlds. One should think that time is more su btle than the intuitive ”dot” that Newton put in the fluxion equations. Perhaps we are now, then, simply correcting a flaw made three h undred years ago. A flaw that Nature pointed to us, when it was clear th e failure of classical mechanics in the short scale. But how did come to exist the conditions to such failure? Why d id geometry need to be reborn in the XVIIth century? Why did the mathemati cians so little information, so that the mistake has a high probability2to happen? If calculus, or ”indivisilia”, were linked to atomism alrea dy in the old age, it could be a sort of explanation. Archimedes explain that De mocritus was the first finding ”without mathematical proof” the volume of the c one. And with Democritean science there was political problems already i n Roman times: A leftist scholar, Farrington[12], claims that political s tability was thought to reside in some platonic tricks, lies, proposed in ”Republ ic”: a solid set of unskeptic faith going up the pyramid until the divine celest ial gods. Epicurus is seen as a fighter for freedom, putting at risk social stabilit y. Against Plato ideals, 2And, by the way, Why there was not the slightest notion of prob ability in the old math- ematics texts, so they were unable even to consider it? 5Epicurus casts in his help the Ionian learning, including De mocritean mathe- matics and physics. According Farrington, if government as pires to platonic republic, it must control or suppress such kind of mathemati cs and physics. No surprise, if this is true, that the man who understood the fl oating bodies and the centers of gravity, who stated the foundations and in tegration, the pro- cess of mechanical discovery and mathematical rigor, who wa s fervently trans- lated by Viviani and Barrow, was killed and dismissed. To be b uried without a name, it could have been Archimedes’ own wish. But to get his books left out of the copy process for centuries until there were extinc t, that is a different thing. ”Only a Greek copy of the Floating Bodies extant, found at Con stantinople. See here the palimpsest, the math almost cleared, a orthodox liturgy, perhaps St John Chrisostom, wrote above instead.” ”Let me to pass the pages, and here you have, the only known ver sion of Archimedes letter about the Mechanical Method. Read only by three persons, perhaps four, since it was deleted in the Xth century. Was thi s reading the goal of Barrow in orient?” And even then, is it the same? It has been altered, the last occ asion in this century, when someone painted four evangelists over the Met hod. Hmm. Last Connes report [7] quotes the Floating Bodies princ iple, doesn’t it?. More and more associations. Stop! And recall. Had our research been different if we had been fully aware of th e indivisilia problems, if we had tried hard for rigour? Perhaps. Only in th e XVI, res- cued Apollonius and Archimedes, the new mathematics re-tak en the old issue. And, as we have seen, in a dark atmosphere. Enough to confuse t hem and go into classical mechanics instead of deformed mechanics. In stead of quantum mechanics. The matter of copernicanism has been usually presented as a p olitical issue. Brecht made a brilliant sketch of it while staying in Copenha gen with some friends, physicists which become themselves caught in the d ark side of our own century. We suspect that the matter of atomism also has suffer ed because of this, and now it appears that Differential Geometry itself ha s run across a world of troubles since the assesination of his founder in Sicily t wo milenia ago. The truth has been blocked again and again by the status quo, by th e ”real world” preferring tales of stable knowledge to inquisitive minds l earning to crawl across, and with, the doubt. If the goal of the Christmas lectures is to move young people t o start a career in science, here is our statement: it is for the honour of huma n spirit. It is because understanding, reading the book of nature, we calm o ur mind. Call it ataraxia, athambia, or simply tranquility. But we have been mistaken, wronged, delayed. The world has tr icked, out- raged, raped us. When we have been wronged, should we not to revenge? Then our main motivation is here: when reality is a lie, the song of science must be a song of vengeance. A man in Syracuse has been killed, all our milenia-old family has been dishonoured. Every mother, every child, eve ry man in Sicily knows then the word. Vendetta. Go to your blackboards, my children, and sing the song. Just t o clear any trace of pain in the soul . 6References [1] V.I. Arnol’d, Huygens and Barrow, Newton and Hooke , Birkhauser (1990) [2] B. Brecht, The Life Of Galileo , tr. by D.L.Vesey [3] A. Connes, Gravity coupled with matter and foundation of non commutative geometry, Comm. Math. Phys. ,182(1996) [4] A. Connes and D. Kreimer, Hopf Algebras, renormalization and quantum field theory , hep-th/9808042 [5] A. Connes and D.Kreimer, Lessons from Quantum Field Theory [6] A. Connes and D. Kreimer, ”Renormalization in quantum fie ld theory and the Riemann Hilbert Problem”, hep-th/9909126 [7] A. Connes and D. Kreimer, ”Renormalization in quantum fie ld theory and the Riemann Hilbert Problem I: the Hopf algebra structure of gra phs and the main theorem”, hep-th/9912092 [8] Dirac P.A.M., Proc. Cambridge Phil. Soc ,30, 150 (1934) [9] A. Dimakis, C. Tzanakis, Dinamical Evolution in NC discrete phase space and the derivation of Classical Kinetic Equations , math-ph/9912016 [10] J.S. Dowker, Path Integrals and Ordering Rules, J. Math. Phys. 17(1976) [11] K. Elsner, Elektroschwaches Modell und Standardmodell in der nichtko mmuta- tiven Geometry , Diplomarbeit, Marbug (1999) [12] B. Farrington, Science and Politics in the Ancient World [13] M. Feingold et al, Before Newton. The life and times of Isaac Barrow , Cambridge University Press (1990) [14] Roy R. Gould, Am. J. Phys. ,63, n. 2 (1995) [15] J.M. Gracia-Bondia, J. Varilly and H. Figueroa, Elements of Noncommutative Geometry , Birkhauser, Boston, (2000) [16] B. Iochum, T. Krakewski, P. Martinetti, Distances in finite spaces from NC ge- ometry hep-th/9912217 [17] Pietro Redondi, Galileo heretico , Alianza Universidad, 1990 [18] A. Rivero, Some conjectures looking for a NCG theory , hep-th/9804169 [19] W. Siegel, Fields , hep-th/9912205 [20] W.T. Shaw, Principles of Derivatives Modelling in a Symbolic Algebra E nviron- ment, Mathematica notebook [21] J. Varilly, NonCommutative Geometry and Quantization hep-th/9912171 [22] R. Wulkenhaar, Introduction to Hoft Algebras in renormalization and nonco m- mutative geometry , hep-th/9912221 [23] R. Wulkenhaar, On Feynman graphs as elements of a Hopf algebra , hep- th/9912220 7

arXiv:physics/0001035v1 [physics.space-ph] 17 Jan 2000What Dimensions Do the Time and Space Have: Integer or Fracti onal? L.Ya.Kobelev Department of Physics, Urals State University Lenina Ave., 51, Ekaterinburg 620083, Russia Electronic address: leonid.kobelev@usu.ru A theory of time and space with fractional dimensions (FD) of time and space ( dα, α=t,r) defined on multifractal sets is proposed. The FD is determine d (using principle of minimum the functionals of FD) by the energy densities of Lagrangians of known physical fields. To describe behaviour of functions defined on multifractal sets the gene ralizations of the fractional Riemann- Liouville derivatives Dd(t) tare introduced with the order of differentiation (depending on time and coordinate) being equal the value of fractional dimension. Fordt=const the generalized fractional derivatives (GFD) reduce to ordinary Riemann-Liouville in tegral functionals, and when dtis close to integer, GFD can be represented by means of derivatives of integer order. For time and space with fractional dimensions a method to investigate the gene ralized equations of theoretical physics by means of GFD is proposed. The Euler equations defined on mul tifractal sets of time and space are obtained using the principle of the minimum of FD functio nals. As an example, a generalized Newton equation is considered and it is shown that this equat ion coincide with the equation of classical limit of general theory of relativity for dt→1. Several remarks concerning existence of repulsive gravitation are discussed. The possibility of ge ometrization all the known physical fields and forces in the frames of the fractal theory of time and spac e is demonstrated. 01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p. I. INTRODUCTION The problem concerning the nature of space and time is one of the most interesting problems of the modern physics. Are the space and time continuous? Why is time irreversible? What dimensions do space and time have? How is the nature of time in the equations of modern physics is reflected? Different approaches (quantum grav- ity, irreversible thermodynamics, synergetics and others ) provide us with different answers to these questions. In this paper the hypothesis about a nature of time and space based on an ideas of the fractal geometry [1] is offered. The corresponding mathematical methods this hypothesis makes use of are based on using the idea about fractional dimensions (FD) as the main characteristics of time and space and in connection with this the gener- alization of the Riemann-Liouville fractional derivative s are introduced. The method and theory are developed to describe dynamics of functions defined on multifractal sets of time and space with FD. Following [2], we will consider both time and space as an only material fields existing in the Universe and gener- ating all other physical fields. Assume that each of them consists of a continuous, but not differentiable bounded set of small elements. Let us suppose a continuity, but not a differentiability, of sets of small time intervals (fro m which time consist) and small space intervals (from which space consist). First, let us consider set of small time in- tervalsSt(for the set of small space intervals the way of reasoning is similar). Let time be defined on multifrac- tal set of such intervals (determined on the carrier of a measure Rn t). Each of intervals of this set (further weuse the approximation in which the description of each multifractal interval of these sets will be characterized b y middle time moment tand refer to each of these intervals as ”points”) is characterized by global fractal dimension (FD)dt(r(t),t), and for different intervals FD are differ- ent( because of the time dependence and spatial coordi- nates dependence of dt). For multifractal sets St(orSr) each set is characterized by global FD of this set and by local FD of this set( the characteristics of local FD of time and space sets in this paper we do not research). In this case the classical mathematical calculus or frac- tional (say, Riemann - Liouville) calculus [3] can not be applied to describe a small changes of a continuous func- tion of physical values f(t), defined on time subsets St, because the fractional exponent depends on the coordi- nates and time. Therefore, we have to introduce integral functionals (both left-sided and right-sided) which are suitable to describe the dynamics of functions defined on multifractal sets (see [1]). Actually, this functionals ar e simple and natural generalization the Riemann-Liouville fractional derivatives and integrals: Dd a+,tf(t) =/parenleftbiggd dt/parenrightbiggn/integraldisplayt af(t′)dt′ Γ(n−d(t′))(t−t′)d(t′)−n+1(1) Dd b−,tf(t) = (−1)n/parenleftbiggd dt/parenrightbiggn/integraldisplayb tf(t′)dt′ Γ(n−d(t′))(t′−t)d(t′)−n+1 (2) where Γ(x) is Euler’s gamma function, and aandbare some constants from [0 ,∞). In these definitions, as usu- ally,n={d}+ 1 , where {d}is the integer part of d 1ifd≥0 (i.e.n−1≤d < n ) andn= 0 ford <0. Functions under the integral sign we will consider to be generalized functions defined on the space of finite func- tions [4]. Similar expressions can be written down for GFD of functions f(r,t) with respect to spatial variables r, withf(r,t) being defined on the elements of set Sr whose dimension is dr. For an arbitrary f(t) it is useful to expand the gen- eralized function 1 /(t−t′)ε(t′)under the integral sign in (1)-(2) into a power series in ε(t′) whend=n+ε,ε→+0 and write Dd a+,tf(t) =/parenleftbiggd dt/parenrightbiggn/integraldisplayt af(t′) Γ(n−d(t′))(t−t′)(3) ×(1 +ε(t′)ln(t−t′) +...)dt′ Dd b−,tf(t) = (−1)n/parenleftbiggd dt/parenrightbiggn/integraldisplayb tf(t′) Γ(n−d(t′))(t′−t)(4) ×(1 +ε(t′)ln(t′−t) +...)dt′ Taking into account that all functions here are real func- tions and 1 /t=P(1/t)±πiδ(t), singular integrals here can be defined through the rule /integraldisplayt 0f(t′) t−t′dt′=af(t) (5) whereais a real regularization factor. A good agreement of (3)-(4) with the exact values given by expressions (1)- (2) can be obtained at large time by fitting the value of a. Instead of usual integrals and usual partial deriva- tives, in the frames of multifractional time hypothe- sis it is necessary to use GFD operators to describe small alteration of physical variables. These function- als reduce to ordinary integrals and derivatives if space and time dimensions are taken to be integer, and coin- cide with the Riemann-Liouville fractional operators if di=const. If fractional dimension can be represented as di=n+εi(r(t),t),|ε| ≪1, it is also possible to reduce GFD to ordinary derivatives of integer order. Here we show this only for the case when d= 1−ε<1 D1+ε 0+f(t) =∂ ∂t/integraldisplayt 0ε(τ)f(τ)dτ Γ(1 +ε(τ))(t−τ)1−ε(τ)(6) ≈∂ ∂t/integraldisplayt 0ε(τ)f(τ)dτ (t−τ)1−ε(τ) Though for ε/ne}ationslash= 0 the last integral is well defined and is real-valued, expanding it in power series in εleads to singular integrals like (5) A=/integraldisplayt 0ε(t′)f(t′) t−t′dt′ To regularize this integral we will consider it to be defined on the space of finite main functions ϕ(t′)/2πiand take the real part of the common regularization procedureA=aε(t)f(t) (7) Thus we obtain D1+ε 0+f(t) =∂ ∂tf(t) +∂ ∂t[aε(r(t),t)f(t)] (8) whereais a regularization parameter. For the sake of independence of GFD from this constant it is useful in the following to choose βi(on whichεdepends linearily) proportional to a−1. It can be shown that for large tthe exact expressions for the terms in (1)-(2) proportional to εare very close to the approximate expression given by (8) provided a special choice for the parameter αis is made (t=t0+ (t−t0), t−t0≪t0, α∼lnt∼lnt0) II. EQUATIONS OF PHYSICAL THEORIES IN MULTIFRACTAL TIME AND SPACE Equations describing dynamics of physical fields, par- ticles and so on can be obtained from the principle of minimum of fractional dimensions functionals. To do this, introduce functionals of fractional dimensions of space and time Fα(...|dα(r)), α=t,r. These function- als are quite similar to the free energy functionals,but now it is fractional dimension (FD) that plays the role of an order parameter (see also [2]). Assume further that FDdαis determined by the Lagrangian densities Lα,i,(i= 1,2,...,α =t,r) of all the fields ψα,i, describ- ing the particles and Φ α,idescribing the interactions in the point ( r) dα=dα[Lα,i(r,t)] (9) Equations that govern dαbehavior can be found by min- imizing this functional and lead to the Euler’s equations written down in terms of GFD defined in (1)-(2) Ddα +,Lα,i(x)dα−Ddα −,xDdα +,L′ α,i(x)dα= 0 (10) Substitution in this equation GFD for usual derivatives and specifying the choice for Fdependence on dαand relations between dαandLα(the latter can correspond to the well known quantum field theory Lagrangians) makes possible to write down the functional dependence F[L] in the form (a,b,c are unknown functions of Lor constants, L0is infinitely large density of the measure carrier Rn energy) F(...|dα) =/integraldisplay dLα/braceleftbigg1 2[a(Lα)∂dα ∂Lα]2 +b(Lα) 2(Lα−Lα,0)d2 α+c(Lα)dα/bracerightbigg (11) or F(...|dα) =/integraldisplay d4Lα/braceleftbigg1 2[a(Lα)∂dα ∂Lα]2 +b(Lα) 2(Lα−Lα,0)d2 α+1 4c(Lα)d4 α/bracerightbigg (12) 2The equations that determine the value of fractional di- mension follow from taking the variation of (11)-(12) and read ∂ ∂L/parenleftbigg a(L)∂dt,α ∂L/parenrightbigg +b(L)(L−L0)dα+c(L)d2 α= 0 (13) or ∂ ∂Lα/parenleftbigg a(Lα)∂dt,α ∂Lα/parenrightbigg +b(Lα)(Lα−L0,α)d2 α+c(Lα)d4 α= 0 (14) For nonstationary processes one have to substitute the time derivative of dαinto the right-hand side of Eqs.(13)- (14). Neglecting the diffusion of dαprocesses in the space with energy densities given by the Lagrangians Lwe can defineLα−Lα,0=˜Lα≪Lα,0with ˜Lαhaving sense of over vacuum energy density and for the simplest case (13) gives ( α=t, Lt,i≡Li) dt=˜Lt= 1 +/summationdisplay iβiLi(t,r,Φi,ψi) (15) More complicated dependencies of dαonLα,iare consid- ered in [2]. Note that relation (15) (and similar expres- sion fordrdoes not contain any limitations on the value ofβiLi(t,r,Φi,ψi) unless such limitations are imposed on the corresponding Lagrangians, and therefore dtcan reach any whatever high or small value. The principle of fractal dimension minimum, consist- ing in the requirement for Fαvariations to vanish under variation with respect to any field, in this theory produce the principle of energy minimum (for any type of frac- tional dimension dependency on the Lagrangian densi- ties). It allows to receive Euler’s-like equations with gen - eralized fractional derivatives for functions f(y(x),y′(x)), that describe behaviour of physical value fdepending on physical variables yand their generalized fractional derivatives y′=Ddα +,xf δFt,yi∼δdt,yi= 0 (16) δyidα(f) =δyiLα,i(f) = 0, α =r,t (17) Ddα +,yi(x)f−Ddα −,xDdα +,y′ i(x)f= 0 (18) The boundary conditions will have the form Ddα +,y′ i(x)f/vextendsingle/vextendsingle/vextendsinglex1 x0= 0 (19) In these equations the variables xstand for either tor r(the latter takes into account fractality of spatial di- mensions), yi={Φi,ψi},(i= 1,2,...), Lα,iare the La- grangian densities of the fields and particles. Here fcan be of any mathematical nature (scalar, vector, tensor, spinor, etc.), and modification of these equations for func- tionsfof more complicated structure does not encounterany principal difficulties. As Lagrangians Lα,ione can choose any of the known in the theoretical physics La- grangians of fields and their sums, taking into account interactions between different fields. From Eq.(17) it is possible to obtain generalizations of all known equations of physics (Newton, Shroedinger, Dirac, Einstein equations and etc.), and the similar equa- tions for fractional space dimensions ( α=r). Such generalized equations extend the application of the cor- responding theories for the cases when time and space are defined on multifractal sets, i.e. these equations would describe dynamics of physical values in the time and the space with fractional dimensions.The Minkowski- like space-time with fractional (fractal) dimensions for the casedt∼1 can be defined on the flat continuous Minkowski space-time (that is, the measure carrier is the Minkowski space-time R4). These equations can be re- duced to the well known equations of the physical theo- ries for small energy densities, or, which is the same, for small forces ( dt→1) if we neglect the corrections arising due to fractality of space and time dimensions (a number of examples from classical and quantum mechanics and general theory of relativity were considered in [2]).For statistical systems of many classical particles the GFD help to describe an influence of fractal structures arising in systems on behavior of distribution functions. III. GENERALIZED NEWTON EQUATIONS Below we write down the modified Newton equations generated by the multifractal time field in the presence of gravitational forces only Ddt(r,t) −,tDdt(r,t) +,tr(t) =Ddr +,rΦg(r(t)) (20) Ddr −,rDdr +,rΦg(r(t)) +b2 g 2Φg(r(t)) =κ (21) In (21) the constant b−1 gis of order of the size of the Uni- verse and is introduced to extend the class of functions on which generalized fractional derivatives concept is ap- plicable. These equations do not hold in closed systems because of the fractality of spatial dimensions, and there- fore we approximate fractional derivatives as Ddr 0+≈ ∇. The equations complementary to (20)-(21) will be given in the next paragraph. Now we can determine dtfor the distances much larger than gravitational radius r0(for the problem of a body’s motion in the field of spherical- symmetric gravitating center) as (see (11) and [5] for more details) dt≈1 +βgΦg (22) Neglecting the fractality of spatial dimensions and the contribution from the term with b−1 g, and taking βg= 2c−2), from the energy conservation law (approximate 3since our theory and mathematical apparatus apply only to open systems) we obtain /bracketleftbigg 1−2γM c2r/bracketrightbigg/parenleftbigg∂r(t) ∂t/parenrightbigg2 +/bracketleftbigg 1−2γM c2r/bracketrightbigg r2/parenleftbigg∂ϕ(t) ∂t/parenrightbigg2 −2mc2 r= 2E (23) Here we used the approximate relation between gener- alized fractional derivative an usual integer-order deriv a- tive (10) with a= 0.5 and notations corresponding to the conventional description of motion of mass mnear grav- itating center M. The value a= 0.5 follows from the regularization method used and alters if we change the latter. Eq.(23) differs from the corresponding equation in general theory of relativity by presence of additional term in the first square brackets. This term describes ve- locity alteration during gyration and is negligible while perihelium gyration calculations. If we are to neglect it, Eq.(23) reduces to the corresponding classical limit of equations of general relativity equation. For large energy densities (e.g., gravitational field at r<r 0) Eqs.(7) con- tain no divergences [2], since integrodifferential operato rs of generalized fractional diferentiation reduce to genera l- ized fractional integrals (see (1)). Note, that choosing for fractional dimension drin GFD Ddr 0+Lagrangian dependence in the form Lr,i≈Lt,igives for (17) additional factor of 0 .5 in square brackets in (23) and it can be compensating by fitting factor βg. IV. FIELDS ARISING DUE TO THE FRACTALITY OF SPATIAL DIMENSIONS (”TEMPORAL” FIELDS) If we are to take into account the fractality of spatial dimensions ( dx/ne}ationslash= 1, dy/ne}ationslash= 1, dz/ne}ationslash= 1), Eqs.(17)-(19), we arrive to a new class of equations describing certain phys- ical fields (we shall call them ”temporal” fields) generated by the space with fractional dimensions. These equations are quite similar to the corresponding equations that ap- pear due to fractality of time dimension and were given earlier. In Eqs.(10)-(12) we must take x=r,/;α=r and fractal dimensions dr(t(r),r) will obey (14) with t being replaced by r. For example, for time t(r(t),t)) and potentials Φ g(t(r),r) and Φ e(t(r),r) (analogues of the gravitational and electric fields) the equations analogous to Newton’s will read (here spatial coordinates play the role of time) Ddr(r,t) −,rDdr(r,t) +,rt(r) =Ddt +,t/parenleftbig Φg(t(r)) +erm−1 rΦe(t(r))/parenrightbig (24) Ddt −,tDdt +,tΦg(t(r)) +b2 gt 2Φg(t(r)) =κr (25)Ddt −,tDdt +,tΦe(t(r)) +b2 et 2Φe(t(r)) =er (26) These equations should be solved together with the gen- eralized Newton equations (20)-(21) for r(t(r),r). With the general algorithm proposed above, it is easy to obtain generalized equations for any physical theory in terms of GFD. From these considerations it also fol- lows that for every physical field originating from the time with fractional dimensions there is the correspond- ing field arising due to the fractional dimension of space. These new fields were referred to as ”temporal fields” and obey Eqs.(14)-(16) with x=r, α=r. Then the question arises, do these equations have any physical sense or can these new fields be discovered in certain experiments? I wont to pay attention on the next fact: if Lt,i≈Lr,ino new fields are generated. This is the case when fractal dimension of time and space dt,/brcan not be divided on dt+dr¯, the time and space fractal sets can not be divided too.The FD time and space are common and defined by value given by Li(the latter can be chosen in the form of usual Lagrangians in the known theories). V. CAN REPULSIVE GRAVITATIONAL FORCES EXIST? In general theory of relativity no repulsive gravita- tional forces are possible without a change of the Rie- mann space curvature (metric tensor changes). But in the frames of multifractal time and space model, even when we can neglect the fractality of spatial coordinates, from (24) it follows (for spherically-symmetric mass and electric charge distributions) mr∂2t(r) ∂r2=∂ ∂t/summationdisplay iΦi(t)≈ −mrkr c2t2±e2 ct2(27) with accuracy of the order of b2. Heremris the ana- logue of mass in the time space and corresponds to spa- tial inertia of object alteration with time changing (it is possible that mrcoincides with ordinary mass up to a dimensional factor). Eq.(27) describes the change of the time flow velocity from space point to space point depending on the ”temporal” forces and indicates that in the presence of physical fields time does not flow uni- formly in different regions of space, i.e the time flow is irregular and heterogeneous (see also Chapter 5 in [2]). Note, that introducing equations like (27) in the time space is connected with the following from our model consequences (see (17)-(19)) about equivalence of time and space and the possibility to describe properties of time (a real field generating all the other fields except ”temporal”) by the methods used to describe the char- acteristics of space. Below we will show that taking into consideration usual gravitational field in the presence of its ”temporal” analogues gives way to the existence of gravitational repulsion proportional to the third power 4of velocity. Indeed, the first term in the right-hand side of (27) is the analogue of gravity in the space of time (”temporal” field). Neglecting fractional corrections to the dimensions and taking into account both usual and ”temporal” gravitation, Newton equations have the form md2r dt2=Fr+Ft=−γmM r2+mrkr ct2/parenleftbiggdr dt/parenrightbigg3 (28) The criteria for the velocity, dividing the regions of at- traction and repulsion reads /parenleftbiggdr dt/parenrightbigg3 =/parenleftbiggγmM r¯2/parenrightbigg−1mrkr ct2(29) Herer(t) must also satisfy Eq.(20). Introducing gravita- tional radii r0andt0(the latter is the ”temporal” gravi- tational radius, similar to the conventional radius r0), we can rewrite (23) as follows /vextendsingle/vextendsingle/vextendsingle/vextendsingledr dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle=c3/radicalBigg ct2 r2t0 r0(30) In the last two expressions ris the distance from a body with massmto the gravitating center, tis the time differ- ence between the points where the body and the gravitat- ing center are situated, mr=m/c, κ r=κt/c. If we ad- mit thatr0andt0are related to each other as r0=t0/c, the necessary condition for the dominance of gravitation repulsion will be c < rt−1. It is not clear whether this criteria is only a formal consequence of the theory or it has something to do with reality and gravitational re- pulsion does exist in nature. What is doubtless, that in the frames of multifractal theory of time and space it is possible to introduce (though, may be, only formally) dynamic gravitational forces of repulsion (as well as re- pulsive forces of any other nature, including nuclear). VI. THE GEOMETRIZATION OF ALL PHYSICAL FIELDS AND FORCES The multifractal model of time and space allows to con- sider the fractional dimensions of time dtand spacedr(or undivided FD dtras the source of all physical fields (see (11)) (including, in particular, the case when flat (not fractal) Minkowski space-time R4is chosen as the mea- sure carrier). From this point of view, all physical fields are consequences of fractionality (fractality) of time and space dimensions. So all the physical fields and forces are exist in considered model of multifractal geometry of time and space as far as the multifractal fields of time and space are exists. Within this point of view, all phys- ical fields are real as far as our model of real multifractal fields of time and space correctly predicts and describes the physical reality. But since in this model all the fields are determined by the value of fractal dimension of time and space, they appear as geometrical characteristics oftime and space (12-14). Therefore there exists a com- plete geometrization of all physical fields, based on the idea of time and space with (multi)fractional dimensions, the hypothesis about minimum of functional of fractal dimensions and GFD calculus used in this model. The origin of all physical fields is the result and consequence of the appearing of the fractional dimensions of time and space. One can say that a complete geometrization of all the fields that takes place in our model of fractal time and space is the consequence of the inducing (and describing by GFD) composed structure of multifractal time and space as the multifractal sets of multifractal subsets St andSr¯with global and local FD. The fractionality of spa- tial dimensions dralso leads to a new class of fields and forces (see (17)-(19) with α=r). For the special case of integer-valued dimensions ( dt= 1, dr= 3) the mul- tifractal sets of time and space StandSrcoincide with the measure carrier R4. From (14) it follows then that neither particles nor fields exist in such a world. Thus the four-dimensional Minkowski space becomes an ideal physical vacuum (for FD dα>1 the exponent of Rnhas valuen >4). On this vacuum, the multifractal sets of time and space ( StandSr¯) are defined with their frac- tional dimensions, and it generates our world with the physical forces and particles. Now the following question can be asked: what is the reason for the dependence in the considered model of fractal theory of time and space of fractionality of di- mensions on Lagrangian densities? One of the simplest hypothesis seems to assume that the appearing of frac- tional parts in the time and space dimensions with de- pendence on Lagrangian densities originates from certain deformations or strains in the spatial and time sets of the measure carrier caused by the influence of the real time field on the real space field and vise versa (generat- ing of physical fields caused by deformations of complex manifolds defined in twistor space is well known [6]). As- suming then that multifractal sets StandSrare complex manifolds (complex-valued dimensions of time and spa- tial points can be compacted), deformation, for example, of complex-valued set Stunder the influence of the spatial points setSrwould result in appearing of spatial energy densities in time dimension, that is generating of physical fields (see [6]). Fractional dimensions of space appearing (under the influence of set Stdeformations) yields new class of fields and forces (or can also not yield). It can be shown also that for small forces (e.g., for gravity - at distances much larger then gravitational radius) gener- alized fractional derivatives (1)-(2) can be approximated through covariant derivatives in the effective Riemann space [2] and covariant derivatives of the space of the standard model in elementary particles theory [7] (with the corrections taking into account fields generating and characterizing the openness of the world in whole [8,9]). All this allows to speak about natural insertion of the offered mathematical tools of GFD, at least for ε≪1, in the structure of all modern physical theories (note here, that the theory of gravitation as the theory of real fields 5with a spin 2 is invented in [10]). Note also, that number of problems within the framework of the theory of multi- fractal time and space (classical mechanics, nonrelativis - tic and relativistic quantum mechanics) were considered in [2,9,11]. VII. CONCLUIONS In our model we postulate the existence of multifractal space and time and treat vacuum as Rnspace which is the measure carrier for the sets of multifractal time and space. Fractionality of time dimension leads then to ap- pearing of space-time energy densities L(r(t),t), that is generating of the known fields and forces, and fraction- ality of space dimensions gives new time-space energy densitiesL(t(r),r) and a new class of ”temporal” fields. Note, that the roles of dtanddrin distorting accord- ingly space and time dimensions is relative and can be interchanged. Apparently, one can consider the ”united” dimensiondt,r- the dimension of undivided onto time and space multifractal continuum in which time and co- ordinates are related to each other by relations like those for Minkowski space, not using the approximate relation utilized in this paper dtt,r=dt+dr. Moreover in some cases it seems to be even impossible to separate space and time variables, and then dtanddrcan be chosen to be equal to each other, i.e., there would be only one frac- tional dimension dt=drx,y,z= 1 +/summationtextβiLi(r(t),t;t(r),r) describing the whole space-time. In this case one would have to calculate generalized fractional derivatives from the same Lagrangians, and new ”temporal” fields will not be generated. The considered model of multifractal time and space offers a new look (both in mathematical and philosophi- cal senses) onto the properties of space and time and their description and onto the nature of all the fields they gen- erate. This gives way to many interesting results and conclusions, and detailed discussion of several problems can be found in [2]. Here we restrict ourselves with only brief enumerating of the most important ones. a) The model does not contradict to the existing phys- ical theories. Moreover, it reduces to them when the potentials and fields are small enough, and gives new predictions (free of divergencies in most cases) for not small fields. Though, the question about applicability of the proposed relation between fractal dimension and Lagrangian densities still remains open. b) We consider time and space to be material fields which are the basis of our material Universe. In such a Universe there exist absolute frames of reference, and all the conservation laws are only good approximations valid for fields and forces of low energy density since the Universe is an open system, defined on certain mea- sure carrier (the latter probably being the 4-dimensional Minkowski space). Smallness of fractional corrections to the value of time dimension in many cases (e.g., on theEarth’s surface it is about dt−1∼10−12) makes possible to neglect it and use conventional models of the physics of closed systems. c) The model allows to consider all fields and forces of the real world as a result of the geometrization of time and space (may be more convenient the term ”fractaliza- tion” of time and space) in the terms of fractal geometry. It is fractional dimensions of time and space that generate all fields and forces that exist in the world. The model in- troduces a new class of physical fields (”temporal” fields), which originates from the fractionality of dimensions of space. These fields are analogous to the known physi- cal fields and forces and can arise or not arise depending on certain conditions. Thus the presented model of time and space is the first theory that includes all forces in single theory in the frames of fractal geometry. Repeat once more: the model allows to consider all the fields and forces of the world as the result of geometrization in- cluding them in FD of time and space. It is non-integer dimensions of time and space that produce the all ob- servable fields. The new class of fields naturally comes into consideration, originating solely from the fractiona l- ity of space dimensions and with the equations similar to those of the usual fields. The presented model of space and time is the first theory that allows to consider all physical fields and forces in terms of a unique approach. d) Basing on the multifractal model of time and space, one can develop a theory of ”almost inertial” systems [11,13,14] which reduces to the special theory of relativit y when we neglect the fractional corrections to the time dimension. In such ”almost inertial” frames of reference motion of particles with any velocity becomes possible. e) On the grounds of the considered fractal theory of time and space very natural but very strong conclusion can be drawn: all the theory of modern physics is valid only for weak fields and forces, i.e. in the domain where fractional dimension is almost integer with fractional cor - rections being negligibly small. f) The problem of choosing the proper forms of defor- mation that would define appearing of fractional dimen- sions also remains to be solved. So far there is no clear understanding now which type of fractal dimensions we must use,dtanddrordt,r. Obviously, solving numerous different problems will depend on this choice as the result of different points of view on the nature of multifractal structures of time and space. The author hopes that new ideas and mathematical tools presented in this paper will be a good first step on the way of investigations of fractal characteristics of tim e and space in our Universe. [1] Mandelbrot B. The fractal geometry of nature (W.H. Freeman, New York, 1982) 6[2] Kobelev L.Ya. Fractal theory of time and space , (Konross, Ekaterinburg, 1999) (in Russian) [3] S.G.Samko , A.A.Kilbas , O.I.Marichev, Fractional Inte- grals and Derivatives - Theory and Applications (Gordon and Breach, New York, 1993) [4] I.M.Gelfand, G.E.Shilov, Generalized Functions (Aca- demic Press, New York, 1964) [5] Kobelev L.Ya. Generalized Riemann-Liouville fractional derivatives for multifractal sets , Urals State University, Dep. v VINITI, 25.11.99, No.3492-B99 [6] Penrose R.J., J.Math. Phys., 8, 345 (1967) [7] Kobelev L.Ya., Multifractal time and space, covariant derivatives and gauge invariance , Urals State University, Dep. v VINITY, 09.08.99, No.2584-B99 (in Russian); [8] Yu.L.Klimontovich Statistical Theory of Open Systems (Kluwer, Dordrecht, 1995) [9] Kobelev L.Ya, Multifractal time and irreversibility in dy- namic systems , Urals State University, Dep. v VINITY, 09.08.99, No.2584-99 (in Russian) [10] A.A.Logunov, M.A.Mestvirishvili, Theoretical and Mathematical Physics, 110, 1 (1997) (in Russian) [11] Kobelev L.Ya., Multifractality of Time and Special The- ory of Relativity , Urals State University, Dep. v VINITY 19.08.99, No.2677-99 (in Russian) [12] Kobelev L.Ya., Fractal Theory of Time and Space , Ural State Univ., Dep.v VINITY 22.01.99, No.189-99 (in Rus- sian) [13] Kobelev L.Ya., Can a Particle Velocity Exceed the Speed of Light in Empty Space? gr-qc/0001042, 15 Jan 2000 [14] Kobelev L.Ya., Physical Consequences of Moving Faster than Light in Empty Space. gr-qc/0001043, 15 Jan 2000 7

arXiv:physics/0001036v1 [physics.gen-ph] 17 Jan 2000D.M.Chi physics/0001036 WAVE ESSENCE OF PARTICLES D. M. Chi1 Center for MT&Anh, Hanoi, Vietnam. Abstract We state several ideas based on the view-point of particle be haviour of matter to explain wave character of photon and elementary pa rticles. By using Newton’s suggestion of light ray, we clarify integrally the behaviour of light “wave”. And “wave” character of particles is also explained by the view-point of particle. 1 Introduction Today everybody believes that the matter has both particle c haracter and wave character. Following belief up to now, wave is understood as continuous change of any quantity in period in space and in time with its immanent cause. With th e view-point of particle the periodic property puzzled everybody, and it is looked as a presence of wave. Now, we think that there is still an other way to understand na ture more accurately and more consistently: we use still Newton’s ideas “along wi th ray of light there must be a manifestation of some periodicity” to explain wave phenome na of the light and elementary particles. The article is organized as follows. In Section 2, we show an e xplanation intuitionally of interference of the light. And the interferential phenom enon of electron is illustrated in Section 3. Conclusion is given in Section 4. 2 Interferential picture of the light The electromagnetic field can exit independently and so it in cludes invariant structures (particles). The electromagnetic field has periodicity and so this periodicity either goes with particle by anyway or manifests in the distribution of p articles in space and this squadron of particles fly along a fixed ray, ◦—–λ—–◦− →c ◦oooooo– – – – ◦oooooo– – – – ◦oooooo– – – – ◦oooooo– – –→ 1Present address: 13A Doi Can, Hanoi, Vietnam. 1 c/circlecopyrt1999 D.M.ChiD.M.Chi physics/0001036 This picture is similar to the so-called wave. Let us use this imagination of the light to explain how the int erferential picture is created. Suppose that there is a gun, and after each fixed interval of ti me ∆tit shoots one ball. Balls are alike in all their aspect. They fly with the sam e velocityvin environment without resistance force and their gravities are ignorable . The “wave-length” - i.e. the distance between two balls - is λ=v.∆t. All balls are electrized weak charge of the same sign, and they are covered with a sticky glue envelope so that electrostatically propulsive force between them cann ot win stickiness of glue envelope when they touch together with small sufficient meeting angle. A target with two parallel interstices in the vertical is set square with the axis of the gun and is also electrized weak charge but different from the sign of balls. The distance betw een two interstices is not too large in comparison with the wave-length λ. And assume that sticky glue of balls does not effect on the target. Thus, probability in order that any ball flies through one int erstice or the other is identical. (The interstices are enough large that balls can fly through easily). After flying past, balls are changed flying direction with all possible angles in the horizontal plane with definite probability distribution. With such initial conditions, balls are in ‘dephasing’ with each other and, after over- coming the target, balls that do not fly through the same inter stice are able to meet together with a some non-zero propability. If meeting angle is enough small in order that stickiness has effect, two some balls will couple together to be a system, then change direction and fly on the bisector of meeting angle. If behind the target and distant from the target a space d≫λ, a reception screen is set parallel to the target, then on this screen we will harv est falling point locations of single balls and couple balls. Argumentation and calculation show that single balls (miss ed interference) form a monotonous background of falling points, and couple balls ( caused by interference) fall concentratively and create definite veins on the background , depending on dephasing degree of interfered balls. Hence, from Newton’s idea and using quantities such as “wave -length”, “dephase”, and so on results obtained is fitted in ones calculated using w ave behaviour. Furthermore, they explain why, when amount of balls is not enough large (th e time to do experiment is short), the picture of falling points seems chaotic, rand um. Only with a large number of balls (the time to do experiment is long), interferential veins are really clear. This is one that, if using wave behaviour, is impossible to explain. Thus, if we consider the light as a system of particles that, i n the most rudimentary level, are similar above balls: they are attractable togeth er and radiated periodically, then the interference of the light is nothing groundless or d ifficult to understand when we refuse to explain it by using wave behaviour. Moreover, wi thout wave conception the 2D.M.Chi physics/0001036 phenomena of the light is very bright, clear, and more unitar y. Such a view-point of the light requires to imagine again many problems, simultaneously brings about new effects that need to prove in experiment. If there is a source that radiates continously separately li ght pulses of a fixed thickness and with a fixed distance between pulses (Fig. 1), we can carry out a following experiment (Fig. 2). FREQUENCY TRANSFER Figure 1: Frequency transfer. With two continously radiative sources we direct light puls es together with an inter- secting angle α. Interference presents only in the area ABCD of Fig. 1. If the light is wave, then to see interference we should set a photographi c plate in the area ABCD, because outside this area interference is impossible to pre sent. Two light sources are independent from each other, the stable condition of interf erential veins is not ensured, position of vains is changed incessantly, and the consequen ce is that on the photographic plate we cannot obtain interferential veins. But if the light is particle, then we always obtain interfere ntial veins (in suitable polarization condition) though the photographic plate is s et inside or outside the area ABCD. Carrying out experiments according to this diagram we can ch eck interferential ability of the light of different wave-lengths. Given two radiative s ources of different frequencies by that way, we are able to see whether photons of different fre quencies is identical. A consequence of particle behaviour is that we can make mirro r-holography with any reappear light source. This is drawn from the phenomenon tha t two photons interfered together change direction and fly on the the bisector of meeti ng angle. 3D.M.Chi physics/0001036 A B C Dα λ λ K Figure 2: Interference of the light. In the conception of particle, frequency of a light wave is un derstood as number of photons radiated in a unit of time to a definite direction. The photo-electric effect, therefore, is understood as follows: the more number of comi ng photons that collide to electrons in a unit of time is, the higher energy that electro ns gain is. If momenta of all photons are of the same value, then it is possible that number of photons electrons gain is proportional to light frequency, and thus momentum of eac h photon is proportional to Plank’s constant /planckover2pi1. With such understanding, energy that electron gain from in terfered photons is higher than one from non-interferential photons . With wave behaviour and imagined that atom sends spherical w aves, then the photo- electric effect is impossible to explain as shown. But if thou ght that atom radiate directly light, then because radiation is a wave process, in radiatio n process atom does not keep still a place in space, thus it is difficult to maintain that all energy quanta is transmitted out space in a definite ray. Situation is the same for absorbin g process of energy quanta. For instance, electron is impossible to keep still a place to await absorbing all energy quanta then moves to other position. 4D.M.Chi physics/0001036 Thus, there is not any firm basis to say that energy is absorbed piecemeal quanta E=/planckover2pi1ν. 3 Interferential picture of elementary particle Let us consider interference of electron. First of all we can confirm that there would not be any experime nt we gain interference of electron if we used conditions as already stated for the li ght. Because with actual experience electrons are not like as above balls with sticki ness that lack of this ability there is not any presence of interferential couples. Interference of electron is completely different, if using t he word “interference”. Suppose that there is an electron flying to a block of matter ma de up from particles heavier very much than electron. Each heavy particle in the b lock of matter is a scattering center. Because of interaction, after flying out of influenti al region of scattering center electron is changed direction with an angle αin comparison with initial direction. Assume that the deviation angle αis dependent on the aim distance ρobeying on the lawaorbas on the figures. ϕα ρ e-, v α= =π=−=2ϕ Figure 3: Scattering of electron. α α0 ρ ρ0α ρα α0 ρρ0α ρ ∆ρε a.b. Figure 4:α-dependence of the aim distance ρ. 5D.M.Chi physics/0001036 Derivative of αwith respect to ρis equal to 0 at ρ=ρ0. If all values of the aim distance ρis the same probability for e, then the probability in order that the particle eis deviated from the coming direction an angle ( α0) is “infinitely large” in comparison with any other angle:∂ρ ∂α/vextendsingle/vextendsingle α=α0=∞. It is necessary to say that α0is inversely proportional to the momentum of e. The higher the momentum of eis, the higher the direction conservability of its momentum vector is, then the smaller the deviation angle αis. (That is just the basis of Broblie relation.) We bring in a quantity ǫcalled the maximum divergence that measurement is accept- able, that border rays deviated in comparison with the angle αa valueǫbelongs to still that angleα. Thus, we can establish a function of probability density ha ving the following form, (Fig. 5), αP(α) α Figure 5: Probability density. The greater the sharpness of this distribution is, the great er the contrast of probability densities between the angle αand its neighbouring angles. Call average distance between two scattering centers 2 R, then influence space of any center (in the field of the aim distance) is πR2. The space, where the particle eis falled into and deviated an angle ( α±ǫ), is propor- tional toπ[(ρ+ ∆ρ)2−(ρ−∆ρ)2]≈4πρ∆ρ. The probability in order that escatters into the angle αis4πρ∆ρ πR2=4ρ∆ρ R2(Fig. 6). The probability density that escatters into the angle αin a some direction is4ρ∆ρ R21 2π= 4ρ∆ρ πR2=P(α). However, using the above method to estimate the probability density with all direc- 6D.M.Chi physics/0001036 Rρ+∆ρρ∆ρ Figure 6: tions in the space for all scattering processes of particles is very completed and unwieldy. For this reason, here we are only interested “relative” prob abilities of variety directions, namely events: the probability in order that escatters into the angle α0is “infinitely”2 large in comparison with that for all other angles. In that co rrelation, all deviation an- gles that are different from α0should be ignored because they form only a monotonous background. This has not influence on the qualitative accura cy of the law. Consideri-th scattering experiment (each experiment there is only on eeattended with a constant momentum, the same for all experiments.) Assume that all scatterings are elastic. The scattering pro cess ofeis easy to imagine as Figure 7. After the 1-th scattering, ecan be deviated with any angle and fly with any direction, but all directions with highest probability ( Pα0=∞) form a cone with the top at the scattering center and an arrangement angle (2 α0). In the 2-nd scattering, the directions with highest probabi lity ofe’s trajectory form the cones with angles (0, 2(2 α0)). This is explained as follows. Scattering centers in the block of matter distribute at rand om fore’s trajectory and this random is always maintained by thermic fluctuations, in elastic scatterings, ... So, on the conic surface (1-2) there forms a brim - the locus of proba bility of 2-nd scattering centers, is plotted as the brim (2-2) in the figure. At each poi nt of the brim there exist 2-nd scattering centers with some probability. Thus, in the 2-nd scattering at each point of the brim there fo rms a new probability 2It means ǫ→0. 7D.M.Chi physics/0001036 e mv1 22 33 44 Figure 7: Diffraction of electron. cone, similar to fomation at the 1-th scattering center. The se probability cones interfere with each other forming two collective cones with open angle s 0 and 2(2 α0) respectively. Indeed, if we put a spherical surface of enough large radius a nd center coincident with the 1-th scattering center, then cones intersect the spheri cal surface and form circles as given in Figure 8. For simplification, we are not concerned with curvature of th e spherical surface. The probability in order that the particle falls into any point o f circles is identical. Let us find the probability density of e’s falling points on this surface. In the figure, it is clear that this density is proportional to the ratio∆ℓ ∆S, where ∆ℓis the total length of circles in an elementary surface ∆ S(dS). Solve this problem in the pole coordinate, the equation of ci rcles in this coordinate is ρ=Ccosϕ±/radicalBig R2−C2sin2ϕ, (1) dρ=−Csinϕ/parenleftBigg 1±Ccosϕ/radicalbig R2−C2sin2ϕ/parenrightBigg dϕ, (2) dS∼=ρdρdϕ, ∆ℓ∼=2/radicalbig dρ2+ρ2dϕ, MP=∆ℓ ∆S=2/radicalbig dρ2+ρ2dϕ ρdρdϕ. (3) By differentiating of MPoverρthen puting this derivative zero or substituting values ofρanddρinto (3) we obtain MP→ ∞ whenϕ→0. 8D.M.Chi physics/0001036 3 3c RρρϕdSdϕ Figure 8: Thus, at maximum and minimum values of ρthe probability density MPis infinitely large (in comparison with other values of ρ). The surface dSis equivalent to the volume angle dΩ, and ∆ℓis equivalent to the probability in order that the particle scatters into that vo lume angle. Therefore, we obtain that in the 2-nd scattering all possibl e directions with highest probability of trajectories form two cones with open angles 0 and 2(2α0). With similar argument we realize that at the 3-rd scattering the directions with highest probability of e’s trajectories form cones with open angles 3(2 α0), 1(2α0), 1(2α0),−1(2α0). Generally, up to the n-th scattering there form probability cones of possible dir ections ofe’s trajectories with open angles n(2α0), (n−2)(2α0), ..., (n−2m)(2α0). Put a spherical surface as a catching screen with its axis coi ncident to the initial direction of particle before coming to the target, and its ra dius much larger than the thickness of the target. Hence, the probability cones forming at the last scattering intersect the catching screen and form circle brims - These are locus of falling points with e’s maximum probability. Ifnis even, we obtain an even number of brims; nis odd, we have an odd number of brims. And in all times of experiment ealways scatters ntimes, then on the catching screen we obtain either an even number or odd number of brims, depending on even or odd value ofn. Of course, in one time of experiment, nhas only value. Thus, the probability intensity of obtained brims after total experiment Σ is depe ndent not only on decreasing 9D.M.Chi physics/0001036 law from inner to outer but also on frequence of n, i.e. on the ratioin Σ;inis frequency (number of occuring times of n) in total experiment Σ. It is difficult to define this frequency for every possible valu e ofn. However, at the most rudimentary, it is sure that spectrum of n’s values is not large3and probability in order that nhas even or odd value is identical, then spectrum of differact ion brims is complete from (0) to ( nmax). From conditions of experiment we realize that the larger the matter density of the target is the tickness is, the larger nmaxis. Here, once again we find out that the differaction picture is on ly clear since the number of particles taking in scattering are enough much. If they ar e is too little, on the screen we see only a chaotic distribution of e’s marks, but on the contrary, if they are too much and the catching screen is a photographic plate, then all points on the screen are saturated and differaction brims are hidden. The scattering of light particles in a radial field is describ ed in mechanics as follows ϕ=/integraldisplayr r0M mr2dr/radicalBig 2 m/parenleftbig E−M2 2mr2−U(r)/parenrightbig, whereϕis the angle made by radius vector of particle’s position on t he trajectory and radius vector of particle’s extremal point ( /vector r0),Mis the momentum of the particle, Eis the energy and mis mass of the particle, and U(r) is the potential of the field. If the potential of the field U(r) has the form U(r) =±A r, then from the above formula we can express ϕas a function of the aim distance ρsince the upper bound approaches to infinity, thus the deviation angle α=π−2ϕis also a function of ρ. Calculations give the result that the derivative of ϕ(ρ) with respect to ρis not equal to zero at any position. That proves that the condition to hav e “wave-like” differaction is not satisfied. If in fact the potential of nuclear field had the form as above, then the ability in order that electron would fall into nucleus has a very large probab ility. This is not compatible with the fact4. We can suppose a supplement as: far from the attracting cente r a distance athere is a surface L. This surface changes trajectory of scattering particles. Because it is not absolutely hard (in the present region of the surface, th e potential field has some form), there happens a slippery effect of particle on the surf ace. This softens variation of deviation angle of trajectory α, and then αis still a continous function of ρ. Thus, the momentum of scattering particle is unsurpassed a some va lue in order to unbreak elasticity of the surface. In summary, with this supplement, αis a function of ρwith dependence as Figure 9. whereρ0≪aand hence, the condition to have the “wave-like” differactio n is satisfied. 3We are only interested particles that overcome through the t arget. 4To avoid this there was a quantum mechanics. 10D.M.Chi physics/0001036 ρ ρ0α Figure 9: 4 Conclusion Thus, we show in rather detail some ideas based on the particl e behavour of matter. By using Newton’s model of light ray we have explained rather co mpletely “wave” behaviours of the light. The polarization of the light is a effect of parti cle behaviour: two photons interfere with each other forming a system with axisymmetry . The experiment set as Figure 1 is important. Carrying on this experiment will take part in confirmation of light’s particle behaviour. Its detailed results will open up new ideas, and new directions of research. The wave character of elementary particle, electron is typi cal, can be explained by pure particle behaviour. This give us a similarity between t he wave function in quantum mechanics and the vector function of particle behaviour. The motion of any particle can be expressed as a vector, whose direction points out particle’s motion direction at a given point of trajectory, and whose module expresses probability amplitude of particle flying in that direction. Thus, the probability trajectory of particle is completely able to expressed as a vector funct ion ψ=A(α(t))eiα(t), α(t)is the deviation angle in comparison with the initial direct ion, is a function of time; A(α) is the probability of particle flying with the angle α. But this does not mean that the particle expressed as above wi ll have a really wave 11D.M.Chi physics/0001036 character. If we find a way to express the value of α(t)by the value of energy-momentum of scattering particle, parameters of scattering environmen t, and simplification: α(t)is con- tinously independent of time but discontinously dependent on time (due to the fact that scattering centers are discontinous), then the vector func tion is not basically different from the wave function in quantum mechanics. Doing with appr opriate operators for probability function, we can obtain correlative quantitie s. Hence, from the natural idea of particle behaviour of matter we can discover further natural phenomena. One of the most host problems today is infl ation of the universe affirmed from the red-shift of Doppler’s effect. However, if th e space between observer and light source is vacuum, then the explanation of the red-s hift based on Doppler’s effect is fully satisfactory. But in fact the universe is fille d with gravitational fields, macrometric and micrometric objects as stars and clusters. Therefore, these problems had been re-examined in further detail by us, with considera tion actual influences of interstellar environment on frequency shift. This is only r ealized by foundation of particle behaviour of the light. The existence of cosmic dusts as moti onal scattering centers is an essential condition to happen scattering-interference pr ocesses of the light when it flies through the interstellar environment. In turn, these scatt ering-interference processes lead to the shift of light frequency. Our results give a reliable c onfirmation that the red-shift is not exhibit of the inflation of the universe. Acknowledgments The present article was supported in part by the Advanced Res earch Project on Natural Sciences of the MT&A Center. References [1] D. M. Chi, The Equation of Causality , (1979), (available in web site: www.mt- anh.com-us.com). 12

arXiv:physics/0001037v1 [physics.acc-ph] 17 Jan 2000IIT-HEP-99/3 Fermilab-Conf-00/019 physics/0001037 Muon Collider/Neutrino Factory: Status and Prospects⋆ Daniel M. Kaplan1 Illinois Institute of Technology, Chicago, IL 60616 and Fermi National Accelerator Laboratory, Batavia, IL 60510 for the Neutrino Factory and Muon Collider Collaboration Abstract During the 1990s an international collaboration has been st udying the possibility of constructing and operating a high-energy high-luminosi tyµ+µ−collider. Such a machine could be the approach of choice to extend our discove ry reach beyond that of the LHC. More recently, a growing collaboration is explor ing the potential of a stored-muon-beam “neutrino factory” to elucidate neutrin o oscillations. A neutrino factory could be an attractive stepping-stone to a muon coll ider. Its construction, possibly feasible within the coming decade, could have subs tantial impact on neu- trino physics. 1 Introduction The Neutrino Factory and Muon Collider Collaboration (NFMC C) [1] is en- gaged in an international R&D project to establish the feasi bility of a high- energy µ+µ−collider and a stored-muon-beam “neutrino factory.” As a he avy lepton, the muon offers important advantages over the electr on for use in a high-energy collider: ⋆Invited talk presented at the 7th International Conference on Instrumentation for Colliding-Beam Physics , Hamamatsu, Japan, Nov. 15–19, 1999. 1E-mail: kaplan@fnal.gov Preprint submitted to Elsevier Preprint 17 February 2014(1) Radiative processes are highly suppressed, allowing us e of recycling accel- erators. This reduces the size and cost of the complex. It als o allows use of a storage ring, increasing luminosity by a factor ≈103over a one-pass collider. (2) In the Standard Model and many of its extensions, use of a h eavy lep- ton increases the cross section for s-channel Higgs production by a factor mµ2/me2= 4.3×104, opening a unique avenue for studying the dynam- ics of electroweak symmetry breaking [2]. More generally, a sensitivity advantage may be expected in any model that seeks to explain m ass generation [3]. (3) Beam-beam interactions make high luminosity harder to a chieve as the energy of an e+e−linear collider is increased, an effect that is negligible for muon colliders [4]. The small size anticipated for a muon collider is indicated i n Fig. 1, which compares various proposed future accelerators. Unlike oth er proposals, muon colliders up to√s= 3 TeV fit comfortably on existing Laboratory sites. Be- yond this energy neutrino-induced radiation (produced by n eutrino interac- tions in surface rock), which increases as E3, starts to become a significant hazard, and new ideas or sites where the neutrinos break grou nd in unin- habited areas would be required. A muon collider facility ca n provide many ancillary benefits (physics “spinoffs”) and can be staged to p rovide interesting physics opportunities even before a high-energy collider i s completed. These include experiments [5] with intense meson and muon beams pr oduced using the high-flux ( ≈2×1022protons/year) proton source, as well as neutrino beams of unprecedented intensity and quality, discussed be low in Sec. 7. Ref. [6] is a comprehensive summary of the status of muon coll ider research as of≈1 year ago. While stored muon beams have been discussed since ≈1960 [7], and muon colliders since 1968 [8], only in recent years has a p ractical approach to the realization of a µ+µ−collider been devised. The key concept that may allow a muon collider to become a reality is ionization cooli ng [9–11]. Muons may be copiously produced using collisions of multi-GeV pro tons with a target to produce pions, which then decay in a focusing “capture cha nnel.” However, those muons occupy a large emittance (phase-space volume) a nd are unsuit- able for injection directly into an accelerator. Muon-beam cooling is needed to reduce the emittance by a sufficient factor but must be carri ed out in a time short compared to the ≈2µs muon lifetime. While other beam-cooling methods are too slow, as discussed below, simulations show t hat ionization cooling can meet these requirements. 2Fig. 1. Sizes of various proposed high-energy colliders as c ompared with FNAL and BNL sites. A muon collider with√sup to 3 TeV fits easily on existing sites. 2 Proton Source, Targetry, and Capture Muon-collider luminosity estimates (see Table 1) have been made under the assumption that a 4MW proton beam may safely strike a target r epresenting 2–3 hadronic interaction lengths, which is tilted at a 100–1 50mr angle with respect to the solenoidal focusing field (see Fig. 2). Design studies are ongo- ing to demonstrate this in detail, and BNL-E951 at the AGS wil l test this experimentally within the next few years [12]. Ideas being e xplored include a liquid-metal jet target (Fig. 2) and a “bandsaw” target (Fig . 3). In neutrino- factory scenarios the beam power requirement is eased to 1MW , making a graphite target also a possibility. About 10% of the beam ene rgy is dissipated in the target. A target with comparable dissipation is being designed for the Spallation Neutron Source at Oak Ridge National Laboratory [13]. The re- quired 4MW proton source, while beyond existing capability , is the subject of ongoing design studies at Brookhaven [14] and Fermilab [15] and is compara- ble in many respects to machines proposed for spallation neu tron sources [16]. For efficient capture of the produced low-energy pions, the ta rget is located within a 20T solenoidal magnetic field to be produced using a s uperconducting solenoid with a water-cooled copper-coil insert (Fig. 2). T he captured pions and their decay muons proceed through a solenoidal field that decreases adia- 3Fig. 2. Liquid-jet pion-production target with solenoidal capture/decay channel. Fig. 3. “Bandsaw” target concept. batically to 1.25T. Simulations show that ≈0.6 pions/proton are captured in such a channel for proton energy in the range 16–24GeV. The resulting muon bunches, while very intense, feature a la rge energy spread, 4Fig. 4. Event distributions vs. kinetic energy and longitud inal distance after phase rotation. which must be decreased for acceptance into a cooling channe l. This can be accomplished via radio-frequency (RF) “phase rotation,” b y which the energy of the low-energy muons is raised and that of the high-energy muons lowered. This brings a substantial fraction ( ≈60%) of the muons into a narrow energy range at the expense of increasing the bunch length (Fig. 4). The large trans- verse size of the beam at this point necessitates low-freque ncy (30–60MHz) RF cavities. An alternative under investigation is use of an induction linac. If the phase-rotation accelerating gradient is sufficiently high (≈4–5MV/m), a significant portion of the pions can be phase-rotated befor e they decay, al- lowing muon polarization as high as ≈50%. Otherwise the polarization is nat- urally≈20% [17]. Muon polarization can be exploited in a variety of p hysics studies [6] as well as in a neutrino factory [18]. It can also p rovide a ∼10−6 fill-to-fill relative calibration of the beam energy [19] (ne eded e.g. to measure the width of the Higgs). 3 Ionization Cooling The goal of beam cooling is to reduce the normalized six-dime nsional emit- tance of the beam. The significance of emittance for accelera tor design is that it places limits on how tightly the beam can be focused and det ermines the 5

arXiv:physics/0001038v1 [physics.class-ph] 18 Jan 2000Dirac–Maxwell Solitons C. Sean Bohun1)and F. I. Cooperstock2) 1)Department of Mathematics and Statistics, University of Vi ctoria, P.O. Box 3045, Victoria, B.C., Canada V8W 3P4 2)Department of Physics and Astronomy, University of Victori a, P.O. Box 3055, Victoria, B.C., Canada V8W 3P6 ABSTRACT Detailed analysis of the coupled Dirac-Maxwell equations a nd the structure of their solutions is presented. Numerical so lu- tions of the field equations in the case of spherical symmetry with negligible gravitational self-interaction reveal th e exis- tence of families of solitons with electric field dominance t hat are completely determined by the observed charge and mass of the underlying particles. A soliton is found which has the charge and mass of the electron as well as a charge radius of10−23m. This is well within the present experimentally determined upper limit of ≃10−18m. Properties of these particles as well as possible extension to the work herein ar e discussed. 1999 PACS numbers: 03.65.Ge, 11.10.Lm, 12.20.Ds 1 Introduction Through the years, a number of authors have attempted to avoi d the problems inherent in the point-particle model by focussing upon finite soliton-like structures. Fields interacting non- linearly provide the binding without invoking any phenomen ological elements. Einstein and Rosen[1] pointed out many years ago that particles should be contained within a field theory and not exist as independent entities. Rosen[2] made consid erable progress in implementing such a program in a gauge-invariant manner by minimally coup ling a scalar field to the Maxwell field. However, the soliton solutions yielded negat ive masses. Later[3], neutral quantized particle states of positive mass were found and a m ore complicated model invoking up to three scalar fields coupled to the Maxwell field was shown to be capable of modeling the known massive leptons[4]. However, the particles were s pinless and the view then was that a subsequent quantization of the theory would induce sp in. In 1991, one of the present authors[5] suggested an alternat ive route to elementary particle modelling, namely as solitons of Dirac–Maxwell theory. Sin ce Dirac–Maxwell theory had been so successful in describing electron spin and magnetic moment, predicting the existence of the positron and refining the energy levels in interacting systems such as hydrogen, itseemed reasonable that this might successfully extend to a s elf-interacting soliton structure to model the elementary particles themselves. Spin would al ready exist in such a model via the spinor structure of the wave function. Shortly thereaft er, such solitons were found and their properties studied[6]. A few years later, Lisi[7] ind ependently discovered some of the results in[6]. Recently, there has been a revival of interes t in this field and in particular, the issue of gravitational coupling in the Dirac–Maxwell syste m has been considered[8]. However, there was the misconception that gravitation was a necessar y ingredient for the creation of the soliton. In this paper, we develop the essential results in[5] and[6] and discuss the role of gravitation in soliton structure. The experimental inputs are the respe ctive masses of the electron, muon and tau, their charge and as a constraint, the upper limit to t heir size which is ≃10−16cm. The plan of the paper is as follows: in sec. 2, we set out the ess ential coupled Dirac–Maxwell equations to be solved. The structure of the Dirac wave funct ion in spherical coordinates is given and particularized to the case of electric field domina nce. The equation is separated in sec. 3 and we contrast the standard treatment in which a pot ential function is imposed such as in the case of hydrogen and the present case of the soli ton where the derivation of the potential is part of the problem. The formal structure of the potential in terms of the Green’s function is given. It is shown that there do exist sph erically symmetric potentials for appropriate choices of quantum numbers. In sec. 4, the spherically symmetric energy-momentum tenso r is derived. The relationship between the parameters in the Dirac equation and the physica lly measured quantities is discussed and the expression for the spatial spread of the so liton is given. The various constraints including singularity avoidance lead to the re quired boundary conditions for the problem. In sec. 5, the results are presented. New variables of conven ience for numerical integration are introduced. The parameters leading to twenty ground sta te solitons are listed. It is found that there is a critical range which leads to solitons within the experimentally observed upper limit to the size of the electron. Excited states are present ed and the mass ratios are found. In the final section 6, the essential achievements as well as t he limitations of the results are discussed. It is stressed that the solitons have been fou nd without the requirement of significant gravitational interaction and it is conjectu red that gravity will be significant for Dirac–Maxwell solitons when e/m≃1 in units for which G=c= 1. In cgs units, this is 2.58×10−4esugm−1. By contrast, the e/mratio for the electron is 2 .04×1021or 5.27×1017esugm−1in cgs units. 22 Derivation of the Equations The field equations are obtained from the Lagrangian of quant um electrodynamics[9] L=i¯hc¯ψγµ∂µψ−mc2¯ψψ−1 16πFµνFµν−e¯ψγµψAµ (1) whereψ= (ψ1,ψ2,ψ3,ψ4)Tis the Dirac spinor, ¯ψ=ψ†γ0= (ψ∗ 1,ψ∗ 2,−ψ∗ 3,−ψ∗ 4),Aµ= (ϕ,A) is the electromagnetic four-vector potential and Fµν=∂µAν−∂νAµis the Maxwell tensor. Theγµare 4×4 Hermitian anticommuting matrices of the unit square γ0=/parenleftigg I0 0−I/parenrightigg , γk=/parenleftigg 0σk −σk0/parenrightigg , k = 1,2,3 whereIis the unit 2 ×2 matrix and the σkare the Pauli matrices σ1=/parenleftigg 0 1 1 0/parenrightigg , σ2=/parenleftigg 0−i i0/parenrightigg , σ3=/parenleftigg 1 0 0−1/parenrightigg . Variation with respect to Aµand¯ψrespectively, yield the field equations Fµν ,ν=−4π¯ψγµψ (2) i¯hcγµ∂µψ−mc2ψ−eγµψAµ= 0. (3) Ifψis chosen to be an energy eigenstate with energy Eand one chooses a static charge distribution with a four-vector potential of the form Aµ=/parenleftig φ(r,θ,ϕ),Ak(r,θ,ϕ)/parenrightig , k = 1,2,3 then the equations (2)-(3) are reduced to /bracketleftig −i¯hcα· ∇+α4mc2−eα·A+eφ−E/bracketrightig ψ= 0 (4) ∇2φ=−4πeψ†ψ (5) ∇×(∇×A) = 4πeψ†αψ (6) whereαk=γ0γk. In spherical coordinates, ( x,y,z) = (rsinθcosϕ,rsinθsinϕ,rcosθ), the Dirac wave func- tion has the structure[10] ψ(r,θ,ϕ) [j=l+1/2]= /radicalig l−m 2l+1gYm l/radicalig l+m+1 2l+1gYm+1 l −i/radicalig l+m 2l−1fYm l−1 i/radicalig l−m−1 2l−1fYm+1 l−1 , ψ(r,θ,ϕ) [j=l−1/2]= /radicalig l+m+1 2l+1gYm l −/radicalig l−m 2l+1gYm+1 l −i/radicalig l−m+1 2l+3fYm l+1 −i/radicalig l+m+2 2l+3fYm+1 l+1 (7) 3wheref=f(r),g=g(r) and the {Ym l(θ,ϕ)}l,mis the set of orthonormal spherical harmonics defined for l= 0,1,...,m=−l,−l+ 1,...,l and Ym l(θ,ϕ) =/radicaligg 2l+ 1 4π(l−m)! (l+m)!Pm l(cosθ)eimϕ. (8) In addition, mis an integer such that −j≤m+ 1/2≤j; (m+ 1/2)¯his thez-component of the total angular momentum. Consider the spinor with j= 1/2,l= 0 andm= 0 which implies from the above representation (7) 4πψ†ψ=f(r)2+g(r)2 4πψ†αψ= 2f(r)g(r)sinθ(−sinϕ,cosϕ,0)T. Resolving equations (5) and (6) into spherical coordinates gives ∇2φ=−e/parenleftig f(r)2+g(r)2/parenrightig ∇×(∇×A)/vextendsingle/vextendsingle/vextendsingle ˆr= 0 ∇×(∇×A)/vextendsingle/vextendsingle/vextendsingleˆθ= 0 ∇×(∇×A)/vextendsingle/vextendsingle/vextendsingle ˆϕ= 2ef(r)g(r)sinθ. Therefore, a four-vector potential of the form Aµ= (φ(r),−Aϕ(r,θ)sinϕ,A ϕ(r,θ)cosϕ,0) should be chosen where the components satisfy d2φ dr2+2 rdφ dr=−e/parenleftig f(r)2+g(r)2/parenrightig (9) ∂2Aϕ ∂r2+2 r∂Aϕ ∂r+cotθ r2∂Aϕ ∂θ+1 r2∂2Aϕ ∂θ2−Aϕ r2sin2θ=−2ef(r)g(r)sinθ. (10) Since the right hand side of equation (10) is nonzero, the the ory can only be exact if Aϕis nonzero. However at this point we will impose the assumption of electric field dominance and hence the dominance of φoverAorf(r) dominance over g(r). For the validity of the approximation A=0, one radial component of the spinor must dominate over the other so that fg≪f2+g2. It will be demonstrated that such objects do exist within the non-linear field. With this ap- proximation the equations to solve reduce to a Dirac equatio n coupled to a Poisson equation: /bracketleftig −i¯hα· ∇+α4mc2+V(r)/bracketrightig ψ=Eψ (11) 4∇2V=−4πe2ψ†ψ. (12) With these facts in mind, we now turn to the separation of the s tationary Dirac equa- tion (11) with respect to a general central potential and the derivation of the form of ψ†ψ for a general set of quantum numbers. 3 Separation of the Equation The separation procedure follows that given in Bethe and Sal peter[10]. First one introduces quantum numbers landj;lis the orbital angular momentum quantum number as well as being an integer ≥0;jis the total angular momentum quantum number and can assume just the two values l+ 1/2 andl−1/2, (but only +1 /2 forl= 0). The forms assumed by the four components of ψare given explicitly in (7). The explicit form of the Dirac equation (11) for the four comp onents of the wave function is: ∂ψ3 ∂z+∂ψ4 ∂x−i∂ψ4 ∂y−i ¯hc/bracketleftig E−V(r)−mc2/bracketrightig ψ1= 0 (13) ∂ψ4 ∂z−∂ψ3 ∂x−i∂ψ3 ∂y+i ¯hc/bracketleftig E−V(r)−mc2/bracketrightig ψ2= 0 (14) ∂ψ1 ∂z+∂ψ2 ∂x−i∂ψ2 ∂y−i ¯hc/bracketleftig E−V(r) +mc2/bracketrightig ψ3= 0 (15) ∂ψ2 ∂z−∂ψ1 ∂x−i∂ψ1 ∂y+i ¯hc/bracketleftig E−V(r) +mc2/bracketrightig ψ4= 0. (16) Therefore, by inserting the assumed wave functions (7) into (13)-(16) and using identities similar to (A.6) we find that the following two coupled equati ons between fandghold: 1 ¯hc/bracketleftig E−V(r) +mc2/bracketrightig f(r)−/bracketleftbiggdg dr+1 +κ rg(r)/bracketrightbigg = 0 (17) 1 ¯hc/bracketleftig E−V(r)−mc2/bracketrightig g(r) +/bracketleftbiggdf dr+1−κ rf(r)/bracketrightbigg = 0 (18) where the new quantum number κis defined as κ=/braceleftigg −l−1 forj=l+ 1/2 (l= 0,1,...) l forj=l−1/2 (l= 1,2,...).(19) These equations are valid for all spherically symmetric pot entialsV(r) =V(r) and together they replace expression (11). At this point, the standard procedure is to specify an extern al spherically symmetric potential, an example of which is the electrostatic potenti al energy of the proton-electron 5interaction. That is, simply V(r) =−Ze2 r, which is the fundamental solution of Laplace’s equation[11 ] ∇2V= 4πZe2δ3(r) (20) whereδ3(r) is a three dimensional Dirac delta function centered at the origin. This is consistent with the far range1behaviour that we expect to find for the self-field of the fermi on since, when we compare (20) with (12) we see that the fermion i s treated as an object without structure through the equality, ψ†ψ=−δ3(r). There is one additional problem that must be explored, namel y how to couple relation (5) to (17)-(18). This will be achieved in three parts. First , we find the Green’s function for the equation (5). Second, we find an analytic form for the prob ability density ψ†ψusing (7). Once this equation is known, we can proceed to the third step w hich is to find V(r) by forming the convolution of the Green’s function of step one, with the probability density of step two. The potential V(r) satisfies the Poisson equation (12) and by assuming that the solution is sufficiently regular, this can be converted to an integral e quation[12] V(r) =−4πe2/integraldisplay G(r,r′)ψ†(r′)ψ(r′)dr′(21) whereG(r,r′) is the Green’s function of the Laplacian operator in three d imensions G(r,r′) =−1 4π1 |r−r′|=−∞/summationdisplay l=01 2l+ 1rl < rl+1 >l/summationdisplay m=−lYm l(θ,ϕ)Ym∗ l(θ′,ϕ′). (22) With the Green’s function determined, we can turn our attent ion to the probability den- sity. This is accomplished by using a pair of identities for t he associated Legendre functions2 (1−µ2)/parenleftig Pm+1 l/parenrightig2=/bracketleftbig(l−m)µPm l−(l+m)Pm l−1/bracketrightbig2, (23) (1−µ2)/parenleftig Pm+1 l−1/parenrightig2=/bracketleftbig(l+m)µPm l−1−(l−m)Pm l/bracketrightbig2(24) together with the definition of the spherical harmonics (8). The resulting expression for the charge density of the Dirac particle is given by ψ†ψ=f2+g2 2l+ 1/bracketleftbigg (l−m)/vextendsingle/vextendsingle/vextendsingleYm+1 l/vextendsingle/vextendsingle/vextendsingle2+ (l+m+ 1)|Ym l|2/bracketrightbigg (25) 1By far range, we mean those distances much larger than the Boh r radius r≫¯h2/me2. 2Both Eqs. (23)-(24) follow directly from Eqs. (8.5.1) and (8 .5.3) of Abramowitz & Stegun[13]. 6whenj=l+ 1/2 and ψ†ψ=f2+g2 2l+ 1/bracketleftbigg (l+m+ 1)/vextendsingle/vextendsingle/vextendsingleYm+1 l/vextendsingle/vextendsingle/vextendsingle2+ (l−m)|Ym l|2/bracketrightbigg (26) whenj=l−1/2. Therefore by using (21), (22) and (25), one obtains the expre ssion V(r) = −4πe2/integraldisplay G(r,r′)ψ†(r′)ψ(r′)dr′ =4πe2 2l′+ 1/integraldisplay∞/summationdisplay l=0rl < rl+1 >l/summationdisplay m=−l1 2l+ 1Ym l(θ,ϕ)Ym l(θ′,ϕ′)/bracketleftig f(r′)2+g(r′)2/bracketrightig ×/bracketleftbigg (l′−m′)/vextendsingle/vextendsingle/vextendsingleYm′+1 l′(θ′,ϕ′)/vextendsingle/vextendsingle/vextendsingle2+ (l′+m′+ 1)/vextendsingle/vextendsingle/vextendsingleYm′ l′(θ′,ϕ′)/vextendsingle/vextendsingle/vextendsingle2/bracketrightbigg r′2dr′d(cosθ′)dϕ′ for the case j=l+ 1/2. Similarly, with the use of (26), it can be shown that the pot ential V(r) takes the form V(r) =4πe2 2l′+ 1/integraldisplay∞/summationdisplay l=0rl < rl+1 >l/summationdisplay m=−l1 2l+ 1Ym l(θ,ϕ)Ym l(θ′,ϕ′)/bracketleftig f(r′)2+g(r′)2/bracketrightig ×/bracketleftbigg (l′+m′+ 1)/vextendsingle/vextendsingle/vextendsingleYm′+1 l′(θ′,ϕ′)/vextendsingle/vextendsingle/vextendsingle2+ (l′−m′)/vextendsingle/vextendsingle/vextendsingleYm′ l′(θ′,ϕ′)/vextendsingle/vextendsingle/vextendsingle2/bracketrightbigg r′2dr′d(cosθ′)dϕ′ for the case j=l−1/2. It is to be noted that the primed indices ( l′,m′) correspond to the angular momentum of the particle, while the unprimed indice s run over the complete set of permissible angular momentum quantum numbers. By performi ng the angular integration of the above formulae, one can immediately conclude that bot h of the above integrals vanish except when m= 0 andl= 0,2,...,2l′. This implies that V(r) = 4πe22(l′−j′) 2l′+ 1l′/summationdisplay n=0Y0 2n(θ,ϕ) 4n+ 1/integraldisplay∞ r′=0r2n < r2n+1 >/bracketleftig f(r′)2+g(r′)2/bracketrightig r′2dr′ ×/bracketleftig (κ′+m′+ 1)/an}b∇acketle{tl′,m′+ 1|Y0 2n|l′,m′+ 1/an}b∇acket∇i}ht+ (κ′−m′)/an}b∇acketle{tl′,m′|Y0 2n|l′,m′/an}b∇acket∇i}ht/bracketrightig (27) where the cases j=l±1/2 have been combined by the application of the definition of κ′. Expression (27) replaces the equation (12). When written in this form, it is clearly seen that the potential V(r) is not in general spherically symmetric. Table 1 lists the p otential (27) forl′= 0,1 and illustrates the fact that there exists spherically sym metric states with l′/ne}ationslash= 0. A localized solution of this model must satisfy the field equa tions (17) and (18) for fand gand a given energy Ewhere the potential is given by the expression (27). Moreove r, it is required that the total probability /an}b∇acketle{tψ|ψ/an}b∇acket∇i}ht=4/summationdisplay i=1/an}b∇acketle{tψi|ψi/an}b∇acket∇i}ht=/integraldisplay∞ 0/parenleftig f2+g2/parenrightig r2dr<∞. 7Since the equations which describe the spatial evolution of the wave function (17)-(18) were derived under the assumption that the potential, V(r), is spherically symmetric, they are not valid for an extended Dirac particle in an arbitrary s tate of angular momentum. We have shown that there do exist certain choices of landmwhere the probability density is spherically symmetric and it is these cases in which our prim ary interest lies. We can conclude that with the spinor representation given by (7), there are essentially three differential equations to be solved simultaneously. E quations (17)-(18) specify the spatial evolution of the wave function and equation (27) refl ects the spatial extent of the self-field of the particle. A strategy for solving these intr insically non-linear equations, as well as a few of their interesting properties, will be explor ed in the following sections. 4 Boundary Conditions From the previous section we have found that the equations to be satisfied for a self-interacting fermion are equations (17)-(18) and ∇2V=−4πe22(l−j) 2l+ 1/parenleftig f2+g2/parenrightig/bracketleftbigg (κ+m+ 1)/vextendsingle/vextendsingle/vextendsingleYm+1 l(θ,ϕ)/vextendsingle/vextendsingle/vextendsingle2+ (κ−m)|Ym l(θ,ϕ)|2/bracketrightbigg (28) where we have combined the j=l±1/2 cases by using the definition of κ. Since we have assumed that the potential Vin equations (17)-(18) is spherically symmetric, this nece ssarily restricts the choice of landm. Assume from this point on that landmare chosen to satisfy this criterion. Consequently, equation (28) becomes ∇2V=−e2/parenleftig f2+g2/parenrightig . (29) Since the soliton asires as a coupling between Dirac and Maxw ell fields, the energy Ethat appears in the Dirac equation is not the total energy of the pa rticle. The total energy can be obtained by calculating the T0 0component of the energy-momentum tensor. For our field, the Lagrangian is given by equation (1) where Aµis the vector potential of the electromagnetic field. One generates the symmetric energy-momentum tensor d irectly from the Lagrangian in the form[14] Tµν=∂L ∂gµν−gµν 2L. (30) Applying (30) to (1) yields Tµν=/bracketleftbiggi¯hc 2/parenleftbig¯ψγµ∂νψ+¯ψγν∂µψ/parenrightbig−1 4πFαµFβνgαβ−e 2/parenleftbig¯ψγµψAν+¯ψγνψAµ/parenrightbig/bracketrightbigg −gµν 2/bracketleftbigg i¯hc¯ψγαgαβ∂βψ−mc2¯ψψ−1 8πFαβFαβ−e¯ψγαψAβgαβ/bracketrightbigg . 8Further simplification gives T0 0=Eψ†ψ+1 8π/parenleftbiggdφ dr/parenrightbigg2 . This yields an expression for the total energy, Etot, of Etot=/integraldisplay T0 0dVol =E/integraldisplay∞ 0/parenleftig f2+g2/parenrightig r2dr+1 2/integraldisplay/parenleftbiggdφ dr/parenrightbigg2 r2dr (31) wheredVolis an infinitesimal volume element. This total energy should be associated with the observed mass of the particle as Etot=mc2. There is still sufficient freedom remaining to set lim r→∞V(r) = 0 because the spinor is invariant under the transformatio n V→V+β;E→E+β for any real-valued β. The massmand the charge ethat appear in the Dirac equation are not necessarily the experimentally measured quantities just as the charge that appears at a vertex of a Feynman graph is not the experimentally measured charge of the parti cle. Because of this, we will replace the min (17)-(18) by the symbol µ. In addition, the ein (29) will be replaced by anǫ. The symbols mandewill be reserved for the physically observed quantities. Wi th these substitutions, we convert to a set of variables whereb y equations (17), (18) and (31) are independent of any physical constants. The particular t ransformation chosen is f=ηF, g =ηG, r =¯hx µc, E=λµc2, V=µc2U whereη2=µ3c4/ǫ2¯h2. These redefined variables have the following dimensions in terms of length (L): [x] =L0; [λ] =L0; [F] =L−3/2; [G] =L−3/2; [U] =L0. This yields the transformed equations: [λ−U(x) + 1]F(x)−/bracketleftbiggdG dx+1 +κ xG(x)/bracketrightbigg = 0 (32) [λ−U(x)−1]G(x) +/bracketleftbiggdF dx+1−κ xF(x)/bracketrightbigg = 0 (33) ∇2U+/parenleftig F2+G2/parenrightig = 0 (34) where ∇2is now the Laplacian with respect to the xcoordinate. The mass of the soliton comes from the transformed version of the total energy expre ssion (31), mc2=¯hµc3 ǫ2/bracketleftigg λ/integraldisplay∞ 0/parenleftig F2+G2/parenrightig x2dx+1 2/integraldisplay∞ 0/parenleftbiggdU dx/parenrightbigg2 x2dx/bracketrightigg (35) 9and the total charge is given as the integral of the charge den sity e=ǫ/integraldisplay ρdVol=¯hc ǫ/integraldisplay∞ 0/parenleftig F2+G2/parenrightig x2dx. (36) We will show that if the charge ǫis replaced by e, that thefcomponent of the spinor greatly dominates the gcomponent. By substituting ǫ=eand choosing a value for m, the value ofµcan be determined numerically once the spatial extent of the soliton is known. In this case, the expectation value of the radius of the particl e becomes /an}b∇acketle{tr/an}b∇acket∇i}ht=¯hc µc2/an}b∇acketle{tx/an}b∇acket∇i}ht=e2 µc2¯hc e2/integraldisplay ∇2Ux3dx /integraldisplay ∇2Ux2dx=e2 µc2/integraldisplay ∇2Ux3dx /bracketleftbigg/integraldisplay ∇2Ux2dx/bracketrightbigg2. To stay within the current experimental bounds of the mean ch arge radius, this value must be less than rexpwhich is ≤10−18min the case of an electron. Hence, /integraldisplay∞ 0∇2Ux3dx≤rexp reµ me/bracketleftbigg/integraldisplay∞ 0∇2Ux2dx/bracketrightbigg2 wherereis the classical electron radius re=e2/mec2. Since we know that Uhas zero slope at x= 0 and that it must behave as N/xfor large argument ( Nis the amount of enclosed charge), we assume as a first approxi mation, that Ucan be represented as the electrostatic potential produced by a sphere of radius R0with uniform charge density. Therefore, U(r) =  N R0/bracketleftigg 3 2−1 2r2 R2 0/bracketrightigg forr<R 0 N rforr≥R0.(37) With this representation, one finds that /an}b∇acketle{tr/an}b∇acket∇i}ht=9 4remeR0 µN which means that since 0 </an}b∇acketle{tr/an}b∇acket∇i}ht<rexp, we can conclude that 0<R0 µN<4 9rexp reme≃3.088×10−4c2/MeV in the case of the electron. LetR0be defined as the effective range of the non-Coulombic behavio ur of the potential energy so that for x > R 0,U(x)∼N/x. SinceUis a solution to a Poisson equation with 10a negative definite charge density, U(0) must be larger than U(R0). This can be quickly verified by considering the opposite. If U(0)<U(R0) then there exists some r∈(0,R0) such thatU′(r) = 0. Therefore integrating (34) from 0 to r, one obtains r2U′(r) = 0 = −/integraldisplayr 0/parenleftig F2+G2/parenrightig x2dx which is clearly a contradiction. To determine the initial values of FandG, one simply eliminates either ForGfrom (32- 33), sayF, which leads to a second order equation for the other, namely , G′′+PG′+QG= 0 where both PandQare functions of U,U′,κ,λandx. To avoid a singularity in the potentialU(x), it must be both bounded and have zero slope in a neighbourho od of the origin. Moreover, both FandGare bounded in this same neighbourhood. From this, it is easy to verify that F(0) =  0 κ=−1 arbitrary κ= +1 0 ∀otherκ,(38) G(0) =  arbitrary κ=−1 0 κ= +1 0 ∀otherκ.(39) Furthermore, by examining the indicial equation, it can be s hown that no fractional powers exist in a power series solution of either ForGabout the origin x= 0. Summarizing the boundary conditions: U(x)<U(0)<∞, x ∈[0,∞), U′(0) = 0, together with the conditions (38), (39). For the case κ=−1, the initial values of U,Gand the energy λare determined by the requirement that the wave function ψ, and hence both FandG, vanish exponentially as x→ ∞. 5 Results In the search for numerical solutions it was specified that κ=−1 andλ= 1 giving the set of differential equations dG dx= [2 −U(x)]F(x) dF dx=−2 xF(x) +U(x)G(x) ∇2U=−F2−G2. 11To find a soliton, the values of F(0),G(0) are specified and a search is made for the value ofU(0) whereby lim x→∞xF(x) = 0 and lim x→∞xG(x) = 0. Only values of G(0)>0 are considered because the equations are symmetric under th e transformation G→ −G, F→ −F,U→U. In a neighbourhood of a ground state soliton, the radial pro bability density is numerically seen to have a single well-defined min imum value for x >0. The choice ofκ=−1 gives the initial condition F(0) = 0. The choice of λ= 1 is simply a numerical convenience. Outside the neighbour hood of a soliton it is expected that the potential will behave as U(x)∼A+B/xfor largex. The value ofλshould have been chosen so that the asymptotic behaviour of t he potential U(x) is purely Coulombic in nature. By defining a shifted potentia l˜U(x) =U(x)−limx→∞U(x), this value of λmust satisfy 1 −U(x) =λ−˜U(x). Therefore after a soliton is found the value ofλis given as λ= 1−limx→∞U(x). In addition, the starting value of ˜U(x) is given by ˜U(0) =U(0) +λ−1. Using the redefined value of λthe observed charge and mass of the particle are compared to the values used in the Lagrangian by using the expressions (36) and (35) respectively. By defining P=/integraldisplay∞ 0/parenleftig F2+G2/parenrightig x2dx, X=/integraldisplay∞ 0/parenleftig F2+G2/parenrightig x3dx, E=λP+1 2/integraldisplay∞ 0/parenleftbiggdU dx/parenrightbigg2 x2dx, the charge ratio ǫ/eis given as ǫ e=¯hc e2P=P α whereαis the fine structure constant. The mass ratio µ/m=P2/αEand the expectation value for the radius of the soliton is /an}b∇acketle{tr/an}b∇acket∇i}ht=/integraltext(f2+g2)r3dr/integraltext(f2+g2)r2dr=¯h µc/integraltext(F2+G2)x3dx/integraltext(F2+G2)x2dx=re/parenleftbiggme m/parenrightbiggEX P3. Both of the quantities PandXare positive. However, depending upon the value of λ,E could be positive, negative or even zero if the electromagne tic and “bare mass” terms in the energy exactly cancel. A negative value for Ewill give an unphysical negative value for the observed radius /an}b∇acketle{tr/an}b∇acket∇i}ht. Because of this ambiguity, both the value of /an}b∇acketle{tr/an}b∇acket∇i}htand the particle width ∆r=/radicalbig /an}b∇acketle{tr2/an}b∇acket∇i}ht − /an}b∇acketle{tr/an}b∇acket∇i}ht2are presented. Tables 2 and 3 respectively list the numerica l parameters and the observed properties of a number of ground state parti cles found where mwas taken to be the observed mass of the electron me. Figure 1 illustrates the radial behaviour of FandGfor the case ǫ=e(i= 1). It is to be noted that for x >0,Fis much larger than Gand as a consequence, FG≪F2+G2. In fact,Gis so small that it resembles a straight line along the xaxis. This supports the argument that the four-vector potential can be reasonably a pproximated with only a radial A0component. 12The characteristics of a typical soliton with ǫ/ne}ationslash=eis illustrated with the choice ǫ/e= 454.8 (i= 19). In this case the potential plays a much more dominant ro le in holding the particle together than in the case ǫ=e. However, since in this case the approximation of FG≪F2+G2is violated, one would have to solve the full model (equation s (9)-(10)) to properly analyse this situation. This would be a far more com plicated problem. Figure 2 illustrates the radial components of this spinor and it show s that the magnitude of Gis now comparable to the magnitude of F. Table 3 also shows that the choice of ǫ= 389.0e, µ= 2.360×1012me(i= 15) yields a soliton with an expectation value for the radiu s of 5.05×10−23m. This size is well within the present experimentally determ ined upper limit for the electron radius of ≃10−18m. These equations also exhibit excited states. The nthexcited state of our field is charac- terized through the functions Fn(x),Gn(x) andUn(x) for which the Gncomponent crosses the abscissa n+ 1 times while the Fncomponent crosses it ntimes. Once the ground state solution is found, the value of µcan be determined through equation (36). The corresponding nthexcited state is that excited state with the same observed ch arge ratio,ǫ/e, as the ground state. Therefore, in this interpretation of the theory, the ratio of the mass of the nthexcited state to the ground state is given by the expression mn m0=µ/m 0 µ/m n=λn/integraldisplay∞ 0/parenleftig F2 n+G2 n/parenrightig x2dx+1 2/integraldisplay∞ 0/parenleftbiggdUn dx/parenrightbigg2 x2dx λ/integraldisplay∞ 0/parenleftig F2+G2/parenrightig x2dx+1 2/integraldisplay∞ 0/parenleftbiggdU dx/parenrightbigg2 x2dx. Figure 3 shows radial probability density of the first three s tates for the case G(0) = 1. Each of these solitons has a different value of ǫ/e. Figure 4 illustrates the behaviour of the mass ratio, µ/m, as a function of the charge ratio ǫ/efor the ground state and the first two excited states. For each class of particles there is a charge ratio where the electromagnetic and bare mass compo nents of the energy balance making E= 0. At this value of ǫ/e, the mass ratio µ/m→ ∞. At charge ratios less than this critical value the mass ratio is negative whereas charg e ratios above this critical value result in a positive value of µ/m. There is numerical evidence that each class of particles has an upper bound for the charge ratio. Above this maximum ch arge ratio we were unable to find any solutions such that lim x→∞xF(x)→0 or lim x→∞xG(x)→0. This necessarily restricts the definition of the mass ratio defined above. Figu re 4 also illustrates the fact that at moderate charge ratios, the electromagnetic field does no t contain an appreciable amount of the particle energy resulting in the behaviour |µ/m| ≃ǫ/e. The mass ratios of the first and second excited states with res pect to the ground state solutions are shown in figure 5. This ratio is only defined up to a maximum value of ǫ/esince beyondǫ/e≃550, a ground state fails to exist. For excited states, this m aximum admissible charge ratio increases. This implies that for a fixed value of ǫ/ethere may not exist a ground state solution, but there will be arbitrarily many excited s tates. As is readily apparent from 13figure 5, the only appreciable mass splitting occurs for larg e charge ratios. However, it is precisely for large charge ratios where our approximation t hatFG≪F2+G2breaks down. 6 Concluding Remarks We have seen that spherically symmetric Dirac-Maxwell soli tons can be constructed and with a charge and mass to model the electron successfully. Howeve r, it should be noted that the higher energy excited states of this form did not yield the la rge mass separations of the muon and tau relative to the electron in this model. The search thu s far has been restricted to spherical solitons. It is conceivable that a relaxation of t his restriction or some other change in conditions would increase the mass splitting. In any even t, we have shown that Dirac- Maxwell solitons exist and are capable of modelling an elect ron where the charge-to-mass ratio is the observed ≃1021in units in which G=c= 1. Furthermore, we have found a charge-to-mass ratio that simultaneously yields the obser ved charge and mass of the electron as well as exhibiting a degreee of compactification that is we ll within the current experimental upper limit. Finster et al.[8] have considered Einstein–Di rac–Maxwell (EDM) solitons and concluded that it is the interaction with gravitation which is responsible for the existence of bound states. However, we see here that bound states exist with negligible gravitational interaction. While the e/mratio at which significant gravitational coupling sets in is yet to be determined for EDM solitons, it is our conjecture that thi s will be so at the same level that was found earlier in the case of minimally coupled scalar int eraction[4], namely for e/m≃1. The known fundamental charged particles of nature, on the ot her hand have enormous e/m ratios. REFERENCES [1] Einstein, A. & Rosen, N. (1935). Physical Review, 48, 73-77. [2] Rosen, N. (1939). Physical Review, 55, 94-101. [3] Rosen, N. & Rosenstock, H. B. (1952). Physical Review, 85(2), 257-259. [4] Cooperstock, F. I. & Rosen, N. (1989). International Jou rnal of Theoretical Physics, 28(4), 423-440. [5] Cooperstock, F. I. (1991). The Electron: New Theory and Experiment , Eds. D. Hestenes and A. Weingartshofer, Kluwer Academic. [6] Bohun, C. S. (1991). A Self-Consistent Dirac–Maxwell Field of Solitons , MSc. Thesis, University of Victoria. 14[7] Lisi, A. G. (1995). Journal of Physics A: Mathematical an d General, 28, 5385-5392. [8] Finster, F., Smoller, J. and Yau, S-T., preprint gr-qc/9 801079 [9] Griffiths, D. J. (1987). Introduction to Elementary Particles . New York: Harper and Row. [10] Bethe, H. A. & Salpeter, E. E. (1957). Quantum Mechanics of One- and Two- Electron Atoms . Berlin: Springer-Verlag. [11] Evans, L. C. (1998). Partial Differential Equations: Graduate Studies in Mathem atics, vol 19 . American Mathematical Society, Providence, Rhode Island . p. 22. [12] Brezzi, F. & Markowich, P. A. (1991). Mathematical Meth ods in the Applied Sciences, 14, 35-61. [13] Abramowitz, M & Stegun, I. A. (1964). Handbook of mathematical functions, with for- mulas, graphs, and mathematical tables . National Bureau of Standards, United States Department of Commerce. [14] Landau, L. D. & Lifshitz E. M. (1971). The classical theory of fields (4th ed.). New York: Pergamon Press. A Derivatives of f(r)Ym l(θ, ϕ) In the Dirac wave equation, all of the derivatives are with re spect to Cartesian coordinates. We can change to a spherical polar representation via the tra nsformation x=rsinθcosϕ y=rsinθsinϕ z=rcosθ. By applying the chain rule, it is trivial to show that this cha nges the first order partial derivatives via ∂ ∂x= sinθcosϕ∂ ∂r+cosθcosϕ r∂ ∂θ−sinϕ rsinθ∂ ∂ϕ(A.1) ∂ ∂y= sinθsinϕ∂ ∂r+cosθsinϕ r∂ ∂θ+cosϕ rsinθ∂ ∂ϕ(A.2) ∂ ∂z= cosθ∂ ∂r−sinθ r∂ ∂θ. (A.3) If the functions ψj(j= 1,...,4) from expression (7) are substituted into (13)-(16), and i f one uses the formulas given in Bethe and Salpeter[10] for the derivatives of a function of the 15formf(r)Ym l(θ,ϕ) with respect to x,y, andz, one finds a coupled pair of first order ordinary equations for f(r) andg(r). For example, in order to calculate ∂ ∂z[f(r)Ym l(θ,ϕ)], we first require the identities cosθPm l(cosθ) =1 2l+ 1/bracketleftbig(l−m+ 1)Pm l+1(cosθ) + (l+m)Pm l−1(cosθ)/bracketrightbig(A.4) sinθd dθPm l(cosθ) =1 2l+ 1/bracketleftbigl(l−m+ 1)Pm l+1(cosθ)−(l+ 1)(l+m)Pm l−1(cosθ)/bracketrightbig,(A.5) which can both be verified through the use of Rodrigues’ formu la Pm l(µ) =(−1)m 2ll!(1−µ2)m/2dl+m dµl+m(µ2−1)l. WritingYm l(θ,ϕ) as a function of Pm lby using (8) gives the relationship ∂ ∂z[f(r)Ym l(θ,ϕ)] =/radicaligg 2l+ 1 4π(l−m)! (l+m)!eimϕ/bracketleftbigg cosθPm l(cosθ)df dr−sinθd dθPm l(cosθ)f r/bracketrightbigg . By substituting (A.4-A.5) in the above, collecting terms, a nd applying the definition of Ym l(θ,ϕ) once again, one obtains the simplification ∂ ∂z[f(r)Ym l(θ,ϕ)] =/radicaligg (l−m+ 1)(l+m+ 1) (2l+ 1)(2l+ 3)Ym l+1(θ,ϕ)/bracketleftbiggdf r−l rf/bracketrightbigg +/radicaligg (l−m)(l+m) (2l−1)(2l+ 1)Ym l−1(θ,ϕ)/bracketleftbiggdf dr+l+ 1 rf/bracketrightbigg . (A.6) Similar relationships for∂ ∂x±i∂ ∂ycan be found in Bethe and Salpeter3, but there is a very elegant way to derive these operators by applying the Wigner –Eckart theorem. First, we evaluate the matrix element /an}b∇acketle{tl0|∇0|l0/an}b∇acket∇i}htof the gradient operator, which is an example of a vector operator. Specifically, ∇0=∂ ∂z,∇±=∓1√ 2/parenleftbigg∂ ∂x±i∂ ∂y/parenrightbigg . Since ∇0f(r)Y0 l=l+ 1/radicalbig(2l+ 1)(2l+ 3)Y0 l+1/bracketleftbiggdf dr−l rf/bracketrightbigg +l/radicalbig(2l−1)(2l+ 1)Y0 l−1/bracketleftbiggdf dr+l+ 1 rf/bracketrightbigg 3See formula (A.38) and (A.39) respectively in Bethe and Salp eter. 16for the special case of (A.6) where m= 0, we have /an}b∇acketle{tl′0|∇0|l0/an}b∇acket∇i}ht=l+ 1/radicalbig (2l+ 1)(2l+ 3)/bracketleftbiggdf dr−l rf/bracketrightbigg δl′ l+1+l/radicalbig (2l−1)(2l+ 1)/bracketleftbiggdf dr+l+ 1 rf/bracketrightbigg δl′ l−1. Now, we are at a point where we can use the Wigner–Eckart theor em. By inspection, the general matrix element is given by /an}b∇acketle{tl′m′|∇µ|l m/an}b∇acket∇i}ht= (−1)l′−m′/parenleftigg l′1l −m′µ m/parenrightigg /an}b∇acketle{tl′||∇||l/an}b∇acket∇i}ht = (−1)m′/parenleftigg l′1l −m′µ m/parenrightigg /parenleftigg l′1l 0 0 0/parenrightigg/an}b∇acketle{tl′0|∇0|l0/an}b∇acket∇i}ht. After evaluating the 3 −jsymbols, one can quickly verify the following equations. ∂ ∂z[(f(r)Ym l(θ,ϕ)] =/radicaligg (l−m+ 1)(l+m+ 1) (2l+ 1)(2l+ 3)Ym l+1(θ,ϕ)/bracketleftbiggdf dr−l rf/bracketrightbigg +/radicaligg (l−m)(l+m) (2l−1)(2l+ 1)Ym l−1(θ,ϕ)/bracketleftbiggdf dr+l+ 1 rf/bracketrightbigg (A.7) /bracketleftbigg∂ ∂x+i∂ ∂y/bracketrightbigg [f(r)Ym l(θ,ϕ)] =/radicaligg (l+m+ 1)(l+m+ 2) (2l+ 1)(2l+ 3)Ym+1 l+1(θ,ϕ)/bracketleftbiggdf dr−l rf/bracketrightbigg −/radicaligg (l−m−1)(l−m) (2l−1)(2l+ 1)Ym+1 l−1(θ,ϕ)/bracketleftbiggdf dr+l+ 1 rf/bracketrightbigg (A.8) /bracketleftbigg∂ ∂x−i∂ ∂y/bracketrightbigg [f(r)Ym l(θ,ϕ)] = −/radicaligg (l−m+ 1)(l−m+ 2) (2l+ 1)(2l+ 3)Ym−1 l+1(θ,ϕ)/bracketleftbiggdf dr−l rf/bracketrightbigg +/radicaligg (l+m−1)(l+m) (2l−1)(2l+ 1)Ym−1 l−1(θ,ϕ)/bracketleftbiggdf dr+l+ 1 rf/bracketrightbigg .(A.9) Linear combinations of (A.8) and (A.9) yield the derivative s with respect to xandy. 17States |l′,m′,j′/an}b∇acket∇i}ht Corresponding Potential V(r) |0,0,1/2/an}b∇acket∇i}ht,|0,−1,1/2/an}b∇acket∇i}ht e2I0 |1,0,1/2/an}b∇acket∇i}ht,|1,−1,1/2/an}b∇acket∇i}ht e2I0 |1,1,3/2/an}b∇acket∇i}ht,|0,−2,3/2/an}b∇acket∇i}hte2/bracketleftig I0−3 2/parenleftbig3cos2θ′−1/parenrightbigI2/bracketrightig |1,0,3/2/an}b∇acket∇i}ht,|0,−1,3/2/an}b∇acket∇i}hte2/bracketleftig I0+3 2/parenleftbig3cos2θ′−1/parenrightbigI2/bracketrightig Table 1: The Dirac–Maxwell particle self-field potential. The self-field potential energy for a Dirac–Maxwell particl e in the states l= 0,1where Il(r) =/integraldisplayr 0/bracketleftbig f(r′)2+g(r′)2/bracketrightbigr′l+2 rl+1dr′+/integraldisplay∞ r/bracketleftbig f(r′)2+g(r′)2/bracketrightbigrl r′l−2dr′. 18iG(0) U(0) λxmax 13.864×10−94.8879852 ×10−6−0.99999712 8342.9 21.0×10−89.2122047 ×10−6−0.99999457 6103.6 31.0×10−59.2122842 ×10−4−0.99945701 688.37 41.0×10−44.2760987 ×10−3−0.99748078 330.61 51.0×10−31.9850763 ×10−2−0.98833078 158.92 65.0×10−35.8063758 ×10−2−0.96604870 81.570 71.0×10−29.2196988 ×10−2−0.94634242 77.061 85.0×10−22.6992683 ×10−1−0.84654287 50.139 91.0×10−14.2884212 ×10−1−0.76093227 38.686 10 2.0×10−16.8134130 ×10−1−0.63095887 33.509 11 3.0×10−18.9315883 ×10−1−0.52683528 23.495 12 4.0×10−11.0821393 ×100−0.43716613 26.964 13 4.1×10−11.1001061 ×100−0.42878460 27.544 14 4.2×10−11.1179261 ×100−0.42049493 26.049 15G∗(0) 1.1309487 ×100−0.41445154 24.146 16 4.3×10−11.1356039 ×100−0.41229414 24.382 17 5.0×10−11.2557044 ×100−0.35715757 24.156 18 6.0×10−11.4178526 ×100−0.28421512 22.584 19 1.0×1001.9913670 ×100−0.03776277 20.438 20 2.0×1003.1519761 ×100+0.42244841 18.125 Table 2: Numerical parameters for a set of various ground sta te particles. For each particle, the value of G(0)is selected and one searches for the value of U(0) +λthat gives a bounded solution. The physical parameters are computed fr om the solution defined on x∈[0,xmax]. G∗(0) = 0.4273589430. 19iǫ/e µ/m /an}b∇acketle{tr/an}b∇acket∇i}ht ∆r 11.000 −1.000×100−6.61×10−82.31×10−8 21.371 −1.371×100−3.51×10−81.22×10−8 313.74 −1.374×101−3.51×10−101.22×10−10 429.55 −2.955×101−7.53×10−112.63×10−11 563.46 −6.462×101−1.60×10−115.61×10−12 6107.6 −1.135×102−5.30×10−121.87×10−12 7134.7 −1.479×102−3.21×10−121.14×10−12 8222.1 −2.991×102−9.07×10−133.31×10−13 9271.3 −4.504×102−4.64×10−131.73×10−13 10326.2 −8.687×102−1.87×10−137.23×10−14 11359.7 −1.838×103−7.51×10−142.99×10−14 12383.6 −9.658×103−1.27×10−145.16×10−15 13385.6 −1.538×104−7.89×10−153.21×10−15 14387.6 −3.666×104−3.28×10−151.34×10−15 15389.0 +2.360×1012+5.05×10−232.06×10−23 16389.5 +1.032×105+1.15×10−154.71×10−16 17401.8 +3.998×103+2.79×10−141.16×10−14 18416.4 +1.818×103+5.68×10−142.39×10−14 19454.8 +6.800×102+1.21×10−135.35×10−14 20498.7 +3.305×102+1.78×10−138.59×10−14 Table 3: Corresponding observable quantities for a set of va rious ground state particles. The values /an}b∇acketle{tr/an}b∇acket∇i}htand∆rare measured in meters and are computed from a soliton defined onx∈[0,xmax]. For this calculation it is assumed that m=me. 20x G /( x /) F /( x /)F /; G /(arb/. units/) 00.20.40.60.81 1000 2000 3000 4000 5000 6000 Figure 1:FandGcomponents of the wave function for the case ǫ/e= 1. Shown here is the radial dependence of the FandGcomponents of the soliton. Note that Gis much smaller than F.Gis barely discernible above the xaxis. 21x G /( x /)F /( x /)F /; G /(arb/. units/) 00.20.40.60.81 2 4 6 8 10 12 14 Figure 2:FandGcomponents of the wave function for the case ǫ/e= 454.8. Shown here is the radial dependence of the FandGcomponents of the soliton. The magnitudes of F andGare now comparable in contrast to the case when ǫ=e. 22n /= /0 n /= /1 n /= /2r /(fm/)x/2 j / n/( x /) j/2 /(arb/. units/) 00.20.40.60.81 1 2 3 4 5 Figure 3: Excited states of the theory. This figure shows the radial probability density of the first t hree particle state in the case G(0) = 1 . These particles have different charge ratios ǫ/e. There exist excited states beyond the ones illustrated. 23log/1/0 /BnZr/ne /
log /1/0/ // n m/ / n /=/0 n /=/1 n /=/2 0123456 0.5 1 1.5 2 2.5 3 3.5 Figure 4: Dependence of the mass ratio as a function of the cha rge ratio. Shown is the dependence of the mass ratio µ/mas a function of the charge ratio ǫ/efor the ground state and first two excited state solitons. For each class of p articles, there is a maximum charge ratio beyond which no solitons were found. 24/nemnm/0 n /= /2n /= /1 -10-8-6-4-20 100 200 300 400 500 Figure 5: Mass ratios of the first and second excited states wi th respect to the ground state. There are essentially two regions of interest. For moderate charge ratios, the value of mn/m0≃1 with the mass ratio of the excited state n= 2slightly larger than for the n= 1state. Beyond the point where the ground state mass ratio becomes unbounded, the mas s ratios begin to split. In this region, the|m1/m0|ratio exceeds the |m2/m0|ratio. In this region the approximation FG≪F2+G2is no longer valid. 25

arXiv:physics/0001039v1 [physics.gen-ph] 19 Jan 2000Localized Superluminal Solutions to Maxwell Equations propagating along a normal-sized waveguide(†) Michel Zamboni Rached Dep.to de F’isica, Universidade Estadual de Campinas, SP, B razil. and Erasmo Recami Facolt` a di Ingegneria, Universit` a Statale di Bergamo, Da lmine (BG), Italy; INFN—Sezione di Milano, Milan, Italy; and DMO–FEEC and CCS, State University of Campinas, Campinas, S .P., Brazil. Abstract – We show that localized (non-evanescent) solutions to Maxwe ll equations exist, which propagate without distortion along normal wav eguides with Superluminal speed. PACS nos.: 03.50.De ; 41.20.Jb ; 03.30.+p ; 03.40.Kf ; 14.80. -j . Keywords: Wave-guides; Localized solutions to Maxwell equ ations; Superluminal waves; Bessel beams; Limited-dispersion beams; Electromagnetic wavelets; X-shaped waves; Evanescent waves; Electromagnetism; Microwaves; Optics; Classical physics; General physics; Special relativity (†)Work partially supported by CAPES (Brazil), and by INFN, MUR ST and CNR (Italy). 11. – Introduction: Localized solutions to the wave equation s Since 1915 Bateman[1] showed that Maxwell equations admit ( besides of the ordinary planewave solutions, endowed in vacuum with speed c) of wavelet-type solutions, en- dowed in vacuum with group-velocities 0 ≤v≤c. But Bateman’s work went practically unnoticed. Only few authors, as Barut et al.[2] followed suc h a research line; inciden- tally, Barut et al. constructed even a wavelet-type solutio n travelling with Superluminal group-velocity[3] v >c. In recent times, however, many authors discussed the fact th at all (homogeneous) wave equations admit solutions with 0 < v < ∞: see, e.g., Donnelly & Ziolkowski[4], Esposito[4], Vaz & Rodrigues[4]. Most of those authors confi ned themselves to investi- gate (sub- or Super-luminal) localized non-dispersive solutions in vacuum: namely, those solutions that were called “undistorted progressive waves ” by Courant & Hilbert. Among localized solutions, the most interesting appeared to be th e so-called “X-shaped” waves, which —predicted even by Special Relativity in its extended version[5]— had been mathe- matically constructed by Lu & Greenleaf[6] for acoustic wav es, and by Ziolkowski et al.[7], and later Recami[8], for electromagnetism. Let us recall that such “X-shaped” localized solutions are S uperluminal (i.e., travel withv>c in the vacuum) in the electromagnetic case; and are “super-s onic” (i.e., travel with a speed larger than the sound-speed in the medium) in the acoustic case. The first authors to produce X-shaped waves experimentally were Lu & Greenleaf[9] for acoustics, and Saari et al.[10] for optics. Notwithstanding all that work, still it is not yet well under stood what solutions (let us now confine ourselves to Maxwell equations and to electrom agnetic waves) have to enter into the play in many experiments. 2. – About evanescent waves Most of the experimental results, actually, did not refer to the abovementioned local- ized, sub- or Super-luminal, solutions, which in vacuum are expected to propagate rigidly (or almost rigidly, when suitably truncated). The experime nts most after fashion are, on 2the contrary, those measuring the group-velocity of evanescent waves [cf., e.g., refs.11,12]. In fact, both Quantum Mechanics[13] and Special Relativity [5] had predicted tunnelling wavepackets (tunnelling photons too) and/or evanescent wa ves to be Superluminal. For instance, experiments[12] with evanescent waves trave lling down an under- sizedwaveguide revealed that evanescent modes are endowed with Superlumin al group- velocities[14]. A problem arises in connection with the experiment[15] with two “barriers” 1 and 2 (i.e., segments of undersized waveguide). In fact, it has been found that for suitable frequency bands the wave coming out from barrier 1 goes on with practically in finite speed, crossing the intermediate normal-sized waveguide 3 in zero time. Even if this can be theoretically understood by looking at the relevant t ransfer function (see the computer simulations, based on Maxwell equations only, in r efs.[16,17]), it is natural to wonder what are the solutions of Maxwell equations that can travel with Supe rluminal speed in a normal waveguide (where one normally meets ordinary propagating — and not evanescent— modes)... Namely, the dispersion relation in undersized guides is ω2−k2=−Ω2, so that the standard formula v≃dω/dkyields av >c group-velocity[17,18]. However, in normal guides the dispersion relation becomes ω2−k2= +Ω2, so that the same formula yields valuesv <c only. We are going to show that actually localized solutions to Max well equations propa- gating with v >c do exist even in normal waveguides; but their group-velocit yvcannot be given#1by the approximate formula v≃dω/dk. One of the main motivations of the present note is just contributing to the clarification of this question. 3. – About some localized solutions to Maxwell equations. Let us start by considering localized solutions to Maxwell e quations in vacuum. A theorem by Lu et al.[19] showed how to start from a solution holding in theplane (x,y) for constructing a threedimensional solution rigidly movi ng along the z-axis with Super- #1Let us recall that the group-velocity is well defined only whe n the pulse has a clear bump in space; but it can be calculated by the approximate, elementary rela tion v≃dω/dkonlywhen some extra conditions are satisfied (namely, when ωas a function of kis also clearly bumped). 3luminal velocity v. Namely, let us assume that ψ(ρ;t), with ρ≡(x,y), is a solution of the 2-dimensional homogeneous wave equation: /parenleftBig ∂2 x+∂2 y−1 c2∂2 t/parenrightBig ψ(ρ;t) = 0. (1) By applying the transformation ρ→ρsinθ;t→t−(cosθ/c)z, the angle θbeing fixed, with 0 <θ<π/ 2, one gets[19] that ψ(ρsinθ;t−(cosθ/c)z) is a solution to the threedimensional homogeneous wave-equation /parenleftBig ∇2−1 c2∂2 t/parenrightBig ψ/parenleftbigg ρsinθ;t−cosθ cz/parenrightbigg = 0. (2) The mentioned theorem holds for the vacuum case, and in gener al is not valid when introducing boundary conditions. However we discovered th at, in the case of a bidimen- sional solution ψvalid on a circular domain of the ( x,y) plane, such that ψ= 0 for |ρ|= 0, the transformation above leads us to a (three-dimensional) localized solution rigidly trav- elling with Superluminal speed v=c/cosθinside a cylindrical waveguide ; even if the waveguide radius rwill be no longer a, butr=a/sinθ > a . We can therefore obtain an undistorted Superluminal solution propagating down cyl indrical (metallic) waveguides for each (2-dimensional) solution valid on a circular domai n. Let us recall that, as well- known, any solution to the scalar wave equation corresponds to solutions of the (vectorial) Maxwell equations (cf., e.g., ref.[8] and refs. therein). For simplicity, let us put the origin O at the center of the cir cular domain C, and choose a 2-dimensional solution that be axially symmetric ψ(ρ;t), withρ=|ρ|, and with the initial conditions ψ(ρ;t= 0) =φ(ρ), and∂ψ/∂t =ξ(ρ) att= 0. Notice that, because of the transformations ρ=⇒ρsinθ (3a) t=⇒t−cosθ cz , (3b) the more the initial ψ(ρ;t) is localized at t= 0, the more the (threedimensional) wave ψ(ρsinθ;t−(cosθ/c)zwill be localized around z=vt. It should be also emphasized 4that, because of transformation (3b), the velocity cgoes into the velocity v=c/cosθ>c . Let us start with the formal choice φ(ρ) =δ(ρ) ρ;ξ(ρ)≡0. (4) In cylindrical coordinates the wave equation (1) becomes /parenleftBigg1 ρ∂ρρ∂ρ−1 c2∂2 t/parenrightBigg ψ(ρ;t) = 0, (1’) which exhibits the assumed axial symmetry. Looking for fact orized solutions of the type ψ(ρ;t) =R(ρ)·T(t), one gets the equations ∂2 tT=−ω2Tand (ρ−1∂ρ+∂2 ρ+ω2/c2)R= 0, where the “separation constant” ωis a real parameter, which yield the solutions T=Acosωt+Bsinωt (5) R=C J0(ω cρ), where quantities A,B,C are real constants, and J0is the ordinary zero-order Bessel function (we disregarded the analogous solution Y0(ωρ/c) since it diverges for ρ= 0). Finally, by imposing the boundary condition ψ= 0 atρ=a, one arrives at the base solutions ψ(ρ;t) =J0(kn aρ) (Ancosωnt+Bnsinωnt) ;k≡ω ca , (6) the roots of the Bessel function being kn=ωna c. The general solution for our bidimensional problem (with ou r boundary conditions) will therefore be the Fourier-type series Ψ2D(ρ;t) =/summationtext∞ n=1J0(kn aρ) (Ancosωnt+Bnsinωnt). (7) 5The initial conditions (4) imply that/summationtextAnJ0(knρ/a) =δ(ρ)/ρ, and/summationtextBnJ0(knρ/a) = 0, so that all Bnmust vanish, while An= 2[a2J2 1(kn)]−1; and eventually one gets: Ψ2D(ρ;t) =/summationtext∞ n=1/parenleftBigg2 a2J2 1(kn)/parenrightBigg J0(kn aρ) cosωnt . (8) , whereωn=knc/a. Let us explicitly notice that we can pass from such a formal so lution to more physical ones, just by considering a finite number Nof terms. In fact, each partial expansion will satisfy (besides the boundary condition) the second initia l condition ∂tψ= 0 fort= 0, while the first initial condition gets the form φ(ρ) =f(ρ), wheref(ρ) will be a (well) localized function, but no longer a delta-type function. Ac tually, the “localization” of φ(ρ) increases with increasing N. We shall come back to this point below. 4. – Localized waves propagating Superluminally down (norm al-sized) waveg- uides. We have now to apply transformations (3) to solution (8), in o rder to pass to threedimensional waves propagating along a cylindrical (m etallic) waveguide with radius r=a/sinθ. We obtain that Maxwell equations admit in such a case the sol utions Ψ3D(ρ,z;t) =/summationtext∞ n=1/parenleftBigg2 a2J2 1(kn)/parenrightBigg J0(kn aρsinθ) cos/bracketleftBiggkncosθ a(z−c cosθt)/bracketrightBigg (9) whereωn=knc/a, which are sums over different propagating modes. Such solutions propagate, down the waveguide, rigidly with Superluminal velocity#2 v=c/cosθ. Therefore, (non-evanescent) solutions to Maxwell equati ons exist, that are waves propagating undistorted along normal waveguides with Superluminal speed (even if in normal-sized waveguides the dispersion relation for e ach mode, i.e. for each term of the Fourier-Bessel expansion, is the ordinary “subluminal ” one,ω2/c2−k2 z= +Ω2). It is interesting that our Superluminal solutions travel ri gidly down the waveguide: #2Let us stress that each eq.(9) represents a multimodal (butlocalized ) propagation, as if the geo- metric dispersion compensated for the multimodal dispersi on. 6this is at variance with what happens for truncated (Superlu minal) solutions[7-10], which travel almost rigidly only along their finite “field depth” an d then abruptly decay. Finally, let us consider a finite number of terms in eq.(8), at t= 0. We made a few numerical evaluations: let us consider the results for N= 22 (however, similar results can be already obtained, e.g., for N= 10). The first initial condition of eq.(4), then, is no longer a delta function, but results to be the (bumped) bidim ensional wave represented in Fig.1. The threedimensional wave, eq.(9), corresponding to it, i. e., with the same finite numberN= 22 of terms, is depicted in Fig.2. It is still an exact soluti on of the wave equation, for a metallic (normal-sized) waveguide with rad iusr=a/sinθ, propagating rigidly with Superluminal group-velocity v=c/cosθ; moreover, it is now a physical solution. In Fig.2 one can see its central portion, while in F ig.3 it is shown the space profile along z, fort= const., of such a propagating wave. Acknowledgements – The authors are grateful to Flavio Fontana (Pirelli Cavi, I taly) for having suggested the problem, and to Hugo E. Hern´ andez- Figueroa (Fac. of Electric Engineering, UNICAMP) and Amr Shaarawi (Cairo University) for continuous scientific collaboration. Thanks are also due to Antˆ onio Chaves Maia N eto for his kind help in the numerical evaluations, and to Franco Bassani, Carlo Bec chi, Rodolfo Bonifacio, Ray Chiao, Gianni Degli Antoni, Roberto Garavaglia, Gershon Ku rizki, Giuseppe Marchesini, Marcello Pignanelli, Andrea Salanti, Abraham Steinberg an d Jacobus Swart for stimulat- ing discussions. 7Figure Captions Fig.1 – Shape of the bidimensional solution of the wave equation val id on the circular domainρ≤a;a= 0.1 mm of the ( x,y) plane, for t= 0, corresponding to the sum of N= 22 terms in the expansion (8). It is no longer a delta functio n, but it is still very well peaked. By choosing it as the initial condition, instea d of the first one of eqs.(4), one gets the threedimensional wave depicted in Figs.2 and 3. The normalization condition is such that |Ψ2D(ρ= 0;t= 0)|2= 1. Fig.2 – The (very well localized) threedimensional wave correspon ding to the initial, bidimensional choice in Fig.1. It propagates rigidly (alon g the normal-sized circular waveguide with radius r=a/sinθ) with Superluminal speed v=c/cosθ. Quantity ηis defined asη≡(z−c cosθt). The normalization condition is such that |Ψ3D(ρ= 0;η= 0)|2= 1. Fig.3 – The shape along z, att= 0, of the threedimensional wave whose main peak is shown in Fig.2. 8References [1] H.Bateman: Electrical and Optical Wave Motion (Cambridge Univ.Press; Cambridge, 1915). [2] A.O.Barut and H.C.Chandola: Phys. Lett. A180 (1993) 5. See also A.O.Barut: Phys. Lett.A189 (1994) 277, and A.O.Barut et al.: refs.[3]. [3] A.O.Barut and A.Grant: Found. Phys. Lett. 3 (1990) 303; A.O.Barut and A.J.Bracken: Found. Phys. 22 (1992) 1267. See also refs.[14,19,20] below. [4] R.Donnelly and R.W.Ziolkowski: Proc. Royal Soc. London A440 (1993) 541 [cf. also I.M.Besieris, A.M.Shaarawi and R.W.Ziolkowski: J. Math. Phys. 30 (1989) 1254]; S.Esposito: Phys. Lett A225 (1997) 203; W.A.Rodrigues Jr. and J.Vaz Jr., Adv. Appl. Cliff. Alg. S-7 (1997) 457. [5] See, e.g., E.Recami: “Classical tachyons and possible a pplications,” Rivista Nuovo Cimento 9 (1986), issue no.6, pp.1-178; and refs. therein. [6] Jian-yu Lu and J.F.Greenleaf: IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 39 (1992) 19. [7] R.W.Ziolkowski, I.M.Besieris and A.M.Shaarawi: J. Opt. Soc. Am. A10 (1993) 75. [8] E.Recami: “On localized ‘X-shaped’ Superluminal solut ions to Maxwell equations”, Physica A 252 (1998) 586. [9] Jian-yu Lu and J.F.Greenleaf: IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 39 (1992) 441. [10] P.Saari and K.Reivelt: “Evidence of X-shaped propagat ion-invariant localized light waves”, Phys. Rev. Lett. 79 (1997) 4135. See also H.S˜ onajalg, M.R¨ atsep and P.Saari : Opt. Lett. 22 (1997) 310; Laser Phys. 7 (1997) 32). 9[11] A.M.Steinberg, P.G.Kwiat and R.Y.Chiao: Phys. Rev. Lett. 71 (1993) 708, and refs. therein; Scient. Am. 269 (1993) issue no.2, p.38. Cf. also R.Y.Chiao, A.E.Kozhek in, G.Kurizki: Phys. Rev. Lett. 77 (1996) 1254; Phys. Rev. A53 (1996) 586. [12] A.Enders and G.Nimtz: J. de Physique-I 2(1992) 1693; 3(1993) 1089; 4(1994) 1; H.M.Brodowsky, W.Heitmann and G.Nimtz: J. de Physique-I 4(1994) 565; Phys. Lett. A 222(1996) 125; Phys. Lett. A 196(1994) 154; G.Nimtz and W.Heitmann: Prog. Quant. Electr. 21(1997) 81. [13] See V.S.Olkhovsky and E.Recami: Phys. Reports 214 (1992) 339, and refs. therein; V.S.Olkhovsky et al.: J. de Physique-I 5 (1995) 1351-1365; T.E.Hartman: J. Appl. Phys. 33(1962) 3427. [14] Cf. A.P.L.Barbero, H.E.Hern´ andez-Figueroa and E.Re cami: “On the propagation speed of evanescent modes” [LANL Archives # physics/981100 1], submitted for pub., and refs. therein. Cf. also E.Recami, H.E.Hern´ andez F., an d A.P.L.Barbero: Ann. der Phys.7(1998) 764. [15] G.Nimtz, A.Enders and H.Spieker: J. de Physique-I 4 (1994) 565; “Photonic tun- nelling experiments: Superluminal tunnelling”, in Wave and Particle in Light and Matter – Proceedings of the Trani Workshop, Italy, Sept.1992 , ed. by A.van der Merwe and A.Garuccio (Plenum; New York, 1993). [16] H.M.Brodowsky, W.Heitmann and G.Nimtz: Phys. Lett. A222 (1996) 125. [17] R.Garavaglia: Thesis work (Dip. Sc. Informazione, Uni versit` a statale di Milano; Milan, 1998; G.Degli Antoni and E.Recami supervisors). [18] E.Recami and F.Fontana: “Special Relativity and Super luminal motions”, submitted for publication. 10[19] J.-y.Lu, H.-h.Zou and J.F.Greenleaf: IEEE Transactions on Ultrasonics, Ferro- electrics and Frequency Control 42 (1995) 850-853. 11-0.01 -0.005 0.005 0.01rHmL0.20.40.60.81¨Y2 D¨2-0.005 -0.0025 0 0.0025 0.005hHmL -0.01-0.00500.0050.01 rHmL00.250.50.751 ¨Y3 D¨2 -0.005 -0.0025 0 0.0025 0.005hHmL -0.02 -0.01 0.01 0.02hHmL0.20.40.60.81¨Y3 D¨2

arXiv:physics/0001040v1 [physics.soc-ph] 19 Jan 2000Black-Scholes option pricing within Itˆ o and Stratonovich conventions J. Perell´ oa, J. M. Porr` aa,b, M. Monteroaand J. Masolivera aDepartament de F´ ısica Fonamental, Universitat de Barcelo na, Diagonal, 647, 08028-Barcelona, Spain bGaesco Bolsa, SVB, S.A., Diagonal, 429, 08036-Barcelona, S pain Abstract Options are financial instruments designed to protect inves tors from the stock mar- ket randomness. In 1973, Fisher Black, Myron Scholes and Rob ert Merton proposed a very popular option pricing method using stochastic differ ential equations within the Itˆ o interpretation. Herein, we derive the Black-Schol es equation for the option price using the Stratonovich calculus along with a comprehe nsive review, aimed to physicists, of the classical option pricing method based on the Itˆ o calculus. We show, as can be expected, that the Black-Scholes equation is indep endent of the interpreta- tion chosen. We nonetheless point out the many subtleties un derlying Black-Scholes option pricing method. 1 Introduction An European option is a financial instrument giving to its own er the right but not the obligation to buy (European call) or to sell (Euro pean put) a share at a fixed future date, the maturing time T, and at a certain price called exercise or striking price xC. In fact, this is the most simple of a large variety of contracts that can be more sophisticated. One of those pos sible extensions is the American option which gives the right to exercise the o ption at any time until the maturing time. In a certain sense, options are a security for the investor thus avoiding the unpredictable consequences of o perating with risky speculative stocks. The trading of options and their theoretical study have been known for long, although they were relative obscure and unimportant financi al instruments until the early seventies. It was then when options experime nted an spectacular development. The Chicago Board Options Exchange, created i n 1973, is the first attempt to unify options in one market and trade them on o nly a few Preprint submitted to Elsevier Preprint 23 September 2013stock shares. The market rapidly became a tremendous succes s and led to a series of innovations in option trading [1]. The main purpose in option studies is to find a fair and presuma bly riskless price for these instruments. The first solution to the proble m was given by Bachelier in 1900 [2], and several option prices were propos ed without be- ing completely satisfactory [3]. However, in the early seve nties it was finally developed a complete option valuation based on equilibrium theoretical hy- pothesis for speculative prices. The works of Fisher Black, Myron Scholes [4] and Robert Merton [5] were the culmination of this great effor t, and left the doors open for extending the option pricing theory in many wa ys. In addi- tion, the method has been proved to be very useful for investo rs and has helped to option markets to have the importance that they hav e nowadays in finance [1,3]. The option pricing method obtains the so-called Black-Scho les equation which is a partial differential equation of the same kind as the diffu sion equation. In fact, it was this similarity that led Black and Scholes to obt ain their option price formula as the solution of the diffusion equation with t he initial and boundary conditions given by the option contract terms. Inc identally, these physics studies applied to economy have never been disrupte d and there still is a growing effort of the physics community to understand the dynamics of finance from approaches similar to those that tackle compl ex systems in physics [6–10]. The economic ideas behind the Black-Scholes option pricing theory translated to the stochastic methods concepts are as follows. First, th e option price de- pends on the stock price and this is a random variable evolvin g with time. Second, the efficient market hypothesis [11], i.e., the market incorporates in- stantaneously any information concerning future market ev olution, implies that the random term in the stochastic equation must be delta -correlated. That is: speculative prices are driven by white noise [6,12] . It is known that any white noise can be written as a combination of the derivat ive of the Wiener process and white shot noise [13]. In this framework, the Bla ck-Scholes option pricing method was first based on the geometric Brownian moti on [4,5], and it was lately extended to include white shot noise [14,15]. As is well known, any stochastic differential equation (SDE) driven by a state dependent white noise, such as the geometric Brownian motio n, is meaningless unless an interpretation of the multiplicative noise term i s given. Two interpre- tations have been presented: Itˆ o [16] and Stratonovich [17 ]. To our knowledge, all derivations of the Black-Scholes equation starting fro m a SDE are based on the Itˆ o interpretation. A possible reason is that mathem aticians prefer this interpretation over the Stratonovich’s one, being the latt er mostly preferred among physicists. Nonetheless, as we try to point out here, I tˆ o framework is 2perhaps more convenient for finance being this basically due to the peculiar- ities of trading (see Sect. 4). In any case, as Van Kampen show ed some time ago [18] no physical reason can be attached to the interpreta tion of the SDE modelling price dynamics. However, the same physical proce ss results in two different SDEs depending on the interpretation chosen. In sp ite of having dif- ferent differential equations as starting point, we will sho w that the resulting Black-Scholes equation is the same regardless the interpre tation of the mul- tiplicative noise term, and this constitutes the main resul t of the paper. In addition, the mathematical exercise that represents this t ranslation into the Stratonovich convention provides a useful review, special ly to physicists, of the option pricing theory and the “path-breaking” Black-Sc holes method. The paper is divided in 5 sections. After the Introduction, a summary of the differences between Itˆ o and Stratonovich calculus is devel oped in Section 2. The following section is devoted to explain the market model assumed in Black-Scholes option pricing method. Section 4 concentrat es in the deriva- tion of the Black-Scholes equation using both Itˆ o and Strat onovich calculus. Conclusions are drawn in Section 5, and some technical detai ls are left to appendices. 2 Itˆ o vs. Stratonovich It is not our intention to write a formal discussion on the diff erences between Itˆ o and Stratonovich interpretations of stochastic differ ential equations since there are many excellent books and reviews on the subject [13 ,18–20]. However, we will summarize those elements in these interpretations t hat change the treatment of the Black-Scholes option pricing method. In al l our discussion, we use a notation that it is widely used among physicists. The interpretation question arises when dealing with a mult iplicative stochas- tic differential equation, also called multiplicative Lang evin equation, ˙X=f(X) +g(X)ξ(t), (1) where fandgare given functions, and ξ(t) is Gaussian white noise, that is, a Gaussian and stationary random process with zero mean and de lta correlated. Alternatively, Eq. (1) can be written in terms of the Wiener p rocess W(t) as dX=f(X)dt+g(X)dW(t), (2) where dW(t) =ξ(t)dt. When gdepends on X, Eqs. (1) and (2) have no mean- ing, unless an interpretation of the multiplicative term g(X)ξ(t) is provided. 3These different interpretations of the multiplicative term must be given be- cause, due to the extreme randomness of white noise, it is not clear what value of Xshould be used even during an infinitesimal timestep dt. According to Itˆ o, that value of Xis the one before the beginning of the timestep, i.e., X=X(t), whereas Stratonovich uses the value of Xat the middle of the timestep: X=X(t+dt/2) =X(t) +dX(t)/2. Before proceeding further with the consequences of the abov e discussion, we will first give a precise meaning of the differential of random processes driven by Gaussian white noise and its implications. Obviously, th e differential of any random process X(t) is defined by dX(t)≡X(t+dt)−X(t). (3) On the other hand, the differential dX(t) of any random process is equal (in the mean square sense) to its mean value if its variance is, at least, of order dt2[13]:/angbracketleft[dX(t)− /angbracketleftdX(t)/angbracketright]2/angbracketright=O(dt2). We observe that from now on all the results of this paper must be interpreted in the mean square s ense. The mean square limit relation can be used to show that |dW(t)|2=dt[20]. We thus have from Eq. (2) that |dX|2=|g(X)|2dt+O(dt2), (4) and we symbolically write dX(t) =O/parenleftBig dt1/2/parenrightBig . (5) Let us now turn our attention to the differential of the produc t of two random processes since this differential adopts a different express ion depending on the interpretation (Itˆ o or Stratonovich) chosen. In accordan ce to Eq. (3), we define d(XY)≡[(X+dX)(Y+dY)]−XY. (6) This expression can be rewritten in many different ways. One p ossibility is d(XY) =/parenleftBigg X+dX 2/parenrightBigg dY+/parenleftBigg Y+dY 2/parenrightBigg dX, (7) but it is also allowed to write the product as d(XY) =XdY +Y dX+dXdY. (8) 4Therefore, we say that the differential of a product reads in t he Stratonovich interpretation when d(XY)≡XSdY+YSdX, (9) where XS(t)≡X(t+dt/2) =X(t) +dX(t)/2, (10) and similarly for YS(t). Whereas we say that the differential of a product follows the Itˆ o interpretation when d(XY)≡XIdY+YIdX+dXdY, (11) where XI(t)≡X(t), (12) andYI(t)≡Y(t). Note that Eq. (9) formally agrees with the rules of calcu- lus while Eq. (11) does not. Note also that Eqs. (9) and (11) ca n easily be generalized to the product of two functions, U(X) and V(X), of the random process X=X(t). Thus d(UV) =U(XS)dV(X) +V(XS)dU(X), (13) where XSis given by Eq. (10), and dV(X) =V(X+dX)−V(X) with an analogous expression for dU(X). Within Itˆ o convention we have d(UV) =U(X)dV(X) +V(X)dU(X) +dU(X)dV(X). (14) Let us now go back to Eq. (1) and see that one important consequ ence of the above discussion is that the expected value of the multip licative term, g(X)ξ(t), depends on the interpretation given. In the Itˆ o interpre tation, it is clear that /angbracketleftg(X)ξ(t)/angbracketright= 0 because the value of X(and, hence the value of g(X)) anticipates the jump in the noise. In other words, g(X) is independent ofξ(t). On the other hand, it can be proved that within the Stratono vich framework the average of the multiplicative term reads g(X)g′(X)/2 where the prime denotes the derivative [20]. The zero value of the a verage /angbracketleftg(X)ξ(t)/angbracketright makes Itˆ o convention very appealing because then the deter ministic equation for the mean value of Xonly depends on the drift term f(X). In this sense, note that any multiplicative stochastic differential equat ion has different ex- pressions for the functions f(X) and g(X) depending on the interpretation 5chosen. In the Stratonovich framework, a SDE of type Eq. (2) c an be written as dX=f(S)(XS)dt+g(S)(XS)dW(t), (15) where XS=X+dX/2. In the Itˆ o sense we have dX=f(I)(XI)dt+g(I)(XI)dW(t), (16) where XI=X. Note that f(S)andf(I)are not only evaluated at different val- ues of Xbut are also different functions depending on the interpreta tion given, and the same applies to g(S)andg(I). One can easily show from Eq. (10) and Eqs. (15)-(16) that, after keeping terms up to order dt, the relation between fSandfIis [20] f(I)(X) =f(S)(X)−1 2g(S)(X)∂g(S)(X) ∂X, (17) while the multiplicative functions g(S)andg(I)are equal g(I)(X) =g(S)(X). (18) Conversely, it is possible to pass from a Stratonovich SDE to an equivalent Itˆ o SDE [20]. Note that the difference between both interpre tation only affects the drift term given by the function fwhile the function gremains unaffected. In addition, we see that for an additive SDE, i.e., when gis independent of X, the interpretation question is irrelevant. Finally, a crucial difference between Itˆ o and Stratonovich interpretations ap- pears when a change of variables is performed on the original equation. Then it can be proved that, using Stratonovich convention, the st andard rules of calculus hold, but new rules appear when the equation is unde rstood in the Itˆ o sense. From the point of view of this property, the Strat onovich criterion seems to be more convenient. For the sake of completeness, we remind here what are the rules of change of variables in each interpretat ion. Let h(X, t) be an arbitrary function of Xandt. In the Itˆ o sense, the differential of h(X, t) reads [20] dh=∂h(X, t) ∂XdX+/bracketleftBigg∂h(X, t) ∂t+1 2g2(X, t)∂2h(X, t) ∂X2/bracketrightBigg dt, (19) 6whereas in the Stratonovich sense, we have the usual express ion [20] dh=∂h(XS, t) ∂XSdX+∂h(XS, t) ∂tdt, (20) where ∂h(XS, t) ∂XS=∂h(X, t) ∂X/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle X=XS, andXSis given by Eq. (10). Equation (19) is known as the Itˆ o’s lema and it is extensivel y used in mathe- matical finance books [12,21–25]. The information on the properties of the Itˆ o and Stratonovi ch interpretation of SDE contained in this brief summary is sufficient to follow t he derivations of the next sections. 3 Market model Option pricing becomes a problem because market prices or in dexes change randomly. Therefore, any possible calculation of an option price is based on a model for the stochastic evolution of the market prices. The first analysis of price changes was given one hundred years ago by Bachelier wh o, studying the option pricing problem, proposed a model assuming that pric e changes behave as an ordinary random walk [2]. Thus, in the continuum limit ( continous time finance [25]) speculative prices obey a Langevin equati on. In order to include the limited liability of the stock prices, i.e., prices cannot be negative, Osborne proposed the geometric or log-Brownian motion for d escribing the price changes [26]. Mathematically, the market model assum ed by Osborne can be written as a stochastic equation of type Eq. (2): dR(t) =µdt+σdW(t), (21) where R(t) is the so-called return rate after a period t. Therefore, dR(t) is the infinitessimal relative change in the stock share price X(t) (see Eq. (22) below), µis the average rate per unit time, and σ2is the volatility per unit time of the rate after a period t,i.e.,/angbracketleftdR/angbracketright=µdtand/angbracketleft(dR−/angbracketleftdR/angbracketright)2/angbracketright=σ2dt. There is no need to specify an interpretation (Itˆ o’s or Stratonov ich’s) for Eq. (21) because σis constant and we are thus dealing with an additive equation . The rate is compounded continuously and, therefore, an initial priceX0becomes after a period t: X(t) =X0exp[R(t)]. (22) 7This equation can be used as a change of variables to derive th e SDE for X(t) given that R(t) evolves according to Eq. (21). However, as it becomes multi - plicative, we have to attach the equation to an interpretati on. Indeed, using Stratonovich calculus (see Eq. (20)), it follows that X(t) evolves according to the equation dX=µXSdt+σXSdW(t), (23) where XS=X+dX/2. In the Itˆ o sense (see Eq. (19)), the equation for X(t) becomes dX=/parenleftBig µ+σ2/2/parenrightBig Xdt+σXdW (t). (24) Therefore, the Langevin equation for X(t) is different depending on the sense it is interpreted. The main objective of this paper is to show that no matter which equation is used to derive the Black-Scholes equation the final result turns out to be the same. Before proceeding further, we point out that the average ind ex price after a timetis/angbracketleftX(t)/angbracketright=X0exp(µ+σ2/2)t, regardless the convention being used. In fact, the independence of the averages on the interpretat ion used holds for moments of any order [18–20]. 4 The Black-Scholes equation There are several different approaches for deriving the Blac k-Scholes equation starting from the stochastic differential equation point of view. These different derivations only differ in the way the portfolio ( i.e., a collection of different assets for diversifying away financial risk) is defined [4,25 ,27,28]. In order to get the most general description of the concepts underlying in t he Black-Scholes theory, our portfolio is similar to the one proposed by Merto n [27], and it is based on one type of share whose price is the random process X(t). The portfolio is compounded by a certain amount of shares, ∆, a nu mber of calls, Ψ, and, finally, a quantity of riskless securities (or bonds) Φ. We also assume that short-selling, or borrowing, is allowed. Specifically, we o wn a certain number of calls worth Ψ Cdollars and we owe ∆ X+ΦBdollars. In this case, the value Pof the porfolio reads P= ΨC−∆X−ΦB, (25) where Xis the share stock price, Cis the call price to be determined, and B is the bond price whose evolution is not random and is describ ed according to 8the value of r, the risk-free interest rate ratio. That is dB=rBdt. (26) The so-called “portfolio investor’s strategy” [22] decide s the quantity to be invested in every asset according to its stock price at time t. This is the reason why the asset amounts ∆ ,Ψ,and Φ are functions of stock price and time, although they are “nonanticipating” functions of the stock price. This somewhat obscure concept is explained in the Appendix A. All derivations of Black-Scholes equation assume a “frictionless market”, th at is, there are no transaction costs for each operation of buying and selling [ 4]. According to Merton [27] we assume that, by short-sales, or b orrowing, the portfolio (25) is constrained to require net zero investmen t, that is, P= 0 for any time t[29]. Then, from Eq. (25) we have C=δnX+φnB, (27) where, δn≡∆/Ψ and φn≡Φ/Ψ are respectively the number of shares per call and the number of bonds per call. As we have mentioned abo ve,δnand φnare nonanticipating functions of the stock price (see Appen dix A). Note that Eq. (27) has an interesting economic meaning, since tel ls us that having a call option is equivalent to possess a certain number, δnandφn, of shares and bonds thus avoiding any arbitrage opportunity [29]. Equati on (27), which is called “the replicating portfolio” [12,22,23], is the star ting point of our deriva- tion that we separate into two subsections according to Itˆ o or Stratonovich interpretations. 4.1 The Black-Scholes equation derivation (Itˆ o) We need first to obtain, within the Itˆ o interpretation, the d ifferential of the call price C. This is done in the Appendix B and we show there that dC=δdX+φdB+Xdδ n+Bdφ n+O(dt3/2), (28) where the relationship between δ,φandδn,φnis given in Appendix A ( cf. Eq. (A.1)). We assume we follow a “self-financing strategy” [ 28], that is, vari- ations of wealth are only due to capital gains and not to the wi thdrawal or infusion of new funds. In other words, we increase [decrease ] the number of shares by selling [buying] bonds in the same proportion. We t hen have (see Appendix A for more details) Xdδ n=−Bdφ n, (29) 9and Eq. (28) reads dC=δdX+φdB. (30) Moreover, from Eqs. (26)-(27) one can easily show that φdB=r(C−δX)dt+O(dt3/2), (cf.Eq. (5) and Eq. (A.1) of Appendix A). Therefore, dC=δdX+r(C−δX)dt+O(dt3/2). (31) On the other hand, since the call price Cis a function of share price Xand time,C=C(X, t), and Xobeys the (Itˆ o) SDE (24), then dCcan be evaluated from the Itˆ o lemma (19) with the result dC=/parenleftBigg∂C ∂t+1 2σ2X2∂2C ∂X2/parenrightBigg dt+∂C ∂XdX. (32) Substituting Eq. (31) into Eq. (32) yields /parenleftBigg δ−∂C ∂X/parenrightBigg dX=/bracketleftBigg∂C ∂t−r(C−δX) +1 2σ2X2∂2C ∂X2/bracketrightBigg dt. (33) Note that this is an stochastic equation because of its depen dence on the Wiener process enclosed in dX. We can thus turn Eq. (33) into a deterministic equation that will give the call price functional dependenc e on share price and time by equating to zero the term multiplying dX. This, in turn, will determine the “investor strategy”, that is the number of shares per cal l, the so called “delta hedging”: δ=∂C(x, t) ∂x. (34) The substitution of Eq. (34) into Eq. (33) results in the Blac k-Scholes equation: ∂C ∂t=rC−rx∂C ∂x−1 2(σx)2∂2C ∂x2. (35) A final observation, in Eqs. (34)-(35) we have set X=x, since, as explained above, Eq. (35) gives the functional dependence of the call p riceConXand tregardless whether the share price Xis random or not. 104.2 The Black-Scholes equation derivation (Stratonovich) Let us now derive the Black-Scholes equation, assuming that the underly- ing asset obeys the Stratonovich SDE (23). In the Appendix B w e present part of this derivation using the concept of nonanticipatin g function within the Stratonovich interpretation. Nevertheless, here we pe rform an alternative derivation that uses the Itˆ o interpretation as starting po int. We thus begin with Eq. (31) that we write in the form dC=δ(X, t)dX(t) +r[C(X, t)−δ(X, t)X]dt+O(dt3/2). (36) Now, we have to express the function δwithin Stratonovich interpretation. Note that X=XS−dX/2. Hence δ(X, t) =δ(XS−dX/2, t), whence δ(X, t) =δ(XS, t)−1 2∂δ(XS, t) ∂XSdX+O(dX2). (37) Analogously C(X, t) =C(XS, t)+O(dX). Therefore, from Eqs. (36)-(37) and taking into account Eq. (4) we have dC=δ(XS, t)dX+/bracketleftBigg rC(XS, t)−rXSδ(XS, t) −1 2σ2X2 S∂δ(XS, t) ∂XS/bracketrightBigg dt+O(dt3/2).(38) On the other hand, dCwill also be given by Eq. (20) dC=∂C(XS, t) ∂tdt+∂C(XS, t) ∂XSdX, (39) From these two equations we get /bracketleftBigg δ(XS, t)−∂C(XS, t) ∂XS/bracketrightBigg dX=/bracketleftBigg∂C(XS, t) ∂t−rC(XS, t) +rXSδ(XS, t) +1 2σ2X2 S∂δ(XS, t) ∂XS/bracketrightBigg dt. (40) Again, this equation becomes non-stochastic if we set δ(XS, t) =∂C(XS, t) ∂XS. (41) 11In this case, the combination of Eqs. (40)-(41) agrees with E q. (35). Although the call price is evaluated at a different value of the share pr ice, this is irrele- vant for the reason explained right after Eq. (35). Therefor e, the Stratonovich calculus results in the same call price formula and equation than the Itˆ o cal- culus. We have used the stochastic differential equation technique in order to derive the option price equation. However, this is only one of the po ssible routes. Another way, which was also proposed in the original paper of Black and Sc- holes [4], uses the Capital Asset Pricing Model (CAPM) [30] w here, adducing equilibrium reasons in the asset prices, it is assumed the eq uality of the so- called “Sharpe ratio” of the stock and the option respective ly. The Sharpe ratio of an asset can be defined as its normalized excess of ret urn, therefore CAPM assumption applied to option pricing reads [25] α−r σ=αC−r σC, where α=/angbracketleftdX/X /angbracketright,σ2= Var( dX/X ),αC=/angbracketleftdC/C/angbracketright, and σ2 C= Var( dC/C ). From this equality it is quite straightforward to derive the Black-Scholes equa- tion [4,25]. As remarked at the end of Sect. 3, moments are ind ependent of the interpretation chosen, we thus clearly see the equivale nce between Itˆ o and Stratonovich calculus for the Black-Scholes equation deri vation. 4.3 The Black-Scholes formula for the European call For the sake of completeness, let us now finish the paper by sho rtly deriving from Eq. (35) the well-known Black-Scholes formula. Note th at the Black- Scholes equation is a backward parabolic differential equat ion, we therefore need one “final” condition and, in principle, two boundary co nditions in order to solve it [31]. In fact, Black-Scholes equation is defined o n the semi-infinite interval 0 ≤x <∞. In this case, since C(x, t) is assumed to be sufficiently well behaved for all x, we only need to specify one boundary condition at x= 0 (see [24] and [31]), although we specify below the boundar y condition atx=∞as well. We also note that all financial derivatives (options of any ki nd, forwards, futures, swaps, etc...) have the same boundary conditions b ut different initial or final condition [23]. Let us first specify the boundary cond itions. We see from the multiplicative character of Eq. (2) that if at some t ime the price X(t) drops to zero then it stays there forever. In such a case, it is quite obvious that the call option is worthless: C(0, t) = 0. (42) 12On the other hand, as the share price increases without bound ,X→ ∞, the difference between share price and option price vanishes, si nce option is more and more likely to be exercised and the value of the option wil l agree with the share price, that is, limx→∞C(x, t) x= 1. (43) In order to obtain the “final” condition for Eq. (35), we need t o specify the following two parameters: the expiration or maturing time T, and the striking or exercise price xCthat fixes the price at which the call owner has the right to buy the share at time T. If we want to avoid arbitrage opportunities, it is clear that the value of the option Cof a share that at time Tis worth x dollars must be equal to the payoff for having the option [2]. T his payoff is either 0 or the difference between share price at time Tand option striking price, that is, max( x−xC,0). Hence, the “final” condition for the European call is C(x, t=T) = max( x−xC,0). (44) In the Appendix C we show that the solution to the problem give n by Eq. (35) and Eqs. (42)-(44) is C(x, t) =xN(d1)−xCe−r(T−t)N(d2), (45) (0≤t≤T), where N(z) =1√ 2πz/integraldisplay −∞e−u2/2du, is the probability integral, d1=ln(x/xc) + (r+σ2/2)(T−t) σ√ T−t, and d2=d1−σ√ T−t. 5 Conclusions We have updated the option pricing theory from the point of vi ew of a physi- cist. We have centered our analysis of option pricing to the B lack-Scholes equation and formula for the European call, extensions to ot her kind of op- tions can be straightforward in many cases and are found in se veral good 13finance books [21–25]. We have reviewed Black-Scholes theor y using Itˆ o cal- culus, which is standard to mathematical finance, with a spec ial emphasis in explaining and clarifying the many subtleties of the calcul ation. Nevertheless, we have not limit ourselves only to review option pricing, bu t to derive, for the first time to our knowledge, the Black-Scholes equation usin g the Stratonovich calculus which is standard to physics, thus bridging the gap between mathe- matical finance and physics. As we have proved, the Black-Scholes equation obtained usin g Stratonovich calculus is the same as the one obtained by means of the Itˆ o ca lculus. In fact, this is the result we expected in advance because Itˆ o a nd Stratonovich conventions are just different rules of calculus. Moreover, from a practical point of view, both interpretations differ only in the drift t erm of the Langevin equation and the drift term does not appear in the Black-Scho les equation and formula. But, again, we think that this derivation is still i nteresting and useful for all the reasons explained above. Acknowledgements This work has been supported in part by Direcci´ on General de Investigaci´ on Cient´ ıfica y T´ ecnica under contract No. PB96-0188 and Proj ect No. HB119- 0104, and by Generalitat de Catalunya under contract No. 199 8 SGR-00015. A Nonanticipating functions and self-financing strategy The functionals φnandδnrepresenting normalized asset quantities are nonatic- ipating functions with respect to the stock price X. This means that these functionals are in some way independent of X(t) implying a sort of causality in the sense that unknown future stock price cannot affect the present port- folio strategy. The physical meaning of this translated to fi nancial markets is: first buy or sell according to the present stock price X(t) and right after the portfolio worth changes with variation of the prices dX,dB, anddC. In other words, the investor strategy does not anticipate the stock price ch ange[3,23]. Therefore, in the Itˆ o sense, the functionals δnandφnrepresenting the number of assets in the portfolio solely depend on the share price right before timet, i.e., they do not depend on X(t) but on X(t−dt) =X−dX. That is, δn(X, t)≡δ(X−dX, t), (A.1) 14and similarly for φn(recall that all equalities must be understood in the mean square sense explained in Sect. 2). The expansion of Eq. (A.1) yields (see Eq. (5)) δn(X, t) =δ(X, t)−∂δ(X, t) ∂XdX+O(dt), but from the Itˆ o lema (19) we see that ∂δ(X, t) ∂XdX=dδ(X, t) +O(dt), and finally δn(X, t) =δ(X, t)−dδ(X, t) +O(dt). (A.2) Analogously, δ(X, t) =δn(X, t) +dδn(X, t) +O(dt), (A.3) and a similar expresion for φ(X, t). As to the self-financing strategy, Eq. (29), we observe that δ(X, t+dt) is the number of shares we have at time t+dt, while δ(X−dX, t) is that number at time t. Therefore, X(t)dδ(X−dX, t) = [δ(X, t+dt)−δ(X−dX, t)]X(t) is the money we need or obtain for buying or from selling share s at time t. Analogously, B(t)dφ(X−dX, t) is the money, needed or obtained at time t, coming from bonds. If we follow a self-financing strategy, bo th quantities are equal but with different sign, i.e., X(t)dδ(X−dX, t) =−B(t)dφ(X−dX, t) (A.4) which agrees with Eq. (29). B The differential of the option price Let us derive the differential of the call price, dC, using either Itˆ o and Stratono- vich interpretations. The starting point for both derivati ons is the replicating portfolio, Eq. (27), C(X, t) =X(t)δn(X, t) +B(t)φn(X, t). (B.1) 15Taking into account the Itˆ o product rule Eq. (11), we have dC= [δn(X, t) +dδn(X, t)]dX+[φn(X, t) +dφn(X, t)]dB +X(t)dδn(X, t) +B(t)dφn(X, t), which, after using Eq. (A.3), reads dC=δ(X, t)dX+φ(X, t)dB+X(t)dδn(X, t) +B(t)dφn(X, t) +O(dt3/2), and this agrees with Eq. (28). Within the Stratonovich calculus, the differential of Eq. (B .1) reads dC=XS(t)dδn+B(t)dφn+δn(XS, t)dX+φn(XS, t)dB. (B.2) From Eq. (A.1) we have δn(XS, t) =δ(XS, t)−∂δ(XS, t) ∂XSdX+O(dX2), (B.3) and analogously for φn. Substituting Eq. (B.3) into Eq. (B.2), and taking into account Eqs. (4)-(5), (10) and (26) we obtain dC= [X(t) +dX/2]dδn+B(t)dφn+δ(XS, t)dX +/bracketleftBigg rB(t)φ(XS, t)−σ2X2 S∂δ(XS, t) ∂XS/bracketrightBigg dt+O(dt3/2). But from Eq. (A.1) and the self-financing strategy (A.4), we s ee that X(t)dδn+ B(t)dφn= 0. Hence dC=1 2dXdδ n+δ(XS, t)dX +/bracketleftBigg rB(t)φ(XS, t)−σ2X2 S∂δ(XS, t) ∂XS/bracketrightBigg dt+O(dt3/2).(B.4) The substitution of the Stratonovich rule Eq. (20), dδn=∂δn(XS, t) ∂XSdX+∂δn(XS, t) ∂tdt, yields 16dC=δ(XS, t)dX+ [rB(t)φ(XS, t) −1 2σ2X2 S∂δ(XS, t) ∂XS/bracketrightBigg dt+O(dt3/2), (B.5) where we have taken into account Eq. (4) and the fact that ∂δn/∂X S= ∂δ/∂X S+O(dt1/2). Eq. (B.5) agrees with Eq. (38) and the rest of the deriva- tion is identical to that of the main text. C Solution to the Black-Scholes equation In this appendix we outline the solution to the Black-Schole s equation (35) under conditions (42)-(44). We first transform Eq. (35) into a forward parabolic equation with constant coefficients by means of the change of variables z= ln(x/xC), t′=T−t. (C.1) We have ∂C ∂t′=−rC(z, t′) +/parenleftbigg r−1 2σ2/parenrightbigg∂C ∂z+1 2σ2∂2C ∂z2, (C.2) (−∞< z < ∞,0< t′< T). Moreover, the definition of a new dependent variable: u(z, t′) = exp/bracketleftbigg −1 2/parenleftbigg 1−2r σ2/parenrightbigg z+1 8σ2/parenleftbigg 1 +2r σ2/parenrightbigg (T−t′)/bracketrightbigg C(z, t′),(C.3) turns Eq. (C.2) into the ordinary diffusion equation in an infi nite medium ∂u ∂t′=1 2σ2∂2u ∂z2, (C.4) with a constant diffusion coefficient given by σ2/2, and initial condition: u(z,0) =xCexp/bracketleftbigg −1 2/parenleftbigg 1−2r σ2/parenrightbigg z +1 8σ2/parenleftbigg 1 +2r σ2/parenrightbigg T/bracketrightbigg max (ez−1,0). (C.5) 17The solution of problem (C.4)-(C.5) is standard and reads [3 1] u(z, t′) =1√ 2πσ2t′∞/integraldisplay −∞u(y,0)e−(z−y)2/2σ2t′dy. (C.6) If we substitute the initial condition (C.5) into the right h and side of this equation and undo the changes of variables we finally obtain t he Black-Scholes formula Eq. (45). References [1] J.C. Cox, M. Rubinstein, Option Markets, Prentice-Hall , New-Jersey, 1985. [2] L. 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arXiv:physics/0001041v1 [physics.comp-ph] 20 Jan 2000Sun Microsystems’AutoClient and management of computer fa rms atBaBar A.V.Telnov1,2, S.Luitz3, T.J.Pavel3,O.H. Saxton3,M.R. Simonson3 1LawrenceBerkeleyNationalLaboratory,USA 2UniversityofCaliforniaatBerkeley,USA 3StanfordLinearAcceleratorCenter,USA Abstract Modern HEP experiments require immense amounts of computin g power. In the BaBar experimentatSLAC,mostofitisprovidedbySolarisSPARCsy stems.AutoClient ,aproduct of Sun Microsystems, was designed to make setting up and mana ging large numbers of So- laris systems more straightforward. AutoClient machines keep all filesystems, except swap, on a server and employ CacheFS to cache them onto a local disk, which makes them Field Replaceable Units with performance of stand-alone systems . We began exploring the tech- nology in Summer 1998, and currently operate online, recons truction, analysis and console AutoClientfarmswiththetotalnumberofnodesexceeding40 0. Althoughthetechnologyhas beenavailablesince 1995,it hasnot beenwidelyused,andth e availabledocumentationdoes notadequatelycovermanyimportantdetailsof AutoClient installationandmanagement. This paper discusses various aspects of our experience with AutoClient , including tips and tricks, performance and maintainability, scalability and server r equirements, existing problems and possiblefutureenhancements. This paper has been submitted to Proceedings of the Conferen ce on Computing in High EnergyandNuclearPhysics(CHEP 2000),February7-11,2000 ,Padova,Italy. Keywords: farm,sun,solaris, autoclient,cachefs,babar,slac,fru, fieldreplaceableunit 1 Introduction Having to administer a large number of workstations has long been a headache to system ad- ministrators working for big businesses, universities and research laboratories. In the case of UNIX-style operating systems, system maintenance tasks ca n very rarely be delegated to the end users, which means that the whole burden of taking care of the company’s computers falls on the system administrators. Unless some kind of special system m aintenance scheme is devised, the required administration effort scaleslinearly withthenu mber ofmachines andasynchronous ma- jor software upgrade is essentially impossible (unless, of course, the company employs an army of sysadmins and the upgrade takes place during the Christma s shutdown). TheBa¯Barexperiment1, which began operation at the Stanford Linear Accelerator C enter in 1999, requires an immense amount of computing power, most of which is currently provided by Solaris SPARC systems. The 300+ -node Analysis and Prompt Reconstruction farm — and especially the 79-node Online Data Flow farm — perform tasks that are critical for experiment operation and demand minimal downtime in case of a hardware m alfunction, operating system crash, or a software upgrade. Without using any special scripts or system administration tools, setting up a Sun Solaris stand-alone “from scratch” and configuring it to make the bes t use of the SLAC computing en- 1Ba¯Barhomepage: http://www.slac.stanford.edu/BFROOT/ .vironment (AFS, NIS, AMD, etc.) and to conform to certain sec urity standards ( ssh,sudo, sendmail ,disabling telnetandrlogin,applying patches) isatask that takes at least 10hours, of which 3 to 5 hours require active administrator involveme nt.Tailor— a collection of sys- tem administration tools developed by the SLAC Computing Se rvices over the past 10 years for many flavors and versions of UNIX — greatly simplifies integra tion of a UNIX workstation into the SLAC environment and saves a lot of time: the process take s 3 to 5 hours with periodical administrator involvement totalling about 1hour per machi ne. Butevenwiththehelpof tailor,managingsuchalargenumberofmachinesisaformidable task, so in Summer 1998 we started looking for ways to make dep loyment or replacement of So- laris machines faster and system management more straightf orward. The three techniques that we have come up with are 1) “non-invasive” hard drive cloning with the help of the Diskless Clienttechnology2, which allows us to make a fully functional copy of a ‘ tailored’ Solaris stand-alone system inabout 20minutes (including changing identity information); 2) Acombina- tion oftailorand Sun’s JumpStart3, which requires a network installation server with custom install/finish scripts that allow tailoring to take place without administrator intervention (this technique of making a Solaris stand-alone takes about 1 hour plus 30 to 60 minutes to initialize the AFScache); and 3) Sun Microsystems’ AutoClient , which will be discussed in the rest of this article. 2 Overview of AutoClient4 Sun Microsystems AutoClient andAdminSuite products were designed to centralize and simplify administration of a large number of Solaris workstations. T o understand what an AutoClient is, let’s compare it withastand-alone system and adiskless cli ent: TableI:AutoClient vs.DisklessClient andstand-alonesystems SystemType LocalFile Systems Local Swap?RemoteFile Systems Network UseRelative Performance Stand-alone Systemroot (/),/usr,/opt, /export/homeYes -none- (meaning“not necessary”)Low High Diskless Client-none- No root (/),swap,/usr,/opt, /homeHigh Low AutoClient Systemcachedroot ( /), cached /usr, cached /optYes root (/),/usr,/opt,/home Low High AutoClient allowsustokeepallclients’ filesystems,except swap,ontheAutoClientServer andlocallycache root(/)andthesharedread-only /usrand/optusingthe CacheFS technology, which isthe most important component of AutoClient . CacheFS cachesfilesthathavebeenaccessedbytheAutoClient,sotha tsubsequentrequests to the same files get referenced to the cache rather than being sent to the server. A cache consis- tencycheck isperformed every24hours byacronjobrunning o ntheAutoClient, onreboot, orat request. Allwritesimmediately updatethebackfilesystem o ntheserver, unless theAutoClient is configured as ‘disconnectable’ and the server is temporaril y unavailable. This consistency check policy relies on the assumption that the cached file systems d o not get changed from the server side except in rare cases by the administrator who explicitl y requests a consistency check after he 2The process isdescribed indetail in[1],section 5. 3TheJumpStart technologyisdescribedinthe SPARC:InstallingSolarisSoftware manualfromSunMicrosystems. 4For amore detaileddiscussion, please refer to[1]and [2].isdone. Specificfilesor directories can be packedinto theCacheFS cache, whichguarantees that theywillalways beinthecacheandwillnot bepurged iftheca chebecomes full. Thisfeature can beparticularly useful with‘disconnectable’ clients. In a nutshell, an AutoClient system has all the advantages of a diskless client (with the e x- ception of not needing ahard drive) whilenot putting aheavy loadonthenetwork andpossessing performance closely matching that of astand-alone system. Since AutoClients do not require any swap space on the server and share /usrand/opt filesystems5, each AutoClient requires only about 40 MB of space on the ser ver for its root file system. In order to backup each AutoClient system, we only need to backup the server. We also can manipulate AutoClient root file systems (read log fil es, apply patches, etc.) directly from the server. AutoClients can be configured to be disconnectable , which means that they will continue to function using their cached filesystems while th e server is temporarily unavailable. AutoClients can be halted and rebooted remotely; they reboo t directly from the cache, so the network traffic during a system-wide reboot is limited to a ca che consistency check. Since no persistent information is stored on the AutoClient itself, it can be considered a field-replaceable unit(FRU).ReplacingafailedunitordeployinganewAutoClienttakes justafewminutes6. Most of the management tasks normally associated with stand-alo ne Solaris systems are thus almost completely eliminated. The quintessence of the centralized administration model, of which AutoClient is a key component, is asignificant reduction of the cost of manageme nt — that is, if everything works as advertised. 3 Ba¯Bar’s experience with AutoClient We started experimenting with AutoClient in June 1998, about a year before Ba¯Bartook its first e+e−collision data. Our first AutoClient server and two dozen or s o AutoClients were Ultra- 5’s with a 270 MHz UltraSPARC-IIi CPU, 128 MB RAM and a 4.3 GB 5, 400 rpm EIDE HDD running Solaris2.6HW3/98; weused thisprototype AutoClie nt farm asconsole andOnlineData Flow machines during the Winter 1998/99 Ba¯Barcosmic ray run. In order to speed up creation of additional AutoClients, wedeveloped a set of scripts tha t ‘clone’ the root file system of a fully configured AutoClient, modify identity-related files in /export/root/ clientname , and make necessary adjustments tothe server configuration files— all inabout one minute. Although theprocess of configuring theserver andthefirstfu lly functional client wasquite bumpy (mostly having to do with getting proper patches insta lled, see [1]), we were satisfied withthefarm’sperformance anddecided tousethe AutoClient technology onall Ba¯Barcomputer farmsat SLAC.Atthistime(January 2000), thereare 309Auto Clients7on6AutoClient Servers8 in the Analysis and Prompt Reconstruction farm, which is loc ated in the SCS building, and 100 AutoClients9and 1 AutoClient Server10in the Online Data Flow and console farms, which are located inthe IR-2 building that houses the Ba¯Bardetector. So,wehavebeenoperatingover400AutoClientsunderrealli feconditions(runningonline, 5The AutoClient Server canalsobe configured toserve its own /usrand/opttothe AutoClients. 6Plus,if AFSis used, the time requiredtobuilda new local AFS cache. 7Currently, Ultra 5’s with a 333 MHz UltraSPARC-IIi CPU, a 9.1 GB EIDE HDD and 256 MB RAM, soon to be replaced with rack-mountable Netra t1’s with a 440 MHz Ultra SPARC IIi CPU, two 9.0 GB 10,000 rpm SCSI HDDs and256 MB RAM. 8Ultra2’s withtwo296 MHz UltraSPARCIICPUsand 9.0GB SCSIHD Ds. 9Mostly Ultra5’s witha 333 MHz UltraSPARC-IIiCPU,a 9.1GB EI DEHDDand 512MB RAM. 10AnEnterprise450serverwithfour296MHzUltraSPARC-IICPU s,2GBRAM,two4.2GBSCSIHDDsandfour 188 GBBaydel RAID Level3 arrays.prompt reconstruction and analysis jobs around the clock at close to 100% capacity) for about 8 months. Overall, the farms performed their goals very well. However, the required management effort turned out to be much bigger than we expected, primari ly because of a bug in CacheFS, which has been identified by Sun Microsystems. A fix for this bu g is reportedly available for Solaris 7, but Sun still has not been able to come up with a fix fo r Solaris 2.6 that Ba¯Baris currently using. Thebugleadstocachecorruption and,occasionally, todisa ppearance offilesontheclient’s root file system during power outages or if connection to the s erver is lost due to a server reboot or crash or a network outage, probably only if the AutoClient was in the process of writing into a file. Suchaccidents have sofar occurred about one aweek, and each timeabout 15-20% of Auto- Clientshadtobemanuallyrebootedwiththe boot -f commandthatforcescachereconstruction; sometimes an AutoClient had to be recloned. This means that a fter an outage the status of each AutoClient has to be checked manually or allAutoClients have to be rebooted with boot -f — either way, this takes alot of time. It also turns out that while in most cases a Solaris stand-alo ne system does not have to be rebooted after a patch is applied to it, AutoClients often do , the reason being differences in the UFSandNFSfilelockingmechanisms. An AutoClient systemhastobeidlebeforepatchingtakes place — otherwise running applications can crash; a global f arm outage has to be scheduled to patch/usr. Wehaveundertakenseveralmeasurestominimizetheimpacto ftheoutages: theAutoClient Serversandnetwork equipment at SCShavebeen connected toU PSpower; thenetwork topology has been modified to remove path redundancies that under cert ain circumstances can lead tobrief periods of network unavailability. We have also realized th at putting the responsibility of being the AutoClient Server on the main IR-2 server was a big mistak e because it often crashed due a kernel memoryleak or had to berebooted. 3.1 Conclusion Atthispoint,wearequitedisappointedbyourexperiencewi thAutoClient ,andunlessSunfixesthe CacheFS buginthenearestfuture,wewillreplaceAutoClientsinthe SCSfarmwithSolarisstand- alonesystemswhichwillbenet-booted fromanet-install se rver using JumpStart andtailor;the recovery strategy in this case would be to reinstall. We are f ar from certain whether we will completely drop the AutoClient technology and think that it has a great potential, so we want to trythemoreclassical approach andseehowthemanagement ef fort comparestousing AutoClient . ThemaingoalofourpresentationatCHEP2000wastomaketheH ighEnergyPhysicscommunity awareofAutoClient ’sexistence, itsprosandcons,andourexperiencewithit—a ndletyoudecide whether you want totry it out or not. Wehope that this goal has been achieved. References 1 A.V. Telnov, “Management of computer farms at BaBar”, BaBa r Note #446, March 29, 1999. This note can bedownloaded from [2]. 2 More information on the use of AutoClient in theBa¯Barexperiment along with a collection of Solaris-related documentation in PDF forma t can be found at http://www.slac.stanford.edu/BFROOT/www/Computing/E nvironment/Admin/ AutoClient/AutoClient.html .

arXiv:physics/0001042 20 Jan 2000 1WAVELET ANALYSIS OF SOLAR ACTIVITY Stefano Sello Thermo – Fluid Dynamics Research Center Enel Research Via Andrea Pisano, 120 56122 PISA - ITALY Topic Note Nr. USG/12012000 (V1.1) ABSTRACT Using wavelet analysis approach, the temporal variations of solar activity on time scales ranging from days to decades are examined from the daily time series of sunspot numbers. A hierarchy of changing complex periods is careful detected and related cycles compared with results from recent similar analyses. A general determination of the main Schwabe cycle length variations is also suggested on the basis of the wavelet amplitude distribution extracted from the local wavelet power map. INTRODUCTION The multiscale evolution of the solar magnetic fields is extremely complicated and this is also transferred to the Sun global activity. The spatial and temporal patterns appearing on the surface of the Sun reflect the underlying dynamo mechanisms which drive the turbulent plasma motions inside the convection region. There exist now many strong indications that the related dynamics is well described, at least for a limited range of scales, by a low dimensional chaotic system [1],[2],[3]. The traditional way to record the variations of solar activity is to observe the sunspot numbers which provide an index of the activity of the whole visible disk of the Sun. These numbers are determined each day and are calculated counting both isolated clusters of sunspots, or sunspot groups, and distinct spots. The current determination of the international sunspot number, Ri, from the Sunspot Index Data Center of Bruxelles, results from a statistical elaboration of the data deriving from an international network of more than twenty five observing stations. Despite sunspot numbers are an indirect measure of the actual physics of the Sun photosphere, they have the great advantage of covering various centuries of time. This property results very crucial when we focus the attention on complex variations of scales and cycles related to the solar magnetic activity.2Many previous works based on standard correlation and Fourier spectral analyses of the sunspot numbers time series, reveal a high energy content corresponding to the Schawbe (~11 years) and a lower energy long term modulation corresponding to the Gleissberg (~100 years) cycles. A clear peak is also evident near the 27 days solar rotation as viewed from Earth. More careful statistical analyses show that these periodicities are in fact strongly changing both in amplitude and in scale, showing clear features of transient phenomena and intermittency [1]. The non stationary feature of those time series is well pointed out by the temporal behaviour of the related high level variance (2nd order moment). Wavelet analysis offers an alternative to Fourier based time series analysis, especially when spectral features are time dependent. By decomposing a time series into time-frequency space we are able to determine both the dominant modes and how they vary with time. Multiscale analysis based on the wavelet approach, has now been successfully used for many different physical applications, including geophysical processes, bio- medicine, and flow turbulence [4]. A high development for astrophysical applications resulted as a consequence of an extension of the wavelet approach to unevenly sampled time series, proposed by Foster in 1996 [5]. There are relatively few and recent works documented in literature, related to applications of wavelet analysis to solar activity, the first one dated back to 1993 by Ochadlick [6]. In the paper the author used a wavelet approach to detect "subtle" variations in the solar cycle period from yearly means of sunspot numbers. Multiscaling and intermittency features have been well analysed through the use of a continuous Morlet wavelet transform on the monthly sunspot numbers time series in the work of Lawrence et al. (1995) [1]. More recently, different authors used longer daily sunspot numbers time series to investigate more accurately new subtle periodicities and their evolutions [7],[8]. The main aim of the present paper is to add a further contribution to the wavelet analyses of solar activity through the application of the Foster projection method (with the estimation of confidence levels), to unevenly sampled time series derived from daily international sunspot numbers (1818-1999) (SIDC) [9], and from less noisy daily group sunspot numbers (1610-1995) (Hoyt and Schatten) [10]. WAVELET ANALYSIS Fourier analysis is an adequate tool for detecting and quantifying constant periodic fluctuations in time series. For intermittent and transient multiscale phenomena, the wavelet transform is able to detect time evolutions of the frequency distribution. The continuous wavelet transform represents an optimal localized decomposition of time series, x(t), as a function of both time t and frequency (scale) a, from a convolution integral:3where ψ is called an analysing wavelet if it verifies the following admissibility condition: where: is the related Fourier transform. In the definition, a and τ denote the dilation (scale factor) and translation (time shift parameter), respectively. We define the local wavelet spectrum: where k0 denotes the peak frequency of the analysing wavelet ψ. From the local wavelet spectrum we can derive a mean or global wavelet spectrum, Pω(k): which is related to the total energy E of the signal x(t) by: The relationship between the ordinary Fourier spectrum PF(ω) and the mean wavelet spectrum Pω(k) is given by:∫+∞ ∞−−= )()(1),(* 2/1attdtxaaWτψ τ ∞< =∫+∞ −2 0^1)(ωψωωψ d c ∫+∞ ∞−−=tietdtωψ ωψ )( )(^ 0 ),(21),(2 0 0≥ = ktkkWkctkP ψω ∫+∞ ∞−= ),( )( tkdtPkPω ω ∫+∞ = 0)(kdkP Eω4indicating that the mean wavelet spectrum is the average of the Fourier spectrum weighted by the square of the Fourier transform of the analysing wavelet ψ shifted at frequency k. Here we used the family of complex analysing wavelets consisting of a plane wave modulated by a Gaussian (called Morlet wavelet) [11]: where ω0 is the non dimensional frequency here taken to be equal to 6 in order to satisfy the admissibility condition. For a more comprehensive and detailed description of the wavelet formalism see references [11], [12]. Following Foster (1996) [5] here we consider an extension of the above wavelet formalism in order to correct handle irregularly sampled time series. The wavelet transform is viewed as a suitable weighted projection onto three trial functions giving the Weighted Wavelet Z transform and the Weighted Wavelet Amplitudes. For all the mathematical details of this formulation and its applications we refer to Foster (1996) and Haubold (1998) papers [5],[12]. Many applications of the wavelet analysis suffered from an apparent lack of quantitative evaluations especially by the use of arbitrary normalization and the lack of statistical significance test in order to estimate the reliability of results. Here we used power spectrum normalization and significance levels following the approach suggested by Torrence et al. (1997). We first assume an appropriate background spectrum and we suppose that different realizations of the considered physical process will be randomly distributed about this expected background spectrum. The actual spectrum is compared against this random distribution. In the present work we assumed as background spectrum a red noise spectrum modeled through a univariate lag-1 autoregressive process: where zn is derived from a Gaussian white noise and α is the lag-1 autocorrelation here estimated by:2 00^ )()(1)( ∫∞+ =kkPdkckPFωψωω ψω 2/ 4/12 0)(ηηωπηψ− −= eei n n n zxx + =−1α5where α1 and α2 are the lag-1 and lag-2 autocorrelations of the considered time series. The discrete normalized Fourier power spectrum of this red noise is: and the following null hypothesis is defined for the wavelet power spectrum: we assume the red noise spectrum as the mean power spectrum of the time series; if a peak in the wavelet power spectrum is significantly (here at 95% confidence level) above this background spectrum, then it can be considered to be a real feature of the time series (see [11]). SOLAR ACTIVITY: SUNSPOT NUMBERS We considered here two records of solar activity: 1) the daily international number of sunspots from SIDC archive covering the time interval: 1818-1999 and consisting of 66474 observations; 2) the daily number of sunspots groups visible on the Sun surface between 1610 and 1995 recently made available by Hoyt and Schatten [10], and consisting of 111358 ob servations. These time series are irregularly sampled especially in the first part of the records. The importance of the last time series results in its more internally self- consistency and in its lesser noise level than that of the Wolf sunspot numbers. It uses only the number of sunspot groups observed, rather than groups and individual sunspots. The group sunspot numbers use 65941 ob servations from 117 observers active before 1874 not used by Wolf in its original time series. The noise contained in this time series is considerably less than the noise contained in the Wolf sunspot numbers. Of course the numerical computation of the wavelet spectrum is here very expensive and time consuming due to the high number of data contained in the record. Moreover, recently Ballester et al. (1999) [8], analyzing the daily group sunspot numbers with a wavelet approach, detect local episodes of the periodicity near 158 days (varying from 140 to 170 days) around the maximum of solar cycle 2 and around the maxima of solar cycles 16-21. They claimed that the presence of that periodicity in the group sunspot numbers confirms that it is caused by a periodic emergence of magnetic flux. A near 154 days periodicity in the Sun was first reported on gamma ray flares, Rieger et al. 1984 [13], and in other related parameters but it does not seemed to be a persistent periodicity. It is22 1ααα+= )/2cos(211 22 NkPkπααα −+−=6well known that solar flares are huge explosions on the surface of the Sun, involving time scales of a few minutes and they are thought to be formed as magnetic fields structures related to sunspots are twisted, releasing energy by magnetic reconnections. During solar cycle 21 a periodicity between 152, 158 d ays in the occurrence rate of energetic flares was detected by the Solar Maximum Mission satellite. If we are able to confirm a temporal and frequency agreement in the wavelet analysis of the group sunspot numbers, we can support the emergence of magnetic flux in the solar photosphere as the common factor relating the two phenomena. - Daily International Sunspot Numbers Figure 1 shows the results of the wavelet analysis applied to the series of daily international sunspot numbers. The upper part shows the original time series in its natural units (red line) and the monthly smoothed sunspot numbers (green line) generally used for solar cycle predictions. Time is here expressed in years. The central part shows the amplitudes (WWA) of the wavelet local power spectrum in terms of an arbitrary colour contour map. Red higher values are strong energetic contributions to power spectrum, while blue lower values are weak energetic contributions. Horizontal time axis corresponds to the axis of time series and vertical scale (frequency) axis is for convenience expressed in log values of cycles per year-1. Thus the range analyzed is between 148 years (value -5) and 2.5 days (value 5). The right part shows the mean global wavelet power spectrum (red line) obtained with a time integration of the local wavelet power spectrum, and the 5% significance level using a red noise autoregressive lag-1 background spectrum with α=0.96 (green line). It is possible to distinguish two main regions corresponding to sharp Schwabe (~11 years) and broad Gleissberg (~100 years) with dominant energetic contribution above the red noise background spectrum. In particular it is well evident the complex evolution of the Schwabe cycle both for scales and amplitudes. The irregular finite extension in the map represents a continuous interaction of frequencies. It is interesting to note the presence of a non persistent significant region around a period of 28 years starting from 1950. In this map there are not clear evidences of episodes related to 158 days periodicity (value 0.8). In fact, for periods below 3 years there are not significant contributions in the global energy spectrum, suggesting dominant stochastic motions [1] with little exceptions. Note the complex shape regions around maxima of cycles related to 27 days solar rotation, as viewed from Earth.7Figure 1 - Daily Group Sunspot Numbers The original record of daily group sunspot numbers was limited between 1653 and 1995 because of the lack of reliability for observations before 1653. The total number of observation is then reduced to 104108. The lower global level of noise in this series is evidenced by a reduction of the variance of about 25%. On the other hand, here the red noise autoregressive lag-1 background spectrum was obtained with α=0.85, i.e. with more features typical of white noise. Figure 2 shows the results of the wavelet computation in the same way of previous analysis. 8Figure 2 The high frequency range explored was limited up to 49 days period, due to computational restrictions. As in the previous analysis, only two dominant peaks were detected, corresponding to the Schwabe and the Gleissberg cycles. Here the longest cycle was detected more precisely, near 110 years. The interesting part of this wavelet analysis is related to the long Maunder minimum ending at 1715 and the next weaker minimum between 1795 and 1820. The evolution of the main cycle near 11 years, as derived from both the wavelet analyses, results quite similar. There is a strong increase of the energy contribution for recent cycles, with a continuous irregular variation of the region related to this periodicity, currently shifted toward high value frequencies. Moreover, a detailed inspection of the wavelet map between 140 and 170 days periods, shows again the absence of significant contributions localised near a 158 days periodicity for the group sunspot numbers. Thus, our wavelet analysis does not support the correlation between the periodicity detected in some high energy solar flares and the periodic emergence of magnetic flux associated to sunspot groups, as indicated in [8]. 9- Solar Cycle Length The long term variations of the solar cycle length have been widely studied and its importance is mainly due to their suggested correlation to global climate [14]. Recently Mursula et al. [15], proposed a new method to determine the solar cycle length based on a difference between the median activity times of two successive sunspot cycles. The advantage of this method, with respect to a conventional approach, is that the median times are almost independent of how the minima or maxima are determined and thus the computation results more accurate. More recently, Fligge et al. [16], proposed a more objective and general cycle length determination using a continuous wavelet transform. Here we propose an alternative general method to determine time variation of the main solar cycle, based on the amplitudes of the wavelet map for sunspot group numbers (Figure 2). The main idea is to extract a characteristic frequency (or period) for each time, from an irregular shaped region related to the main solar cycle. From the global wavelet power spectrum we selected a suitable range of frequencies, including the irregular region related to the Schwabe cycle: ω0 =-3 (20 years), ω1 =-1.6 (5 years). We then define a characteristic frequency of the wavelet map, at a given time t, as: where w(W(ω,t)) are proper weights derived from the intensity of the local wavelet power map. This characteristic frequency gives a wavelet based evaluation of the evolution of the periods related to the selected range of frequencies. When we apply the above relation to the main Schwabe cycle we obtain the result shown in Figure 3. The central part of the Figure shows the behaviour of the characteristic periods or lengths of the solar cycle. For a comparison, we show also the solar cycle lengths obtained with the median method (blue line) (from [15]). The qualitative behaviour is quite similar, even if the wavelet method gives, in general, lower values. The accuracy in the determination of time variations of the solar cycle lengths is very high for the wavelet method which is completely independent of the arbitrary choice of exact times of sunspots minima or maxima. The long term evolution of the solar cycle length is then confirmed to be in good agreement with the climate changes [15], where high periods are related to weak solar activity and low periods are related to strong solar activity, as well depicted in the most recent cycles.∫∫ = 1 01 0 )),(()),(( )(~ ω ωω ω ωωω ωω ω tWwdtWwd t10Figure 3 CONCLUSIONS Wavelet analysis approach allows a refined investigation of the temporal variations of solar activity on time scales ranging from days to decades. Considering the daily time series of international sunspot numbers and group sunspot numbers, we analyzed a hierarchy of changing complex periods. In particular, we detected statistically significant periodicities and we excluded a clear evidence of the existence of a near 158 days periodicity correlated to high energy solar flares. A general and accurate determination of the main Schwabe cycle length variations was also suggested on the basis of the wavelet amplitude distribution, extracted from the local wavelet power map. 11REFERENCES [1] Lawrence, J.K., Cadavid, A.C., Ruzmaikin, A.A. (1995) ApJ, 455,366 [2] Tobias, S.M., Weiss, N.O., Kirk, V. (1995) Mont.Not.R.Astron. 502,273,1150 [3] Qin, Z. (1996) Astron. Astrophys. 310,646 [4] Farge, M., Kevlahan, N., Perrier, V., Goirand, E. (1996) Proc. IEEE, 84, 4 [5] Foster, G. (1996) Astron. J. 112, 4 [6] Ochadlick, A. R., Kritikos, H.N., Giegengack, R. (1993) Geophys. Res. Lett. 20,14 [7] Frick, P., Galyagin, D., Hoyt, D. V., Nesme-Ribes, R., Schatten, K. H., Sokoloff, D., Zakhorov, V. (1997) Astron. Astrophys. 328,670 [8] Ballester, J.L., Oliver, R., Baudin, F. (1999) ApJ. 522,2 [9] Sunspot Index Data Center Bruxelles http://www.oma.be/KSB-ORB/SIDC/index.html [10] Hoyt, D. V., Schatten, K. H., (1998) Solar Phys. 181,2 [11] Torrence, C., Compo, G. P., (1998) Bull. 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arXiv:physics/0001043v1 [physics.space-ph] 20 Jan 20003-Dextentof themainionospherictrough —a casestudy Mikael Hedin1,IngemarH ¨aggstr¨om1,AstaPellinen-Wannberg1,LailaAndersson1, UrbanBr ¨andstr¨om1,Bj¨ornGustavsson1,˚Ake Steen1,AssarWestman2,Gudmund Wannberg2,TonyvanEyken3,Takehiko Aso4,CynthiaCattell5,DaveKlumpar6 and CharlesW. Carlson7 1SwedishInstituteofSpacePhysics,Box812,S-98128Kiruna ,Sweden. 2EISCATScientific Association,Box812,S-98128Kiruna,Swe den. 3EISCATScientific Association,Postboks432,N-9170Longye arbyen,Norway. 4NationalInstituteofPolarResearch,1-9-10Kaga,Itabash i-ku,Tokyo173,Japan. 5SchoolofPhys. andAstr.,Univ. ofMinnesota,MN 55455,USA. 6SpaceSciencesLaboratory,Univ. ofCalifornia,Berkeley, CA 94720,USA. 7Lockheed-MartinPaloAltoResearchLabs,CA 94304,USA. Abstract TheEISCATradar system hasbeen used forthefirsttimein afou r-beam meridional mode. The FASTsatellite and ALISimaging system isusedin conjunctio n to supporttheradar data, which was usedto identify amain ionospheric trough. With this lar gelatitude coverage thetrough was passed in 21 2hours period. Its 3-dimensional structure is investigated and discussed. Itis found thattheshapeiscurvedalongtheauroraloval, andthatthet roughiswiderclosertothemidnight sector. The position of the trough coincide rather well with various statistical models and this trough isfound to be atypical one. 1 Introduction The main ionospherictroughis a typical featureof the sub-a uroralionosphericF-region, where it is manifest as a substantial depletion in electron c oncentration. It is frequently observed in the nighttime sector, just equatorward of the au roral zone. This trough is oftenreferredtoasthe“mainionospherictrough”or“mid-l atitudetrough”todistinguish fromtroughsinotherlocations. Thepolaredgeofthetrough isco-locatedwiththe auro- ral zone. The equatorward boundaryis less distinct, consis ting of a graduallyincreasing amount of electrons, towards the plamasphere, which could b e called the normal iono- sphere. Inside the trough,there are E-fields present, givingrise to westward ion convec- tion. Extensivereviewsofmodellingandobservationsofth emainionospherictroughare givenbyMoffettandQuegan(1983)andRodgeretal. (1992). The present study was performed as a part of International Au roral Study (IAS). The goalforIASwas toprovidesimultaneousobservationsfromg roundandspaceof auroral processes. An importantpart ofIAS wasthe FAST (Carlsonet a l., 1998)satellite. FAST was planned with significant ground-base support, i.e.,control station and supporting scientificinstrument,inAlaska,butasallpolarorbitings atellitesalsopassesovernorthern Scandinavia,itiswellsuitedforcoordinationwithground -basedinstrumentthereaswell. The most important, besidesthe EISCAT radars, is ALIS (Br¨ a ndstr¨ omand Steen, 1994), thecameranetworkinnorthernSwedenforauroralimaging. 2 The 4-beamEISCAT radarconfiguration Previous radar experiments have used different scanning pa tterns to determine the to- pographyofthetrough;CollisandH¨ aggstr¨ om(1988)useda widelatitudescan(EISCAT commonprogramexperimentCP-3),CollisandH¨ aggstr¨ om(1 989)usedasmall4-position scan(CP-2),andJonesetal.(1997)usedacombinationofawi descantofindthetrough, and a narrow scan to observe the structure. Also a single posi tion tri-static radar mode (CP-1)hasbeenusedbyH¨ aggstr¨ omandCollis(1990). These methodsallgiveatrade-off2 between time and space resolution—the more scan points the l onger time before repeti- tion. Hereweuseforthefirst timealltheEISCATradarsinafour-be amconfigurationclose tothemeridianplanetogetawideareaofobservationwithou tlosingtimeresolution. The EISCAT Svalbard Radar (ESR) (Wannberget al., 1997)is field a ligned (elevation81.5◦) atinvariantlatitude(ILAT)75.2◦N.Themainlandtri-staticUHFsystem(Folkestadetal., 1983)is also field aligned (elevation77.4◦) at ILAT 66.3◦N. Between these, the Tromsø VHFradarisusedina“split-beammode”,withtheeasternant ennapanelspointingnorth (70◦elevation) and the western panels pointing vertical. The da ta is split into two sets, VHF-N(north)andVHF-V (vertical). The combined “meta-radar” has a huge fan-like observation a rea around 70◦N–80◦N ingeographiclatitude. 3 Characteristicsoftheobservedtrough The radar observations of the trough in question are shown in figure 1. The trough is seen as the clear decrease in electron density, and we determ ine the time in UT for the radarspassingunderthetroughtobe1710–1720,1820–1900, 1845–1925and1900–1940 respectively. The more prominentdensity increase for the T romsø sites after 2000UT is nottheedgeofthe trough,rathertypicalF-regionblobs,po lewardofthetrough. In figure 1 we see that the troughextendsthroughall of the F-r egion,but not in the E- region. Note that white colorisbothhighest densityandnou sable data, butgenerallyno dataisduetolowdensityandthisisseentobethecaseherein thetrough. Inthenorthern part of the trough, we can actually see some typical weak ioni zation in the E-region, interpretedasdiffuseaurora. We compare the actual location of the trough minimum with pre dicted positions from modelsbyCollisandH¨ aggstr¨ om(1988);K¨ ohnleinandRait t(1977);RycroftandBurnell (1970) respectively1—all linear in SLT (solar local time) and Kp. For the present day, Kpvaluesare 1−beforeand1oafter1800UT.Infigure2,thesemodelvaluesare shown together with the actual position of the whole trough as dete rmined from ESR, FAST, VHF-N, VHF-V and UHF respectively(fromhighto low latitude ). We see that the devi- ationsfromthemodelpredictionsaresubstantial,andconc ludethatthelinearmodelsare notadequateforuseoverawiderangeinlatitudeandtime. Th isisnotsurprisingbecause the fits used to construct the equationsall had a big spread, e ven though the correlations werequitegood. A morerecent studybased on satellite data is made byKarpach evet al. (1996),where they use both latitude and longitude to make a statistical mo del not restricted to linear relations. In this model, the time used is magnetic local tim e (MLT), which is reason- able because the trough structure is governedby the earth ma gnetic field. They use both linear and non-linear time dependency, and find that for midn ight hours, the difference is rather small, compared to the large spread in the data, but favours the non-linear time dependency. From the start time of when each radar beam enters the trough, as seen in figure 1, we can estimate the trough apparent southward speed to be aro und 12 km/min from Longyearbyen to Tromsø, and around 4 km/min between the Trom sø beams. Assum- ing that the trough appears to pass over ESR and between the ma inland radars with the respectivespeed,wecanmakeacrudeestimateofthevertica lshapeofthetroughandthe width: Fromtheplotinfigure1,weestimatethatthesouthwar d(early)wall,takenasthe border of blue and green color, has an inclination of 35◦for the VHF-N beam and 21◦ for VHF-V. This should be comparedwith the direction of the m agnetic field at Tromsø, zenithangle13◦southwards. Ifwe assume the troughwall to be fieldaligned,t he antici- patedinclinationis 33◦(20+13)forVHF-N and13◦forVHF-V, in goodagreementwith the roughestimate. In the plotof the field alignedbeams, ESR andUHF, the troughwall appears vertical, which means it is field-aligned. The passi ng time for Longyearbyenis 10 minutes, which, usingthe abovespeeds, givesa width of 12 0 km, andfor the Tromsø 1with Rycroft and Burnell (1970) changed to invariant latitu de as suggested by K¨ ohnlein and Raitt (1977)3 Fig. 1.EISCAT electron density (ESR and UHF) and raw electron densi ty (VHF-N and VHF-V) plots for 970314. The electron concentration (in m−3) is colour coded, with UT on horizontal and height (km) on vertical axis.4 181920212223606264666870727476Invariant latitude ( °N) Solar local timeCH KR RB Fig. 2.The thick lines show the coordinates in ILAT and SLT (solar lo cal time) of the actual pass of the trough as determined from (high to low latitude) ESR, FAST, V HF-N, VHF-V and UHF. The dash-dotted line is an estimate of the trough boundaries, extrapolated to lat er time from these direct measurements. The diffuse aurora observed with ALIS is indicated by the small spot in th e latest part of the plot, just above the border of the estimated trough boundary. The solid lines show predict ions for location of trough minima from models discussed in the text. beamsthe time is 40 minutes, which gives a width of 160 km, bot h in north-southdirec- tion. This assumes the trough to consist of two straight part s, one extending from over LongyearbyenandTromsø,theotherovertheTromsøbeams,in steadoftheactualarc-like shape as seen in figure 2, but it will anyway givean indication of the size. The east-west size canbe estimatedbyusingtheearthrotation,thisgives a velocityof6.1km/min,and a sizeof61kmforLongyearbyenandaround10km/minwith size 400kmoverTromsø. The actual width, measured perpendicular over the trough ex tent is then 54 km for the part passing over Longyearbyen and 149 km for the part passin g over Tromsø. That is, the trough radial width is actually broader equatorwards, c loser to magnetic midnight. Thisisshowninfigure2. The FAST satellite does not carry sounding instruments, so t rough signatures are not as obvious to detect. The signature used is the precipitatio n north of the trough and the electricfieldassociatedwiththeion( E×B)drift(absentsouthofthetrough). Itisknown fromearlierstudies(CollisandH¨ aggstr¨ om,1988)thatth ereisawestwardionconvection inthetrough,whichisabsentoutside. Thesatellitepasses overthetrough1742–1743UT asshownin figure2. During the night, the auroral imaging system ALIS was not ope rating continuously becausetherewasnosignificantauroraandit waspartlyclou dy. However,somepictures were taken at relevant times, and they show faint diffuse aur ora around the time when ALIS passes under the trough poleward boundary. If we plot th e position of ALIS at the time of diffuse aurora, it located just in the northern pa rt of the trough region as extrapolatedfromtheothermoredirectmeasurementsin figu re2. Ifwecomparetheobservedtroughwiththerelevanttypicalf eaturesofthemid-latitude trough as described in Moffett and Quegan (1983), we can say t hat the present trough is quitetypical.5 4 Conclusions The EISCAT facility has been used in a new four-beam meridion al mode. The ESR and UHFsystemarefield-aligned,andtheVHFsystemissplitintw obeamsinbetween. This gives a very wide latitude range for observation, well suite d to study ionospheric struc- turesmovingoverlargerangeinshorttime. Withthisconfigu ration,notimeresolutionis lost. Inthiscase,themainionospherictroughhasbeenobserveda shighas75◦NILATdown to 66◦N ILAT. The observedtroughis a quitetypical one–it hasall t he commonfeatures knownfromearlierstudies. Thepresentobservationismade whenthe Kpindexwaslow andstable,andsothetroughwasalsoquitenon-dramatic,bu tthismeansthatthetroughis ratherstationaryoverthebigareaofobservation. Thisrul esoutsignificanttimevariations ofthetroughposition,aswouldhavebeenthecaseinamoreac tiveenvironment,enabling investigationovera relativelylongperiodastheearthmov esunderit. Ifwecalculatetheapparentsouthwardmotionofthetrough, thespeedisover10km/min betweenLongyearbyenandTromsø,butabout4 km/minbetween the Tromsøsites. This is consistent if the trough is really an oval shape passing ov er ground, so that it passes Longyearbyenclose to perpendicular,but at Tromsølatitud esthe directionismuchmore oblique, thereby the apparentsouthward motion is much slow er. The trough is also seen to be wider towardsmagnetic midnight. The earlier proposed linear equationsfor trough motionsareshownnottobevalidoverthelatituderangeinqu estion. Coordinated studies poses substantial difficulties, most o f which are not scientific but rather administrative or probabilistic by nature. Neverth eless, if one measurement (EIS- CAT in this case) is good,the otherscan often be used to extra ct some extra information insupport. Acknowledgement. We gratefully acknowledge assistance of the EISCAT staff. T he EISCAT Scientific Asso- ciation is supported by France (CNRS), Germany (MPG), Unite d Kingdom (PPARC), Norway (NFR),Sweden (NFR), Finland (SA) and Japan(NIPR). References Br¨ andstr¨ om, U. and Steen, ˚A., ALIS - a new ground-based facility for auroral imaging in northern scandinavia, in Proceedings of ESA Symposium on European Rocke and BalloonProgrammes ,ESA SP-355,ESA,1994. Carlson,C.W.,Pfaff,R.F.,andWatzin,J.G.,Thefastauror alsnapshotmission, Geophys. Res.Lett.,25,2013–2016,1998. Collis, P. N. and H¨ aggstr¨ om, I., Plasma convection and aur oral precipitation processes associatedwiththemainionospherictroughathighlatitud es,J.Atmos.Terr.Phys. ,50, 389–404,1988. Collis, P. N. and H¨ aggstr¨ om, I., High resolution measurem ents of the main ionospheric troughusingEISCAT, Adv.SpaceRes. ,9,545–548,1989. Folkestad,K., Hagfors,T., and Westerlund, S., EISCAT: An u pdateddescriptionof tech- nicalcharacteristicsandoperationalcapabilities, RadioSci. ,18,867–879,1983. H¨ aggstr¨ om,I.andCollis,P.,Ioncompositionchangesdur ingF-regiondensitydepletions inthepresenceofelectricfieldsataurorallatitudes, J.Atmos.Terr.Phys. ,52,519–529, 1990. Jones, D. G., Walker, I. K., and Kersley, L., Structure of the polward wall of the trough and the inclination of the geomagnetic field above the EISCAT radar,Ann. Geophys. , 15,740–746,1997. Karpachev, A. T., Deminov, M. G., and Afonin, V. V., Model of t he mid-latitude iono- spheric trough on the base of cosmos-900 and intercosmos-19 satellites data, Adv. SpaceRes. ,18,6221–6230,1996.6 K¨ ohnlein,W.andRaitt,W.J.,Positionofthemid-latitude troughinthetopsideionosphere asdeducedfromESRO 4observations, Planet.SpaceSci. ,25,600–602,1977. Moffett,R.J.andQuegan,S.,Themid-latitudetroughinthe electronconcentrationofthe ionosphericF-layer: a review of observationsand modeling ,J. Atmos. Terr. Phys. ,45, 315–343,1983. Rodger, A. S., Moffett, R. J., and Quegan, S., The role of ion d rift in the formation of ionisation troughs in the mid- and high-latitude ionospher e—a review, J. Atmos. Terr. Phys.,54,1–30,1992. Rycroft, M. J. and Burnell, S. J., Statistical analysis of mo vement of the ionospheric troughandtheplasmapause, J.Geophys.Res. ,75,5600–5604,1970. Wannberg, G., Wolf, I., Vanhainen, L.-G., Koskenniemi, K., R¨ ottger, J., Postila, M., Markkanen,J., Jacobsen, R., Stenberg, A., Larsen, R., Elia ssen, S., Heck, S., and Hu- uskonen, A., The EISCAT Svalbard radar: A case study in moder n incoherent scatter radarsystem design, RadioSci. ,32,2283–2307,1997.

arXiv:physics/0001044v1 [physics.optics] 20 Jan 2000Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect Luca Matone1, Matteo Barsuglia, Fran¸ cois Bondu2, Fabien Cavalier, Henrich Heitmann2, Nary Man2 Laboratoire de l’Acc´ el´ erateur Lin´ eaire, Universit´ e Paris-Sud, Bat 208, 91405 Orsay (France) Abstract We here present an investigation of the ringing effect observ ed on the VIRGO mode-cleaner prototype MC30. The results of a numerical cal culation show how a simple empirical formula can determine the cavity expansio n rate from the oscilla- tory behavior. We also show how the simulation output can be a djusted to estimate the finesse value of the suspended cavity. 1 Introduction Interferometric gravitational wave detectors, like VIRGO [1], LIGO [2], GEO [3] and TAMA [4], make use of suspended Fabry-Perot cavities for their proper- ties: spatial filtering of the laser beam, optical path ampli fication and power recycling to name a few. Due to the suspension system, the mot ion of each mirror is dominated by oscillations at the pendulum’s funda mental resonance, typically below 1 Hzand with amplitudes as large as tens of a laser wave- length. A VIRGO mode-cleaner prototype MC30 [5], which operated in O rsay for sev- eral years, consists of a 30 mtriangular Fabry-Perot cavity with two possible finesse values, 100 or 1600, depending on the incoming laser b eam polariza- tion state. The two-stage suspension system, together with a local damping 1Corresponding author. Present address: MS 18-34 LIGO, Calt ech, Pasadena CA 91125 (USA); tel.(626) 395-2071; fax (626) 304-9834; lmato ne@ligo.caltech.edu 2Pre.address: Observatoire Cˆ ote d’Azur, B.P. 4229, 06304 N ice Cedex 4 (France) Preprint submitted to Elsevier Preprint 2 February 2008control system, results in a residual RMS displacement valu e, for each mirror, of 0.8µm. Prior to the lock acquisition, the cavity length sweeps the o ptical resonance at different rates of expansion. If the relative velocity bet ween the mirrors is constant, the DC transmitted power delineates the Airy peak as a function of time, easily observed for the optical system with F= 100. However, in the case of F= 1600, a deformation of the Airy peak, similar to a ringing, w as observed (see fig.(1)). Both [6,7], and references therein, discuss this phenomeno n. Briefly, this effect arises once the cavity sweeps the optical resonance in a time τswof the order of or less than the cavity storage time τst= 2L0F/c π, where L0= 30mis the cavity length and cthe speed of light. This effect is observed when the rate of expansion is so high th at, as resonance is approached, the cavity doesn’t have enough time to comple tely fill itself. It is the beating between the incoming laser field and the evolvi ng stored field that gives rise to this oscillatory behavior. The goal of this letter is to present the informations which c an be extracted from such deformation, in particular, the finesse of the cavi ty and the relative mirror speed. 2 The ringing effect in Fabry-Perot cavities The model used for this study is shown in fig.(2). Assuming a ne gligible mirror displacement for times of the order of the round trip time of l ightτ= 2L0/c= 0.2µs, the stored field Ψ 1(t) at time tcan be written as Ψ1(t) =t1Ψin+r2 1exp(−2i k L) Ψ1(t−τ) (1) where r1andt1denote the amplitude reflectivity and transmittivity of eac h mirror, Ψ inis the incoming laser field, and Lis the cavity length. Assuming that the cavity expands at a constant rate v, we can write L=L0+v tand solve eq.(1) iteratively, for different velocities vand finesse F. Fig.(3) shows the Airy peak for three velocities: v= 0 (static approximation), v= 1λ/s, and v= 2.6λ/s, with F= 4000. The curve labeled static, corre- sponding to v= 0, was generated by neglecting the travel time of light, i.e the cavity has reached its equilibrium point at each step. The tw o other curves, on the other hand, were simulated according to the dynamical model here presented. Notice how the main peak height decreases, its wi dth increases and 2its position shifts ahead of the resonance. These changes ar e greater for larger velocities. 3 The speed measurement We would now like to discuss a property of the ringing effect ob served from the simulation runs. Fig.(3) graphs the stored power as a fun ction of cavity length for a given finesse and for different values of velocity . We can now plot the stored power as a function of time, setting the velocity t o a fixed value, but varying the finesse. One example is given in fig.(4). The to p graph of this figure shows the stored power as a function of time, for an expa nsion rate set tov= 10λ/s, for three different finesse values: F= 1000, 2000, and 3000. The bottom graph is the curves’ time derivative. From these p lots, we remark a particular characteristic of the phenomenon: the positio n of the minima and maxima, with the exception of the main peak, are almost indep endent from the finesse value. Furthermore, going back to fig.(3), we can now note that the de rivative zeros depend only on the relative mirror velocity. The simulation output shown in fig.(4) not only shows how the derivative zeros are independe nt, at least to first approximation, from the finesse, but it also shows a part icular regularity in the spacing between the minima and maxima. The upper graph of fig.(5) shows the simulated stored power of a cavity with F= 3500, expanding at a rate 10 λ/s. Let the position of the curve’s derivative zeros, tzero, be labeled by the index n, so that, for the first zero, positioned at tzero≃5.07ms,n= 0, for the second zero, located at tzero≃5.103ms,n= 1 and so on. Then, the bottom graph of fig.(5) shows the plot of in dexnas a function of time. We remark that the n-th zero of the derivative is a quadratic function of the zero crossing time tzero:n∝t2 zero. By fitting the simulation outputs to the expression nzero=p1+p2tzero+p3t2 zerowhere p1,2,3are fitting parameters, we empirically found that the coefficient p3can be written as p3=cv/λL where Lis the cavity length and vis the cavity expansion rate (an example is shown in the bottom graph of fig.(5)). Therefor e, an estimate of coefficient p3would also give us an estimate of the relative velocity v. Fig.(6) shows the results of a fit on a measured event. Notice h ow the parabolic behavior is in agreement with the experimental points, resu lting in a measured speed of 12 .8±10−2µm/s. 34 The Finesse measurement Once the speed is extracted, it is possible to fit the measurem ents with the simulation’s output to find the remaining parameter: the fine sse. The fit results of a set of measurements, six of which are shown in fig.(7), led to a mean finesse ofF= 1554 ±160, a value later confirmed by a measurement of the cavity pole. The ten percent precision on the finesse measurement is most probably due to the alignment state of the cavity, as suggested by simu lation studies. 5 Conclusion An analysis of the optical ringing effect, observed on the VIR GO mode-cleaner prototype in Orsay, was here presented. We investigated a me thod to extract, from the oscillatory behavior, both the relative mirror spe ed and the finesse of the system. The numerical results showed how the position of the oscilla tions’ minima and maxima, when plotted as a function of time, weakly depend on t he finesse value and are completely determined by the cavity expansion rate a s the resonance is being crossed. In particular, we showed how a simple empir ical formula can determine the cavity expansion rate by observing these mini ma and maxima. Once the speed was reconstructed, it was possible to fit the me asurements with the simulation’s output and estimate the cavity’s finesse to F= 1554 ±160. The present letter gives an alternative method to the finesse measurement for a suspended cavity. The simplicity in the velocity reconstr uction algorithm may be useful for a future control system capable of guiding t he cavity into lock. References [1] C. Bradaschia et al, Nucl. Intrum. Meth. Phys. Res. A 289,518 (1990). [2] A. Abramovici et al, Science 256,325 (1992). [3] K. Danzmann et al, Internal Report MPQ 190(1994). [4] N.Kanda et al, Proceedings of second Workshop of Gravitational Wave Data Analysis , Orsay 1997 [5] M.Barsuglia et al., submitted to Rev.Sci.In 4[6] J.P´ erˆ ome et al., J.Opt.Soc.Am.B ,142811 (1997) [7] S. T’Jampens, Rapport de Stage de Licence au LAL (1996) 5Figure Captions Fig.1: The observed ringing effect on the transmitted DC power of the MC30 prototype. The transmitted power is shown as a function of ti me as the cavity length sweeps the optical resonance at an unknown rate. Fig.2: The model used for the study of the MC30 ringing effect. Fig.3: The calculated Fabry-Perot transmitted power, with F= 4000, as a function of cavity length ∆ Las the resonance is swept at v= 0 (static ap- proximation), v= 1λ/s, and v= 2.6λ/s. In the figure, ∆ L= 0 corresponds to resonance. Fig.4: The calculated stored power as a function of time, wit h a fixed ex- pansion rate set to v= 10λ/s, for different finesse values: F= 1000, 2000, and 3000. Top graph: the stored power. Bottom graph: the stor ed power time derivative. Fig.5: The simulated stored power of a Fabry-Perot, expandi ng at a constant ratev= 10λ/s, withF= 3500. Top graph: the stored power as a function of time. Bottom graph: the index n, corresponding to the n-th derivative zero, as a function of time. The curve is fit to the expression n=p1+p2t+p3t2. Notice that p3=cv/λL = 100[1 /ms2]. Fig.6: Fit results for the mirror relative velocity reconst ruction.On the left: The measured DC transmitted power. On the right: the plot of tzeroas a func- tion of index n. The error bars correspond to half of the oscilloscope’s sam pling time. The reconstructed speed is 12 .8±10−2µm/s. Fig.7: The observed ringing effect for the MC30 prototype: me asurements and fits. The finesse and velocity values are shown for each graph. TDCin arbitrary units. 6Fig. 1. Matone et al., Phys.Lett.A 7ΨΨL ΨT 1IN Fig. 2. Matone et al., Phys.Lett.A 8Fig. 3. Matone et al., Phys.Lett.A 9Fig. 4. Matone et al., Phys.Lett.A 10Fig. 5. Matone et al., Phys.Lett.A 11Fig. 6. Matone et al., Phys.Lett.A 12Fig. 7. Matone et al., Phys.Lett.A 13

arXiv:physics/0001045v1 [physics.atom-ph] 20 Jan 2000Large-scale Breit-Pauli R-matrix calculations for transi tion probabilities of Fe V Sultana N. Nahar and Anil K. Pradhan Department of Astronomy, The Ohio State University Columbus, Ohio 43210, U.S.A. February 9, 2008 Abstract Ab initio theoretical calculations are reported for the ele ctric (E1) dipole allowed and inter- combination fine structure transitions in Fe V using the Brei t-Pauli R-matrix (BPRM) method. We obtain 3865 bound fine structure levels of Fe V and 1 .46×106oscillator strengths, Ein- stein A-coefficients and line strengths. In addition to the re lativistic effects, the intermediate coupling calculations include extensive electron correla tion effects that represent the complex con- figuration interaction (CI). For bound-bound transitions t he BPRM method, based on atomic collision theory, entails the computation of the CI wavefun ctions of the atomic system as an (electron + target ion) complex. The target ion Fe VI is repre sented by an eigenfunction expan- sion of 19 fine structure levels dominated by the spectroscop ic configuration 3 d3, and a number of correlation configurations. Fe V bound levels are obtaine d with angular and spin symmetries SLπandJπof the (e + Fe VI) system such that 2 S+1 = 5,3,1, L≤10,J≤8. The bound levels are obtained as solutions of the Breit-Pauli (e + ion) Hamilt onian for each Jπ, and are designated according to the ‘collision’ channel quantum numbers. A maj or task has been the identification of these large number of bound fine structure levels in terms o f standard spectroscopic desig- nations. A new scheme, based on the analysis of quantum defec ts and channel wavefunctions, has been developed. The identification scheme aims particul arly to determine the completeness 1of the results in terms of all possible bound levels with n≤10, l≤n−1, for applications to analysis of experimental measurements and plasma modeling . Sample results are presented and the accuracy of the results is discussed. A comparison of the dipole length and velocity oscillator strengths is presented, indicating an uncertainty of 10-20 % for most transitions. 1. Introduction Transition probabilities of heavy elements, particularly the iron group, are of great importance in astrophysical and laboratory sources. Fuhr et al. [1] hav e compiled data from a number of available sources. However, the accuracy and the extent of t hese data is largely inadequate for many general applications such as the calculation of local t hermodynamic equilibrium (LTE) stellar opacities [2,3], and radiative levitation and acce lerations of heavy elements [4]. Among the particular applications including Fe V as a prominent sp ectral constituent are the the non- LTE models of Fe V spectra in hot stars [5], and the observed ex treme ultraviolet Fe V emission from young white dwarfs [6]. For example, currently availab le data for Fe V fails to account for the observed opacity of iron in the XUV region where obser vations of newly formed hot and young white dwarfs clearly show Fe V lines [6]. In all of these applications it is highly desirable to have as complete a dataset of radiative transition probab ilities as possible. While the twin problems of completeness and accuracy pose a challenge to th e theoretical methods, they are of interest not only in various applications but may also be of u se in the analysis of experimental measurements of observed energy levels of complex atomic sy stems from the iron group. The Opacity Project (OP) [7,2] and the Iron Project (IP) [8] l aid the foundation for large- scale theoretical calculations using ab intio methods. The R-matrix method [9], based on atomic collision theory techniques and adapted for the OP [10] and t he IP [8], has proven to be very efficient for these calculations. Whereas the OP calculation s were all in the LS coupling approx- imation, with no relativistic effects included, the subsequ ent IP work is in intermediate coupling using the Breit-Pauli extension of the R-matrix method [8]. While most the IP work has con- centrated on collisional calculations, recent works have e xtended the BPRM method to radiative bound-bound and bound-free calculations for transition pr obabilities [11], photoionization [12], and (electro-ion) recombination [13]. The first comprehens ive BPRM calculation of fine structure 2transition probabilities was carried out for the highly cha rged ions Fe XXIV and Fe XXV [11] that are of special interest in X-ray astronomy. Very good ag reement was found with existing results available for a limited number of transitions but us ing very accurate theoretical methods including relativistic and QED effects [14,15], thus establ ishing the achievable accuracy for the BPRM calculations. However those He-like and Li-like atomi c systems are relatively simple, and the electron correlation effects relatively weak, compared to the low ionization stages of iron group elements. The present work attempts to enlarge the sco pe of the possible BPRM calcula- tions to include the iron group elements, as well as to solve s ome outstanding problems related to level identifications in ab initio theoretical calculations using collision theory methods. Unlike atomic structure calculations, where the electroni c configurations are pre-specified and the levels identified, the bound levels calculated by col lision theory methods adopted in the OP and the IP need to be identified since only the channel quant um numbers are known for the bound states corresponding to the (e + ion) Hamiltonian o f a given total angular and spin symmetry SLπorJπ. The precise correspondence between the channels of the col lision complex, and the bound levels, must therefore be determined. The prob lem is non-trivial for complex atoms and ions with many highly mixed levels due to configurat ion interaction. In the OP work, carried out in LS coupling, this problem was solved by an anal ysis based on quantum defects and the numerical components of wavefunctions in the region out side the R-matrix boundary (that envelops the target ion orbitals). The present work extends that treatment to the analysis of fine structure levels computed in intermediate coupling. In add ition, considerable effort is devoted to the determination of the completeness of the set of comput ed bound levels; comparison with the expected levels derived from all possible combination o f angular and spin quantum numbers reveals the missing levels. The general procedure could be a pplied to spectroscopic measurements and the analysis of observed levels of a given atomic system b y comparison with the theoretical predictions. 2. Theory The general theory for the calculation of bound states in the close coupling (CC) approximation of atomic collision theory, using the R-matrix method, is de scribed by Burke and Seaton [16] and 3Seaton [17]. The application to the Opacity Project work is d escribed by Seaton [7], Berrington etal.[10], and Seaton etal.[2]. The relativistic extensions of the R-matrix method in t he Breit- Pauli approximation are discussed by Scott and Taylor [18], and the computational details by Berrington, Eissner, and Norrington [19]. The application to the Iron Project work is outlined in Hummer etal.[8]. In the present work we describe the salient features of the th eory and computations as they pertain to large-scale BPRM calculations for complex atomi c systems. Identification of fine structure energy levels is discussed in detail. Following standard collision theory nomenclature, we refe r to the (e + ion) complex in terms of the ’target’ ion, with N bound electrons, and a ’free’ elec tron that may be either bound or continuum. The total energy of the system is either negative or positive; negative eigenvalues of the (N + 1)-electron Hamiltonian correspond to bound states of the (e + ion) system. In the coupled channel or close coupling (CC) approximation the wa vefunction expansion, Ψ( E), for a total spin and angular symmetry SLπorJπ, of the (N+1) electron system is represented in terms of the target ion states as: ΨE(e+ion) =A/summationdisplay iχi(ion)θi+/summationdisplay jcjΦj, (1) where χiis the target ion wave function in a specific state SiLiπior level Jiπi, andθiis the wave function for the (N+1)th electron in a channel labeled as SiLi(Ji)πik2 iℓi(SLπ) [Jπ];k2 i(=ǫi) is the incident kinetic energy. In the second sum the Φ j’s are correlation wavefunctions of the (N+1) electron system that (a) compensate for the orthogonality c onditions between the continuum and the bound orbitals, and (b) represent additional short-ran ge correlation that is often of crucial importance in scattering and radiative CC calculations for eachSLπ. The functions Ψ( E) are given by the R-matrix method in an inner region r≤a. These are bounded at the origin and contain radial functions that sati sfy a logarithmic boundary condition atr=a[20]. In the outer region r > a the inner region functions are matched to a set of linearly independent functions that correspond to all possible (e + i on) channels of a given symmetry SLπorJπ. The outer region wavefunctions are computed for all channe ls, (CtStLtπt)ǫl, where Ctis the target configuration, and used to determine the indivi dual channel contributions (called “channel weights”). 4In the relativistic BPRM calculations the set of SLπare recoupled to obtain (e + ion) levels with total Jπ, followed by diagonalisation of the (N+1)-electron Hamilt onian, HBP N+1Ψ=EΨ. (2) The BP Hamiltonian is HBP N+1=HN+1+Hmass N+1+HDar N+1+Hso N+1, (3) where HN+1is the nonrelativistic Hamiltonian, HN+1=N+1/summationdisplay i=1  −∇2 i−2Z ri+N+1/summationdisplay j>i2 rij  , (4) and the additional terms are the one-body terms, the mass cor rection term, the Darwin term and the spin-orbit term respectively. Spin-orbit interact ion,Hso N+1, splits the LS terms into fine- structure levels labeled by Jπ, where Jis the total angular momentum. Other terms of the Breit-interaction [22], HB=/summationdisplay i>j[gij(so + so′) +gij(ss′)], (5) representing the two-body spin-spin and the spin-other-or bit interactions are not included. The positive and negative energy states (Eq. 1) define contin uum or bound (e + ion) states, E=k2>0−→continuum (scattering )channel E=−z2 ν2<0−→bound state,(6) where νis the effective quantum number relative to the core level. If E <0 then all continuum channels are ‘closed’ and the solutions represent bound sta tes. Determination of the quantum defect ( µ(ℓ)), defined as νi=n−µ(ℓ) where νiis relative to the core level SiLiπi, is helpful in establishing the ℓ-value associated with a given channel (level). At E<0 a scattering channel may represent a bound state at the prop er eigenvalue of the Hamiltonian (Eq. 2). A large number of channels are consider ed for the radiative processes of Fe V. Each SL πor Jπsymmetry is treated independently and corresponds to a larg e number of channels. Therefore, the overall configuration interact ion included in the total (e + ion) wavefunction expansion is quite extensive. This is the main advantage of the CC method in representing electron correlation accurately. 5a) Level identification and coupling schemes The BPRM calculations in intermediate coupling employ the p air-coupling representation Si+Li−→Ji Ji+ℓ−→K K+ s−→J, (7) where the ‘i’ refers to the target ion level and ℓ, sare the orbital angular momemtum (partial wave) and spin of the additional electron. According to desi gnations of a collision complex, a channel is fully specified by the quantum numbers (SiLiJi)πiǫiℓiK s[Jπ] (8) The main problem in identification of the fine structure level s stems from the fact that the bound levels are initially given only as eigenvalues of the ( e + ion) Hamiltonian of a given sym- metry Jπ. Each level therefore needs to be associated with the quantu m numbers characterizing a given collision channel. Subsequently, three main parame ters are to be determined: (i) the parent or the target ion level, (ii) the orbital, effective an d principal quantum numbers ( l, ν, n ) of the (N+1)th electron, and (iii) the symmetry, SLπ. The task is relatively straightforward for simple few-electron atomic systems. For example, in a recen t work Nahar and Pradhan [11] have calculated a large number of transition probabilities for L i-like Fe XXIV and He-like Fe XXV, where the problem of level identification is trivial, compar ed to the present work, since the bound levels are well separated in energy and in ν. However when a number of mixed bound levels fall within a given interval ( ν, ν+ 1), for the same Jπ, the quantum numbers and the magnitude of the components in all associated channels must be analyse d. A scheme for identification of levels is developed (discussed later) that rests mainly on a n analysis of quantum defects of the bound levels and their orbital angular momenta, and the perc entage of the total wavefunction in all channels of a given Jπ. Following level identification, further work is needed to en able a direct correspondence with standard spectroscopic designations that follow different coupling schemes, such as between LS andJJ, appropriate for atomic structure calculations as, for exa mple, in the NIST tables of observed energy levels [1]. The correspondence provides th e check for completeness of calculated set of levels or the levels missing. The level identification procedure involves considerable ma- 6nipulation of the bound level data and, although it has been e ncoded for general applications, still requires analysis and interpretation of problem case s of highly mixed levels that are difficult to identify. b) Oscillator strengths and transition probabilities The oscillator strength (or photoionization cross section ) is proportional to the generalized line strength defined, in either length form or velocity form, by t he equations SL=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBigg Ψf|N+1/summationdisplay j=1rj|Ψi/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (9) and SV=ω−2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBigg Ψf|N+1/summationdisplay j=1∂ ∂rj|Ψi/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (10) In these equations ωis the incident photon energy in Rydberg units, and ΨiandΨfare the wave functions representing the initial and final states respect ively. The boundary conditions satisfied by a bound state with negative energy correspond to exponent ially decaying partial waves in all ‘closed’ channels, whilst those satisfied by a free or contin uum state correspond to a plane wave in the direction of the ejected electron momentum ˆkand ingoing waves in all open channels. Using the energy difference, Eji, between the initial and final states, the oscillator streng th, fij, for the transition can be obtained from Sas fij=Eji 3giS, (11) and the Einstein’s A-coefficient, Aji, as Aji(a.u.) =1 2α3gi gjE2 jifij, (12) where αis the fine structure constant, and gi,gjare the statistical weight factors of the initial and final states, respectively. In terms of c.g.s. unit of tim e, Aji(s−1) =Aji(a.u.) τ0, (13) where τ0= 2.4191−17s is the atomic unit of time. 73. Computations The target wavefunctions of Fe VI were obtained by Chen and Pr adhan [21] from an atomic structure calculation using the Breit-Pauli version of the SUPERSTRUCTURE program [22], intended for electron collision calculations with Fe VI usi ng the Breit-Pauli R-matrix method. Present work employs their optimized target of 19 fine struct ure levels [21] corresponding to the 8-term LSbasis set of 3 d3(4F,4P,2G,2P,2D2,2H,2F,2D1). The set of correlation configurations used were 3 s23p63d24s, 3s23p63d24d, 3s3p63d4, 3p63d5, 3s23p43d5, and 3 p63d44s. The values of the scaling parameter in the Thomas-Fermi pote ntial for each orbital of the target ion are given in Ref. [21]. Table I lists the 19 fine structure e nergy levels of Fe VI used in the eigenfunction expansion where the energies are the observe d ones. Most bound levels in low ionization stages correspond to the level of excitation of t he parent ion involving the first few excited states. The criterion remains the accuracy of the ta rget represetation that constitute the core ion states. The (N+1) electron configurations, Φ j, which meet the orthogonality condition for the CC expansion (the second term of the wavefunction, Eq . (1)) are given below Table I. The same set of configurations is used for all the states consi dered in this work. STG1 of the BPRM codes computes the one- and two-electron radial integr als using the one-electron target orbitals generated by SUPERSTRUCTURE. The number of contin uum basis functions is 12. The present calculations are concerned with all possible bo und levels with n≤10, ℓ≤n−1. These correspond to total (e + Fe VI) symmetries ( SLπ) with (2 S+ 1) = 1,3,5 and L= 0 - 10 (even and odd parities). The intermediate coupling calcula tions are carried out on recoupling theseLSsymmetries in a pair-coupling representation, Eq. 6, in sta ge RECUPD. The computer memory requirement for this stage is the maximum, since it ca rries out angular algebra of dipole matrix elements of a large number of fine structure lev els. The (e + Fe VI) Hamiltonian is diagonalized for each resulting Jπin STGH. The negative eigenvalues of the Hamiltonian correspond to the bound levels of Fe V, that are found accordi ng to the procedure described below. 8a) Calculation of bound levels The eigenenergies of the Hamiltonian for each Jπare determined with a numerical search on an effective quantum number mesh, with an interval ∆ ν, using the code STGB. In the relativistic case, the number of Rydberg series of levels increases consi derably from those in LScoupling due to splitting of the target states into their fine structure co mponents. This results in a large number of fine structure levels in comparatively narrow energy band s. A mesh with ∆( ν) = 0.01 is usually adequate to scan for LSterm energies; however, it is found to be of insufficient resol ution for fine structure energy levels. The mesh needs to be finer by an order of magnitude, i.e., ∆( ν) = 0.001, so as not to miss out any significant number of bound levels. Th is considerably increases the computational requirements for the intermediate coupling calculations of bound levels over the LS coupling case by orders of magnitude. The calculations ta ke up to several CPU hours per Jπin order to determine the corresponding eigenvalues. All bo und levels of total J≤8, of both parities, are considered. However, a further search with an even finer ∆ νreveals that a few levels are still missing for some Jπsymmetries. b) Procedure for level identification The energy levels in the BPRM approximation (from STGB) are i dentified by Jπalone. This is obviously insufficient information to identify all associ ated quantum numbers of a level from among a large set levels for each Jπ, typically a few hundred for Fe V. A sample set of energy levels for J= 2, even parity, obtained from the BPRM calculations is pres ented in Table II. The table shows energies and effective quantum number νg, as calculated relative to the ground level (3 d3 4F3/2) of the core ion Fe VI. The complexity of the calculations, an d that of level identification, may be gauged from the fact that 30 of these le vels have nearly the same νg. Further, the νgdo not in general correspond to the actual effective quantum n umber of the Fe V level since it may belong to an excited parent level, and not t he ground level, of Fe VI. A scheme has been developed to identify the levels with compl ete spectroscopic information consisting of (CtStLtJtπtℓ[K]s)J π, (14) and also to designate the levels with a possible SLπsymmetry. The designation of the SLπ, 9from the identifications denoted above, is generally ambigu ous since the collision channels are all in intermediate coupling. However, in most cases we are able to carry through the identification procedure to the LSterm designation. An advantage of identification is that it g reatly facilitates the completeness check for all possible LS terms and locate a ny missing levels. A computer code PRCBPID has been developed to identify all the quantum numbe rs relevant to the Jπand the LSterm assignments. Identification is carried out for all the l evels belonging to a Jπsymmetry at a time. The components of the total wavefunction of a given fine struc ture energy level span all closed ”collision” channels ( CtStLt(Jt)πt)ǫl. Each channel contains the information of the relevant core and the outer electron angular momentum. The “channel weigh ts”, mentioned earlier, determine the magnitude of the wavefunction in the outer R-matrix regi on of each channel evaluated in STGB. A bound level may be readily assigned to the quantum num bers of a given channel provided the corresponding channel weight (in percentage t erms) dominates the other channels. The number of channels can be large especially for complex io ns. For Fe V, for example, each level with J > 2 corresponds to several hundred channels. As the first step i n the level identi- fication scheme we isolate the two most dominant channels by c omparing all channel percentage weights. The reason is that the largest channel percentage w eight may not uniquely determine the identifications since the channel weights are evaluated from the outer region contributions (r > a); the inner region contributions are unknown. Also, many le vels are often heavily mixed and no assignment for the dominant channel may be made. The program, PRCBPID, sorts out the duplicate identificatio ns in all the levels of the Jπ symmetry. Two levels with the same configuration and set of qu antum numbers can actually be two independent levels due to outer electron spin additio n/ subtraction to/from the parent spin angular momentum, i.e. St±s=S. The identical pair of levels are tagged with positive and negative signs indicating higher and lower multiplicit y respectively. The lower energies are normally assigned with the higher spin multiplicity. Howev er, the energies and effective quantum numbers ( ν) of levels of higher and lower spin multiplicity can be very c lose to each other, in which case the spin multiplicity assignment may be uncertai n. One important identification criterion is the analysis of th e quantum defect, µ, or the effective quantum number, ν, of the outer or the valence electron. The principle quantum number, n, 10of the outer electron of a level is determined from its ν, and a Rydberg series of levels can be identified from the effective quantum number. Hence, in the id entification procedure, νof the lowest member (level with the lowest principal quantum numb er of the valence electron) of a Rydberg series is determined from quantum defect analysis o f all the computed levels for each partial wave l. The lowest partial wave has the highest quantum defect. A ch eck is maintained to differentiate the quantum defect of a′s′electron with that of an equivalent electron state which has typically a large value in the close coupling calculatio ns. The principle quantum number, n, of the lowest member of the series is determined from the orb ital angular momentum of the outer electron and the target or core configuration. Once νandµ=n−νof the lowest member are known, the n-values of all levels can be assigned for each paritial wave, l. The relevant Rydberg series of levels is also identified from the levels th at have the same symmetry, Jπ, core configuration, CtStLtπtand outer electron orbital angular momentum l, but different νthat differs between successive levels by ∼1. While the ν(n ℓ) are more accurate for the higher members of the series, they are more approximate for the lowe st ones. The quantum defect of a given partial wave ℓalso varies slightly with different parent core levels and fin alSLJsymmetries. Of the two most dominant channels the proper one for each boun d level is determined based on several criteria. There are cases when more than two level s are found to have identical identifications. These levels are checked individually for proper identification. Often a swap of identifications is needed between the two sets of dominant ch annels since the second dominating channel is more likely to be associated with the given level, consistent with all other criteria. In some cases the most dominant channel (largest percentage weight in the outer region) may correspond to comparatively larger νfor the partial wave ℓ, than to a reasonable νfor the second channel, indicating that the identification should corresp ond to the second channel. In a few cases a level is found not to correspond to any of the tw o dominant channels, predetermined from the channel weights. At the same time oft en a level is found to be missing in the same energy range. In such case the level is assigned to a c hannel of lower percentage weight that has a reasonable core configuration and term, nlquantum numbers for the outer electron and effective quantum number that match the missing level. There are a number of levels belonging to equivalent-electr on configurations and require different identification criteria from those of the Rydberg s tates. These levels usually have: (i) a 11number of approximately equal channel weights, and (ii) qua ntum defects that are larger than that of the lowest partial wave, or an inconsistent νthat does not match with any reasonable n. Once these levels are singled out, they are identified with t he possible configurations of the core level, augmented by one electron in the existing orbita l sub-shell. These low-lying levels are often assigned to those identified from the small experiment ally available set of observed levels. The levels that can not be identified in the above procedure, s uch as by swapping of channels, or maching to a missing level, are assumed to belong to mixed s tates. These are not analysed futher by quantum defects. Two additional (and related) problems, as mentioned above, are addressed in the identifica- tion work: (A) standard LScoupling designation, SLπ, and (B) the completeness check for the set of all fine structure components within an LSmultiplet. Identification according to collision channel quantum numbers is not quite sufficient to establish a direct correspondence with the standard spectroscopic notation employed in atomic struct ure calculations, or in the compiled databases such as those by the U.S. National Institute for St andards and Technology (NIST). The possible set of SLπs of a level is obtained from the target term, StLtπt, and the valence electron angular momentum, l, at the first occurance of the level in the set. The total spin multiplicity of the level is defined according to the energy l evel position as discussed above. For example, the core 3 d3(4Fe) combining with a 4 delectron forms the terms5(P, D, F, G, H )eand 3(P, D, F, G, H )e(Table IV) where the quintet for each Lshould be lower than the triplet. To each LSsymmetry, SLπ, of the set belongs a set of predetermined J-levels. The set of total J-values of same spin multiplicity is then calculated from all possible LSterms, equal to |L+S|. The program sorts out all calculated fine stucture levels wi th the same configuration, but with different sets of JtandJ, e.g. ( CtStLtJtπtnℓ)J π(including the sign for the upper or lower spin multiplicity), compares them with the predeterm ined set, and groups them together. Thus a correspondence is made between the set of SLπand the calculated fine structure levels of same configuration. In addition to the correspondence between the two sets, the p rogram PRCBPID also calcu- lates the possible set of SLπ’s for each single J-level in above group. In the set of SLπs, the total spin is fixed while the angular momentum, L, varies. In the above example for the quintets, 5(P, D, F, G, H )e, each J=1 level is assigned to a possible set of terms,5(P, D, F ) (Table IV). 12However, these levels can be futher identified uniquely foll owing Hund’s rule that the term with the larger angular momentum, L, is the lower one, i.e., the first or the lowest J=1 level should correspond to5F, the second one to5Dand the last one to5P. The completeness of sets of fine structure levels with respec t to the LSterms are checked. As mentioned above, PRCBPID determines the possible sets of SLπfrom the target term and va- lence electron angular momentum of a level at its first occura nce and calculates the total J-values of the set of LSterms. The number of these J-values, Nlv, is compared with that of calculated levels, Ncal to check the completeness. For example, for the above case of5(P, D, F, G, H )ein Table IV discussed above,5Pcan have J= 1,2,3,5Dcan have J= 0,1,2,3,4, and so on, giving a total of 23 fine structure levels for this set of LSterms. The one J= 0 level belongs to5D, the three J= 1 levels belong to5(P, D, F ), and so on. All 23 levels of this set are found in the computed levels (Table IV), thus making the computed set complete. This procedure, in addition to finding the link between the two diiferent coupli ng schemes, enables an independent counting of the number of levels obtained, and ascertains mi ssing or mis-identified levels. c) Transition probabilities The oscillator strengths (f-values) and transition probab ilites (A-values) for bound-bound fine structure level transitions in Fe V are calculated for level s up to J≤8. Computations are carried out using STGBB of the BPRM codes. The f-values are initially calculated by the program STGBB w ith level designations given byJπonly. However, the transitions may be fully described follo wing the level identifications as described in the previous section. Work is in progress to ide ntify all the transitions with proper quantum numbers, configurations and possible SLπ’s. A subset of the large number of transitions has been processe d with complete identifications. Among these transitions are those that correspond to the exp erimentally observed levels [23]. As these levels have been identified, their oscillator stren ghts could be sorted out from the file of f-values. Another subsidiary code, PRCBPRAD, is developed to reprocess the transition proba- bilities where the calculated transition energies are repl aced by the observed ones for improved accuracy. The computation time required for the BPRM calculations was orders of magnitude longer 13compared to oscillator strengths calculations in LS coupli ng, as carried out under the OP for example. The time excludes that needed for creating the nece ssary bound state wavefunctions and calculating dipole matrix elements using the R-matrix p ackage of codes. Computations are carried out for one or a few pairs of symmetries at a time requi ring several hours of CPU time on the Cray T94. The memory requirement was over 30 MWords. 4. Results and discussion Theoretical spectroscopic data are calculated on a large-s cale with relativistic fine structure included in an ab initio manner, and ensuring completeness in terms of obtaining nea rly all possible energy levels and transition probabilities for Fe V for the total angular symmetries considered. The results are described below. a) Energy Levels We have calculated 3,865 fine structure bound levels, with 0 ≤J≤8, for Fe V. Following level identification, as explained in the previous section, the en ergy levels are arranged according to ascending order in energy. The present energies are compared with the relatively small set of experimentally observed levels compiled by NIST [23] in Table III. All 179 observed le vels are obtained and identified. Asterisks attached to levels in Table III indicate an incomp lete set of observed levels correspond- ing to the LSterm. Often in experimental measurements the weak lines are not observed. The theoretical datasets on the other hand are usually complete . We find some discrepancies regarding the identification of a c ouple of levels in the NIST tabulation. The J= 2 level at 2.9395 Ry identified in the NIST table as 3 d3(4P)4p(5So)2, from the maximum leading percentage, may have been misident ified. Present analysis for the completeness of a set of fine structure levels belonging to a t erm indicates it as an extra level for the given configuration and that the possible LSterms for this level are 3 d3(2D2)4p(3PDFo), possibly3Fo. Similarly the NIST identification for the J=3 level at 2.8968 Ry is 3 d3(2P)4p(3Do)3, from the maximum leading percentage. Present calculations however assign the level to possible LS terms, 3 d3(2D2)4p(3DFo), and most likely to3Do. 14In the computed set of fine structure levels the observed leve ls are usually the ones with the lowest energy in each subset of Jπ. The lowest calculated levels are the 34 levels of the ground configuration 3 d4of Fe V, in agreement with the observed ones. The agreement be tween the observed and calculated energies for these levels is within 1%. The calculated energies agree to about 1% with the measured ones for most of the observed level s. Although the energies are exoected to be highly accurate, but the uncertainty in the ca lculations is not comparable to that in spectroscopic observations (of the order of few wavenumb ers). Employing the completeness procedure the computed fine stru cture levels are tabulated, according to the two sets of cross-correlating quantum numb ers: one according to the collision channels identified as ( CtStLtJtπtnℓ[K]s)J π, and the other according to the complete set of J-values for each multiplicity (2 S+ 1),Landπ. A subset of the complete table of fine structure levels is presented in Table IV. (The complete table will be a vailable electronically). Each set of levels is grouped by the possible set of LSterms followed by the levels of same configuration, core term, total spin multiplicity and parity, and with diffe rentJ-values. The header for each group contains the total number of possible J-levels, Nlv, total spin multiplicity, parity, and all possible Lvalues formed from the core and the outer electron. The possi bleJ-values for each SLπare given within parentheses next to each Lvalue. The two sets of quantum numbers are compared. The levels that may be missing or mis- identified are thereby checked out. The number of computed le vels,Ncal, is compared with that expected from angular and spin couplings, Nlv. For most of the configurations the set of levels is complete except for the high lying ones. The comparison de tects missing levels. An example is shown in the the set of 3 d32(2D)5d3(S, P, D, F, G )ein Table IV where one level with Je= 4 is missing. In Table IV, the effective quantum number νis specified alongwith other quantum numbers for each level. The consistency in ν=z√ (E−Et), where Etis the corresponding target energy, for each set of levels may be noted. The possible SLπs for each level are given in the last column. The levels with a single possible term only are uniquely defined. However, those with two or multiple term assignments can be defined uniquely applying Hund’s rule that the higher Lcorresponds to the lower energy of same Jπas explained in the previous section (we note that Hund’s rul e may not always apply 15in cases of strong CI). There are 112 levels of odd parity that we could not properly i dentify. Some of these levels are given in Table IV. These levels could be equivalent electron levels of configuration, 3 p5(2Po)3d5. The 16 LS terms of 3 d5, which are2D1,2P3,2D3,2F3,2G3,2H3,4P5,4F5,2S5,2D5,2F5, 2G5,2I5,4D5,4G5, and6S5, in combination with the parent core2Po, form 88 LS terms with 31 singlets, 43 triplets, 13 quintets and 1 septet. The numbe r of fine structure levels from these terms exceed the 112 computed levels that have not been ident ified. This new procedure of cross-correlation between two coupli ng schemes thus provides a pow- erful check on the completeness and level identification, an d is expected to be of use in further BPRM work on complex atomic systems. b) Transition Probabilities The oscillator strengths (f-values) and transition probab ilites (A-values) for fine structure level transitions in Fe V are obtained for J≤8. The allowed ∆ J= 0,±1 transitions include both the dipole allowed (∆ S= 0,±1) and the intercombination (∆ S/negationslash= 0) transitions. The total number of computed transition probabilities is well over a million , approximately 1 .46×106. For most allowed pairs of Jπsymmetries, there are about 103−105transitions. As explained in the previous section, a subset of the encoded transitions have been processed to present them with proper identifications. These correspo nd to the levels that have been observed. A sample of these is presented in Table V. In all of t he f-values presented the calculated transition energy has been replaced by the observed one, usi ng the BPRM line strengths ( S) which are energy independent. Since measured energies in ge neral have smaller uncertainties than the calculated ones, this replacement improves the acc uracy of the oscillator strengths. The transitions among the 179 observed levels correspond to 372 7 oscillator strengths. (The complete set of transition probabilities will be available electron ically.) The f-values in Table V have been reordered to group the trans itions of the same multiplet together. This enables a check on the completeness of the set of transitions. As this table corresponds to transitions among observed levels only, the completeness depends on the set of observed levels belonging to the LS terms. For the dipole all owed transitions, the LS multiplets are also given at the end of jj′transitions. 16To our knowledge, no measured f-values for Fe V are available for comparison. Current NIST compilation [1] contains no f-values for any allowed transi tion. On the other hand, Fe V oscillator strengths for a large number of transitions were obtined in t he close coupling approximation under the OP [24] and the IP [25]. Both of these datasets are no n-relativistic calculations in LS coupling and do not compute fine structure transitions. Fawcett has [26] carried out semi-empirical relativistic atomic structure calculatio ns for fine structure transitions in Fe V. Comparison of the present f-values is made with the previous ones in Table VI, showing va rious degrees of agreement. Present values agree within 10% with t hose by Fawcett for a number of fine structure transitions of multiplets, 3 d4(5D)→3d3(4F)4p(5Do), and 3 d4(5D)→3d3(4P)4p(5Po), and the disagreement is large with other as well as with those of 3d4(5D)→3d3(4F)4p(5Fo). The agreement of the present LSmultiplets with the others is good for transitions 3 d4(5D)→ 3d3(4F)4p(5Fo,5Do,5Po). More detailed comparisons will be made at the completion o f this work. The procedure of substitution of experimental for calculat ed energies provides an indication of uncertainties in the calculated f-values. The difference between the f-values obtained using the calculated transition energies and the observed ones is only a few percents ( <5%) for most of the allowed transitions. The difference is usually larger fo r the intercombination transitions which have lower transition probabilities. In atomic structure c alculations, it is possible to re-adjust eigenenergies of the Hamiltonian to match the observed ones and then use the wavefunctions to obtain the transition probabilities. Such a re-adjustment is not carried out in the BPRM calcu- lations of bound states, which are entirely ab initio, with t he associated advantage of consistent uncertainties for most transitions considered. To obtain an estimate of the accuracy of the wavefunctions em ployed in the length and the velocity formulations, we plot, for example, the gf-values for transitions ( J= 1)e−(J= 2)o and (J= 3)e−(J= 4)oin Fig. 1. The top panel contains over 13,300 transitions bet ween the pair of symmetries ( J= 1)e−(J= 2)o, and the bottom panel contains over 20,200 transitions between the pair ( J= 3)e−(J= 4)o. The plots show practically no dispersion for the strongest transitions with gf≈5−10, and some dispersion around 10-20% for others with gf < 3. Up togf < 0.1 the dispersion in length and velocity remains around the 10 -20% level for most of the transitions, although the number of outlying transitio ns increases with decreasing gf. Given 17the large number of points in the figures, the relatively low d ispersion of gfLandgfVindicates that the f-values ( gfdivided by the statistical weight factor, 2 J+1) for most of the transitions withgf∼1 should be within 20% uncertainty. The fL’s are usually more accurate than fV’s since the asymptotic region wavefunctions are more accurat ely represented in the close coupling calculations using the R-matrix method. In general the intercombination transitions are weaker tha n the dipole allowed ones; the f-values can be orders of magnitude lower. The BP Hamiltonia n in the present work (Eq. 2) does not include the two-body spin-spin and spin-other-orb it terms of Breit interaction [22]. A discussion of these terms is given by Mendoza etal.in a recent IP paper [27]. Their study on the intercombination transitions in C-like ions shows that the effect of the two-body Breit terms, relative to the one-body operators, decreases with Z such th at for Z = 26 the computed A-values with and without the two-body Breit terms differ by less than 0 .5 %. However, the differences towards the neutral end of the C-sequence is up to about 20%. I t may therefore be expected that for Fe V the weaker intercombination f-values may also be sys tematically affected to a similar extent (the uncertainties in the dipole allowed f-values sh ould be much less). Further studies of the Breit interaction in complex atoms are needed to ascerta in this effect more precisely. Several aspects of the present work are targets for future st udies, such as atomic structure calculations to study the effect of configuration interactio n and relativistic effects on different types of transitions, and a detailed quantum defect analysi s along interacting Rydberg series of levels in intermediate coupling. These studies should prov ide information on the accuracy of particular type of transitions and groups of levels, as well as address general problems in the analysis of complex spectra. 5. Conclusion The present work is the first study of large-scale transition probabilities computed using the ac- curate BPRM method for a highly complex ion. Some of the resul ts obtained herein are expected to form the basis for future computational spectroscopy of h eretofore intractable complex atomic systems using efficient collision theory methods. The comput ational procedures developed for such undertakings are described, and illustrative results are presented from the ab initio Breit- 18Pauli R-matrix calculations for Fe V. Detailed analysis for the identification of over 3,800 fine structure levels of Fe V is carried out using a combination of methods that include quantum de- fect theory. Further work on the analysis of relativistic qu antum defects in intermediate coupling is planned. Following the completion of all computations and identifica tions, the dataset of approx- imately 1.5 million oscillator strengths will be described in another publication with a view towards astrophysical and laboratory applications. In ord er to complete the dataset for practical applications calculations are also in progress for the forb idden electric quadrupole and magnetic dipole transition probabilties using the atomic structure program SUPERSTRUCTURE. The newly acquired theoretical capability to obtain an esse ntially complete description of radiative transitions for an atomic system should enable se veral new advances such as: (a) the synthesis of highly detailed monochromatic opacity spectr a [2], (b) the simulation of “quasi- continuum” line spectra from iron ions [28], (c) high resolu tion spectral diagnostics of iron in laboratory fusion and astrophysical sources, and (d) the an alysis of experimentally measured spectra of complex iron ions. Acknowledgements This work was supported partially by the U.S. National Scien ce Foundation (AST-9870089) and the NASA Astrophysical Theory Program. The computational w ork was carried out at the Ohio Supercomputer Center in Columbus Ohio. References 1. Fuhr, J.R., Martin, G.A., Wiese, W.L., J. Phys. Chem. Ref. Data17, Suppl No. 4 (1988) 2. Seaton, M.J., Yu, Y., Mihalas, D., Pradhan, A.K., MNRAS 266. 805 (1994). 3. Rogers F.J. and Iglesias C.A., Science 263, 50 (1994) 4. Chayer, P., Fontaine, G., and Wesemael, F., Astrophys.J. 99, 189 (1995). 5. Becker, S.R. and Butler, K., Astron. Astrophys. ,265, 647 (1992) 196. Vennes, S., Astrophysics in the Extreme Ultraviolet (Ed: Stuart Bowyer and Roger F. Malina), Kluwer, p. 185; (1996); Pradhan, A.K., Ibid., p. 569 7. Seaton M.J. 1987, J.Phys.B 20, 6363. 8. Hummer, D.G., Berrington, K.A., Eissner, W., Pradhan, A. K., Saraph, H.E., Tully, J.A., Astron. Astrophys. 279, 298 (1993) 9. Burke, P.G., Hibbert, A., and Robb, Journal Of Physics B 4, 153 (1971) 10. Berrington K.A., Burke P.G., Butler K., Seaton M.J., Sto rey P.J., Taylor K.T. and Yu Yan, J.Phys.B 206379 (1987). 11. Nahar S.N. and Pradhan, A.K., Astron. Astrophys. Suppl. Ser. 135, 347 (1999). 12. Zhang, H.L., Physical Review A 57, 2640 (1998) 13. Zhang H.L., Nahar S.N., and Pradhan A.K., J. Phys. B 32, 1459 (1999). 14. Johnson W.R., Liu Z.W., and Sapirstein J., At. Data Nucl. Data64, 279 (1996). 15. Yan Z-C, Tambasco M., and Drake G.W.F., Phys. Rev. A 57, 1652 (1998) 16. Burke P.G. and Seaton M.J., Journal Of Physics B 17, L683 (1984) 17. Seaton M.J. J.Phys.B 18, 2111 (1985) 18. Scott N.S., Taylor K.T., Comput. Phys. Commun. 25, 347 (1982) 19. Berrington K.A., Eissner, W., Norrington P.H. Comput. P hys. Commun. 92, 290 (1995) 20. Burke, P.G. and Berrington, K.A., Atomic and Molecular Processes, an R-matrix Approach , Institute of Physics Publishing, Bristol (1993) 21. Chen, G.X. and Pradhan, A.K., Journal Of Physics B 32, 1809 (1999a); Astron. Astrophys. Suppl. 136, 395 (1999b) 22. Eissner, W., Jones, M., Nussbaumer, H., Comput. Phys. Co mmun 8, 270 (1974); W. Eiss- ner, J. Phys. IV (Paris) C1, 3 (1991), Eissner W. (in preparat ion, 1999) 23. Sugar, J. and Corliss, C., J. Phys. Chem. Ref. Data 14, Sup pl.2(1985) 24. Butler, K. (unpublished); data are available through th e OP database, TOPbase (W. Cunto, C. Mendoza, F. Ochsenbein, C.J. Zeippen, Astron. Astrophys 275, L5 (1993)) 25. Bautista M.A., A&A Suppl. Ser. 119, 105 (1996) 26. Fawcett, B.C. At. Data Nucl. Data Tables 41, 181 (1989) 27. Mendoza C., Zeippen C.J. and Storey P.J., A&A Suppl.Ser. 135, 159 (1999) 28. Beirsdorfer, P. Lepson, J.K., Brown, G.V., Utter, S.B., Kahn, S.M., Liedahl, D.A., and 20Mauche, C.W., Astrophys. J. 519, L185 (1999) Figure captions: 1. Comprarison of gfLversus gfVfor bound-bound fine structure level transitions in Fe V obtained in BPRM approximation. 21Table I. The 19 fine strucuture levels of Fe IV in the close coupling eig enfunction expansion of Fe V. List of configurations, Φj, in the second sum of Ψis given below the table. Term JtEt(Ryd) Term JtEt(Ryd) 3d3(4F) 3/2 0.0 3 d3(2P) 1/2 0.241445 3d3(4F) 5/2 0.004659 3 d3(2D2) 5/2 0.259568 3d3(4F) 7/2 0.010829 3 d3(2D2) 3/2 0.260877 3d3(4F) 9/2 0.018231 3 d3(2H) 9/2 0.261755 3d3(4P) 1/2 0.170756 3 d3(2H) 11/2 0.266116 3d3(4P) 3/2 0.172610 3 d3(2F) 7/2 0.421163 3d3(4P) 5/2 0.178707 3 d3(2F) 5/2 0.424684 3d3(2G) 7/2 0.187871 3 d3(2D1) 5/2 0.653448 3d3(2G) 9/2 0.194237 3 d3(2D1) 3/2 0.656558 3d3(2P) 3/2 0.238888 Φj:, 3s23p63d4, 3s23p63d34s, 3s23p63d34p, 3s23p63d34d, 3s23p53d44s, 3s23p53d44d, 3s23p53d34s, 3p53d54s4d, 3p63d54p, 3s23p43d6, 3s23p43d54p, 3s3p63d34s4d, 3p63d34s4p2, 3s23p43d44p2, 3s23p43d44s2, 3s3p63d44s, 3s23p53d5, 3p63d6, 3p63d34s24d, 3s3p63d44d Table II. Sample set of calculated energy levels (in z2-scale) of J= 2. E(Ry) νg E(Ry) νg -1.492946E-01 2.58808 -1.455232E-01 2.62140 -1.421287E-01 2.65252 -1.383809E-01 2.68820 -1.381112E-01 2.69083 -1.373195E-01 2.69857 -1.355085E-01 2.71654 -1.307289E-01 2.76576 -1.202324E-01 2.88396 -1.185430E-01 2.90444 -9.421634E-02 3.25789 -9.408765E-02 3.26012 -9.385297E-02 3.26419 -9.375868E-02 3.26584 -9.098725E-02 3.31520 -9.071538E-02 3.32016 -8.971136E-02 3.33869 -8.651911E-02 3.39973 -8.627750E-02 3.40448 -8.536568E-02 3.42262 -8.458038E-02 3.43847 -8.448129E-02 3.44049 -8.420175E-02 3.44619 -8.371297E-02 3.45624 -8.338890E-02 3.46295 -8.317358E-02 3.46743 -8.306503E-02 3.46969 -8.225844E-02 3.48666 -8.180410E-02 3.49633 -8.135593E-02 3.50595 -7.919621E-02 3.55343 -7.827940E-02 3.57418 -7.815480E-02 3.57703 -7.770277E-02 3.58742 -7.708061E-02 3.60186 -7.691196E-02 3.60581 -7.594121E-02 3.62879 -7.370981E-02 3.68330 -7.315088E-02 3.69735 -7.138001E-02 3.74293 -7.028267E-02 3.77204 -6.977748E-02 3.78567 -6.955230E-02 3.79179 -6.845060E-02 3.82218 -6.765945E-02 3.84446 -6.721368E-02 3.85719 -6.518968E-02 3.91661 -6.265635E-02 3.99501 -6.190890E-02 4.01905 -6.111942E-02 4.04492 -5.831623E-02 4.14100 22Table III. Comparison of calculated and observed energy levels of Fe V. Configuration Term J E(cal) E(expt) 3d4 5D 4 5.5493 5.5015 3d4 5D 3 5.5542 5.5058 3d4 5D 2 5.5580 5.5094 3d4 5D 1 5.5607 5.5119 3d4 5D 0 5.5621 5.5132 3d423P 2 5.3247 5.2720 3d423P 1 5.3389 5.2856 3d423P 0 5.3471 5.2940 3d4 3H 6 5.3074 5.2805 3d4 3H 5 5.3111 5.2833 3d4 3H 4 5.3143 5.2860 3d423F 4 5.3043 5.2674 3d423F 3 5.3064 5.2686 3d423F 2 5.3076 5.2693 3d4 3G 5 5.2581 5.2359 3d4 3G 4 5.2614 5.2384 3d4 3G 3 5.2651 5.2415 3d421G 4 5.2006 5.1798 3d4 3D 3 5.1950 5.1794 3d4 3D 2 5.1945 5.1782 3d4 3D 1 5.1928 5.1767 3d4 1I 6 5.1852 5.1713 3d421S 0 5.1700 5.1520 3d421D 2 5.1353 5.0913 3d4 1F 3 5.0476 5.0326 3d413P 2 4.9756 4.9495 3d413P 1 4.9663 4.9398 3d413P 0 4.9616 4.9352 3d413F 4 4.9719 4.9460 3d413F 3 4.9706 4.9449 3d413F 2 4.9712 4.9453 3d411G 4 4.8830 4.8636 3d411D 2 4.6609 4.6581 3d411S 0 4.4302 4.4093 3d3(4F)4s5F 5 3.7161 3.7964 3d3(4F)4s5F 4 3.7228 3.8025 3d3(4F)4s5F 3 3.7282 3.8077 3d3(4F)4s5F 2 3.7324 3.8116 3d3(4F)4s5F 1 3.7352 3.8143 3d3(4F)4s3F 4 3.6222 3.7194 3d3(4F)4s3F 3 3.6311 3.7277 3d3(4F)4s3F 2 3.6381 3.7344 3d3(4P)4s5P 3 3.5483 3.6402 3d3(4P)4s5P 2 3.5532 3.6453 3d3(4P)4s5P 1 3.5554 3.6475 3d3(2G)4s3G 5 3.5090 3.6038 3d3(2G)4s3G 4 3.5130 3.6076 3d3(2G)4s3G 3 3.5158 3.6101 3d3(4P)4s3P 2 3.4595 3.5663 3d3(4P)4s3P 1 3.4677 3.5738 3d3(4P)4s3P 0 3.4706 3.5763 3d3(2G)4s1G 4 3.4636 3.5673 3d3(2P)4s3P 2* 3.4330 3.5583 3d3(2P)4s3P 1* 3.4515 3.5575 3d3(2D2)4s3D 3 3.4337 3.5399 3d3(2D2)4s3D 2 3.4528 3.5394 3d3(2D2)4s3D 1 3.4405 3.5468 3d3(2H)4s3H 6 3.4364 3.5346 3d3(2H)4s3H 5 3.4392 3.5370 23Table III. continues. Configuration Term J E(cal) E(expt) 3d3(2H)4s3H 4 3.4398 3.5377 3d3(2P)4s1P 1 3.4015 3.5131 3d3(2D2)4s1D 2 3.3877 3.5027 3d3(2H)4s1H 5 3.3894 3.4965 3d3(2F)4s3F 4 3.2711 3.3841 3d3(2F)4s3F 3 3.2695 3.3822 3d3(2F)4s3F 2 3.2682 3.3806 3d3(2F)4s1F 3 3.2280 3.3468 3d3(4F)4p5Go6 3.1534 3.1594 3d3(4F)4p5Go5 3.1631 3.1700 3d3(4F)4p5Go4 3.1710 3.1787 3d3(4F)4p5Go3 3.1773 3.1858 3d3(4F)4p5Go2 3.1821 3.1912 3d3(4F)4p5Fo1* 3.1495 3.1644 3d3(4F)4p5Do4 3.1359 3.1498 3d3(4F)4p5Do3 3.1433 3.1559 3d3(4F)4p5Do2 3.1479 3.1609 3d3(4F)4p5Do1 3.1408 3.1540 3d3(4F)4p5Do0 3.1469 3.1565 3d3(2D1)4s3D 3 3.0074 3.1581 3d3(2D1)4s3D 2 3.0058 3.1564 3d3(2D1)4s3D 1 3.0047 3.1551 3d3(4F)4p5Fo5* 3.1191 3.1343 3d3(4F)4p5Fo4* 3.1243 3.1391 3d3(4F)4p5Fo3* 3.1300 3.1443 3d3(4F)4p5Fo2* 3.1350 3.1496 3d3(4F)4p3Do3 3.1170 3.1331 3d3(4F)4p3Do2 3.1258 3.1401 3d3(4F)4p3Do1 3.1309 3.1439 3d3(2D1)4s1D 2 2.9636 3.1210 3d3(4F)4p3Go5 3.0737 3.0973 3d3(4F)4p3Go4 3.0812 3.1035 3d3(4F)4p3Go3 3.0872 3.1083 3d3(4F)4p3Fo4 3.0449 3.0716 3d3(4F)4p3Fo3 3.0520 3.0779 3d3(4F)4p3Fo2 3.0586 3.0836 3d3(4P)4p5Po3 2.9842 3.0078 3d3(4P)4p5Po2 2.9903 3.0151 3d3(4P)4p5Po1 2.9937 3.0195 3d3(4P)4p5Do4 2.9526 2.9792 3d3(4P)4p5Do3 2.9600 2.9883 3d3(4P)4p5Do2 2.9733 2.9912 3d3(4P)4p5Do1 2.9747 3.0058 3d3(4P)4p5Do0 2.9797 3.0094 3d3(4P)4p3Po2 2.9622 3.0038 3d3(4P)4p3Po1 2.9635 2.9911 3d3(4P)4p3Po0 2.9676 2.9941 3d3(2G)4p3Ho6 2.9484 2.9739 3d3(2G)4p3Ho5 2.9591 2.9863 3d3(2G)4p3Ho4 2.9655 2.9942 3d3(2G)4p3Go5 2.9289 2.9613 3d3(2G)4p3Go4 2.9353 2.9662 3d3(2G)4p3Go3 2.9412 2.9726 3d3(2G)4p3Fo4 2.9238 2.9583 3d3(2G)4p3Fo3 2.9262 2.9540 3d3(2G)4p3Fo2 2.9329 2.9567 3d3(2P)4p3Po1 2.9326 2.9439 3d3(2P)4p3Po0 2.9458 2.9413 3d3(2G)4p1Go4 2.9068 2.9430 24Table III. continues. Configuration Term J E(cal) E(expt) 3d3(4P)4p5So! 2 2.9231 2.9395 3d3(2G)4p1Fo3 2.9067 2.9382 3d3(4P)4p5So2 2.8995 2.9379 3d3(2P)4p3Po2 2.8605 2.8668 3d3(2G)4p1Ho5 2.9006 2.9354 3d3(2P)4p3Do3 2.8895 2.9117 3d3(2P)4p3Do2 2.8944 2.9169 3d3(2P)4p3Do1 2.8790 2.9274 3d3(2H)4p3Ho6 2.8793 2.9143 3d3(2H)4p3Ho5 2.8830 2.9180 3d3(2H)4p3Ho4 2.8834 2.9189 3d3(2D2)4p1Po1 2.9031 2.9073 3d3(2D2)4p3Fo4* 2.8653 2.8922 3d3(2D2)4p3Fo2* 2.9144 2.9055 3d3(2P)4p3So1 2.8621 2.9052 3d3(4P)4p3Do3 2.8593 2.9030 3d3(4P)4p3Do2 2.8749 2.8991 3d3(4P)4p3Do1 2.8478 2.8991 3d3(2P)4p3Do! 3* 2.8718 2.8968 3d3(2H)4p3Io7 2.8414 2.8780 3d3(2H)4p3Io6 2.8487 2.8872 3d3(2H)4p3Io5 2.8537 2.8938 3d3(2D2)4p3Do3 2.8366 2.8713 3d3(2D2)4p3Do2 2.8325 2.8761 3d3(2D2)4p3Do1 2.8842 2.8826 3d3(2H)4p1Go4 2.8325 2.8746 3d3(2H)4p1Ho5 2.8244 2.8696 3d3(2D2)4p3Po1* 2.8313 2.8652 3d3(2D2)4p3Po0* 2.8300 2.8623 3d3(2D2)4p1Fo3 2.8238 2.8593 3d3(2H)4p3Go5 2.8039 2.8496 3d3(2H)4p3Go4 2.8035 2.8483 3d3(2H)4p3Go3 2.8049 2.8476 3d3(2H)4p1Io6 2.8012 2.8489 3d3(4P)4p3So1 2.7705 2.8282 3d3(2D2)4p1Do2 2.7815 2.8184 3d3(2P)4p1Po1 2.7822 2.8161 3d3(2F)4p3Fo4 2.7083 2.7556 3d3(2F)4p3Fo3 2.7105 2.7577 3d3(2F)4p3Fo2 2.7109 2.7585 3d3(2F)4p3Go5 2.6652 2.7150 3d3(2F)4p3Go4 2.6672 2.7190 3d3(2F)4p3Go3 2.6695 2.7229 3d3(2F)4p3Do3 2.6648 2.7129 3d3(2F)4p3Do2 2.6705 2.7050 3d3(2F)4p3Do1 2.6559 2.7003 3d3(2F)4p1Do2 2.6592 2.7097 3d3(2F)4p1Go4 2.6165 2.6775 3d3(2F)4p1Fo3 2.6172 2.6742 3d3(2D1)4p3Do3 2.4918 2.5249 3d3(2D1)4p3Do2 2.4938 2.5278 3d3(2D1)4p3Do1 2.4936 2.5285 3d3(2D1)4p1Do2 2.4581 2.5074 3d3(2D1)4p3Fo4 2.4499 2.4876 3d3(2D1)4p3Fo3 2.4532 2.4935 3d3(2D1)4p3Fo2 2.4510 2.4938 3d3(2D1)4p3Po2 2.4196 2.4649 3d3(2D1)4p3Po1 2.4141 2.4580 3d3(2D1)4p3Po0 2.4116 2.4546 3d3(2D1)4p1Fo3 2.4058 2.4518 3d3(2D1)4p1Po1 2.3394 2.3924 25Table IV. Sample table of calculated and identified fine strucuture ene rgy levels of Fe V. Nlv=total number of levels expected for the possible LSterms (specified as 2S+ 1,π, and set of Lwith J-values within paratheses), formed from the target term and lof the outer electron, and Ncal = number of calculated levels. SLπin last column=possible LSterms for each level. Ct StLtπt Jtnl J E(cal) ν SLπ Nlv= 5, 5,e: F ( 5 4 3 2 1 ) 3d3 (4Fe) 3/2 4s 1 -3.73515E+00 2.59 5 F e 3d3 (4Fe) 5/2 4s 2 -3.73238E+00 2.59 5 F e 3d3 (4Fe) 5/2 4s 3 -3.72820E+00 2.59 5 F e 3d3 (4Fe) 7/2 4s 4 -3.72275E+00 2.59 5 F e 3d3 (4Fe) 9/2 4s 5 -3.71610E+00 2.59 5 F e Ncal= 5 : set complete Nlv= 3, 3,e: F ( 4 3 2 ) 3d3 (4Fe) 3/2 4s 2 -3.63808E+00 2.62 3 F e 3d3 (4Fe) 7/2 4s 3 -3.63107E+00 2.62 3 F e 3d3 (4Fe) 9/2 4s 4 -3.62225E+00 2.62 3 F e Ncal= 3 : set complete Nlv= 3, 1,o: P ( 1 ) D ( 2 ) F ( 3 ) 3d3 1 (2De) 3/2 4p 2 -2.45812E+00 2.83 1 D o 3d3 1 (2De) 5/2 4p 3 -2.40581E+00 2.86 1 F o 3d3 1 (2De) 5/2 4p 1 -2.33944E+00 2.89 1 P o Ncal= 3 : set complete Nlv= 23, 5,e: P ( 3 2 1 ) D ( 4 3 2 1 0 ) F ( 5 4 3 2 1 ) G ( 6 5 4 3 2 ) H ( 7 6 5 4 3) 3d3 (4Fe) 3/2 4d 3 -2.37021E+00 3.25 5 PDFGH e 3d3 (4Fe) 5/2 4d 4 -2.36647E+00 3.25 5 DFGH e 3d3 (4Fe) 5/2 4d 5 -2.36189E+00 3.25 5 FGH e 3d3 (4Fe) 3/2 4d 1 -2.35988E+00 3.25 5 PDF e 3d3 (4Fe) 7/2 4d 6 -2.35651E+00 3.25 5 GH e 3d3 (4Fe) 5/2 4d 2 -2.35541E+00 3.26 5 PDFG e 3d3 (4Fe) 9/2 4d 7 -2.35041E+00 3.25 5 H e 3d3 (4Fe) 5/2 4d 1 -2.34932E+00 3.26 5 PDF e 3d3 (4Fe) 9/2 4d 3 -2.34736E+00 3.25 5 PDFGH e 3d3 (4Fe) 7/2 4d 2 -2.34633E+00 3.26 5 PDFG e 3d3 (4Fe) 3/2 4d 2 -2.34397E+00 3.27 5 PDFG e 3d3 (4Fe) 7/2 4d 4 -2.34329E+00 3.26 5 DFGH e 3d3 (4Fe) 5/2 4d 3 -2.34092E+00 3.26 5 PDFGH e 3d3 (4Fe) 9/2 4d 3 -2.33989E+00 3.26 5 PDFGH e 3d3 (4Fe) 9/2 4d 5 -2.33822E+00 3.26 5 FGH e 3d3 (4Fe) 7/2 4d 5 -2.33234E+00 3.27 5 FGH e 3d3 (4Fe) 9/2 4d 6 -2.32699E+00 3.26 5 GH e 3d3 (4Fe) 3/2 4d 4 -2.28772E+00 3.31 5 DFGH e 3d3 (4Fe) 3/2 4d 0 -2.28673E+00 3.31 5 D e 3d3 (4Fe) 7/2 4d 1 -2.28265E+00 3.30 5 PDF e 3d3 (4Fe) 9/2 4d 2 -2.27468E+00 3.30 5 PDFG e 3d3 (4Fe) 7/2 4d 3 -2.26346E+00 3.31 5 PDFGH e 3d3 (4Fe) 9/2 4d 4 -2.25835E+00 3.31 5 DFGH e Ncal= 23 : set complete Nlv= 15, 3,e: P ( 2 1 0 ) D ( 3 2 1 ) F ( 4 3 2 ) G ( 5 4 3 ) H ( 6 5 4 ) 3d3 (4Fe) 3/2 4d 1 -2.35634E+00 3.26 3 PD e 3d3 (4Fe) 5/2 4d 2 -2.35219E+00 3.25 3 PDF e 3d3 (4Fe) 7/2 4d 3 -2.34872E+00 3.25 3 DFG e 3d3 (4Fe) 5/2 4d 4 -2.33701E+00 3.27 3 FGH e 3d3 (4Fe) 5/2 4d 5 -2.28113E+00 3.31 3 GH e 3d3 (4Fe) 3/2 4d 3 -2.28060E+00 3.31 3 DFG e 3d3 (4Fe) 7/2 4d 4 -2.27417E+00 3.31 3 FGH e 3d3 (4Fe) 9/2 4d 6 -2.27323E+00 3.30 3 H e 3d3 (4Fe) 3/2 4d 2 -2.26789E+00 3.32 3 PDF e 3d3 (4Fe) 7/2 4d 5 -2.26632E+00 3.31 3 GH e 3d3 (4Fe) 5/2 4d 0 -2.24585E+00 3.33 3 P e 3d3 (4Fe) 5/2 4d 1 -2.24483E+00 3.33 3 PD e 3d3 (4Fe) 7/2 4d 2 -2.24278E+00 3.33 3 PDF e 3d3 (4Fe) 7/2 4d 3 -2.23973E+00 3.33 3 DFG e 3d3 (4Fe) 9/2 4d 4 -2.23571E+00 3.33 3 FGH e Ncal= 15 : set complete Nlv= 13, 5,e: P ( 3 2 1 ) D ( 4 3 2 1 0 ) F ( 5 4 3 2 1 ) 3d3 (4Pe) 1/2 4d 1 -2.16405E+00 3.27 5 PDF e 3d3 (4Pe) 3/2 4d 2 -2.16298E+00 3.27 5 PDF e 3d3 (4Pe) 3/2 4d 3 -2.16165E+00 3.27 5 PDF e 3d3 (4Pe) 3/2 4d 1 -2.15901E+00 3.27 5 PDF e 3d3 (4Pe) 3/2 4d 4 -2.15825E+00 3.27 5 DF e 3d3 (4Pe) 5/2 4d 2 -2.15694E+00 3.27 5 PDF e 3d3 (4Pe) 5/2 4d 3 -2.15521E+00 3.27 5 PDF e 3d3 (4Pe) 5/2 4d 5 -2.15477E+00 3.27 5 F e26Table IV. continues. Ct StLtπt Jtnl J E(cal) ν SLπ Nlv= 13, 5,e: P ( 3 2 1 ) D ( 4 3 2 1 0 ) F ( 5 4 3 2 1 ) 3d3 (4Pe) 1/2 4d 3 -2.13511E+00 3.29 5 PDF e 3d3 (4Pe) 5/2 4d 1 -2.13294E+00 3.29 5 PDF e 3d3 (4Pe) 1/2 4d 2 -2.11203E+00 3.31 5 PDF e 3d3 (4Pe) 5/2 4d 4 -2.09728E+00 3.31 5 DF e 3d3 (4Pe) 3/2 4d 0 -2.08337E+00 3.33 5 D e Ncal= 13 : set complete Nlv= 9, 3,e: P ( 2 1 0 ) D ( 3 2 1 ) F ( 4 3 2 ) 3d3 (4Pe) 5/2 4d 2 -2.13414E+00 3.29 3 PDF e 3d3 (4Pe) 3/2 4d 3 -2.10788E+00 3.31 3 DF e 3d3 (4Pe) 3/2 4d 1 -2.08179E+00 3.33 3 PD e 3d3 (4Pe) 5/2 4d 2 -2.07934E+00 3.46 3 PDF e 3d3 (4Pe) 5/2 4d 1 -2.07801E+00 3.33 3 PD e 3d3 (4Pe) 5/2 4d 3 -2.07780E+00 3.33 3 DF e 3d3 (4Pe) 5/2 4d 0 -2.07641E+00 3.33 3 P e 3d3 (4Pe) 5/2 4d 4 -2.07512E+00 3.33 3 F e 3d3 (4Pe) 3/2 4d 2 -2.04510E+00 3.36 3 PDF e Ncal= 9 : set complete Nlv= 15, 3,e: D ( 3 2 1 ) F ( 4 3 2 ) G ( 5 4 3 ) H ( 6 5 4 ) I ( 7 6 5 ) 3d3 (2Ge) 7/2 4d 3 -2.16065E+00 3.26 3 DFG e 3d3 (2Ge) 7/2 4d 5 -2.15935E+00 3.26 3 GHI e 3d3 (2Ge) 7/2 4d 6 -2.15443E+00 3.27 3 HI e 3d3 (2Ge) 7/2 4d 4 -2.15035E+00 3.27 3 FGH e 3d3 (2Ge) 9/2 4d 7 -2.14872E+00 3.27 3 I e 3d3 (2Ge) 9/2 4d 3 -2.14865E+00 3.27 3 DFG e 3d3 (2Ge) 9/2 4d 5 -2.14663E+00 3.27 3 GHI e 3d3 (2Ge) 9/2 4d 5 -2.13490E+00 3.28 3 GHI e 3d3 (2Ge) 9/2 4d 6 -2.13050E+00 3.30 3 HI e 3d3 (2Ge) 9/2 4d 2 -2.11451E+00 3.29 3 DF e 3d3 (2Ge) 7/2 4d 1 -2.10743E+00 3.30 3 D e 3d3 (2Ge) 7/2 4d 2 -2.10505E+00 3.30 3 DF e 3d3 (2Ge) 7/2 4d 3 -2.09600E+00 3.31 3 DFG e 3d3 (2Ge) 9/2 4d 4 -2.08773E+00 3.31 3 FGH e 3d3 (2Ge) 9/2 4d 4 -1.95461E+00 3.41 3 FGH e Ncal= 15 : set complete Nlv= 5, 1,e: D ( 2 ) F ( 3 ) G ( 4 ) H ( 5 ) I ( 6 ) 3d3 (2Ge) 7/2 4d 5 -2.15108E+00 3.27 1 H e 3d3 (2Ge) 7/2 4d 4 -2.13926E+00 3.27 1 G e 3d3 (2Ge) 9/2 4d 3 -2.11077E+00 3.29 1 F e 3d3 (2Ge) 7/2 4d 6 -2.10182E+00 3.27 1 I e 3d3 (2Ge) 7/2 4d 2 -2.09282E+00 3.30 1 D e Ncal= 5 : set complete Nlv= 9, 3,e: P ( 2 1 0 ) D ( 3 2 1 ) F ( 4 3 2 ) 3d3 (2Pe) 1/2 4d 1 -2.11449E+00 3.26 3 PD e 3d3 (2Pe) 3/2 4d 4 -2.10214E+00 3.27 3 F e 3d3 (2Pe) 1/2 4d 2 -2.08472E+00 3.28 3 PDF e 3d3 (2Pe) 1/2 4d 3 -2.08116E+00 3.28 3 DF e 3d3 (2Pe) 3/2 4d 2 -2.07662E+00 3.29 3 PDF e 3d3 (2Pe) 3/2 4d 3 -2.05165E+00 3.30 3 DF e 3d3 (2Pe) 3/2 4d 0 -1.94013E+00 3.39 3 P e 3d3 (2Pe) 3/2 4d 1 -1.93293E+00 3.39 3 PD e 3d3 (2Pe) 3/2 4d 2 -1.89853E+00 3.42 3 PDF e Ncal= 9 : set complete Nlv= 3, 1,e: P ( 1 ) D ( 2 ) F ( 3 ) 3d3 (2Pe) 3/2 4d 1 -2.08815E+00 3.26 1 P e 3d3 (2Pe) 1/2 4d 2 -1.95699E+00 3.41 1 D e 3d3 (2Pe) 3/2 4d 3 -1.94667E+00 3.38 1 F e Ncal= 3 : set complete Nlv= 5, 1,e: S ( 0 ) P ( 1 ) D ( 2 ) F ( 3 ) G ( 4 ) 3d3 2 (2De) 3/2 4d 3 -2.06726E+00 3.28 1 F e 3d3 2 (2De) 3/2 4d 1 -1.94012E+00 3.37 1 P e 3d3 2 (2De) 5/2 4d 2 -1.92280E+00 3.38 1 D e 3d3 2 (2De) 3/2 4d 4 -1.88850E+00 3.46 1 G e 3d3 2 (2De) 5/2 4d 0 -1.84426E+00 3.45 1 S e Ncal= 5 : set complete 27Table IV. continues. Ct StLtπt Jtnl J E(cal) ν SLπ Nlv= 13, 3,e: S ( 1 ) P ( 2 1 0 ) D ( 3 2 1 ) F ( 4 3 2 ) G ( 5 4 3 ) 3d3 2 (2De) 5/2 5d 1 -1.05446E+00 4.36 3 SPD e 3d3 2 (2De) 5/2 5d 5 -1.04247E+00 4.38 3 G e 3d3 2 (2De) 3/2 5d 3 -1.04179E+00 4.38 3 DFG e 3d3 2 (2De) 3/2 5d 1 -1.03332E+00 4.40 3 SPD e 3d3 2 (2De) 5/2 5d 4 -1.03014E+00 4.40 3 FG e 3d3 2 (2De) 5/2 5d 2 -1.02889E+00 4.40 3 PDF e 3d3 2 (2De) 3/2 5d 0 -1.02130E+00 4.42 3 P e 3d3 2 (2De) 5/2 5d 1 -1.01682E+00 4.43 3 SPD e 3d3 2 (2De) 5/2 5d 3 -1.01420E+00 4.43 3 DFG e 3d3 2 (2De) 3/2 5d 2 -1.00936E+00 4.44 3 PDF e 3d3 2 (2De) 3/2 5d 2 -9.92578E-01 4.47 3 PDF e 3d3 2 (2De) 5/2 5d 3 -9.81500E-01 4.49 3 DFG e Ncal= 12 , Nlv= 13 : set incomplete, level missing: 4 Odd Parity, Equivalent-electron & unidentifiable levels (s ee discussion) 3p53d5 1 -1.23559E+00 3p53d5 2 -1.22406E+00 3p53d5 6 -1.21757E+00 3p53d5 5 -1.21176E+00 3p53d5 4 -1.20632E+00 3p53d5 3 -1.20599E+00 3p53d5 3 -1.20156E+00 3p53d5 2 -1.19775E+00 3p53d5 4 -1.15378E+00 3p53d5 3 -1.13582E+00 3p53d5 2 -1.13085E+00 3p53d5 5 -1.12662E+00 3p53d5 2 -1.12076E+00 3p53d5 4 -1.11088E+00 3p53d5 0 -1.10428E+00 3p53d5 2 -1.09132E+00 3p53d5 7 -1.06928E+00 3p53d5 6 -1.05100E+00 3p53d5 1 -1.04259E+00 3p53d5 3 -1.02720E+00 3p53d5 4 -9.99298E-01 3p53d5 1 -9.99223E-01 3p53d5 3 -9.98823E-01 3p53d5 5 -9.92970E-01 28Table V. Transition probabilties of Fe V among observed fine structur e levels. Ci Cf SiLiπiSfLfπfgigf Ei Ef f A fi (Ry) (Ry) ( s−1) 3d4−3d3(4F)4p5De 5Fo1 3 5.5132 3.1644 2.154E-01 3.18E+09 3d4−3d3(4F)4p5De 5Fo3 3 5.5119 3.1644 3.790E-04 1.68E+07 3d4−3d3(4F)4p5De 5Fo3 5 5.5119 3.1496 1.358E-03 3.65E+07 3d4−3d3(4F)4p5De 5Fo5 3 5.5094 3.1644 4.617E-02 3.40E+09 3d4−3d3(4F)4p5De 5Fo5 5 5.5094 3.1496 5.967E-02 2.67E+09 3d4−3d3(4F)4p5De 5Fo5 7 5.5094 3.1443 1.462E-02 4.69E+08 3d4−3d3(4F)4p5De 5Fo7 5 5.5058 3.1496 6.895E-03 4.30E+08 3d4−3d3(4F)4p5De 5Fo7 7 5.5058 3.1443 5.889E-02 2.64E+09 3d4−3d3(4F)4p5De 5Fo9 7 5.5015 3.1443 1.966E-03 1.13E+08 3d4−3d3(4F)4p5De 5Fo7 9 5.5058 3.1391 3.262E-02 1.14E+09 3d4−3d3(4F)4p5De 5Fo9 9 5.5015 3.1391 5.139E-02 2.30E+09 3d4−3d3(4F)4p5De 5Fo9 11 5.5015 3.1343 7.548E-02 2.78E+09 LS 25 35 5.5055 3.1451 1.068E-01 3.42E+09 3d4−3d3(4F)4p5De 5Do1 3 5.5132 3.1540 5.515E-03 8.22E+07 3d4−3d3(4F)4p5De 5Do3 1 5.5119 3.1565 6.255E-02 8.36E+09 3d4−3d3(4F)4p5De 5Do3 3 5.5119 3.1540 3.888E-02 1.74E+09 3d4−3d3(4F)4p5De 5Do3 5 5.5119 3.1609 1.360E-01 3.62E+09 3d4−3d3(4F)4p5De 5Do5 3 5.5094 3.1540 1.704E-02 1.27E+09 3d4−3d3(4F)4p5De 5Do5 5 5.5094 3.1609 1.372E-02 6.08E+08 3d4−3d3(4F)4p5De 5Do5 7 5.5094 3.1559 1.087E-01 3.45E+09 3d4−3d3(4F)4p5De 5Do7 5 5.5058 3.1609 4.155E-02 2.57E+09 3d4−3d3(4F)4p5De 5Do7 7 5.5058 3.1559 4.936E-02 2.19E+09 3d4−3d3(4F)4p5De 5Do9 7 5.5015 3.1559 2.644E-02 1.50E+09 3d4−3d3(4F)4p5De 5Do7 9 5.5058 3.1498 7.311E-02 2.54E+09 3d4−3d3(4F)4p5De 5Do9 9 5.5015 3.1498 1.168E-01 5.19E+09 LS 25 25 5.5064 3.1551 1.541E-01 6.84E+09 3d4−3d3(4F)4p5De 3Do1 3 5.5132 3.1439 5.744E-02 8.63E+08 3d4−3d3(4F)4p5De 3Do3 3 5.5119 3.1439 6.807E-03 3.07E+08 3d4−3d3(4F)4p5De 3Do3 5 5.5119 3.1401 3.147E-02 8.53E+08 3d4−3d3(4F)4p5De 3Do5 3 5.5094 3.1439 4.980E-03 3.73E+08 3d4−3d3(4F)4p5De 3Do5 5 5.5094 3.1401 2.987E-07 1.35E+04 3d4−3d3(4F)4p5De 3Do5 7 5.5094 3.1331 1.149E-02 3.72E+08 3d4−3d3(4F)4p5De 3Do7 5 5.5058 3.1401 6.830E-03 4.30E+08 3d4−3d3(4F)4p5De 3Do7 7 5.5058 3.1331 4.107E-03 1.86E+08 3d4−3d3(4F)4p5De 3Do9 7 5.5015 3.1331 3.671E-03 2.13E+08 3d4−3d3(4P)4p5De 5Po1 3 5.5132 3.0195 8.420E-02 1.40E+09 3d4−3d3(4P)4p5De 5Po3 3 5.5119 3.0195 6.281E-02 3.13E+09 3d4−3d3(4P)4p5De 5Po3 5 5.5119 3.0151 2.114E-02 6.35E+08 3d4−3d3(4P)4p5De 5Po5 3 5.5094 3.0195 2.926E-02 2.43E+09 3d4−3d3(4P)4p5De 5Po5 5 5.5094 3.0151 4.831E-02 2.41E+09 3d4−3d3(4P)4p5De 5Po5 7 5.5094 3.0078 6.221E-03 2.23E+08 3d4−3d3(4P)4p5De 5Po7 5 5.5058 3.0151 5.555E-02 3.88E+09 3d4−3d3(4P)4p5De 5Po7 7 5.5058 3.0078 3.105E-02 1.56E+09 3d4−3d3(4P)4p5De 5Po9 7 5.5015 3.0078 8.781E-02 5.64E+09 LS 25 15 5.5071 3.0126 8.610E-02 7.17E+09 3d4−3d3(4P)4p5De 5Do1 3 5.5132 3.0058 4.401E-03 7.41E+07 3d4−3d3(4P)4p5De 5Do3 1 5.5119 3.0094 4.902E-04 7.40E+07 3d4−3d3(4P)4p5De 5Do3 3 5.5119 3.0058 7.201E-04 3.63E+07 3d4−3d3(4P)4p5De 5Do3 5 5.5119 2.9912 2.402E-03 7.35E+07 3d4−3d3(4P)4p5De 5Do5 3 5.5094 3.0058 1.502E-03 1.26E+08 3d4−3d3(4P)4p5De 5Do5 5 5.5094 2.9912 2.248E-03 1.14E+08 3d4−3d3(4P)4p5De 5Do5 7 5.5094 2.9883 1.474E-03 5.38E+07 3d4−3d3(4P)4p5De 5Do7 5 5.5058 2.9912 2.675E-03 1.90E+08 3d4−3d3(4P)4p5De 5Do7 7 5.5058 2.9883 1.048E-03 5.33E+07 3d4−3d3(4P)4p5De 5Do9 7 5.5015 2.9883 2.846E-06 1.86E+05 3d4−3d3(4P)4p5De 5Do7 9 5.5058 2.9792 1.408E-03 5.61E+07 3d4−3d3(4P)4p5De 5Do9 9 5.5015 2.9792 4.558E-03 2.33E+08 LS 25 25 5.5064 2.9892 4.731E-03 2.41E+08 3d4−3d3(4P)4p5De 3Po1 3 5.5132 2.9911 8.365E-05 1.42E+06 3d4−3d3(4P)4p5De 3Po3 1 5.5119 2.9941 1.179E-03 1.80E+08 3d4−3d3(4P)4p5De 3Po3 3 5.5119 2.9911 4.467E-04 2.28E+07 3d4−3d3(4P)4p5De 3Po3 5 5.5119 3.0038 4.331E-04 1.31E+07 3d4−3d3(4P)4p5De 3Po5 3 5.5094 2.9911 1.298E-05 1.10E+06 3d4−3d3(4P)4p5De 3Po5 5 5.5094 3.0038 9.745E-05 4.91E+06 3d4−3d3(4P)4p5De 3Po7 5 5.5058 3.0038 1.164E-04 8.19E+06 3d4−3d3(2P)4p5De 3Po1 3 5.5132 2.9439 1.035E-03 1.83E+07 3d4−3d3(2P)4p5De 3Po3 1 5.5119 2.9413 2.456E-08 3.91E+03 3d4−3d3(2P)4p5De 3Po3 3 5.5119 2.9439 4.505E-04 2.39E+07 29Table V. continues. Ci Cf SiLiπiSfLfπfgigf Ei Ef f A fi (Ry) (Ry) ( s−1) 3d4−3d3(2P)4p5De 3Po3 5 5.5119 2.8668 2.963E-05 9.99E+05 3d4−3d3(2P)4p5De 3Po5 3 5.5094 2.9439 6.224E-04 5.48E+07 3d4−3d3(2P)4p5De 3Po5 5 5.5094 2.8668 2.903E-05 1.63E+06 3d4−3d3(2P)4p5De 3Po7 5 5.5058 2.8668 3.718E-05 2.91E+06 3d4−3d3(2D2)4p5De 1Po1 3 5.5132 2.9073 1.306E-08 2.38E+02 3d4−3d3(2D2)4p5De 1Po3 3 5.5119 2.9073 1.655E-06 9.02E+04 3d4−3d3(2D2)4p5De 1Po5 3 5.5094 2.9073 9.168E-06 8.31E+05 3d4−3d3(2D2)4p5De 3Do1 3 5.5132 2.8826 2.062E-06 3.82E+04 3d4−3d3(2D2)4p5De 3Do3 3 5.5119 2.8826 5.025E-06 2.79E+05 3d4−3d3(2D2)4p5De 3Do3 5 5.5119 2.8761 1.691E-05 5.66E+05 3d4−3d3(2D2)4p5De 3Do5 3 5.5094 2.8826 2.087E-07 1.93E+04 3d4−3d3(2D2)4p5De 3Do5 5 5.5094 2.8761 1.929E-05 1.07E+06 3d4−3d3(2D2)4p5De 3Do5 7 5.5094 2.8713 3.481E-05 1.39E+06 3d4−3d3(2D2)4p5De 3Do7 5 5.5058 2.8761 1.622E-06 1.26E+05 3d4−3d3(2D2)4p5De 3Do7 7 5.5058 2.8713 2.264E-02 1.26E+09 3d4−3d3(2D2)4p5De 3Do9 7 5.5015 2.8713 1.661E-05 1.19E+06 3d4−3d3(2D2)4p5De 3Do7 9 5.5058 2.8761 6.490E-10 2.80E+01 3d4−3d3(2D2)4p5De 3Do9 9 5.5015 2.8761 4.627E-08 2.56E+03 3d4−3d3(2P)4p5De 3Do1 3 5.5132 2.9274 1.222E-03 2.19E+07 3d4−3d3(2P)4p5De 3Do3 3 5.5119 2.9274 8.954E-04 4.80E+07 3d4−3d3(2P)4p5De 3Do3 5 5.5119 2.9169 2.958E-06 9.60E+04 3d4−3d3(2P)4p5De 3Do5 3 5.5094 2.9274 3.739E-04 3.34E+07 3d4−3d3(2P)4p5De 3Do5 5 5.5094 2.9169 9.819E-06 5.30E+05 3d4−3d3(2P)4p5De 3Do5 7 5.5094 2.9117 3.148E-06 1.22E+05 3d4−3d3(2P)4p5De 3Do7 5 5.5058 2.9169 9.881E-06 7.45E+05 3d4−3d3(2P)4p5De 3Do7 7 5.5058 2.9117 1.265E-03 6.84E+07 3d4−3d3(2P)4p5De 3Do9 7 5.5015 2.9117 1.136E-05 7.87E+05 3d4−3d3(2P)4p5De 3So1 3 5.5132 2.9052 4.138E-06 7.54E+04 3d4−3d3(2P)4p5De 3So3 3 5.5119 2.9052 1.659E-08 9.05E+02 3d4−3d3(2P)4p5De 3So5 3 5.5094 2.9052 6.255E-07 5.68E+04 3d4−3d3(4P)4p5De 3Do1 3 5.5132 2.8991 4.318E-06 7.90E+04 3d4−3d3(4P)4p5De 3Do3 3 5.5119 2.8991 1.262E-06 6.92E+04 3d4−3d3(4P)4p5De 3Do3 5 5.5119 2.8991 9.987E-07 3.29E+04 3d4−3d3(4P)4p5De 3Do5 3 5.5094 2.8991 2.563E-06 2.34E+05 3d4−3d3(4P)4p5De 3Do5 5 5.5094 2.8991 3.397E-08 1.86E+03 3d4−3d3(4P)4p5De 3Do5 7 5.5094 2.9030 6.662E-05 2.60E+06 3d4−3d3(4P)4p5De 3Do7 5 5.5058 2.8991 1.275E-06 9.74E+04 3d4−3d3(4P)4p5De 3Do7 7 5.5058 2.9030 1.589E-02 8.65E+08 3d4−3d3(4P)4p5De 3Do9 7 5.5015 2.9030 7.636E-05 5.32E+06 3d4−3d3(2D2)4p5De 3Po1 3 5.5132 2.8652 3.020E-05 5.67E+05 3d4−3d3(2D2)4p5De 3Po3 1 5.5119 2.8623 1.555E-05 2.63E+06 3d4−3d3(2D2)4p5De 3Po3 3 5.5119 2.8652 1.906E-05 1.07E+06 3d4−3d3(2D2)4p5De 3Po5 3 5.5094 2.8652 6.600E-06 6.18E+05 3d4−3d3(2P)4p5De 1Po1 3 5.5132 2.8161 2.397E-06 4.67E+04 3d4−3d3(2P)4p5De 1Po3 3 5.5119 2.8161 2.381E-06 1.39E+05 3d4−3d3(2P)4p5De 1Po5 3 5.5094 2.8161 3.000E-06 2.91E+05 3d4−3d3(4P)4p5De 3So1 3 5.5132 2.8282 6.009E-05 1.16E+06 3d4−3d3(4P)4p5De 3So3 3 5.5119 2.8282 4.589E-05 2.65E+06 3d4−3d3(4P)4p5De 3So5 3 5.5094 2.8282 2.678E-05 2.58E+06 3d4−3d3(2F)4p5De 3Do1 3 5.5132 2.7003 4.619E-06 9.78E+04 3d4−3d3(2F)4p5De 3Do3 3 5.5119 2.7003 6.420E-07 4.08E+04 3d4−3d3(2F)4p5De 3Do3 5 5.5119 2.7050 3.933E-09 1.49E+02 3d4−3d3(2F)4p5De 3Do5 3 5.5094 2.7003 4.515E-08 4.77E+03 3d4−3d3(2F)4p5De 3Do5 5 5.5094 2.7050 6.185E-06 3.91E+05 3d4−3d3(2F)4p5De 3Do5 7 5.5094 2.7129 7.193E-07 3.23E+04 3d4−3d3(2F)4p5De 3Do7 5 5.5058 2.7050 1.201E-07 1.06E+04 3d4−3d3(2F)4p5De 3Do7 7 5.5058 2.7129 6.082E-05 3.81E+06 3d4−3d3(2F)4p5De 3Do9 7 5.5015 2.7129 6.629E-06 5.32E+05 30Table VI. Comparison of present f-values with the earlier ones. Ci Cj SiLiπiSjLjπj2Ji+ 1 Ii2Jj+ 1 Ijfij(P) fij(others ) 3d4 -3d3(4F)4p 5D0 5F1 1 1 3 1 0.2154 0.163a 3d4 -3d3(4F)4p 5D0 5F1 3 1 3 1 3.790E-04 3d4 -3d3(4F)4p 5D0 5F1 3 1 5 3 0.00136 3d4 -3d3(4F)4p 5D0 5F1 5 1 3 1 0.04617 0.0126a 3d4 -3d3(4F)4p 5D0 5F1 5 1 5 3 0.05967 0.0596a 3d4 -3d3(4F)4p 5D0 5F1 5 1 7 3 0.01462 0.0138a 3d4 -3d3(4F)4p 5D0 5F1 7 1 5 3 0.006895 0.0274a 3d4 -3d3(4F)4p 5D0 5F1 7 1 7 3 0.05889 0.0544a 3d4 -3d3(4F)4p 5D0 5F1 9 1 7 3 0.001966 0.00756a 3d4 -3d3(4F)4p 5D0 5F1 7 1 9 3 0.03262 0.0414a 3d4 -3d3(4F)4p 5D0 5F1 9 1 9 3 0.05139 0.03a 3d4 -3d3(4F)4p 5D0 5F1 9 1 11 2 0.07548 0.0686a 3d4 -3d3(4F)4p 5D0 5F1 25 35 0.107 0.0804b,0.0915c 3d4 -3d3(4F)4p 5D0 5D1 1 1 3 2 0.00551 0.041a 3d4 -3d3(4F)4p 5D0 5D1 3 1 1 1 0.06255 0.0607a 3d4 -3d3(4F)4p 5D0 5D1 3 1 3 2 0.03888 0.0343a 3d4 -3d3(4F)4p 5D0 5D1 3 1 5 2 0.1360 0.1257a 3d4 -3d3(4F)4p 5D0 5D1 5 1 3 2 0.01704 0.0532a 3d4 -3d3(4F)4p 5D0 5D1 5 1 5 2 0.01372 0.0092a 3d4 -3d3(4F)4p 5D0 5D1 5 1 7 2 0.1087 0.1006a 3d4 -3d3(4F)4p 5D0 5D1 7 1 5 2 0.04155 0.0247a 3d4 -3d3(4F)4p 5D0 5D1 7 1 7 2 0.04936 0.0517a 3d4 -3d3(4F)4p 5D0 5D1 9 1 7 2 0.02644 0.0222a 3d4 -3d3(4F)4p 5D0 5D1 7 1 9 2 0.07311 0.0588a 3d4 -3d3(4F)4p 5D0 5D1 9 1 9 2 0.1168 0.130a 3d4 -3d3(4F)4p 5D0 5D1 25 25 0.1541 0.1708b,0.192c 3d4 -3d3(4P)4p 5D0 5P1 1 1 3 4 0.08420 0.076a 3d4 -3d3(4P)4p 5D0 5P1 3 1 3 4 0.06281 0.057a 3d4 -3d3(4P)4p 5D0 5P1 3 1 5 6 0.02114 0.019a 3d4 -3d3(4P)4p 5D0 5P1 5 1 3 4 0.02926 0.0266a 3d4 -3d3(4P)4p 5D0 5P1 5 1 5 6 0.04831 0.0442a 3d4 -3d3(4P)4p 5D0 5P1 5 1 7 7 0.00622 0.0054a 3d4 -3d3(4P)4p 5D0 5P1 7 1 5 6 0.05555 0.0499a 3d4 -3d3(4P)4p 5D0 5P1 7 1 7 7 0.03105 0.0264a 3d4 -3d3(4P)4p 5D0 5P1 9 1 7 7 0.08782 0.0758a 3d4 -3d3(4P)4p 5D0 5P1 25 15 0.0861 0.076b,0.0893c a Fawcett (1989), b Butler (TOPbase), c Bautista (1996) 31

arXiv:physics/0001046v1 [physics.atom-ph] 21 Jan 2000Role of negative-energy states and Breit interaction in cal culation of atomic parity-nonconserving amplitudes. A. Derevianko Institute for Theoretical Atomic and Molecular Physics Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138 (November 23, 2013) It is demonstrated that Breit and negative-energy state contributions reduce the 2 .5σdeviation [S.C. Bennett and C.E. Wieman, Phys. Rev. Lett. 82, 2484 (1999)] in the value of the weak charge of133Cs from the Standard Model predic- tion to 1.7 σ. The corrections are obtained in the relativis- tic many-body perturbation theory by combining all-order Coulomb and second-order Breit contributions. The correc- tions to parity-nonconserving amplitudes amount to 0.6% in 133Cs and 1.1% in223Fr. The relevant magnetic-dipole hy- perfine structure constants are modified at the level of 0.3% in Cs, and 0.6% in Fr. Electric-dipole matrix elements are affected at 0.1% level in Cs and a few 0.1% in Fr. PACS: 31.30.Jv, 12.15.Ji, 11.30.Er Atomic parity-nonconserving (PNC) experiments com- bined with accurate atomic structure calculations pro- vide constrains on “new physics” beyond the Standard Model of elementary-particle physics. Compared to high- energy experiments or low-energy epscattering exper- iments, atomic single-isotope PNC measurements are uniquely sensitive to new isovector heavy physics [1]. Presently, the PNC effect in atoms was most precisely measured by Boulder group in133Cs [2]. They de- termined ratio of PNC amplitude EPNCto the tensor transition polarizability βfor 7S1/2−6S1/2transition with a precision of 0.35%. In 1999, Bennett and Wie- man [3] accurately measured tensor transition polariz- ability β, and by combining the previous theoretical de- terminations of the EPNC[4,5] with their measurements, they have found a value of the weak charge for133Cs QW=−72.06(28) expt(34)theorwhich differed from the prediction [6] of the Standard Model QW=−73.20(13) by 2.5 standard deviations. They also reevaluated the precision of the early 1990s atomic structure calcula- tions [4,5], and argued that the uncertainty of the pre- dicted EPNCis 0.4%, rather than previously estimated 1%. This conclusion has been based on a much bet- ter agreement of calculated and recently accurately mea- sured electric-dipole amplitudes for the resonant transi- tions in alkali-metal atoms. In view of the reduced uncertainty, the purpose of this Letter is to evaluate contributions from negative-energy states (NES) and Breit interaction. It will be demon- strated that these contributions correct theoretical EPNC and the resultant value of the weak charge by 0.6% in Cs. It is worth noting, that due to the smallness of thesecontributions at what had been believed to be a 1% the- oretical error in Cs, the previous calculations have either omitted [5], or estimated the contributions from Breit interaction only partially [4]. The main focus of the pre- viousab initio calculations has been correlation contribu- tion from the residual Coulomb interaction (i.e. beyond Dirac-Hartree-Fock level). In both calculations impor- tant chains of many-body diagrams have been summed to all orders of perturbation theory. This Letter also reports correction due to NES and Breit interaction for EPNCin francium. The interest in Fr stems from the fact that analogous PNC amplitude is 18 times larger in heavier223Fr compared to Cs [7]. The measurement of atomic PNC in Fr is pursued by Stony Brook group [8]. The quality of theoretical atomic wave-functions at small radii is usually judged by comparing calculated and experimental hyperfine-structure magnetic-dipole con- stants A, and in the intermediate region by comparing electric-dipole matrix elements. It will be demonstrated that the corresponding corrections to all-order Coulomb values are at the level of a few 0.1%. The PNC amplitude of nS1/2→n′S1/2transition can be represented as a sum over intermediate states mP1/2 EPNC=/summationdisplay m/angbracketleftn′S|D|mP1/2/angbracketright/angbracketleftmP1/2|HW|nS/angbracketright EnS−EmP1/2 +/summationdisplay m/angbracketleftn′S|HW|mP1/2/angbracketright/angbracketleftmP1/2|D|nS/angbracketright En′S−EmP1/2. (1) The overwhelming contribution from parity-violating in- teractions arises from the Hamiltonian HW=GF√ 8QWρnuc(r)γ5, (2) where GFis the Fermi constant, γ5is the Dirac ma- trix, and ρnuc(r) is the nuclear distribution. To be consistent with the previous calculations the ρnuc(r) is taken to be a Fermi distribution with the “skin depth” a= 2.3/(4 ln3) fm and the cutoff radius c= 5.6743 fm for133Cs as in Ref. [4], and c= 6.671 fm for223Fr as in Ref. [7]. The PNC amplitude is customarily ex- pressed in the units of 10−11i(−QW/N), where Nis the number of neutrons in the nucleus ( N= 78 for 133Cs and N= 136 for223Fr ). Atomic units are used throughout the Letter. The results of the calcu- lations for133Cs are EPNC=−0.905×10−11i(−QW/N), 1Ref. [4] and EPNC=−0.908×10−11i(−QW/N), Ref. [5]. The former value includes a partial Breit contribution +0.002×10−11i(−QW/N), and the latter does not. Both calculations are in a very close agreement if the Breit con- tribution is added to the value of Novosibirsk group. The reference many-body Coulomb value EC PNC=−0.9075×10−11i(−QW/N) (3) is determined as an average of the two, with Breit con- tribution removed from the value of Notre Dame group. For223FrEPNC= 15.9×10−11i(−QW/N), Ref. [7], this value does not include Breit interaction. Ab initio relativistic many-body calculations of wave- functions, like coupled-cluster type calculations of [4,9 ], to avoid the “continuum dissolution problem” [10], start from the no-pair Hamiltonian derived from QED [11]. Theno-pair Hamiltonian excludes virtual electron- positron pairs from the resulting correlated wave- function. If the no-pair wave-functions are further used to obtain many-body matrix elements, the negative- energy state (NES) contribution is missing already in the second order. Recently, it has been shown that the magnetic-dipole transition amplitude both in He-like ions [12] and alkali-metal atoms [13] can be strongly af- fected by the NES correction. The enhancement mecha- nism is due to vanishingly small lowest-order values and also due to mixing of large and small components of a Dirac wavefunction by magnetic-dipole operator. For the NES ( E <−mc2) the meaning of large and small com- ponents is reversed, i.e., small component is much larger than large component. The mixing of large positive- energy component with small component of NES results in much larger one-particle matrix elements, than in the no-pair case. The 2 mc2energy denominators lessen the effect, but, for example, the Rb 5 S1/2−6S1/2magnetic- dipole rate is reduced by a factor of 8 from the no-pair value by the inclusion of NES [13]. The inclusion of Breit interaction also becomes important and the size of the correction is comparable to the Coulomb contribu- tion. Just as in the case of magnetic-dipole operator, the Dirac matrix γ5in the weak Hamiltonian Eq. (2) mixes large and small components of wavefunctions. Similar mixing occurs in the matrix element describing interac- tion of an electron with nuclear magnetic moment (hy- perfine structure constant A). As demonstrated below, the relative effect for these operators is not as strong as in the magnetic-dipole transition case, since the low- est order matrix elements are nonzero in the nonrela- tivistic limit, but is still important. It is worth noting that the problem of NES does not appear explicitly in the Green’s function [5] or mixed-parity [4] approaches; however the correction due to Breit interaction still has to be addressed. The NES Coulomb corrections have to be taken into account explicitly in the “sum-over-states” method [4], employing all-order many-body values ob- tained with the no-pair Hamiltonian. The analysis is based on the VN−1Dirac-Hartree-Fockpotential wave-functions, with the valence and virtual or- bitals mcalculated in the “frozen” potential of core or- bitals a. The second-order correction to a matrix element of one-particle operator Zbetween two valence states w andvis represented as Z(2) wv=/summationdisplay i/negationslash=vzwibiv ǫv−ǫi+/summationdisplay i/negationslash=wbwiziv ǫw−ǫi+ +/summationdisplay mazam(/tildewidegwmva+/tildewidebwmva) ǫa+ǫv−ǫm−ǫw+/summationdisplay ma(/tildewidegwavm+/tildewidebwavm)zma ǫa+ǫw−ǫm−ǫv.(4) This expression takes into account the residual (two- body) Coulomb, gijkl, and two-body bijkland one-body bij=/summationtext a˜biajaBreit interaction. Static form of Breit interaction is used in this work. The tilde denotes an- tisymmetric combination ˜bijkl=bijkl−bijlk. Subscript iranges over both core and excited states. Note that summation over states iandmincludes negative-energy states. The NES correction to PNC amplitudes arises in two circumstances, directly from the sum in Eq. (1) and in the values of electric-dipole and weak interaction ma- trix elements. If the length-gauge of the electric-dipole operator is used, the direct contribution of NES in the amplitude Eq. (1) is a factor of 10−13smaller than the total amplitude, and will be disregarded in the following. The numerical summations are done using 100 positive- and 100 negative-energy wavefunctions in a B-spline rep- resentation [14] obtained in a cavity with a radius of 75 a.u. The breakdown of second-order corrections to matrix elements of weak interaction for Cs is given in Table I. The all-order values from Ref. [4] are also listed in the ta- ble to fix the relative phase of the contributions. The ma- trix elements are each modified at 0.6-0.7% level. Most of the correction arises from positive-energy Breit con- tribution, negative-energy states contribute at a smaller but comparable level. The contributions from NES due to one-body and two-body Breit interaction are almost equal ( B(1) −≈B(2) −) and, in addition, B(1) +≈2B(2) +. The same relations hold also in francium. The corrections to the relevant length-form matrix elements are overwhelm- ingly due to the one-body Breit interaction, and are at 0.1% level. For example, the all-order reduced matrix element /angbracketleft6S1/2||D||6P1/2/angbracketright= 4.478 Ref. [9] is increased by 0.005, bringing the total 4.483 into an excellent agree- ment with experimental value [15] 4.4890(65). Generally, the corrections reduce absolute values of the weak inter- action matrix elements, and increase absolute values of the dipole matrix elements, therefore, their net contribu- tions to EPNChave an opposite sign. Matrix elements of weak interaction are affected more strongly, because of the sampling of wave-function in the nucleus, where relativity is important. As demonstrated in Ref. [4], the four lowest-energy va- lencemP1/2states contribute 98% of the sum in Eq. (1), and for the purposes of this work, limiting the sums to only these states is sufficient. The corrections to EPNC 2are calculated first by replacing the weak interaction ma- trix elements with the relevant second-order contribu- tions and at the same time using all-order dipole matrix elements, and second by taking all-order HWmatrix ele- ments, and replacing Dwith the appropriate correction. In both cases the experimental energies are used in the denominators. The needed all-order matrix elements for Cs are tabulated in Ref. [4]. The summary of corrections toEPNCis presented in Table II. The modifications in the weak interaction matrix element provide a dominant correction. The contribution due to NES in the Coulomb part is insignificant, and is already effectively included in the reference many-body Coulomb value EC PNC, Eq. (3). The reference value EC PNCis modified by the Breit con- tributions by 0.6%, almost two times larger than the un- certainty in the Boulder experiment [2]. The modified value is EC+B PNC(133Cs) = −0.902(36) ×10−11i(−QW/N). A 0.4% uncertainty had been assigned to the above re- sult following analysis of Bennett and Wieman [3]. When EC+B PNCis combined with the experimental values of tran- sition polarizability β[3] and EPNC/β[2], one obtains for the weak charge QW(133Cs) = −72.42(28) expt(34)theor. This value differs from the prediction of the Standard Model QW=−73.20(13) by 1.7 σ, versus 2.5 σdiscussed in Ref. [3], where σis calculated by taking uncertainties in quadrature. The only previous calculation of Breit contribution to PNC amplitude in Cs has been performed by the Notre Dame group [4], using mixed-parity Dirac-Hartree-Fock formalism. The one-body Breit interaction has been in- cluded on equal footing with the DHF potential, but the linearized modification to one-body Breit potential due toHW(VPNC−HFBin notation of Ref. [4]) has been omit- ted. It is straightforward to demonstrate that because of this omission, the comparable contribution from two- bodypart of the Breit interaction has been disregarded. In units of 10−11i(−QW/N), the result of the present calculation for one-body Breit contribution is 0.003 ver- sus 0.002 in Ref. [4]. Such disagreement is most prob- ably caused by different types of correlation contribu- tion included in the two approaches. Treating one-body Breit together with the DHF potential effectively sums the many-body contributions from one-body Breit inter- action to all orders, and presents the advantage of the scheme employed in Ref. [4]. However, the dipole matrix elements and energies in the sum Eq. (1) are effectively included at the DHF level in the formulation of Ref. [4], in contrast to high-precision all-order values employed in the present work. The difference between the two val- ues can be considered as a theoretical uncertainty in the value of the Breit correction. Clearly more work needs to be done to resolve the discrepancy. The accuracy of the present analysis can be improved if the one-body Breitinteraction is embodied in DHF equations, and the many- body formulation starts from the resulting basis. How- ever, to improve present second-order treatment of the two-body part of the Breit interaction, higher orders of perturbation theory have to be considered. Apparently the most important contribution would arise from terms linearized in the Breit interaction, i.e. diagrams contain - ing one matrix elements of the Breit interaction and the rest of the residual Coulomb interaction. The Breit and NES corrections to PNC amplitude in heavier Fr are more pronounced. The223Fr PNC amplitude 15.9 ×10−11i(−QW/N) from Ref. [7] is re- duced by 1.1%. Using all-order dipole matrix elements from Ref. [9], the following corrections due to modifica- tions in the hWare found (in units of 10−11i(−QW/N)): one-body Breit B(1) ±=−0.131, two-body Breit B(2) ±= −0.053, and the C−correction, implicitly included in Ref. [7], is −0.003. As in the case of Cs, the all-order no-pair Coulomb result [9] for reduced matrix element /angbracketleft7P1/2||D||7S1/2/angbracketright= 4.256 is increased by inclusion of the Breit interaction and NES by 0 .0011, a 0.3% modifica- tion, leading to a much better agreement with experi- mental value 4.277(8) [16]. The modification of the EPNC due to corrections in the dipole matrix elements is much smaller than in the case of hW. At present there is no tabulation of accurate matrix elements of weak interac- tion for Fr, and the influence on EPNCdue to the Breit contribution in dipole matrix elements is estimated from average of the modification of individual dipole matrix elements 0.2%. The netresult decreases the reference Coulomb value for223Fr [7] by 0.18 ×10−11i(−QW/N), and the corrected value is EC+B PNC(223Fr) = 15 .7×10−11i(−QW/N). Finally, it is worth discussing Breit and NES contribu- tions to hyperfine-structure magnetic-dipole constants A for the states involved into PNC calculations. The all- orderno-pair Coulomb values in the recent work [9] have been corrected using a similar second-order formulation; no details of the calculation have been given. The ex- plicit contributions listed in Table III will be useful for correcting ab initio many-body Coulomb values. The ta- ble presents the contributions for two lowest valence S1/2 andP1/2states. The calculations are performed using a model of uniformly magnetized nucleus with a magneti- zation radius Rmgiven in the table. One finds that the additional terms reduce values calculated in the no-pair Coulomb-correlated approach. For Cs the corrections are of order 0.2% for 6 S1/2, 0.1% for 7 S1/2, and 0.3% for 6P1/2and 7P1/2. The relative contributions to hyperfine constants in heavier Fr are larger, accounting for 0.5% of the total value for 7 S1/2, 0.4% for 8 S1/2, and 0.6% for 7P1/2and 8P1/2. This work demonstrates that the Breit and NES con- tributions are comparable to the remainder of Coulomb correlation corrections unaccounted for in modern rela- tivistic all-order many-body calculations and hence have 3to be systematically taken into account. In particu- lar, the Breit interaction contributes 0.6% to parity- nonconserving amplitudes in Cs and 1.1% in Fr. The correction for Cs is almost twice the experimental un- certainty and reduces the recently determined [3] 2.5 σ deviation in the value of weak charge from the Stan- dard Model prediction to 1.7 σ. Both hyperfine constants and electric-dipole matrix elements are affected at a few 0.1%. By including NES and Breit correction, the no-pair Coulomb all-order dipole matrix elements [9] for resonant transitions are brought into an excellent agreement with the accurate experimental values. This work was supported by the U.S. Department of Energy, Division of Chemical Sciences, Office of Energy Research. The author would like to thank W.R. Johnson for useful discussions and H.R. Sadeghpour for sugges- tions on manuscript. TABLE I. Contributions to matrix elements of the weak interaction for133Cs in units 10−11i(−QW/N). All-order no-pair values are from Blundell et al. [4].C−is the cor- rection from negative-energy states for the residual Coulo mb interaction, and B(1) ±andB(2) ±are positive/negative energy state contributions from one-body and two-body Breit inter - action. Notation x[y] =x×10y. nall-order C− B(1) + B(1) − B(2) + B(2) − δ/angbracketleftnP1/2|hW|6S1/2/angbracketright 6 5.62[-2] -9.05[-6] -3.10[-4] 5.64[-5] -1.54[-4] 5.73[-5 ] 7 3.19[-2] -5.41[-6] -1.82[-4] 3.37[-5] -9.21[-5] 3.43[-5 ] 8 2.15[-2] -3.71[-6] -1.23[-4] 2.31[-5] -6.32[-5] 2.35[-5 ] 9 1.62[-2] -2.86[-6] -9.27[-5] 1.79[-5] -4.87[-5] 1.81[-5 ] δ/angbracketleft7S1/2|hW|nP1/2/angbracketright 6 2.72[-2] -4.74[-6] -1.53[-4] 2.96[-5] -8.06[-5] 3.00[-5 ] 7 1.54[-3] -2.84[-6] -8.96[-5] 1.77[-5] -4.83[-5] 1.80[-5 ] 8 1.04[-3] -1.95[-6] -6.06[-5] 1.21[-5] -3.31[-5] 1.23[-5 ] 9 0.78[-3] -1.50[-6] -4.56[-5] 9.36[-6] -2.55[-5] 9.51[-6 ] TABLE II. Summary of corrections to PNC amplitude in133Cs due to Breit interaction and negative-energy states. LineδHWlists contributions due to modifications in the weak interaction matrix elements, and δDdue to corrections in the electric dipole matrix elements. See the Table I caption for the explanation of columns. The units are 10−11i(−QW/N) , andx[y] =x×10y. C− B(1) ± B(2) ± δEPNC δHW 0.0002 0.0042 0.0019 0.0063 δD -2.6[-10] -0.0008 1.3[-6] -0.0008 Total 0.0002 0.0034 0.0019 0.0055TABLE III. Contributions to hyperfine-structure con- stants in MHz. Column “Expt” lists experimental values, where available, and δAgives the total of the contributions from negative-energy states and Breit interaction. See the Ta- ble I caption for the explanation of other columns. Notation x[y] means x×10y. state Expt C−B(1) +B(1) −B(2) +B(2) − δA 133Cs, gI= 0.73789 , R m= 5.6748 fm 6S1/22298.2 0.11 -8.14 0.25 3.50 -0.35 -4.64 7S1/2545.90(9) 0.03 -1.80 0.07 0.96 -0.097 -0.83 6P1/2291.89(9) -6.1[-4] -1.58 0.25 0.73 -0.27 -0.87 7P1/294.35 -2.2[-4] -5.43 0.09 0.26 -0.098 -0.29 211Fr, gI= 0.888, R m= 6.71 fm 7S1/28713.9(8) 0.07 -66.7 -0.08 19.8 -0.54 -47.4 8S1/21912.5(1.3) 0.02 -12.8 -0.02 5.08 -0.14 -7.88 7P1/21142.0(3) -5.6[-3] -10.8 1.23 3.62 -0.95 -6.90 8P1/2362.91a-2.0[-3] -3.61 0.44 1.29 -0.34 -2.23 aAll-order many-body calculations Ref. [9]. 4[1] M.J. Ramsey-Musolf, Phys. Rev. C 60, 015501 (1999). [2] C.S. Wood, S.C. Bennett, D. Cho, B. P. Masterson, J.L. Roberts, C.E. Tanner, and C.E. Wieman, Science 275, 1759 (1997). [3] S.C. Bennett and C.E. Wieman, Phys. Rev. Lett. 82, 2484 (1999). [4] S. A. Blundell, W.R. Johnson, and J. Sapirstein, Phys. Rev. Lett. 65, 1411 (1990); Phys. Rev. D 45, 1602 (1992). [5] V.A. Dzuba, V.V. Flambaum, O.P. Sushkov, Phys. Lett. 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arXiv:physics/0001047v1 [physics.plasm-ph] 22 Jan 2000Path Integral Monte Carlo Calculation of the Deuterium Hugo niot B. Militzer and D. M. Ceperley Department of Physics National Center for Supercomputing Applications University of Illinois at Urbana-Champaign, Urbana, IL 618 01 (February 2, 2008) Restricted path integral Monte Carlo simulations have been used to calculate the equilibrium properties of deuterium for two densities: 0 .674 and 0 .838 gcm−3(rs= 2.00 and 1 .86) in the temperature range of 10 000 K ≤T≤1000 000 K. Using the calculated internal energies and pressures we estimate the shock hugoniot and compare with re cent Laser shock wave experiments. We study finite size effects and the dependence on the time step of the path integral. Further, we compare the results obtained with a free particle nodal rest riction with those from a self-consistent variational principle, which includes interactions and bo und states. PACS Numbers: 71.10.-w 05.30.-d 02.70.Lq I. INTRODUCTION Recent laser shock wave experiments on pre- compressed liquid deuterium [1,2] have produced an un- expected equation of state for pressures up to 3.4 Mbar. It was found that deuterium has a significantly higher compressibility than predicted by the semi-empirical equation of state based on plasma many-body theory and lower pressure shock data (see SESAME model [3]). These experiments have triggered theoretical efforts to understand the state of compressed hydrogen in this range of density and temperature, made difficult be- cause the experiments are in regime where strong cor- relations and a significant degree of electron degeneracy are present. At this high density, it is problematic even to define the basic units such as molecules, atoms, free deuterons and electrons. Conductivity measurements [2] as well as theoretical estimates [4,5] suggest that in the experiment, a state of significant but not complete met- alization was reached. A variety of simulation techniques and analytical mod- els have been advanced to describe hydrogen in this par- ticular regime. There are ab initio methods such as re- stricted path integral Monte Carlo simulations (PIMC) [6,7,5] and density functional theory molecular dynamics (DFT-MD) [8,9]. Further there are models that min- imize an approximate free energy function constructed from known theoretical limits with respect to the chemi- cal composition, which work very well in certain regimes. The most widely used include [10,11,4]. We present new results from PIMC simulations. What emerges is a relative consensus of theoretical calculation s. First, we performed a finite size and time step study us- ing a parallelized PIMC code that allowed simulation of systems with NP= 64 pairs of electrons and deuterons and more importantly to decrease the time step from τ−1= 106K toτ−1= 8·106K. More importantly, we studied the effect of the nodal restriction on the hugoniot.II. RESTRICTED PATH INTEGRALS The density matrix of a quantum system at temper- ature kBT= 1/βcan be written as a integral over all pathsRt, ρ(R0,Rβ;β) =1 N!/summationdisplay P(±1)P/contintegraldisplay R0→PRβdRte−S[Rt].(1) Rtstands for the entire paths of Nparticles in 3 dimen- sional space Rt= (r1t, . . .,rNt) beginning at R0and connecting to PRβ.Plabels the permutation of the par- ticles. The upper sign corresponds to a system of bosons and the lower one to fermions. For non-relativistic parti- cles interacting with a potential V(R), the action of the pathS[Rt] is given by, S[Rt] =/integraldisplayβ 0dt/bracketleftBigg m 2/vextendsingle/vextendsingle/vextendsingle/vextendsingledR(t) ¯hdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +V(R(t))/bracketrightBigg + const .(2) One can estimate quantum mechanical expectation val- ues using Monte Carlo simulations [12] with a finite num- ber of imaginary time slices Mcorresponding to a time stepτ=β/M. For fermionic systems the integration is complicated due to the cancellation of positive and negative contribu- tions to the integral, ( the fermion sign problem ). It can be shown that the efficiency of the straightforward imple- mentation scales like e−2βNf, where fis the free energy difference per particle of a corresponding fermionic and bosonic system [13]. In [14,13], it has been shown that one can evaluate the path integral by restricting the path to only specific positive contributions. One introduces a reference point R∗on the path that specifies the nodes of the density matrix, ρ(R,R∗, t) = 0. A node-avoiding path for 0 < t≤βneither touches nor crosses a node: ρ(R(t),R∗, t)/negationslash= 0. By restricting the integral to node- avoiding paths, 1ρF(Rβ,R∗;β) =/integraldisplay dR0ρF(R0,R∗; 0)/contintegraldisplay R0→Rβ∈Υ(R∗)dRte−S[Rt],(3) (Υ(R∗) denotes the restriction) the contributions are positive and therefore PIMC represents, in principle, a solution to the sign problem. The method is exact if the exact fermionic density matrix is used for the restriction. However, the exact density matrix is only known in a few cases. In practice, applications have approximated the fermionic density matrix, by a determinant of single particle density matrices, ρ(R,R′;β) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ1(r1,r′ 1;β). . . ρ 1(rN,r′ 1;β) . . . . . . . . . ρ1(r1,r′ N;β). . . ρ 1(rN,r′ N;β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(4) This approach has been extensively applied using the free particle nodes [13], ρ1(r,r′, β) = (4 πλβ)−3/2exp/braceleftbig −(r−r′)2/4λβ/bracerightbig (5) withλ= ¯h2/2m, including applications to dense hydro- gen [6,7,5]. It can be shown that for temperatures larger than the Fermi energy the interacting nodal surface ap- proaches the free particle (FP) nodal surface. In addi- tion, in the limit of low density, exchange effects are negli- gible, the nodal constraint has a small effect on the path and therefore its precise shape is not important. The FP nodes also become exact in the limit of high density when kinetic effects dominate over the interaction poten- tial. However, for the densities and temperatures under consideration, interactions could have a significant effect on the fermionic density matrix. To gain some quantitative estimate of the possible ef- fect of the nodal restriction on the thermodynamic prop- erties, it is necessary to try an alternative. In addition to FP nodes, we used a restriction taken from a variational density matrix (VDM) that already includes interactions and atomic and molecular bound states. The VDM is a variational solution of the Bloch equa- tion. Assume a trial density matrix with parameters qi that depend on imaginary time βandR′, ρ(R,R′;β) =ρ(R, q1, . . ., q m). (6) By minimizing the integral: /integraldisplay dR/parenleftbigg∂ρ(R,R′;β) ∂β+Hρ(R,R′;β)/parenrightbigg2 = 0 ,(7) one determines equations for the dynamics of the param- eters in imaginary time: 1 2∂H ∂/vector q+↔ N˙/vector q= 0 where H≡/integraldisplay ρHρ dR.(8) The normalization matrix is:Nij= lim q′→q∂2 ∂qi∂q′ j/bracketleftbigg/integraldisplay dRρ(R, /vector q;β)ρ(R, /vector q′;β)/bracketrightbigg .(9) We assume the density matrix is a Slater determinant of single particle Gaussian functions ρ1(r,r′, β) = (πw)−3/2exp/braceleftbig −(r−m)2/w+d/bracerightbig (10) where the variational parameters are the mean m, squared width wand amplitude d. The differential equa- tion for this ansatz are given in [15]. The initial condi- tions at β−→0 are w= 2β,m=r′andd= 0 in order to regain the correct FP limit. It follows from Eq. 7 that at low temperature, the VDM goes to the lowest energy wave function within the variational basis. For an isolated atom or molecule this will be a bound state, in contrast to the delocalized state of the FP density matrix. A further discussion of the VDM properties is given in [15]. Note that this discussion concerns only the nodal restriction. In performing the PIMC simulation, the complete potential between the interacting charges is taken into account as discussed in detail in [12]. 0 50000 100000 150000 T (K)−2−1.5−1−0.500.5EVDM−Efree (eV) rS=1.86 rS=2.00 FIG. 1. Difference in the internal energy from PIMC sim- ulations with VDM and FP nodes vs. temperature using NP= 32 and τ−1= 2·106K. Simulations with VDM nodes lead to lower internal energies than those with FP nodes as shown in Fig. 1. Since the free energy Fis the integral of the internal energy over temperature, one can conclude that VDM nodes yield to a smaller Fand hence, are the more ap- propriate nodal surface. For the two densities considered here, the state of deu- terium goes from a plasma of strongly interacting but un-bound deuterons and electrons at high Tto a regime at low T, which is characterized by a significant elec- tronic degeneracy and bound states. Also at decreasing T, one finds an increasing number of electrons involved in long permutation cycles. Additionally for T≤15 625 K, molecular formation is observed. Comparing FP and VDM nodes, one finds that VDM predicts a higher molec- ular fraction and fewer permutations hinting to more lo- calized electrons. 2III. SHOCK HUGONIOT The recent experiments measured the shock velocity, propagating through a sample of pre-compressed liquid deuterium characterized by an initial state, ( E0,V0,p0) withT= 19.6 K and ρ0= 0.171 g/cm3. Assuming an ideal planar shock front, the variables of the shocked ma- terial ( E,V,p) satisfy the hugoniot relation [16], H=E−E0+1 2(V−V0)(p+p0) = 0 . (11) We set E0to its exact value of −15.886eV per atom [17] andp0= 0. Using the simulation results for pandE, we calculate H(T, ρ) and then interpolate Hlinearly at constant Tbetween the two densities corresponding to rs= 1.86 and 2 to obtain a point on the hugoniot in the (p, ρ) plane. (Results at rs= 1.93 confirm the function is linear within the statistical errors). The PIMC data forp,E, and the hugoniot are given in Tab. I. 0.70 0.75 0.80 ρ (gcm−3)0123456p (Mbar)VDM PIMC free, N=32, τ−1=106K PIMC free, N=32, τ−1=2*106K PIMC free, N=32, τ−1=8*106K PIMC free, N=64, τ−1=2*106K PIMC VDM, N=32, τ−1=106K PIMC VDM, N=32, τ−1=2*106K FIG. 2. Comparison hugoniot function calculated with PIMC simulations of different accuracy: FP nodes with NP=32 ( △forτ−1= 106K reported in [5], for τ−1= 2·106K,▽forτ−1 F= 8·106K and τ−1 B= 2·106K) and NP=64 ( 2forτ−1= 2·106K) as well as with VDM nodes and NP=32 (◦forτ−1= 106K and •forτ−1= 2·106K). Begin- ning at high pressures, the points on each hugoniot correspo nd to the following temperatures 125 000 ,62 500 ,31 250 ,15 625 , and 10 000 K. The dashed line corresponds to a calculation using the VDM alone. In Fig. 2, we compare the effects of different approx- imations made in the PIMC simulations such as time stepτ, number of pairs NPand the type of nodal re- striction. For pressures above 3 Mbar, all these approx- imations have a very small effect. The reason is that PIMC simulation become increasingly accurate as tem- perature increases. The first noticeable difference occurs atp≈2.7Mbar, which corresponds to T= 62 500 K. At lower pressures, the differences become more and morepronounced. We have performed simulations with free particle nodes and NP= 32 for three different values of τ. Using a smaller time step makes the simulations com- putationally more demanding and it shifts the hugoniot curves to lower densities. These differences come mainly from enforcing the nodal surfaces more accurately, which seems to be more relevant than the simultaneous im- provements in the accuracy of the action S, that is the time step is constrained more by the Fermi statistics than it is by the potential energy. We improved the efficiency of the algorithm by using a smaller time step τFfor eval- uating the Fermi action than the time step τBused for the potential action. Unless specified otherwise, we used τF=τB=τ. At even lower pressures not shown in Fig. 2, all of the hugoniot curves with FP nodes turn around and go to low densities as expected. As a next step, we replaced the FP nodes by VDM nodes. Those results show that the form of the nodes has a significant effect for pbelow 2 Mbar. Using a smaller τalso shifts the curve to slightly lower densities. In the region where atoms and molecules are forming, it is plausible that VDM nodes are more accurate than free nodes because they can describe those states [15]. We also show a hugoniot derived on the basis of the VDM alone (dashed line). These results are quite reasonable considering the approximations (Hartree-Fock) made in that calculation. Therefore, we consider the PIMC simu- lation with the smallest time step using VDM nodes ( •) to be our most reliable hugoniot. Going to bigger system sizesNP= 64 and using FP nodes also shows a shift towards lower densities. 0.4 0.6 0.8 1.0 1.2 ρ (gcm−3)01234p (Mbar)Experiment [1] Experiment [2] SESAME Linear mixing DFT−MD PACH Saumon PIMC FIG. 3. Comparison of experimental and several theoreti- cal Hugoniot functions. The PIMC curve was calculated with VDM nodes, τ−1= 2·106K, and 32 pairs of electrons and deuterons. Fig. 3 compares the Hugoniot from Laser shock wave experiments [1,2] with PIMC simulation (VDM nodes, 3τ−1= 2·106K) and several theoretical approaches: SESAME model by Kerley [3] (thin solid line), linear mixing model by Ross (dashed line) [4], DFT-MD by Lenosky et al. [8] (dash-dotted line), Pad´ e approxima- tion in the chemical picture (PACH) by Ebeling et al. [11] (dotted line), and the work by Saumon et al. [10] (thin dash-dotted line). The differences of the various PIMC curves in Fig. 2 are small compared to the deviation from the experimen- tal results [1,2]. There, an increased compressibility wit h a maximum value of 6 ±1 was found while PIMC pre- dicts 4 .3±0.1, only slightly higher than that given by the SESAME model. Only for p >2.5Mbar, does our hugo- niot lie within experimental errorbars. In this regime, the deviations in the PIMC and PACH hugoniot are rel- atively small, less than 0 .05 gcm−3in density. In the high pressure limit, the hugoniot goes to the FP limit of 4-fold compression. This trend is also present in the experimental findings. For pressures below 1 Mbar, the PIMC hugoniot goes back lower densities and shows the expected tendency towards the experimental values from earlier gas gun work [18,19] and lowest data points from [1,2]. For these low pressures, differences between PIMC and DFT-MD are also relatively small.IV. CONCLUSIONS We reported results from PIMC simulations and per- formed a finite size and time step study. Special emphasis was put on improving the fermion nodes where we pre- sented the first PIMC results with variational instead of FP nodes. We find a slightly increased compressibility of 4.3±0.1 compared to the SESAME model but we cannot reproduce the experimental findings of values of about 6 ±1. Further theoretical and experimental work will be needed to resolve this discrepancy. ACKNOWLEDGMENTS The authors would like to thank W. Magro for the col- laboration concerning the parallel PIMC simulations and E.L. Pollock for the contributions to the VDM method. This work was supported by the CSAR program and the Department of Physics at the University of Illinois. We used the computational facilities at the National Center for Supercomputing Applications and Lawrence Liver- more National Laboratories. TABLE I. Pressure pand internal energy per atom Efrom PIMC simulations with 32 pairs of electrons and deutero ns. For T≥250 000 K, we list results from simulations with FP nodes and τ−1 F= 8·106K and τ−1 B= 2·106K, otherwise with VDM nodes and τ−1= 2·106K. T(K) p(Mbar ),rs= 2 E(eV),rs= 2 p(Mbar) ,rs= 1.86 E(eV),rs= 1.86 ρHug(gcm−3) pHug(Mbar) 1000000 53.79 ±0.05 245.7 ±0.3 66.85 ±0.08 245.3 ±0.4 0.7019 ±0.0001 56.08 ±0.05 500000 25.98 ±0.04 113.2 ±0.2 32.13 ±0.05 111.9 ±0.2 0.7130 ±0.0001 27.48 ±0.04 250000 12.12 ±0.03 45.7 ±0.2 14.91 ±0.03 44.3 ±0.2 0.7242 ±0.0001 12.99 ±0.02 125000 5.29 ±0.04 11.5 ±0.2 6.66 ±0.02 11.0 ±0.1 0.7300 ±0.0003 5.76 ±0.02 62500 2.28 ±0.04 -3.8 ±0.2 2.99 ±0.04 -3.8 ±0.2 0.733 ±0.001 2.54 ±0.03 31250 1.11 ±0.06 -9.9 ±0.3 1.58 ±0.07 -9.7 ±0.3 0.733 ±0.003 1.28 ±0.05 15625 0.54 ±0.05 -12.9 ±0.3 1.01 ±0.05 -12.0 ±0.2 0.721 ±0.004 0.68 ±0.04 10000 0.47 ±0.05 -13.6 ±0.3 0.80 ±0.08 -13.2 ±0.4 0.690 ±0.007 0.51 ±0.05 [1] I. B. Da Silva et al. Phys. Rev. Lett. ,78:783, 1997. [2] G. W. Collins et al. Science ,281:1178, 1998. [3] G. I. Kerley. Molecular based study of fluids. page 107. ACS, Washington DC, 1983. [4] M. Ross. Phys. Rev. B ,58:669, 1998. [5] B. Militzer, W. Magro, and D. Ceperley. Contr. Plasma Physics ,391-2:152, 1999. [6] C. Pierleoni, D.M. Ceperley, B. Bernu, and W.R. Magro. Phys. Rev. Lett. ,73:2145, 1994. [7] W. R. Magro, D. M. Ceperley, C. Pierleoni, and B. Bernu. Phys. Rev. Lett. ,76:1240, 1996. [8] T. J. Lenosky, S. R. Bickham, J. D. Kress, and L. A. Collins. Phys. Rev. B , 61:1, 2000. [9] G. Galli, R.Q. Hood, A.U. Hazi, and F. Gygi. in press, Phys. Rev. B , 1999.[10] D. Saumon, G. Chabrier, and H. M. Van Horn. Astro- phys. J. , 99 2:713, 1995. [11] W. Ebeling and W. Richert. Phys. Lett. A ,108:85, 1985. [12] D. M. Ceperley. Rev. Mod. Phys. , 67:279, 1995. [13] D. M. Ceperley. Monte carlo and molecular dynam- ics of condensed matter systems. Editrice Compositori, Bologna, Italy, 1996. [14] D. M. Ceperley. J. Stat. Phys. , 63:1237, 1991. [15] B. Militzer and E. L. Pollock. in press, Phys. Rev. E , 2000. [16] Y. B. Zeldovich and Y. P. Raizer. Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena . Aca- demic Press, New York, 1966. [17] W. Kolos and L. Wolniewicz. J. Chem. Phys. , 41:3674, 1964. [18] W.J. Nellis and A.C. Mitchell et al. J. Chem. Phys. , 79:1480, 1983. [19] N.C.Holmes, M. Ross, and W.J.Nellis. Phys. Rev. B , 52:15835, 1995. 4

arXiv:physics/0001048 23 Jan 2000High-resolution path-integral de velopment of financial options Lester Ingber Lester Ingber Research POB 06440 Sears T ower,Chicago, IL 60606 and DRWInv estments LLC 311 S Wacker Dr ,Ste 900, Chicago, IL 60606 ingber@ingber.com, ingber@alumni.caltech.edu ABSTRACT The Black-Scholes theory of option pricing has been considered for man yyears as an important but v ery approximate zeroth-order description of actual market beha vior.We generalize the functional form of the dif fusion of these systems and also consider multi-f actor models including stochastic v olatility.Daily Eurodollar futures prices and implied v olatilities are fit to determine e xponents of functional behavior of diffusions using methods of global optimization, Adapti ve Simulated Annealing (ASA), to generate tight fits across mo ving time windows of Eurodollar contracts. These short-time fitted distributions are then de veloped into long-time distributions using a rob ust non-Monte Carlo path-integral algorithm, PATHINT,t ogenerate prices and deri vativescommonly used by option traders. Ke ywords: options; eurodollar; volatility; path integral; optimization; statistical mechanicsHigh-resolution path-integral ... -2- L ester Ingber 1. INTRODUCTION 1.1. Backgr ound There always is much interest in de veloping more sophisticated pricing models for financial instruments. In particular ,there currently is much interest in impro ving option pricing models, particularly with respect to stochastic variables [1-4]. The standard Black-Scholes (BS) theory assumes a lognormal distribution of market prices, i.e., a diffusion linearly proportional to the market price. However, manytexts include outlines of more general diffusions proportional to an arbitrary power of the market price [5]. The above aspects of stochastic volatility and of more general functional dependencies of dif fusions are most often “swept under the rug” of a simple lognormal form. Experienced traders often use their ownintuition to put volatility “smiles” into the BS theoretical constant coefficient in the BS lognormal distribution to compensate for these aspects. It is generally acknowledged that since the mark et crash of 1987, markets ha ve been increasingly difficult to describe using the BS model, and so better modelling and computational techniques should be used traders[6], although in practice simple BS models are the rule rather than the e xception simply because the yare easy to use [7]. Toalarge extent, previous modelling that has included stochastic volatility and multiple factors has been dri venmore by the desire to either delv einto mathematics tangential to these issues, or to deal only with models that can accommodate closed-form algebraic expressions. W ed onot see much of the philosoph yi nthe literature that has long dri venthe natural sciences: to respect first ra wdata, secondly models of ra wdata, and finally the use of numerical techniques that do not e xcessively distort models for the sak eo fease of analysis and speed of computation. Indeed, very often the re verse set of priorities is seen in mathematical finance. 1.2. Our Approach We hav eaddressed the abo ve issues in detail within the frame work of a previously de veloped statistical mechanics of financial markets (SMFM) [8-13]. Our approach requires three sensible parts. Part one is the formulation of the model, which to some extent also in volves specification of the specific market(s) data to be addressed. Part twoisthe fitting ofHigh-resolution path-integral ... -3- L ester Ingber the model to specific market data. Part three is the use of the resulting model to calculate option prices and their Greeks (partial deri vativeso fthe prices with respect to their independent variables), which are used as risk parameters by traders. Each part requires some specific numerical tuning to the mark et under consideration. The first part was to de velop the algebraic model to replace/generalize BS, including the possibility of also addressing ho wt ohandle data regions not previously observed in trading. This is not absurd; current BS models perform integrals that must include a much influence from f at tails that include data regions neverseen or likely to be seen in real-world mark ets. There are some issues as to whether we should tak eseriously the notion that the market is strongly dri venb ysome element of a “self-fulfilling prophesy” by the BS model [14], but in an ycase our models ha ve parameters to handle a wide range of possible cases that might arise. We hav edeveloped twoparallel tracks starting with part one, a one-factor and a tw o-factor model. The two-factor model includes stochastic v olatility.Atfirst we sensed the need to de velop this tw o-factor model, and we no wsee that this is at the least an important benchmark against which to judge the w orth of the one-factor model. The second part was to fit the actual ra wdata so we can come up with real distrib utions. Some tests illustrated that standard quasi-linear fitting routines, could not get the proper fits, and so we used a more powerful global optimization, Adapti ve Simulated Annealing (ASA) [15]. Tuning and selection of the time periods to perform the fits to the data were not tri vial aspects of this research. Practical decisions had to be made on the time span of data to be fit and ho wt oaggregate the fits to get sensible “f air values” for reasonable standard deviations of the exponents in the diffusions. The third part was to de velop Greeks and risk parameters from these distributions without making premature approximations just to ease the analysis. Perhaps someday ,simple approximations and intuitions similar to what traders no wuse for BS models will be a vailable for these models, but we do not think the best approach is to start out with such approximations until we first see proper calculations, especially in this uncharted territory .When it seemed that Cox-Ross-Rubenstein (CRR) standard tree codes (discretized approximations to partial differential equations) [16] were not stable for general exponents, i.e., for other than the lognormal case, we turned to a P ATHINT code de veloped a decade ago for some hard nonlinear multifactor problems [17], e.g., combat analyses[18], neuroscience[19,20], andHigh-resolution path-integral ... -4- L ester Ingber potentially chaotic systems [21,22]. In 1990 and 1991 papers on financial applications, it was mentioned howthese techniques could be used for stochastic interest rates and bonds [9,10]. The modifications required here for one-factor European and then American cases went surprisingly smoothly; we still had to tune the meshes, etc. The two-factor model presented a technical problem to the algorithm, which we have reasonably handled using a combination of selection of the model in part one and a reasonable approach to de veloping the meshes. 1.3. Outline of Paper Section 1 is this introduction. Section 2 describes the nature of Eurodollar (ED) futures data and the evidence for stochastic v olatility.Section 3 outlines the algebra of modelling options, including the standard BS theory and our generalizations. Section 4 outlines the three equi valent mathematical representations used by SMFM; this is required to understand the de velopment of the short-time distribution that defines the cost function we deri ve for global optimization, as well as the numerical methods we ha ve dev eloped to calculate the long-time e volution of these short-time distrib utions. Section 5outlines ASA and explains its use to fit short-time probability distributions defined by our models to the Eurodollar data; we offer the fitted e xponent in the diffusion as a ne wimportant technical indicator of market behavior.Section 6 outlines P ATHINT and explains its use to de velop long-time probability distributions from the fitted short-time probability distributions, for both the one-factor and tw o-factor tracks. Section 7describes ho ww euse these long-time probability distributions to calculate European and American option prices and Greeks; here we gi ve numerical tests of our approach to BS CRR algorithms. Section 8i sour conclusion. 2. DAT A 2.1. Eurodollars Eurodollars are fix ed-rate time deposits held primarily by o verseas banks, but denominated in US dollars. The yare not subject to US banking regulations and therefore tend to ha ve a tighter bid-ask spread than deposits held in the United States [23].High-resolution path-integral ... -5- L ester Ingber 2.2. Futur es The three-month Eurodollar futures contract is one of the most acti vely traded futures markets in the world. The contract is quoted as an inde xwhere the yield is equal to the Eurodollar price subtracted from 100. This yield is equal to the fixed rate of interest paid by Eurodollar time deposits upon maturity and is expressed as an annualized interest rate based on a 360-day year .The Eurodollar futures are cash settled based on the 90-day London Interbank Of fer Rate (LIBOR). A“notional” principal amount of $1 million, is used to determine the change in the total interest payable on a hypothetical underlying time deposit, but is ne veractually paid or recei ved[23]. Currently a total of 40 quarterly Eurodollar futures contracts (or ten years w orth) are listed, with expirations annually in March, June, September and December. 2.3. Options on Futures The options traded on the Eurodollar futures include not only 18 months of options expiring at the same time as the underlying future, but also various short dated options which themselves expire up to one year prior to the expiration of the underlying futures contract. 2.4. Front/Back Month Contracts Forpurposes of risk minimization, as discussed in a previous paper[4], traders put on spreads across a v ariety of option contracts. One common example is to trade the spread on contracts e xpiring one year apart, where the future closer to expiration is referred to as the front month contract, and the future expiring one year later is called the back month. The a vailability of short dated or “mid-curv e” options which are based on an underlying back month futures contract, but expire at the same time as the front month, allo wone to trade the v olatility ratios of the front and back month futures contracts without having to tak ethe time differences in option expirations into consideration. We studied the volatilities of these types of front and back month contracts. Here, we gi ve analyses with respect only to quarterly data longer than six months from expiration.High-resolution path-integral ... -6- L ester Ingber 2.5. Stochastic Volatility Beloww edev elop two-factor models to address stochastic v olatility.Inaprevious paper ,w ehav e performed empirical studies of Eurodollar futures to support the necessity of dealing with these issues [4]. 3. MODELS 3.1. Random walk model The use of Brownian motion as a model for financial systems is generally attrib uted to Bachelier [24], though he incorrectly intuited that the noise scaled linearly instead of as the square root relative tothe random log-price v ariable. Einstein is generally credited with using the correct mathematical description in a larger physical context of statistical systems. However, sev eral studies imply that changing prices of man ymarkets do not follo warandom walk, that the ymay have long-term dependences in price correlations, and that the ymay not be efficient in quickly arbitraging ne w information [25-27]. Arandom walk for returns, rate of change of prices o verprices, is described by a Langevin equation with simple additi ve noiseη,typically representing the continual random influx of information into the market. ˙Γ= −γ1+γ2η, ˙Γ=dΓ/dt, <η(t)>η=0,<η(t),η(t′)>η=δ(t−t′), ( 1) whereγ1andγ2are constants, and Γis the logarithm of (scaled) price. Price, although the most dramatic observable, may not be the only appropriate dependent v ariable or order parameter for the system of markets [28]. This possibility has also been called the “semistrong form of the ef ficient mark et hypothesis” [25]. The generalization of this approach to include multi variate nonlinear nonequilibrium markets led to amodel of statistical mechanics of financial markets (SMFM) [8].High-resolution path-integral ... -7- L ester Ingber 3.2. Black-Scholes (BS) Theory The standard partial-differential equation used to formulate most v ariants of Black-Scholes (BS) models describing the market value of an option, V,is ∂V ∂t+1 2σ2S2∂2V ∂S2+rS∂V ∂S−rV=0, ( 2) whereSis the asset price, and σis the standard deviation, or volatility of S,andris the short-term interest rate. The solution depends on boundary conditions, subject to a number of interpretations, some requiring minor transformations of the basic BS equation or its solution. Forexample, the basic equation can apply to a number of one-dimensional models of interpretations of prices gi ventoV,e.g., puts or calls, and to S,e.g., stocks or futures, dividends, etc. Forinstance, if Vis set toC,acall on an European option with e xercise price Xwith maturity at T, the solution is C(S,t)=SN(d1)−Xe−r(T−t)N(d2), d1=ln(S/X)+(r+1 2σ2)(T−t) σ(T−t)1/2, d2=ln(S/X)+(r−1 2σ2)(T−t) σ(T−t)1/2.( 3) In practice, the v olatility σis the least kno wn parameter in this equation, and its estimation is generally the most important part of pricing options. Usually the volatility is gi veni nayearly basis, baselined to some standard, e.g., 252 trading days per year ,or360 or 365 calendar days. Therefore, all values of volatility gi veni nthe graphs in this paper ,based on daily data, w ould be annualized by multiplying the standard de viations of the yields by √252=15. 87.We hav eused this factor to present our implied volatilities as daily mo vements. 3.3. Some KeyIssues in Deri vation of BS The basic BS model considers a portfolio in terms of delta(Δ), Π=V−ΔS,( 4)High-resolution path-integral ... -8- L ester Ingber in a market with Gaussian-Mark ovian (“white”) noise Xand drift µ, dS S=σdX+µdt,( 5) whereV(S,t)inherits a random process from S, dV=σS∂V ∂SdX+  µS∂V ∂S+1 2σ2S2∂2V ∂S2+∂V ∂t  dt.( 6) This yields dΠ=σ ∂V ∂S−Δ dX+  µS∂V ∂S+1 2σ2S2∂2V ∂S2+∂V ∂t−µΔS  dt.( 7) The expected risk-neutral return of Πis dΠ=rΠdt=r(V−ΔS)dt.( 8) OptionsVon futures Fcan be deri ved, e.g., using simple transformations to tak ecost of carry into consideration, such as F=Ser(T−t),( 9) and setting dΠ=rV dt.( 10) The corresponding BS equation for futures Fis ∂V ∂t+1 2σ2F2∂2V ∂S2−rV=0. ( 11) At least tw oadvantages are present if Δis chosen such that Δ=∂V ∂S.( 12) Then, the portfolio can be instantaneously “risk-neutral, ”interms of zeroing the coefficient of X,a swell as independent of the direction of market, in terms of zeroing the coef ficient of µ.For the abo ve example ofV=C, Δ=N(d1). ( 13)High-resolution path-integral ... -9- L ester Ingber Other trading strategies based on this simple model use similar constructs as risk parameters, e.g., gamma(Γ),theta(Θ),vega(ϒ),rho(ρ)[5], Γ=∂2Π ∂S2, Θ=∂Π ∂t, ϒ=∂Π ∂σ, ρ=∂Π ∂r.( 14) The BS equation, Eq. (2), may be written as Θ+rSΔ+1 2(σS)2Γ=rf.( 15) 3.4.SxModels Our two-factor model includes stochastic volatility σof the underlying S, dS=µdt+σF(S,S0,S∞,x,y)dzS, dσ=νdt+εdzσ, <dzi>=0,i={S,σ}, <dzi(t)dzj(t′)>=dtδ(t−t′),i=j, <dzi(t)dzj(t′)>=ρdtδ(t−t′),i≠j, F(S,S0,S∞,x,y)=    S, SxS1−x 0, SyS1−x 0Sx−y ∞,S<S0 S0≤S≤S∞ S>S∞,( 16) whereS0andS∞are selected to lie outside the data region used to fit the other parameters, e.g., S0=1 andS∞=20 for fits to Eurodollar futures which historically ha ve a very tight range relati ve tootherHigh-resolution path-integral ... -10- L ester Ingber markets. Wehav eused the Black-Scholes form F=SinsideS<S0to obtain the usual benefits, e.g., no negative prices as the distribution is naturally excluded from S<0 and preservation of put-call parity . Put-call parity for European options is deri vedquite independent of an ymathematical model of options [5]. In its simplest form, it is gi venby c+Xe−r(T−t)=p+S,( 17) wherec(p)i sthe fair price of a call (put), Xis the strik eprice,ris the risk-free interest rate, tis the present time, Tis the time of expiration, and Sis the underlying mark et. Wehav etakeny=0, a normal distribution, to reflect total ignorance of markets outside the range of S>S∞.The one-factor model just assumes a constant σ.Iti soften noted that BS models incorrectly include untenable contributions from largeSregions because of their fat tails [29]. (Ifwe wished to handle ne gative interest rates, ED prices > 100, we would mo ve shift theS=0axis to some S<0value.) We found that the abrupt, albeit continuous, changes across S0especially for x≤0did not cause anysimilar effects in the distributions e volved using these diffusions, as reported belo w. The formula for pricing an option P,derivedi naB lack-Scholes generalized frame work after factoring out interest-rate discounting, is equi valent to using the form dS=µSdt+σF(S,S0,S∞,x,y)dzS, dσ=νdt+εdzσ.( 18) We experimented with some alternati ve functional forms, primarily to apply some smooth cutof fs across the abo ve three regions of S.For example, we used F′,afunctionFdesigned to re vert to the lognormal Black-Scholes model in se veral limits, F′(S,S0,S∞,x)=SC0+(1−C0)((SxS1−x 0C∞+S0(1−C∞))), C0=exp  − S S0|1−x| 1+|1−x| |2−x|+1  , C∞=exp  − S S∞ 2  , S→∞,x≠1limF′(S,S0,S∞,x)=S0=constant ,High-resolution path-integral ... -11- L ester Ingber S→0+limF′(S,S0,S∞,x)= x→1limF′(S,S0,S∞,x)=S.( 19) However, our fits were most sensiti ve tothe data when we permitted the central region to be pure Sxusing Fabove. 3.4.1. VariousF(S,x)Diffusions Fig. 1 givesexamples of F(S,S0,S∞,x,y)dzSforxin{-1, 0, 1, 2 }.The other parameters are S=5,S0=0. 5,S∞=20,y=0. Fig. 1. 4. STATISTICAL MECHANICS OF FINANCIAL MARKETS (SMFM) 4.1. Statistical Mechanics of Large Systems Aggregation problems in nonlinear nonequilibrium systems typically are “solved” (accommodated) by having ne wentities/languages de veloped at these disparate scales in order to efficiently pass information back and forth. This is quite different from the nature of quasi-equilibrium quasi-linear systems, where thermodynamic or cybernetic approaches are possible. These approaches typically fail for nonequilibrium nonlinear systems. Manysystems are aptly modeled in terms of multi variate differential rate-equations, kno wn as Langevin equations, ˙MG=fG+ˆgG jηj,(G=1,...,Λ),(j=1,...,N), ˙MG=dMG/dt, <ηj(t)>η=0,<ηj(t),ηj′(t′)>η=δjj′δ(t−t′), ( 20) wherefGand ˆgG jare generally nonlinear functions of mesoscopic order parameters MG,jis a microscopic inde xindicating the source of fluctuations, and N≥Λ.The Einstein con vention of summing overrepeated indices is used. Vertical bars on an inde x, e.g., |j|, imply no sum is to be taken on repeatedHigh-resolution path-integral ... -12- L ester Ingber indices. Viaasomewhat length y, albeit instructi ve calculation, outlined in se veral other papers [8,10,30], involving an intermediate deri vation of a corresponding F okker-Planck or Schr ¨odinger-type equation for the conditional probability distrib utionP[M(t)|M(t0)], the Langevin rate Eq. (20) is de veloped into the more useful probability distribution for MGat long-time macroscopic time e ventt=(u+1)θ+t0,in terms of a Stratonovich path-integral o vermesoscopic Gaussian conditional probabilities [31-35]. Here, macroscopic variables are defined as the long-time limit of the e volving mesoscopic system. The corresponding Schr ¨odinger-type equation is [33,34] ∂P/∂t=1 2(gGG′P),GG′−(gGP),G+V, gGG′=kTδjkˆgG jˆgG′ k, gG=fG+1 2δjkˆgG′ jˆgG k,G′, [...],G=∂[...]/∂MG.( 21) This is properly referred to as a F okker-Planck equation when V≡0. Note that although the partial differential Eq. (21) contains information re gardingMGas in the stochastic dif ferential Eq. (20), all references to jhave been properly a veraged over. I.e., ˆgG jin Eq. (20) is an entity with parameters in both microscopic and mesoscopic spaces, b utMis a purely mesoscopic variable, and this is more clearly reflected in Eq. (21). The path integral representation is gi veni nterms of the “Feynman” Lagrangian L. P[Mt|Mt0]dM(t)=∫...∫DMexp(−S)δ[M(t0)=M0]δ[M(t)=Mt], S=k−1 Tmint t0∫dt′L, DM= u→∞limu+1 ρ=1Πg1/2 GΠ(2πθ)−1/2dMG ρ, L(˙MG,MG,t)=1 2(˙MG−hG)gGG′(˙MG′−hG′)+1 2hG ;G+R/6−V,High-resolution path-integral ... -13- L ester Ingber hG=gG−1 2g−1/2(g1/2gGG′),G′, gGG′=(gGG′)−1, g=det(gGG′), hG ;G=hG ,G+ΓF GFhG=g−1/2(g1/2hG),G, ΓF JK≡gLF[JK,L]=gLF(gJL,K+gKL,J−gJK,L), R=gJLRJL=gJLgJKRFJKL, RFJKL=1 2(gFK,JL−gJK,FL−gFL,JK+gJL,FK)+gMN(ΓM FKΓN JL−ΓM FLΓN JK). ( 22) Mesoscopic variables ha ve been defined as MGin the Langevin and F okker-Planck representations, in terms of their de velopment from the microscopic system labeled by j.The Riemannian curvature term R arises from nonlinear gGG′,which is a bona fide metric of this space [33]. Even if a stationary solution, i.e.,˙MG=0, is ultimately sought, a necessarily prior stochastic treatment of ˙MGterms givesrise to these Riemannian “corrections. ”Evenfor a constant metric, the term hG ;Gcontributes toLfor a nonlinear meanhG.Vmay include terms such as T′ΣJT′GMG,where the Lagrange multipliers JT′Gare constraints onMG,which are advantageously modeled as extrinsic sources in this representation; the ytoo may be time-dependent. Forour purposes, the abo ve Feynman Lagrangian defines a kernel of the short-time conditional probability distribution, in the curved space defined by the metric, in the limit of continuous time, whose iteration yields the solution of the previous partial differential equation Eq. (21). This differs from the Lagrangian which satisfies the requirement that the action is stationary to the first order in dt—the WKBJ approximation, b ut which does not include the first-order correction to the WKBJ approximation as does the Feynman Lagrangian. This latter Lagrangian differs from the Feynman Lagrangian, essentially by replacing R/6 above byR/12 [36]. In this sense, the WKBJ Lagrangian is more useful for some theoretical discussions [37]. Ho wever, the use of the Fe ynman Lagrangian coincides with the numerical method we present here using the P ATHINT code.High-resolution path-integral ... -14- L ester Ingber Using the variational principle, JTGmay also be used to constrain MGto regions where the yare empirically bound. More complicated constraints may be af fixed toLusing methods of optimal control theory [38]. With respect to a steady state P,when it exists, the information gain in state Pis defined by ϒ[P]=∫...∫DM′Pln (P/P), DM′=DM/dMu+1.( 23) In the economics literature, there appears to be sentiment to define Eq. (20) by the Ito, rather than the Stratono vich prescription. It is true that Ito integrals ha ve Martingale properties not possessed by Stratonovich integrals [39] which leads to risk-neural theorems for mark ets [40,41], butthe nature of the proper mathematics should e ventually be determined by proper aggre gation of relati vely microscopic models of mark ets. Itshould be noted that virtually all in vestigations of other physical systems, which are also continuous time models of discrete processes, conclude that the Stratono vich interpretation coincides with reality ,when multiplicati ve noise with zero correlation time, modeled in terms of white noise ηj,is properly considered as the limit of real noise with finite correlation time [42]. The path inte gral succinctly demonstrates the difference between the two: The Ito prescription corresponds to the prepoint discretization of L,whereinθ˙M(t)→Mρ+1−MρandM(t)→Mρ.The Stratonovich prescription corresponds to the midpoint discretization of L,wherein θ˙M(t)→Mρ+1−Mρand M(t)→1 2(Mρ+1+Mρ). Interms of the functions appearing in the F okker-Planck Eq. (21), the Ito prescription of the prepoint discretized Lagrangian, LI,i srelatively simple, albeit decepti vely so because of its nonstandard calculus. LI(˙MG,MG,t)=1 2(˙MG−gG)gGG′(˙MG′−gG′)−V.( 24) In the absence of a nonphenomenological microscopic theory ,the difference between a Ito prescription and a Stratonovich prescription is simply a transformed drift [36]. There are se veral other advantages to Eq. (22) o verEq. (20). Extrema and most probable states of MG,<<MG>>,are simply deri vedb yavariational principle, similar to conditions sought in pre vious studies [43]. In the Stratonovich prescription, necessary ,albeit not sufficient, conditions are gi venby δGL=L,G−L,˙G:t=0,High-resolution path-integral ... -15- L ester Ingber L,˙G:t=L,˙GG′˙MG′+L,˙G˙G′¨MG′.( 25) Forstationary states, ˙MG=0, and∂L/∂MG=0defines <<MG>>,where the bars identify stationary variables; in this case, the macroscopic v ariables are equal to their mesoscopic counterparts. [Note that L isnotthe stationary solution of the system, e.g., to Eq. (21) with ∂P/∂t=0. However, insome cases [44], Lis a definite aid to finding such stationary states.] Manytimes only properties of stationary states are examined, but here a temporal dependence is included. E.g., the ˙MGterms inLpermit steady states and their fluctuations to be in vestigated in a nonequilibrium conte xt. Note that Eq. (25) must be deri vedfrom the path integral, Eq. (22), which is at least one reason to justify its de velopment. 4.2. Corr elations Correlations between variables are modeled e xplicitly in the Lagrangian as a parameter usually designated ρ(not to be confused with the Rho Greek calculated for options). This section uses a simple two-factor model to de velop the correspondence between the correlation ρin the Lagrangian and that among the commonly written Weiner distributions dz. Consider coupled stochastic differential equations dr=fr(r,l)dt+ˆgr(r,l)σ1dz1, dl=fl(r,l)dt+ˆgl(r,l)σ2dz2, <dzi>=0,i={1, 2}, <dzi(t)dzj(t′)>=dtδ(t−t′),i=j, <dzi(t)dzj(t′)>=ρdtδ(t−t′),i≠j, δ(t−t′)=  0,, 1,t≠t′, t=t′,(26) where <.> denotes expectations. These can be rewritten as Langevin equations (in the Ito ˆprepoint discretization) dr/dt=fr+ˆgrσ1(γ+n1+sgnργ−n2),High-resolution path-integral ... -16- L ester Ingber dl/dt=gl+ˆglσ2(sgnργ−n1+γ+n2), γ±=1 √2[1±(1−ρ2)1/2]1/2, ni=(dt)1/2pi,( 27) wherep1andp2are independent [0,1] Gaussian distributions. The equivalent short-time probability distribution, P,for the abo ve set of equations is P=g1/2(2πdt)−1/2exp(−Ldt), L=1 2F†gF, F= dr/dt−fr) dl/dt−fl) , g=det(g), k=1−ρ2.( 28) g,the metric in {r,l}-space, is the in verse of the co variance matrix, g−1=  (ˆgrσ1)2 ρˆgrˆglσ1σ2ρˆgrˆglσ1σ2 (ˆglσ2)2  .( 29) The above also corrects previous papers which inadvertently dropped the sgn factors in the above [9,10,17]. 5. ADAPTIVE SIMULATED ANNEALING (ASA) FITS 5.1. ASA Outline The algorithm de veloped which is no wcalled Adapti ve Simulated Annealing (ASA) [45] fits short- time probability distributions to observed data, using a maximum likelihood technique on the Lagrangian. This algorithm has been de veloped to fit observed data to a theoretical cost function o veraD-dimensional parameter space[45], adapting for varying sensitivities of parameters during the fit. The ASA code canHigh-resolution path-integral ... -17- L ester Ingber be obtained at no charge, via WWW from http://www.ingber.com/ or via FTP from ftp.ingber.com [15]. 5.1.1. General description Simulated annealing (SA) was de veloped in 1983 to deal with highly nonlinear problems[46], as an extension of a Monte-Carlo importance-sampling technique de veloped in 1953 for chemical ph ysics problems. It helps to visualize the problems presented by such comple xsystems as a geographical terrain. Forexample, consider a mountain range, with tw o“parameters, ”e.g., along the North−South and East−West directions. We wish to find the lowest v alleyi nthis terrain. SA approaches this problem similar to using a bouncing ball that can bounce o vermountains from v alleyt ovalley. Westart at a high “temperature, ”where the temperature is an SA parameter that mimics the ef fect of a fast moving particle in a hot object lik eahot molten metal, thereby permitting the ball to mak every high bounces and being able to bounce o verany mountain to access an yvalley, giv enenough bounces. As the temperature is made relati vely colder,the ball cannot bounce so high, and it also can settle to become trapped in relatively smaller ranges of valleys. We imagine that our mountain range is aptly described by a “cost function. ”Wedefine probability distributions of the tw odirectional parameters, called generating distributions since the ygenerate possible valleys or states we are to e xplore. W edefine another distribution, called the acceptance distrib ution, which depends on the difference of cost functions of the present generated v alleyw eare to explore and the last sa vedlowest valley. The acceptance distribution decides probabilistically whether to stay in a ne w lower valleyo rt ob ounce out of it. All the generating and acceptance distrib utions depend on temperatures. In 1984[47], it w as established that SA possessed a proof that, by carefully controlling the rates of cooling of temperatures, it could statistically find the best minimum, e.g., the lowest v alleyo four example abo ve.This was good news for people trying to solv ehard problems which could not be solv ed by other algorithms. The bad news was that the guarantee was only good if the ywere willing to run SA forever. In1987, a method of fast annealing (FA) was de veloped [48], which permitted lowering the temperature exponentially f aster,thereby statistically guaranteeing that the minimum could be found in some finite time. However, that time still could be quite long. Shortly thereafter ,Very Fast Simulated Reannealing (VFSR) was de veloped in 1987 [45], nowcalled Adapti ve Simulated Annealing (ASA),High-resolution path-integral ... -18- L ester Ingber which is exponentially faster than FA. ASA has been applied to man yproblems by man ypeople in man ydisciplines [49-51]. The feedback of man yusers regularly scrutinizing the source code ensures its soundness as it becomes more flexible and powerful. 5.1.2. Mathematical outline ASA considers a parameter αi kin dimension igenerated at annealing-time kwith the range αi k∈[Ai,Bi], ( 30) calculated with the random variable yi, αi k+1=αi k+yi(Bi−Ai), yi∈[−1, 1].( 31) The generating function gT(y)isdefined, gT(y)=D i=1Π1 2(|yi|+Ti)ln(1+1/Ti)≡D i=1Πgi T(yi), ( 32) where the subscript ionTispecifies the parameter index, and the k-dependence in Ti(k)for the annealing schedule has been dropped for brevity .Its cumulati ve probability distribution is GT(y)=y1 −1∫...yD −1∫dy′1...dy′DgT(y′)≡D i=1ΠGi T(yi), Gi T(yi)=1 2+sgn (yi) 2ln(1+|yi|/Ti) ln(1+1/Ti).( 33) yiis generated from a uifrom the uniform distribution ui∈U[0, 1], yi=sgn (ui−1 2)Ti[(1+1/Ti)|2ui−1|−1] . (34) It is straightforward to calculate that for an annealing schedule for Ti Ti(k)=T0iexp(−cik1/D), ( 35)High-resolution path-integral ... -19- L ester Ingber aglobal minima statistically can be obtained. I.e., ∞ k0Σgk≈∞ k0Σ[D i=1Π1 2|yi|ci]1 k=∞.( 36) Control can be taken o verci,such that Tfi=T0iexp(−mi)whenkf=expni, ci=miexp(−ni/D), ( 37) wheremiandnican be considered “free” parameters to help tune ASA for specific problems. ASA has o ver100 OPTIONS a vailable for tuning. Afew important ones were used in this project. 5.1.3. Reannealing Wheneverdoing a multi-dimensional search in the course of a comple xnonlinear physical problem, inevitably one must deal with different changing sensiti vities of the αiin the search. At an ygiv en annealing-time, the range o verwhich the relati vely insensiti ve parameters are being searched can be “stretched out” relati ve tothe ranges of the more sensiti ve parameters. This can be accomplished by periodically rescaling the annealing-time k,essentially reannealing, e very hundred or so acceptance- ev ents (or at some user-defined modulus of the number of accepted or generated states), in terms of the sensitivities sicalculated at the most current minimum value of the cost function, C, si=∂C/∂αi.( 38) In terms of the lar gestsi=smax,adefault rescaling is performed for each kiof each parameter dimension, whereby a ne windexk′iis calculated from each ki, ki→k′i, T′ik′=Tik(smax/si), k′i=((ln(Ti0/Tik′)/ci))D.( 39) Ti0is set to unity to begin the search, which is ample to span each parameter dimension.High-resolution path-integral ... -20- L ester Ingber 5.1.4. Quenching Another adapti ve feature of ASA is its ability to perform quenching in a methodical f ashion. This is applied by noting that the temperature schedule abo ve can be redefined as Ti(ki)=T0iexp(−cikQi/D i), ci=miexp(−niQi/D), ( 40) in terms of the “quenching factor” Qi.The sampling proof fails if Qi>1as kΣD Π1/kQi/D= kΣ1/kQi<∞.( 41) This simple calculation shows ho wthe “curse of dimensionality” arises, and also gi vesapossible wayo fliving with this disease. In ASA, the influence of large dimensions becomes clearly focussed on the exponential of the power of kbeing 1/D,a sthe annealing required to properly sample the space becomes prohibiti vely slow. So, if resources cannot be committed to properly sample the space, then for some systems perhaps the ne xt best procedure may be to turn on quenching, whereby Qican become on the order of the size of number of dimensions. The scale of the power of 1/ Dtemperature schedule used for the acceptance function can be altered in a similar f ashion. Ho wever, this does not affect the annealing proof of ASA, and so this may used without damaging the sampling property. 5.2.x-Indicator of Market Contexts Our studies of contexts of markets well recognized by option traders to ha ve significantly dif ferent volatility behavior sho wthat the exponents xare reasonably faithful indicators defining these dif ferent contexts. We feel the tw o-factor model is more accurate because the data indeed demonstrate stochastic volatility [4]. We also note that the tw o-factorx’s are quite robust and uniform when being fit by ASA across the last fe wyears. This is not true of the one-factor ASA fitted x’s unless we do not use the Black- Scholes σas a parameter ,but rather calculate as historical volatility during all runs. Some results of tw o- factor studies and one-factor studies using a Black-Scholes σhave been reported elsewhere [13].High-resolution path-integral ... -21- L ester Ingber Sinceσis not widely traded and arbitraged, to fit the tw o-factor model, we calculate this quantity as an historical v olatility for both its prepoint and postpoint v alues. Some previous studies used a scaled implied volatility (which is calculated from a BS model). We use a standard deviation σ′, σ′=StdDev( (dS/F(S,S0,S∞,x,y))). ( 42) In the one-f actor model, it does not mak egood numerical sense to ha ve two free parameters in one term, i.e., σandx,a sthese cannot be fit v ery well within the variance the data. Instead, one method is to takeguidance from the tw o-factor results, to set a scale for an ef fectiveσ,and then fit the parameter x. Another method it apply the abo ve StdDeva saproxy for σ.Some moti vation for this approach is gi ven by considering collapsing a tw o-factor stochastic v olatility model in one-factor model: The one-f actor model nowhas an integral o verthe stochastic process in its diffusion term. The is integral is what we are approximating by using the standard deviation of a moving windo wofthe data. 6. PATH-INTEGRAL (P ATHINT) DEVELOPMENT 6.1. PATHINT Outline The fits described abo ve clearly demonstrate the need to incorporate stochastic volatility in option pricing models. If one-f actor fits are desired, e.g., for ef ficiencyo fcalculation, then at the least the exponent of price xshould be permitted to freely adapt to the data. In either case, it is required to de velop afull set of Greeks for trading. To meet these needs, we ha ve used a path-integral code, P ATHINT, described belo w, with great success. At this time, the tw o-factor code takes too long to run for daily use, butitprovestob eag ood weekly baseline for the one-factor code. The PATHINT algorithm de velops the long-time probability distribution from the Lagrangian fit by the first optimization code. Arobust and accurate histogram-based (non-Monte Carlo) path-inte gral algorithm to calculate the long-time probability distribution has been de veloped to handle nonlinear Lagrangians [18-20,22,52-54], The histogram procedure recognizes that the distrib ution can be numerically approximated to a high degree of accurac ya ssum of rectangles at points Miof height Piand width ΔMi.For convenience, just consider a one-dimensional system. The abo ve path-integral representation can be rewritten, for each of its intermediate integrals, asHigh-resolution path-integral ... -22- L ester Ingber P(M;t+Δt)=∫dM′[g1/2 s(2πΔt)−1/2exp(−LsΔt)]P(M′;t)=∫dM′G(M,M′;Δt)P(M′;t), P(M;t)=N i=1Σπ(M−Mi)Pi(t), π(M−Mi)=    0, (Mi−1 2ΔMi−1)≤M≤(Mi+1 2ΔMi), 1, otherwise ,,( 43) which yields Pi(t+Δt)=Tij(Δt)Pj(t), Tij(Δt)=2 ΔMi−1+ΔMi∫Mi+ΔMi/2 Mi−ΔMi−1/2dM∫Mj+ΔMj/2 Mj−ΔMj−1/2dM′G(M,M′;Δt). ( 44) Tijis a banded matrix representing the Gaussian nature of the short-time probability centered about the (varying) drift. Care must be used in de veloping the mesh in ΔMG,which is strongly dependent on the diagonal elements of the diffusion matrix, e.g., ΔMG≈(Δtg|G||G|)1/2.( 45) Presently,this constrains the dependence of the co variance of each variable to be a nonlinear function of that variable, albeit arbitrarily nonlinear ,i norder to present a straightforward rectangular underlying mesh. Belo ww eaddress ho ww ehav ehandled this problem in our tw o-factor stochastic-volatility model. Fitting data with the short-time probability distribution, ef fectively using an integral o verthis epoch, permits the use of coarser meshes than the corresponding stochastic differential equation. The coarser resolution is appropriate, typically required, for numerical solution of the time-dependent path- integral: By considering the contrib utions to the first and second moments of ΔMGfor small time slices θ, conditions on the time and variable meshes can be deri ved[52]. The time slice essentially is determined byθ≤L−1,whereLis the “static” Lagrangian with dMG/dt=0, throughout the ranges of MGgiving the most important contrib utions to the probability distrib utionP.The variable mesh, a function of MG,is optimally chosen such that ΔMGis measured by the co variancegGG′,orΔMG≈(gGGθ)1/2.High-resolution path-integral ... -23- L ester Ingber If the histogram method is further de veloped into a trapezoidal integration, then more accurac ycan be expected at more comple xboundaries [53]. Such problems does not arise here, and 6−7 significant figure accurac yiseasily achie vedprovided great care is taken to de velop the mesh as described abo ve. Forexample, after setting the initial-condition discretized delta function MSUPG at the prepoint of an interval, the mesh going forward in MGis simply calculated stepwise using ΔMG=(gGGθ)1/2.( 46) However, going backw ards inMG,a niterative procedure was used at each step, starting with an estimate from the prepoint and going forward again, until there was no mismatch. That much care is required for the mesh was observed in the original Wehner-Wolfer paper [52]. It is important to stress that v ery good numerical accurac yi srequired to get very good Greeks required for real-w orld trading. Manyauthors develop very efficient numerical schemes to get reasonable prices to 2 or 3 significant figures, but these methods often are not very good to enough significant figures to get good precision for the Greeks. Typical Monte Carlo methods are notorious for gi ving such poor results after very long computer runs. In particular ,w ed on ot believe that good Greeks required for trading can be obtained by using meshes obtained by other simpler algorithms [55]. The PATHINT algorithm in its present form can “theoretically” handle an yn-factor model subject to its diffusion-mesh constraints. In practice, the calculation of 3-f actor and 4-factor models likely will wait until giga-hertz speeds and giga-byte RAM are commonplace. 6.2. Development of Long-Time Probabilities The noise determined empirically as the dif fusion of the data is the same, independent of xwithin our approach. Therefore, we scale different exponents such that the the diffusions, the square of the “basis-point volatilities” (BPV), are scaled to be equi valent. Then, there is not a v ery drastic change in option prices for different e xponents xfor the strik eXset to the Sunderlying, the at-the-mone y(ATM) strike. Thisis not the case for out of the mone y(OTM) or in the mone y(ITM) strikes, e.g., when exercising the strik ewould generate loss or profit, resp. This implies that current pricing models are not radically mispricing the markets, b ut there still are significant changes in Greeks using more sophisticated models.High-resolution path-integral ... -24- L ester Ingber 6.3. Dependence of Probabilities on Sandx Fig. 2 givesexamples of the short-time distribution e volved out to T=0. 5year forxin{-1, 0, 1, 2},with 500 intermediate epochs/foldings, and BS σ=0. 0075.Each calculation scales σby multiplying byS/F(S,S0,S∞,x,y). Fig. 2. Fig. 3 givesa nexample of a two-factor distribution e volved out to T=0. 5year forx=0. 7. Fig. 3. 6.4. Two-Factor Volatility and P ATHINT Modifications In our tw o-factor model, the mesh of Swould depend on σand cause some problems in an y PATHINT grid to be de veloped inS-σ. Forsome time we ha ve considered ho wt ohandle this generic problem for n-factor multi variate systems with truly multi variate diffusions within the frame work of PATHINT.Inone case, we ha ve taken advantage of the Riemannian in variance of the probability distribution as discussed abo ve,t otransform to asystem where the dif fusions ha ve only “diagonal” multiplicati ve dependence [19]. Howe ver, this leads to cumbersome numerical problems with the transformed boundary conditions [20]. Another method, not yet fully tested, is to de velop a tiling of diagonal meshes for each f actorithat often are suitable for of f- diagonal regions in an n-factor system, e.g., ΔMi k=2mi kΔMi 0, ΔMi 0≈√ g|i||i| k0Δt,( 47) where the mesh of v ariableiat a givenpoint labeled by kis an exponentiation of 2, labeled by mi k;the integral powermi kis determined such that it gi vesagood approximation to the diagonal mesh gi venby the one-factor P ATHINT mesh conditions, in terms of some minimal mesh ΔMi 0,throughout regions of the Lagrangian giving most important contrib utions to the distribution as predetermined by a scan of theHigh-resolution path-integral ... -25- L ester Ingber system. This tiling of the kernel is to be used together with interpolation of intermediate distributions. The results of our study here are that, after the at-the-mone yBPV are scaled to be equi valent, there is not a very drastic change in the one-f actor ATM Greeks de veloped belo w. Therefore, while we ha ve not at all changed the functional dependence of the Lagrangian on Sandσ,w ehav edetermined our meshes using a diffusion for the Sequation as σ0F(S,S0,S∞,x,y), where σ0is determined by the same BPV-equi valent condition as imposed on the one-factor models. This seems to work very well, especially since we ha ve taken ourσequation to be normal with a limited range of influence in the calculations. Future work yet has to establish a more definiti ve distribution for σ. 7. CALCULA TION OF DERIV ATIVES 7.1. Primary Use of Probabilities For European Options We can use PATHINT to de velop the distribution of the option v alue back in time from e xpiration. This is the standard approach used by CRR, explicit and implicit Crank-Nicolson models, etc [56]. ForEuropean options, we also tak eadvantage of the accurac yo fPATHINT enhanced by normalizing the distribution as well as the kernel at each iteration (though in these studies this w as not required after normalizing the k ernel). Therefore, we have calculated our European option prices and Greeks using the most elementary and intuiti ve definition of the option’ spriceV[57], which is the expected value V=<max[z(S−X), 0]>,  z=1, z=−1,call put,( 48) whereXis the strik eprice, and the expected value < .> istaken with respect to the risk-neutral distribution of the underlying mark etS.Itshould be noted that, while the standard approach of developing the option price deli vers at the present time a range of underlying values for a gi venstrike, our approach deli vers a more practical complete range of strikes (as man ya s50−60 for Eurodollar options) for a givenunderlying at the present time, resulting in a greatly enhanced numerical ef ficiency. The risk- neutral distribution is ef fectively calculated taking the drift as the cost-of-carry btimesS,using the abo ve arguments leading to the BS formula. We hav edesigned our codes to use parameters risk-free-rate rand cost-of-carry bsuch thatHigh-resolution path-integral ... -26- L ester Ingber b=r,cost of carry on nondividend stock , b=r−q,cost of carry on stock paying dividend yield q, b=0, cost of carry on future contract , b=r−rf,cost of carry on currency with rate rf,( 49) which is similar to ho wgeneralized European BS codes use bandr[58]. Using this approach, the European price VEis calculated as VE=<< max[ [z(e(b−r)TST−e−rTX), 0]]>>. ( 50) The American price VAmust be calculated using a different k ernel going back in time from expiration, using as “initial conditions” the option values used in the abo ve average. This kernel is the transposed matrix used for the European case, and includes additional drift and “potential” terms due to the need to de velop this back in time. This can be understood as requiring the adjoint partial dif ferential equation or a postpoint Lagrangian in real time. That is, a forward equation for the conditional probability distrib utionP[M(t+dt),t+dt|M(t),t]is ∂P/∂t=1 2(gGG′P),GG′−(gGP),G+V,( 51) where the partial deri vativeswith respect to MGact on the postpoint MG(t+dt). Abackward equation for the conditional probability distribution P[M(t+dt),t+dt|M(t),t]is ∂P/∂t=1 2gGG′P,GG′−gGP,G+V,( 52) where the partial deri vativeswith respect to MGact on the prepoint MG(t). Theforward equation has a particularly simple form in the mathematically equi valent prepoint path integral representation. Above,w ehav edescribed ho wthe forward distribution at present time T0is evolved using PATHINT to the time of e xpiration, P(T), e.g., using a path-integral k ernelKfolded overnepochs, where it is folded with the function O, V=<O(T)P(T)>=<O((KnP(t0)))>, O(T)=max[ [z(e(b−r)TST−e−rTX), 0]], ( 53)High-resolution path-integral ... -27- L ester Ingber to determine the European v alue at the present time of the calls and puts at different strik evaluesX.An equivalent calculation can be performed by using the backward equation, expressed in terms of the “equivalent” kernel K†acting on O, V=<O(T)P(T)>=<((KnO(T)) )P(t0)>. ( 54) It is convenient to use the simple prepoint representation for the Lagrangian, so the backward kernel is first re-expressed as a forward kernel by bringing the dif fusions and drifts inside the partial deri vatives, giving a transformed adjoint kernel K†. The above mathematics is easily tested by calculating European options going forwards and backwards. For American options, while performing the backwards calculation, at each point of mesh, the options price e volving backwards from Tis tested and calculated as Onew=max[(S−X), (e−rdtOold)] . (55) The Greeks {Δ,Γ,Θ}are directly taken of fthe final de veloped option at the present time, since the MG mesh is a vailable for all deri vatives. Weget excellent results for all Greeks. Note that for CRR trees, only one point of mesh is at the present time, so Δrequires moving back one epoch and Γrequires mo ving back twoepochs, unless the present time is pushed back additional epochs, etc. 7.2. PATHINT Baselined to CRR and BS The CRR method is a simple binomial tree which in a specific limit approaches the BS partial differential equation. It has the virtues of being f ast and readily accommodates European and American calculations. Ho wever, itsuffers a number of well-known numerical problems, e.g., a systematic bias in the tree approximation and an oscillatory error as a function of the number of intermediate epochs/iterations in its time mesh. Some Greeks lik e{Δ,Γ,Θ}can be directly tak en offthe tree used for pricing with reasonable approximations (at epochs just before the actual current time). The first problem for American options can be alleviated somewhat by using the variant method [5], CRRvariant=CRRAmerican−CRREuropean+BS . (56) The second problem can be alleviated somewhat by a veraging runs of niterations with runs of n+1 iterations [59]. This four -fold increase of runs is rarely used, though perhaps it should be more often. Furthermore, if increased accurac yi nprice is needed in order to tak enumerical deri vatives, typicallyHigh-resolution path-integral ... -28- L ester Ingber 200−300 iterations should be used for e xpirations some months a way, not 30−70 as too often used in practice. When taking numerical deri vativesthere can arise a need to tune increments tak en for the differentials. F or some Greeks lik eΔandΓthe size of the best differentials to use may vary with strik es that have different sensitivities to parts of the underlying distrib ution. One method of b uilding in some adaptive flexibility across man ysuch strikes is to increase the order of terms used in taking numerical derivatives. (Thiswasnot required for results presented here.) Forexample, it is straightforw ard to verify that, while the central difference df dx=f(x+dx)−f(x−dx) 2dx(57) is typically good to o(((dx)3)), df dx=−f(x+2dx)+8f(x+dx)−8f(x−dx)+f(x−2dx) 12dx(58) is typically good to o(((dx)5)).Similarly,while d2f dx2=f(x+dx)−2f(x)+f(x−dx) dx dx(59) is typically good to ( (( dx)4)), d2f dx2=−d(x+2dx)+16f(x+dx)−30f(x)+16f(x−dx)−f(x−2dx) dx dx(60) is typically good to ( (( dx)6)). Table 1 givesa nexample of baselining our one-factor P ATHINT code to the CRR and BS codes using the abo ve safeguards for an option whose American price is the same as its European counterpart, a typical circumstance [58]. In the literature, the CRR code is most often tak en as the benchmark for American calculations. We took the number of intermediate epochs/points to be 300 for each calculation. Parameters used for this particular ATM call are T=0. 5years,r=0. 05,b=0,σ=0. 10. Table 1.High-resolution path-integral ... -29- L ester Ingber Tests with American CRR and American P ATHINT lead to results with the same de grees of accuracy. 7.3. Two-Factor P ATHINT Baselined to One-Factor P ATHINT Previous papers and tests ha ve demonstrated that the tw o-factor PATHINT code performs as expected. The code was de veloped with only a fe wlines to be changed for running an yn-factor problem, i.e., of course after coding the Lagrangian specific to a gi vensystem. Tests were performed by combining twoone-factor problems, and there is no loss of accurac y. Howev er, here we are making some additional mesh approximations as discussed abo ve toaccommodate σin theSdiffusion. This seems quite reasonable, but there is no sure test of the accurac y. Weindeed see that the ATM results are very close acrossx’s,similar to our ATM comparisons between BS and our one-factor P ATHINT results for v arious x’s,where again scaling is performed to ha ve all models used the same BPV (using the σ0procedure for the mesh as described abo ve for the two-factor model). The logical e xtension of Greeks for the tw o-factor model is to de velop derivativeso fprice with respect to ρandεinσvolatility equation. However, wedid not find a bona fide two-factor proxy for the one-factor ϒ,the derivative ofprice with respect to the one-f actorσconstant. W eget very good A TMϒ comparisons between BS and our one-factor models with v ariousx’s.Wetried simply multiplying the noise in the tw o-factor stochastic v olatility in the price equation by a parameter with deviations from 1 to get numerical deri vativeso fPATHINT solutions, and this g av esomewhat good agreement with the A TM BPV-scaled BS ϒwithin a couple of significant figures. Perhaps this is not too surprising, especially giventhe correlation substantial ρbetween the price and volatility equations which we do not neglect. 8. CONCLUSIONS The results of our study are that, after the at-the-mone ybasis-point volatilities are scaled to be equivalent, there is only a v ery small change in option prices for different e xponents x,both for the one- factor and tw o-factor models. There still are significant differences in Greeks using more sophisticated models, especially for out-of-the-mone yoptions. This implies that current pricing models are not radically mispricing the markets.High-resolution path-integral ... -30- L ester Ingber Our studies point to contexts of mark ets well recognized by option traders to ha ve significantly different volatility beha vior.Suppression of stochastic volatility in the one-factor model just leaks out into stochasticity of parameters in the model, e.g., especially in x,unless additional numerical methods are emplo yed, e.g., using an adapti ve standard de viation. Our studies sho wthat the tw o-factor exponents xare reasonably f aithful indicators defining these different conte xts. The x-exponents in the tw o-factor fits are quite stable. As such, especially the tw o-factorxcan be considered as a “context indicator” o vera longer time scale than other indicators typically used by traders. The tw o-factor fits also e xhibit differences due to the σparameters, including the ρcorrelations, in accord with the sense traders ha ve about the nature of changing volatilities across this time period. Modern methods of de veloping multi variate nonlinear multiplicati ve Gaussian-Mark ovian systems are quite important, as there are man ysuch systems and the mathematics must be diligently e xercised if such models are to f aithfully represent the data the ydescribe. Similarly ,sophisticated numerical techniques, e.g., global optimization and path inte gration are important tools to carry out the modeling and fitting to data without compromising the model, e.g., by unwarranted quasi-linear approximations. Three quite different systems ha ve benefited from this approach: The large-scale modeling of neocortical interactions has benefited from the use of intuiti ve constructs that yet are faithful to the comple xalgebra describing this multiple-scaled comple xsystem. Forexample, canonical-momenta indicators ha ve been successfully applied to multi variate financial markets. It is clear that ASA optimization and P ATHINT path-integral tools are very useful to de velop the algebra of statistical mechanics for a large class of nonlinear stochastic systems encountered in finance. However, italso is clear that each system typically presents its own non-typical unique character and this must be included in an ysuch analysis. Avirtue of this statistical mechanics approach and these associated tools is the yappear to be flexible and robust to handle quite different systems. ACKNOWLEDGMENTS Ithank Donald W ilson for his support and Jennifer Wilson for collaboration running ASA and PATHINT calculations. Implied volatility and yield data w as extracted from the MIM database of Logical Information Machines (LIM).High-resolution path-integral ... -31- L ester Ingber REFERENCES 1. 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W olfer,Numerical e valuation of path integral solutions to F okker-Planck equations. III. Time and functionally dependent coefficients, Phys. Rev. A35,1795-1801 (1987). 55. M. Rosa-Clot and S. Taddei, A path integral approach to deri vative security pricing: II. Numerical methods, INFN, Firenze, Italy ,(1999). 56. P.Wilmott, S. Howison, and J. De wynne,The Mathematics of Financial Derivatives ,Cambridge U Press, Cambridge, (1995). 57. L.Ingber,Asimple options training model, Mathl. Computer Modelling 30,167-182 (1999).High-resolution path-integral ... -35- L ester Ingber 58. E.G. Haug,The Complete Guide to Option Pricing F ormulas,McGraw-Hill, Ne wYork, NY, (1997). 59. M. Broadie and J. Detemple, Recent advances in numerical methods for pricing deri vative securities, in Numerical Methods in F inance,(Edited by L.C.G Rogers and D. Talay), pp. 43-66, Cambridge Uni versity Press, Cambridge, UK, (1997).High-resolution path-integral ... -36- L ester Ingber FIGURE CAPTIONS FIG. 1.(a)F(S,S0,S∞,x,y)forx=1, the Black-Scholes case. The other parameters are S=5, S0=0. 5,S∞=20,y=0. (b)F(S,S0,S∞,x,y)forx=0, the normal distrib ution. (c) F(S,S0,S∞,x,y) forx=−1. (d)F(S,S0,S∞,x,y)forx=2. FIG. 2.The short-time probability distribution at time T=0. 5years for x=1, the (truncated) Black-Scholes distrib ution. The short-time probability distribution at time T=0. 5years for x=0, the normal distrib ution. The short-time probability distribution at time T=0. 5years for x=−1. Theshort- time probability distribution at time T=0. 5years for x=2. FIG. 3.Atwo-factor distribution e volved out to T=0. 5year forx=0. 7.High-resolution path-integral ... -37- L ester Ingber TABLE CAPTIONS Table 1. Calculation of prices and Greeks are gi venfor closed form BS (only v alid for European options), binomial tree CRR European,CRRAmerican,CRRvariant,and PATHINT.Asverified by calculation, the American option would not be e xercised early ,sothe PATHINT results are identical to the European option. The CRRAmericandiffers somewhat from the CRR Europeandue to the discrete nature of the calculation. All CRR calculations include a veraging over300 and 301 iterations to minimize oscillatory errors.High-resolution path-integral ... -Figure 1 - Lester Ingber 02468101214161820 02468101214161820F(S) S(a) x = 1 x = 1 0.050.10.150.20.250.30.350.40.450.5 02468101214161820F(S) S(b) x = 0 x = 0 00.050.10.150.20.250.30.350.40.450.5 02468101214161820F(S) S(c) x = -1 x = -1 0100200300400500600700800 02468101214161820F(S) S(d) x = 2 x = 2High-resolution path-integral ... -Figure 2 - Lester Ingber 00.10.20.30.40.50.60.70.8 3.544.555.566.57Long-Time Probability Price(a) x = 1 x = 1 00.10.20.30.40.50.60.70.8 33.544.555.566.57Long-Time Probability Price(b) x = 0 x = 0 00.10.20.30.40.50.60.70.8 33.544.555.566.5Long-Time Probability Price(c) x = -1 x = -1 00.10.20.30.40.50.60.70.8 3.544.555.566.57Long-Time Probability Price(d) x = 2 x = 2High-resolution path-integral ... -Figure 3 - Lester Ingber Two-Factor Probability 3.7544.254.54.7555.255.55.7566.25Price 0.160.1650.170.1750.180.1850.190.195 Volatility0100Long-Time ProbabilityHigh-resolution path-integral ... -Table 1 - Lester Ingber Greek BS CRREuropean CRRAmerican CRRvariant PATHINT Price 0.138 0.138 0.138 0.138 0.138 Delta 0.501 0.530 0.534 0.506 0.501 Gamma 1.100 1.142 1.159 1.116 1.100 Theta -0.131 -0.130 -0.132 -0.133 -0.131 Rho -0.0688 -0.0688 -0.0530 -0.0530 -0.0688 Ve ga1 .375 1.375 1.382 1.382 1.375

arXiv:physics/0001049 23 Jan 2000STATISTICAL MECHANICS OF NEOCORTICAL INTERACTIONS: REACTION TIME CORRELATES OF THE gFA CTOR Lester Ingber Lester Ingber Research PO Box 06440 Wacker Dr PO Sears T ower Chicago, IL 60606 and DRWInv estments LLC 311 S Wacker Dr Ste 900 Chicago, IL 60606 ingber@ingber.com, ingber@alumni.caltech.edu http://www.ingber.com/ Psycholoquy Commentary on ThegFactor: The Science of Mental Ability by Arthur Jensen ABSTRACT :Astatistical mechanics of neuronal interactions (SMNI) is explored as pro viding some substance to a ph ysiological basis of the gfactor.Some specific elements of SMNI, previously used to develop a theory of short-term memory (STM) and a model of electroencephalograph y(EEG) are k ey to providing this basis. Specifically ,Hick’sLaw,a nobserved linear relationship between reaction time (RT) and the information storage of STM, in turn correlated to a R T-grelationship, is deri ved. KEYWORDS: short term memory; nonlinear; statisticalBasis for the gfactor - 2- L ester Ingber 1. Introduction 1.1. Context of Review My specific interest in taking on this re viewo f“ThegFactor” by Arthur Jensen (AJ) is to see if some anatomical and/or physiological processes at the columnar le velo fneuronal interactions can account for the gfactor. From circa 1978 through the present, a series of papers on the statistical mechanics of neocortical interactions (SMNI) has been de veloped to model columns and regions of neocorte x, spanning mm to cm of tissue. Most of these papers ha ve dealt explicitly with calculating properties of short-term memory (STM) and scalp EEG in order to test the basic formulation of this approach. SMNI deri vesaggregate behavior of experimentally observ ed columns of neurons from statistical electrical-chemical properties of synaptic interactions. While not useful to yield insights at the single neuron le vel, SMNI has demonstrated its capability in describing large-scale properties of short-term memory and electroencephalographic (EEG) systematics (Ingber ,1982; Ingber ,1983; Ingber ,1984; Ingber ,1991; Ingber,1994; Ingber ,1995a; Ingber & Nunez, 1995; Ingber ,1996a; Ingber ,1997). 1.2. Errors In Simple Statistical Approaches Imust assume that AJ faced very difficult problems in choosing just ho wmuch technical details to give inhis broad text, e.g., discussing the extent of expert statistical analyses that ha ve been brought to bear upon the gfactor.Howev er, I dosee reason to criticize some general features of the simple statistical algorithms presented, especially those that o verlap with my own mathematical and physics expertise. The simple approach to factor analysis initiated on page 23, X=t+e,( 1) whereeis the residual “noise” of fitting the v ariableXto the independent v ariablet,has some serious flaws not addressed by additional material presented thereafter .For example, in this context, I find the arguments in the long footnote 16 on pages 101-103 unconvincing, but I agree with its conclusion: But the question is mainly of scientific interest, and a really satisfactory answer ... will become possible only as part and parcel of a comprehensi ve theory of the nature of g.... The distribution of obtained measurements should conform to the characteristics of the distribution dictated by theoretical considerations. Ithink it clear that an ysuch “theoretical considerations” must themselves be well tested ag ainstBasis for the gfactor - 3- L ester Ingber experimental evidence at each spatial-temporal scale purported to be modeled. It must be understood that there is a quite explicit model being assumed here of the real w orld — that of a simple normal Gaussian process. The real issue in man yphysical/biological systems is that most often the real multi variable world is much more aptly described by something like X=tX(X,Y)+sX(X,Y)eX (2.a) Y=tY(X,Y)+sY(X,Y)eY.( 2.b) When the t’s ands’s are constants, then simple statistics can determine their values and cross-correlations between the s’s. Simple statistical methods can e vend oO Ki ft het’s are relatively simple quasi-linear parameterized functions. Such simple methods fail quite miserably if the t’s are highly nonlinear functions, especially if care is not taken to emplo ysophisticated optimization algorithms. The most terrible flaws often occur because, for the sak eo fmaking life easier for the analyst, an ymodel faithful to the real system is butchered and sacrificed, and the s’s are taken to be constants. In general, there can be a lot of “signal” in the (generally nonlinear) functionality of the “noise” terms. In general, no degree of fancyquasi-linear statistical analysis can substitute for a proper theory/model of the real system. In general, the proper treatment of the problem is quite difficult, which is of course no excuse for poor treatment. The solution in man ydisciplines is to go a le velo rtwo deeper in some Reductionist sense, to de velop plausible models at the top scale being analyzed. Indeed, this was the call I sa wand responded to in the advertisement for reviewers of the work by AJ: Commentary is in vited from psychometricians, statisticians, geneticists, neuropsychologists, psychoph ysiologists, cogniti ve modellers, e volutionary psychologists and other specialties concerned with cogniti ve abilities, their measurement, and their cogniti ve and neurobiological basis. In this conte xt, the successes of SMNI and its agreement with general STM observations are due to processing stochastic nonlinearities of the forms described abo ve.Attempts to a void dealing with these nonlinearities, deri vedfrom lower-levelsynaptic and neuronal acti vity,hav enot been as successful as SMNI in detailing STM (Ingber ,1995b). 2. SMNI Description of Short-Term Memory (STM) Since the early 1980’s, a series of papers on the statistical mechanics of neocortical interactions (SMNI) has been de veloped to model columns and regions of neocorte x, spanning mm to cm of tissue.Basis for the gfactor - 4- L ester Ingber Most of these papers ha ve dealt explicitly with calculating properties of short-term memory (STM) and scalp EEG in order to test the basic formulation of this approach (Ingber ,1981; Ingber ,1982; Ingber , 1983; Ingber ,1984; Ingber ,1985a; Ingber ,1985b; Ingber ,1986; Ingber & Nunez, 1990; Ingber ,1991; Ingber,1992; Ingber ,1994; Ingber & Nunez, 1995; Ingber ,1995a; Ingber ,1995b; Ingber ,1996b; Ingber , 1997; Ingber ,1998). This model was the first physical application of a nonlinear multi variate calculus developed by other mathematical physicists in the late 1970’ s(Graham, 1977; Langouche et al,1982). 2.1. Statistical Aggregation SMNI studies ha ve detailed a physics of short-term memory and of (short-fiber contribution to) EEG phenomena (Ingber ,1984), in terms of MGfirings, where Grepresents EorI,MErepresents contributions to columnar firing from excitatory neurons, and MIrepresents contributions to columnar firing from inhibitory neurons. About 100 neurons comprise a minicolumn (twice that number in visual cortex); about 1000 minicolumns comprise a macrocolumn. Amesocolumn is de veloped by SMNI to reflect the con vergence of short-ranged (as well as long-ranged) interactions of macrocolumnar input on minicolumnar structures, in terms of synaptic interactions taking place among neurons (about 10,000 synapses per neuron). The SMNI papers gi ve more details on this deri vation. In this SMNI de velopment, a Lagrangian is explicitly defined from a deri vedprobability distribution of mesocolumnar firings in terms of the MGand electric potential v ariables, ΦG.Sev eral examples ha ve been givent oillustrate ho wthe SMNI approach is complementary to other models. For example, a mechanical string model w as first discussed as a simple analog of neocortical dynamics to illustrate the general ideas of top down and bottom up interactions (Nunez, 1989; Nunez & Srini vasan, 1993). SMNI wasapplied is to this simple mechanical system, illustrating ho wmacroscopic variables are derivedfrom small-scale variables (Ingber & Nunez, 1990). 2.2. STM Stability and Duration SMNI has presented a model of STM, to the extent it offers stochastic bounds for this phenomena during focused selecti ve attention. This 7±2“rule” is well verified by SMNI for acoustical STM (Ingber , 1984; Ingber ,1985b; Ingber ,1994), transpiring on the order of tenths of a second to seconds, limited to the retention of 7 ±2items (Miller ,1956). The 4±2“rule” also is well v erified by SMNI for visual or semantic STM, which typically require longer times for rehearsal in an h ypothesized articulatory loop of individual items, with a capacity that appears to be limited to 4 ±2(Zhang & Simon, 1985). SMNI has detailed these constraints in models of auditory and visual corte x(Ingber,1984; Ingber ,1985b; Ingber ,Basis for the gfactor - 5- L ester Ingber 1994; Ingber & Nunez, 1995). Another interesting phenomenon of STM capacity e xplained by SMNI is the primac yversus recencyeffect in STM serial processing (Ingber ,1985b), wherein first-learned items are recalled most error-free, with last-learned items still more error-free than those in the middle (Murdock, 1983). The basic assumption is that a pattern of neuronal firing that persists for man yτcycles,τon the order of 10 msec, is a candidate to store the “memory” of activity that g av erise to this pattern. If several firing patterns can simultaneously exist, then there is the capability of storing se veral memories. The short-time probability distribution deri vedfor the neocorte xi sthe primary tool to seek such firing patterns. The highest peaks of this probability distribution are more likely accessed than the others. Theyare more readily accessed and sustain their patterns ag ainst fluctuations more accurately than the others. The more recent memories or ne wer patterns may be presumed to be those having synaptic parameters more recently tuned and/or more acti vely rehearsed. It has been noted that e xperimental data on velocities of propagation of long-ranged fibers (Nunez, 1981; Nunez, 1995) and deri vedvelocities of propag ation of information across local minicolumnar interactions (Ingber ,1982) yield comparable times scales of interactions across minicolumns of tenths of asecond. Therefore, such phenomena as STM likely are inextricably dependent on interactions at local and global scales. 2.3. SMNI Correlates of STM and EEG Previous SMNI studies ha ve detailed that maximal numbers of attractors lie within the ph ysical firing space of MG,consistent with experimentally observed capacities of auditory and visual short-term memory (STM), when a “centering” mechanism is enforced by shifting background noise in synaptic interactions, consistent with experimental observations under conditions of selecti ve attention (Mountcastle et al,1981; Ingber ,1984; Ingber ,1985b; Ingber ,1994; Ingber & Nunez, 1995). This leads to all attractors of the short-time distribution lying along a diagonal line in MGspace, effectively defining anarrowparabolic trough containing these most likely firing states. This essentially collapses the 2 dimensional MGspace down to a 1 dimensional space of most importance. Thus, the predominant physics of short-term memory and of (short-fiber contribution to) EEG phenomena tak es place in a narrow“parabolic trough” in MGspace, roughly along a diagonal line (Ingber ,1984). Using the po wer of this formal structure, sets of EEG and e vokedpotential data, collected to investigate genetic predispositions to alcoholism, were fitted to an SMNI model to extract brain “signatures” of short-term memory (Ingber ,1997; Ingber ,1998). These results give strong quantitati veBasis for the gfactor - 6- L ester Ingber support for an accurate intuiti ve picture, portraying neocortical interactions as having common algebraic or physics mechanisms that scale across quite disparate spatial scales and functional or beha vioral phenomena, i.e., describing interactions among neurons, columns of neurons, and regional masses of neurons. Forfuture work, I ha ve described ho wbottom-up neocortical models can be de veloped into eigenfunction expansions of probability distrib utions appropriate to describe short-term memory in the context of scalp EEG (Ingber ,2000). The mathematics of eigenfunctions are similar to the top-do wn eigenfunctions de veloped by some EEG analysts, albeit the yhav edifferent physical manifestations. The bottom-up eigenfunctions are at the local mesocolumnar scale, whereas the top-do wn eigenfunctions are at the global regional scale. However, these approaches ha ve regions of substantial o verlap (Ingber & Nunez, 1990; Ingber ,1995a), and future studies may expand top-down eigenfunctions into the bottom-up eigenfunctions, yielding a model of scalp EEG that is ultimately e xpressed in terms of columnar states of neocortical processing of attention and short-term memory. An optimistic outcome of future w ork might be that these EEG eigenfunctions, baselined to specific STM processes of individuals, could be a more direct correlate to estimates of gfactors. 2.4. Complexity in EEG In Chapter 6, Biological Correlates of g,AJputs these issues in perspecti ve: First, psychometric tests were ne verintended or devised to measure an ything other than behavioral variables. ... at this point most explanations are still conjectural. However, most of the chapter f alls back to similar too-simple statistical models of correlations between measured variables and behavioral characteristics. The sections on Biological Correlates of gdealing with EEG recordings are incomplete and at a leveltoo superficial relati ve most of the other parts of the book. The introduction of “complexity” as a possible correlate to IQ is based on f addish studies that do not ha ve any theoretical or e xperimental support (Nunez, 1995). If such support does transpire, most likely it will be de veloped on the shoulders of more complete stochastic models and much better EEG recordings. 3. SMNI STM Correlates of the gFactorBasis for the gfactor - 7- L ester Ingber 3.1. High vs LowgCategories The outline of test categories giving rise to high versus lo wgloadings on page 35 pro vides me with some immediate candidates for further in vestigations of a physiological basis for the gfactor.Ithink a good working hypothesis is that these tw ocategories are marked by having the high gloadings more correlated than the lo wgloadings to statistical interactions among the peaks of the probability distributions described abo ve inthe context of SMNI’ stheory of STM. The high gcategories clearly require relati vely more processing of se veral items of information than do the low gcategories. This seems to be a similar correlation as that dra wn by Spearman, as described by AJ, in ha ving “education of relations” and “education of correlates” more highly correlation with high gcategories than the “apprehension of experience.” 3.2. Mechanisms of HighgCategories From the SMNI perspecti ve,control of selecti ve attention generally is highly correlated with high utilization of STM, e.g., to tune the “centering mechanism. ”This seems similar to Spearman’ scorrelation of “mental energy” with high gcategories. There are se veral mechanisms that might distinguish ho wwell individuals might perform on high g category tasks. The ability to control the “centering mechanism” is required to sustain a high degree of statistical processes of multiple most probable states of information. The particular balance of general chemical-electrical activity directly shapes the distribution of most probable states. Forsome tasks, processing across relati vely more most probable states might be required; for other tasks processing among larger focussed peaks of most probable states might be more important. 3.3. Hick’ sLaw — Linearity of R TvsSTM Information The SMNI approach to STM gi vesareasonable foundation to discuss R Tand items in STM storage. These previous calculations support the intuiti ve description of items in STM storage as peaks in the 10-millisecond short-time (Ingber ,1984; Ingber ,1985b) as well as the se veral-second long-time (Ingber,1994; Ingber & Nunez, 1995) conditional probability distribution of correlated firings of columns of neurons. These columnar firing states of STM tasks also were correlated to EEG observations of ev okedpotential activities (Ingber ,1997; Ingber ,1998). This distribution is explicitly calculated by respecting the nonlinear synaptic interactions among all possible combinatoric aggre gates of columnar firing states (Ingber ,1982; Ingber ,1983).Basis for the gfactor - 8- L ester Ingber The RTnecessary to “visit” the states under control during the span of STM can be calculated as the mean time of “first passage” between multiple states of this distribution, in terms of the probability P as an outer inte gral∫dt(sum) overrefraction times of synaptic interactions during STM time t,and an inner inte gral∫dM(sum) taken o verthe mesocolumnar firing states M(Risken, 1989), which has been explicitly calculated to be within observed STM time scales (Ingber ,1984), RT=−∫dt t∫dMdP dt.( 3) As demonstrated by previous SMNI STM calculations, within tenths of a second, the conditional probability of visiting one state from another P,can be well approximated by a short-time probability distribution expressed in terms of the previously mentioned Lagrangian Las P=1 √ (2πdtg)exp(−Ldt), ( 4) wheregis the determinant of the co variance matrix of the distrib utionPin the space of columnar firings. This expression for RTcan be approximately rewritten as RT≈K∫dt∫dM PlnP,( 5) whereKis a constant when the Lagrangian is approximately constant o verthe time scales observ ed. Since the peaks of the most lik elyMstates of Pare to a very good approximation well-separated Gaussian peaks (Ingber ,1984), these states by be treated as independent entities under the inte gral. This last expression is essentially the “information” content weighted by the time during which processing of information is observed. The calculation of the heights of peaks corresponding to most lik ely states includes the combinatoric f actors of their possible columnar manifestations as well as the dynamics of synaptic and columnar interactions. In the approximation that we only consider the combinatorics of items of STM as contributing to most likely states measured by P,i.e., thatPmeasures the frequenc yo foccurrences of all possible combinations of these items, we obtain Hick’ sLaw,the observed linear relationship of R Tversus STM information storage, first discussed by AJ in Chapter 8. Forexample, when the bits of information are measured by the probability Pbeing the frequenc yo faccessing a gi vennumber of items in STM, the bits of information in 2, 4 and 8 states are gi vena sapproximately multiples of ln 2o fitems, i.e., ln 2, 2ln2 and 3ln 2,resp. (The limit of taking the log arithm of all combinations of independent items yields a constant times the sum o verpilnpi,wherepiis the frequenc yo foccurrence of item i.)Basis for the gfactor - 9- L ester Ingber 4. Conclusion The book written by AJ provides moti vation to explore a more fundamental basis of the gfactor.I have examined this work in the narro wfocus of some specific elements of SMNI, pre viously used to develop a theory of STM and a model of EEG. Ihav efocussed on ho wbottom-up SMNI models can be de veloped into eigenfunction e xpansions of probability distributions appropriate to describe STM. This permits R Tt ob ec alculated as an expectation value o verthe STM probability distrib ution of stored states, and in the good approximation of such states being represented by well separated Gaussian peaks, this yield the observed linear relationship of RTversus STM information storage. This SMNI STM approach also suggests se veral other studies that can be performed in the conte xt of examining an underlying basis for the gfactor. REFERENCES Graham, R. (1977) Co variant formulation of non-equilibrium statistical thermodynamics. Z. Physik B26:397-405. Ingber,L.(1981) Tow ards a unified brain theory. J. Social Biol. Struct. 4:211-224. Ingber,L.(1982) Statistical mechanics of neocortical interactions. I. Basic formulation. Physica D 5:83-107. [URL http://www.ingber.com/smni82_basic.ps.gz] Ingber,L.(1983) Statistical mechanics of neocortical interactions. Dynamics of synaptic modification. Phys. Rev. A28:395-416. [URL http://www.ingber.com/smni83_dynamics.ps.gz] Ingber,L.(1984) Statistical mechanics of neocortical interactions. Deri vation of short-term-memory capacity. Phys. Rev. A29:3346-3358. [URL http://www.ingber.com/smni84_stm.ps.gz] Ingber,L.(1985a) Statistical mechanics of neocortical interactions. EEG dispersion relations. IEEE Tr ans. Biomed. Eng. 32:91-94. [URL http://www.ingber.com/smni85_eeg.ps.gz] Ingber,L.(1985b) Statistical mechanics of neocortical interactions: Stability and duration of the 7+−2 rule of short-term-memory capacity .Phys. Re v. A 31:1183-1186. [URL http://www.ingber.com/smni85_stm.ps.gz] Ingber,L.(1986) Statistical mechanics of neocortical interactions. Bull. Am. Phys. Soc. 31:868. Ingber,L.(1991) Statistical mechanics of neocortical interactions: A scaling paradigm applied to electroencephalograph y. Phys. Re v. A 44:4017-4060. [URL http://www.ingber.com/smni91_eeg.ps.gz]Basis for the gfactor - 10 - Lester Ingber Ingber,L.(1992) Generic mesoscopic neural netw orks based on statistical mechanics of neocortical interactions. Phys. Rev. A45:R2183-R2186. [URL http://www.ingber.com/smni92_mnn.ps.gz] Ingber,L.(1994) Statistical mechanics of neocortical interactions: P ath-integral evolution of short-term memory. Phys. Rev. E49:4652-4664. [URL http://www.ingber.com/smni94_stm.ps.gz] Ingber,L.(1995a) Statistical mechanics of multiple scales of neocortical interactions, In: Neocortical Dynamics and Human EEG Rhythms ,ed. P.L. Nunez. Oxford Uni versity Press, 628-681. [ISBN 0-19-505728-7. 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Modelling 13:83-95.Basis for the gfactor - 11 - Lester Ingber Ingber,L .&Nunez, P.L. (1995) Statistical mechanics of neocortical interactions: High resolution path- integral calculation of short-term memory .Phys. Re v. E51:5074-5083. [URL http://www.ingber.com/smni95_stm.ps.gz] Langouche, F ., Roekaerts, D. & Tirapegui, E. (1982)Functional Inte gration and Semiclassical Expansions .Reidel, Dordrecht, The Netherlands. Miller,G.A. (1956) The magical number se ven, plus or minus two. Psychol. Re v.63:81-97. Mountcastle, V .B., Andersen, R.A. & Motter ,B.C. (1981) The influence of attenti ve fixation upon the excitability of the light-sensiti ve neurons of the posterior parietal corte x.J. Neurosci.1:1218-1235. Murdock, B.B., Jr .(1983) A distrib uted memory model for serial-order information. Psychol. Rev. 90:316-338. Nunez, P.L. (1981) Electric Fields of the Brain: The Neurophysics of EEG .Oxford Uni versity Press, London. Nunez, P .L. (1989) To wards a physics of neocortex, Vol. 2, In: Advanced Methods of Physiolo gical System Modeling ,ed. V.Z. Marmarelis. Plenum, 241-259. Nunez, P.L. (1995) Neocortical Dynamics and Human EEG Rhythms .Oxford Uni versity Press, Ne w York, NY. Nunez, P.L. & Srini vasan, R. (1993) Implications of recording strate gy for estimates of neocortical dynamics with electroencephalograph y.Chaos3:257-266. Risken, H.(1989)The Fokker-Planc kEquation: Methods of Solution and Applications .Springer-Verlag, Berlin. Zhang, G. & Simon, H.A. (1985) STM capacity for Chinese words and idioms: Chunking and acoustical loop hypotheses. Memory & Cognition 13:193-201.

arXiv:physics/0001050 23 Jan 2000STATISTICAL MECHANICS OF NEOCORTICAL INTERACTIONS: EEG EIGENFUNCTIONS OF SHOR T-TERM MEMORY Lester Ingber Lester Ingber Research PO Box 06440 Wacker Dr PO Sears T ower Chicago, IL 60606 and DRWInv estments LLC 311 S Wacker Dr Ste 900 Chicago, IL 60606 ingber@ingber.com, ingber@alumni.caltech.edu Behavioral and Brain Sciences Commentary on To ward a Quantitati ve Description of Large-Scale Neo-Cortical Dynamic Function and EEG by Paul Nunez ABSTRACT :This paper focuses on ho wbottom-up neocortical models can be de veloped into eigenfunction expansions of probability distrib utions appropriate to describe short-term memory in the conte xt of scalp EEG. The mathematics of eigenfunctions are similar to the top-down eigenfunctions de veloped by Nunez, albeit the yhav edifferent ph ysical manifestations. The bottom-up eigenfunctions are at the local mesocolumnar scale, whereas the top-do wn eigenfunctions are at the global regional scale. However, asdescribed in several joint papers, our approaches ha ve regions of substantial o verlap, and future studies may expand top-do wn eigenfunctions into the bottom-up eigenfunctions, yielding a model of scalp EEG that is ultimately e xpressed in terms of columnar states of neocortical processing of attention and short-term memory. KEYWORDS: EEG; short term memory; nonlinear; statistical 1. INTRODUCTION 1.1. Categorization of Experimental Spatial-Temporal EEG Manyreasonable theoretical studies of synaptic-lik eo rneuron-likestructures ha ve attempted experimental confirmation. However, most often these in vestigations do not necessarily address the “spatial-temporal” scales the ypurport to describe. Manyinv estigators would lik et osee more work on experimental design/tests of local-global interactions correlated to behavioral states at specific scales. For example, if an EEG could reasonably be correlated to a resolution of 3−5 cm within a time scale of 1−3 msec, then e xperiments should be attempted to test if specific states of attentional information-processing are highly correlated within this specific spatial-temporal range. In this context, the work of P aul Nunez (PN) stresses resolution of EEG data within specific spatial- temporal scales, giving us candidate data for such correlations. It is most important for researchers to deal with the details of experimental evidence, not just pay homage to its existence. 1.2. Theor etical Descriptions of Spatial-Temporal EEG The theoretical frame work givenb yP Ne ncompasses global and local neuronal columnar acti vity, giving a primary role to global acti vity.His work has generated interest in other in vestigators to tak e similar approaches to describing neocortical activity (Jirsa & Haken, 1996).... statistical mechanics ... -2- L ester Ingber Other investigators, myself included, gi ve a primary role to local acti vity,immersed in global circuitry.Inthis context, PN brings to BBS commentary a sound frame work in which to further analyze the importance of considering multiple scales of neocortical activity. 1.3. Generality of Eigenfunction Expansion Here, I add my own emphasis this subject, to address ho weigenfunction expansions of models of brain function, similar to those performed by PN to describe ho wglobal models of w av ephenomenon can be used ef fectively to describe EEG, can be applied to probability distributions of short-term memory fits to EEG data. In the following description, emphasis is placed on o verlap and collaboration with the w ork of PN, especially in areas wherein local and global interactions are required to detail models of neocortical interactions giving rise to EEG phenomena. 2. SMNI Description of Short-Term Memory (STM) Since the early 1980’ s, a series of papers on the statistical mechanics of neocortical interactions (SMNI) has been de veloped to model columns and regions of neocortex, spanning mm to cm of tissue. Most of these papers ha ve dealt explicitly with calculating properties of short-term memory (STM) and scalp EEG in order to test the basic formulation of this approach (Ingber ,1981; Ingber ,1982; Ingber , 1983; Ingber ,1984; Ingber ,1985a; Ingber ,1985b; Ingber ,1986; Ingber & Nunez, 1990; Ingber ,1991; Ingber,1992; Ingber ,1994; Ingber & Nunez, 1995; Ingber ,1995a; Ingber ,1995b; Ingber ,1996; Ingber , 1997; Ingber ,1998). This model was the first ph ysical application of a nonlinear multi variate calculus developed by other mathematical physicists in the late 1970’ s(Graham, 1977; Langouche et al,1982). 2.1. Statistical Aggregation SMNI studies ha ve detailed a physics of short-term memory and of (short-fiber contrib ution to) EEG phenomena (Ingber ,1984), in terms of MGfirings, where Grepresents EorI,MErepresents contributions to columnar firing from e xcitatory neurons, and MIrepresents contributions to columnar firing from inhibitory neurons. About 100 neurons comprise a minicolumn (twice that number in visual cortex); about 1000 minicolumns comprise a macrocolumn. Amesocolumn is de veloped by SMNI to reflect the con vergence of short-ranged (as well as long-ranged) interactions of macrocolumnar input on minicolumnar structures, in terms of synaptic interactions taking place among neurons (about 10,000 synapses per neuron). The SMNI papers gi ve more details on this deri vation. In this SMNI de velopment, a Lagrangian is explicitly defined from a deri vedprobability distribution of mesocolumnar firings in terms of the MGand electric potential v ariables, ΦG.A mechanical string model, as first discussed by PN as a simple analog of neocortical dynamics (Nunez, 1989; Nunez & Srini vasan, 1993), is deri vedexplicitly for neocortical interactions using SMNI (Ingber & Nunez, 1990). In addition to providing o verlap with current EEG paradigms, this defines a probability distribution of firing acti vity,which can be used to further in vestigate the existence of other nonlinear phenomena, e.g., bifurcations or chaotic behavior ,i nbrain states. 2.2. STM The SMNI calculations are of minicolumnar interactions among hundreds of neurons, within a macrocolumnar extent of hundreds of thousands of neurons. Such interactions tak eplace on time scales of severalτ,whereτis on the order of 10 msec (of the order of time constants of cortical p yramidal cells). This also is the observed time scale of the dynamics of STM. SMNI hypothesizes that columnar interactions within and/or between re gions containing man ymillions of neurons are responsible for these phenomena at time scales of se veral seconds. That is, the nonlinear e volution at finer temporal scales givesabase of support for the phenomena observ ed at the coarser temporal scales, e.g., by establishing mesoscopic attractors at man ymacrocolumnar spatial locations to process patterns in larger regions.... statistical mechanics ... -3- L ester Ingber 3. SMNI Description of EEG 3.1. EEG Regional Circuitry of STM Local Firings Previous calculations of EEG phenomena (Ingber ,1985a), sho wthat the short-fiber contribution to theαfrequencyand the mo vement of attention across the visual field are consistent with the assumption that the EEG ph ysics is deri vedfrom an average overthe fluctuations of the system. I.e., this is described by the Euler-Lagrange equations deri vedfrom the variational principle possessed by LΦ,which yields the string model described abo ve (Ingber,1982; Ingber ,1983; Ingber ,1988). 3.2. Indi vidual EEG Data The 1996 SMNI project used e vokedpotential (EP) EEG data from a multi-electrode array under a variety of conditions, collected at se veral centers in the United States, sponsored by the National Institute on Alcohol Abuse and Alcoholism (NIAAA) project (Zhang et al,1995). The earlier SMNI 1991 study used only a veraged EP data. After fits were performed on a set of training data (Ingber ,1997), the parameters for each subject were used to generate CMI for out-of-sample testing data for each subject (Ingber ,1998). The results illustrate that the CMI gi ve enhanced patterns to exhibit differences between the alcoholic and control groups of subjects. 4. EEGEigenfunctions of STM 4.1. STM Eigenfunctions The study fitting individual EEG data to SMNI parameters within STM-specific tasks can no wbe recast into eigenfunction expansions yielding orthogonal memory traces fit to indi vidual EEG patterns. This development was described in the first SMNI papers. The clearest picture that illustrates ho wthis eigenfunction expansion is achie vedi si n(Ingber & Nunez, 1995) wherein, within a tenth of a second, there are stable multiple non-o verlapping Gaussian- type peaks of an e volving probability distribution with the same STM constraints used in the NIH EEG study abo ve.These peaks can be simply modeled as a set of orthogonal Hermite polynomials. 4.2. Expansion of Global EEG Eigenfunctions into Local STM Eigenfunctions In the “classical” limit defined by the variational Euler -Lagrange equations described abo ve,we have demonstrated ho wthe local SMNI theory reduces to a string model similar to the global model of PN (Ingber & Nunez, 1990). One reasonable approach to de veloping the EEG eigenfunctions in the approach gi venb yP Ni st oa pply the variational deri vativesdirectly to the SMNI STM Hermite polynomials, thereby yielding a model of scalp EEG that is ultimately e xpressed in terms of states of neocortical processing of attention and short-term memory. 5. CONCLUSION PN has been a k ey exponent of realistic modeling of realistic neocorte xfor manyyears, evenwhen it was not as popular as neural network modeling of “to ybrains.”Inthe process of insisting on dealing with aspects of models of neocortical systems that could be e xperimentally v erified/ne gated, he has contributed to a rich approach to better understanding the nature of experimental EEG. His approach has led him to stress appreciation of neocorte xa sfunctioning on multiple spatial- temporal scales, and he has collaborated with other in vestigators with different approaches to these multiple scales of interactions. His approach has been to formulate a top-do wn global model appropriate to the scale of scalp EEG phenomena, which includes some important local mesocolumnar features. This approach has been sufficiently robust to o verlap with and enhance the understanding of other bottom-up approaches such as I ha ve described, wherein mesocolumnar models appropriate to the scale of STM are scaled up to global regions appropriate to EEG.... statistical mechanics ... -4- L ester Ingber In this paper ,Ihav efocussed on ho wbottom-up SMNI models can be de veloped into eigenfunction expansions of probability distributions appropriate to describe STM in the context of EEG. The mathematics of eigenfunctions are similar to the top-down eigenfunctions de veloped by PN, albeit the y have different physical manifestations. The bottom-up eigenfunctions are at the local mesocolumnar scale, whereas the top-down eigenfunctions are at the global regional scale. Future studies may e xpand top-down eigenfunctions into the bottom-up eigenfunctions, yielding a model of scalp EEG that is ultimately e xpressed in terms of columnar states of neocortical processing of attention and short-term memory. REFERENCES Graham, R. (1977) Co variant formulation of non-equilibrium statistical thermodynamics. Z. Physik B26:397-405. Ingber,L.(1981) Tow ards a unified brain theory. J. Social Biol. Struct. 4:211-224. Ingber,L.(1982) Statistical mechanics of neocortical interactions. I. Basic formulation. Physica D 5:83-107. [URL http://www.ingber.com/smni82_basic.ps.gz] Ingber,L.(1983) Statistical mechanics of neocortical interactions. Dynamics of synaptic modification. Phys. Rev. A28:395-416. [URL http://www.ingber.com/smni83_dynamics.ps.gz] Ingber,L.(1984) Statistical mechanics of neocortical interactions. Deri vation of short-term-memory capacity. Phys. Rev. A29:3346-3358. [URL http://www.ingber.com/smni84_stm.ps.gz] Ingber,L.(1985a) Statistical mechanics of neocortical interactions. EEG dispersion relations. IEEE Tr ans. Biomed. Eng. 32:91-94. [URL http://www.ingber.com/smni85_eeg.ps.gz] Ingber,L.(1985b) Statistical mechanics of neocortical interactions: Stability and duration of the 7+−2 rule of short-term-memory capacity .Phys. Re v. A 31:1183-1186. [URL http://www.ingber.com/smni85_stm.ps.gz] Ingber,L.(1986) Statistical mechanics of neocortical interactions. Bull. Am. Phys. Soc. 31:868. Ingber,L.(1988) Mesoscales in neocorte xand in command, control and communications (C3)systems, In:Systems with Learning and Memory Abilities: Proceedings, University of Paris 15-19 J une 1987,ed. J. Delacour & J.C.S. Levy .Elsevier, 387-409. Ingber,L.(1991) Statistical mechanics of neocortical interactions: A scaling paradigm applied to electroencephalograph y. Phys. Re v. A 44:4017-4060. 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[URL http://www.ingber.com/smni95_stm40hz.ps.gz] Ingber,L.(1996) Statistical mechanics of neocortical interactions: Multiple scales of EEG, In: Fr ontier Science in EEG: Continuous Waveform Analysis (Electr oencephal. clin. Neurophysiol. Suppl. 45) , ed. R.M. Dasheif f&D.J. Vincent. Else vier, 79-112. [Invited talk to Frontier Science in EEG Symposium, Ne wOrleans, 9 Oct 1993. ISBN 0-444-82429-4. URL http://www.ingber.com/smni96_eeg.ps.gz] Ingber,L.(1997) Statistical mechanics of neocortical interactions: Applications of canonical momenta indicators to electroencephalograph y.Phys. Re v. E 55:4578-4593. [URL http://www.ingber.com/smni97_cmi.ps.gz]... statistical mechanics ... -5- L ester Ingber Ingber,L.(1998) Statistical mechanics of neocortical interactions: T raining and testing canonical momenta indicators of EEG. Mathl. Computer Modelling 27:33-64. [URL http://www.ingber.com/smni98_cmi_test.ps.gz] Ingber,L .&N unez, P.L. 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(1995) Ev ent related potentials during object recognition tasks. Brain Res. Bull. 38:531-538.

arXiv:physics/0001051 23 Jan 2000Canonical Momenta Indicators of Financial Markets and Neocortical EEG Lester Ingber Lester Ingber Research P.O. Box 857, McLean, Virginia 22101, U.S.A. ingber@ingber.com, ingber@alumni.caltech.edu Abstract—A paradigm of statistical mechanics of financial mark ets (SMFM) is fit to multi variate financial markets using Adapti ve Simulated Annealing (ASA), a global optimization algorithm, to perform maximum likelihood fits of Lagrangians defined by path integrals of multi variate conditional pr obabilities. Canonical momenta ar ethereby derivedand used as technical indicators in a r ecursive ASA optimization process to tune trading rules. These trading rules ar ethen used on out-of-sample data, to demonstrate that they can profit from the SMFM model, to illustrate that these markets ar elikely not efficient. This methodology can be extended to other systems, e.g., electr oencephalograph y. This approach to complex systems emphasizes the utility of blending an intuiti ve and powerful mathematical-physics formalism to generate indicators which ar e used by AI-type rule-based models of management. 1. Introduction Over a decade ago, the author published a paper suggesting the use of ne wly developed methods of multivariate nonlinear nonequilibrium calculus to approach a statistical mechanics of financial mark ets (SMFM) [1]. These methods were applied to interest-rate term-structure systems [2,3]. Still, for some time, the standard accepted paradigm of financial markets has been rooted in equilibrium processes [4]. There is a current effort by man yto examine nonlinear and nonequilibrium processes in these mark ets [5],and this paper reinforces this point of vie w. Another paper gi vessome earlier 1991 results using this approach [6]. There are se veral issues that are clarified here, by presenting calculations of a specific trading model: (A) It is demonstrated ho wmultivariate markets might be formulated in a nonequilibrium paradigm. (B) It is demonstrated that numerical methods of global optimization can be used to fit such SMFM models to data. (C) A v ariational principle possessed by SMFM permits deri vation of technical indicators, such as canonical momenta, that can be used to describe deviations from most lik ely evolving states of the multi variate system. (D) These technical indicators can be embedded in realistic trading scenarios, to test whether the ycan profit from nonequilibrium in markets. Section 2 outlines the formalism used to de velop the nonlinear nonequilibrium SMFM model. Section 3 describes application of SMFM to SP500 cash and future data, using Adapti ve Simulated Annealing (ASA) [7] to fit the short-time conditional probabilities de veloped in Section 2, and to establish trading rules by recursi vely optimizing with ASA, using optimized technical indicators de veloped from SMFM. These calculations were briefly mentioned in another ASA paper [8]. Section 4describes similar applications, no wi nprogress, to correlating customized electroencephalographic (EEG) momenta indicators to physiological and behavioral states of humans. Section 5 is a brief conclusion. 2. SMFM Model 2.1. Random walk model The use of Brownian motion as a model for financial systems is generally attrib uted to Bachelier [9], though he incorrectly intuited that the noise scaled linearly instead of as the square root relati ve tothe random log-price variable. Einstein is generally credited with using the correct mathematical description in a lar ger physical conte xt of statistical systems. However, sev eral studies imply that changing prices of man ymarkets do not follo warandom walk, that the ymay have long-term dependences in price correlations, and that the ymay not be efficient in quickly arbitraging ne winformation [10-12]. Arandom w alk for returns, rate of change of prices o verprices, is described by a Langevin equation with simple additi ve noiseη,typically representing the continual random influx of information into the market. ˙Γ= −γ1+γ2η, ˙Γ=dΓ/dt, <η(t)>η=0,<η(t),η(t′)>η=δ(t−t′), (1) whereγ1andγ2are constants, and Γis the logarithm of (scaled) price. Price, although the most dramatic observable, may not be the only appropriate dependent v ariable or order parameter for the system of mark ets [13]. This possibility has also been called the “semistrong form of the efficient market hypothesis” [10]. It is necessary to e xplore the possibilities that a gi venmarket evolves in nonequilibrium, e.g., e volving irreversibly,aswell as nonlinearly ,e.g.,γ1,2may be functions of Γ.Irreversibility,e.g., causality[14] and nonlinearity [15], ha ve been suggested as processes necessary to tak einto account in order to understand mark ets, butmodern methods of statistical mechanics no wprovide a more explicit paradigm to consistently include these processes in bona fide probability distrib utions. Reserv ations have been expressed about these earlier models at theCanonical momenta indicators of financial mark ets - 2- L ester Ingber time of their presentation [16]. Developments in nonlinear nonequilibrium statistical mechanics in the late 1970’ sand their application to a variety of testable ph ysical phenomena illustrate the importance of properly treating nonlinearities and nonequilibrium in systems where simpler analyses prototypical of linear equilibrium Bro wnian motion do not suffice [17]. 2.2. Statistical mechanics of large systems Aggregation problems in nonlinear nonequilibrium systems, e.g., as defines a mark et composed of man y traders [1], typically are “solved” (accommodated) by having ne wentities/languages de veloped at these disparate scales in order to efficiently pass information back and forth [18,19]. This is quite different from the nature of quasi- equilibrium quasi-linear systems, where thermodynamic or c ybernetic approaches are possible. These approaches typically fail for nonequilibrium nonlinear systems. These newmethods of nonlinear statistical mechanics only recently ha ve been applied to comple xlarge-scale physical problems, demonstrating that observ ed data can be described by the use of these algebraic functional forms. Success was gained for large-scale systems in neuroscience, in a series of papers on statistical mechanics of neocortical interactions [20-30], and in nuclear ph ysics [31-33]. This methodology has been used for problems in combat analyses [19,34-37]. These methods ha ve been suggested for financial mark ets [1],applied to a term structure model of interest rates [2,3], and to optimization of trading [6]. 2.3. Statistical development When other order parameters in addition to price are included to study mark ets, Eq. (1) is accordingly generalized to a set of Langevin equations. ˙MG=fG+ˆgG jηj,(G=1,...,Λ),(j=1,...,N), ˙MG=dMG/dΘ, <ηj(Θ)>η=0,<ηj(Θ),ηj′(Θ′)>η=δjj′δ(Θ−Θ′), (2) wherefGandˆgG jare generally nonlinear functions of mesoscopic order parameters MG,jis a microscopic inde x indicating the source of fluctuations, and N≥Λ.The Einstein con vention of summing o verrepeated indices is used. Vertical bars on an inde x, e.g., |j|, imply no sum is to be taken on repeated indices. Θis used here to emphasize that the most appropriate time scale for trading may not be real time t. Viaasomewhat length y, albeit instructi ve calculation, outlined in se veral other papers [1,3,25], in volving an intermediate deri vation of a corresponding F okker-Planck or Schr ¨odinger-type equation for the conditional probability distrib utionP[M(Θ)|M(Θ0)],the Langevin rate Eq. (2) is de veloped into the probability distrib ution forMGat long-time macroscopic time e ventΘ=(u+1)θ+Θ0,i nterms of a Stratonovich path-integral o ver mesoscopic Gaussian conditional probabilities [38-40]. Here, macroscopic variables are defined as the long-time limit of the e volving mesoscopic system. The corresponding Schr ¨odinger-type equation is [39,41] ∂P/∂Θ =1 2(gGG′P),GG′−(gGP),G+V, gGG′=kTδjkˆgG jˆgG′ k,gG=fG+1 2δjkˆgG′ jˆgG k,G′, [...],G=∂[...]/∂MG. (3) This is properly referred to as a F okker-Planck equation when V≡0.Note that although the partial differential Eq. (3) contains equi valent information re gardingMGas in the stochastic differential Eq. (2), all references to jhave been properly a veraged over. I.e.,ˆgG jin Eq. (2) is an entity with parameters in both microscopic and mesoscopic spaces, but Mis a purely mesoscopic variable, and this is more clearly reflected in Eq. (3). The path integral representation is gi veninterms of the Lagrangian L. P[MΘ|MΘ0]dM(Θ)=∫...∫DMexp(−S)δ[M(Θ0)=M0]δ[M(Θ)=MΘ], S=k−1 TminΘ Θ0∫dΘ′L,Canonical momenta indicators of financial mark ets - 3- L ester Ingber DM= u→∞limu+1 ρ=1Πg1/2 GΠ(2πθ)−1/2dMG ρ, L(˙MG,MG,Θ)=1 2(˙MG−hG)gGG′(˙MG′−hG′)+1 2hG ;G+R/6−V, hG=gG−1 2g−1/2(g1/2gGG′),G′, gGG′=(gGG′)−1,g=det(gGG′), hG ;G=hG ,G+ΓF GFhG=g−1/2(g1/2hG),G, ΓF JK≡gLF[JK,L]=gLF(gJL,K+gKL,J−gJK,L), R=gJLRJL=gJLgJKRFJKL, RFJKL=1 2(gFK,JL−gJK,FL−gFL,JK+gJL,FK)+gMN(ΓM FKΓN JL−ΓM FLΓN JK). (4) Mesoscopic variables ha ve been defined as MGin the Langevin and F okker-Planck representations, in terms of their development from the microscopic system labeled by j.The Riemannian curvature term Rarises from nonlinear gGG′,which is a bona fide metric of this parameter space [39]. 2.4. Algebraic complexity yields simple intuiti ve results It must be emphasized that the output need not be confined to comple xalgebraic forms or tables of numbers. BecauseLpossesses a variational principle, sets of contour graphs, at dif ferent long-time epochs of the path-inte gral ofPoverits variables at all intermediate times, gi ve a visually intuiti ve and accurate decision-aid to vie wthe dynamic e volution of the scenario. Forexample, this Lagrangian approach permits a quantitati ve assessment of concepts usually only loosely defined. “Momentum” =ΠG=∂L ∂(∂MG/∂Θ), “Mass”gGG′=∂2L ∂(∂MG/∂Θ)∂(∂MG′/∂Θ), “Force”=∂L ∂MG, “F=ma ”:δL=0=∂L ∂MG−∂ ∂Θ∂L ∂(∂MG/∂Θ), (5) whereMGare the variables and Lis the Lagrangian. These physical entities pro vide another form of intuiti ve,but quantitati vely precise, presentation of these analyses. Forexample, daily newspapers use this terminology to discuss the movement of security prices. Here, we will use the canonical momenta as indicators to de velop trading rules. 2.5. Fitting parameters The short-time path-integral Lagrangian of a Λ-dimensional system can be de veloped into a scalar “dynamic cost function,” C,interms of parameters, e.g., generically represented as C(˜α), C(˜α)=LΔΘ +Λ 2ln(2πΔΘ)−1 2lng, (6) which can be used with the ASA algorithm [7], originally called Very Fast Simulated Reannealing (VFSR)[42], to find the (statistically) best fit of parameters. The cost function for a gi vensystem is obtained by the product of P’s overall data epochs, i.e., a sum of C’s isobtained. Then, since we essentially are performing a maximum lik elihood fit, the cost functions obtained from somewhat different theories or data can provide a relati ve statistical measure of their likelihood, e.g., P12∼exp(C2−C1). If there are competing mathematical forms, then it is advantageous to utilize the path-inte gral to calculate the long-time e volution ofP[19,35]. Experience has demonstrated that the long-time correlations deri vedfrom theory ,Canonical momenta indicators of financial mark ets - 4- L ester Ingber measured against the observ ed data, is a viable and expedient way of rejecting models not in accord with observ ed evidence. 2.6. Numerical methodology ASA [42] fits short-time probability distributions to observed data, using a maximum likelihood technique on the Lagrangian. This algorithm has been de veloped to fit observ ed data to a theoretical cost function o veraD- dimensional parameter space [42], adapting for varying sensitivities of parameters during the fit. Simulated annealing (SA) was de veloped in 1983 to deal with highly nonlinear problems [43], as an extension of a Monte-Carlo importance-sampling technique de veloped in 1953 for chemical physics problems. It helps to visualize the problems presented by such comple xsystems as a geographical terrain. Forexample, consider amountain range, with tw o“parameters, ”e.g., along the North−South and East−West directions. We wish to find the lowest v alleyinthis terrain. SA approaches this problem similar to using a bouncing ball that can bounce o ver mountains from v alleyt ovalley. Westart at a high “temperature, ”where the temperature is an SA parameter that mimics the effect of a fast moving particle in a hot object lik eahot molten metal, thereby permitting the ball to makevery high bounces and being able to bounce o verany mountain to access an yvalley, giv enenough bounces. As the temperature is made relati vely colder,the ball cannot bounce so high, and it also can settle to become trapped in relatively smaller ranges of valleys. We imagine that our mountain range is aptly described by a “cost function. ”Wedefine probability distributions of the tw odirectional parameters, called generating distributions since the ygenerate possible v alleys or states we are to e xplore. W edefine another distribution, called the acceptance distribution, which depends on the difference of cost functions of the present generated v alleyw eare to explore and the last sa vedlowest valley. The acceptance distribution decides probabilistically whether to stay in a ne wlower valleyo rt obounce out of it. All the generating and acceptance distributions depend on temperatures. In 1984[44], it w as established that SA possessed a proof that, by carefully controlling the rates of cooling of temperatures, it could statistically find the best minimum, e.g., the lowest v alleyo four example abo ve.This was good news for people trying to solv ehard problems which could not be solv ed by other algorithms. The bad ne ws wasthat the guarantee was only good if the ywere willing to run SA fore ver. In1987, a method of fast annealing (FA) was de veloped [45], which permitted lowering the temperature exponentially f aster,thereby statistically guaranteeing that the minimum could be found in some finite time. However, that time still could be quite long. Shortly thereafter ,i n1987 the author de veloped Very Fast Simulated Reannealing (VFSR)[42], no wcalled Adapti ve Simulated Annealing (ASA), which is exponentially faster than F A. Itis used world-wide across man y disciplines [8], and the feedback of man yusers regularly scrutinizing the source code ensures the soundness of the code as it becomes more flexible and powerful [46]. ASA has been applied to man yproblems by man ypeople in man ydisciplines [8,46,47]. The code is available via anonymous ftp from ftp.ingber .com, which also can be accessed via the world-wide web (WWW) as http://www.ingber.com/. 3. Fitting SMFM to SP500 3.1. Data processing Forthe purposes of this paper ,i tsuffices to consider a tw o-variable problem, SP500 prices of futures, p1,and cash,p2.(Note that in a previous paper [6], these tw ovariables were inadvertently incorrectly re versed.) Data included 251 points of 1989 and 252 points of 1990 daily closing data. Time between data was taken as real time t, e.g., a weekend added tw odays to the time between data of a Monday and a previous Friday. It was decided that relati ve data should be more important to the dynamics of the SMFM model than absolute data, and an arbitrary form was de veloped to preprocess data used in the fits, Mi(t)=pi(t+Δt)/pi(t), (7) wherei={1, 2}={futures, cash },andΔtwasthe time between neighboring data points, and t+Δtis the current trading time. The ratio served to served to suppress strong drifts in the absolute data. 3.2. ASA fits of SMFM to data Tw osource of noise were assumed, so that the equations of this SMFM model are dMG dt=2 G′=1ΣfG G′MG′+2 i=1ΣˆgG iηi,G={1, 2}. (8) The 8 parameters, {fG G′,ˆgG i}were all taken to be constants. As discussed pre viously,the path-integral representation w as used to define an ef fective cost function. Minimization of the cost function was performed using ASA. Some experimentation with the fitting process led to a scheme whereby after sufficient importance-sampling, the optimization w as shunted o vert oaquasi-local code, theCanonical momenta indicators of financial mark ets - 5- L ester Ingber Broyden-Fletcher-Goldf arb-Shanno (BFGS) algorithm[48], to add another decimal of precision. If ASA w as shunted o vertoo quickly to BFGS, then poor fits were obtained, i.e., the fit stopped in a higher local minimum. Using 1989 data, the parameters fG G′were constrained to lie between -1.0 and 1.0. The parameters ˆgG iwere constrained to lie between 0 and 1.0. The values of the parameters, obtained by this fitting process were: f1 1= 0.0686821, f1 2=−0.068713, ˆg1 1=0.000122309, ˆg1 2=0.000224755, f2 1=0.645019, f2 2=−0.645172, ˆg2 1= 0.00209127, ˆg2 2=0.00122221. 3.3. ASA fits of trading rules Asimple model of trading was de veloped. T wo time-weighted moving a verages, of wide and narro w windows, awandanwere defined for each of the tw omomenta variables. During each newepoch ofaw,always using the fits of the SMFM model described in the pre vious section as a zeroth order estimate, the parameters {fG G′,ˆgG i}were refit using data within each epoch. Av eraged canonical momenta, i.e., using Eq. (5), were calculated for each ne wset ofawandanwindows. Fluctuation parameters ΔΠG wandΔΠG n,were defined, such that anychange in trading position required that there w as some reasonable information outside of these fluctuations that could be used as criteria for trading decisions. No trading was performed for the first fe wdays of the year until the momenta could be calculated. Commissions of $70 were paid e very time a ne wtrade of 100 units was tak en. Thus, there were 6 trading parameters used in this example, {aw,an,ΔΠG w,ΔΠG n}. The order of choices made for daily trading are as follo ws. A0represents no positions are open and no trading is performed until enough data is g athered, e.g., to calculate momenta. A1represents entering a long position, whether from a w aiting or a short position, or a current long position was maintained. This was performed if the both wide-windo wand narrow-windowaveraged momenta of both cash and futures prices were both greater than theirΔΠG wandΔΠG nfluctuation parameters. A− 1represents entering a short position, whether from a w aiting or a long position, or a current short position was maintained. This was performed if the both wide-windo wand narrow-windo waveraged momenta of both cash and futures prices were both less than their ΔΠG wandΔΠG nfluctuation parameters. 3.4. In-sample ASA fits of trading rules Forthe data of 1989, recursi ve optimization was performed. The trading parameters were optimized in an outer shell, using the ne gative ofthe net yearly profit/loss as a cost function. This could ha ve been weighted by something lik ethe absolute value of maximum loss to help minimize risk, but this was not done here. The inner shell of optimization fine-tuning of the SMFM model was performed daily o verthe current awepoch. At first, ASA and shunting o vert oBFGS was used for each shell, but it was realized that good results could be obtained using ASA and BFGS on the outer shell, and just BFGS on the inner shell (al ways using the ASA and BFGS deri vedzeroth order SMFM parameters as described abo ve). Thus,recursive optimization was performed to establish the required goodness-of-fit, and more efficient local optimization was used only in those instances where it could replicate the global optimization. This is expected to be quite system dependent. The trading-rule parameters were constrained to lie within the following ranges: awintegers between 15 and 25,anintegers between 3 and 14, ΔΠG wandΔΠG nbetween 0 and 200. The trading parameters fit by this procedure were:aw=18,an=11,ΔΠ1 w=30.3474,ΔΠ2 w=98.0307,ΔΠ1 n=11.2855,ΔΠ2 n=54.8492. The summary of results w as: cumulati ve profit = $54170, number of profitable long positions = 11, number of profitable short positions = 8, number of losing long positions = 5, number of losing short positions = 6, maximum profit of an ygiv entrade = $11005, maximum loss of an ytrade = −$2545, maximum accumulated profit during year =$54170, maximum loss sustained during year = $0. 3.5. Out-of-sample SMFM trading The trading process described abo ve was applied to the 1990 out-of-sample SP500 data. Note that 1990 w as a“bear” market, while 1989 was a “b ull” mark et. Thus, these twoyears had quite different o verall contexts, and this wasbelievedtoprovide a stronger test of this methodology than picking tw oyears with similar contexts. The inner shell of optimization was performed as described abo ve for 1990 as well. The summary of results was: cumulati ve profit = $28300, number of profitable long positions = 10, number of profitable short positions = 6, number of losing long positions = 6, number of losing short positions = 10, maximum profit of an ygiv entrade = $6780, maximum loss of an ytrade = −$2450, maximum accumulated profit during year = $29965, maximum loss sustained during year = −$5945. Tables of results are a vailable as file markets96_momenta_tbl.txt.Z in http://www.ingber.com/MISC.DIR/ and ftp.ingber.com/MISC.DIR. Only one variable, the futures SP500, was actually traded, albeit the code can accommodate trading on multiple mark ets. There is more le verage and liquidity in actually trading the futures mark et. Themultivariable coupling to the cash market entered in three important ways: (1) The SMFM fits were to the coupled system, requiring a global optimization of all parameters in both markets to define the time e volution of the futures mark et. (2) The canonical momenta for the futures mark et is in terms of the partial deri vative ofthe full Lagrangian; the dependenc yo nthe cash market enters both as a function of the relati ve value of the off-diagonal to diagonal terms inCanonical momenta indicators of financial mark ets - 6- L ester Ingber the metric, as well as a contribution to the drifts and diffusions from this mark et. (3)The canonical momenta of both markets were used as technical indicators for trading the futures market. 3.6. Reversing data sets The same procedures described abo ve were repeated, but using the 1990 SP500 data set for training and the 1989 data set for testing. Forthe training phase, using 1990 data, the parameters fG G′were constrained to lie between -1.0 and 1.0. The parameters ˆgG iwere constrained to lie between 0 and 1.0. The values of the parameters, obtained by this fitting process were: f1 1=0.0685466, f1 2=−0.068571, ˆg1 1=7.5236810−6,ˆg1 2=0.000274467, f2 1=0.642585,f2 2= −0.642732, ˆg2 1=9.3076810−5,ˆg2 2=0.00265532. Note that these values are quite close to those obtained abo ve when fitting the 1989 data. The trading-rule parameters were constrained to lie within the following ranges: awintegers between 15 and 25,anintegers between 3 and 14, ΔΠG wandΔΠG nbetween 0 and 200. The trading parameters fit by this procedure were:aw=11,an=8,ΔΠ1 w=23.2324,ΔΠ2 w=135.212,ΔΠ1 n=169.512,ΔΠ2 n=9.50857, The summary of results w as: cumulati ve profit = $42405, number of profitable long positions = 11, number of profitable short positions = 8, number of losing long positions = 7, number of losing short positions = 6, maximum profit of an ygiv entrade = $8280, maximum loss of an ytrade = −$1895, maximum accumulated profit during year = $47605, maximum loss sustained during year = −$2915. Forthe testing phase, the summary of results was: cumulati ve profit = $35790, number of profitable long positions = 10, number of profitable short positions = 6, number of losing long positions = 6, number of losing short positions = 3, maximum profit of an ygiv entrade = $9780, maximum loss of an ytrade = −$4270, maximum accumulated profit during year = $35790, maximum loss sustained during year = $0. Tables of results are a vailable as file markets96_momenta_tbl.txt.Z in http://www.ingber.com/MISC.DIR/ and ftp.ingber.com/MISC.DIR. 4. Extrapolations to EEG 4.1. Customized Momenta Indicators of EEG These techniques are quite generic, and can be applied to a model of statistical mechanics of neocortical interactions (SMNI) which has utilized similar mathematical and numerical algorithms [20-23,25,26,29,30,49]. In this approach, the SMNI model is fit to EEG data, e.g., as pre viously performed [25]. This develops a zeroth order guess for SMNI parameters for a gi vensubject’straining data. Next, ASA is used recursi vely to seek parameterized predictor rules, e.g., modeled according to guidelines used by clinicians. The parameterized predictor rules form an outer ASA shell, while regularly fine-tuning the SMNI inner -shell parameters within a moving windo w(one of the outer-shell parameters). The outer-shell cost function is defined as some measure of successful predictions of upcoming EEG e vents. In the testing phase, the outer-shell parameters fit in the training phase are used in out-of-sample data. Again, the process of regularly fine-tuning the inner-shell of SMNI parameters is used in this phase. If these SMNI techniques can find patterns of such such upcoming activity some time before the trained e ye of the clinician, then the costs of time and pain in preparation for sur gery can be reduced. This project will determine inter-electrode and intra-electrode activities prior to spik eactivity to determine lik ely electrode circuitries highly correlated to the onset of seizures. This can only do better than simple a veraging or filtering of such acti vity, as typically used as input to determine dipole locations of activity prior to the onset of seizures. If a subset of electrode circuitries are determined to be highly correlated to the onset of seizures, then their associated regions of acti vity can be used as a first approximate of underlying dipole sources of brain acti vity affecting seizures. This first approximate may be better than using a spherical head model to deduce such a first guess. Such first approximates can then be used for more realistic dipole source modeling, including the actual shape of the brain surface to determine likely localized areas of diseased tissue. These momenta indicators should be considered as supplemental to other clinical indicators. This is ho wthey are being used in financial trading systems. 5. Conclusion Acomplete sample scenario has been presented: (a) de veloping a multi variate nonlinear nonequilibrium model of financial markets; (b) fitting the model to data using methods of ASA global optimization; (c) deri ving technical indicators to express dynamics about most lik ely states; (d) optimizing trading rules using these technical indicators; (e) trading on out-of-sample data to determine if steps (a)−(d) are at least sufficient to profit by the knowledge gained of these financial markets, i.e., these markets are not efficient. Just based the models and representati ve calculations presented here, no comparisons can yet be made of an y relative superiority of these techniques o verother models of mark ets and other sets of trading rules. Rather,this exercise should be viewed as an explicit demonstration (1) that financial markets can be modeled as nonlinear nonequilibrium systems, and (2) that financial markets are not efficient and that the ycan be properly fit andCanonical momenta indicators of financial mark ets - 7- L ester Ingber profitably traded on real data. Canonical momenta may offer an intuiti ve yet detailed coordinate system of some comple xsystems, which can be used as reasonable indicators of ne wand/or strong trends of beha vior,upon which reasonable decisions and actions can be based. Adescription has been gi veno faproject in progress, using this same methodology to customize canonical momenta indicators of EEG to human behavioral and physiological states [50]. References [1] L.Ingber,“Statistical mechanics of nonlinear nonequilibrium financial mark ets,”Math. Modelling 5(6), pp. 343-361, 1984. [2] L. Ingber,“Statistical mechanical aids to calculating term structure models, ”Phys. Re v. 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Phys. 13,pp. 1723-1726, 1972. [39] R.Graham, “P ath-integral methods on nonequilibrium thermodynamics and statistics, ”inStochastic Processes in Nonequilibrium Systems ,(Edited by L. Garrido, P .Seglar and P.J. Shepherd), pp. 82-138, Springer ,New York, NY,1978. [40] F.Langouche, D. Roekaerts, and E. T irapegui, “Short deri vation of Feynman Lagrangian for general dif fusion process,” J. Phys. A113,pp. 449-452, 1980. [41] F.Langouche, D. Roekaerts, and E. T irapegui, “Discretization problems of functional integrals in phase space,”Phys. Rev. D20,pp. 419-432, 1979. [42] L.Ingber,“Very fast simulated re-annealing,” Mathl. Comput. Modelling 12(8), pp. 967-973, 1989. [43] S. Kirkpatrick, C.D. Gelatt, Jr., and M.P .Vecchi, “Optimization by simulated annealing, ”Science 220(4598), pp. 671-680, 1983. [44] S.Geman and D. Geman, “Stochastic relaxation, Gibbs distribution and the Bayesian restoration in images, ” IEEE Trans. Patt. Anal. Mac. Int. 6(6), pp. 721-741, 1984. [45] H.Szu and R. Hartle y, “Fast simulated annealing,” Phys. Lett. A 122(3-4), pp. 157-162, 1987. [46] L.Ingber,“Adaptive simulated annealing (ASA): Lessons learned, ”Control and Cybernetics 25(1), pp.(to be published), 1996. [47] M. Wofsey, “Technology: Shortcut tests v alidity of complicated formulas, ”The Wall Street J ournal 222(60), pp. B1, 1993. [48] D.F .Shanno and K.H. Phua, “Minimization of unconstrained multi variate functions, ”ACMTrans. Mathl. Software 2,pp. 87-94, 1976. [49] L. Ingber,“Statistical mechanics of neocortical interactions. I. Basic formulation, ”Physica D 5,pp. 83-107, 1982. [50] L. Ingber,“Canonical momenta indicators of neocortical EEG, ”inPhysics Computing 96 (PC96) ,PC96, Krakow,Poland, 1996.

arXiv:physics/0001052 23 Jan 2000Statistical mechanics of neocortical interactions: Canonical momenta indicators of electroencephalography Lester Ingber Lester Ingber Resear ch,P.O. Box 857, McLean, V A22101 ingber@ingber.com, ingber@alumni.caltech.edu Aseries of papers has de veloped a statistical mechanics of neocortical interactions (SMNI), deri ving aggregate behavior of experimentally observ ed columns of neurons from statistical electrical-chemical properties of synaptic interactions. While not useful to yield insights at the single neuron le vel, SMNI has demonstrated its capability in describing large-scale properties of short-term memory and electroencephalographic (EEG) systematics. The necessity of including nonlinear and stochastic structures in this de velopment has been stressed. Sets of EEG and e vokedpotential data were fit, collected to in vestigate genetic predispositions to alcoholism and to extract brain “signatures” of short- term memory .Adaptive Simulated Annealing (ASA), a global optimization algorithm, was used to perform maximum likelihood fits of Lagrangians defined by path inte grals of multi variate conditional probabilities. Canonical momenta indicators (CMI) are thereby deri vedfor individual’sEEG data. The CMI give better signal recognition than the ra wdata, and can be used to advantage as correlates of behavioral states. These results gi ve strong quantitati ve support for an accurate intuiti ve picture, portraying neocortical interactions as having common algebraic or ph ysics mechanisms that scale across quite disparate spatial scales and functional or beha vioral phenomena, i.e., describing interactions among neurons, columns of neurons, and regional masses of neurons. PA CSNos.: 87.10.+e, 05.40.+j, 02.50.-r ,02.70.-cStatistical Mechanics of Neocortical ... -2- Lester Ingber 1. INTRODUCTION Amodel of statistical mechanics of neocortical interactions (SMNI) has been de veloped [1-20], describing large-scale neocortical acti vity on scales of mm to cm as measured by scalp EEG, with an audit trail back to minicolumnar interactions among neurons. There are se veral aspects of this modeling that should be further in vestigated: to explore the robustness of the model, the range of e xperimental paradigms to which it is applicable, and further de velopment that can increase its spatial resolution of EEG. The underlying mathematical physics used to de velop SMNI gi vesrise to a natural coordinate system faithful to nonlinear multi variate sets of potential data such as measured by multi-electrode EEG, canonical momenta indicators (CMI) [20-22]. Recent papers in finance [21,22] and in EEG systems [20] have demonstrated that CMI gi ve enhanced signal resolutions o verraw data. The basic philosoph yofSMNI is that good physical models of comple xsystems, often detailed by variables not directly measurable in man yexperimental paradigms, should of fer superior descriptions of empirical data beyond that a vailable from black-box statistical descriptions of such data. Forexample, good nonlinear models often of fer sound approaches to relati vely deeper understandings of these systems in terms of synergies of subsystems at finer spatial-temporal scales. In this context, a generic mesoscopic neural network (MNN) has been de veloped for dif fusion-type systems using a confluence of techniques dra wn from the SMNI, modern methods of functional stochastic calculus defining nonlinear Lagrangians, adapti ve simulated annealing (ASA) [23], and parallel- processing computation, to de velop a generic nonlinear stochastic mesoscopic neural netw ork (MNN) [14,19]. MNN increases the resolution of SMNI to minicolumnar interactions within and between neocortical regions, a scale that o verlaps with other studies of neural systems, e.g., artificial neural networks (ANN). In order to interface the algebra presented by SMNI with experimental data, se veral codes ha ve been developed. A keytool is adapti ve simulated annealing (ASA), a global optimization C-language code [23-27]. Ov er the years, this code has e volved to a high degree of robustness across man y disciplines. Ho wever, there are o ver100 OPTIONS a vailable for tuning this code; this is as e xpected for anysingle global optimization code applicable to man yclasses of nonlinear systems, systems whichStatistical Mechanics of Neocortical ... -3- Lester Ingber typically are non-typical. Section 2 gi vesthe background used to de velop SMNI and ASA for the present study .Appendix A gi ves more detail on ASA rele vant to this paper .Section 3 gi vesthe mathematical de velopment required for this study .Section 4 describes the procedures used. Section 5 presents conclusions. 2. BACKGROUND 2.1. EEG The SMNI approach de velops mesoscopic scales of neuronal interactions at columnar le vels of hundreds of neurons from the statistical mechanics of relati vely microscopic interactions at neuronal and synaptic scales, poised to study relati vely macroscopic dynamics at regional scales as measured by scalp electroencephalograph y(EEG). Rele vant experimental confirmation is discussed in the SMNI papers at the mesoscopic scales as well as at macroscopic scales of scalp EEG. The derivedfirings of columnar activity,considered as order parameters of the mesoscopic system, de velop multiple attractors, which illuminate attractors that may be present in the macroscopic re gional dynamics of neocorte x. SMNI proposes that models to be fitted to data include models of activity under each electrode, e.g., due to short-ranged neuronal fibers, as well as models of acti vity across electrodes, e.g., due to long-ranged fibers. These influences can be disentangled by SMNI fits. The SMNI approach is complementary to other methods of studying nonlinear neocortical dynamics at macroscopic scales. Forexample, EEG and magnetoencephalograph y(MEG) data ha ve been expanded in aseries of spatial principal components, a Karhunen-Loe ve expansion. The coefficients in such expansions are identified as order parameters that characterize phase changes in cogniti ve studies [28,29] and epileptic seizures [30,31]. Ho wever, the SMNI CMI may be considered in a similar context, as providing a natural coordinate system that can be sensiti ve toexperimental data, without assuming av erages overstochastic parts of the system that may contain important information. Theoretical studies of the neocortical medium ha ve inv olved local circuits with postsynaptic potential delays [32-35], global studies in which finite velocity of action potential and periodic boundary conditions are important [36-39], and nonlinear nonequilibrium SMNI. The local and the global theories combineStatistical Mechanics of Neocortical ... -4- Lester Ingber naturally to form a single theory in which control parameters effect changes between more local and more global dynamic beha vior [39,40], in a manner somewhat analogous to localized and e xtended w av e- function states in disordered solids. Plausible connections between the multiple-scale statistical theory and the more phenomenological global theory have been proposed [12]. Experimental studies of neocortical dynamics with EEG include maps of magnitude distribution o verthe scalp [37,41], standard Fourier analyses of EEG time series[37], and estimates of correlation dimension [42,43]. Other studies ha ve emphasized that man yEEG states are accurately described by a fe wcoherent spatial modes exhibiting comple xtemporal behavior [28-31,37,39]. These modes are the order parameters at macroscopic scales that underpin the phase changes associated with changes of physiological state. Forextracranial EEG, it is clear that spatial resolution, i.e., the ability to distinguish between tw odipole sources as their distance decreases, is dif ferent from dipole localization, i.e., the ability to locate a single dipole [39]. The development of methods to impro ve the spatial resolution of EEG has made it more practical to study spatial structure. Forexample, high resolution methods provide apparent spatial resolution in the 2-3 cm range [44]. Dipole resolution may be as good as se veral mm[45]. Some algorithms calculate the (generally non-unique) in verse-problem of determining cortical sources that are weighted/filtered by volume conductivities of concentric spheres encompassing the brain, cerebrospinal fluid, skull, and scalp. Astraightforw ard approach is to calculate the surface Laplacian from spline fits to the scalp potential distrib ution, yielding estimates similar to those obtained using concentric spheres models of the head [44]. Other measuring techniques, e.g., MEG, can provide complementary information. These methods ha ve their strengths and weaknesses at various spatial-temporal frequencies. These source localization methods typically do not include in their models the syner gistic contrib utions from short-ranged columnar firings of mm spatial extent and from long-ranged fibers spanning cm spatial extent. The CMI study presented here models these syner gistic short-ranged and long-ranged interactions. This is elaborated on in the conclusion. 2.2. Short-T erm Memory (STM) The development of SMNI in the context of short-term memory (STM) tasks leads naturally to the identification of measured electric scalp potentials as arising from e xcitatory and inhibitory short-rangedStatistical Mechanics of Neocortical ... -5- Lester Ingber and excitatory long-ranged fibers as the ycontribute to minicolumnar interactions [12,13]. Therefore, the SMNI CMI are most appropriately calculated in the context of STM experimental paradigms. It has been been demonstrated that EEG data from such paradigms can be fit using only physical synaptic and neuronal parameters that lie within experimentally observed ranges [13,20]. The SMNI calculations are of minicolumnar interactions among hundreds of neurons, within a macrocolumnar extent of hundreds of thousands of neurons. Such interactions tak eplace on time scales of severalτ,whereτis on the order of 10 msec (of the order of time constants of cortical p yramidal cells). This also is the observed time scale of the dynamics of STM. SMNI hypothesizes that columnar interactions within and/or between re gions containing man ymillions of neurons are responsible for these phenomena at time scales of se veral seconds. That is, the nonlinear e volution at finer temporal scales givesabase of support for the phenomena observ ed at the coarser temporal scales, e.g., by establishing mesoscopic attractors at man ymacrocolumnar spatial locations to process patterns in larger regions. SMNI has presented a model of STM, to the extent it offers stochastic bounds for this phenomena during focused selecti ve attention [4,6,15,46-48], transpiring on the order of tenths of a second to seconds, limited to the retention of 7 ±2items [49]. These constraints exist e venfor apparently e xceptional memory performers who, while the ymay be capable of more ef ficient encoding and retrie valo fSTM, and while the ymay be more efficient in ‘ ‘chunking’ ’larger patterns of information into single items, nevertheless are limited to a STM capacity of 7 ±2items [50]. Mechanisms for various STM phenomena have been proposed across man yspatial scales [51]. This “rule” is v erified for acoustical STM, as well as for visual or semantic STM, which typically require longer times for rehearsal in an h ypothesized articulatory loop of indi vidual items, with a capacity that appears to be limited to 4 ±2[52]. SMNI has detailed these constraints in models of auditory and visual corte x[4,6,15,16]. Another interesting phenomenon of STM capacity explained by SMNI is the primac yversus recenc y effect in STM serial processing[6], wherein first-learned items are recalled most error -free, with last- learned items still more error-free than those in the middle [53]. The basic assumption being made is that apattern of neuronal firing that persists for man yτcycles is a candidate to store the ‘ ‘memory’ ’of activity that g av erise to this pattern. If se veral firing patterns can simultaneously exist, then there is the capability of storing se veral memories. The short-time probability distribution deri vedfor the neocorte x is the primary tool to seek such firing patterns. The deepest minima are more likely accessed than theStatistical Mechanics of Neocortical ... -6- Lester Ingber others of this probability distribution, and these v alleys are sharper than the others. I.e., theyare more readily accessed and sustain their patterns ag ainst fluctuations more accurately than the others. The more recent memories or ne wer patterns may be presumed to be those having synaptic parameters more recently tuned and/or more acti vely rehearsed. It has been noted that e xperimental data on velocities of propagation of long-ranged fibers [37,39] and derivedvelocities of propag ation of information across local minicolumnar interactions[2] yield comparable times scales of interactions across minicolumns of tenths of a second. Therefore, such phenomena as STM likely are inextricably dependent on interactions at local and global scales. 2.2.1. SMNI &ADP Aproposal has been advanced that STM is processed by information coded in approximately 40-Hz (approximately 2.5 foldings of τ)bursts per stored memory ,permitting up to se vensuch memories to be processed serially within single w av eso flower frequencies on the order of 5 to 12 Hz [54]. Toaccount for the observed duration of STM, the ypropose that observed after-depolarization (ADP) at synaptic sites, affected by the action of relati vely long-time acting neuromodulators, e.g., acetylcholine and serotonin, acts to re gularly “refresh” the stored memories in subsequent oscillatory c ycles. A recent study of the action of neuromodulators in the neocorte xsupports the premise of their effects on broad spatial and temporal scales [55], b ut the ADP model is much more specific in its proposed spatial and temporal influences. SMNI does not detail an yspecific synaptic or neuronal mechanisms that might refresh these most-lik ely states to reinforce multiple short-term memories [18]. Ho wever, the calculated e volution of states is consistent with the observ ation that an oscillatory subcycle of 40 Hz may be the bare minimal threshold of self-sustaining minicolumnar firings before the ybegin to degrade [16]. The mechanism of ADP details a specific synaptic mechanism that, when coupled with additional proposals of neuronal oscillatory cycles of 5−12 Hz and oscillatory subcycles of 40 Hz, can sustain these memories for longer durations on the order of seconds. By itself, ADP does not provide a constraint such as the 7±2rule. The ADP approach does not address the observ ed random access phenomena of these memories, the 4 ±2rule, the primac yversus recenc yrule, or the influence of STM in observed EEG patterns.Statistical Mechanics of Neocortical ... -7- Lester Ingber SMNI and ADP models are complementary to the understanding of STM. MNN can be used to o verlap the spatial scales studied by the SMNI with the finer spatial scales typically studied by other relati vely more microscopic neural netw orks. At this scale, such models as ADP are candidates for providing an extended duration of firing patterns within the microscopic networks. 2.2.2. PATHINT Apath-integral C-language code, P ATHINT,calculates the long-time probability distribution from the Lagrangian, e.g., as fit by the ASA code. Arobust and accurate histogram-based (non-Monte Carlo) path- integral algorithm to calculate the long-time probability distrib ution had been de veloped to handle nonlinear Lagrangians [56-58], which was extended to two-dimensional problems [59]. PATHINT was developed for use in arbitrary dimensions, with additional code to handle general Neumann and Dirichlet conditions, as well as the possibility of including time-dependent potentials, drifts, and dif fusions. The results of using P ATHINT to determine the e volution of the attractors of STM gi ve overall results consistent with previous calculations [15,16]. 2.3. ASA In order to maintain some audit trail from large-scale regional activity back to mesoscopic columnar dynamics, desirable for both academic interest as well as practical signal enhancement, as fe w approximations as possible are made by SMNI in de veloping synaptic interactions up to the le velof regional activity as measured by scalp EEG. This presents a formidable multi variate nonlinear nonequilibrium distribution as a model of EEG dynamics, a concept considered to be quite tentati ve by research panels as late as 1990, until it was demonstrated ho wfits to EEG data could be implemented [13]. In order to fit such distributions to real data, ASA has been de veloped, a global optimization technique, a superior variant of simulated annealing [24]. This wastested using EEG data in 1991 [13], using an early and not as fle xible version of ASA, very fast reannealing (VFSR) [24]. Here, this is tested on more refined EEG using more sensiti ve CMI to portray results of the fits [20]. ASA [23] fits short-time probability distributions to observ ed data, using a maximum likelihood technique on the Lagrangian. This algorithm has been de veloped to fit observed data to a theoretical cost functionStatistical Mechanics of Neocortical ... -8- Lester Ingber overaD-dimensional parameter space[24], adapting for v arying sensitivities of parameters during the fit. Appendix A contains details of ASA rele vant to its use in this paper. 2.4. Complementary Research 2.4.1. Chaos Giventhe context of studies in comple xnonlinear systems [60], the question can be asked: What if EEG has chaotic mechanisms that o vershadowthe above stochastic considerations? The real issue is whether the scatter in data can be distinguished between being due to noise or chaos [61]. Inthis regard, several studies ha ve been proposed with re gard to comparing chaos to simple filtered (colored) noise [60,62]. Since the existence of multiplicati ve noise in neocortical interactions has been deri ved, then the pre vious references must be generalized, and further in vestigation is required to decide whether EEG scatter can be distinguished from multiplicati ve noise. Arecent study with realistic EEG w av eequations strongly suggests that if chaos exists in a deterministic limit, it does not survi ve inmacroscopic stochastic neocorte x[63]. I.e., it is important to include stochastic aspects, as arise from the statistics of synaptic and columnar interactions, in an yrealistic description of macroscopic neocortex. 2.4.2. Other Systems Experience using ASA on such multi variate nonlinear stochastic systems has been gained by similar applications of the approach used for SMNI. From 1986-1989, these methods of mathematical physics were utilized by a team of scientists and of ficers to develop mathematical comparisons of Janus computer combat simulations with e xercise data from the National Training Center (NTC), de veloping a testable theory of combat successfully baselined to empirical data [59,64-68]. This methodology has been applied to financial mark ets [21,69-71], de veloping specific trading rules for S&P 500 to demonstrate the robustness of these mathematical and numerical algorithms.Statistical Mechanics of Neocortical ... -9- Lester Ingber 3. MATHEMATICAL DEVELOPMENT Fitting a multi variate nonlinear stochastic model to data is a necessary ,but not sufficient procedure in developing newdiagnostic softw are. Even an accurate model fit well to real data may not be immediately useful to clinicians and experimental researchers. To fill this void, the powerful intuiti ve basis of the mathematical physics used to de velop SMNI has been utilized to describe the model in terms of rigorous CMI that provide an immediate intuiti ve portrait of the EEG data, faithfully describing the neocortical system being measured. The CMI gi ve anenhanced signal o verthe rawdata, and gi ve some insights into the underlying columnar interactions. 3.1. CMI, Information, Energy In the first SMNI papers, it w as noted that this approach permitted the calculation of a true nonlinear nonequilibrium “information” entity at columnar scales. With reference to a steady state P(˜M)for a short-time Gaussian-Mark ovian conditional probability distrib utionPof variables ˜M,when it exists, an analytic definition of the information g ainˆϒin state˜P(˜M)overthe entire neocortical volume is defined by [72,73] ˆϒ[˜P]=∫...∫D˜M˜Pln(˜P/P),DM=(2πˆg2 0Δt)−1/2u s=1Π(2πˆg2 sΔt)−1/2dMs,( 1) where a path inte gral is defined such that all intermediate-time values of ˜Mappearing in the folded short- time distrib utions˜Pare integrated o ver. This is quite general for an ysystem that can be described as Gaussian-Mark ovian [74], e veni fonly in the short-time limit, e.g., the SMNI theory. As time e volves, the distrib ution likely no longer beha vesi naG aussian manner ,and the apparent simplicity of the short-time distribution must be supplanted by numerical calculations. The Feynman Lagrangian is written in the midpoint discretization, for a specific macrocolumn corresponding to M(ts)=1 2[M(ts+1)+M(ts)] . (2) This discretization defines a co variant Lagrangian LFthat possesses a v ariational principle for arbitrary noise, and that e xplicitly portrays the underlying Riemannian geometry induced by the metric tensor gGG′, calculated to be the in verse of the co variance matrix gGG′.Using the Einstein summation con vention,Statistical Mechanics of Neocortical ... -10- Lester Ingber P=∫...∫DMexp −u s=0ΣΔtLFs , DM=g1/2 0+(2πΔt)−Θ/2u s=1Πg1/2 s+Θ G=1Π(2πΔt)−1/2dMG s, ∫dMG s→NG ι=1ΣΔMG ιs,MG 0=MG t0,MG u+1=MG t, LF=1 2(dMG/dt−hG)gGG′(dMG′/dt−hG′)+1 2hG ;G+R/6−V, (...),G=∂(...) ∂MG, hG=gG−1 2g−1/2(g1/2gGG′),G′, gGG′=(gGG′)−1, gs[MG(ts),ts]=det(gGG′)s,gs+=gs[MG s+1,ts], hG ;G=hG ,G+ΓF GFhG=g−1/2(g1/2hG),G, ΓF JK≡gLF[JK,L]=gLF(gJL,K+gKL,J−gJK,L), R=gJLRJL=gJLgJKRFJKL, RFJKL=1 2(gFK,JL−gJK,FL−gFL,JK+gJL,FK)+gMN(ΓM FKΓN JL−ΓM FLΓN JK), ( 3) whereRis the Riemannian curvature, and the discretization is explicitly denoted in the mesh of MG ιsbyι. IfMis a field, e.g., also dependent on a spatial v ariablexdiscretized by ν,then the v ariablesMG sis increased to MGν s,e.g., as prescribed for the macroscopic neocorte x. ThetermR/6 inLFincludes aStatistical Mechanics of Neocortical ... -11- Lester Ingber contribution of R/12 from the WKB approximation to the same order of ( Δt)3/2[75]. Aprepoint discretization for the same probability distribution Pgivesamuch simpler algebraic form, M(ts)=M(ts), L=1 2(dMG/dt−gG)gGG′(dMG′/dt−gG′)−V,( 4) butthe Lagrangian Lso specified does not satisfy a variational principle useful for moderate to lar ge noise; its associated variational principle only provides information useful in the weak-noise limit [76]. The neocorte xpresents a system of moderate noise. Still, this prepoint-discretized form has been quite useful in all systems e xamined thus f ar,simply requiring a somewhat finer numerical mesh. Note that although integrations are indicated o verahuge number of independent variables, i.e., as denoted by dMGν s,the physical interpretation af forded by statistical mechanics makes these systems mathematically and physically manageable. It must be emphasized that the output need not be confined to comple xalgebraic forms or tables of numbers. Because LFpossesses a v ariational principle, sets of contour graphs, at different long-time epochs of the path-integral of P,integrated overall its variables at all intermediate times, gi ve a visually intuitive and accurate decision aid to vie wthe dynamic e volution of the scenario. Forexample, as gi ven in Table 1, this Lagrangian approach permits a quantitati ve assessment of concepts usually only loosely defined. These physical entities provide another form of intuiti ve,but quantitati vely precise, presentation of these analyses [68,77]. In this study ,the above canonical momenta are referred to canonical momenta indicators (CMI). In a prepoint discretization, where the Riemannian geometry is not e xplicit (but calculated in the first SMNI papers), the distributions of neuronal acti vitiespσiis developed into distributions for activity under an electrode site Pin terms of a Lagrangian Land threshold functions FG, P= GΠPG[MG(r;t+τ)|MG(r′;t)]= σjΣδ  jEΣσj−ME(r;t+τ)  δ  jIΣσj−MI(r;t+τ)  N jΠpσj ≈ GΠ(2πτgGG)−1/2exp(−NτLG)=(2πτ)−1/2g1/2exp(−NτL),Statistical Mechanics of Neocortical ... -12- Lester Ingber Concept Lagrangian equivalent Momentum ΠG=∂LF ∂(∂MG/∂t) Mass gGG′=∂LF ∂(∂MG/∂t)∂(∂MG′/∂t) Force∂LF ∂MG F=ma δLF=0=∂LF ∂MG−∂ ∂t∂LF ∂(∂MG/∂t) TABLE 1.Descriptive concepts and their mathematical equi valents in a Lagrangian repre- sentation. L=T−V,T=(2N)−1(˙MG−gG)gGG′(˙MG′−gG′), gG=−τ−1(MG+NGtanhFG),gGG′=(gGG′)−1=δG′ Gτ−1NGsech2FG,g=det(gGG′), FG=VG−v|G| G′T|G| G′ ((π[(v|G| G′)2+(φ|G| G′)2]T|G| G′))1/2, T|G| G′=a|G| G′NG′+1 2A|G| G′MG′+a†|G| G′N†G′+1 2A†|G| G′M†G′+a‡|G| G′N‡G′+1 2A‡|G| G′M‡G′, a†G G′=1 2A†G G′+B†G G′,A‡I E=A‡E I=A‡I I=B‡I E=B‡E I=B‡I I=0,a‡E E=1 2A‡E E+B‡E E,(5) where no sum is taken o verrepeated |G|,AG G′andBG G′are macrocolumnar -averaged interneuronal synaptic efficacies, vG G′andφG G′are averaged means and variances of contrib utions to neuronal electric polarizations, NGare the numbers of excitatory and inhibitory neurons per minicolumn, and the v ariables associated with MG,M†GandM‡Grelate to multiple scales of acti vities from minicolumns, between minicolumns within re gions, and across regions, resp. The nearest-neighbor interactions Vcan be modeled in greater detail by a stochastic mesoscopic neural netw ork [14]. The SMNI papers gi ve moreStatistical Mechanics of Neocortical ... -13- Lester Ingber detail on this deri vation. In terms of the abo ve variables, an energy or Hamiltonian density Hcan be defined, H=T+V,( 6) in terms of the MGandΠGvariables, and the path inte gral is no wdefined overall theDMGas well as overtheDΠGvariables. 3.2. Nonlinear String Model Amechanical-analog model the string model, is deri vedexplicitly for neocortical interactions using SMNI [12]. In addition to providing o verlap with current EEG paradigms, this defines a probability distribution of firing acti vity,which can be used to further in vestigate the existence of other nonlinear phenomena, e.g., bifurcations or chaotic behavior ,i nbrain states. Previous SMNI studies ha ve detailed that maximal numbers of attractors lie within the physical firing space of MG,consistent with experimentally observed capacities of auditory and visual STM, when a “centering” mechanism is enforced by shifting background conducti vities of synaptic interactions, consistent with e xperimental observations under conditions of selecti ve attention [4,6,15,16,78]. This leads to an effect of having all attractors of the short-time distribution lie along a diagonal line in MG space, effectively defining a narro wparabolic trough containing these most likely firing states. This essentially collapses the 2 dimensional MGspace down to a 1 dimensional space of most importance. Thus, the predominant ph ysics of short-term memory and of (short-fiber contribution to) EEG phenomena takes place in a narro w‘‘parabolic trough’ ’inMGspace, roughly along a diagonal line [4]. Here, G represents EorI,MErepresents contributions to columnar firing from excitatory neurons, and MI represents contributions to columnar firing from inhibitory neurons. The object of interest within a short refractory time, τ,approximately 5 to 10 msec, is the Lagrangian Lfor a mesocolumn, detailed abo ve. τLcan vary by as much as a f actor of 105from the highest peak to the lowest v alleyinMGspace. Therefore, it is reasonable to assume that a single independent firing v ariable might offer a crude description of this ph ysics. Furthermore, the scalp potential Φcan be considered to be a function of this firing variable. (Here, ‘‘potential’ ’refers to the electric potential, not the potential term in the Lagrangian above.) Inan abbreviated notation subscripting the time-dependence,Statistical Mechanics of Neocortical ... -14- Lester Ingber Φt−<<Φ>>=Φ(ME t,MI t)≈a(ME t−<<ME>>)+b(MI t−<<MI>>), ( 7) whereaandbare constants, and < <Φ>> and <<MG>> represent typical minima in the trough. In the context of fitting data to the dynamic variables, there are three effecti ve constants, {a,b,φ}, Φt−φ=aME t+bMI t.( 8) The mesoscopic probability distrib utions,P,are scaled and aggre gated overthis columnar firing space to obtain the macroscopic probability distribution o verthe scalp-potential space: PΦ[Φ]=∫dMEdMIP[ME,MI]δ[Φ−Φ′(ME,MI)] . (9) In the prepoint discretization, the postpoint MG(t+Δt)moments are gi venby m≡<Φν−φ>=a<ME>+b<MI>=agE+bgI, σ2≡<(Φν−φ)2>−<Φν−φ>2=a2gEE+b2gII,( 10) where the MG-space drifts gG,and diffusionsgGG′,are givenabove.Note that the macroscopic drifts and diffusions of the Φ’s are simply linearly related to the mesoscopic drifts and diffusions of the MG’s.For the prepoint MG(t)firings, the same linear relationship in terms of {φ,a,b}is assumed. Forthe prepoint ME(t)firings, advantage is taken of the parabolic trough deri vedfor the STM Lagrangian, and MI(t)=cME(t), ( 11) where the slope cis set to the close approximate value determined by a detailed calculation of the centering mechanism [15], AE EME−AE IMI≈0. ( 12) This permits a complete transformation from MGvariables to Φvariables. Similarly,a sappearing in the modified threshold f actorFG,each regional influence from electrode site µ acting at electrode site ν,giv enb yafferent firings M‡E,i staken asStatistical Mechanics of Neocortical ... -15- Lester Ingber M‡E µ→ν=dνME µ(t−Tµ→ν), ( 13) wheredνare constants to be fitted at each electrode site, and Tµ→νare the delay times estimated abo ve for inter-electrode signal propagation, based on anatomical knowledge of the neocorte xand of velocities of propagation of action potentials of long-ranged fibers, typically on the order of one to se veral multiples ofτ=5msec. Some terms in which ddirectly affects the shifts of synaptic parameters BG G′when calculating the centering mechanism also contain long-ranged ef ficacies (in verse conducti vities)B∗E E′. Therefore, the latter were kept fix ed with the other electrical-chemical synaptic parameters during these fits. Future fits will experiment taking the T’s asparameters. This defines the conditional probability distribution for the measured scalp potential Φν, Pν[Φν(t+Δt)|Φν(t)]=1 (2πσ2Δt)1/2exp(−LνΔt), Lν=1 2σ2(˙Φν−m)2.( 14) The probability distribution for all electrodes is taken to be the product of all these distributions: P= νΠPν,L= νΣLν.( 15) Note that the belief in the dipole or nonlinear-string model is being in voked. Themodel SMNI, deri ved forP[MG(t+Δt)|MG(t)], is for a macrocolumnar -averaged minicolumn; hence it is expected to be a reasonable approximation to represent a macrocolumn, scaled to its contribution to Φν.Hence,Lis used to represent this macroscopic re gional Lagrangian, scaled from its mesoscopic mesocolumnar counterpart L.Howev er, the above expression for Pνuses the dipole assumption to also use this expression to represent se veral to man ymacrocolumns present in a re gion under an electrode: A macrocolumn has a spatial extent of about a mm. It is often argued that typically se veral macrocolumns firing coherently account for the electric potentials measured by one scalp electrode [79]. Then, this model is being tested to see if the potential will scale to a representati ve macrocolumn. The results presented here seem to confirm that this approximation is in fact quite reasonable.Statistical Mechanics of Neocortical ... -16- Lester Ingber The parabolic trough described abo ve justifies a form PΦ=(2πσ2Δt)−1/2exp(−Δt 2σ2∫dx LΦ), LΦ=α 2|∂Φ/∂t|2+β 2|∂Φ/∂x|2+γ 2|Φ|2+F(Φ), ( 16) whereF(Φ)contains nonlinearities a wayfrom the trough, σ2is on the order of Ngiventhe derivation of Labove,and the inte gral overxis taken o verthe spatial region of interest. In general, there also will be terms linear in ∂Φ/∂tand in∂Φ/∂x.(This corrects a typo that appears in se veral papers [12,13,17,19], incorrectly giving the order of σ2as 1/N.The orderNwasfirst derived[13] from σ2being expressed as asum overtheEandIdiffusions gi venabove.) Previous calculations of EEG phenomena [5], sho wthat the short-fiber contribution to the αfrequency and the mo vement of attention across the visual field are consistent with the assumption that the EEG physics is deri vedfrom an average overthe fluctuations of the system, e.g., represented by σin the abo ve equation. I.e., this is described by the Euler -Lagrange equations deri vedfrom the variational principle possessed by LΦ(essentially the counterpart to force equals mass times acceleration), more properly by the ‘‘midpoint-discretized’ ’FeynmanLΦ,with its Riemannian terms [2,3,11]. 3.3. CMI Sensitivity In the SMNI approach, “information” is a concept well defined in terms of the probability eigenfunctions of electrical-chemical activity of this Lagrangian. The path-integral formulation presents an accurate intuitive picture of an initial probability distrib ution of patterns of firings being filtered by the (exponential of the) Lagrangian, resulting in a final probability distribution of patterns of firing. The utility of a measure of information has been noted by other in vestigators. For example, there ha ve been attempts to use information as an inde xo fEEG activity [80,81]. These attempts ha ve focused on the concept of “mutual information” to find correlations of EEG acti vity under different electrodes. Other investigators have looked at simulation models of neurons to extract information as a measure of complexity of information processing [82]. Some other investigators have examined the utility of the energy density as a viable measure of information processing STM paradigms [83].Statistical Mechanics of Neocortical ... -17- Lester Ingber The SMNI approach at the outset recognizes that, for most brain states of late latenc y, atleast a subset of regions being measured by se veral electrodes is indeed to be considered as one system, and their interactions are to be explicated by mathematical or physical modeling of the underlying neuronal processes. Then, it is not rele vant to compare joint distributions o vera set of electrodes with mar ginal distributions o verindividual electrodes. In the conte xt of the present SMNI study ,the CMI transform co variantly under Riemannian transformations, but are more sensiti ve measures of neocortical acti vity than other in variants such as the energy density ,effectively the square of the CMI, or the information which also ef fectively is in terms of the square of the CMI (essentially path inte grals overquantities proportional to the energy times a f actor of an exponential including the ener gy as an ar gument). Neither the energy or the information gi ve details of the components as do the CMI. EEG is measuring a quite oscillatory system and the relati ve signs of such activity are quite important. The information and energy densities are calculated and printed out after ASA fits along with the CMI. 4. SMNI APPLICATIONS T OINDIVIDUAL EEG 4.1. Data EEG spontaneous and e vokedpotential (EP) data from a multi-electrode array under a v ariety of conditions w as collected at se veral centers in the United States, sponsored by the National Institute on Alcohol Abuse and Alcoholism (NIAAA) project. The earlier 1991 study used only a veraged EP data [84]. These experiments, performed on carefully selected sets of subjects, suggest a genetic predisposition to alcoholism that is strongly correlated to EEG AEP responses to patterned targets. It is clear that the author is not an expert in the clinical aspects of these alcoholism studies. It suffices for this study that the data used is clean ra wEEG data, and that these SMNI, CMI, and ASA techniques can and should be used and tested on other sources of EEG data as well. Each set of results is presented with 6 figures, labeled as [ {alcoholic | control },{stimulus 1 | match | no- match},subject,{potential | momenta }], abbreviated to {a|c}_{1|m|n}_subject.{pot | mom }where match or no-match was performed for stimulus 2 after 3.2 sec of a presentation of stimulus 1 [84]. DataStatistical Mechanics of Neocortical ... -18- Lester Ingber includes 10 trials of 69 epochs each between 150 and 400 msec after presentation. Foreach subject run, after fitting 28 parameters with ASA, epoch by epoch a verages are de veloped of the ra wdata and of the multivariate SMNI CMI. It was noted that much poorer fits were achie vedwhen the “centering” mechanism [4,6], dri ving multiple attractors into the physical firing regions bounded by MG≤±NG,was turned offand the denominators in FGwere set to constants, confirming the importance of using the full SMNI model. All stimuli were presented for 300 msec. Forexample, c_m_co2c0000337.pot is a figure. Note that the subject number also includes the {alcoholic | control }tag, but this tag was added just to aid sorting of files (as there are contribution from co2 and co3 subjects). Each figure contains graphs superimposed for 6 electrode sites (out of 64 in the data) which ha ve been modeled by SMNI using a circuitry gi veni nTable 2 of frontal sites (F3 and F4) feeding temporal (sides of head T7 and T8) and parietal (top of head P7 and P8) sites, where odd-numbered (e ven-numbered) sites refer to the left (right) brain. 4.2. ASA Tuning Athree-stage optimization was performed for each of 60 data sets in {a_n, a_m, a_n, c_1, c_m, c_n }of 10 subjects. As described pre viously,each of these data sets had 3-5 parameters for each SMNI electrode-site model in {F3, F4, T7, T8, P7, P8 },i.e., 28 parameters for each of the optimization runs, to be fit to o ver400 pieces of potential data. Foreach state generated in the fit, prior to calculating the Lagrangian, tests were performed to ensure that all short-ranged and long-ranged firings lay in their ph ysical boundaries. When this test failed, the generated state was simply excluded from the parameter space for further consideration. This is a standard simulated-annealing technique to handle comple xconstraints. 4.2.1. First-Stage Optimization The first-stage optimization used ASA, version 13.1, tuned to gi ve reasonable performance by e xamining intermediate results of se veral sample runs in detail. Table 3 givesthose OPTIONS changed from their defaults. (See Appendix A for a discussion of ASA OPTIONS.) The ranges of the parameters were decided as follo ws. The ranges of the strength of the long-range connectivities dνwere from 0 to 1. The ranges of the {a,b,c}parameters were decided by usingStatistical Mechanics of Neocortical ... -19- Lester Ingber Site Contrib utions From Time Delays (3.906 msec) F3 − − F4 − − T7 F3 1 T7 T8 1 T8 F4 1 T8 T7 1 P7 T7 1 P7 P8 1 P7 F3 2 P8 T8 1 P8 P7 1 P8 F4 2 TABLE 2. Circuitry of long-ranged fibers across most rele vant electrode sites and their assumed time-delays in units of 3.906 msec. minimum and maximum values of MGandM‡Gfirings to keep the potential v ariable within the minimum and maximum values of the experimentally measured potential at each electrode site. Using the abo ve ASA OPTIONS and ranges of parameters, it w as found that typically within se veral thousand generated states, the global minimum w as approached within at least one or tw osignificant figures of the ef fective Lagrangian (including the pref actor). This estimate was based on final fits achievedwith hundreds of thousands of generated states. Runs were permitted to continue for 50,000 generated states. This v ery rapid con vergence in these 30-dimensional spaces was partially due to the invocation of the centering mechanism. Some tests with SMNI parameters of fthe diagonal in MG-space, as established by the centering mechanism, confirmed that ASA con verged back to this diagonal, but requiring man ymore generated states. Of course, an examination of the Lagrangian shows this tri vially,a snoted in previous papers [3,4],Statistical Mechanics of Neocortical ... -20- Lester Ingber OPTIONS Def ault Stage 1Use Limit_Acceptances 10000 25000 Limit_Generated 99999 50000 Cost_Precision 1.0E-18 1.0E-9 Number_Cost_Samples 5 3 Cost_Parameter_Scale_Ratio 1.0 0.2 Acceptance_Frequenc y_Modulus 100 25 Generated_Frequenc y_Modulus 10000 10 Reanneal_Cost 1 4 Reanneal_P arameters 1 0 SMALL_FLO AT 1 .0E-18 1.0E-30 ASA_LIB F ALSE TR UE QUENCH_COST F ALSE TR UE QUENCH_PARAMETERS F ALSE TR UE COST_FILE TR UE F ALSE NO_PARAM_TEMP_TEST F ALSE TR UE NO_COST_TEMP_TEST F ALSE TR UE TIME_CALC F ALSE TR UE ASA_PRINT_MORE F ALSE TR UE TABLE 3. ASA OPTIONS changes from their defaults used in stage one optimization. wherein the Lagrangian v alues were on the order of 105τ−1,compared to 10−2−10−3τ−1along the diagonal established by the centering mechanism. 4.2.2. Second-Stage Optimization The second-stage optimization was in vokedt ominimize the number of generated states that w ould have been required if only the first-stage optimization were performed. Table 4 givesthe changes made in theStatistical Mechanics of Neocortical ... -21- Lester Ingber OPTIONS from stage one for stage two. OPTIONS Stage 2Changes Limit_Acceptances 5000 Limit_Generated 10000 User_Initial_P arameters TR UE User_Quench_P aram_Scale[.] 30 TABLE 4. ASA OPTIONS changes from their use in stage one for stage tw ooptimization. The final stage-one parameters were used as the initial starting parameters for stage tw o. (Athigh annealing/quenching temperatures at the start of an SA run, it typically is not important as to what the initial values of the the parameters are, pro vided of course that the ysatisfy all constraints, etc.) The second-stage minimum of each parameter w as chosen to be the maximum lower bound of the first-stage minimum and a 20% increase of that minimum. The second-stage maximum of each parameter w as chosen to be the minimum upper bound of the first-stage maximum and a 20% decrease of that maximum. Extreme quenching was turned on for the parameters (not for the cost temperature), at v alues of the parameter dimension of 30, increased from 1 (for rigorous annealing). This w orked very well, typically achieving the global minimum with 1000 generated states. Runs were permitted to continue for 10000 generated states. 4.2.3. Third-Stage Optimization The third-stage optimization used a quasi-local code, the Bro yden-Fletcher-Goldf arb-Shanno (BFGS) algorithm [85], to gain an extra 2 or 3 figures of precision in the global minimum. This typically took several hundred states, and runs were permitted to continue for 500 generated states. Constraints were enforced by the method of penalties added to the cost function outside the constraints. The BFGS code typically got stuck in a local minimum quite early if in vokedjust after the first-stage optimization. (There neverwas a reasonable chance of getting close to the global minimum using theStatistical Mechanics of Neocortical ... -22- Lester Ingber BFGS code as a first-stage optimizer .) These fits were much more efficient than those in a pre vious 1991 study [13], where VFSR, the precursor code to ASA, w as used for a long stage-one optimization which wasthen turned o vertoBFGS. 4.3. Results Figs. 1-3 compares the CMI to ra wdata for an alcoholic subject for the a_1, a_m and a_n paradigms. Figs. 4-6 gi vessimilar comparisons for a control subject for the c_1, c_m and c_n paradigms. The SMNI CMI give better signal to noise resolution than the ra wdata, especially comparing the significant matching tasks between the control and the alcoholic groups, e.g., the c_m and a_m paradigms. The CMI can be processed further as is the ra wdata, and also used to calculate “energy” and “information/entrop y” densities.Statistical Mechanics of Neocortical ... -23- Lester Ingber 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-0.20-0.15-0.10-0.050.000.050.100.150.200.250.30momenta (1/uV)F3 F4 P7 P8 T7 T8 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-20.0-18.0-16.0-14.0-12.0-10.0-8.0-6.0-4.0-2.00.02.04.06.08.010.0potentials (uV)F3 F4 P7 P8 T7 T8 FIG. 1.Forthe initial-stimulus a_1 paradigm for alcoholic subject co2a0000364, plots are giveno factivities under 6 electrodes of the CMI in the upper figure, and of the electric potential in the lower figure.Statistical Mechanics of Neocortical ... -24- Lester Ingber 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-0.30-0.25-0.20-0.15-0.10-0.050.000.050.100.150.20momenta (1/uV)F3 F4 P7 P8 T7 T8 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-20.0-18.0-16.0-14.0-12.0-10.0-8.0-6.0-4.0-2.00.02.04.0potentials (uV)F3 F4 P7 P8 T7 T8 FIG. 2.Forthe match second-stimulus a_m paradigm for alcoholic subject co2a0000364, plots are gi veno factivities under 6 electrodes of the CMI in the upper figure, and of the electric potential in the lower figure.Statistical Mechanics of Neocortical ... -25- Lester Ingber 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-0.40-0.30-0.20-0.100.000.100.200.300.40momenta (1/uV)F3 F4 P7 P8 T7 T8 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-15.0-13.0-11.0-9.0-7.0-5.0-3.0-1.01.03.05.07.09.0potentials (uV)F3 F4 P7 P8 T7 T8 FIG. 3.Forthe no-match second-stimulus a_n paradigm for alcoholic subject co2a0000364, plots are gi veno factivities under 6 electrodes of the CMI in the upper figure, and of the electric potential in the lower figure.Statistical Mechanics of Neocortical ... -26- Lester Ingber 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-0.20-0.15-0.10-0.050.000.050.100.150.20momenta (1/uV)F3 F4 P7 P8 T7 T8 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-10.0-8.0-6.0-4.0-2.00.02.04.06.08.010.012.014.0potentials (uV)F3 F4 P7 P8 T7 T8 FIG. 4.Forthe initial-stimulus c_1 paradigm for control subject co2c0000337, plots are giveno factivities under 6 electrodes of the CMI in the upper figure, and of the electric potential in the lower figure.Statistical Mechanics of Neocortical ... -27- Lester Ingber 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-0.20-0.15-0.10-0.050.000.050.100.150.20momenta (1/uV)F3 F4 P7 P8 T7 T8 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-20.0-18.0-16.0-14.0-12.0-10.0-8.0-6.0-4.0-2.00.02.04.0potentials (uV)F3 F4 P7 P8 T7 T8 FIG. 5.Forthe match second-stimulus c_m paradigm for control subject co2c0000337, plots are giveno factivities under 6 electrodes of the CMI in the upper figure, and of the electric potential in the lower figure.Statistical Mechanics of Neocortical ... -28- Lester Ingber 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-0.20-0.15-0.10-0.050.000.050.100.150.200.250.30momenta (1/uV)F3 F4 P7 P8 T7 T8 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 time (sec)-15.0-13.0-11.0-9.0-7.0-5.0-3.0-1.01.03.05.0potentials (uV)F3 F4 P7 P8 T7 T8 FIG. 6.Forthe no-match second-stimulus c_n paradigm for control subject co2c0000337, plots are gi veno factivities under 6 electrodes of the CMI in the upper figure, and of the electric potential in the lower figure.Statistical Mechanics of Neocortical ... -29- Lester Ingber Similar results are seen for other subjects. Acompressed tarfile for 10 control and 10 alcoholic subjects, atotal of 120 PostScript figures, can be retrie vedvia WWW from http://www.ingber.com/MISC.DIR/smni97_eeg_cmi.tar .Z, or as file smni97_ee g_cmi.tar.Z via FTP from ftp.ingber.com in the MISC.DIR directory. 5. CONCLUSIONS 5.1. CMI and Linear Models It is clear that the CMI follo wthe measured potential v ariables closely .Inlarge part, this is due to the prominent attractors near the firing states MGbeing close to their origins, resulting in moderate threshold functions FGin these re gions. This keeps the term in the drifts proportional to tanh FGnear its lo west values, yielding v alues of the drifts on the order of the time deri vativeso fthe potentials. The dif fusions, proportional to sech FG,also do not fluctuate to very large values. However, when the feedback among potentials under electrode sites are strong, leading to enhanced (nonlinear) changes in the drifts and diffusions, then these do cause relati vely largest signals in the CMI relative tothose appearing in the ra wpotentials. Thus, these effects are strongest in the c_m sets of data, where the control (normal) subjects demonstrate more intense circuitry interactions among electrode sites during the matching paradigm. These results also support independent studies of primarily long-ranged EEG acti vity,that have concluded that EEG man ytimes appears to demonstrate quasi-linear interactions [39,86]. Ho wever, itmust be noted that this is only true within the confines of an attractor of highly nonlinear short-ranged columnar interactions. It requires some effort, e.g., global optimization of a robust multi variate stochastic nonlinear system to achie ve finding this attractor .Theoretically ,using the SMNI model, this is performed using the ASA code. Presumably ,the neocortical system utilizes neuromodular controls to achie ve this attractor state [55,78], as suggested in early SMNI studies [3,4]. 5.2. CMI Features Essential features of the SMNI CMI approach are: (a) A realistic SMNI model, clearly capable of modeling EEG phenomena, is used, including both long-ranged columnar interactions across electrodeStatistical Mechanics of Neocortical ... -30- Lester Ingber sites and short-ranged columnar interactions under each electrode site. (b) The data is used ra wfor the nonlinear model, and only after the fits are moments (a verages and v ariances) taken of the deri vedCMI indicators; this is unlik eother studies that most often start with a veraged potential data. (c) A no veland sensitive measure, CMI, is used, which has been sho wn to be successful in enhancing resolution of signals in another stochastic multi variate time series system, financial mark ets [21,22]. As was performed in those studies, future SMNI projects can similarly use recursi ve ASA optimization, with an inner -shell fitting CMI of subjects’ EEG, embedded in an outer -shell of parameterized customized clinician’ sAI-type rules acting on the CMI, to create supplemental decision aids. Canonical momenta offers an intuiti ve yet detailed coordinate system of some comple xsystems amenable to modeling by methods of nonlinear nonequilibrium multi variate statistical mechanics. These can be used as reasonable indicators of ne wand/or strong trends of beha vior,upon which reasonable decisions and actions can be based, and therefore can be be considered as important supplemental aids to other clinical indicators. 5.3. CMI and Source Localization Global ASA optimization, fitting the nonlinearities inherent in the syner gistic contributions from short- ranged columnar firings and from long-ranged fibers, mak es it possible to disentangle their contrib utions to some specific electrode circuitries among columnar firings under re gions separated by cm, at least to the degree that the CMI clearly offer superior signal to noise than the ra wdata. Thus this paper at least establishes the utility of the CMI for EEG analyses, which can be used to complement other EEG modeling techniques. In this paper ,aplausible circuitry was first hypothesized (by a group of e xperts), and it remains to be seen just ho wmanymore electrodes can be added to such studies with the goal being to have ASA fits determine the optimal circuitry. It is clear that future SMNI projects should integrate current modeling technologies together with the CMI. For example, one approach for adding CMI to this set of tools would be to use source-localization techniques to generate simulated macrocolumnar cortical potentials (ef fectively a best fit of source- generated potentials to ra wscalp data) to determine the CMI. The CMI then can provide further disentanglement of short-ranged and long-ranged interactions to determine most lik ely circuit dynamics. Since source localization often is a non-unique process, this may pro vide an iterati ve approach to aid finerStatistical Mechanics of Neocortical ... -31- Lester Ingber source localization. That is, SMNI is a nonlinear stochastic model based on realistic neuronal interactions, and it is reasonable to assume that the deri vedCMI add much additional information to these localization analyses. 5.4. SMNI Features Sets of EEG data taken during selecti ve attention tasks ha ve been fit using parameters either set to experimentally observed values, or ha ve been fit within e xperimentally observed v alues. The ranges of columnar firings are consistent with a centering mechanism deri vedfor STM in earlier papers. These results, in addition to their importance in reasonably modeling EEG with SMNI, also ha ve a deeper theoretical importance with respect to the scaling of neocortical mechanisms of interaction across disparate spatial scales and beha vioral phenomena: As has been pointed out pre viously,SMNI has gi ven experimental support to the deri vation of the mesoscopic probability distribution, illustrating common forms of interactions between their entities, i.e., neurons and columns of neurons, respecti vely.The nonlinear threshold f actors are defined in terms of electrical-chemical synaptic and neuronal parameters all lying within their experimentally observed ranges. It also was noted that the most likely trajectories of the mesoscopic probability distribution, representing a verages overcolumnar domains, gi ve a description of the systematics of macroscopic EEG in accordance with e xperimental observ ations. It has been demonstrated that the macroscopic re gional probability distribution can be deri vedt ohav esame functional form as the mesoscopic distribution, where the macroscopic drifts and dif fusions of the potentials described by the Φ’s are simply linearly related to the (nonlinear) mesoscopic drifts and diffusions of the columnar firing states gi venb ytheMG’s.Then, this macroscopic probability distribution gi vesareasonable description of experimentally observed EEG. The theoretical and experimental importance of specific scaling of interactions in the neocorte xhas been quantitati vely demonstrated on individual brains. The explicit algebraic form of the probability distribution for mesoscopic columnar interactions is dri venb yanonlinear threshold factor of the same form taken to describe microscopic neuronal interactions, in terms of electrical-chemical synaptic and neuronal parameters all lying within their e xperimentally observed ranges; these threshold factors lar gely determine the nature of the drifts and diffusions of the system. This mesoscopic probability distrib ution has successfully described STM phenomena and, when used as a basis to deri ve the most lik elyStatistical Mechanics of Neocortical ... -32- Lester Ingber trajectories using the Euler -Lagrange variational equations, it also has described the systematics of EEG phenomena. In this paper ,the mesoscopic form of the full probability distribution has been tak en more seriously for macroscopic interactions, deriving macroscopic drifts and dif fusions linearly related to sums of their (nonlinear) mesoscopic counterparts, scaling its v ariables to describe interactions among re gional interactions correlated with observed electrical acti vities measured by electrode recordings of scalp EEG, with apparent success. These results gi ve strong quantitati ve support for an accurate intuiti ve picture, portraying neocortical interactions as having common algebraic or ph ysics mechanisms that scale across quite disparate spatial scales and functional or beha vioral phenomena, i.e., describing interactions among neurons, columns of neurons, and regional masses of neurons. 5.5. Summary SMNI is a reasonable approach to extract more ‘ ‘signal’’out of the ‘ ‘noise’’i nEEG data, in terms of physical dynamical variables, than by merely performing re gression statistical analyses on collateral variables. T olearn more about comple xsystems, inevitably functional models must be formed to represent huge sets of data. Indeed, modeling phenomena is as much a cornerstone of 20th century science as is collection of empirical data [87]. It seems reasonable to speculate on the e volutionary desirability of de veloping Gaussian-Mark ovian statistics at the mesoscopic columnar scale from microscopic neuronal interactions, and maintaining this type of system up to the macroscopic re gional scale. I.e., this permits maximal processing of information [73]. There is much work to be done, b ut modern methods of statistical mechanics ha ve helped to point the way to promising approaches. APPENDIX A: ADAPTIVE SIMULATED ANNEALING (ASA) 1. General Description Simulated annealing (SA) w as developed in 1983 to deal with highly nonlinear problems [88], as an extension of a Monte-Carlo importance-sampling technique de veloped in 1953 for chemical ph ysics problems. It helps to visualize the problems presented by such comple xsystems as a geographical terrain. Forexample, consider a mountain range, with tw o“parameters, ”e.g., along the North−South andStatistical Mechanics of Neocortical ... -33- Lester Ingber East−West directions, with the goal to find the lowest v alleyi nthis terrain. SA approaches this problem similar to using a bouncing ball that can bounce o vermountains from v alleyt ovalley. Start at a high “temperature, ”where the temperature is an SA parameter that mimics the effect of a fast moving particle in a hot object lik eahot molten metal, thereby permitting the ball to mak every high bounces and being able to bounce o verany mountain to access an yvalley, giv enenough bounces. As the temperature is made relati vely colder,the ball cannot bounce so high, and it also can settle to become trapped in relatively smaller ranges of valleys. Imagine that a mountain range is aptly described by a “cost function. ”Define probability distributions of the twodirectional parameters, called generating distributions since the ygenerate possible v alleys or states to e xplore. Define another distribution, called the acceptance distribution, which depends on the difference of cost functions of the present generated v alleyt ob eexplored and the last sa vedlowest valley. The acceptance distribution decides probabilistically whether to stay in a ne wlower valleyo rt obounce out of it. All the generating and acceptance distributions depend on temperatures. In 1984 [89], it w as established that SA possessed a proof that, by carefully controlling the rates of cooling of temperatures, it could statistically find the best minimum, e.g., the lowest v alleyo four example abo ve.This was good news for people trying to solv ehard problems which could not be solv ed by other algorithms. The bad news was that the guarantee was only good if the ywere willing to run SA forever. In1987, a method of fast annealing (FA) was de veloped [90], which permitted lowering the temperature exponentially f aster,thereby statistically guaranteeing that the minimum could be found in some finite time. However, that time still could be quite long. Shortly thereafter ,Very Fast Simulated Reannealing (VFSR) was de veloped [24], no wcalled Adapti ve Simulated Annealing (ASA), which is exponentially faster than FA. ASA has been applied to man yproblems by man ypeople in man ydisciplines [26,27,91]. The feedback of manyusers regularly scrutinizing the source code ensures its soundness as it becomes more fle xible and powerful. The code is available via the world-wide web (WWW) as http://www .ingber.com/ which also can be accessed anonymous FTP from ftp.ingber.com.Statistical Mechanics of Neocortical ... -34- Lester Ingber 2. Mathematical Outline ASA considers a parameter αi kin dimension igenerated at annealing-time kwith the range αi k∈[Ai,Bi], ( A.1) calculated with the random variable yi, αi k+1=αi k+yi(Bi−Ai), yi∈[−1, 1].( A.2) The generating function gT(y)isdefined, gT(y)=D i=1Π1 2(|yi|+Ti)ln(1+1/Ti)≡D i=1Πgi T(yi), ( A.3) where the subscript ionTispecifies the parameter index, and the k-dependence in Ti(k)for the annealing schedule has been dropped for brevity .Its cumulati ve probability distribution is GT(y)=y1 −1∫...yD −1∫dy′1...dy′DgT(y′)≡D i=1ΠGi T(yi), Gi T(yi)=1 2+sgn (yi) 2ln(1+|yi|/Ti) ln(1+1/Ti).( A.4) yiis generated from a uifrom the uniform distribution ui∈U[0, 1], yi=sgn (ui−1 2)Ti[(1+1/Ti)|2ui−1|−1] . (A.5) It is straightforward to calculate that for an annealing schedule for Ti Ti(k)=T0iexp(−cik1/D), ( A.6)Statistical Mechanics of Neocortical ... -35- Lester Ingber aglobal minima statistically can be obtained. I.e., ∞ k0Σgk≈∞ k0Σ[D i=1Π1 2|yi|ci]1 k=∞.( A.7) Control can be taken o verci,such that Tfi=T0iexp(−mi)whenkf=expni, ci=miexp(−ni/D), ( A.8) wheremiandnican be considered “free” parameters to help tune ASA for specific problems. 3. ASAOPTIONS ASA has o ver100 OPTIONS a vailable for tuning. Afew are most rele vant to this project. 3.1. Reannealing Wheneverdoing a multi-dimensional search in the course of a comple xnonlinear physical problem, inevitably one must deal with different changing sensitivities of the αiin the search. At an ygiv en annealing-time, the range o verwhich the relati vely insensiti ve parameters are being searched can be “stretched out” relati ve tothe ranges of the more sensiti ve parameters. This can be accomplished by periodically rescaling the annealing-time k,essentially reannealing, e very hundred or so acceptance- ev ents (or at some user-defined modulus of the number of accepted or generated states), in terms of the sensitivities sicalculated at the most current minimum value of the cost function, C, si=∂C/∂αi.( A.9) In terms of the lar gestsi=smax,adefault rescaling is performed for each kiof each parameter dimension, whereby a ne windexk′iis calculated from each ki, ki→k′i, T′ik′=Tik(smax/si),Statistical Mechanics of Neocortical ... -36- Lester Ingber k′i=((ln(Ti0/Tik′)/ci))D.( A.10) Ti0is set to unity to begin the search, which is ample to span each parameter dimension. 3.2. Quenching Another adapti ve feature of ASA is its ability to perform quenching in a methodical f ashion. This is applied by noting that the temperature schedule abo ve can be redefined as Ti(ki)=T0iexp(−cikQi/D i), ci=miexp(−niQi/D), ( A.11) in terms of the “quenching factor” Qi.The sampling proof fails if Qi>1as kΣD Π1/kQi/D= kΣ1/kQi<∞.( A.12) This simple calculation shows ho wthe “curse of dimensionality” arises, and also gi vesapossible way of living with this disease. In ASA, the influence of large dimensions becomes clearly focussed on the exponential of the power of kbeing 1/D,a sthe annealing required to properly sample the space becomes prohibitively slow. So, if resources cannot be committed to properly sample the space, then for some systems perhaps the next best procedure may be to turn on quenching, whereby Qican become on the order of the size of number of dimensions. The scale of the power of 1/ Dtemperature schedule used for the acceptance function can be altered in a similar fashion. Ho wever, this does not affect the annealing proof of ASA, and so this may used without damaging the sampling property. 3.3. Self Optimization If not much information is known about a particular system, if the ASA defaults do not seem to w ork very well, and if after a bit of experimentation it still is not clear ho wt oselect values for some of the ASA OPTIONS, then the SELF_OPTIMIZE OPTIONS can be very useful. This sets up a top le velsearch onStatistical Mechanics of Neocortical ... -37- Lester Ingber the ASA OPTIONS themselv es, using criteria of the system as its own cost function, e.g., the best attained optimal value of the system’ scost function (the cost function for the actual problem to be solv ed) for each gi venset of top le velOPTIONS, or the number of generated states required to reach a gi ven value of the system’ scost function, etc. Since this can consume a lot of CPU resources, it is recommended that only a fe wASA OPTIONS and a scaled do wn system cost function or system data be selected for this OPTIONS. Even if good results are being attained by ASA, SELF_OPTIMIZE can be used to find a more ef ficient set of ASA OPTIONS. Self optimization of such parameters can be v ery useful for production runs of complexsystems. 3.4. Parallel Code It is quite dif ficult to directly parallelize an SA algorithm[26], e.g., without incurring very restricti ve constraints on temperature schedules [92], or violating an associated sampling proof [93]. However, the fattail of ASA permits parallelization of de veloping generated states prior to subjecting them to the acceptance test [14]. The ASA_PARALLEL OPTIONS pro vide parameters to easily parallelize the code, using various implementations, e.g., PVM, shared memory ,etc. The scale of parallelization afforded by ASA, without violating its sampling proof, is gi venb yatypical ratio of the number of generated to accepted states. Several experts in parallelization suggest that massi ve parallelization e.g., on the order of the human brain, may tak eplace quite f ar into the future, that this might be somewhat less useful for man yapplications than previously thought, and that most useful scales of parallelization might be on scales of order 10 to 1000. 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arXiv:physics/0001053 23 Jan 2000Statistical mechanics of neocortical interactions: Training and testing canonical momenta indicators of EEG Lester Ingber Lester Ingber Resear ch,POBox 06440, Wac kerDrP O-S earsTower,Chicago, IL 60606-0440 and DRWInvestments LLC, Chicago Mercantile Exchang eCenter,3 0SWackerDrSte 1516, Chicago, IL 60606 ingber@ingber.com, ingber@alumni.caltech.edu Abstract—A series of papers has de veloped a statistical mechanics of neocortical interactions (SMNI), deriving aggre gate behavior of experimentally observ ed columns of neurons from statistical electrical- chemical properties of synaptic interactions. While not useful to yield insights at the single neuron le vel, SMNI has demonstrated its capability in describing large-scale properties of short-term memory and electroencephalographic (EEG) systematics. The necessity of including nonlinear and stochastic structures in this de velopment has been stressed. Sets of EEG and e vokedpotential data were fit, collected to in vestigate genetic predispositions to alcoholism and to extract brain “signatures” of short- term memory .Adaptive Simulated Annealing (ASA), a global optimization algorithm, was used to perform maximum likelihood fits of Lagrangians defined by path inte grals of multi variate conditional probabilities. Canonical momenta indicators (CMI) are thereby deri vedfor individual’sEEG data. The CMI give better signal recognition than the ra wdata, and can be used to advantage as correlates of behavioral states. These results gi ve strong quantitati ve support for an accurate intuiti ve picture, portraying neocortical interactions as having common algebraic or ph ysics mechanisms that scale across quite disparate spatial scales and functional or beha vioral phenomena, i.e., describing interactions among neurons, columns of neurons, and re gional masses of neurons. This paper adds to these pre vious investigations twoimportant aspects, a description of ho wthe CMI may be used in source localization, and calculations using previously ASA-fitted parameters in out-of-sample data. Keywords—Electroencephalograph y, EEG, Simulated Annealing, Statistical MechanicsStatistical Mechanics of Neocortical ... -2- L ester Ingber 1. INTRODUCTION Amodel of statistical mechanics of neocortical interactions (SMNI) has been de veloped [1-21], describing large-scale neocortical acti vity on scales of mm to cm as measured by scalp EEG, with an audit trail back to minicolumnar interactions among neurons. This paper adds to these pre vious investigations twoimportant aspects, a description of ho wthe CMI may be used in source localization, and calculations using previously ASA-fitted parameters in out-of-sample data. The underlying mathematical physics used to de velop SMNI gi vesrise to a natural coordinate system faithful to nonlinear multi variate sets of potential data such as measured by multi-electrode EEG, canonical momenta indicators (CMI) [20,22,23]. Recent papers in finance [22,23] and in EEG systems [21] ha ve demonstrated that CMI gi ve enhanced signal resolutions o verraw data. The basic philosoph yo fSMNI is that good physical models of comple xsystems, often detailed by variables not directly measurable in man yexperimental paradigms, should offer superior descriptions of empirical data beyond that a vailable from black-box statistical descriptions of such data. Forexample, good nonlinear models often offer sound approaches to relati vely deeper understandings of these systems in terms of synergies of subsystems at finer spatial-temporal scales. In this conte xt, a generic mesoscopic neural network (MNN) has been de veloped for dif fusion-type systems using a confluence of techniques dra wn from the SMNI, modern methods of functional stochastic calculus defining nonlinear Lagrangians, adapti ve simulated annealing (ASA) [24], and parallel- processing computation, to de velop MNN [14,19]. MNN increases the resolution of SMNI to minicolumnar interactions within and between neocortical regions, a scale that o verlaps with other studies of neural systems, e.g., artificial neural networks (ANN). In order to interface the algebra presented by SMNI with experimental data, se veral codes ha ve been developed. A keytool is adapti ve simulated annealing (ASA), a global optimization C-language code [24-28]. Ov er the years, this code has e volved to a high degree of robustness across man y disciplines. Ho wever, there are o ver100 OPTIONS a vailable for tuning this code; this is as e xpected for anysingle global optimization code applicable to man yclasses of nonlinear systems, systems which typically are non-typical. Section 2 gi vesthe background used to de velop SMNI and ASA for the present study .Appendix A gi ves more detail on ASA rele vant to this paper .Section 3 gi vesthe mathematical de velopment required forStatistical Mechanics of Neocortical ... -3- L ester Ingber this study .Section 4 describes the procedures used. Section 5 discusses source localization in the conte xt of global and local EEG theories. Section 6 presents conclusions. 2. BACKGROUND 2.1. EEG The SMNI approach de velops mesoscopic scales of neuronal interactions at columnar le vels of hundreds of neurons from the statistical mechanics of relati vely microscopic interactions at neuronal and synaptic scales, poised to study relati vely macroscopic dynamics at re gional scales as measured by scalp electroencephalograph y(EEG). Rele vant experimental confirmation is discussed in the SMNI papers at the mesoscopic scales as well as at macroscopic scales of scalp EEG. The derivedfirings of columnar activity,considered as order parameters of the mesoscopic system, de velop multiple attractors, which illuminate attractors that may be present in the macroscopic re gional dynamics of neocorte x. SMNI proposes that models to be fitted to data include models of activity under each electrode, e.g., due to short-ranged neuronal fibers, as well as models of acti vity across electrodes, e.g., due to long-ranged fibers. These influences can be disentangled by SMNI fits. The SMNI approach is complementary to other methods of studying nonlinear neocortical dynamics at macroscopic scales. Forexample, EEG and magnetoencephalograph y(MEG) data ha ve been expanded in aseries of spatial principal components, a Karhunen-Loe ve expansion. The coefficients in such expansions are identified as order parameters that characterize phase changes in cogniti ve studies [29,30] and epileptic seizures [31,32]. Ho wever, the SMNI CMI may be considered in a similar context, as providing a natural coordinate system that can be sensiti ve toexperimental data, without assuming av erages overstochastic parts of the system that may contain important information. Theoretical studies of the neocortical medium ha ve inv olved local circuits with postsynaptic potential delays [33-36], global studies in which finite velocity of action potential and periodic boundary conditions are important[37-40], and nonlinear nonequilibrium SMNI. The local and the global theories combine naturally to form a single theory in which control parameters effect changes between more local and more global dynamic beha vior [40,41], in a manner somewhat analogous to localized and extended w av e- function states in disordered solids.Statistical Mechanics of Neocortical ... -4- L ester Ingber Plausible connections between the multiple-scale statistical theory and the more phenomenological global theory have been proposed [12]. Experimental studies of neocortical dynamics with EEG include maps of magnitude distribution o verthe scalp [38,42], standard Fourier analyses of EEG time series [38], and estimates of correlation dimension [43,44]. Other studies ha ve emphasized that man yEEG states are accurately described by a fe wcoherent spatial modes exhibiting comple xtemporal behavior [29-32,38,40]. These modes are the order parameters at macroscopic scales that underpin the phase changes associated with changes of physiological state. Forextracranial EEG, it is clear that spatial resolution, i.e., the ability to distinguish between tw odipole sources as their distance decreases, is dif ferent from dipole localization, i.e., the ability to locate a single dipole [40]. The development of methods to impro ve the spatial resolution of EEG has made it more practical to study spatial structure. Forexample, high resolution methods pro vide apparent spatial resolution in the 2-3 cm range [45]. Dipole resolution may be as good as se veral mm[46]. Some algorithms calculate the (generally non-unique) in verse-problem of determining cortical sources that are weighted/filtered by volume conductivities of concentric spheres encompassing the brain, cerebrospinal fluid, skull, and scalp. Astraightforw ard approach is to calculate the surface Laplacian from spline fits to the scalp potential distrib ution, yielding estimates similar to those obtained using concentric spheres models of the head [45]. Other measuring techniques, e.g., MEG, can provide complementary information. These methods ha ve their strengths and weaknesses at various spatial-temporal frequencies. These source localization methods typically do not include in their models the synergistic contrib utions from short-ranged columnar firings of mm spatial extent and from long-ranged fibers spanning cm spatial extent. The CMI study presented here models these syner gistic short-ranged and long-ranged interactions. 2.2. Short-T erm Memory (STM) The development of SMNI in the context of short-term memory (STM) tasks leads naturally to the identification of measured electric scalp potentials as arising from excitatory and inhibitory short-ranged and excitatory long-ranged fibers as the ycontribute to minicolumnar interactions [12,13]. Therefore, the SMNI CMI are most appropriately calculated in the context of STM experimental paradigms. It has been been demonstrated that EEG data from such paradigms can be fit using only ph ysical synaptic andStatistical Mechanics of Neocortical ... -5- L ester Ingber neuronal parameters that lie within experimentally observed ranges [13,20]. The SMNI calculations are of minicolumnar interactions among hundreds of neurons, within a macrocolumnar e xtent of hundreds of thousands of neurons. Such interactions tak eplace on time scales of severalτ,whereτis on the order of 10 msec (of the order of time constants of cortical p yramidal cells). This also is the observ ed time scale of the dynamics of STM. SMNI hypothesizes that columnar interactions within and/or between regions containing man ymillions of neurons are responsible for these phenomena at time scales of se veral seconds. That is, the nonlinear e volution at finer temporal scales givesabase of support for the phenomena observ ed at the coarser temporal scales, e.g., by establishing mesoscopic attractors at man ymacrocolumnar spatial locations to process patterns in larger regions. SMNI has presented a model of STM, to the extent it offers stochastic bounds for this phenomena during focused selecti ve attention [4,6,15,47-49], transpiring on the order of tenths of a second to seconds, limited to the retention of 7 ±2items [50]. These constraints exist e venfor apparently e xceptional memory performers who, while the ymay be capable of more ef ficient encoding and retrie valo fSTM, and while the ymay be more efficient in ‘ ‘chunking’ ’larger patterns of information into single items, nevertheless are limited to a STM capacity of 7 ±2items [51]. Mechanisms for various STM phenomena have been proposed across man yspatial scales [52]. This “rule” is verified for acoustical STM, as well as for visual or semantic STM, which typically require longer times for rehearsal in an h ypothesized articulatory loop of individual items, with a capacity that appears to be limited to 4 ±2[53]. SMNI has detailed these constraints in models of auditory and visual corte x[4,6,15,16]. Another interesting phenomenon of STM capacity explained by SMNI is the primac yversus recenc y effect in STM serial processing [6], wherein first-learned items are recalled most error-free, with last- learned items still more error-free than those in the middle [54]. The basic assumption being made is that apattern of neuronal firing that persists for man yτcycles is a candidate to store the ‘ ‘memory’ ’of activity that g av erise to this pattern. If se veral firing patterns can simultaneously exist, then there is the capability of storing se veral memories. The short-time probability distribution deri vedfor the neocorte x is the primary tool to seek such firing patterns. The deepest minima are more likely accessed than the others of this probability distrib ution, and these v alleys are sharper than the others. I.e., the yare more readily accessed and sustain their patterns against fluctuations more accurately than the others. The more recent memories or newer patterns may be presumed to be those ha ving synaptic parameters moreStatistical Mechanics of Neocortical ... -6- L ester Ingber recently tuned and/or more acti vely rehearsed. It has been noted that experimental data on velocities of propag ation of long-ranged fibers[38,40] and derivedvelocities of propagation of information across local minicolumnar interactions [2] yield comparable times scales of interactions across minicolumns of tenths of a second. Therefore, such phenomena as STM likely are inextricably dependent on interactions at local and global scales. 2.2.1. SMNI &ADP Aproposal has been advanced that STM is processed by information coded in approximately 40-Hz (approximately 2.5 foldings of τ)bursts per stored memory ,permitting up to se vensuch memories to be processed serially within single w av eso flower frequencies on the order of 5 to 12 Hz [55]. Toaccount for the observ ed duration of STM, the ypropose that observed after-depolarization (ADP) at synaptic sites, affected by the action of relati vely long-time acting neuromodulators, e.g., acetylcholine and serotonin, acts to regularly “refresh” the stored memories in subsequent oscillatory c ycles. A recent study of the action of neuromodulators in the neocorte xsupports the premise of their effects on broad spatial and temporal scales[56], but the ADP model is much more specific in its proposed spatial and temporal influences. SMNI does not detail an yspecific synaptic or neuronal mechanisms that might refresh these most-lik ely states to reinforce multiple short-term memories [18]. Ho wever, the calculated e volution of states is consistent with the observation that an oscillatory subcycle of 40 Hz may be the bare minimal threshold of self-sustaining minicolumnar firings before the ybegin to degrade [16]. The mechanism of ADP details a specific synaptic mechanism that, when coupled with additional proposals of neuronal oscillatory c ycles of 5−12 Hz and oscillatory subcycles of 40 Hz, can sustain these memories for longer durations on the order of seconds. By itself, ADP does not provide a constraint such as the 7±2rule. The ADP approach does not address the observed random access phenomena of these memories, the 4 ±2rule, the primac yversus recenc yrule, or the influence of STM in observ ed EEG patterns. SMNI and ADP models are complementary to the understanding of STM. MNN can be used to o verlap the spatial scales studied by the SMNI with the finer spatial scales typically studied by other relati vely more microscopic neural netw orks. At this scale, such models as ADP are candidates for providing anStatistical Mechanics of Neocortical ... -7- L ester Ingber extended duration of firing patterns within the microscopic networks. 2.2.2. PATHINT Apath-integral C-language code, P ATHINT,calculates the long-time probability distribution from the Lagrangian, e.g., as fit by the ASA code. Arobust and accurate histogram-based (non-Monte Carlo) path- integral algorithm to calculate the long-time probability distrib ution had been de veloped to handle nonlinear Lagrangians[57-59], which w as extended to two-dimensional problems [60]. PATHINT was developed for use in arbitrary dimensions, with additional code to handle general Neumann and Dirichlet conditions, as well as the possibility of including time-dependent potentials, drifts, and dif fusions. The results of using P ATHINT to determine the e volution of the attractors of STM gi ve overall results consistent with previous calculations [15,16]. 2.3. ASA In order to maintain some audit trail from large-scale regional activity back to mesoscopic columnar dynamics, desirable for both academic interest as well as practical signal enhancement, as fe w approximations as possible are made by SMNI in de veloping synaptic interactions up to the le velof regional activity as measured by scalp EEG. This presents a formidable multi variate nonlinear nonequilibrium distrib ution as a model of EEG dynamics, a concept considered to be quite tentati ve by research panels as late as 1990, until it was demonstrated ho wfits to EEG data could be implemented [13]. In order to fit such distributions to real data, ASA has been de veloped, a global optimization technique, a superior v ariant of simulated annealing [25]. This wastested using EEG data in 1991[13], using an early and not as flexible version of ASA, very fast reannealing (VFSR) [25]. Here, this is tested on more refined EEG using more sensiti ve CMI to portray results of the fits [20]. ASA [24] fits short-time probability distributions to observ ed data, using a maximum likelihood technique on the Lagrangian. This algorithm has been de veloped to fit observed data to a theoretical cost function overaD-dimensional parameter space[25], adapting for v arying sensitivities of parameters during the fit. Appendix A contains details of ASA rele vant to its use in this paper.Statistical Mechanics of Neocortical ... -8- L ester Ingber 2.4. Complementary Research 2.4.1. Chaos Giventhe context of studies in comple xnonlinear systems [61], the question can be ask ed: What if EEG has chaotic mechanisms that o vershadowthe above stochastic considerations? The real issue is whether the scatter in data can be distinguished between being due to noise or chaos [62]. Inthis regard, several studies ha ve been proposed with re gard to comparing chaos to simple filtered (colored) noise [61,63]. Since the existence of multiplicati ve noise in neocortical interactions has been deri ved, then the pre vious references must be generalized, and further in vestigation is required to decide whether EEG scatter can be distinguished from multiplicati ve noise. Arecent study with realistic EEG w av eequations strongly suggests that if chaos exists in a deterministic limit, it does not survi ve inmacroscopic stochastic neocorte x[64]. I.e., it is important to include stochastic aspects, as arise from the statistics of synaptic and columnar interactions, in an yrealistic description of macroscopic neocortex. 2.4.2. Other Systems Experience using ASA on such multi variate nonlinear stochastic systems has been gained by similar applications of the approach used for SMNI. From 1986-1989, these methods of mathematical physics were utilized by a team of scientists and of ficers to develop mathematical comparisons of Janus computer combat simulations with e xercise data from the National Training Center (NTC), de veloping a testable theory of combat successfully baselined to empirical data [60,65-69]. This methodology has been applied to financial mark ets [22,23,70-72], de veloping specific trading rules for S&P 500 to demonstrate the robustness of these mathematical and numerical algorithms. 3. MATHEMATICAL DEVELOPMENT Fitting a multi variate nonlinear stochastic model to data is a necessary ,but not sufficient procedure in developing newdiagnostic softw are. Even an accurate model fit well to real data may not be immediately useful to clinicians and experimental researchers. To fill this void, the powerful intuiti ve basis of theStatistical Mechanics of Neocortical ... -9- L ester Ingber mathematical physics used to de velop SMNI has been utilized to describe the model in terms of rigorous CMI that provide an immediate intuiti ve portrait of the EEG data, faithfully describing the neocortical system being measured. The CMI gi ve anenhanced signal o verthe rawdata, and gi ve some insights into the underlying columnar interactions. 3.1. CMI, Information, Energy In the first SMNI papers, it was noted that this approach permitted the calculation of a true nonlinear nonequilibrium “information” entity at columnar scales. With reference to a steady state P(˜M)for a short-time Gaussian-Mark ovian conditional probability distrib utionPof variables ˜M,when it exists, an analytic definition of the information g ainˆϒin state˜P(˜M)overthe entire neocortical volume is defined by [73,74] ˆϒ[˜P]=∫...∫D˜M˜Pln(˜P/P),DM=(2πˆg2 0Δt)−1/2u s=1Π(2πˆg2 sΔt)−1/2dMs,( 1) where a path integral is defined such that all intermediate-time values of ˜Mappearing in the folded short- time distrib utions˜Pare integrated o ver. This is quite general for an ysystem that can be described as Gaussian-Mark ovian [75], e veni fonly in the short-time limit, e.g., the SMNI theory. As time e volves, the distribution likely no longer beha vesi naG aussian manner ,and the apparent simplicity of the short-time distrib ution must be supplanted by numerical calculations. The Fe ynman Lagrangian is written in the midpoint discretization, for a specific macrocolumn corresponding to M(ts)=1 2[M(ts+1)+M(ts)] . (2) This discretization defines a co variant Lagrangian LFthat possesses a variational principle for arbitrary noise, and that e xplicitly portrays the underlying Riemannian geometry induced by the metric tensor gGG′, calculated to be the in verse of the co variance matrix gGG′.Using the Einstein summation con vention, P=∫...∫DMexp −u s=0ΣΔtLFs , DM=g1/2 0+(2πΔt)−Θ/2u s=1Πg1/2 s+Θ G=1Π(2πΔt)−1/2dMG s,Statistical Mechanics of Neocortical ... -10- L ester Ingber ∫dMG s→NG ι=1ΣΔMG ιs,MG 0=MG t0,MG u+1=MG t, LF=1 2(dMG/dt−hG)gGG′(dMG′/dt−hG′)+1 2hG ;G+R/6−V, (...),G=∂(...) ∂MG, hG=gG−1 2g−1/2(g1/2gGG′),G′, gGG′=(gGG′)−1, gs[MG(ts),ts]=det(gGG′)s,gs+=gs[MG s+1,ts], hG ;G=hG ,G+ΓF GFhG=g−1/2(g1/2hG),G, ΓF JK≡gLF[JK,L]=gLF(gJL,K+gKL,J−gJK,L), R=gJLRJL=gJLgJKRFJKL, RFJKL=1 2(gFK,JL−gJK,FL−gFL,JK+gJL,FK)+gMN(ΓM FKΓN JL−ΓM FLΓN JK), ( 3) whereRis the Riemannian curvature, and the discretization is explicitly denoted in the mesh of MG ιsbyι. IfMis a field, e.g., also dependent on a spatial v ariablexdiscretized by ν,then the v ariablesMG sis increased to MGν s,e.g., as prescribed for the macroscopic neocorte x. ThetermR/6 inLFincludes a contribution of R/12 from the WKB approximation to the same order of ( Δt)3/2[76]. Aprepoint discretization for the same probability distribution Pgivesamuch simpler algebraic form, M(ts)=M(ts), L=1 2(dMG/dt−gG)gGG′(dMG′/dt−gG′)−V,( 4) butthe Lagrangian Lso specified does not satisfy a variational principle useful for moderate to lar ge noise; its associated v ariational principle only provides information useful in the weak-noise limit [77]. The neocorte xpresents a system of moderate noise. Still, this prepoint-discretized form has been quite useful in all systems examined thus f ar,simply requiring a some what finer numerical mesh. Note thatStatistical Mechanics of Neocortical ... -11- L ester Ingber although integrations are indicated o verahuge number of independent variables, i.e., as denoted by dMGν s,the physical interpretation afforded by statistical mechanics mak es these systems mathematically and physically manageable. It must be emphasized that the output need not be confined to comple xalgebraic forms or tables of numbers. Because LFpossesses a v ariational principle, sets of contour graphs, at different long-time epochs of the path-inte gral ofP,integrated overall its variables at all intermediate times, gi ve a visually intuitive and accurate decision aid to vie wthe dynamic e volution of the scenario. Forexample, as gi ven in Table 1, this Lagrangian approach permits a quantitati ve assessment of concepts usually only loosely defined [69,78]. In this study ,the above canonical momenta are referred to canonical momenta indicators (CMI). Table 1 In a prepoint discretization, where the Riemannian geometry is not e xplicit (but calculated in the first SMNI papers), the distributions of neuronal acti vitiespσiis developed into distributions for activity under an electrode site Pin terms of a Lagrangian Land threshold functions FG, P= GΠPG[MG(r;t+τ)|MG(r′;t)]= σjΣδ  jEΣσj−ME(r;t+τ)  δ  jIΣσj−MI(r;t+τ)  N jΠpσj ≈ GΠ(2πτgGG)−1/2exp(−NτLG)=(2πτ)−1/2g1/2exp(−NτL), L=T−V,T=(2N)−1(˙MG−gG)gGG′(˙MG′−gG′), gG=−τ−1(MG+NGtanhFG),gGG′=(gGG′)−1=δG′ Gτ−1NGsech2FG,g=det(gGG′), FG=VG−v|G| G′T|G| G′ ((π[(v|G| G′)2+(φ|G| G′)2]T|G| G′))1/2, T|G| G′=a|G| G′NG′+1 2A|G| G′MG′+a†|G| G′N†G′+1 2A†|G| G′M†G′+a‡|G| G′N‡G′+1 2A‡|G| G′M‡G′, a†G G′=1 2A†G G′+B†G G′,A‡I E=A‡E I=A‡I I=B‡I E=B‡E I=B‡I I=0,a‡E E=1 2A‡E E+B‡E E,(5)Statistical Mechanics of Neocortical ... -12- L ester Ingber where no sum is taken o verrepeated |G|,AG G′andBG G′are macrocolumnar -averaged interneuronal synaptic efficacies, vG G′andφG G′are averaged means and variances of contributions to neuronal electric polarizations, NGare the numbers of excitatory and inhibitory neurons per minicolumn, and the v ariables associated with MG,M†GandM‡Grelate to multiple scales of activities from minicolumns, between minicolumns within regions, and across regions, resp. The nearest-neighbor interactions Vcan be modeled in greater detail by a stochastic mesoscopic neural netw ork [14]. The SMNI papers gi ve more detail on this deri vation. In terms of the abo ve variables, an energy or Hamiltonian density Hcan be defined, H=T+V,( 6) in terms of the MGandΠGvariables, and the path inte gral is no wdefined overall theDMGas well as overtheDΠGvariables. 3.2. Regional Propagator Relevant to the issue of source localization is ho wSMNI describes individual head shapes. The algebra required to treat these issues was de veloped in the earliest SMNI papers [2,3]. Define the Λ-dimensional spatial vector ˜Msat timets, ˜Ms={Mν s=Ms(rν);ν=1,...,Λ}, Mν s={MGν s;G=E,I}.( 7) Formacroscopic space-time considerations, mesoscopic ρ(spatial extent of a minicolumn) and τscales are measured by dranddt.Inthe continuum limits of randt, MGν s→MG(r,t),˙MGν s→dMG/dt, (MG,ν+1−MGν)/(rν+1−rν)→∇rMG.( 8) The previous de velopment of mesocolumnar interactions via nearest-neighbor deri vative couplings permits the regional short-time propagator ˜Pto be developed in terms of the Lagrangian L[79]: ˜P(˜M)=(2πθ)−Λ/2∫d˜MgΛ/2exp[−N˜S(˜M)]˜P(˜M),Statistical Mechanics of Neocortical ... -13- L ester Ingber ˜S=mint+θ t∫dt′L[˙M(t′),M(t′)], L=ΛΩ−1∫d2rL,( 9) whereΩis the area of the region considered, and ΛΩ−1∫d2r=ΛΩ−1∫dx dy=ρ→0 Λ→∞limΛ ν=1Σ.( 10) With˜Pproperly defined by this space-time mesh, a path-inte gral formulation for the regional long-time propagator at time t=(u+1)θ+t0is developed: ˜P[˜M(t)]d˜M(t)=∫...∫D˜Mexp(−Nt t0∫dt′L), ˜P[˜M(t0)]=δ(˜M−˜M0), D˜M=u+1 s=1ΠΛ ν=1ΠE,I GΠ(2πθ)−1 2(gν s)1/4dMGν s.( 11) Note that, e venforNτL≈1,Nt t0∫dt′Lis very large for macroscopically large time ( t−t0)and macroscopic size Λ,demonstrating ho wextrema of Ldefine peaked maximum probability states. This derivation can be viewed as containing the dynamics of macroscopic causal irre versibility,whereby˜Pis an unstable fixed point about which deviations from the extremum are greatly amplified [80]. Thus, anybrain surface is mapped according to its Λminicolumns, wherein its geometry is e xplicitly developed according the the discretized path inte gral according to d˜Moverthe minicolumnar -discretized surface mapped by d2r. 3.3. Nonlinear String Model Amechanical-analog, the string model, is deri vedexplicitly for neocortical interactions using SMNI [12]. In addition to pro viding overlap with current EEG paradigms, this defines a probability distribution of firing acti vity,which can be used to further in vestigate the existence of other nonlinear phenomena, e.g., bifurcations or chaotic behavior ,i nbrain states.Statistical Mechanics of Neocortical ... -14- L ester Ingber Previous SMNI studies ha ve detailed that maximal numbers of attractors lie within the physical firing space of MG,consistent with experimentally observed capacities of auditory and visual STM, when a “centering” mechanism is enforced by shifting background conducti vities of synaptic interactions, consistent with e xperimental observations under conditions of selecti ve attention [4,6,15,16,81]. This leads to an effect of having all attractors of the short-time distribution lie along a diagonal line in MG space, effectively defining a narro wparabolic trough containing these most likely firing states. This essentially collapses the 2 dimensional MGspace down to a 1 dimensional space of most importance. Thus, the predominant ph ysics of short-term memory and of (short-fiber contribution to) EEG phenomena takes place in a narro w‘‘parabolic trough’ ’inMGspace, roughly along a diagonal line [4]. The object of interest within a short refractory time, τ,approximately 5 to 10 msec, is the Lagrangian Lfor a mesocolumn, detailed abo ve.τLcan vary by as much as a factor of 105from the highest peak to the lowest valleyinMGspace. Therefore, it is reasonable to assume that a single independent firing v ariable might offer a crude description of this ph ysics. Furthermore, the scalp potential Φcan be considered to be a function of this firing v ariable. (Here, ‘‘potential’ ’refers to the electric potential, not the potential term in the Lagrangian abo ve.) Inan abbreviated notation subscripting the time-dependence, Φt−<<Φ>>=Φ(ME t,MI t)≈a(ME t−<<ME>>)+b(MI t−<<MI>>), ( 12) whereaandbare constants, and < <Φ>> and <<MG>> represent typical minima in the trough. In the context of fitting data to the dynamic variables, there are three effecti ve constants, {a,b,φ}, Φt−φ=aME t+bMI t.( 13) Accordingly ,there is assumed to be a linear relationship (about minima to be fit to data) between the MG firing states and the measured scalp potential Φν,a tagiv enelectrode site νrepresenting a macroscopic region of neuronal activity: Φν−φ=aME+bMI,( 14) where{φ,a,b}are constants determined for each electrode site. In the prepoint discretization, the postpoint MG(t+Δt)moments are gi venby m≡<Φν−φ>=a<ME>+b<MI>=agE+bgI, σ2≡<(Φν−φ)2>−<Φν−φ>2=a2gEE+b2gII,( 15)Statistical Mechanics of Neocortical ... -15- L ester Ingber where the MG-space drifts gG,and diffusionsgGG′,are givenabove.Note that the macroscopic drifts and diffusions of the Φ’s are simply linearly related to the mesoscopic drifts and diffusions of the MG’s.For the prepoint MG(t)firings, the same linear relationship in terms of {φ,a,b}is assumed. The mesoscopic probability distrib utions,P,are scaled and aggre gated overthis columnar firing space to obtain the macroscopic probability distribution o verthe scalp-potential space: PΦ[Φ]=∫dMEdMIP[ME,MI]δ[Φ−Φ′(ME,MI)] . (16) The parabolic trough described abo ve justifies a form PΦ=(2πσ2)−1/2exp(−Δt 2σ2∫dx LΦ), LΦ=α 2|∂Φ/∂t|2+β 2|∂Φ/∂x|2+γ 2|Φ|2+F(Φ), ( 17) whereF(Φ)contains nonlinearities a wayfrom the trough, σ2is on the order of 1/ Ngiventhe derivation ofLabove,and the integral o verxis taken o verthe spatial region of interest. In general, there also will be terms linear in ∂Φ/∂tand in∂Φ/∂x. Previous calculations of EEG phenomena [5], showthat the short-fiber contribution to the αfrequency and the mo vement of attention across the visual field are consistent with the assumption that the EEG physics is deri vedfrom an average overthe fluctuations of the system, e.g., represented by σin the abo ve equation. I.e., this is described by the Euler-Lagrange equations deri vedfrom the variational principle possessed by LΦ(essentially the counterpart to force equals mass times acceleration), more properly by the ‘‘midpoint-discretized’ ’FeynmanLΦ,with its Riemannian terms[2,3,11], Hence, the v ariational principle applies, 0=∂ ∂t∂LΦ ∂(∂Φ/∂t)+∂ ∂x∂LΦ ∂(∂Φ/∂x)−∂LΦ ∂Φ.( 18) The result is α∂2Φ ∂t2+β∂2Φ ∂x2+γΦ−∂F ∂Φ=0. ( 19) If there exist regions in neocortical parameter space such that β/α=−c2,γ/α=ω2 0, 1 α∂F ∂Φ=−Φf(Φ), ( 20)Statistical Mechanics of Neocortical ... -16- L ester Ingber andxis taken to be one-dimensional, then the nonlinear string is reco vered. Terms linear in ∂Φ/∂tand in ∂Φ/∂xinLΦcan makeother contributions, e.g., giving rise to damping terms. The path-integral formulation has a utility beyond its deterministic Euler-Lagrange limit. This has been utilized to explicitly examine the long-time e volution of systems, to compare models to long-time correlations in data [60,68]. This use is being e xtended to other systems, in finance [71,72] and in EEG modeling as described here. Forthe prepoint ME(t)firings, advantage is taken of the parabolic trough deri vedfor the STM Lagrangian, and MI(t)=cME(t), ( 21) where the slope cis set to the close approximate v alue determined by a detailed calculation of the centering mechanism [15], AE EME−AE IMI≈0. ( 22) This permits a complete transformation from MGvariables to Φvariables. Similarly,a sappearing in the modified threshold f actorFG,each regional influence from electrode site µ acting at electrode site ν,giv enb yafferent firings M‡E,i staken as M‡E µ→ν=dνME µ(t−Tµ→ν), ( 23) wheredνare constants to be fitted at each electrode site, and Tµ→νare the delay times estimated abo ve for inter-electrode signal propag ation, based on anatomical knowledge of the neocorte xand of velocities of propag ation of action potentials of long-ranged fibers, typically on the order of one to se veral multiples ofτ=5msec. Some terms in which ddirectly affects the shifts of synaptic parameters BG G′when calculating the centering mechanism also contain long-ranged ef ficacies (in verse conducti vities)B∗E E′. Therefore, the latter were kept fix ed with the other electrical-chemical synaptic parameters during these fits. Future fits will experiment taking the T’s asparameters. This defines the conditional probability distribution for the measured scalp potential Φν, Pν[Φν(t+Δt)|Φν(t)]=1 (2πσ2Δt)1/2exp(−LνΔt), Lν=1 2σ2(˙Φν−m)2.( 24)Statistical Mechanics of Neocortical ... -17- L ester Ingber The probability distribution for all electrodes is taken to be the product of all these distributions: P= νΠPν,L= νΣLν.( 25) Note that the belief in the dipole or nonlinear-string model is being in voked. Themodel SMNI, deri ved forP[MG(t+Δt)|MG(t)], is for a macrocolumnar -averaged minicolumn; hence it is expected to be a reasonable approximation to represent a macrocolumn, scaled to its contribution to Φν.Hence,Lis used to represent this macroscopic re gional Lagrangian, scaled from its mesoscopic mesocolumnar counterpart L.Howev er, the above expression for Pνuses the dipole assumption to also use this e xpression to represent se veral to man ymacrocolumns present in a re gion under an electrode: A macrocolumn has a spatial extent of about a mm. Ascalp electrode has been sho wn, under extremely f avorable circumstances, to ha ve a resolution as small as se veral mm, directly competing with the best spatial resolution attributed to MEG [46]. It is often argued that typically se veral macrocolumns firing coherently account for the electric potentials measured by one scalp electrode [82]. Then, this model is being tested to see if the potential will scale to a representati ve macrocolumn. The results presented here seem to confirm that this approximation is in fact quite reasonable. Future projects will de velop SMNI to include higher resolution minicolumnar -minicolumnar dynamics using stochastic MNN, described abo ve [14]. 3.4. CMI Sensitivity In the SMNI approach, “information” is a concept well defined in terms of the probability eigenfunctions of electrical-chemical activity of this Lagrangian. The path-integral formulation presents an accurate intuitive picture of an initial probability distrib ution of patterns of firings being filtered by the (exponential of the) Lagrangian, resulting in a final probability distribution of patterns of firing. The utility of a measure of information has been noted by other in vestigators. For example, there ha ve been attempts to use information as an inde xo fEEG activity [83,84]. These attempts ha ve focused on the concept of “mutual information” to find correlations of EEG acti vity under different electrodes. Other investigators have looked at simulation models of neurons to e xtract information as a measure of complexity of information processing [85]. Some other investigators have examined the utility of the energy density as a viable measure of information processing STM paradigms [86].Statistical Mechanics of Neocortical ... -18- L ester Ingber The SMNI approach at the outset recognizes that, for most brain states of late latenc y, atleast a subset of regions being measured by se veral electrodes is indeed to be considered as one system, and their interactions are to be e xplicated by mathematical or physical modeling of the underlying neuronal processes. Then, it is not rele vant to compare joint distributions o veraset of electrodes with mar ginal distributions o verindividual electrodes. In the context of the present SMNI study ,the CMI transform co variantly under Riemannian transformations, but are more sensiti ve measures of neocortical activity than other in variants such as the energy density ,effectively the square of the CMI, or the information which also ef fectively is in terms of the square of the CMI (essentially path integrals o verquantities proportional to the energy times a f actor of an exponential including the energy as an ar gument). Neither the energy or the information gi ve details of the components as do the CMI. EEG is measuring a quite oscillatory system and the relati ve signs of such activity are quite important. The information and energy densities are calculated and printed out after ASA fits along with the CMI. 4. SMNI APPLICATIONS T OINDIVIDUAL EEG 4.1. Data EEG spontaneous and e vokedpotential (EP) data from a multi-electrode array under a variety of conditions was collected at se veral centers in the United States, sponsored by the National Institute on Alcohol Ab use and Alcoholism (NIAAA) project. The earlier 1991 study used only a veraged EP data [87]. These experiments, performed on carefully selected sets of subjects, suggest a genetic predisposition to alcoholism that is strongly correlated to EEG AEP responses to patterned targets. It is clear that the author is not an e xpert in the clinical aspects of these alcoholism studies. It suffices for this study that the data used is clean ra wEEG data, and that these SMNI, CMI, and ASA techniques can and should be used and tested on other sources of EEG data as well. Each set of results is presented with 6 figures, labeled as [ {alcoholic | control },{stimulus 1 | match | no- match},subject,{potential | momenta }], abbreviated to {a|c}_{1| m | n}_subject.{pot | mom }where match or no-match w as performed for stimulus 2 after 3.2 sec of a presentation of stimulus 1 [87]. Data includes 10 trials of 69 epochs each between 150 and 400 msec after presentation. Foreach subject run,Statistical Mechanics of Neocortical ... -19- L ester Ingber after fitting 28 parameters with ASA, epoch by epoch a verages are de veloped of the ra wdata and of the multivariate SMNI CMI. It was noted that much poorer fits were achie vedwhen the “centering” mechanism [4,6], dri ving multiple attractors into the physical firing regions bounded by MG≤±NG,was turned offand the denominators in FGwere set to constants, confirming the importance of using the full SMNI model. All stimuli were presented for 300 msec. Forexample, c_m_co2c0000337.pot is a figure. Note that the subject number also includes the {alcoholic | control }tag, but this tag was added just to aid sorting of files (as there are contrib ution from co2 and co3 subjects). Each figure contains graphs superimposed for 6 electrode sites (out of 64 in the data) which ha ve been modeled by SMNI using a circuitry gi veni nTable 2 of frontal sites (F3 and F4) feeding temporal (sides of head T7 and T8) and parietal (top of head P7 and P8) sites, where odd-numbered (e ven-numbered) sites refer to the left (right) brain. Table 2 4.2. ASA Tuning Athree-stage optimization was performed for each of 60 data sets in {a_n, a_m, a_n, c_1, c_m, c_n }of 10 subjects. As described pre viously,each of these data sets had 3-5 parameters for each SMNI electrode-site model in {F3, F4, T7, T8, P7, P8 },i.e., 28 parameters for each of the optimization runs, to be fit to o ver400 pieces of potential data. Foreach state generated in the fit, prior to calculating the Lagrangian, tests were performed to ensure that all short-ranged and long-ranged firings lay in their ph ysical boundaries. When this test failed, the generated state was simply excluded from the parameter space for further consideration. This is a standard simulated-annealing technique to handle comple xconstraints. 4.2.1. First-Stage Optimization The first-stage optimization used ASA, version 13.1, tuned to gi ve reasonable performance by e xamining intermediate results of se veral sample runs in detail. Table 3 givesthose OPTIONS changed from their defaults. (See Appendix A for a discussion of ASA OPTIONS.)Statistical Mechanics of Neocortical ... -20- L ester Ingber Table 3 The ranges of the parameters were decided as follo ws. The ranges of the strength of the long-range connectivities dνwere from 0 to 1. The ranges of the {a,b,φ}parameters were decided by using minimum and maximum values of MGandM‡Gfirings to keep the potential v ariable within the minimum and maximum v alues of the experimentally measured potential at each electrode site. (This corrects a typo in a previous paper [21], where these 3 parameters were referred to as {a,b,c}.) Using the abo ve ASA OPTIONS and ranges of parameters, it w as found that typically within se veral thousand generated states, the global minimum was approached within at least one or tw osignificant figures of the ef fective Lagrangian (including the pref actor). This estimate w as based on final fits achievedwith hundreds of thousands of generated states. Runs were permitted to continue for 50,000 generated states. This very rapid con vergence in these 30-dimensional spaces w as partially due to the invocation of the centering mechanism. Some tests with SMNI parameters of fthe diagonal in MG-space, as established by the centering mechanism, confirmed that ASA con verged back to this diagonal, but requiring man ymore generated states. Of course, an examination of the Lagrangian shows this tri vially,a snoted in pre vious papers [3,4], wherein the Lagrangian values were on the order of 105τ−1,compared to 10−2−10−3τ−1along the diagonal established by the centering mechanism. 4.2.2. Second-Stage Optimization The second-stage optimization was in vokedt ominimize the number of generated states that would ha ve been required if only the first-stage optimization were performed. Table 4 givesthe changes made in the OPTIONS from stage one for stage tw o. Thefinal stage-one parameters were used as the initial starting parameters for stage tw o. (Athigh annealing/quenching temperatures at the start of an SA run, it typically is not important as to what the initial v alues of the the parameters are, provided of course that theysatisfy all constraints, etc.) The second-stage minimum of each parameter was chosen to be the maximum lower bound of the first-stage minimum and a 20% increase of that minimum. The second- stage maximum of each parameter was chosen to be the minimum upper bound of the first-stageStatistical Mechanics of Neocortical ... -21- L ester Ingber maximum and a 20% decrease of that maximum. Table 4 Extreme quenching w as turned on for the parameters (not for the cost temperature), at values of the parameter dimension of 30, increased from 1 (for rigorous annealing). This w orked very well, typically achieving the global minimum with 1000 generated states. Runs were permitted to continue for 10000 generated states. 4.2.3. Third-Stage Optimization The third-stage optimization used a quasi-local code, the Bro yden-Fletcher-Goldf arb-Shanno (BFGS) algorithm [88], to gain an extra 2 or 3 figures of precision in the global minimum. This typically took several hundred states, and runs were permitted to continue for 500 generated states. Constraints were enforced by the method of penalties added to the cost function outside the constraints. The BFGS code typically got stuck in a local minimum quite early if in vokedjust after the first-stage optimization. (There neverwas a reasonable chance of getting close to the global minimum using the BFGS code as a first-stage optimizer .) These fits were much more efficient than those in a previous 1991 study [13], where VFSR, the precursor code to ASA, w as used for a long stage-one optimization which wasthen turned o vertoBFGS. Table 5 givesthe 28 ASA-fitted parameters for each of the 3 e xperimental paradigms for each alcoholic and control subject. Table 5 4.3. Testing Data When the parameters of a theory of a physical system posses clear relationships to observed ph ysical entities, and the theory fits e xperimental phenomenon while the parameters stay within e xperimentally determined ranges of these entities, then generally it is conceded that the theory and its parameters ha veStatistical Mechanics of Neocortical ... -22- L ester Ingber passed a reasonable test. It is ar gued that this is the case for SMNI and its parameters, and this approach sufficed for the first study of the present data [21], just as SMNI also has been tested in previous papers. When a model of a ph ysical system has a relati vely phenomenological nature then often such a model is best tested by first “training” its parameters on one set of data, then seeing to what degree the same parameters can be used to match the model to out-of-sample “testing” data. Forexample, this w as performed for the statistical mechanics of financial mark ets (SMFM) project[70-72], applied to trading models [22,23]. The SMFM projects similarly use ASA and the algebra presented here for this SMNI project. In the present project, there exists barely enough data to additionally test SMNI in this training v ersus testing methodology .That is, when first e xamining the data, it was decided to to try to find sets of data from at least 10 control and 10 alcoholic subjects, each set containing at least 10 runs for each of the 3 experimental paradigms, as reported in a previous paper [21]. When reviewing this data, e.g., for the example of the one alcoholic and the one control subject which were illustrated in graphs in that pre vious paper,i twas determined that there exists 10 additional sets of data for each subject for each paradigm, except for the c_n case of the no-match paradigm for the control subject where only 5 additional out-of- sample runs e xist. For this latter case, to keep the number of runs sampled consistent across all sets of data, e.g., to keep the relati ve amplitudes of fluctuations reasonably meaningful, 5 runs of the pre vious testing set were joined with the 5 runs of the present training set to fill out the data sets required for this study. 4.4. Results Figs. 1-3 compares the CMI to ra wdata for an alcoholic subject for the a_1, a_m and a_n paradigms, for both the training and testing data. Figs. 4-6 gi vessimilar comparisons for a control subject for the c_1, c_m and c_n paradigms. The SMNI CMI clearly gi ve better signal to noise resolution than the ra wdata, especially comparing the significant matching tasks between the control and the alcoholic groups, e.g., the c_m and a_m paradigms, in both the training and testing cases. The CMI can be processed further as is the rawdata, and also used to calculate “energy” and “information/entropy” densities.Statistical Mechanics of Neocortical ... -23- L ester Ingber Figures 1-6 Similar results are seen for other subjects. Acompressed tarfile of additonal results for 10 control and 10 alcoholic subjects, using the training data and the testing data, including tables of ASA-fitted parameters and 60 files containing 240 PostScript graphs, can be retrie vedvia WWW from http://www.ingber.com/MISC.DIR/smni97_eeg_cmi.tar .Z, or as file smni96_ee g_cmi.tar.Z via FTP from ftp.ingber.com in the MISC.DIR directory. After the abo ve training-testing methodology is applied to more subjects, it will then be possible to perform additional statistical analyses to seek more abbre viated measures of differences between alcoholic and control groups across the 3 experimental paradigms. 4.5. Availability of Codes and Data The ASA code can be downloaded at no charge from http://www .ingber.com/ under WWW or from ftp://ftp.ingber .com under FTP .Amirror site for the home page is http://www .alumni.caltech.edu/˜ingber/ under WWW. This ASA applications to EEG analysis is one of se veral ASA applications being prepared for the SPEC (Standard Performance Evaluation Corporation) CPU98 suite. The goal of the program is to elicit benchmarks representing important applications in v arious technical fields. When SPEC publishes its CDROM, the SMNI CMI codes will be included. It is extremely difficult for modelers of nonlinear time series, and EEG systems in particular ,t oget access to large sets of ra wclean data. The data used for this study is no wpublicly available, as described in http://www.ingber.com/smni_ee g_data.html under WWW or ftp://ftp.ingber.com/MISC.DIR/smni_ee g_data.txt under FTP .Data is being pro vided as is, and may be deleted without notice, or mo vedt oother archi veswith the only notice gi veni nthat file. There are 11,075 typical gzip’ dfiles in the 122 tar’ ddirectories; each file is about 65K. The entire set of data is about 700 MBytes in this compressed format. Forconvenience, there are subsets of data a vailable that were used for this study.Statistical Mechanics of Neocortical ... -24- L ester Ingber 5. SOURCE LOCALIZATION 5.1. Global and Local Dynamics The SMNI vie wi sthat, for purposes of regional EEG acti vity,short-ranged interactions can be treated statistically ,while long-ranged interactions can be incorporated more specifically by fitting e xplicit circuitries. A different approach is taken by present global approaches, which include some statistics of long-range circuitries to de velop a wav etheory overall or most of neocorte x[38,40,89]. The SMNI approach can provide specific fitted long-ranged circuits as points to insert into such global approaches, e.g., to use as specific “data” points to fit the global w av es. Forexample, a leading global theory[38,40] consists of three coupled equations, tw olinear and one nonlinear.The first tw odescribe the number of synaptic e vents in a volume as being linearly related to the number of action potentials in the brain v olume (with a space-dependent time delay). These equations can be represented in either (space, time) or (w av enumber,frequency) dimensions. Most of the complication ultimately comes from the third equation, relating action potentials fired to synaptic input. This is where a better (nonlinear) description is needed. SMNI can provide this as it is articulated in terms of Lagrangians and probability densities in real space and real time, e.g., of a field of neuronal columns throughout neocorte x. Interactions among these columns is in terms of local and global columnar firings of afferent con vergent and efferent di verging fibers. Synaptic activities are an intermediary calculation in SMNI. These complementary global and local approaches can be integrated, e.g., EEG Data -> local theory -> columnar firings -> global theory Forexample, in one approach, Laplacians of EEG potentials o verapre-determined brain surface could be used as v ariables input into SMNI to fit neuronal-synaptic parameters via the joint action A(the time integral overthe Lagrangian L)integrated overthe brain surf ace. These firings would then be transformed via global theory into synaptic acti vities, from which dispersion relations deduce global dynamics [38,40].Statistical Mechanics of Neocortical ... -25- L ester Ingber 5.2. Approaches to Source Localization As described abo ve,SMNI stresses that is natural to describe regional macroscopic dynamics in terms of mesocolumnar interactions, i.e., a true “field” of firings is de veloped, which has been mathematically articulated in the SMNI papers as defining the re gional propagator in the path inte gralD˜Moverthe space-time v olumedx−dtof the brain, in terms of the Lagrangian Leach volume point. The space volumedxis where the head shape explicitly enters the calculation. Forexample, fitting Lto actual data might require discretization at electrode sites, as performed in this study ,where actually the cost function in terms of Lto be fit contains the v olume elements in dx.Ifa3-D fit were being done, then dxwould be the head volume; if a 2-D surface fit were being done, dxwould be the head/brain surf ace. Inpractice, we must rely on other methods, e.g., MRI, to determine the head shape to articulate the shape spanned by dx.Ahigher resolution algorithm, at the le velo fminicolumns, shows ho wshort-ranged as well as long- ranged interactions among mesocolumns within and between regions can be naturally included in the SMNI theory [14]. Such fits can be performed after other source localization techniques are applied to the data, e.g., head- volume or Laplacian techniques. This still is very important, as identification of source(s), especially those that may be nonstationary ,isnot sufficient to describe their dynamical interaction. The fitted SMNI dynamics then can provide the dynamical description to predict future e volution of the interactions among sources, e.g., to fill in gaps in data that might aid correlation with behavioral states. The SMNI dynamics can more directly be part of source localization algorithms. Forexample, if M/dtis identified as the the mesocolumnar current ( Mis essentially the number of neuronal firings within the timeτof about 5 msec; dtis the time resolution of the EEG, typically on this order or somewhat less), then we can identify M=B∇2Φ,where∇2is the Laplacian and Bis a constant included in the fit, similar to the relationship between MandΦin this present project. Then, the Laplacian of Φwould be input into the SMNI action A,and the parameters of the model would be fit as performed here. The ASA global optimization of the highly nonlinear SMNI finds the best fit among the combined local-global interactions algebraically described in the SMNI action A.The “curvatures” (second deri vatives) of the parameters about the global minimum, automatically returned by ASA, gi ve a covariance matrix of the goodness of fit about the global minimum.Statistical Mechanics of Neocortical ... -26- L ester Ingber To include some global dynamics in the language of present global theories, e.g., if some subset of Legendre-decomposed Φare deemed to be most important, their coefficients can be considered parameters, and the fit can find the degree of which partial w av esare most likely present in the data. It is important to stress that the abo ve approach recognizes that both local and global neuronal dynamics, head shapes, etc., enter as nonlinear stochastic e vents in the real brain, and this approach can match these ev ents, but it requires global optimization to get the specificity to fit real individual brains. 6. CONCLUSIONS 6.1. CMI and Linear Models It is clear that the CMI follo wthe measured potential v ariables closely .Inlarge part, this is due to the prominent attractors near the firing states MGbeing close to their origins, resulting in moderate threshold functions FGin these re gions. This keeps the term in the drifts proportional to tanh FGnear its lo west values, yielding v alues of the drifts on the order of the time deri vativeso fthe potentials. The dif fusions, proportional to sech FG,also do not fluctuate to very large values. However, when the feedback among potentials under electrode sites are strong, leading to enhanced (nonlinear) changes in the drifts and diffusions, then these do cause relati vely largest signals in the CMI relative tothose appearing in the ra wpotentials. Thus, these effects are strongest in the c_m sets of data, where the control (normal) subjects demonstrate more intense circuitry interactions among electrode sites during the matching paradigm. These results also support independent studies of primarily long-ranged EEG acti vity,that have concluded that EEG man ytimes appears to demonstrate quasi-linear interactions [40,90]. Ho wever, itmust be noted that this is only true within the confines of an attractor of highly nonlinear short-ranged columnar interactions. It requires some effort, e.g., global optimization of a robust multi variate stochastic nonlinear system to achie ve finding this attractor .Theoretically ,using the SMNI model, this is performed using the ASA code. Presumably ,the neocortical system utilizes neuromodular controls to achie ve this attractor state [56,81], as suggested in early SMNI studies [3,4].Statistical Mechanics of Neocortical ... -27- L ester Ingber 6.2. CMI Features Essential features of the SMNI CMI approach are: (a) A realistic SMNI model, clearly capable of modeling EEG phenomena, is used, including both long-ranged columnar interactions across electrode sites and short-ranged columnar interactions under each electrode site. (b) The data is used ra wfor the nonlinear model, and only after the fits are moments (a verages and v ariances) taken of the deri vedCMI indicators; this is unlik eother studies that most often start with a veraged potential data. (c) A no veland sensitive measure, CMI, is used, which has been sho wn to be successful in enhancing resolution of signals in another stochastic multi variate time series system, financial mark ets [22,23]. As was performed in those studies, future SMNI projects can similarly use recursi ve ASA optimization, with an inner -shell fitting CMI of subjects’ EEG, embedded in an outer -shell of parameterized customized clinician’ sAI-type rules acting on the CMI, to create supplemental decision aids. Canonical momenta offers an intuiti ve yet detailed coordinate system of some comple xsystems amenable to modeling by methods of nonlinear nonequilibrium multi variate statistical mechanics. These can be used as reasonable indicators of ne wand/or strong trends of beha vior,upon which reasonable decisions and actions can be based, and therefore can be be considered as important supplemental aids to other clinical indicators. 6.3. CMI and Source Localization Global ASA optimization, fitting the nonlinearities inherent in the syner gistic contributions from short- ranged columnar firings and from long-ranged fibers, mak es it possible to disentangle their contrib utions to some specific electrode circuitries among columnar firings under re gions separated by cm, at least to the degree that the CMI clearly offer superior signal to noise than the ra wdata. Thus this paper at least establishes the utility of the CMI for EEG analyses, which can be used to complement other EEG modeling techniques. In this paper ,aplausible circuitry was first hypothesized (by a group of e xperts), and it remains to be seen just ho wmanymore electrodes can be added to such studies with the goal being to have ASA fits determine the optimal circuitry.Statistical Mechanics of Neocortical ... -28- L ester Ingber 6.4. SMNI Features Sets of EEG data taken during selecti ve attention tasks ha ve been fit using parameters either set to experimentally observed values, or ha ve been fit within e xperimentally observed v alues. The ranges of columnar firings are consistent with a centering mechanism deri vedfor STM in earlier papers. These results, in addition to their importance in reasonably modeling EEG with SMNI, also ha ve a deeper theoretical importance with respect to the scaling of neocortical mechanisms of interaction across disparate spatial scales and beha vioral phenomena: As has been pointed out pre viously,SMNI has gi ven experimental support to the deri vation of the mesoscopic probability distribution, illustrating common forms of interactions between their entities, i.e., neurons and columns of neurons, respecti vely.The nonlinear threshold f actors are defined in terms of electrical-chemical synaptic and neuronal parameters all lying within their experimentally observed ranges. It also was noted that the most likely trajectories of the mesoscopic probability distribution, representing a verages overcolumnar domains, gi ve a description of the systematics of macroscopic EEG in accordance with e xperimental observ ations. It has been demonstrated that the macroscopic re gional probability distribution can be deri vedt ohav esame functional form as the mesoscopic distribution, where the macroscopic drifts and dif fusions of the potentials described by the Φ’s are simply linearly related to the (nonlinear) mesoscopic drifts and diffusions of the columnar firing states gi venb ytheMG’s.Then, this macroscopic probability distribution gi vesareasonable description of experimentally observed EEG. The theoretical and experimental importance of specific scaling of interactions in the neocorte xhas been quantitati vely demonstrated on individual brains. The explicit algebraic form of the probability distribution for mesoscopic columnar interactions is dri venb yanonlinear threshold factor of the same form taken to describe microscopic neuronal interactions, in terms of electrical-chemical synaptic and neuronal parameters all lying within their e xperimentally observed ranges; these threshold factors lar gely determine the nature of the drifts and diffusions of the system. This mesoscopic probability distrib ution has successfully described STM phenomena and, when used as a basis to deri ve the most lik ely trajectories using the Euler -Lagrange variational equations, it also has described the systematics of EEG phenomena. In this paper ,the mesoscopic form of the full probability distribution has been tak en more seriously for macroscopic interactions, deriving macroscopic drifts and dif fusions linearly related to sums of their (nonlinear) mesoscopic counterparts, scaling its v ariables to describe interactions among re gionalStatistical Mechanics of Neocortical ... -29- L ester Ingber interactions correlated with observed electrical acti vities measured by electrode recordings of scalp EEG, with apparent success. These results gi ve strong quantitati ve support for an accurate intuiti ve picture, portraying neocortical interactions as having common algebraic or ph ysics mechanisms that scale across quite disparate spatial scales and functional or beha vioral phenomena, i.e., describing interactions among neurons, columns of neurons, and regional masses of neurons. 6.5. Summary SMNI is a reasonable approach to extract more ‘ ‘signal’’out of the ‘ ‘noise’’i nEEG data, in terms of physical dynamical variables, than by merely performing re gression statistical analyses on collateral variables. T olearn more about comple xsystems, inevitably functional models must be formed to represent huge sets of data. Indeed, modeling phenomena is as much a cornerstone of 20th century science as is collection of empirical data [91]. It seems reasonable to speculate on the e volutionary desirability of de veloping Gaussian-Mark ovian statistics at the mesoscopic columnar scale from microscopic neuronal interactions, and maintaining this type of system up to the macroscopic re gional scale. I.e., this permits maximal processing of information [74]. There is much work to be done, b ut modern methods of statistical mechanics ha ve helped to point the way to promising approaches. APPENDIX A: ADAPTIVE SIMULATED ANNEALING (ASA) 1. General Description Simulated annealing (SA) w as developed in 1983 to deal with highly nonlinear problems [92], as an extension of a Monte-Carlo importance-sampling technique de veloped in 1953 for chemical ph ysics problems. It helps to visualize the problems presented by such comple xsystems as a geographical terrain. Forexample, consider a mountain range, with tw o“parameters, ”e.g., along the North−South and East−West directions, with the goal to find the lowest v alleyi nthis terrain. SA approaches this problem similar to using a bouncing ball that can bounce o vermountains from v alleyt ovalley. Start at a high “temperature, ”where the temperature is an SA parameter that mimics the effect of a fast moving particle in a hot object lik eahot molten metal, thereby permitting the ball to mak every high bounces and beingStatistical Mechanics of Neocortical ... -30- L ester Ingber able to bounce o verany mountain to access an yvalley, giv enenough bounces. As the temperature is made relati vely colder,the ball cannot bounce so high, and it also can settle to become trapped in relatively smaller ranges of valleys. Imagine that a mountain range is aptly described by a “cost function. ”Define probability distributions of the twodirectional parameters, called generating distributions since the ygenerate possible v alleys or states to e xplore. Define another distribution, called the acceptance distribution, which depends on the difference of cost functions of the present generated v alleyt ob eexplored and the last sa vedlowest valley. The acceptance distribution decides probabilistically whether to stay in a ne wlower valleyo rt obounce out of it. All the generating and acceptance distributions depend on temperatures. In 1984 [93], it w as established that SA possessed a proof that, by carefully controlling the rates of cooling of temperatures, it could statistically find the best minimum, e.g., the lowest v alleyo four example abo ve.This was good news for people trying to solv ehard problems which could not be solv ed by other algorithms. The bad news was that the guarantee was only good if the ywere willing to run SA forever. In1987, a method of fast annealing (FA) was de veloped [94], which permitted lowering the temperature exponentially f aster,thereby statistically guaranteeing that the minimum could be found in some finite time. However, that time still could be quite long. Shortly thereafter ,Very Fast Simulated Reannealing (VFSR) was de veloped [25], no wcalled Adapti ve Simulated Annealing (ASA), which is exponentially faster than FA. ASA has been applied to man yproblems by man ypeople in man ydisciplines [27,28,95]. The feedback of manyusers regularly scrutinizing the source code ensures its soundness as it becomes more fle xible and powerful. The code is available via the world-wide web (WWW) as http://www .ingber.com/ which also can be accessed anonymous FTP from ftp.ingber.com. 2. Mathematical Outline ASA considers a parameter αi kin dimension igenerated at annealing-time kwith the range αi k∈[Ai,Bi], ( A.1) calculated with the random variable yi, αi k+1=αi k+yi(Bi−Ai),Statistical Mechanics of Neocortical ... -31- L ester Ingber yi∈[−1, 1].( A.2) The generating function gT(y)isdefined, gT(y)=D i=1Π1 2(|yi|+Ti)ln(1+1/Ti)≡D i=1Πgi T(yi), ( A.3) where the subscript ionTispecifies the parameter index, and the k-dependence in Ti(k)for the annealing schedule has been dropped for brevity .Its cumulati ve probability distribution is GT(y)=y1 −1∫...yD −1∫dy′1...dy′DgT(y′)≡D i=1ΠGi T(yi), Gi T(yi)=1 2+sgn (yi) 2ln(1+|yi|/Ti) ln(1+1/Ti).( A.4) yiis generated from a uifrom the uniform distribution ui∈U[0, 1], yi=sgn (ui−1 2)Ti[(1+1/Ti)|2ui−1|−1] . (A.5) It is straightforward to calculate that for an annealing schedule for Ti Ti(k)=T0iexp(−cik1/D), ( A.6) aglobal minima statistically can be obtained. I.e., ∞ k0Σgk≈∞ k0Σ[D i=1Π1 2|yi|ci]1 k=∞.( A.7) Control can be taken o verci,such that Tfi=T0iexp(−mi)whenkf=expni, ci=miexp(−ni/D), ( A.8) wheremiandnican be considered “free” parameters to help tune ASA for specific problems.Statistical Mechanics of Neocortical ... -32- L ester Ingber 3. ASAOPTIONS ASA has o ver100 OPTIONS a vailable for tuning. Afew are most rele vant to this project. 3.1. Reannealing Wheneverdoing a multi-dimensional search in the course of a comple xnonlinear physical problem, inevitably one must deal with different changing sensitivities of the αiin the search. At an ygiv en annealing-time, the range o verwhich the relati vely insensiti ve parameters are being searched can be “stretched out” relati ve tothe ranges of the more sensiti ve parameters. This can be accomplished by periodically rescaling the annealing-time k,essentially reannealing, e very hundred or so acceptance- ev ents (or at some user-defined modulus of the number of accepted or generated states), in terms of the sensitivities sicalculated at the most current minimum value of the cost function, C, si=∂C/∂αi.( A.9) In terms of the lar gestsi=smax,adefault rescaling is performed for each kiof each parameter dimension, whereby a ne windexk′iis calculated from each ki, ki→k′i, T′ik′=Tik(smax/si), k′i=((ln(Ti0/Tik′)/ci))D.( A.10) Ti0is set to unity to begin the search, which is ample to span each parameter dimension. 3.2. Quenching Another adapti ve feature of ASA is its ability to perform quenching in a methodical f ashion. This is applied by noting that the temperature schedule abo ve can be redefined as Ti(ki)=T0iexp(−cikQi/D i), ci=miexp(−niQi/D), ( A.11) in terms of the “quenching factor” Qi.The sampling proof fails if Qi>1asStatistical Mechanics of Neocortical ... -33- L ester Ingber kΣD Π1/kQi/D= kΣ1/kQi<∞.( A.12) This simple calculation shows ho wthe “curse of dimensionality” arises, and also gi vesapossible way of living with this disease. In ASA, the influence of large dimensions becomes clearly focussed on the exponential of the power of kbeing 1/D,a sthe annealing required to properly sample the space becomes prohibitively slow. So, if resources cannot be committed to properly sample the space, then for some systems perhaps the next best procedure may be to turn on quenching, whereby Qican become on the order of the size of number of dimensions. The scale of the power of 1/ Dtemperature schedule used for the acceptance function can be altered in a similar fashion. Ho wever, this does not affect the annealing proof of ASA, and so this may used without damaging the sampling property. 3.3. Self Optimization If not much information is known about a particular system, if the ASA defaults do not seem to w ork very well, and if after a bit of experimentation it still is not clear ho wt oselect values for some of the ASA OPTIONS, then the SELF_OPTIMIZE OPTIONS can be very useful. This sets up a top le velsearch on the ASA OPTIONS themselv es, using criteria of the system as its own cost function, e.g., the best attained optimal value of the system’ scost function (the cost function for the actual problem to be solv ed) for each gi venset of top le velOPTIONS, or the number of generated states required to reach a gi ven value of the system’ scost function, etc. Since this can consume a lot of CPU resources, it is recommended that only a fe wASA OPTIONS and a scaled do wn system cost function or system data be selected for this OPTIONS. Even if good results are being attained by ASA, SELF_OPTIMIZE can be used to find a more ef ficient set of ASA OPTIONS. Self optimization of such parameters can be v ery useful for production runs of complexsystems. 3.4. Parallel Code It is quite dif ficult to directly parallelize an SA algorithm[27], e.g., without incurring very restricti ve constraints on temperature schedules [96], or violating an associated sampling proof [97]. However, theStatistical Mechanics of Neocortical ... -34- L ester Ingber fattail of ASA permits parallelization of de veloping generated states prior to subjecting them to the acceptance test [14]. The ASA_PARALLEL OPTIONS pro vide parameters to easily parallelize the code, using various implementations, e.g., PVM, shared memory ,etc. The scale of parallelization afforded by ASA, without violating its sampling proof, is gi venb yatypical ratio of the number of generated to accepted states. Several experts in parallelization suggest that massi ve parallelization e.g., on the order of the human brain, may tak eplace quite f ar into the future, that this might be somewhat less useful for man yapplications than previously thought, and that most useful scales of parallelization might be on scales of order 10 to 1000. Depending on the specific problem, such scales are common in ASA optimization, and the ASA code can implement such parallelization. ACKNOWLEDGMENTS Ithank Paul Nunez at Tulane Uni versity for se veral discussions on issues of source localization. Data was collected by Henri Be gleiter and associates at the Neurodynamics Laboratory at the State Uni versity of NewYork Health Center at Brooklyn, and prepared by Da vid Chorlian. 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Forthe initial-stimulus a_1 paradigm for alcoholic subject co2a0000364, each figure gi vesdata under 6 electrodes marked in the le gends. The left hand figures represent data for the training calculations; the right hand figures represent data for the testing calculations. The top figures represent av erages over1 0runs of ra wevokedpotential data; the bottom figures represent a verages over10 calculations of canonical momenta indicators using this data. Figure 2. Forthe match second-stimulus a_m paradigm for alcoholic subject co2a0000364, each figure givesdata under 6 electrodes mark ed in the le gends. Descriptions of the four figures are contained in the legend for Figure 1. Figure 3. Forthe no-match second-stimulus a_n paradigm for alcoholic subject co2a0000364, each figure givesdata under 6 electrodes mark ed in the le gends. Descriptions of the four figures are contained in the legend for Figure 1. Figure 4. Forthe initial-stimulus c_1 paradigm for control subject co2c0000337, each figure gi vesdata under 6 electrodes marked in the le gends. Descriptions of the four figures are contained in the legend for Figure 1. Figure 5. Forthe match second-stimulus c_m paradigm for control subject co2c0000337, each figure givesdata under 6 electrodes mark ed in the le gends. Descriptions of the four figures are contained in the legend for Figure 1. Figure 6. Forthe no-match second-stimulus c_n paradigm for control subject co2c0000337, each figure givesdata under 6 electrodes mark ed in the le gends. Descriptions of the four figures are contained in the legend for Figure 1.Statistical Mechanics of Neocortical ... -44- L ester Ingber TABLE CAPTIONS Table 1.Descriptive concepts and their mathematical equi valents in a Lagrangian representation. These physical entities pro vide another form of intuiti ve,but quantitati vely precise, presentation of these analyses. Table 2.Circuitry of long-ranged fibers across most rele vant electrode sites and their assumed time- delays in units of 3.906 msec. Table 3. ASA OPTIONS changes from their defaults used in stage one optimization. Table 4. ASA OPTIONS changes from their use in stage one for stage tw ooptimization. Table 5.Parameters fit by ASA are gi ven, as described in the text, for 3 sets of data for alcoholic subject co2a0000364 and for control subject co2c0000337.Statistical Mechanics of Neocortical ... -Figure 1 - Lester Ingber -12-10-8-6-4-20246810 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Train a_1_co2a0000364 F3 F4 P7 P8 T7 T8 -10-8-6-4-202468 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Test a_1_co2a0000364 F3 F4 P7 P8 T7 T8 -0.2-0.15-0.1-0.0500.050.10.150.20.25 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Train a_1_co2a0000364 F3 F4 P7 P8 T7 T8 -0.15-0.1-0.0500.050.10.150.20.25 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Test a_1_co2a0000364 F3 F4 P7 P8 T7 T8Statistical Mechanics of Neocortical ... -Figure 2 - Lester Ingber -16-14-12-10-8-6-4-2024 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Train a_m_co2a0000364 F3 F4 P7 P8 T7 T8 -16-14-12-10-8-6-4-20246 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Test a_m_co2a0000364 F3 F4 P7 P8 T7 T8 -0.25-0.2-0.15-0.1-0.0500.050.10.150.2 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Train a_m_co2a0000364 F3 F4 P7 P8 T7 T8 -0.3-0.2-0.100.10.20.30.40.5 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Test a_m_co2a0000364 F3 F4 P7 P8 T7 T8Statistical Mechanics of Neocortical ... -Figure 3 - Lester Ingber -14-12-10-8-6-4-20246 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Train a_n_co2a0000364 F3 F4 P7 P8 T7 T8 -12-10-8-6-4-20246 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Test a_n_co2a0000364 F3 F4 P7 P8 T7 T8 -0.3-0.2-0.100.10.20.3 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Train a_n_co2a0000364 F3 F4 P7 P8 T7 T8 -0.3-0.2-0.100.10.20.30.4 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Test a_n_co2a0000364 F3 F4 P7 P8 T7 T8Statistical Mechanics of Neocortical ... -Figure 4 - Lester Ingber -8-6-4-2024681012 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Train c_1_co2c0000337 F3 F4 P7 P8 T7 T8 -10-8-6-4-202468 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Test c_1_co2c0000337 F3 F4 P7 P8 T7 T8 -0.2-0.15-0.1-0.0500.050.10.150.2 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Train c_1_co2c0000337 F3 F4 P7 P8 T7 T8 -0.25-0.2-0.15-0.1-0.0500.050.10.150.2 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Test c_1_co2c0000337 F3 F4 P7 P8 T7 T8Statistical Mechanics of Neocortical ... -Figure 5 - Lester Ingber -16-14-12-10-8-6-4-2024 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Train c_m_co2c0000337 F3 F4 P7 P8 T7 T8 -16-14-12-10-8-6-4-202 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Test c_m_co2c0000337 F3 F4 P7 P8 T7 T8 -0.15-0.1-0.0500.050.10.150.2 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Train c_m_co2c0000337 F3 F4 P7 P8 T7 T8 -0.2-0.15-0.1-0.0500.050.10.15 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Test c_m_co2c0000337 F3 F4 P7 P8 T7 T8Statistical Mechanics of Neocortical ... -Figure 6 - Lester Ingber -14-12-10-8-6-4-2024 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Train c_n_co2c0000337 F3 F4 P7 P8 T7 T8 -12-10-8-6-4-2024 0.10.150.20.250.30.350.40.45Φ (µV) t (sec)Test c_n_co2c0000337 F3 F4 P7 P8 T7 T8 -0.2-0.15-0.1-0.0500.050.10.150.20.25 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Train c_n_co2c0000337 F3 F4 P7 P8 T7 T8 -0.15-0.1-0.0500.050.10.150.20.250.3 0.10.150.20.250.30.350.40.45Π (1/µV) t (sec)Test c_n_co2c0000337 F3 F4 P7 P8 T7 T8Statistical Mechanics of Neocortical ... -Table 1 - Lester Ingber Concept Lagrangian equivalent Momentum ΠG=∂LF ∂(∂MG/∂t) Mass gGG′=∂LF ∂(∂MG/∂t)∂(∂MG′/∂t) Force∂LF ∂MG F=ma δLF=0=∂LF ∂MG−∂ ∂t∂LF ∂(∂MG/∂t)Statistical Mechanics of Neocortical ... -Table 2 - Lester Ingber Site Contrib utions From Time Delays (3.906 msec) F3 − − F4 − − T7 F3 1 T7 T8 1 T8 F4 1 T8 T7 1 P7 T7 1 P7 P8 1 P7 F3 2 P8 T8 1 P8 P7 1 P8 F4 2Statistical Mechanics of Neocortical ... -Table 3 - Lester Ingber OPTIONS Def ault Stage 1Use Limit_Acceptances 10000 25000 Limit_Generated 99999 50000 Cost_Precision 1.0E-18 1.0E-9 Number_Cost_Samples 5 3 Cost_Parameter_Scale_Ratio 1.0 0.2 Acceptance_Frequenc y_Modulus 100 25 Generated_Frequenc y_Modulus 10000 10 Reanneal_Cost 1 4 Reanneal_P arameters 1 0 SMALL_FLO AT 1 .0E-18 1.0E-30 ASA_LIB F ALSE TR UE QUENCH_COST F ALSE TR UE QUENCH_PARAMETERS F ALSE TR UE COST_FILE TR UE F ALSE NO_PARAM_TEMP_TEST F ALSE TR UE NO_COST_TEMP_TEST F ALSE TR UE TIME_CALC F ALSE TR UE ASA_PRINT_MORE F ALSE TR UEStatistical Mechanics of Neocortical ... -Table 4 - Lester Ingber OPTIONS Stage 2Changes Limit_Acceptances 5000 Limit_Generated 10000 User_Initial_P arameters TR UE User_Quench_P aram_Scale[.] 30Statistical Mechanics of Neocortical ... -Table 5 - Lester Ingber Parameter c_1 c_n c_m a_1 a_n a_m F3 a -0.281 0.174 -0.346 0.245 -0.342 0.139 b 0.970 -0.848 1.045 0.606 -0.312 -0.645 <φ>- 0.297 -2.930 -2.922 -0.015 1.127 -0.260 F4 a -0.255 0.243 0.356 -0.401 0.127 -0.066 b 0.799 -0.765 -0.906 -0.303 -0.625 0.778 <φ>- 0.935 -4.296 -5.557 0.427 2.047 -0.650 T7 a -0.784 0.181 0.592 0.267 -0.146 0.232 b -0.552 -1.035 0.500 -0.973 1.005 -0.961 <φ>1 .902 -4.622 -6.080 -2.143 -3.023 -6.085 d1(F3) 0.325 0.993 0.579 0.306 0.898 0.589 d1(T8) 0.358 0.070 0.437 0.056 0.288 0.325 T8 a 0.959 0.781 0.755 0.892 -0.348 -0.398 b 0.624 0.989 0.785 0.701 1.295 1.355 <φ>1 .417 -6.822 -8.742 -2.928 -2.638 -6.005 d1(F4) 0.441 0.101 0.468 0.814 0.138 0.255 d1(T7) 0.376 0.074 0.383 0.613 0.851 0.308 P7 a 0.830 0.778 0.273 0.166 -0.262 0.385 b 0.371 0.498 -1.146 -1.083 1.006 0.720 <φ>2 .832 -4.837 -5.784 -5.246 -6.395 -9.980 d1(T7) 0.291 0.257 0.153 0.157 0.472 0.081 d1(P8) 0.217 0.079 0.717 0.945 0.826 0.186 d2(F3) 0.453 0.604 0.401 0.867 0.517 0.096 P8 a 0.350 0.300 0.618 0.155 -0.534 0.130 b -1.238 -1.378 0.545 -0.942 -0.338 -0.795 <φ>1 .772 -7.231 -8.866 -5.641 -7.539 -9.869 d1(T8) 0.297 0.844 0.645 0.547 0.864 0.706 d1(P7) 0.809 0.634 0.958 0.314 0.862 0.374 d2(F4) 0.569 0.600 0.487 0.426 0.282 0.722

arXiv:physics/0001054 24 Jan 2000%A L. Ingber %A P.L. Nunez %T Statistical mechanics of neocortical interactions: High resolution path-integral calculation of short-term memory %J Phys. Re v. E %V 51 %N 5 %D 1995 Statistical mechanics of neocortical interactions: High resolution path-integral calculation of short-term memory Lester Ingber Lester Ingber Resear ch,P.O. Box 857, McLean, Virginia 22101 ingber@alumni.caltech.edu and Paul L. Nunez Department of Biomedical Engineering ,Tulane University ,New Orleans, Louisiana 70118 pln@bmen.tulane.edu We present high-resolution path-integral calculations of a pre viously de veloped model of short-term memory in neocorte x. These calculations, made possible with supercomputer resources, supplant similar calculations made in L. Ingber ,Phys. Rev. E49,4652 (1994), and support coarser estimates made in L. Ingber,Phys. Rev. A29,3346 (1984). We also present a current experimental conte xt for the rele vance of these calculations using the approach of statistical mechanics of neocortical interactions, especially in the context of electroencephalographic data. PA CSNos.: 87.10.+e, 05.40.+j, 02.50.-r ,02.70.-cStatistical Mechanics of Neocortical ... -2- Ingber and Nunez I. INTRODUCTION This paper describes a higher -resolution calculation of a similar calculation performed in a recent paper [1], using supercomputer resources not a vailable at that time, and are of the quality of resolution presented in a different system using the same path-inte gral code P ATHINT [2]. Amore detailed description of the theoretical basis for these calculations can be found in that paper ,and in pre vious papers in this series of statistical mechanics of neocortical interactions (SMNI) [3-18]. The SMNI approach is to de velop mesoscopic scales of neuronal interactions at columnar le vels of hundreds of neurons from the statistical mechanics of relati vely microscopic interactions at neuronal and synaptic scales, poised to study relati vely macroscopic dynamics at re gional scales as measured by scalp electroencephalograph y(EEG). Rele vant experimental data are discussed in the SMNI papers at the mesoscopic scales, e.g., as in this paper’ scalculations, as well as at macroscopic scales of scalp EEG. Here, we demonstrate that the deri vedfirings of columnar acti vity,considered as order parameters of the mesoscopic system, de velop multiple attractors, which illuminate attractors that may be present in the macroscopic regional dynamics of neocortex. The SMNI approach may be complementary to other methods of studying nonlinear neocortical dynamics at macroscopic scales. Forexample, EEG and magnetoencephalograph ydata have been expanded in a series of spatial principal components (Karhunen-Loe ve expansion). The coefficients in such expansions are identified as order parameters that characterize phase changes in cogniti ve studies [19,20] and epileptic seizures [21,22], which are not considered here. The calculations gi venhere are of minicolumnar interactions among hundreds of neurons, within a macrocolumnar extent of hundreds of thousands of neurons. Such interactions tak eplace on time scales of severalτ,whereτis on the order of 10 msec (of the order of time constants of cortical p yramidal cells). This also is the observ ed time scale of the dynamics of short-term memory .Wehypothesize that columnar interactions within and/or between regions containing man ymillions of neurons are responsible for phenomena at time scales of se veral seconds. That is, the nonlinear e volution as calculated here at finer temporal scales gi vesabase of support for the phenomena observed at the coarser temporal scales, e.g., by establishing mesoscopic attractors at manymacrocolumnar spatial locations to process patterns at larger regions domains. This moti vates us toStatistical Mechanics of Neocortical ... -3- Ingber and Nunez continue using the SMNI approach to study minicolumnar interactions across macrocolumns and across regions. For example, this could be approached with a mesoscopic neural network using a confluence of techniques dra wn from SMNI, modern methods of functional stochastic calculus defining nonlinear Lagrangians, adapti ve simulated annealing (ASA)[23], and parallel-processing computation, as previously reported [16]. Other developments of SMNI, utilizing coarser statistical scaling than presented here, have been used to more directly interf ace with EEG phenomena, including the spatial and temporal filtering observed experimentally [14,15,17,18]. Section II presents a current e xperimental and theoretical context for the rele vance of these calculations. W estress that neocortical interactions tak eplace at multiple local and global scales and that aconfluence of experimental and theoretical approaches across these scales v ery likely will be required to improve our understanding of the physics of neocortex. Section III presents our current calculations, summarizing 10 CPU days of Con vex 120 supercomputer resources in se veral figures. These results support the original coarser arguments gi venin SMNI papers a decade ago [6,8]. Section IV presents our conclusions. II. EXPERIMENTAL AND THEORETICAL CONTEXT A. EEG studies EEG provides a means to study neocortical dynamic function at the millisecond time scales at which information is processed. EEG provides information for cogniti ve scientists and medical doctors. Amajor challenge for this field is the inte gration of these kinds of data with theoretical and e xperimental studies of the dynamic structures of EEG. Theoretical studies of the neocortical medium ha ve inv olved local circuits with postsynaptic potential delays [24-27], global studies in which finite velocity of action potential and periodic boundary conditions are important [28-31], and nonlinear nonequilibrium statistical mechanics of neocorte xt odeal with multiple scales of interaction [3-18]. The local and the global theories combine naturally to form a single theory in which control parameters ef fect changes between more local and more global dynamicStatistical Mechanics of Neocortical ... -4- Ingber and Nunez behavior [31,32], in a manner somewhat analogous to localized and e xtended w av e-function states in disordered solids. Recently,plausible connections between the multiple-scale statistical theory and the more phenomenological global theory were proposed [14]. Experimental studies of neocortical dynamics with EEG include maps of magnitude distribution o verthe scalp [29,33], standard F ourier analyses of EEG time series[29], and estimates of correlation dimension [34,35]. Other studies ha ve emphasized that manyEEG states are accurately described by a fe wcoherent spatial modes exhibiting comple xtemporal behavior [19-22,29,31]. These modes are the order parameters at macroscopic scales that underpin the phase changes associated with changes of physiological state. The recent de velopment of methods to impro ve the spatial resolution of EEG has made it more practical to study spatial structure. The ne whigh resolution methods provide apparent resolution in the 2-3 cm range, as compared to 5-10 cm for con ventional EEG [36]. EEG data were obtained in collaboration with the Swinburne Centre for Applied Neurosciences using 64 electrodes o verthe upper scalp. These scalp data are used to estimate potentials at the neocortical surf ace. The algorithms mak e use of general properties of the head volume conductor .Astraightforw ard approach is to calculate the surface Laplacian from spline fits to the scalp potential distrib ution. This approach yields estimates similar to those obtained using concentric spheres models of the head [36]. Here we report on data recorded from one of us (P.L.N), while a wake and relaxed with closed e yes (the usual alpha rh ythm). The resulting EEG signal has dominant power in the 9-10 Hz range. We Fourier transformed the 64 data channels and passed Fourier coefficients at 10 Hz through our Laplacian algorithm to obtain cortical F ourier coef ficients. In this manner the magnitude and phase structure of EEG was estimated. Atypical Laplacian magnitude and phase plot for 1 sec of EEG is sho wn in Fig. 1. This structure w as determined to be stable on 1-min time scales; that is a verages over1min exhibit minimal minute to minute changes when the psychological/ph ysiological state of the brain is held fix ed. By contrast, the structure is quasi-stable on 1-sec time scales. To showthis we calculated magnitude and phase templates based on an a verage over3min. Wethan obtained correlation coefficients by comparing magnitudes and phases at each electrode position for one second epochs of data with the templates. In this manner we determined that the structure is quasi-stable on 1 sec time scales. That is, correlation coefficients vary from second to second o vermoderate ranges, as sho wn in Fig. 2. Another interestingStatistical Mechanics of Neocortical ... -5- Ingber and Nunez aspect of these data is the periodic behavior of the correlation coefficients; magnitudes and phases undergo large changes roughly e very 6 sec and then return to patterns that more nearly match templates. We hav epreviously considered ho wmesoscopic activity may influence the very large scale dynamics observ ed on the scalp [14]. Insome limiting cases (especially those brain states with minimal cognitive processing), this mesoscopic influence may be sufficiently small so that macroscopic dynamics can be approximated by a quasi-linear “fluid-lik e” representation of neural mass action [28-31]. In this approximation, the dynamics is crudely described as standing w av esi nthe closed neocortical medium with periodic boundary conditions. Each spatial mode may exhibit linear or limit cycle behavior at frequencies in the 2−20 Hz range with mode frequencies partly determined by the size of the corte xand the action potential velocity in corticocortical fibers. The phase structure shown in Fig. 1 may sho wthe nodal lines of such standing w av es. B. Short-term memory SMNI has presented a model of short-term memory (STM), to the extent it of fers stochastic bounds for this phenomena during focused selecti ve attention [1,6,8,37-39], transpiring on the order of tenths of a second to seconds, limited to the retention of 7 ±2items [40]. This is true e venfor apparently exceptional memory performers who, while the ymay be capable of more efficient encoding and retrie val of STM, and while the ymay be more efficient in ‘ ‘chunking’ ’larger patterns of information into single items, ne vertheless are limited to a STM capacity of 7 ±2items [41]. Mechanisms for various STM phenomena ha ve been proposed across man yspatial scales [42]. This “rule” is v erified for acoustical STM, but for visual or semantic STM, which typically require longer times for rehearsal in an hypothesized articulatory loop of individual items, STM capacity appears to be limited to 4 ±2[43]. Another interesting phenomenon of STM capacity e xplained by SMNI is the primac yversus recencyeffect in STM serial processing, wherein first-learned items are recalled most error-free, with last- learned items still more error-free than those in the middle [44]. The basic assumption being made is that apattern of neuronal firing that persists for man yτcycles is a candidate to store the ‘ ‘memory’ ’of activity that g av erise to this pattern. If se veral firing patterns can simultaneously exist, then there is the capability of storing se veral memories. The short-time probability distribution deri vedfor the neocorte x is the primary tool to seek such firing patterns.Statistical Mechanics of Neocortical ... -6- Ingber and Nunez It has been noted that experimental data on velocities of propagation of long-ranged fibers [29,31] and derivedvelocities of propag ation of information across local minicolumnar interactions[4] yield comparable times scales of interactions across minicolumns of tenths of a second. Therefore, such phenomena as STM likely are inextricably dependent on interactions at local and global scales, and this is assumed here. III. PRESENT CALCULATIONS A. Probability distribution and the Lagrangian As described in more detail in a previous paper[1], the short-time conditional probability of changing firing states within relaxation time τof excitatory ( E)and inhibitory ( I)firings in a minicolumn of 110 neurons (twice this number in the visual neocorte x) is givenb ythe following summary of equations. The Einstein summation con vention is used for compactness, whereby an yindexappearing more than once among f actors in an yterm is assumed to be summed o ver, unless otherwise indicated by vertical bars, e.g., | G|. Themesoscopic probability distrib utionPis givenb ythe product of microscopic probability distrib utionspσi,constrained such that the aggre gate mesoscopic excitatory firings ME= Σj∈Eσj,and the aggre gate mesoscopic inhibitory firings MI=Σj∈Iσj. P= G=E,IΠPG[MG(r;t+τ)|MG(r′;t)] = σjΣδ  j∈EΣσj−ME(r;t+τ)  δ  j∈IΣσj−MI(r;t+τ)  N jΠpσj ≈ GΠ(2πτgGG)−1/2exp(−NτLG), ( 1) where the final form is deri vedusing the fact that N>100.Grepresents contributions from both EandI sources. This defines the Lagrangian, in terms of its first-moment drifts gG,its second-moment dif fusion matrixgGG′,and its potential V′,all of which depend sensiti vely on threshold factors FG, P≈(2πτ)−1/2g1/2exp(−NτL),Statistical Mechanics of Neocortical ... -7- Ingber and Nunez L=(2N)−1(˙MG−gG)gGG′(˙MG′−gG′)+MGJG/(2Nτ)−V′, V′= GΣV′′G G′(ρ∇MG′)2, gG=−τ−1(MG+NGtanhFG), gGG′=(gGG′)−1=δG′ Gτ−1NGsech2FG, g=det(gGG′), FG=(VG−a|G| G′v|G| G′NG′−1 2A|G| G′v|G| G′MG′) {π[(v|G| G′)2+(φ|G| G′)2](a|G| G′NG′+1 2A|G| G′MG′)}1/2, aG G′=1 2AG G′+BG G′,( 2) whereAG G′andBG G′are macrocolumnar -averaged interneuronal synaptic ef ficacies, vG G′andφG G′are av eraged means and variances of contrib utions to neuronal electric polarizations, and nearest-neighbor interactions V′are detailed in other SMNI papers [4,6].MG′andNG′inFGare afferent macrocolumnar firings, scaled to efferent minicolumnar firings by N/N∗∼10−3,whereN∗is the number of neurons in a macrocolumn. Similarly ,AG′ GandBG′ Ghave been scaled by N∗/N∼103to keepFGinvariant. This scaling is for con venience only .For neocortex, due to chemical independence of excitatory and inhibitory interactions, the diffusion matrix gGG′is diagonal. The above dev elopment of a short-time conditional probability for changing firing states at the mesoscopic entity of a mesocolumn (essentially a macrocolumnar a veraged minicolumn), can be folded in time overand overb ypath-integral techniques de veloped in the late 1970s to process multi variate Lagrangians nonlinear in their drifts and dif fusions [45,46]. This is further de veloped in the SMNI papers into a full spatial-temporal field theory across regions of neocortex.Statistical Mechanics of Neocortical ... -8- Ingber and Nunez B. PATHINT algorithm The PATHINT algorithm can be summarized as a histogram procedure that can numerically approximate the path integral to a high de gree of accurac ya sasum of rectangles at points Miof height Pi and width ΔMi.For convenience, just consider a one-dimensional system. The path-inte gral representation described abo ve can be written, for each of its intermediate integrals, as P(M;t+Δt)=∫dM′[g1/2 s(2πΔt)−1/2exp(−LsΔt)]P(M′;t) =∫dM′G(M,M′;Δt)P(M′;t), P(M;t)=N i=1Σπ(M−Mi)Pi(t), π(M−Mi)=    1, (Mi−1 2ΔMi−1)≤M≤(Mi+1 2ΔMi), 0, otherwise .(3) This yields Pi(t+Δt)=Tij(Δt)Pj(t), Tij(Δt)=2 ΔMi−1+ΔMi∫Mi+ΔMi/2 Mi−ΔMi−1/2dM∫Mj+ΔMj/2 Mj−ΔMj−1/2dM′G(M,M′;Δt). ( 4) Tijis a banded matrix representing the Gaussian nature of the short-time probability centered about the (possibly time-dependent) drift. Care must be used in de veloping the mesh in ΔMG,which is strongly dependent on the diagonal elements of the diffusion matrix, e.g., ΔMG≈(Δtg|G||G|)1/2.( 5) Presently,this constrains the dependence of the co variance of each variable to be a nonlinear function of that variable, albeit arbitrarily nonlinear ,i norder to present a straightforward rectangular underlying mesh.Statistical Mechanics of Neocortical ... -9- Ingber and Nunez Aprevious paper [1] attempted to circumvent this restriction by taking adv antage of pre vious observations [6,8] that the most likely states of the “centered” systems lie along diagonals in MGspace, a line determined by the numerator of the threshold factor ,essentially AE EME−AE IMI≈0, ( 6) where for neocorte xAE Eis on the order of AE I.Along this line, for a centered system, the threshold f actor FE≈0, andLEis a minimum. However, looking at LI,inFIthe numerator ( AI EME−AI IMI)is typically small only for small ME,since for the neocortex AI I<<AI E. C. Further considerations for high-resolution calculation However, sev eral problems plagued these calculations. First, and lik ely most important, is that it wasrecognized that a Sun workstation was barely able to conduct tests at finer mesh resolutions. This became apparent in a subsequent calculation in a dif ferent system, which could be processed at finer and finer meshes, where the resolution of peaks w as much more satisf actory [2]. Second, it was difficult, if not impossible gi venthe nature of the algorithm discussed abo ve,t odisentangle an ypossible sources of error introduced by the approximations based on the transformation used. The main issues to note here are that the physical boundaries of firings MG=±NGare imposed by the numbers of excitatory and inhibitory neurons per minicolumn in a gi venregion. Physically,firings at these boundaries are unlikely in normal brains, e.g., unless the yare epileptic or dead. Numerically, PATHINT problems with SMNI diffusions and drifts arise for large MGat these boundaries: (a) SMNI has regions of relati vely small dif fusionsgGG′at the boundaries of MGspace. As the ΔMGmeshes are proportional to ( gGG′Δt)1/2,this could require P ATHINT to process relati vely small meshes in these otherwise ph ysically uninteresting domains, leading to kernels of size tens of millions of elements. These small diffusions also lead to large Lagrangians which imply relati vely small contributions to the conditional probabilities of firings in these domains. (b) At the boundaries of MGspace, SMNI can ha ve large negative drifts,gG.This can cause anomalous numerical problems with the Neumann reflecting boundary conditions taken at all boundaries. Forexample, if gGΔtis sufficiently large and ne gative,neg ative probabilities can result. Therefore, thisStatistical Mechanics of Neocortical ... -10- Ingber and Nunez would require quite small Δtmeshes to treat properly ,affecting the ΔMGmeshes throughout MGspace. Aquite reasonable solution is to cut of fthe drifts and diffusions at the edges by Gaussian factors Γ, gG→gGΓ, gGG′→gGG′Γ+(1−Γ)NG/τ, Γ= G=E,IΠexp[−(MG/NG)2/C]−exp(−1/C) 1−exp(−1/C),( 7) whereCis a cutoffparameter and the second term of the transformed diffusion is weighted by NG/τ,the value of the SMNI diffusion at MG=0. Avalue ofC=0. 2wasfound to gi ve good results. However, the use of this cutof frendered the diffusions approximately constant o vertheEand theI firing states, e.g., on the order of NG.Therefore, here the diffusions were taken to be these constants. While a resolution of Δt=0. 5τwastaken for the pre vious PATHINT calculation[1], here a temporal resolution of Δt=0. 01τwasnecessary to get well-de veloped peaks of the e volving distrib ution for time epochs on the order of se veralτ.Asdiscussed in the Appendix of an earlier paper [6], such a finer resolution is quite physically reasonable, i.e., e venbeyond anynumerical requirements for such temporal meshes. That is, defining θin that previous study to be Δt,firings ofMG(t+Δt)for 0≤Δt≤τ arise due to interactions within memory τas far back as MG(t+Δt−τ). Thatis, the mesocolumnar unit expresses the firings of af ferentsMG(t+τ)a ttimet+τas having been calculated from interactions MG(t)a ttheτ-averaged efferent firing time t.With equal lik elihood throughout time τ,any oftheN* uncorrelated efferent neurons from a surrounding macrocolumn can contrib ute to change the minicolumnar mean firings and fluctuations of their Nuncorrelated minicolumnar af ferents. Therefore, forΔt≤τ,a tleast to resolution Δt≥τ/Nand to order Δt/τ,i ti sreasonable to assume that efferents ef fect achange in afferent mean firings of Δt˙MG=MG(t+Δt)−MG(t)≈ΔtgGwith varianceΔtgGG.Indeed, columnar firings (e.g., as measured by a veraged evokedpotentials) are observed to be faithful continuous probabilistic measures of individual neuronal firings (e.g., as measured by poststimulus histograms) [47]. When this cutof fprocedure is applied with this temporal mesh, an additional physically satisfying result is obtained, whereby the ΔMGmesh is on the order of a firing unit throughout MGspace. TheStatistical Mechanics of Neocortical ... -11- Ingber and Nunez interesting physics of the interior region as discussed in pre vious papers is still maintained by this procedure. D. Four models of selecti ve attention Three representati ve models of neocorte xduring states of selecti ve attention are considered, which are effected by considering synaptic parameters within experimentally observed ranges. Amodel of dominant inhibition describes ho wminicolumnar firings are suppressed by their neighboring minicolumns. Forexample, this could be effected by de veloping nearest-neighbor mesocolumnar interactions [5], but the a veraged effect is established by inhibitory mesocolumns (IC) by settingAI E=AE I=2AE E=0. 01N*/N.Since there appears to be relati vely littleI—Iconnectivity ,w eset AI I=0. 0001N*/N.The background synaptic noise is taken to be BE I=BI E=2BE E=10BI I=0. 002N*/N. As minicolumns are observed to ha ve∼110 neurons (the visual corte xappears to ha ve approximately twice this density) [48] and as there appear to be a predominance of EoverIneurons [29], we take NE=80 andNI=30. Assupported by references to e xperiments in early SMNI papers, we tak e N*/N=103,JG=0(absence of long-ranged interactions), VG=10 mV,|vG G′|=0. 1mV,andφG G′=0. 1 mV.Iti sdiscovered that more minima of Lare created, or “restored, ”i fthe numerator of FGcontains terms only in MG,tending to center the Lagrangian about MG=0. Ofcourse, an ymechanism producing more as well as deeper minima is statistically f avored. However, this particular centering mechanism has plausible support: MG(t+τ)=0i sthe state of af ferent firing with highest statistical weight. That is, there are more combinations of neuronal firings σj=±1yielding this state more than an yother MG(t+τ); e.g.,∼2NG+1/2(πNG)−1/2relative tothe states MG=±NG.Similarly, M*G(t)i sthe state of efferent firing with highest statistical weight. Therefore, it is natural to explore mechanisms that f avor common highly weighted efferent and af ferent firings in ranges consistent with f avorable firing threshold factorsFG≈0. The centering effect of the IC model of dominant inhibition, labeled here as the IC ′model, is quite easy for the neocorte xt oaccommodate. F or example, this can be accomplished simply by readjusting the synaptic background noise from BG EtoB′G E,Statistical Mechanics of Neocortical ... -12- Ingber and Nunez B′G E=VG−(1 2AG I+BG I)vG INI−1 2AG EvG ENE vG ENG(8) for bothG=EandG=I.This is modified straightforwardly when re gional influences from long-ranged firingsM‡Eare included [15]. In general,BG EandBG I(and possibly AG EandAG Idue to actions of neuromodulators and JGorM‡Econstraints from long-ranged fibers) are a vailable to force the constant in the numerator to zero, gi ving an extra degree(s) of freedom to this mechanism. (IfB′G Ewould be negative, this leads to unphysical results in the square-root denominator of FG.Here, in all e xamples where this occurs, it is possible to instead find positi veB′G Ito appropriately shift the numerator of FG.) Inthis context, it is experimentally observed that the synaptic sensitivity of neurons eng aged in selecti ve attention is altered, presumably by the influence of chemical neuromodulators on postsynaptic neurons [49]. By this centering mechanism, the model FG IC′is obtained FE IC′=0. 5MI−0. 25ME π1/2(0. 1MI+0. 05ME+10. 4)1/2, FI IC′=0. 005MI−0. 5ME π1/2(0. 001MI+0. 1ME+20. 4)1/2.( 9) The other ‘ ‘extreme’ ’o fnormal neocortical firings is a model of dominant e xcitation, effected by establishing e xcitatory mesocolumns (EC) by using the same parameters {BG G′,vG G′,φG G′,AI I}as in the IC model, b ut setting AE E=2AI E=2AE I=0. 01N*/N.Applying the centering mechanism to EC, B′E I=10. 2andB′I I=8. 62.This yields FE EC′=0. 25MI−0. 5ME π1/2(0. 05MI+0. 10ME+17. 2)1/2, FI EC′=0. 005MI−0. 25ME π1/2(0. 001MI+0. 05ME+12. 4)1/2.( 10)Statistical Mechanics of Neocortical ... -13- Ingber and Nunez Nowi tisnatural to examine a balanced case intermediate between IC and EC, labeled BC. This is accomplished by changing AE E=AI E=AE I=0. 005N*/N.Applying the centering mechanism to BC, B′E E=0. 438andB′I I=8. 62.This yields FE BC′=0. 25MI−0. 25ME π1/2(0. 050ME+0. 050MI+7. 40)1/2, FI BC′=0. 005MI−0. 25ME π1/2(0. 001MI+0. 050ME+12. 4)1/2.( 11) Afourth model, similar to BC ′,for the visual neocorte xi sconsidered as well, BC ′_VIS, where NG is doubled. FE BC′_VIS=0. 25MI−0. 25ME π1/2(0. 050ME+0. 050MI+20. 4)1/2, FI BC′_VIS=0. 005MI−0. 25ME π1/2(0. 001MI+0. 050ME+26. 8)1/2.( 12) E. Results of calculations Models BC ′,EC′,and IC′were run at time resolutions of Δt=0. 01τ,resulting in firing meshes of ΔME=0. 894427 (truncated as necessary at one end point to fall within the required range of ±NE), and ΔMI=0. 547723. To besure of accurac yi nthe calculations, off-diagonal spreads of firing meshes were taken as±5. Thislead to an initial four -dimensional matrix of 179 ×110×11×11=2382 490 points, which was cut down to a kernel of 2 289 020points because the of f-diagonal points did not cross the boundaries. Reflecting Neumann boundary conditions were imposed by the method of images, consisting of a point image plus a continuous set of images leading to an error function [50]. A Convex 120 supercomputer was used, but there were problems with its C compiler ,s ogcc version 2.60 w as built and used. Runs across several machines, e.g., Suns, Dec workstations, and Crays, checked reproducibility of this compiler on this problem. It required about 17 CPU min to build the kernel, and about 0.45 CPU min for eachΔt-folding of the distribution.Statistical Mechanics of Neocortical ... -14- Ingber and Nunez Formodel BC′_VIS, the same time resolution and of f-diagonal range was taken, resulting in firing meshes of ΔE=1. 26491 andΔMI=0. 774597, leading to a k ernel of size 4 611 275 elements. It required about 34 CPU min to build the kernel, and about 0.90 CPU min for each Δtfolding of the distribution. An initial δ-function stimulus was presented at ME≈MI≈0for each model. The subsequent dispersion among the attractors of the systems gi vesinformation about the pattern capacity of this system. Data was printed e very 100 foldings, representing the e volution of one unit of τ.For run BC ′,data were collected for up to 50 τ,and for the other models data were collected up to 30 τ. As pointed out in Sec. II, long-ranged minicolumnar circuitry across re gions and across macrocolumns within regions is quite important in the neocorte xand this present calculation only represents a model of minicolumnar interactions within a macrocolumn. Therefore, only the first fe wτ foldings should be considered as having much physical significance. Figure 3(a) shows the e volution of model BC ′after 100 foldings of Δt=0. 01,or one unit of relaxation time τ.Note the existence of ten well de veloped peaks or possible trappings of firing patterns. The peaks more distant from the center of firing space would be e vensmaller if the actual nonlinear diffusions were used, since the yare smaller at the boundaries, increasing the Lagrangian and diminishing the probability distrib ution. Ho wever, there still are tw oobvious scales. If both scales are able to be accessed then all peaks are a vailable to process patterns, but if only the larger peaks are accessible, then the capacity of this memory system is accordingly decreased. This seems to be able to describe the “7±2” rule. Figure 3(b) shows the e volution after 500 foldings at 5 τ;note that the integrity of the different patterns is still present. Figure 3(c) shows the e volution after 1000 foldings at 10 τ;note the deterioration of the patterns. Figure 3(d) shows the e volution after 3000 foldings at 30 τ;note that while the original central peak has survi ved, nowmost of the other peaks ha ve been absorbed into the central peaks and the attractors at the boundary. Figure 4(a) shows the e volution of model EC ′after 100 foldings of Δt=0. 01,or one unit of relaxation time τ.Note that, while ten peaks were present at this time for model BC ′,now there are only four well de veloped peaks, of which only tw oare quite strong. Figure 4(b) shows the e volution after 1000 foldings at 10 τ;note that only the tw opreviously prominent peaks are no wbarely distinguishable.Statistical Mechanics of Neocortical ... -15- Ingber and Nunez Figure 5(a) shows the e volution of model IC ′after 100 foldings of Δt=0. 01,or one unit of relaxation time τ.While similar to model BC ′,here too there are ten peaks within the interior of firing space. Ho wever, quite contrary to that model, here the central peaks are much smaller and therefore less likely than the middle and the outer peaks (the outer ones prone to being diminished if nonlinear diffusions were used, as commented on abo ve), suggesting that the original stimulus pattern at the origin cannot be strongly contained. Figure 5(b) shows the e volution after 100 foldings at 10 τ;note that only the attractors at the boundaries are still represented. Figure 6(a) shows the e volution of model BC ′_VIS after 100 foldings of Δt=0. 01,or one unit of relaxation time τ.Incomparison to model BC ′,this model exhibits only six interior peaks, with three scales of relati ve importance. If all scales are able to be accessed, then all peaks are a vailable to process patterns, but if only the lar ger peaks are accessible, then the capacity of this memory system is accordingly decreased. This seems to be able to describe the “4 ±2” rule for visual memory .Figure 6(b) shows the evolution after 100 foldings at 10 τ;note that these peaks are still strongly represented. Also note that no wother peaks at lower scales are clearly present, numbering on the same order as in the BC ′ model, as the strength in the original peaks dissipates throughout firing space, but these are much smaller and therefore much less probable to be accessed. As seen in Fig. 6c, similar to the BC ′model, by 15 τ, only the original tw olarge peaks remain prominent. IV.CONCLUSION Experimental EEG results are a vailable for re gional interactions and the evidence supports attractors that can be considered to process short-term memory under conditions of selecti ve attention. There are man ymodels of nonlinear phenomena that can be brought to bear to study these results. There is not much experimental data a vailable for large-scale minicolumnar interactions. However, SMNI offers a theoretical approach, based on experimental data at finer synaptic and neuronal scales, that develops attractors that are consistent with short-term memory capacity .The duration and the stability of such attractors lik ely are quite dependent on minicolumnar circuitry at regional scales, and further study will require more intensi ve calculations than presented here [16]. We hav epresented a reasonable paradigm of multiple scales of interactions of the neocorte xunder conditions of selecti ve attention. Presently ,global scales are better represented e xperimentally ,but theStatistical Mechanics of Neocortical ... -16- Ingber and Nunez mesoscopic scales are represented in more detail theoretically .Wehav eoffered a theoretical approach to consistently address these multiple scales [14-16], and more a phenomenological macroscopic theory [28-32] that is more easily compared with macroscopic data. We expect that future e xperimental efforts will offer more knowledge of the neocorte xa tthese multiple scales as well. ACKNOWLEDGMENTS We acknowledge the use of the Tulane Uni versity Con vex 120 supercomputer for all calculations presented in this paper .Wethank Richard Silberstein of the Centre for Applied Neurosciences for recording PLN’ sEEG data.Statistical Mechanics of Neocortical ... -17- Ingber and Nunez FIGURE CAPTIONS FIG. 1.Magnitude (upper) and phase (lower) at 9 Hz of 1 sec of alpha rhythm is sho wn. The plots represent estimates of potential on the cortical surface calculated from a 64 channel scalp recording (average center-to center electrode spacing of about 2.7 cm). The estimates of cortical potential w av e forms were obtained by calculating spatial spline functions at each time slice to obtain analytic fits to scalp potential distrib utions. Surf ace Laplacian w av eforms were obtained from second spatial deri vatives (in the tw osurface tangent coordinates). Magnitude and phase were obtained from temporal F ourier transforms of the Laplacian w av eforms. This particular Laplacian algorithm yields estimates of cortical potential that are similar to in verse solutions based on four concentric spheres modes of the head. The Laplacian appears to be rob ust with respect to noise and head model errors [36]. The dark and the lighter shaded regions are 90 °out of phase, suggesting quasi-stable phase structure with re gions separated by a fewcentimeters 180 °out of phase (possible standing w av es). Datarecorded at the Swinburne Centre for Applied Neurosciences in Melbourne, Australia. FIG. 2. Changes of alpha rhythm correlation coef ficients based on comparisons of magnitude (solid line) and phase (dashed line) plots of successi ve 1-sec epochs of alpha rh ythm compared with spatial templates based on a verages over3 min of data (similar to Fig. 1). The data sho waquasistable structure with major changes in magnitude or phase about e very 6 sec, after which the structure tends to return to the template structure. FIG. 3. Model BC ′:(a) the evolution at τ,(b) the evolution at 5 τ,(c) the evolution at 10 τ,and (d) the evolution at 30 τ. FIG. 4. Model EC ′:(a) the evolution at τand (b) the e volution at 10 τ. FIG. 5. Model IC ′:(a) the evolution at τand (b) the e volution at 10 τ. FIG. 6. Model BC ′_VIS: (a) the e volution at τ,(b) the evolution at 10 τ,and (c) the e volution at 15τ.Statistical Mechanics of Neocortical ... -18- Ingber and Nunez REFERENCES [1] L.Ingber,‘‘Statistical mechanics of neocortical interactions: P ath-integral evolution of short-term memory,’’Phys. Rev. E49,4652-4664 (1994). [2] L.Ingber,‘‘Path-inte gral evolution of multi variate systems with moderate noise, ’’Phys. Rev. E51, 1616-1619 (1995). [3] L.Ingber,‘‘Tow ards a unified brain theory ,’’J. Social Biol. Struct. 4,211-224 (1981). 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Ingber and Nunez Figure3b Statistical Mechanics of Neocortical ... Ingber and Nunez Figure3c Statistical Mechanics of Neocortical ... Ingber and Nunez Figure3d Statistical Mechanics of Neocortical ... Ingber and Nunez Figure4a Statistical Mechanics of Neocortical ... Ingber and Nunez Figure4b Statistical Mechanics of Neocortical ... Ingber and Nunez Figure5a Statistical Mechanics of Neocortical ... Ingber and Nunez Figure5b Statistical Mechanics of Neocortical ... Ingber and Nunez Figure6a Statistical Mechanics of Neocortical ... Ingber and Nunez Figure6b Statistical Mechanics of Neocortical ... Ingber and Nunez Figure6c

arXiv:physics/0001056v1 [physics.gen-ph] 24 Jan 2000EXACT SOLUTION OF THE RESTRICTED THREE-BODY SANTILLI-SHILLADY MODEL OF H2 MOLECULE A.K.Aringazin1and M.G.Kucherenko2 1Karaganda State University, Karaganda 470074 Kazakstan ascar@ibr.kargu.krg.kz 2Orenburg State University, 13 Pobedy Ave., Orenburg 460352 Russia rphys@osu.ac.ru December 1999 Abstract In this paper, we study the exact solution of the restricted i so- chemical model of H2molecule with fixed nuclei recently proposed by Santilli and Shillady in which the two electrons are assum ed to be bonded/correlated into a quasiparticle called the isoelectronium . Under the conditions that: 1) the isoelectronium is stable; 2) the effective size of the isoelectronium is ignorable, in compar ison to in- ternuclear distance; and 3) the two nuclei are at rest, the Sa ntilli- Shillady model of the H2molecule is reduced to a restricted three- bodysystem essentially similar to a neutral version of the H+ 2ion, which, as such, admits exact solution. Our main result is tha t the restricted three-body Santilli-Shillady approach to H2is capable to fit the experimental binding energy, at the isoelectronium mas s param- eterM= 0.308381 me, although under optimal internuclear distance about 19.6% bigger than the conventional experimental valu e, indi- cating an approximate character of the three-body model. 01 Introduction In this paper, we study isochemical model of the H2molecule recently in- troduced by R. M. Santilli and D. D. Shillady [1], which is cha racterized by the conventional H2model set up plus a short-range attractive Hulten potential interaction between the two electrons originati ng from the deep overlapping of their wave functions at mutual distances of t he order of 1 fm; see also [2]. If one assumes that this attractive potential i s strong enough to overcome Coloumb repulsion between the two electrons, th ey can form electron-electron system called isoelectronium . The isoelectronium is char- acterized by ”bare” mass M= 2me, as a sum of masses of two constituent electrons, charge −2e, radius about 10−11cm, and null magnetic moment. The used Hulten potential contains two real parameters, one of which is the isoelectronium correlation length parameter rc, which can be treated as an effective radius of isoelectronium. The main structural difference between the Santilli-Shilla dy isochemical model and the conventional quantum chemical model of the H2molecule, is that the former admits additional nonlinear, nonlocal, and nonpotential, thus nonunitary effects due to the deep overlapping of the wavepac kets of valence electrons at short distances, which are responsible for the strong molecu- lar bond. In a first nonrelativistic approximation, Santill i and Shillady [1] derived the following characteristics of the isoelectroni um: total rest mass M= 2me, charge −2e, magnetic moment zero, and radius 6 .84323×10−11cm. The valueM= 2meof the rest mass was derived via the assumption of a contact, nonpotential interactions due to the mutual wave-overlapping suffi- ciently strong to overcome the repulsive Coulomb force. The nonpotential character of the bond was then responsible for the essential lack of binding energy in the isoelectronium, and the resulting value M= 2me. However, the authors stressed in [1] that the isoelectronium is expected to have a non-null binding energy, and, therefore, a rest mass smaller than 2 me. One argument presented in [1] is that, when coupled in singlet at very shor t distances, the two electrons eventually experience very strong attractive forces of magnetic type, due to the two pairs of opposing magnetic polarities, r esulting in a bond. The potential origin of the bond then implies the exist ence of a bind- ing energy, resulting in a rest mass of the isoelectronium sm aller than 2 me. Also, in the subsequent paper [3], Santilli pointed out that the isoelectronium can at most admit a small instability. 1As a result of a correlation/bonding between the two electro ns, Santilli and Shillady were able to reach, for the first time, represent ations of the binding energy and other characteristics of H2molecule which are accurate to theseventh digit , within the framework of numerical Hartree-Fock approach toH2molecule viewed as a four-body system with fixed nuclei, and w ith the use of Gaussian screened Coloumb potential taken as an appro ximation to the Hulten potential [1]. On the other hand, the above mentioned strong short-range ch aracter of the electron-electron interaction suggests the use of appr oximation of sta- bleisoelectronium of ignorably small size, in comparison to the internuclear distance [1]. Indeed, under these two assumptions one can re duce the con- ventional four-body structure of the H2molecule to a three-body system (the two electrons are viewed as a single point-like particle). F urthermore, in the Born-Oppenheimer approximation, i.e. at fixed nuclei, we ha ve arestricted three-body system, the Shr¨ odinger equation for which admi tsexact analytic solution. So, we have the original four-body Santilli-Shillady model ofH2molecule, and the three-body Santilli-Shillady model of H2, which is an approximation to it. The former is characterized by, in general, unstable i soelectronium and, thus, sensitivity to details of the electron-electron inte raction, while the latter deals with a single point-like particle (stable isoelectro nium of ignorable size) moving around two fixed nuclei. Clearly, the three-body Santilli-Shillady model of H2molecule can be viewed asH+ 2ion like system. For the sake of brevity and to avoid confusio n with theH+ 2ion itself, we denote H2molecule, viewed as the restricted three- body system, as ˆH2. Note that ˆH2is a neutral H+ 2ion like system. The quantum mechanical problem of the restricted H+ 2ion like systems, associated differential equation, and its exact analytic so lution have been studied in the literature by various authors [6]-[14]. In this paper we present the exact analytic solution of the ab ove indi- cated restricted three-body Santilli-Shillady isochemic al model of the hydro- gen molecule, study its asymptotic behavior, and analyze th e ground state energy, presenting numerical results in the form of tables a nd plots. Our analysis is based on the analytical results obtained for tho roughly studied H+ 2ion. In Sec. 2, we review some features of the four-body Santilli- Shillady model ofH2necessary for our study, and introduce our separation of var iables in 2the Schr¨ odinger equation under the assumption that the iso electronium is a stable quasiparticle of ignorable size. In Sec. 3, we review the exact analytic solution of the H+ 2ion like systems (which includes the ˆH2system), and study their asymptotic behavior at large and small distances between the two nuclei. In Sec. 4, we use the preceding solution to find the binding ene rgy of ˆH2system. We then develop a scaling method and use Ritz’s varia tional approach to check the results. Both the cases of the isoelect ronium ”bare” massM= 2meand of variable mass parameter, M=ηmehave been studied. All the data and basic results of this Section have been colle cted in Table 1. In Sec. 5, we introduce a preliminary study on the applicatio n of Ritz’s variational approach to the general four-body Santilli-Sh illady model of H2, where the isoelectronium is an unstable composite particle, in which case the model re-acquires its four-body structure, yet preserves a strong bond- ing/correlation between the electrons. In the Appendix, we present the results of our numerical calc ulations of the ground state energy of H+ 2ion and of ˆH2system, for different values of the isoelectronium mass parameter M, based on their respective exact solutions, in the form of tables and plots. Our main result is that the restricted three-body Santilli-Shillady isochem- ical model of the hydrogen molecule does admit exact analyti c solution capable of an essentially exact representation of the binding energ y, although under internuclear distance about 19.6% bigger than the conventi onal experimental value. The mass parameter Mof isoelectronium has been used here to fit the experimental value of the binding energy, with the result M= 0.308381me (i.e. about 7 times less than the ”bare” mass M= 2me). In this paper, we assume that some defect of mass effect may have place leading t o decrease of the ”bare” mass M= 2me. We also note that the value M= 0.308381meimplies a binding energy of about 1.7 MeV, which is admittedly rather large. Recent studies by Y. Rui [4] on the correct force law among spinning charges have indi cated the exis- tence of a critical distance below which particles with the s ame charge attract each others. If confirmed, these studies imply that the repul sive Coulomb force itself between two electrons in singlet coupling can b e attractive at a sufficiently small distance, thus eliminating the need to pos tulate an attrac- tive force sufficiently strong to overcome the repulsive Coul omb force. As a result, a binding energy in the isoelectronium structure of the order of 1.7 3MeV cannot be excluded on grounds of our knowledge at this tim e. Clearly, however, that due to the current lack of dynamical d escription of the above mentioned defect of mass, and the obtained resul t that the predicted internuclear distance is about 19.6% bigger than the experimental value, our study is insufficient to conclude that the isoelect ronium is perma- nently stable, and one needs for additional study on the four -body Santilli- Shillady isochemical model of H2, which is conducted in a subsequent paper by one of the authors [5]. 2 Santilli-Shillady model of H2molecule 2.1 General equation The Santilli-Shillady iso-Shr¨ odinger’s equation for H2molecule with short- range attractive Hulten potential between the two electron s can be reduced to the following form [1]: /parenleftBigg −¯h2 2m1∇2 1−¯h2 2m2∇2 2−V0e−r12/rc 1−e−r12/rc+e2 r12(2.1) −e2 r1a−e2 r2a−e2 r1b−e2 r2b+e2 R/parenrightBigg |φ/angb∇acket∇ight=E|φ/angb∇acket∇ight, whereV0andrcare positive constants, and Ris distance between nuclei a andb. By using vectors of center-of-mass system of electrons 1 an d 2,/vector raand /vector rb, originated at nuclei aandb, respectively, we have r1a=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector ra−m2 m1+m2/vector r12/vextendsingle/vextendsingle/vextendsingle/vextendsingle, r 2a=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector ra+m1 m1+m2/vector r12/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2.2) r1b=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector rb−m2 m1+m2/vector r12/vextendsingle/vextendsingle/vextendsingle/vextendsingle, r 2b=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector rb+m1 m1+m2/vector r12/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (2.3) (for electrons we have m1=m2=me). The Lagrangian of the system can be written L=m1˙r2 1 2+m2˙r2 2 2−V(r12)−W(r1a,r1b,r2a,r2b,R), (2.4) 4Here,Vis the potential energy of interaction between the electron s 1 and 2, V(r12) =e2 r12−V0e−r12/rc 1−e−r12/rc, (2.5) andWis the potential energy of interaction between electrons an d nuclei, and between two nuclei, W(r1a,r1b,r2a,r2b,R) =−e2 r1a−e2 r2a−e2 r1b−e2 r2b+e2 R. (2.6) Notice that ˙ r1= ˙r1a= ˙r1b, and ˙r2= ˙r2a= ˙r2b, because/vector r1a=/vector r1b+/vectorRand /vector r2a=/vector r2b+/vectorR, where/vectorRis constant vector. Similarly, /vector ra=/vector rb+/vectorR, /vector r a=m1/vector r1a+m2/vector r2a m1+m2, /vector r b=m1/vector r1b+m2/vector r2b m1+m2. (2.7) Then, Lagrangian (2.4) can be rewritten as L=L(ra,rb,r12), L=M˙r2 a 2+m˙r2 12 2−V(r12)−W(ra,rb,r12,R). (2.8) Here,M=m1+m2is the total mass of the electrons, and m=m1m2/(m1+ m2) is the reduced mass. Corresponding generalized momenta ta ke the form /vectorPM=∂L ∂˙/vector rA=M˙/vector rA. /vector p m=∂L ∂˙/vector r12=m˙/vector r12. (2.9) The system reveals axial symmetry, with the axis connecting two nuclei. Also, for identical nuclei we have reflection symmetry in res pect to the plane perpendicular to the above axis and lying on equal distances from the two nuclei. 2.2 Separation of variables Santilli and Shillady [1] then assume that, as a particular c ase under study in this paper (not to be confused with the general four-body c ase), the two valence electrons of the H2molecule can form a stable quasi-particle of small size due to short-range attractive Hulten potential, such t hat r12≪ra, r 12≪rb. (2.10) 5Therefore, we can ignore r12in Eqs.(2.2) and (2.3), /vector r1a≈/vector r2a≈/vector ra, /vector r 1b≈/vector r2b≈/vector rb. (2.11) The Hamiltonian of the system then becomes ˆH=ˆP2 M 2M+ˆp2 m 2m+V(r12) +W(ra,rb,R), (2.12) where W(ra,rb,R) =−2e2 ra−2e2 rb+2e2 R. (2.13) In this approximation, it is possible to separate the variables ra,bandr12. Namely, inserting |φ/angb∇acket∇ight=ψ(ra,rb,R)χ(r12) into the equation [1] /parenleftBigg −¯h2 2M∇2 ab−¯h2 2m∇2 12−V0e−r12/rc 1−e−r12/rc+e2 r12−2e2 ra−2e2 rb+e2 R/parenrightBigg |φ/angb∇acket∇ight=E|φ/angb∇acket∇ight (2.14) we obtain −¯h2 2M∇2 abψ ψ−¯h2 2m∇2 12χ χ+V(r12) +W(ra,rb,R)−E= 0. (2.15) By separating the variables, we have the following two equat ions: −¯h2 2m∇2 12χ+V(r12)χ=εχ. (2.16) and −¯h2 2M∇2 abψ+W(ra,rb,R)ψ= (E−ε)ψ. (2.17) In this way, under approximation (2.10), the original four- body problem is reduced to a three-body problem characterized by two differe ntial equations: 1) Equation (2.16), which describes the electron-electron system forming the bound quasi-particle state called isoelectronium, wit h ”bare” total mass M= 2meand charge −2e. This equation will not be studied in this paper, since we assume that the isoelectronium is permanently stab le. 2) Equation (2.17), which is the structural equation of the r estricted three- body Santilli-Shillady isochemical model ˆH2, in which the stable isoelectro- nium with ”bare” mass M= 2me, charge −2e, null magnetic moment and 6ignorable size orbits around the two nuclei, hereon assumed to have infinite mass (the Born-Oppenheimer approximation). This paper is devoted to the study of the exact analytic solut ion of the latter equation, and its capability to represent the experi mental data on the binding energy, bond length, and other characteristics of the hydrogen molecule. 3 Exact solution for H+ 2ion like system In this Section, we present analytical solution of the Schr¨ odinger equation forH+ 2ion-like systems in Born-Oppenheimer approximation, we an alyze the associated recurrence relations, and asymptotic behav ior of the solutions at large and small distances between the two nuclei. As it was indicated [1], this problem arises when Santilli-Shillady model of H2is reduced to the restricted three-body problem characterized by Eq. (2.17), which possesses exact solution under appropriate separation of variables. 3.1 Differential equations In Born-Oppenheimer approximation, i.e., at fixed nuclei, t he equation for H+ 2ion-like system for a particle of mass Mand charge qis ∇2ψ+ 2M(E+q ra+q rb)ψ= 0. (3.1) In spheroidal coordinates, x=ra+rb R,1<x< ∞, (3.2) y=ra−rb R,−1<y< 1, (3.3) ϕ,0<ϕ< 2π, (3.4) whereRis a fixed separation distance between the nuclei aandb, and ∇2=4 R2(x2−y2)/parenleftBigg∂ ∂x(x2−1)∂ ∂x+∂ ∂y(1−y2)∂ ∂y/parenrightBigg (3.5) +1 R2(x2−1)(1−y2)∂2 ∂ϕ2. 7We then have from Eq.(3.1) /bracketleftBigg∂ ∂x(x2−1)∂ ∂x+∂ ∂y(1−y2)∂ ∂y+x2−y2 4(x2−1)(1−y2)∂2 ∂ϕ2(3.6) +MER2 2(x2−y2) + 2MqRx/bracketrightBigg ψ= 0. Here, we have used 1 ra+1 rb=4 Rx x2−y2. (3.7) Obviously, Equation (3.6) can be separated by the use of the representation ψ=f(x)g(y)eimϕ, (3.8) under which we have two second-order ordinary differential equations, d dx/parenleftBigg (x2−1)d dxf/parenrightBigg −/parenleftBigg λ−2MqRx −MER2 2x2+m2 x2−1/parenrightBigg f= 0,(3.9) d dy/parenleftBigg (1−y2)d dyg/parenrightBigg +/parenleftBigg λ−MER2 2y2−m2 1−y2/parenrightBigg g= 0, (3.10) whereλis a separation constant (cf. [6]). So, the problem is to iden tify solutions for fandg. 3.2 Recurrence relations By introducing the re-formulations f→(x2−1)m/2f, (3.11) g→(1−y2)m/2g, (3.12) to handle singularities at x=±1 andy=±1 in Eqs.(3.9) and (3.10), respectively, we reach the following final form of the equati ons to be solved: (x2−1)f′′+ 2(m+ 1)f′−(λ+m(m+ 1)−˜ax−c2x2)f= 0 (3.13) and (1−y2)g′′−2(m+ 1)g′+ (λ−m(m+ 1)−c2y2)g= 0, (3.14) 8where we have denoted c2=MER2 2,˜a= 2MqR. (3.15) We shall look for solutions in the form of power series. Subst ituting the power series f=/summationdisplay fkxk, (3.16) g=/summationdisplay gkyk, (3.17) into Eqs. (3.13) and (3.14), we obtain the recurrence relati ons, c2fn−2+ ˜afn−1−(λ−(m+n)(m+n+ 1))fn (3.18) −(n+ 1)(n+ 2)fn+2= 0 and c2gn−2−(λ−(m+n)(m+n+ 1))gn−(n+ 1)(n+ 2)gn+2= 0,(3.19) from which coefficients fkandgkmust be found. Here, f0andg0are fixed by normalization of the general solution. Note that the recurr ence relation (3.18) contains term 2 MqRf n−1raised from the linear term 2 MqRx in Eq.(3.9). In the next two Sections we consider some particular cases of interest prior to going into details of the general solution. These pa rticular solutions are important for the study of the general case. 3.3 The particular case R= 0 In the particular case R= 0, the two nuclei are superimposed, so that the system is reduced to a helium-like system, d dx/parenleftBigg (x2−1)d dxf/parenrightBigg −/parenleftBigg λ+m2 x2−1/parenrightBigg f= 0, (3.20) d dy/parenleftBigg (1−y2)d dyg/parenrightBigg +/parenleftBigg λ−m2 1−y2/parenrightBigg g= 0. (3.21) From recurrence relations (3.18) and (3.19) we obtain the fo llowing particular recurrence sequences, (λ−(m+n)(m+n+ 1))fn−(n+ 1)(n+ 2)fn+2= 0 (3.22) 9and (λ−(m+n)(m+n+ 1))gn−(n+ 1)(n+ 2)gn+2= 0, (3.23) which are equivalent to each other, and can be stopped by putting the sepa- ration constant λ= (m+n)(m+n+ 1) =l(l+ 1), (3.24) withm=−l,...,l. This gives us well known solution for gin terms of Legendre polynomials, g= (1−y2)m/2d dymPl(y), (3.25) wherem=|m|, and Pl=1 2ll!dl dyl(y2−1)l. (3.26) The solution is the well known spherical harmonic function Ylm=NlmPm l(y)eimϕ, (3.27) with normalization constant Nlm=/radicaltp/radicalvertex/radicalvertex/radicalbt(l−m)!(2l+ 1) (l+m)!4π. (3.28) This solution corresponds to the case of an ellipsoid degene rated into a sphere, and we can put y= cosθfor identification with the angular spherical coordi- nates (θ,ϕ). Equation in xcorresponds to the radial part of the well known solution expressed in terms of Laguerre polynomials. 3.4 The particular case q= 0 In the particular case of zero charge, q= 0, we have from Eqs.(3.9) and (3.10) d dx/parenleftBigg (x2−1)d dxf/parenrightBigg −/parenleftBigg λ−c2x2+m2 x2−1/parenrightBigg f= 0, (3.29) d dy/parenleftBigg (1−y2)d dyg/parenrightBigg +/parenleftBigg λ−c2y2−m2 1−y2/parenrightBigg g= 0. (3.30) 10One can see that these equations originate straightforward ly also from the standard wave equation ∇2ψ+k2ψ= 0, in the spheroidal coordinates ( x,y,ϕ ). Recurrence relations (3.18) and (3.19) then become c2fn−2−(λ−(m+n)(m+n+ 1))fn−(n+ 1)(n+ 2)fn+2= 0 (3.31) and c2gn−2−(λ−(m+n)(m+n+ 1))gn−(n+ 1)(n+ 2)gn+2= 0,(3.32) which are equivalent to each other. A general solution for fis given by linear combinations of radial spheroidal functionsR(p) mn(c,x) of first,p= 1, and second, p= 2, kind [7], R(p) mn(c,x) =  ∞′/summationdisplay r=0,1(2m+r)! r!dmn r  −1/parenleftBiggx2−1 x2/parenrightBiggm/2 × (3.33) ×∞′/summationdisplay r=0,1ir+m−n(2m+r)! r!dmn rZ(p) m+r(cx), where, Z(1) n(z) =/radicalbiggπ 2zJn+1/2(z), (3.34) Z(2) n(z) =/radicalbiggπ 2zYn+1/2(z), (3.35) andJn+1/2(z) andYn+1/2(z) are Bessel functions of first and second kind, respectively. The sum in (3.33) is made over either even or od d values of r depending on the parity of n−m. Asymptotics of R(1) mn(c,x) andR(2) mn(c,x) are R(1) mn(c,x)cx→∞−→1 cxcos/bracketleftbigg cx−1 2(n+ 1)π/bracketrightbigg , (3.36) R(2) mn(c,x)cx→∞−→1 cxsin/bracketleftbigg cx−1 2(n+ 1)π/bracketrightbigg . (3.37) Particularly, to have well defined limit at x= 0 we should use only spheroidal function of first kind, R(1) mn(c,x), because Bessel function of second kind, Yn(z), has logarithmic divergence at z= 0. 11General solution for gis given by linear combination of angular spheroidal functions of first and second kind [7], S(1) mn(c,y) =∞′/summationdisplay r=0,1dmn r(c)Pm m+r(y), (3.38) S(2) mn(c,y) =∞′/summationdisplay r=−∞dmn r(c)Qm m+r(y), (3.39) wherePm n(y) andQm n(y) are the associated Legendre polynomials of first and second kind, respectively. Expressions for radial and angular spheroidal functions, a nd correspond- ing eigenvalues λ, for particular values of mandn, are presented in Ref. [7]. Coefficients dmn k(c) are calculated with the help of the following recurrence relation: αkdk+2+ (βk−λmn)dk+γkdk−2= 0, (3.40) where αk=(2m+k+ 2)(2m+k+ 1)c2 (2m+ 2k+ 3)(2m+ 2k+ 5), (3.41) βk= (m+k)(m+k+ 1) +2(m+k)(m+k+ 1)−2m2−1 (2m+ 2k−1)(2m+ 2k+ 3)c2,(3.42) γk=k(k−1)c2 (2m+ 2k−3)(2m+ 2k−1). (3.43) The calculation is made by the following procedure. First, o ne calculates Nm r, Nm r+2=γm r−λmn−βm r Nmr(r≥2), (3.44) Nm 2=γm 0−λmn;Nm 3=γm 1−λmn, (3.45) γm r= (m+r)(m+r+1)+1 2c2/bracketleftBigg 1−4m2−1 (2m+ 2r−1)(2m+ 2r+ 3)/bracketrightBigg (r≥0). (3.46) Second, one calculates the fractions d0/d2randd1/d2p+1with the use of d0 d2r=d0 d2d2 d4···d2r−2 d2r, (3.47) 12d1 d2p+1=d1 d3d3 d5···d2p−1 d2p+1, (3.48) and Nm r=(2m+r)(2m+r−1)c2 (2m+ 2r−1)(2m+ 2r+ 1)dr dr−2(3.49) The coefficients d0, for evenr, andd1, for oddr, are determined via the normalization of the solution. 3.5 The general case In this Section, we consider the general solution of our basi c equations (3.9) and (3.10). To have more general set up, we consider the case o f different charges of nuclei, Z1andZ2. This leads to appearance of additional linear in yterm in Eq.(3.10), so that both the ordinary differential equ ations become of similar structure. Also, we restrict consideration by an alyzing discrete spectrum, i.e. we assume that the energy E <0. Let us denote p=R 2√ −2E, a =R(Z2+Z1), b=R(Z2−Z1). (3.50) Then, Eqs.(3.9) and (3.10), for the general case of different charges of nuclei, can be written as d dx/parenleftBigg (x2−1)d dxfmk(p,a;x)/parenrightBigg +/parenleftBigg −λ(x) mk−p2(x2−1) +ax−m2 x2−1/parenrightBigg fmk(p,a;x) = 0, (3.51) d dy/parenleftBigg (1−y2)d dygmq(p,b;y)/parenrightBigg +/parenleftBigg λ(y) mq−p2(1−y2) +by−m2 1−y2/parenrightBigg gmq(p,b;y) = 0, (3.52) where we assume that the solutions obey |fmk(p,a; 1)|<∞,limx→∞fmk(p,a;x) = 0,|gmq(p,b;±1)|<∞.(3.53) 13The eigenvalues λin Eqs.(3.51) and (3.52) should be equal to each other, λ(x) mk(p,a) =λ(y) mq(p,b). (3.54) The general solution ψ(x,y,ϕ ) of Eq.(3.6) is represented in the following factorized form: ψkqm(x,y,ϕ ;R) =Nkqm(p,a,b)fmk(p,a;x)gmq(p,b;y)exp(±imϕ)√ 2π.(3.55) The normalization coefficients Nkqm(p,a,b) in Eq.(3.55) are represented with the help of derivatives of the eigenvalues, λ(x) mk(p,a) andλ(y) mq(p,b), namely, N2 kqm(p,a,b) =16p R3 ∂λ(y) mq(p,b) ∂p−∂λ(x) mk(p,a) ∂p −1 . (3.56) For a given indices k,q,m, and fixed values of Z1,Z2, andR, the discrete energy spectrum Ecan be determined from Eq.(3.54). This equation has unique solution, p=pkqm(a,b). Then, by solving the relation stemming from (3.50) pkqm(R(Z2+Z1),R(Z2−Z1)) =R 2√ −2E (3.57) in respect to E, we can find the discrete spectrum of energy, Ej(R) =Ekqm(R,Z 1,Z2). (3.58) Number of zeroes, k,q, andm, of the functions g(y),f(x), and exp ±imϕare the angular, radial and azimuthal quantum numbers, respect ively. However, instead ofk,q, andmone can use their linear combinations, namely, N= k+q+m+ 1 is main quantum number and l=q+mis orbital quantum number. To construct the general solution u(z), which is called Coloumb spheroidal function [8] (csf), in terms of angular csfg(y) andradialcsff(x), let us, again, use the form which accounts for singularities at the p ointsz=±1 and z=∞, u(z) = (1−z2)m/2exp[−p(1±z)]v(z). (3.59) Then, we represent v(z) as an expansion, v(z) =∞/summationdisplay s=0as(p,b,λ)ws(z), (3.60) 14in some set of basis functions ws(z). Now, the complexity of the recurrence relations depends on t he basis. In the preceding sections, where the particular cases, R= 0 andq= 0, have been considered, we used a power series representation. One can try other forms of the representation as well. For a good choice of the b asis functions ws(z), we can obtain three-term recurrence relation of the form αsas+1−βsas+γsas−1= 0, (3.61) whereαs,βs, andγsare some polynomials in p,b, andλ. Then, using the tridiagonal matrix ˆAconsisting of the coefficients αs,βs, andγsentering Eq.(3.61), we can write down the equation to find out eigenval uesλ(x) mk(p,a) andλ(y) mq(p,b). Namely, detˆA=F(p,b,λ) = 0. (3.62) The matrix ˆAhas a tridiagonal form. This leads directly to one-to-one co r- respondence between det ˆAand the infinite chain fraction , F(p,b,λ) =β0−α0γ1 β1−α1γ2 β2−...αNγN+1 βN+1−...=β0−α0γ1 β1−α1γ2 β2−...≃QN PN. (3.63) In numerical computations, this relation allows one to find o ut eigenvalues λin an easier way due to simpler algorithm provided by the chai n fraction. Consequently, one can compute the energy and coefficients asof the expan- sion of eigenfunctions g(y) andf(x) by using the chain fraction. The result of this approach in constructing of the solutions depends on the convergence of the chain fraction. Analysis of the converge nce can be made from a general point of view. Sufficient conditions of the conv ergence of the chain (3.63), and of the expansion (3.60), are the following two relations: /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleαs−1γs βs−1βs/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1 4,as+1 as|s→∞∼βs 2αs 1−/parenleftBigg 1−4αsγs β2s/parenrightBigg1/2 . (3.64) Further analysis of the convergence depends on specific choi ce of the basis functionsus(z). 15(i) Series expansion, vs(z) =zs. In this case, the radius, Zv, of conver- gence is Zv= lims→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleas as+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (3.65) Particularly, when as+1/as→0 ats→ ∞ the series (3.60) converges at any z. (ii) For the choice of basis function vs(z) in the form of orthogonal poly- nomials, the sufficient condition for convergence of Fourier series (3.60) is /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleas as+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle s→∞≤1−1 s. (3.66) Below, we consider separately angular and radial csfentering the general solution. 3.5.1 The angular Coloumb spheroidal function For the angular Coloumb spheroidal function ( acsf), it is natural to choose the basis functions vs(y) in the form of associated Legendre polynomials, Pm s+m(y). Indeed, they form complete system in the region y∈[−1,1], and reproduce acsf atp=b= 0 (see Sec. 3.3). Inserting of the expansion gmq(p,b;y) =∞/summationdisplay s=0csPm s+m(y) (3.67) into Eq.(3.52) entails five-term recurrence relation. Howe ver, this relation, which is sometimes used, is not so suitable as the three-term relation. This is because the determinant of the corresponding pentadiago nal matrix can not be represented as a chain fraction. Nevertheless, in the caseb= 0, i.e. forZ1=Z2, this five-terms recurrence relation is reduced to two three -terms recurrence relations, separately for even ( c−2= 0,c0= 1) and odd ( c−1= 0, c1= 1) solutions of Eq.(3.52) presented in previous Section. For the general case b/negationslash= 0, the expansions of g(p,b;y), handling singular- ities at the points y=±1 andy=∞, respectively, as considered by Baber and Hasse [9], are gmq(p,b;y) = exp[ −p(1 +y)]∞/summationdisplay s=0csPm s+m(y), (3.68) 16gmq(p,b;y) = exp[ −p(1−y)]∞/summationdisplay s=0c′ sPm s+m(y), (3.69) These expansions yield three-terms recurrence relation, ρscs+1−κscs+δscs−1= 0, c −1= 0, (3.70) where the coefficients for the case of expansion (3.68) have th e following form: ρs=(s+ 2m+ 1)[b−2p(s+m+ 1)] 2(s+m) + 3, κs= (s+m)(s+m+ 1)−λ, (3.71) δs=s[b+ 2p(s+m)] 2(s+m)−1. To estimate convergence of these expansions, one can use the above made estimation of the convergence, with the following replacem ents:αs→ρs, βs→κs,γs→δs, andas→cs. For the expansion (3.79) we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleρs−1δs κs−1κs/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle s→∞∼/parenleftbiggp s/parenrightbigg2 , (3.72) i.e., atp >1, convergence takes place only at s >2p. We should take into account this condition when choosing minimal number of terms in the chain fraction (3.73) which is sufficient to calculate λ, to a required accuracy. The recurrence relation for the coefficients c′ sof the expansion (3.69) differs from that of Eq.(3.70) by the replacement p→ −pin formulas (3.71). Clearly, this replacement does not change the form of the cha in fraction, F(y)(p,b,λ) =κ0−ρ0δ1 κ1−ρ1δ2 κ2−··· (3.73) So, in both the cases, (3.68) and (3.69), the eigenvalues λcan be found from one and the same equation, F(y)(p,b,λ) = 0. (3.74) In practical calculations with the help of this algorithm, t he infinite chain fraction (3.63) is, of course, replaced by the finite one,F(y) N+1(p,b,λ), in which 17one retains a sufficiently big number Nof terms. Typically, N >10 provides very good accuracy. So, the eigenvalues are computed as the r oots of the polynomial QN+1(p,b,λ) of degree N+ 1, namely, F(y) N+1(p,b,λ) =QN+1(p,b,λ) PN+1(p,b,λ). (3.75) Such a representation allows one to exclude singularities, associated to zeroes of the polynomial PN+1(p,b,λ), from Eq.(3.74). Further, from the definitions (3.63) and (3.75) we obtain the following recurrence relati on for the polyno- mialQk(p,b,λ): Qk+1=Qk¯κN−k−Qk−1¯ρN−k¯δN−k+1, Q −1= 0, Q 0= 1, (3.76) with the use of which one can find QN+1. Here, the coefficients ¯ κs, ¯ρs, and ¯δs differ from that of Eq.(3.71) by the factor (1 + κ2 s)−1/2. This factor does not change the recurrence relation (3.70). However, it makes po ssible to avoid accumulating of big numbers at intermediate computations. Indeed, from Eq.(3.71) for κsit follows that the leading coefficients of the polynomials Qkwould behave as k4k, for example, for k= 4 we would have 416, if we would not made the above mentioned renormalization of the co efficientsρs, κs, andδs. The eigenvalue is found as an appropriate root of the polyno mial QN+1(p,b,λ(y)). Clearly, for big N, there is no way to represent in general the roots ofQN+1analytically so one is forced to use numerical computations . In the numerical computations, to pick up the appropriate ei genvalue λ(y) mq(p,b) amongN+1 roots of the polynomial QN+1it is necessary to choose some starting value of λ. For example, one can put the starting value at the pointp=b= 0, where λ(y) mq(0,0) = (q+m)(q+m+ 1). The first step is to increase discretely p→p+ ∆pandb→b+ ∆bbeginning from the starting pointp=b= 0, at fixed values of mandq, and the second step is to find λ(y) mq(p+ ∆p,b+ ∆b) with the help of Eq.(3.74). Repeating these steps one can findλ(y) mqnumerically as a function of pandbin some interval of interest. Also, asymptotics of λwhich will be studied in Sec. 3.6 are of much help here to choose the appropriate root. For example, for b= 0 andN= 5 we obtain numerically from the determinant of the tridiagonal matrix consist- ing of the coefficients defined by Eq.(3.71), with −κson the main digonal, andρsandδson the upper and lower adjacent diagonals respectively, the 18polynomial, detˆA= 0.003λ6−0.2λ5+(0.2p2+5.5)λ4−(6.3p2+56)λ3+(1.5p4+66p2+231)λ2 (3.77) −(19p4+ 226p2+ 277)λ+p6+ 44p4+ 186p2. Only one of its six roots has asymptotics, λ|p→0= 0.667p2−0.0148p4+O(p5), (3.78) which reproduces, to a good accuracy, the asymptotics (3.12 4). So, this is the desired root to be used in subsequent calculations. Also , observe the decrease of the numerical coefficients at higher degrees of λwhich control the convergence. Note that, at p≫1, theacsf is concentrated around the points y=±1 so that expansion (3.68) converges slowly. In this case one u ses another, more appropriate, expansions, gmq(p,b;y) = (1−y2)m/2exp[−p(1 +y)]∞/summationdisplay s=0cs(1 +y)s, (3.79) gmq(p,b;y) = (1−y2)m/2exp[−p(1−y)]∞/summationdisplay s=0c′ s(1−y)s. (3.80) Evidently, expansion (3.79) converges faster in the region [−1,0] while the expansion (3.80) converges faster in the region [0 ,1]. Here, the coefficients cs of the expansion (3.79) obey the three-term recurrence sequ ence (3.70), with ρs= 2(s+ 1)(s+m+ 1), κs=s(s+ 1) + (2s+m+ 1)(2p+m) +b−λ, (3.81) δs=b+ 2p(s+m). It is remarkable to note that expansions (3.79) and (3.80) co nverge at any y, and the corresponding chain fractions (3.63) converge at an ypsince cs+1 cs|s→∞∼2p s,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleρs−1δs κs−1κs/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle s→∞∼4p s. (3.82) Similarly, the coefficients c′ sobey the same relation, with the replacement b→ −bin Eq.(3.81). 19In practical calculations, one can use a combination of expa nsions (3.68) and (3.79). Namely, the procedure is: from expansion (3.68) one finds eigen- values while the eigenfunctions are calculated from to Eq.( 3.79). Of course, both solutions (3.79) and (3.80) should be sewed, for exampl e, at the point y= 0, because the recurrence relations do not determine, in th is case, a general normalization of the coefficients csandc′ s. Particularly, the sewing condition, which defines the normalization of csandc′ s, has the form /summationdisplay s=0cs=/summationdisplay s=0c′ s. (3.83) To derive the asymptotics of acsf and its eigenvalues we can use an expansion in Laguerre polynomials, gmq(p,b;y) = (1−y2)m/2exp[−p(1±y)]∞/summationdisplay s=0csLm s+m(2p(1±y)),(3.84) Lm n(z) =ezz−m n!dn dzn(e−zzn+m). (3.85) The insertion of this expansion into Eq.(3.52) and the use of the differential equation for Laguerre polynomials, zd2 dz2Lm n(z) + (1−z+m)d dzLm n(z) +nLm n(z) = 0 (3.86) yield recurrence relation (3.70). For the case of positive s ign in Eq.(3.84), we should put ρs=−(s+m+ 1)/parenleftBigg s+ 1 +b 2p/parenrightBigg , κs=−(2s+m+1)/parenleftBigg s+m+ 1 +b 2p−2p/parenrightBigg +(s+m)(m+1)+b−λ,(3.87) δs=−s/parenleftBigg s+m+b 2p/parenrightBigg . 203.5.2 The radial Coloumb spheroidal function The radial Coloumb spheroidal function ( rcsf) obviously should be written in a form suitable to handle singularities at the points x= 1 andx=∞, namely, fmk(p,a;x) = (1−x2)m/2exp[−p(x−1)]f(x). (3.88) So, the equation for f(x) takes the form (x2−1)f′′(x)+[−2p(x2−1)+2(m+1)x]f′(x)+[−λ+m(m+1)+2pσx]f(x) = 0, (3.89) where we have denoted σ=a 2p−(m+ 1). In the case when the expansion f(x) =/summationdisplay s=0asus(x) (3.90) implies a three-terms recurrence relation, the eigenvalue sλ(x) mk(p,a) can be found from the chain fraction equation, F(x)(p,a;λ) = 0. (3.91) Also, the expansion which is of practical use has been consid ered by Jaffe [10]. In this case, the expansion series (3.90) becomes f(x) = (x+ 1)σ/summationdisplay s=0asχs, (3.92) whereχ= (x−1)/(x+ 1) is Jaffe’s variable. By inserting (3.92) into the equation for the function f(x), we get recurrence relation (3.61), where the coefficients are αs= (s+ 1)(s+m+ 1), βs= 2s2+ (2s+m+ 1)(2p−σ)−a−m(m+ 1) +λ= (3.93) = 2s(s+ 2p−σ)−(m+σ)(m+ 1)−2pσ+λ, γs= (s−1−σ)(s−m−1−σ). Also, for Jaffe series expansion, we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleαs−1γs βs−1βs/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle s→∞=1 4/parenleftbigg 1−4p s/parenrightbigg +O/parenleftBiggp2 s2/parenrightBigg , (3.94) 21i.e., the chain fraction converges at p >0. One can see also that the Jaffe expansion converges at any x. In addition, the function f(x) can be expanded in associated Laguerre polynomials, f(x) = (x+ 1)σ∞/summationdisplay s=0asLm s+m(¯x),¯x= 2p(x−1). (3.95) In this case,the recurrence relation is of three-terms form , and the coefficients are αs=−(s+m+ 1)/bracketleftBigga 2p−(s+ 1)/bracketrightBigg = (s+m+ 1)(s−m−σ), βs=−(2s+m+ 1)/bracketleftBigga 2p−(s+m+ 1)/bracketrightBigg + (3.96) +2p(2s+m+ 1)−(s+m)(m+ 1)−a+λ, γs=−s/bracketleftBigga 2p−(s+m)/bracketrightBigg =s(s−1−σ). As to numerical computation of the eigenvalues λ(x), expansions (3.92) and (3.95) are equivalent because the chain fraction depends, i n fact, only on βs andαsγs+1. Indeed, by comparing Eq.(3.93) and Eq.(3.96), one can easi ly see that in both cases βsandαsγs+1are the same. Evidently, it then follows that the associated chain fractions are equivalent to each o ther. However, we should note that Jaffe’s recurrence sequence, in general, is more stable, while Laguerre expansion (3.95) is more suitab le to find out the asymptotics of fmk(p,a;x). Also, we note that the associated ”radial” polynomials QN+1, the root λ(x)(p,a) of which should be found, contain a much bigger number of ter ms, in comparison to the ”angular” case. So, practically finding of radial eigenvalues is much harder than that of angular eigenvalues. At equal charges of nuclei, Z1=Z2, the equation for g, and the recurrence relation for gk, are the same as they are in the particular case q= 0 considered in Sec. 3.4. A general solution for gis then given by acsf (3.38) and (3.39), with coefficients dmn rgiven by recurrence relation (3.40). In the reminder of this Section we would like to note that, in g eneral, solving the recurrence relations can be made equivalent to s olving associated 22ordinary differential equations by making the ztransform. In many cases theztransform helps to solve recurrence relations. Namely, one defines the function Z(z) =∞/summationdisplay s=0as zn(3.97) associated to the coefficients asentering Eq.(3.61) viewed as a function of discrete variable s. Forαs,βs, andγsgiven by Eq.(3.93) we obtain from Eq.(3.61) z(z−1)2Z′′+/bracketleftBig (1−m)z2+ 2(2p−σ−1)z+ 2σ+m+ 1/bracketrightBig Z′+ (3.98) +/bracketleftBigg (σ+m)(m+ 1) + 2pσ−λ+(m+σ)σ z/bracketrightBigg Z= 0. For the coefficients cs, we define Y(z) =∞/summationdisplay s=0cs zn. (3.99) and forρs,κs, andδsgiven by Eq.(3.81) we obtain from Eq.(3.70) z(z−1)2Y′′+/bracketleftBig (τ−2)z2+ 2(2p−τ+ 1)z+τ/bracketrightBig Y′+ (3.100) +/bracketleftbigg (m(m+ 1)−mτ)z−2p(m+ 1) + (τ+m)(m+ 1) +b−λ−τ z/bracketrightbigg Z− −m(m−τ+ 1)c0z= 0, where we have denoted τ=b 2p+m+ 1. (3.101) If one has solved these differential equations for Z(z) andY(z), then, by making the inverse ztransform, one can find the expansion coefficients as andcs(and thus the general solution of the problem). 3.6 Asymptotics of csfand their eigenvalues To analyze the exact solution, which is of rather complicate dnonclosed form (infinite series) given in the previous Sections, it is much i nstructive to derive its asymptotics, which can be represented in a closed form. I n this Section, we present the asymptotics at large ( R→ ∞) and small ( R→0) distances between the nuclei, with a particular attention paid to the g round state. 233.6.1 Asymptotics at R→ ∞ For increasing distances Rbetween the nuclei, at fixed quantum numbers k, q, andm, we have increasing values of the parameters p,a, andb, p= (−2E)1/2R/2→ ∞, a= (Z2+Z1)R→ ∞, b= (Z2−Z1)R→ ±∞. (3.102) Let us introduce the notation α=a 2p=Z2+Z1√ −2E, β=b 2p=Z2−Z1√ −2E(3.103) and assume that α∼1 andβ∼1. acsfatR→ ∞. Let us consider the asymptotic expansion of acsf. In this case, the equation for the Whittaker function, Mκ,µ(y), builds ansatz around the poles y=±1. Here, the solution is constructed in two overlapping interv als,D−= [−1,y1] andD+= [y2,1], withy2<y1. Then, the asymptotics of acsfgmq(p,2pβ;y) in the interval D−have the form gmq(p,2pβ;y) =d− Γ(m+ 1) 2Γ/parenleftBig κ+1+m 2/parenrightBig Γ/parenleftBig κ+1−m 2/parenrightBig 1/2 × ×Mκ,m/2/parenleftBig 2p(1 +y) + 2(κ+β) ln1−y 2/parenrightBig √1−y2[1 +O(p−1)], y∈ D−.(3.104) while in the interval D+it is gmq(p,2pβ;y) =d+ Γ(m+ 1) 2Γ/parenleftBig κ′+1+m 2/parenrightBig Γ/parenleftBig κ′+1−m 2/parenrightBig 1/2 × ×Mκ′,m/2/parenleftBig 2p(1−y) + 2(κ′+β) ln1+y 2/parenrightBig √1−y2[1 +O(p−1)], y∈ D+.(3.105) Here, the coefficients d−andd+(d2 −+d2 += 1) are defined by the relations d−=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinπ(2κ′−m−1) sinπ(2κ−m−1) + sinπ(2κ′−m−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/2 × (3.106) 24×sgn/bracketleftBigg−cosπ(κ−(m+ 1)/2) sinπ(κ′−(m+ 1)/2)/bracketrightBigg d+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinπ(2κ−m−1) sinπ(2κ−m−1) + sinπ(2κ′−m−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/2 . (3.107) rcsfatR→ ∞. Now, let us consider the asymptotic expansion of rcsf. The replacements x→ −y,p→ −p,α→ −βconvert the radial equation around the point x= 1 to the angular equation around the point y=−1. Thus, the corresponding asymptotics of rcsf are directly related to the above found asymptotics of acsf. Thercsf, normalized to the first order in p, has the form fmk(p,2pα;x) =1 m!/bracketleftBigg2(k+ +m)! k!(x2−1)/bracketrightBigg1/2 × ×Mκ,m/2(2p(x−1)) [1 +O(p−1)]. (3.108) Since the first index of Whittaker function in Eq.(3.108) is κ=k+(m+1)/2, the function can be expressed in terms of Laguerre polynomia ls. Energy at R→ ∞. In the limit R→ ∞, the Coloumb two-center problem is evidently reduced to two separate problems of Coloumb centers, with the charges Z1andZ2. Each of the atoms, eZ1andeZ2, is characterized by a set of parabolic quantum numbers, [n,n1,n2,m] and [n′,n′ 1,n′ 2,m], which are related to each other by the relations n=n1+n2+m+ 1, n′=n′ 1+n′ 2+m+ 1. (3.109) The number kof zeroes of rcsfcoincides with the number n1, for the angular functions of the left center, eZ1, and with the number n′ 1, for the angular functions of the right center, eZ2. A series expansion in inverse power of Rcan be obtained in the form E[nn1n2m](Z1,Z2,R) =−Z2 1 2n2−Z2 R+3Z2n∆ 2R2Z1−Z2n2 2R3Z2 1(6∆2−n2+1)+ (3.110) 25+Z2n3 16R4Z4 1[Z1∆(109∆2−39n2−9m2+ 59)−Z2n(17n2−3∆2−9m2+ 19)]+ +ε5 R5+ε6 R6+O/parenleftbigg1 R7/parenrightbigg , where ∆ = n1−n2, andε5,6are defined via the expressions, ε5=−n3 64Z3 1[nZ(1065∆4−594n2∆2+ 1230∆2−234m2∆2+ 9m4+ (3.111) +33n4−18n2m2−18m2+105−138n2)+4n2 Z∆(21∆2−111n2+63m2−189)]. ε6=−n4 64Z4 1[nZ∆(−2727∆4+ 2076n2∆2−5544∆2+ 1056m2∆2−93m4− (3.112) −273n4+ 78n2m2+ 450m2−1533 + 1470 n2) + 2n2 Z(−207∆4+ 1044n2∆2+ +2436∆2−576∆2m2−42n2+ 371−162m2+ 42m2n2−89n4+ 15m4)+ +2n3 Z∆(3∆2−69n2−117−33m2)], wherenZ=nZ2/Z1. Eq.(3.110) gives the multipole expansion in the electrosta tic energy of the interaction between the atom eZ1and the far-distant charge Z2(so called eZ1-terms). Note that expansion (3.110) can be obtained by ordinary pert urbation techniques as well. Indeed, the degrees of Z1display the orders of the mul- tipole moment of the atom eZ1. The series of terms corresponding to the other atom, eZ2, is obtained from Eq.(3.110) with the use of self-evident replacements, Z1↔Z2,n→n′, ∆→∆′, andn2→n′ 2. Finally, the energy of the ground state 1 sσgof the molecular ion, for which Z1=Z2= 1 (equal charges of nuclei), can be written, to a high accura cy, as E1000(1,1,R) =−1 2−9 4R4−15 2R6−213 4R7−7755 64R8−1733 2R9−86049 16R10−O/parenleftbigg1 R11/parenrightbigg . (3.113) 263.6.2 Asymptotics at R→0 Energy at R→0. In the case of positive total charge, Z=Z1+Z2>0, and atR→0, we can use perturbative approach to Z1eZ2problem, without using a separation of variables. Namely, the Hamiltonian of the system Z1eZ2is represented as the sum ˆH=ˆHUA+ˆW=ˆP2 2m−Z1 r1−Z2 r2. (3.114) The operator ˆHUAis usually chosen as the Hamilton operator of the so called united atom , ˆHUA=ˆP2 2m−Z rc, (3.115) which is placed on the z-axis at the point z=z0, z0=/parenleftbigg −1 2+Z2 Z/parenrightbigg R=/parenleftbigg1 2+Z1 Z/parenrightbigg R. (3.116) The point (0 ,0,z0) is called center of charges due to the fact that it lies at the distances R1=Z2 ZRandR2=Z1 ZR, (3.117) from the left and right atoms, respectively. We choose a spherical coordinate system, ( rc,ϑc,ϕ), with the origin at point (0,0,z0), and the angle ϑcmeasured from z-axis. Then, the eigenstates ψUA Nlmof the operator ˆHUAare ψUA Nlm(/vector rc) =RNl(rc)Ym l(ϑc,ϕ), (3.118) while the eigenvalues are given by EUA Nlm=−Z2 2N2. (3.119) The matrix WNl′m′ Nlmof the perturbation operator ˆWis diagonal on the func- tionsψUA Nlm(/vector rc) of the atom if z0is defined by Eq.(3.116). Below, the first two terms of the expansion of energy in powers of Rare given, ENlm(Z1,Z2,R) = 27−Z2 2N2−2Z1Z2[l(l+ 1)−3m2] N3l(l+ 1)(2l−1)(2l+ 1)(2l+ 3)(ZR)2+O((ZR)3).(3.120) For the ground state of the Z1eZ1system, with equal values of the charges, one can find the following expression for the energy, up to the second order of perturbation: E(2) 000(Z1,Z1,R) = Z2/bracketleftbigg −1 2+1 6(ZR)2−1 6(ZR)3+43 2160(ZR)4−1 36(ZR)5lnZR+···/bracketrightbigg . (3.121) csfatR→0. For fixed quantum numbers we have, at R→0, p= (−2E)1/2R/2→0, a= (Z2+Z1)R→0, b= (Z2−Z1)R→0. (3.122) Let us denote α=a 2p=Z2+Z1√ −2E=σ+m+ 1, β=b 2p=Z2−Z1√ −2E. (3.123) In this notation, the energy is E=−Z2/(2α)2. Let us consider asymptotics ofcsfof the ground state of the molecular ion. acsfatR→0. The power series expansion of acsfg00(p,2pβ;y) in small parameter pcan be obtained by expanding it in the Legendre polynomials. For the eigenvalue λ(y) 00(p,2pβ), we then get λ(y) 00(p,2pβ) = (1−β2)/bracketleftbigg2 3p2−2 135p4(1 + 11β2) +O(p6)/bracketrightbigg . (3.124) rcsfatR→0. To expand rcsff00(p,2p(1 +σ);x), atp→0,σ=O(p2), we use Jaffe’s expansion, f00(p,2p(1 +σ);x) = exp ( −px)(1 +x)σ∞/summationdisplay s=0asχs, χ=x−1 x+ 1,(3.125) whereas’s obey three-term recurrence relation with the coefficients (3.93). 28For the eigenvalue λ(x) 00(p,2p(1 +σ)), we get λ(x) 00(p,2p(1 +σ)) =σ(1 + 2p) +σ2(1 + 4pln4pγ) +o(p5). (3.126) rcsfof the ground state of the molecular ion, Z1=Z2= 1, can be presented as f00(p,2p(1 +σ);x) = exp ( −px)(1 +x)σ[1 +σ2Li2(χ) +o(p4)],(3.127) whereχ= (x−1)/(x+1) is Jaffe’s variable and Li 2(χ) is dilogarithm function, Li2(χ) =∞/summationdisplay n=1χn n2=−χ/integraldisplay 0ln(1−ξ)dξ ξ. (3.128) The ground state energy is defined as a function of three param eters,Z1,Z2, andR, λ(y) 00(p,2pβ) =λ(x) 00(p,2p(1 +σ)) (3.129) Combining Eqs.(3.124), (3.126) and (3.129), we get series e xpansion for the ground state energy of the Z1eZ2system in the form E000(Z1,Z2,R) = −1 2Z2+2 3Z1Z2(ZR)2−2 3Z1Z2(ZR)3+2 5Z1Z2/parenleftbigg 1−64Z1Z2 27Z2/parenrightbigg (ZR)4− −8 45Z1Z2/bracketleftbigg5Z1Z2 Z2ln (2RZγ) + 1−199Z1Z2 12Z2/bracketrightbigg (ZR)5+o((ZR)5).(3.130) Comparing Eqs.(3.121) and (3.130) we see that the terms prop ortional to (ZR)2and (ZR)3coincide. The next order corrections makes a difference; Eq.(3.121) obtained by the second-order perturbation is of less accuracy. A practically achieved accuracy of the first-order perturbat ion (3.120) and of Eq.(3.130) is the same; at ZR< 0.1, the discrepancy is not bigger than 1%, and becomes sharply smaller with the increase of the paramet erZR. 3.6.3 Quasiclassical asymptotics AtR→ ∞, foreZ1solutions we have λ(x) mk(p,2pα) =−2p(2κ−α)−κ(2κ−α−m)+ (3.131) 29+κ 2p(2κ2−3κα+α2−m2) +o(p−2), λ(y) mq(p,2pβ) = 2p(2χ+β)−χ(2χ+ 2β−m)− (3.132) −χ 2p(2χ2+ 3χβ+β2−m2) +o(p−2). From the equality λ(x) mk=λ(y) mq, we get the expansion for Ej(R) which coincides with the asymptotics (3.110), up to terms of the order of R−2. In the limit R→0, the following expansions are justified, λ(x) mk=/bracketleftBigga 2p−(k+ 1/2)/bracketrightBigg2 +O(p2), (3.133) λ(y) mq= (l+ 1/2)2+p2 2−b2 8(l+ 1/2)2+O(p4). (3.134) and we get, by using the equation λ(x)=λ(y), ENlm(R) =−1 2/parenleftbiggZ1+Z2 N/parenrightbigg2 −R2Z1Z2(Z1+Z2)2 4N3(l+ 1/2)5[(l+1/2)2−3m2].(3.135) Expression (3.135) for the energy coincides with the asympt otics (3.120), up toO(l−2). Clearly, an accuracy of the quasiclassical Eqs.(3.131)- (3.135) becomes higher for a greater number of zeroes of the solution s,kandq. However, even for the ground state, k=q=m= 0, these equations give a good approximation for the energy in both limiting cases, R→0 and R→ ∞. Also, we note that corresponding numerical calculations s howed that for the intermediate values of Rthe termsEj(R) can be determined within the quasiclassical approach with accuracy of about 5 %, or more [11]. 4 Scaling method and binding energy of three- body Santilli-Shillady isochemical model ˆH2 To find the ground state energies of H+ 2andˆH2, we use computations of the 1sσterms ofH+ 2ion and of ˆH2based on the above presented exact csf solution by solving the corresponding equations λ(x)=λ(y). 30The angular and radial eigenvalues λare found as solutions of the equa- tions containing infinite chain fractions presented in prev ious Sections which should be interrupted and then solved numerically, to a requ ired accuracy. Our primary interest is the study of Santilli-Shillady mode lˆH2system. However, we present the results for H+ 2ion as well to check our calculations and to use them in the scaling method described below. Also, the reader should keep in mind that we are primarily int erested in ascertaining whether there exist a non-zero value of Rfor which a fully stable and point-like isoelectronium permits an exact representa tion of the binding energy ofH2molecule. It should be noted that the ground state electronic energy is obtained as a function of the parameter Rdue to Eq.(3.58). By adding to it the internuclear potential energy 1 /R, we obtain the totalground state energy of the system, so that at some value R=Rmin, the total energy Enecessarily has aminimum , if the system is stable. This is the way to determine uniquely the internuclear distance under an exact representation of the total energy E. To have an independent check of the result for the total groun d state en- ergy of ˆH2(with the stable and point-like isoelectronium) obtaining from the exact solution, we develop a scaling method based on the orig inal Schr¨ odinger equation for H+ 2ion like system. Namely, it appears that one is able to cal- culate the ground state energy as a function of RforanyH+ 2ion like system with equal charges of nuclei, Z1=Z2, provided that one knows the ground state energy as a function of Rfor theH+ 2ion itself. It should be pointed out that the scaling method does not depend on the obtained solut ion because it reflects, in fact, the scaling properties of the Schr¨ odin ger equation itself. In addition, we use below Ritz’s variational approach to H+ 2like systems to find out the approximate value of the ground state energy of ˆH2, as well as to check the result provided by the exact solution, and to d emonstrate the accuracy of the variational approximation. Our general remark is that in both approaches, the exact solu tion and Ritz’s variational solution, we use Born-Oppenheimer appr oximation (fixed nuclei). Clearly, taking into account the first order correc tion, i.e., zero harmonic oscillations of the nuclei in H2around their equilibrium positions, we achieve greater accuracy. But we still have a significant inaccuracy in the value of diss ociation energy due to the fact that H2system has the lightest possible nuclei (two 31single protons). To estimate this inaccuracy, one can invoke Morse’s potenti al customarily used for diatomic molecules. In particular, the analysis fo rH2molecule shows that the ground state energy of harmonic oscillations of the nuclei receives 1.4% correction due to the first anharmonic term. 4.1 Exact representation of binding energies of H+ 2ion and three-body ˆH2system Thecsfbased computations for H+ 2ion were presented, for example, by Teller [12], Bates et al. [13], and Wind [14], and we do not repeat this study here for brevity, while we shall just describe the procedure and present our final numerical results in Appendix, Table 2. In particular, Teller presented a plot of the resulting func tionE1sσ(R) which provides a good accuracy, and Wind used the exact solut ion to present a table of energy values in seven decimal places for distance values ofRup to 20 a.u. in steps of 0.05 a.u. However, these results cannot be used directly in our case since the repulsive potential between the nuclei, 1 /R, has not been accounted for, and, as the main reason, we have the isoelectr onium instead of one single electron. In the Appendix, we present the results obtained from the csfbased recurrence relations by numerical calculations for ordina ryH+ 2ion and for ˆH2system, at M= 2me. These results are presented in Tables 2 and 4. Tables 3 and 5 have been derived from Tables 2 and 4, respectiv ely, by simple adding internuclear potential energy 1 /R, to obtain the totalenergy of the system. In Table 2, we present the 1 sσgelectronic term of H+ 2. In Table 3, we present the total energy of H+ 2. In Table 4, we present the 1 sσgterm of ˆH2, at the mass M= 2me. In Table 5, we present the total energy of ˆH2, at the massM= 2me. Also, in Table 6, we present the total minimal energies of ˆH2andoptimal distancesR, for various values of the isoelectronium mass parameter, M=ηme. All the data of these Tables are purely theoretical and, additionally, we plot them in Figures 1–8, for the reade r convenience. Figures 6 and 8 give more detailed view on the interval 0 .26<η < 0.34. The analysis of the data in Tables 2–5 is simple. Namely, one s hould iden- tify the minimal value of the energy in each Table. One can use Figures 1–4 32for visual identification of the minima, and then turn to the c orresponding Tables 2–5, to reach a higher numerical accuracy. Let us consider, as an example, Table 2. One can see that the en ergy minimum for H+ 2is E1sσ=−2.0 a.u. atR= 0 a.u. (4.1) Note thatE1sσis the electronic energy, i.e. the internuclear repulsion h as not been taken into account here. Our remark is that this ener gy value corresponds to He+ion due to the fact that two H+ 2nuclei are superimposed and formHenucleous at R= 0. Let us consider now Table 3. In this Table, one can find the line |2.0| − 0.602634 |corresponding to visual minimal value of the energy. To iden tify a more precise value of the minimal energy, one should use the interpolation of the data. This gives us the minimum of the total energy of H+ 2, E=−0.6026346 a.u. at R= 1.9971579 a.u. (4.2) This theoretical value represents rather accurately the kn own experimental valueEexper[H+ 2] =−0.6017 a.u. [15] for H+ 2ion, thus establishing the valid- ity of our csfbased calculations. For completeness, we note that the expe ri- mental dissociation energy of H+ 2ion isDexper[H+ 2]≃0.0974 a.u. = 2 .65 eV, and the internuclear distance Rexper[H+ 2]≃2.00 a.u. = 1.0584˚A. Let us now consider Table 4. One can see that the energy minimu m for ˆH2, atM= 2me, is E1sσ=−16.0 a.u. atR= 0 a.u. (4.3) Note thatE1sσonly yields the isoelectronium’s energy, i.e. the internuc lear repulsion has not been taken into account here. Our remark is that this energy value corresponds to the Heatom, where the two electrons form a stable point-like isoelectronium of mass M= 2me. Let us consider now Table 5 which is of striking interest for o ur study. In this Table, one can find the line |0.250| −7.61428940411169996 |cor- responding to the visual minimal value of the energy. To iden tify a more precise value of the minimal energy, one should use the inter polation of the data given in this Table. This gives us the minimum of the tota l energy of ˆH2, E=−7.617041 a.u. at R= 0.258399 a.u. (4.4) 33This theoretical value is in quite good agreement with the pr eceding the- oretical result by Santilli and Shillady obtained via struc turally different variational numerical method, Evar=−7.61509174 at Rvar= 0.2592 (see third column of Table 1 in Ref.[1]). It is quite naturally to o bserve that this variational energy is a bit higher (by 0 .002 a.u.) than the above one obtained from the exact solution (as it is expected to be for any variat ional solution). However, this exact theoretical value (4.4) does not meet th e experimental valueEexper[H2] =−1.17 a.u. [15] known for H2molecule. Indeed, adopted approximation that the isoelectronium is point-like, stab le, and has mass M= 2meleads us to the theoretical value (4.4) while the known exper imental value,Eexper[H2] =−1.17 a.u., differs much from it. Essentially the same conclusion is due to numerical program SASLOBE by Santilli and Shillady [1], where Gaussian screened Colou mb potential in- teraction between the electrons, rather than the stable poi nt-like isoelec- tronium approximation, has been used to achieve final precis e fit ofE= −1.174474 a.u., with the obtained bond length R= 1.4011 a.u., at the iso- electronium correlation length rc= 0.01125 a.u. (see Table 1 in Ref.[1]). We discuss on this issue in Sec. 5. Our remark is that, due to Table 5, the experimental value E=−1.17 a.u. is fitted by the distance R= 0.072370 a.u. However, this energy value is not minimal and thus can not be ascribed reasonable physical treatment in Table 5. Our conclusion from the above analysis is that we have two mai n possi- bilities to overcome this sharp discrepancy between our the oretical and the experimental binding energy values which has place at M= 2me: 1. Consider unstable isoelectronium, i.e. the four-body Sa ntilli-Shillady model ofH2molecule; 2. Treat the mass Mof isoelectronium as a free parameter, instead of fixing it to M= 2me, assuming thus some defect of mass discussed in Introduction, in order to fit the experimental data on H2molecule. The first possibility will be considered in a subsequent pape r because it needs in application of different technique, while the secon d possibility can be studied within the three-body Santilli-Shillady model under consideration to which we turn below. 34In the next Section, we develop simple formalism allowing on e to deal with the mass and charge of isoelectronium viewed as free paramet ers, and arrive at the conclusion (see Table 6) that the restricted three-bo dy Santilli-Shillady model ofH2molecule is capable to fit the experimental binding energy, w ith the total mass of isoelectronium equal to M= 0.308381me, although with the internuclear distance about 19.6% bigger than the exper imental value. 4.2 The scaling method In order to relate the characteristics of H+ 2ion like system to that of thor- oughly studied H+ 2ion, we develop scaling method based on the Schr¨ odinger equation. The neutral ˆH2system with stable point-like isoelectronium is an example of H+ 2ion like system in which we are particularly interested here . Below, we develop scaling method for the case of arbitrary ma ss and charge of the particle. Let us write the Schr¨ odinger equation for a particle of the r escaled charge e→ −ζe, (4.5) (we turn here from e=−1 to−e= 1 representation), and the rescaled mass me→ηme, (4.6) withequal charges of nuclei, + eZ1= +eZ2= +eZ, /bracketleftbigg −¯h2 2ηme∇2 r−ζZe2 ra−ζZe2 rb+Z2e2 Rab/bracketrightbigg ψ=Eψ. (4.7) whereηandζare scaling parameters, and Rabis distance between the nuclei. The condition Z1=Z2is an essential point to stress here because owing to which we can successfully develop the scaling method. We int roduce the unit of length, r0=1 ηζZrB≡1 ηζZ¯h2 mee2, (4.8) whererBis Bohr’s radius. Dividing Eq.(4.7) by ζZe2, and multiplying it by r0, we get /bracketleftbigg −¯h2 ηζZm ee2r01 2∇2 r−r01 ra−r01 rb+r0/parenleftBiggZ ζ/parenrightBigg1 Rab/bracketrightbigg ψ=r0E ζZe2ψ. (4.9) 35We introduce dimensionless entities ρ=r/r0,ρa=ra/r0,ρb=rb/r0, and R=Rab/r0. Then, Laplacian in Eq.(4.9) becomes r2 0∇2 r=∇2 ρ. Further, introducing unit of energy, E0=ηmeζ2Z2e4 ¯h2≡ηζ2Z2mee4 ¯h2, (4.10) we have dimensionless energy ε=E/E 0so that Eq.(4.9) can be rewritten as /bracketleftbigg −1 2∇2 ρ−1 ρa−1 ρb+1 (ζ ZR)/bracketrightbigg ψ=εψ. (4.11) Note that, at η= 1,ζ= 1, andZ= 1, the constants r0(η,ζ,Z ) and E0(η,ζ,Z ) reproduce ordinary atomic units, r0(1,1,1) =rB=¯h2 mee2, E 0(1,1,1) = 2EB=mee4 ¯h2, (4.12) and we recover the case of H+ 2ion. On the other hand, in terms of dimen- sionless entities the original Schr¨ odinger equation for H+ 2ion is /bracketleftbigg −1 2∇2 ρ−1 ρa−1 ρb+1 R/bracketrightbigg ψ0=ε(R)ψ0, (4.13) whereR=Rab/rB. Comparison of Eq.(4.11) and Eq.(4.13) shows that by puttingR= (ζ/Z)Rin Eq.(4.11), we obtain the equation, /bracketleftbigg −1 2∇2 ρ−1 ρa−1 ρb+1 R/bracketrightbigg ψ=ε(R)ψ, (4.14) which identically coincides with the original Eq.(4.13). T he difference is that Eq.(4.14) is treated in terms of the rescaled units,r0(η,ζ,Z ) andE0(η,ζ,Z ), instead of the ordinary Bohr’s units, rBandEB. As the result, we have one and the same form of Schr¨ odinger equation for any H+ 2like system charac- terized by equal charges of nuclei. This makes a general grou nd to calculate some characteristic entity of any H+ 2like system when one knows its value forH+ 2ion. Particularly, one can easily derive RabandEfor the system with arbitrary parameters η,ζ, andZfrom their values, Rab[H+ 2] andE[H+ 2] = 2EB−ε(R), 36obtained for H+ 2ion (for which η= 1,ζ= 1, andZ= 1). Indeed, since for arbitraryη,ζ, andZ R=ζ ZR=ζ ZRab r0=ζ ZRab rBηζZ, (4.15) we can establish the following relationship between the dis tances correspond- ing to arbitrary ZζZsystem and H+ 2ion, Rab=R[H+ 2] ηζ2. (4.16) It is remarkable to note that the dependence on Zdisappeared in Eq.(4.16). In the case of isoelectronium of mass M= 2meand charge −2e, we have η= 2 andζ= 2, so that Rab=R[H+ 2] 8. (4.17) Also, the energy E(R) ofZζZsystem and energy ε(R) ofH+ 2ion are related to each other according to the equation, E(R) =ηζ2Z2/parenleftBiggmee4 ¯h2/parenrightBigg ε(R). (4.18) 4.2.1 The case M= 2me So, in the case of isoelectronium of mass M= 2meand charge −2e, we get E(Rab) = 8ε(R). (4.19) Note however that the factor ζ/Z= 2 arised due to R= (ζ/Z)Ris hidden here so that in order to calculate the values of E(Rab) andRabfromε(R) andRrespectively one should multiply εby 8 andRby 1/4. As the result, in accordance with the scaling method the poin ts can be calculated due to the following rule: (R, E)→(R, E + 1/R)→(R/4,8E)→(R/4,8E+ 4/R),(4.20) for Tables 2 →3→4→5. One can easily check numerically that these prop- erties indeed hold true for the presented Tables. Thus, the s caling method 37can be used instead of the independent numerical calculations for ˆH2system if one has the data for H+ 2ion. It is highly important to note here that the energy minimum in Table 3 is notrescaled to the energy minimum in Table 5 due to the absence of energy scaling between these Tables; see Eq.(4.20), from which one can observe that (8E+ 4/R) can not be expressed as n(E+ 1/R), wherenis a number. So one needs to identify minimum in Table 5 independently (afte r calculating all the points), rather than direct rescale the minimum from Table 3 to try to get minimum for Table 5. 4.2.2 The case M=ηme For a more general case of isoelectronium mass, M=ηme, and charge −2e, we should keep the following sequence of calculations: (R, E)→(R, E + 1/R)→(R 2η,4ηE)→(R 2η,4ηE+2η R). (4.21) starting from Table 2 to obtain, at the last step, the table of values (similar to Table 5) from which we should extract a minimal value of the energy and the corresponding optimal distance, at each given value of mass η. The result of the analysis of a big number of such tables is collected in Tab le 6, where the interval 0.26< η < 0.34 appears to be of interest; M=η, in atomic units. Plots of the data of Table 6 are presented in Figures 5 and 7 (Fi gures 6 and 8 give more detailed view on the interval of interest) show th at Emin(M)≃ −3.808M, R opt(M)≃0.517 M, (4.22) to a good accuracy. Note that Emin(M) unboundedly decreases with the increase ofM(there is no local minimum), so we can use a fit, instead of the minimization in respect with M. From this Table, we obtain the following final fit of the binding energy for the restricted three-body Santilli-Shillady model ofH2molecule: M= 0.308381me, E =−1.174475 a.u. , R = 1.675828 a.u. ,(4.23) where the mass parameter Mof the isoelectronium has been varied in order to meet the experimental energy Eexper[H2] =−1.174474 a.u. = −31.9598 eV. 38Using this value of mass, M= 0.308381me, we computed the total energy as a function of the internuclear distance R, and depicted it in Figure 9 to illustrate that R= 1.675828 a.u. indeed corresponds to a minimal value of the energy. Note that the predicted optimal distance R= 1.675828 a.u. = 0.886810 ˚Aappears to be about 19.6% bigger than the conventional exper i- mental value Rexper[H2] = 1.4011 a.u. = 0 .742˚A. This rather big (19.6%) discrepancy can not be ascribed to th e Born- Oppenheimer approximation used in this paper since it gives relatively small uncertainty in the energy value, even in the case of H2molecule. We stress here that in the Born-Oppenheimer approximation, the three -body problem (the Schr¨ odinger equation) can be given exact solution owi ng to separation of the electronic and nuclear degrees of freedom while the fu ll three-body problem (accounting for the wave functions of the nuclei, et c.) can not be solved exactly. In a strict consideration, we should calculate the dissociation energy ofH2 molecule,D= 2E0−E−Enucl, to make comparison to the experimental value, Dexper[H2]≃0.164 a.u. = 4 .45 eV [15]. Here, E0=−0.5 a.u. = −13.606 eV is the ground state energy of separate H-atom and Enuclis the energy of zero mode harmonic oscillations of the nuclei, with the expe rimental value Enucl exper[H2]≃0.01 a.u. = 0 .27 eV [15]. One can see that the zero mode energyEnucl(which is taken to be Enucl= 0, in the Born-Oppenheimer approximation) is estimated to be less than 1% of the predict edE. The leading anharmonic correction to the harmonic oscillation energy is estimated to be 1.4% of Enucl, i.e. it is of the order of 0 .00014 a.u. = 0 .004 eV, in the case of H2molecule. So, in total the Born-Oppenheimer approximation makes only up to 1% uncertainty, which is obviously insufficie nt to treat the predictedR= 1.675828 a.u. as an acceptable value, from the experimental point of view. Note that, at the given M, we can not ”fix” Rto be equal to the de- sired experimental value Rexper= 1.4011 a.u. unless we shift Eto some nonminimal value, which is, as such, meaningless. Conversely, if we wou ld fit experimental Rexperby varying M, we were obtain Eminbetween −1.52 a.u. and−1.33 a.u. (see Table 6), which is much deviated from the experim en- talEexper. In other words, the relation between EandR, governed by the Schr¨ odinger equation, is such that at some value of Rthere is a minimum of Eso thatRis not some kind of free parameter here since the system tends to minimize its own energy. In accordance to the exact soluti on of the model, 39our single free parameter, M, can not provide us with the exact fit of both the experimental values, EexperandRexper. Thus, we arrive at the conclusion that the three-body Santil li-Shillady model ofH2molecule yields the result (4.23), which indicates that the as- sumption of stable point-like isoelectronium builds a crud eapproximation to the general (four-body) Santilli-Shillady model. This m eans that we are forced to possess that the isoelectronium is not stable point-like quasi-particle, to meet the experimental data on H2molecule. 4.3 Variational solution In studying H+ 2ion like systems, one can use Ritz variational approach to obtain the value of the ground state energy as well. This appr oach assumes analytical calculations, which are easier than that used in finding the above exact solution but they give approximate value of the energy. It is helpful in making simplified analysis of the system. This can be made for the general case of isoelectronium total mass, which eventually underg oes some ”defect” while its ”bare” total mass is assumed to be M= 2me. Ritz variational solu- tion of the H+ 2like problem yields, of course, similar result for the energ y of ˆH2. Below, we present shortly results of our calculations. How ever, we stress that the variational solution is given here just to make some support to the exact solution, and to see the typical order of the variation al approximation. Using hydrogen ground state wave function and one-paramete r Ritz vari- ation, we obtain the following expression for the energy of H+ 2like system: E(ρ) =−1 2e2 a0+e2 a01 ρ(1 +ρ)e−2ρ+ (1−2 3ρ2)e−ρ 1 + (1 +ρ+1 3ρ2)e−ρ, (4.24) whereρ=R/a0is variational parameter. For the general case of mass m and charge q=ζe, Eq.(4.24) can be rewritten in the following form: E(ρ,ζ) =Me4ζ2 ¯h2/parenleftbigg −1 2+F(ρ)/parenrightbigg , (4.25) where F(ρ) =1 ρ(1 +ρ)e−2ρ+ (1−2 3ρ2)e−ρ 1 + (1 +ρ+1 3ρ2)e−ρ. (4.26) 40Numerically, the function F(ρ) reaches minimum at the value ρ= 2.5, which should be used in the above expression for E(ρ,ζ). So, putting ζ= 1 we obtain the variational value of H+ 2ion energy, E(ρ) =−0.565. Note, to make a comparison, that we have the value Eexact=−0.6026 due to the exact solution (4.2), and the value Eexper[H+ 2] =−0.6017 as the experimental value of the energy of H+ 2ion. Thus, the optimal distance between the protons in H+ 2ion isRm=a0ρ= 2.5 a.u., and the obtained variational energyEis slightly higher than both the values EexactandEexper, as it is normally expected to be in the variational approach. Now, we should replace electron by isoelectronium to describe the associated ˆH2model. Substituting M= 2meandζ= 2, we see that the r.h.s. of Eq.(4.25) contains overall fact or 8, in comparison to the H+ 2ion case (M=meandζ= 1), ˜E(ρ) = 8|2EB|/parenleftbigg −1 2+F(ρ)/parenrightbigg , (4.27) The function F(ρ) remains the same, and its minimum is reached again at ρ= 2.5. Then, energy of H2molecule due to Eq.(4.27) is ˜E(ρ) =−8|2EB|0.565 = −4.520 a.u. This value should be compared with the one given by Eq .(4.4). Below, we collect the above mentioned data and results of thi s Section in Table 1. E, a.u. R, a.u. H+ 2ion, exact theory ( N=16) -0.6026346 1.9971579 H+ 2ion, experiment [15] -0.6017 2.00 3-body ˆH2,M=2me, exact theory ( N=16) -7.617041 0.258399 3-body ˆH2,M=2me, var. theory [1] -7.61509174 0.2592 3-body ˆH2,M=0.381 me, exact theory ( N=16) -1.174475 1.675828 4-body H2,rc=0.01125 a.u., var. theory [1] -1.174474 1.4011 H2, experiment -1.174474 1.4011 Table 1: The total ground state energy Eand the internuclear distance R. 415 Concluding remarks In this paper we have shown that the restricted three-body Sa ntilli-Shillady isochemical model of the hydrogen molecule admits an exact a nalytic solu- tion capable of representing the molecular binding energy i n a way accu- rate to the sixth digit, E=−1.174475 a.u., and the internuclear distance R= 1.675828 a.u., which is about 19.6% bigger than the convention al ex- perimental value, Rexper[H2] = 1.4011 a.u. We should emphasize that the presented exact analytical sol ution includes infinite chain fractions. They still need numerical computa tion to reach the characteristic values of H+ 2ion like systems, such as the ground state energy, with the understanding that these values can be reached with any needed accuracy. For example, at the lengths of the chain fractions N= 100 and N= 50 for the angular and radial eigenvalues, one achieves acc uracy of the ground state energy of about 10−12. The general (four-body) Santilli-Shillady isochemical mo del ofH2cannot be, apparently, solved exactly, even in Born-Oppenheimer a pproximation, so that Ritz variational approach can be applied here to get the approximate values of the ground state energy and corresponding internu clear distance. Ritz variational approach is a good instrument to analyze fe w-body prob- lems, and restricted H2molecule is such a system. It is wellknown that the variational solution of the ordinary model of H2molecule includes rather com- plicated analytical calculations of the molecular integra ls, with the hardest part of work being related to the exchange integral. Particu larly, evaluated exchange integral for H2molecule is expressed in terms of a special func- tion (Sugiura’s result, 1927). It is quite natural to expect that even more complications will arise when dealing with the Hulten poten tial. The reason to consider the general four-body Santilli-Shil lady model of H2 molecule, after the made analysis of H+ 2like system approximate approach to it, is that the stable point-like isoelectronium ˆH2model-based theoreti- cal prediction does not meet the experimental data on H2molecule, for the ”bare” isoelectronium mass M= 2me, although we achieved essentially ex- act representation of the binding energy taking M= 0.308381me. Also, this stable point-like isoelectronium (three-body) model does not account for es- sential effect existing in the general (four-body) model. Th is effect is related to the potential barrier between the region associated to th e attractive Hulten potential,r12< r0, and the region associated to repulsive Coloumb poten- 42tial,r12>r0, wherer0is the distance between the electrons at which Hulten potential is equal to Coloumb potential, V(r12) = 0; see Eq.(2.5). Char- acteristics of the barrier can be extracted from the functio nV(r12). The barrier is finite for the used values of the parameters V0andrcso that the electrons penetrate it. The two 1 selectrons are thus simultaneously in two regimes, the first is strongly correlated regime due to short -range attractive Hulten potential (isoelectronium) and the second is weakly correlated regime due to the ordinary Coloumb repulsion. Also, there exist a tr ansient regime corresponding to the region about the equilibrium point, r12≃r0, i.e. inside the barrier. Schematically, one could thought of that the el ectrons are, for instance, 10% in the isoelectronium regime, 1% in the transi ent regime, and 89% in the Coloumb regime. We stress that in the three-body ap proach to H2molecule considered in this paper we have 100% for the isoele ctronium regime. Numerical computation by Santilli and Shillady [1] based on Gaussian transform techniques and SASLOBE computer program has show nexcel- lentagreement of the general four-body model with experimental data on H2molecule. They used Gaussian screened Coloumb potential as an ap- proximation to the Hulten potential. It would be instructiv e to use Ritz variational approach, which deals with analytical calculations, in studying the four-body Santilli-Shillady model of H2molecule. One can try it first for the Gaussian screened Coloumb potential, or exponential sc reened Coloumb potential in which case there is a hope to achieve exact analy tical evalua- tion of the Coloumb and exchange integrals. Being a different approach, this would give a strong support to the numerical results on the gr ound state energy obtained by Santilli and Shillady. Also, having anal ytical set up one can make qualitative analysis of the four-body Santilli-Sh illady model of H2 molecule. However, we should to note that these potentials, being approxi- mations to the Hulten potential, will yield some approximat e models, with corresponding approximate character of the results. 43Appendix We useN= 16 power degree approximation, the polynomials Q(x) NandQ(y) N, to find both the radial, λ(x)(p,a), and angular, λ(y)(p,b), eigenvalues of the csf.QN’s are obtained by the use of recurrence relations (3.76) and defini- tions of the coefficients αs,βs,γs,ρs,κs, andδs, where we put b= 0, i.e. Z1=Z2= 1, and quantum number m= 0. Each of the two polynomials has 16 roots for λfrom which we select one root which is appropriate due to its asymptotic behavior at R→0. Numerical solution of the equation λ(x)(p,a) =λ(y)(p,b) gives us the list of values of the electronic ground state energyE(R) =E1sσ(R), which corresponds to 1 sσgterm of the H+ 2ion, as a function of the distance Rbetween the nuclei. Table 2 presents the result, where no interpolation has been used. Numerical computatio n of each point in Table 2 took about 88 sec on ordinary Pentium desktop compu ter. Below, we present some useful numerical values enabling one to convert atomic units, at which me=e= ¯h= 1, to the other units. Note also that for the energy 1 a.u. ≡1 hartree, and for the length 1 a.u. ≡1 bohr. Atomic units in terms of the other units 1 a.u. of mass, me 9.10953 ·10−28gramms 1 a.u. of charge, e 1.60219 ·10−19Coloumbs 1 a.u. of action, ¯ h 1.05459 ·10−27erg·sec 1 a.u. of length,¯h2 mee20.529177 ·10−8cm 1 a.u. of energy,mee4 ¯h2 27.2116 eV 1 a.u. of time,¯h3 mee4 2.41888 ·10−17sec 1 a.u. of velocity, e2/¯h2.18769 ·108cm/sec α=e2 ¯hc1/137.0388 Conversion of the energy units a.u. eV Kcal·mole cm−1 a.u. 1 27.212 6.2651·1022.1947·105 eV 3.6749·10−21 23.061 8065.48 Kcal·mole 1.5936·10−34.3364 ˙10−21 3.4999·102 cm−14.5563 ·10−61.2398 ˙10−42.8573·10−31 44R, a.u. E(R), a.u. R, a.u. E(R), a.u. 0.0 -1.99999999761099225 2.01 -1.10013870349441877 0.1 -1.97824134920757046 2.05 -1.09030214496519061 0.2 -1.92862028526774320 2.1 -1.07832542203506842 0.3 -1.86670393395684293 2.2 -1.05538508113994433 0.4 -1.80075405253452878 2.3 -1.03371349485820318 0.5 -1.73498799160719041 2.4 -1.01322030525887973 0.6 -1.67148471440012302 2.5 -0.99382351101203490 0.7 -1.61119720301672586 2.6 -0.97544858094023219 0.8 -1.55448006449772595 2.7 -0.95802766004904907 0.9 -1.50138158334624467 2.8 -0.94149886061738322 1.0 -1.45178630031448951 2.9 -0.92580563147803989 1.1 -1.40550252191841256 3.0 -0.91089619738235434 1.2 -1.36230785783351171 3.1 -0.89672306127076382 1.3 -1.32197139010318509 3.2 -0.88324255989570446 1.4 -1.28426925496894185 3.3 -0.87041447461742614 1.5 -1.24898987186705512 3.4 -0.85820167794222435 1.6 -1.21593722446146546 3.5 -0.84656982450508629 1.7 -1.18493139974611416 3.6 -0.83548707392472181 1.8 -1.15580915764590441 3.7 -0.82492384412924800 1.9 -1.12842156954614458 3.8 -0.81485259165546253 1.95 -1.11533575206408963 3.9 -0.80524772550750985 1.99 -1.10514450160298682 4.0 -0.79608496995054425 2.0 -1.10263415348745197 8.0 -0.62757022044109352 Table 2: The electronic energy of H+ 2ion (see Fig. 1). Minimum of the energy E1sσ(R) isE1sσ=−1.9999999976 a.u. at R= 0, which reproduces the known value E1sσ=−2 a.u. to a very high accuracy. Moreover, one can compare Table 2 and the table of Ref.[14] to see that each energy value in Table 2 does reproduce Wind’s result up to sev en decimal places. This means that our numerical calculations are corr ect. We remark that Wind used N= 50 approximation and presented seven decimal places while we use N= 16 approximation and present seventeen decimal places. Alas, there is no need to keep such a high accu racy, and also Wind mentioned that even N= 10 approximation gives the same result, up to seven digits. By making 16th-order interpolation of the points in Table 2 a nd adding to it the potential of interaction between the nuclei, 1 /R, we obtain the list of values of the total energy presented in Table 3. It reveals th e only minimum of the total energy, E(R) +R−1=Emin=−0.6026346 a.u. at the distance 45R=Ropt= 1.9971579 a.u. R, a.u. E(R) +R−1, a.u. R, a.u. E(R) +R−1, a.u. 0.1 +8.02176 2.1 -0.602135 0.2 +3.07138 2.2 -0.600840 0.3 +1.46663 2.3 -0.598931 0.4 +0.69924 2.4 -0.596554 0.5 +0.26501 2.5 -0.593824 0.6 -0.004818 2.6 -0.590833 0.7 -0.182626 2.7 -0.587657 0.8 -0.304480 2.8 -0.584356 0.9 -0.390270 2.9 -0.580978 1.0 -0.451786 3.0 -0.577563 1.1 -0.496412 3.1 -0.574142 1.2 -0.528975 3.2 -0.570743 1.3 -0.552741 3.3 -0.567384 1.4 -0.569984 3.4 -0.564084 1.5 -0.582323 3.5 -0.560856 1.6 -0.590937 3.6 -0.557709 1.7 -0.596696 3.7 -0.554654 1.8 -0.600254 3.8 -0.551695 1.9 -0.602106 3.9 -0.548837 2.0 -0.602634 4.0 -0.546085 Table 3: The total energy of H+ 2ion (see Fig. 2). The results collected in Table 4 have been obtained directly by numerical calculations with the use of replacements p2→2p2anda→4a, wherep andaare defined by Eq.(3.50), in the coefficients αs,βs,γs,ρs,κs, and δsof the recurrence relations. These replacements have been m ade due to Eq.(3.15), with the mass parameter M= 2 and the charge parameter q=−2, corresponding to the stable point-like isoelectronium of m assM= 2meand charge −2e. In addition, it turns out that Table 4 can be derived directl y from Table 2 by the use of rescalements R→R/4 andE→8E. This remarkable property is confirmed by the scaling method developed in Sec. 4.2, and proves that the scaling method is correct. By adding 1 /Rto the isoelectronic energy values of Table 4 we obtain Table 5 showing the total energy of theˆH2 system, at the mass M= 2me. The minimum of the total energy is found E(R) +R−1=Emin=−7.617041 a.u. at R=Ropt= 0.258399 a.u. 46R, a.u. E(R), a.u. R, a.u. E(R), a.u. 0.00 -16.0000000000000008 0.50 -8.82107371596060652 0.05 -15.4289613540288446 0.55 -8.44308064942978475 0.10 -14.4060324481253427 0.60 -8.10576243870545276 0.15 -13.3718774940028106 0.65 -7.80358864752256309 0.20 -12.4358407555134897 0.70 -7.53199088488360768 0.225 -12.0110526341241952 0.75 -7.28716957910914597 0.25 -11.6142894041116995 0.80 -7.06594047783391143 0.275 -11.2440221722385613 0.85 -6.86561342351093717 0.30 -10.8984629511501962 0.90 -6.68389659171409977 0.35 -10.2741538838779300 0.95 -6.51882073404630535 0.40 -9.72749779604447084 1.00 -6.36867910416260141 0.45 -9.24647351730713928 Table 4: The isoelectronium energy of the three-body ˆH2system, at the mass M= 2me(see Fig. 3). R, a.u. E(R), a.u. R, a.u. E(R), a.u. 0.010 +84.0273769895390465 0.150 -6.70521082733614370 0.015 +50.7305619003569852 0.200 -7.43584075551348888 0.020 +34.1124961901906909 0.250 -7.61428940411169996 0.025 +24.1713076006601701 0.300 -7.56512961781686321 0.030 +17.5720885190207170 0.350 -7.41701102673507239 0.035 +12.8849357295675638 0.400 -7.22749779604447084 0.040 +9.39442987714794597 0.450 -7.02425129508491696 0.045 +6.70277469374505585 0.500 -6.82107371596060652 0.050 +4.57103864597115538 0.550 -6.62489883124796641 0.055 +2.84698239022676524 0.600 -6.43909577203878669 0.060 +1.42896579635860376 0.650 -6.26512710906102388 0.065 +0.24650856005471055 0.700 -6.10341945631217797 0.070 -0.75082137484248256 0.750 -5.95383624577581205 0.075 -1.60007720159364552 0.800 -5.81594047783391143 0.080 -2.32910343012903808 0.850 -5.68914283527564279 0.085 -2.95923562411244667 0.900 -5.57278548060298906 0.090 -3.50710131449860362 0.950 -5.46618915509893721 0.095 -3.98585308299203866 1.000 -5.36867910416260141 0.100 -4.40603244812534278 Table 5: The total energy of the three-body ˆH2system, at the mass M= 2me (see Fig. 4). 47M, a.u. Emin(M), a.u. Ropt(M), a.u. 0.10 -0.380852 5.167928 0.15 -0.571278 3.445291 0.20 -0.761704 2.583964 0.25 -0.952130 2.067171 0.26 -0.990215 1.987664 0.27 -1.028300 1.914050 0.28 -1.066385 1.845688 0.29 -1.104470 1.782044 0.30 -1.142556 1.722645 0.307 -1.169215 1.683367 0.308 -1.173024 1.677899 0.308381 -1.174475 1.675828 0.309 -1.176832 1.672471 0.31 -1.180641 1.667073 0.32 -1.218726 1.614977 0.33 -1.256811 1.566041 0.34 -1.294896 1.519981 0.35 -1.332982 1.476553 0.40 -1.523408 1.291982 0.45 -1.713834 1.148428 0.50 -1.904260 1.033585 0.75 -2.856390 0.689058 1.00 -3.808520 0.516792 1.25 -4.760650 0.413434 1.50 -5.712780 0.344529 1.75 -6.664910 0.295310 2.00 -7.617040 0.258396 Table 6: The minimal total energy Eminand the optimal internuclear distance Roptof the three-body ˆH2system as functions of the mass Mof the stable point-like isoelectronium (see Figs. 5–8). Table 6 presents result of calculations of the minimal total energies and corresponding optimal distances, at various values of the isoelectronium mass parameterM=ηme(M=η, in atomic units). We have derived some 27 tables (such as Table 5) from Table 2 by the scaling method acc ording to Eq.(4.21), and find minimum of the total energy in each table, together with the corresponding optimal distance. Then we collected all the obtained energy minima and optimal distances in Table 6. With the four th order in- terpolation/extrapolation, the graphical representatio ns of Table 6 show (see Figures 5–8) that the minimal total energy behaves as Emin(M)≃ −3.808M, 48and the optimal distance behaves as Ropt(M)≃0.517/M, to a good accu- racy. One can see that at M= 2mewe haveEmin(M) =−7.617040 a.u. andRopt(M) = 0.258396 a.u., which recover the earlier obtained values Emin=−7.617041 a.u. and Ropt= 0.258399 a.u. of Table 5, to a high ac- curacy, thus showing once again correctness of the used scal ing method. In fact, the values of EandRforM= 1.50me,M= 1.75me, andM= 2.00me in Table 6 have been obtained by extrapolation so they are not as much ac- curate as they are in Table 5. However, this is not of much impo rtance here because we use them only to check the results of the scaling me thod. The main conclusion following from Table 6 is that the mass pa rameter valueM= 0.308381mefits the energy value Emin(M) =−1.174475 a.u., with the corresponding Ropt(M) = 1.675828 a.u., which appears to be about 19.6% bigger than the experimental value Rexper[H2] = 1.4011 a.u. The total energy as a function of internuclear distance, for this valu e of mass, M= 0.308381me, is shown in Figure 9 to illustrate that the obtained optimal distanceRopt= 1.675828 a.u. corresponds to a minimal value of the total energy. 490 1 2 3 4 R, a.u.-2-1.8-1.6-1.4-1.2-1-0.8E, a.u. Figure 1: The electronic energy E(R) ofH+ 2ion as a function of the inter- nuclear distance R. 1 1.5 2 2.5 3 3.5 4 R, a.u.-0.6-0.58-0.56-0.54-0.52E+R-1, a.u. Figure 2: The total energy E(R) +R−1ofH+ 2ion as a function of the internuclear distance R. 500 0.2 0.4 0.6 0.8 1 R, a.u.-16-14-12-10-8E, a.u. Figure 3: The isoelectronium energy E(R) of the ˆH2system as a function of the internuclear distance R, at the isoelectronium mass M= 2me. 0 0.2 0.4 0.6 0.8 1 R, a.u.-7.5-7-6.5-6-5.5-5-4.5E+R-1, a.u. Figure 4: The total energy E(R)+R−1of the ˆH2system as a function of the internuclear distance R, at the isoelectronium mass M= 2me. 510 0.5 1 1.5 2 Mass M, a.u.-7-6-5-4-3-2-10Emin, a.u. Figure 5: The minimal total energy Emin(M) of the ˆH2system as a function of the isoelectronium mass M. 0.26 0.28 0.3 0.32 0.34 Mass M, a.u.-1.3-1.25-1.2-1.15-1.1-1.05-1Emin, a.u. Figure 6: The minimal total energy Emin(M) of the ˆH2system as a function of the isoelectronium mass M. More detailed view. 520 0.5 1 1.5 2 Mass M, a.u.012345Ropt, a.u. Figure 7: The optimal internuclear distance Ropt(M) of the ˆH2system as a function of the isoelectronium mass M. 0.26 0.28 0.3 0.32 0.34 Mass M, a.u.1.61.71.81.9Ropt, a.u. Figure 8: The optimal internuclear distance Ropt(M) of the ˆH2system as a function of the isoelectronium mass M. More detailed view. 531 1.2 1.4 1.6 1.8 2 R, a.u.-1.17-1.16-1.15-1.14E+R-1, a.u. Figure 9: The total energy E(R)+R−1of the ˆH2system as a function of the internuclear distance R, at the isoelectronium mass M= 0.308381me. 54References [1] R. M. Santilli and D. D. Shillady, Int. J. Hydrogen Energy 24(1999), 943-956. [2] R. M. Santilli and D. D. Shillady, Hadronic J. 21(1998), 633-714; 21 (1998), 715-758; 21(1998), 759-788. [3] R. M. Santilli, Hadronic J. 21(1998), 789-894. [4] Y. Rui, Hadronic J. 22(1999), in press. [5] A. K. Aringazin, Hadronic J. 22(1999), in press. [6] L. D. Landau and E. M. Lifschitz, Vol. 3, Quantum Mechanic s (Moscow, 1963). [7] M. Abramowitz and I. A. Stegun, Handbook of mathematical func- tions with formulas, graphs and mathematical tables. Natio nal Bureau of Standards Applied Math. Series - 55(1964) 830 pp. [8] I. V. Komarov, L. I. Ponomarev, S. Yu. Slavyanov, Spheroi dal and Coloumb spheroidal functions (Nauka, Moscow, 1976) (in Rus sian). [9] W. G. Baber and H. R. Hasse, Proc. Cambr. Phil. Soc. 31(1935) 564- 581. [10] G. Jaffe, Z. Physik, 87(1934) 535-544. [11] S. S. Gerstein and L. I. Ponomarev, Mesomolecular proce sses induced by µ−-mesons, in: Muon physics, Eds. C. S. Wu and V. Hughes (Academ ic Press, New York, 1975). [12] E. Teller, Z. Phys. 61(1930) 458-480. [13] D. R. Bates, K. Ledsham, A. L. Stewart. Wave functions of the hydrogen molecular ion, Phil. Trans. Roy. Soc. (London), 246(1953) 215-240; D. R. Bates and T. R. Carson. Exact wave functions of HeH2+, Proc. Roy. Soc. (London), A 234 (1956) 207-217. [14] H. Wind, J. Chem. Phys. 42(1965) 2371-2373. [15] Z. Flugge, Practical Quantum Mechanics, Vols. 1, 2 (Spr inger-Verlag, Berlin, 1971). 55

arXiv:physics/0001057v1 [physics.gen-ph] 24 Jan 2000ON VARIATIONAL SOLUTION OF THE FOUR-BODY SANTILLI-SHILLADY MODEL OF H2MOLECULE A.K.Aringazin Karaganda State University, Karaganda 470074 Kazakstan ascar@ibr.kargu.krg.kz December 1999 Abstract In this paper, we apply Ritz variational approach to a new iso - chemical model of H2molecule suggested by Santilli and Shillady. We studied Gaussian, Vg, and exponential, Ve, screened Coloumb poten- tialapproximations , as well as the original Hulten potential, Vh, case. Both the Coloumb and exchange integrals have been calculate d only forVeowing to Gegenbauer expansion while for VgandVhcases we achieved analytical results only for the Coloumb integrals . We con- clude that the Ve-based model is capable to fit experimental data on H2molecule in confirmation of the results of numerical HFR appr oach by Santilli and Shillady. Also, we achieved the energy-base d estima- tion of the weight of the isoelectronium phase which is appea red to be of the order of 1%...6%, for the case of Ve. However, we note that this isnotthe result corresponding to the original Santilli-Shillad y model, which is based on the Hulten potential Vh. An interesting result is that in order to prevent divergency of the Coloumb integral f orVhthe correlation length parameter rcshould run discrete set of values. We used this condition in our Ve-based model. 01 Introduction In this paper, we consider the four-body Santilli-Shillady isochemical model ofH2molecule [1, 2, 3] characterized by additional short-range attractive Hulten potential between the electrons. This potential is a ssumed to lead to bound state of electrons called isoelectronium. The rest ricted three-body Santilli-Shillady model (stable and point-like isoelectr onium) ofH2has been studied in ref. [4], in terms of exact solution. For the mass o f isoelectronium M= 2me, this solution implied much lower energy than the experimen tal one so we varied the mass and obtained that M= 0.308381mefits the experi- mental binding energy, Eexper[H2] =−1.174474 a.u. up to six decimal places, although at bigger value of the internuclear distance, R= 1.675828 a.u. in contrast to Rexper[H2] = 1.4011 a.u. We realize that the three-body model is capable to represent the binding energy but it is only some approxima- tion to the four-body model, and one should study the general four-body hamiltonian of the Santilli-Shillady model as well. In the present paper, we use Ritz variational approach to the four-body Santilli-Shillady isochemical model of H2molecule, i.e. without restriction that the isoelectronium is stable and point-like particle, in order to find the ground state energy and bond length of the H2molecule. In Sec. 2, we analyze some features of the four-body Santilli -Shillady isochemical model of H2molecule. In Sec. 3, we apply Ritz variational approach to the four-bod y Santilli- Shillady model of H2molecule. We calculate Coloumb integral for the cases of Hulten potential (Sec. 3.1.1), exponential screen ed Coloumb po- tential (Sec. 3.1.2), and Gaussian screened Coloumb potent ial (Sec. 3.1.3). Owing to Gegenbauer expansion, exchange integral has been c alculated for the case of exponential screened potential, with some appro ximation made (Sec. 3.1.4). Exchange integrals for the Hulten potential a nd the Gaussian screened Coloumb potential have not been derived, and requi re more study. We present main details of calculations of the Coloumb and ex change inte- grals which have been appeared to be rather cumbersome, espe cially in the case of Hulten potential. In Sec. 3.2, we make numerical fitting of the variational ener gy for the case of exponential screened Coloumb potential Ve. Also, we estimate the weight of the isoelectronium phase. However, we use all the i mportant results of the analysis made for the Hulten potential Vh. 11) We conclude that the Ve-based model with the one-level isoelectronium iscapable to fit the experimental data on H2molecule (both the binding energyEand the bond length R). This is in confirmation of the results of numerical HFR approach (SASLOBE routine) to the Vg-based model of ref. [1]. 2) One of the interesting implications of the Ritz variation al approach to the Hulten potential case is that the correlation length par ameterrc, entering the Hulten potential, and, as a consequence, the variationa l energy, should run discrete set of values during the variation. In other wor ds, this means that only some fixed values of the effective radius of the one-level isoelectronium are admitted , in the original Santilli-Shillady model, within the frame work of the Ritz approach. This highly remarkable property is speci fic to the Hulten potentialVhwhile it is absent in the Ve, orVg-based models. 3) Also, we achieved an estimation of the weight of the isoelectronium phase for the case of Ve-based model which is appeared to be of the order of 1%...6%. This weight has been estimated from the energy contribution, related to the exponential screened potential Ve, in comparison to the con- tribution related to the Coloumb potential. 4) Another general conclusion is that the effective radius of the isoelec- troniumrcshould be less that 0 .25 a.u. We note that the weight of the phase does not mean directly a ti me share between the two regimes, i.e., 1...6% of time for the pure iso electronium regime, and 99...94% of time for the decoupled electrons reg ime. This means instead relative contribution to the total energy provided by the potential Ve and by the usual Coloumb potential between the electrons, re spectively. As a consequence, the weight of the isoelectronium phase, whic h can be thought of as a measure of stability of the isoelectronium, may be 1. Different from the obtained 1...6% when calculated for som e other char- acteristics of the molecule, e.g., for a relative contribut ion of the pure isoelectronium to the total magnetic moment of the H2molecule; 2. Different from the obtained 1...6% for the case of the origi nal Hulten potentialVh. So, the result of the calculation made in this paper is not the final result implied by the general four-body Santilli-Shillady model o fH2molecule since the latter model is based on the Hulten potential Vh. This paper can be viewed only as a preliminary study to it. However, we have mad e some 2essential advance in analyzing the original Hulten potenti al case (Sec. 3.1.1), which we have used in the Ve-based model. Below, we describe the procedure used in Sec. 3 in a more detai l. In Ritz variational approach, the main problem is to calculate analytically so called molecular integrals. The variational molecular ene rgy, in which we are interested in, is expressed in terms of these integrals; see Eq.(3.2). These integrals arise when using some wave function basis (usuall y it is a simple hydrogen ground state wave functions) in the Schr¨ odinger e quation for the molecule. The main idea of the Ritz approach is to introduce p arameters into the wave function, and vary them, together with the internuc lear distance parameterR, to achieve a minimum of the molecular energy. In the case under study, we have two parameters, γandρ, whereγenters hydrogen- like ground state wave function (3.10), and ρ=γRmeasures internuclear distance. These parameters should be varied (analytically or numerically) in the final analytical expression of the molecular energy, aft er the calculation is made for the associated molecular integrals. However, the four-body Santilli-Shillady model of H2molecule suggests additional, Hulten potential interaction between the elec trons. The Hulten potential contains two parameters, V0andrc, whereV0is a general factor, andrcis a correlation length parameter which can be viewed as an eff ective radius of the isoelectronium; see Eq. (3.23). Thus, we have f our parameters to be varied, γ,ρ,V0, andrc. The introducing of Hulten potential leads to modification of some molecular integrals, namely, of the C oloumb and exchange integrals; see Eqs. (3.5) and (3.7). The other mole cular integrals remain the same as in the case of usual model of H2, and we use the known analytical results for them. So, we should calculate the ass ociated Coloumb and exchange integrals for the Hulten potential to get the va riational energy analytically. In fact, calculating of these integrals, whi ch are six-fold ones, constitutes the main problem here. Normally, Coloumb integ ral, which can be performed in bispherical coordinates, is much easier tha n the exchange one, which is performed in bishperoidal coordinates. Calculation of the Coloumb integral for Hulten potential, Vh, appeared to be rather nontrivial (Sec. 3.1.1). Namely, we used bisphe rical coordinates, and have faced several special functions, such as polylogar ithmic function, Riemannζ-function, digamma function, and Lerch function, during th e cal- culation. Despite the fact that we see no essential obstacle s to calculate this six-fold integral, we stopped the calculation after fifth st ep because sixth (the 3last) step assumes necessity to calculate it separately for each integer value of λ−1≡(2γrc)−1, together with the need to handle very big number of terms. During the calculations, we were forced to use the condition thatλ−1should take integer values in order to prevent divergency of the Col oumb integral for Hulten potential. Namely, some combination of terms con taining Lerch functions gives a finite value only if this condition holds. T his condition is specific to Hulten potential. Note also that we can not get gen eral form of a final expression for the Coloumb integral for Hulten pote ntial because Lerch functions entering the intermediate expression (aft er the fifth step, see Eq.(3.80)) can be integrated over only for a concrete numeri cal value of their third argument. In order to proceed with the Santilli-Shillady approach, we invoke to two different simplified potentials, the exponential screened C oloumb potential, Ve, and the Gaussian screened Coloumb potential, Vg, instead of the Hulten potentialVh. They both mimic well Hulten potential at short and long range asymptotics, and each contains two parameters, for wh ich we use the notation,Aandrc. In order to reproduce the short range asymptotics of Hulten potential the parameter Ashould have the value A=V0rc, for both the potentials. The Coloumb integrals for these two potenti als have been calculated exactly (Secs. 3.1.2 and 3.1.3) owing to the fact that they are much simpler than the Hulten potential. Particularly, we no te that the final expression of the Coloumb integral for Vgcontains only one special function, the error function erf( z), while for Veit contains no special functions at all. Having these results we turned next to the most hard part of wo rk: the exchange integral. Usually, to calculate it one has to use bi spheroidal coordi- nates, and needs in an expansion of the potential in some orth ogonal polyno- mials, such as Legendre polynomials, in bispheroidal coord inates. Here, only the exponential screened potential Veis known to have such an expansion but it is formulated, however, in terms of bispherical coordina tes (the Gegenbauer expansion). Accordingly, we calculated exactly the exchan ge integral for Ve, atzerointernuclear separation, R= 0, at which case one can use bispherical coordinates. After that, we recovered partially the Rdependence using the standard result for the exchange integral for Coloumb poten tial (Sugiura’s result). Thus, we achieved some approximate expression of t he exchange integral for the case of Ve. So, all the subsequent results correspond to the Ve-based model. Inserting obtained Ve-based Coloumb and exchange integrals into the to- 4tal molecular energy expression, we get the final analytical expression con- taining four parameters, γ,ρ,A, andrc. Prior to going into details of the energy minimization for the Ve-based (approximate) model, we analyze the set of parameters, and the conditions which we derived in the original Hulten potential case. (1) From the analysis of Hulten potential, we see (Sec. 2.1) t hat the existence of a bound state of two electrons (which is proper i soelectronium) leads to the following relationship between the parameters for the case of oneenergy level of the electron-electron system: V0= ¯h2/(2mr2 c). So, using the above mentioned relation A=V0rcwe haveA= 1/rc≡2γ/λ, in atomic units (¯h=me=c= 1). Thus note that, in this paper, we confined our consideration to the case of one-level isoelectronium. (2) From the analysis of the Coloumb integral for Hulten pote ntial, we see (Sec. 3.1.1) that the condition, λ−1=integer number , should hold, and one can use it as well. We use the above two conditions, coming from the Hulten poten tial anal- ysis, in the energy minimization calculations for the case o f ourVe-based model. The first condition diminishes the number of independ ent parame- ters by one (they become three, γ,ρ, andλ) while the second condition means a discretization of the λparameter, λ−1= 4,5,6,...Here, we used the con- ditionλ−1>3 which we obtained during the calculation of the Coloumb integral for Ve. With the above set up, we minimized the total molecular energ y of the Ve-based model. Numerical analysis shows that the λdependence does not reveal any minimum, in the interval of interest, 4 <λ−1<60, while we have a minimum of the energy at some values of γandρ. So, we calculated the energy minima for different values of λ, in the interval of interest, 4 <λ−1< 60. Results are presented in Tables 2 and 3. One can see that th e binding energy decreases with the increase of the parameter rc, which corresponds to the effective radius of the isoelectronium. The following remarks are in order. (i) Note that the discrete character of rcdoes not mean that the iso- electronium is some kind of a multilevel system, with differe nt effective radia of isoelectronium assigned to the levels. We remind th at we treat the isoelectronium as one-level system due to the above mentioned relation V0= ¯h2/(2mr2 c). In fact, this means that there is a set of one-level isoelec - tronia of different fixed effective radia from which we should s elect only one, 5to fit the experimental data. (ii) The use of the exponential screened potential Vecan only be treated as someapproximation to the original Hulten potential, and, thus, to the original Santilli-Shillady model of H2molecule. So, the numerical results obtained in Sec. 3.2 are valid only within this approximation. Hulten potential makes a difference (one can see this, e.g., by comparing Sec. 3.1.1 a nd Sec. 3.1.2), and it is worth to be investigated more closely by, for exampl e, combination of analytical and numerical methods. (iii) The results obtained in ref. [1] are based on the Gaussi an screened Coloumb potential Vgapproximation, to which the present work gives sup- port in the form of exact analytical calculation of the Colou mb integral for Vg (Sec. 3.1.3). Also, the present work gives possibility to ma ke a comparative analysis of ref. [1], due to some similarity of the used poten tials,VeandVg. (iv) Both the Coloumb integrals, for VeandVg, reveal a minimum in respect with λ= 2γrc, i.e. in respect with rc(see Figures 6 and 9) since min- imization in Ritz parameter γis made independently. In principle, this gives us an opportunity to minimize the total molecular energy Emolwith respect torc. However, there are two reasons that we can not provide this m inimiza- tion. First, these minima correspond to rather large values ofrc, namely, rc≥1 a.u. forVe(Fig. 6), and rc>2 a.u. forVg(Fig. 9). Of course, this is not an obstacle to do minimization but we note that we gener ally assume that the effective radius of the isoelectronium rcis much less than the inter- nuclear distance, rc≪R=Rexper[H2] = 1.4011 a.u. Second, and the main, reason is that for the exponential screened potential case ( Sec. 3.1.2) the pa- rameterλshould be less than 1/3 to provide convergency of the associa ted Coloumb integral. Typically, γ≃1.2, from which we obtain the condition rc=λ/2γ <0.2 a.u. Also, for the Hulten potential case (Sec. 3.1.1), we obtainedλ <1/2, and hence rc<0.25 a.u. This means that, in fact, it is impossible to reach finite minimum of the total molecular energy Emolin re- spect withrcsince the Coloumb integrals blow up, at rc>0.25 a.u., leading thus to infinite total energy Emol. So, in our approach we arrive at a strict theoretical conclusion that the effective radius of the isoe lectronium rcmust be less than 0 .25 a.u. Clearly, this supports our assumption that rcis much less than the internuclear distance R. 62 Santilli-Shillady model and the barrier In this Section, we consider the general four-body Santilli -Shillady model [1] ofH2molecule, in Born-Oppenheimer approximation (i.e. at fixed nu- clei). Shr¨ odinger equation for H2molecule with the additional short range attractive Hulten potential between the electrons is of the following form: /parenleftBigg −¯h2 2m1∇2 1−¯h2 2m2∇2 2−V0e−r12/rc 1−e−r12/rc+e2 r12(2.1) −e2 r1a−e2 r2a−e2 r1b−e2 r2b+e2 R/parenrightBigg |ψ/an}b∇acket∇i}ht=E|ψ/an}b∇acket∇i}ht, whereRis distance between the nuclei aandb. Interaction between the two electrons in the model is due to t he potential V(r12) =VC(r12) +Vh(r12) =e2 r12−V0e−r12/rc 1−e−r12/rc, (2.2) wherer12is distance between the electrons, V0andrcare real positive pa- rameters. Here, first term, VC, is usual repulsive Coloumb potential, and the second term, Vh, is an attractive Hulten potential. Extrema of V(r12) are defined by the equation V′(r12) =−e2 r2 12+V0 rcer12/rc (er12/rc−1)2= 0. (2.3) In the limit r12→ ∞, potentialV(r12)∼e2/r12=VC(r12). Series expansion ofV(r12) atr12→0 is V(r12)|r12→0=e2−V0rc r12+V0 2−V0 12rcr12+O(r3 12). (2.4) In general, there is relationship of Hulten potential to Ber noulli polynomials Bn(x). Namely, Bernoulli polynomials are defined due to sexs es−1=∞/summationdisplay n=0Bn(x)sn n!, (2.5) and we can reproduce Hulten potential, es 1−es=−1 s∞/summationdisplay n=0Bn(1)sn n!, (2.6) 7takings=−r12/rc. First five Bernoulli coefficients are B0(1) = 1, B1(1) =1 2, B2(1) =1 6, B3(1) = 0, B4(1) =−1 30.(2.7) Eq.(2.6) means expansion of Hulten potential with the use of Bernoulli coef- ficients. Eq.(2.4) implies that to have an attraction nearr12= 0, which is necessary for forming of isoelectronium, we should put the condition V0rc>e2. (2.8) We note that, in view of the asymptotics (2.4), Q=√V0rccan be thought of asHulten charge of the electrons. Under this condition, V(r12) has one maximum at the point defined by Eq.(2.3). This is the equilibrium point at which the Coloumb potential is equal to the Hulten potential. So, we have barrier ( B) separating two asymp- totic regions, ( A)r→0 and (C)r→ ∞, with Coloumb-like attraction and Coloumb-like repulsion, respectively. In the region A, attractive Hulten potential Vhdominates, and there- fore two electrons form bound state (isoelectronium), whil e in the region C Coloumb repulsion VCdominates, and they are separated. This separation is limited by the size of the neutral molecule. For example, ass uming that H2 molecule is in the ground state we have r≤rmol= 3.46 bohrs, where we have assumed that separation between the protons is R= 1.46 bohrs = 0 .77r12A. Existence of the bound state of the electrons and of the barri erBis a novel feature provided by the model. The asymptotic states, in regionsA andC, pertube each other due to the barrier effect in region B. 2.1 Region A In the case V0rc≫e2(2.9) we can ignore Coloumb repulsion VC, and region Ais a Hulten region, |Vh| ≫ |VC|; see Eq.(2.4). Then, exact solution of one-particle Schr¨ o dinger equation with Hulten potential Vh, where wave function has the boundary conditions ψ(0) = 0 and ψ(∞) = 0 (see [5], problem 68), can be used to establish the relation between the parameters V0andrc, and to estimate rc. 8Energy spectrum for Hulten potential is given by En=−V0/parenleftBiggβ2−n2 2nβ/parenrightBigg2 , n= 1,2,.... (2.10) where β2=2mV0 ¯h2r2 c, (2.11) andmis mass of the particle. Assuming that there is only oneenergy level, namely, ground state n= 1, we obtain the condition β2= 1, (2.12) which can be rewritten as rc= ¯h/radicalBigg 1 2mV0. (2.13) Note that this state is characterized by approximately zeroenergy,E1= 0, due to Eq.(2.10); strictly speaking, β2must be bigger but close to 1 in Eq.(2.12). We should to note that the number of energy levels for Hulten p otential is always finite due to Eq.(2.10). Assumption that there are m ore than one energy levels in the bound state of two electrons, i.e. that β >1, leads to drastical decrease of ground level energy E1<0, and relatively small increase of characteristic size of isoelectronium in the ground stat e. As the conclusion, the model implies ”quantization” of the d istance be- tween two electrons, r=r12, namely, forming of relatively small quasiparticle (isoelectronium) characterized by total mass M= 2me, chargeq=−2e, spin zero,s= 0, and small size in the ground one-level state. This quasip article, as a strongly correlated system of two electrons, moves in th e potential of two protons of H2molecule, and one can apply methods developed for H+ 2 ion, with electron replaced by isoelectronium, to calculat e approximate en- ergy spectrum of H2[4]. However, this quasiparticle is not stable, being a quasi-stationary state, due to finite height and width of the barrierB. So, we must take into account effects of both regions BandCto obtain correct energy spectrum of H2molecule, within the framework of the model. 92.2 Region B Quasiclassically, due to smooth shape of the barrier, and be cause of expo- nential decrease of wave functions inside the barrier, elec trons are not much time in region B, so we can ignore this transient phase in subsequent consid- eration. We should to point out that the existence of the bound state in the region Aand repulsion in the region Cunavoidably leads to existence of the barrier. 2.3 Region C In general, region Cis infinite, rC< r < ∞, whererCis the distance between two electrons at which the Hulten potential is much s maller than the Coloumb potential, |Vh| ≪ |VC|. In this region, electrons are not strongly correlated, in co mparison to that in region A. Here, correlation is due to usual overlapping, Coloumb repulsion, exchange effects, and Coloumb attraction to prot ons. Shortly, we have the usual set up as it for the standard model of H2molecule. Discarding, for a moment, effects coming from the considerat ion of regions AandB, we have finite motion of the electrons in region C. Namely, in the ground state of H2, the distance between electrons is confined by r=rmol= 3.46 bohrs. We restrict consideration by the ground state of H2molecule. Due to this finiteness of the region C,r<r mol, two electrons on the same orbit have constant probability to penetrate the barrier to form strongly correlated system, isoelectronium, and vice verca. 2.4 Model of decay of isoelectronium Below, we assume that the isoelectronium undergoes decay, a nd the resulting two electrons are separated by sufficiently large distance, i n the final state. This leads us to consideration of the effective life-time of isoelectronium . To estimate the order of the life-time, we use ordinary formula for radioactive α- decay since the potential V(r) is of the same shape, with very sharp decrease atr<r maxand Coloumb repulsion at r>r max. This quisiclassical model is a crude approximation because in fact the electrons do not lea ve the molecule. Moreover, we have the two asymptotic regimes simultaneousl y, with some distribution of probability, and it would be more justified h ere to say on 10frequency of the decay-formation process. However, due to o ur assumption of small size of isoelectronium, in comparison to the molecu le size, we can study an elementary process of decay separately, and use the notion of life- time. Decay constant is λ=¯hD0 2mr2maxexp/braceleftBigg −4πZe2 ¯h/radicalbiggm 2E+4e ¯h/radicalBig Zmr max/bracerightBigg , (2.14) where we put, in atomic units, ¯h= 1, e= 1, m= 1/2, Z= 1, rmax= 0.048, E= 1. (2.15) Here,E= 1 a.u. = 27 .212 eV is double kinetic energy of the electron on first Bohr’s orbit, a0= 0.529r12A, that corresponds approximately to maximal relative kinetic energy of two electrons in ground state of H2, andm= 1/2 is reduced mass of two electrons. We obtain the following numerical estimation for the life-t ime of isoelec- tronium: 1/λ=D0·1.6·10−17sec, (2.16) i.e. it is of the order of 1 atomic unit of time, τ= 2.42·10−17sec. For lower values of the relative energy E, we obtain longer lifetimes; see Table 2. The quasiclassical model for decay we are using here is the fo llowing. Particle of reduced mass m= 1/2 penetrate the barrier B. This means a decay of isoelectronium. In the center of mass of electrons s ystem, electrons undergo Coloumb repulsion and move in opposite directions r eceiving equal speed so that at large distances, r≫rmax, each of them have some kinetic energy. This energy can be given approximate upper estimati on using linear velocity of electron on first Bohr’s orbit, v= 2.19·106cm/sec, since electrons are in the ground level of H2molecule (this is the effect of the nuclei). This upper estimation corresponds to assumption of zero velocit y of the center of mass in respect to protons which we adopt here. Kinetic ene rgy of the particle of reduced mass is then double kinetic energy of ele ctron, in center of mass system. As the conclusion, in the framework of the model, H2molecule can be viewed as a mixed state of H+ 2ion like system, i.e. strongly correlated phase (Hulten phase), when electrons form isoelectronium, and st andard model of 11EnergyE, a.u. eV Lifetime,D0·sec 2 54.4 2.6·10−18 1 27.2 1.6·10−17 0.5 13.6 2.2·10−16 0.037 1 5.1·10−6 0.018 0.5 4.0 0.0018 0.1 3.1·10+25 Table 1: Lifetime of isoelectronium. Eis relative kinetic energy of the elec- trons, at large distances, r≫rmax, in the center of mass system. H2, i.e.weakly correlated phase (Coloumb phase), when electrons are sepa- rated by large distance, r>r max. Note that, as it has been mentioned above, we ignore the transient phase (inside the barrier) in this consideration. Ev- idently, the (statistical) weight of each phase depends on t he characteristics of the potential V(r12). For extremally high barrier, only one of the phases could be r ealized with some energy spectra in each phase, namely, either spectrum o fH+ 2ion like system (with electron replaced by isoelectronium), or usua l spectrum of H2 molecule (without Hulten potential), respectively. For high but finite barrier, each phase receives perturbatio n, and their (ground) energy levels split to two levels corresponding to simultaneous re- alization of both the phases. Note that the value Vmaxis indeed high, Vmax∼500 eV, under the given values of the parameters. In general, existence of the strongly correlated phase (iso electronium) leads to increase of the predicted dissociation energy, D, ofH2molecule. Indeed, the mutual infuence of the regions AandCdecreases the ground energy level EofH2due to the above mentioned splitting. The general formula for Dis D= 2E0−(E+1 2¯hω), (2.17) where 2E0=−1 is total energy of two separated Hatoms, and1 2¯hωis zero mode energy of the protons in H2. So, decreasing of E <0 causes increase ofD. It is remarkable to note that experimental data give dissoci ation energy 12Dexper[H2] = 4.45 eV forH2molecule (see, e.g. [5] and references therein) while theoretical predictions within the standard model ar eD= 2.90 eV (Heitler-London), D= 3.75 eV (Flugge), and D= 4.37 eV (Hylleraas). We observe that improvement of the variational approximation gives better fits but still it gives lower values (about 2% lower) partially due to the fact that variational technique used there predicts generally bigge r value (upper limit) for the ground energy. Below, we use the same Ritz variational technique as it had be en used by Heitler, London and Hylleraas but the feature of the model is the existence of additional attractive short range potential between the el ectrons suggested by Santilli and Shillady. 3 Variational solution for ground state en- ergy ofH2molecule In the limiting case of large distances between the nuclei, R→ ∞, we have the total wave function of the electrons in the form |ψ/an}b∇acket∇i}ht=f(ra1)f(rb2)±f(rb1)f(ra2), (3.1) where the first term corresponds to the case when electron 1 is placed close to nucleusaandf(ra1) is wave function of the corresponding separate Hatom while the second term corresponds to the case when electron 1 is placed close to nucleusb. Symmetrized combination (′+′sign) corresponds to antiparal- lel spins of the electrons 1 and 2, and, as the result of the usu al analysis, leads to attraction between the Hatoms. Below, we use this symmetrized representation of the total wave function as the approximat ion to exact wave function. 3.1 Analytical calculations By using Ritz variational approach and representation (3.1 ), we obtain from the Schr¨ odinger equation (2.1) the energy of H2molecule in the following form (cf. [5]), Emol= 2A+A′S 1 +S2−2(C+ES)−(C′+E′) 1 +S2+1 R, (3.2) 13where S=/integraldisplay dv f∗(ra1)f(rb1) (3.3) is overlap integral, C=/integraldisplay dv1 rb1|f(ra1)|2, (3.4) C′=/integraldisplay dv1dv2/parenleftBigg1 r12−V0e−r12/rc 1−e−r12/rc/parenrightBigg |f(ra1)|2|f(rb2)|2, (3.5) are Coloumb integrals, E=/integraldisplay dv1 ra1f∗(ra1)f(rb1), (3.6) E′=/integraldisplay dv1dv2/parenleftBigg1 r12−V0e−r12/rc 1−e−r12/rc/parenrightBigg f∗(ra1)f(rb1)f∗(ra2)f(rb2) (3.7) are exchange integrals, A=/integraldisplay dv f∗(ra1)/parenleftbigg −1 2∇2 1−1 ra1/parenrightbigg f(ra1) (3.8) and A′=/integraldisplay dv f∗(ra1)/parenleftbigg −1 2∇2 1−1 rb1/parenrightbigg f(rb1). (3.9) We use atomic units, e= 1,m1=m2=me= 1. Quite natural choice is that the wave functions in Eq.(3.1) a re taken in the form of hydrogen ground state wave function, f(r) =/radicalBigg γ3 πe−γr, (3.10) whereγis Ritz variational parameter ( γ=1 for the proper hydrogen wave function), and r=ra1,rb1,ra2,rb2. With the help of γwe should make better approximation to an exact wave function of the ground state. Namely, we should calculate all the integrals presented above analyti cally, and then vary the parameters γaway from the value γ= 1 andRin some appropriate region, say 1 < R < 2, to minimize the energy (3.2). As the energy min- imum will be identified the found value of the parameter Rcorresponds to 140 1 2 3 4 5 6 r, a.u.00.20.40.60.81S, a.u. Figure 1: The overlap integral Sas a function of ρ, Eq. (3.11). Here, ρ=γR, whereγis Ritz parameter and Ris the internuclear distance. optimal distance between the nuclei. This value should be co mpared to the experimental value of R. All the molecular integrals (3.3)-(3.9), except for the Hul ten potential parts in (3.5) and (3.7), are wellknown and can be calculated exactly; see, e.g. [5]. Namely, they are S=/parenleftbigg 1 +ρ+1 3ρ2/parenrightbigg e−ρ, (3.11) C ≡ C C=γ ρ(1−(1 +ρ)e−2ρ), (3.12) C′ C≡ C′ |V0=0=γ ρ/parenleftBig 1−(1 +11 8ρ+3 4ρ2+1 6ρ3)e−2ρ/parenrightBig , (3.13) E ≡ E C=γ(1 +ρ)e−ρ, (3.14) E′ C≡ E′ |V0=0=γ/parenleftbigg5 8+23 20ρ−3 5ρ2−1 15ρ3/parenrightbigg e−2ρ+6γ 5h(ρ) ρ, (3.15) h(ρ) =S2(ρ)(lnρ+C)− S2(−ρ)E1(4ρ) + 2S(ρ)S(−ρ)E1(2ρ),(3.16) 150 1 2 3 4 5 6 r, a.u.00.20.40.60.81CC, a.u. Figure 2: The Coloumb integral CCas a function of ρ, Eq. (3.12). 0 1 2 3 4 5 6 r, a.u.0.20.30.40.50.6CC', a.u. Figure 3: The Coloumb integral C′ Cas a function of ρ, Eq. (3.13). 160 1 2 3 4 5 6 r, a.u.00.10.20.30.40.50.6EC', a.u. Figure 4: The exchange integral E′ Cas a function of ρ, Eq. (3.15). E1(ρ) =∞/integraldisplay ρe−t tdt, (3.17) A=−1 2γ2+γ(γ−1),A′=−1 2γ2S+γ(γ−1)E, (3.18) whereCis Euler constant, and we have denoted ρ=γR, (3.19) which can be taken as a second Ritz variational parameter in a ddition to γ. The most hard part of work here is the exchange integral (3.1 5), which was calculated for the first time by Sugiura (1927), and conta ins one special function, the exponential integral function E1(ρ). Our problem is thus to calculate analytically the Hulten pot ential parts of the Coloumb integral (3.5) and of the exchange integral (3 .7), and then vary all the Ritz variational parameters in order to minimiz e the ground state energy (3.2), Emol(parameters) = minimum . (3.20) In general, we have four parameters in our problem, Emol=Emol(γ,ρ,V 0,rc), with the first two parameters characterizing inverse radius of electronic orbit 17and the internuclear distance, respectively, and the last t wo parameters com- ing from the Hulten potential. However, assuming that the is oelectronium is characterized by oneenergy level, i.e. β= 1, we have the relation (2.13) betweenV0andrcso that we are left with threeindependent parameters, say, Emol=Emol(γ,ρ,r c). In fact, we have three independent parameters for any fixed number βof the levels due to the general relation (2.11), V0=β2¯h2 2mr2 c, β= 1,2,.... (3.21) Behavior of the energy Emolas a function of γandρis more or less clear owing to known variational analysis of the standard model of H2molecule. Namely,Emolreveals a local minimum at some values of γandρ. Thus, we should closely analyze the rcdependence of the energy which is specific to the Santilli-Shillady model of H2molecule. Below, we turn to the Coloumb integral for the Hulten potenti al. 3.1.1 Coloumb integral for Hulten potential To calculate the Hulten part of the Coloumb integral (3.5) we use spher- ical coordinates, ( rb2,θ2,ϕ2), when integrating over second electron, and (rb1,θ1,ϕ1), when integrating over first electron. The integral is C′ h= 4π2π/integraldisplay 0dθ1∞/integraldisplay 0drb1π/integraldisplay 0dθ2∞/integraldisplay 0drb2Vh(r12)/parenleftBigγ3 πe−2γrb2rb22sinθ2/parenrightBig ×(3.22) ×/parenleftBigγ3 πe−2γ√ rb12+R2−rb12Rcosθ1rb12sinθ1/parenrightBig , where Hulten potential is Vh(r12) =V0e−√ rb22+rb12−2rb2rb1cosθ2/rc 1−e−√ rb22+rb12−2rb2rb1cosθ2/rc. (3.23) Here, we have used ra1=/radicalBig rb12+R2−rb12Rcosθ1, 18r12=/radicalBig rb22+rb12−2rb2rb1cosθ2, and the fact that integrals over azimuthal angles ϕ1andϕ2give us 4π2. First, we integrate over coordinates of second electron, I= 2ππ/integraldisplay 0dθ2∞/integraldisplay 0drb2Vh(r12)/parenleftBigγ3 πe−2γrb2rb22sinθ2/parenrightBig . (3.24) Integration over θ2gives us I=∞/integraldisplay 0drb2(I1+I2+I3+I4+I5), (3.25) where I1=−4γ3e−2γrb2rb22, (3.26) I2=−2γ3rcrb2 rb1/radicalBig (rb1−rb2)2e−2γrb2ln(1−e√ (rb1−rb2)2/rc), (3.27) I3=−2γ3rcrb2 rb1/radicalBig (rb1+rb2)2e−2γrb2ln(1−e√ (rb1+rb2)2/rc), (3.28) I4= 2γ3rc2rb2 rb1e−2γrb2Li2(e√ (rb1−rb2)2/rc), (3.29) I5= 2γ3rc2rb2 rb1e−2γrb2Li2(e√ (rb1+rb2)2/rc), (3.30) and Li2(z) =∞/summationdisplay k=1zk k2=0/integraldisplay zln(1−t) tdt (3.31) is dilogarithm function. Now, we turn to integrating over rb2. ForI1we have ∞/integraldisplay 0drb2I1=−1. (3.32) InI2, we should keep ( rb1−rb2) to be positive so we write down two separate terms, ∞/integraldisplay 0drb2I2=I21+I22≡rb1/integraldisplay 0drb2I2(rb2<rb1) +∞/integraldisplay rb1drb2I2(rb2>rb1).(3.33) 19In these two integrals, I21andI22, we change variable rb2toxandy, respec- tively, x= (rb1−rb2)/rc, r b1/rc<x< 0, y= (rb2−rb1)/rc,0<y< ∞, (3.34) in order to simplify integrating. In terms of these variable s, we have I21=0/integraldisplay rb1/rcdx2γ3rc3(x−rc rb1x2)e−2γ(rb1−rcx)ln(1−ex), (3.35) I22=−∞/integraldisplay 0dy2γ3rc3(y+rc rb1y2)e−2γ(rb1+rcy)ln(1−ey). (3.36) We are unable to perform these integrals directly. To calcul ate these inte- grals we use method of differentiating in parameter. Namely, we use simpler integrals, L1=/integraldisplay dx e2γrcxln(1−ex) (3.37) and L2=/integraldisplay dy e−2γrcyln(1−ey), (3.38) and differentiate them in parameter rcto reproduce I21andI22. (One can use parameter γfor this purpose, or introduce an independent parameter putting it to one after making calculations, with the same re sult.) Namely, by using definitions of L1andL2we have I21= 2γ3rc3(1 2d drcL1−rc 4rb1d2 drc2L1)|x=0 x=rb1/rc, (3.39) I22=−2γ3rc3(−1 2d drcL2−rc 4rb1d2 drc2L2)|y=∞ y=0. (3.40) Now, the problem is to calculate indefinite integrals, L1andL2, which make basis for further algebraic calculations. After making the calculations, we have L1=1 4γ2rc2e2γrc/parenleftBig 2γrc(Φ(ex,1,2γrc) + ln(1 −ex))−1/parenrightBig (3.41) 20and L2=−1 4γ2rc2e−2γrc/parenleftBig 2γrc(Φ(ey,1,−2γrc) + ln(1 −ey)) + 1/parenrightBig ,(3.42) where Φ(z,s,a) =∞/summationdisplay k=0zk (a+k)s, a+k/ne}ationslash= 0, (3.43) isLerch function , which is a generalization of polylogarithm function Lin(z) and Riemann ζ-function. Particularly, Li2(z) = Φ(z,2,0). Also, we note that the Lerch function arises when dealing with Fermi-Dira c distribution, e.g., ∞/integraldisplay 0dkks ek−µ+ 1=eµΓ(s+ 1)Φ( −eµ,s+ 1,1). (3.44) Below, we will need in derivatives of Lerch function Φ( z,s,a) in third argu- ment. By using the definition (3.43) we obtain directly d daΦ(z,s,a)≡Φ′(z,s,a) =−sΦ(z,s+ 1,a), (3.45) d2 da2Φ(z,s,a)≡Φ′′(z,s,a) =s(s+ 1)Φ(z,s+ 2,a). (3.46) Inserting (3.41) and (3.42) into (3.39) and (3.40) we get I21=1 4γrb1/parenleftBig e−2γ(rb1−rcx)(3 + 2γ(rb1−(2 +γrb1)rcx+γrc2x2)−(3.47) −2γrc((1 +γ(rb1−2(1 +γrb1)rcx+ 2γrc2x2))[Φ(ex,1,2γrc) + ln(1 −ex)]+ +2γrc(−(1 +γ(rb1−2rcx))Φ′(ex,1,2γrc) +γrcΦ′′(ex,1,2γrc)))/parenrightBig |x=0 x=rb1/rc, I22=1 4γrb1/parenleftBig e−2γ(rb1+rcx)(3 + 2γ(rb1+ (2 +γrb1)rcx+γrc2x2)+ (3.48) +2γrc((1 +γ(rb1+ 2(1 +γrb1)rcx+ 2γrc2x2))[Φ(ex,1,−2γrc) + ln(1 −ex)]+ +2γrc((1 +γ(rb1+ 2rcx))Φ′(ex,1,−2γrc) +γrcΦ′′(ex,1,−2γrc)))/parenrightBig |y=∞ y=0. Now, we have to use the above derivatives (3.45) and (3.46) of Lerch function to obtain final expressions for I21andI22. Then, we should take the limits 21x→rb1/rc,x→0, andy→0,y→ ∞ , respectively. The endpoints x=rb1/rcandy=∞can be inserted easily, with the endpoint y=∞ yielding zero, while the limits x→0 andy→0 require some care because of the presence of some divergent terms. To collect all the terms, we sum up I21and (−1)I22given by (3.47) and (3.48), put x=y, and take common limit x→0, inserting x= 0 for polynomial and exponential (welldefined) terms. We get I21−I22|x→0= =−1 2rb1/parenleftBig rce−2γrb1(2γrc(1 +γrb1)[Φ(ex,2,2γrc)−Φ(ex,2,−2γrc)]+ (3.49) +4γ2rc2[Φ(ex,3,2γrc) + Φ(ex,3,−2γrc)]+ +(1 +γrb1)[Φ(ex,1,2γrc) + Φ(ex,1,−2γrc)−2 ln(1−ex)]/parenrightBig |x→0 The limits of Lerch functions of second, Φ( ex,2,±2γrc), and third, Φ(ex,3,±2γrc), order, at x→0, are welldefined while each of the terms in B(2γrc)≡lim x→0[Φ(ex,1,2γrc) + Φ(ex,1,−2γrc)−2 ln(1−ex)] (3.50) isdivergent since Lerch function of first order, Φ( ex,1,±2γrc), increases un- boundedly at x→0. We will analyze this limit below, to identify the con- dition at which the divergencies cancel each other. Now, we c ollect all the terms obtaining final result for the integral in the form ∞/integraldisplay 0drb2I2=1 4rb1/parenleftBig1 γ(−3 + 2γrb1+ 8γ3rc3Φ(erb1/rc,3,2γrc)+ (3.51) +2γrc(1−γrc)[Φ(erb1/rc,1,2γrc) + 2γrcΦ(erb1/rc,2,2γrc) + ln(1 −erb1/rc)])− −2rce−2γrb1(1 +γrb1){B(2γrc) + 2γrc[ζ(2,2γrc)−ζ(2,−2γrc)]+ +4γ2rc2[ζ(3,2γrc) +ζ(3,−2γrc)]}/parenrightBig , where ζ(s,a) =∞/summationdisplay k=11 (a+k)s, a+k/ne}ationslash= 0, (3.52) 22is generalized Riemann ζ-function. The values of ζ(2,±2γrc) andζ(3,±2γrc) entering (3.51) are welldefined. For example, at γ= 1.4 andrc= 0.0048, we have ζ(2,±2γrc)≃5537, ζ(3,±2γrc)≃2462. (3.53) Now, we turn to close consideration of the limit (3.50) enter ing (3.51). Let us calculate it for the particular value 2 γrc= 1/100. Using expansion of each term of Baroundx= 0, we obtain B(1 100) = lim s→1/bracketleftBig 100−1 Γ(1 100){100Γ(101 100)(C+ ln(1 −s) +ψ(1 100))}−(3.54) −1 99Γ(99 100){100Γ(199 100)(C+ ln(1 −s) +ψ(99 100))}+ 2 ln(1 −s) +O(1−s)/bracketrightBig , where we have denoted, for brevity, s=ex, ψ(z) =∞/summationdisplay n=01 z+n=Γ′(z) Γ(z)(3.55) is digamma function, Γ( z) is Euler gamma function, and Cis Euler constant. Using elementary properties of gamma function we obtain fro m Eq.(3.54) B(1 100) = 100 −2C−ψ(1 100)−ψ(99 100), (3.56) so one can see that the logarithmic divergent terms cancel ea ch other, and the limit is welldefined for 2 γrc= 1/100. The same is true for any integer value of k=1 2γrc(3.57) while at noninteger kthe limitB(1 k) blows up. Generalizing the above par- ticular result (3.56), we can write down B(1 k) =k−2C−ψ(1 k)−ψ(1−1 k), (3.58) for any integer k>2. This highly remarkable result means that to have finite value of the Coloumb integral we should use the condition that λ−1≡(2γrc)−1=k 23is an integer number. Recalling that typically γ≃1.5 andrc≃0.01 we have the integer number k≃30. Now, we turn to the next integral, I3. It is similar to I2so that we present the final expression, ∞/integraldisplay 0drb2I3=1 4γrb2/parenleftBig 3+2γrb1+2γrc(1+γrb1)[Φ(erb1/rc,1,−2γrc)+ln(1 −erb1/rc)] (3.59) −4γ2rc2(1 +γrb1)Φ(erb1/rc,2,−2γrc) + 8γ3rc3Φ(erb1/rc,3,−2γrc)/parenrightBig . The integral I4is more complicated, ∞/integraldisplay 0drb2I4=I41+I42, (3.60) where I41=rb1/integraldisplay 0drb22γ3rc2rb2 rb1e−2γrb2Li2(e(rb1−rb2)/rc), (3.61) I42=∞/integraldisplay rb1drb22γ3rc2rb2 rb1e−2γrb2Li2(e(rb2−rb1)/rc). (3.62) Introducing variables x= (rb1−rb2)/rc, y= (rb2−rb1)/rc, (3.63) we rewrite the integrals in the form I41=0/integraldisplay rb1/rcdx2γ3rc3e2γ(rb1−rcx)[Li2(ex)−rc rb1xLi2(ex)], (3.64) I42=−∞/integraldisplay 0dy2γ3rc3e2γ(rb1+rcy)[Li2(ey) +rc rb1yLi2(ey)]. (3.65) In the r.h.s. of I41, the first term can be calculated directly in terms of Lerch function while the second term can be obtained from the first term by 24differentiating it in the parameter, for which we choose agai nrc. Namely, the basic integral, which we will use to calculate I41, is M1=0/integraldisplay x0dx e2γrcxLi2(ex), (3.66) for which we have M1=1 24γ3rc3Γ(2γrc)/parenleftBig 3(e2γrcx0−1)Γ(2γrc)+ (3.67) +Γ(1 + 2γrc)(γrcπ2−3C−3ψ(2γrc))− −3e2γrcx0Γ(1 + 2γrc)(Φ(ex0,1,2γrc) + ln(1 −ex0) + 2γrcLi2(ex0))/parenrightBig . We use this result in the first term of I41. Differentiating M1given by Eqs. (3.66) and (3.67) in rc, we reproduce the second term of I41, up to a factor. So, collecting these results and inserting x0=rb1/rcwe obtain after some algebra I41=−1 12rb1Γ(1 + 2γrc)/parenleftBig e−2γrc[rc(9 + 4π2γ3rc2rb1)Γ(2γrc)+ (3.68) Γ(1 + 2γrc)(6Crc−3rb1−6Cγr crb1−π2γrc2−6(γrb1−1)rcψ(2γrc)− −6γrc2ψ′(2γrc))] + 3(2rb1Γ(1 + 2γrc)−3rcΓ(2γrc)+ +2rcΓ(1 + 2γrc)[(1−2γrb1)Φ(erb1/rc,1,2γrc) +γrcΦ(erb1/rc,2,2γrc)+ +(1−2γrb1) ln(1−erb1/rc) +γ(1−4γrb1)rcψ′(erb1/rc)]/parenrightBig , whereψ′(z) =dψ(z)/dzis derivative of digamma function. To calculate I42we use a similar method. However, care should be exerted when taking limit y→0. The basic integral, which we will use to calculate I42, is M2=−/integraldisplay dy e2γrcyLi2(e−y), (3.69) where we have replaced y→ −yso that the endpoints will be due to 0 < y<−∞. The result for M2is M2=1 8γ3rc3e2γrcy(1 + 2γrceyΦ(ey,1,1 + 2γrc) + 2γrcln(1−e−y)−(3.70) 25−4γ2rc2Li2(e−y)). We should insert here the endpoints y= 0 andy=−∞. In the limit y→ −∞ ,M2is zero. In the limit y→0, we have Li2(e−y)|y→0=π2 6(3.71) and, assuming that k= 1/(2γrc) is an integer number, Φ(ey,1,1 + 2γrc) + ln(1 −e−y)|y→0=−(1 2γrc+C+ψ(2γrc)).(3.72) Thus, M2|y=−∞ y=0=1 12γ2rc2(3C+π2γrc+ 3ψ(2γrc)), (3.73) for integer k. We should point out that, in the case of noninteger k,M2 increases unboundedly at y→0. Using this result in I42, we obtain I42=−rc 12rb1e−2γrb1/parenleftBig 6C(1 +γrb1) +π2γrc+ 2π2γ2rcrb1+ (3.74) +6(1 +γrb1)ψ(2γrc)−6γrcψ′(2γrc)/parenrightBig . Summing up I41given by (3.68) and I42given by (3.74), we get ∞/integraldisplay 0drb2I4==1 4rb1Γ(1 + 2γrc)/parenleftBig 3rcΓ(2γrc) +e−2γrc[−rb1Γ(1 + 2γrc)+ (3.75) +rc(−3Γ(2γrc) + 4Γ(1 + 2 γrc)(−(1 +γrb1)(C+ψ(2γrc) +γrcψ′(2γrc)))]− −2rcΓ(1 + 2γrc)[Φ(erb1/rc,1,2γrc) + ln(1 −erb1/rc) +γrc(Φ(erb1/rc,2,2γrc)+ +Li2(erb1/rc))]/parenrightBig . The integral I5is similar to I4so that we present the final expression, ∞/integraldisplay 0drb2I5=−1 8γrb1/parenleftBig 3 + 4γrc[e−rb1/rc(Φ(e−rb1/rc,1,1 + 2γrc)+ (3.76) +Φ(e−rb1/rc,2,1 + 2γrc)) + ln(1 −erb1/rc)−γrcψ′(erb1/rc)]/parenrightBig . 26Now, we are in a position to sum up all the calculated integral sI1,...,I 5, and obtain, due to (3.25), the following final expression for the Coloumb integral over coordinates of second electron, I(rb1) =−(1 2+5 8γrb1)e−2γrb1+ (3.77) +1 2γrc/bracketleftBig π(1 +1 γrb1)ctg(2γrcπ)e−2γrb1−1 γrb1e−rb1/rcΦ(e−rb1/rc,1,1 + 2γrc)+ +Φ(erb1/rc,1,−2γrc)−Φ(erb1/rc,1,2γrc) +1 γrb1Φ(erb1/rc,1,−2γrc)/bracketrightBig + +γ2rc2/bracketleftBig −1 2γrb1e−rb1/rcΦ(e−rb1/rc,2,−2γrc)−Φ(erb1/rc,2,−2γrc)− −Φ(erb1/rc,2,2γrc) +1 γrb1(1 2Φ(erb1/rc,2,2γrc)−Φ(erb1/rc,2,−2γrc))+ +1 γrb1e−2γrb1ψ′(2γrc) +e−2γrb1(1 +1 γrb1)(ζ(2,−2γrc)−ζ(2,2γrc))/bracketrightBig + +2 γrb1γ3rc3/bracketleftBig Φ(erb1/rc,3,2γrc) + Φ(erb1/rc,3,−2γrc)− −e−2γrb1(ζ(3,2γrc) +ζ(3,−2γrc))/bracketrightBig , where we have collected the terms due to power degrees of rc. It should be stressed that here (2 γrc)−1is assumed to be an integer number. The above expression represents the Hulten part of the electrostatic potential caused by charge distribution of the second electron. Next step is to integrate (3.77) over the coordinates of firstelectron, C′ h= 2ππ/integraldisplay 0dθ1∞/integraldisplay 0drb1I(rb1)γ3 πe−2γ√ rb12+R2−rb12Rcosθ1rb12sinθ1.(3.78) Prior to that, we denote λ= 2γrc=1 k, r=γrb1, (3.79) 27and rewrite Eq. (3.77) in a more compact form, I(r) =−(1 2+5 8r)e−2r+1 4λ/bracketleftBig π(1+1 r)ctg(πλ)e−2r−1 re−2r/λΦ(e−2r/λ,1,1+λ)+ (3.80) +Φ(e2r/λ,1,−λ)−Φ(e2r/λ,1,λ) +1 rΦ(e2r/λ,1,−λ)/bracketrightBig + +λ2 4/bracketleftBig −1 2re−2r/λΦ(e−2r/λ,2,−λ)−Φ(e2r/λ,2,−λ)−Φ(e2r/λ,2,λ)+ 1 r(1 2Φ(e2r/λ,2,λ)−Φ(e2r/λ,2,−λ)+e−2rψ′(λ))+e−2r(1+1 r)(ζ(2,−λ)−ζ(2,λ))/bracketrightBig +λ3 4r/bracketleftBig Φ(e2r/λ,3,λ) + Φ(e2r/λ,3,−λ)−e−2r(ζ(3,λ) +ζ(3,−λ))/bracketrightBig . SinceI(r) does not depend on θ1one can easily integrate over θ1in Eq.(3.78), and then change variable rb1tor=γrb1, obtaining C′ h=1 2ρ∞/integraldisplay 0dr I(r)/bracketleftBig (1 + 2/radicalBig (ρ−r)2)re−2√ (ρ−r)2−(1 + 2(ρ+r))re−2(ρ+r)/bracketrightBig , (3.81) whereρ=γR. Again, we should use separate intervals to keep ( ρ−r) to be positive, namely, we rewrite C′ has C′ h=J1+J2+J3, (3.82) where J1=1 2ρρ/integraldisplay 0dr I(r)(1 + 2ρ−2r)re−2(ρ−r), (3.83) J2=1 2ρ∞/integraldisplay ρdr I(r)(1 + 2r−2ρ)re−2(r−ρ), (3.84) J3=−1 2ρ∞/integraldisplay 0dr I(r)(1 + 2ρ+ 2r)re−2(ρ+r). (3.85) Now, we are ready to make integration over the last remaining variable,r, to obtain complete analytical expression of the Coloumb integ ral for the Hulten potential. 28However, Lerch functions entering Eq.(3.80) make obstacle to do integral (3.82) for a general case because they have different functio nal form for dif- ferent values of the parameter λ. So, each of the above integrals J1,2,3should be calculated independently for every numerical value of λ. Moreover, for the values of interest, e.g., λ= 1/30,eachLerch function is expressed in the form of sum of elementary functions with too big number of non trivial terms to handle them (incomplete Euler beta function arises here) . So, the integral cannot be reliably calculated even for a single value of λ, within the interval of interest, λ= 1/30,1/31,...,1/100. Also, elementary analysis shows that we can not implement the assumption of small rcinto Eq.(3.80), to use first order approximation in rc. Indeed, Lerch functions in (3.80) contain rcboth in first and third argument so that their asymptotics at rc→0 make no sense. Thus, we stop here further calculation of the Coloumb integr alC′ hgetting, however, as our main result the fact that (2 γrc)−1should be integer number, in the variational approach to the model, to have finite energ y of the ground state. We consider this as very interesting result deservin g rather involved calculations made above. Also, we have a detailed technical view on the problems which arise when dealing with molecular integrals with the Hulten potential . Practically, this means that there is a very little hope that the exchange integral (3.7), which is structurally much more complicated than the above consid ered Coloumb one, can be calculated exactly for the case of Hulten potenti al. Because of these difficulties, below we use appropriate simplified poten- tials, instead of Hulten potential, to have some analytical set up for the variational analysis of the Santilli-Shillady model. Clea rly, by this we go to some approximation to the original Santilli-Shillady mode l. 3.1.2 Coloumb integral for exponential screened Coloumb po ten- tial We use simple function to mimic Hulten potential. Namely, we approximate the general potential (2.2) by V(r12) =VC+Ve=e2 r12−Ae−r12/rc r12, (3.86) 29whereAandrcare positive parameters. It has similar behavior both at sho rt and long distances. Indeed, at long distances, r12→ ∞, we can ignore Veand the behavior is solely due to the Coloumb potential while its series expansion about the point r12= 0 (short distances) is V(r12)|r12→0=e2−A r12+A rc−A 2rcr12+O(r2 12). (3.87) Here, we should put A=V0rcto have the same coefficient at r−1 12in the r12→0 asymptotics as it is in the case of Hulten potential; see Eq. (2.4). Using Eq.(3.21) we have A=V0rc=β2¯h2 2mrc, β= 1,2,..., (3.88) whereβis a number of energy levels of isoelectronium. Taking β= 1 we have, in atomic units (¯ h= 1,m=me/2 = 1/2), A=1 rc. (3.89) Below, we calculate the Coloumb integral (3.5), with the exp onential screened Coloumb potential Vedefined by Eq.(3.86), C′ E=/integraldisplay dv1dv2/parenleftBigge2 r12−Ae−r12/rc r12/parenrightBigg |f(ra1)|2|f(rb2)|2, (3.90) Below, we present some details of calculation of the Coloumb integral (3.90). Apart from the case of Hulten potential considered in Sec. 3. 1.1, it appears that this integral can be calculated in terms of elementary f unctions. The integral we are calculating is C′ e=/integraldisplay dv1dv2Ae−r12/rc r12|f(ra1)|2|f(rb2)|2, (3.91) where f(r) =/radicalBigg γ3 πe−γr, (3.92) 30anddv1anddv2are volume elements for the first and second electron, respec - tively. We use spherical coordinates. In spherical coordin ates (rb2,θ2,ϕ2), with polar axis directed along the vector /vector rb1, we have r12=/radicalBig rb12+rb22−2rb1rb2cosθ2. (3.93) We use these coordinates when integrating over second elect ron. In spherical coordinates ( rb1,θ1,ϕ1), with polar axis directed along the vector /vectorR, we have ra1=/radicalBig rb12+R2−2rb1Rcosθ1. (3.94) We use these coordinates when integrating over first electro n. First, we integrate over angular coordinates of second elec tron, I1=2π/integraldisplay 0dϕ2π/integraldisplay 0dθ2Ae−r12/rc r12γ3 πe−2γrb2rb22sinθ2, (3.95) wherer12is defined by (3.93). It is relatively easy to calculate this i ntegral, I1=2Aγ3rc rb1e−2γrb2/parenleftbigg e−√ (rb2−rb1)2/rc−e−√ (rb2+rb1)2/rc/parenrightbigg . (3.96) Further, integrating on radial coordinate rb2must be performed in separate intervals, I2=rb1/integraldisplay 0drb2I1(rb2<rb1) +∞/integraldisplay rb1drb2I1(rb2>rb1), (3.97) where/radicalBig (rb1−rb2)2=/braceleftBigg rb1−rb2, r b2<rb1, rb2−rb1, r b2>rb1,(3.98) with the result I2=4Aγ3rc2(4γrc2(e−rb1/rc−e−2γrb1) +rb1e−2γrb1(1−4γ2rc2)) rb1(1−4γ2rc2)2.(3.99) Now, we turn to integrating over coordinates of first electro n, (rb1,θ1,ϕ1), I3=π/integraldisplay 0dθ12π/integraldisplay 0dϕ1I2γ3 πe−2γra1rb12sinθ1, (3.100) 31wherera1is defined by (3.94). We obtain after tedious calculations I3=2Aγ4rc2 R(1−4γ2rc2)2e−2γ(√ (R−rb1)2+rb1+√ (R+rb1)2)−rb1/rc(3.101) ×/bracketleftbigg e2γ√ (R+rb1)2−2γe2γ√ (R+rb1)2/radicalBig (R−rb1)2+ 2γe2γ√ (R−rb1)2/radicalBig (R+rb1)2/bracketrightbigg ×/bracketleftBig erb1/rc(4γ(1 +γrb1)rc2−rb1)−4γrc2e2γrb1/bracketrightBig . Again, we must further integrate in rb1by separate intervals, I4=R/integraldisplay 0drb1I3(rb1<R) +∞/integraldisplay Rdrb1I3(rb1>R)≡I41+I42, (3.102) obtaining after rather tedious calculations I41=Ae−6γRγ2rc2 12R(1−4γ2rc2)4× (3.103) /bracketleftBig 96e2γR−R/rcγ3/parenleftBig (4γ(2γRr c+R+rc)+1)(1 −2γrc)2+e4γR(2γrc+1)2(4γrc−1)/parenrightBig rc3 −3e4γR+ 3γ/parenleftBig −64γ5(γR(8γR+ 13) + 4)rc6+ 16γ3(γR(24γR+ 31) + 9)rc4 −4γ(γR(24γR+ 23) + 6)rc2+R(8γR+ 5)/parenrightBig −e4γRγ× /parenleftBig 64γ5(γR(4γR(2γR+9)+57)+36) rc6−48γ3(γR(4γR(2γR+7)+21) −9)rc4 +12γ(γR(4γR(2γR+ 5) + 1) −6)rc2−R(4γR(2γR+ 3)−3) + 3)/parenrightBig + 3/bracketrightBig , I42=−Ae−6γRγ2rc2 4R(1−2γrc)2(1 + 2γrc)4× (3.104) ×/bracketleftBig 1−e4γR−128γ6R2rc5+γ/parenleftBig (5−3e4γR)R−4(e4γR−1)rc/parenrightBig + +16γ5Rrc3/parenleftBig −8R+ (16e2γR−R/rc+ 3e4γR−13)rc/parenrightBig + +16e−R/rcγ4rc3/parenleftBig (8e2γR+3e4γ+R/rc−13eR/rcR−4(e4γR−1)(2e2γR−eR/rc)rc/parenrightBig + +4γ2/parenleftBig 2R2−(3e4γR−5)Rrc+ 3(e4γR−1)rc2/parenrightBig + 32+32γ3rc/parenleftBig R2−Rrc+e−R/rc(e4γR−1)(2eR/rc−e2γR)rc2)/parenrightBig/bracketrightBig . In calculating I42, we put the condition 6γrc<1, (3.105) which is necessary to prevent divergency at the endpoint rb1=∞. Collecting the above two integrals we obtain I4≡ C′ e=−Aγ3rc2 6R(1−4γ2rc2)4/bracketleftBig e−2γR/parenleftBig −R(3 + 2γR(3 + 2γR)) (3.106) +12γ2Rrc2(5 + 2γR(5 + 2γR))−48γ4Rrc4(15 + 2γR(7 + 2γR)) +64γ5rc6(24 +γR(33 + 2γR(9 + 2γR)))/parenrightBig −1536γ5rc6e−R/rc/bracketrightBig . Thus, we have finally for the Coloumb integral for exponentia l screened Coloumb potential, C′ e=−Aλ2 8(1−λ2)4γe−2ρ ρ/bracketleftbigg −(ρ+ 2ρ2+4 3ρ3) + 3λ2(5ρ+ 10ρ2+ 4ρ3) (3.107) −λ4(15ρ+ 14ρ2+ 4ρ3) +λ6(8 + 11ρ+ 6ρ2+4 3ρ3−8e2ρ−2ρ λ)/bracketrightbigg . Here, we have used notation λ= 2γrc, and alsoλ<1/3 due to Eq.(3.105). 330 1 2 3 4 5 6 r, a.u.-0.006-0.005-0.004-0.003-0.002-0.0010Ce', a.u. Figure 5: The Coloumb integral C′ eas a function of ρ, Eq. (3.107), at λ= 1/37. Here,ρ=γR, whereRis the internuclear distance, and λ= 2γrc, wherercis the correlation length parameter. 0 1 2 3 4 rc, a.u.-0.09-0.08-0.07-0.06-0.05-0.04-0.03Ce,, a.u. Figure 6: The Coloumb integral C′ eas a function of rc, Eq. (3.107), at ρ= 1.67. Forrc>0.2 a.u., the regularized values of C′ eare presented. 340 0.02 0.04 0.06 0.08 0.1 rc, a.u.-0.0175-0.015-0.0125-0.01-0.0075-0.005-0.00250Ce,, a.u. Figure 7: The Coloumb integral C′ eas a function of rc, Eq. (3.107), at ρ= 1.67. More detailed view. The total Coloumb integral is C′ E=C′ C− C′ e, (3.108) where C′ Cis wellknown Coloumb potential part given by Eq.(3.13). Below, we turn to the other potential, Gaussian screened Col oumb po- tential, considered by Santilli and Shillady [1]. The Colou mb integral for this potential can be calculated exactly, and the result con tains one special function, the error function erf( z). 3.1.3 Coloumb integral for Gaussian screened Coloumb poten tial In this Section, we calculate the Coloumb integral for the ca se of Gaussian screened potential. Namely, we approximate the general pot ential (2.2) by [1] V(r12) =VC+Vg=e2 r12−Ae−r2 12/c r12, (3.109) 35whereAandc=rc2are positive parameters. At long distances, r12→ ∞, we can ignore Vgwhile its series expansion about the point r12= 0 is V(r12)|r12→0=e2−A r12+A cr12+O(r2 12). (3.110) Here, we should put A=V0rcto have the same coefficient at r−1 12in the r12→0 asymptotics as it is in the case of Hulten potential; see Eq. (2.4). The Coloumb integral is C′ G=/integraldisplay dv1dv2/parenleftBigge2 r12−Ae−r2 12/c r12/parenrightBigg |f(ra1)|2|f(rb2)|2. (3.111) The integral we are calculating is C′ g=/integraldisplay dv1dv2Ae−r2 12/c r12|f(ra1)|2|f(rb2)|2, (3.112) where notation and coordinate system are due to Sec. 3.1.2. F irst, we inte- grate over angular coordinates of second electron, I1=2π/integraldisplay 0dϕ2π/integraldisplay 0dθ2Ae−r2 12/c r12γ3 πe−2γrb2rb22sinθ2, (3.113) wherer12is defined by (3.93). We have I1=Aγ3√πce−2γrb2 rb1 erf(/radicalBigg (rb1+rb2)2 c)−erf(/radicalBigg (rb1−rb2)2 c) ,(3.114) where erf(z) =2√πz/integraldisplay 0e−t2dt (3.115) is error function. Further, integrating on radial coordina terb2must be per- formed in separate intervals, I2=rb1/integraldisplay 0drb2I1(rb2<rb1) +∞/integraldisplay rb1drb2I1(rb2>rb1), (3.116) 36where/radicalBig (rb1−rb2)2=/braceleftBigg rb1−rb2, r b2<rb1, rb2−rb1, r b2>rb1,(3.117) with the result I2=−Aγ√ce−2γrb1−rb12/c 4rb1/parenleftBigg 4γ√c(erb12/c−e2γrb1)) (3.118) +√πerb12/c+cγ2/bracketleftBigg (1 + 2γ(rb1−cγ))(erfc(rb1−cγ√c) + 2erfc(√cγ)−2) +e4γrb1(2γ(rb1+cγ)−1)erfc(rb1+cγ√c)/bracketrightBigg/parenrightBigg , where erfc( z) = 1−erf(z). Now, we turn to integrating over coordinates of first electron, ( rb1,θ1,ϕ1), I3=π/integraldisplay 0dθ12π/integraldisplay 0dϕ1I2γ3 πe−2γra1rb12sinθ1, (3.119) wherera1is defined by (3.94). We obtain after tedious calculations I3=−A√cγ2 8Re−rb12 c−2γ(√ (R−rb1)2+2rb1+√ (R+rb1)2)(3.120) ×/parenleftBigg e2γ(√ (R−rb1)2+rb1)−e2γ(√ (R+rb1)2+rb1)−2γe2γ(√ (R+rb1)2+rb1)/radicalBig (R−rb1)2 +2γe2γ(√ (R−rb1)2+rb1)/radicalBig (R+rb1)2/parenrightBigg ×/bracketleftBigg√πerb12 c+cγ2/parenleftBigg (1 + 2γrb1−2cγ2)(2erfc(γ√c) + erfc(rb1−γc√c)−2) +(1 + 2γrb1+ 2cγ2)e4γrb1erfc(rb1+γc√c)/parenrightBigg/bracketrightBigg . Again, we must further integrate in rb1by separate intervals, I4=R/integraldisplay 0drb1I3(rb1<R) +∞/integraldisplay Rdrb1I3(rb1>R). (3.121) 37First, we replace the endpoint rb1=∞by finite value rb1= Λ to avoid divergencies at intermediate calculations. After straigh tforward but tedious calculations we obtain rather long expression so that we do n ot represent it here noting however that the following integrals are used du ring the calcula- tions:/integraldisplay erf(z)dz=e−z2 √π+zerf(z), (3.122) /integraldisplay zerf(z)dz=ze−z2 2√π−1 4erf(z) +1 2z2erf(z), (3.123) /integraldisplay e−azerf(z)dz=−1 ae−azerf(z) +1 aea2/4erf(a 2+z), (3.124) /integraldisplay ze−azerf(z)dz=−1 a√πe−az−z2−1 a2e−az(1 +az)erf(z) (3.125) −1 2a2(a2−1)ea2/4erf(a 2+z), /integraldisplay e−az−bz2dz=√π 2√ bea2/(4b)erf(a+ 2bz 2√ b), (3.126) /integraldisplay ze−az−bz2dz=−1 2be−az−bz2−a√π 4b3/2ea2/(4b)erf(a+ 2bz 2√ b). (3.127) Using lim Λ→∞erf(Λ) = 1 and replacing welldefined exponentially decreasi ng terms by zero, we obtain some finite terms and big number (abou t fourty) of Λ dependent terms, which are unbounded at Λ → ∞. All the divergent terms totally cancel each other so the final expression turns out to be automatically finite. As the result, we obtain the Coloumb integral for Gaussian sc reened Coloumb potential in the following form: C′ g=Aγκe−2ρ 96ρ/bracketleftBigg −(60 + 96ρ+ 48ρ2)κ+ (32 + 48ρ)κ3−16κ5(3.128) +/parenleftBig (60 + 16ρ2)κ−32κ3+ 16κ5/parenrightBig e2ρ−ρ2 κ2 +√πeκ2/parenleftBigg (30ρ+8ρ3−36ρκ2+24ρκ4)(2erf(κ)−erfc(ρ κ−κ)−e4ρerfc(ρ κ+κ)) 380 1 2 3 4 5 6 r, a.u.-0.003-0.0025-0.002-0.0015-0.001-0.00050Cg,, a.u. Figure 8: The Coloumb integral C′ gas a function of ρ, Eq. (3.128), at 2 κ= λ= 1/37. Here,ρ=γR, whereRis the internuclear distance, and λ= 2γrc, wherercis the correlation length parameter. +(15+24ρ2−(18+24ρ2)κ2+12κ4−8κ6)(2erf(κ)−erfc(ρ κ−κ)+e4ρerfc(ρ κ+κ))/parenrightBigg/bracketrightBigg , where we have used notation κ=γ√c=γrc=λ 2. (3.129) The total Coloumb integral is C′ G=C′ C− C′ g, (3.130) where C′ Cis given by Eq.(3.13). 390 1 2 3 4 rc, a.u.-0.1-0.09-0.08-0.07-0.06-0.05-0.04Cg,, a.u. Figure 9: The Coloumb integral C′ gas a function of rc, Eq. (3.128), at ρ= 1.67. 0 0.02 0.04 0.06 0.08 0.1 rc, a.u.-0.008-0.006-0.004-0.0020Cg,, a.u. Figure 10: The Coloumb integral C′ gas a function of rc, Eq. (3.128), at ρ= 1.67. More detailed view. 403.1.4 Exchange integral Our general remark is that all calculations for the above Coloumb integrals are made in spherical coordinates, which correspond to sphe rical symme- try of the charge distributions of both 1 selectrons, |ψ(ra1)|2and|ψ(ra2)|2, each moving around one nucleus. One can use prolate spheroid al coordi- nates, which are exploited sometimes when integrating over coordinates of last electron, but we have encountered the same problem of bi g number of terms in the intermediate expressions, with no advantage in comparison to the use of spherical coordinates. Unlike to Coloumb integral, calculation of exchange integral should be made in the spheroidal coordinates, which correspond to sph eroidal symme- try of charge distributions of the electrons, ψ∗(ra1)ψ(rb1) andψ∗(ra2)ψ(rb2), each moving around boththe nuclei,aandb. Calculation of the exchange integral, E′=/integraldisplay dv1dv2V(r12)f∗(ra1)f(rb1)f∗(ra2)f(rb2), (3.131) essentially depends on the form of the potential V(r12) in the sense that the integration can be made only in spheroidal coordinates, ( x1,y1,ϕ1) and (x2,y2,ϕ2), and one should use an expansion of V(r12) in the associated Legendre polynomials. For the usual Coloumb potential, V(r12) =r−1 12, it is rather long (about 12 pages to present the main details) and nontrivial calcula tion, where Neu- mann expansion in terms of associated Legendre polynomials , in spheroidal coordinates, is used (celebrated result by Sugiura, see Eq. (3.15)). In general, any analytical square integrable function can b e expanded in associated Legendre polynomials. However, in direct calcu lating of the ex- pansion coefficients by means of integral of the function with Legendre poly- nomials, one meets serious problems even for simple functio ns. Practically, one uses, instead, properties of special functions to deriv e such expansions. We mention that there is Gegenbauer expansion [6], having in a particular case the form [5] eikr12 r12=1 r1r2∞/summationdisplay l=0/radicalBigg 2l+ 1 4πi kjl(kr1)n(1) l(kr2)Yl,0(θ12), (3.132) wherejl(z) andn(1) l(z) are spherical Bessel and spherical Hankel functions of first kind, respectively, θ12is angle between vectors /vector r1and/vector r2, andr1=|/vector r1|, 41r2=|/vector r2|;r1<r2. Spherical harmonics Yl,0(θ12) can be rewritten in terms of Legendre polynomials due to the summation theorem. We note that this expansion can be used, at k=i/rc, to reproduce ex- ponential screened potential, Ve(r12), and to calculate associated exchange integral (3.131) but, however, it concerns spherical (not spheroidal) coordi- nates, (r1,θ1,ϕ1) and (r2,θ2,ϕ2). For Hulten potential Vh(r12), exponential screened potential, Ve(r12), and Gaussian screened potential, Vg(r12), which are of interest in this paper, we have no such an expansion in spheroidal coordinates. To stress that this is not only the problem of changing coordinate system, we men tion that the solution of usual 3-dimensional wave equation, ∆ ψ+k2ψ= 0, is given by function ei/vectork/vector r/r, in spherical coordinates, to which one can apply Gegen- bauer expansion, while in spheroidal coordinates its solut ion is represented by complicated function containing infinite series of recur rent coefficients [7]; see also [4], Sec. 3.4. As the result, we have no possibility t o calculate exactly exchange integrals for these non-Coloumb potentials. In order to obtain approximate expression for the exchange integral for the case of the above non-Coloumb potentials, we make analysis o f asymptotics of the standard exchange integral (i.e. that for the Coloumb potential), Eq.(3.15). It is easy to derive that E′ C|ρ→∞∼e−2ρ, (3.133) at long distances between the nuclei, and E′ C|ρ=0=5 8γ, (3.134) in the case of coinciding nuclei. At r−1 c→0, we should have the same asymptotics for exchange integral for each of the above non- Coloumb poten- tials because these potentials behave as Coloumb potential atr−1 c→0. In both the limiting cases, ρ→ ∞ andρ= 0, the exchange integral for the non-Coloumb potentials is simplified, and one can use spheri cal coordinates since the two-center problem is reduced to one-center probl em. We consider two limiting cases. a)ρ=∞case. This case is trivial because exchange integral tends to zero due to lack of overlapping of the wave functions of two Hatoms. 42b)ρ= 0case. In this case , we have ra1=rb1=r1andra2=rb2=r2so that Eq.(3.131) becomes E′=/integraldisplay dv1dv2V(r12)|f(r1)|2|f(r2)|2, (3.135) One can see that this is the case of Heatom with two electrons in the ground state. Evidently, in terms of our anzatz (3.1) we have comple te overlapping of the wave functions. Even the above mentioned simplification of the exchange inte gral and use of spherical coordinates does not enable us to calculate straightforwardly the integral (3.135) for the non-Coloumb potentials, Vh,Ve, orVg; the integrands are still too complicated. This indicates that we should use expansion of these potentials in Legendre polynomials, in spherical coordina tes, to perform the integrals. Only exponential screened potential Veis given such an expansion here. Namely, this is Gegenbauer expansion (3.132), owing t o which we can calculate the exchange potential for exponential screened potentialVe, to which we turn below. Exchange integral for the exponential screened Coloumb pot en- tialVe, atρ= 0. The integral is E′ E|ρ=0≡(E′ C− E′ e)|ρ=0=5 8γ−/integraldisplay dv1dv2Ae−r12/rc r12|f(r1)|2|f(r2)|2,(3.136) where we have used Eq.(3.134) for the usual Coloumb potentia l part of the integral. In the Gegenbauer expansion (3.132), we assume k=i/rcto repro- duce the potential Ve(r12). Since the wave functions f(r1) andf(r2) given by Eq.(3.10) do not depend on the angles, only l= 0,m= 0 term of the expan- sion (3.132) contributes to the exchange integral (3.136) d ue to orthogonality of Legendre polynomials. Using j0(z) = sinz, n 0(z) =−ieiz, Y 0,0=/radicalBigg 1 4π, (3.137) we thus have Aeikr12 r12→/braceleftBiggA kr1r2sinkr1eikr2, r 1<r2, A kr1r2sinkr2eikr1, r 1>r2,(3.138) 43Then the exchange integral (3.136) is written as E′ E|ρ=0=5 8γ−∞/integraldisplay 04πr2 2dr2/bracketleftBigr2/integraldisplay 04πr2 1dr1A kr1r2sinkr1eikr2γ3 πe−2γr1γ3 πe−2γr2 (3.139) +∞/integraldisplay r24πr2 1dr1A kr1r2sinkr2eikr1γ3 πe−2γr1γ3 πe−2γr2/bracketrightBig , where 4πr2 1and 4πr2 2are volume factors. The two above integrals over r1can be easily calculated, with the result 16Aγ6r2 (k2+ 4γ2)2/bracketleftBig 4γei(k+2iγ)r2+1 kei(k+4iγ)/parenleftBig (k2−4γ2) sinkr2−4kγcoskr2(3.140) −(k2+ 4γ2)(kcoskr2+ 2γsinkr2)/parenrightBig/bracketrightBig and −16Aγ6r2 k(k+ 2iγ)2/parenleftBig 1 + (2γ−ik) sinkr2ei(k+4iγ)/parenrightBig . (3.141) Summing up these terms and integrating over r2we get after some algebra E′ E|ρ=0=5 8γ+Aγ3 2(k+ 2iγ)4(k2+ 8ikγ−20γ2). (3.142) Inserting k=i rc, (3.143) to reproduce the potential Ve, and denoting λ= 2γrcwe write down our final result, E′ E|ρ=0=5 8γ−γAλ2 8(1 +λ)4(1 + 4λ+ 5λ2). (3.144) Note that, at r−1 c→0, i.e. atλ→ ∞, we have E′ E|ρ=0=5 8γ−5 8Aγ (3.145) that is in agreement with the value (3.134). We should to emph asize here that Eq.(3.144) is exact result for the exchange integral E′ E, atρ= 0. 440 1 2 3 4 5 6 r, a.u.-0.006-0.005-0.004-0.003-0.002-0.0010Ee', a.u. Figure 11: The exchange integral E′ eas a function of ρ, Eq. (3.146), at λ= 1/37. Here,ρ=γR, whereRis the internuclear distance, and λ= 2γrc, wherercis the correlation length parameter. Next step is to implement ρdependence into (3.144) following to natural criteria. To restore partially ρdependence in the exchange integral (3.144), we use exact result (3.15), and write down for the ρdependent exchange integral the following approximate expression: E′ E=E′ C− E′ e≈ E′ C−Aλ2 (1 +λ)4(1 8+1 2λ+5 8λ2)8 5E′ C, (3.146) where E′ Cis standard exact exchange integral for Coloumb potential g iven by Eq.(3.15) while the approximate λdependent part arised from our potential Ve. 450 1 2 3 4 5 6 rc, a.u.-0.07-0.06-0.05-0.04-0.03-0.02-0.010Ee,, a.u. Figure 12: The exchange integral E′ eas a function of rc, Eq. (3.146), at ρ= 1.67. 0 0.02 0.04 0.06 0.08 0.1 rc, a.u.-0.02-0.015-0.01-0.0050Ee,, a.u. Figure 13: The exchange integral E′ eas a function of rc, Eq. (3.146), at ρ= 1.67. More detailed view. 46We have a good accuracy of the approximation (3.146). Indeed , exchange integrals make sensible contribution to the total molecula r energy at deep overlapping of the wave functions, S >0.5, and we have made calculation just for the case of complete overlapping, S= 1, with the necessary asymp- totic factor, e−2ρ, provided by E′ C(ρ). Note that at λ→ ∞, the term E′ eof Eq.(3.146) becomes AE′ C, as it should be because at λ→ ∞ (no screening) we haveVe→A/r12. In addition, although there is no possibility to restore completely ρdependence for the second term in r.h.s. of Eq.(3.146), we ha ve got information on λdependence, which is of mostinterest here. 3.2 Numerical calculations for the Ve-based model In this Section, we consider the case of exponential screene d potential Ve= Ae−r12/rc/r, for which we have calculated all the needed molecular integ rals. TheH2molecule energy, due to Eq.(3.2), is written as Emol(γ,R,A,r c) = 2A+A′S 1 +S2−2(C+ES)−(C′ C− C′ e+E′ C− E′ e) 1 +S2+1 R, (3.147) where the specific terms are the Coloumb integral C′ egiven by Eq.(3.107) and the exchange integral E′ egiven by (3.146). We should find extremum of Emol as a function of our basic parameters, γ,R,A, andrc. We are using notation ρ=γRandλ= 2γrcso that our four parameters are γ,ρ,A, andλ. In general, the number of energy levels of isoelectronium can a lso be viewed as a parameter of the model. However, we restrict our conside ration by the one-level case, β2= 1; see Sec. 2.1. 3.2.1 Minimization of the energy First, we analyze the Adependence of Emol. Due to Eq.(3.89), for one-level isoelectronium we have A=r−1 c, that can be identically rewritten as A=2γ λ. (3.148) Thus theAdependence converts to γandλdependence. This is the conse- quence of consideration of the Hulten potential interactio n for the electron pair made in Sec. 2.1. 47Second, we turn to γdependence. Due to (3.148), the Adependent parts, C′ eandE′ e, acquire additional γfactor and thus become γ2dependent. The other molecular integrals depend on γlinearly so that we define accordingly, ¯C=1 γC,¯E=1 γE,¯C′ C=1 γC′ C,¯E′ C=1 γE′ C,¯C′ e=1 γ2C′ e,¯E′ e=1 γ2E′ e. (3.149) Inserting the computed integrals AandA′into (3.147) we have Emol(γ,ρ,λ ) =−aγ+bγ2, (3.150) where a(ρ,λ) =2 + 2¯C+ 4S¯E −¯C′ C−¯E′ C 1 +S2−1 ρ(3.151) and b(ρ,λ) =S2−1−2S¯E+¯C′ e+¯E′ e 1 +S2. (3.152) The value of γcorresponding to an extremum of Emolis found from the equationdEmol/dγ= 0, which gives the optimal value γopt=a 2b. (3.153) Inserting this into (3.150) we get the extremal value of Emol, Emol(ρ,λ) =−a2 4b. (3.154) Using definitions of aandbwe have explicitly γopt=1−2ρ+S2+ρ(−2¯C −4S¯E+¯C′ C+¯C′ C) 2ρ(−1 +S2−2S¯E′ C+¯C′e+¯E′e)(3.155) and Emol(ρ,λ) =(1−2ρ+S2+ρ(−2¯C −4S¯E+¯C′ C+¯C′ C))2 4ρ2(1 +S2)(−1 +S2−2S¯E′ C+¯C′e+¯E′e). (3.156) Next, we turn to the extremum in the parameter ρ. Theρdependence, as well as the λdependence, of Emolis essentially nonalgebraic so that we are forced to use numerical calculations. 481 1.5 2 2.5 3 r, a.u.-1.175-1.15-1.125-1.1-1.075-1.05-1.025-1E, a.u. Figure 14: The total energy E=Emolas a function of ρ, Eq. (3.156), at λ= 1/60,1/40,1/20,1/10,1/5. The lowest plot corresponds to λ= 1/5 (ρ=γR,λ= 2γrc). It appears that the λdependence does not reveal any local energy mini- mum while the ρdependence does. Below, we use the condition, λ−1=in- teger number , obtained during the calculation of the Coloumb integral wi th Hulten potential Vh; see Eq.(3.57). Although there is obviously no necessity to keep this condition for the case of exponential screened p otentialVe, we consider it as a prescription for allowed values of λ. Since theλdependence of the energy has no minimum we can use fitting of the predicted energy Emol(λ) to the experimental value by varying λ. This allows us to estimate the value of the parameter λ, and thus the value of the effective radius of the isoelectronium rc=λ/2γopt. 3.2.2 Fitting of the energy and the bond length The procedure is the following. We fix some numerical value of λ, and iden- tify minimal value of Emol(ρ,λ), given by Eq.(3.156), in respect with the parameterρ. This gives us minimal energy and corresponding optimal val ue ofρ, at some fixed value of λ. Then, we calculate γoptby using Eq.(3.155), and use obtained values of ρoptandγoptto calculate values of Roptandrc. 49We calculated minimal values of Emolinρ, for a wide range of integer values ofλ−1. The results are presented in Tables 2 and 3, and Figures 15 and 16. One can see that the energy Emoldecreases with the increase of rc (proportional to size of isoelectronium), as it was expecte d to be. We note that all the presented values of Emolin Tables 2 and 3 are lower than that,Evar mol=−1.139 a.u., obtained via two-parametric Ritz variational approach to the standard model of H2(see, e.g., [5]), which is the model without the assumption of short-range attractive potentia l between the elec- trons. This means that the Ve-based model gives better prediction than the one of the standard model, for any admitted value of the effect ive radius of isoelectronium rc>0. Indeed, the standard prediction Evar mol=−1.139 a.u. is much higher than the experimental value Eexper[H2] =−1.174474 a.u. 3.2.3 The results of fitting Best fit of the energy Emol. Due to Table 2 (see also Fig. 15), the experimental value, Eexp[H2] = −1.174...−1.164 a.u. (here we take 0.9% uncertainty of the experimental value) is fitted by rc= 0.0833...0.0600 a.u., (3.157) i.e.λ= 1/5...1/7, with the optimal distance, Ropt= 1.3184...1.3441 a.u. We see that the predicted Roptappeared to be about 6% less than the ex- perimental value Rexper[H2] = 1.4011 a.u. We assign this discrepancy to the approximation we have made for the exchange integral (3.146 ). Below we fit Ropt, to estimate the associated minimal energy. Best fit of the internuclear distance R. Due to Table 2 (see also Fig. 16), the experimental value of th e inter- nuclear distance, Rexp= 1.4011 a.u., is fitted by rc= 0.0115 a.u., with the corresponding minimal energy Emin=−1.144 a.u., which is about 3% big- ger than the experimental value. Again, we assign this discr epancy to the approximation we have made for the exchange integral (3.146 ), and take rc= 0.0115 a.u., (3.158) 50i.e.λ= 1/37, as the result of our final fit noting that (a) in ref. [1] the v alue rc= 0.0112 a.u. has been used to make exact numerical fit of the energ y, with corresponding R= 1.40 a.u., and (b) we have less discrepancy. The weight of the pure isoelectronium phase. To estimate the weight of the pure isoelectronium phase, whi ch can be viewed as a measure of stability of the pure isoelectronium s tate, we use the above obtained fits and the fact that this phase makes contrib ution to the total molecular energy via the Coloumb and exchange integra ls. According to Eq.(3.147), the isoelectronium phase display s itself only by the termPe≡ |C′ e(γ,ρ,λ ) +E′ e(γ,ρ,λ )|while the Coloumb phase displays itself by the corresponding term PC≡ |C′ C(γ,ρ) +E′ C(γ,ρ)|. Putting the total sumPC+Pe= 1, i.e.PC+Peis 100%, the weights are defined simply by WC=PC PC+Pe, W e=Pe PC+Pe, (3.159) The weight for the best fit of R. At the values λ= 1/37 (i.e.rc= 0.0115),γ= 1.1706, andρ= 1.6320, for which we have minimal Emol=−1.144 and optimal R= 1.40, we get the numerical values of the weights, We= 0.84% (3.160) for the pure isoelectronium phase, and WC= 100% −We= 99.16% for the Coloumb phase. The weight for the best fit of Emol. At the values λ= 1/5 (i.e.rc= 0.0833 a.u.), γ= 1.2005, andρ= 1.5827, for which we have minimal Emol=−1.173 a.u. and optimal R= 1.318 a.u., we obtain We= 6.16%, W C= 93.84%. (3.161) From the above two cases, one can see that the weight of pure is oelectronium phase is estimated to be We≃1...6%, (3.162) for the predicted variational energy Emol=−1.143...−1.173 a.u. The biggest possible weight. Note that in our Ve-based model the biggest allowed value of λisλ= 1/4 51(i.e.rc= 0.1034) because λ <1/3, to avoid divergency of the Coloumb integral Ce. For this value of λ, we obtain minimal Emol=−1.182 a.u. and optimalR= 1.297 a.u. This value corresponds to the biggest possible weight of the pure isoelectronium phase, We= 7.32%, (3.163) within our approximate model. The following three remarks are in order. (i) We consider the existence of this upper limit, We≤7.32%, as a highly remarkable implication of our Ve-model noting however that it may be artifact of the use of the exponential screened Coloumb potential. (ii) Another remarkable implication is due to the condition ,λ−1=integer number , obtained for the case of Hulten potential. One can see from T able 2 that the energy Emolvaries discretely with the discrete variation of λ−1. This means that there is no possibility to make a “smooth fit”. For e xample, at λ= 1/5, we have Emol=−1.173, and the nearest two values, λ= 1/4 and λ= 1/6, give usEmol=−1.182 andEmol=−1.167, respectively. Therefore, owing to the above condition the model becomes more predicitive . (iii) Numerical calculation shows that the formal use of the exact Coloumb integral C′ g, given by Eq.(3.128), of the Gaussian screened Coloumb potential, instead of C′ e, in Eq.(3.147) gives us approximately the same fits. Namely, the best fit of the energy is achieved at λ= 1/5, withrc= 0.1042, optimal R= 1.323, and minimal Emol=−1.172. Also, the best fit of R= 1.40 is atλ= 1/29, for which rc= 0.0147 and minimal Emol=−1.144. Here, we have used the same exchange integral as it is for the case of exponential screened potential so these fits have been presented just for a comparison with our basic fits, and to check the results. Note that for the case of Gaussian screened Coloumb integral we have no restriction on the allo wed values of λ. Analysis shows that, at big values of λ, e.g. atλ>4, the integral C′ g, given by (3.128), rapidly oscillates in the region of small ρ(ρ<0.5). This means that when the correlation length rcbecomes comparable to the internuclear distance an effect of instability of the molecule arises. Thi s can be viewed as a natural criterium to fix the upper limit of λ. Normally, we use the values λ<1, for which case there are no any oscillations of C′ g(see Fig. 9). 52λ−1rc, a.u. Ropt, a.u. Emin, a.u. 40.10337035071618050 1.297162129235449 -1.181516949656805 50.08329699109108888 1.318393698326879 -1.172984902150024 60.06975270534273319 1.333205576478603 -1.167271240301846 70.05999677404817234 1.344092354783681 -1.163188554065554 80.05263465942162049 1.352417789644028 -1.160130284706318 90.04688158804756491 1.358984317233049 -1.157755960428922 100.04226204990365446 1.364292909163710 -1.155860292450436 110.03847110142927672 1.368671725082009 -1.154312372623724 120.03530417706681329 1.372344384866235 -1.153024886026671 130.03261892720535206 1.375468373051375 -1.151937408039373 140.03031323689615631 1.378157728092548 -1.151006817317425 150.02831194904031777 1.380497017045902 -1.150201529091051 160.02655851947236431 1.382550255552670 -1.149497886394651 170.02500959113834722 1.384366780045693 -1.148877823925501 180.02363136168905809 1.385985219224291 -1.148327310762828 190.02239708901865092 1.387436244558651 -1.147835285349041 200.02128533948435381 1.388744515712491 -1.147392910500336 210.02027873303335994 1.389930082626193 -1.146993041730378 220.01936302821907175 1.391009413196452 -1.146629840949675 230.01852644434336641 1.391996158084790 -1.146298491232105 240.01775915199935013 1.392901727808297 -1.145994983116511 250.01705288514774330 1.393735733699196 -1.145715952370148 260.01640064219648127 1.394506328745493 -1.145458555325045 270.01579645313764336 1.395220473843219 -1.145220372020229 280.01523519631632570 1.395884147817973 -1.144999330178493 290.01471245291356761 1.396502514589167 -1.144793644973560 300.01422439038752817 1.397080057337240 -1.144601770891686 Table 2: The total minimal energy Eminand the optimal internuclear distance Roptas functions of the correlation length rc. The exponential screened Coloumb potential Vecase (see Figures 15 and 16). 53λ−1rc, a.u. Ropt, a.u. Emin, a.u. 310.01376766836566138 1.397620687025853 -1.144422362947838 320.01333936209977966 1.398127830817745 -1.144254245203342 330.01293689977547854 1.398604504597664 -1.144096385030938 340.01255801083612469 1.399053372836414 -1.143947871939897 350.01220068312791624 1.399476798299823 -1.143807900045981 360.01186312715793131 1.399876883556063 -1.143675753475045 370.01154374612489787 1.400255505817128 -1.143550794143290 390.01095393745919852 1.400954915288619 -1.143320213707519 400.01068107105944273 1.401278573036792 -1.143213620508321 410.01042146833640030 1.401586548200467 -1.143112256673494 420.01017418516195214 1.401879953246168 -1.143015746732479 430.00993836493541500 1.402159797887369 -1.142923750307661 440.00971322867044429 1.402427000676349 -1.142835958109381 450.00949806639934841 1.402682399061957 -1.142752088467028 460.00929222969498477 1.402926758144872 -1.142671884314343 470.00909512514431396 1.403160778323019 -1.142595110561057 480.00890620863525624 1.403385101987775 -1.142521551794315 490.00872498034101540 1.403600319405678 -1.142451010262626 500.00855098030451296 1.403806973898863 -1.142383304102633 510.00838378454080327 1.404005566419838 -1.142318265775268 520.00822300158793934 1.404196559601683 -1.142255740683024 530.00806826944722482 1.404380381352424 -1.142195585944305 540.00791925286251402 1.404557428052374 -1.142137669304475 550.00777564089552400 1.404728067404676 -1.142081868166104 560.00763714476025456 1.404892640982100 -1.142028068723488 570.00750349588477794 1.405051466507240 -1.141976165188595 580.00737444417302681 1.405204839898059 -1.141926059097351 590.00724975644291090 1.405353037106507 -1.141877658686723 600.00712921502024112 1.405496315774223 -1.141830878334298 Table 3: The total minimal energy Eminand the optimal internuclear distance Roptas functions of the correlation length rc. The exponential screened Coloumb potential Vecase (see Figures 15 and 16). 540 0.02 0.04 0.06 0.08 0.1 rc, a.u.-1.18-1.17-1.16-1.15Emin, a.u. Figure 15: The total minimal energy Eminas a function of the correlation lengthrc. The exponential screened Coloumb potential Vecase (see Tables 2 and 3). 0 0.02 0.04 0.06 0.08 0.1 rc, a.u.1.31.321.341.361.381.4Ropt, a.u. Figure 16: The optimal internuclear distance Roptas a function of the cor- relation length rc. The exponential screened Coloumb potential Vecase (see Tables 2 and 3). 55References [1] R. M. Santilli and D. D. Shillady, Int. J. Hydrogen Energy 24(1999), 943-956. [2] R. M. Santilli and D. D. Shillady, Hadronic J. 21(1998), 633-714; 21 (1998), 715-758; 21(1998), 759-788. [3] R. M. Santilli, Hadronic J. 21(1998), 789-894. [4] A. K. Aringazin and M. G. Kucherenko, Hadronic J. 22(1999) (in press). [5] Z. Flugge, Practical Quantum Mechanics, Vols. 1, 2 (Spri nger-Verlag, Berlin, 1971). [6] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonski, Qua ntum Theory of Angular Momentum (Leningrad, 1975) (in Russian). [7] M. Abramowitz and I. A. Stegun, Handbook of mathematical func- tions with formulas, graphs and mathematical tables. Natio nal Bureau of Standards Applied Math. Series - 55(1964) 830 pp. 56

arXiv:physics/0001058v1 [physics.atom-ph] 25 Jan 2000Calculation of positron binding to silver and gold atoms V. A. Dzuba, V. V. Flambaum, and C. Harabati School of Physics, The University of New South Wales, Sydney 2052, Australia (January 12, 2014) Abstract Positron binding to silver and gold atoms was studied using a fullyab initio relativistic method, which combines the configuration inte raction method with many-body perturbation theory. It was found that the silver atom forms a bound state with a positron with binding energy 123 ( ±20%) meV, while the gold atom cannot bind a positron. Our calculations reveal th e importance of the relativistic effects for positron binding to heavy atoms . The role of these effects was studied by varying the value of the fine structure c onstant α. In the non-relativistic limit, α= 0, both systems e+Ag and e+Au are bound with binding energies of about 200 meV for e+Ag and 220 meV for e+Au. Relativistic corrections for a negative ion are essentiall y different from that for a positron interacting with an atom. Therefore, the calc ulation of electron affinities cannot serve as a test of the method used for positro n binding in the non-relativistic case. However, it is still a good test o f the relativistic calculations. Our calculated electron affinities for silver (1.327 eV) and gold (2.307 eV) atoms are in very good agreement with correspondi ng experimental values (1.303 eV and 2.309 eV respectively). Typeset using REVT EX 1I. INTRODUCTION Positron binding by neutral atoms has not been directly obse rved yet. However, intensive theoretical study of the problem undertaken in the last few y ears strongly suggests that many atoms can actually form bound states with a positron (se e, e.g. [1–8]). Most of the atoms studied so far were atoms with a relatively small value of the nuclear charge Z. It is important to extend the study to heavy atoms. The main obst acle in this way is the rapid rise of computational difficulties with increasing num ber of electrons. However, as we show in this paper, an inclusion of relativistic effects is al so important. The role of these effects in positron binding to atoms has not been truly apprec iated. Indeed, one can say that due to strong Coulomb repulsion a positron cannot penet rate to short distances from the nucleus and remains non-relativistic. However, the pos itron binding is due to interaction with electrons which have large relativistic corrections t o their energies and wave functions. The binding energy is the difference between the energies of a neutral atom and an atom bound with a positron. This difference is usually small. On th e other hand, relativistic contributions to the energies of both systems are large and t here is no reason to expect they are the same and cancel each other. Therefore, some rela tivistic technique is needed to study positron binding by heavy atoms. For both light and heavy atoms the main difficulty in calculati ons of positron interaction comes from the strong electron-positron Coulomb attractio n. This attraction leads to vir- tual positronium (Ps) formation [9]. One can say that it give s rise to a specific short-range attraction between the positron and the atom, in addition to the usual polarizational poten- tial which acts between a neutral target and a charged projec tile [1,9–11]. This attraction cannot be treated accurately by perturbations and some all- order technique is needed. In our earlier works [1,9–11] we used the Ps wave function expli citly to approximate the vir- tual Ps-formation contribution to the positron-atom inter action and predicted e+Mg,e+Zn, e+Cd and few other bound states. The same physics may also expla in the success of the stochastic variational method in positron-atom bound stat e calculations (see, e.g. [3] and Refs. therein). In this approach the wave function is expand ed in terms of explicitly cor- related Gaussian functions which include factors exp( −αr2 ij) with inter-particle distances rij. Using this method Ryzhikh and Mitroy obtained positron bou nd states for a whole range of atoms (Be, Mg, Zn, Cu, Ag, Li, Na, K, etc.). This metho d is well suited for few- particle systems. Its application to heavier systems is don e by considering the Hamiltonian of the valence electrons and the positron in the model potent ial of the ionic core. However, for heavier atoms, e.g., Zn, the calculation becomes extrem ely time consuming [5], and its convergence cannot be ensured. Another non-perturbative technique is the configuration in teraction (CI) method widely used in standard atomic calculations. This method was appli ed to the positron-copper bound state in [6]. In this work the single-particle orbitals of th e valence electron and positron are chosen as Slater-type orbitals, and their interaction with the Cu+core is approximated by the sum of the Hartree-Fock and model polarization potentia ls. The calculation shows slow convergence with respect to the number of spherical harmoni cs included in the CI expansion, Lmax= 10 being still not sufficient to extrapolate the results reli ably toLmax→ ∞. In their more recent work the same authors applied the CI meth od to a number of systems consisting of an atom and a positron. These include P sH,e+Cu,e+Li,e+Be,e+Cd 2and CuPs. In spite of some improvements to the method they sti ll regard it as a “tool with which to perform preliminary investigations of positron bi nding” [12]. In our previous paper we developed a different version of the C I method for the positron- atom problem [13]. The method is based on the relativistic Ha rtree-Fock method (RHF) and a combination of the CI method with many body perturbatio n theory (MBPT). This method was firstly developed for pure electron systems [14] a nd its high effectiveness was demonstrated in a number of calculations [15–17]. In the pap er [13] it was successfully applied to the positron binding by copper. There are several important advances in the technique compared to the standard non-relativistic CI met hod which make it a very effective tool for the investigation of positron binding by heavy atom s. 1. The method is relativistic in the sense that the Dirac-Har tree-Fock operator is used to construct an effective Hamiltonian for the problem and to c alculate electron and positron orbitals. 2.B-splines [18] in a cavity of finite radius Rwere used to generate single-particle basis sets for an external electron and a positron. The B-spline technique has the remarkable property of providing fast convergence with respect to the n umber of radial functions included into the calculations [19,20]. Convergence can be further controlled by varying the cavity radius Rwhile the effect of the cavity on the energy of the system is tak en into account analytically [13]. Convergence was clearly ac hieved for the e+Cu system in Ref. [13] and for the e+Ag ande+Au systems as presented below. 3. We use MBPT to include excitations from the core into the eff ective Hamiltonian. This corresponds to the inclusion of the correlations between co re electrons and external particles (electron and positron) and of the effect of screen ing of the electron-positron interaction by core electrons. These effects are also often c alled the polarization of the core by the external particles. We include them in a fully ab initio manner up to the second order of the MBPT. In the present paper we apply this method to the problem of pos itron binding by silver and gold atoms. Using a similar technique we also calculate e lectron affinities for both these atoms. Calculations for negative ions serve as a test of the t echnique used for positron-atom binding. We also study the role of the relativistic effects in neutral silver and gold, silver and gold negative ions and silver and gold interacting with a pos itron. This is done by varying the value of the fine structure constant αtowards its non-relativistic limit α= 0. II. THEORY A detailed description of the method was given in Ref. [13]. W e briefly repeat it here emphasizing the role of the relativistic effects. We use the r elativistic Hartree-Fock method in theVN−1approximation to obtain the single-particle basis sets of e lectron and positron orbitals and to construct an effective Hamiltonian. The two-particle electron-positron wave function is given by the CI expansion, Ψ(re,rp) =/summationdisplay i,jCijψe i(re)ψp j(rp), (1) 3whereψe iandψp jare the electron and positron orbitals respectively. The ex pansion coeffi- cientsCijare determined by the diagonalization of the matrix of the eff ective CI Hamiltonian acting in the Hilbert space of the valence electron and the po sitron, HCI eff=ˆhe+ˆhp+ˆhep, ˆhe=cαp+ (β−1)mc2−Ze2 re+VN−1 d−ˆVN−1 exch+ˆΣe, ˆhp=cαp+ (β−1)mc2+Ze2 rp−VN−1 d+ˆΣp, (2) ˆhep=−e2 |re−rp|+ˆΣep, where ˆheandˆhpare the effective single-particle Hamiltonians of the elect ron and positron, andˆhepis the effective electron-positron two-body interaction. A part from the relativistic Dirac operator, ˆheandˆhpinclude the direct and exchange Hartree-Fock potentials of the core electrons,VN−1 dandVN−1 exch, respectively. The additional ˆΣ operators account for correlations involving core electrons. Σ eand Σ pare single-particle operators which can be considered as a self-energy part of the correlation interaction between a n external electron or positron and core electrons. These operators are often called “correlat ion potentials” due to the analogy with the non-local exchange Hartree-Fock potential. Σ eprepresents the screening of the Coulomb interaction between external particles by core ele ctrons (see [13,14] for a detailed discussion). To study the role of the relativistic effects we use the form of the operators heandhpin which the dependence on the fine structure constant αis explicitly shown. Single-particle orbitals have the form ψ(r)njlm=1 r/parenleftBigg fn(r)Ω(r/r)jlm iαgn(r)˜Ω(r/r)jlm/parenrightBigg . (3) Then the RHF equations (hi−ǫn)ψi n= 0,(i=e,p) take the following form f′ n(r) +κn rfn(r)−[2 +α2(ǫn−ˆV)]gn(r) = 0 (4) g′ n(r)−κn rgn(r) + (ǫn−ˆV)fn(r) = 0, whereκ= (−1)l+j+1/2(j+ 1/2) andVis the effective potential which is the sum of the Hartree-Fock potential and correlation potential Σ: ˆV=−Ze2 re+VN−1 d−ˆVN−1 exch+ˆΣe,- for an electron , ˆV=Ze2 rp−VN−1 d+ˆΣp,- for a positron . (5) 4The non-relativistic limit can be achieved by reducing the v alue ofαin (4) toα= 0. The relativistic energy shift in atoms with one external ele ctron can also be estimated by the following equation [21] ∆n=En ν(Zα)2/bracketleftBigg1 j+ 1/2−C(Z,j,l)/bracketrightBigg , (6) whereEnis the energy of an external electron, νis the effective principal quantum number (En=−0.5/ν2a.u.). The coefficient C(Z,j,l) accounts for many-body effects. Note that formula (6) is based on the specific expression for the electr on density in the vicinity of the nucleus and therefore is not applicable for a positron. III. SILVER AND GOLD NEGATIVE IONS We calculated electron affinities of silver and gold atoms mos tly to test the technique used for positron-atom binding. The calculation of a negative io n Ag−or Au−is a two-particle problem technically very similar to positron-atom binding . The effective Hamiltonian of the problem has a form similar to (2) HCI eff=ˆhe(r1) +ˆhe(r2) +ˆhee, ˆhee=e2 |re−rp|+ˆΣee, where ˆΣeerepresents the screening of the Coulomb interaction betwee n external electrons by core electrons (see Refs. [14,13] for detailed discussio n). Electron affinity is defined when an electron can form a bound state with an atom. In this case th e difference between the energy of a neutral atom and the energy of a negative ion is cal led the electron affinity to this atom. Energies of Ag, Ag−, Au, Au−obtained in different approximations and corresponding electron affinities are presented in Table I to gether with experimental data. The energies are given with respect to the cores (Ag+and Au+). Like in the case of Cu− [13] the accuracy of the Hartree-Fock approximation is very poor. The binding energies of the 5selectron in neutral Ag and the 6 selectron in neutral Au are underestimated by about 21% and 23% respectively, while the negative ions are u nbound. Inclusion of either core-valence correlations (Σ) or valence-valence correla tions (CI) does produce binding but the accuracy is still poor. Only when both these effects are in cluded the accuracy for the electron affinities improves significantly becoming 20% for A g−and 11% for Au−. Further improvement can be achieved by introducing numerical facto rs before ˆΣeto fit the lowest s,p anddenergy levels of the neutral atoms. These factors simulate t he effect of higher-order correlations. Their values are fs= 0.88,fp= 0.97,fd= 1.08 for the Ag atom and fs= 0.81, fp= 1,fd= 1.04 for the Au atom in the s,panddchannels, respectively. As is evident from Table I, the fitting of the energies of neutral atoms also significantly improves electron affinities. It is natural to assume that the same procedure sho uld work equally well for the positron-atom problem. Results of other calculations of the electron affinities of si lver and gold are presented in Table II together with the experimental values. 5IV. POSITRON BINDING TO SILVER AND GOLD AND THE ROLE OF RELATIVISTIC EFFECTS As for the case of copper [13] we have performed calculations for two different cavity radii R= 30a0andR= 15a0. For a smaller radius convergence with respect to the number of single-particle basis states is fast. However, the effect of the cavity on the converged energy is large. For a larger cavity radius, convergence is slower a nd the effect of the cavity on the energy is small. When the energy shift caused by the finite cavity radius is taken into account both calculations come to the same value of the posit ron binding energy. Table III illustrates the convergence of the calculated energies of e+Ag ande+Au with respect to the maximum value of the angular momentum of single-particle or bitals. Energies presented in the table are two-particle energies (in a.u.) with respect t o the energies of Ag+and Au+. The number of radial orbitals nin each partial wave is fixed at n= 16. Fig. 1 shows the convergence of the calculated energy with respect to nwhen maximum momentum of the single-particle orbitals was fixed at L= 10. The cavity radius in both cases was R= 30a0. Table III and Fig. 1 show that even for a larger cavity radius, convergence was clearly achieved. Table III also shows the convergence in different a pproximations, namely with and without core-valence correlations (Σ). One can see that while inclusion of Σ does shift the energy, the convergence is not affected. Table IV shows how positron binding by silver and gold is form ed in different approxi- mations. This table is very similar to Table I for the negativ e ions except there is no RHF approximation for the positron binding. Indeed, the RHF app roximation for the negative ions means a single-configuration approximation: 5 s2for Ag−and 6s2for Au−. These con- figurations strongly dominate in the two-electron wave func tion of the negative ions even when a large number of configurations are mixed to ensure conv ergence. In contrast, no sin- gle configuration strongly dominates in the positron bindin g problem. Therefore we present our results in Table IV starting from the standard CI approxi mation. In this approximation positron is bound to both silver and gold atoms. However, the inclusion of core-valence correlations through the introduction of the Σ e, Σpand Σ epoperators shifts the energies significantly. In the case of gold, the e+Au system becomes unbound when all core-valence correlations are included. As was discussed in our previous paper [13] the dominating fa ctor affecting the accuracy of the calculations is higher-order correlations which mos tly manifest themself via the value of the Σ operator. An introduction of the fitting parameters a s described in the previous section can be considered as a way to simulate the effect of hig her-order correlations. Also, the energy shift caused by the fitting can be considered as an e stimation of the uncertainty of the calculations. This shift is 0.00240 a.u. in the case of silver and 0.00023 a.u. in the case of gold (see Table IV). Note that these values are consid erably smaller than energy shifts for the silver and gold negative ions (0.00854 a.u. an d 0.00921 a.u. respectively, see Table I). This is because of the cancellation of the effects of the variation of Σ eand Σ p. In particular, for gold it is accidentally very small. One can s ee that even if the value of 0.00240 a.u. is adopted as an upper limit of the uncertainty of the cal culations, the e+Ag system remains bound while the e+Au system remains unbound. However, the actual accuracy might be even higher. We saw that the fitting procedure signifi cantly improves the accuracy of the calculations for the silver and gold negative ions. It is natural to assume that the 6same procedure works equally well for the positron binding p roblem. The final result for the energy of positron binding by the silver atom as presente d in Table IV is 0.00434 a.u. This result does not include the effect of the finite cavity siz e. When this effect is taken into account, by means of the procedure described in Ref. [13 ], the binding energy becomes 0.00452 a.u. or 123 meV. If we adopt the value of 0.00240 a.u as an estimation of the uncertainty of the result, then the accuracy we can claim is a bout 20%. The calculation of the positron binding by copper [13], silv er and gold reveal an inter- esting trend. All three atoms have very similar electron str ucture. However the positron binding energy for silver (123 meV) is considerably smaller than that for copper (170 meV [13]) while gold atoms cannot bind positrons at all. We belie ve that this trend is caused by relativistic effects. An argument that the positron is alway s non-relativistic does not look very convincing because electrons also contribute to the bi nding energy. Relativistic effects are large for heavy atoms and electron contributions to the p ositron binding energy could be very different in the relativistic and non-relativistic l imits. Indeed, we demonstrated in Ref. [21] that the relativistic energy shift considerably c hanges the values of the transition frequencies in Hg+ion and sometimes even changes the order of the energy levels . If we use formula (6) with the contribution of the many-body effect sC= 0.6, as suggested in Ref. [21], to estimate the relativistic energy shift for neu tral Au then the result is -0.037 a.u. This is about an order of magnitude larger than the energ y difference between Au and e+Au. If the relativistic energy shift in e+Au is different from that in Au then the positron binding energy may be strongly affected. To study the role of the relativistic effects in positron bind ing in more detail we performed the calculations for Ag, Ag−,e+Ag, Au, Au−ande+Au in the relativistic and non-relativistic limits. The latter corresponds to the zero value of the fine st ructure constant α(see Section II). The results are presented in Table V. One can see that the actual relativistic energy shift for neutral Au is even bigger than is suggested by formu la (6) with C= 0.6. The shift is 0.0805 a.u. which corresponds to C= 0.08. Formula (6) with C= 0.08 also reproduces the relativistic energy shift for neutral Ag. Th e relativistic energy shift for an atom with a positron is of the same order of magnitude but a lit tle different in value. This difference turned out to be enough to affect the positron bindi ng energy significantly. In particular, the e+Au system which is unbound in relativistic calculations bec omes bound in the non-relativistic limit with binding energy 0.0080 a.u o r 218 meV. In the case of silver, the positron binding energy is considerably higher in the no n-relativistic limit. It is 0.0073 a.u. or 199 meV. It is interesting to compare this value with t he value of 150 meV obtained by Mitroy and Ryzhikh using the non-relativistic stochasti c variational method [4]. Since the convergence was achieved in both calculations the remai ning difference should probably be attributed to the different treatment of the core-valence correlations. We use many-body perturbation theory for an accurate calculation of the Σ ope rator which accounts for these correlations. Mitroy and Ryzhikh use an approximate semi-e mpirical expression for the Σ operator which is based on its long-range asymptotic behavi or. Note that the relativistic energy shift for negative ions is also large. However electron affinities are less affected. This is because electron affinitie s are many times larger than positron binding energies and therefore less sensitive to t he energy shift. Apart from that there is a strong cancellation between relativistic energy shifts in the negative ion and neutral atom. This means in particular that the calculation of the el ectron affinities cannot serve as a 7test of a non-relativistic method chosen for the positron bi nding problem. However, it is still a good test of the relativistic calculations. Note also that our calculated relativistic energy shifts for neutral and negative silver and gold are in very go od agreement with calculations performed by Schwerdtfeger and Bowmaker by means of relativ istic and non-relativistic versions of the quadratic configuration interaction method (see Table VI and Ref. [24]). The authors are grateful to G. F. Gribakin for many useful dis cussions. 8REFERENCES [1] V. A. Dzuba, V. V. Flambaum, G. F. Gribakin, and W. A. King, Phys. Rev. A 52, 4541 (1995). [2] G. G. Ryzhikh and J. Mitroy, Phys. Rev. Lett. 79, 4124 (1997); J. Phys. B. 31, L265 (1998); J. Phys. B. 31, 3465 (1998); J. Phys. B. 31, L401 (1998); J. Phys. B. 31, 4459 (1998). [3] G. G. Ryzhikh, J. Mitroy, and K. Varga, J. Phys. B. 31, 3965 (1998). [4] G. G. Ryzhikh and J. Mitroy, J. Phys. B. 31, 5013 (1998). [5] J. Mitroy and G. G. Ryzhikh, J. Phys. B. 32, 1375 (1999). [6] J. Mitroy, and G. G. Ryzhikh, J. Phys. B. 32, 2831 (1999). [7] K. Strasburger and H. Chojnacki, J. Chem. Phys. 108, 3218 (1998). [8] J. Yuan, B. D. Esry, T. Morishita, and C. D. Lin, Phys. Rev. A58, R4 (1998). [9] V. A. Dzuba, V. V. Flambaum, W. A. King, B. N. Miller, and O. P. Sushkov, Phys. Scripta T 46, 248 (1993). [10] G. F. Gribakin and W. A. King, J. Phys. B 27, 2639 (1994). [11] V. A. Dzuba, V. V. Flambaum, G. F. Gribakin, and W. A. King , J. Phys. B 29, 3151 (1996). [12] M. W. J. Bromley, J. Mitroy, and G. G. Ryzhikh, unpublish ed. [13] V. A. Dzuba, V. V. Flambaum, G. F. Gribakin, and C. Haraba ti, Phys. Rev. A 60, 3641 (1999). [14] V. A. Dzuba, V. V. Flambaum, and M. G. Kozlov, Phys. Rev. A 54, 3948 (1996); JETP Letters, 63, 882 (1996). [15] M. G. Kozlov and S. G. Porsev, JETF 84, 461 (1997). [16] V. A. Dzuba and W. R. Johnson, Phys. Rev. A 57, 2459 (1998). [17] M. G. Kozlov and S. G. Porsev, Opt. Spectrosc. 87, 352 (1999). [18] C. deBoor, A Practical Guide to Splines ( Springer, New York, 1978). [19] J. E. Hansen, M. Bentley, H. W. van der Hart, M. Landtman, G. M. S. Lister, Y.-T. Shen, and N. Vaeck, Phys. Scr. T47, 7 (1993). [20] J. Sapirstein and W. R. Johnson, J. Phys. B 29, 5213 (1996). [21] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, Phys. Rev. A 59, 230 (1999). [22] C. E. Moore, Atomic Energy Levels , Natl. Bur. Stand. Circ. No. 467 (U.S. GPO, Wash- ington, DC, 1958), Vol. III [23] T. M. Miller CRC Handbook of chemistry and Physics , Editor-in Chief D. R. Lide and H. P. R. Frederikse (Boca Raton, Florida, CRC Press, 1993). [24] P. Schwerdtfeger and G. A. Bowmaker, J. Chem. Phys. 100, 4487 (1994). [25] P. Neogrady, V. Kello, M. Urban, and A. J. Sadrej, Int. J. Quantum Chem. 63, 557 (1997). [26] E. Eliav, U. Kaldor, and Y. Ishikawa, Phys. Rev. A 491724 (1994). [27] U. Kaldor and B. A. Hess, Chem. Phys. Lett. 230, 229 (1994). [28] H. Hotop and W. C. Lineberger, J. Phys. Chem. Ref. Data 14, 731 (1975). 9TABLES TABLE I. Ground state energies (in a.u.) of silver, gold and t heir negative ions calculated in different approximations Neutral atom Negative ion Electron affinitya Silver RHFb-0.22952 -0.20156 -0.02795 RHF + Σc-0.27990 -0.30231 0.02241 CId-0.22952 -0.25675 0.02722 CI +Σ ee-0.28564 -0.33560 0.04996 CI +Σ e+ Σeef-0.28564 -0.34298 0.05734 CI + fΣe+ Σeeg-0.27841 -0.32721 0.04880 Experimenth-0.27841 -0.32626 0.04784 Gold RHFb-0.27461 -0.26169 -0.01292 RHF + Σc-0.34900 -0.41046 0.06146 CId-0.27461 -0.31369 0.03908 CI +Σ ee-0.35536 -0.43913 0.08376 CI +Σ e+ Σeef-0.35536 -0.44943 0.09407 CI + fΣe+ Σeeg-0.33903 -0.42389 0.08486 Experimenth-0.33903 -0.42386 0.08483 aNegative affinity means no binding. bRelativistic Hartree-Fock; a single-configuration approx imation, no core-valence correlations are included. cSingle-configuration approximation, core-valence correl ations are included by means of MBPT. dStandard CI method. eSelf-energy part of core-valence correlations are include d by adding the Σ eoperator to the CI Hamiltonian. fCI+MBPT method, self-energy and screening correlations ar e included by Σ operators while valence-valence correlations are included by configuratio n interaction. gΣein different waves are taken with factors to fit energies of a ne utral atom. hReferences [22,23]. 10TABLE II. Electron affinities of Ag and Au (eV). Comparison wit h other calculations and experiment. Ag Au Ref. Method Theory 1.008 1.103 [24] Non-relativistic quadratic configuration interaction method 1.199 2.073 [24] Relativistic quadratic configuration inte raction method 1.254 2.229 [25] Relativistic coupled cluster method 1.022 [4] Non-relativistic stochastic variational method 2.28 [26] Fock-space relativistic coupled-cluster method 2.26 [27] Fock-space coupled-cluster method with Douglas- Kroll transformation (relativistic) 1.327 2.307 Present work Experiment 1.303 2.309 [28] 11TABLE III. Convergence of the calculation of the energies of e+Ag and e+Au with respect to the number of included partial waves (a.u.) Lmax CIaCI +ΣbCI + fΣc e+Ag 0 −0.2232729 −0.2800223 −0.2729038 1 −0.2271709 −0.2838360 −0.2749591 2 −0.2309207 −0.2868375 −0.2765124 3 −0.2350823 −0.2895691 −0.2780571 4 −0.2388315 −0.2916800 −0.2793784 5 −0.2419251 −0.2932381 −0.2804487 6 −0.2443218 −0.2943470 −0.2812678 7 −0.2460745 −0.2951085 −0.2818603 8 −0.2472812 −0.2956100 −0.2822647 9 −0.2480477 −0.2959189 −0.2825199 10 −0.2484749 −0.2960829 −0.2826596 11 −0.2486698 −0.2961444 −0.2827143 12 −0.2487554 −0.2961682 −0.2827367 13 −0.2487928 −0.2961778 −0.2827459 14 −0.2488090 −0.2961817 −0.2827498 e+Au 0 −0.2684049 −0.3500447 −0.3330163 1 −0.2706582 −0.3526602 −0.3339500 2 −0.2719813 −0.3539745 −0.3344564 3 −0.2732705 −0.3550481 −0.3348765 4 −0.2743905 −0.3558030 −0.3351787 5 −0.2753222 −0.3563289 −0.3353973 6 −0.2760539 −0.3566883 −0.3355525 7 −0.2765943 −0.3569283 −0.3356590 8 −0.2769686 −0.3570837 −0.3357294 9 −0.2772074 −0.3571791 −0.3353733 10 −0.2773390 −0.3572293 −0.3357972 11 −0.2773925 −0.3572449 −0.3358049 12 −0.2774146 −0.3572505 −0.3358078 13 −0.2774239 −0.3572527 −0.3358091 14 −0.2774278 −0.3572536 −0.3358095 aStandard CI method. bCI+MBPT method, both core-valence and valence-valence cor relations are included. cΣ is taken with fitting parameters as explained in the text. 12TABLE IV. Positron binding by silver and gold calculated in d ifferent approximations (all energies are in a.u.) Neutral atom Atom with e+∆a Silver CI -0.22952 -0.24881 0.01929 CI +Σ e+ Σp -0.28564 -0.29618 0.01054 CI +Σ e+ Σp+ Σep -0.28564 -0.28843 0.00279 CI + fΣe+fΣp+ Σep -0.27841 -0.28275 0.00434 Gold CI -0.27461 -0.27743 0.00282 CI +Σ e+ Σp -0.35536 -0.35725 0.00189 CI +Σ e+ Σp+ Σep -0.35536 -0.35191 -0.00345 CI + fΣe+fΣp+ Σep -0.33903 -0.33581 -0.00322 aPositron binding energy. Negative energy means no binding. TABLE V. Energies (in a.u.) of Ag, Ag−,e+Ag, Au, Au−ande+Au with respect to the energy of the core in relativistic and non-relativistic cases Neutral Negative Atom with Electron Positron binding atom ion a positron affinity energya Silver Non-relativistic -0.2558 -0.2974 -0.2640 0.0416 0.0073 Relativistic -0.2784 -0.3272 -0.2827 0.0488 0.0043 ∆ 0.0226 0.0298 0.0187 -0.0072 0.0030 Gold Non-relativistic -0.2537 -0.3040 -0.2665 0.0503 0.0080 Relativistic -0.3390 -0.4239 -0.3358 0.0849 -0.0032 ∆ 0.0853 0.1199 0.0693 -0.0346 0.0112 aPositive energy means bound state TABLE VI. Comparison of the relativistic energy shift with o ther calculations (energies are in a.u.) Atom/Ion Present work Schwerdtfeger and Bowmakera Ag 0.0226 0.0200 Ag−0.0072 0.0070 Au 0.0853 0.0714 Au−0.0346 0.0357 aQuadratic configuration interaction method, Ref. [24] 13FIGURES 0 5 10 15 20-0.29-0.28-0.27-0.26 FIG. 1. Energy of e+Ag as a function of the number of radial electron and positron basis functions in each partial wave ( Lmax= 10) in the cavity with R= 30a0. Dashed line represents the energy of neutral silver. 140 5 10 15 20-0.34-0.335-0.33-0.325 FIG. 2. Same as Fig. 1 but for e+Au. 15

24/01/00 14:26 G. Van Hooydonk 1Gauge symmetry, chirality and parity violation in four-particle systems: Coulomb's law as a universal molecular function. G. Van Hooydonk, Ghent University, Department of Library Sciences/Department of Physical and Inorganic Chemistry, Rozier 9, B-9000 Ghent (Belgium) E-mail: guido.vanhooydonk@rug.ac.be Abstract. Following recent work in search for a universal function (Van Hooydonk, Eur. J. Inorg. Chem., 1999 , 1617), we test four symmetric ± anRn potentials for reproducing molecular potential energy curves ( PECs). Classical gauge symmetry is broken, which results in generic left-right asymmetric PECs for 1/R potentials. A pair of symmetric perturbed Coulomb potentials is in accordance with the shape of observed PECs. For a bond, a four-particle system, charge inversion (parity violation, atom chirality) is the key to explain this shape generically. A parity adapted Hamiltonian reduces from ten to two terms and to a soluble Bohr-like formula, the Kratzer potential (1-R e/R)2. The result is similar to the combined action of spin and wave functional symmetry effects upon the Hamiltonian in the Heitler-London theory. The corresponding analytical perturbed Coulomb function varies simply with (1-R e/R) and scales attractive and repulsive branches of PECs for 13 bonds H 2, HF, LiH, KH, AuH, Li 2, LiF, KLi, NaCs, Rb 2, RbCs, Cs 2 and I2 in a single straight line. Turning points for 13 bonds are reproduced with an absolute deviation of 0,3 % (0,007 Å) for about 400 points at both branches. For 230 points at the repulsive side, the deviation is 0,2 % (0,003 Å). Available turning points for I 2 are in need of revision. This universal molecular scaling function is the classical electrostatic perturbed Coulomb law, which reduces the complex four-particle system to a central force system on one nucleon. The Kratzer function relates to two central force systems, one on each nucleon. A minimum of parameters is required and even the ab initio zero molecular parameter function gives PECs of acceptable quality, just using atomic ionisation energies. The function can be used as a model potential for inverting levels and gives a first principle's comparison of short- and long-range interactions, of importance for the study of cold atoms. The theory may be tested with wave-packet dynamics: femto-chemistry applied to the crossing of covalent and ionic curves. We anticipate this scale and shape invariant scheme applies to smaller scales in nuclear and high- energy particle physics. For larger gravitational scales (Newton 1/R potentials), problems with super- unification are discussed. Reactions between hydrogen and anti-hydrogen, feasible in the near future, will probably produce normal H 2. 1. Introduction Ehrenfest's theorem (1927) states that in the limit quantum mechanical expectation values behave classically. The most classical form of physics is elementary statics ( Stevin, 1605) and is directly linked to Euclidean geometry. Symmetry, statics and geometry are scale-invariant and of fundamental importance for describing particle interactions. Symmetry is independent of dynamics (Gross, 1996). The effects of symmetry are discrete, permanent or time-invariant: parity, mirror symmetry and left-right asymmetry ( chirality, handedness) and show clearly in polyatomic molecules ( Kellman, 1996, Dunitz, 1996). Until 1956 is was generally accepted that parity was never violated but the discovery of parity violation in the weak interactions (Lee and Yang, 1956) led to the new physics, culminating in the Standard Model and beyond. But parity is also violated in atoms and in polyatomic molecules and the significance of this observation can not be underestimated. According to Bouchiat and Bouchiat (1997) low energy physics still has a role to play in the exploration of the Standard Model. Looking for systems where symmetry is broken is an important issue in physics, in particular SUSY ( Gel'fand and Likhtman, 1971, Wess and Zumino, 1974, Witten, 1981, 1982). This powerful new tool for physics between Fermi- and Planck-scales (Lopez, 1997) found applications at the Bohr-scale ( Kostelecky, 1992, Cooper, 1993, Lévai, 1994, Roy and Varshni, 1991, Blado, 1996, Dutt et al., 1995, Mukherjee et al., 1995, Guerin, 1996) and even in biology ( Bahsford et al., 1998). SUSY uses algebraic quadratic super- potentials in the framework of quantum mechanics. The basis for SUSY was laid with the method of factorisation ( Dirac, 1935, Schrödinger, 1940). It was developed by Infeld and Hull (1951) with a major interest24/01/00 14:26 G. Van Hooydonk 2in oscillator- and generalised Kepler systems, important for molecular spectroscopy. Algebraic schemes apply to domains varying from molecular to particle and nuclear physics ( Alhassid et al, 1983, Iachello, 1981, Cooper, 1993, Lévai, 1994). Nevertheless, SUSY might remain a mathematical artefact. Desperately breaking SUSY is the motto of today’s physics (Lopez, 1997, Poppitz, 1997, Poppitz and Trivedi, 1998). However, symmetry effects are scale-invariant. If parity is violated in real systems (atoms and molecules at the Bohr-scale, sub-atomic particles at the Fermi-scale), other examples must exist in nature. Parity violating effects in atoms and in polyatomic molecules are small in terms of energy. Sophisticated experimental and computational methods are required to disclose the mechanisms (for atoms: Bouchiat and Bouchiat, 1997; for molecules: Bakasov et al., 1998). Chiral molecules consist of four atoms (particles) not in a linear alignment. But any diatomic bond can be considered as a four-particle system with the four particles not in a linear alignment either. This makes a diatomic bond a theoretical candidate for observing parity violation. The total charge of the system is zero and it is symmetric with respect to charge but not with respect to mass. Four-particle systems are of fundamental interest (Richard, 1994, Abdel-Raouf et al., 1998, Benslama et al., 1998) because of the quark- antiquark model and the prospects on hydrogen/anti-hydrogen reactions ( Armour and Zeman, 1999, Russell, 1999). The H2 molecule, two electrons and two protons, remains the standard to test any theory on four-particle systems but it does not show parity violation. Partly due to its ordinary scale, the system got less attention in recent years, since Heitler and London (1927) solved the problem of chemical bonding 80 years ago. The quality of their potential energy curve (PEC) for H 2 was poor but James and Coolidge (1933) soon succeeded in calculating a better one. Theoretical physics has evolved drastically ever since but theoretical chemistry remained focused upon developing better computational methods ( Pople, 1999). For H 2, an exact PEC was computed 30 years ago ( Kolos and Wolniewicz, 1968). The next molecules in the Periodic Table are LiH and Li2, which was also studied extensively ( Hessel and Vidal, 1979). Femto-chemistry, an application of wave- packet dynamics ( Garraway and Suominen, 1995) gave a new impetus to the study of PECs at the critical distance, where ionic and covalent curves cross (Rose et al, 1988). Here, quasi-classical approximations are used to describe long-range phenomena ( Aquilanti et al., 1997, Garraway and Suominen, 1995, Remacle and Levine, 1999, Hutchinson et al., 1999). Long-range potentials explain the physics of ultra-cold atoms ( Zemke and Stwalley, 1999, Wang et al., 1997, Marinescu et al., 1994, Hajigeorgiou and Leroy, 1999, Stwalley and Wang, 1999). Fitting PECs at long-range requires accurate potentials and is a delicate matter, given the small energy differences. In this respect, the Dunham series ( Dunham, 1932) is not useful at all, since the series does not converge. The Morse-function (Morse, 1929) is only reliable when thoroughly adapted ( Hajigeorgiou and Leroy, 1999) and lacks a theoretical basis. Therefore a universal first principle's potential is badly needed as a reference for inverting observed levels into PECs and for studying long-range behaviour in particular. If a universal PEC is available, long-range behaviour must be assessable from short-range behaviour (the repulsive branch of a PEC). A correct quantitative evaluation of the long-range behaviour ( Côté and Dalgarno, 1999) is also of importance to test QED (Quantum Electro-Dynamics) as in trapped deuterium (Schmidt- Kaler et al., 1992). Finding a first principle's relation for short- and long-range atomic interactions is still a challenge, although the problem of calculating PECs can theoretically be considered as solved. Nevertheless, there is the question24/01/00 14:26 G. Van Hooydonk 3whether or not a universal potential or a species independent PEC exists, the Holy Grail of Molecular Spectroscopy (Tellinghuisen et al, 1989). This function should rationalise the behaviour of spectroscopic constants, account for the shape invariance of PECs and lead to global scaling. Exactly here, the Heitler- London theory can not give a simple straightforward answer . Symmetry effects in particle systems show in PECs. Typically, a two- centre two-electron bond gives rise to two PECs (fermion behaviour): one for a repulsive triplet state, another for the stable singlet state (sigma-state). The two branches of the singlet-state PEC are not symmetrical with respect to the minimum but show left-right asymmetry ( Herrick and O'Connor, 1998). In terms of the algebra of 1/R-potentials and according to convention, attraction follows -1/R, repulsion +1/R but this elementary symmetry is broken. The attractive branch has a finite asymptote at R = ∞, the repulsive branch almost invariantly goes to infinity at R = 0. Therefore, many empirical asymmetrical 1/R-potentials were suggested ( Varshni, 1957, Steele et al., 1968, Varshni and Shukla, 1963). Most are successful for related (ionic) molecules. The invariant shape and the asymmetric chiral behaviour of singlet PECs point towards a universal function. Shape invariance indicates that two-dimensional scaling should be possible ( Varshni, 1957, Calder and Ruedenberg, 1968, Jenc, 1990, 1996, Graves and Parr, 1985, Tellinghuisen et al., 1989, Van Hooydonk, 1999) but in practice it is not. Therefore the final solution is doomed to be generic, i.e. hidden in first principles, but this truly universal, perfectly scalable ab initio function still remains to be found. Claims have been made that three, probably four or even more molecular parameters are needed for a universal function ( Varshni 1957, Graves and Parr, 1985). The 3 standard parameters are the equilibrium inter-nuclear distance Re, the dissociation energy D e and the force constant ke. But we showed recently that even a two- parameter function can have universal character (Van Hooydonk, 1999). Unfortunately, the H-L theory can not help to solve this problem, as it is impossible to derive analytically a universal function from this theory. Traditionally, we have a right to expect that the complete theory contain well behaving empirical relations found previously. This is not so. Despite the many good empirical 1/R functions available, the H-L theory can not at all predict whether a function A will perform better than B and why this is so. We therefore wonder if the H-L theory is really complete. With a universal function f(x), all observed PECs must reduce to a perfectly symmetric and linearly scalable V- shape with a slope equal to one. Algebraically, the two branches of different PECs should reduce to a single straight line with variable x. Despite all previous efforts, we show that molecular PECs are quantitatively dominated by the universal Coulomb 1/R-potential. Using classical gauge symmetry, we prove that the universal molecular function derives from a pair of symmetric Coulomb potentials. This symmetry has to be broken by a perturbation mechanism that must fit into the classical Hamiltonian. A well known yet overlooked form of chiral symmetry can achieve this, i.e. the handedness of atoms (their mirror symmetry). But a static Coulomb law seems much too old-fashioned and even inappropriate, as no dynamics is involved. Yet this law contains interesting continuous, discrete and scaling ingredients: - continuous: the 1/R-dependence, - discrete: 1/R is a power law: only a positive world is allowed ; a symmetry (parity) effect : attraction and repulsion; a species independent unit of charge, and24/01/00 14:26 G. Van Hooydonk 4- scaling: the asymptote of a Coulomb system is determined by R e, which makes it perfectly suited for two- dimensional scaling (scale invariance). In addition, Coulomb's law is valid both on the micro- and macro-scale and explains both the short and long rang interaction of charged particles. This issue is of central importance for a quantitative assessment of short- and long-range behaviour ( Côté and Dalgarno, 1999). The present contribution deals with many different applications of Coulomb's first principle's law in chemistry and physics. We follow a classical procedure. Elementary steps show what kind of symmetry is generically broken and why Coulomb's law can indeed be a universal molecular function, which allows perfect scaling. Useful references are Bouchiat and Bouchiat (1997) for parity violation in atoms, Varshni (1957), Tellinghuisen et al. (1989) for PECs, Bakasov et al. (1998) for parity breaking in molecules, Garraway and Suominen (1995) for femto-chemistry. Recent work by Hajigeorgiou and Leroy (1999), Stwalley and Wang (1999), Côté and Dalgarno (1999) reviews long-range potentials. Section 2 gives a summary of observed PECs and empirical potentials and the effects of gauge symmetry in general. Sections 3-9 contain the elementary steps to arrive at theoretical Coulomb-based PECs: gauge symmetry for discrete and continuous elements of Coulomb's law, perturbation theory, and the parity violation adaptations for the classical Hamiltonian of a four-particle system. Section 10 gives the theoretical results on the universal function, whereby a generic perturbation is identified. The results are confronted with experiment in section 11. Here we show how well a zero molecular parameter function fits experimental PECs and how 13 different PECs can be brought back into a single straight line. Section 12 discusses generic effects of charge inversion. Sections 13-14 deal with other consequences. 2. Observed PECs: potentials and scaling 2.1. Potentials for a four-particle system. Asymptotes. Series expansions Denoting the lepton-nucleon system in atom X as (a ,1) and in atom Y as (b,2) the interatomic potential V(R), deriving from the Hamiltonian H XY, is V(R) = H XY - (HX + HY) = -e2/R1b - e2R2b + e2/R12 + e2/Rab (1a) The asymptote is assumed to be the atomic dissociation limit or D e = -V(Re). This assumption is probably not true. V(R) consists of 4 potentials, only 1 is related to the internuclear separation R 12, the standard variable for PECs. No information is available about this potential or its character, except that it is zero at infinite internuclear separation, which is trivial. There is no hint as to triplet- singlet splitting. To decide whether V(R) is basically attractive or repulsive, data for the singlet-state at equilibrium are available but the minimum must be supposed to be generic. In first order R 1b = R2b, R12 = Rab and R12 = Rab  2R1b = 2R2b at the minimum R e. V(Re) is then in a good approximation equal to V(Re) = - 2e2/R12 or, due to nucleon-lepton attractions, V(R) is attractive between R e and  as expected. V(Re) is two times the asymptote of two charges in a Coulomb model, 2Ryd or 220000 cm-1. This value points to the absolute well depth H XY, the asymptote IE X + IEY + De (IEX is the ionisation energy of atom X), rather than to the atomic dissociation limit D e. In fact, the maximum value for the dissociation energy D e of bonds between two monovalent atoms is about 50000 cm-1. If true, the H-L Hamiltonian would be in error by 400 % in an24/01/00 14:26 G. Van Hooydonk 5otherwise legitimate approximation based upon available equilibrium data, which is impossible. As a result, Coulson (1959) said that quantum mechanics weighs the captain of a ship by weighing the ship when he is and when he is not on board . This means one has to solve the Hamiltonian first and then subtract the atomic energies. This is a cumbersome procedure, since observed PECs, the result of V(R) in (1a), are shape invariant with only R 12 as a variable. In an ionic approximation, the potential V'(R) is V'(R) = H XY -(HX+ + HY-) = - e2/R2a - e2/R2b + e2/R12 (1b) As with (1a) singlet-triplet splitting is not obtained but the R-dependence is more specific. The well depth for an ionic system is IE X + EAX +e2/Re, if EAX is the electron affinity of X. We get e2/Re = IEX - EAX + De (1c) the classical ionic bond energy. At large R, this approximation (1b) gives R 2a = R2b  R12 or V'(R>>R e) = - e2/R12 (1d) which starts off as an ionic Coulomb attraction at the asymptote, although interchanging an inter-nuclear with a nucleon-lepton term is rather artificial. With the same conventions as for (1a), (1b) leads to V'(Re) = - 3e2/R12 about 3Ryd or 330000 cm-1, larger than the covalent one, because a repulsive term in (1a) is suppressed. It seems that V(Re) refers to a well depth of order 2Ryd >> D e. Then V’(Re) must be about half as large, 1Ryd or the ionic asymptote (1c), situated between 0 and the absolute well depth. Things go wrong when extrapolating this long-range ionic behaviour (1d) to the minimum. The only conclusion possible is that potential V(R) has an asymptote of order Ryd, in any case larger than D e. But no information is found about the minimum, the existence of a triplet state or the shape of the singlet PEC. Bonding is secured by nucleon-electron interactions, conforming to the H-L theory, which leads to the cumbersome procedure referred to above. Just in the ionic one case, a classical picture (1d) emerged, which would lead to an acceptable asymptote of 1Ryd. Unfortunately, this is of restricted validity (just applicable at large R). The behaviour of Coulomb PECs is illustrated in Fig. 1, where for comparison the semi-empirical RKR-curve (Rydberg, 1931, 1933, Klein, 1932, Rees, 1947) for H 2 is included ( Weissman et al., 1963). We use the Coulomb asymptote 116400 cm-1, twice of which is about the absolute well depth (order 1 hartree). The minimum derives from an algebraic Coulomb law |1-0,74144/R|, which seems like an artefact (see further below). The PEC with asymptote D e (38283 cm-1) is computed similarly. In comparison with the RKR, the slopes of Coulomb PECs are much too large and the curvatures have the wrong sign. This is as far as one can go with Coulomb’s law, the only first principle’s law available for a system of four charged particles. Fig. 1 is the reference for this work. The situation is completely hopeless if one tries to explain bonding in H 2 with a static Coulomb law. Nevertheless, for ionic bonds, such as alkali-halides, the middle PEC in Fig. 1 is a good first order approximation for the PEC away from the minimum , and remains useful for calculating ionic curves ( Russon et al., 1997). The H-L theory accounts for the lower PEC for H 2 in Fig.1, not a Coulomb law situation but it can not account for ionic bonds obeying the Coulomb PEC in the middle of Fig. 1. Coulomb's law can account for bonds obeying this middle PEC but it can not account for the lower PEC for H 2. This dilemma led to an almost 100 year old compromise: there are two kinds of bonds, covalent and ionic . But exactly this compromise is in24/01/00 14:26 G. Van Hooydonk 6contradiction with spectroscopic evidence, the good empirical Coulomb 1/R potentials available, the invariant shape of PECs and the dependence of the harmonic frequency on 1/R for both ionic and non-ionic bonds (Van Hooydonk, 1999). This evidence is not covered by the H-L theory. In fact, at least molecular spectroscopy shows that there is no spectroscopic distinction whatsoever between covalent and ionic bonds: they behave alike when properly scaled (Van Hooydonk, 1981, 1999). A curious result is that this spectroscopic information invariantly points towards Coulomb's law and its asymptote (middle PEC in Fig. 1) as being valid for all bonds, ionic and covalent as well . Therefore, molecular spectroscopy indicates that the H-L theory may not be complete indeed. Additional empirical evidence can be found in the many persistent studies on scaled or reduced potentials. The question of bonding may be solved in principle and calculating PECs may no longer be a problem, this kind of empirical research is going strong for decades, see Frost and Musulin (1954), Varshni (1957), Steele et al. (1968), Jenc (1988), Zavitsas (1992), Tellinghuisen et al. (1988), Graves and Parr (1986), Jhung et al. (1989) and Van Hooydonk (1999). One is entitled to do so: a universal potential lies probably hidden in the molecular spectra but the H-L theory is unable to identify this function. The most universal potential imaginable is a first principles local, static, mass-less Coulomb 1/R potential: exactly this type of potential is among the favourites in empirical approximations. But a simple one term Coulomb law is much too rigid and probably not flexible enough. It is commonly generalised using a power series. This has disadvantages ( Varshni, 1957), but the popular Dunham potential (1932) is of this type. Consider the three series V(R) = anRn + an+1Rn+1 + an+3Rn+2 + an+3Rn+3 + …. V(Re) = anRen + an+1Ren+1 + an+3Ren+2 + an+3Ren+3 + …. V(R)-V(Re) = an(Rn-Ren) + an+1(Rn+1-Ren+1) + an+3(Rn+2-Ren+2) + an+3(Rn+3-Ren+3) + ... (1e) The third series gives a different picture than (1f), a series expansion in (R-R e) V(R-Re) = a’n (R-Re)n + a’n+1(R-Re)n+1 + a’n+2(R-Re)n+2 + a’n+3(R-Re)n+3 + … (1f) Only for n = 1 the corresponding two terms in (1e) and (1f) are identical. Starting a function at any n does not lead to loss of generality. In practice, it is convenient to use n = 2 in (1f) to obtain oscillator models ( Varshni, 1957, Dunham, 1932). But, the variables in (1e) and (1f) can be scaled in two mathematically equivalent ways. With the Dunham-variable d = (1-R/R e) (1g) and n = 2, potential (1f) starts off at V(R) = a’ 2Re2d2(1h) and (1e) starts with Ren((R/Re)n-1). Using the Kratzer (1920) variable k = (1-R e/R) (1i) and n = 2 also, potential (1f) starts off at V(R) = a’ 2R2k2(1j) identical with (1h). Using k, (1e) starts with Rn((Re/R)n-1). Both (1h) and (1j) imply harmonic oscillator behaviour in function of (R e-R)2 in the (1f) expansion. The distinction between (1e) and (1f) and between d and k may seem subtle but it is not: it produces different short- and long-range behaviour, see below. A closed formula like Coulomb's, i.e. n = -1 in (1e) and neglecting all other terms, is by all means more challenging and24/01/00 14:26 G. Van Hooydonk 7interesting, since convergence problems are avoided, although both (1e) and (1f) are more flexible. We choose for analytical rigour instead of flexibility from the start and use a few single terms in (1e) as a starting point in our analysis. This choice must lead to mathematical, physical and chemical problems, as all coefficients in (1e) and (1f) are known to be species dependent. 2.2. Empirical potentials and scaling. Short- and long-range behaviour. Observed PECs. Benchmarks The classical Born- Landé (1918) function V(R) = -e2/R + B/Rn(2a) for a singlet PEC uses 2 non-consecutive terms in (1e) with exponents -1 and -9 (n is accessible through compressibility measurements). Function (2a) gives reasonable PECs for ionic bonds without computational difficulties, which is amazing if we recall the complexity of quantum-mechanical calculations and compare the analytical forms of (1a) and (2a). Its eigenvalue is about 0,9e2/Re, close to the generic Coulomb asymptote, see section 2.1. Any PEC generated by (2a) is close to the middle PEC in Fig. 1. But this is not the end of the story. The predictions of (2a) for spectroscopic constants are reasonably accurate for ionic and covalent bonds (Van Hooydonk, 1982, 1999). Kratzer (1920) introduced an even more intriguing potential, with n in (2a) equal to 2 and for which the wave equation can be solved ( Fues, 1926). The Kratzer potential V(R) = -Ae2/R + B/R2(2b) uses 2 consecutive terms in (1e) and can directly be rewritten in reduced form U(R)/(Ae2/2Re) = V(R)/(Ae2/2Re) +1 = (1-R e/R)2(2c) (for references see Znojil (1999) and Van Hooydonk (1999)). This strange potential (2c) has always been overshadowed by Morse's (1929). It is a generalised Kepler system ( Infeld and Hull, 1951) but it also mixes atomic and molecular behaviour. With A = 1 and R e = 1r, where r is the atomic radius, it is an atomic potential , a generalisation of the Bohr equation. This shows after taken the first derivative in function of R. With A = 2 and Re = 2r, (2c) is a molecular potential . These generic aspects of Kratzer’s potential are discussed further below. In practice, around Re, the RHS of (2c) secures the PEC shows an-harmonic oscillator behaviour with left-right asymmetry, always better than a harmonic oscillator. The repulsive quadratic term (R e/R) 2 refers to the kinetic energy of interacting particles, the Planck- Bohr quantum condition for central force systems. A generalisation of (2c) due to Varshni (1957) is consistent with the spectroscopic constants of hundreds of bonds (Van Hooydonk, 1999). Varshni introduced an exponent v for R e/R U(R)/(e2/2Re) = V(R)/(e2/Re) +1 = (1-(R e/R)v)2(2d) This two parameter (R e and v) Kratzer-Varshni-potential (2d) almost behaves like a universal function and is superior to Morse's three parameter potential (Van Hooydonk, 1999). Morse's and (in part) Dunham's oscillator models (1h) would be perfect if there was left-right symmetry in PECs. In Dunham's case, deviations from left- right asymmetry leads to the cumbersome series of Dunham coefficients, all needed to get only moderate agreement with observed PECs and a bad convergence. Moreover, the elementary connection with the energy consequences of 1/R-potentials seems to be lost (Van Hooydonk, 1999). Exactly these form the basis of interactions at the Bohr/Fermi-scale. Conversely, the invariantly observed left-right asymmetry of singlet PECs at Re gives an idea about the nature of the interactions.24/01/00 14:26 G. Van Hooydonk 8The Morse potential is W(R) = D e (1-e-d)2(2e) with  a species dependent constant. Dunham's is W(R) = a0d2(1 + a1d + a2d2 +...) (2f) where a n are the so-called Dunham coefficients, related to the spectroscopic constants. This function can never converge at large R. For long-range interactions the situation gets more complex in general, since these lead to PECs described by functions like V(R) = D e -Cn/Rn, with n >> 1 and Cn a parameter. Many empirical fitting procedures have been presented in the literature but some lead to 'pathological' behaviour ( Coxon and Hajigeorgiou, 1991). The best known are the Ogilvie-Tipping anharmonic oscillator ( Ogilvie, 1988), the generalised Morse oscillator ( Coxon and Hajigeorgiou, 1990, 1991), the modified Morse oscillator ( Hedderich et al., 1993) and the modified Lennard-Jones oscillator ( Hajigeorgiou and Le Roy, 1999), the latter being a mixture of Morse and Kratzer potential elements. Most of these fitting procedures are based upon the Morse potential, which is inferior to Kratzer's when it comes to rationalise the behaviour of the lower order spectroscopic constants (Van Hooydonk, 1999). The Morse-function is confined to the observed dissociation limit D e, which guarantees it will always converge to unity for RKRs scaled with D e. Fig. 1b illustrates the observed situation. RKRs reduced with D e are shown for 13 bonds (details are given below) in function of the reduced distance R/R e. Although both the y- and x-axis are scaled consistently, the PECs do certainly not coincide. Nevertheless, the shape invariance referred to above shows clearly and needs to be explained. The Morse function f(R) in (2e) is a better measure for the x-axis. If the Morse function is universal, as claimed by Jhung et al. (1989), all PECs of Fig. 1b should reduce to a perfectly symmetric V- shape with perpendicular legs with the Morse function at the x-axis. The actual result is shown in Fig. 1c. The required symmetric V-shape is only obtained in the vicinity of Re. At the attractive side, the agreement seems satisfactory, although this is for the larger part due to the fact that the asymptote of Morse' s function is D e. Despite this 'imposed' asymptote the different PECs do not collapse into a single line, although all have a slope near unity. It would appear that Morse's function describes long-range behaviour rather well but it does not. This moderate agreement is not confirmed by the data at the repulsive branch. Here the slopes show large divergences. At the extreme short side, 'turn over' points appear. This leads to a strange almost contradictory situation: the 'simpler' repulsive branches are not well reproduced by Morse's function, whereas the description of the complex attractive side, where long-range potentials interfere, is better. Nevertheless, Fig. 1c sets the standard for other analytical potentials. Morse's function (2e) is exponential in R and uses three parameters, R e, De and a0 to get only moderate agreement with experiment. This shows even better in Fig. 1d, a linear plot of algebraic attractive and repulsive branches against the algebraic Morse function. The relative good V-shape in Fig. 1c is not confirmed, since a straight line is not obtained. This imposes restraints on the universal character of the 3-parameter Morse-function and sets a clear benchmark for other scaling approaches. Morse's function is more complex than Coulomb's but it certainly does not result in perfect scaling. It can not properly account for the 'simple' repulsive branches, which are more suited for scaling (Gardner and Von Szentpaly, 1999).24/01/00 14:26 G. Van Hooydonk 9With this experimental background and these benchmarks in mind, we start from scratch. We eliminate all parameters and test the only parameter-less first principles potential available: Coulomb's. A few symmetric single-term Rn-potentials appearing in the series (1e) will illustrate the procedure. Two-dimensional scaling is essential. The consequences of the next and ultimate step i.e. perfect symmetry between attraction and repulsion, the basis of Coulomb's law, is a challenge. But the major problem with a pair of symmetric potentials is V(R) = 0 (annihilation) for all R in (2a) and (2b). But working with a closed formula, devoid of any parameter, may lead to the simplest analytical form possible for a universal potential and to that single symmetry element in the Hamiltonian, needed to account generically for shape invariant asymmetrical molecular PECs. 2.3. Classical gauge symmetry and two-dimensional scaling Consider two particles with identical masses m interacting through a potential V(R), for which the classical Hamiltonian H 12 reads H12 = ½mv2 + ½mv2 + V(R) (3a) The reduced mass  would be equal to 0,5m. The particles interact through a single term in (1e) V(R) = anRn(3b) with n an integer, a n a potential dependent constant (with dimensions energy times length-n) and R the particle separation. Let n be equal to - 2, -1, 1 or 2 (n = 0 leads to a one-dimensional PEC). The x-axis R is scaled by a characteristic distance for the system, R = mRe, giving V(R) = anRen mn(4) A pair of symmetric potentials (attractive negative, repulsive positive) can be created using a parity operator P 2 = 1 or P = ±1. With (-1 )t this gives V+(R) = anRen mn = -V-(R) (5a) V±(R) = (-1 )t V(R) (5b) where the exponent t, the type of interaction, is equal to 0 (even) for repulsion and 1 (uneven) for attraction. The zero reference point is free (gauge-symmetry). Adding a constant C gives W(R) = V(R) + C (6) and by virtue of (5a) W+(R) = -W -(R) (7) independent of the value of C. Scaling C on the y-axis in terms of a typical energy for the system C = anRen = V(Re) (8a) leads to two scaling operations on an equal and consistent basis (two-dimensional scaling, gauge symmetry, see Introduction). C is the asymptote and is determined by C = |V() - V(R e)| (8b) which shows why its sign is a matter of convention. The levels ±C are symmetrically distributed around the original x-axis, the gap being |2C|. Using a second parity operator (-1 )g for C leads to four different symmetric states for (3b) potentials W(R) = ± anRen ± (anRen)Rn/Ren = C (-1)g + C (-1)t Rn/Ren24/01/00 14:26 G. Van Hooydonk 10W(R)/C = w (m) = (-1 )g (1 + (-1)t-g Rn/Ren) w(m) = (-1)g (1 + (-1)t-g mn) (9a) if the exponent g for the gauge is equal to 0 (even) or 1 (uneven). Result (9) represents the algebraic effect of classical gauge symmetry for potentials: it is a well-known generic result, independent of the analytical form of the potential. The R- or m-dependence is extremely simple and is the same for all four states and the scheme is scale invariant. The distinction between the four states is only due to one algebraic symmetry operator, parity. The meaning of the four states is discussed below for a Coulomb potential. These scaling and symmetry operations artificially create two symmetric isospectral worlds W < 0 and W > 0 but only one can be real. There are mathematical techniques to arrive at the same results without negative worlds, see below. Classically, parity is never violated. If the two worlds remain separated and PECs do or can not cross, parity is not violated and convention is sufficient. But, two symmetric t - g = 1 states (asymptote and interaction have opposite signs), always cross at R = R e or at W(R) = 0. For mass-less particles and for systems with m 1 = m2, symmetry may never be broken and no stable states are produced. States with t - g = 0 can never cross. The two symmetric states with t = 0 and t = 1 around + C and - C will never cross. Symmetry breaking for two crossing t - g = 1 states is necessary, independent of n in (3b). If a perturbation P is present for these two t - g = 1 states, scaling P with C gives p = P/C and applying the non- crossing rule to a symmetric pair gives in first order w'(m) = ((1 - mn)2 + p2)1/2 - p (9b) as a generic perturbed function. A 'classical' definition of p is needed, as is its dependence on m. For a pair of algebraic potentials, crossing of symmetric states at the minimum is generic and the breaking of this symmetry must therefore also be generic. This can only be achieved by finding a generic perturbation. The physical origin of this must be found. If so, perfect scaling is theoretically obtained. 2.4. Symmetry of attraction and repulsion The symmetry of attraction - Rn and repulsion + Rn in (5) does not lead to simple solutions for N-particle systems. It is more practical to use different n-values or analytical expressions for attraction and repulsion, which is contrary to their algebraic symmetry. This is a characteristic of many empirical approaches. A typical one consists in choosing two terms in (1e), each having a different sign. For instance, (2a) and (2b) both use n = -1 for attraction but n-values different from -1 for repulsion, i.e. -9 and -2 respectively. Quantum mechanics, where a 1/R-dependence is invariantly applied for attraction and repulsion, shows that calculations can get complicated, see (1a) and Fig. 1a on account of this basic symmetry. Obtaining theoretical PECs is difficult in contrast with (2a) and (2b). Some of the difficulties in trying to find solutions for N-particle (particle- antiparticle) systems have been studied before (Richard, 1992, 1993, 1994, Abdel-Raouf, 1992, Abdel-Raouf et al, 1998). In this work, this algebraic symmetry is respected throughout, as it is a first principles element of Coulomb's law, conforming to gauge symmetry, and secures that scaling can not lead to distortions when going from the left to the right branch of PECs. 2.5. Theoretical shape invariant PECs from a pair of symmetric anRn potentials and symmetry breaking24/01/00 14:26 G. Van Hooydonk 11At this stage, the shape of PECs generated by closed formula algebraic functions (9a) can only be discussed in the ad hoc hypothesis that crossing of t - g = 1 curves is avoided, see (9b). The five major invariant characteristics of PECs are: (a) triplet-singlet splitting occurs at large R; (b) the triplet-state is repulsive (unstable state); (c) the singlet-state is attractive (stable state) and shows a minimum with asymmetric left and right branches at either side of the minimum; (d) the left branch is repulsive with a very large slope and does not reach a finite asymptote (it reaches infinity at R = 0 or at least becomes extremely large before eventually reaching the united atom energy at R=0 or even lower); (e) the right branch is attractive with a slope less than in the left branch and reaches a finite asymptote at R = ∞ (for other minor characteristics, see Varshni, 1957). The general behaviour of (3b) in terms of (9) must be discussed referring to these characteristics (a)-(e). For the singlet-state, a reference PEC is generated with the reduced Kratzer potential (2c). Crossing is avoided by using a small constant ad hoc perturbation p2 = 0.1 with C = 1 instead of the generic perturbation needed in (9b) but not yet identified. Results are shown in Fig. 2 and Fig. 3 . The morphology of PECs deriving from n > 0 potentials in Fig. 2a (n = 1) and 2b (n = 2) is not consistent with patterns (a)-(e). Although a splitting and a minimum is generated, the shape of the bonding PEC is wrong (reversed) at the tree level. Even the n = 1 case in Fig. 2a, the better of the two with respect to the Kratzer PEC, leads to an opposite picture. The situation gets worse for the n = 2 potential in Fig. 2b, although this is exactly the potential for the harmonic oscillator. This n = 2 potential is used frequently as a model in a variety of symmetry problems (see Witten, 1981, 1982, for a classical example in SUSY). The left branch instead of the right would give the finite asymptote at R = 0. The right instead of the left branch goes to infinity for large R. This apparent reversed left-right asymmetry simply calls for a switch from n > 0 to n < 0 potentials. In addition, it seems unlikely that any justifiable non-crossing scheme (perturbation) would improve the situation with respect to the Kratzer-reference PEC, see Fig. 2a and 2b. But perturbation alone can not remedy the wrong left-right asymmetry inherent to n > 0 potentials . Part of this so- called wrong behaviour has to do with convention: 1/R as a variable instead of R for n = 1 would give the correct result, see below. Fig. 2a is a Landau- Zener model: curves 2 and 3 are diabatic levels, curve 6 is (one of the two) adiabatic levels in (9a). The morphology of PECs generated with n < 0 potentials in Fig. 3a (n = -1) and Fig. 3b (n = -2) is in line with (a)-(e) and conforming to the Kratzer singlet PEC. At the tree level, the n = -1 potential performs better than the n = -2 potential, if the Kratzer potential is used as a reference. As above, avoiding crossings is illustrated in Fig. 3a and 3b. In Fig. 3a, the resulting PEC is close to the Kratzer-approximation for larger R but gets automatically worse in the left branch. For the repulsive (triplet) state, the generic shape is almost correct by definition in the case of an n = -1 potential. At the tree level, a pair of symmetric 1/R-potentials leads to the correct shape of PECs in all aspects (a)-(e). Fig. 3a is a Demkov-Kunike model: also here curve 2 and 3 are diabatic levels whereas 6 is (one of the two) adiabatic levels in (9a). Splitting, the shape of the triplet state, the existence of a minimum for the singlet state and the left- right asymmetry at the minimum all secure that, in first order, a pair of symmetric 1/R potentials is at work in cases for which Kratzer's potential gives the reference PEC. This general and very consistent model is one of the main reasons why people remain interested in empirical 1/R potentials (Fig. 3a)24/01/00 14:26 G. Van Hooydonk 12hoping one of them has universal characteristics. Parameters give the flexibility needed to apply the potential to more than one bond. We choose for rigour, by eliminating parameters, to find out what a truly generic one-term potential is capable off. The main difficulty of scaling in molecular spectroscopy is to find a (generic) species independent variable which would allow a smooth transition from Fig. 3a (1/x-situation) to the perfect V-shape of Fig. 2a (x-situation) using the same experimental data, see above Fig. 1c-e. In addition, closed formula potentials, conforming to observations (a)-(e), as in Fig. 3a, cause problems for the H-L theory, bound to the complex potential V(R) in (1a). The internuclear 1/R potential in the present scheme is just one out of the four potentials in (1a). Exactly this important internuclear potential in (1a) is repulsive instead of attractive. 2.6. Symmetry breaking and gauge symmetry The n = -1 potential is a power law, confined to a positive world. However, the convention is that two symmetric ± solutions are possible (attraction and repulsion). This can only be achieved by shifting the zero of the system, C in (9), and allowing worlds wherein attraction and repulsion are possible at the same time (classical gauge symmetry and a characteristic of potential theories). The maximum range R for attraction is then automatically governed by the asymptote C . Therefore, at the zero, the symmetry of attraction and repulsion must be broken in order for the conventions to remain valid. For n = -1 potentials, finite reference points have been replaced by asymptotes. The n = -2 case runs similar. The next problem with the two n < 0 worlds is scaling, since asymptotes are now the reference points, needed for scaling. Mathematically, there is no substantial difference between n = 1 and n = -1 cases, since we can always replace x with 1/x. The use of 1/x is more a tradition (Newton, Coulomb), see above. With respect to scaling, the information contained in PECs is more manageable with x as a variable rather than with 1/x, especially with respect to the asymptotes. Conflicting situations can occur for systems subject to two different laws. In classical and wave mechanics, n = 2 behaviour is needed for the kinetic energy, n = -1 behaviour for the potential. Conventions about the real world must be in line with these two behaviours, if both are allowed. If a conflict occurs, the symmetry of the behaviour that causes the conflict must be broken immediately at the critical point where the conflict occurs. Determining what exactly this symmetry breaking effect means in terms of the physics of the interactions between particles is the next problem to be dealt with . Gauge symmetry must also be confronted with the Hamiltonian to find out if this symmetry breaking effect can be incorporated. In the H-L theory, there is no direct link within the Hamiltonian to symmetry breaking effects leading to (a)-(e) as observed in PECs. 3. Algebraic Coulomb 1/R-potentials. A new degree of freedom. Perfect Coulomb scaling Consider a system of two charges, interacting through a Coulomb 1/R-potential, see Fig. 3a. The charge symmetry is not (yet) broken by the (identical) particle masses giving V±(R) = ± e2/R (10a) in which, according to convention, the + sign refers to charges with equal sign (repulsion), the - sign to charges with opposite sign (attraction).24/01/00 14:26 G. Van Hooydonk 13The problem case is V(R) = - e2/R + e2/R = 0 (10b) when referring to (2a) and (2b), for any R. Fig. 4 represents the general shape of these classical Coulomb PECs, which can not show a minimum, since they never cross ( fermion-behaviour). Due to charge invariance, the four states ++, +-, -+ and - - reduce to two degenerate pairs, +-, -+ and ++, --. Generalising (10a) is possible using the procedure of section 2.3 W(R) = V ±(R) ± C = ± (e2/Re)(1 ± Re/R) W(R)/(e2/Re) = w (m) = ±(1 ± R e/R) w(m) = ± (1 ± 1/m) = (-1)g (1 + (-1)t-g/m) (11) The Coulomb asymptote C can be divided or multiplied with any constant without loosing generality, since R e is multiplied or divided by the same constant. Result (11) is illustrated in Fig. 5. Referring to Fig. 3a, Coulomb forces imply a new degree of freedom . The symmetry of attraction and repulsion resides in the interaction of two charges. Any system of two interacting charges obeys charge invariance. This principle secures that, within a given world, the energy of the system is not altered when the signs of two interacting charges are interchanged, say from +- to -+ for attraction or from + + to - - for repulsion. But, a transition from a positive world into a negative one as in Fig. 5 affects all symmetry aspects. At the minimum, attraction (+-) changes into repulsion (- -), if the original conventions hold. A symmetry breaking effect would explain the minimum, the asymmetric branches and the different slopes in observed PECs automatically. But there is no reason whatsoever for the interaction of two charges in the real world to suddenly change from attraction into repulsion. The effect of the two worlds is that, in contrast with (9), the four states in (11) are classically identified and further diversified, i.e. ++, +-, -- and -+, exactly in this order when starting from asymptote +C, see Fig. 5 and the arrow notation therein. As a result, Fig. 5 is more complex, since the sigma bonding state is now described, on account of gauge symmetry, by + - and - - states, whereas the two triplet states are ++ and - +. But the repulsive (- -) state +1/R - C, is also the attractive state in the negative world and would cross the (+-) attractive state -1/R + C of the positive world at R e/R = 1. But two states with the same basic gauge symmetry -attractive or t - g = 1- are not allowed to cross and perturbation must be invoked as in (9b). Looking at Fig. 5, the - - attractive state +1/R - C in the negative world is a repulsive state with a downshift with respect to the repulsive state +1/R + C in the positive world, the shift being equal to |2C|. For a system of two charges, all this may seem meaningless. Charge invariance allows a shift from + - to - + for attractive states without energy implications. But gauge symmetry overrules charge invariance: now there are two not degenerate attractive (repulsive) states + - and - - (- + and ++) with totally different energies away from R e (Fig. 5). But we know that switching fermion chiralities simply corresponds with switching a sign in the Hamiltonian, which is the result of charge conjugation combined with the particle-hole transformation (Neuberger, 1999) . Which sign(s) must be switched will be demonstrated below. It is important to realise that PECs generated by gauge symmetry (Fig. 3a) are observed for systems consisting of four particles, such as chemical bonds. Therefore, we must find out how the gauge symmetry dictated scheme can be translated into a physical law in these four-particle systems.24/01/00 14:26 G. Van Hooydonk 14As in SUSY, the problem is to find a real N-particle system, where this mathematical artefact applies. For a system of two charges, it is impossible, in the absence of external effects, to imagine any perturbation, if self-perturbation is excluded. Positronium and protonium systems are extreme but straightforward examples. The 1/R attraction is used in full and can not, at the same time, perturb itself. The origin of the minimum remains a mystery and must be considered as generic, since only an electrostatic 1/R potential is used to describe the system (dynamics is not -yet- involved) but we know R e derives from classical equilibrium conditions, involving dynamics. Up till now only a generic ad hoc V(Re) is available as a Coulomb asymptote C = e2/Re, also present in (2a) and (2b) and this derives naturally from gauge symmetry (parity). Classical equilibrium conditions in two particle central force systems respect the virial theorem and lead to an asymptote, (1/2) e2/Re = C/2, although multiplying e2/Re with any constant (as the virial ½) does not alter Coulomb scaling. Returning to two-dimensional scaling, (11) applies to any asymptote Cb = C/b, where b is a constant, see Introduction. The R-dependent part of (11) will still vary as R e/R = 1/m for any b-value. As a result, w (m) may be multiplied with any constant without loosing generality or universality. The b-values will affect the shape of unscaled PECs, as illustrated in Fig. 6a, b and c. Fig. 6a shows that, with the same Coulomb 1/R potential, a smaller asymptote reduces the slopes of the branches and increases R e-values. Scaling R with R e (Fig. 6b) draws the attention exclusively to asymptote differences. Perfect two-dimensional scaling results in a reduced universal PEC as in Fig. 6c for any b-value. If b is very small, physics at the Fermi- and, in the limit, even at the Planck-scale is obtained. If b is very large, we would eventually get physics at the Newton-scale, see below. Fig. 6c results from perfect two-dimensional scaling, the key being the universal character of the 1/R potential, its closed analytical form and gauge-symmetry. These three Figures represent the major issue of scaling, symmetry and the n-dependence for R in (1e) or (3b). Even for seemingly weak interactions as in Fig. 6a (large Re-values, small slopes) or as in Fig. 6b (small asymptotes), it is tempting but not always necessary to invoke a different n-dependence on R in the potential (1e) to account for these effects. But if gauge symmetry is applicable, a unique universal Coulomb 1/R law applies to all cases, see Fig. 6c. The dissociation products in a Coulomb system must be charged particles. This is not trivial: in practice, observed molecular PECs are usually scaled with the atomic dissociation limit D e, where the dissociation products are two neutral atoms. Theoretically, these can never represent the natural eigen-value of a true Coulomb system. Exactly the dissociation limit of four-particle systems (chemical bonds and the long-range behaviour of atomic interactions) cause severe problems when interpreting RKRs. As a matter of fact, Fig. 1b clearly shows that D e is certainly not a Coulomb scaling asymptote in the strict sense as the 1/R behaviour is not reproduced. The long-range behaviour is usually described by terms in (1e) of the form Cn/Rn, where n is large, see above. The range can be estimated by means of the Le Roy radius (Le Roy, 1973). Then it is of primordial importance to describe as exactly as possible the normal R-dependence of particles in this region, about which wave-packet dynamics ( femto-chemistry) could provide us with new information. With a universal function it may be possible to scale long-range behaviour too, which, to the best of our knowledge, is not yet possible. We applied this generic Coulomb scheme to see whether the exclusive use of R and Re leads to a generally consistent physical picture. Fig. 6d shows the behaviour of 13 observed RKRs in an algebraic Coulomb scheme24/01/00 14:26 G. Van Hooydonk 15with gauge symmetry (with a small perturbation added at the minimum). It is readily verified that this extremely simple 1/R and 2C-gauge system reproduces the general shape of observed molecular PECs. At the repulsive side, Coulomb's law is closely followed in the whole range. At the attractive side, the branches do not follow symmetrically Coulomb's law, probably due to long range forces. A perturbation is needed to shift the minima of the PECs upwards by a small amount, as in Fig. 3a. This gives a better correspondence with Coulomb 1/R behaviour at both branches. The underlying perturbation mechanism is very important, but must be identified. Fig. 6d reveals that PECs may indeed validly subject to a Coulomb scaling process (Van Hooydonk, 1999). The only exception seems to be the RKR for I 2, as it does not follow the general trend, especially when compared with Li 2, since bot have approximately the same R e-values of 2,67 Å. It appears that, if we scale locally all observed PECs with D e, all RKR-data-points will be contained between 0 and 1, irrespective of the value of Re. To get a scaled result, all PECs reduced with D e must somehow also be rescaled using relative R e values. For instance, bringing up (scaling) the Rb 2 RKR to the position of that of H 2 in Fig. 6d does not depend on the value of D e in the first place but on R e. How to achieve this is illustrated below. 4. Problems with generic shape invariant PECs and a four-particle Hamiltonian. The switch Reconstructing qualitatively the shape invariance of observed PECs is no problem, as long as symmetry is broken: gauge symmetry and a pair of symmetric 1/R-potentials seems sufficient (see Fig. 3a and 6d) and guarantees Coulomb scaling for PECs. With respect to quantitative aspects (slopes and curvatures), problems are met if we confront this simple picture with the complex four-particle Hamiltonian H = ½ m 1 v2 + ½ m2 v2 + ½ ma v2 - e2/R1a + ½ mb v2 - e2/R2b - e2/R1b - e2/R2a + e2/Rab + e2/R12 Not less than 10 terms appear, 4 of which are kinetic energies and 6 are Coulomb potentials. We must select the signs to switch (see above) in the 6 charge conjugated terms. This complex Hamiltonian generates problems for a 1/R scheme dictated by elementary gauge-symmetry: 1. Prove that, like in (5), introducing a parity operator (a switch) in this Hamiltonian suffices to reproduce quantitatively shape invariant PECs. In the H-L theory, both splitting and the asymmetric minimum result from the combined (external) symmetry effects of wave functions and electron spin. The symmetry breaking effect we need must be a generic consequence of a physical 1/R process. 2. Prove that the resulting PEC is in a good first order approximation, obtained by a pair of Coulomb 1/R potentials . The H-L Hamiltonian contains 6 potential energy terms, only one describes the internuclear separation R, needed to construct PECs and is repulsive. At least this term needs a switch, a parity operator . The asymptote must be identified. 3. Find a suited, realistic and even generic perturbation to break the symmetry and to apply the non-crossing rule for the intersecting pair of t - g = 1 states in Fig. 3a, a problem in its own right. A perturbation does not leave the asymptote unaffected. Alternatively, we can look for lower states with the same symmetry, intersecting the generic state away from the minimum, which avoids crossing at the minimum. 4. Not the least, find a real and observable N-particle system , where these theoretical deductions apply. It seems impossible to avoid composite particle systems, be it alone to bring more complexity into the rigid analytical form of a simple Coulomb potential for a two-particle system.24/01/00 14:26 G. Van Hooydonk 165. A theoretical four-particle system. Coulomb-potentials and charge-distributions. Perturbation. The Born-Oppenheimer (B-O) approximation 5.1. Symmetric perturbed Coulomb 1/R-potentials for a theoretical four- particle system A system with only two charges ( two fermions ) with particles of equal mass must be ruled out, see above. The next neutral system possible consists of at least four charges and must be subdivided in such a way that a Coulomb interaction between two point-like sub-particles emerges. The sub-particles must have a global mass asymmetry and carry opposite charges. Partitioning the four-particle system into two neutral subsystems, consisting of two neutral particles each, can never produce the Coulomb interaction e2/R needed. Therefore, the only possible and ideal partitioning leads to two asymmetric charged subsystems: a charged composite particle X± interacting with an oppositely charged not composite particle Y±. The total charge of the composite X± particle being ±e implies that the number of charged particles in X is 3. These are confined to the X domain and their mutual separation is close to R e/2, negligible at large R. The masses of X± and Y± must be nearly equal to secure that the charge symmetry will not be broken by the X-Y mass difference. As in section 2.1, the 3 charges in X are labelled as (1,a ,b) and the one in Y as (2). Since the total charge is zero, two pairs of fermions with opposite charges ( a,b) and (1, 2) are needed. One pair resides within X, the other is distributed between X and Y. We can distinguish two charge distributions in the fermion pairs forming the neutral four-particle system: (a) fermion pairs (2 leptons, 2 nucleons) have equal charges: +e2 (++ and --, or vice versa); (b) fermion pairs have opposite charges: -e2 (+- and -+, or vice versa). For the classical case (a) in this theoretical four-particle system, partitioned in this asymmetric way, the long rang interaction V(R) is equal to (1b), the standard ionic model discussed in Section 2.1 . For case (b) we get instead V(R) = - e2/R12 + e2/R2a - e2/R2b (12) When compared with (1b), the three terms in (12) are the same but two of them have switched signs. The last two terms are equal in magnitude, the fermion pair ( a,b) being confined to X. A small perturbation V 12 at large R results from (12) V(R) = - e2/R12 + (e2/R2a - e2/R2b) = - e2/R12 + V12 (12a) Apart from two signs, potential (12a) has the same structure as (1b). Therefore, only this theoretical four- particle system, case (b), would give rise to the situations already depicted in Fig. 3a and 5: a pair of two perturbed Coulomb potentials, if we only could justify the switch in sign for this part of the Hamiltonian. The standard non-crossing rule would simply lead to W(R) = (C2k2 + V2 12) 1/2 - V12 (12b) in agreement with (9b) and used for obtaining Fig. 3a in the ad hoc hypothesis that V 12 is a small constant . Fig. 7 gives the comparison of the results obtained with the generic potentials (12b) for V 12 = 0,1 and 0,35 respectively and with the Dunham, Born- Landé and original Kratzer potentials of section 2.2. Reminding Fig. 1, the PEC for a theoretical four-particle system, (12b) may be close to reality, if the Born- Landé/Kratzer PECs24/01/00 14:26 G. Van Hooydonk 17are useful references. The famous oscillator PEC (n = 2 potential), the leading term in the Dunham expansion, is also shown in Fig. 7 (curve 1) but is completely out of range, at both short and long ranges. 5.2. Can the real four-particle system consist of ions interacting through Coulomb's law? The switch Although the PEC generated by the present scheme seems to have prospects, see Fig. 7, composite asymmetric systems like those described in Section 5.1 may seem exotic. Fortunately, case (a) resembles systems very common in chemistry: two ions interacting through Coulomb's law. An anion is like the composite particle, say F- and a cation the non-composite particle, say H+. Anions consist of three particles: one positive nucleon and two negative valence electrons. Strictly spoken a division into a molecular ion XY± and a lepton e± is also possible. But in that case the charge symmetry is spontaneously broken by the large mass difference, which must be avoided. In fact, this essentially atom-like interaction will generate an atom-like spectrum. The masses of the partners, anion and cation, can not be identical. The mass difference is at maximum, when the nucleon masses are identical and small. For H it is 2/1836 or about 10-3. This difference leads to a small perturbation and to symmetry breaking as required. Selecting the case of ionic bonding would secure that the non-crossing rule applies on account of (12), if only the signs of the terms in (12) were in agreement with those in (1b). But if all this can be rationalised, many PECs would be available for testing. The interactions confined to the X-domain do not vary with R. Hence e2/R1a, e2/R1b and e2/Rab are frozen in first approximation, an IIM (Ions In Molecules) approach (Van Hooydonk, 1999). But since the theoretical system (12a) gives different signs for the potential than that for the classical ion-pair (1b), an ionic approximation seems to be forbidden on account of symmetry effects connected with charges, which reduces to a switch problem. These were discussed in Section 3 and we will now show how to remedy generically this symmetry- or parity-anomaly. 5.3. Confrontation with the Hamiltonian The B-O approximation freezes nucleons and implies central force systems. Reduced masses appear. The eight terms in an order related to a standard AIM (Atoms In Molecules) approach are H12 = (½av2 - e2/R1a +½bv2 - e2/R2b) - e2/R1b - e2/R2a + e2/Rab + e2/R12 (13a) The first 4 are the (intra-) atomic terms , the last 4 form a perturbation of the atomic states leading to V(R) in (1a). The inter-nuclear term is not dominant and is repulsive instead of attractive . With an IIM (Ions In Molecules) approach the eight terms in (13a) are rearranged as H12 =(½av2 +½bv2 - e2/R1a - e2/R1b + e2/Rab) + e2/R12 - e2/R2a - e2/R2b (13b) The first 5 terms belong to the domain of the anion X±, the latter 3 to the cation-anion interaction V’(R) in (1d). Although the relative influence of the internuclear term in (13b) is more pronounced than in (13a), it remains repulsive, not in agreement PECs either. To get an ionic interaction, dominated by - e2/R12 (case b. of the Section 5.1) the following identity could be artificially imposed - e2/R12 = e2/R12 - e2/R2a - e2/R2b leading to e2/R12 = (1/2)( e2/R2a + e2/R2b) (13c)24/01/00 14:26 G. Van Hooydonk 18The consequence of trying to reproduce an ionic attraction - e2/R12 in this way automatically rules out the possibility of finding a small perturbation term, which vanishes due to (13c). A way out is considering one of the nucleon-electron interactions in (13b), say - e2/R2a, to act as a pseudo-ionic attraction or - e2/R2a + (e2/R12 - e2/R2b) (13d) Rewriting the last 3 terms in (13b) accordingly gives the almost correct picture needed for the theoretical scheme (12a) and the correct perturbation result (12b). In fact the two terms between brackets in (13d) can be considered as a perturbation V 12 of the pseudo-ionic interaction - e2/R2a but it will only be small at large R . This solution is artificial: we interchanged nucleon-nucleon repulsion and nucleon-electron attraction to get pseudo- ionic attraction. This artificial mechanism for going from cas (a) to case (b) in Section 5.1 does not give the correct switch. The charge symmetry of the system is broken, which is not allowed. Nevertheless, this solution would allow to apply the non-crossing rule as in (9b) or (12b), if only the internal consistency of the model was not broken so drastically as in (13d). But all evidence of observed molecular PECs points out that attraction depends naturally on 1/R 12 and not on a nucleon-electron potential as in (13d). In addition, the scheme must be valid for all R, not only for R>> R e. 6. Intra-atomic charge inversion. A generic switch (parity operator) to apply gauge-symmetry to a chemical bond There is a relatively simple way to realise the transition from case (a) to case (b) in Section 5.1. This makes the sign in the ionic potential (1b) correspond with that of theoretical model (12), restores the broken internal symmetry in (13d) and secures at the same time that V 12 will remain small at all R-values . By definition, the main interaction must be attractive and must depend on R 12, not on R 2a or R2b. For the perturbation to be small and in order to retain the internal symmetry of all six interactions in (13a) and (13b), V 12 should be related to a difference e2/R2a - e2/R2b instead of to an asymmetrical solution (13d). In this hypothesis |V12| ~ |e2/R2a - e2/R2b|<< |e2/R12| (13e) the non-crossing rule could be applied correctly as in (12b) and the internal charge-symmetry of the four- particle system is restored. Although the geometry of the four-particle system and the reference axis are unknown, it is difficult to describe exactly which interactions could switch sign generically. Yet, making the signs of (1b) generically congruent with those in (12) has an unexpected consequence in connection with the principle of charge invariance, discussed in Section 3. The difference in (13e) can only be obtained in any geometrical arrangement of the four particles , when the two electrons (leptons) in the anion have opposite charges, which is contrary to convention and especially to the standard H-L bonding theory. The next consequence of charge invariance is that this charge inversion also applies to the nucleons . This provides exactly the key for going from case (a) to case (b) charge distributions, discussed in Section 5.1. We remind that the conventional case (a) has dominated theoretical chemistry in the post H-L era ( Pople, 1999). The possibility that case (b) might be involved in theoretical chemistry has never been considered. But due to the principle of charge invariance, case (b) would never have changed the energy of an atom, since the strength of the intra-atomic interaction does not change when going from +- to -+. Intra-atomic charge24/01/00 14:26 G. Van Hooydonk 19inversion can only show when two ions interact according to Coulomb's law : then a ++ interaction can change into a +- interaction of the same character, as discussed in Section 3. This intra-atomic charge inversion provides us exactly with the generic switch, the single parity operator we wanted above. The effect of a charge inversion in one of the two atoms is different for a parallel and an anti-parallel alignment of the charges within the two atoms . A simple example of this effect is the interaction of two magnets (dipoles). Attraction and repulsion are immediately felt when turning one of the magnets upside down when bringing them together. The effect of this generic switch, not available in the H-L theory, is not dependent on the shape or geometry of the four-particle system: it is a generic chiral symmetry effect implied in gauge symmetry and only one algebraic operator is needed to describe it, a parity operator, as expected. The mathematical symmetry breaking effect, inherent to t - g = 1 Coulomb states has now become a real physical even generic effect. (Footnote remark: The effect of this chiral symmetry has been brilliantly visualised by the Belgian surrealist Magritte in a painting showing a man seeing his back when looking in a mirror). The only way to make the Hamiltonian of the four-particle system in a chemical bond congruent with the recipe provided with gauge-symmetry (see Fig. 3a and 5), is to adapt it for parity violation: a repulsive nucleon- nucleon or lepton-lepton H-L state ++ (+C + 1/R 12) becomes attractive + - (+C - 1/R 12). The opposite applies to two of the four nucleon-lepton interactions. This is needed to get charged dissociation products at large R (fermion behaviour), in agreement with the generic gauge-symmetry based Coulomb scheme, see conclusion of Section 3. All these preliminary considerations led us to very specific and clear benchmarks for an alternative way to explain chemical bonding or the stability of 4-particle systems in general. The first criterion to justify this rather drastic measure will be the agreement with experimental PECs (published RKRs): the theory should result in a really universal function, whereby the ultimate challenge is to reduce all 13 PECs in Fig. 1b to a single straight line. The second is the character of this theoretical result, which depends upon the number of parameters required or allowed to fit theory with experiment ( Pople, 1999). If this number is zero, the ab initio status applies. The third is an objective quantitative criterion for the acceptance of ab initio methods and was clearly set by Pople (1999) in his recent review about the evolution in quantum chemistry since 1927: heats of formation should be reproduced with an accuracy of 1 kcal/ mol or about 400 cm-1. In terms of molecular spectroscopy this is a relatively wide margin when level energies are under discussion but it is a clear quantitative benchmark for theoretical approaches. 7. Intra-atomic charge inversion in the Hamiltonian leads to symmetric 1/R potentials and shape invariant PECs Permutational symmetry requirements and their effect upon the wave function in the classical H-L theory (antisymmetry for fermions and symmetry for bosons) are now taken over by charges and charge distributions. Mirror symmetry for the two charges in a boson produces a switch which transforms an attraction into a repulsion or vice versa. This gives fermion behaviour for a pair of bosons (see Fig. 5). As a result, 4 out of 624/01/00 14:26 G. Van Hooydonk 20potential energies in Hamiltonian (13a) change sign (are switched), if charge inversion, atom handedness or atomic chirality in one atom or boson is introduced (Van Hooydonk, 1985). We get HXY =(½av2 - e2/R1a +½bv2 - e2/R2b) + (-1)t (- e2/R1b - e2/R2a + e2/Rab + e2/R12) = HX + HY + (-1)t V(R) = H X + HY ±V(R) (14a) with the switch t, as in (5), the type of interaction determined by charge inversion in X or Y. But (14a) can not simply be generalised as in (9a) or (11). For writing down a system like (9a) or (11) the real asymptote C = V() - V(Re) is needed, see (8). This asymptote derives from the internal mechanics of the system. The dissociation products of systems obeying Coulomb's law must be charged particles like ions, not neutral species like atoms. The asymptote of a Coulomb system can therefore never be equal to D e, unless, during the long-range interaction, the 'ionic' interaction transforms smoothly into an atomic interaction. If so, the shape of observed PECs and the curvatures must contain a (hidden) message that this conversion has taken place. Evidence points out that the true asymptote can be more than 5 times as large than D e, see Section 2.1. This rather extra-ordinary scaling hypothesis, stating that only an ionic Coulomb asymptote e2/Re is effective for scaling has been detected recently (Van Hooydonk, 1999). Only V(Re) = HXY (Re) - HXY () (14b) can give the real value of the asymptote. For the attractive branch of a molecular PEC, the conventional value for w (m) would be W(R)/D e = w (m) = 1 – R e/R = 1 – 1/m (14c) which proves to be fallacious (see above and also Fig. 1b). But an intermediate ( virial) asymptote, situated at about half the well depth (in any case larger than D e), gives W(R)/IE X = w (m) = 1 – R e/R = 1 – 1/m (14d) This fits in a generic Coulomb scheme and leads to different prospects (see Section 2.1, middle PEC in Fig. 1). Unfortunately, it is difficult to imagine, given the complexity of the H-L theory, that this extremely simple static Coulomb picture (14d) could ever be applicable to the internal dynamics of real molecules and would be the basis of chemical bonding. Nevertheless, it gives us a hint for a universal potential fitting in a Coulomb model with gauge symmetry, in a form ± (1-1/m) we wanted and roughly in accordance with observation, see Fig. 6d. When t = 1, the Hamiltonian (14a) can directly be rewritten, if the small perturbation is neglected, as HXY = HX- + HY+ - e2/R12  - (IEX + EAX)+ 0 - e2/R12 (14e) the simple one-term generic Coulomb solution, nearly conforming to (9a) or (11). We first discuss some implications of (14e) and of intra atomic charge inversion in the Hamiltonian. 8. Reducing the 10 term four-particle Hamiltonian to a Kratzer potential, a Bohr-like equation and a Coulomb potential Formally in both AIM and IIM approaches, the singlet Hamiltonian with charge inversion t = 1 (14a) has a generic advantage over (13a), since the 4 nucleon-lepton interactions pair-wise get opposite signs instead of being attractive all four (Van Hooydonk, 1985). Unfortunately in the covalent (AIM) approach , the 2 nucleon-24/01/00 14:26 G. Van Hooydonk 21lepton pairs refer to different domains. For large R, R 1a << R1b and R2b << R2a, so they can not cancel for all R. Cancelling these terms is only allowed at R = R e. In the ionic approximation (IIM), the switch t = 1 has a different effect H = HX+Y- = (½av2 + ½bv2 - e2/R1a + e2/R1b - e2/Rab) - e2/R2a + e2/R2b - e2/R12 = HX+ + HY- - V'(R) (15a) the basis of (14e). The advantage of using an ionic instead of a covalent model now shows, since R 1a = R1b and R2b = R2a for all R . The 4 nucleon-lepton interactions cancel pair-wise for any R in an ionic model. This leads to a first simplification of Hamiltonian (15a) to four terms, applicable for all R H = (½av2 +½bv2 - e2/Rab) - e2/R12 = -IEX - EAX - e2/R12  Hpos - e2/R12 (15b) The 3 terms between brackets correspond with the anion, a positronium-like system with Hamiltonian Hpos. The last term is the nucleon attraction, a protonium-like system. This is the ionic Coulomb interaction , we needed to construct shape invariant observed PECs. We get H - Hpos = - e2/R12 = V(R) exactly of the Coulomb form required in our generic model, see section 3, if the small V 12 is neglected. In general, four-particle systems are supposed to be stable (Richard, 1992, 1993, 1994, Abdel-Raouf et al, 1998), although trying to solve these systems requires a number of hypotheses, one being the dissociation products (the asymptote) of the system. A B-O approximation in H 2 with charge inversion is simply a positronium-protonium system (15b) or a hydrogen- antihydrogen system. If our derivations are valid, we can easily compute a PEC for this system, see below. Depending on the (unknown) equilibrium geometry of the 4-particle system, different static cases can be distinguished one of which was already discussed by Luck (1957) without using charge inversion. If Rab is perpendicular to R 12, (15b) is in first order equal to -0,5IE H - e2/R12, the 1st term being the Bohr- solution for a positronium-like system Hpos (the reduced mass for positronium is m e/2). In any case, this is a species independent (universal) molecular 1/R potential , having R e = 2rB if rB is the Bohr-length, the asymptote being 0.5Ryd, about 54800 cm-1. This is of the correct order of magnitude for bonds between monovalent atoms, see Section 2.1. In this model, the two leptons in the positronium part would act as a classical Watt- regulator on the protonium system. If Rab is parallel or antiparallel to R 12, we leave the B-O-approximation and get, under the conditions above, that H = (½m av2 +½mbv2 - e2/Rab) + (½m 1v2 +½m2v2 - e2/R12) an explicit positronium-protonium system in the case of hydrogen. If R12  Rab, we can substitute this 2-component system with two hydrogen-like systems and get H  2(½mev2 + ½mpv2 - e2/R) (15c) if me is the electron and mp the proton mass. Deviations occur from the value of 2 in (15c), if the static alignment of Rab and R12 is parallel or anti-parallel. It is also different, for intermediate cases. Nevertheless, R 12  Rab is a reasonable assumption and both will be equal to about 2r B. The sequence of the particles -alternating or not- is important in a not perpendicular arrangement. These are all classical situations (Luck, 1957) but we can never solve this 4-particle system exactly. For a positronium and a protonium (Kong and Ravndal, 1998) system, the reduced mass equals m/2. This charge-mass symmetry will be broken in the two hydrogen-like24/01/00 14:26 G. Van Hooydonk 22systems, where a B-O approximation reappears. The leptons in (15c) are to be redistributed between the nucleons. If applicable, (15c) reduces to only 2 terms H  2(½v2 - e2/2r) (15d) which, as in Bohr's theory, can be solved, if these would be central force systems. Since this is not certain at all, hydrogen-like systems can be expected to obey the Planck- Bohr quantum condition, meaning that ½ v2 in (15d) varies as B/R2. Instead of (15d) we now get (Van Hooydonk, 1985) H  2(-e2/R + B/R2) (15e) nothing else than the empirical Kratzer potential (2b) in Section 2.2. The first derivative of (15e) at R = R e yields B = e2Re/2. This leads to the reduced molecular form (2c) of the Kratzer potential for A = 2 , an extremely useful molecular potential (Van Hooydonk, 1999). The result is the Kratzer PEC W(R) /e2/Re = (1- R e/R)2 = k2(15f) These derivations show why the Kratzer potential can be useful for both atomic and molecular spectroscopy , see Section 2.2. In an ionic charge inverted model, the 2-term Hamiltonian (15d) is a good theoretical approximation for the original classical 10-term four-particle Hamiltonian. The Kratzer potential (15f) can safely be applied in the whole range 0 < R < . Its use is not restricted to Re, where an an-harmonic oscillator results, see Fig. 3a. The strange thing is that a near Bohr solution (15d) for atoms produces an ionic Coulomb solution with asymptote e2/Re for molecules in (15f). We do not need perturbation for the Kratzer potential which produces the required shape for a PEC, see above. These results were presented some time ago in a different version (Van Hooydonk, 1985). Reminding that R e  2rB (see above), the universal asymptote in this case obeys C = -H  -2(- 0,5e2/2rB) = e2/2rB = IEH = 13,595 eV = 109600 cm-1 (15g) or 1Ryd, see (1d), which, as in the perpendicular case above, can never be equal to D e. For 1 , this is 116400 cm-1. If this universal asymptote has any meaning, it must appear when scaling PECs. Nevertheless, the m-dependence in a Kratzer potential is quadratic and different from that in a gauge-symmetry based generic scheme, where it is linear. Using (14e) and (15g), the generic solution from the total well depth of bond XX is HXX(Re) = -(IE X + EAX + e2/2rX) = - (2IE X + De) = -(IEX + EAX + e2/Re) and, without perturbation , HXX(R) - HXX(Re) = e2/Re((1-Re/R)2)1/2 = IEX((1- 1/m)2)1/2 = k (15h) independent of D e, as required and consistent with guess (14d) but different from (15f). Moreover V12 = e2/R2a - e2/R2b(<< e2/R12) (16) is the important perturbation term, exactly as required, see (13e) and this is in line with the ionic charge inverted Hamiltonian. All conditions to apply (12b) are now satisfied. But it appears we found not one but two generic solutions: (15h) a Coulomb solution varying as k = (1-1/m) for which perturbation is required, and (15f) a Kratzer solution varying as k2 =(1-1/m )2 for which no perturbation is needed, except at the crossing with the atomic dissociation limit. The difference in the physics of the two model potentials has to do with the 'central force' character of a 4- particle system. With k2 two central force subsystems appear centred on each of the nucleons. With k, a central24/01/00 14:26 G. Van Hooydonk 23force system is assigned to only one nucleon, i.e. in the anion. The k2-potential is closer to the H-L theory than the k-potential, which relates to simple hyper-classical ionic bonding. Charge inversion (a generic switch in the H-L Hamiltonian) leads to a straightforward introduction of polarisation effects as the major contribution to V 12, since the leptons have opposite charges. Without charge inversion polarisation occurs only between nuclei and leptons, since the leptons have equal charges. This section shows that even a static Coulomb based generic model (15b) or (15e) can theoretically be relevant for chemical bonding. It justifies part of the criticism given above on the H-L theory. This theory does not contain any well behaving empirical 1/R potential out of the many available in the literature. The relatively small but fundamental adaptation (charge inversion, chiral behaviour of a two- fermion atomic system) opens the way for really getting first hand theoretical, generic, approximations from parity-adapted Hamiltonians in perfect agreement with these earlier empirical findings about 1/R-potentials. This result is consistent with spectroscopic evidence offered by the generalised Kratzer-Varshni potential (2c) and by 1/R potentials (Van Hooydonk, 1999). Here the H-L theory does not give an explanation, on the contrary: it immediately led to the total rejection of ionic bonding models, the very basis of the Coulomb k-potential. The last attempt to explain chemical bonding with electrostatics was given by Luck (1957), more than 40 years ago. But if the present theoretical derivations are confirmed by experimental RKR-curves, it seems that even an hyper-classical static Coulomb law is theoretically capable of coping with PECs for the so-called dynamic process of bond formation and if so, the scaling problem is automatically solved (see Fig. 6a-c). Both solutions (15f) and (15h) even indicate that a zero molecular parameter function can exist: this means that, theoretically, molecular PECs can be constructed just using atomic data. Then the PEC of H 2 must have a connection with the Rydberg, the connection being of universal form k = (1-1/m) or k2 = (1-1/m )2. When confronting these derivations with experiment, the number of parameters required for fitting the data will determine the ab initio character of the final result. 9. Applying the non-crossing rule at the minimum for states governed by 1/R potentials. Perturbation. Consequences for the asymptote In order to perform the calculations, we first need to identify the perturbation for the k-potential. For Kratzer's solution (15f), no perturbation is needed, unless for the crossing with the atomic dissociation limit. The generic 1/R solutions, (14e) or (15h) can now be used in combination with the generic perturbation (13e) or (16). Reminding (1e), this is still a one-term solution for V(R). To avoid crossing, we need the sum and the difference of two diabatic potentials W1 + W2 = -e2/R12 + C +e2/R12 - C = 0 (17a) W1 - W2 = 2C(1 - R e/R12) = 2C(1 - R e/R) = 2C(1-1/m) (17b) with R12 = R. The classical result for the non-crossing rule gives two adiabatic potentials W(R)/C = w (m) = ±((1 - 1/m) 2 + (V12/C) 2) 1/2 (17c) with V12 a small perturbation (13e) or (16) and used already above, see (9a), (12b) and Fig. 3a. The perturbation in (17c) affects the asymptote. But the most important effect of perturbation around the minimum is that it affects curvature at the minimum (the force constant) and the curvature of the branches, see Fig. 2 and24/01/00 14:26 G. Van Hooydonk 243. In this respect, only perturbation would solve some of the difficulties met above, see Fig. 1 and Fig. 6a-c. In any case, with V 12 small and constant, symmetry will always be broken. The perturbation must be confined to the region where the two attractive t - g = 1 curves intersect, the minimum. 9.1. Symmetric 1/R potentials at the minimum Around the minimum, a generic Coulomb solution 1-1/m can be expanded as W(Re)/C = |k| = |1 - 1/m| = |1-1/(1 + xp)| (18a) if m = 1 + xp with x > or < 0. This secures that the PEC remains shape invariant and (quasi-) harmonic around the minimum depending on the analytical form of x. By definition, xp must be small, when it remains coupled to W(R)/C as a scaled unit of energy for the system. It must also be related to a scaled difference in a distance away from the minimum |R- Re|, see (1f) in Section 2.1. For scaling this distance, the variables (1g) or (1i) are available, related as k = d/(1+d) or d = k/(1-k) (18b) since Re/R = 1/m = R e/(R - Re + Re) = 1/(1 + d). The transition from variable k to d causes algebraic effects not visible when using 1/m = R e/R but we do not discuss these here. In general |R - Re| = Re|d| = R|k| (18c) irrespective of the functional dependence of xp on exponent p. Using k or d for x, expansion gives W(Re)/C = |xp /(1 + xp)|  |xp (1 - xp …)| = | xp - x2p +...| (18d) This leads to the Dunham expansion if |x| = |d|, see also the general discussion in Section 2.1 and (1f). But the |x| = |k| solution is equally valid and probably even better, since it is asymmetric again and will respect the left- right asymmetry at the minimum more explicitly. If (18a) is expanded in function of k, this leads to self- replication (continued fractions) and to all consequences thereof ( Fibonacci-like series, chaos/fractal behaviour, Freeman et al., 1997). Continued fractions have been discussed recently by Molski (1999). Since an expansion of W(Re)/C in (18d) must give positive values by definition, the minimum value of exponent p should be 2 in a non algebraic context. This is always a quasi-harmonic dependence on (m-1 )2 or an an-harmonic dependence on (1-1/m)2, in both cases an oscillator presentation confined to m = 1, the only case where |x|= |d| = |k|. A solution with k is valid for all R. A perturbation, confined to the minimum, can be supposed to vary in an oscillator mode. With the exponent p = 2 and using (18d), we can write V12 (R) = V12(Re) (1 - x2/(1 + x2)) = V12(Re)/(1 + x2) (18e) x = k = (R-R e)/R = 1 - 1/m (18f) This relation has the disadvantage of not converging to zero for large m. It gives 0,5 V12(Re) at infinite separation, and normalisation is required. Here, we use a simplified expression V12 (R) = V12(Re)(1 - x2)= V12(Re)(1 - k2) (18g) deriving from (18e) in first order, which vanishes at k = 1. The value of the exponent p in (18c) remains somewhat arbitrary: maybe it is 2 but it could be any other integral. We remark that both (18e) and (18g) for the perturbation are still closed form analytical functions. In an algebraic context , as with a pair of generic Coulomb potentials, the p = 2 case is not the only one possible. In the Landau- Zener formalism (Landau, 1932, Zener, 1932), the non-crossing rule leads to avoided24/01/00 14:26 G. Van Hooydonk 25crossings away from the minimum. Applying it at the minimum, where the intersecting states W(R) and asymptote are almost perpendicular is allowed, see Fujikawa and Suzuki (1997). The behaviour of k, k2 and d2 for the leading terms is illustrated in Fig. 8. The H2-RKR data points are included (level energies scaled with the Dunham asymptote, 78580 cm –1). The leading term in the Dunham variable d2 is completely wrong (see also Fig. 7) although this is exactly the oscillator representation used so frequently in theoretical physics ( Witten, 1981, 1982). The unperturbed Coulomb potential using |k| is not performing well either (see Fig. 1 and 7). The Kratzer potential leads to an acceptable result. The perturbed Coulomb potential is calculated with a perturbation of 0,35C (see below). In comparison with Fig. 1a, the progress is considerable, qualitatively and even quantitatively . 9.2. Intersecting states. The asymptote problem. Varshni’s fifth potential. The attractive branch of W(R) operates between C and 0. A singlet Kratzer state C(1-1/m)2 = Ck2 can never cross a generic singlet state C(1-1/m) = Ck starting from the same asymptote. Their 'crossing' points are m =  and m = 1. The m-values for intersection points mb with a fixed asymptote below C, say Cb = C/b, can easily be calculated. With b > 1, i.e. lower lying asymptotes, such as D e, are generated. Kratzer’s potential gives Ck2 = C/b, with an intersection point mb = b/(b -1), a generic potential gives m’b = b/(b-1). Their ratio m’b/mb = b/(b+1) shows that the generic potential will intersect a lower asymptote Cb closer to the minimum than m’b < mb for all b. Theoretically, it is possible that the generic asymptote is not identical with Kratzer's. Then, crossing between the two states can occur away from the minimum. But this only applies when the generic asymptote is lower than Kratzer’s. In this case, the intersection points are obtained using Ck2 = (C/b)k, which is equal to Ck = C/b = Cb (19a) and gives the same solution m’b as for the generic unperturbed potential crossing an asymptote Cb. The other case where intersection can occur is with a perturbed generic potential. Let both generic and Kratzer potentials start at any asymptote Cb. Then W(R) = Cb|kq| (19b) The generic potential, q = 1, starting from the absolute well depth (see section 2.1) is generated with b = 1/2. The original Kratzer potential (2c), q = 2, starts from the ionic dissociation limit would with b very close if not identical to 1. Next, applying a constant perturbation C/b for a generic potential starting from an asymptote C gives, using (17c) Wq(R) = C((k2 + 1/b2)½-1/b) (20a) If b is large, 1/b can be subtracted to obtain zero at m = 1, to account for the asymptote shift C – V 12 = C(1- 1/b). Without perturbation , the attractive branch obeys W(R) = C (1 - 1/m) - C/b = C (1 - 1/m - 1/b) = C(1 - (m+b)/mb) (20b) a simplification of (20a), i.e. neglecting powers of b higher than 1, but this does not lead to the smooth curvatures at the minimum needed (see Fig. 6a-c). The Kratzer result (15f) can be factorised as24/01/00 14:26 G. Van Hooydonk 26W(R)/C = (1-1/m)(1-1/m) (20c) Let a generic potential start at a reduced asymptote C(1 - 1/b) = C – V 12 leading, without perturbation, to W(R) = C(1 - 1/b)(1 - 1/m) = C(1 – (m+b)/mb + 1/mb) (20d) as illustrated in Fig. 6a-c. If the reduced asymptote C(1 - 1/b) is a running asymptote, i.e. if b is a variable just like m, the result is a potential bearing a striking resemblance with Kratzer’s reduced potential (20c), whereby the starting value of asymptote C remains unchanged. This is a similar picture as that obtained in (18d). Nevertheless, (20b) and (20d) can both be of help in interpreting the Varshni exponent v in (2d). In the present model, a running asymptote means it is not necessary to apply a perturbation due to lower asymptotes. The result is W(R) = C(((1-1/b)(1-1/m))2)½(21a) No normalisation is required and this potential is a pseudo generalised Kratzer-Varshni potential. However, Fig. 6a-b show this does not lead to the correct curvatures around the minimum. Allowing for an m-dependent perturbation V 12 (18g), means (20a) must be replaced by W(R) = C((k2 + ((1/b)(1 - k2))2)½-1/b) (21b) with the exponent in (18a) p = 2. Fig. 9 shows how perturbed generic potentials (21b) behave with respect to the generalised Kratzer-Varshni potential (2d), known to be consistent with the spectroscopic constants (Van Hooydonk, 1999). Three v-values, equal to 0.7, 1.0 and 2.0 are used with an asymptote C/2 and these potentials are compared with a perturbed generic scheme based upon the asymptote C, using (21b) with b-values: 1,5, 2, 3, 6, 10 and infinity. The case v = 0.7 typically applies to homonuclear bonds, whereas v = 2.0 and larger applies to heteropolar bonds. For v = 0.7 (covalent bonds), the generalised Kratzer is almost identical with the generic state with b = 3/2. The original Kratzer state is exactly the same as the generic perturbed by its virial asymptote with b=2 (see also below). For v = 2, the attractive side coincides with the generic b = 4 state up till m = 2. Large b-values will give the generic potential almost the shape of a Born- Landé potential or, in the limit, the Coulomb potential again. The greatest deviations are found at the repulsive branch, but these are difficult to show graphically. The efficient generalisation of the Kratzer potential, suggested a long time ago by Varshni (1957) turns out to be an elegant artefact to reproduce perturbed generic 1/R potentials if these would occur in practice , which remains to be determined. Conversely, extrapolating these results to observed PECs, Fig. 6-7 and the closeness of the two kinds of potentials show that generic 1/R Coulomb potentials can most probably cope with observed molecular PECs. In fact, Varshni’s potential (2d) leads to very consistent results for hundreds of bonds (Van Hooydonk, 1999). The intersection of generic and Kratzer states with a lower fixed asymptote, such as the atomic dissociation limit De is also of interest, given the developments in femto-chemistry in particular. The perturbation around the critical distance being small and confined to that region, we get WPEC(R) = ½((W(R) + D e) – ((W(R) – D e)2 + V’2(R))1/2) (22) with W(R) given by (21b) for the generic function or by (20c) for Kratzer's.24/01/00 14:26 G. Van Hooydonk 27Even a crude approximation V'(R) = 0 for (22) will not influence the result drastically, provided W(R) is accurate enough, especially at long range. If valid, this justifies our previous conclusion that the usual constraint for a molecular function W() = De can even be disregarded (Van Hooydonk, 1999). Femto-chemistry could provide us with information about that specific region where an ionic potential crosses the covalent asymptote. We consider this asymptote as fixed in a good first approximation since the Coulomb interactions between neutral atoms are not based upon a strong e2/R law but refer to smoother polarisation terms (Aquilanti et al, 1997). Covalent interactions can be described as a function of atomic polarisabilities, having a stabilising effect only between the long-range contours (higher turning points) of an ionic Coulomb PEC (see for instance Aquilanti et al, 1997, especially their Fig. 1). We therefore expect femto-chemistry applied to the critical distance of covalent bonds (crossing of ionic X+X- and non-ionic XX curves) can provide conclusive evidence for a generic static bonding scheme, see also below. The situation is not that simple however: finding a universal potential is essential for understanding the behaviour of interacting particles at short- and long-range. Accurate long-range potential energy curves are extremely important for studying ultra- cold atoms and related phenomena ( Zemke and Stwalley, 1999, Wang et al., 1997, Marinescu et al., 1994, Stwalley and Wang, 1999). In this respect, the phenomena at the 'critical' distance are themselves also critical. We will show below that this critical region is almost perfectly scalable in the relatively small portion of the total PEC, i.e. around 10 to 20 % from the crossing point. 10. Generic low parameter universal Coulomb function. To finalise quantitatively the Coulomb scheme and to conserve its ab initio character, non-empirical generic perturbations or b-values in (21b) are required. Two such solutions are easily found. (a) First (21b) may be rewritten as W(R)b/C + 1 = (1 + k2(b2-2) + k4)1/2(22c) With b = 2 or V12(Re) = C/2, the virial asymptote is the generic perturbation. The first generic result is therefore, as expected, the Kratzer potential 2W(R)/C = k2(22d) see (15f). This is consistent with the Hamiltonian, see section 8 and already illustrated in Fig. 9. Applying a perturbation equal to the virial asymptote C/2 to a Coulomb 1/R potential leads to the (quadratic) Kratzer potential. This is not really a surprise when looking at Fig. 3a and 5. The gap is equal to 2C and covers the two worlds by definition. The Kratzer potential is confined to half the total gap or C, i.e. the positive world. For an attractive Coulomb potential to stop at the zero energy in this picture, it is necessary that it be confined to C. All this derives from classical gauge-symmetry. Replacing W(R) in (22d) by Kk2, where K is the Kratzer asymptote, leads to K = C/2 (22e) suggesting that in this case C, the generic asymptote, is the absolute well depth of a bond, the total gap. This is in agreement with an earlier guess (14d). Also, this result makes it quite acceptable that the most probable value of the exponent p in (18c) is 2, see section 9.1.24/01/00 14:26 G. Van Hooydonk 28 (b) A second generic b-value is obtained by equating the perturbed generic and Kratzer potentials. This unusual procedure corresponds with a redistribution of the central force character within the different subsystems of a four-particle system (see above). If a non-trivial solution can be found, this leads to generic b- values in terms of K and C. If P is the perturbation, we get Kk2 = (C2k2 + (P(1 – kp))2)1/2 – P (22f) This leads to P = C/b =|C - K| for k = 1 and to a trivial P = P for k = 0, irrespective of the value of the exponent p in (18b). Working (22f) out analytically gives a relation between P and the asymptotes C and K. This implies a generic value for the exponent p also. We rewrite (22f) as a quadratic relation in P P2kp(kp - 2) - 2PKk2 -K2k4 + C2k2= 0 (22g) After dividing by non-zero k2, the solutions P = (K/(kp-2(kp-2))(1 ± (1 - (kp-2(kp-2)((C/K)2 – k2))1/2) (22h) are obtained. For k = 1, one solution is like C – K above. But a generic non trivial solution also results when the exponent p = 2 , since then kp-2 = k0 = 1 for any k or P = (K/(k2-2)(1 ± (1 - ((k2-2)((C/K )2 – k2))1/2) (22i) When C = K, this gives the non trivial P (or b) solution P = (K/2)(-1 + (1 + 2(C/K )2)1/2 = 0,5K( 3 – 1) = 0,366025 K (22j) and solves the problem about truly generic perturbations or generic b-values, which are found to be equal to 2 and to 2,7321 (close to e). For k = 1, (22i) leads to P = -K(1 ± C/K) (22k) or P = -K - C or P = C - K as above. The second generic, i.e. Coulomb, solution is now found to be w(k) = W(R)/C = (k2 + (0,366025(1-k2))2)1/2 – 0,366025 = kgen (22l) if both the Kratzer and generic potential refer to the same asymptote C. The only variable in (22l) is a number 1/m = R e/R and therefore the RHS of (22l) can be called a generic species independent universal variable kgen. But W(R) in (22l) is always below the original Kratzer result and leads to a more stable system. With P(R) = P/(1+k2) instead of P(1-k2), section 9.1, the same value 0.366025 is obtained. Reminding (1e) and (1f) and our choice for a single term in (1e), a simple Coulomb potential produces, unlike the Dunham expansion, closed formulae for PECs: the Kratzer potential k2 and a more complicated one kgen (22l), still a closed formula. For larger b-values (small perturbations) no generic solutions are available. For these ionic cases, empirical approximations will have to be used to account for the PECs, if they do not obey a generic w (m) solution. If the virial asymptote acts as a constant perturbation for W(R), a normalised asymptote 5/2 appears, indicating a possible interference of Euclid's golden number (Van Hooydonk, 1987) and, eventually chaos/fractal behaviour, suggested by Freeman et al. (1997), see also above. Finally, taking into account the -constant- atomic dissociation limit D e a theoretical PEC obeys PECtheo = 0,5(W(R) + D e) – 0,5((W(R)-D e)2 + (V’’(R))2)1/2(22m) like (22), with W(R) either equal to the Kratzer (22d) or generic potential (22l). In this report we test (22m) mainly in the hypothesis V’’(R) = 0, see Section 9, a reasonable working hypothesis. Only experiment will decide if these two generic functions, both depending solely on R in a closed analytical formula, correspond24/01/00 14:26 G. Van Hooydonk 29with observed PECs within reasonable limits. The constraints on PECs generated by functions (22d) and (22l) are stringent, since their analytical form is extremely simple. Both are, at this stage, one parameter (R) functions. Just one molecular parameter Re is needed if this can not be computed from atomic data. Smooth transitions from a two central force systems (k2) to a one central force system ( kgen) approach in function of k and (1-k) are easily made and can be written down using the equations above. Nevertheless, at this stage we leave out computations based upon hybrid potentials, which will always give results very close to the starting potentials. The benchmark, Morse's, is in Fig. 1c-d. Apart from reducing all 13 PECs to a single straight line, the ultimate challenge is to calculate reasonable PECs with a zero molecular parameter function. 11. Results and discussion The 13 PECs (RKR or IPA) used are for the bonds H 2 and I2 (RKR, Weismann et al., 1963), HF (RKR, Fallon et al., 1960, Di Leonardo and Douglas, 1973, IPA by Coxon and Hajigeorgiou, 1990), LiH (IPA, Chan et al., 1986), KH (Hussein et al., 1986), AuH (PEC data by Le Roy's method, Seto et al., 1999), Li 2 (RKR, Kusch and Hessel, 1977; IPA, Hessel and Vidal, 1979) , KLi (Bednarska et al., 1998), NaCs (Diemer et al., 1984), Rb 2 (Amiot et al., 1985), RbCs (Fellows et al., 1999), Cs 2 (Weickenmeier et al., 1985) and a theoretical PEC for LiF (Padé approximant, Jordan et al., 1974, Kratzer-Varshni-type, Van Hooydonk, 1982). We will use here the Morse PEC for LiF instead. Some of these data were used above for Fig. 1b-d. We did not always use all published IPA data available, which are sometimes very detailed. The number of data-points is over 500, including the 13 minimum values, or about 40 per bond . These date have been used in constructing (part of) the Fig. 1c-d and 6d. 11.1. The asymptote problem Seven asymptotes are available to interpret PECs using a universal function f(R). For a bond XX these are: 1. the absolute asymptote (absolute well depth), covalent approximation (AIM): C C(abs) = 2IE X + De; 2. the absolute asymptote, ionic approximation (IIM): C I(abs) = IE X + EAX + e2/Re; 3. the generic asymptote G = e2/Re (C in this text). For homonuclear bonds, R e is equal to 2r X, where rX is the covalent radius of atom X (see above). If true, asymptotes 2 and 3 are available from atomic data and would not be molecular parameters ; 4. the ionic asymptote I = e2/Re = IEX – EAX + De, nearly equal to G in a reasonable first approximation, although this is not a priori certain (Van Hooydonk, 1999). The main difference between G and I is the dependence on De, which is to be avoided for a generic solution. Asymptote 2 is equal to asymptote 3 only if EA X = De (Van Hooydonk, 1982, 1999). In this case, all molecular asymptotes 1-4 for a bond XX would be available from atomic data X, IEX = e2/Re and EA X = De; 5. the atom (species independent) Coulomb asymptote in terms of the internuclear separation (in Å) and invariantly equal to 116431 cm–1, the basis of 4 and 5 and corresponds with 1 Ryd; 6. the covalent asymptote, the atomic dissociation limit D e (apparently the first real molecular parameter met unless EA X = De) and finally 7. the Dunham-asymptote or the first Dunham-coefficient: A = a 0 = 2 e/4Be = 0,5keR2 e deriving mathematically from the zeroth order spectroscopic constants e and Be. For the molecules H 2, HF, Li 2, LiH and LiF data collected earlier for asymptotes 3, 4 and 6 (Van Hooydonk, 1982) are now completed with asymptotes 1, 2, 5 and 7 and are given in Table 1. Only the Dunham asymptote24/01/00 14:26 G. Van Hooydonk 30is confined to the minimum in a specific mathematical way. Species dependent asymptotes show large divergences. Dunham's asymptote values are strange: they are either below (H 2), almost equal to (Li 2, LiH) or above the generic/ionic one (HF, LiF). There is no regularity in asymptote ratio’s either. This species dependence shows why it is so difficult to find a universal function f(R). Table 1. Six species dependent asymptotes (in cm-1) for five bonds BondR(Å) Cc = IEX + IEX + DeCI = IEX + EAX +e2/ReG = C = e2/ReI = IEX - EAX + DeCovalent D eDunham A H20,7414 257584 272264 156453 141878 38283 79580 HF0,9168 300041 295090 126526 131633 49406 204308 Li22,6729 95558 94991 43399 46909 8612 45902 LiF1,5639 232580 242674 74175 64173 48122 154008 LiH1,5957 173415 188404 72695 57681 20292 65747 The 7 asymptotes, all candidates for scaling RKRs, each have their own pros and cons. The data in Table 1 illustrate why the asymptote problem had to be discussed in detail, especially in connection with scaling semi- empirically constructed RKRs. If D e would be the unique reference asymptote, w(m) should vary between 0 and 1 invariantly for all bonds. Using larger asymptotes X compresses the maximum value of 1 to D e/X, leading to different slopes and curvatures, as outlined above. 11.2. PECs for H 2, Li2 and Cs 2 from a zero molecular parameter function If the generic solution that the ionisation energy of an atom X generically determines the molecular PEC of the X2 molecule is true (see Section 8), it must be possible to calculate X 2-PECs with a zero molecular parameter function just using atomic X-data. If moreover D e is equal to EA X, the complete PEC becomes available, including the dissociation limit. If the present scheme is really universal and of first principle's nature, it must apply to the simplest bonds H 2, LiH and Li 2 in the first place. This sounds impossible, given the complex procedure to get solutions for the 4-particle Hamiltonian in the H-L theory. The case of H 2 is a standard example, see above. The Li 2 molecule has been studied extensively in the past as it is the lightest molecule in the Periodic Table after H 2. A review on theoretical and experimental studies on Li 2 is given by Hessel and Vidal (1978) of interest also because of the convergence problems with the Dunham expansion. The earliest attempts for understanding bonding in this molecule go back to Delbrück (1930) using the H-L method. Bond LiH is treated in the next Section 11.3. The atomic data are IE H 13.595 eV and IELi 5.3917 eV, giving theoretically Re(H2) = 1.0572 Å and R e(Li2) = 2.6731 Å. With these atomic values, the Kratzer potential predicts PECs obeying 13.595(1-1.0572/R )2 and 5.392(1-2.6731/R)2. The same asymptotes are used for the generic function kgen. The theoretical PECs deriving from these atomic data are presented in Fig. 10a for H2 and Fig. 10b for Li2 and Cs2 with R/R e on the x-axis to make the minima of theoretical PECs and RKRs coincide. The curve for Cs 2 (IECs 3,8939 eV) is included in Fig. 10b, since, unlike the RKR for Li 2, the turning points go near D e. The mean % deviation for all 108 turning points is 11.4 %. In these very important 'simple' cases, the agreement between the zero molecular parameter and observed PECs is rather astonishing. For H2, Fig. 10a, the largest deviations are found at the repulsive branch, where experiment shows that the asymptote generated by the present method is close to the ionisation potential of H (see below, Table 3). We24/01/00 14:26 G. Van Hooydonk 31wonder if these RKR turning points are not in need of revision. For the attractive branch, the agreement is astonishingly good also at long-range close to the dissociation limit, which intersects the theoretical curve. This simple first principles 'atomic' PEC for H 2, deriving from Kratzer's generalisation of Bohr's formula, is much closer to the observed one than the PEC originally calculated by Heitler and London (1927). Exactly this poor H-L PEC for H 2 is at the origin of quantum chemistry, as we know it today, see also Introduction and Pople (1999). For Li2 and Cs2, Fig. 10b, the agreement is better at both branches. Here the 'atomic' PECs are available up to infinity as alkali-metals have electron affinities very close to their D e-values. Quantitative details for Li 2 are given below. Before discussing the intermediate case of LiH, results of the same quality are obtained for all bonds between elements of the first Column in the Periodic Table if we use (IE X+IEY)/2 as a first order approximation for the asymptote of an XY bond. In Fig. 10c , observed level energies are plotted versus those obtained in this zero molecular parameter approximation for 8 bonds Li 2, LiH, KH, KLi, NaCs, Rb 2, RbCs and Cs 2. The differences in cm-1 are also shown. Only the repulsive branch of KH deviates from the general trend. The slope is close to unity and the goodness of fit is relatively high as indicated in Fig. 10c. The average deviation for the 310 turning points of 8 bonds is 9,42 % (for Cs 2 11,3, RbCs 10,8, Rb 2 9,0, NaCs 12,2, KLi 9,3, KH 7,5 and LiH 6,3, including long-range situations where applicable). Table 2 gives the details of the results for Li 2. The Kratzer results are collected in Columns 2-4. The accuracy for the 30 turning points of the Kuch and Hessel (1977) RKR, calculated in this zero molecular parameter approximation is 2.5 %, impressive be it not of spectroscopic accuracy. The clearly visible inflection points are consistent with an expected crossing with the atomic dissociation limit at larger R, a feature inherent to the Kratzer potential (see above). Columns 2-4 represent molecular ab initio results acquired with atomic data only (zero molecular parameter function). In this case, the EALi value is very close to D e as it is also for several other alkali-metals. This assures the complete PEC is available from atomic data only using (22m) with EA X as a substitute for D e. We multiplied IELi with kgen for the ionic k-model and plotted the calculated levels versus the RKR. Fitting gives a scaling constant of 0,935596. This leads to the results given in Columns 5-7 in Table 2. Deviations for the zero parameter Coulomb k-function (0,6 %) are smaller than for the k2-potential (2,5 %). The corresponding G(v) curve (not shown) is within 1 % of the experimental value for the 14 levels. The balance between the levels varies between 98,9 and 100,1 %, usually a problem for Morse's potential. This analysis is consistent with the results of Hessel and Vidal (1978), who extended the range to v = 18 using the Inverted Perturbation Approach. This IPA PEC is claimed to be more accurate than a RKR (differences vary from 0,1 cm-1 to 5 cm-1 for inner- and outermost turning points). The dependence of the IPA for Li 2 on the Kratzer and generic variables leads to a goodness of fit close almost equal to 1. The asymptote obtained by fitting the data by a linear equation through the origin is 45246,2 cm-1, only 1,42 % lower than the Dunham asymptote 45902 cm-1 in Table 1. Up till R = 4 Å, the correlation between generic PEC and IPA data-points is y = 0,97993x with R2 = 0,9999156, which makes our approach reliable. Despite the fact the agreement is not exactly 100 %, the zero molecular parameter function and its ab initio character suggest that conventional inversion procedures do not imply absolute certainty about the real PEC. In24/01/00 14:26 G. Van Hooydonk 32the extreme case, we might even claim various available turning points are in error with the same absolute deviations as given in Table 2. This is certainly a matter of further research. The first hurdle for this theory and the Coulomb based function kgen, deriving from gauge symmetry has been taken: it is possible to calculate realistic PECs from atomic data only. Or, PECs show definite Coulomb behaviour, dictated by atomic parameters, in agreement with our derivations above. But this also implies that Coulomb based scaling (Fig. 6a-d) must apply to RKRs. If the present generic function indeed leads to an acceptable scaling scheme for RKRs of different bonds into a single one, preferably a single straight line, the Coulomb approach must be universally valid. This may ultimately lead to more reliable PECs obtained by inverting observed level energies. Table 2. Li2 PEC calculated (cm-1) with a zero molecular parameter Kratzer potential (columns 2-4) and a generic Coulomb potential (columns 5-7). RKR Kratzer PEC DiffAbs % Generic PEC DiffAbs % 4525 4694.8 169.8 3.75 4514,81 -9,97 0,22 4252 4390.7 138.7 3.26 4246,58 -5,15 0,12 3972 4082.4 110.4 2.78 3971,81 -0,65 0,02 3687 3770.0 83.0 2.25 3690,43 3,32 0,09 3396 3454.3 58.3 1.72 3402,80 7,00 0,21 3099 3135.3 36.3 1.17 3108,87 10,22 0,33 2796 2813.4 17.4 0.62 2808,53 12,79 0,46 2487 2488.8 1.8 0.07 2501,86 14,67 0,59 2173 2161.8 -11.2 0.52 2188,84 15,77 0,73 1853 1832.7 -20.3 1.10 1869,46 16,00 0,86 1528 1501.8 -26.2 1.72 1543,69 15,28 1,00 1198 1169.4 -28.6 2.39 1211,54 13,54 1,13 862 835.7 -26.3 3.05 872,85 10,59 1,23 521 501.3 -19.7 3.78 527,91 6,65 1,27 175 166.7 -8.3 4.75 176,91 1,88 1,07 0 0.0 0.0 0.00 0,00 0,00 0,00 175 165.7 -9.3 5.32 176,70 1,66 0,95 521 494.2 -26.8 5.14 522,00 0,74 0,14 862 820.7 -41.3 4.79 859,37 -2,89 0,34 1198 1145.9 -52.1 4.35 1189,99 -8,00 0,67 1528 1470.2 -57.8 3.78 1514,78 -13,64 0,89 1853 1794.1 -58.9 3.18 1834,29 -19,17 1,03 2173 2117.9 -55.1 2.54 2149,25 -23,82 1,10 2487 2441.9 -45.1 1.81 2460,17 -27,02 1,09 2796 2766.6 -29.4 1.05 2767,66 -28,08 1,00 3099 3092.3 -6.7 0.22 3072,15 -26,49 0,85 3396 3419.5 23.5 0.69 3374,22 -21,58 0,64 3687 3748.5 61.5 1.67 3674,34 -12,77 0,35 3972 4079.9 107.9 2.72 3973,07 0,61 0,02 4252 4414.0 162.0 3.81 4270,87 19,13 0,45 4525 4751.6 226.6 5.01 4568,35 43,57 0,96 Mean % 2.549 0,639 Although all the above results are unprecedented, the zero molecular parameter solution delivers a continuous PEC, not aware of turning points, related to vibrational levels, obeying quantum mechanics. However, Fues (1926) pointed out a long time ago how to solve the wave equation exactly for the Kratzer potential, which closes the circle. At this stage, it seems unavoidable that the nature of bonding in covalent molecules is24/01/00 14:26 G. Van Hooydonk 33basically ionic, conforming to gauge-symmetry and that PECs obey Coulomb's law, with an asymptote deriving simply from atomic characteristics. It seems all three criteria given at the end of Section 6 are met. 11.3. The procedure: LiH and the performance of the variables d, d2, k, k2 and kgen If the zero molecular parameter function works for H 2 and Li2, it must also apply to LiH, the third member of the critical series for an ab initio approach. The global data for LiH were already incorporated in Fig. 10c but we now use this bond as a test case for the general scaling procedure with various variables. The first 4 correspond roughly with the four potentials anRn in (3b) discussed above, the fifth is the generic variable (22l ). Fig. 11a gives an algebraic plot of all five variables versus the LiH-RKR, known almost to 100% of the atomic dissociation limit. We use 23 levels (46 turning points). It is obvious from Fig. 11a that a linear fit through the origin can never produce a smooth relation for d, k and even d2, the basis of conventional oscillator models (Morse, Dunham) and so widely used in theoretical models ( Witten, 1981, 1982). For the variables of the Coulomb approach k2 and kgen linear relations are detected, with slopes producing asymptote values close to the Dunham, ionic and generic asymptote values. The goodness of fit is over 0,98 for Kratzer's variable and over 0,99 for the generic Coulomb variable. This proves the present procedure extends to this important bond also (see Section 11.2). In addition, we observe diverging points at long-range (about 5) probably symmetrically followed by 5 turn over points at the short range but which are less visible. Turn over points are more clearly visible with the Kratzer than with the generic Coulomb variable. Despite this, all 46 turning points are within 1,85 % (or 0,068 ) of the RKR for the generic variable fit, which is close to but not of spectroscopic accuracy provided this RKR is correct. Fig. 11b gives the results for 18 levels and their 36 turning points leaving out extreme turning points. The asymptote for the generic variable is within 0,086 % of the generic asymptote e2/Re. In this case, the accuracy of the generic Coulomb potential model is simply impressive. Turning points are within 0,54 % of R (corresponding with an average deviation of 0,012 ) the largest deviations still being found at the long-range attractive branch. For the repulsive branch the deviation reduces to 0,0047 , even closer to spectroscopic accuracy. The quantitative possibilities of this simple non-empirical ab initio model potential (22l) are obvious. We can easily calculate the turning points from observed level energies. Due to the nature of the generic function, this will always produce a perfect balance. Since no parameters are involved, this may form the basis of an alternative inversion process. This example for LiH clearly shows that spectroscopic accuracy can be obtained with Coulomb behaviour. Apart from the theoretical consequences these results have, they also have practical implications for the determination of turning points by an inversion process. In fact, we are now at a stage where we could even suggest to revise published turning points. Mainly for practical reasons we will adhere to these published turning points and derive 'operational' asymptotes in the framework of a Coulomb based scaling scheme by using a similar analysis as the one illustrated in Fig. 11a and 11b. This means that any asymptote obeying Coulomb's law and its scaling power (see Fig. 6a-c) may be used. Asymptotes not obeying 1/R behaviour must lead to deviations in the scaling procedure.24/01/00 14:26 G. Van Hooydonk 3411.4 Determining the asymptote for PECs: the universal function, 1st scaling procedure, classical view The most direct way to find the asymptote for a universal PEC is to plot the observed energy of the levels against the variable or the function and fit the data with a linear equation as exemplified in section 11.3. This procedure is generalised in Fig. 12a for 7 bonds with smaller R e-values and Fig. 12b for 6 bonds with large Re-values. Here all RKR- or IPA-data are plotted against kgen. The 99 long-range points (at > 50% of D e) are shown but have not been included in determining the fit. The remaining 409 turning points cover about 75% of the total PEC. The asymptote values obtained are shown with the goodness of fit (typically 0.99 or better). In general these operational asymptotes are close to but not equal to the Dunham asymptotes. This can be interpreted in two ways as indicated above: either the R e values must be (slightly) adapted or the published turning points would need revision. This is a general problem when discussing PECs constructed with inverting model potentials. In the case of scaling, slight deviations are allowed provided the general trend of inversion technique used is obeyed and physically or chemically meaningful. As indicated above, we choose, mainly for practical reasons, to calculate operational asymptotes instead of recalculating all the turning points, which is mathematically equivalent. Table 3. Average operational asymptote values computed from RKR/ kgen and RKR/k2 for 407 points for 13 bonds in cm-1 (the 13 minima are not taken into account). Deviations (%) of theoretical level energies calculated with kgen and asymptotes (from graphical fit or pivot table) from observed ones. Bond Branch DataAuH* Cs2H2HF I2KH KLi Li2LiHNaCs Rb2RbCs LiFTotal attr #58 10 5 2 9 6 15 7 8 17 12 11 2162 Abs% level energy (fit)10,31 6,45 13,86 25,01 55,01 16,93 1,33 0,57 3,40 5,51 3,00 5,13 1,42 10,0 RKR/k2 156898 35997 88920 186973 250894 65543 41121 44881 65662 39064 34967 35482 156433 99833 StdDev of RKR/k2 9684 4530 2811 5521 22632 3070 1590 1226 649 2081 2004 2956 379 RKR/k gen 159633 37010 98872 199185 176613 69352 42769 46723 70995 40276 35789 36378 162090 98162 StdDev of RKR/k gen 11295 3965 8261 2151132794 432 684 272 2332 1257 1320 2255 3226 Abs% level energy (pivot) 5,86 6,94 6,61 0,76 29,85 0,47 1,34 0,45 2,64 2,65 3,08 5,00 1,41 5,55 rep # 36 28 15 5 20 14 15 10 23 26 19 19 2232 Abs% level energy (fit) 14,73 1,03 3,42 5,49 6,71 3,72 1,89 0,97 3,34 1,56 1,59 1,49 7,13 4,53 RKR/k2 204221 33542 63727 215346 382163 71635 42142 43328 60441 40997 35548 35100 144335 103976 StdDev of RKR/k2 17234 1127 4636 1192 23639 1379 904 1564 2130 1266 1297 1297 6891 RKR/k gen 206479 35888 82117 243113 389978 80054 43794 46138 71436 43112 37123 37014 152224 109487 StdDev of RKR/k gen 15942 1080 2832 15868 28935 3647 26 369 2773 440 139 593 915 Abs% level energy (pivot) 5,67 1,34 2,88 5,32 6,70 3,83 0,05 0,69 3,27 0,87 0,33 1,14 0,43 2,73 #94 38 20 7 29 20 30 17 31 43 31 30 4394 Asymptote from graphical fit175155 36038 84673 248983 390257 81088 42965 46552 73207 42457 36534 36736 163075 116580 Average RKR/k2 175021 34188 70025 207240 341424 69807 41632 43968 61788 40233 35323 35240 150384 102273 Average RKR/k gen 177574 36183 86306 230562 323761 76843 43281 46379 71322 41991 36607 36781 157157 104831 Abs% level energies (fit) 12,0 2,5 6,0 11,1 21,7 7,7 1,6 0,8 3,4 3,1 2,1 2,8 4,3 6,8 Abs% level energies (pivot) 5,8 2,8 3,8 4,0 13,9 2,8 0,7 0,6 3,1 1,6 1,4 2,6 0,9 3,9 Ratio abs % energy levels 2,1 0,9 1,6 2,8 1,6 2,7 2,3 1,4 1,1 2,0 1,5 1,1 4,7 1,7 * PEC computed by Le Roy's potential. Although this 'empirical' PEC is reproduced qualitatively in an acceptable way, the picture is slightly distorted. Further work is needed. Without these AuH-data, the absolute % deviations at the attractive side are 5,38 %, at the repulsive side 2,19 %. For the global results on all remaining data points the errors are respectively 5,14 and 3,29 %. To check the generic procedure further we calculated average RKR/k2- and RKR/ kgen-values for the same 395 data-points used in Fig. 12a and 12b. The results are given in Table 3 together with standard deviations for the asymptote values and the resulting deviations for the level energies, calculated with asymptotes obtained from the graphical fitting procedure and the pivot table results for RKR/ kgen. The average asymptote values are all24/01/00 14:26 G. Van Hooydonk 35close to Dunham's (see above). The major difference resides between asymptotes for repulsive and attractive branches. Pertinent examples are AuH, HF, H 2 and I2, the last three are older RKR-curves. Absolute deviations for level energies returned by this method are given for each bond and for each branch. Various published or computed turning points quite exactly match those predicted by our model potential (KH, KLi, Li2, NaCs and RbCs) with an error of about 1 %, which justifies our decision not to recalculate the turning points. If the published tutning points are exact, the only molecular parameter that could account for the deviations in asymptote values is R e, since e and reduced masses are well known. The R e-value is decisive for obtaining turning points. Even a very small shift in R e (1 % or about 0,01 ) can have a drastic influence upon the turning points calculated by semi-empirical approaches (RKR, IPA). This problem is discussed further below. Overall deviations are collected in the last rows of Table 3, starting with the headers Abs% level energies (fit) and Abs% level energies (pivot) . For five bonds, the average deviations are equal to or smaller than 1,6 %. The worst results are obtained for I 2 (as to be expected from Fig. 6d above) and AuH (see also footnote of Table3). The ratio of absolute deviations by the two methods is given in the last row. On average (last column), the pivot approximation (Table 3) returns level energies with an accuracy about 70 % greater than with the graphical fitting procedure. The accuracy for the attractive side (deviation 5,55 %) is less than for the repulsive side (deviation 2,73%) leading to an overall deviation of 3,9 % for the level energies in 75 % of the complete PEC for 13 bonds. In general, the differences between the averaged asymptotes in Table 3 are minor and are not of that order to influence the basic scaling results we want. The final 'averaged' value for the asymptote at the repulsive side is almost equal to the exact Coulomb asymptote (or 1 Ryd), see Table 3, last column. This is strange but conforms to the general idea about the importance of Coulomb's law in molecular spectroscopy. We refer to Fig. 6d, where this is the unique reference asymptote for all bonds at the repulsive side. For the attractive side, this standard value is reached for about 80 %, meaning either that more turn over points are present or that the turning points at this side are too large in general. These results seem to point out that, given the definition and the correctness of the Dunham asymptote values, the published turning points of some RKRs may indeed be in need of revision. Finer details for the short range behaviour are observed, i.e. turn over points, in agreement with the analysis by Zemke and Stwalley (1999) in the case of NaK at v=60 near the dissociation limit. Having at our disposal a 'generic' function, it is straightforward to detect turn over levels. The empirical function of Gordon and Von Szentpaly (1999) claims a high accuracy for repulsive branches of PECs but it does not detect these important turn over points at short-range. This is certainly a topic for further investigation as similar effects must also be visible at the attractive branch, where long-range behaviour appears (see Fig. 11a for LiH). For the long-range, deviations set in at a particular R-value, maybe around the critical distance or at the Le Roy radius, which we could now only empirically estimate with reference to the goodness of fit in the asymptote finding process. Fig. 13a gives a graphical illustration of all the data points in Table 3. Quantitative details about the level energy accuracy are given in Table 3. At the attractive side and in cases where the RKR/IPA reaches D e, it appears that atomic asymptotes must be considered as 'deviations' from Coulomb scaling. In fact, at the repulsive side Coulomb behaviour is respected throughout in a species independent way. We can not but conclude that D e is indeed not an asymptote conforming to Coulomb scaling, as argued above.24/01/00 14:26 G. Van Hooydonk 36This is clearly visible in Fig. 13b where 506 theoretical and observed level energies are plotted against k = 1- R/Re. The atomic asymptotes appear as side branches of the general reduced PECs. We easily verify that anharmonic oscillator behaviour is nicely obeyed , i.e. a near quadratic dependence on the Kratzer variable k. Of all 13 bonds, only H 2 seems to obey Kratzer k2-behaviour at long range with the asymptote deriving from the graphical procedure. Using a higher value (closer to the atomic ionisation energy) will give kgen behaviour at this branch, see also Fig. 10a. Fig. 13b must be compared with the general scaled result in Fig. 1b to notice the effect of our procedure. The data-points for AuH (Seto et al., 1999) are hardly visible. The three coinage metal PECs AuH, AgH and CuH (Hajigeorgiou and Le Roy, 1999, Seto et al., 1999) show a very similar behaviour in comparison with the generic Coulomb variable, but are slightly distorted, especially at the minimum, where the largest % deviations are found. Since these are unusually large deviations, we left out the 100 coinage metal PEC-data for AuH, which reduces the absolute deviation to almost half of the total or 3,3 % for the 300 turning points for the remaining 12 largely different bonds. The relative large number of turning points for one bond AuH in the data range (about 100) would otherwise have distorted the whole picture for the remaining bonds (quantitative results are given in footnote of Table 3). Whether or not our theoretical conclusions given above that the atomic dissociation limit is not necessary to reveal the nature of the universal function, must become more apparent when we scale PECs with D e, i.e. test De for Coulomb scaling as in Fig. 6a-c. In essence, the results of this section are in support of the general theory outlined above, in particular with the generic and completely ab initio kgen variable in (22l) leading to acceptable results in the zero molecular parameter approach (see 11.2) even without using (or even knowing) a0. These results are in support of the intra-atomic charge inversion technique. 11.5. Effects of the first 1st step in scaling: V-shaped or linear reduced PECs coinciding at R e Up till now, scaling effects of RKRs are usually presented in the classical form RKR(R) or RKR(R/R e) as in Fig. 1b above. The algebraic linear approach is more illustrative and, to our knowledge, unprecedented. Using the generic ab initio Coulomb function, it is a simple matter to present V-shaped PECs with perpendicular legs or even linear PECs (see the Fig. 2a version of Fig. 3a). We reduce the levels with a(piv) at the y-axis and use the generic Coulomb variable kgen on the x-axis. For all 506 points for 13 bonds these V-shaped PECs are given in Fig. 14a . The agreement at the repulsive branch is again impressive, since all data collapse into a single line (left branch of the V). At the attractive side, the same effect is noticed but the various atomic dissociation limits are clearly visible, when the RKR extends to that region, as in Fig. 13b. This result must be compared with the Morse equivalent in Fig. 1c above. The consistency of our generic Coulomb function shows even better when, as in Fig. 2a, the results are converted algebraically into a linear form ( Fig. 14b ). It is not difficult (now) to ascertain that the attractive branch is simply an algebraic continuation of the repulsive branch, the essence of a Coulomb approach (see above). It illustrates the perfect mathematical symmetry between attraction and repulsion, which proves our point about Coulomb scaling for PECs and charge inversion in full detail. The reference point here is the equilibrium distance, the origin of all V's. When compared with Fig. 1d, Fig. 14b is almost an astonishing result as so many people during so many years have been trying to find a universal function. Atomic dissociation24/01/00 14:26 G. Van Hooydonk 37limits, looked from the Coulomb physics of a 4-particle system are misleading, a trompe l'oeil effect (Van Hooydonk, 1999). We must now try to find out if the complete curves are scalable, i.e. from origin to asymptote (identical V's with scaled legs). 11.6. The 2nd step: scaling from asymptotes The second step of the scaling process can not but deal with the 'chemical' asymptotes D e. Normally, scaling simply with D e gives rise to reduced PECs varying from 0 to 1 (all legs of the V's would then be equal). We already noticed, see Fig. 6d, that for this process Re values (generic asymptotes) are unavoidable. Unfortunately, the transition from ionic to covalent behaviour falls outside the range covered by our generic approach (see Fig. 13b) and we must use an indirect method to zoom in on this part of PECs. A universal function f(R) can be rescaled from any viewpoint, if the (Coulomb) scaling mechanism is not distorted. The disadvantage of the classical view using (a) D e for scaling RKRs and (b) R e for scaling R, as in Fig. 1b, is that it is probably distorted. There is no simple smooth or scaling relation between D e and Re (determining the Coulomb asymptote), the main conclusion of this report. If we want to check the long-range forces for Coulomb behaviour, we must rescale the PECs accordingly, i.e. use the generic Coulomb potential e2/Re instead of the atomic asymptote as a starting point (see Fig. 6d on the general Coulomb behaviour of bonds). In practice, this would correspond with setting the atomic limit equal to 1 and shifting the data for true Coulomb behaviour, which we are now able to do. In the scaled picture Fig. 14a, all V's coinciding at the origin, the picture is completely Coulombic, atomic dissociation limits appear as disturbances. Therefore, it is of interest to see how the bonds behave when their asymptotes are aligned. The asymptote values in Fig. 14 are equal to D e/a0. Shifting the asymptotes by their differences (the maximum D e/a0(piv) value is 0,46620 for H 2, for the 15 turning points at the attractive branch of H 2 we used the ionisation energy as asymptote) gives the results presented in Fig. 15a . The global picture hardly shows diverging behaviour near the atomic dissociation limit. This means that, when local scaling at individual R e-values is retained, no great anomalies seem to appear. However, the insert in Fig. 15a shows that this is not so in reality. The upper clustering in this insert shows that I 2 is behaving abnormally, the two other bonds clustering here are H 2 and LiH. The lower clustering set encompasses the remaining bonds, for which the PEC extends to D e. The clustering of MM bonds (M alkali) seems real, KH is slightly above the MM clustering line, HF and LiF are both slightly below. Nevertheless, the global picture in Fig. 15a would suggest that the mechanism governing the transition from the ionic PEC to the atomic dissociation limit is probably very similar, even scalable, for all bonds. In this critical region, the only mechanism applicable is obvious and is the same for all bonds: it consists of charge transfers from two ionic bonding partners to two (weakly interacting) atomic systems. However, the good agreement in the global picture Fig. 15a is in part due to the fact that the legs of the V-shaped PECs are not scaled yet for Coulomb behaviour, which we must also try to remedy. Scaling RKRs directly with De makes all legs of the V-s, regardless of the value of Re, equal to 1 if the RKR extends to D e. The question is whether or not this conforms with Coulomb behaviour as in Fig. 6a-d were the effects of Coulomb scaling on PECs are illustrated. This is the most intriguing case, since scaling RKRs with De is the standard procedure (see the Sutherland parameter) and the basis of Morse's function, see Fig. 1d. No24/01/00 14:26 G. Van Hooydonk 38linear shifts have to be applied. The question whether or not Coulomb behaviour is respected when using D e for scaling can now be answered in more detail. The result is shown in Fig. 15b and this must be compared with the benchmark Morse solution in Fig. 1d. We remark that in principle a perfect scaling procedure can not be destroyed by the choice of any Coulomb asymptote (see above: any asymptote obeying the same 1/R law will reproduce invariantly the same scaled results, see Fig. 6c, Fig. 14 and Fig. 15). However, the use of D e as a scaling asymptote does not obey this principle. In fact, scaling with D e shows an expected duality: Coulomb scaling is retained at short range, even with D e, which proves a unique Coulomb law is active at the repulsive side, but at the attractive branch the situation gets distorted more than in Fig. 15a. The atomic dissociation limit is not suited for scaling this branch of a PEC when speaking in terms of a Coulomb process. This may seem trivial (as pointed out above, since D e refers to two neutral particles) but exactly this asymptote has -up till now- invariantly been used for scaling RKRs in molecular spectroscopy. D e can not act as a scaling asymptote for a process of interacting charged particles dissociating into neutral particles, unless it would be itself a scaled form (a projection) of the asymptote consisting of charged particles. This latter possibility can not be excluded a priori but remains to be proven. Fig. 15b is nevertheless more consistent than the Morse benchmark Fig. 1d. At long range, we observe roughly three clusters: HH and LiH; HF, LiF and KH; all MM ( M=alkali metal); I2 is (again) a clear exception in all respects. The only thing that is clear now is that, at long range, there is no simple 1/R dependence, governed by D e as an asymptote. Only not too far from equilibrium, Coulomb behaviour towards the asymptote is retained with D e. It may be possible of course that D e(R) may be a scaling function in some cases at attractive branches (Morse behaviour), but this relation has not yet been found. In fact, we are convinced long range behaviour will have to be explained by the interplay between long range forces of type Cn/Rn (see above) and Coulomb 1/R behaviour and with the charge transfer process in the critical region. We remind the basis of our approach is essentially RKR = s(e2/Re)kgen = a0kgen where s is a species dependent parameter. After multiplying with R e we get (RKR)R e = Rea0kgen which should reflect more appropriately the Coulomb scaling process illustrated in Fig. 6d: it is an attempt to scale the length of the V-legs according to the position of the bond in the global W(R) Coulomb field. The asymptote is now DeRe, having a maximum for LiF at 75258 cm-1. Fig. 15c gives the corresponding results. As in Fig. 15a, the first clustering of lines refers to H 2 and LiH, the second to all other bonds except I 2 (an exception also at the repulsive branch) and the Morse RKR for LiF. Leaving out these latter two cases, it appears that scaling RKRs with a Coulomb asymptote e2/Re indeed leads to more consistent results for the complete PEC than scaling with D e, also the essential conclusion of our previous analysis of the constants (Van Hooydonk, 1999). This leads to the interesting prospect that the long-range situation in PECs may be derived in an elementary way from their short-range behaviour, which is an unprecedented result also. Finally, Fig. 15d gives observed level energies in function of the theoretical ones, both shifted by (49406 - D e) where 49406 is the largest D e-value in the set for HF. The anomalous I 2 set has been left out. Clustering now extends throughout the complete data range, except near the asymptote. The few diverging points here belong to only24/01/00 14:26 G. Van Hooydonk 39two molecules HF and LiF. This scaling result from the asymptote is by far the best we could obtain up till now and it does not use D e as a Coulomb scaling factor (Fig. 15b). Diverging points are now confined to the relatively small triangle anchored to the intersection of ionic and covalent curves. It strengthens our idea that scaling long-range behaviour must be possible, as it relies upon the same charge-transfer process, needed in all cases to convert ionic states to atomic states. If this is valid, we can 'safely' put V''(R) = 0 in equation (22m) in first order. For all data points, we applied this simple working hypothesis for the complete PECs. This effortless procedure gives an absolute deviation of about 4% for all data points with the function (22m) in a zero perturbation approximation, which is almost the same deviation as that found for all points below 50 % of D e mentioned in Table 3. The corresponding graph is shown in Fig. 16, whereby the values obtained with (22m) are easily retraced. We remark that these data refer to 100% of the observed RKRs for 13 different bonds. On the whole, this absolute deviation for energy levels is well within the criteria for a function to be universal when it comes to reproduce PECs (Van Hooydonk, 1999, Varshni, 1998, private communication). The slope is very close to unity and the goodness of fit is almost 0,999. When looking at previous scaling attempts, also this is an unprecedented result. 11.7. Turning points in RKRs and in PECs deriving from a Coulomb based scheme The accuracy of the turning points depends solely on the accuracy of the model potential. This delicate matter was recognised a long time ago. Using an RKR as an observed PEC -the working hypothesis of our present analysis- remains a matter of belief, faith or trust, as no real benchmark solution is available. In other words, there is no alternative for checking the present theory. The dissatisfaction with conventional RKRs and the Dunham expansion ultimately led to alternate inversion methods (IPA) to arrive at PECs. Table 4. Agreement between 394 RKR and Coulomb turning points below 50% of D e. SideData Bond AuH CsCs HH HF IIKH KLi LiLi LiH LiFNaCs RbRb RbCs Total attr# 58 10 5 2 9 6 15 7 8 2 17 12 11 162 abs dev in % 0,346 0,583 1,899 0,133 1,716 0,0813 0,16478 0,0464 0,514 0,1571 0,281 0,29 0,431 0,4432 in Angstrom 0,006 0,034 0,020,0017 0,0507 0,0026 0,00706 0,0016 0,012 0,003 0,014 0,0153 0,024 0,0134 rep# 36 28 15 5 20 14 15 10 23 2 26 19 19 232 abs dev in % 0,218 0,082 0,565 0,7009 0,3319 0,4472 0,00310 0,06716 0,497 0,0381 0,076 0,0244 0,079 0,2205 in Angstrom 0,003 0,003 0,003 0,0047 0,0078 0,0075 0,00009 0,00141 0,005 0,0005 0,002 0,0008 0,003 0,0034 # 94 38 20 7 29 20 30 17 31 4 43 31 30 394 Average abs dev in % 0,297 0,214 0,898 0,5386 0,7615 0,3375 0,08394 0,0586 0,501 0,0976 0,157 0,1272 0,208 0,3121 Idem in Angstrom 0,005 0,011 0,007 0,0038 0,0211 0,006 0,00357 0,0015 0,007 0,0018 0,007 0,0064 0,011 0,0075 Especially here, the H-L theory is not very useful either, as the number of approximations to be made in sophisticated quantum mechanical calculations to obtain a theoretical PEC is innumerable ( Pople, 1999). We remind the techniques underlying the computations for four-particle systems, studied by Richard and Abdel- Raouf (cited above), where about 300 parameters are needed ( Hylleraas type approximation) to get theoretical results with a reliable CL. For all these reasons, we adopted the same pragmatic procedure as above to cope with this important issue. We calculated the turning points by our generic scheme for all data points below 50% of D e using the pivot a 0-24/01/00 14:26 G. Van Hooydonk 40values and the observed level energies in cm-1 (see Table 3). The results are shown in Fig. 17 and in Table 4, where deviations in italics are minima and those underlined are maxima in their category. The average 'deviation' in the turning points is 0,31 % and shows the over-all agreement is acceptable, although the present Coulomb model potential is extremely simple. For the attractive branches, deviations are in general 'large', i.e. 0,44 %. For the repulsive branches the global deviations are 2 times smaller in % (0,22 %) and much smaller in  in comparison with the attractive branches. This latter agreement brings us closer to spectroscopic accuracy, as the 0,001  barrier or even lower can be reached. 11.8 Specific bonds and systematics - AuH is one of the three coinage metal hydrides studies by Seto et al., 1999 using the Le Roy fitting technique to extract the complete PEC from a number of observed levels. It is not difficult to obtain a complete PEC using the generic model potential. The results are shown in Fig. 18. The agreement near Re only seems satisfactory from the plot: in reality very large % deviations are found. At the short range a slight divergence (in %) is noticed. At long range, the differences are more pronounced and are similar for all three coinage metal PECs (not shown). We used (22m) to compute the long range PEC with the generic function only. - HF is an intriguing molecule and its PEC has been studied extensively (see Coxon and Hajigeorgiou, 1990). With its companions DF and TF, it is suited to study BOB (breaking of the Born Oppen heimer approximation). As an illustration of our procedure, we used an hybrid potential k2 (rep) + kgen(att) and used a graphical fitting technique to determine the asymptotes in each case. We get 216756,46 cm-1 for the repulsive Kratzer branch and 197861,72 cm-1 for the attractive generic branch, giving an average value of 202309,1 cm-1 within 1 % of the Dunham asymptote 204308 cm-1 (see Table 1). The resulting PEC, using (22m) is given in Fig. 19. The agreement is acceptable. The average deviation for all 40 turning points, including those at long range (the triangle), is a modest 1,8 %. - I2 is a clear exception in this series of 13 bonds. The R e value compares with that of Li 2 but the two PECs are completely different, as easily verified in Fig. 6d. The PEC for Li 2 is consistent with a Coulomb model but that for I2 is not (the Dunham asymptote for I 2 is exceptionally large). The atomic data are largely different, IE I = 10,45 eV whereas IELi is only 5,4 eV. This suggests the R e value for I 2 is long by about 1 Å. In Fig. 6d, the PEC for I2 should correspondingly be shifted towards lower R-values, where the shape of the reported RKR would be in line the shape predicted by a Coulomb model. As a matter of fact, I 2 is the only case out of 13 where the RKR does not fit into the gauge-symmetry based bonding model that led us to Fig. 6d. Analysing the PECs for the other halogens can be of help. - For LiF we used a Morse-curve (Van Hooydonk, 1982), since the two alternatives available, a Padé approximant (Jordan et al., 1974) and a Kratzer PEC (Van Hooydonk, 1982) do not show asymptote behaviour near De. The Kratzer PEC would be too close to the generic scheme presented here. 11.9 Intermediate conclusion At this stage (the usefulness of the zero molecular parameter function and the scaling results), the pertinent question is about the agreement between our model potential and the available inversion techniques (depending on iterative processes and higher order WKB terms). This is certainly a matter of further investigation. A24/01/00 14:26 G. Van Hooydonk 41correct scaling procedure can not but lead to the scaling of different PECs into a single straight line. To the best of our knowledge, only the present Coulomb scheme can achieve this with reasonable success. These two kinds of results provide us with ample evidence that our earlier conclusions (Van Hooydonk, 1982, 1983, 1985, 1999) are essentially correct. The hyper-classical Coulomb law and its inherent scaling capacity are important for rationalising the abundant data in molecular spectroscopy. But the main conclusion of this section with quantitative results about PECs from atomic data and about scaling is that intra-atomic charge inversion ( chirality, parity violation) at the Bohr-scale is a reality. A previous report on the same issue (Van Hooydonk, 1985) lacked the generic Coulomb function for calculating PECs and the experimental confirmation presented here. In fact, the only reaction on this first attempt to include the generic effect of intra-atomic charge inversion in a theory for the chemical bond came from the late Pauling (1985), who wrote that 'charge- symmetry is broken by the electron-proton mass difference' (see further below). 12. Generic effects of intra-atomic charge inversion Table 5a. Matrix of 16 states for two interacting atoms (absolute charge distributions ) Intra-atomic charge distributions in atoms X 1 and X2 and the sign of interatomic interactions (nucleon charge between brackets) different states for any molecule X 1X2 (homonuclear bonds only) World > 0 World < 0 X1 X1 (+)+ (+)- (-)+ (-)- World > 0 X2 (+)+ (+)+|(+)+ (+)-|(+)+ (-)+|(+)+ (-)-|(+)+ Total charge +4 +2 +2 0 (+)- (+)+|(+)- (+)-|(+)- (-)+|(+)- (-)-|(+)- Total charge +2 0 0 -2 World < 0 X2 (-)+ (+)+|(-)+ (+)-|(-)+ (-)+|(-)+ (-)-|(-)+ Total charge +2 0 0 -2 (-)- (+)+|(-)- (+)-|(-)- (-)+|(-)- (-)-|(-)- Total charge 0 -2 -2 -4 Heitler and London (1927) could not foresee the power of a direct internal (algebraic) correlation between fermion-boson symmetry and potential energy in a four-particle Hamiltonian. This link shows at the Bohr- scale, operates when going from atoms to molecules by means of a generalised Bohr-formula and is measurable in eV. The scale invariance has interesting prospects. The energetic effects of a bonding process are easily described by a static Coulomb law (with charge inversion in one atom or boson) and may provide us with a simple alternative to the H-L theory, all other things remaining equal. The two approximations use the same Pauli-matrices. Since the energetic effect of spin is small, not of order eV, spin symmetry effects must operate on wave functions (H-L theory). It is not difficult to apply the charge inversion technique to four-particle systems in general, giving 16 theoretically possible states, equally distributed over two symmetrical worlds, a chemical eight-fold way. Allowing for the principle of charge inversion, the four particles, arranged in a pair of two atoms, lead to a number of forbidden or allowed Hamiltonians, mainly on account of the total charge of the four-particle system . These are not discussed here in full, since their characteristics can be24/01/00 14:26 G. Van Hooydonk 42derived from the formulae above. Table 5a gives the matrix for atom combinations. Only the neutral states in the centre correspond with allowed states. Neutral states with 0 are H-L states. For these states to become bonding, the combined symmetry effects of electron-spin and of electron wave functions must be invoked. Neutral states with 0 lead to a PEC deriving from parity-violation adapted Hamiltonians, i.e. without needing spin and wave function symmetry effects but using charge inversion instead. Table 5b gives the matrix for ionic interactions in the generic Coulomb scheme for bonding. Singlet- states in bold result from charge inversion. Doublets in Table 5b do not arise from charge inversion but from lepton-rotation. At the real asymptote, a four-particle Coulomb system must not be divided into two interacting boson pairs (two neutral atoms) but into two interacting fermion pairs. This generates splitting within the Hamiltonian, which is then soluble classically, see Section 8. Each world has one H-L state and two degenerate states with intra-atomic charge inversion (parity violating states in the conventional way). These are shown with a total zero charge in bold in Table 5b. Table 5b . Matrix of 16 states for two interacting ions (absolute charge distributions) Intra-ionic charge distributions in anion and cation and the signs in homonuclear bonds (nucleon charges between brackets) Anion states +(+)- and -(+)+ do not imply charge inversion, only rotation is involved. World <0 anionWorld >0 anion +(+)+ +(+)- -(+)+ +(-)+ -(+)- +(-)- -(-)+ -(-)-- inversion =rotation Inversion =rotation Total; mean charge*+3; +1 +1; +1/3 +1; +1/3 +1; +1/3 -1; -1/3 -1; -1/3 -1; -1/3 -3, -1 World > 0 Cation (+) Bond +(+)+(+) +(+)-(+) -(+)+(+) +(-)+(+) -(+)-(+) +(-)-(+) -(-)+(+) -(-)-(-) Total charge +4 +2 +2 +2 0 0 0 -2 Remark HL state World < 0 Cation (-) Bond +(+)+(-) +(+)-(-) -(+)+(-) +(-)+(-) -(+)-(-) +(-)-(-) -(-)+(-) -(-)-(-) Total charge +2 0 0 0 -2 -2 -2 -4 HL state * mean charges are discussed below We have restricted the present analysis to bonds between mono- valent atoms but it is straightforward to apply the principle of charge inversion to polyvalent atoms. Consider a bond between two divalent atoms in the 2nd or 6th Column of the Periodic Table. The classical atom has a charge distribution - (++)-, where the charges between brackets relate to the nucleon. Inverting charges in only one lepton- nucleon pair gives -(+-)+ and in both gives +(--)+. Interactions between two divalent atoms can proceed in two ways either as: a. -(+-)+/-(+-)+ in which one pair is inverted in each atom or as: b. - (++)-/+(--)+, whereby two pairs are inverted in one atom. In terms of Coulomb's law, type b. bonding will lead to strong nucleon-nucleon bonding, whereas type a. bonding will be (very) weak, since only lepton-lepton interactions can generate a bond. Atoms of the 6th Column (O 2,S2,...) form strong bonds indeed, whereas those of the 2nd Column (Be 2, Mg2, Ca2,...) form weak bonds.24/01/00 14:26 G. Van Hooydonk 4313. Born- Oppenheimer (B-O) approximations BOB, breaking of the B-O approximation, shows when applying a Dunham type expansion to isotopomers. The B-O-approximation freezes nucleons and considers two central force systems (reduced masses). Temporarily freezing leptons instead leads to an inverted B-O and gives H12 = (½m1v2 +½m2v2 - e2/R12) - e2/Rab (23a) using the charge inverted, parity violating Hamiltonian. The further treatment runs exactly as in section 8 in the case of the B-O approximation. The minimal solution ( Rab perpendicular to R 12) for vibrating nucleons leads to the generic force constant and the harmonic frequency  2R = m12R/2 = e2/Re2 taking into account that the reduced mass equals m 1/2. With m 12 = ke, the force constant, we get ke = 2e2/Re3(23b) This expression has been used successfully for scaling the Dunham asymptote in the parameter t, proposed by Varshni and Shuckla some time ago (1963) for ionic bonds and which leads to acceptable results for all kinds of bonds (Van Hooydonk, 1982). Depending on the (unknown) alignment of the particles, i.e. Rab relative to R 12, different values are obtained for ke, conforming to the analysis in Section 8. Instead of freezing nucleons or leptons, one can freeze nucleon and lepton motion temporarily at the same time, which corresponds with mass annihilation and gives a static mass-less four-particle system. The charge inversion technique here leaves only two electrostatic terms in the Hamiltonian H = - e2/R12 - e2/Rab (23c) which brings us directly at the top of the absolute well depth. Exactly as above, and depending whether or not the 2 two-particle subsystems are perpendicular, starting at the top of the well depth, the Coulomb potential (23c) is -2e2/R12. Starting half way, with one subsystem fixed at an intermediate asymptote, gives HX-X+ = -e2/Rab Exactly as for the chemical bonds studied here, it can be expected that these mass-less systems are stable. This is also the essential and general conclusion of Richard (1994) and Abdel-Raouf et al. (1998), although the charge inversion technique is not used in their works. But their conclusions are similar: insoluble four-particle systems appear to be stable, given a particular charge-distribution . The experimental evidence collected above about charge inversion in molecules can be used in atoms too, although it will not be detectable in two-particle systems. But, theoretically, an atomic Hamiltonian H X can now be generalised as in (9a) but, due to the large mass difference between an electron and a proton, charge symmetry is broken almost by definition ( Pauling's remark, cited above). Its effect can only be observed through the interactions between two atoms (bonds). The atomic Hamiltonian H = T + V = (1/2 )mv2 - e2/r would be, referring to (9a) or (11), a t - g = 1 state (interaction and asymptote have opposite signs), just one of the four states allowed by gauge symmetry. The 4 atomic Coulomb states are theoretically24/01/00 14:26 G. Van Hooydonk 44W(r) = (-1)g C(1 + (-1)t-g re/r) (23d) of which two belong to a real, the other two to an imaginary world. But (23d) does not lead to an atomic PEC, it results in a molecular Coulomb PEC | IEX(1-Re/R)| with R e = 2re. The classical equilibrium condition gives the absolute gap, 2C = e2/re = mve2 and in the algebraic scheme used here, equilibrium is obtained half way this gap (the virial theorem) or at C = e2/2re = mve2/2 (see Fig. 3a and 5). Using the charge inversion technique, boson-boson interactions can be rewritten as fermion-fermion interactions. Due to the appearance of intra-atomic charge inversion, the analysis of PECs at the Bohr-scale is ideally suited for examining the behaviour of (composite) particle-antiparticle systems. Then, the interaction of a composite particle (atom, positronium, protonium) with its antiparticle (anti-atom, i.e. a charge inverted atom, anti-positronium, anti- protonium), follows exactly the generic scheme proposed here. Reference systems are the positronium molecule (Richard, 1994) and, of course, the much underestimated hydrogen molecule itself, see Introduction. At the molecular level , the mystery surrounding the absence or presence of anti-matter may be solved, if our derivations are valid. Both kinds of matter are present in exactly the same amount in bonds since, for example, H 2 = (H, AH) where AH is an anti-hydrogen atom (a charge inverted hydrogen atom, Van Hooydonk, 1985). This is the more intriguing since decisive experiments are planned to reveal the mechanism of anti-hydrogen reactions, as relatively large amounts of anti-hydrogen can now be produced ( Armour and Zeman, 1999 and references therein). For a variety of reasons, the outcome of these future experiments is of crucial interest. If applying the charge inversion technique to atoms from the start is valid in chemistry and leads to Coulomb scaling (as demonstrated here) it should apply at all scales in physics since it is scale-invariant. Compactifying H2 to smaller R-values must lead to a similar composite but smaller system, the atom D. Then, the ionic version of a deuterium atom would have as constituent particles a proton and an anti-proton, surrounded by two oppositely charged leptons. This entity must be related to H 2. In the above, the ionic variant (XY)+ and e- was avoided above, since it would lead to an atomic spectrum. Schemes deriving from classical gauge symmetry like those shown in Fig. 3a for a gap of 2C will have repercussions when similar systems with larger or smaller gaps are considered. For large differences between the gaps, fine structures could be generated generically, leading to double well PECs. Extrapolating these results to nuclear physics and neglecting this possible generic fine structure, the so-called Z-instability of nuclei can now be avoided. Only two types of stable nuclear ground states would remain: Z-even ( singlets) and Z- uneven (triplets). Singlets are stable with packings similar to those in crystals (solid-state physics). Triplets are stable too whatever the value of Z. These consequences are as observed ( Bohr and Mottelson, 1969). There would be no need to consider a separate class of nuclear forces to hold nuclei together since intra-nuclear repulsions roughly of order (Z2/4) are replaced by attractions of the same order of magnitude. This makes nuclei in their ground state naturally stable instead of theoretically unstable. Here, the transition from repulsive states in nuclei on account of positive Z-values to attractive states is similar to the situation met above for bonds. Intra-atomic charge inversion secures that the intra-nuclear repulsion changes into an attraction (see Sections 3 and 6). On the chemical level, uneven Z-nuclei would act as monovalent atoms, whereas even Z- nuclei would behave as noble gases. This is confirmed by experiment: molecules like Be 2 and Ca 2 are very unstable and have small dissociation energies (Van Hooydonk, 1999) just like noble gases (see also Section 12).24/01/00 14:26 G. Van Hooydonk 45To understand classically the stability of composite stable neutral systems in nature, they must be partitioned in oppositely charged mass-asymmetric subsystems (or particles) if Coulomb's law applies universally. Neutral mass-asymmetrical molecules in a dielectric medium such as water (reducing the power of Coulomb's law by a factor of 80) dissociate in a charge-symmetric but mass-asymmetric ion pair. The only concurrent 1/R law available, Newton's, uses particle masses instead of charges in the coefficient an of R-dependent potentials such as (3b). 14. Further consequences 14.1. Decisive experimental evidence The results in Section 11.2 show that also covalent X 2 bonds have a critical region where there is interplay between a bond X 2, atoms X and ions X+ and X- since D e is not a Coulomb asymptote. To determine the intersection point of the ionic potential and a nearly fixed De, the classical result for the critical distance IEX + EAX + e2/Rcrit = 2IEX leads to e2/Rcrit = IEX - EAX In the case of Li 2, this gives a critical distance of about 3,3 Å. We can calculate the perturbation in the right branch between W(R) and the lower lying asymptote at 2IE Li (i.e. De) provided EALi = De. The complete PEC can then be computed using (22m). Using our generic function we find different intersection points ( Rcrit). For Li2, the intersection at the left branch occurs at 1,815  and at 5,069  for the attractive branch. On account of atomic long-range interactions varying with Cn/Rn, this critical distance at the attractive side can be different. But these crossing points open the way for a possible experimental proof of the generic scheme based upon electrostatics . Femto-chemistry can help to find out ( i) whether or not the atomic dissociation limit is a scaled form of the ionic asymptote, which seems unlikely but can not be excluded or (ii) if crossing of ionic and atomic states is really avoided. As in the case of NaI (Rose et al, 1988), femto-chemistry can be of assistance to verify which species are present in this critical region of covalent molecules X 2 like Li2. If ions Li - and Li+ are found experimentally, this would be conclusive evidence in favour of an electrostatic approximation to chemical bonding. 14.2. Universal Coulomb function as an ab initio model potential. Scaling the Dunham coefficients. Chaos and fractal behaviour: an open question? Coulomb's law in a zeroth order approximation with V 12 = 0 can never account for the (lower order) spectroscopic constants as it leads to meaningless F and G Varshni-parameters, based upon Dunham's analysis. This is not so for the V 12  0 solution, the perturbed Coulomb function. The corresponding derivations were not made. Maybe this is not even necessary, since the generalised Kratzer-Varshni potential can rationalise the lower order spectroscopic constants quite easily (Van Hooydonk, 1999). Other generalisations are possible. The shift from the generic variable R to R + r, accompanied by a similar shift for R e, where r is relatively small and constant, can not be excluded (Van Hooydonk, 1983, 1999). The introduction of a small r might lead to a consistent extension to a united atom model and even to high-energy (nuclear) physics for R close to zero (Van24/01/00 14:26 G. Van Hooydonk 46Hooydonk, 1983). On the other hand, Varshni-like generalisations (v  1) can not be excluded, such as (1/m )x, with x not too different from 1 (see below). Dunham's expansion is a poor approximation to the reality of Coulomb based PECs. Ionic asymptotes must be used to scale Dunham-coefficients. The relation between the Dunham variable d and the Coulomb/ Kratzer variable k = d/(1+d) is at the origin of this process. An expansion of the Kratzer potential in function of the Dunham variable leads to ideal Dunham coefficients a n = (-1)nn (Van Hooydonk, 1983), in agreement with observation in a relatively large number of cases (Van Hooydonk, 1999). The fluctuations in the Dunham series increase linearly with n and illustrate the convergence problems associated with Dunham's series. The Dunham expansion must be abandoned as many people realised ( Molski, 1999, Lemoine et al., 1988, Bahnmaier et al., 1989, Urban et al., 1989, 1990, 1991, Maki et al., 1989, 1990). Gauge symmetry leads to a very simple generic and effective model potential. It derives from first principles and does not contain parameters. It can be expected that fitting observed levels with this scheme automatically leads to a better inversion procedure to construct PECs, transferable from bond to bond, a vacuum in the now available methods. Further work must reveal the universal character of this process. If so, it can not be excluded that we would end with chaos and fractal behaviour. Given the results obtained here and the possibility of functions varying as (1/m )x, it seems interesting to analyse the behaviour of power laws with fractional exponents like n/3, with n an integer, especially in connection with the non Coulomb D e(R) dependence. We already constructed Kratzer-Varshni type potentials of this type, some of which are very close to the generic potential used here. Above, we remarked the self-replication of expansions in terms of k. Molski (1999) recently reported on the interest of potentials with continued fractions and March et al. (1997) suggested chaos or fractal behaviour of the lower order spectroscopic constants. They observed that G varies with F4/3, a formula we already tested on a large scale (Van Hooydonk, 1999). Further research along these lines remains interesting, reminding we suggested a long time ago that Euclid's golden number might interfere in the physics of interacting charges (Van Hooydonk, 1987). Above, we mentioned an asymptote wherein this strange number also appears. The generic perturbation depends on the square root of 3, which may point in the same direction. Its (geometrical?) origin must be understood. Fig. 20 gives the Coulomb asymptote and D e in function of Re for the 13 bonds with a power law fit. A fractional exponent appears for D e almost as expected. But our data about 400 bonds (Van Hooydonk, 1999) do not allow a generalisation of the power law for D e in Fig. 20. A new attempt to smoothly generalise Kratzer's potential was recently proposed by Hall and Saad (1998). We did not yet investigate the possibility that Coulomb's law may be adapted (generalised) with a slowly varying maybe exponential R-dependence, as in Yukawa's potential. 14.3. Charge inversion: a generic consequence of Coulomb’s law. Holes. Cooper pairs. Fractional charges If the present scheme is scale invariant and if the charge inversion Coulomb mechanism is generic, there must be examples in other domains where analogous effects might have been observed already. - In solid state physics, the concept of holes, bearing a positive unit of charge, has long been accepted and is now an essential part of the theory. With the charge inversion technique, holes can be considered as positive electrons, positrons . Also, ionic bonding models have an excellent reputation in solid state physics (we refer to24/01/00 14:26 G. Van Hooydonk 47the Born- Landé potential very close to a Coulomb potential). The hole-concept gives circumstantial evidence for this scheme. A similar situation applies to Cooper pairs and to the absence of isotope effects in high-T C superconductors (Van Hooydonk, 1989). - As suggested above, the present scheme must also be applicable to reactions between hydrogen and anti- hydrogen, to be observed in the years to come. The outcome of this reaction, according to the present scheme, should be a normal hydrogen molecule. - Part of the mechanism leading to the quark concept and the quark-anti-quark interaction scheme to explain the stability of (composite) elementary particles, in the first instance baryons, bears an analogy with the present scheme. Quarks are elementary particles with ±1/3 of the unit of charge. For the present approach to make sense in the simplest possible case two singly charged ions of comparable masses are required, one of which must be a composite particle, consisting of three charged sub-units. A unit of charge ±e for 3 particles leads to an average of ±e/ 3, see Table 5b. At the Bohr- scale, the mass difference between the constituent fermions is of the order of 10-3. Nevertheless, fractional charges equal to ±e/3 in the present scheme do not exist in reality as they are averaged values. 14.4. Charge inversion and theoretical chemistry. Reactivity. Charge alternation - Molecular wave functions consist of a linear combination of atomic orbitals (LCAO) of Slater-type (ST)- or of (quadratic) Gauss-type (GT) in a SCF-HF procedure to describe bonding with a classical not parity adapted H-L Hamiltonian . Using ST-AO's with an exponential 1/R dependence typical for Coulomb's law (implying a central force system) is equivalent with the a priori assumption that the atomic dissociation limit is the natural asymptote for the Hamiltonian. This convention is now proved to be incapable to get a scaling procedure for PECs, see above. Coulomb's law and the generic or ionic asymptote are the necessary elements in achieving universal scaling. Unfortunately, ionic wave functions and a parity adapted Hamiltonian, the better of choices in the present analysis, have thus far never been used to our knowledge to construct molecular PECs below and beyond the atomic dissociation limit. Ionic contributions are commonly classified only as (very) small perturbations of the H-L scheme ( Weinbaum, 1933). But the present interpretation of symmetry effects in bonding is a generic consequence of using a pair of symmetric Coulomb potentials and gauge symmetry. Conversely, the H-L theory even seems unnecessarily complicated when it comes to understand the mechanism leading to bond formation . The complexity of standard quantum-chemical calculations (SCF, HF using atomic STO or GTOs in a LCAO) is a natural consequence of the H-L theory, its Hamiltonian and the conventions about fermion charges ( Pople, 1999). The H-L scheme can only produce theoretical PECs through a rather cumbersome procedure. On the basis of the present results, we predict that very accurate theoretical PECs can be obtained without too much effort in the quantum-chemical way by using a charge inversion adapted Hamiltonian and ionic instead of atomic wave functions (one central force system approximation). The atomic dissociation limit can be described by higher order Coulomb interactions in function of atom polarisability, a standard practice in modern physics ( Aquilanti et al., 1997, Côté and Dalgarno, 1999). These covalent potentials, varying with Cn/Rn with large n, were approximated in this work even by means of a fixed lower asymptote, D e.24/01/00 14:26 G. Van Hooydonk 48The apparatus of wave mechanics remains necessary in the present scheme. If the bonding procedure were really electrostatic, homonuclear bonds in an ionic approximation would show a dipole moment, which is not so. Hence, a homonuclear bond X 1X2 has the wave mechanical hybrid ionic counterpart (X 1+X2- + X1-X2+), a straightforward effect of permutational symmetry. For bonds of intermediate polarity, the weights of the two possible ionic structures are different from 1/2 (Van Hooydonk, 1999). - Ionic bonding implies a smooth picture for chemical reactivity. For common displacement reactions AB + CD ↔ AD + CB a simple electrostatic model has advantages. Electrostatic approximations in the field of chemical reactivity are numerous ( see for instance Van Hooydonk, 1975, 1976). - A challenging example is aromatic reactivity with its many empirical rules, believed to be a purely covalent problem, soluble only in an approximation of the H-L model such as Hückel’s MO theory (1930). These rules are easily accounted for by Coulomb electrostatics and charge alternation (Van Hooydonk and Dekeukeleire, 1983). - If the present Coulomb approach is valid, an ionic approximation implicitly contains the important Coulomb principle of charge alternation , an additional consequence of Coulomb’s law, not yet discussed. Charge alternation can account for a number of details in organic reactivity (Van Hooydonk and Dekeukeleire, 1991, Klein, 1983, 1989). 14.5. Kratzer-Bohr: a consistency or a dilemma The peculiarities of Kratzer's potential with respect to Bohr's were discussed above (see also Van Hooydonk, 1983, 1984). Kratzer's function is more general than Bohr's as it opens the way to the molecular level. Some of the continuous R/Re-, m-, or l- dependences (with l = m - 1) in Kratzer's potential, rewritten in function of m and d, are remarkably similar mathematically with the dependence on discrete quantum numbers in Bohr's theory (Van Hooydonk, 1984). This can hardly be a coincidence. 14.6. Possible consequences for the other 1/R potential: Newton's The scale and shape invariance of the present scheme implies that it stands irrespective of the nature of a n in (1e) or (3b). The most typical consequence of algebraic 1/R potentials is that they remain valid whatever the scale (Newton, Bohr, Fermi or Planck, see Fig. 6a-b). If gauge symmetry is universal, gravitational forces can be considered as attractive and repulsive too. This can not yet be proven with gravitational PECs, because these are not available. Compactifying systems towards smaller (Planck-) or blowing them up to larger (Newton-) scales should have no impact whatsoever on the general features of 1/R potential governed systems and their PECs (see Fig. 6a-c). This aspect of shape- and scale invariance is a challenge in physics and in SUSY, especially for super-unification . If a gravitational PEC can be found with a shape like the one generated by a Coulomb 1/R potential, systems with positive and negative masses will have to be allowed in exactly the same way as for positive and negative charges. Newton's law has the right structure already. Yet, ±m 1m2 unlike ±e 1e2 states seem to be forbidden. But, if atoms can be annihilated in a molecule, this can not be done properly, if negative masses are not allowed for algebraically. What a negative mass really means is a different question. It could be a mathematical artefact related to another intrinsic property of matter ( not mass ), similar to the24/01/00 14:26 G. Van Hooydonk 49mathematical distinction between worlds, between variables like x and 1/x or between gaps, as discussed above. Qualifying gravitation as weak by a factor of 1040 led to the exploration of physics at the Plank-scale, where the effect of interacting masses would compare with that of interacting charges (the definition of the Planck-mass). At comparable distances, a unit of charge has a mass about 1020 times a unit of mass . The charge/mass ratio for the electron is about 1020. The centre of a system of particles with m 1 and m2, each with a unit of charge e and in equilibrium, must be governed by charges, as these have the larger mass. The centre must be in the middle of the distance, separating the two charges, since |e 1e2/(e1 + e2)| = e/2. This corresponds with e 1/r1 = e2/r2 if r1 + r2 = r. If masses determine the centre, m 1/r1 = m2/r2 applies. If charges are equal but masses differ by a factor of 2000, the centre should be displaced in favour of the heavier particle. The total mass of a charged particle with mass m is m(1020 + 1) = e(1 + 1/1020), with mass 2000m it is m(1020 + 2000) = e(1+2000/1020). The particles’ distances r 1 and r2 from the centre then obey e1(1+1/1020)r2 = e2(1+2000/1020)r1. The difference with respect to the centre of charges is negligible, as 1 part in 1017 is beyond the accuracy of any experiment. But the opposite is observed: the centre of the H-atom is not determined by charges, it is determined by masses. Charge symmetry is apparently broken by particle mass (the essence of Pauling's remark, 1985) but this leads to question marks about the meaning of the factor 1040. This huge factor does not respect two-dimensional scaling either. Only asymptotes or a n/Re-values can be used for scaling, the main conclusion of our results. The factor x = 1040 results from comparing coefficients an as appearing in (1e) or (3b), i.e. xGm1m2 = e2(24) This does not take into account the different scales (R e-values or asymptotes) the two different an might refer to. If so, the scale ratio itself might be at the origin of this large scaling factor. Therefore (24) is meaningless, as long as the exact form of the complete potentials leading to (24) is not known. The hypothesis that these different an values apply to the same scale is in clear contradiction with experiment as the centre of mass of a simple H-atom clearly shows. A generalised form of (24) is Gm1m2/Rx = e2/rBohr (25) How the scaling potential Gm/R x must be determined is another problem. But the factor 1040 as it stands now can certainly not be used as such to determine the hierarchy of forces (Van Hooydonk, 1999a). 16. Conclusion If the 13 bonds used here are a representative sample for observed PECs, intriguing conclusions can be made. Gauge symmetry and the discrete (symmetry) aspects of a first principle's electrostatic Coulomb scaling power law in real particle systems have not been fully exploited in the past. Solutions for molecular PECs based on this law are in agreement with experiment with a more than reasonable confidence level. The universal function, the Holy Grail of Spectroscopy , seems to be very close, if not identical with, a derivative of a scaling Coulomb law, characterised by a dependence on (1-1/m), where m is a number. The shape and scale invariance of PECs and especially the observed left-right asymmetry around the minimum seen in this context indicate that intra-atomic charge inversion must be allowed in the Hamiltonian. Coulomb's law, simple electrostatics, is24/01/00 14:26 G. Van Hooydonk 50much more powerful than hitherto believed and, if true, we have been wrong-footed by nature (Van Hooydonk 1999). Cancelling the nucleon-lepton interactions in the Hamiltonian of four-particle systems (bonds), a generic consequence of intra-atomic charge inversion, is confirmed by experiment. A Coulomb function is an ab initio scaling model potential and can probably be used as a first principle's guide line for inverting energy levels. One important aspect of quantum behaviour, i.e. the appearance of integral numbers, is not discussed in this report: these solutions are all known and are obtained by standard wave mechanics. But the shape of a PEC for N-particle systems (N > 2) belongs to the Coulomb domain and some points can now be clarified. The first concerns the H-L theory and its interpretation of chemical bonding by means of 'exchange forces': this exotic exchange process reduces to intra-atomic charge inversion in one of the two atoms and to resonance between two (degenerate) ionic structures. The underlying static central force system is an acceptable working model and is nothing else than classical 'ionic' bonding, applying to covalent bonds as well. Despite the historical evolution, ionic bonding schemes remain of fundamental physical importance in the context of N-particle systems. It is a pity it took so long to show that the contributions of Davy (18th century) and Berzelius (1835, 19th century), almost 200 years old, are essentially correct. With us, only Kossel (1916) and Luck (1957) found enough evidence to defend ionic or electrostatic approximations to bonding in this century, although these contributions are all against the establishment in the post H-L era. The charge inversion technique (Van Hooydonk, 1985), based upon atom chirality can play a role in the future because of its generic character, its simplicity (the magnet metaphor) and the scale- and shape-invariant PECs it leads to. With respect to symmetry and parity breaking, many efforts in SUSY are based on the mathematical harmonic oscillator, a poor physical model. With Bouchiat and Bouchiat (1997), we hope that a feedback with the Standard Model is possible. Cosmology and super-unification are at stake. The question about anti-matter may even be a false one. Whether physical processes are continuous or discrete is still uncertain and this is a long-standing discussion. When we try to measure a continuous 1/R-dependence we use a discrete measure, light. But a continuous physical 1/R process can be described with a static mass-less Coulomb law and its implicit discrete elements (power law, gauge- symmetry, species independent charge) and its perfect scaling ability. Acknowledgements We are in debt to various colleagues (correspondence, e-mails, pre- and reprints): RL Hall, JR Le Roy, M Molski, WC Stwalley, YP Varhsni and L Von Szentpaly. Coinage metal PEC-data were kindly provided by JR Le Roy. 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Van Hooydonk 54Fig 1b Observed PECs (RKR/De) for 13 bonds versus R/R e 0,000,200,400,600,801,001,20 0,00 1,00 2,00 3,00 4,00 R/ReRKR/DeRbCs RbRb NaCs LiH LiF LiLi KLi KH II HF HH CsCs AuH24/01/00 14:26 G. Van Hooydonk 55Fig 1c Observed PECs (RKR/D e) for the same 13 bonds (V- shape) versus Morse function 0,000,200,400,600,801,001,20 -1,50 -1,00 -0,50 0,00 0,50 1,00 1,50 Morse functionRKR/D e24/01/00 14:26 G. Van Hooydonk 56Fig 1d Benchmark: observed PECs (RKR/D e) for the same bonds (straight line) versus Morse function -1,00-0,500,000,501,00 -1,50 -1,00 -0,50 0,00 0,50 1,00 1,50 Morse functionRKR/D e24/01/00 14:26 G. Van Hooydonk 57Fig. 2a Potential n = 1 -4-3-2-101234 0 1 2 3 4 R/ReW(R)/C 1: +1+R; 2: +1-R; 3: -1+R; 4: -1-R; 5: Kratzer; 6: perturbed Coulomb1 23 45624/01/00 14:26 G. Van Hooydonk 58Fig. 2b Potential n = 2 -4-3-2-101234 0 1 2 3 4 R/ReW(R)/C 1:+1+R2; 2:+1-R2; 3: -1+R2; 4: -1-R2; 5: Kratzer; 6: perturbed Coulomb51 23 4624/01/00 14:26 G. Van Hooydonk 59Fig. 3a Potential n = -1 -4-3-2-101234 0 1 2 3 4 R/ReW(R)/C 1: +1+1/R; 2: +1-1/R; 3: -1+1/R; 4: -1-1/R; 5: Kratzer; 6:perturbed Coulomb1(triplet) 23 45624/01/00 14:26 G. Van Hooydonk 60Fig. 3b Potential n = -2 -4-3-2-101234 0 1 2 3 4 R/ReW(R)/C+1+1/R2 +1-1/R2-1+1/R2 -1-1/R2perturbed Kratzer24/01/00 14:26 G. Van Hooydonk 61Fig. 4 Coulomb's law -6-4-20246 0 1 2 3 4 5 RV(R) (+,+ or -,-): +1/R (+,- or -,+): -1/R24/01/00 14:26 G. Van Hooydonk 62Fig. 5 Gauge symmetry and Coulomb's law -4-2024 0 1 2 3 4 Re/RV(R) (+,+): +1/R + C (-,+): -1/R-C (-,-): +1/R - C (+,-): -1/R + C+ + + -- - - +24/01/00 14:26 G. Van Hooydonk 63Fig. 6a.Coulomb PECs for various asymptotes versus R 05101520 0 1 2 3 4 5 6 RC 20 C 15 C 10 C 5 C 224/01/00 14:26 G. Van Hooydonk 64Fig. 6b. Coulomd PECs for several asymptotes versus reduced R 05101520 0 1 2 3 4 5 6 R/ReC 20 C 15 C 10 C 5 C 224/01/00 14:26 G. Van Hooydonk 65Fig. 6c. Scaled Coulomb asymptotes at asymptote 10 versus reduced R: generic result of universal Coulomb scaling 05101520 0 1 2 3 4 R/ReC20 C15 C10 C5 C224/01/00 14:26 G. Van Hooydonk 66Fig. 6d Coulomb scheme/gauge symmetry (-) and observed PECs (o) versus R (13 bonds)(log-scale) 100001000001000000 0,1 1 10 100 RW( R) +e2/Re Cs2RbCs Rb2NaCsKLiI2(sharp)H2 HF LiF AuHLiH KH Li224/01/00 14:26 G. Van Hooydonk 67Fig. 7 Dunham, Born-Landé, Kratzer and Generic variables 0,00,20,40,60,8 0 1 2 3 4 R/ReVariables1 2 3 4 5 1: Dunham, 2: Generic (perturbation 0,1), 3: Born-Landé, 4: Kratzer, 5: Generic (perturbation 0,35) 1324/01/00 14:26 G. Van Hooydonk 68Fig. 8 Comparison of Dunham, Kratzer and generic functions w(m) with experimental data for H 2 0,00,20,40,60,81,0 0 1 2 3 R/Rew(m)21 3 4 o RKR H 2 1: Dunham, 2: Coulomb (unperturbed), 3: Kratzer, 4: Coulomb (perturbation 0,35)24/01/00 14:26 G. Van Hooydonk 69Fig. 9 Perturbed generic and generalised Kratzer- Varshni functions 0,00,20,40,60,81,0 0 1 2 3 4 5 6 R/Rew(m)(infinite) (10) (6) (3) v = 2 v = 1(2) v = 0,7 (1,5) Generic: full-line, b-values between brackects; Kratzer: dashed lines24/01/00 14:26 G. Van Hooydonk 70Fig. 10a. PEC for H 2 from atomic data 01000020000300004000050000 0 0,5 1 1,5 2 2,5 3 3,5 R/ReLevel energy (cm-1)o RKR --- generic long dashes Kratzer24/01/00 14:26 G. Van Hooydonk 71Fig. 10b. PECs for Li 2 and Cs 2 from atomic data 01000200030004000500060007000 0,5 1 1,5 2 2,5 R/ReLevel energy (cm-1)Li2 Cs2o RKR ---- generic long dashes Kratzer24/01/00 14:26 G. Van Hooydonk 72Fig. 10c Observed versus level energies computed from atomic data for 8 bonds (300 data points) Linear fit y = 0,9775x (dashed) R2 = 0,995 -25000-20000-15000-10000-50000500010000150002000025000 -25000 -12500 0 12500 25000 Level energy from atomic data (algebraic)Observed level energy (algebraic)- Difference in cm -1 + Level energies24/01/00 14:26 G. Van Hooydonk 73Fig. 11a. Variables versus level energies, IPA for LiH Kratzer y = 0,0000176774415x R2 = 0,98829 (long dashes) asymptote 56569,3 cm-1Generic y = 0,000014091195x R2 = 0,99481 (short dashes) asymptote 70966,3 cm-1 -1,0-0,50,00,51,01,5 -30000 -20000 -10000 0 10000 20000 30000 Level energies IPA LiHVariable or w(m) Asymptotes (cm-1) Dunham 65747, Generic 72790,4 Ionic 57681d d2 k k2 kgen (+)24/01/00 14:26 G. Van Hooydonk 74Fig. 11b Kratzer and Generic functions versus IPA for LiH less 10 extreme turning points Kratzer y = 0,000016670860x R2 = 0,99861 (long dashes) asymptote 59984,91 cm -1 Generic y = 0,000013726387x R2 = 0,99966 (short dashes) asymptote 72852,91 cm-1 -0,4-0,3-0,2-0,100,10,20,30,4 -20000 -15000 -10000 -5000 05000 10000 15000 20000 Level energies IPA LiHVariable or w(m)k2 kgen (+)24/01/00 14:26 G. Van Hooydonk 75Fig. 12a General fitting procedure for 6 bonds (small R e) and I2 to determine the asymptote -60000-40000-200000200004000060000 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 kgen (algebraic)RKR or IPA (algebraic) LiH LiF KH II HF KK AuH Extreme turning pointsBond; relation; fit H2; y = 84673x, R2 0,9958 HF; y = 248983x; R2 0,993 LiH; y = 73207x; R2 0,9994 KH; y = 81088x; R2 0,9957 AuH; Y = 175155x; R2 0,9949 LiF; y = 163075x; R2 0,9992 I2; y = 390257x; R2 0,962724/01/00 14:26 G. Van Hooydonk 76Fig. 12b General fitting procedure for 6 bonds (large R e) to determine the asymptote -7500-5000-25000250050007500 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 0,25 0,3 kgen (algebraic)RKR or IPA (algebraic) RbCs RbRb NaCs LiLi KLi CsCs Extreme turning pointsBond; relation; fit Li2; y = 46553x; R2 1,0000 KLi; y = 42965x; R2 0,9996 NaCs; y = 42457x; R2 0,998 Rb2; y = 36534x; R2 0,9989 RbCs; y = 36736x; R2 0,9989 Cs2; y = 36038x; R2 0,999624/01/00 14:26 G. Van Hooydonk 77Fig. 13a About 400 theoretical level energies from asymptotes obtained with graphical fitting procedure and pivot table results versus observed level energies Trendline fom graphical fit (full-line) y = 0,99655 x R2 = 0,99575 Trendline from pivot asymptote data (dashed line) y = 0,97103x R2 = 0,99900 -60000-40000-20000020000400006000080000 -6000 0-5000 0-4000 0-3000 0-2000 0-1000 0010000 20000 30000 40000 50000 Observed level energiesTheoretical level energies+ From pivot asymptoteso From graphical fit24/01/00 14:26 G. Van Hooydonk 78Fig. 13b Reduced RKRs, Kratzer and Generic variables versus k (all data) 0,000,100,200,300,400,50 -1,0 -0,8 -0,5 -0,3 0,0 0,3 0,5 0,8 1,0 kReduced RKRs, k2 and k gen+ RKR/asymptote (from graphical fit) Kratzer k2 (dashed line) Generic (full line)H M224/01/00 14:26 G. Van Hooydonk 79Fig 14a. Reduced RKRs, V-shape 0,00,10,20,30,40,50,6 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 kgenReduced RKRs, k genkgen (full line)24/01/00 14:26 G. Van Hooydonk 80Fig. 14b Rduced RKRs versus k gen, linear form -0,6-0,4-0,200,20,40,6 -0,6 -0,4 -0,2 00,2 0,4 0,6 kgenReduced RKRs, k genkgen (full line)24/01/00 14:26 G. Van Hooydonk 81Fig. 15a RKR/A(piv) versus k gen both shifted with 0,46620-D e/A(piv) for 13 bonds -0,6-0,4-0,20,00,20,40,6 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 kgen + 0,46620-D e/A(piv)RKR/A(piv)+0,46620-D e/A(piv)RbCs RbRb NaCs LiH LiF LiLi KLi KH II HF HH CsCs AuH 0,400,450,50 0,40 0,45 0,5024/01/00 14:26 G. Van Hooydonk 82Fig. 15b Observed RKR/D e versus theoretical k genA/De -1,5-1,0-0,50,00,51,01,5 -3,0 -2,0 -1,0 0,0 1,0 2,0 3,0 Theoretical RKR/D eObserved RKR/D eRbCs RbRb NaCs LiH LiF LiLi KLi KH II HF HH CsCs AuH y = x24/01/00 14:26 G. Van Hooydonk 83Fig. 15c Observed RKR*R e+75258-D eRe versus theoretical -20000-1000001000020000300004000050000600007000080000 -20000 020000 40000 60000 80000 100000 120000 TheoreticalObservedRbCs RbRb NaCs LiH LiF LiLi KLi KH II HF HH CsCs AuH24/01/00 14:26 G. Van Hooydonk 84Fig. 15d Observed versus theoretical level energies +49406-D e for 12 bonds (not I 2) -60000-40000-200000200004000060000 -60000 -40000 -20000 0 20000 40000 60000 80000 TheoreticalObservedred line: equation (22m)24/01/00 14:26 G. Van Hooydonk 85Fig. 16 Theoretical level energies (-) versus observed using (22m) for all 13 bonds, complete range Trend-line (dashed) y = 0,9987x R2 = 0,9982 -60000-40000-200000200004000060000 -60000 -40000 -20000 0 20000 40000 60000 Observed level energyTheoretical level energy24/01/00 14:26 G. Van Hooydonk 86Fig. 17 Theoretical turning points (394 at <50 % of De) versus observed (published) in Angstrom Linear fit (dashed line) y = 0,999121x R2 = 0,999871 01234567 0 1 2 3 4 5 6 7 Observed (published)Theoretical24/01/00 14:26 G. Van Hooydonk 87Fig. 18 Complete PEC for AuH: Generic (22m) and Le Roy functions 050001000015000200002500030000 1 1,5 2 2,5 3 3,5 4 RLevel energiesLe RoyGenericLe Roy Gen24/01/00 14:26 G. Van Hooydonk 88Fig. 19 Observed and theoretical RKRs for HF 0100002000030000400005000060000 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 kLevel energiesRKR (IPA): full line Generic RKR with hybrid function (22m): dashed line24/01/00 14:26 G. Van Hooydonk 89Fig. 20 Power law for asymptotes Coulomb and D e + Generic: y = 116143x-1 R2 = 1 o De: y = 43435x-1,5601 R2 = 0,9307 0300006000090000120000150000180000 0 1 2 3 4 5 ReGeneric asymptote, D e

arXiv:physics/0001060 25 Jan 2000%A L. Ingber %T Path-integral e volution of multi variate systems with moderate noise %J Phys. Re v. E %V 51 %N 2 %P 1616-1619 %D 1995 Path-integral e volution of multi variate systems with moderate noise Lester Ingber Lester Ingber Resear ch,P.O. Box 857, McLean, V A22101 ingber@alumni.caltech.edu Anon Monte Carlo path-integral algorithm that is particularly adept at handling nonlinear Lagrangians is extended to multi variate systems. This algorithm is particularly accurate for systems with moderate noise. PA CSNos.: 02.70.Rw ,05.40.+j, 02.50.-rPath-integral e volution of multi variate systems -2- Lester Ingber PATHINT is a non Monte Carlo histogram C computer code de veloped to e volveann-dimensional system (subject to machine constraints) based on a generalization of an algorithm demonstrated by Wehner and W olfer to be extremely robust for nonlinear Lagrangians with moderate noise [1-3]. PATHINT was recently used in a neuroscience study [4], where it w as observed ho wdifficult it can be ev eni ntwo dimensions to get good resolution because of CPU constraints on Sun SPARC 2 and 10MP machines. Here it appears that the resolution is quite satisfactory on these machines. The system selected for this paper to illustrate the use of P ATHINT is the classical analog of a quantum system. “Quantum chaos” was a term used to describe the observ ation of chaos in the classical trajectories of the Hamilton’ sequations of motion, ˙p=−∂H ∂q, ˙q=∂H ∂p,( 1) where a Hamiltonian of the form H=1 2p2+ˆΦ(q), ˆΦ=1 2(∇Φ)2−1 2D∇2Φ,( 2) with a coef ficientDwasconsidered in a qdimension ≥2, representing double this dimension in the corresponding phase space of qandp[5]. In ref. [5] a potential Φwasconsidered, Φ=2x4+3 5y4+εxy(x−y)2,( 3) in a classical Fokker-Planck system ∂Pt=∇.(P∇Φ)+D 2∇2P.( 4) Aslight change of notation is used in this paper .Theyposed the question as to just what properties suchPath-integral e volution of multi variate systems -3- Lester Ingber classical systems might possess? This paper does not at all deal with the quantum system described abo ve,but it does deal in detail with the associated classical system, and it does gi ve some answers to the abo ve posed question. This study should be considered as illustrating a particular numerical approach that promises to be quite useful in studying the e volution of such classical systems. The results obtained can be considered as “experimental” data on the exact re gion of such classical transformations. The results here are ne gative with respect to an yunusual or interesting activity in the parameter region observed in the quantum mechanical calculation. This should at least help other in vestigators who might tend to focus on this region in the classical system, based on the results obtained for the associated quantum system. This paper computes the path integral of the classical system in terms of its Lagrangian L. P[qt|qt0]dq(t)=∫...∫Dqexp  −mint t0∫dt′L  δ((q(t0)=q0))δ((q(t)=qt)) , Dq= u→∞limu+1 ρ=1Πg1/2 iΠ(2πΔt)−1/2dqi ρ, L(˙qi,qi,t)=1 2(˙qi−gi)gii′(˙qi′−gi′), gii′=(gii′)−1, g=det(gii′). ( 5) Here the diagonal diffusion terms are gxx=gyy=Dand the drift terms are gi=−∂Φ/∂qi.Ifthe diffusions terms are not constant, then there are additional terms [6]. The histogram procedure recognizes that the distrib ution can be numerically approximated to a high degree of accurac yb ysums of rectangles of height Piand width Δqiat points qi.For convenience, just consider a one-dimensional system. The abo ve path-integral representation can be rewritten, for each of its intermediate integrals, asPath-integral e volution of multi variate systems -4- Lester Ingber P(x;t+Δt)=∫dx′[g1/2(2πΔt)−1/2exp(−LΔt)]P(x′;t) =∫dx′G(x,x′;Δt)P(x′;t), P(x;t)=N i=1Σπ(x−xi)Pi(t), π(x−xi)=    1, (xi−1 2Δxi−1)≤x≤(xi+1 2Δxi), 0, otherwise .(6) This yields Pi(t+Δt)=Tij(Δt)Pj(t), Tij(Δt)=2 Δxi−1+Δxi∫xi+Δxi/2 xi−Δxi−1/2dx∫xj+Δxj/2 xj−Δxj−1/2dx′G(x,x′;Δt). ( 7) Tijis a banded matrix representing the Gaussian nature of the short-time probability centered about the (possibly time-dependent) drift. This histogram procedure was extended to tw odimensions using a matrix Tijkl[7]. Explicit dependence of Lon timetalso can be included without complications. Care must be used in de veloping the mesh in Δqi,which is strongly dependent on the diagonal elements of the diffusion matrix, e.g., Δqi≈(Δtgii)1/2.( 8) This constrains the dependence of the co variance of each v ariable to be a (nonlinear) function of that variable in order to present a straightforward rectangular underlying mesh. Since integration is inherently a smoothing process [8], fitting the data with the short-time probability distribution, ef fectively using an inte gral overthis epoch, permits the use of coarser meshes than the corresponding stochastic differential equation. Forexample, the coarser resolution is appropriate, as typically required, for a numerical solution of the time-dependent path inte gral. ByPath-integral e volution of multi variate systems -5- Lester Ingber considering the contributions to the first and second moments, conditions on the time and v ariable meshes can be deri ved[1]. The time slice essentially is determined by Δt≤L−1,whereLis the uniform Lagrangian, respecting ranges giving the most important contributions to the probability distrib utionP. ThusΔtis roughly measured by the diffusion divided by the square of the drift. Such calculations are useful in man ydisciplines, e.g., some financial instruments [8,9]. Monte Carlo algorithms for path inte grals are well known to ha ve extreme difficulty in e volving nonlinear systems with multiple optima[10], but this algorithm does very well on such systems. The PATHINT code was tested against the test problems gi veninprevious one-dimensional systems[1,2], and it w as established that the method of images for both Dirichlet and Neumann boundary conditions is as accurate as the boundary element methods for the systems in vestigated. Two-dimensional runs were tested by using cross products of one-dimensional examples whose analytic solutions are known. Attempts were made to process the same system considered for the quantum case [5]. Therefore, the diffusion was taken to be D=0. 2.Since the yselected an harmonic oscillator basis for their eigenvalue study,i swas assumed that natural boundary conditions are appropriate for this study ,and ranges of xandywere tested to ensure that this was reasonable. Aband of three units on each side of the short-time distribution was sufficient for these runs. Forε≤5, the range of xwastaken to be±3and the range ofywastaken to be ±6. Amesh ofΔt=0. 1wasreasonable to calculate the e volution of this system for 0. 1≤ε≤0. 5.The quantum study observed chaos at ε≥0. 14,butthe classical system appears to be v ery stable with a single peak in its probability density up through ε=0. 5.The time mesh was tested by performing se veral calculations at time meshes of Δt=0. 01on a Sun SPARC 10MP .All other calculations reported here were performed on a Sun SPARC 2. Figures 1(a) and 1(b) sho wthe evolution of the distribution for ε=0.1,t=1 0and 100, i.e., after 100 and 1000 foldings of the path integral, respecti vely.The distribution starts at single peaks at the imposed initial condition, a δfunction at the origin, i.e., x=−0. 0302andy=−0. 0603with this mesh, and swell out their stable structures within t<0.03. Note the stability o verthe duration of the calculation. A similar stability was noted for ε<0.55, as illustrated in Figures 2(a) and 2(b) for ε=0.3 and 0.5 after 1000 foldings.Path-integral e volution of multi variate systems -6- Lester Ingber Forvalues of ε>0.5,amesh ofΔt=0. 05wasused to investigate the onset of instabilities which were noted for higher εwith coarser meshes. To two significant figures, this first occurs at ε=0. 55.As εincreases, so does the peak spread quite early along a diagonal in the e volving distrib ution. As the time of the calculation increased, there w as concern that the boundaries were being approached. Therefore, for these runs, the range of xwasincreased to ±5and the range of ywasincreased to ±10. Figure 3 illustrates the e volution at time t=5. 0(100 foldings) and t=100 (2000 foldings) with ε=0.55. Figure 4 illustrates the early structure for ε=0.6 att=1 5after 300 foldings of Δt=0. 05. The PATHINT algorithm utilized here will be used to explore the time e volution of other F okker- Planck systems in the presence of moderate noise. As mentioned abo ve,such problems arise in man y systems ranging from neocortical interactions to financial mark ets. Also, we can no waccurately e xamine long-time correlations of chaotic models as multiplicati ve noise is increased to moderate and strong levels; manychaotic models do not include such le vels of noise as is found in the systems the yare attempting to model. Aproject is no wunderway under an a ward of Cray computer time from the Pittsburgh Supercomputing Center through the National Science F oundation (NSF), the Parallelizing ASA and PATHINT Project (PAPP) further de veloping this code for large systems [11].Path-integral e volution of multi variate systems -7- Lester Ingber FIGURE CAPTIONS FIG. 1. Probability density for ε=0. 1,forΔt=0. 1(a) after 100 foldings ( t=10) and (b) after 1000 foldings ( t=100). FIG. 2. Probability density after 1000 foldings of Δt=0. 1(t=100) for εset to (a) 0.3 and (b) 0.5. FIG. 3. Probability density for εset to 0.55 using Δt=0. 05,(a) att=5after 100 foldings, and (b) att=100 after 2000 foldings. As discussed in the text, the ranges of xandyforε>0.5 were increased. FIG. 4. Probability density at t=15, after 300 foldings of Δt=0. 05,forε=0.6.Path-integral e volution of multi variate systems -8- Lester Ingber REFERENCES [1] M.F .Wehner and W.G. 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[7] L.Ingber,H .Fujio, and M.F .Wehner,‘‘Mathematical comparison of combat computer models to exercise data, ’’Mathl. Comput. Modelling 15,65-90 (1991). [8] L. Ingber,‘‘Statistical mechanical aids to calculating term structure models, ’’Phys. Rev. A42, 7057-7064 (1990). [9] L. Ingber,M.F.Wehner,G.M. Jabbour ,and T.M. Barnhill, ‘ ‘Application of statistical mechanics methodology to term-structure bond-pricing models, ’’Mathl. Comput. Modelling 15,77-98 (1991). [10] K.Binder and D. Stauf fer,‘‘A simple introduction to Monte Carlo simulations and some specialized topics,’’ inApplications of the Monte Carlo Method in Statistical Physics ,ed. by K. Binder (Springer-Verlag, Berlin, 1985), p. 1-36. [11] .Path-integral e volution of multi variate systems Lester Ingber Figure1a Path-integral e volution of multi variate systems Lester Ingber Figure1b Path-integral e volution of multi variate systems Lester Ingber Figure2a Path-integral e volution of multi variate systems Lester Ingber Figure2b Path-integral e volution of multi variate systems Lester Ingber Figure3a Path-integral e volution of multi variate systems Lester Ingber Figure3b Path-integral e volution of multi variate systems Lester Ingber Figure4

arXiv:physics/0001061v1 [physics.ed-ph] 25 Jan 2000FERMI–PUB–00/027–T EFI–2000–3 February 21, 2014 Teaching Symmetry in the Introductory Physics Curriculum Christopher T. Hill1,2 and Leon M. Lederman1,3 1Fermi National Accelerator Laboratory P.O. Box 500, Batavia, Illinois, 60510 2The University of Chicago Enrico Fermi Institute, Chicago, Illinois 3The Illinois Math-Science Academy Aurora, Illinois, USA Modern physics is largely defined by fundamental symmetry pr inciples and N¨ oether’s Theorem. Yet these are not taught, or rarely ment ioned, to be- ginning students, thus missing an opportunity to reveal tha t the subject of physics is as lively and contemporary as molecular biology, and as beautiful as the arts. We prescribe a symmetry module to insert into the curriculum, of a week’s length.1 Introduction Symmetry is a crucial concept in mathematics, chemistry, an d biology. Its definition is also applicable to art, music, architecture and the innum erable patterns designed by nature, in both animate and inanimate forms. In modern physi cs, however, symmetry may be the most crucial concept of all. Fundamental symmetry principles dictate the basic laws of physics, control stucture of matter, and define the fundamental forces in nature. Some of the most famous mathematicians and physicists had th is to say about sym- metry: •“I aim at two things: On the one hand to clarify, step by step, t he philosophic- mathematical significance of the idea of symmetry and, on the other, to display the great variety of applications of symmetry in the arts, in inorganic and organic nature.” — Hermann Weyl [1]. •“Special relativity emphasizes, in fact is built on, Lorent z symmetry or Lorentz invariance, which is one of the most crucial concepts in 20th Century Physics.” — C. N. Yang (Nobel Laureate in Physics) [2]. •“Symmetry is fascinating to the human mind; everyone likes o bjects or patterns that are in some way symmetrical.... but we are most interest ed in the symmetries that exist in the basic laws themselves.” — Richard P. Feynman (Nobel Laureate in Physics) [3]. •“I heave the basketball; I know it sails in a parabola, exhibi ting perfect symmetry, which is interrupted by the basket. Its funny, but it is alway s interrupted by the basket.” — Michael Jordan (retired Chicago Bull) [4]. Today we understand that all of the fundamental forces in nat ure are unified under one elegant symmetry principle. We revere the fundamental s ymmetries of nature and we have come to intimately appreciate their subtle consequenc es. As we will see, to succumb to a crack-pot’s invention requiring us to give up the law of e nergy conservation would be to give up the notion of a symmetry principle, that time flows w ith no change in the laws 1of physics. Symmetry controls physics in a most profound way , and this was the ultimate lesson of the 20th century. Yet, even a sampling of the crucial role of symmetry in physic s by beginning students is completely omitted, not only from the high school curricu lum, but in the standard first year college calculus-based physics course. It does not app ear in the Standards. It is possible, nonetheless, to incorporate some of the unde rlying ideas of symmetry and its relationship to nature into the beginning courses in physics and mathematics, at the high school and early college level. They really are no t that difficult. When the elementary courses are spiced with these ideas, they beg in to take on some of the dimensions of a humanities or fine arts study: Symmetry is one of the most beautiful concepts, and its expression in nature is perhaps the most st unning aspect of our physical world. What follows is a description of a high school module that int roduces the key ideas which, in many examples, ties physics to astrophysics, biol ogy and chemistry. It also reveals some of the modern thinking in a conversational way. We are experimenting in the classroom, in Saturday Morning Physics at Fermilab, and els ewhere in the implementation of this approach. A lot more material that cannot be presented in this brief art icle, can be found at our symmetry website, www.emmynoether.com . We will continually update our website as our educational experiment in Symmetry proceeds . We encourage you, and your students, to visit it. And don’t hesitate to send us sugg estions, comments, and even kindly worded complaints. 2 What is Symmetry? When a group of students is asked to define “symmetry” the answ ers they give are gen- erally all correct. For example, to the question: “what is sy mmetry?” we hear some of the following: •“its like when the sides of an equilateral triangle are all th e same, or when the angles are all the same...” •“things are in the same proportion to each other... ” 2•“things that look the same when you see them from different poi nts of view ... ” From the many diverse ways of describing symmetry, one quick ly gets to agreement with the scientists’ definition: “Symmetry is an invariance of an object or system to a set of ch anges (trans- formations).” In simpler language, a thing ( a system ) is said to possess a symmetry if one can make a change ( a transformation ) in the system such that, after the change, the thing appears exactly the same ( is invariant ) as before. Let us consider some examples. 2.1 Translations in Space A physical system can simply be moved from one place to anothe r place in space. This is called a “spatial translation”. Consider a classroom pointer. Usually it is a wooden stick of a fixed length, about 1 meter. We can translate the pointer freely in space. Do its p hysical properties change as we perform this translation? Clearly they do not. The phys ical material, the atoms, the arrangement of atoms into molecules, into the fibrous mat erial that is wood, etc., do not vary in any obvious way when we translate the pointer. Thi s is a symmetry: it is a statement that the laws of physics themselves are symmetric al under translations of the system in space. Any equation we write describing the quarks, leptons, atoms, molecules, stresses and bulk moduli, electrical resistance, etc., of o ur pointer must itselfbe invariant under translation in space. For example, we can easily write a formula for the “length” of the pointer that is independent of where the pointer is located in space (we leav e this as an exercise, or see our website). Such a formula contains the information that t he length of the pointer, a physical measure of the pointer, doesn’t change under tran slations in space. Or, put another way, the formula is “invariant under spatial transl ations.” While this would be a simple example, the (highly nontrivial) assumption is that all correct equations in physics are translationally invariant! Thus, if we have a physics laboratory in which all kinds of experiments are carried out and all sorts of laws of nature are discovered and tested, 3the symmetry dictates that the same laws will be true if the la boratory itself is moved (translated ) to another location in space. The implications of this “oh so simply stated” symmetry are p rofound. It is a state- ment about the nature of space. If space had at very short dist ances the structure of, e.g., a crystal, then moving from a lattice site to a void woul d change the laws of nature within the crystal. The hypothesis that space is translationally invariant is equivalent to the statement that one point in space is equivalent to any oth er point, i.e. the symmetry is such that translations of any system or, equivalently, th e translation of the coordinate system, does not change the laws of nature. We emphasize that this is a statement about space itself ; one piece of space is as good as another! We say that space is s mooth or ho- mogeneous (Einstein called it a “continuum”). Equivalentl y, the laws and the equations that express these laws are invariant to translations, i.e. , possess translational symmetry. Now, one can get confused in applying translational invaria nce. Consider an ex- periment to study the translational symmetry of the electri c charge by measuring, e.g., acceleration of electrons in a cathode ray tube. If there wer e, outside of the laboratory, a huge magnet, then the experimental results would change whe n the tube is moved around inside of the lab. This is not, however, a violation of transl ational symmetry, because we forgot to include the magnet in the move. If we live in a region of space with intrinsic magnetic fields, then we might detect dependence upon positi on, and the symmetry would not seem to exist. However, it is our belief that flat space is s mooth and homogeneous. The most profound evidence comes later. 2.2 Translations in Time The physical world is actually a fabric of events. To describ e events we typically use a 3-dimensional coordinate system for space, but we also nee d an extra 1-dimensional coordinate system for time. This is achieved by building a cl ock. The time on the clock, together with the 3-dimensional position of somethi ng, forms a four coordinate thing ( x, y, z, t ), called an “event” (Note: we always assume that the clock is ideally located at the position of the event, so we don’t get confused about how long it takes for light to propagate from the face of a distant clock to the l ocation of the event, etc.). Some examples of events: (i) We can say that there was the even t of the firecracker explosion at ( xf, yf, zf, tf), (ii) The N.Y. Yankees’ third baseman hits a fast pitch at 4(xH, yH, zH, tH), (iii) Niel Armstrong’s foot first touched the surface of th e Moon at the event ( xM, yM, zM, tM). Now we have the important symmetry hypothesis of physics: The laws of physics, and thus all correct equations in physics, are invariant und er translations in time . That is, to all of our fabric of events, such as the events we descri bed above, we can just shift every time coordinate by an overall common constant. Mathem atically, we replace every timetifor every phsyical event by a new value ti+T. The T’s cancel in all correct physics equations; the equations are all time translationally invariant! Time, we believe, is also smooth and homogeneous. Indeed, the constancy of the basic parameters of physics, e. g., electric charge, electron mass, Planck’s constant, the speed of light, etc., over vast distances and times has been established in astronomical and geological observations t o a precision of approaching 10−8 over the entire age of Universe [2]. The laws of physics appea r to be constant in time. The experimental evidence is very strong! 2.3 Rotations A sphere (or a spherical system, or MJ’s basketball) can be ro tated about any axis that passes through the center of the sphere. The rotation angle c an be anything we want, so let’s take it to be 63o. After this rotation (often called an “operation” or “trans formation”) the appearance of the sphere is not changed. We say that the sp here is “invariant” under the “transformation” of rotating it about the axis by 6 3o. Any mathematical description we use of the sphere will also be unchanged (inva riant) under this rotation. There are an infinite number of symmetry operations that we ca n perform upon the sphere. Furthermore, there is no “smallest” nonzero rotati on that we can perform; we can perform “infinitesimal” rotations of the sphere. We say t hat the symmetry of the sphere is “continuous”. Consider again our classroom pointer. We can rotate the poin ter freely in space. Do its physical properties change as we perform this rotation? Clearly they do not. This too is a symmetry: it is a statement that the laws of physics thems elves are symmetrical under rotations in space. Under rotations in free space the length of our classroom pointer, R, doesn’t change. We could actually perform a mathematical rotation about the origin of our coordinate 5system in which we have written a formula for the length of a po inter. We would find that the formula doesn’t change (just the coordinates, the t hings the formula acts upon, do; this isn’t hard to see, and we do it on the website). We say t hat the length of the pointer is invariant under rotations. Indeed, it is our firm b elief that the laws of physics, and thus all correct equations in physics, are invariant und er rotations in space . This is, again, based upon experimental data. It is a statement about the nature of space; space is said to be isotropic , that is, all directions of space are equivalent. In summary: The laws of physics are invariant under spatial and temporal translations, and rota- tions in space . Needless to say, there are many additional symmetries, some of which we will discuss later. 3 Symmetries of the Laws of Physics and Emmy N¨ oether’s Theorem In 1905, a mathematician named Emmy (Amalie) N¨ oether, Fig. (1), proved the following theorem: For every continuous symmetry of the laws of physics, there m ust exist a conservation law. For every conservation law, there must exist a continuous sy mmetry. Thus, we have a deep and profound connection between a symmet ry of the laws of physics, and the existence of a corresponding conservation law. In presenting N¨ oether’s theorem at this level we usually state it without proof (A fai rly simple proof can be given if the student is familiar with the action principle; it can, however, be motivated with simple examples, as we do below). Conservation laws, like the conservation of energy, moment um and angular momen- tum (these are the most famous), are studied in high school. T hey are usually presented as consequences of Newton’s Laws (which is true). We now see f rom N¨ oether’s theorem that they emerge from symmetry concepts far deeper than Newt on’s laws. 6Now, as we have stated above, it is an experimental fact that t he laws of physics are invariannt under the symmetry of spatial translations. Thi s is a strong statement. What is the physical consequence of this? Thus comes the amazing t heorem of Emmy N¨ oether, which states, in this case: The conservation law corresponding to space translational symme- try is the Law of Conservation of Momentum. So, we learn in senior physics class that the total momentum o f an isolated system remains constant. The ith element of the system has a momentu m in Newtonian physics of the form: /vector pi=m/vector viand the total momentum is just the sum of all of the elements, /vectorPtotal=/vector p1+/vector p2+...+/vector pN (3.1) for a system of N elements. N¨ oether’s theorem states that /vectorPtotalis conserved, i.e., it does not change in time, no matter how the various particles inter act, because the interactions are determined by laws that don’t depend upon where the whole system is located in space! Note that momentum is, and must be, a vector quantity (hence t he little arrow, /vector, over the stuff in the equations). Why? Because momentum is ass ociated with translations in space, and the directions you can translate (move) a physi cal system form a vector! So, if you remember the N¨ oether theorem, you won’t forget th at momentum is a vector when taking an SAT test! Turning it around, the validity of the Law of Conservation of Momentum as an obser- vational fact, via N¨ oether’s theorem, supports the hypoth esis that space is homogeneous, i.e., possessing translational symmetry. The more we verif y the law of conservation of momentum, and it has been tested literally trillions of time s in laboratories all over the world, at all distance scales, the more we verify the idea tha t space is homogeneous, and not some kind of crystal lattice! We have also stated above the laws of physics are invariant un der translations in time. What conservation law then follows by N¨ oether’s Theo rem? Surprise! It is nothing less than the law of conservation of energy: The conservation law corresponding to time translational s ymmetry is the Law of Conservation of Energy. 7Since the constancy of the total energy of a system is extreme ly well tested experi- mentally, this tells us that nature’s laws are invariant und er time translations. Here is a cute example of how time invariance and energy conse rvation are inter- related. Consider a water tower that can hold a mass Mof water and has a height of Hmeters. Assume that the gravitational constant, which dete rmines the acceleration of gravity, is g, on every day of the week, except Tuesday when it is a smaller v alueg′< g. Now, we run water down from the water tower on Monday through a turbine (a fancy water wheel) generator which converts the potential energy MgH to electrical current to charge a large storage battery, Fig.(2). We’ll assume 100 % efficiencies for everything, because we are physicists. This is Monday’s job. For Tuesday ’s job we pump the water back up to H, using the battery power that we accumulated from Monday’s j ob to run the pump. But now the g′value is smaller than gand the work done is Mg′H, which is now much less than the energy we got from Monday’s job. This leave s us with M(g−g′)H extra energy still in the battery, which we can sell to a local power company to live on until next Monday. This is a perpetual motion machine! It pro duces energy for us, and we can convert that to cash. It does not conserve energy because we cooked up false laws of physics, in this case gravity, that are not time translation ally invariant! Hence, we violated a precept of N¨ oether’s Theorem. (Can you come up with simila r cute example of violating momentum conservation by making the laws of physics spatial ly inhomogeneous?) We also live in a world where the laws of physics are rotationa lly invariant: The conservation law corresponding to rotational symmetry is the Law of Conservation of Angular Momentum. Conservation of angular momentum is often demonstrated in l ecture by what is usu- ally called “the 3 dumbbell experiment”. The instructor sta nds on a rotating table, his hands outstretched, with a heavy dumbbell in each hand (who i s the third dumbbell?), Fig.(3). He turns slowly, and then brings his hands (and dumb bells) close to his body, Fig.(4). His rotation speed (angular velocity) speeds up su bstantially. What is kept con- stant is the angular momentum, J, the product of I, the moment of inertia, times the angular velocity ω. By bringing his dumbbells in close to his body, Iis decreased. But J, the angular momentum, must be conserved, so ωmust increase. Skaters do this trick all the time. 8Atoms, elementary particles, etc., all have angular moment um. The intrinsic angular momentum of an elementray particle is called spin. In any rea ction or collision, the final angular momentum must be equal to the initial angular moment um. Like our planet earth, particles spin and execute orbits and both motions have asso ciated angular momentum. Data over the past 70 or so years confirms conservation of this quantity on the macroscopic scale of people and their machines and on the microscopic sca le of particles. And now, (thanks to Emmy) we learn that these data imply that space is i sotropic; All directions in space are equivalent. The translational and rotational symmetries of space and ti me need not have existed. That they do is the way nature is. These are some of the actual p roperties of the basic concepts we use to describe the world: space and time. 4 Beyond We have described how the fundamental conservation laws of e veryday physics follow from the continuous symmetry properties of space and time. There are, however, many other conservation laws that are not usually studied in a first year physics course. A simple example is the conservation of electric charge in all reacti ons. The total electric charge in an isolated system is a constant in time. For example, proces ses like: electron−→neutrino0+ photon0(4.2) (where superscripts denote charges) in which case electric charge could completely disap- pear, are forbidden. On the other hand, processes like this o ne do occur: electron−+ proton+→neutron0+ neutrino0(4.3) Since the final state is electrically neutral, the negative e lectric charge of the electron must be identically equal and opposite to that of the proton t o an infinite number of significant figures. Indeed, we can place a large quantity of H ydrogen gas into a container and observe to a very high precision that Hydrogen atoms (whi ch are just bound states ofe−+p+) are electrically neutral. This conservation law, by N¨ oether’s Theorem, also arises f rom a profound symmetry of nature called “ gauge symmetry.” This is an example of an abstract symmetry that 9does not involve space and time. In the late 20th century we ha ve come to realize that all of the forces in nature are controlled by such gauge symme tries. Gauge symmetry is very special, and it actually leads us to the complete theory of electrons and photons, known as (quantum) electrodynamics, which has been tested t o 10−12precision. This is the most accurate and precise theory of nature that humans have ever constructed. Perhaps the most stunning result of the 20th Century has been the understanding that all known forces in nature are described by gauge symmetries . Einstein’s Special Theory of Relativity is all about relati ve motion, and is based upon a fundamental symmetry principle about motion itself. This is a statement that the laws of physics must be the same for all observers independen t of their state of uniform motion. This symmetry principle of Relativity can be expres sed in a way that shows that it is a generalization of the concept of a rotation. The t ime interval between two events that occur at the same point in space is called the “pro per time.” The proper time can be expressed, like the length of our pointer, in such a way that it is invariant under motion1A formula can be written that relates the coordinate systems of two observers moving relative to one another, in terms of their relative ve locity v. The formula is called a “Lorentz Transformation” and it mixes time and spac e, much like a rotation in thexyplane mixes xandy. Like a rotation, it leaves the proper time invariant. Thus motion is sort-of like a rotation in space and time! Unfortun ately, we must send you off to a textbook (or our website) on Special Relativity to learn about all of the miraculous effects that occur as a consequence of this. Relativity is an e xpansion of our understanding of the deep and profound symmetries of nature. Other extremely important symmetries arise at the quantum l evel. A simple example is the replacement of one atom, say a Hydrogen atom sitting in a molecule, by another Hydrogen atom. This is a symmetry because all Hydrogen atoms are exactly the same, oridentical , in all respects. There are no warts or moles or identifying b ody markings on Hydrogen atoms, or any other atomic scale particle for tha t matter, such as electrons, protons, quarks, etc. The effects of the symmetry associated with exchanging positions or motions of identical atoms or electrons or quarks, has profo und effects upon the structure of matter, from the internal structure of a nucleus of an atom , to the properties of everyday 1A moving observer sees the two events at different points in sp ace; the formula for proper time involves the spatial separation of the events divided by c, the speed of light; this differs from Newtonian physics in which the proper time would be independent of the s patial separation of events. 10materials like metals, insulators and semiconductors, to t he external stucture of a neutron star or white dwarf star. This identical particle symmetry e xplains nothing less than the “Periodic Table of the Elements,” i.e., how the motion and di stributions of the electrons are organized within the atoms as we go from Hydrogen to Urani um! All of chemistry is controlled by the interactions of electromagnetism togeth er with the symmetry of identical particles. In a one week long module we can develop these and other import ant symmetries, such as mirror symmetry, time reversal symmetry and the symm etry between matter and anti-matter (we can even explain why antimatter exists, fro m the symmetry of Relativ- ity and N¨ oether’s theorem!). Conservation laws that are as sociated with more abstract symmetries, such as “quark color,” and “supersymmetry,” ca n also be illustrated and discussed. Indeed, this leads us to the frontier of theoreti cal physics, e.g., superstrings, M-theory, and the deeply disturbing open questions, such as “(why) is the Cosmological Constant zero?” As we progressively proceed to the deepest f oundations of the structure of matter, energy, space and time, we must be more descriptiv e for our beginning students, but we become more thrilled, enchanted and excited by the fun damental symmetries that control the structure and evolution of our Universe. Let this provocative conclusion close this very incomplete survey of the role of sym- metry in physics. Please visit www.emmynoether.com for a much expanded version of this brief letter. Acknowledgements We wish to thank Shea Ferrel for Figures (3) and (4). 11Bibiliography 1.Symmetry , Hermann Weyl (Princeton Science Library, Reprint edition , Princeton, 1989). 2. C. N. Yang, in Proceedings of the First Int’l Symposium on Symmetries in Su batomic Physics , ed. W-Y. Pauchy Huang, Leonard Kisslingler, v. 32, no. 6-11 (Dec. 1994) 143 3. R. P. Feynman, The Feynman Lectures on Physics , (Addison-Wesley Pub. Co., 1963). 4. M. Jordan, private communication . 5. For limits on time dependence of fundamental constants se e, e.g., F.W. Dyson, in: Aspects of Quantum Theory , eds. A. Salam and E.P. Wigner (Cambridge Univ. Press, Cambridge, 1972) 213; in: Current Trends in the the Theory of Fields eds. J.E. Lannutti and P.K. Williams (American Institute of Phys ics, New York, 1978) 163. See also, C. T. Hill, P. J. Steinhardt, M. S. Turner Phys.Lett. B252 ,1990, 343, and references therein. 6. see, e.g. Women in Mathematics , L.M. Osen, MIT Press (1974) 141. 12Figure 1: Emmy Noether, pronounced like “mother.” Born, 188 2, she practiced at G¨ ottingen where the great mathematicians Hilbert and Klei n and the physicists Heisen- berg and Schr¨ oedinger were professors. Fleeing the rise of Naziism, she spent her last few years in the U.S. at Bryn Mawr and the Institute for Advanc ed Study at Princeton. She died in 1935 [6]. Emmy N¨ oether was one of the greatest mat hematicians of the 20th century. 13Figure 2: Water is drained through turbine generator on days when the gravitational acceleration is g′> gand energy is produced and sold to power company. On days when gravitational acceleration is g < g′the water is pumped back up into the tower at a reduced cost in energy. Hence net energy is available fro m the system if gis time dependent. 14Figure 3: The Professor with dumbells rotates slowly when hi s arms are outstretched. Figure 4: Pulling the dumbells close to his body reduces the m oment of inertia, but angular momentum is conserved, hence the Professor rotates faster. 15

arXiv:physics/0001062v1 [physics.gen-ph] 26 Jan 2000Further Effects of Varying G B.G. Sidharth∗ Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India) Abstract The correct perihelion precession was recently deduced wit hin the frame work of a time varying Gravitational constant G. Here, we de- duce also the observed gravitational bending of light and fla ttening of galactic rotational curves. 1 Introduction In a recent communication[1] we saw that it is possible to acc ount for the precession of the perihelion of Mercury, for example, only i n terms of the time varying universal constant of gravitation G. It may be m entioned that Dirac had argued[2] that a time varying G could be reconciled with General Relativity and the perihelion precession by considering a s uitable redefinition of units. We will now show that it is also possible to account f or the bending of light on the one hand and on the other, the flat galactic rota tion curves without invoking dark matter, with the same time variation o f G. 2 Bending of Light It may also be mentioned that some varying G cosmologies have been re- viewed by Narlikar and Barrow, while a fluctuational cosmolo gy with the 0∗Email:birlasc@hd1.vsnl.net.in; birlard@ap.nic.in 1above G variation has been considered by the author [3, 4, 5, 6 ] and [7]. We start by observing that, as is well known, the bending of li ght can be deduced in Newtonian theory also, though the amount of bendi ng is half of that predicted by General Relativity[8, 9, 10, 11]. In this c ase the equations for the orbit of a particle of mass mare used in the limit m→0 with due justification. A quick way of obtaining the result is to obser ve that we have the well known orbital equations[1, 12]. 1 r=GM L2(1 +ecosΘ) (1) where Mis the mass of the central object, Lis the angular momentum per unit mass, which in our case is bc,bbeing the impact parameter or minimum approach distance of light to the object, and ethe eccentricity of the trajectory is given by e2= 1 +c2L2 G2M2(2) For the bending of light, if we substitute in (1), r=±∞, and then use (2) we get α=2GM bc2(3) αbeing the deflection or bending of the light. This is half the G eneral Relativistic value. We also note that the effect of time variation is given by (cf.r ef.[1]) G=G0(1−t t0), r=r0(1−t t0) (4) where t0is the present age of the universe and tis the time elapsed from the present epoch. Using (4) the well known equation for the trajectory is given by (Cf.[13],[12],[14]) u” +u=GM L2+ut t0+ 0/parenleftbiggt t0/parenrightbigg2 (5) where u=1 rand primes denote differenciation with respect to Θ. The first term on the right hand side represents the Newtonian contribution 2while the remaining terms are the contributions due to (4). T he solution of (5) is given by u=GM L2/bracketleftbigg 1 +ecos/braceleftbigg/parenleftbigg 1−t 2t0/parenrightbigg Θ +ω/bracerightbigg/bracketrightbigg (6) where ωis a constant of integration. Corresponding to −∞< r <∞in the Newtonian case we have in the present case, −t0< t < t 0, where t0is large and infinite for practical purposes. Accordingly the analog ue of the reception of light for the observer, viz., r= +∞in the Newtonian case is obtained by taking t=t0in (6) which gives u=GM L2+ecos/parenleftbiggΘ 2+ω/parenrightbigg (7) Comparison of (7) with the Newtonian solution obtained by ne glecting terms ∼t/t0in equations (4),(5) and (6) shows that the Newtonian Θ is rep laced byΘ 2, whence the deflection obtained by equating the left side of ( 6) or (7) to zero, is cosΘ/parenleftbigg 1−t 2t0/parenrightbigg =−1 e(8) where eis given by (2). The value of the deflection from (8) is twice th e Newtonian deflection given by (3). That is the deflection αis now given not by (3) but by α=4GM bc2, which is the correct General Relativistic Formula. 3 Galactic Rotation The problem of galactic rotational curves is well known (cf. ref.[8]). We would expect, on the basis of straightforward dynamics that the ro tational velocities at the edges of galaxies would fall off according to v2≈GM r(9) whereas it is found that the velocities tend to a constant val ue, v∼300km/sec (10) 3This has lead to the hypothesis of as yet undetected dark matt er, that is that the galaxies are more massive than their visible material co ntent indicates. We observe that from (4) it can be easily deduced that a≡(¨ro−¨r)≈1 to(t¨ro+ 2˙ro)≈ −2ro t2 o(11) as we are considering infinitesimal intervals tand nearly circular orbits. Equation (11) shows (Cf.ref[1] also) that there is an anomal ous inward accel- eration, as if there is an extra attractive force, or an addit ional central mass. So, GMm r2+2mr t2 o≈mv2 r(12) From (12) it follows that v≈/parenleftBigg2r2 t2o+GM r/parenrightBigg1/2 (13) From (13) it is easily seen that at distances within the edge o f a typical galaxy, that is r <1023cmsthe equation (9) holds but as we reach the edge and beyond, that is for r≥1024cmswe have v∼107cmsper second, in agreement with (10). Thus the time variation of G given in equation (4) explains ob servation with- out taking recourse to dark matter. References [1] B.G. Sidharth, ”Effects of Varying G” to appear in Nuovo Ci mento B. [2] P.A.M. Dirac, ”Directions in Physics”, Wiley-Intersci ence, New York, 1978, p.79. [3] J.V. Narlikar, Foundations of Physics, Vol.13. No.3, 19 83. [4] J.D. Barrow and Paul Parsons, Physical Review D, Vol.55, No.4, 1997. [5] B.G. Sidharth, Int.J.Mod.Phys.A, 13 (15), 1998, p.2599 ff. [6] B.G. Sidharth, Int.J.Th.Phys. 37(4), 1998, pp.1307ff. 4[7] B.G. Sidharth, ”Instantaneous Action at a distance in a h olistic uni- verse”, Invited submission to, ”Instantaneous action at a d istance in Modern Physics: Pros and Contra”, Eds., A. Chubykalo and R. S mirnov- Rueda, ”Nova Science Books and Journals”, New York, 1999. [8] J.V. Narlikar, ”Introduction to Cosmology”, Cambridge University Press, Cambridge, 1993. [9] H.H. Denman, Am.J.Phys. 51(1), 1983, 71. [10] M.P. Silverman, Am.J.Phys. 48 , 1980, 72. [11] D.R. Brill and D. Goel, Am.J.Phys. 67(4), 1999, 317. [12] H. Goldstein, ”Classical Mechanics”, Addison-Wesley , Reading, Mass., 1966. [13] P.G. Bergmann, ”Introduction to the Theory of Relativi ty”, Prentice- Hall (New Delhi), 1969, p248ff. [14] H. Lass, ”Vector and Tensor Analysis”, McGraw-Hill Boo k Co., Tokyo, 1950, p295 ff. 5

Superluminal Near-field Dipole Electromagnetic Fields William D. Walker KTH-Visby Cramérgatan 3SE-621 57 Visby, Sweden Email: bill@visby.kth.se 1 Introduction The purpose of this paper is to present mathematical evidence that electromagnetic near-field waves and wave groups, generated by an oscillating electric dipole, propagate much faster thanthe speed of light as they are generated near the source, and reduce to the speed of light at aboutone wavelength from the source. The speed at which wave groups propagate (group speed) isshown to be the speed at which both modulated wave information and wave energy densitypropagate. Because of the similarity of the governing partial differential equations, two otherphysical systems (magnetic oscillating dipole, and gravitational radiating oscillating mass) arenoted to have similar results. 2 Analysis of electric dipole 2.1 General solution Numerous textbooks present solutions of the electromagnetic fields generated by an oscillatingelectric dipole 1,3. One simple and elegant solution solves the inhomogeneous second order “superpotential” wave equation1(pp. 254 - 260) . The electromagnetic fields can then be derived from the Hertz vector (Z). Figure 1. Spherical coordinant system used in problem: PDE (superpotential wave equation): oZ

 (1) Solution: /G01/G02 reCosZ okri R
  4)( /G01/G02 reSinZ okri 
  /G014)(
 0 /G02Z (2) Defining equations: ZxC 0 RC 0 /G01C
/G01/G02kri oeikrrSinC
 14)( 2
  /G02 (3) Field calculations: CxE tC cB o21 where: oBH  (4)Erz xH /G01 E /G02/G01 y/G02 /G03rPresented at: International Workshop ”Lorentz Group,CPT and Neutrinos” Zacatecas, Mexico, June 23-26, 19992Resultant electrical and magnetic field components for an oscillating electric dipole 
/G01/G02kri or ekri rCosE
 1 2)( 3
   
/G01/G02kri oekrikr rSinE

2 31 4)( 
  /G01 (5)
/G01/G02krieikr rSinH

24)(   /G03 (6) It should be noted that this solution is only valid for distances (r) much greater than the dipole length (d o). In the region next to the source (r ~ d o) can not be modeled as a sinusoid: tSin. Instead it must be modeled as a sinusoid inside a dirac delta function: 
 tSindro 
. The solution to this hyper-near-field problem can be calculated using the Lienard-Wiehart potentials13, 14, 18. 2.2 Lines of electric force analysis Traditionally the electric lines of force can be determined from the relation that a line element(ds) crossed with the electric field is zero. The resulting partial differential equation can then besolved yielding the classical result. Resultant Equation:   0   /G02 /G02 dSinrCdrSinrC r (7) Solution: 
 ConsttkrTankrCosSin kr

/G01 1 2 211 (8) A contour plot of this solution (Eq. 8) reveals the classical radiating oscillating electric dipole field pattern (Fig. 2). Careful examination of the pattern reveals that the wavelength of thegenerated fields are larger in the nearfield (a) and reduce to a constant wavelength after the fieldshave propagated about one wavelength from the source (b). The speed of the fields (phase speed= c ph) near the source can then be concluded to propagate faster than the speed of light from the relation that wave speed (c ph) is equal to the wavelength ( /G01) multiplied by the frequency (f), which is constant. phcf /G01 /G02 /G01 /G03 /G04 /G03phc if /G01 Figure 2. Mathematica animation contour plot of above solution /G01 /G01 /G02/G03 /G04 /G01 /G05 /G02 /G02/G06 /G07 /G08/G03 /G09 /G01 /G04 /G01 /G01/G03 /G0A /G01 /G02 /G01 /G09/G03 /G0B /G01 /G0C/G0D /G02 /G0E/G0F /G03 /G09/G03 /G10 /G01 /G0B /G01 /G04/G03 /G09/G11 /G01 /G12/G13/G14/G15 /G02 /G02 /G01/G04 /G10/G14 /G05 /G07/G0C /G03 /G02 /G03 /G02 /G16/G17/G18 /G02 /G0A/G19 /G03 /G07/G0C /G02 /G16/G17/G18 /G02 /G0B/G15 /G04 /G10/G14 /G03 /G1A/G14/G04/G0A/G1B/G11 /G02 /G10/G14 /G03/G03 /G03 /G14 /G01 /G12/G13/G14/G15 /G02 /G1C/G07/G0C /G03 /G1D/G07 /G0C /G03 /G03/G0A /G19 /G01 /G1A/G14/G04/G0A/G1B/G11 /G02 /G1D /G01 /G1C /G03 /G03 /G1A/G11/G0F/G1E/G1B/G15/G1F /G02 /G16/G17/G11/G15/G17/G20/G14/G0E/G21/G17/G15 /G02 /G09/G11/G22 /G06 /G1C/G22 /G06/G23/G06/G02/G22 /G05 /G07 /G22 /G06 /G1D/G22 /G04 /G0C/G22 /G0C /G07 /G22 /G0E/G21/G17/G15/G0E/G17/G0F/G11/G15/G18 /G04/G05 /G02/G06/G06 /G03 /G22 /G06 /G15/G22 /G06/G22 /G05 /G02 /G0A /G07 /G22 /G16/G17/G11/G15/G17/G20/G14/G12/G19/G1B/G24/G0F/G11/G25 /G04/G05 /G26/G1B/G21/G18/G1F/G22 /G16/G17/G11/G15/G17/G20/G14/G18 /G04/G05 /G06 /G04 /G23/G27/G22 /G04 /G23/G28/G22 /G04 /G23/G29/G22 /G04 /G23/G05/G22 /G04 /G23/G02/G22 /G23/G02/G22 /G23/G05/G22 /G23/G29/G22 /G23/G28/G22 /G23/G27 /G07 /G22 /G1A/G18/G2A/G1F/G04/G15/G2B/G1B/G15/G0F/G17 /G04/G05 /G05 /G01 /G0C /G03Mathematica code used to generate animation
abfinal/G01 /G02 /G01/G01 /G01/G02/G03 /G04 /G04/G02/G03 /G05 /G05/G02/G03 /G06/G07/G05/G07/G04/G01/G04/G05 aba > b Radial Field (Er) Transverse Field (E /G01) t = 03The Mathematica code (Ver. 3.0) shown above generates 24 plots of the propagating electric field at different isolated moments in time. Mouse clicking any of the frames in Mathematicaanimates the plot, revealing that the ovals of constant electric field enlarge as they propagateaway from the source (located at r = 0). It is also interesting to note that as the electric field lines are generated, some of the electric lines of force very near the dipole (~ /G01 /G01/10 wavelength) appear to propagate only a short distance and then reverse and propagate back into the source. Figure 3. Mathematica animation of contour plot of electric lines of force near source 3 Analysis of phase speed and group speed 3.1 Phase calculation The general form of the electromagnetic fields (ref Eq. 5,6) generated by a dipole is: /G01/G02tkrieiyxField/G04 /G01   (9) If the source is modeled as tCos, the resultant generated field is: 
 t CosMagtphkrCosMagField  
 
   (10) where: 22xMag y  It should be noted that the formula describing the phase is dependant on the quadrant of the complex vector.  /G09/G0A /G0D /G01 xyTankr1 1  /G09/G0A  /G0D 
/G01 221 2yxxCoskr  (11) 3.2 Definition and calculation of wave phase speed Phase speed can be defined as the speed at which a wave composed of one frequency propagates. The phase speed (c ph) of an oscillating field of the form Sin t kr() /G01 /G01, in which kk r /G02(,) /G01, can be determined by setting the phase part of the field to zero, differentiating the resultant equation, and solving for tr /G01 /G01./G01/G02 /G03 /G01/G02 /G04 /G01/G02 /G05 /G01/G02 /G06 /G01/G02 /G07/G08/G01 /G02/G06/G08/G01 /G02/G04/G01/G01/G02 /G04/G01/G02 /G06  /G01  /G02xy/G01/G02 /G03 /G01/G02 /G04 /G01/G02 /G05 /G01/G02 /G06 /G01/G02 /G07/G08/G01 /G02/G06/G08/G01 /G02/G04/G01/G01/G02 /G04/G01/G02 /G06 /G01/G02 /G03 /G01/G02 /G04 /G01/G02 /G05 /G01/G02 /G06 /G01/G02 /G07/G08/G01 /G02/G06/G08/G01 /G02/G04/G01/G01/G02 /G04/G01/G02 /G06 /G01/G02 /G03 /G01/G02 /G04 /G01/G02 /G05 /G01/G02 /G06 /G01/G02 /G07/G08/G01 /G02/G06/G08/G01 /G02/G04/G01/G01/G02 /G04/G01/G02 /G06
Signal peek propagates with speed equal to phase speed (c ph) Sinusoidal signal40)( 
krtt 0 /G01/G02/G02 /G02/G02/G03/G02/G02/G03 /G04tr rkrtrk /G01 (12) Differentiating kr /G02 /G03 /G02 with respect to r yields: rkrkr /G01/G01 /G01/G01/G02/G01 /G01 /G02 (13) Combining these results and using kco yields: rkcrco ph/G01/G01/G02 /G03 /G02 /G03/G01 /G02/G02/G01/G03 (14) 3.3 Definition and calculation of wave group speed In some physical systems the wave phase speed is a function of frequency. In these systems whenwaves composed of different frequencies propagate, the wave group (wave envelope) propagatesat a different speed (group speed) than the individual waves. The group speed is also known tobe the speed at which wave energy and wave information propagate 6(pp.268-269), 10(p.123). The group speed of an oscillating field of the form Sin t kr()
, in which kk r(,), can be calculated by considering two Fourier components of a wave group: )() () () (22 11 krtSinkrtSinrktSinrktSin

 

    (15) in which: 
12 2, kkk
12 2, 12 2, kkk12 2 The group speed (c g) can then be determined by setting the phase part of the modulation component of the field to zero, differentiating the resultant equation, and solving for tr: 0) (  
krtt 0

tr rkrtrk   rkrktrcg      (16) Differentiating kr /G04 /G02 /G03 /G04 /G02 with respect to r yields: rkrkr   

 (17) Combining these results and using the relation kco yields: 1 /G01   

  rrcg 1212 lim1/G01 /G01  
 
  /G01/G01 krc rc og small  /G01/G02 (18)Peak of modulation part of signal propagates with speed equal to group speed (c g) Modulated sinusoidal53.4 Radial electric field (E r) Applying the above phase and group speed relations (Eq. 14, 18) to the radial electrical field (E r) component (Eq. 5) yields the following results: Figure 4 kry /G01 /G02 1 /G02x (19)

3 11 31kr krTankr kr/G05 /G06 /G05 /G01 /G01/G01/G02/G02 (20)
okro kroph c krc krcc 1212)(11 /G03/G03 /G01/G01/G06 /G06/G07/G07 /G08/G09 /G0A/G0A /G0B/G0C/G0D /G01 (21)
 okrph kro g cc krkrkrcc 1 14 222 3)()(3)(1 /G03/G03 /G01/G01/G06 /G06/G0D/G0D/G01 (22) Figure 5 Figure 6 3.5 Transverse electric field (E /G01) Applying the above phase and group speed relations (Eq. 14, 18) to the transverse electrical field (E /G03) component (Eq. 5) yields the following results: Figure 7 kry
 21krx
 (23)


/G01/G01 /G02/G03 /G04/G04 /G05/G06 /G07 /G08/G08/G08 /G09/G01 4 22 1 11 krkrkrCoskr /G01 (24)



/G07/G07 /G08/G09 /G0A/G0A /G0B/G0C /G0D /G05/G0D /G05/G014 24 2 21 krkrkrkrccoph (25)
 8 6 4 224 2 )()()(7)(6)()(1 krkrkrkrkrkrcco g/G07 /G08 /G07 /G08/G07 /G08/G09 (26) Figure 8 Figure 9/G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G01/G09/G05/G06/G06/G09/G02/G06/G09/G03/G06/G09/G04/G06/G09/G05/G01 /G01 /G01 /G01 /G02/G01/G02 /G03 /G01 /G04/G05 /G06 /G02 coEr /G01 /G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G01/G06/G02/G0A/G03/G0B/G04/G0C/G05/G0D/G06/G01/G01 /G03 /G04 /G01 /G01 /G02/G01/G02 /G03 /G02/G03 /G04/G05 /G06 /G02 coEr /G01 /G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G01 /G0E/G06 /G0E/G02 /G0E/G0A /G0E/G03 /G0E/G0B/G06/G02/G0A/G03/G0B/G01 /G03 /G04 /G01 /G01 /G02/G01 /G01 /G03 /G02/G03 /G04/G05 /G06 /G02 co /G01E /G02 -co /G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G01 /G0E/G06 /G0E/G02 /G0E/G0A /G0E/G03 /G0E/G0B/G06/G02/G0A/G03/G0B/G01 /G01 /G01 /G01 /G02/G01 /G02 /G03 /G01 /G04/G05 /G06 /G02 co /G01E /G02 -coE /G01 Cg vs kr/G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G01/G06/G01/G01/G02/G01/G01/G0A/G01/G01/G03/G01/G01/G0B/G01/G01/G04/G01/G01/G01 /G02 /G02 /G03 /G04 /G03/G01/G02 /G07 /G08 /G09 /G05 /G0A /G01 /G01 /G02 /G04/G05 /G06 /G02 co Er /G01/G01 90 deg /G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G0E/G06/G01/G01/G01/G06/G01/G01/G02/G01/G01/G0A/G01/G01/G03/G01/G01/G0B/G01/G01/G04/G01/G01/G01 /G02 /G02 /G03 /G04 /G03/G01 /G01 /G07/G08 /G09 /G05/G0A /G01 /G01 /G02 /G04/G05 /G06 /G02 co /G01E /G02
180 deg63.6 Transverse magnetic field (H /G03) Applying the above phase and group speed relations (Eq. 14, 18) to the transverse magnetic field (H /G04) component (Eq. 6) yields the following results: Figure 10 1 /G01 /G02y krx /G01 /G02 (27) /G01/G01 /G02/G03 /G04/G04 /G05/G06 /G07/G08/G08 /G09/G01 21 )(1krkrCoskr /G01 (28) /G07/G07 /G08/G09 /G0A/G0A /G0B/G0C/G0D /G012)(11krccoph (29)
 4 222 )()(3)(1 krkrkrcco g/G0D/G0D/G01 (30) Figure 11 Figure 12 4 Graphical evidence of superluminal phase and group speed 4.1 Superluminal near-field phase speed of radial electric field To demonstrate the superluminal near-field phase velocity of the longitudinal electric field, the calculated phase and amplitude functions can be inserted into a cosine signal and the fieldamplitude can then be plotted in the near field as a function of space (r) at several isolatedmoments in time (t), (Fig. 14). A field propagating at the speed of light (shown as a dashed line)is also included in the plot for reference. The following parameters are used in the subsequent plots: 1m wavelength ( ), 300GHz signal frequency (f), 3.3ns signal period (T). The following Mathematica code (Fig. 13) is used to generate these plots: Figure 13. Mathematica code used to generate plots/G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G01/G06/G02/G0A/G03/G0B/G04/G0C/G05/G0D/G06/G01/G01 /G03 /G04 /G01 /G01 /G02/G0B /G02 /G03 /G02/G03 /G04/G05 /G06 /G02 co /G01H /G01 /G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G01/G09/G05/G06/G06/G09/G02/G06/G09/G03/G06/G09/G04/G06/G09/G05/G01 /G01 /G01 /G01 /G02/G0B /G02 /G03 /G01 /G04/G05 /G06 /G02 co /G01H /G01 /G01 /G01 /G02/G03 /G04 /G01 /G05 /G02 /G02/G06 /G07 /G08/G03 /G09 /G01 /G04 /G01 /G01/G03 /G0A /G01 /G02 /G01 /G09/G03 /G0B /G01 /G0C/G0D /G02 /G0E/G0F /G03 /G09/G03 /G10 /G01 /G0B /G01 /G04/G03 /G1A/G11/G0F/G1E/G1B/G15/G1F /G02 /G0E/G21/G17/G15 /G02/G06 /G02 /G01 /G14/G07 /G0C /G02 /G12/G13/G14/G15 /G02 /G02 /G01 /G14/G07 /G0C /G03 /G10/G07 /G0C /G03 /G02 /G16/G17/G18 /G02 /G0B/G15 /G04 /G10/G14 /G03 /G1A/G14/G04/G0A/G1B/G11 /G02 /G10/G14 /G03/G03 /G22/G0C /G06 /G02 /G16/G17/G18 /G02 /G0B/G15 /G04 /G10/G14 /G03/G07 /G22 /G06 /G14/G22 /G06/G23/G02/G22 /G05 /G02 /G01 /G07 /G22 /G0E/G21/G17/G15/G0E/G17/G0F/G11/G15/G18 /G04/G05 /G2C/G06/G06/G22 /G0E/G21/G17/G15/G2B/G1B/G11/G25/G1F /G04/G05 /G06 /G04 /G2C/G06/G22 /G2C/G06 /G07/G03 /G22 /G06 /G15/G22 /G06/G22 /G05 /G02 /G0A /G07/G03 /G03/G01 /G02 /G03 /G04 /G05 /G06/G01 /G07/G08/G09/G0A/G01/G01/G06/G01/G01/G02/G01/G01/G0B/G01/G01/G03/G01/G01/G0C/G01/G01/G04/G01/G01/G01 /G02 /G02 /G03 /G04 /G03/G0B /G01 /G07/G08 /G09 /G05/G0A /G01 /G02 /G02 /G04/G05 /G06 /G02 co /G01H /G01
180 deg7Figure 14. Er vs Space – Cosinusoidal Signal Tt 230 Tt 235 Tt 2311
Tt2317 The longitudinal field (shown as a solid line in the plot above) is observed to propagate away from the source, which is located at r = 0. As it propagates away from the source, the oscillation amplitude decays rapidly (1/r3) near the source (r < ), and decreases more slowly (1/r2) in the farfield (r > ) (ref Eq. 5). A field propagating at the speed of light (shown as a dashed line in the plot above) is also included in the plot for reference. Both signals start together in phase. The longitudinal field is seen to propagate faster than the light signal initially when it is generated atthe source. After propagating about one wavelength the longitudinal electric field is observed toslow down to the speed of light, resulting in a final relative phase difference of 90 degrees. Inorder to see the effect more clearly the signals can be plotted with the amplitude part of thefunction set to unity (Fig. 15). Figure 15. E r (Normalised) vs Space – Cosinusoidal Signal )(230Tt )(235Tt )(2311Tt )(2317Tt It is also instructive to plot the signals as a function of time (t) for several positions (r) away from the source (Fig 16). At the source (r = 0) both signals are observed to be in phase. Furtheraway from the source the longitudinal field signal is observed to shift 90 degrees, indicating thatit arrives earlier in time. The plots shown below are normalized for clarity, but it should be notedthat the signals have the same form even if the amplitude part of the function were included. Theonly difference is the vertical scaling of the plot. From these plots it can also be seen that the longitudinal field propagates much faster than the speed of light near the source (r < ), and reduces to the speed of light at about one wavelength from the source (r  ), resulting in a final relative phase difference of 90 degrees between the longitudinal field (shown as a solid line), and the field propagating at the speed of light (shown as dashed line)./G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G06 /G01/G08/G04 /G01/G01/G04/G01/G06/G01/G0B/G01/G01 /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G06 /G01/G08/G04 /G01/G01/G04/G01/G06/G01/G0B/G01/G01 /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G06 /G01/G08/G04 /G01/G01/G04/G01/G06/G01/G0B/G01/G01 /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G06 /G01/G08/G04 /G01/G01/G04/G01/G06/G01/G0B/G01/G01 /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 Erco /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G03/G08/G01 /G02/G07/G01/G01/G02 /G07/G03/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G03/G08/G01 /G02/G07/G01/G01/G02 /G07/G03/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G03/G08/G01 /G02/G07/G01/G01/G02 /G07/G03/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G03/G08/G01 /G02/G07/G01/G01/G02 /G07/G03/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 Erco8Figure 16. Er (Normalised) vs Time – Cosinusoidal Signal
 /G01230/G05r
 /G01235/G05r
 /G012311/G05r
 /G012317/G05r 4.2 Superluminal near-field group speed of radial electric field To demonstrate the superluminal near-field group propagation speed of the longitudinal field, the calculated phase and amplitude functions can be inserted into the spectral components of anamplitude modulated cosine signal, and the field amplitude can then be plotted as a function ofspace (r) at several isolated moments in time (t), (Fig. 18). To demonstrate this technique thegroup propagation (shown as a solid line) is compared to the phase speed propagation (shown as a dashed line) of waves of the form: tkrCos 
. Note that the phase component (kr) is independent of frequency. This result is known to produce group waves and phase waves that both propagate at the speed of light. This can be seen by using (Eq. 14): since kr /G05 /G02 ok ph crc /G09 /G09 /G0A/G0A/G09 /G0B/G01/G01/G02 . Using (Eq. 18):
o r g c c  /G01 /G02 /G02/G021 2 /G04/G01 . The following parameters are used in the following plots: Carrier part of signal - 1m wavelength ( c), 300MHz signal frequency (fc), 3.3ns signal period (Tc)  , Modulation part of the signal - m wavelength, 30MHz signal frequency (fm), 33.3ns signal period (Tm) . Note that the phase relation of both signals are the same at different isolated moments in time and that both signals propagate away from the source at the same speed. The following Mathematica code (Fig. 17) is used to generate these plots: Figure 17. Mathematica code was used to generate these plots Figure 18. Light Phase (Cosine Wave) and Group (AM Signal) vs Space
cT10230t /G05 /G01 cT t /G06 /G0510235 cT t /G06 /G05102311 cT t /G06 /G05102317/G01 /G04 /G06 /G0B /G0C /G03/G01 /G09/G08/G04/G08/G03/G01/G03/G04/G07 /G08 /G0C /G0D /G06 /G0E/G0F /G09 /G10/G11/G08 /G12 /G13 /G14 /G15/G13 /G06/G0D/G0A /G06 /G16 /G0B /G17 /G02/G04/G18 /G0F /G0E /G08 /G12 /G16 /G06 /G07 /G0D /G0C /G0D /G02 /G01 /G04 /G06 /G0B /G0C /G03/G01 /G09/G08/G04/G08/G03/G01/G03/G04/G07 /G08 /G0C /G0D /G06 /G0E/G0F /G09 /G10/G11/G08 /G12 /G13 /G14 /G15/G13 /G06/G0D/G0A /G06 /G16 /G0B /G17 /G02/G04/G18 /G0F /G0E /G08 /G12 /G16 /G06 /G07 /G0D /G0C /G0D /G02 /G01 /G04 /G06 /G0B /G0C /G03/G01 /G09/G08/G04/G08/G03/G01/G03/G04/G07 /G08 /G0C /G0D /G06 /G0E/G0F /G09 /G10/G11/G08 /G12 /G13 /G14 /G15/G13 /G06/G0D/G0A /G06 /G16 /G0B /G17 /G02/G04/G18 /G0F /G0E /G08 /G12 /G16 /G06 /G07 /G0D /G0C /G0D /G02 /G01 /G04 /G06 /G0B /G0C /G03/G01 /G09/G08/G04/G08/G03/G01/G03/G04/G07 /G08 /G0C /G0D /G06 /G0E/G0F /G09 /G10/G11/G08 /G12 /G13 /G14 /G15/G13 /G06/G0D/G0A /G06 /G16 /G0B /G17 /G02/G04/G18 /G0F /G0E /G08 /G12 /G16 /G06 /G07 /G0D /G0C /G0D /G02 cg cph/G01 /G04 /G01 /G03/G01/G01/G02/G06 /G01 /G03/G01/G01/G02/G0B /G01 /G03/G01/G01/G02/G0C /G01 /G03/G01/G01/G02/G03 /G01 /G03/G01/G01/G03 /G0D /G01 /G0E /G02/G08/G03/G08/G01 /G02/G07/G01/G01/G02 /G07/G03/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G19 /G08 /G05 /G0A /G02 /G01 /G04 /G01 /G03/G01/G01/G02/G06 /G01 /G03/G01/G01/G02/G0B /G01 /G03/G01/G01/G02/G0C /G01 /G03/G01/G01/G02/G03 /G01 /G03/G01/G01/G03 /G0D /G01 /G0E /G02/G08/G03/G08/G01 /G02/G07/G01/G01/G02 /G07/G03/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G19 /G08 /G05 /G0A /G02 /G01 /G04 /G01 /G03/G01/G01/G02/G06 /G01 /G03/G01/G01/G02/G0B /G01 /G03/G01/G01/G02/G0C /G01 /G03/G01/G01/G02/G03 /G01 /G03/G01/G01/G03 /G0D /G01 /G0E /G02/G08/G03/G08/G01 /G02/G07/G01/G01/G02 /G07/G03/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G19 /G08 /G05 /G0A /G02 /G01 /G04 /G01 /G03/G01/G01/G02/G06 /G01 /G03/G01/G01/G02/G0B /G01 /G03/G01/G01/G02/G0C /G01 /G03/G01/G01/G02/G03 /G01 /G03/G01/G01/G03 /G0D /G01 /G0E /G02/G08/G03/G08/G01 /G02/G07/G01/G01/G02 /G07/G03/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G19 /G08 /G05 /G0A /G02 Er co /G1A/G2D /G01 /G16/G17/G18 /G02 /G2E/G04 /G15 /G03 /G02 /G04 /G02 /G03 /G16/G17/G18 /G02 /G2E/G1E /G15 /G03/G05 /G03 /G2A/G19/G02 /G01 /G10/G04 /G1C/G03 /G2A/G19/G0C /G01 /G04 /G10/G04 /G04 /G10/G1E /G05 /G1C/G03 /G2A/G19/G05 /G01 /G04 /G10/G04 /G03 /G10/G1E /G05 /G1C/G03 /G1A/G2D/G02 /G01 /G0A/G14/G0F/G25/G2B/G1F/G24/G20/G04/G1F /G02 /G1A/G2D /G03 /G03 /G1A/G2D/G0C /G01 /G02 /G01 /G0C /G02 /G04 /G0C /G16/G17/G18 /G02 /G15/G2E /G04 /G04 /G2A/G19/G02 /G03 /G03 /G16/G17/G18 /G02 /G15/G2E /G04 /G04 /G15/G2E /G1E /G04 /G2A/G19/G0C /G03 /G03 /G16/G17/G18 /G02 /G15/G2E /G04 /G03 /G15/G2E /G1E /G04 /G2A/G19/G05 /G03/G05 /G03 /G01 /G01 /G02/G03 /G04 /G01 /G05 /G02 /G02/G06 /G07 /G08/G03 /G09/G04 /G01 /G04 /G01 /G01/G03 /G09/G1E /G01 /G09/G04 /G01 /G02/G06/G03 /G0A /G01 /G02 /G01 /G09/G04/G03 /G2E/G04 /G01 /G0C/G0D /G02 /G0E/G0F /G03 /G09/G04/G03 /G2E/G1E /G01 /G0C/G0D /G02 /G0E/G0F /G03 /G09/G1E /G03 /G10/G04 /G01 /G2E/G04 /G01 /G04/G03 /G10/G1E /G01 /G2E/G1E /G01 /G04/G03 /G1A/G11/G0F/G1E/G1B/G15/G1F /G02 /G0E/G21/G17/G15 /G02/G06 /G1A/G2D/G0C/G22 /G0C /G02 /G16/G17/G18 /G02 /G2E/G04 /G15 /G04 /G10/G04 /G1C /G03/G07 /G22 /G06 /G1C/G22 /G06/G22 /G02/G06 /G02 /G01 /G07/G03 /G22 /G06 /G15/G22 /G06/G22 /G02/G06 /G02 /G0A /G07/G03 /G039Plotting the signals as a function of time for several spatial positions from the source also shows that group and phase signals travel at the same speed and remain in phase as they propagate. Figure 19. Light Phase (Cosine Wave) and Group (AM Signal) vs Time
c r /G09 10230
c r /G09 10230
c r /G09 10230
c r /G09 10230 The superluminal near-field group propagation speed of the longitudinal electric field can also be demonstrated in the same way as in the above example. The calculated phase function for thefield can be inserted in into the spectral components of an amplitude modulated cosine signal andthe field amplitude (shown as a solid line in the plot below) can then be plotted as a function ofspace (r) at several isolated moments in time (t), (Fig. 21, 22). An amplitude modulated fieldpropagating at the speed of light (shown as a dashed line) is also included in the plot forreference (envelope propagates at speed of light). Note that for this reference signal both thephase speed and the group speed are equal to the speed of light (ref Fig. 18, 19). The followingmathematica code (Fig. 20) is used to generate the plots below. The same signal parameters usedin the previous example are used in the calculation. Figure 20. Mathematica code used to generate plots Figure 21 . Er (Normalized) vs Space – AM Signal
cT t  10230
cT t  10235
cT t  102311
cT t  102317/G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G07 /G08 /G0C /G0D /G06 /G0E/G0F /G09 /G10/G11/G08 /G12 /G13 /G14 /G15/G13 /G06/G0D/G0A /G06 /G16 /G0B /G17 /G02/G04/G18 /G0F /G0E /G08 /G12 /G16 /G06 /G07 /G0D /G0C /G0D /G14 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G07 /G08 /G0C /G0D /G06 /G0E/G0F /G09 /G10/G11/G08 /G12 /G13 /G14 /G15/G13 /G06/G0D/G0A /G06 /G16 /G0B /G17 /G02/G04/G18 /G0F /G0E /G08 /G12 /G16 /G06 /G07 /G0D /G0C /G0D /G14 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G07 /G08 /G0C /G0D /G06 /G0E/G0F /G09 /G10/G11/G08 /G12 /G13 /G14 /G15/G13 /G06/G0D/G0A /G06 /G16 /G0B /G17 /G02/G04/G18 /G0F /G0E /G08 /G12 /G16 /G06 /G07 /G0D /G0C /G0D /G14 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G07 /G08 /G0C /G0D /G06 /G0E/G0F /G09 /G10/G11/G08 /G12 /G13 /G14 /G15/G13 /G06/G0D/G0A /G06 /G16 /G0B /G17 /G02/G04/G18 /G0F /G0E /G08 /G12 /G16 /G06 /G07 /G0D /G0C /G0D /G14 cg cph /G01 /G04 /G06 /G0B /G0C /G03/G01 /G0F /G01 /G0A /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G0E /G0F/G06 /G10/G0A /G01 /G04 /G06 /G0B /G0C /G03/G01 /G0F /G01 /G0A /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G0E /G0F/G06 /G10/G0A /G01 /G04 /G06 /G0B /G0C /G03/G01 /G0F /G01 /G0A /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G0E /G0F/G06 /G10/G0A /G01 /G04 /G06 /G0B /G0C /G03/G01 /G0F /G01 /G0A /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G0E /G0F/G06 /G10/G0A Erco/G1A/G2D /G01 /G16/G17/G18 /G02 /G2E/G04 /G15 /G03 /G02 /G04 /G02 /G03 /G16/G17/G18 /G02 /G2E/G1E /G15 /G03/G05 /G03 /G2A/G19/G02 /G01 /G10/G04 /G1C/G03 /G2A/G19/G0C /G01 /G04 /G10/G04 /G04 /G10/G1E /G05 /G1C/G03 /G2A/G19/G05 /G01 /G04 /G10/G04 /G03 /G10/G1E /G05 /G1C/G03 /G2A/G19/G02/G02 /G01 /G10/G04 /G1C /G03 /G1A/G14/G04/G0A/G1B/G11 /G02 /G04 /G10/G04 /G1C /G03 /G03 /G2A/G19/G0C/G0C /G01 /G04 /G10/G04 /G04 /G10/G1E /G05 /G1C /G03 /G1A/G14/G04/G0A/G1B/G11 /G02 /G04 /G04 /G10/G04 /G04 /G10/G1E /G05 /G1C /G03 /G03 /G2A/G19/G05/G05 /G01 /G04 /G10/G04 /G03 /G10/G1E /G05 /G1C /G03 /G1A/G14/G04/G0A/G1B/G11 /G02 /G04 /G04 /G10/G04 /G03 /G10/G1E /G05 /G1C /G03 /G03 /G1A/G2D/G02 /G01 /G0A/G14/G0F/G25/G2B/G1F/G24/G20/G04/G1F /G02 /G1A/G2D /G03 /G03 /G1A/G2D/G0C /G01 /G02 /G01 /G0C /G02 /G04 /G0C /G16/G17/G18 /G02 /G15/G2E /G04 /G04 /G2A/G19/G02 /G03 /G03 /G16/G17/G18 /G02 /G15/G2E /G04 /G04 /G15/G2E /G1E /G04 /G2A/G19/G0C /G03 /G03 /G16/G17/G18 /G02 /G15/G2E /G04 /G03 /G15/G2E /G1E /G04 /G2A/G19/G05 /G03/G05 /G03 /G1A/G2D/G05 /G01 /G02 /G01 /G0C /G02 /G04 /G0C /G16/G17/G18 /G02 /G15/G2E /G04 /G04 /G2A/G19/G02/G02 /G03 /G03 /G16/G17/G18 /G02 /G15/G2E /G04 /G04 /G15/G2E /G1E /G04 /G2A/G19/G0C/G0C /G03 /G03 /G16/G17/G18 /G02 /G15/G2E /G04 /G03 /G15/G2E /G1E /G04 /G2A/G19/G05/G05 /G03/G05 /G03 /G01 /G01 /G02/G03 /G04 /G01 /G05 /G02 /G02/G06 /G07 /G08/G03 /G09/G04 /G01 /G04 /G01 /G01/G03 /G09/G1E /G01 /G09/G04 /G01 /G02/G06/G03 /G0A /G01 /G02 /G01 /G09/G04/G03 /G2E/G04 /G01 /G0C/G0D /G02 /G0E/G0F /G03 /G09/G04/G03 /G2E/G1E /G01 /G0C/G0D /G02 /G0E/G0F /G03 /G09/G1E /G03 /G10/G04 /G01 /G2E/G04 /G01 /G04/G03 /G10/G1E /G01 /G2E/G1E /G01 /G04/G03 /G1A/G11/G0F/G1E/G1B/G15/G1F /G02 /G0E/G21/G17/G15 /G02/G06 /G1A/G2D/G05/G22 /G1A/G2D/G0C /G07 /G22 /G06 /G1C/G22 /G06/G22 /G02/G06 /G02 /G01 /G07 /G22 /G0E/G21/G17/G15/G0E/G17/G0F/G11/G15/G18 /G04/G05 /G2C/G06/G06 /G03 /G22 /G06 /G15/G22 /G06/G22 /G02/G06 /G02 /G0A /G07/G03 /G0310Figure 22. Zoom of E r (Normalized) vs Space – AM Signal
cTt 230/G05
cTt 235/G05
cTt 2311/G05
cTt 2317/G05 The above plots show an amplitude modulated longitudinal field group packet (shown as a solid line) propagating away from the source, at r = 0 (group maxima marked by vertical arrow). Apropagating speed of light group wave (shown as a dashed line) is also provided for reference.The group maxima of the amplitude modulated longitudinal field is observed to propagate to theright side of the plot before the group maxima of the speed of light wave. These series of plotsclearly demonstrate that the longitudinal group wave propagates much faster than the speed of light near the source ( r < c/G09). After propagating about one carrier wavelength from the source (r  c) the modulation part of longitudinal field (envelope of solid line) reduces to the speed of light, resulting in a final relative phase difference of 90 degrees (relative to the carrier signal)between the longitudinal field, and the field propagating at the speed of light. It is also very instructive to plot the field amplitude of the amplitude modulated longitudinal wave (shown as a solid line in plot below) as a function of time (t), for several positions awayfrom the source (r), (Fig. 23, 24). As before, an amplitude modulated wave traveling with lightspeed is also plotted for reference (shown as a dashed line). At the source (r = 0) both signals areobserved to be in phase. Further away from the source the modulated longitudinal field signal isobserved to shift to the left, indicating that the modulation part of longitudinal field (envelope)arrives earlier in time. The plots shown below are normalized for clarity, but it should be notedthat the signals have the same form even if the amplitude part of the function were included. Theonly difference is the vertical scaling of the plot. From these plots it can be seen that themodulation part of longitudinal field (envelope of solid line) propagates much faster than themodulated light speed signal (envelope of dashed line propagates at speed of light) near the source (r <  c). After propagating about one carrier wavelength from the source (r /G01 c) the modulation part of longitudinal field (envelope of solid line) reduces to the speed of light,resulting in a final relative phase difference of 90 degrees (relative to the carrier signal) betweenthe longitudinal field, and the field propagating at the speed of light. Figure 23. E r (Normalized) vs Time – AM Signal c r /G09 10230 c r /G09 10235 c r /G09 102311 c r /G09 102317/G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G1A/G04 /G04/G05 /G01 /G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G1A/G04 /G04/G05 /G01 /G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G1A/G04 /G04/G05 /G01 /G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G09 /G01 /G0A /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G1A/G04 /G04/G05 /G01 /G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G0C/G0D /G0E /G0F/G06 /G10/G0A /G02 Erco /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G19 /G08 /G05 /G0A /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G19 /G08 /G05 /G0A /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G19 /G08 /G05 /G0A /G01 /G01/G02 /G07 /G03 /G03/G02 /G07 /G04 /G04/G02 /G07 /G05 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G01/G02 /G01 /G03/G04/G02 /G05 /G06/G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G19 /G08 /G05 /G0A coEr11/G01 /G01/G02 /G04 /G01/G02 /G06 /G01/G02 /G0B /G01/G02 /G0C /G03 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G1A/G04 /G04/G05 /G01 /G01/G02 /G01 /G03 /G04 /G02/G05/G06 /G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G19 /G08 /G05 /G0A /G01 /G01/G02 /G04 /G01/G02 /G06 /G01/G02 /G0B /G01/G02 /G0C /G03 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G1A/G04 /G04/G05 /G01 /G01/G02 /G01 /G03 /G04 /G02/G05/G06 /G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G19 /G08 /G05 /G0A /G01 /G01/G02 /G04 /G01/G02 /G06 /G01/G02 /G0B /G01/G02 /G0C /G03 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G1A/G04 /G04/G05 /G01 /G01/G02 /G01 /G03 /G04 /G02/G05/G06 /G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G19 /G08 /G05 /G0A /G01 /G01/G02 /G04 /G01/G02 /G06 /G01/G02 /G0B /G01/G02 /G0C /G03 /G0D /G01 /G0F/G03 /G01/G01 /G03/G0E /G02/G08/G04/G08/G03/G01/G03/G04/G01 /G02 /G01 /G03 /G04 /G05 /G06 /G07 /G08 /G09 /G01 /G0A /G0B /G02/G1A/G04 /G04/G05 /G01 /G01/G02 /G01 /G03 /G04 /G02/G05/G06 /G07 /G08 /G09 /G0A /G0B /G02 /G0C/G0D /G19 /G08 /G05 /G0A /G01/G02cr /G012317/G01 /G01/G02cr /G012311/G01 /G01/G02cr /G01235/G01 /G01/G02cr /G01230/G01coEr t (x10-8 s) t (x10-8 s) t (x10-8 s) t (x10-8 s)Zoom – Er (Normalized ) vs Time Zoom – Er (Normalized ) vs Time Zoom – Er (Normalized ) vs Time Zoom – Er (Normalized ) vs Time Figure 24. Zoom of Er (Normalized) vs Time – AM Signal cr /G09230
cr /G09235
cr /G092311
cr /G092317 The first frame of the plot shows the two wave groups starting in phase at the source (r = 0). The following frames show that as the two waves propagate away from the source, the group maxima(marked by a vertical arrow) of the amplitude modulated longitudinal arrives earlier in time thanthe group maxima of a light speed amplitude modulated signal, thus demonstrating that the groupspeed of an amplitude modulated longitudinal electrical field is much faster than the speed of light near the source (r < c/G09), and reduces to the speed of light after it has propagated about one wavelength from the source (r /G01c/G09), resulting in a final phase difference of 90 degrees. 5 Relation between group speed and propagation speed of information and energy Several authors have indicated that wave group speed (propagation speed of wave envelope) is also the speed at which wave energy and wave information propagates 6(pp.268-269), 10(p.123). One intuitive way to understand this is to mathematically amplitude modulate and demodulate apropagating wave, thereby transmitting and detecting information. An amplitude modulatedwave can be mathematically modeled as follows:   tCostCosmSigAMc m   1 (31) where (m) is the modulation frequency, (c) is the carrier frequency, and (m) is the index of modulation. Using trigonometric identities the AM signal can be shown to be:   ] [2][] [2t CosmtCost CosmSigAMmc c mc /G01 /G01 /G01 /G01 /G01 /G01 /G01 /G01 /G02 /G03 (32) Wave propagation can then be modelled by inserting the calculated phase relation of the wave into the spectral components of the modulated signal.   ] [2][] [23 2 1 /G02 /G01 /G01 /G02 /G01 /G02 /G01 /G01 /G01 /G01 /G01 /G01 /G01 /G01 /G02 /G03 t CosmtCost CosmSigAMmc c mc (33) If the longitudinal electrical field is used to transmit the signal, then the near-field phase (r < c/G09), relations for the spectral components are (ref Eq. 20): where
 133 33
cmr c
/G02/G03 233 33/G01cr c
 333 33cmr c (34)12The speed at which the wave group (envelope) propagates can be determined by squaring the resultant modulated signal (Eq. 35) and noting the phase shift ( m /G04) of the modulation component (m) of the resultant signal. Performing this computation on the longitudinal electrical field and using trigonometric identities yields:

2 3 2 1 ] [21][] [21 /G02/G03/G04 /G05/G06/G07/G08 /G08 /G08 /G08 /G08 /G08 /G09 /G0A /G01 /G02 /G02 /G01 /G02 /G01 /G02 /G02 t Cos tCost CosDetSigAMmc c mc = (35) The resulatant phase shift of the modulation component is: 3 2phph m
/G04 . Substituting the phase relations (ref Eq. 34) yields:   32 2 3 3333 crm mc cm crdom   /G04/G04 
 /G03 /G04/G04 (36) The speed at which the modulation envelope (information) propagates can then be calculated using (Eq. 14). Performing this computation on the above result yields the same answer as whencalculated using the group speed (ref Eq. 22): ph c m mc ccrdm m g c rc rc rc mc 31 3 33223 2 2 23 30   
 /G05/G05/G03 /G04/G04      /G04 /G04 /G04 (37) Note that this model can also be used to show that the near-field energy density generated by an electric dipole propagates at the group speed, since the energy density (w) of an electromagneticfield is known to be equal to the sum of the squares of each field component 5(p.127): 2 2 2 2 2 2/G02 /G01

B E Ewo o ro   (38) 6 Proposed experiments to measure superluminal near-field phase and group speed of the longitudinal electric field The previous mathematical arguments have indicated that the group speed of a propagating near- field longitudinal electric field is much faster than the speed of light for propagation distances less than one wavelength (r < ). The approximate form of the group speed, of the near-field longitudinal electric field (ref Eq. 22) is:  23 23 3 rc krcco o g  (39) The following experiment is proposed to measure the near-field group speed of a propagating near-field longitudinal electric field. It is suggested that an amplitude modulated signal beinjected into one end of a parallel plate capacitor and detected on the other side of the capacitorby an amplitude demodulator. The phase difference between the resultant demodulated signal/G01 /G02/G01/G01 /G02/G03/G04/G05 /G01 /G02 /G06/G07/G01 /G02 /G02/G08 /G09 /G0A /G02 /G01/G01 /G02/G0B /G03/G04/G05 /G01 /G06/G07/G01 /G02 /G06/G07/G02 /G02 /G08/G09 /G0B /G02 /G01/G01 /G02/G0B /G03/G04/G05 /G01 /G06/G07/G01 /G01 /G06/G07/G0C /G02 /G02/G08 /G09 /G0A /G02 /G08/G09 /G0B /G02 /G01 /G01 /G02/G0B /G03/G04/G05 /G01 /G06/G07/G01 /G02 /G06/G07/G0C /G01 /G08/G09 /G0B /G02 /G01/G01 /G02/G0B /G03/G04/G05 /G01 /G06/G07/G01 /G01 /G06/G07/G02 /G02 /G02/G08 /G09 /G0A /G01 /G08/G09 /G0B /G02+O[m2]13and the original modulation signal should then be measured. The gap distance of the capacitor plates should then be increased and the phase difference should then be measured again. The group speed can then be determined by entering the value of the modulation frequency ( ), the measured change in phase ( ), and the change in capacitor plate gap distance (d o) in the following relation:  om gdc (40) Because the phase speed of the longitudinal electric field is much faster than the speed of light, the expected phase change may not be easy to measure. In the nearfield, at distances less thanone tenth wavelength (relative to the carrier frequency), the group speed of the longitudinalelectric field is approximately (ref Eq. 22): 223 3ocgdcc  (41) 332 3cd cdocm gom     (42) Note that using a high modulation frequency and an even higher carrier frequency can increase the observed phase change. If a 50MHz modulation frequency and a 500MHz carrier frequencywere used to generate the amplitude modulation signal then a 1mm change in capacitor platedistance would generate a 2x10 -5 deg phase change. Note that these frequencies correspond to a 6m far-field modulation electrical wavelength and a 0.6m carrier wavelength. This phase changewould be very difficult to measure but it may be possible using a high phase sensitivity lock-intechnique developed by the author 14. 7 Superluminal wave propagation in other physical systems 7.1 Electromagnetic fields generated by a magnetic dipole. The electromagnetic fields generated by an oscillating current loop has been shown by severalauthors to have a similar form to the electric dipole 1(p261) 3(pp. 623-625, 601). The only difference between the solutions is that electric and magnetic fields are reversed: Resultant electrical and magnetic fields for an oscillating magnetic dipole 
/G01/G02kri o r ekrirCosmH
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/G01/G02kri o ooeikr rSinmE

 /G09/G0A  /G0D221 4)(  
 /G03 (44)14Consequently, all the analysis performed on the electric dipole in the previous pages can also be applied to this system. Specifically, the near-field phase and group speeds of this system are alsoconcluded to be much faster than the speed of light. 7.2 Gravitational fields generated by an oscillating mass Mathematical analysis of an the gravitational fields generated by a vibrating mass reveals that forweak gravitational fields, along the axis of vibration, the oscillating gravitational dipole ismodeled with the a partial differential equation similar to that of the oscillating electric dipole.One significant difference between the two systems is that in order to conserve momentum, avibrating mass must be accompanied with another oscillating mass oscillating with the samefrequency but with opposite phase. The effect of the second vibrating mass is to cancel thepropagating gravitational fields in the far field. However, near the source the second vibratingmass does not contribute significantly to the resultant propagating gravitational fields, and can beneglected in the modelling 18. Using the Einstein relation: GG cT/G01/G02 /G01/G02/G03/G058 4 (45) Along the axis of vibration, for small masses and low velocities, the Einstein equation reduces to: oo o oT cG rtc2 22 22 24 1 /G0A 
 /G09/G0A  /G0D
(46) The only non-vanishing term in the energy momentum tensor to order /G032 is: 
  2 21)(    OtSindrcTo oo 
/G06 in which: oo cd (47) Solving the partial differential equation yields17:  21 11 11   /G0A O cltCoscltSinrK o oo      /G09/G0A  /G0D 

 /G09/G0A  /G0D 

 (48) in which: Ko = -m G , d ro,  /G09/G0A  /G0D 

 oocltSindrl (49) The gravitational field (g) can then be calculated using the relation /G0A rg
:


  .. 1 21 12 223 2 2hh O OtSin rd rd rmGgo o   

       /G09/G0A /G0D/G05    (50)15Since      /G09/G0A  /G0D 2 oo ph cdOcr   , solving this relation for the phase speed of the longitudinal gravitational field yields:      /G09/G0A  /G0Do o ophc drOcc 2 (51) Therefore it is concluded that near the source, along the axis of vibration, the longitudinal gravitational phase speed is much faster than the speed of light. The group velocity has beenshown to be 1 12/G01 /G01   /G09/G0A /G0D    r rcg  (ref Eq. 18). Inserting the relation for the phase speed (ref Eq. 14) yields: 1 /G01      /G09/G0A  /G0D  phgcc . Inserting the order estimate of the phase speed (Eq. 51) yields:      /G09/G0A  /G0Do o ogc drOcc 22 (52) From the above results (Eq. 51, 52) it is concluded that near the source both the phase speed and the group speed of the longitudinal gravitational field, along the axis of vibration of the mass, aremuch faster than the speed of light. 8 Conclusion This paper has provided mathematical evidence that electromagnetic near-field waves and wavegroups, generated by an oscillating electric dipole, propagate much faster than the speed of lightas they are generated near the source, and reduce to the speed of light at about one wavelengthfrom the source. The speed at which wave groups propagate (group speed) has been shown to bethe speed at which both the wave energy density and modulated wave information propagate.Because of the similarity of the governing partial differential equations, two other physicalsystems (magnetic oscillating dipole, and gravitational radiating oscillating mass) have beenshown to have similar near-field superluminal results.169 References 1 W. Panofsky, M. Philips, Classical electricity and magnetism, Addison-Wesley Pb. Co., (1962) 2 J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, (1975) 3 P. Lorrain, D. Corson, Electromagnetic fields and waves, W. H. Freeman and Company, (1970) 4 R. Feynman, R. Leighton, M. Sands, Feynman lectures on physics, Vol. 2, Ch. 21, Addison-Wesley Pb. Co., (1964) 5 D. Towne, Wave phenomena, Dover Pub., New York (1967) 6 F. Crawford, Waves Berkeley physics course, Vol. 3, McGraw-Hill Pub. Co. (1968) 7 W. Elmore, M. Heald, Physics of waves, Dover books, New York, (1969) 8 M. Heald, J. Marion, Classical Electromagnetic radiation, Saunders Pb. Co. (1995) 9 K. Graff, Wave motion in elastic solids, Dover pub., New York, (1975) 10 W. Gough, J. Richards, R. Williams, Vibrations and waves, Prentice Hall pub., (1996) 11 I. Main, Vibrations and waves in physics, Cambridge Univ. Press, (1994) 12 H. Pain, The physics of vibrations and waves, John Wiley & Sons, (1993) 13 W. Walker, Superluminal propagation speed of longitudinally oscillating electrical fields, Conference on causality and locality in modern physics, Kluwer Acad pub., (1998) 14 W. Walker, J. Dual, Phase speed of longitudinally oscillating gravitational fields, E Amaldi conference on gravitational waves, (1997) 15 L. Brillouin, wave propagation and group velocity, Academic press, (1960) 16 W. Walker J. Dual, Experiment to measure the propagation speed of gravitational interaction, Virgo Conference on gravitational waves sources and detectors, World Scientific, (1997) 17 N. Straumann, General Relativity and relativistic astrophysics, texts and monographs in physics, Springer Verlag, Berlin, Heidellberg, pp. 218-219 (1991) 18 W. Walker, Gravitational interaction studies, Dissertation ETH No. 12289, Zurich, Switzerland, (1997)

arXiv:physics/0001064v1 [physics.gen-ph] 27 Jan 2000Space-time geometry of quantum dielectrics Ulf Leonhardt School of Physics and Astronomy, University of St Andrews, N orth Haugh, St Andrews, Fife, KY16 9SS, Scotland Physics Department, Royal Institute of Technology (KTH), L indstedtsv¨ agen 24, S-10044 Stockholm, Sweden Light experiences dielectric matter as an effective gravita tional field and matter experiences light as a form of gravity as well. Light and matter waves see each ot her as dual space–time metrics, thus establishing a unique model in field theory. Actio et reactio are governed by Abraham’s energy– momentum tensor and equations of state for quantum dielectr ics. 03.75.-b ,03.50.-z ,04.20.-q I. INTRODUCTION A moving dielectric medium appears to light as an ef- fective gravitational field [1–4]. The medium alters the way in which an electromagnetic field perceives space and time, formulated most concisely in Gordon’s effec- tive space–time metric [1] gF αβ=gαβ+/parenleftbigg1 εµ−1/parenrightbigg uαuβ. (1) We allow for a back–ground metric gαβ, mostly to have the convenience of choosing arbitrary coordinates, but also for the possible inclusion of a genuine gravitational field. Gordon’s metric (1) depends on the dielectric properties of the medium, on the permittivity εand on the magnetic permeability µ, as well as on the four– dimensional flow uαof the medium (the local four– velocity). The product εµis the square of the refractive index and the prefactor 1 −(εµ)−1is known as Fresnel’s dragging coefficient [5–7] (in Fresnel’s days the part of the ether that the moving medium is able to drag [6]). In the limit of geometrical optics [8], light rays are zero– geodesic lines with respect to Gordon’s metric [1–4]. In the special case of a medium at rest, this result is equiv- alent to Fermat’s principle [8] and to the formulation of geometrical optics as a non–Euclidean geometry in space [9]. Light sees dielectric matter as an effective space–time metric. How does matter see light? In atom optics [10], the traditional role of light and matter is reversed: Atomic de–Broglie waves are subject to atom–optical in- struments made of light. Light acts on matter waves in a similar way as matter acts on light. This paper indicates that an atomic matter wave experiences an electromag- netic field as the effective metric gA αβ= (1−aLF)gαβ−bTF αβ (2) with a=1 mc2ρ/parenleftbigg ε+1 µ−2/parenrightbigg , b=1 mc2ρ/parenleftbigg ε−1 µ/parenrightbigg .(3) HereLFis the Lagrangian of the free electromagnetic field, defined in Eq. (8), and TF αβis the free–field energy– momentum tensor (10). As usual, cdenotes the speed oflight in vacuum and mis the mass of a single dielectric atom. In the definition (3), ρcan be regarded as the prob- ability density of the atomic de–Broglie wave, for most practical purposes. (Strictly speaking, mc2ρdescribes the total enthalpy density of the matter wave, includ- ing the rest energy as the lion’s share.) Throughout this paper we employ SI units and use the Landau–Lifshitz convention [11] of general relativity (with the exception of using greek space–time and latin space indices). To de- rive the result (2) with the dielectric parameters (3) we postulate that the interaction between light and matter takes on the general form of a metric. Then we demon- strate the consistency of this idea with previous knowl- edge, and in particular with Gordon’s metric (1). The metric (2) indicates that the energy–momentum of light curves directly the space–time of a dielectric matter wave. Under normal circumstances the deviation from the back–ground geometry is very small, see Eqs. (2) and (3), because the ratio between the electromagnetic energy and the atomic rest energy mc2is typically an extremely small number. In the Newtonian limit of gen- eral relativity [11], the gravitational correction to a flat Minkowski space–time is tiny as well, because the cor- rection is proportional to the ratio between the potential energy and mc2of a test particle. For weak gravitational fields and low test–particle velocities, general relativit y is an equivalent formulation of Newtonian physics that agrees in all predicted effects and yet establishes a rad- ically different physical interpretation. Similarly, give n the current state of the art in atom optics, the idea that light curves the space–time for matter waves is an equiv- alent formulation of the known light forces, i.e. of the dipole force and of the recently investigated R¨ ontgen in- teraction [12]. However, one can conceive of significantly enhancing the dielectric properties of matter waves [4] using similar methods as in the spectacular demonstra- tions of slow light [13]. Loosely speaking, a large effective dielectric constant εcould counteract the rest energy mc2 in the relations (3). In this way one could use light to build atom–optical analogues of astronomical objects on Earth, for example a black hole made of light. 1II. ELECTROMAGNETIC FIELDS A. Field tensors Let us first agree on the definitions of the principal electromagnetic quantities in SI units in general relativ- ity. We employ the space–time coordinates xµ= (ct,x). The electromagnetic four–potential is Aν= (U,−cA). (4) The electromagnetic field–strength tensor is constructed as Fµν≡DµAν−DνAµ=∂µAν−∂νAµ (5) using the covariant derivatives Dµwith respect to the back–ground metric gµν. As is well known [11], in the definition (5) of Fµνon a possibly curved space–time, we have been able to replace the Dµby ordinary partial derivatives ∂µ≡∂/∂xµ. The field–strength tensor reads in local–galilean coordinates (in a local Minkowski frame) Fµν= 0ExEyEz −Ex0−cBzcBy −EycBz0−cBx −Ez−cBycBx0 . (6) It will become useful at a later stage of this enterprise to introduce a four–dimensional formulation Hµνof the dielectric DandHfields, Hµν= 0−Dx−Dy−Dz Dx0−Hz/c H y/c DyHz/c 0−Hx/c Dz−Hy/c H x/c 0 , (7) here defined in local–galilean coordinates. B. Quadratic field tensors In dielectric media, induced atomic dipoles constitute an interaction between light and matter that is quadratic in the electromagnetic field–strength tensor [14]. Let us therefore list a set of linearly independent second–rank tensors that are quadratic in Fµν. The most elemen- tary one is the product of the metric tensor gµνwith the scalar Lagrangian LFof the free electromagnetic field [11]. This Lagrangian is LF=−ε0 4FαβFαβ=−ε0 4gαα′gββ‘FαβFα′β′,(8) or, in local–galilean coordinates, LF=ε0 2/parenleftbig E2−c2B2/parenrightbig . (9) Another quadratic second–rank tensor is the free electro- magnetic energy–momentum tensor [11]TF µν=ε0FµαgαβFβν−LFgµν, (10) or, in local–galilean coordinates, TF µν=/parenleftbigg I−S/c −S/c σ/parenrightbigg , Tµν F=/parenleftbigg IS/c S/c σ/parenrightbigg (11) with I=ε0 2/parenleftbig E2+c2B2/parenrightbig ,S=ε0c2E∧B, σ=ε0/bracketleftbigg/parenleftbiggE2 2+c2B2 2/parenrightbigg 1−E⊗E−c2B⊗B/bracketrightbigg .(12) HereIdenotes the intensity, Sis the Poynting vector, andσis Maxwell’s stress tensor. The symbols ∧and⊗ denote the three–dimensional vector and tensor product, respectively. We can only form second–rank tensors from FαβFα′β′ by some contraction. Consequently, the linear combi- nations of the two elementary tensors LFgµνandTF µν form the complete class of second–rank tensors that are quadratic in the field strengths Fµν. III. CLASSICAL ATOMS A. Postulates Consider a classical atom in an electromagnetic field. The atom is point–like, has a mass m, and can sustain in- duced electric and magnetic dipoles. In the restframe of the atom the dipoles respond to the square of the electric field strength, E2, and to the magnetic B2, respectively. How does a dielectric atom experience the electromag- netic field when the atom is moving? Let us postulate that the atom sees the field as an effec- tive metric. Consequently, according to general relativit y [11], the action S0of the atom is S0=−mc/integraldisplay ds , ds2=gA µνdxµdxν. (13) Let us further postulate that the metric of the atom, gA µν, is quadratic in the electromagnetic field strengths. Any metric is a second–rank tensor. Hence, we obtain from Sec. IIB the general form (2) mentioned in the Introduc- tion. B. Properties A metric of the structure (2) has nice mathematical properties. In particular, the contravariant metric ten- sorgµν A(the inverse of gA µν) takes on a simple analytic expression, √−gAgµν A=√−g[(1−aLF)gµν+bTµν F] (14) 2with gA≡det(gA µν) (15) and √−gA=√−g/bracketleftbigg (1−aLF)2−b2 4TF αβTαβ F/bracketrightbigg ,(16) as one verifies in local–galilean coordinates, with the re- lationTF αβTαβ F=ε2 0[(E2−c2B2)2+ 4c2(E·B)2]. C. Non–relativistic limit So far, we have not seen how the metric theory (2) and (13) is related to the model of a moving induced dipole. Let us consider the non–relativistic limit of velocities lo w compared with the speed of light. This limit corresponds to a motion in an inertial frame close to a restframe co– moving with the atom. We also regard the electromag- netic field energy to be weak compared with the atomic rest energy mc2. We neglect any genuine gravitational field, and obtain in cartesian coordinates ds=/radicalig (1−aLF)(c2dt2−dx2)−bTFµνdxµdxν ≈/radicalig c2dt2−dx2−/parenleftbig aLF+bTF 00/parenrightbig c2dt2 ≈/parenleftbigg 1−v2 2c2−aLF+bTF 00 2/parenrightbigg cdt (17) withv=dx/dt. Consequently, we can write the action S0as S0=−mc/integraldisplay ds≈/integraldisplay/parenleftbig −mc2+L0/parenrightbig dt (18) with the non–relativistic Lagrangian L0=m 2v2+αE 2E2+αB 2c2B2(19) and αE=a+b 2ε0mc2, α B=b−a 2ε0mc2, a=αE−αB ε0mc2, b=αE+αB ε0mc2. (20) The Lagrangian L0describes indeed a non–relativistic atom with electric and magnetic polarizibility αEand αB, respectively. In this way we have verified that the metric theory (2) and (13) agrees with the physical pic- ture of traveling dipoles and, simultaneously, we have been able to express the coefficients aandbof the metric (2) in terms of atomic quantities.IV. MATTER WAVES A. Postulate Gordon has shown [1] that an electromagnetic field experiences dielectric matter as the effective metric (1). Here we postulate that also the opposite is true: A di- electric matter wave sees the electromagnetic field as a metric, and in particular as the metric (2) that we have motivated for traveling dipoles in Sec. III. We demon- strate the consistency of this idea with Gordon’s theory in Sec. V. Let us model the matter wave as, fittingly, a complex Klein–Gordon scalar ψin an effectively curved space–time. The action SAof the atom wave ψis SA=/integraldisplay LA√−gd4x (21) in terms of the Klein–Gordon Lagrangian [15] LA=/radicalbigggA g/bracketleftbigg1 2mgµν A(−i¯h∂µψ∗)(i¯h∂νψ)−mc2 2ψ∗ψ/bracketrightbigg =/radicalbigggA g/bracketleftbigg¯h2 2m(Dµ Aψ∗)(DA µψ)−mc2 2ψ∗ψ/bracketrightbigg (22) where we have employed the covariant derivatives DA µ with respect to the effective metric (2). The action (21) is minimal if the matter wave ψobeys the Klein–Gordon equation DA µDµ Aψ+m2c2 ¯h2ψ= 0, (23) or, written explicitly [11], 1√−gA∂µ/parenleftbig√−gAgµν A∂νψ/parenrightbig +m2c2 ¯h2ψ= 0. (24) Equation (24) together with the functions (14) and (16) and the parameters (20) describes how atomic matter waves respond to electromagnetic fields. B. R¨ ontgen limit Let us prove explicitly that the Klein–Gordon La- grangian (22) contains the known light forces in the limit of relatively low velocities (compared with c) and of weak fields (compared with mc2). We separate from the atomic wave function ψthe notorious rapid oscillations due to the rest energy mc2by defining ϕ≡ψexp/parenleftbigg imc2 ¯ht/parenrightbigg . (25) We neglect gravity and obtain in cartesian coordinates 3LA≈√−gA/bracketleftbigg1 2g00 A/parenleftbig mc2ϕ∗ϕ+i¯hϕ∗˙ϕ−i¯h˙ϕ∗ϕ/parenrightbig +i¯hc 2g0k A(ϕ∗∂kϕ−ϕ∂kϕ∗) −¯h2 2mgkl A(∂kϕ∗)(∂lϕ)−mc2 2ϕ∗ϕ/bracketrightbigg ≈mc2 2ϕ∗ϕ/parenleftbig 1−aLF+bT00 F/parenrightbig +i¯h 2(ϕ∗˙ϕ−˙ϕ∗ϕ) +i¯hc 2bT0k F(ϕ∗∂kϕ−ϕ∂kϕ∗) −¯h2 2m(∇ϕ∗)·(∇ϕ)−mc2 2(1−2aLF)ϕ∗ϕ. =i¯h 2(ϕ∗˙ϕ−˙ϕ∗ϕ)−¯h2 2m(∇ϕ∗)·(∇ϕ) +/parenleftigαE 2E2+αB 2c2B2/parenrightig ϕ∗ϕ +αE+αB 2m(E∧B)·i¯h(ϕ∗∇ϕ−ϕ∇ϕ∗).(26) This result agrees with the R¨ ontgen Lagrangian of Ref. [16] in the limit of weak fields and, consequently, de- scribes indeed the known non–resonant light forces in- cluding the R¨ ontgen interaction [12]. C. Dielectric flow Accelerated by light forces, an atomic matter wave will form a probability current that appears as a dielectric flow. Let us calculate the flow from the phase Sof the wave function, ψ=|ψ|eiS. (27) We introduce wµ≡ −¯h mcgµν A∂νS , (28) and obtain from the Klein–Gordon equation (24) the con- servation law of the four–dimensional probability current , DA µ/parenleftbig |ψ|2wµ/parenrightbig =1√−gA∂µ/parenleftbig√−gA|ψ|2wµ/parenrightbig = 0.(29) In the absence of electromagnetic forces, wµdescribes the local four–velocity of a free matter wave. In the presence of a field, we introduce the dielectric flow uµby normal- izingwµto unity with respect to the back–ground metric gµν, uµ≡wµ w, w≡/radicalbig gµνwµwν. (30) We define two densities, ̺andρ, as ̺≡ |ψ|2w/radicalbigggA g, ρ≡̺w . (31)We obtain from the conservation law (29) 1√−g∂µ/parenleftbig√−g̺uµ/parenrightbig =Dµ(̺uµ) = 0. (32) Consequently, ̺is the scalar probability density of the atomic de–Broglie wave. For most practical purposes the two densities ̺andρare identical, because wis unity to a very good approximation. The difference between ̺andρis subtle: In Sec. VE we show that mc2ρis the total enthalpy density of the dielectric matter wave, with the rest–energy density mc2̺as the lion’s share. D. Hydrodynamic limit As has been mentioned, the objective of this paper is the proof that the metric interaction (2) between matter waves and light is compatible with the known theory of dielectrics [1,14]. When a matter wave or, more likely, a macroscopic condensate of identical matter waves reaches the status of a dielectric it behaves like a quantum fluid. In this macroscopic limit the de–Broglie density varies over significantly larger ranges than the de–Broglie wave length (the same applies to frequencies), and a hydro- dynamic approach has become extremely successful [17]. Let us approximate i¯h∂νψ≈ −ψ¯h∂νS . (33) We obtain from the Klein–Gordon Lagrangian (22) the hydrodynamic approximation LA=/radicalbigggA g|ψ|2/bracketleftbigg¯h2 2mgµν A(∂µS)(∂νS)−mc2 2/bracketrightbigg .(34) Let us consider the Euler–Lagrange equations derived from the hydrodynamic Lagrangian (34). We obtain from the∂µSdependence of LAthe dielectric flow (32) and from a variation with respect to |ψ|2the dielectric Hamilton–Jacobi equation gµν A(∂µS)(∂νS) =m2c2 ¯h2, (35) or, in terms of the four–vector wµof Eq. (28), gA µνwµwν= 1. (36) In the hydrodynamic limit the wµvector represents a four–velocity that is normalized with respect to the ef- fective metric (2). We also see that the hydrodynamic Lagrangian (34) vanishes at the actual minimum that corresponds to the physical behavior of a dielectric mat- ter wave. 4V. QUANTUM DIELECTRICS A. Actio et reactio In the previous section we considered a dielectric mat- ter wave in a given electromagnetic field. Gordon [1] studied the opposite extreme — an electromagnetic field in a given dielectric medium. Let us address here an intermediate regime of actio et reactio where light acts on matter as well as matter acts on light. Such a phys- ical regime, characterizing a quantum dielectric, occurs for example when a Bose–Einstein condensate of an al- kali vapor [17] interacts non–resonantly with light [18]. If we were able to arrive at Gordon’s metric (1) from our starting point (2) we were inclined to take this as evidence that our approach is right. To include the dynamics of the electromagnetic field we add the free–field Lagrangian LFto the atomic LA in hydrodynamic approximation (34), L=LF+LA, (37) and regard the electromagnetic field as a dynamic ob- ject that is subject to the principle of least action. We could also easily include other interactions by additional terms in LAsuch as the atomic collisions within a Bose– Einstein condensate [17] by a Gross–Pitaevskii term. Let us consider the field variation δFL=δFLF+/radicalbigggA g|ψ|2¯h2 2m(∂µS)(∂νS)δFgµν A +/radicalbiggg gALAδF/radicalbigggA g. (38) As has been mentioned in Sec. IVD, the atomic La- grangian LAvanishes at the minimum of the action, in the hydrodynamic limit. We utilize that δFgµν A=−gµα Agνβ AδFgA αβ, (39) and obtain, using Eqs. (28-31), δFL=δFLF−mc2 2ρuαuβδFgA αβ. (40) The variation of the Lagrangian with respect to the field determines via the Euler–Lagrange equations the field dynamics. Can we cast δFLin the role of a dielectric? B. Effective Lagrangian The principal mathematical artifice of this paper is an effective Lagrangian that is designed to agree with Lunder field variations, and that describes a dielectric medium, LEFF≡LF+mc2 2ρ/parenleftbig gαβ−gA αβ/parenrightbig uαuβ(41)with δFL=δFLEFF. (42) Note that the two field variations in the relation (42) differ in a subtle way: On the left–hand side, δFabbre- viates the total variation with respect to the electromag- netic field, whereas on the right–hand side of Eq. (42) we treatε,µ, anduαas being fixed, despite their hidden dependence on the field due to the relations (28-31). We show explicitly in Sec. VD that LEFFis indeed the desired Lagrangian of light in a dielectric medium. Here we note that LEFFmay metamorphose into a multitude of forms. For example, we introduce the permittivity ε and the magnetic permeability µin terms of elementary atomic quantities and in accordance with the parameters (3) mentioned in the Introduction ε= 1 +αE ε0ρ ,1 µ= 1−αB ε0ρ . (43) In this way we obtain directly from Eqs. (2) and (3) LEFF=1 2/bracketleftbigg/parenleftbigg ε+1 µ/parenrightbigg LF+/parenleftbigg ε−1 µ/parenrightbigg uαuβTF αβ/bracketrightbigg .(44) We can also express the effective Lagrangian as LEFF=1 µLF+ε0εµ−1 2µFα′β′Fαβuαuα′gββ′,(45) due to the definition (10) of the free–field energy– momentum tensor, or we may perform further manip- ulations, utilizing the relations Fα′β′Fαβuαuα′gββ′=Fα′β′Fαβgαα′uβuβ′, Fα′β′Fαβuαuα′uβuβ′= 0, (46) due to the symmetry of the back–ground metric gαβand the anti–symmetry of the field–strength tensor Fαβ. C. Gordon’s metric Quite remarkably, one can express the effective La- grangian in the form [1] LEFF=−ε0 4µFαβF(α)(β)(47) with F(α)(β)≡gαα′ Fgββ′ FFα′β′ (48) and gαβ F=gαβ+ (εµ−1)uαuβ. (49) The effective Lagrangian appears as the free electromag- netic Lagrangian in a curved space–time with metric (49). 5A short exercise proves that gαβ Fis the inverse of gF αβ, i.e., as the notation suggests it, the contravariant metric tensor with respect to the covariant gF αβ. Consequently, we have indeed arrived at Gordon’s space–time geometry of light in moving media, starting from our metric (2), which supports the validity of our postulates. Note that Gordon’s space–time geometry is not com- pletely perfect [1]. The metrics (1) and (49) depend only on the square of the refractive index, εµ, whereas a di- electric medium is characterized by two dielectric con- stantsεandµ, in general. What is the imperfection in the Lagrangian (47)? In order to describe a density in general relativity, and in particular a Lagrangian den- sity, we must consider the determinant of the metric that describes the scaling of space and time. Gordon [1] calcu- lated the determinant by employing co–moving medium coordinates, with the result gF≡det/parenleftbig gF αβ/parenrightbig =g εµ. (50) Hence we obtain the effective action SEFF=/integraldisplay LEFF√−gd4x =−ε0 4/integraldisplay /radicalbiggε µFαβF(α)(β)√−gFd4x (51) that may deviate from the perfect SF=−ε0 4/integraldisplay FαβF(α)(β)√−gFd4x (52) whenε/µvaries significantly. However, when the density profile of the quantum liquid varies smoothly compared with the wave length of light we can neglect the variation ofε/µ. Ultracold atoms or Bose–Einstein condensates [17] are usually in this regime that is also compatible with the hydrodynamic behavior of the quantum liquid. D. Maxwell’s equations The first group of Maxwell’s equations follows from the structure (5) of the field–strength tensor Fµν. The Euler– Lagrange equations of the effective Lagrangian (47) yield the second group [1,14], DαHαβ= 0 or∂α/parenleftbig√−gHαβ/parenrightbig = 0 (53) with the constitutive equations Hαβ=ǫ0 µF(α)(β). (54) In local–galilean coordinates we can represent Hαβin terms (7) of the dielectric DandHfields in SI units. In this way we find yet another physically meaningful expression for the effective Lagrangian,LEFF=−1 4FαβHαβ=E·D 2−B·H 2, (55) which is indeed the explicit form of the Lagrangian for the electromagnetic field in a linear dielectric. Equation (54) is equivalent [1] to Minkowski’s constitu- tive equations in a moving medium [14,19]. In the limit of low velocities we recover the familiar relations D=ε0εE andµH=ε0c2B, and, via Eq. (43), D≈(ε0+αE̺)E,H≈(ε0−αB̺)c2B,(56) assuming a weak field when ρ≈̺. Relativistic first– order corrections lead to the constitutive equations de- rived in Ref. [16] that describe, for example, the R¨ ontgen effect [20] or lead to Fresnel’s light drag [6] measured in Fizeau’s experiment [7]. In case of a smooth dielectric density we can regard ε/µas a constant, and obtain from Maxwell’s equations ∂α/parenleftig√−gFF(α)(β)/parenrightig = 0 orDF αF(α)(β)= 0.(57) Light experiences the quantum dielectric as the space– time metric (1), i.e. as an effective gravitational field. E. Energy–momentum tensor According to Antoci and Mihich [21] Gordon [1] has already settled the notorious debate about Minkowski’s [19] versus Abraham’s [22] energy–momentum tensor in Abraham’s favor. However, in his paper [1], Gordon as- sumed the dielectric properties of the medium ε,µ, and uα, as preassigned quantities. Having done so, the de- rived energy–momentum tensor is valid if and only if the dielectric quantities are constants, i.e. in the case of a uniform medium, because the conservation of energy and momentum presupposes the homogeneity of space–time, according to Noether’s theorem. If one tries to deter- mine the energy and momentum of the electromagnetic field in an inhomogeneous medium one must not consider the dielectric properties as given functions, but rather as being generated by a physical object, such as the quan- tum dielectric studied in this paper. In short, one should take into account actio et reactio , and in particular the back action of the medium (an effect seen experimentally [23]). Does Abraham’s tensor have significance beyond uniform media? Let us determine the energy–momentum tensor via the royal road of general relativity, as a variation of the La- grangian with respect to the back–ground metric [11], Tµν=−2√−gδ(√−gL) δgµν=−2δL δgµν−Lgµν.(58) A metric variation δgof the Lagrangian gives, in analogy with Eq. (40) and the considerations in Sec. VB, 6δgL=δgLF−mc2 2ρuαuβδggA αβ =δgLEFF−mc2 2ρuαuβδggαβ. (59) We recall that LAvanishes in the hydrodynamic limit. Consequently, we arrive at the total energy–momentum tensor in the form Tµν=−2δLEFF δgµν−LFgµν+mc2ρuµuν.(60) We represent this expression as the sum Tµν=Tµν A+Tµν EFF (61) with the atomic component Tµν A=mc2ρuµuν−pgµν, (62) p=LF−LEFF=1 4Fαβ/parenleftbig Hαβ−ε0Fαβ/parenrightbig ,(63) and Tµν EFF=−2δLEFF δgµν−LEFFgµν. (64) We are inclined to interpret the tensor (64) as the ef- fective energy–momentum tensor of the electromagnetic field in the presence of a dielectric medium. The atomic tensor (62) appears as the energy– momentum of a fluid under the dielectric pressure (63). In the limit of low flow velocities the pressure approaches −ε0(αEE2+αBc2B2)̺/2, according to Eqs. (55) and (56). In this limit, atomic dipoles with positive αEand αBare attracted towards increasing field intensities. We also see from the atomic energy–momentum tensor (62) that a dielectric fluid possesses the total enthalpy density mc2ρ=mc2w̺, including the relativistic rest energy. In this way we have found an interpretation for the density ρthat appears at the prominent place (3). To calculate the enthalpy, we express the effective Lagrangian (41) in terms of the norm w. We use the definition (30) of the four–velocity uαand the normalization (36) of the wα, and obtain p=LF−LEFF=mc2̺ 2/parenleftbigg1 w−w/parenrightbigg , (65) or, by inversion, mc2ρ=mc2w̺=/radicalbig m2c4̺2+p2−p . (66) This equation describes how the enthalpy density de- pends on the pressure and on the dielectric density. On the other hand, Eq. (63) quantifies the pressure that de- pends on the dielectric density and flow, and on the elec- tromagnetic field as an external quantity. We may in- terpret the two formulas (63) and (66) as the equations of state for the quantum dielectric. The density of thefluid’s internal energy is the difference between enthalpy density and pressure [24] ǫ=/radicalbig m2c4̺2+p2−2p . (67) We see that the internal energy approaches mc2+ ε0(αEE2+αBc2B2) in the limit of a slow flow and a low dielectric pressure. Atomic dipoles with positive αE andαBseem to gain internal energy in the presence of an electromagnetic field. Let us turn to the energy–momentum tensor of the field. The effective Lagrangian LEFFcharacterizes a medium with preassigned dielectric functions εandµ, i.e. Gordon’s case [1]. Consequently [1], the effective energy–momentum tensor of the electromagnetic field is Abraham’s [22] Tµν EFF=Tµν Ab=Tµν Mk−(εµ−1)uµΩν, (68) with Minkowski’s tensor [19], Tµν Mk=HµαFαβgβν+1 4HαβFαβgµν, (69) corrected by the Ruhstrahl [22] Ων=Fαα′uα′uβ/parenleftbig Hαβuν+Hβνuα+Hναuβ/parenrightbig .(70) In locally co–moving galilean coordinates or in a medium at rest, the spatial component of the Ruhstrahl is propor- tional to the Poynting vector (hence the name), Ων=/parenleftbigg 0,E∧H c/parenrightbigg . (71) In this case the effective energy–momentum tensor of the field takes the form Tµν Ab=/parenleftbigg IS/c S/c σ/parenrightbigg (72) with intensity I, Poynting vector S, and stress tensor σ I=E·D 2+B·H 2,S=E∧H, σ=/parenleftbiggE·D 2+B·H 2/parenrightbigg 1−E⊗D−B⊗H.(73) We see that Abraham’s tensor describes indeed the effec- tive energy–momentum of the electromagnetic field, even in the general case of a non–uniform medium that is able to move under the pressure of light forces. VI. CREDO Light experiences dielectric matter as an effective grav- itational field [1–4] and matter experiences light as a form of gravity as well. Light and matter see each other as dual space–time metrics, a unique model in field theory, 7to the knowledge of the author. We have solidified this mental picture by postulating the idea and demonstrat- ing its striking consistency with the theory of dielectrics [1,14]. It would be interesting to see whether our model can be derived directly from first principles. In pass- ing, we have determined the energy–momentum tensor that governs actio et reactio of electromagnetic fields in quantum dielectrics. The tensor is Abraham’s [22] plus the energy–momentum of the medium characterized by a dielectric pressure and an enthalpy density. Our idea may serve as a guiding line for understanding the effects of slow light [13] on matter waves. Here one can conceive of creating light fields that appear to atoms as quasi–astronomical objects. The holy grail in this field would be the creation of a black hole made of light. Light and matter interact with each other as if both were gravitational fields, and light and matter are gen- uine quantum fields in Nature. A distinct quantum regime of dielectrics has been prepared in the labora- tories where Bose–Einstein condensates of alkali vapors [17] interact non–resonantly with light quanta, but has never been viewed as an analogue of quantum gravity, to the knowledge of the author. Sound in superfluids [25] and in alkali Bose–Einstein condensates [26] has been considered as a quantum field in a curved space–time, as being able to emit the acoustic analogue of Hawking radiation [27]. However, the quantum sound still propa- gates in a classical medium, in contrast to light quanta in a quantum dielectric. In many respects, we have rea- sons to hope that Bose–Einstein condensates may serve as testable prototype models for quantum gravity. ACKNOWLEDGEMENTS I am very grateful to Sir Michael Berry, Ignacio Cirac, Carsten Henkel, Susanne Klein, Rodney Loudon, Paul Piwnicki, Stig Stenholm, and Martin Wilkens for conver- sations on moving media. I acknowledge the generous support of the Alexander von Humboldt Foundation and of the G¨ oran Gustafsson Stiftelse. [1] W. Gordon, Ann. Phys. (Leipzig) 72, 421 (1923). [2] Pham Mau Quan, C. R. Acad. Sci. (Paris) 242, 465 (1956); Archive for Rational Mechanics and Analysis 1, 54 (1957/58). [3] U. Leonhardt and P. Piwnicki, Phys. Rev. A 60, 4301 (1999). [4] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 (2000). [5] C. Møller, The Theory of Relativity (Oxford University Press, Oxford, 1972). [6] A. J. Fresnel, Ann. Chim. Phys. 9, 57 (1818).[7] H. Fizeau, C. R. Acad. Sci. (Paris) 33, 349 (1851). [8] M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999). [9] E. Bortolotti, Rend. R. Acc. Naz. Linc., 6a, 4, 552 (1926). [10] See e.g. P. Berman (ed.), Atom Interferometry (Aca- demic, San Diego, 1997). [11] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1975). [12] H. Wei, R. Han, and X. Wei, Phys. Rev. Lett. 75, 2071 (1995), see also Refs. [16] and [20], and M. Babiker, E. A. Power, and T. Thirunamachandran, Proc. Roy. Soc. A 332, 187 (1973); M. Babiker, J. Phys. B 17, 4877 (1984); C. Baxter, M. Babiker, and R. Loudon, Phys. Rev. A 47, 1278 (1993); V. Lembessis, M. Babiker, C. Baxter, and R. Loudon, ibid.48, 1594 (1993); M. Wilkens, Phys. Rev. A49, 570 (1994); Phys. Rev. Lett. 72, 5 (1994); ibid.81, 1533 (1998); G. Spavieri, ibid.81, 1533 (1998); 82, 3932 (1999); Phys. Rev. A 59, 3194 (1999). [13] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999); M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovsev, E. S. Fry, and M. O. Scully, Phys. Rev. Lett.82, 5229 (1999); D. Budiker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, ibid.83, 1767 (1999). [14] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984). [15] L. D. Landau and E. M. Lifshitz, Quantum Electrody- namics (Pergamon, Oxford, 1982). [16] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 82, 2426 (1999). [17] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. String ari, Rev. Mod. Phys. 71, 463 (1999). [18] U. Leonhardt, T. Kiss, and P. Piwnicki, Euro. Phys. J. D7, 413 (1999). [19] H. Minkowski, Nachr. d. K. Ges. d. Wiss. zu G¨ ott., Math.-Phys. Kl. 53 (1908). [20] W. C. R¨ ontgen, Ann. Phys. Chem. 35, 264 (1888). [21] S. Antoci and L. Mihich, Nuovo Cimento B 112, 991 (1997); Euro. Phys. J. D 3, 205 (1998). [22] M. Abraham, Rend. Circ. Matem. Palermo 28, 1 (1909); 30, 33 (1910). [23] A. Ashkin and J. M. Dziedzic, Phys. Rev. Lett. 30, 139 (1973). [24] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Perg- amon, Oxford, 1987). [25] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981); Phys. Rev. D 51, 2827 (1995); T. A. Jacobson, ibid.44, 1731 (1991); T. A. Jacobson and G. E. Volovik, ibid.58, 064021 (1998); N. B. Kopnin and G. E. Volovik, JETP Lett.67, 140 (1998); T. A. Jacobson and G. E. Volovik, ibid.,68, 874 (1998); G. E. Volovik, ibid.,69, 705, (1999); M. Visser, Class. Quantum Grav. 15, 1767 (1998). [26] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Sci- ence (submitted). [27] S. M. Hawking, Nature 248, 30 (1974); Commun. math. Phys.43, 199 (1975). 8

arXiv:physics/0001065v1 [physics.acc-ph] 27 Jan 2000Beam Coupling Impedances of Small Discontinuities∗ Sergey S. Kurennoy SNS Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract A general derivation of the beam coupling impedances produc ed by small discontinuities on the wall of the vacuum chamber of an accelerator is reviewed. A collecti on of analytical formulas for the impedances of small obstacles is presented. 1 Introduction A common tendency in design of modern accelerators is to mini mize beam-chamber coupling impedances to avoid beam instabilities and reduce heating. Even contri butions from tiny discontinuities like pumping holes have to be accounted for because of their large number. Numerical time-domain methods are rather involved and time-consuming for small obstacles, especial ly for those protruding inside the beam pipe. This makes analytical methods for calculating the impedances of small discontinuities very important. A discontinuity on the wall of a beam pipe — a hole, cavity, pos t, mask, etc. — is considered to be small when its typical size is small compared to the size of th e chamber cross section and the wavelength. Analytical calculations of the coupling impedances of smal l obstacles are based on the Bethe theory of diffraction of electromagnetic waves by a small hole in a meta l plane [1]. The method’s basic idea is that the hole in the frequency range where the wavelength is large com pared to the typical hole size, can be replaced by two induced dipoles, an electric and a magnetic one. The di pole magnitudes are proportional to the beam fields at the hole location, with the coefficients called polar izabilities [2]. The fields diffracted by a hole into the vacuum chamber can be found as those radiated by these effe ctive electric and magnetic dipoles, and integrating the fields gives us the coupling impedances. Fol lowing this path, the impedances produced by a small hole on the round pipe were calculated first in [3]. Since essentially the same idea works for any small obstacle , the method can be extended for arbitrary small discontinuities on the pipe with an arbitrary-shaped cross section. Analytical expressions for the coupling impedances of various small discontinuities on th e wall of a cylindrical beam pipe with an arbitrary single-connected cross section have been obtained in Refs. [3]-[9]. All the dependence on the discontinuity shape enters only th rough its polarizabilities. Therefore, the problem of calculating the impedance contribution from a gi ven small discontinuity was reduced to finding its electric and magnetic polarizabilities, which can be do ne by solving proper electro- or magnetostatic problem. Useful analytical results have been obtained for v arious axisymmetric obstacles (cavities and irises) [10], as well as for holes and slots: circular [1] and elliptic [2] hole in a zero-thickness wall, circular [11] and elliptic [12] hole in a thick wall, various slots (some re sults are compiled in [13]), and for a ring-shaped cut [14]. The impedances of various protrusions (post, mask , etc) have been calculated in [15]. This text is organized as follows. In Sections 2 and 3 a genera l derivation of the coupling impedances for a small discontinuity is given, and then in Sect. 4 the tra pped modes are discussed. This part mostly follows Ref. [9]. In Sect. 5 we collected for the reader conve nience some practical formulas for the coupling ∗Lectures presented at the US Particle Accelerator School in Tucson, AZ on January 17-21, 2000, as a part of R.L. Gluck- stern’s course ”Analytic Methods for Calculating Coupling Impedances” 1impedances of various small discontinuities. In a compact f orm, many of these formulas are included in the handbook [16]. 2 Fields 2.1 Beam Fields Let us consider an infinite cylindrical pipe with an arbitrar y cross section Sand perfectly conducting walls. Thezaxis is directed along the pipe axis, a hole (or another small discontinuity like a post, cavity or mask) is located at the point ( /vectorb,z= 0), and its typical size hsatisfiesh≪b. To evaluate the coupling impedance one has to calculate the fields induced in the chamber by a give n current. If an ultrarelativistic point charge qmoves parallel to the chamber axis with the transverse offset /vector sfrom the axis, the fields harmonics /vectorEb,/vectorHb produced by this charge on the chamber wall without disconti nuity would be Eb ν(/vector s,z;ω) =Z0Hb τ(/vector s,z;ω) (1) =−Z0qeikz/summationdisplay n,mk−2 nmenm(/vector s)∇νenm(/vectorb), whereZ0= 120πOhms is the impedance of free space, and k2 nm,enm(/vector r) are eigenvalues and orthonormalized eigenfunctions (EFs) of the 2D boundary problem in S: /parenleftbig ∇2+k2 nm/parenrightbig enm= 0 ;enm/vextendsingle/vextendsingle ∂S= 0. (2) Here/vector∇is the 2D gradient in plane S;k=ω/c; ˆνmeans an outward normal unit vector, ˆ τis a unit vector tangent to the boundary ∂Sof the chamber cross section S, and{ˆν,ˆτ,ˆz}form a right-handed basis. The differential operator ∇νis the scalar product ∇ν= ˆν·/vector∇. The eigenvalues and EFs for circular and rectangular cross sections are given in the Appendix. Let us introduce the following notation for the sum in Eq. (1) eν(/vector s) =−/summationdisplay gk−2 geg(/vector s)∇νeg(/vectorb) (3) whereg={n,m}is a generalized index. From a physical viewpoint, this is ju st a normalized electrostatic field produced at the hole location by a filament charge displa ced from the chamber axis by distance /vector s. It satisfies the normalization condition /contintegraldisplay ∂Sdl eν(/vector s) = 1, (4) where integration goes along the boundary ∂S, which is a consequence of the Gauss law. It follows from the fact that Eq. (3) gives the boundary value of /vector eν(/vector s)≡ −/vector∇Φ(/vector r−/vector s), where Φ( /vector r−/vector s) is the Green function of boundary problem (2): ∇2Φ(/vector r−/vector s) =−δ(/vector r−/vector s). For the symmetric case of an on-axis beam in a circular pipe of radius bfrom Eq. (4) immediately follows eν(0) = 1/(2πb). It can also be derived by directly summing up the series in Eq. (3) for this particular case. 2.2 Effective Dipoles and Polarizabilities At distances lsuch thath≪l≪b, the fields radiated by the hole into the pipe are equal to thos e produced by effective dipoles [1, 2] Pν=−χε0Eh ν/2;Mτ= (ψττHh τ+ψτzHh z)/2; Mz= (ψzτHh τ+ψzzHh z)/2, (5) 2where superscript ’ h’ means that the fields are taken at the hole. Polarizabilitie sψ,χare related to the effective ones αe,αmused in [2, 3] as αe=−χ/2 andαm=ψ/2, so that for a circular hole of radius ain a thin wallψ= 8a3/3 andχ= 4a3/3 [1]. In general, ψis a symmetric 2D-tensor, which can be diagonalized. If the hole is symmetric, and its symmetry axis is parallel to ˆz, the skew terms vanish, i.e. ψτz=ψzτ= 0. In a more general case of a non-zero tilt angle αbetween the major symmetry axis and ˆ z, ψττ=ψ⊥cos2α+ψ/bardblsin2α , ψτz=ψzτ= (ψ/bardbl−ψ⊥)sinαcosα , (6) ψzz=ψ⊥sin2α+ψ/bardblcos2α , whereψ/bardblis the longitudinal magnetic susceptibility (for the exter nal magnetic field along the major axis), andψ⊥is the transverse one (the field is transverse to the major axi s of the hole). When the effective dipoles are obtained, e.g., by substituting beam fields (1) i nto Eqs. (5), one can calculate the fields in the chamber as a sum of waveguide eigenmodes excited in the chamb er by the dipoles, and find the impedance. This approach has been carried out for a circular pipe in [3], and for an arbitrary chamber in [5]. The polarizabilities for various types of small discontinuiti es are discussed in detail in Section 5. 2.3 Radiated Fields The radiated fields in the chamber can be expanded in a series i n TM- and TE-eigenmodes [2] as /vectorF=/summationdisplay nm/bracketleftBig A+ nm/vectorF(E)+ nmθ(z) +A− nm/vectorF(E)− nmθ(−z)/bracketrightBig + (7) /summationdisplay nm/bracketleftBig B+ nm/vectorF(H)+ nmθ(z) +B− nm/vectorF(H)− nmθ(−z)/bracketrightBig , where/vectorFmeans either /vectorEor/vectorH, superscripts ’ ±’ denote waves radiated respectively in the positive (+, z>0) or negative ( −,z <0) direction, and θ(z) is the Heaviside step function. The fields F(E) nmof{n,m}th TM-eigenmode in Eq. (7) are expressed [2] in terms of EFs (2) E∓ z=k2 nmenmexp(±Γnmz) ;H∓ z= 0 ; /vectorE∓ t=±Γnm/vector∇enmexp(±Γnmz) ; (8) /vectorH∓ t=ik Z0ˆz×/vector∇enmexp(±Γnmz), where propagation factors Γ nm= (k2 nm−k2)1/2should be replaced by −iβnmwithβnm= (k2−k2 nm)1/2for k>k nm. For given values of dipoles (5) the unknown coefficients A± nmcan be found [3, 5] using the Lorentz reciprocity theorem A± nm=anmMτ±bnmPν, (9) with anm=−ikZ0 2Γnmk2nm∇νeh nm;bnm=1 2ε0k2nm∇νeh nm. (10) The fieldsF(H) nmof the TE nm-eigenmode in Eq. (7) are H∓ z=k′2 nmhnmexp(±Γ′ nmz) ;E∓ z= 0 ; /vectorH∓ t=±Γ′ nm/vector∇hnmexp(±Γ′ nmz) ; (11) /vectorE∓ t=−ikZ0ˆz×/vector∇hnmexp(±Γ′ nmz), with propagation factors Γ′ nm= (k′2 nm−k2)1/2replaced by −iβ′ nm=−i(k2−k′2 nm)1/2whenk>k′ nm. Here EFshnmsatisfy the boundary problem (2) with the Neumann boundary c ondition ∇νhnm|∂S= 0, andk′2 nm 3are corresponding eigenvalues, see in Appendix. The TE-mod e excitation coefficients in the expansion (7) for the radiated fields are B± nm=±cnmMτ+dnmPν+qnmMz, (12) where cnm=1 2k′2nm∇τhh nm;qnm=1 2Γ′nmhh nm; dnm=−ik 2Z0ε0Γ′nmk′2nm∇τhh nm. (13) 2.4 Fields near Hole with Radiation Corrections A more refined theory should take into account the reaction of radiated waves back on the hole. Adding corrections to the beam fields (1) due to the radiated waves in the vicinity of the hole gives Eν=Eb ν+ψzτΣ′ xZ0Hτ+ψzzΣ′ xZ0Hz 1−χ(Σ1−Σ′ 1), (14) Hτ=Hb τ+ψτz(Σ2−Σ′ 2)Hz 1−ψττ(Σ2−Σ′ 2), (15) Hz=χΣ′ xEν/Z0+ψzτΣ′ 3Hτ 1−ψzzΣ′ 3, (16) where (s={n,m}is a generalized index) Σ1=1 4/summationdisplay sΓs/parenleftbig ∇νeh s/parenrightbig2 k2s; Σ2=k2 4/summationdisplay s/parenleftbig ∇νeh s/parenrightbig2 Γsk2s; Σ′ 1=k2 4/summationdisplay s/parenleftbig ∇τhh s/parenrightbig2 Γ′sk′2s; Σ′ 2=1 4/summationdisplay sΓ′ s/parenleftbig ∇τhh s/parenrightbig2 k′2s; Σ′ x=ik 4/summationdisplay shh s∇τhh s Γ′s; Σ′ 3=1 4/summationdisplay sk′2 s/parenleftbig hh s/parenrightbig2 Γ′s. (17) Since this consideration works at distances larger than h, one should restrict the summation in Eq. (17) to the values of s={n,m}such thatksh≤1 andk′ sh≤1. 3 Impedance 3.1 Longitudinal Impedance The generalized longitudinal impedance of the hole depends on the transverse offsets from the chamber axis /vector sof the leading particle and /vectortof the test particle, and is defined [6] as Z(k;/vector s,/vectort) =−1 q/integraldisplay∞ −∞dze−ikzEz(/vectort,z;ω), (18) where the longitudinal field Ez(/vectort,z;ω) is taken along the test particle path. The displacements fr om the axis are assumed to be small, s≪bandt≪b. The impedance Z(k;/vector s,/vectort) includes higher multipole longitudinal impedances, and in the limit s,t→0 gives the usual monopole one Z(k) =Z(k; 0,0). To calculate Ez(/vectort,z;ω), 4we use Eq. (7) with coefficients (9) and (12) in which the correc ted near-hole fields (14)-(16) are substituted [a dependence on /vector senters via beam fields (1)]. It yields Z(k;/vector s,/vectort) =−ikZ0eν(/vector s)eν(/vectort) 2× (19) ×/bracketleftbiggψττ 1−ψττ(Σ2−Σ′ 2)+ψ2 τzΣ′ 3−χ 1−χ(Σ1−Σ′ 1)/bracketrightbigg , whereeν(/vector r) is defined above by Eq. (3). In practice, we are usually inter ested only in the monopole term Z(k) =Z(k; 0,0), and will mostly use Eq. (19) with replacement eν(/vector s)eν(/vectort)→˜e2 ν, where ˜eν≡eν(0). In deriving Eq. (19) we have neglected the coupling terms bet weenEν,HτandHz, cf. Eqs. (14)-(16), which contribute to the third order of an expansion discusse d below, and also have taken into account that ψτz=ψzτ. For a small discontinuity, polarizabilities ψ,χ=O(h3), and they are small compared to b3. If we expand the impedance (19) in a perturbation series in polarizabili ties, the first order gives Z1(k) =−ikZ0˜e2 ν 2(ψττ−χ), (20) that is exactly the inductive impedance obtained in [5] for a n arbitrary cross section of the chamber. For a particular case of a circular pipe, from either direct summa tion in (1) or applying the Gauss law, one gets ˜eν= 1/(2πb). Substituting that into Eq. (20) leads to a well-known resu lt [3, 4]: Z(k) =−ikZ0ψττ−χ 8π2b2=−ikZ0αe+αm 4π2b2, (21) where we recall that two definitions of the polarizabilities are related as αe=−χ/2 andαm=ψττ/2. From a physical point of view, keeping only the first order ter m (20) corresponds to dropping out all radiation corrections in Eqs. (14)-(16). These correction s first reveal themselves in the second order term Z2(k) =−ikZ0˜e2 ν 2/bracketleftbig ψ2 ττ(Σ2−Σ′ 2) +ψ2 τzΣ′ 3 (22) +χ2(Σ′ 1−Σ1)/bracketrightbig , which at frequencies above the chamber cutoff has both a real a nd imaginary part. The real part of the impedance is ReZ2(k) =k3Z0˜e2 ν 8/braceleftBigg ψ2 τz</summationdisplay sk′2 s/parenleftbig hh s/parenrightbig2 k2β′s(23) +ψ2 ττ/bracketleftBigg</summationdisplay s/parenleftbig ∇νeh s/parenrightbig2 βsk2s+</summationdisplay sβ′ s/parenleftbig ∇τhh s/parenrightbig2 k2k′2s/bracketrightBigg +χ2/bracketleftBigg</summationdisplay sβs/parenleftbig ∇νeh s/parenrightbig2 k2k2s+</summationdisplay s/parenleftbig ∇τhh s/parenrightbig2 β′sk′2s/bracketrightBigg/bracerightBigg , where the sums include only a finite number of the eigenmodes p ropagating in the chamber at a given frequency, i.e. those with ks<kork′ s<k. The dependence of ReZ on frequency is rather complicated; it has sharp peaks near t he cutoffs of all propagating eigenmodes of the chamber, and increases in ave rage with the frequency increase. Well above the chamber cutoff, i.e. when kb≫1 (but still kh≪1 to justify the Bethe approach), this dependence can be derived as follows. If the waveguide cross section Sis a simply connected region, the average number n(k) of the eigenvalues ks(ork′ s) which are less than k, forkb≫1, is proportional to k2[17]: n(k)≃S 4πk2+O(k), 5whereSis the area of the cross section. Using this property, and tak ing into account that ∇νeh s∝kseh s, and∇τhh s∝k′ shh s, we replace sums in the RHS of Eq. (23) by integrals as/summationtext< s→/integraltextkdkd dkn(k). It turns out that all sums in Eq. (23) have the same asymptotic behavior, b eing linear in k, and as a result, ReZ∝k4. Obtaining the exact coefficient in this dependence seems rath er involved for a general S, but it can be easily done for a rectangular chamber, see in Appendix B. The result is ReZ=Z0k4˜e2 ν 12π(ψ2 ττ+ψ2 τz+χ2). (24) Remarkably, the same answer (for ψτz= 0) has been obtained in Ref. [3] simply by calculating the en ergy radiated by the dipoles into a half-space. The physical reas on for this coincidence is clear: at frequencies well above the cutoff the effective dipoles radiate into the wavegu ide the same energy as into an open half-space. Strictly speaking, the real part of impedance is non-zero ev en below the chamber cutoff, due to radiation outside. In the case of a thin wall, ReZ below the cutoff can be estimated by Eq. (24), and twice that fo r high frequencies, kb≫1. For a thick wall, the contribution of the radiation outsid e toReZis still given by Eq. (24), but with the outside polarizabilities substitute d, and it decreases exponentially with the thickness increase [4]. The real part of the impedance is related to the power Pscattered by the hole into the beam pipe as ReZ= 2P/q2. These energy considerations can be used as an alternative w ay for the impedance calculation. The radiated power is P=/summationdisplay s/bracketleftBig A2 sP(E) s+B2 sP(H) s/bracketrightBig , where we sum over all propagating modes in both directions, a ndPsmeans the time-averaged power radiated insth eigenmode: P(E) s=kβsk2 s/(2Z0) andP(H) s=Z0kβ′ sk′2 s/2. Substituting beam fields (1) into Eqs. (9)-(13) for the coeffic ientsAsandBsand performing calculations gives us exactly the result (23). Such an alternative deriva tion of the real part has been carried out in Ref. [8] for a circular pipe with a symmetric untilted hole ( ψτz= 0). Our result (23) coincides, in this particular case, with that of Reference [8]. It is appropriate to mentio n also that in this case at high frequencies the series has been summed approximately [8] using asymptotic e xpressions for roots of the Bessel functions, and the result, of course, agrees with Eq. (24). One should note that the additional ψ2 τz-term in Eq. (23) is important in some particular cases. For example, this skew term gives a leading contribution to ReZfor a long and slightly tilted slot, because ψτz can be much larger than ψττin this case, since ψ/bardbl≫ψ⊥, cf. Eqs. (6). 3.2 Transverse Impedance We will make use of the expression for the generalized longit udinal impedance Z(k;/vector s,/vectort), Eq. (19). According to the Panofsky-Wenzel theorem, the transverse impedance c an be derived as /vectorZ⊥(k;/vector s,/vectort) =/vector∇Z(k;/vector s,/vectort)/(ks), see, e.g., [6] for details. This way leads to the expression /vectorZ⊥(k;/vector s,/vectort) =−iZ0edip ν(/vector s)/vector∇eν(/vectort) 2s× (25) ×/bracketleftbiggψττ 1−ψττ(Σ2−Σ′ 2)+ψ2 τzΣ′ 3−χ 1−χ(Σ1−Σ′ 1)/bracketrightbigg , whereedip ν(/vector s) =/vector s·/vector∇eν(/vector s). Going to the limit s→t→0, we get the usual dipole transverse impedance /vectorZ⊥(k) =−iZ01 2(d2 x+d2 y)/vector adcos(ϕb−ϕd)× (26) ×/bracketleftbiggψττ 1−ψττ(Σ2−Σ′ 2)+ψ2 τzΣ′ 3−χ 1−χ(Σ1−Σ′ 1)/bracketrightbigg . 6Herex,yare the horizontal and vertical coordinates in the chamber c ross section; dx≡∂xeν(0),dy≡∂yeν(0); ϕb=ϕs=ϕtis the azimuthal angle of the beam position in the cross-sect ion plane;/vector ad=/vector axcosϕd+/vector aysinϕd is a unit vector in this plane in direction ϕd, which is defined by conditions cos ϕd=dx//radicalBig d2x+d2y, sinϕd= dy//radicalBig d2x+d2y. It is seen from Eq. (26) that the angle ϕdshows the direction of the transverse-impedance vector/vectorZ⊥and, therefore, of the beam-deflecting force. Moreover, the value ofZ⊥is maximal when the beam is deflected along this direction and vanishes when the beam o ffset is perpendicular to it. The equation (26) includes the corrections due to waves radi ated by the hole into the chamber in exactly the same way as Eq. (19) for the longitudinal impedance. If we expand it in a series in the polarizabilities, the first order of the square brackets in (26) gives ( ψττ−χ), and the resulting inductive impedance is [5]: /vectorZ⊥(k) =−iZ01 2(d2 x+d2 y)/vector adcos(ϕb−ϕd)(ψττ−χ). (27) For a circular pipe, dx= cosϕh/(πb2) anddy= sinϕh/(πb2), whereϕhis the azimuthal position of the discontinuity (hole). As a result, ϕd=ϕh, and the deflecting force is directed toward (or opposite to) the hole. Note that in axisymmetric structures the beam-defl ecting force is directed along the beam offset; the presence of an obstacle obviously breaks this symmetry. For a general cross section, the direction of the deflecting force depends on the hole position in a complic ated way, see [5] for rectangular and elliptic chambers. The transverse impedance of a discontinuity on the wall of a c ircular pipe has a simple form [3]: /vectorZ⊥circ(k) =−iZ0ψττ−χ 2π2b4/vector ahcos(ϕb−ϕd) =−iZ0αm+αe π2b4/vector ahcos(ϕb−ϕd), (28) where/vector ahis a unit vector in the direction from the chamber axis to the d iscontinuity, orthogonal to ˆ z. ForM(M≥3) holes uniformly spaced in one cross section of a circular b eam pipe, a vector sum of M expressions (28) gives the transverse impedance as /vectorZ⊥circ M (k) =−iZ0αm+αe π2b4M 2/vector ab, (29) where/vector ab=/vector s/|/vector s|is a unit vector in the direction of the beam transverse offset . One can see that the deflecting force is now directed along the beam offset, i.e. some kind of t he axial symmetry restoration occurs. The maximal value of Z⊥forMholes which are uniformly spaced in one cross-section is onl yM/2 times larger than that for M= 1. Moreover, the well-known empirical relation Z⊥= (2/b2k)Z, which is justified only for axisymmetric structures, holds in this case also. The second order term in Eq. (26) includes ReZ⊥, cf. Sect. 3.1 for the longitudinal impedance. 4 Trapped Modes So far we considered the perturbation expansion of Eq. (19) i mplicitly assuming that correction terms O(ψ) andO(χ) in the denominators of its right-hand side (RHS) are small c ompared to 1. Under certain conditions this assumption is incorrect, and it leads to some non-pertu rbative results. Indeed, at frequencies slightly below the chamber cut-offs, 0 < ks−k≪ks(or the same with replacement ks→k′ s), a single term in sums Σ′ 1, Σ2, or Σ′ 3becomes very large, due to very small Γ s= (k2 s−k2)1/2(or Γ′ s) in its denominator, and then the “corrections” ψΣ orχΣ can be of the order of 1. As a result, one of the denominators o f the RHS of Eqs. (19) can vanish, which corresponds to a resonance of t he coupling impedance. On the other hand, vanishing denominators in Eqs. (14)-(16) mean the existenc e of non-perturbative eigenmodes of the chamber with a hole, since non-trivial solutions E,H∝ne}ationslash= 0 exist even for vanishing external (beam) fields Eb,Hb= 0. These eigenmodes are nothing but the trapped modes studied i n [18] for a circular waveguide with a small discontinuity. In our approach, one can easily derive param eters of trapped modes for waveguides with an arbitrary cross section. 74.1 Frequency Shifts Let us for brevity restrict ourselves to the case ψτz= 0 and consider Eq. (15) in more detail. For Hb= 0 we have Hτ/bracketleftBigg 1−ψττk2/parenleftbig ∇νeh s/parenrightbig2 4Γsk2s+.../bracketrightBigg = 0, (30) wheres≡ {nm}is the generalized index, and ...mean all other terms of the series Σ 2,Σ′ 2. At frequency Ω s slightly below the cutoff frequency ωs=kscof the TM s-mode, the fraction in Eq. (30) is large due to small Γsin its denominator, and one can neglect the other terms. Then the condition for a non-trivial solution Hτ∝ne}ationslash= 0 to exist is Γs≃1 4ψττ/parenleftbig ∇νeh s/parenrightbig2. (31) In other words, there is a solution of the homogeneous, i.e., without external currents, Maxwell equations for the chamber with the hole, having the frequency Ω s<ωs— thesth trapped TM-mode. When Eq. (31) is satisfied, the series (7) is obviously dominated by the sin gle termAsFE s; hence, the fields of the trapped mode have the form [cf. Eq. (8)] Ez=k2 sesexp(−Γs|z|) ; Hz= 0 ; /vectorEt= sgn(z)Γs/vector∇esexp(−Γs|z|) ; (32) Z0/vectorHt=ikˆz×/vector∇esexp(−Γs|z|), up to some arbitrary amplitude. Strictly speaking, these ex pressions are valid at distances |z|>bfrom the discontinuity. Typically, ψττ=O(h3) and ∇νeh s=O(1/b), and, as a result, Γ sb≪1. It follows that the field of the trapped mode extends along the vacuum chamber ove r the distance 1 /Γs, large compared to the chamber transverse dimension b. The existence of the trapped modes in a circular waveguide wi th a small hole was first proved in [18], and conditions similar to Eq. (31) for this particular case were obtained in [18, 19], using the Lorentz reciprocity theorem. From the general approach presented here for the wa veguide with an arbitrary cross section, their existence follows in a natural way. Moreover, in such a derivation, the physical mechanism of this phenomenon becomes quite clear: a tangential magnetic field induces a magnetic moment on the hole, and the induced magnetic moment supports this field if the resona nce condition (31) is satisfied, so that the mode can exist even without an external source. One should al so note that the induced electric moment Pν is negligible for the trapped TM-mode, since Pν=O(Γsb)Mτ, as follows from Eq. (32). The equation (31) gives the frequency shift ∆ ωs≡ωs−Ωsof the trapped sth TM-mode down from the cutoffωs∆ωs ωs≃1 32k2sψ2 ττ/parenleftbig ∇νeh s/parenrightbig4. (33) In the case of a small hole this frequency shift is very small, and for the trapped mode (32) to exist, the width of the resonance should be smaller than ∆ ωs. Contributions to the resonance width come from energy dissipation in the waveguide wall due to its finite conductiv ity, and from energy radiation inside the waveguide and outside, through the hole. Radiation escaping through t he hole is easy to estimate [18], and for a thick wall it is exponentially small, e.g., [4]. The damping rate d ue to a finite conductivity is γ=P/(2W), where Pis the time-averaged power dissipation and Wis the total field energy in the trapped mode, which yields γs ωs=δ 4k2s/contintegraldisplay dl(∇νes)2, (34) whereδis the skin-depth at frequency Ω s, and the integration is along the boundary ∂S. The evaluation of the radiation into the lower waveguide modes propagating in the chamber at given frequency Ω sis also straightforward [8], if one makes use of the coefficients of mo de excitation by effective dipoles on the hole, Eqs. (9)-(13). The corresponding damping rate γR=O(ψ3) is small compared to ∆ ωs. For instance, if there 8is only one TE p-mode with the frequency below that for the lowest TM s-mode, like in a circular waveguide (H11has a lower cutoff than E 01), γR ∆ωs=ψττβ′ p k′2p/parenleftbig ∇νhh s/parenrightbig2, (35) whereβ′ p≃(k2 s−k′2 p)1/2becausek≃ks. One can easily see that denominator [1 −χ(Σ1−Σ′ 1)] in Eq. (14) does not vanish because singular terms in Σ′ 1have a “wrong” sign. However, due to the coupling between EνandHz, a non-trivial solution Eν,Hz∝ne}ationslash= 0 of simultaneous equations (14) and (16) can exist, even when Eb= 0. The corresponding condition has the form Γ′ nm≃1 4/bracketleftBig ψzzk′2 nm/parenleftbig hh nm/parenrightbig2−χ/parenleftbig ∇τhh nm/parenrightbig2/bracketrightBig , (36) which gives the frequency of the trapped TE nm-mode, provided the RHS of Eq. (36) is positive. 4.2 Impedance The trapped mode (32) gives a resonance contribution to the l ongitudinal coupling impedance at ω≈Ωs Zs(ω) =2iΩsγsRs ω2−(Ωs−iγs)2, (37) where the shunt impedance Rscan be calculated as that for a cavity with given eigenmodes, e.g. [6], Rs=σδ/vextendsingle/vextendsingle/integraltext dzexp(−iΩsz/c)Ez(z)/vextendsingle/vextendsingle2 /integraltext Swds|Hτ|2. (38) The integral in the denominator is taken over the inner wall s urface, and we assume here that the power losses due to its finite conductivity dominate. Integrating in the numerator one should include all TM-modes generated by the effective magnetic moment on the hole using E qs. (9)-(13), in spite of a large amplitude of only the trapped TM smode. While all other amplitudes are suppressed by factor Γ sb≪1, their contributions are comparable to that from TM s, because this integration produces the factor Γ qbfor any TM qmode. The integral in the denominator is obviously dominated by TM s. Performing calculations yields Rs=Z0˜e2 νψ3 ττks/parenleftbig ∇νeh s/parenrightbig4 8δ/contintegraltext dl(∇νes)2, (39) where ˜eν=eν(0) is defined by Eq. (3). Results for a particular shape of the chamber cross section c an be obtained from the equations above by substituting the corresponding eigenfunctions (see Appen dix). One should note that typically the peak value Rsof the impedance resonance due to one small hole is rather small except for the limit of a perfectly conducting w all,δ→0 — indeed, Rs∝(h/b)9b/δ, andh≪b. However, for many not-so-far separated holes, the resultin g impedance can be much larger. The trapped modes for many discontinuities on a circular waveguide has b een studied in Ref. [19], and the results can be readily transferred to the considered case of an arbitrary s hape of the chamber cross section. In particular, it was demonstrated that the resonance impedance in the extr eme case can be as large as N3times that for a single discontinuity, where Nis the number of discontinuities. It strongly depends on the distribution of discontinuities, or on the distance between them if a regula r array is considered. After the trapped modes in beam pipes with small holes were pr edicted theoretically [18, 19], their existence was proved by experiments with perforated wavegu ides at CERN [20]. 95 Analytical Formulae for Some Small Discontinuities For reader convenience, in this section we collected analyt ical expressions for the coupling impedances of various small discontinuities. The expressions give the in ductive part of the impedance and work well at frequencies below the chamber cutoff, and, in many cases, eve n at much higher frequencies. However, there can also exist resonances of the real part at frequencies nea r the cutoff for holes and cavities due to the trapped modes, as was shown in [18, 19], and the real part of th e impedance due to the energy radiated into the beam pipe should be taken into account at frequencies abo ve the cutoff, see [7, 9, 21]. It is worth noting that both the longitudinal and transverse impedance are proportional to the same combination of polarizabilities, αe+αm, for any cross section of the beam pipe. Here we use the effecti ve polarizabilities αe,αmas defined in [2]; they are related to the magnetic susceptibi lityψand the electric polarizability χof an obstacle as αe=−χ/2 andαm=ψ/2. The real part of the impedance is proportional toα2 e+α2 m, and is usually small compared to the reactance at frequenci es below the chamber cutoff. While the impedances below are written for a round pipe, more resul ts for the other chamber cross sections, the impedance dependence on the obstacle position on the wall an d on the beam position can be found in [5, 7, 9, 21, 22]. The longitudinal impedance of a small obstacle on the wall of a cylindrical beam pipe with a circular transverse cross section of radius Ris simply [3] (up to notations, it is the same Eq. (21) above) Z(k) =−ikZ0αe+αm 4π2R2, (40) whereZ0= 120πOhms is the impedance of free space, k=ω/cis the wave number, and αe,αmare the electric and magnetic polarizabilities of the discontinui ty. The polarizabilities depend on the obstacle shape and size. The transverse dipole impedance of the discontinuity for th is case is /vectorZ⊥(ω) =−iZ0αm+αe π2R4/vector ahcos(ϕh−ϕb), (41) where/vector ahis the unit vector directed to the obstacle in the chamber tra nsverse cross section containing it, ϕh andϕbare azimuthal angles of the obstacle and beam in this cross se ction. 5.1 Holes and Slots For a circular hole with radius ain a thin wall, when thickness t≪a, the polarizabilities are αm= 4a3/3, αe=−2a3/3, so that the impedance Eq. (40) takes a simple form Z(k) =−ikZ0a3 6π2R2, (42) and similarly for Eq. (41). For the hole in a thick wall, t≥a, the sum ( αm+αe) = 2a3/3 should be multiplied by a factor 0.56, see [4, 11]. There are also analy tical expressions for polarizabilities of elliptic holes in a thin wall [2], and paper [12] gives thickness corre ctions for this case. Surprisingly, the thickness factor for (αm+αe) exhibits only a weak dependence on ellipse eccentricity ε, changing its limiting value for the thick wall from 0.56 for ε= 0 to 0.59 for ε= 0.99. For a longitudinal slot of length land widthw,w/l≤1, in a thin wall, useful approximations have been obtained [13]: for a rectangular slot αm+αe=w3(0.1814−0.0344w/l) ; and for a rounded end slot αm+αe=w3(0.1334−0.0500w/l) ; substituting of which into Eqs. (40)-(41) gives the impedan ces of slots. Figure 1 compares impedances for different shapes of pumping holes. 100 2 4 6 8 1000.511.522.5 l/wZRectangular Rounded end Elliptic Figure 1: Slot impedance versus slot length lfor fixed width win units of the impedance of the circular hole with diameter w. 5.2 Annular Cut The polarizabilities of a ring-shaped cut in the wall of an ar bitrary thickness have been calculated in [14]. Such an aperture can serve as an approximation for a electrod e of a button-type beam position monitor, for a thin wall, or a model of a coax attached to the vacuum chamber , when the wall thickness is large. If aand bdenote the inner and outer radii of the annular cut, a≤b≪R, the magnetic susceptibility of a narrow (w=b−a≪b) annular slot in a thin plate is ψ≃π2b2a ln(32b/w)−2. (43) For a narrow annular gap in the thick wall the asymptotic beha vior isψ≃2πb2w. The approximation (43) works well for narrow gaps, w/b≤0.15, while the thick wall result is good only for w/b≤0.05. Analytical results for the electric polarizability of a nar row annular cut are: for a thin wall χ≃π2w2(b+a)/8, (44) and for a thick wall χ≃w2(b+a). (45) These estimates work amazingly well even for very wide gaps, up tow/b≥0.85. The electric polarizability depends on the wall thickness rather weakly. The difference ( ψ−χ)/b3= 2(αm+αe)/b3for an annular cut, calculated by variational methods in [14 ], is plotted in Fig. 2 for a few values of the wall thickness. 5.3 Protrusions For a protrusion inside the beam pipe having the shape of a hal f ellipsoid with semiaxis ain the longitudinal direction (along the chamber axis), bin the radial direction, and cin the azimuthal one, with a,b,c≪R, 11ψ−χ w/b0.2 0.4 0.6 0.8 100.511.522.5 Figure 2: Difference of the polarizabilities (in units of b3) of an annular cut versus its relative width w/b for different thicknesses of the wall t= 0;w/2;w; 2w, andt≫w(from top to bottom). The dotted line corresponds to the circular hole in a thin wall, ( ψ−χ)/b3= 4/3. the polarizabilities are [15] αe=2πabc 3Ib, (46) and αm=2πabc 3(Ic−1), (47) where Ib=abc 2/integraldisplay∞ 0ds (s+b2)3/2(s+a2)1/2(s+c2)1/2, (48) andIcis given by Eq. (48) with bandcinterchanged. In the particular case a=c,b=hwe have an ellipsoid of revolution, and the polarizabilitie s are expressed in terms of the hypergeometric function 2F1: αe=2πa2h 2F1(1,1; 5/2; 1−h2/a2), (49) and αm=2πa2h 2F1(1,1; 5/2; 1−a2/h2)−3. (50) 5.3.1 Post In the limit a=c≪h, corresponding to a pinlike obstacle, we get a simple expres sion for the inductive impedance of a narrow pin (post) of height hand radius a, protruding radially into the beam pipe:1 Z(k)≃ −ikZ0h3 6πR2(ln (2h/a)−1). (51) 1One could use the known result for the induced electric dipol e of a narrow cylinder parallel to the electric field [23]. It w ill only change ln(2 h/a)−1 in Eq. (51) to ln(4 h/a)−7/3. 122 4 6 8 h/a010203040 F Figure 3: Function F≡(αe+αm)/Vversus aspect ratio h/afor a pinlike obstacle (solid line). The electric contribution is short dashed, the magnetic one is long dashe d, and the dotted line shows the asymptotic form used in Eq. (51). The factor F≡(αe+αm)/V, whereV= 2πa2h/3 is the volume occupied by the obstacle, is plotted in Fig. 3 versus the ratio h/a. The figure also shows comparison with the asymptotic approx imation given by Eq. (51). 5.3.2 Mask One more particular case of interest is h=a, i.e. a semispherical obstacle of radius a. From Eqs. (49)-(50) the impedance of such a discontinuity is Z(k) =−ikZ0a3 4πR2, (52) which is 3π/2 times that for a circular hole of the same radius in a thin wal l, cf. Eq. (42). Another useful result that can be derived from the general so lution, Eqs. (46)-(48), is the impedance of a mask intended to intercept synchrotron radiation. We pu tb=c=h, so that our model mask has the semicircular shape with radius hin its largest transverse cross section. Then the integral i n Eq. (48) is reduced to Ib=Ic=1 32F1/parenleftbigg 1,1 2;5 2; 1−h2 a2/parenrightbigg , and we can further simplify the result for two particular cas es. The first one is the thin mask, a≪h, in which case αe≃8h3/3 , and again it dominates the magnetic term,αm≃ −V=−2πah2/3. The coupling impedance for such an obstacle — a half disk of radiushand thickness 2a,a≪h, transverse to the chamber axis — is therefore Z(k) =−ikZ02h3 3π2R2/bracketleftbigg 1 +/parenleftbigg4 π−π 4/parenrightbigga h+.../bracketrightbigg , (53) where the next-to-leading term is shown explicitly. 13Z a/h2 4 6 800.20.40.60.81 Figure 4: Impedance Zof a mask (in units of that for a semisphere with the same depth , Eq. (52) with a=h) versus its length. The narrow-mask approximation, Eq. (53 ), is short dashed, and the long-mask one, Eq. (54), is long dashed. In the opposite limit, h≪a, which corresponds to a long (along the beam) mask, the leadi ng terms αe≃ −αm≃4πah2/3 cancel each other. As a result, the impedance of a long mask w ith lengthl= 2aand heighth,h≪l, is Z(k)≃ −ikZ04h4 3πR2l/parenleftbigg lnl h−1/parenrightbigg , (54) which is relatively small due to the “aerodynamic” shape of t his obstacle, in complete analogy with results for long elliptic slots [3, 4, 13]. Figure 4 shows the impedance of a mask with a given semicircul ar transverse cross section of radius h versus its normalized half length, a/h. The comparison with the asymptotic approximations Eqs. (5 3) and (54) is also shown. One can see, that the asymptotic behavior (54) starts to work well only for very long masks, namely, when l= 2a≥10h. Figure 4 demonstrates that the mask impedance depends rath er weakly on the length. Even a very thin mask ( a≪h) contributes as much as 8 /(3π)≃0.85 times the semisphere (a=h) impedance, Eq. (52), while for long masks the impedance dec reases slowly: at l/h= 20, it is still 0.54 of that for the semisphere. In practice, however, the mask has usually an abrupt cut towa rd the incident synchrotron radiation, so that it is rather one-half of a long mask. From considerati ons above one can suggest as a reasonable impedance estimate for such a discontinuity the half sum of t he impedances given by Eqs. (53) and (54). This estimate is corroborated by 3D numerical simulations u sing the MAFIA code, at least, for the masks which are not too long. In fact, the low-frequency impedance s of a semisphere and a half semisphere of the same depth — which can be considered as a relatively short rea listic mask — were found numerically to be almost equal (within the errors), and close to that for a lo nger half mask. From these results one can conclude that a good estimate for the mask impedance is given simply by Eq. (52). 145.4 Axisymmetric Discontinuities Following a similar procedure one can also easily obtain the results for axisymmetric irises having a semi- elliptic profile in the longitudinal chamber cross section, with depth b=hand length 2 aalong the beam. For that purpose, one should consider limit c→ ∞ in Eq. (48) to calculate the effective polarizabilities ˜ αeand ˜αmper unit length of the circumference of the chamber transver se cross section. The broad-band impedances of axisymmetric discontinuities have been studied in [10], and the longitudinal coupling impedance is given by Z(k) =−ikZ0˜αe+ ˜αm 2πR, (55) quite similar to Eq. (40). As c→ ∞, the integral Ic→0, andIbis expressed in elementary functions as Ib=1 22F1/parenleftbigg 1,1 2; 2; 1−h2 a2/parenrightbigg =a a+h. It gives us immediately ˜αe=π 2h(h+a); ˜αm=−π 2ah , (56) and the resulting impedance of the iris of depth hwith the semielliptic profile is simply Z(k) =−ikZ0h2 4R, (57) which proves to be independent of the iris thickness a. The same result has been obtained using another method [24], and also directly by conformal mapping in [15], following the general method of [10]. Using a conformal mapping, one can readily obtain an answer a lso for irises having the profile shaped as a circle segment with the chord of length salong the chamber wall in the longitudinal direction, and opening angle 2 ϕ, where 0 ≤ϕ≤π. The impedance of such an exotic iris, expressed in terms of i ts height h=s(1−cosϕ)/(2 sinϕ): Z(k) = −ikZ0h2 2R(1−cosϕ)2× (58) ×/bracketleftbiggϕ(2π−ϕ) 3(π−ϕ)2sin2ϕ−2ϕ−sin2ϕ 2π/bracketrightbigg . Again, the impedance is proportional to h2, but the coefficient now depends (in fact, rather weakly) on ϕ. A few useful results for low-frequency impedances of axisym metric cavities and irises with a rectangular, trapezoidal and triangular transverse profile have been obt ained in [10] using conformal mapping to calculate the electric polarizability. The low-frequency impedance of the small short pill-box who se lengthgis not large than depth his Z(ω) =−ikZ01 2πR/parenleftbigg gh−g2 2π/parenrightbigg , (59) The low-frequency impedance of the shallow enlargement tak es the form Z(ω) =−ikZ0h2 2π2R(2 ln(2πg/h) + 1), (60) whereg≫h, but still less than R. The low-frequency impedance of a small step of depth h≪Ris Z(ω) =−ikZ0h2 4π2R(2 ln(2πR/h) + 1). (61) 150 0.1 0.2 0.3 0.4 0.500.10.20.30.40.50.60.70.80.91 νZ Figure 5: Impedance Zof a transition versus its slop, in units of that for the abrup t step (ν= 1/2, Eq. (61)) with the same height. The inductance produced by the transition with the slope ang leθ=πνhas the form: Z(ω) =−iZ0kh2 2π2R/braceleftbigg ln/bracketleftbigg πν/parenleftbiggb h−2 cotπν/parenrightbigg/bracketrightbigg +3 2−γ−ψ(ν)−π 2cotπν−1 2ν/bracerightbigg , (62) whereγ= 0.5772...is Euler’s constant, ψ(ν) is the psi-function and the transition is assumed to be shor t compared to the chamber radius, i.e. transition length l=hcotπν≪R. The ratio of this inductance to that of the abrupt step ( ν= 1/2, Eq. (61)) with the same height is plotted in Fig. 5 as a funct ion of the slope angle. The impedance of a thin (or deep) iris, g≪h, has the form Z(ω) =−ikZ01 4R/bracketleftbigg h2+gh π(ln(8πg/h)−3)/bracketrightbigg . (63) This formula works well even for rather large h, whenhis close toR. More generally, the low-frequency impedance of the iris hav ing a rectangular profile with an arbitrary aspect ratio is Z(ω) =−ikZ0gh 2πRF/parenleftbiggh g/parenrightbigg , (64) where function F(x) is plotted in Fig. 6. The impedances of discontinuities having a triangle-shape d cross section with height (depth) hand base galong the beam are given below. When g≪h, the low-frequency impedance of a triangular enlargement is Z(ω) =−ikZ01 4πR/parenleftbigg gh−g2 π/parenrightbigg , (65) and that of a triangular iris is Z(ω) =−ikZ01 4R/bracketleftbigg h2+2gh π(1−ln 2)/bracketrightbigg . (66) 160 1 2 3 4 5012345678910 h/gF(h/g)g = 2 mm g = 4 mm g = 8 mm Figure 6: The inductance dependence on the aspect ratio of a r ectangular iris, cf. Eq. (64). The marks show numerical results for a beam pipe with radius R= 2 cm, for comparison. For the case of shallow triangular perturbations, h≪g < R , both the enlargement and contraction of the chamber have the same inductance, Z(ω) =−ikZ0h22 ln2 π2R, (67) which is independent of g. Appendix Circular Chamber For a circular cross section of radius bthe eigenvalues knm=µnm/b, whereµnmismth zero of the Bessel functionJn(x), and the normalized EFs are enm(r,ϕ) =Jn(knmr)/radicalbig NEnm/braceleftbiggcosnϕ sinnϕ/bracerightbigg , (68) withNE nm=πb2ǫnJ2 n+1(µnm)/2, whereǫ0= 2 andǫn= 1 forn∝ne}ationslash= 0. For TE-modes, k′ nm=µ′ nm/bwith J′ n(µ′ nm) = 0, and hnm(r,ϕ) =Jn(k′ nmr)/radicalbig NHnm/braceleftbiggcosnϕ sinnϕ/bracerightbigg , (69) whereNH nm=πb2ǫn(1−n2/µ′2 nm)J2 n(µ′ nm)/2. In this case ˜ eν= 1/(2πb), which also follows from the Gauss law, and the formula for the inductive impedance takes an esp ecially simple form, cf. [3, 4]. 17Rectangular Chamber For a rectangular chamber of width aand height bthe eigenvalues are knm=π/radicalbig n2/a2+m2/b2with n,m= 1,2,..., and the normalized EFs are enm(x,y) =2√ absinπnx asinπmy b, (70) with 0 ≤x≤aand 0 ≤y≤b. Let a hole be located in the side wall at x=a, y=yh. From Eq. (3) after some algebra follows ˜eν=1 bΣ/parenleftBiga b,yh b/parenrightBig , (71) where Σ(u,v) =∞/summationdisplay l=0(−1)lsin[π(2l+ 1)v] cosh[π(2l+ 1)u/2](72) is a fast converging series; the behavior of Σ( u,v) versusvfor different values of the aspect ratio uis plotted in Ref. [5]. 6 Summary A review of calculating the beam coupling impedances of smal l discontinuities was presented. We also collected some analytical formulas for the inductive contr ibutions due to various small obstacles to the beam coupling impedances of the vacuum chamber. An importance of understanding these effects can be illustra ted by the following example. An original design of the beam liner for the LHC vacuum chamber anticipat ed a circular liner with many circular holes of 2-mm radius, providing the pumping area about 5% of the lin er surface. Their total contribution to the low-frequency coupling impedance was calculated [3] to be |Z/n|= 0.53 Ω,|Z⊥|= 20MΩ/m, which was close or above (for Z⊥) the estimated instability threshold. A modified liner desi gn had about the same pumping area provided by rounded-end slots 1 .5×6 mm2, which were placed near the corners of the rounded-square cross section of the liner [7]. As a resul t of these changes, the coupling impedances were reduced by more than an order of magnitude, 30-50 times: |Z/n|= 0.017 Ω,|Z⊥|= 0.4MΩ/m. Now the pumping slots are not among the major contributors to the impedance budget of the machine. One should mention that these notes do not include more recen t developments, in particular, results for coaxial structures, frequency corrections for polariz abilities, etc. Some of these new results and proper references can be found in [16]. References [1] H.A. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66, 163 (1944). [2] R.E. Collin, Field Theory of Guided Waves (IEEE Press, NY, 1991). [3] S.S. Kurennoy, “Coupling impedance of pumping holes,” P art. Acc. 39, 1 (1992); also CERN Report SL/91-29(AP)rev, Geneva (1991) [4] R.L. Gluckstern, “Coupling impedance of a single hole in a thick-wall beam pipe,” Phys. Rev. A 46, 1106 and 1110 (1992). 18[5] S.S. Kurennoy, “Beam interaction with pumping holes in v acuum-chamber walls,” in Proceed. of EPAC (Berlin, 1992), p.871; Report IHEP 92-84, Protvino, 1992. [6] S.S. Kurennoy, “Beam-chamber coupling impedance. Calc ulation methods,” CERN Report SL/91- 31(AP), Geneva (1991); also Phys. Part. Nucl. 24, 380 (1993). [7] S.S. Kurennoy, “Impedance issues for LHC beam screen,” P art. Acc. 50, 167 (1995). [8] G.V. Stupakov, “Coupling impedance of a long slot and an a rray of slots in a circular vacuum chamber,” Phys. Rev. E 51, 3515 (1995). [9] S.S. Kurennoy, R.L. Gluckstern, and G.V. Stupakov, “Cou pling impedances of small discontinuities: A general approach,” Phys. Rev. E 52, 4354 (1995). [10] S.S. Kurennoy and G.V. Stupakov, “A new method for calcu lation of low frequency coupling impedance,” Part. Acc. 45, 95 (1994). [11] R.L. Gluckstern and J.A. Diamond, “Penetration of field s through a circular hole in a wall of finite thickness,” IEEE Trans. MTT 39, 274 (1991). [12] B. Radak and R.L. Gluckstern, “Penetration of fields thr ough an elliptical hole in a wall of finite thickness,” IEEE Trans. MTT 43, 194 (1995). [13] S.S. Kurennoy, “Pumping slots: coupling impedance cal culations and estimates,” Report SSCL-636, Dallas (1993) (unpublished). [14] S.S. Kurennoy, “Polarizabilities of an annular cut in t he wall of an arbitrary thickness,” IEEE Trans. MTT44, 1109 (1996). [15] S.S. Kurennoy, “Beam coupling impedances of obstacles protruding into a beam pipe,” Phys. Rev. E 55, 3529 (1997). [16]Handbook of Accelerator Physics and Engineering , A.W. Chao and M. Tigner, Eds. (World Scientific, Singapore, 1999), Sect. 3.2. [17] P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, NY, 1953), §6.3. [18] G.V. Stupakov and S.S. Kurennoy, “Trapped electromagn etic modes in a waveguide with a small dis- continuity,” Phys. Rev. E 49, 794 (1994). [19] S.S. Kurennoy, “Trapped modes in waveguides with many s mall discontinuities,” Phys. Rev. E 51, 2498 (1995). [20] F. Caspers and T. Scholz, “Measurement of trapped modes in perforated waveguides,” Part. Acc. 51, 251 (1995). [21] S.S. Kurennoy and Y.H. Chin, “Impedances and power loss es due to pumping slots in B-factories,” Part. Acc. 52, 201 (1996). [22] S.S. Kurennoy, “Impedances and Power Losses for an Off-A xis Beam”, in Proceed. of EPAC (Barcelona, 1996), p.1449. [23] L.D. Landau and I.M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1960), §3. [24] R.L. Gluckstern and S.S. Kurennoy, “Coupling impedanc es of axisymmetric irises and cavities with semi-elliptical profile in a beam pipe,” Phys. Rev. E 55, 3533 (1997). 19

arXiv:physics/0001066v1 [physics.bio-ph] 28 Jan 2000January 2000 UTAS-PHYS-99-20 ADP-00-05/T393 The genetic code as a periodic table: algebraic aspects J. D. Bashford Centre for the Structure of Subatomic Matter, University of Adelaide Adelaide SA 5005, Australia P. D. Jarvis School of Mathematics and Physics, University of Tasmania GPO Box 252-21, Hobart Tas 7001, Australia Abstract The systematics of indices of physico-chemical properties of codons and amino acids across the genetic code are examined. Using a simple numeric al labelling scheme for nucleic acid bases, A= (−1,0),C= (0,−1),G= (0,1),U= (1,0), data can be fitted as low order polynomials of the 6 coordinates in the 64- dimensional codon weight space. The work confirms and extends the recent studies by Sie mion of the conformational parameters. Fundamental patterns in the data such as codon p eriodicities, and related harmonics and reflection symmetries, are here associated wi th the structure of the set of basis monomials chosen for fitting. Results are plotted us ing the Siemion one step mutation ring scheme, and variants thereof. The connection s between the present work, and recent studies of the genetic code structure using dynam ical symmetry algebras, are pointed out.1 Introduction and main results Fundamental understanding of the origin and evolution of th e genetic code[1] must be grounded in detailed knowledge of the intimate relationshi p between the molecular bio- chemistry of protein synthesis, and the retrieval from the n ucleic acids of the proteins’ stored design information. However, as pointed out by Lacey and Mullins[2], although ‘the nature of an evolutionary biochemical edifice must reflect .. .its constituents, ...properties which were important to prebiotic origins may not be of relev ance to contemporary sys- tems’. Ever since the final elucidation of the genetic code, t his conviction has led to many studies of the basic building blocks themselves, the amino a cids and the nucleic acid bases. Such studies have sought to catalogue and understand the spe ctrum of physico-chemical characteristics of these molecules, and of their mutual cor relations. The present work is a contribution to this programme. Considerations of protein structure point to the fundament al importance of amino acid hydrophilicity and polarity in determining folding an d enzymatic capability, and early work (Woese[3], Volkenstein[4], Grantham[5]) conce ntrated on these aspects; Weber and Lacey[6] extended the work to mono- and di-nucleosides. Jungck[7] concluded from a compilation of more than a dozen properties that correlati ons between amino acids and their corresponding anticodon dinucleosides were stronge st on the scale of hydrophobic- ity/hydrophilicity or of molecular volume/polarity (for a comprehensive review see[2]). Subsequent to this early work, using statistical sequence i nformation, conformational indices of amino acids in protein structure have been added t o the data sets[8]. Recently Siemion[10] has considered the behaviour of these paramete rs across the genetic code, and has identified certain periodicities and pseudosymmetries present when the data is plotted in a certain rank ordering called ‘one step mutation rings’, being generated by a hierarchy of cyclic alternation of triplet base letters(see [11]). Th e highest level in this hierarchy is the alternation of the second base letter, giving three majo r cycles based on the families U,CandA, each sharing parts of the Gfamily. The importance of the second base in relation to amino acid hydrophilicity is in fact well known[ 6, 2, 12], and the existence of three independent correlates of amino acid properties, aga in associated with the U,Cand Afamilies, has also been statistically established by princ ipal component analysis[13]. Given the existence of identifiable patterns in the genetic c ode in this sense, it is of some interest and potential importance to attempt to descri be them more quantitatively. Steps along these lines were taken by Siemion[14]. With a lin ear rank ordering of amino acids according to ‘mutation angle’ πk/32,k= 1, . . .,64 along the one step mutation rings, parameter Pαwas reasonably approximated by trigonometric functions wh ich captured the essential fluctuations in the data. The metaphor of there being some quantity k(such as the Siemion mutation angle), allowing the genetic code to be arranged in a way which best re flects its structure, in analogy with elemental atomic number Zand the chemical periodic table, is an extremely powerful one. The present paper takes up this idea, but in a mo re flexible way which does not rely on a single parameter. Instead, a natural labelling scheme is used which is directly related to the combinatorial fact of the triplet base codon s tructure of the genetic code, and the four letter base alphabet. Indeed, any bipartite lab elling system which identifies each of the four bases A, C, G, U , extends naturally to a composite labelling for codons,and hence amino acids. We choose for bases two coordinates as A= (−1,0),C= (0,−1), G= (0,1),U= (1,0), so that codons are labelled as ordered sextuplets, for ex ample ACG = (−1,0,0,−1,0,1). In quantitative terms, any numerical indices of amino acid o r codon properties, of physico-chemical or biological nature, can then be modelle d as some functions of the coordinates of this codon ‘weight space’. Because of other a lgebraic approaches to the structure of the genetic code, we take polynomial functions (for simplicity, of as low order as possible). This restriction does not at all exclude the possibility of periodicities and associated symmetry patterns in the data. In fact, as eac h of the six coordinates takes discrete values 0 ,±1, appropriately chosen monomials can easily reproduce suc h effects (with coefficients to be fitted which reflect the relativ e strengths of various different ‘Fourier’ components). Quite simply, the directness of a li near rank ordering, as in the measure kof mutation angle, which necessitates Fourierseries analy sis of the data, is here replaced by a more involved labelling system, but with numer ical data modelled as simple polynomial functions. The main results of our analysis are as follows. In §2 the labelling scheme for nucleic acid bases is introduced, leading to 4N-dimensional ‘weight spaces’ for length NRNA strands: in particular, 16-dimensional for N= 2, and 64-dimensional for the sextuplet codon labelling ( N=3). For N= 2 the dinucleoside hydrophilicity, hydrophobicity, and free energy of formation are considered. Displayed as linea r plots (or bar charts) on a ranking from 1 to 16, the data have obvious symmetry properti es, and corresponding basis monomials are identified, resulting in good fits. Only four co ordinates are involved for these 16 part data sets (see table 1). Moving in §3 on to codon properties as correlated to those of amino acids, the method of Siemion using amino acid r anking by mutation angle is briefly reviewed. It is shown that the trigonometric appro ximation of Siemion [14] to the Chou-Fasman[9] conformational parameters is effectively a four parameter function which allows for periodicities of 32/5, 8, 32/3 and 64 codons. Agai n, simple basis monomials having the required elements of the symmetry structure of Pαare identified, leading to a reasonable (four parameter) fit. Results are displayed as S iemion mutation angle plots. Pβis treated in a similar fashion. The method established in §§2 and 3 is then applied in §4 to other amino acid properties, including hydrophilicity and Grantham polarity. It is clearly shown that appropriate polynomial functions can be fitted to most of them (amino acid data is summarised in table 2). In§5 some concluding remarks, and outlook for further developm ent of these ideas are given. It is emphasised that, while the idiosyncracies of re al biology make it inappropriate to regard this type of approach as anything but approximate, nonetheless there may be some merit in a more rigorous follow up to establish our concl usions in a statistically valid way. This is particularly interesting in view of the ap pendix, §A. This gives a brief review of algebraic work based on methods of dynamical symme tries in the analysis of the excitation spectra of complex systems (such as atoms, nucle i and molecules), which has recently been proposed to explain the origin and evolution o f the genetic code. Specifically, it is shown how the labelling scheme adopted in the paper aris es naturally in the context of models, based either on the Lie superalgebra A5,0∼sl(6/1), or the Lie algebra B6∼ so(13), or related semisimple algebras. The origins and natur e of the polynomial functions adopted in the paper, and generalisations of these, are also discussed in the algebraiccontext. The relationship of the present paper to the dynami cal symmetry approach is also sketched in §5 below. 2 Codon systematics Ultimately our approach involves a symmetry between the 4 he terocyclic bases U,C,A,G commonly occurring in RNA. A logical starting point then, is to consider the physical properties of small RNA molecules. Dinucleosides and dinuc leotides in particular are relevant in the informational context of the genetic code an d anticode, and moreover are the building blocks for larger nucleic acids (NA’s). Wha t follows in this section is a numerical study of some properties of NA’s consisting of 2 ba ses, while in later sections NA’s with 3 bases (i.e. codons and anticodons) are considere d in the context of the genetic code as being correlated with properties of amino acids. As mentioned in the introduction, we give each NA base coordi nates in a two-dimensional ‘weight space’, namely A= (−1,0),C= (0,−1),G= (0,1),U= (1,0) with the axes labelled d, mrespectively∗. Dinucleosides and dinucleotides are therefore associate d with four coordinates ( d1, m1, d2, m2),e. g. AC= (−1,0,0,−1) with subscripts referring to the first and second base positions. The physical properties of nucleic acids we choose to fit to ar e the relative hydrophilic- itiesRfof the 16 dinucleoside monophosphates as obtained by Weber a nd Lacey[6], the relative hydrophobicity Rxof dinucleotides as calculated by Jungck[7] from the mononu - cleotide data of Garel et al. [15], and the 16 canonical (Cric k-Watson) base pair stacking parameters of Turner[16] et al. used to compute the free heat of formation ∆ G0 37of formation of duplex RNA strands at 37◦Centigrade. It should be noted that the quantity Rxwas computed as the product of mononu- cleotide values under the assumption that this determines t he true dinucleotide values to within 95 %. A result of this is that Rxis automatically the same for dinucleotides 5′−XY−3′and 5′−Y X−3′, (naturally the same holds for molecules with the reverse orientation), thus Rxis at best an approximate symmetry. The 16 Turner free energy parameters are a subset of a larger n umber of empirically determined thermodynamic “rules of thumb” (see for example Turner[17]), developed to predict free heats of formation of larger RNA and DNA molecul es. The possibility that these rules have an underlying group-theoretic structure i s a consideration for a future work. The least-squares fit to the most recent values of the Turner p arameters[16] is shown in figure 1 and is given by ∆G0 37(d1, m1, d2, m2) =−3.21−0.05(d1−d2)−1.025(d2 1+d2 2) + 0.175(m1−m2) (1) which could be qualitatively compared to the tentative “fit” obtained in [18] of eigenvalues to the older version of the Turner [17] free energy parameter s: −ǫ(j1, j2, q1, q2) =−0.9−0.5(j1(j1+ 1) + j2(j2+ 1))−0.5(q1+q2) (2) ∗The choice ( ±1,±1) and ( ±1,∓1) for the four bases simply represents a 45◦rotation of the adopted scheme, which turns out to be more convenient for our purpose s. The nonzero labels at each of the four base positions are given by the mnemonic ‘ diamond’.The latter eigenvalues were obtained from a consideration o f “polarity spin”, a theoretical property postulated for the bases G and C whereby these bases made contributions of different sign when placed in adjacent sites of an RNA chain, i n an attempt to account for nearest-neighbour stacking effects. In eq.(1) d2 icorresponds roughly to the Ai 1label anddito the spin qi. In particular the equal coefficients of the d2 ianddiand the relative minus sign of the dishould be noted in connection with equation (2). Moreover eq .(1) accounts for stacking effects of A and U through the miterms. Encouraged by this we attempt a fit using the same monomials to RfandRxwith varying success, as shown in figure 2 where Rf(d1, m1, d2, m2) = 0.191−0.087(d2 1−d2 2) + 0.09d1+ 0.107d2−0.053m1−0.077m2(3) Rx(d1, m1, d2, m2) = 0.3278−0.1814(d1+d2) + 0.093(d2 1+d2 2) + 0.0539(m1+m2) (4) The values for RfandRxare seen to be fairly anti-correlated, thus fits using the sam e monomials for each seems appropriate. 3 Amino acid conformational parameters As a case study for amino acid properties (as opposed to their correlated codon properties in§2 above) we consider the structural conformational paramet ersPαandPβ, which have been discussed by Siemion[10]. In [14] a quantity k,k= 1, . . .,64 was introduced which defined the so-called ‘mutation angle’ πk/32 for a particular assignment of codons (and hence of amino acids) in rank ordering. This is a modification of the four ring ordering used above for plots (expanded from 16 to 64 points), and aris es from a certain hierarchy of one step base mutations. It assigns the following kvalues to the NN′YandNN′R codons† 1 3 5 7 9 11 13 15 GGR GAR GAY AAY AAR CAR CAY UAY 17 19 21 23 25 27 29 31 UAR UGR UGY UCY UCR GCR GCY ACY 33 35 37 39 41 43 45 47 ACR CCR CCY CGY CGR CUR CUY UUY 49 51 53 55 57 59 61 63 GUR GUY AUY AUR AGR AGR AGY GGY wherein (as in the ‘four ring’ scheme) the third base alterna tes as . . .−G, A−U, C−C, U− A, G−. . .for purine-pyrimidine occurrences . . .−R−Y−Y−R−. . .. This ‘mutation ring’ ordering corresponds to a particular trajectory around the diamond shaped representation of the genetic code (figure 3), which is pictured in figure 4 ([1 0]) where nodes have been labelled by amino acids. Inspecting the trends of assigned Pαvalues for the amino acids ordered in this way, a suggestive 8 codon periodicity, and a plausible additional C2rotation axis about a spot in †Individual codons are labelled so that these Y,Rpositions are at the midpoints of thier respective k intervals. Thus GGR occupies 0 ≤k≤2, with nominal k= 1 and codons k(GGA) = 0.5, k(GGC) = 1.5the centre of the diagram, have be identified[10]. Figure 5 gi ves various fits to this data, as follows. Firstly, consideration of the modulation of the peaks and troughs of the period 8 component, on either side of the centre at k= 0, leads to the trigonometric function[14] Pα S(k) = 1.0−[0.32 + 0 .12 cos(kπ 16)] cos(kπ 4)−.09 sin(kπ 32) (5) where the parameters are estimated simply from the degree of variation in their heights (and 0 .44 = 0 .32 + 0 .12 is the average amplitude). Least squares fitting of the sam e data in fact leads to a similar function, Pα L(k) = 1.02−[0.22 + 0 .21 cos(kπ 16)] cos(kπ 4) +.005 sin(kπ 32). (6) From the point of view of Fourier series, however, the amplit ude modulation of the codon period 8 term in Pα SorPα Lmerely serves to add extra beats of period 32/5 and 32/3 of equal weight 0.06; an alternative might then be to allow diffe rent coefficients. This gives instead the fitted function Pα F(k) = 1.02−0.22 cos(kπ 4)−0.11 cos(3kπ 16)−.076 cos(5kπ 16) (7) which has no sin(kπ 32) term, but is almost indistinguishable from equation (6) ab ove (note that 0 .22 + 0 .21≃0.22 + 0 .11 + 0 .07≃0.32 + 0 .12 = .44). In figure 5 the Pαdata is displayed as a bar chart along with Pα S, and Pα Fabove; as can be seen, both fits show similar trends, and both have difficulty in reproducing the da ta around the first position codons of the Cfamily in the centre of the diagram (see caption to figure 5). Basing the systematics of the genetic code on numerical base labels, as advocated in the present work, a similar analysis to the above trigonomet ric functions is straighfor- ward, but now in terms of polynomials over the six codon ( i.e.trinucleotide) coordinates (d1, m1, d2, m2, d3, m3). There is no difficulty in establishing basic 8-codon period ic func- tions; combinations such as3 2d3−1 2m3(with values −3 2,−1 2,1 2,+3 2onA, G, C, U ), or more simply the perfect Y/Rdiscriminator d3−m3(with values −1,+1 on R,Yrespectively) can be assumed. Similarly, terms such as d1±m1have period 16, and d2±m2have period 64. The required modulation of the 8 codon periods can also be regained by including in the basis functions for fitting a term such as d2 2, and finally an enhancement of the Cring family boxes GCN,CCN is provided by the cubic term m1m2(m2−1). The resulting least squares fitted function is Pα 6(d1, m1, d2, m2, d3, m3) = 0.86 + 0 .24d2 2+ 0.21m1m2(m2−1)−0.02(d3−m3)−0.075d2 2(d3−m3) (8) and is plotted against the Pαdata in figure 7. The resulting fit‡is rather insensitive to the weights of d3andm3(allowing unconstrained coefficients in fact results in iden tical weights ±.02 for the linear terms and −.064,+.085 for the d2 2coefficients respectively). It ‡In contrast to the trigonometric fits which are only intended to fit the data for specified codons (indicated by the dots in figure 5), the least squares fit is app lied for the polynomial functions to all 64 data points. See [14] and the captions to figures 5 and 7should be noted that, despite much greater fidelity in the Cring,Pα 6shows similar features to the least squares trigonometric fits Pα LandPα Fin reproducing the 8 codon periodicity less clearly than Pα S(see figure 5). This indicates either that the minimisation i s fairly shallow at the fitted functions (as suggested by the fact that Pα LandPα Fdiffer by less than ±0.01 over one period), or that a different minimisation algorit hm might yield somewhat different solutions. To show the possible range of acceptabl e fits, a second monomial is displayed in figure 7 whose d2 2(d3−m3) coefficient is chosen as −0.2 rather than −0.075. This function plays the role of the original estimate Pα Sof figure 5 in displaying a much more pronounced eight codon periodicity than allowed by the least squares algorithm. The nature of the eight-codon periodicity is related to the m odulation of the con- formational status of the amino acids through the RorYnature of their third codon base[19]. A sharper discriminator[19] of this is the differe ncePα−Pβ, which suggests that a more appropriate basis for identifying numerical tre nds is with Pα−Pβ(the helix forming potential) and Pα+Pβ(generic structure forming potential). Although we have not analysed the data in this way, this is indirectly borne ou t by separate fitting (along the same lines as above) of Pβ, for which nosignificant component of ( d3−m3) is found. A typical five parameter fit, independent of third base coordi nate, is given by Pβ 6(d1, m1, d2, m2, d3, m3) = 1.02 +.26d2+.09d2 1−.19d2(d1−m1)−.1d1m2(m2−1)−.16m2 1m2(m2−1).(9) Figure 8 shows that this function does indeed average over th e third base Y/Rfluctuations evident in the Afamily data. A major component appears to be the dependence o n (d1−m1), that is, on the Y/Rnature of the firstcodon base, responsible for the major peaks and troughs visible on the AandUrings (and reflected in the d2(d1−m1) term). The cubic and quartic terms follow the modulation of the data on the Cring. The suggested pseudosymmetries of the conformational para meters are important for trigonometric functions of the mutation angle, and for poly nomial fits serve to identify leading monomial terms with simple properties. The d2(d1−m1) term in the fit of Pβ 6 above has been noted already in this connection. In the case o fPα, it should be noted that an offset of 2 codons in the position of a possible C2rotation axis (from k= 34, between ACY andACR tok= 32, after GCY) changes the axis from a pseudosymmetry axis (minima coincide with maxima after rotation) to a true symme try axis (as the alignment of minima and maxima is shifted by four codons), necessitati ng fitting by an eight period component which is evenabout k= 32. At the same time the large amplitude changes in theCring appear to require an oddfunction, and are insensitive to whether the C2axis is chosen at k= 32 or k= 34. The terms in Pα 6above have just these properties. 4 Other amino acid properties In this section we move from the biologically measured confo rmational parameters to biochemical indices of amino acid properties. Two of the mos t significant of these are the Grantham polarity[5] and the relative hydrophilicity as ob tained by Weber and Lacey [6]. Variations in chemical reactivity have been considered in [ 11], but are not modelled here.The composite Grantham index incorporates weightings for m olecular volume and molecular weight, amongst other ingredients[5]. From figur e 10 it is evident that a major pattern is a broad 16-codon periodicity (indicative of a ter m linear in d2). Additional smaller fluctuations coincide approximately with the 8-cod on periodicity of the Y/Rna- ture of the third base ( d3−m3dependence). Although there is much complex variation due to the first base, in the interests of simplicity, the foll owing fitted function ignores this latter structure, and provides an approximate (2 param eter) model (see figure 10): G6(d1, m1, d2, m2, d3, m3) = 8.298−2.716d2−0.14(d3−m3). (10) The pattern of amino acid hydrophilicity is also seen to poss ess an 8 codon periodicity. The 4 parameter fitted function considered, which is plotted in fig. 9, is: Rf6(d1, m1, d2, m2, d3, m3) = 0.816−0.038d2−0.043m2+ 0.022(d3−m3) + 0.034(1−d2)d2(d3−m3) (11) As with the case of Grantham polarity, the 8-period extrema m ight be more ’in phase’ with the data if codons were weighted according to usage, aft er the approach of Siemion [14]. 5 Conclusions and outlook In this paper we have studied codon and amino acid correlatio ns across the genetic code starting from the simplest algebraic labelling scheme for n ucleic acid bases (and hence RNA or DNA strands more generally). In §2 several dinucleoside properties have been fitted as quadratic polynomials of the labels, and §3 and§4 have considered amino acid parameters as correlated to codons (trinucleotides), name ly conformational parameters, Grantham polarity and hydrophilicity. In all cases accepta ble algebraic fitting is possible, and various patterns and periodicities in the data are readi ly traced to the contribution of specific monomials in the least squares fit. As pointed out in the appendix, §A, our algebraic approach is a special case of more general dynamical symmetry schemes in which measurable att ributes Hare given as combinations of Casimir invariants of certain chains of emb edded Lie algebras and super- algebras ([18],[20]-[26]). The identification by Jungck[7 ] of two or three major characters, to which all other properties are strongly correlated, woul d similarly in the algebraic description mean the existence of two or three distinct, ‘ma ster’ Hamiltonians H1,H2, H3,. . .(possibly with differing branching chains). In themselves t hese could be abstract and need not have a physical interpretation, but all other pr operties should be highly correlated to them, K=α1H1+α2H2+α3H3. (12) Much has been made of the famous redundancy of the code in prov iding a key to a group theoretical description[20, 22]. In the present fram ework (see also [18, 23]), codon degeneracies take second place to major features such as per iodicity and other systematic trends. Thus for example the noted 8 codon periodicity of the conformational parameter Pαallows the Ycodons for k= 25, UCY, and k= 63, AGY both to be consistentwithser(as the property attains any given value twice per 8 codon per iod, at Y/Rbox k= 24 + 1 = 25, and again 4 periods later at the alternative phase k= 56 + 7 = 63). A related theme is the reconstruction of plausible ancestra l codes based on biochemical and genetic indications of the evolutionary youth of certai n parts of the existing code. For example the anomalous features of arginine, argwhich suggests that it is an ‘intruder’ has led[28] to the proposal of a more ancient code using ornit hineorninstead. This has been supported by the trigonometric fit to Pα[14, 29], as the inferred parameters for orn actually match the fitted function better than argat the k= 41, k= 61 CGR,AGR codons. Such variations could obviously have some influence on the polynomial fitting, but at the present stage have not been implemented§. To the extent that the present analysis has been successful i n suggesting the viabil- ity of an algebraic approach, further work with the intentio n of establishing (12) in a statistically reliable fashion may be warranted. What is ce rtainly lacking to date is any microscopic justification for the application of the techni ques of dynamical symmetry al- gebras (but see [18, 23]). However, it can be considered that in the path to the genetic code, the primitive evolving and self-organising system of information storage and di- rected molecular synthesis has been subjected to ‘optimisa tion’ (whether through error minimisation, energy expenditure, parsimony with raw mate rials, or several such factors). If furthermore the ‘space’ of possible codes has the correct topology (compact and con- vex in some appropriate sense), then it is not implausible th at extremal solutions, and possibly the present code, are associated with special symm etries. It is to support the identification of such algebraic structures that the presen t analysis is directed. Acknowledgements The authors would like to thank Elizabeth Chelkowska for ass istance with Mathematica (c/circlecopyrtWolfram Research Inc) with which the least squares fitting wa s performed, and Ignacy Siemion for correspondence in the course of the work. References [1] S Osawa, T H Jukes, K Watanabe and A Muto, Microbiol Rev 56(1992) 229. [2] J C Lacey and D W Mullins Jr, Origins of Life 13(1983) 3-42. [3] C R Woese, D H Dugre, M Kando and W C Saxinger, Cold Spring Harbour Symp Quant Biol 31(1966) 723. [4] M V Volkenstein, Biochim Biohys Acta 119(1966) 421-24. [5] R Grantham, Science 185(1974) 862-64. [6] A L Weber and J C Lacey Jr, J Mol Evol 11(1978) 199-210. §The polynomial fits are to all 64 codons, not just those with greatest usage. In fact there is n o particular difficulty with argin the Pα 6function (see figure 7).[7] J R Jungck, J Mol Evol 11(1978) 211-24. [8] M Goodman and G W Moore, J Mol Evol 10(1977) 7-47. [9] P Y Chou and G D Fasman, Biochemistry 13(1974) 211-22 , see also G D Fasman in ‘Prediction of Protein Structure and the Principles of Pr otein Conformation’, ed. G D Fasman (New York: Plenum, 1989) pp. 193-316. [10] I Z Siemion, BioSystems 32(1994) 25-35; I Z Siemion, BioSystems 32(1994) 163-70 [11] I Z Siemion and P Stefanowicz, BioSystems 27(1992) 77-84. [12] F J R Taylor and D Coates, BioSystems 22(1989) 177-87. [13] M Sj¨ ostr¨ om and S Wold, J Mol Evol 22(1985) 272-77. [14] I Z Siemion, BioSystems 36(1995) 231-38. [15] J P Garel, D Filliol, and P Mandel, J Chromatog 78(1973) 381-91. [16] D H Mathews, J Sabina, M. Zuker and D.H. Turner, J Mol Biol 288(1999) 911-40. [17] M J Serra and D H Turner, Meth Enzymol 259(1995) 243-61; S M Frier, R Kierzek, J A Jaeger, N Sugimoto, M.H. Caruthers, T Neilson and D H Turne r,Proc Nat Acad Sci USA 83(1986) 9373-77. [18] J D Bashford, I Tsohantjis and P D Jarvis, Phys Lett A 233(1997) 481-88 . [19] I Z Siemion, BioSystems 33(1994) 139-48. [20] J Hornos and Y Hornos, Phys. Rev. Lett. 71(1993) 4401-04. [21] M Schlesinger, R D Kent and B G Wybourne, in Proc 4th Inter national Summer School in Theoretical Physics (Singapore: World Scientific , 1997) pp. 263-82, M Schlesinger and R D Kent, in ‘Group22: Proceedings of the XII International Collo- quium on Group Theoretical Methods in Physics’, eds S P Corne y, R Delbourgo and P D Jarvis, (Boston: International Press, 1999) pp. 152-59. [22] M Forger, Y Hornos and J Hornos, Phys Rev E 56(1997) 7078-82. [23] J D Bashford, I Tsohantjis and P D Jarvis, Proc Nat Acad Sci USA 95(1998) 987-92. [24] M Forger and S Sachse, in ‘Group22: Proceedings of the XI I International Colloquium on Group Theoretical Methods in Physics’, eds S P Corney, R De lbourgo and P D Jarvis, (Boston: International Press, 1999) pp. 147-51, se e also math-ph/9808001, math-ph/9905017. [25] L Frappat, A Sciarrino and P Sorba, Phys Lett A 250(1998) 214-21. [26] P D Jarvis, J D Bashford, in ‘Group22: Proceedings of the XII International Collo- quium on Group Theoretical Methods in Physics’, eds S P Corne y, R Delbourgo and P D Jarvis, (Boston: International Press, 1999) pp. 143-46.[27] M O Bertman and J R Jungck, J Heredity 70(1979) 379-84. [28] T H Jukes, Biochem Biophys Res Comm 53(1973) 709-14. [29] I Z Siemion and P Stefanowicz, Bull Polish Acad Sci 44(1996) 63-69.A Appendix: Dynamical symmetry algebras and ge- netic code structure The radical proposal of Hornos and Hornos[20] to elucidate t he genetic code structure using the methods of dynamical symmetry algebras drew atten tion to the relationship of certain symmetry breaking chains in the Lie algebra C3∼Sp(6) to the fundamental degeneracy patterns of the 64 codons. This theme has been tak en up subsequently using various different Lie algebras[21, 22] and also Lie superalg ebras [18, 23, 24] (see also [25]). In addition to possible insights into the code redundancy, a representation-theoretical description also leads to a code elaboration picture whereb y evolutionary primitive, de- generate assignments of many codons to a few amino acids and l arger symmetry algebras gave place, after symmetry breaking to subalgebras, to the i ncorporation of more amino acids, each with fewer redundant codons. In [18, 23, 26] emphasis was given not to the patterns of codon redundancy as such, but rather to biochemical factors which have been recognise d as fundamental keys to be incorporated in any account of evolution from a primitive co ding system to the present universal one. Among these factors is the primacy of the seco nd base letter over the first and third in correlating with such basic amino acid prop erties as hydrophilicity[3, 4]. Also, the partial purine/pyrimidine dependence of the amin o acid assignments within a family box further underlines the informational content of the third codon base[19] and necessitates a symmetry description which distinguishes t he third base letter. In [23] the amino acid degeneracy was replaced by the weaker condition o f anticodon degeneracy, leading to a Lie superalgebra classification scheme using ch ains of subalgebras of A5,0∼ sl(6/1) (see below for details). A concomitant of any representation theoretical descripti on of the genetic code is the ‘weight diagram’ mapping the 64 codons to points of the weigh t lattice (whose dimension is the rank of the algebra chosen). Reciprocally, the line of reasoning adocated above and applied in [23] to the case of Lie superalgebras suggests thatany description using dynamical symmetry algebras must be compatible with the com binatorial fact of the four- letter alphabet, three-letter word structure of the code . The viewpoint adopted in the present paper is to explore the implications of generic labe lling schemes of this type, independently of the particular choice of algebra or supera lgebra. In particular, as pointed out in §2 above, the weight diagram is supposed to arise from labelli ng each of the three base letters of the codon alphabet with a pair of dichotomic v ariables. Thus the only technical structural requirement for Lie algebras and supe ralgebras compatible with the present work is the existence of a 6 dimensional maximal abel ian (Cartan) subalgebra, and of 64-dimensional irreducible representations whose w eight diagram has the geometry of a six dimensional hypercube in the weight lattice. (The re lationship between the base alphabet and the Z2×Z2Klein four-group has been discussed in [27]). As examples of a Lie algebra and a Lie superalgebra with this structure, we here t ake the case of B6∼SO(13) andA5,0∼sl(6/1) respectively (other examples would be SO(4)3, sl(2/1)3). The orthogonal algebra SO(14) has been suggested[21] as a unifying scheme for vari- ants of the Sp(6) models[20, 22]. However, from the present perspective, it is sufficient to take spinor representations of the rank 6 odd orthogonal a lgebra SO(13) which havedimension 64. Consider the subalgebra chain SO13⊃SO(2) 4×SO9 ⊃SO(2) 4×SO(1) 4×SO(3) 5; SO(3) 5⊃SO(3) 3×SO(3) 2,or SO(3) 5∼Sp(3) 4⊃Sp(3) 2×Sp(3)′ 2, where superscripts indicate base letter. The 64-dimension al representation splits into 4 16-plets at the first breaking stage (the four families label led by second codon base letter, the latter being distinguished as a spinor (1 2,0) + (0 ,1 2) ofSO(2) 4). The same pattern repeats for the first codon base SO(1) 4providing a complete labelling of the 16 family boxes (fixed first and second base letter). The last stage give s two possible alternatives for the third base symmetry breaking: in the first, each famil y box would split into two doublets (1 2,1 2) + (1 2,−1 2) ofSO(3) 3×SO(3) 2, corresponding to a perfect 32 amino acid code 4→2 + 2, or to Y/Rdegeneracy in anticodon usage; in the second case, breaking of Sp(3)′ 2toU(3) 1yields a family box assignment 2 ×(1 2,0) + (0 ,+1 2) + (0,−1 2) coinciding to a 48 amino acid code, 4 →2 + 1 + 1, or to perfect Ydegeneracy and Rsplitting in amino acidusage. In the eukaryotic code, the 4 →2 + 1 + 1 family box pattern of anticodon usage is seen, whereas in the vertebrate mitochondrial code , only partial 4 →2+2 family box splitting of anticodon usage is found (see below). Final ly, the above labels are all (up to normalisation) of the form (0 ,±1) or ( ±1,0) for each base letter (or ( ±1,±1) for the third base for one branching) showing that this group theore tical scheme does indeed give a hypercubic geometry for the codon weight diagram. Thesl(6/1) superalgebra was advocated in a survey of possible Lie sup eralgebras relevant to the genetic code [18, 23], and possesses irreduc ible, typical representations of dimension 64 which share many of the properties of spinor rep resentations of orthogonal Lie algebras (in the family sln/1of Lie superalgebras this class of representations has dimension 2n) and so can be compared with spinors of the even and odd dimens ional Lie algebras of rank n, namely SO2nandSO2n+1respectively). The superalgebra branching chain related to the SO(13) chain described above is sl6/1⊃sl(2) 2×sl4/1 ⊃sl(2) 2×sl(1) 2×sl(3) 2/1; sl(3) 2/1⊃sl1/1or sl(3) 2/1⊃sl2×U1 where the last two steps correspond as above either to family box breaking to Y/Rdoublets (as in many of the anticodon assignments of the vertebrate mi tochondrial code) or to a 4→2 + 1 + 1 pattern (as in the anticodons of the eukaryotic code). The nature of the weight diagram follows from knowledge of the branching in ea ch of the above embeddings. In fact both in the decomposition of the irreducible 64 to fam ilies of 16, and in that of the 16 to family boxes of 4, there are a doublet and two singlet s of the accompanying sl2 2 andsl2 1algebras, so that the diagonal Cartan element (magnetic qua ntum number) hasthe spectrum 0 ,±1 2. A second diagonal label arises because there is also an addi tional commuting U1generator at each stage with value ±1 on the two singlets and 0 on the doublet. Alternatively, the additional label may be taken a s the±1 or 0 shift in the noninteger Dynkin label of the commuting sln/1algebra ( n= 4 and n= 2 respectively). Similar considerations apply to the last branching stage[2 3], so that again the weight diagram has the hypercubic geometry assumed in the text of th e paper. In the dynamical symmetry algebra approach to problems of co mplex spectra, im- portant physical quantities such as the energy levels of the system, and the transition probabilities for decays, are modelled as matrix elements o f certain operators belonging to the Lie algebra or superalgebra. In particular, the Hamil tonian operator which deter- mines the energy is assumed to be a linear combination of a set of invariants of a chain of subalgebras G⊃G1⊃G2⊃ ···T: H=c1Γ1+c2Γ2+···+cTΓT for coefficients cito be determined. For states in a certain representation of t he algebra G, the energy can often be evaluated once the hierarchy of repre sentations of ⊃G1⊃G2⊃ ···Tto which they belong is identified, as the invariants are func tions of the corresponding representation labels. As has been emphasised above, the discussion of fitting of cod on and amino acid prop- erties in the main body of the paper is independent of specific choices of Lie algebras or superalgebras. In fact, the polynomial functions of the 6 co don coordinates may simply be regarded as generalised invariants of the smallest subalgebra common to all cases, namely the 6-dimensional Cartan (maximal abelian) subalge braT(so that there are sev- eral nonzero coefficients cT, with all other cizero). This approach is thus complementary to detailed applications of a chosen symmetry algebra, wher e the coefficients ci(includ- ingcT) might accompany a specific set of Γ i(functions of the whole hierarchy of labels, whose form is fixed, depending on the subalgebra). However, b ecause the weight labels used in the present work already provide an unambiguous iden tification of the 64 states, such functions of any possible additional labels are in prin ciple determined as cases of the general expansions we have been studying. For this reason it is expected that the present work, although deliberately of a generic nature, does indee d confirm the viability of the dynamical symmetry approach.Table 1: Table of dinucleoside properties ∆G0 37(kcal/mol) fit RF fit RX fit GG -3.3 -3.2 0.065 0.0651 0.436 0.436 CG -2.4 -2.8 0.146 0.166 0.326 0.327 UG -2.1 -20.16 0.185 0.291 0.293 AG -2.1 -1.9 0.048 0.007 0.660 0.656 AC -2.2 -2.3 0.118 0.162 0.494 0.548 UC -2.4 -2.4 0.378 0.341 0.218 0.186 CC -3.3 -3.2 0.349 0.321 0.244 0.22 GC -3.4 -3.5 0.193 0.216 0.326 0.328 GU -2.2 -2.3 0.224 0.227 0.291 0.293 CU -2.1 -1.9 0.359 0.332 0.218 0.186 UU -0.9 -1.1 0.389 0.352 0.194 0.151 AU -1.1 -10.112 0.173 0.441 0.514 AA -0.9 -1.1 0.023 -0.04 10.877 UA -1.3 -1.2 0.090 0.139 0.441 0.514 CA -2.1 -20.083 0.119 0.494 0.548 GA -2.4 -2.4 0.035 0.014 0.660 0.656 Table 2: Table of amino acid properties ( Pα,β: conformational parameters; PGr: Grantham polarity; Rf: Relative hydrophilicity; Rx: Relative hydrophobicity) AA AA Pα PβPGrRf ala A1.38 0.79 8.09 0.89 arg R 10.938 10.5 0.88 asn N0.78 0.66 11.5 0.89 asp D1.06 0.66 130.87 cys C0.95 1.07 5.5 0.85 gln Q1.12 110.5 0.82 glu E1.43 0.509 12.2 0.84 gly G0.629 0.869 90.92 his H1.12 0.828 10.4 0.83 ile I0.99 1.57 5.2 0.76 leu L 1.3 1.16 4.9 0.73 lys K1.20 0.729 11.3 0.97 met M1.32 1.01 5.7 0.74 phe F1.11 1.22 5.2 0.52 pro P0.55 0.62 80.82 ser S0.719 0.938 9.19 0.96 thr T0.78 1.33 8.59 0.92 trp W 1.03 1.23 5.4 0.2 tyr Y0.729 1.31 6.2 0.49 val V0.969 1.63 5.9 0.85GGCGUGAGACUCCCGCGUCUUUAUAAUACAGA1.522.533.5-G Figure 1: Least squares fit (curve) to the Turner free energy p arameters (points) at 37◦. Units are in kcal mol−1. GGCGUGAGACUCCCGCGUCUUUAUAAUACAGA0.20.40.60.81RxGGCGUGAGACUCCCGCGUCUUUAUAAUACAGA0.10.20.3Rf Figure 2: Least squares fits for Rf(upper) and Rx(lower). Points are experimental values while the curves are least squares fits.CCC GCGG UG CG UC ACGGA UA CAAA AG A UCU AU GUUU G A C UG A C UGACU GACUG A C UG A C UGACU GACU G A C UG A C UGACU GACU G A C UG A C UGACU GACU Figure 3: ‘Weight diagram’ for the genetic code, arising as t he superposition of two projections of the 6-dimensional space of codon coordinate s onto planes corresponding to coordinates for bases of the first and second codon letters , and an additional one dimensional projection along a particular direction in the space of the third codon base. The orientations of the three projections are chosen to corr espond with the rank ordering of amino acids according to the one-step mutation rings. Figure 4: Siemion’s interpretation of the weight diagram in terms of the rank ordering of ‘one-step mutation rings’.0.50.60.70.80.911.11.21.31.41.5 10 20 30 40 50 60✸✸ ✸✸✸✸ ✸✸✸✸ ✸✸✸ ✸✸ ✸✸ ✸✸ ✸✸✸ ✸ ✸ Figure 5: Estimated trigonometric fit to the Pαconformational parameter as a function of mutation angle k. solid curve: data; dots: estimated fit, parametrised as an a mplitude modulated form (three parameters); points: evaluated fit at preferred codon positions. 0.50.60.70.80.911.11.21.31.41.5 10 20 30 40 50 60✸✸ ✸ ✸✸✸ ✸✸✸ ✸ ✸✸✸ ✸✸ ✸✸ ✸✸ ✸✸✸ ✸ ✸ Figure 6: Least squares trigonometric fit to Pαas a function of k. solid curve: data; dots: least squares fit; diamonds: evaluated fit at preferred codon positions.0.40.60.811.21.41.6 0 10 20 30 40 50 60 70✸ ✸✸ ✸✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸✸ ✸✸ ✸✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ + ++ ++ + + + + ++ + + + + + + ++ + + ++ + + ++ ++ ++ + + ++ + + ++ ++ ++ ++ + + + + ++ + + + + + + ++ + + ++ + Figure 7: Polynomial fits to the Pαconformational parameter as a function of the six codon coordinates. Solid curve: data; diamonds: least squa res fit (4 parameters); crosses: same function, with one coefficient modified to enhance eight c odon periodicity. 0.40.60.811.21.41.61.8 0 10 20 30 40 50 60 70✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ Figure 8: Least squares fit (5 parameters) to the Pβconformational parameter. Solid curve: data; diamonds: least squares fit0.20.30.40.50.60.70.80.91 0 10 20 30 40 50 60 70✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸✸ ✸✸ ✸ Figure 9: Least squares fit (4 parameters) to Weber and Lacey r elative hydrophilicity. Solid curve: data; diamonds: least squares fit. 45678910111213 0 10 20 30 40 50 60 70✸ ✸✸ ✸✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸ ✸ ✸✸ ✸✸ ✸✸ ✸ Figure 10: Least squares fit (2 parameters) to Grantham polar ity. Solid curve: data; diamonds: least squares fit.

arXiv:physics/0001067v1 [physics.acc-ph] 28 Jan 2000Cooling of Particle Beams in Storage Rings E.G. Bessonov, Lebedev Physical Institute AS, Moscow, Russia Abstract Old and new cooling methods are discussed in reference to e±, ion and µ±beams. Contents 1 Introduction 2 2 Three-dimensional radiative cooling of particle beams in storage rings by laser beams 3 2.1 Three-dimensional radiative cooling of ion beams in sto rage rings by broadband lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Three-dimensional radiative cooling of electron beams in storage rings by lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 One-dimensional laser cooling of ion beams in storage rings 7 4 Two-dimensional method of cooling of particle beams in storage rings 8 5 Two-dimensional method of cooling of ion beams in storage rings 12 6 On the ionization cooling of muons in storage rings 13 7 Laser cooling of ion beams trough the e±pair production. 14 8 Radiative cooling of e±beams in linear accelerators 15 9 Conclusion 16 9 References 17 10 Appendix 1 19 11 Appendix 2 20 11 Introduction Different cooling methods were suggested to decrease the emi ttances of charged particle beams in storage rings. Among them methods based on synchrot ron radiation damping [1]- [4], electron cooling [5], laser cooling in traps [6] and in s torage rings [7], [8], ionization cooling [3], [9]-[12] and stochastic cooling [13]. Almost all cooli ng methods are based on a friction of particles in external electromagnetic fields or in media. The friction and corresponding energy losses are determined by the next processes: 1) the spontaneous emission of the electromagnetic radiati on in external fields produced by bending magnets, undulators/wigglers, laser beams et al . (radiative reaction force origi- nating through self-fields), 2) ionization and excitation of atoms of a target at rest inst alled on the orbit of the storage ring, 3) the elastic scattering and the corresponding transfer of the kinetic energy from particles of a being cooled beam to particles of a co-propagating cold b eam ofe−,e+or ions [12], 4) excitation of being cooled ions and emission of photons by these ions through the inelastic intrabeam scattering [14], [15], [16], 5) thee±pair production by photons of a laser beam in fields of being co oled ions [17]. Only stochastic method of cooling is not based on a friction. It consists in the individual observation of particles and acting on them by external cont rol fields introduced in the storage ring and correcting theirs trajectories. A friction originating in a media or in the process of emissio n (scattering) of photons by charged particles in external fields under definite conditio ns leads to a damping of amplitudes of both betatron and phase oscillations of these particles w hen they are captured in buckets of storage rings, i.e. leads to a three-dimensional cooling of particle beams1. In this case particles of a beam loose theirs momentum. At that the fricti on force is parallel to the particle velocity, and therefor momentum losses include bo th transverse and longitudinal ones. Longitudinal momentum losses are compensated by a rad io frequency accelerating system of the storage ring. In time the longitudinal momentu m of a particle tends to a certain equilibrium value as the rate of the momentum loss of the particle is higher/less then the equilibrium one when the value of this momentum higher/l ess then the equilibrium one. The transverse vertical momentum of particles disappears i rreversibly. The transverse radial and longitudinal particle oscillati ons are dispersion coupled through energy losses [4]2. Theirs damping rates can be corrected or redistributed by i nsertion devices placed at straight sections of storage rings to introduce an additional friction of particles3 and by corresponding correction of lattice parameters of th ese rings. The damping rates of both transverse and longitudinal oscillations can be redis tributed by coupling transverse and longitudinal particle oscillations near betatron and sync hro-betatron resonances [3]. In the case of emission of particles in a laser wave the radiat ion friction force appears together with the laser wave pressure force. Friction force is determined by self-fields of the accelerated particles and the pressure force is determi ned by the electromagnetic fields 1This conclusion is valid for all possible types of friction p rocesses and was emphasized in some pioneer papers on damping/cooling (see, e.g. [3, 10]). 2Owing to this coupling radial betatron oscillations can hav e negative damping rate. 3Undulators, laser beams having in general case non-homogen eous density in the radial direction at the position of being cooled particle beam, wedge shaped materi al targets. 2of the laser wave (average of the vector product of the partic le velocity and the magnetic field strength of the laser wave which is not equal to zero when the self-force is taken into account) [18]. The total averaged force can be directed both in the same, in the transverse or in the opposite direction relative to the particle veloci ty. It depends on the direction of the particle velocity relative to the direction of the wave p ropagation. We will consider the relativistic case γ≫1 and conditions corresponding to the interaction angle bet ween vectors of the particle velocity and the direction of the laser beam p ropagation θint≫1/γ, where γ=ε/Mc2is the relativistic factor of the particle, Mparticle rest mass, εthe energy of the particle4. In this case both the radiation friction force and the press ure force are directed in the opposite site to the particle velocity. The process of cooling of particle beams based on friction fo rces has a classical nature and can be described in the framework of classical electrody namics. However such data as atomic and nuclear levels, oscillator strengths, degene racy parameters and so on which determine the corresponding laser wavelengths, cross-sec tions of particle interactions and friction forces have to be used from quantum mechanics. At th e same time the excitation of longitudinal and transverse oscillations of particles in s torage rings has a quantum nature. In this case the quasi-classical description of excitation of these oscillations can be used [4]. Friction forces can have both non-resonance and resonance n ature or can have an energy threshold (e±pair production, interaction of ions with a broadband laser beam having sharp frequency edges), the target can overlap the being cooled be am partly, the degree of over- lapping can depend on time. All these features can be used for cooling. That is why many methods can be suggested which depend on kind of particles of being cooled particle beam, theirs energy and so on. In this paper some peculiarities of one-, two- and three-dim ensional cooling methods based on a friction are discussed in reference to electron, i on and muon storage rings. The main characteristics of cooling methods (damping time, equ ilibrium emittance of the cooled beam) will be presented. 2 Three-dimensional radiative cooling of particle beams in storage rings by laser beams In the method of the three-dimensional radiative laser cool ing of particle beams in buckets of storage rings the laser beam overlap the particle beam on a pa rt of its orbit. We will consider a cooling configuration in which a laser beam is colliding hea d-on with a particle beam in a dispersion-free straight section of a storage ring. Both th e laser beam and the particle beam are focused to a waist at the center of the storage ring. In lim its of this region the particle beam is affected by a friction force through a scattering of la ser photons. Particles lose theirs energy mainly in the process of backward Compton or Rayleigh scattering of laser photons. The radiofrequency accelerating systems of storage rings a re switched on and compensate the radiative losses of the particle energy. We assume that t he incident laser beam has an uniform spectral intensity Iω=dIL/dω=IL/∆ωLin the frequency interval ∆ ωLcentered aroundωL, whereILis the total intensity (power per unit area). In the case of an ion cooling the electronic transitions of no t fully stripped ions or nuclear 4In the case θint<1/γthe particles will be accelerated by the light pressure forc e. 3transitions and broadband laser beam have to be used. When th e ion beam has an angular spread ∆ψaroundψ= 0, and relative energy spread ∆ γaround an average value γ, the full bandwidth required for the incoming laser to interact w ith all ions simultaneously (to shorten the damping time of radial betatron oscillations) i s (∆ω/ω)L= (∆ψ)2/4 + ∆γ/γ, whereωL=ωtr/γ(1 +βz),ωtrthe resonant transition energy in the rest frame of the ion, βz=vz/c,vzthe longitudinal component of the vector of the particle vel ocity. In the case of radiative cooling of electron and positron beams a monochro matic laser beam can be used. The physics of a damping in this method is similar to a synchro tron radiation damping originating from a particle emission in bending magnets of s torage rings. Difference is in the appearance of another regions where the emission of photons takes place, lattice parameters of these regions and in spectral distributions of emitted (s cattered) photons. Equilibrium emittances of particle beams are determined by a product of a damping times and rates of excitation of longitudinal or transverse oscillations of particles in the storage rings. The rate of excitation of particle oscillati ons is determined by a hardness and a power of the emitted radiation and by lattice features o f the storage ring such as its global parameter ”momentum compaction factor α” and its local parameter ”dispersion function” at the regions where the particle emit radiation [ 4]. By analogy with synchrotron radiation damping to shorten a bunch length in a storage ring one can reduce α→0 by manipulating the ring optics [19, 20]. To shorten the transv erse radial emittance one have to use long dispersion-free straight sections filled with st rong wigglers or laser beams (to produce fast damping without additional excitation of beta tron oscillations). In the case of cooling of electron beams the lattice of the storage ring m ust have large radius arcs with strongly focused FODO to produce low quanta excitation by synchrotron radiation in bending magnets of the ring [21]-[23]. 2.1 Three-dimensional radiative cooling of ion beams in sto rage rings by broadband lasers In the three-dimensional laser cooling method ion beams are cooled under conditions of Rayleigh scattering of laser photons when all ions interact with the homogeneous laser beam simultaneously and independently on theirs energy [24]-[2 7]. In this case the average cross- section of the photo-particle interactions σ=πf12reλtr(ω/∆ω)Lis larger then the Compton (Thompson) cross-section σT≃8πr2 e/3≃6.65·10−25cm2by about a factor ( λtr/re)(ω/∆ω)L, which is large, about 106−109for many examples of the practical interest. In the previous expressions the value f12is the oscillator strength, re=e2/mc2the classical electron radius. Assuming that photon-ion interactions take place in disper sion-free straight sections, the damping time of horizontal betatron oscillations τxis the same as the vertical one τy, because a variation in the radiated energy due to a variation in the or bit vanishes. The damping time of amplitudes of betatron and phase ( τǫ) oscillations of ions are τx=τy=τǫ (1 +D)=2ε P, (1) whereP=f∆Nintεlsis the average power of the electromagnetic radiation emitt ed (scat- tered) by the particle, fthe frequency of the particle beam revolution in the storage ring, ∆Nint= (1+β)IsatlintσD/c¯hωL(1+D) the number of interactions of a particle with photons of the laser beam per one collision of the particle with the la ser beam,lintthe length of the 4interaction region of the laser and particle beams, λtr= 2πc/ω tr,D=IL/Isatthe saturation parameter, Isat= [g1/4(g1+g2)](¯hω4 tr/π2c2γγ)(∆ω/ω)Lthe saturation intensity, g1(g2) are the degeneracy factor of state 1(2), εls= ¯hωtrγ= (1 +β)¯hωLγγthe average loss of the particle energy per one event of interaction of the particle with the laser photon. The expression τǫin Eq(1) is specific to the assumption that the spectral inten sity of the laser beam Iω(ω,x,y ) is constant inside its bandwidth and inside the area of the l aser beam occupied by the being cooled ion beam5. Moreover we assume that the length of the ion decay ldec=c2g2βγ/2g1f12reω2 tris much less then the length of the dispersion-free straight section [24]-[27]. The length of the ion decay is de termined by the spontaneous decay timeτsp= 1/Γ21, where Γ 21= 2g1f12reω2 tr/g2cis the probability of the spontaneous photon emission of the excited ion or the natural line width ∆ ω. Usually the relative natural line width (∆ω/ω)nat= 4πf12(g1/g2)(re/λtris less then the line width of a laser beam (∆ ω/ω)L necessary for the three-dimensional ion cooling by broadba nd laser beams and determined by the energy and angular spreads of the ion beams. Otherwise the monochromatic laser beams can be effectively used for the same purpose. The quantum nature of scattering of laser photons leads to an excitation of betatron and phase oscillations of particles. The calculation of equili brium amplitudes is similar to one for the case of ordinary electron storage rings, except that the spectral-angular probability distribution of the scattered photons here is given by that o f the undulator radiation. We found that rms relative energy spread of the particle beam at equilibrium is given by [26] σε ε=/radicalBig 1.4(1 +D)¯hωtr/Mc2. (2) In the present case, where the interaction takes place in a di spersion-free straight section, the excitation of both the horizontal and vertical betatron motions are due only to a small effect that the directions of propagation of emitted photons are not exactly parallel to the vector of the particle momentum. The equilibrium rms horizo ntal particle beam emittance is found to be ǫx=3 20¯hωtr γ2Mc2<βx>. (3) In (3)< β x>is the average horizontal beta function in the interaction r egion. The equilibrium vertical emittance is obtained by replacing βxbyβy. The rms transverse size of the particle beam at the waist σx=√βx·ǫx. Three-dimensional method of laser cooling of bunched ion be ams by narrowband or monochromatic laser beam is possible when radiofrequency c avities are switched on. In this case the two stage cooling can be done. On the first stage the value of the laser frequency have to choo se so that the related to this frequency resonance ion energy was situated above the e quilibrium energy of the storage ring and was higher then the maximum amplitude of oscillatio ns of the energy of ions in the ion beam. Then the laser frequency have to scan so that to m ove the corresponding resonance ion energy εr=Mc2γrto the equilibrium one with some velocity ˙ εr=dεr/dt corresponding to the condition ˙ εr< P max, where the resonance relativistic factor γr= 5The presence of a derivative ∂Iω/∂ωwill lead to another value τǫ[26]. The longitudinal-radial coupling arising in non-zero dispersion straight sections of the sto rage rings leads to a redistribution of the longitudinal and radial damping times when the radial gradient of the lase r beam intensity ∂Iω/∂xis introduced [24, c]. 5[1 + (ωtr/ωL)2]/2(ωtr/ωL),ωL=ωL(t). The damping time of longitudinal ion oscillations and the laser power in this case will be less (the enhanced coo ling) then in the previous case because of ions of the energy less then the equilibrium o ne will not interact with the laser beam and hence will not decrease theirs energy6. The longitudinal cooling in this case corresponds to a version of the one dimensional laser coolin g (see Section 3). The nature of the transverse cooling by the narrowband or mon ochromatic laser beam on the second stage does not differ from the case of the broadband laser cooling. A difference is in conditions of interaction of ion and laser beams. In the case of a broadband laser beam all ions interact with the laser beam simultaneously an d continuously. In the case of a narrow or monochromatic laser beam every ion interacts wit h the laser beam only a part of time when it passes the ion resonance energy in the process of its phase oscillations in a bucket of a storage ring. The minimal damping time of transv erse oscillations, according to (1), is limited by the maximal average power of scattered r adiation that is by saturation intensity and by the useful part of time when the ion and laser beams are in the state of interaction. The saturation intensity and the average po wer of scattered radiation are increased with the increasing of the width of spectrum of the laser beam. It follows that in the case of ion cooling under conditions of a minimal dampi ng time of transverse ion oscillations, i.e. under conditions of continuous interac tion of ion and laser beams along the interaction region and high saturation parameter ( D∼1), we must increase the width of the laser beam simultaneously with its power or, in a limit, by us ing the high power broadband laser. Small value of the transverse damping time is the adva ntage of the broadband laser cooling method in this case of the tree-dimensional laser co oling. The employment of the one-dimensional method of cooling on t he first stage and three- dimensional one on the second stage will permit to shorten th e value of the bandwidth of the laser beam and the power of the laser. Experimentally the version of the longitudinal cooling of a bunched non-relativistic beam of24Mg+ions (kinetic energy ∼100 keV) was observed first in the storage ring ASTRID [28]. The monochromatic laser beam co-propagated with the i on beam (conditions of the ion acceleration) at the regime of scanning of its frequency and under conditions of switched on the accelerating system of the storage ring was used. At th e conditions of such cooling some degree of the radiative transverse cooling could be obs erved. 2.2 Three-dimensional radiative cooling of electron beams in storage rings by lasers A three-dimensional cooling of electron and positron beams based on the backward Compton scattering of laser photons in the dispersion-free straigh t sections of the storage rings can be used [26, 29, 30]. In this case we can use the expressions (1 ) - (3) if we replace the valuesσ→σT,IsatD=ILand accept D= 1. The method is identical to one suggested in papers [21]-[23] where magnetic wigglers were used instead of laser beams7. In this case the excitation of radial betatron oscillations will take place only through the emission of photons of synchrotron radiation from bending magnets of the ring. A t the same time the rate of 6The longitudinal cooling in the method of the three-dimensi onal ion cooling is based on a difference of rates of energy change of ions with positive and negative ene rgy deviations from the equilibrium energy. In our case the unwanted energy change of ions corresponding to theirs negative energy deviation is absent. 7Electromagnetic waves can be considered as objects which be long to a type of undulators/wigglers [31]. 6damping of particle oscillations will be determined by the t otal power emitted both in the form of synchrotron radiation and Compton scattering of las er photons. The electromagnetic radiation emitted by electrons in the process of Compton sca ttering can lead to a significant shortening of damping times and hence equilibrium emittanc es of stored beams if the power of scattered radiation will be much higher then the power of t he synchrotron radiation. * * * The three-dimensional method of the radiative cooling cons idered in a section 2 is not selective one. All particles of a beam interact with the broa dband laser beam simultaneously independently on the energy and the position of particles. T he gradient of the rate of the energy change of particles and the corresponding rate of damping of amplitudes of betatron oscillations are small in this case. This fact lead s to the result that the damping times are approximately twice as higher as the time interval for which particles emit the electromagnetic energy equal to theirs kinetic energy. The damping times and emittances of particle beams can be sho rtened significantly by using selective interaction of particles and laser beams. T he selectivity can be achieved by different ways. Among them there are the ways based on resonan ce and threshold interaction of complicated particles with a laser beam, not full overlap ping of a being cooled particle beam and moving targets, using of broadband laser beams with sharp frequency edges. Below we will consider one- and two-dimensional cooling methods b ased on selective interactions of particle beams and targets. 3 One-dimensional laser cooling of ion beams in storage rings The one-dimensional method of laser cooling is a method of en hanced longitudinal cooling. In this method ions are cooled mainly in the longitudinal pha se space (energy-longitudinal coordinate). A typical version of the one-dimensional meth od of cooling is based on the resonance interaction of unbunched ion beams and homogeneo us monochromatic laser beams. The frequency of the laser wave (photon energy) in this versi on of cooling is chosen such a way that the laser beam interacts first with the most high ene rgy ions of the ion beam. Then the frequency is scanning in the low frequency directio n (frequency chirp) and ions of a lower energy begin to interact with the laser beam and decre ase theirs energy up to the time corresponding to the given minimal energy of the ions in the beam. Then the scanning of the laser frequency is stopped and all ions are gathered at the given energy of the beam. The cooling time of the ion beam in this case is equal to the tim e interval for which first particle emit the energy of the electromagnetic radiation e qual to the initial energy spread σε inof the ion beam and the energy spread of the cooled beam is dete rmined by either the average energy of the scattered photons or the natural line w idth of the laser beam τǫ=σε in P,σε ε=max/parenleftBiggεls Mc2γ,(∆ω ω)nat/parenrightBigg . (4) The cooling time of the ion beam in the method of the one-dimen sional laser cooling ∼ε/σε in∼103÷104times lower then that in the case of the three-dimensional on e. This is the consequence of the selective resonance interaction o f photon and ion beams in the 7one-dimensional method when initial energies of ions are hi gher then some given energy and the minimal energy of the resonance interaction of ion and la ser beams determined by the maximal laser frequency is limited by the given energy. The considered method is one of possible one-dimensional io n cooling methods8. First similar method was used for cooling of non-relativistic ion beams [8], [32]9. Relativistic version of such method was developed in the paper [34]. One-dimensional laser cooling of bunched ion beams by monoc hromatic laser beam is possible when radiofrequency cavities are switched on (see section 2.1) [28]. The broadband laser beam with a sharp low frequency edge can be used as well. In this case the edge frequency must have such value that only ions with energies a bove some given one (in the case of cooling in the bucket - above the equilibrium energy) can be excited [35]. After cooling of ions under conditions of switched off accele rating fields the energy of ions can be increased by eddy electric fields of induction acc elerators located in straight sections of storage rings, phase displacement mechanism or another way to move the orbits of ions to the central part of the storage ring. To gather the i on beam in short bunches the radiofrequency accelerating cavity can be switched on adia batically. * * * One-dimensional method of laser cooling of ion beams in stor age rings based on the resonance interaction of monochromatic laser and ion beams is one of possible methods of enhanced longitudinal cooling. Similar method can be appli ed to electron, ion and muon beams when another targets and selective interactions are u sed. One dimensional laser cooling is highly efficient in the longi tudinal direction, but diffi- cult in the transverse direction unless a special longitudi nal-radial coupling mechanism is introduced (synchro-betatron resonance [3, 36, 24], dispe rsion coupling [9, 16]). In papers [24]-[27] another three-dimensional RIC method is propose d and considered above. Never- theless the quest for new more efficient laser cooling methods remains vital for the cooling of electrons, protons, mesons and both not fully stripped an d fully stripped high current ion beams. Below we would like to discuss one of such methods. 4 Two-dimensional method of cooling of particle beams in storage rings Below the basic principle of the two-dimensional method of c ooling of particle beams in storage rings will be presented. We will consider the proces s of change of amplitudes of betatron oscillations and positions of instantaneous orbi ts in the process of the energy loss of particles when the RF accelerating system of the storage r ing is switched off10[37]. Let us the instantaneous particle orbits are distributed in a regionxη±σx εand am- plitudes of particle radial betatron oscillations are dist ributed in a region σx brelative to 8In the another version of cooling the laser frequency can be c onstant and acceleration of ions in the direction of given resonance energy can be produced by eddy e lectric fields of a linear induction accelerator. Using of induction accelerator is not efficient in high energy (ε >1 Gev/nucleon) storage rings. 9Two laser beams of different frequencies co- and counter-pro pagated with the ion beam can be used in the non-relativistic and moderate relativistic case. In th e coordinate system connected with the ion beam the frequencies of laser waves can be equal and form a standin g wave at the resonance energy [33]. 10Using coupling resonance of betatron oscillations permits to cool particle beam in all three dimensions. 8instantaneous orbits corresponding to a particle energy ε, wherexηis the location of the middle instantaneous orbit of particles of the beam, σx εmean-root square deviation of in- stantaneous orbits from the middle one. The value σxεis determined by the initial energy spreadσε in. At the first stage of the two-dimensional cooling a target T1(laser beam or material medium target) is situated at the orbit region ( xT,xT−a), whereais the target width (see Fig.1). The internal edge of the target has the form of a fl at sharp boundary. The target overlaps only a part of the particle beam so that parti cles with largest amplitudes of betatron oscillations interact with the target. At that the interaction takes place only at the moment when the particle deviation caused by betatron oscil lations have approximately the amplitude value and the deviation is directed toward the tar get. 1 21 2 3 6 7 8 9 10 11 T1T2 1 2✲ y ✻/vector v1❄/vector v2 Fig.1: The scheme of the two-dimensional ion cooling. The ax isyis the equilibrium orbit of the storage ring, 1-1, 2-2 ... the location of the ins tantaneous ion orbit after 0,1,2 ... events of the ion energy loss, T1andT2are targets moving by turns with the velocity /vector v1,2from outside to the equilibrium orbit.❄ ✻ ❄axT3 3✻ In this case immediately after the interaction the position and the direction of the particle momentum stay the same but the instantaneous orbit is displa ced inward in the direction of the target (we suppose that the damping caused by the frict ion leads to decrease of the momentum by the value only and neglect the scattering of the p article in the target). The radial coordinate of the instantaneous orbit and the amplit ude of betatron oscillations are 9decreased at the same value. After every interaction the pos ition of the instantaneous orbit will be nearer and nearer to the target and the amplitude of be tatron oscillations will be smaller and smaller until it will reach some small value. At t he same time the instantaneous orbit will reach the edge of the target. Up to this time the ins tantaneous orbit went in the direction of the target but particle was not moved forward de eper and deeper into the target. At the moment when the instantaneous orbit of a particle will be deepened into the target the amplitude of the particle betatron oscillations will be decreased much. The instantaneous orbit will continue its movement in the target with some velo city ˙xη in. Now the amplitude will be being increased proportionally to the root-mean-sq uare of the number of interactions of the particle with the target that is slowly then the decrea sing of the amplitude in the previous case when it was changed proportionally to the numb er of interactions. This is because of the particle inside the target will interact with the target both with positive and negative deviation from the instantaneous orbit (see Appen dix 1). The particle beam has a set of amplitudes of betatron oscilla tions and instantaneous orbits. To cool all the beam we must move the target radial pos ition in the direction of the particle beam or move instantaneous orbits of the particle b eam in the direction of the target at some velocity v111. This velocity must be much less then the velocity of the inst antaneous orbit crossing the target ˙ xη in. In this case the amplitudes of betatron oscillations will b e in time to be decreased to small values before they enter the tar get. After all particles of the beam will be in the state of interaction with the target then t he target must be removed or the particle beam must be returned to the initial position for a short time. After this all particles will have small amplitudes of betatron oscill ations and increased energy spread. The corresponding spread of instantaneous orbits must be in the limits of the working region of the storage ring. Notice that in general case the velocity of a particle ˙ xηdepends on the position of its instantaneous orbit. When the instantaneous orbit cross th e target and the particle pass the target every turn (the orbit enter the target at the depth hig her then the amplitude of the particle oscillations A) then its velocity ˙ xη inis higher then the velocity ˙ xηcorresponding to the orbit position outside of the beam. This is because of i n the second case particles interact with the target not every turn (see Appendix 2). At the first stage of the two-dimensional cooling of particle beams it is desirable to use the straight section with low-beta and high dispersion functio ns. In this case it is required less events of the photon emission to cool the beam in the transver se direction. This is because of the change of amplitudes of betatron oscillations is equa l to the change of positions of instantaneous orbits of the particle and at the same time in s uch straight section the spread of amplitudes of betatron oscillations is small and the step be tween positions of instantaneous orbits is high. At the second stage of this method we can use a second target fr om the opposite side of the particle beam. At this stage the particle beam will have s mall amplitudes of betatron oscillations. The radial position of the second target T2must be moved in the direction of the particle beam (or instantaneous orbits of the particl e beam must be moved in the direction of the target) with a velocity v2slightly higher then the velocity of the instantaneous particle orbit ˙ xη in. In this case particle orbits will be in time to enter into the target at a distance higher then the residual amplitudes of particle be tatron oscillations without essential 11A kick, decreasing of the value of the magnetic field in bendin g magnets of the storage ring, a phase displacement or eddy electric fields can be used for this purp ose. 10increasing of theirs amplitudes. After the particle beam en ter the target the amplitudes of betatron oscillations will be increased slowly with the num ber of interactions (according to the root-mean-square law). High energy particles first of al l and then particles with smaller energies will interact with the target. This process must la st until the target will reach the instantaneous orbit with the least energy and then the ta rget must be removed or the particle beam instantaneous orbits must be returned to the i nitial position for a short time. At that the energy spread of the particle beam will be decreas ed (1−1/kb2)−1times, where kb2=v2/˙xη in≥1. Such a way we can repeat the cooling process Ntimes and decrease the energy spread of the particle beam ∼(1−1/kb2)−Ntimes. The energy spread of the cooled particle beam will be σε ε>max/parenleftBiggεls Mc2γ,δr R,(∆ω ω)nat/parenrightBigg , (5) whereδris a width of the sharp edge of the target, Rthe average radius of the storage ring. The total damping time of the particle beam is τ=τs+τb, where τb=σεb kb1P, τ s=σε in P, (6) τs,τbare the damping times for the longitudinal and transverse ra dial phase spaces, σεb= σx bε/αRthe energy interval corresponding to the energy spread of th e particle beam whose instantaneous orbits are distributed through the interval of radiiσx b,kb1=v1/˙xη in∼ 0.1÷0.2 the ratio of the velocity of movement of the first target v1to the velocity of movement of the particle instantaneous orbit ˙ xη incaused by the interaction of particles with the target, αthe momentum compaction function. The two-dimensional method of cooling of particle beams is b ased on the longitudinal- radial coupling arising from the storage ring dispersion an d on using of moving targets (or moving instantaneous orbits). The effect is due to the fact th at the amplitudes of betatron oscillations are decreased proportionally to the number of interactions in the target when theirs instantaneous orbits are outside of the target and in creasing proportionally to the root-mean square of the number of interactions when theirs o rbits are inside of the target. This method of cooling is valid in a general case when both res onance and non-resonance selective interactions of particles are used. It is an enhan ced method because of in this method the rates of change of spreads of instantaneous orbit s and amplitudes of betatron oscillations of particles are equal to the rate of change of p ositions of instantaneous orbits in the target which is much higher then the rate of change of ampl itudes of betatron oscillations of particles in the three-dimensional method of cooling. The described process of transverse cooling is based on part icle interactions with external and internal targets. Similar interactions were described in 1956 by O’Neil [9]. However targets in his case were motionless12. They could only transform the longitudinal phase space occupied by the beam to the radial one and back. The volu me in the 4- and 6- dimensional phase space occupied by the beam was not changed . That is why the targets could be used for injection and capture of only one portion of particles. For the purpose of the multi-cycle injection and storage of heavy particles O’Neil, in addition to targets, suggested the ordinary three-dimensional ionization cool ing based on a thin hydrogen target 12Internal target could be rotated out of the medium plane to st op particle movement inward direction and such a way to prevent the beam losses. 11jet situated in the working region of the storage ring (the mu ltiple scattering of particles at the nuclei of the target was not considered). 5 Two-dimensional method of cooling of ion beams in storage rings In the case of two-dimensional ion cooling by the selective r esonance interaction the targets can be laser beams having sharp frequency edges and scanning central frequencies. The second target can be a monochromatic laser beam overlapping the ion beam as a whole and having a sweeping frequency. In this case the ordinary on e-dimensional resonance laser cooling will take place at the second stage of the laser cooli ng. The high and low boundary frequencies of the first and second lasers must correspond to the internal boundary instan- taneous orbits which are located higher and lower of the cent ral instantaneous orbits of the beam. The first target must be a broadband laser beam with the s harp edge. At that we can start from the second stage. Then the first stage and the se cond one must be repeated. Metal screens can be used on the exit of the laser beams from an optical resonator to produce the extracted laser beam with sharp geometrical edg e13. The two dimensional method of laser cooling of particle beam s will work in the case when the RF system of the storage ring is switched on as well. When the RF system of the storage ring is switched on then the s canning of the frequency of the RF system is possible instead of moving targets. Example 1. The two-dimensional cooling of a hydrogenlike beam of207 82Pb+81in the CERN LHC through the backward Rayleigh scattering of photon s of two laser beam targets. The broadband laser beams overlap the ion beam, have sharp fr equency edges and scanning central frequencies. The relevant parameters of LHC and the beam in LHC are: 2 πR= 27 km,f= 1.11·104 Hz,α= 2.94·10−4,γ= 3000,Mc2γ= 575 TeV, σε in/ε= 2·10−4(σε in= 1.15·1011eV), the valueσεb=σε in,σx b=σx ε= 1.2·10−2cm. The relevant characteristics of the hydrogenlike ( f12= 0.416,g1= 1,g2= 3) lead ions are: the transition between the 1 Sground state and the 2 Pexcited state of the particle corresponds to the value of the resonant transition energy ¯ hωtr= 68.7 keV,λtr= 1.8·10−9 sm, (∆ω/ω)nat= 2.72·10−4. The relevant parameters of a laser: the laser wavelength λL= 4πcγ/ω tr= 1080 ˚A, ¯hωL= 11.49 eV, the bandwidth of the laser beam (∆ ω/ω)L= 5·10−4, the rms transverse laser beam size at its waist σL= 1.52·10−2cm, the Rayleigh length zR=πσ2 L/λL= 67.2 cm,lint= 2zR= 135 cm, the power of the laser beam is PL= 400 W. In this case the average energy of the scattered photons <¯hωs>=εls=γ¯hωtr= 206 MeV,Isat= 4.94·1012W/cm2,IL= 5.48·105W/cm2,D= 1.1·10−7≪1,σ= 1.32·10−18 cm2, ∆Nint= 3.52·10−3,P= 8.04·109eV/sec, the longitudinal damping time τs≃14.3 sec., the damping time for the betatron oscillations τb|kb1=0.1≃143 sec. According to (4), (5) the limiting relative energy spread in this case σε/ε≃(∆ω/ω)nat≃2.72·10−4. A monochromatic laser beam can be used in this example becaus e of the natural line width (∆ω/ω)natis approximately equal to the laser line width (∆ ω/ω)Lwhich is determined 13Production of a laser beam with a sharp edge in a resonator is a difficult problem. 12by the large initial energy and angular spreads of the ion bea m (in time these spreads will be decreased and the necessity in the so large value (∆ ω/ω)Lwill disappear). Using of monochromatic laser beams with scanning frequenci es is possible in the case when the RF system of the storage ring is switched both off and o n. The gradient of the laser intensity ∂I/∂x can be introduced in the region of the ion beam by analogy with the three-dimensional ion cooling (see section 2.1). Experime nts [16] confirm this observation. Two-target scheme (Fig. 1) can be considered as a version of t he enhanced scheme based on the longitudinal-radial coupling in the storage ring. 6 On the ionization cooling of muons in storage rings Muons have rather small lifetime ( ∼2.2µs) in theirs rest frame. They can do only about 3·103turns in the strong magnetic field ( ∼10 T). That is why muon beams require enhanced cooling14. Muons have no nuclear interactions with the material mediu m of the target. That is why they have no problems with the inelastic scattering in the target. The material medium target can be used for the muon cooling. T he rate of the momentum loss of muons through the ionization friction in such target is high. The momentum loss is parallel to the particle velocity and therefor includes tra nsverse and longitudinal momentum losses. The element of the irreversibility takes place in th is case. The reacceleration of the muon beam in rf cavities restores only longitudinal momentu m. The transverse emittance is reduced by 1 /ewith as little as 2 εof the total energy exchange. The longitudinal cooling of muons is also possible when the energy losses are increased w ith the increasing of the energy of being cooled particles. Both the natural energy loss func tion for muons ( ε >0.3 GeV) and the wedge-shaped absorber at a non-zero dispersion regi on can be used in this case (in the framework of the three-dimensional method of a particle cooling the transverse cooling is decreased with the increasing of longitudinal cooling th rough the dispersion). Such three-dimensional method of cooling of muon beams disc ussed in the paper of D.V.Neuffer [38] is similar to one intended for the multi-cyc le injection and storage of heavy particles in storage rings suggested by O’Neil [9]. In the la st paper the particle beam was supposed to inject from the outside of the working region of t he ring at radial position a little higher then the position of the second target. The ins tantaneous orbit radii of particles interacting with this target would shrink after injection a nd after definite number of turns would be at radii at or less then the inside edge of the target. At that the injected particles would be drawn aside from the inflector. After that moment the difficulty would appeared with the enhanced increasing of the betatron oscillations ( see Appendix 1). This difficulty could be overcome by involving a first target placed at the inn er limit of the good region of the storage ring. This target would move the instantaneou s orbits of particles inward as soon as betatron oscillations become serious and would then continue reducing the betatron oscillation amplitudes until the first target (foil) itself was rotated out of the medium plane. Then the betatron oscillations would continue to be reduced by a thin hydrogen target jet under condition of the switched on radio frequency accel erating cavities and wait the injection of the next portion of particles. The idea of a more efficient method of the two-dimensional cool ing of particle beams in storage rings can be used in the case of muon cooling as well. T wo material medium targets 14The development of this topic was stimulated by V.Telnov aft er repeated discussions of the paper [37]. 13have to be placed in the inner and external sides of the workin g region of the storage ring. Quick-moving (velocity ∼˙xη in) and quick-disappearing targets must be used (in a time of a few particle revolutions in the storage ring). The using of t he material medium targets will lead to the high rate ionization energy losses of muons in the targets. The kick/bump can be used for the displacement of the instantaneous orbits for th and back with high velocity (instead of the displacement of targets). If the velocity of movement of the instantaneous orbit by a ki ck in the direction of the external target is about the velocity of movement of the orbi t in the target ˙ xη inthen during the stage of longitudinal cooling the instantaneous orbits will be gathered at the inner orbit position in a damping time (4), (5), where Pis the average power of the ionization energy loss. At this moment the kick must be switched off in a short tim e. In this case the muon beams will be high degree cooled. On practice the beam has some initial spread of betatron osci llations. That is why the velocity of movement of the orbit by kick must be increased 1 .2÷2 times to prevent enhanced transverse heating process target for a time when the instan taneous orbits of particles will pass the distance ∼Ain the target. The enhanced transverse muon cooling will take place when th e inner target will be moved in the direction of the muon beam with a velocity ˙ xη T1∼0.1 ˙xη in(see section 4) or the instantaneous orbits will be moved in the direction of th e target. The two-stage cooling process can be repeated several times to increase the degree of cooling. Notice that in this method of cooling the target can have shar p edge and the cooling of particles at the second stage can lead the energy spread of th e beam to the values near to zero when the beam has not betatron oscillations and the velo city of the target v2=xη in (see Appendix 2). 7 Laser cooling of ion beams trough the e±pair pro- duction. The three-dimensional method of laser cooling of ion beams b ased on the nonselective in- teraction of counterpropagating ion and photon beams throu gh thee±pair production were considered in the paper [17]. More effective two-dimensiona l method of cooling based on the threshold phenomena of the e±pair production in Coulomb fields of ions was considered in the paper [37]. Below we will consider the last case. The cross-section of the electron-positron pair productio n has the form σ|¯hω−mec2≤mec2∼π 12Z2αr2 e/parenleftBigg¯hω−2mec2 mec2/parenrightBigg3 , σ|¯hω≫2mec2≃28 9Z2αr2 e[ln2¯hω mec2−109 42−f(αZ)], (7) whereZis the atomic number of the ion, f(ν)/ν2≃1.203−ν2[39]. We see that the cross-section (7) has a threshold photon ener gy ¯hω′ thr= 2mec2in the ion rest frame or ¯ hωthr= 2mec2/(1 +βthr)γthrin the laboratory reference frame, where the ion threshold relativistic factor γthr= [1 + (ω′ thr/ωL)2]/2(ω′ thr/ωL). It means that all ions of the 14energyε>¯hωthrwill interact with the photon beam and will be gathered at the threshold energyεthr=Mc2γthr. This is a version of a method of the one-dimensional laser co oling. Below we will consider an example of the two-dimensional las er cooling of the lead ion beam trough the e±pair production. Example 2. The two-dimensional cooling of the fully stripped ion beam o f lead207 82Pb+82 in the CERN LHC through the e±pair production. The relevant parameters of LHC and the beam in LHC are the same as in the previous example: 2 πR= 27 km,f= 1.11·104Hz,α= 2.94·10−4,γ= 3000,Mc2γ= 575 TeV, σε in/ε= 2·10−4(σε in= 1.15·1011eV), the value σεb=σε in,σx b=σx ε= 1.2·10−2cm. In this method of cooling ¯ hω′ thr= 2mec2= 1.02MeV,λ′ thr= 0.01217 ˚A, ¯hωthr= 2mec2/(1 +β)γ= 170 eV,λthr= 73˚A. The relevant parameters of lasers: the laser wavelength λL= 36.5˚A, ¯hωL= 340 eV, the bandwidth of the laser beam (∆ ω/ω)L<10−4, the rms transverse laser beam size at its waistσL= 1.52·10−2cm, the Rayleigh length zR=πσ2 L/λL= 1989 cm, lint= 2zR= 3978 cm, the power of the laser beam is PL= 400 W. In this case the average energy loss per one event of pair prod uctionεls≃2γmec2= 3.06 GeV,IL= 5.48·105W/cm2,σ= 8.1·10−24cm2, ∆Nint= 2.69·10−9,P= 6.43·1010eV/sec, the longitudinal damping time τs≃1.58·105sec., the damping time for the betatron oscillations ∼10 times higher. According to (5) the limiting relative ener gy spread is σε/ε≃ 5.3·10−6. The average power of the X-ray laser in the considered exampl e is rather high but it can be realized in the future FELs [40]. The X-ray FELs are propos ed to work in the pulsed regime. In this case the damping time or/and the power of the l aser can be decreased essentially (two-four orders) if we will use ion beam gather ed in short bunches separated by a long distances (duty cycle ∼102÷104) and will use interaction regions with more small diameters of ion and photon beams. 8 Radiative cooling of e±beams in linear accelerators The electron beams can be cooled in linear accelerators if ex ternal fields (wigglers, electro- magnetic waves) producing radiation friction forces will b e distributed along the axes of the accelerators. The physics of cooling of electron beams unde r conditions of linear accelera- tion is similar to one in storage rings when external fields ar e created in the dispersion-free straight sections of the rings and when the synchrotron radi ation in bending magnets of the rings can be neglected. The element of the irreversibility t akes place in this case too. The electron momentum losses are parallel to the particle veloc ity and therefor include transverse and longitudinal momentum losses. The reacceleration of th e electron beams in accelerat- ing structures of linear accelerators restores only longit udinal momentum. The transverse emittance is reduced by 1 /ewith as little as 2 εof the total energy exchange. The expected emittances of cooled beams are small in both transverse dire ctions. In the paper [41] the effect of synchrotron radiation damping on the transverse beam emittance for an inverse free-electron laser linear accele rator is studied. In the paper [42] the same effect applied to the case of the radio frequency line ar accelerators is considered. In the paper [43] the case of linear acceleration was investi gated, where a laser beam was used instead of a wiggler. General formulas in this cases are similar. Some peculiarities are 15in the hardness of the emitted radiation. The emittances of b eams are proportional to the hardness of the emitted radiation. The hardness of the backw ard scattered laser radiation is more high. That is why laser beams for damping can be used at sm all (∼100 MeV) electron energies. A strong focusing of electron and laser beams at th e interaction point is necessary in this case. Damping wigglers can be used at high energies (1 0÷100 GeV) and in the limits of long distances along the axis of the accelerator (about so me kilometers). The described method of cooling is the three-dimensional on e. Ion and muon beams can be cooled in linear accelerators as well. The analogues of th e enhanced particle cooling in linear accelerators are possible as well. 9 Conclusion This paper was intend mainly for those who are interested in a cooling of relativistic particle beams15. We have considered different methods of cooling in a single p article approximation, when the interactions between particles are negligible (lo w density beams). The fundamental ideas of cooling of particle beams based on a friction were invented long ago. However only synchrotron radiation damping/cool ing was developed and used for a long time in e±accelerators and storage rings. The development of the e±colliders and the next generations of the synchrotron and undulator radia tion sources have led to the development of storage ring lattices (straight sections wi th high- and low-beta functions, zero momentum compaction factors and large bending radii of storage rings). The necessity in increasing of the luminosity of the p,pcolliders have led to a development of the electron and stochastic methods of cooling of particle beams. The dev elopment of the idea of the inertial confinement fusion have led to a necessity of ion coo ling of non-fully stripped ion beams through the intrabeam scattering [14, a]. The develop ment ofµcolliders have led to development of schemes based on ionization friction and an e nhanced cooling through the longitudinal-transverse dispersion of the storage rings. Laser cooling in gaps was naturally extended to the laser cooling in storage rings. We hope that the development and adoption of different method s of cooling of relativistic particle beams will lead to the next generation of storage ri ngs dedicated to colliders of different particles, new generations of light sources in opt ical to X-ray and γ-ray regions [27], ion fusion [14, a], sources of gravitational radiatio n in IR and more hard regions [44] and so on. 15Nevertheless it can be useful for those who deal with a coolin g of non-relativistic ion beams. 16References [1] D.Bohm, Phys. Rev. v.70, 249, (1946). [2] M.Sands, Phys. Rev., v.97, p. 740 (1955). [3] K.W.Robinson, Phys. Rev., 1958, v.111, No 2, p.373; A.Ho ffman, R.Little, J.M.Peterson et al., Proc. VI Int. Conf. High Energy Accel. 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[34] D.Habs et al., Proc. of the Workshop on Electron cooling and new techniques, Lengardo, Padowa - Italy, 1990, World Scientific, p.122. [35] E.G.Bessonov, Bulletin of the American physical socie ty, Vol.40, No 3, May 1995, p.1196. [36] H.Okamoto, A.M.Sessler, and D.M¨ ohl, Phys. Rev. Lett. 72, 3977 (1994); T.Kihara, H.Okamoto, Y.Iwashita, et al. Phys. Rev. E, v.59, No 3, p. 359 4, (1999). [37] E.G.Bessonov, K.-J Kim, F.Willeke, physics/9812043. 18[38] D.V.Neuffer, Nucl. Instr. Meth., 1994, v.A350, p.24. [39] V.B.Berestetskii, E.M.Lifshitz and L.P.Pitaevskii, Quantum electrodynamics, 2nd ed. (Pergamon Press, New York, 1982). [40] J.Feldhause, B.Sonntag, Synchrotron Radiation News, 1998, v.1, No 1, p.14. [41] A.C.Ting, P.A.Sprangle, Particle accelerators, 1987 , v.22, p.149 [42] N.S.Dikanskii, A.A.Mikhailichenko, Preprint 88-9, B INP, Novosibirsk, 1988; The Proc. VI All-Union Particle Accelerator Conf., v.1, p.419, 1988, Dubna D9-89-52, 1989 [43] V.Telnov, Phys. Rev. Lett., 78 (1997), p.4757; Advance d ICFA Beam dynamics Work- shop on Quantum Aspects of Beam Physics, World Scientific, Ed . Pisin Chen, Monterey, California, USA, 1998, p.173. [44] E.G.Bessonov, Advanced ICFA Beam dynamics Workshop on Quantum Aspects of Beam Physics, World Scientific, Ed. Pisin Chen, Monterey, Ca lifornia, USA, 1998, p.330. 10 Appendix 1 In the smooth approximation the movement of a particle relat ive to its instantaneous orbit position xηis described by the equation x′=Acos(Ωt+ϕ). (8) wherex′=x−xη. The amplitude of betatron oscillations of the particle A0=/radicalBig x′2 0+ ˙x′2 0/Ω2, wherex′ 0is the particle deviation from the instantaneous orbit at the mome ntt0of change of the particle energy in a target (laser beam, material medium), ˙ x′ 0=−Aωsin(ωt+ϕ) the transverse velocity of the particle. After the interaction the instant aneous orbit will be changed. The particle will continue its movement relative to a new orbit. At that its deviation relative to the new orbit will be x′ 0+δxηand the angle will not be changed. The new amplitude will beA1=/radicalBig (x′ 0−δxη)2+ ˙x′2 0/Ω2. The change of the square of the amplitude δ(A)2=A2 1−A2 0=−2x′ 0δxη+ (δxη)2. (9) Whenδxη≪x′ 0< A 0then the second term in (2) can be neglected16. In this case the valueδ(A)2≃ −2x′ 0δxηand the amplitude of betatron oscillations of the particle w ill be 16The damping of particle beam emittances in storage rings is a classical effect. In the framework of classical electrodynamics particles loose theirs energy s moothly. That is why the value ( δxη)2in (9) can be neglected. The averaged value <2Px′ 0δxη>/negationslash= 0 when the gradient of the energy loss ∂P/∂x /negationslash= 0 (see Appendix 2). Both damping and antidamping of betatron oscil lations of particles in storage rings can be in this case [4]. The term < Pδx2 η>leads to the excitation of betatron oscillations only in the case of random energy loss. 19increased by the low δA≃δxηwhenx′ 0≃A. It means that the particle will increase its amplitude of oscillations proportional to the number of pas sagesNof the particle through the target when the instantaneous orbit of the particle is at a di stance of about their amplitude (∼A) away from the target. This is the highest rate of increase of the amplitude of the particle oscillations. It follows that when the target is located at the external sid e of the working region of the storage ring xT>0, the instantaneous orbit position xη< x T, the particle enter the edge of the target under conditions of deviations x′ 0>0 and when the loss of the energy of the particle leads to the decrease of its instantaneous orbi t position ( ∂xη/∂ε > 0) then the amplitude of betatron oscillations of the particle will be i ncreased (heating regime). The rate of heating in this case is maximal (∆ A∼δxη·N). In the opposite case when the target is located at the inner side of the working region of th e storage ring xT<0,xη>xT, andx′ 0<0 the amplitude of betatron oscillations of the particle wil l be decreased (cooling regime). The instantaneous orbits in this case will shrink u nder conditions of maximal rate of cooling (∆ A∼ −δxη·N). The valueδxη=Dx∆p/p, whereDxis the local dispersion function [4], p=Mcβγ the momentum of the particle. It means that the scheme works when the dispersion function Dx/negationslash= 0. The higher Dxthe higher the rate of cooling. The term 2 x′ 0δxηin (9) determine the classical damping (antidamping) proce sses in par- ticle beams of storage rings. The value ( δxη)2determine the excitation processes of particle beams in storage rings. 10 Appendix 2 In general case the value ˙ xηdepends on the distance xT−xηbetween the target and the instantaneous orbit and on the amplitude of the particle oscillations A. The ratio F= F(xη,xT,A) = ˙xη/˙xη inis the probability of the oscillating particle to pass throu gh the target. This probability in the case of the homogeneous target with t he sharp edge is determined by the part of the period of betatron oscillation of the parti cle where the deviation of the particle from the instantaneous orbit is higher then the dis tance between the orbit and the target ( |xη−xT| ≤x′ 0≤A). In the second stage of cooling ( xT>0) the value F=ϕ2/π, whereϕ2=arccos [(xT−xη)/A]. In the first stage of cooling ( xT<0) the value F=ϕ1/π, whereϕ1=π−ϕ2. The value ˙ xη in<0 is given by the thickness of the target and the kind of its material matter or in another case by the intensity of t he laser beam and the length of the interaction region. In the approximation when the random processes of excitatio n of betatron oscillations of particles in a storage ring can be neglected the behavior o f the amplitudes of beta- tron oscillations of these particles is determined by the eq uation∂A2/∂xη= 2< x′ 0> or∂A/∂x η=< x′ 0> /A (see Appendix 1), where < x′ 0>is the particle deviation from the instantaneous orbit averaged through the range of phase s of betatron oscillations corre- sponding to the particle passage of the target ( |x′ 0|>|xη−xT|). In the first stage of cooling the value<x′ 0>=−Asinϕ 2/(π−ϕ2) =sincϕ 1, wheresincϕ 1=sinϕ 1/ϕ1. In the second 20stage of cooling the value < x′ 0>=Asincϕ 2. It follows that the cooling processes at the first and second stages (labels 1,2) are determined by the sys tem of equations ∂A ∂xη=±sincϕ 1,2,∂xη ∂t=˙xη in πϕ1,2, ϕ 2=arccosxT−xη A, ϕ 1=π−ϕ2, (10) where signs + and −are related to the first and the second stages (lower and upper targets)17. First stage of cooling Suppose that the initial spread of amplitudes of betatron os cillations of particles σx b0 does not depend on the position of the initial instantaneous orbit of the beam. Let the internal target is displaced in the direction of the being co oled beam step by step with a pitch ∆xT=σx b0/M, whereM≥2 is the whole number. In this case at the first step of the first stage of cooling the ta rget overlap a pitch ∆ xT of being cooled beam. The instantaneous orbits of this part o f particles after the moment t0 of theirs entry the target will go in the direction of the inte rnal target with a velocity ˙ xη. It follows from the equations (10) that when M≫1,xη−xT≫∆xTthen the value A−(xη−xT)≃const = ∆xT.ϕ1≃/radicalBig 2∆xT/A,sincϕ 1≃1−∆xT/3A. Up to the time when the instantaneous orbit will approach the target to the distance ∼∆xTthe value ∂A/∂x η≃1. When the orbit will enter the target ( ϕ1=π/2) then the value dA/dx ηwill be decreased only π/2 times. When the orbit will be deepened into the target at the depth higher then the amplitude Athen the amplitude will stay constant ( ϕ1=π,∂A/∂x η= 0). It means that the amplitude of betatron oscillations will no t tend to zero but to some finite valueA∼∆xT∼σx b0/M. Really in this case the amplitude of the particle oscillati ons is determined by the equation ∂A/∂x η≃1−∆xT/3Aor by the expression A−A0+ (∆xT/3) ln[(3A−∆xT)/(3A0−∆xT)] =xη−xη0. When the instantaneous orbit reach the target (xη=xT) thenxη−xη0=−A0and the amplitude is determined from the equation A≃(∆xT/3) ln[A0/(A−(∆xT/3)]. Specifically when M= 10 thenA≃1.1∆xT. At the same conditions M≫1,xη−xT≫∆xTthe valueA≃xη−xTand according to (10) the value ∂xη/∂t≃( ˙xη in/π)/radicalBig 2∆xT/A= ( ˙xη in/π)/radicalBig 2∆xT/(xη−xT). It follows that the evolution of the instantaneous orbit is described by the low (xη−xT)3/2−(xη0−xT)3/2= (3√2∆xT/2π)˙xη in(t−t0). According to this low and the condition A0=σx b0=xη0− xTthe particles of the first boundary instantaneous orbit corr esponding to the maximal initial energy of particles of the beam will reach the target (xη=xT) in a time interval ∆t1=π√ 2Mσx b0/3|˙xη in|. When the beam has the spread of instantaneous orbits σε b0 then we must move the target uniformly step by step the distan ceσε b0+σx b0with the 17This result is valid when the thickness of the target lTis much less then the length of the period of betatron oscillations. In this case the particle will inter act with the target not every turn. When the thickness of the target is large (say the target is distributed uniform ly along the orbit in the form of a set of thin foils of the same total thickness and at the definite depth) then the particle will interact with the target every turn (every period of betatron oscillations) but only along a part Fof the length of its trajectory. However this case can be realized technologically only when the targ ets are at rest. The velocity of the instantaneous orbits in general case depend on the azimuth direction. That is why in this case the presented expressions forϕ,˙A, ˙xηandFare valid only for the case when the β-function and the dispersion functions of the storage ring does not depend on azimuth. 21step ∆xT=σx b0/M. In this case the total time of the M-multiple damping of beta tron oscillations of the beam ∆ t= ∆t1(σε b0+σx b0)/∆xTor ∆t=πM√ 2M 3σx b0+σε b0 |˙xη in|. (11) The spread of instantaneous orbits after the first stage of co oling (transverse direction) will be equal σε b1= ∆t· |xη in|or σε b1=πM√ 2M 3(σx b0+σε b0). (12) Hence it follows that at the first stage of cooling the spread o f amplitudes of betatron oscillations is decreased ∼Mtimes and at the same time the spread of instantaneous orbits is increased ∼M√ Mtimes. Notice that we can move the target uniformly with the velocit yv1= ∆xT/∆t1= 3|˙xη in/πM√ 2Minstead of moving it step by step. At the first stage of the two-dimensional cooling of particle beams it is desirable to use the straight section with low-beta and high dispersion func tions. In this case less events of the photon emission is required to cool the beam in the transv erse direction as the change of amplitudes of betatron oscillations is approximately eq ual to the change of positions of instantaneous orbits of the particle and at the same time in s uch straight section we have a small spread of amplitudes of betatron oscillations and hi gh step between positions of instantaneous orbits. Second stage of cooling At the second stage of cooling the target is moving with a velo cityv2<0 from outside of the working region of the storage ring in the direction of t he being cooled particle beam. The instantaneous orbits of particles after the moment t0of theirs entry the target will go in the same direction with a velocity ˙ xη(|˙xη|<|˙xη in|). According to the expression (10) the process of change of positions of instantaneous orbits and a mplitudes of betatron oscillations is described by the system of equations (10), where xT=xT0+v2(t−t0) is the given function of time. From the equations (10) and the expression ∂A/∂x η= [∂A/∂t ]/[∂xη/∂t] the equation is followed ∂A ∂t=˙xη in π/radicalBigg 1−(xT−xη A)2. (13) According to (10), (13) the value ∂A/∂t ≤˙xη in/πfor the arbitrary time and the value ∂xη/∂t≤˙xη in/2 whenxt−xη≥0 that is up to the time t1/2< t0+A0/|v2−˙xη in/2| when the target will reach the instantaneous orbit. For the t imet1/2−t0the target will pass the way ∆ xT<A 0|v2|/|v2−˙xη in/2|, the instantaneous orbit will pass the way ∆ xη< A0|˙xη in|/2|v2−˙xη in/2|and the amplitude of betatron oscillations will be increase d on the value ∆A<A 0|˙xη in|/π|v2−˙xη in/2|. Specifically when v2= 1.5˙xη inthe values ∆ xT= 1.5A0, ∆xη< A 0/2, ∆A < A 0/π. After the target will pass outdistance the instantaneous o rbit then in a time interval t1−t1/2≃t1/2−t0the value ˙xηwill be increased to the value ˙ xη in and the value ∂A/∂t will be decreased to zero. After this moment the value xT−xη<−A 22and the amplitude of betatron oscillations will be constant . Specifically when v2= 1.5˙xη in thent1−t2≃2(t1/2−t0) and at this step the amplitude of betatron oscillations wil l be increased to the value ∆ A<A 0(1 + 1/π)(2/π)≃0.8A0. When the spread of instantaneous orbits is much higher then t he spread of amplitudes of betatron oscillations then the spread of the instantaneous orbits of the particle beam (or the energy spread) will be decreased (1 −1/kb2)−1times, where kb2=v2/˙xη in≥1. Specifically in the previous example the value kb2= 1.5, (1−1/kb2)−1= 3. When the spread of amplitudes of betatron oscillations is ze ro then we can get the parameterkb2= 1. In this case the cooled beam will be monochromatic one (1 −1/kb2)−1→ 0. In general case the value kb2has some optimum. This optimum can be received by more precise numerical calculations. The second stage can be rep eatedNtimes to decrease the energy spread of the beam (1 −1/kb2)−Ntimes. Notice that the equation (10) do not take into account that th e target pass a finite distance for the period T= 1/fof the particle revolution around its orbit in the storage ringδxT=|v2|T. When the value δxT> σ x b0,v2≥ |xin|then all instantaneous orbits of the particle beam can enter the target at the distance xη0−xT> σx b0, that is all at once under conditions ∂A/∂t = 0 (ϕ2=π). This case can be realized easier when we will do a high degree cooling ( M≫1) of the particle beam for the first stage of cooling and when the target will be installed at the straight section with low β- function and high dispersion function at the second stage of cooling. Thus at the first stage of cooling we can have significant degre e of decreasing of amplitudes of betatron oscillations (transverse cooling) and at the sa me time more significant degree of increasing of the spread of instantaneous orbits (longit udinal heating). At the second one we can have significant degree of decreasing of the spread of instantaneous orbits and some degree of increasing of amplitudes of betatron oscilla tions. We hope that successive application of two stages will lead to cooling of the particl e beam in both degrees of freedom even in the case when the interaction ot the particle beam and the target has not the resonance or threshold nature (say muon cooling). The valueσx ε0/˙xη in=σε in/˙Pdetermine the damping time τsin (5). 23

arXiv:physics/0001068v1 [physics.atom-ph] 28 Jan 2000Calculation of the Electron Self Energy for Low Nuclear Char ge Ulrich D. Jentschura,1,2,∗, Peter J. Mohr,1,†, and Gerhard Soff2,‡ 1National Institute of Standards and Technology, Gaithersb urg, MD 20899-0001, USA 2Institut f¨ ur Theoretische Physik, TU Dresden, Mommsenstr aße 13, 01062 Dresden, Germany Abstract We present a nonperturbative numerical evaluation of the on e-photon electron self energy for hydrogenlike ions with low nuclear charge nu mbers Z= 1 to 5. Our calculation for the 1 Sstate has a numerical uncertainty of 0.8 Hz for hydrogen and 13 Hz for singly-ionized helium. Resummation a nd convergence acceleration techniques that reduce the computer time by ab out three orders of magnitude were employed in the calculation. The numerica l results are compared to results based on known terms in the expansion of t he self energy in powers of Zα. PACS numbers 12.20.Ds, 31.30.Jv, 06.20.Jr, 31.15.-p Typeset using REVT EX 1Recently, there has been a dramatic increase in the accuracy of experiments that measure the transition frequencies in hydrogen and deuterium [1,2] . This progress is due in part to the use of frequency chains that bridge the range between o ptical frequencies and the microwave cesium time standard. The most accurately measur ed transition is the 1 S-2S frequency in hydrogen; it has been measured with a relative u ncertainty of 3 .4×10−13or 840 Hz. With trapped hydrogen atoms, it should be feasible to observe the 1 S-2Sfrequency with an experimental linewidth that approaches the 1 .3 Hz natural width of the 2 Slevel [3,4]. Indeed, it is likely that transitions in hydrogen wil l eventually be measured with an uncertainty below 1 Hz [5,6]. In order for the anticipated improvement in experimental ac curacy to provide better values of the fundamental constants or better tests of QED, t here must be a corresponding improvement in the accuracy of the theory of the energy level s in hydrogen and deuterium, particularly in the radiative corrections that constitute the Lamb shift. As a step toward a substantial improvement of the theory, we have carried out a numerical calculation of the one-photon self energy of the 1 Sstate in a Coulomb field for values of the nuclear charge Z= 1,2,3,4,5. This is the first complete calculation of the self energy at lowZand provides a result that contributes an uncertainty of about 0.8 Hz in hy drogen and deuterium. This is a decrease in uncertainty of more than three orders of magn itude over previous results. Among all radiative corrections, the largest by several ord ers of magnitude are the one- photon self energy and vacuum polarization corrections. Of these, the larger and historically most problematic is the self energy. Analytic calculations of the electron self energy at low nuclear charge Zhave extended over 50 years. The expansion parameter in the a nalytic calculations is the strength of the external binding field Zα. This expansion is semi-analytic [i.e., it is an expansion in powers of Zαand ln( Zα)−2]. The leading term was calculated in [7]. It is of the order of α(Zα)4ln(Zα)−2in units of mec2, where meis the mass of the electron. In subsequent work [7–25] higher-order coefficien ts were evaluated. The analytic results are relevant to low- Zsystems. For high Z, the complete one- photon self energy has been calculated without expansion in Zαby numerical methods [26–37]. However, such numerical evaluations at low nuclea r charge suffer from severe loss of numerical significance at intermediate stages of the calc ulation and slow convergence in the summation over angular momenta. As a consequence, the nu merical calculations have been confined to higher Z. Despite these difficulties, the numerical calculations at hi gherZcould be used together with the power-series results to extrapolate to low Zwith an assumed functional form in order to improve the accuracy of the self energy at low Z[30]; up to the present, this approach has provided the most accurate theoretical prediction for t he one-photon self energy of the 1Sstate in hydrogen [38]. However, this method is not completely satisfactory. The ex trapolation procedure gives a result with an uncertainty of 1.7 kHz, but employs a necessa rily incomplete analytic approximation to the higher-order terms. It therefore cont ains a component of uncertainty that is difficult to reliably assess. Termination of the power series at the order of α(Zα)6 leads to an error of 27 kHz. After the inclusion of a result rec ently obtained in [25] for the logarithmic term of order α(Zα)7ln(Zα)−2the error is still 13 kHz. A detailed comparison between the analytic and numerical ap proaches has been inhibited by the lack of accurate numerical data for low nuclear charge . The one-photon problem is 2especially well suited for such a comparison because five ter ms in the Zαexpansion have been checked in independent calculations. The known terms c orrespond to the coefficients A41,A40,A50,A62andA61listed below in Eq. (3). The energy shift ∆ ESEdue to the electron self energy is given by ∆ESE=α π(Zα)4 n3mec2F(Zα), (1) where nis the principal quantum number. For a particular atomic sta te, the dimensionless function Fdepends only on one argument, the coupling Zα. The semi-analytic expansion ofF(Zα) about Zα= 0 gives rise to the following terms: F(Zα) =A41ln(Zα)−2+A40+ (Zα)A50+ (Zα)2 ×/bracketleftBig A62ln2(Zα)−2+A61ln(Zα)−2+GSE(Zα)/bracketrightBig , (2) where GSE(Zα) represents the nonperturbative self-energy remainder fu nction. The first index of the Acoefficients gives the power of Zα[including the ( Zα)4prefactor from Eq. (1)], the second corresponds to the power of the logarithm. F or the 1 Sground state, which we investigate in this Letter, the terms A41andA40were obtained in [7–13]. The correction termA50was found in [14–16]. The higher-order corrections A62andA61were evaluated and confirmed in [17–21]. The results are A41=4 3, A40=10 9−4 3lnk0, A50= 2π/parenleftbigg139 64−ln 2/parenrightbigg , A62=−1, A61=28 3ln 2−21 20. (3) The Bethe logarithm ln k0has been evaluated, e.g., in [39,40] as ln k0= 2.984 128 555 8(3). For our high-accuracy, numerical calculation of F(Zα), we divide the calculation into a high- and a low-energy part (see Ref. [28]). Except for a fur ther separation of the low- energy part into an infrared part and a middle-energy part, w hich is described in [41] and not discussed further here, we use the same integration cont our for the virtual photon energy and basic formulation as in [28]. The numerical evaluation of the radial Green function of the bound electron [see Eq. (A.16) in [28]] requires the calculation of the Whittaker fu nction Wκ,µ(x) (see [42], p. 296) over a very wide range of parameters κ,µand arguments x. Because of numerical cancella- tions in subsequent steps of the calculation, the function Whas to be evaluated to 1 part in 1024. In a problematic intermediate region, which is given appro ximately by the range 15< x < 250, we found that resummation techniques applied to the div ergent asymptotic series of the function Wprovide a numerically stable and efficient evaluation scheme . These techniques follow ideas outlined in [43] and are described i n detail in [41]. For the acceleration of the slowly convergent angular momen tum sum in the high-energy part [see Eq. (4.3) in [29]], we use the combined nonlinear-c ondensation transformation [44]. 3This transformation consists of two steps: First, we apply t he van Wijngaarden condensation transformation [45] to the original series to transform the slowly convergent monotone input series into an alternating series [46]. In the second step, t he convergence of the alternating series is accelerated by the δtransformation [see Eq. (3.14) in [44]]. The δtransformation acts on the alternating series much more effectively than on t he original input series. The highest angular momentum, characterized by the Dirac quant um number κ, included in the present calculation is about 3 500 000. However, even in thes e extreme cases, evaluation of less than 1 000 terms of the original series is required. As a r esult, the computer time for the evaluation of the slowly convergent angular momentum ex pansion is reduced by roughly three orders of magnitude. The convergence acceleration te chniques remove the principal numerical difficulties associated with the singularity of th e relativistic propagators for nearly equal radial arguments. These singularities are present in all QED effects in bound systems, irrespective of the number of photons involved. It is expect ed that these techniques could lead to a similar decrease in computer time in the calculatio n of QED corrections involving more than one photon. In the present calculation, numerical results are obtained for the scaled self-energy func- tionF(Zα) for the nuclear charges Z= 1,2,3,4,5 (see Table 1). The value of αused in the calculation is α0= 1/137.036. This is close to the current value from the anomalous magnetic moment of the electron [47], 1/α= 137 .035 999 58(52) . The numerical data points are plotted in Fig. 1, together wit h a graph of the function determined by the analytically known lower-order coefficien ts listed in Eq. (3). In order to allow for a variation of the fine-structure consta nt, we repeated the calculation with two more values of α, which are 1/α>= 137 .035 999 5 and 1 /α<= 137 .036 000 5 . On the assumption that the main dependence of FonZαis represented by the lower-order terms in (3), the change in F(Zα) due to the variation in αis ∂F(Zα) ∂αδα=−2A41δα α+/bracketleftBig Z A50+ O(αln2α)/bracketrightBig δα for a given nuclear charge Z. Based on this analytic estimate, we expect a variation F(Zα>)−F(Zα0)≈F(Zα0)−F(Zα<)≈ −9×10−9 for the different values of α. This variation is in fact observed in our calculation. E.g. , for the case Z= 2 we find F(2α<) = 8.528 325 061(1) , F(2α0) = 8.528 325 052(1) and F(2α>) = 8.528 325 043(1) . This constitutes an important stability check on the numeri cs and it confirms that the main dependence of Fon its argument is indeed given by the lowest-order analytic coefficients A41andA50. 4In addition to the results for F(Zα0), numerical results for the nonperturbative self- energy remainder function GSE(Zα0) are also given in Table 1. The results for the re- mainder function are obtained from the numerical data for F(Zα0) by direct subtraction of the analytically known terms corresponding to the coefficien tsA41,A40,A50,A62andA61 [see Eqs. (2,3)]. Note that because the dependence of FonZαis dominated by the sub- tracted lower-order terms, we have at the current level of ac curacy GSE(Zα<) =GSE(Zα0) = GSE(Zα>). The numerical uncertainty of our calculaton is 0 .8×Z4Hz in frequency units. A sensitive comparison of numerical and analytic approache s to the self energy can be made by extrapolating the nonperturbative self-energy rem ainder function GSE(Zα) to the point Zα= 0. It is expected that the function GSE(Zα) approaches a constant in the limit Zα→0. This constant is referred to as GSE(0)≡A60. In the analytic approach, much attention has been devoted to the coefficient A60[21–24]. The correction has proven to be difficult to evaluate, and analytic work on A60has extended over three decades. A step-by- step comparison of the analytic calculations has not been fe asible, because the approaches to the problem have differed widely. An additional difficulty i s the isolation of terms which contribute in a given order in Zα, i.e. the isolation of only those terms which contribute to A60(and not to any higher-order coefficients). In order to address the question of the consistency of A60with our numerical results, we perform an extrapolation of our data to the point Zα= 0. The extrapolation procedure is adapted to the problem at hand. We fit GSEto an assumed functional form which corre- sponds to A60,A71andA70terms, with the coefficients to be determined by the fit. We find that our numerical data is consistent with the calculated va lueA60=−30.924 15(1) [24,48]. It is difficult to assess the seventh-order logarithmic term A71, because the extrapolated value for A71is very sensitive to possible eighth-order triple and doubl e logarithmic terms, which are unknown. We obtain as an approximate result A71= 5.5(1.0), and we therefore cannot conclusively confirm the result [25] A71=π/parenleftbigg139 64−ln 2/parenrightbigg = 4.65. Since our all-order numerical evaluation eliminates the un certainty due to higher-order terms, we do not pursue this question any further. The numerical data points of the function GSE(Zα) are plotted in Fig. 2 together with the value GSE(0) = A60=−30.924 15(1). For a determination of the Lamb shift, the dependence of GSEon the reduced mass mrof the system has to be restored. In general, the coefficients in the analytic expansion (2) acquire a facto r (mr/me)3, because of the scaling of the wave function. Terms associated with the anom alous magnetic moment are proportional to ( mr/me)2[49]. The nonperturbative remainder function GSEis assumed to be approximately proportional to ( mr/me)3, but this has not been proved rigorously. Work is currently in progress to address this question [50]. We conclude with a brief summary of the results of this Letter . (i) We have obtained accurate numerical results for the self energy at low nuclea r charge. Previously, severe numerical cancellations have been a problem for these evalu ations. (ii) For a particular example, we have addressed the question of how well semi-ana lytic expansions represent all-order results at low nuclear charge. Our numerical data is consistent with the value A60=−30.924 15(1) [24,48]. (iii) Numerical techniques [44] have bee n developed that reduce the computer time for the problem by about three order s of magnitude. 5The calculation presented here is of importance for the inte rpretation of measurements in hydrogen, deuterium and singly-ionized helium and for th e improvement of the Rydberg constant, because of recent and projected progress in accur acy. In the determination of the Rydberg constant, uncertainty due to the experimentally de termined proton radius can be eliminated by comparing the frequencies of more than one tra nsition [2]. We have shown that an all-order calculation can provide the required accu racy if suitable numerical methods are used. The authors acknowledge helpful discussions with E.J. Weni ger. U.D.J. gratefully ac- knowledges helpful conversations with J. Baker, J. Conlon, J. Devaney and J. Sims, and sup- port by the Deutsche Forschungsgemeinschaft (contract no. SO333/1-2) and the Deutscher Akademischer Austauschdienst. 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Kinoshita (World Scientific, Singapore, 1990), pp. 560–672. [50] K. Pachucki and S. Karshenboim, private communication (1998). 8TABLES TABLE I. Scaled self-energy function and nonperturbartive self-energy remainder function for low-Zhydrogenlike systems. F(Zα0) and GSE(Zα0) Z F (Zα0) GSE(Zα0) 1 10.316 793 650(1) −30.290 24(2) 2 8.528 325 052(1) −29.770 967(5) 3 7.504 503 422(1) −29.299 170(2) 4 6.792 824 081(1) −28.859 222(1) 5 6.251 627 078(1) −28.443 472 3(8) 9FIGURES FIG. 1. The self-energy function F(Zα). The points are the numerical results of this work, the curve is given by the analytically known terms that corre spond to the coefficients listed in Eq. (3). 012345 AtomicNumber Z681012F/LParen1ZΑ/RParen1 681012 FIG. 2. Results for the scaled self-energy remainder functi onGSE(Zα) at low Z. 012345 AtomicNumber Z/Minus31/Minus29.5/Minus28GSE/LParen1ZΑ/RParen1 /Minus31/Minus29.5/Minus28 10

arXiv:physics/0001069v1 [physics.data-an] 28 Jan 2000Nonlinear denoising of transient signals with application to event related potentials A. Efferna,b, K. Lehnertza, T. Schreiberc, T. Grunwalda, P. Davidb, C.E. Elgera aDepartment of Epileptology, University of Bonn, Sigmund-F reud Str. 25, 53105 Bonn, Germany bInstitute of Radiation and Nuclear Physics, University of B onn, Nussallee 11-13, 53115 Bonn, Germany cDepartment of Physics, University of Wuppertal, Gauss-Str asse 20, 42097 Wuppertal Abstract We present a new wavelet based method for the denoising of event related potentials (ERPs), employing techniques recently developed for the pa radigm of deterministic chaotic systems. The denoising scheme has been constructed to be appropriate for short and transient time sequences using circular state spa ce embedding. Its effec- tiveness was successfully tested on simulated signals as we ll as on ERPs recorded from within a human brain. The method enables the study of ind ividual ERPs against strong ongoing brain electrical activity. Keywords: nonlinear denoising, state space, wavelets, cir cular embedding PACS numbers: 05.45.+b 87.22.-q 87.22.Jb Preprint submitted to Elsevier Preprint 2 February 20081 Introduction Theelectroencephalogram (EEG) reflects brain electrical activity owing to both intrinsic dynamics and responses to external stimuli. To ex amine pathways and time courses of information processing under specific condi tions, several exper- iments have been developed controlling sensory inputs. Usu ally, well defined stimuli are repeatedly presented during experimental sess ions (e.g., simple tones, flashes, smells, or touches). Each stimulus is assume d to induce syn- chronized neural activity in specific regions of the brain, o ccurring as potential changes in the EEG. These evoked potentials (EPs) often exhibit multiphasic peak amplitudes within the first hundred milliseconds after stimulus onset. They are specific for different stages of information process ing, thus giving access to both temporal and spatial aspects of neural proces ses. Other classes of experimental setups are used to investigate higher cogni tive functions. For example, subjects are requested to remember words, or perha ps they are asked to respond to specific target stimuli, e.g. by pressing a butt on upon their oc- currence. The neural activity induced by this kind of stimul ation also leads to potential changes in the EEG. These event related potentials (ERPs) can extend over a few seconds, exhibiting peak amplitudes mostl y later than EPs. Deviation of amplitudes and/or moment of occurrence (laten cy) from those of normal EPs/ERPs are often associated with dysfunction of th e central nervous system and thus, are of high relevance for diagnostic purpos es. As compared to the ongoing EEG, EPs and ERPs possess very low p eak am- plitudes which, in most cases, are not recognizable by visua l inspection. Thus, to improve their low signal-to-noise ratio, EPs/ERPs are co mmonly averaged (Figure 1), assuming synchronous, time-locked responses n ot correlated with the ongoing EEG. In practice, however, these assumptions ma y be inaccurate 2and, as a result of averaging, variations of EP/ERP latencie s and amplitudes are not accessed. In particular, short lasting alterations which may provide relevant information about cognitive functions are probab ly smoothed or even masked by the averaging process. Therefore, investigators are interested in single trial analysis , that allows extraction of reliable signal characteristic s out of single EP/ERP sequences [1]. In ref. [2] autoregressive models ( AR) are adopted to EEG sequences recorded prior to stimulation i n order to sub- tract uncorrelated neural activity from ERPs. However, it i s an empirical fact, that external stimuli lead to event-related-desynchroniz aition of the ongoing EEG. Thus, the estimated AR-model might be incorrect. The au thors of [3] applied autoregressive movingaverage ( ARMA ) models to time sequences which were a concatenation of several EP/ERP sequences. In t he case of short signal sequences, this led to better spectral estimat ions than commonly achieved by periodograms. The main restriction is, however , that investigated signals must be linear and stationary, which cannot be stric tly presumed for the EEG. In particular the high model order in comparison to t he signal length shows that AR- and ARMA-models are often inadequate for EP/E RP analy- sis. Other methods have been developed to deal with the nonst ationary and transient character of EPs/ERPs. Woody [4] introduced an it erative method for EP/ERP latency estimation based on common averages. He d etermined the time instant of the best correlation between a template ( EP/ERP average) and single trials by shifting the latter in time. This method corrects a possible latency variability of EPs/ERPs, but its performance highl y depends on the initial choice of templates. The Wiener filter [5,6], on the other hand, uses spectral estimation to reduce uncorrelated noise. This tec hnique, however, is less accurate for EPs/ERPs, because the time course of trans ient signals is lost in the Fourier domain. Thus, DeWeerd [7,8] introduced a time adaptive Wiener filter, allowing better adjustment to signal compone nts of short du- 3ration. The paradigm of orthogonal wave packets ( wavelet transform1) also follows this concept of adopted time-frequency decomposit ion. In addition, the wavelet transform provide several useful properties which make it preferable even for the analysis of transient signals [9–11]: •Wavelets can represent smooth functions as well as singular ities. •The basis functions are local which makes most coefficient bas ed algorithms to be naturally adapted to inhomogeneities in the function. •They have the unconditional basis property to represent a va riety of func- tions implying that the wavelet basis is usually a reasonabl e choice even if very little is known about the signal. •Fast wavelet transform is computationally inexpensive of o rderO(N), where Ndenotes the number of sample points. In contrast, fast Fouri er transform (FFT) requires O(Nlog(N)). •Nonlinear thresholding is nearly optimal for signal recove ry. For that reasons, wavelets became a popular tool for the anal ysis of brain electrical activity [12–15], especially for denoising and classification of single trial EPs/ERPs. Donoho et al.[16] introduced a simple thres holding algorithm to reduce noise in the wavelet domain requiring no assumptio ns about the time course of signals. Nevertheless, high signal amplitud es are in need to distinguish between noise and signal related wavelet coeffic ients in single trials. Bertrand et al. [17] modified the original a posteriori Wiener filter to find accurate filter settings. The authors emphasized better ado ption to transient signal components than can be achieved by corresponding tec hniques in the frequency domain. However, due to the averaging process, th is technique runs 1Continuous wavelet transform: wa,b(Ψ,x(t)) =1√ |a|/integraltext+∞ −∞x(t)Ψ(t−b a)dt w: wavelet coefficient, a: scaling parameter, b: translation parameter, x(t): time series, Ψ: mother wavelet function 4the risk of choosing inadequate filter settings in the case of a high latency variability. The same restriction is valid for discriminan t techniques applied e.g. by Bartink et al. [18,19]. Nevertheless, wavelet based methods enable a more adequate treatment of transient signals than techniqu es applied in the frequency domain. The question of accurate filter settings, however, is still an unresolved problem. To circumvent this problem, we introduce a new method for sin gle trial anal- ysis of ERPs that neither assumes fully synchronized nor sta tionary ERP sequences. The method is related to techniques already deve loped for the paradigm of deterministic chaotic systems, using time dela y embeddings of signals for state space reconstruction and denoising [20]. Schreiber and Ka- plan [21] demonstrated the accuracy of these methods to redu ce measurement noise in the human electrocardiogram (ECG). Heart beats are also of transient character and exhibit relevant signal components in a frequ ency range that compares to ERPs. Unfortunately, ERPs are of shorter durati on as compared to the ECG. Thus, in the case of high dimensional time delay em bedding (in the order of the signal length), we cannot create a sufficient n umber of delay vectors for ERP sequences. To circumvent this problem we rec onstruct ERPs in state-space using circular embeddings, that have turned out to be appro- priate even for signal sequences of short duration. In contr ast to the nonlinear projection scheme described in [20], we do not use singular value decomposi- tion(SVD) to determine clean signals in state space. The reason f or this is threefold. First, estimating relevant signal components u sing the inflexion of ordered eigen-values is not always applicable to EEG becaus e eigen-values may decay almost linearly. In this case, an a priori restriction to a fixed embedding dimension is in need, running the risk either to discard impo rtant signal com- ponents or to remain noise of considerable amplitude if only little is known 5about the signal. Second, SVD stresses the direction of high est variances, so that transient signal components may be smoothed by project ion. Third, the number of signal related directions in state space may alter locally, which is also not concerned by SVD. Instead we calculate wavelet tran sforms of de- lay vectors and determine signal related components by esti mating variances separately for each state-space direction. Scaling proper ties of wavelet bases allow very fast calculation as well as focusing on specific fr equency bands. To confirm the accuracy of our method, we apply it to ERP-like t est signals contaminated with different types of noise. Afterwards, we g ive an example of reconstructed mesial temporal lobe P300 potentials, that w ere recorded from within the hippocampal formation of a patient with focal epi lepsy. 2 Outline of the Method A time series may be contaminated by random noise allowing th e measurement yn=xn+ǫn. If the measured time series is purely deterministic, it is r estricted to a low-dimensional hyper-surface in state space. For the t ransient signals we are concerned with here, we assume this still to be valid. We h ope to identify this direction and to correct ynby simply projecting it onto the subspace spanned by the clean data [22,21]. Technically we realize projections onto noise free subspac es as follows. Let Y= (y1, y2, . . ., y N) denote an observed time sequence. Time-delay embedding of this sequence in a m-dimensional state space leads to state space vectors yn= (yn, . . ., y n−(m−1)τ), where τis an appropriate time delay. In an embedding space of dimension mwe compute the discrete wavelet transform [11,10,9] of all delay vectors in a small neighborhood of a vector ynwe want to correct. Let rn,jwithj= 0, . . ., k denote the indices of the k nearest neighbors of yn, and 6forynitself, i.e. j= 0, and rn,0=n. Thus, the first neighbor distances from ynin increasing order are d(Y)(1) n≡ ||yn−yrn,1||= min r′||yn−yr′||,d(Y)(2) n≡ ||yn−yrn,2||= min r′/negationslash=rn,1||yn−yr′||, etc., where ||y−y′||is the Euclidean distance in state space. Now the important assumption is tha t the clean signal lies within a subspace of dimension d≪m, and that this subspace is spanned by only a few basis functions in the wavelet domain. Let wrn,jdenote the fast wavelet transform [23,24] of yrn,j. Futhermore, let C(k) i(wrn) =/angbracketleftwrn,j/angbracketrightidenote theithcomponent of the centre of mass of wrn, and σ2 n,ithe corresponding variance. In the case of neighbors owing to the signal ( true neighbors ), we can expect the ratio C(k) i(wrn)/σ2 n,ito be higher in signal than in noise related directions. Thus, a discrimination of noise and noise free c omponents in state space is possible. Let ˜wn,i=  wn,i:|C(k) i(wrn)| ≥2λσn,i√k+1 0 : else(1) define a shrinking condition to carry out projection onto a no ise free manifold [16]. The parameter λdenotes a thresholding coefficient that depends on spe- cific qualities of signal and noise. Inverse fast wavelet tra nsform of ˜wnprovides a corrected vector in state space, so that application of our projection scheme to all remaining delay vectors ends up with a set of corrected vectors, out of which the clean signal can be reconstructed. 2.1 Extension to multiple signals of short length LetYl= (yl,1, yl,2, . . ., y l,N) denote a short signal sequence that is repeatedly recorded during an experiment, where l= 1, . . ., L orders the number of rep- etitions. A typical example may be ERP recordings, where eac hYlrepresents an EEG sequence following well defined stimuli. Time-delay e mbeddings of 7these sequences can be written as yl,n= (yl,n. . ., y l,n−(m−1)τ). To achieve a sufficient number of delay vectors even for high embedding dim ensions, we define circular embeddings by yl,n= (yl,n, . . ., y l,1, yl,N, . . ., y l,N−(m−q))∀n < m, (2) so that all delay vectors with indices 1 ≤n≤Ncan be formed. Circular embeddings are introduced as the most attractive choice to h andle the ends of sequences. Alternatives are (i) losing neighbors, (ii) z eropadding, and (iii) shrinking the embedding dimension towards the ends. Howeve r, discontinuities may occur at the edges, requiring some smoothing. For each Ylwe define the smoothed sequence as ys l,n,i=  yl,n,ie−(q−i p)2:i < q yl,n,i :q≤i≤N−q yl,n,ie−(i−(N−q) p)2:i > N −q(3) where qdefines the window width in sample points, pthe steepness of expo- nential damping, and ithe time index. Time-delay embedding of several short sequences leads to a filling of the state space, so that a suffici ent number of nearest neighbors can be found for each point. 2.2 Parameter Selection Appropriate choice of parameters, in particular embedding dimension m, time delay τ, thresholding coefficient λ, as well as the number of neighbors kis important for accurate signal reconstruction in state spac e. Several methods have been developed to estimate “optimal” parameters, depe nding on specific aspects of the given data (e.g., noise level, type of noise, s tationarity, etc.). These assume that the clean signal is indeed low dimensional , an assumption 8we are not ready to make in the case of ERPs. Thus, we approache d the problem of “optimal” parameters empirically. Parameters τandmare not independent from each other. In particular, high embedding dimensions allow small time-delays and vice vers a. We estimated ”optimal” embedding dimensions and thresholding coefficien ts on simulated data by varying mandλfor a fixed τ= 1. To allow fast wavelet transform, we chose mto be a power of 2. Repeated measurements, like in the case of EPs/ERPs, have a m aximum num- ber of true neighbors which is given by kmax=L. In the case of identical signals this is the best choice imaginable. However, real EPs/ERPs m ay alter during experiments, and it seems more appropriate to use a maximum d istance true neighbors are assumed to be restricted to. We define this dist ance by d(y)max=√ 2 LNL,N/summationdisplay l=1,n=1d(y)(L) n,l (4) 3 Model Data 3.1 Generating test signals and noise To demonstrate the effectiveness of our denoising technique and to estimate accurate values for m,λ, andL, we applied it to EP/ERP-like test signals con- taminated with white noise and in-band noise. The latter was generated using phase randomized surrogates of the original signal [25]. Te st signals consisted of 256 sample points and were a concatenation of several Gaus sian functions with different standard deviations and amplitudes. To simul ate EPs/ERPs not fully synchronized with stimulus onset, test signals we re shifted randomly in time (normally deviated, std. dev.: 20 sample points, max . shift: 40 sample 9points). Since even fast components of the test signal exten ded over several sample points, a minimum embedding dimensions m= 16 was required to cover any significant fraction of the signal. The highest emb edding dimension was bounded by the length of signal sequences and the number o f embedded trials, thus allowing a maximum of m= 256. However, if the embedding di- mension is m=N, neighborhood is not longer defined by local characteristic s, and we can expect denoised signals to be smoothed in the case o f multiple time varying components. 3.2 Denoising of test signals LetXl= (xl,1, xl,2, . . ., x l,N) denote the lthsignal sequence of a repeated measurement, Yl= (yl,1, yl,2, . . ., y l,N) the noise contaminated sequence, and ˜Yl= (˜yl,1,˜yl,2, . . .,˜yl,N) the corresponding result of denoising. Then r=1 LL/summationdisplay l=1/radicaltp/radicalvertex/radicalvertex/radicalbt(Yl−Xl)2 (˜Yl−Xl)2(5) defines the noise reduction factor which quantifies signal im provement owing to the filter process. We determined rfor test signals contaminated with white noise, using noise amplitudes ranging from 25% - 150%, and embedding dimension s ranging from 16 - 128 (Figure 2a, Figure 3). Five repetitions for each para meter configu- ration were calculated using 5 embedded trials each. In the c ase of λ≤2, thenoise reduction factor was quite stable against changes of noise levels but depended on embedding dimension mand thresholding coefficient λ. Best per- formance was achieved for 1 .0≤λ≤2.0 (rm=128,λ=2.0 max = 4.7). In the case of λ >4.0, most signal components were rejected, and as a result, the noise re- duction factor rincreased linearly with noise levels, as expected. Figure 2 b and 10Figure 4 depict effects of denoising of 5 test signals contami nated with in-band noise. In comparison to white noise the performance decreas ed, but neverthe- less, enabled satisfactory denoising for 0 .5≤λ≤1.0 (rm=128,λ=1.0 max = 1.6). Within this range, the noise reduction factor rdepended weakly on noise lev- els. Note that the embedding dimension must be sufficiently hi gh (m= 128) to find true neighbors. In order to simulate EPs/ERPs with several time-varying com ponents, we used 5 test signals which were again a concatenation of different G aussian functions, each, however, randomly shifted in time (Figure 2c and Figur e 5). In contrast to test signals with time fixed components, ”optimal” embedd ing dimension depended on the thresholding coefficient λ. Higher values of λrequired lower embedding dimensions and vice versa. Best results were achi eved for 0 .5≤ λ≤2.0 (rm=128,λ=1.0 max = 3.2). Even for high noise levels, the proposed denoising scheme pr eserved finer struc- tures of original test signals in all simulations. Moreover , the reconstructed sequences were closer to the test signals than the correspon ding averages, es- pecially for time varying signals. Power spectra showed tha t denoising took part in all frequency bands and was quite different from commo n low-, or band-pass filtering. Simulation indicated that ”optimal” v alues of the thresh- olding coefficient were in the range 0 .5≤λ≤2.0. Best embedding dimension was found to be m= 128, since the ongoing background EEG can be assumed to be in-band with ERPs. The filter performance was quite stab le against the number of embedded sequences, at least for L= 5,10,20. 4 Real data 114.1 Data Acquisition We analyzed event related potentials recorded intracerebr ally in patients with pharmacoresistent focal epilepsy [26]. Electroencephalo graphic signals were recorded from bilateral electrodes implanted along the lon gitudinal axis of the hippocampus. Each electrode carried 10 cylindrical con tacts of nickel- chromium alloy with a length of 2.5 mm and an intercontact dis tance of 4 mm. Signals were referenced to linked mastoids, amplified wi th a bandpass filter setting of 0.05 - 85.00 Hz (12dB/oct.) and, after 12 bit A/D conver- sion, continuously written to a hard disk using a sampling in terval of 5760 µs. Stimulus related epochs spanning 1480 ms (256 sample points ) including a 200 ms pre-stimulus baseline were extracted from recorded d ata. The mean of the pre-stimulus baseline was used to correct possible ampl itude shifts of the following ERP epoch. In avisual odd-ball paradigm 60 rare (letter < x > , targets) and 240 frequent stimuli (letter < o > , distractors) were randomly presented on a computer monitor once every 1200 ±200ms(duration: 100 ms, probability of occur- rence: 1 ( < x > ) : 5 ( < o > )). Patients were asked to press a button upon each rare target stimulus. This pseudo-random presentatio n of rare stimuli in combination with the required response is known to elicit the mesial tem- poral lobe (MTL) P300 potential in recordings from within th e hippocampal formation [27] (cf. Figure 1). 4.2 Results By simulation, we estimated a range in which ”optimal” param eters of the filter can be expected. However, the quality of denoising ERP sequences could 12not be estimated, because the clean signal was not known a pri ori. A rough estimation of filter performance was only possible by a compa rison to ERP averages. Taking into account results of simulation as well as ERP averages, we estimated λ= 0.6 and m= 128 to be the best configuration. Based on the empirical fact that specific ERP components exhi bit peak am- plitudes within a narrow time range related to stimulus onse t, we defined a maximum allowed time jitter of ±20 sample points ( ≈116ms) true neighbors are assumed to be restricted to. This accelerated the calcul ation time and avoided false nearest neighbors. Figure 6 depicts several E RPs recorded from different electrode contacts within the hippocampal format ion. The number of embedded sequences was chosen as L= 8. Comparing averages, we can ex- pect that the filter extracted the most relevant MTL-P300 com ponents. Even for low amplitude signals reconstruction was possible, exh ibiting higher am- plitudes in single trial data than in averages. As correspon ding power spectra show, the 50 Hz power line was reduced but not eliminated afte r filtering. Especially low amplitude signals showed artifacts based on the 50 Hz power line. 5 Conclusion In this study, we introduced a new wavelet based method for no nlinear noise reduction of single trial EPs/ERPs. We employed advantages of methods de- veloped for the paradigm of deterministic chaotic systems, that allowed de- noising of short and time variant EP/ERP sequences without a ssuming fully synchronized or stationary EEG. Denoising via wavelet shrinkage does not require a priori as sumptions about constrained dimensions, as is usually required for other te chniques (e.g., singu- 13lar value decomposition). Besides, it is more straight forw ard using thresholds depending on means and variances rather than initial assump tions about con- strained embedding dimensions. Moreover, the local calcul ation of thresholds in state space enables focusing on specific frequency scales , which may be ad- vantageous in order to extract signal components located wi thin both narrow frequency bands and narrow time windows. Extension of our denoising scheme to other types of signals s eems to be possi- ble, however, demands further investigations, since ”opti mal” filter parameters highly depend on signal characteristics. In addition, the noise reduction factor rdoes not consider all imaginable features of signals invest igators are possibly interested in, so that other measures may be more advantageo us in specific cases. So far, we have not considered effects of smoothing the edges o f signal se- quences. 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A theory for multiresolution signal decom position: The wavelet representation. IEEE Trans Patt Recog Mach Intel , 11:674, 1989. [25] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer. Testing of nonlinearity in time series: The method of surrogate data .Physica D , 58:77, 1992. [26] T. Grunwald, H. Beck, K. Lehnertz, I. Bl¨ umcke, N. Pezer , M. Kutas, M. Kurthen, H. M. Karakas, D. Van Roost, O. D. Wiestler, and C. E. Elger. Limbic P300s in temporal lobe epilepsy with and without Ammo n’s horn sclerosis. Eur J Neurosci , 11:1899, 1999. [27] A. Puce, R. M. Berkovic, G A. Donnan, and P. F. Baldin. Lim bic P3 potentials, seizure localization, and surgical pathology in temporal l obe epilepsy. Annals of Neurology , 26:377, 1989. 16Figure captions: Fig. 1: Examples of averaged ERPs recorded along the longitu dinal axis of the hippocampal formation in a patient with epilepsy. Randomiz ed presentation of target andstandard stimuli is known to elicit the mesial temporal lobe P300, a negative deflection peaking at about 500 ms after stimulus on set (cf. Sect. 4.1 for more details). Letters (a), (b), and (c) indicate record ings used for single trial analysis (cf. Figure 6). Fig. 2: Results of denoising test signals. Parts a) and b): co ntamination with white noise and in-band noise. Part c): time varying signal c omponents and white noise contamination (see text for more details). Five calculations for each parameter configuration have been executed to determine sta ndard deviations. Fig. 3: Nonlinear denoising applied to white noise contamin ated test signals (5 sequences embedded, each 256 sample points, randomly shift ed in time (std. dev.: 20 sample points, max. shift: 40 sample points), noise amplitude 75%, m= 128, τ= 1,λ= 1.5). Power spectra in arbitrary units. For state space plots we used a time delay of 25 sample points. Fig. 4: Same as Figure 3 but for in-band noise and λ= 0.75. Fig. 5: Same as Figure 3 but for Gaussian functions each rando mly shifted in time and λ= 0.75. Fig. 6: Examples of denoised MTL-P300 potentials (cf. Figur e 1). Power spec- tra in arbitrary units. For state space plots we used a time de lay of 25 sample points. 17

arXiv:physics/0001070v1 [physics.flu-dyn] 29 Jan 2000Interaction of a vortex ring with the free surface of ideal fluid V.P. Ruban∗ Optics and Fluid Dynamics Department, Risø National Laboratory, DK-4000 Roskilde Denmark July 22, 2013 Abstract The interaction of a small vortex ring with the free surface o f a perfect fluid is considered. In the frame of the point ring approximation t he asymptotic expression for the Fourier-components of radiated surface waves is obtained in the case when the vortex ring comes from infinity and has both h orizontal and vertical components of the velocity. The non-conservative corrections to the equations of motion of the ring, due to Cherenkov radiation, are derived. 1 Introduction The study of interaction between vortex structures in a fluid and the free surface is important both from practical and theoretical points of vie w. In general, a detailed investigation of this problem is very hard. Even the theorie s of potential surface waves and the dynamics of vortices in an infinite space taken s eparately still have a lot of unsolved fundamental problems on their own. Only the consideration of significantly simplified models can help us to understand the processes which take place in the combined system. In many cases it is possible to neglect the compressibility o f the fluid as well as the energy dissipation. Therefore the model of ideal homoge neous incompressible fluid is very useful for hydrodynamics. Because of the conser vative nature of this model the application of the well developed apparatus of Ham iltonian dynamics be- comes possible [1] [2]. An example of effective use of the Hami ltonian formalism in hydrodynamics is the introduction of canonical variables f or investigations of poten- tial flows of perfect fluids with a free boundary. V.E.Zakharo v showed at the end of the sixties [3] that the surface shape z=η(x,y,t) and the value of the velocity potentialψ(x,y,t) on the surface can be considered as generalized coordinate and momentum, respectively. ∗Permanent address: L.D.Landau Institute for Theoretical P hysics, 2 Kosygin str., 117334 Moscow, Russia. E-mail: ruban@itp.ac.ru 1It is important to note that a variational formulation of Ham iltonian dynamics in many cases allows to obtain good finite-dimensional approxi mations which reflect the main features of the behavior of the original system. There a re several possibilities for a parameterization of non-potential flows of perfect flui d by some variables with dynamics determined by a variational principle. All of them are based on the con- servation of the topological characteristics of vortex lin es in ideal fluid flows which follows from the freezing-in of the vorticity field Ω(r,t) = curl v(r,t). In particular, this is the representation of the vorticity by Clebsch canon ical variables λandµ[4] [2] Ω(r,t) = [∇λ× ∇µ] However, the Clebsch representation can only describe flows with a trivial topology (see, e.g., [5]). It cannot describe flows with linked vortex lines. Besides, the variables λandµare not suitable for the study of localized vortex structure s like vortex filaments. In such cases it is more convenient to use the param eterization of vorticity in terms of vortex lines and consider the motion of these line s [6],[7], even if the global definition of canonically conjugated variables is impossib le due to topological reasons. This approach is used in the present article to describe the i nteraction of deep (or small) vortex rings of almost ideal shape in the perfect fluid with the free surface. In the case under consideration the main interaction of the vor tex rings with the surface can be described as the dipole-dipole interaction between ” point” vortex rings and their ”images”. Moving rings interact with the surface wave s, leading to radiation due to the Cherenkov effect. Deep rings disturb the surface we akly, so the influence of the surface can be taken into account as some small correct ions in the equations of motion for the parameters of the rings. In Sec.2 we discuss briefly general properties of vortex line dynamics, which fol- low from the freezing-in of the vorticity field. In Sec.3 poss ible simplifications of the model are made and the point ring approximation is introd uced. In Sec.4 the interaction of the ring with its image is considered. In Sec. 5 we calculate the Fourier- components of Cherenkov surface waves radiated by a moving v ortex ring and deter- mine the non-conservative corrections caused by the intera ction with the surface for the vortex ring equations of motion. 2 Vortex lines motion in perfect fluid It is a well known fact that the freezing-in of the vorticity l ines follows from the Euler equation for ideal fluid motion Ωt= curl [ v×Ω],v= curl−1Ω Vortex lines are transported by the flow [1],[4],[8]. They do not appear or disappear, neither they intersect one another in the process of motion. This property of perfect fluid flows is general for all Hamiltonian systems of the hydro dynamic type. For simplicity, let us consider temporally the incompressible fluid without free surface in infinite space. The dynamics of the system is specified by a bas ic Lagrangian L[v], which is a functional of the solenoidal velocity field. The re lations between the ve- locityv, the generalized vorticity Ω, the basic Lagrangian L[v] and the Hamiltonian 2H[Ω] are the following [9]1 Ω= curl/parenleftBiggδL δv/parenrightBigg ⇒ v=v[Ω] (1) H[Ω] =/parenleftBigg/integraldisplay v·/parenleftBiggδL δv/parenrightBigg d3r−L[v]/parenrightBigg/vextendsingle/vextendsingle/vextendsingle v=v[Ω](2) v= curl/parenleftBiggδH δΩ/parenrightBigg (3) and the equation of motion for the generalized vorticity is Ωt= curl [curl ( δH/δΩ)×Ω] (4) This equation corresponds to the transport of frozen-in vor tex lines by the velocity field. In this process all topological invariants [10] of the vorticity field are conserved. The conservation of the topology can be expressed by the foll owing relation [7] Ω(r,t) =/integraldisplay δ(r−R(a,t))(Ω0(a)∇a)R(a,t)da=(Ω0(a)∇a)R(a,t) det/ba∇dbl∂R/∂a/ba∇dbl/vextendsingle/vextendsingle/vextendsingle a=a(r,t)(5) where the mapping R(a,t) describes the deformation of lines of some initial solenoi dal fieldΩ0(r). Here a(r,t) is the inverse mapping with respect to R(a,t). The direction of the vector b b(a,t) = (Ω0(a)∇a)R(a,t) (6) coincides with the direction of the vorticity field at the poi ntR(a,t). The equation of motion for the mapping R(a,t) can be obtained with the help of the relation Ωt(r,t) = curl r/integraldisplay δ(r−R(a,t))[Rt(a,t)×b(a,t)]da, (7) 1For the ordinary ideal hydrodynamics in infinite space the ba sic Lagrangian is LEuler[v] =/integraldisplayv2 2dr ⇒ Ω= curl v The Hamiltonian in this case coincides with the kinetic ener gy of the fluid and in terms of the vorticity field it reads HEuler[Ω] =−1/2/integraldisplay Ω∆−1Ωdr=1 8π/integraldisplay/integraldisplayΩ(r1)·Ω(r2) |r1−r2|dr1dr2 where ∆−1is the inverse Laplace operator. Another example is the basic Lagrangian of Electron Magneto -hydrodynamics which takes into account the magnetic field created by the current of electron fluid through the motionless ion fluid. LEMHD [v] =1 2/integraldisplay v(1−∆−1)vdr ⇒ Ω= curl(1 −∆−1)v HEMHD [Ω] =1 2/integraldisplay Ω(1−∆)−1Ωdr=1 8π/integraldisplay/integraldisplaye−|r1−r2| |r1−r2|Ω(r1)·Ω(r2)dr1dr2 The second example shows that the relation between the veloc ity and the vorticity can be more complex than in usual hydrodynamics. 3which immediately follows from Eq.(5). The substitution of Eq.(7) into the equation of motion (4) gives [11] curlr/parenleftBiggb(a,t)×[Rt(a,t)−v(R,t)] det/ba∇dbl∂R/∂a/ba∇dbl/parenrightBigg = 0 One can solve this equation by eliminating the curl roperator. Using the general relationship between variational derivatives of some func tionalF[Ω] /bracketleftBigg b×curl/parenleftBiggδF δΩ(R)/parenrightBigg/bracketrightBigg =δF δR(a)/vextendsingle/vextendsingle/vextendsingle Ω0(8) it is possible to represent the equation of motion for R(a,t) as follows [(Ω0(a)∇a)R(a)×Rt(a)] =δH[Ω[R]] δR(a)/vextendsingle/vextendsingle/vextendsingle Ω0. (9) It is not difficult to check now that the dynamics of the vortici ty field with topological properties defined by Ω0in the infinite space is equivalent to the requirement of an extremum of the action ( δS=δ/integraltextLΩ0dt= 0) where the Lagrangian is [7] LΩ0=1 3/integraldisplay/parenleftBig [Rt(a)×R(a)]·(Ω0(a)∇a)R(a)/parenrightBig da− H[Ω[R]]. (10) In the simplest case, when all vortex lines are closed it is po ssible to choose new curvilinear coordinates ν1,ν2,ξina-space such that Eq.(5) can be written in a simple form Ω(r,t) =/integraldisplay Nd2ν/contintegraldisplay δ(r−R(ν,ξ,t))Rξdξ. (11) Hereνis the label of a line lying on a fixed two-dimensional manifol dN, andξ is some parameter along the line. It is clear that there is a ga uge freedom in the definition of νandξ. This freedom is connected with the possibility of changing the longitudinal parameter ξ=ξ(˜ξ,ν,t) and also with the relabeling of ν ν=ν(˜ν,t),∂(ν1,ν2) ∂(˜ν1,˜ν2)= 1. (12) Now we again consider the ordinary perfect fluid with a free su rface. To describe the flow entirely it is sufficient to specify the vorticity field Ω(r,t) and the motion of the free surface. Thus, we can use the shape R(ν,ξ,t) of the vortex lines as a new dynamic object instead of Ω(r,t). It is important to note that in the presence of the free surface the equations of motion for R(ν,ξ,t) follow from a variational principle as in the case of infinite space. It has been shown [12] that the Lagrangian for a perfect fluid, with vortices in its bulk and with a free surfac e, can be written in the form L=1 3/integraldisplay Nd2ν/contintegraldisplay ([Rt×R]·Rξ)dξ+/integraldisplay Ψηtdr⊥− H[R,Ψ,η]. (13) The functions Ψ( r⊥,t) andη(r⊥,t) are the surface degrees of freedom for the system. Ψ is the boundary value of total velocity potential, which in cludes the part from vor- tices inside the fluid, and ηis the deviation of the surface from the horizontal plane. 4This formulation supposes that vortex lines do not intersec t the surface anywhere. In the present paper only this case is considered. The Hamiltonian Hin Eq.(13) is nothing else than the total energy of the system expressed in terms of [ R,Ψ,η]. Variation with respect to R(ν,ξ,t) of the action defined by the Lagrangian (13) gives the equation of motion for vortex lines in the form [Rξ×Rt] =δH[Ω[R],Ψ,η] δR. (14) This equation determines only the transversal component of Rtwhich coincides with the transversal component of the actual solenoidal velocit y field. The possibility of solving Eq.(14) with respect to the time derivative Rtis closely connected with the special gauge invariant nature of the H[R] dependence which results in δH δR·Rξ≡0. The tangential component of Rtwith respect to vorticity direction can be taken arbitrary. This property is in accordance with the longitud inal gauge freedom. The vorticity dynamics does not depend on the choice of the tange ntial component. Generally speaking, only the local introduction of canonic al variables for curve dynamics is possible. For instance, a piece of the curve can b e parameterized by one of the three of Cartesian coordinates R= (X(z,t),Y(z,t),z) In this case the functions X(z,t) andY(z,t) are canonically conjugated variables. Another example is the parameterization in cylindrical coo rdinates, where variables Z(θ,t) and (1/2)R2(θ,t) are canonically conjugated. Curves with complicated topological properties need a gene ral gauge free descrip- tion by means of a parameter ξ. It should be mentioned for clarity that the conservation of a ll vortex tube volumes, reflecting the incompressibility of the fluid, is not the cons traint in this formalism. It is a consequence of the symmetry of the Lagrangian (13) with r espect to the relabel- ing (12)ν→˜ν[9]. Volume conservation follows from that symmetry in acco rdance with Noether’s theorem. To prove this statement, we should c onsider such subset of relabelings which forms a one-parameter group of transform ations of the dynamical variables. For small values of the group parameter, τ, the transformations are de- termined by a function of two variables T(ν1,ν2) (with zero value on the boundary ∂N) so that R(ν1,ν2,ξ)→Rτ T(ν1,ν2,ξ) =R/parenleftBigg ν1−τ∂T ∂ν2+O(τ2), ν2+τ∂T ∂ν1+O(τ2), ξ/parenrightBigg (15) Due to Noether’s theorem, the following quantity is an integ ral of motion [13] IT=/integraldisplay Nd2ν/contintegraldisplayδL δRt·∂Rτ T ∂τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle τ=0dξ=1 3/integraldisplay Nd2ν/contintegraldisplay [R×Rξ]·(R2T1−R1T2)dξ 5After simple integrations in parts the last expression take s the form IT=/integraldisplay Nd2ν/contintegraldisplay T(ν1,ν2)([R1×R2]·Rξ)dξ=/integraldisplay NT(ν1,ν2)V(ν1,ν2,t)d2ν (16) where V(ν1,ν2,t)d2νis the volume of an infinitely thin vortex tube with cross-sec tion d2ν. It is obvious that actually the function Vdoesn’t depend on time tbecause the functionT(ν1,ν2) is arbitrary2. 3 Point ring approximation In general case an analysis of the dynamics defined by the Lagr angian (13) is too much complicated. We do not even have the exact expression fo r the Hamiltonian H[R,Ψ,η] because it needs the explicit knowledge of the solution of t he Laplace equation with a boundary value assigned on a non-flat surface . Another reason is the very high nonlinearity of the problem. In this paper we consider some limits where it is possible to s implify the system significantly. Namely, we will suppose that the vorticity is concentrated in several very thin vortex rings of almost ideal shape. For a solitary r ing the perfect shape is stable for a wide range of vorticity distributions through t he cross-section. This shape provides an extremum of the energy for given values of the vol umes of vortex tubes and for a fixed momentum of the ring. As already mentioned, vol ume conservation follows from Noether’s theorem. Therefore some of these qua ntities (those of which are produced by the subset of commuting transformations) ca n be considered as canonical momenta. Corresponding cyclical coordinates de scribe the relabeling (12) of the line markers, which doesn’t change the vorticity field . Actually these degrees of freedom take into account a rotation around the central li ne of the tube. This line represents the mean shape of the ring and we are interest ed in how it behaves in time. For our analysis we don’t need the explicit values of cy clical coordinates, but only the conserved volumes as parameters in the Lagrangian. A possible situation is when a typical time of the interactio n with the surface and with other rings is much larger then the largest period of osc illations corresponding to deviations of the ring shape from perfect one. Under this c ondition, excitations of all (non-cyclical) internal degrees of freedom are small during all the time, and a variational anzats completely disregarding them reflects t he behavior of the system adequately. The circulations Γn=/integraldisplay Nnd2ν of the velocity for each ring don’t depend on time. A perfect r ing is described by the coordinate Rnof the center and by the vector Pn= Γ nSn, where Snis an oriented area of the ring. We use in this work the Cartesian system of co ordinates (x,y,z ), so that the vertical coordinate is z, and the unperturbed surface is at z= 0. The 2If vortex lines are not closed but form a family of enclosed to ri then the relabeling freedom is less rich. In that case one can obtain by the similar way the co nservation laws for volumes inside closed vortex surfaces. Noether’s theorem gives integrals of motion which depend on an arbitrary function of one variable S(ζ), where ζis the label of the tori. 6corresponding components of the vectors RnandPnare Rn= (Xn,Yn,Zn),Pn= (Pxn,Pyn,Pzn) It is easy to verify that the vectors Pnare canonically conjugated momenta for the coordinates Rn. To verify that we can parameterize the shape of each vortex l ine in the following manner R(ξ,t) =M/summationdisplay m=−Mrm(t)eimξ,r−m=¯rm (17) Hererm(t) are complex vectors. Substituting this into the first term o f the La- grangian (13) gives 1 3/contintegraldisplay ([Rt×R]Rξ)dξ= 2πi˙r0([r−1×r1] + 2[r−2×r2] +...)+ +d{...} dt+ 2πi˙r−1[r−1×r2]−2πi˙r1[r1×r−2] +... (18) If we neglect the internal degrees of freedom which describe deviations of the ring from the ideal shape (r−1)2= (r1)2= 0,r2=r−2= 0, ... then the previous statement about canonically conjugated v ariables becomes obvious: Rn=r0n,Pn= 2πΓn·i[r−1n×r1n] (19) Such an approximation is valid only in the limit when sizes of rings are small in comparison with the distances to the surface and the distanc es between different rings/radicalBigg Pn Γn≪ |Zn|,|Rn−Rl|, l /ne}ationslash=n. (20) These conditions are necessary for ensuring that the excita tions of all internal degrees of freedom are small. Obviously, this is not true when a ring a pproaches the surface. In that case one should take into account also the internal de grees of freedom for the vortex lines. The inequalities (20) also imply that vortex rings in the lim it under consideration are similar to point magnet dipoles. This analogy is useful f or calculation of the Hamiltonian for interacting rings. In the main approximati on we may restrict the analysis by taking into account the dipole-dipole interact ion only. It should be mentioned that in some papers (see e.g. [14] and r eferences in that book) the discrete variables identical to RnandPnare derived in a different way and referred as the vortex magnetization variables. In the expression for the Hamiltonian, several simplificati ons can be made. Let us recall that for each moment of time it is possible to decomp ose the velocity field into two components v=V0+∇φ. (21) 7Here the field V0satisfies the following conditions (∇ ·V0) = 0,curlV0=Ω,(n·V0)|z=η= 0. The boundary value of the surface wave potential φ(r) isψ(r⊥). In accordance with these conditions the kinetic energy is decomposed into two p arts and the Hamiltonian of the fluid takes the form H=1 2/integraldisplay z<ηV2 0d3r+1 2/integraldisplay ψ(∇φ·dS) +g 2/integraldisplay η2dr⊥ (22) The last term in this expression is the potential energy of th e fluid in the gravitational field. If all vortex rings are far away from the surface then it s deviation from the horizontal plane is small |∇η| ≪1,|η| ≪ |Zn| (23) Therefore in the main approximation the energy of dipoles in teraction with the sur- face can be described with the help of so called ”images”. The images are vortex rings with parameters Γn,R∗ n= (Xn,Yn,−Zn),P∗ n= (Pxn,Pyn,−Pzn) (24) The kinetic energy for the system of point rings and their ima ges is the sum of the self-energies of rings and the dipole-dipole interaction b etween them. The expres- sion for the kinetic energy of small amplitude surface waves employs the operator ˆkwhich multiplies Fourier-components of a function by the ab solute value kof a two-dimensional wave vector k. So the real Hamiltonian His approximately equal to the simplified Hamiltonian ˜H H ≈ ˜H=1 2/integraldisplay (ψˆkψ+gη2)dr⊥+/summationdisplay nEn(Pn)+ +1 8π/summationdisplay l/negationslash=n3((Rn−Rl)·Pn)((Rn−Rl)·Pl)− |Rn−Rl|2(Pn·Pl) |Rn−Rl|5+ +1 8π/summationdisplay ln3((Rn−R∗ l)·Pn)((Rn−R∗ l)·P∗ l)− |Rn−R∗ l|2(Pn·P∗ l) |Rn−R∗ l|5(25) With the logarithmic accuracy the self-energy of a thin vort ex ring is given by the expression En(Pn)≈Γ2 n 2/radicalBigg Pn πΓnln/parenleftBigg(Pn/Γn)3/4 A1/2 n/parenrightBigg (26) where the small constant Anis proportional to the conserved volume of the vortex tube forming the ring. This expression can easily be derived if we take into account that the main contribution to the energy is from the vicinity of the tube where the velocity field is approximately the same as near a straigh t vortex tube. The logarithmic integral should then be taken between the limit s from the thickness of the tube to the radius of the ring. 8In the relation Ψ = Φ 0+ψthe potential Φ 0is approximately equal to the potential created on the flat surface by the dipoles and their images Φ0(r⊥)≈Φ(r⊥) =−1 2π/summationdisplay n(Pn·(r⊥−Rn)) |r⊥−Rn|3(27) In this way we arrive at the following simplified system descr ibing the interaction of point vortex rings with the free surface ˜L=/summationdisplay n˙RnPn+/integraldisplay ˙η(ψ+ Φ)d2r⊥−˜H[{Rn,Pn},η,ψ] (28) It should be noted that due to the condition (20) the maximum v alue of the velocityV0on the surface is much less then the typical velocities of the vortex rings Pn Z3 n≪Γ3/2 n P1/2 n Therefore the term V2 0/2 in the Bernoulli equation Ψt+V2 0/2 +gη+ small corrections = 0 is small in comparison with the term Ψ t. The Lagrangian (28) is in accordance with this fact because it does not take into account terms like (1 /2)/integraltextV2 0ηd2r⊥in the Hamiltonian expansion. 4 Interaction of the vortex ring with its image Now let us for simplicity consider the case of a single ring. I t is shown in the next section, that for a sufficiently deep ring the interactio n with its image is much stronger than the interaction with the surface waves. So it i s interesting to examine the motion of the ring neglecting the surface deviation. In t his case we have the integrable Hamiltonian for the system with two degrees of fr eedom H=1 64π/parenleftBigg α(P2 x+P2 z)1/4−2P2 z+P2 x |Z|3/parenrightBigg , Z < 0 (29) whereα≈const. The system has integrals of motion Px=p=const, H =E=const so it is useful to consider the level lines of the energy funct ion in the left ( Z,P z)- half-plane taking Pxas the parameter (see the Figure). One can distinguish three regions of qualitatively differen t behavior of the ring in that part of this half-plane where our approximation is va lid (see Eq.(20)). In the upper region the phase trajectories come from infinitely large negative Zwhere they have a finite positive value of Pz. In the process of motion Pzincreases. This behavior corresponds to the case when the ring approaches th e surface. Due to the symmetry of the Hamiltonian (29) there is a symmetric low er region, where the vortex ring moves away from the surface. And there is the midd le region, where Pz changes the sign from negative to positive at a finite value of Z. This is the region of the finite motion. In all three cases the track of the vortex ring bends toward th e surface, i.e. the ring is ”attracted” by the surface. 9ZP_z Figure 1: The sketch of level lines of the function H(Z,P z), Eq.(29). 5 Cherenkov interaction of a vortex ring with sur- face waves When the ring is not very far from the surface and not very slow , the interaction with the surface waves becomes significant. Let us consider t he effect of Cherenkov radiation of surface waves by a vortex ring which moves from t he infinity to the surface. This case is the most definite from the viewpoint of i nitial conditions choice. We suppose that the deviation of the free surface from the hor izontal plane z= 0 is zero att→ −∞ , and we are interested in the asymptotic behavior of fields ηand ψat large negative t. In this situation we can neglect the interaction of the ring with its image in comparison with the self-energy and concen trate our attention on interaction with surface waves only. The ring moves in the ( x,z)-plane with an almost constant velocity. In the main approximation the position Rof the vortex ring is given by the relations R≈Ct,C=C(P) =∂E ∂P= (Cx,0,Cz)∼P P3/2, (30) Cx>0, C z>0, t< 0. The equations of motion for the Fourier-components of ηandψfollow from the Lagrangian (28) ˙ηk=kψk, ˙ψk+gηk=−˙Φk (31) Eliminating ηkwe obtain an equation for ψk ¨ψk+gkψk=−¨Φk (32) 10where Φ kis the Fourier-transform of the function Φ( r⊥). Simple calculations give Φk=e−ikxX 2π/integraldisplayPzZ−Pxx/radicalBig (x2+y2+Z2)3e−i(kxx+kyy)dxdy = =−e−ikxX 2π/parenleftBigg PzD(k|Z|) +iPx |Z|∂ ∂kxD(k|Z|)/parenrightBigg (33) where D(q) =/integraldisplaye−iqαdαdβ/radicalBig (α2+β2+ 1)3= 2πe−|q|(34) Finally, we have for Φ k Φk=/parenleftBiggiPxkx k−Pz/parenrightBigg e−k|Z|−ikxX=/parenleftBiggiPxkx k−Pz/parenrightBigg et(kCz−ikxCx)(35) Due to the exponential time behavior of Φ k(t) it is easy to obtain the expressions forψk(t) andηk(t). Introducing the definition λk=kCz−ikxCx (36) we can represent the answer in the following form ψk(t) =−/parenleftBig iPxkx k−Pz/parenrightBig λ2 k gk+λ2 keλkt=/parenleftbiggP Ck/parenrightbiggλ3 k gk+λ2 keλkt(37) ηk(t) =/parenleftbiggP C/parenrightbiggλ2 k gk+λ2 keλkt(38) The radiated surface waves influence the motion of the vortex ring. The terms produced by the field ηk(t) in the equations of motion for the ring come from the part/integraltext˙ηΦd2r⊥in the Lagrangian (28). Using Eq.(35) for the Fourier-trans form of Φ we can represent these terms as follows δ˙X=/integraldisplayd2k (2π)2˙ηkikx kekZ+ikxX(39) δ˙Z=/integraldisplayd2k (2π)2˙ηkekZ+ikxX(40) δ˙Px=−/integraldisplayd2k (2π)2˙ηk·(ikx)/parenleftBigg Pz+iPxkx k/parenrightBigg ekZ+ikxX(41) δ˙Pz=−/integraldisplayd2k (2π)2˙ηk·k/parenleftBigg Pz+iPxkx k/parenrightBigg ekZ+ikxX(42) We can use Eq.(38) to obtain the nonconservative correction s for time derivatives of the ring parameters from these expressions. It is conveni ent to write down these corrections in the autonomic form δ˙X=/parenleftbiggP C/parenrightbigg/integraldisplayd2k (2π)2/parenleftBiggikx k/parenrightBigg(kCz−ikxCx)3 gk+ (kCz−ikxCx)2e−2k|Z|(43) 11δ˙Z=/parenleftbiggP C/parenrightbigg/integraldisplayd2k (2π)2·(kCz−ikxCx)3 gk+ (kCz−ikxCx)2e−2k|Z|(44) δ˙Px=−/parenleftbiggP C/parenrightbigg2/integraldisplayd2k (2π)2/parenleftBiggikx k/parenrightBigg(kCz−ikxCx)2(C2 zk2+C2 xk2 x) gk+ (kCz−ikxCx)2e−2k|Z|(45) δ˙Pz=−/parenleftbiggP C/parenrightbigg2/integraldisplayd2k (2π)2·(kCz−ikxCx)2(C2 zk2+C2 xk2 x) gk+ (kCz−ikxCx)2e−2k|Z|(46) whereCxandCzcan be understood as explicit functions of Pdefined by the de- pendence C(P) =∂E/∂P. More exact definition of CxandCzas˙Xand˙Zis not necessary. To analyze the above integrals let us first perform there the i ntegration over the angleϕink-space. It is convenient to use the theory of contour integra ls in the complex plane of variable w= cosϕ. The contour γof integration in our case goes clockwise just around the cut which is from −1 to +1. We define the sign of the square root R(w) =√ 1−w2so that its values are positive on the top side of the cut and negative on the bottom side. After introducing the qu antities a=Cz Cx, ω2 k=gk, b k=ωk Cxk=1 Cx/radicalbiggg k(47) we have to use the following relations I1(a,b)≡ −/contintegraldisplay γdw√ 1−w2·w(w+ia)3 b2−(w+ia)2= =π(1 + 2b2) +πi (b−ia)b2 /radicalBig 1−(b−ia)2−(b+ia)b2 /radicalBig 1−(−b−ia)2  (48) I2(a,b)≡i/contintegraldisplay γdw√ 1−w2·(w+ia)3 b2−(w+ia)2= =π 2a+b2 /radicalBig 1−(b−ia)2+b2 /radicalBig 1−(−b−ia)2  (49) J1(a,b)≡i/contintegraldisplay γdw√ 1−w2·w(w+ia)2(w2+a2) b2−(w+ia)2= =−4πab2+π b(b−ia)(a2+ (b−ia)2)/radicalBig 1−(b−ia)2+b(b+ia)(a2+ (b+ia)2)/radicalBig 1−(−b−ia)2  (50) J2(a,b)≡/contintegraldisplay γdw√ 1−w2·(w+ia)2(w2+a2) b2−(w+ia)2= 12=−2π(a2+b2+ 1/2)−πi b(a2+ (b−ia)2)/radicalBig 1−(b−ia)2−b(a2+ (b+ia)2)/radicalBig 1−(−b−ia)2  (51) where the sign of the complex square root should be taken in ac cordance with the previous choice. It can easily be seen that the integrals I2andJ1have resonance structure at a≪1 and |b|<1. This is the Cherenkov effect itself. Now the expressions (43-46) take the form δ˙X=Px (2π)2+∞/integraldisplay 0I1(a,bk)k2e−2k|Z|dk=Px (2π)2/parenleftBiggg C2x/parenrightBigg3 F1/parenleftBigg a,2g|Z| C2x/parenrightBigg (52) δ˙Z=Px (2π)2+∞/integraldisplay 0I2(a,bk)k2e−2k|Z|dk=Px (2π)2/parenleftBiggg C2x/parenrightBigg3 F2/parenleftBigg a,2g|Z| C2x/parenrightBigg (53) δ˙Px=P2 x (2π)2+∞/integraldisplay 0J1(a,bk)k3e−2k|Z|dk=P2 x (2π)2/parenleftBiggg C2x/parenrightBigg4 G1/parenleftBigg a,2g|Z| C2x/parenrightBigg (54) δ˙Pz=P2 x (2π)2+∞/integraldisplay 0J2(a,bk)k3e−2k|Z|dk=P2 x (2π)2/parenleftBiggg C2x/parenrightBigg4 G2/parenleftBigg a,2g|Z| C2x/parenrightBigg (55) Here the functions F1(a,Q)..G2(a,Q) are defined by the integrals F1(a,Q) =+∞/integraldisplay 0I1/parenleftBigg a,1√ξ/parenrightBigg exp (−Qξ)ξ2dξ (56) F2(a,Q) =+∞/integraldisplay 0I2/parenleftBigg a,1√ξ/parenrightBigg exp (−Qξ)ξ2dξ (57) G1(a,Q) =+∞/integraldisplay 0J1/parenleftBigg a,1√ξ/parenrightBigg exp (−Qξ)ξ3dξ (58) G2(a,Q) =+∞/integraldisplay 0J2/parenleftBigg a,1√ξ/parenrightBigg exp (−Qξ)ξ3dξ (59) andQ= 2g|Z|/C2 xis a dimensionless quantity3. The Cherenkov effect is most clear when the motion of the ring is almost horizontal. In this case a→+0, and it is convenient to rewrite these integrals without use of comple x functions F1(+0,Q) =π+∞/integraldisplay 0/parenleftBig ξ2+ 2ξ/parenrightBig exp (−Qξ)dξ−2π1/integraldisplay 0ξdξ√1−ξexp (−Qξ) (60) 3If we consider a fluid with surface tension σ, then two parameters appear: QandT=gσ/C4 x. In that case one should substitute bk→/radicalbig 1/ξ+Tξas the second argument of the functions I1, I2, J1, J2in the integrals (56-59) 13F2(+0,Q) =G1(+0,Q) =−2π+∞/integraldisplay 1ξ3/2dξ√ξ−1exp (−Qξ) (61) G2(+0,Q) =−π+∞/integraldisplay 0/parenleftBig ξ3+ 2ξ2/parenrightBig exp (−Qξ)dξ+ 2π1/integraldisplay 0ξ2dξ√1−ξexp (−Qξ) (62) Here the square root is the usual positive defined real functi on. We see that only resonant wave-numbers contribute to the functions F2andG1, whileF1andG2are determined also by small values of ξwhich correspond to the large scale surface deviation co-moving with the ring. So the effect of the Cheren kov radiation on the vortex ring motion is the most distinct in the equations for ˙Zand˙Px. Especially it is important for Pxbecause the radiation of surface waves is the only reason for change of this quantity in the frame of our approximation. The typical values of Qare large in practical situations. In this limit asymptotic values of the integrals above are F1(+0,Q)≈ −9π 2Q4, G 2(+0,Q)≈18π Q5 F2(+0,Q) =G1(+0,Q)≈ −2π√π·exp(−Q)√Q and δ˙X≈ −9 64πP |Z|3·1 Q, δ ˙Z≈ −1 16√πP |Z|3·Q2+1/2exp(−Q), δ˙Px≈ −1 32√πP2 |Z|4·Q3+1/2exp(−Q), δ ˙Pz≈+9 32πP2 |Z|4·1 Q. It follows from these expressions that the interaction with the surface waves is small in comparison with the interaction between ring and its imag e, ifQ≫1. The corresponding small factors are 1 /QforXandPz, andQ2+1/2exp(−Q) forZ. As against the flat boundary, now Pxis not conserved. It decreases exponentially slowly and this is the main effect of Cherenkov radiation. We see also that the interaction with waves turns the vector Ptowards the surface which results in a more fast boundary approach by the ring tra ck. 6 Conclusions and acknowledgments In this paper we have derived the simplified Lagrangian for th e description of the motion of deep vortex rings under free surface of perfect flui d. We have analyzed the integrable dynamics corresponding to the pure interaction of the single point vortex ring with its image. It was found that there are three types of qualitatively different behaviour of the ring. The interaction of the ring with the su rface has an attractive character in all three regimes. The Fourier-components of r adiated Cherenkov waves were calculated for the case when the vortex ring comes from i nfinity and has both horizontal and vertical components of the velocity. The non -conservative corrections 14to the equations of motion of the ring, due to Cherenkov radia tion, were derived. Due to these corrections the track of the ring bends towards t he surface faster then in the case of flat surface. For simplicity, all calculations in Sec.5 were performed for a single ring. The generalization for the case of many rings i s straightforward. The author thanks professor J.J. Rasmussen for his attentio n to this work and for helpful suggestions. This work was supported by the INTAS (g rant No. 96-0413), the Russian Foundation for Basic Research (grant No. 97-01- 00093), and the Landau Postdoc Scholarship (KFA, Forschungszentrum, Juelich, Ge rmany). References [1] V. I. Arnol’d, Mathematical Methods of Classical Mechanics , 2nd edition (Springer-Verlag, New York, 1989) [Russian original, Nauk a, Moscow, 1974]. [2] V.E.Zakharov and E.A.Kuznetsov, Usp.Fiz.Nauk 167, 1037 (1997). [3] V.E.Zakharov, Prikl. Mekh. Tekh. Fiz., No.2, 86 (1968). [4] H. Lamb, Hydrodynamics , 6th edition (Cambridge University Press, Cambridge, 1932) [Russian translation, Gostekhizdat, Moscow, 1947]. [5] E.A.Kuznetsov and A.V.Mikhailov, Phys. Lett. A 77, 37 (1980). [6] V.Berdichevsky, Phys. Rev. E, 57, 2885 (1998). [7] E.A.Kuznetsov and V.P.Ruban, Pis’ma v ZhETF, 67, 1012, (1998) [JETP Let- ters,67, 1076, (1998)]. [8] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, New York) [Russian original, Nauka, Moscow, 1988]. [9] V.P.Ruban, ZhETF, 116, 563–585 (1999). [JETP, 89, 299, (1999)]. [10] M. I. Monastyrskii and P. V. Sasorov, Zh. Eksp. Teor. Fiz .93, 1210 (1987) [Sov. Phys. JETP 66, 683 (1987)]. [11] E.A.Kuznetsov and V.P.Ruban, Phys. Rev. E ,61, N.1, 831 (2000). [12] E.A.Kuznetsov and V.P.Ruban, ZhETF, 115, 894-919, (1999). [JETP, 88, 492, (1999)]. [13] B.A.Dubrovin, S.P.Novikov, and A.T.Fomenko, Modern Geometry (Nauka, Moscow, 1979). [14] A.J.Chorin, Vorticity and Turbulence , Springer-Verlag, New York, 1994. 15

arXiv:physics/0001071v1 [physics.flu-dyn] 31 Jan 2000Non equilibrium in statistical and fluid mechanics. Ensembles and their equivalence. Entropy driven intermitt ency. Giovanni Gallavotti Fisica, Universit` a di Roma 1 P.le Moro 2, 00185 Roma, Italia Abstract: We present a review of the chaotic hypothesis and discuss its applications to intermittency in statistical mechanics and fluid mechanics proposing a quantitative defi- nition. Entropy creation rate is interpreted in terms of cer tain intermittency phenomena. An attempt to a theory of the experiment of Ciliberto–Laroch e on the fluctuation law is presented. §1. Introduction. A general theory of non equilibrium stationary phenomena ex tending classical thermo- dynamics to stationary non equilibria is, perhaps surprisi ngly, still a major open problem more than a century past the work of Boltzmann (and Maxwell, G ibbs,...) which made the breakthrough towards an understanding of properties of matter based on microscopic Newton’s equations and the atomic model. In the last thirty years, or so, some progress appears to have been achieved since the recognition that non equilibrium statistical mechanics an d stationary turbulence in fluids are closely related problems and, in a sense, in spite of the a pparently very different nature of the equations describing them they are essentially the sa me. The unifying principle, originally proposed for turbulent motions by Ruelle, [Ru78], in the early 970’s, has been extended to statistical mechanics and eventually called the “chaotic hypothesis ”, [GC95]: Chaotic hypothesis: Asymptotic motions of a chaotic system , be it a multi particle system of microscopic particles or a turbulent macroscopic fluid, c an be regarded as a transitive Anosov system for the purposes of computing time averages in stationary states. It may be useful to make a few comments on how this is supposed t o be interpreted. The conclusions that we draw here from the chaotic hypothesi s are summarized in §13 which might be consulted at this point. For a review on the sub ject seen from a different perspective see [Ru99a] §2. Meaning of the chaotic hypothesis. Anosov systems are well understood dynamical systems: they play a paradigmatic role with respect to chaotic systems parallel to the one harmonic oscillators play with respect to orderly motions. They are so simple, and yet very chaotic, that their properties are likely to be the ones everybody develops in thinking about ch aos, even without having any familiarity with Anosov systems which certainly are not (yet) part of the background of most contemporary physicists.1 1Informally a map x→Sxis a Anosov map if at every point xof the bounded phase space Mone can set up a local system of coordinates with origin at x, continuously dependent on xand covariant under the action of Sand such that in this comoving system of coordinates the poin txappears as a hyperbolic fixed point for S. The corresponding continuous time motion, when the evolut ion is x→Stx, t∈R, requires that the local system of coordinates contains the phase space velocity ˙ xas one of the coordinate axes and that the motion transversal to it s eesxas a hyperbolic fixed point: note that a motion in continuous time cannot possibly be hyperbolic in all directions and it has to be neutral in the direction of ˙ xbecause the velocity has to be bounded if Mis bounded, while hyperbolicity would imply exponential growth as either t→+∞ort→ −∞ . Furthermore there should be no equilibrium 2/febbraio/ 2008; 1:06 1In general an Anosov system has asymptotic motions which app roach one out of finitely many invariant closed sets C1,...,C qeach of which contains a dense orbit,:one says that the systems ( Cj,St) are “transitive”. One of them, at least, must be an attracti ve set. To say that “the asymptotic motions form a transitive Anosov system” means that (1) each of the sets Cjwhich is attractive is a smooth surface in phase space and (2) only one of them is attractive: The last “transitivity” assumption is meant to exclude the t rivial case in which there are more than one attractive sets and the system de facto consists of several independent systems. The smoothness of Cjis astrong assumption that means that one does not regard possible lack of smoothness, i.e.fractality, as a really relevant property in systems with large number of degrees of freedom. In any event one could con sider (if necessary) replacing “Anosov systems” with some slightly weaker prope rty like “axiom A” systems which could permit more general asymptotic motions. Here we adhere strictly to the chaotic hypothesis in the stated original form, [GC95]. §3. Basic implications of the chaotic hypothesis and relatio n with the ergodic hypothesis. The chaotic hypothesis boldly extends to non equilibrium th eergodic hypothesis : applied to equilibrium systems, i.e.to systems described by Hamiltonian equations, it implies the latter, [Ga98]. This means that if a Hamiltonian system a t a given energy is as- sumed to verify the chaotic hypothesis, i.e.to be a transitive Anosov system, then for all observables F(i.e.for all smooth functions Fdefined on phase space) T−1/integraldisplayT 0F(Stx)dt− − − − →T→∞/integraldisplay MF(y)µL(dy) (3 .1)3.1 whereµLis the Liouville distribution on the constant energy surfac eM, and (3.1) holds for almost all points x∈M,i.e.forxoutside a set Nof zero Liouville volume on M. Being very general one cannot expect that the chaotic hypoth esis will solve any special problem typical of non equilibrium physics, like “proving” the Fourier’s law of heat conduction, the Ohm’s law of electric conduction or the K41 t heory of homogeneous turbulence. Nevertheless, like the ergodic hypothesis in equilibrium, the chaotic hypothesis accom- plishes the remarkable task of giving us the “statistics” of motions. If Mis the phase space, which we suppose a smooth bounded surface, and t→Stxis the motion starting atx∈M, the time average: T−1/integraldisplayT 0F(Stx)dt− − − − →T→∞/integraldisplay MF(y)µSRB(dy) (3 .2)3.2 of the observable Fexists forxoutside a set Nofzero phase space volume and it is x–independent, thus defining the probability distribution µSRBvia (3.2). Note, in fact, that the probability distribution µSRBdefined by the l.h.s. of (3.2) is uniquely determined (provided it exists): it is usually cal led the “ statistics of the motion ” or the “ SRB distribution ” associated with the dynamics of the system. To appreciate the above property (existence and uniqueness of the statistics) the fol- lowing considerations seem appropriate. points and the periodic points should be dense in phase space . When the system has one or more (the so called “ hysteresis phenomenon” ) attracting sets which do not occupy the whole phase space th e chaotic hypothesis can be interpreted as saying that each at tracting set is a smooth surface on which the time evolution flow (or map) acts as an Anosov flow (map). 2/febbraio/ 2008; 1:06 2An essential feature, and the main novelty, with respect to e quilibrium systems is that non conservative forces may act on the system: this is in fact the very definition of “no n equilibrium system”. Since non conservative forces perform work it is necessary t hat on the system act also other forces that take energy out of it, at least if we wish tha t the system reaches a stationary state, showing a well defined statistics. As a consequence any model of the system must contain, besides non conservative forces which keep it out of equilibrium by establishing “ flows” on it (like a heat flow, a matter flow, ...), also dissipative forces preventing the energy to increase indefinitely and forcing the motion to visit only a finite region of phase space. The dissipation forces, also called “thermostatting force s”, will in general be such that the volume in phase space is no longer invariant under time evolution. Mathematically this means that the divergence −σ(x) of the equations of motion will be not zero and its time average/integraltext Mσ(y)µSRB(dy)def=σ+will be positive or zero as it cannot be negative (“because phase space is supposed bounded”: see [Ru96]). One calls a system “ dissipative ” ifσ+>0 and we expect this to be the case as soon as there are non conservative forces acting on it. We see that if a system is dissipative then its statistics µSRBmustbe concentrated on a set of zero volume inM: this means that µSRBcannot be very simple, and in fact it is somewhat hard to imagine it. If the acting forces depend on a parameter E, “strength of the non conservative forces”, and forE= 0 the system is Hamiltonian we have a rather unexpected situ ation. At E= 0 the chaotic hypothesis and the weaker ergodic hypothesis imply that the statistics µSRBis equal to the Liouville distribution µL; but ifE/\e}atio\slash= 0, no matter how small, itwill not be possible to express µSRBvia some density ρE(y) in the form µSRB(dy) = ρE(y)µL(dy), becauseµSRBattributes probability 1 to a set Nwith zero volume in phase space ( i.e.µL(N) = 0). Nevertheless natura non facit saltus (no discontinuities appear in natural phenomena) so that sets that have probability 1 wi th respect to µSRBmay be all still dense in phase space, at least for Esmall. In fact this is a “ structural stability ” property for systems which verify the chaotic hypothesis (s ee [Ga96c]) The above observations show one of the main difficulties of non equilibrium physics: the unknownµSRBis intrinsically more complex than a function ρE(y) and we cannot hope to proceed in the familiar way we might have perhaps expected from previous experiences: namely to just set up some differential equations for the unkn ownρE(y). Hence it is important that the chaotic hypothesis not only gu arantees us the existence of the statistics µSRBbutalsothat it does so in a “constructive way” giving at the same time formal expressions for the distribution µSRBwhich should possibly play the same role as the familiar formal expressions used in equilibrium statistical mechanics in writing expectations of observables with respect to the microcanon ical distribution µL. For completeness we write a popular expression for µSRB. Ifγis a periodic orbit in phase space, xγa point onγ,T(γ) the period of γthen /integraldisplay F(y)µSRB(dy) = lim T→∞/summationtext γ:T(γ)≤Te−/integraltextT(γ) 0σ(Stxγ)dt/integraltextT(γ) 0F(Stxγ)dt /summationtext γ:T(γ)≤Te−/integraltextT(γ) 0σ(Stxγ)dtT(γ)(3.3)3.3 This is simple in the sense that it does not require, to be form ulated, an even slight understanding of any of the properties of Anosov or hyperbol ic dynamical systems. But in many respects it is not a natural formula: as one can grasp from the fact that it is far from clear that in the equilibrium cases (3.3) is an alt ernative definition of the 2/febbraio/ 2008; 1:06 3microcanonical ensemble ( i.e.of the Liouville distribution µL), in spite of the fact that in this case σ≡0 and (3.3) becomes slightly simpler. To prove (3.3) one first derives alternative and much more use ful expressions for µSRB which, however, require a longer discussion to be formulate d, see [Ga99a], [Ga86c]: the original work is due to Sinai and in cases more general than An osov systems, to Ruelle and Bowen. §4. What can one expect from the chaotic hypothesis? In equilibrium statistical mechanics we know the statistic s of the motions, if the ergodic hypothesis is taken for granted. However this hardly solves the problems of equilibrium physics simply because evaluating the averages is a difficult task which is also model dependent. Nevertheless there are a few general consequenc es that can be drawn from the ergodic hypothesis: the simplest (and first) is embodied in the “ heat theorem ” of Boltzmann. Imagine a system of Nparticles in a box of volume Vsubject to pair interactions and to external forces with potential energy WV, due to the walls and providing the confinement of the particles to the box. Define T=average kinetic energy U=total energy p=average of ∂VWV(4.1)4.1 where the averages are taken with respect to the Liouville di stribution on the surface of energyU. Imagine varying the parameters on which the system depends ( e.g.the energy Uand the volume V) so thatdU,dV are the corresponding variations of U,V, then (dU+pdV)/T= exact (4 .2)4.2 expresses the heat theorem of Boltzmann. It is a consequence of the ergodicity assumption, but it is notequivalent to it as it only involves a relation between a few averages ( U,p,V,T ), see [Bo66], [Bo84], [Ga99a]. Not only it gives us a relation which is a very familiar property o f macroscopic systems, but it also suggests us that even if the ergodic hypothesis is not strictly valid some of its consequences might, still, be regarded as correct. The proposal is to regard the chaotic hypothesis in the same w ay: it is possible to imagine that mathematically speaking the hypothesis is not strictl y valid and that, nevertheless, it yields results which are physically correct for the few ma croscopic observables in which one is really interested in. The ergodic hypothesis implies the heat theorem as a general (“somewhat trivial”) mechanical identity valid for systems of Nparticles with N= 1,2,...,1023,.... For smallNit might perhaps be regarded as a curiosity: such it must have been considered by most readers of the key paper [Bo84] who were possibly misl ed by several examples withN= 1 given by Boltzmann in this and other previous papers. Like the example of the system consisting of one “averaged” Saturn ring, i.e.one homogeneous ring of mass rotating around Saturn with energy U, kinetic energy Tand “volume” V(improbably identified with the strength of the gravitational attractio n!). But for N= 1023it is no longer a curiosity and it is a fundamental law of thermodynam ics in equilibrium: which, therefore, can be regarded on the same footing of a symmetry b eing a direct consequence of the structure of the equations of motion, [Ga99a] appendi ces to Ch.1 and Ch.9. It reflects in macroscopic terms a simple microscopic assumpti on (i.e.Newton’s equations for atomic motions, in this case). 2/febbraio/ 2008; 1:06 4No new consequences of even remotely comparable importance are known to follow from the chaotic hypothesis besides the fact that it implies the v alidity of the ergodic hypoth- esis itself (hence of all its consequences, first of them clas sical equilibrium statistical mechanics). Nevertheless the chaotic hypothesis doeshave some rather general consequences. We mention here the fluctuation theorem . Letσ(x) be the phase space contraction rate and σ+be its SRB average ( i.e.σ+=/integraltext σ(x)µSRB(dx)), letτ >0 and define p(x) =τ−1/integraldisplayτ/2 −τ/2σ(Stx) σ+dx (4.3)4.3 and study the fluctuations of the observable p(x) in the stationary state µSRB. We write πτ(p)dpthe probability that, in the distribution µSRB, the quantity p(x) has actually value between pandp+dpas πτ(p)dp= consteζτ(p)τdp (4.4)4.4 Them lim τ→∞ζτ(p) =ζ(p) exists and is convex in p; and Theorem: (fluctuation theorem) Assume the chaotic hypothesis and sup pose that the dynamics is reversible, i.e. that there is an isometry Iof phase space such that ISt=S−tI, I2= 1 (4 .5)4.5 and that the attracting set is the full phase space.2Then ζ(−p) =ζ(p)−σ+p, for allp (4.6)4.6 whereσ+=µSRB(σ). It should be pointed out that the above relation was first disc overed in an experiment, see [ECM93], where also some theoretical ideas were present ed, correctly linking the result to the SRB distributions theory and to time reversal s ymmetry. Although such hints were not followed by what can be considered a proof, [CG 99], still the discovery has plaid a major role and greatly stimulated further resear ch. The interest of (4.6) is that, in general, it is a relation wit hout free parameters. The above theorem, proved in [GC95] for discrete evolutions (ma ps) and in [Ge97] for con- tinuous time systems (flows), is one among the few general con sequences of the chaotic hypothesis, see [Ga96a], [Ga96b], [Ga99b] for others. §5. Non equilibrium ensembles. Thermodynamic limits. Equiv alence. The chaotic hypothesis gives us, unambiguously, the probab ility distribution µSRBwhich has to be employed to compute averages of observables in stat ionary states. For each value of the parameters on which the system depends w e have, therefore a well defined probability distribution µSRB. Callingα= (α1,...,α p) the parameters andµαthe corresponding SRB distribution we consider the collect ionEof probability 2It is perhaps important to stress that we distinguish betwee nattracting set andattractor : the first is a closed set such that the motions that start close enough to it approach it ever closer; an attractor is a subset of an attracting set that (1) has probability 1 with respect to the statistics µof the motions that are attracted by the attracting set (a notion which makes sense when such statistics exists, but for a zero volume set of initial data, and is unique) and that (2) has the smallest Hausdorff dimension among such probabil ity 1 sets. Hence density of an attracting set in phase space does not mean that the corresponding attra ctor has dimension equal to that of the phase space: it could be substantially lower, see [GC95]. 2/febbraio/ 2008; 1:06 5distributions µαobtained by letting the parameters αvary. We call such a collection an “ensemble ”. For instance αcould be the average energy Uof the system, the average kinetic energy T, the volume V, the intensity Eof the acting non conservative forces, etc Non equilibrium thermodynamics can be defined as the set of re lations that the vari- ations of the parameters αand of other average quantities are constrained to obey as some of them are varied. In equilibrium the heat theorem is an example of such relations. In reversible non equilibria the fluctuation theorem (4.6) i s an example. In non equilibrium systems the equations of motion play a muc h more prominent role than in equilibrium: in fact one of the main properties of equ ilibrium statistical mechanics is that dynamics enters only marginally in the definition of t he statistical distributions of the equilibrium states. The necessity of a reversibility assumption in the fluctuati on theorem already hints at the usefulness of considering the equations of motion thems elves as “parameters” for the ensembles describing non equilibrium stationary state s: we are used to irreversible equations in describing non equilibrium phenomena (like th e heat equation, the Navier Stokes equation, etc) and unless we are able to connect our experiments with rever sible dynamical models we shall be unable to make use of the fluctuat ion theorem. Furthermore it is quite clear that once a system is not in equi librium and thermostat- ting forces act on it, the exact nature of such forces might be irrelevant within large equivalence classes: i.e.it might be irrelevant which particular “cooling device” we use to take heat out of the system. Hence one would like to have a fr ame into which to set up a more precise analysis of such arbitrariness. Therefore we shall set Definition 1: A stationary ensemble Efor a system of particles or for a fluid is the collection of SRBdistributions, for given equations of motion, obtained by v arying the parameters entering into the equations. It can happen that for the same system one can imagine different models . In this case we would like that the models give the same results, i.e.the same averages to the same observables, at least in some relevant limit. Like in the lim it of infinite size in which the number N, the volume Vand the energy Utend to infinity but N/VandU/Vstay constant. Or in the limit in which the Reynolds number Rtends to infinity in the case of fluids. This gives the possibility of giving a precise meaning to the equivalence of different thermostatting mechanisms. We shall declare Definition 2, (equivalence of ensembles): Two thermostatting mechanism s are equiv- alent “in the thermodynamic limit” if one can establish a one to one correspondence between the elements of the ensembles EandE′of SRB distributions associated with the two models in such a way that the same observables, in a certai n class Lof observables, have the same averages in corresponding distributions, at l east when some of the param- eters of the system are sent to suitable limiting values to wh ich we assign the generic name of “thermodynamic limit”. In the following sections we illustrate possible applicati ons of this concept. §6. Drude–Lorentz’ electric conduction models. Understanding of electric conduction is in a very unsatisfa ctory state. It is usually based on linear response theory and very seldom a fundamenta l approach is attempted. Of course this is so for a good reason, because a fundamental a pproach would require imposing an electric field Eon the system and, at the same time, a thermostatting force to keep the system from blowing up and to let it approach a stea dy state with a current JEflowing in it, and then taking the ratio JE/E(with or without taking also the limit 2/febbraio/ 2008; 1:06 6asE→0). However, as repeatedly mentioned, it is an open problem to st udy steady states out of equilibrium. Hence most theories have recourse to linear re sponse where the problem of studying stationary non equilibria does not even arise. The reason why this is unsatisfactory is that as long as we are in principle unable to study stationary non equilibria we are also in principle unable to estimate the size of the approximation and errors of linear response. In spite of many attempts the old theory of Drude, see [Be59], [Se87], seems to be among the few conduction theories which try to establish a conduct ivity theory based on the study of electric current at non zero fields. We imagine a set of obstacles distributed randomly or period ically and among them conduction electrons move, roughly with density of one per o bstacle. The (screened) interactions between the electrons are, at a first approximation, ignored. The collisions between electrons and obstacles (“nuclei”) will take place in the average after the electrons have traveled a distance λ= (ρa2)−1ifρis the nuclei density and a is their radius. Between collisions the electrons, with electric charge e, accelerate in the direction of an imposed field Eincrementing, in that direction, velocity by δv=eEλ mv=eE(ρa2)−1 m/radicalbig kBT/m(6.1)6.1 wherekBis Boltzmann’s constant. At collision they are “thermalize d”: an event that is modeled by giving them a new velocity of size v=/radicalbig kBT/mand a random direction. The latter is the “thermostatting mechanism” which is a, som ewhat rough, description of the energy transfer from electrons to lattice which physi cally corresponds to electrons losing energy in favor of lattice phonons, which in turn are kept at constant temperature by some other thermostatting mechanism which prevents the w ire melting. All things considered the total current that flows will be JE=e2 ρa2√mkBTEdef=χE (6.2)6.2 obtaining Ohm’s law. To the same conclusion we arrive by a different thermostat mod el. We imagine that the electrons move exchanging energy with lattice phonons b ut keeping their total energy constant and equal to NkBT:i.e.2−1/summationtextN j=1m˙x2 j= 3NkBT/2, wherekBis Boltzmann’s constant. There are several forces that can achieve this res ult we select the “Gaussian minimal constraint” force.3This is the force that is required to keep/summationtextm˙x2 jstrictly constant and that is determined by “Gauss least effo rt” principle, see [Ga99a], ch. 9, appendix 4, for instance: as is well known this is, on the i–th particle, a force −α˙xidef=−eE·/summationtext j˙xj/summationtext j˙x2 j˙xi≡ −mE·N J 3NkBT˙xi (6.3)6.3 If there are Nparticles and Nis large it follows that J=N−1e/summationtext j˙xjis essentially constant, see [Ru99b], and each particle evolves, almost in dependently of the others, according to an equation: 3Not because it plays any fundamental role but because it has b een studied by many authors and because it represents a mechanism very close to that proposed by Drud e.We recall, for copleteness, that the effort of a constraint reaction on a motion on which the active force isf(with 3 Ncomponents) and a is the acceleration of the particles (with 3 Ncomponents) and mis the mass is E(a) = ( f−m a)2/m; then Gauss’ principle is that the effort is minimal if ais given the actual value of the acceleration, at fixed space positions and velocities. 2/febbraio/ 2008; 1:06 7m¨xi=eE−ν˙xi (6.4)6.4 between collisions, with a suitably fixed constant ν. If we imagine that the velocity of the particles between collisions changes only by a small quanti ty compared to the average velocity the “friction term” which in the average will be of o rderE2will be negligible except for the fact that its “only” effect will be of insuring that the totalkinetic energy stays constant and the speeds of the particles are constantl y renormalized. In other words this is the same as having continuously collisions between e lectrons and phonons even when there is no collision between electrons and obstacles. Hence the resulting current is the same (if Nis large) as in (6.2). §7. Ensemble equivalence: the example of electric conductio n theories. We have derived three models for the conduction problem, nam ely (1) the classical model of Drude, [Se87], in which at every collision the electron velocity is reset to the average velocity at the given temperature, wi th a random direction, c.f.r. (6.1) and (6.2). (2) the Gaussian model in which the total kinetic energy is ke pt constant by a thermostat force m¨xi=E−mE·J 3kBT˙xi+ “collisional forces′′(7.1)7.1 where 3NkBTis the total kinetic energy (a constant of motion in this mode l). The model has been widely studied and it was introduced by Hoover and Ev ans (see for instance [HHP87] and [EM90]). (3) a “friction model” in which particles independently exp erience a constant friction m¨xi=E−ν˙xi+ “collisional forces′′(7.2)7.2 whereνis a constant tuned so that the average kinetic energy iseNk BT/2. This model was considered in the perspective of the conjectures of ense mble equivalence in [Ga95], [Ga96b]. The first model is a “stochastic model” while the second and th ird are deterministic: the third is “irreversible” while the second is reversible beca use the involution I(xi, vi) = (xi,−vi) anticommutes with the time evolution flow Stdefined by the equation (7.1): ISt=S−tI(as the “friction term” is oddunderI). Letµδ,Tbe the SRB distribution for (7.1) for the stationary state th at is reached starting from initial data with energy 3 NkBT/2. The collection of the distributions µδ,Tas the kinetic energy Tand the density δ=N/V vary, define a “statistical ensemble” Eof stationary distributions associated with the equation (7. 1). Likewise we call ˜ µδ,νthe class of SRB distributions associated with (7.2) which f orms an “ensemble” ˜E. We establish a correspondence between distributions of the ensembles Eand˜E: we say thatµδ,Tand ˜µδ′,νare “corresponding elements” if δ=δ′, T =/integraldisplay1 2(/summationdisplay jm˙x2 j) ˜µδ,ν(dxd˙x) (7 .3)7.3 Then the following conjecture was proposed in [Ga96b]. Conjecture 1: (equivalence conjecture) Let Fbe a “ local observable ”, i.e. an observ- able depending solely on the microscopic state of the electr ons whose positions is inside a fixed box V0. Then, if Ldenotes the local smooth observables 2/febbraio/ 2008; 1:06 8lim N→∞,N/V=δ˜µδ,ν(F) = lim N→∞,N/V=δµδ,T(F)F∈ L (7.4)7.4 ifTandνare related by (7.3). This conjecture has been discussed in [Ga95], sec. 5, and [Ga 96a], see sec. 2 and 5: and in [Ru99b] arguments in favor of it have been developed. Clearly the conjecture is very similar to the equivalence in equilibrium between canonical and microcanonical ensembles: here the friction νplays the role of the canonical inverse temperature and the kinetic energy that of the microcanonic al energy. It is remarkable that the above equivalence suggests equiva lence between a “reversible statistical ensemble”, i.e.the collection Eof the SRB distributions associated with (7.1) and a “irreversible statistical ensemble”, i.e.the collection ˜Eof SRB distributions asso- ciated with (7.2). Furthermore it is natural to consider also the collection E′of stationary distributions for the original stochastic model (1) of Drude, whose elemen tsµ′ ν,Tcan be parameterized by the quantities T, temperature (such that1 2/summationtext jm˙x2 j=3 2NkBT, andN/V=δ). This is an ensemble E′whose elements can be put into one to one correspondence with the elements of, say, the ensemble Eassociated with model (2), i.e.with (7.1): an element µ′ ν,T∈ E′corresponds to µδ,ν∈ EifTverifies (7.3). Then Conjecture 2: Ifµδ,T∈ Eandµ′ δ,ν∈ E′are corresponding elements (i.e. (7.3) holds) then lim N→∞,N/V=δµδ,T(F) = lim N→∞,N/V=δµ′ δ,T(F)F∈ L (7.5)7.5 for all local observables F∈ L. Hence we see that there can be statistical equivalence betwe en a viscous irreversible dissipation model and either a stochastic dissipation mode l or a reversible dissipation model, at least as far as the averages of special observables are concerned. The argument in [Ru99b] in favor of conjecture 1 is that the co efficientαin (6.3) is essentially the average Jof the current over the whole box containing the system of particles,J=N−1e/summationtext j˙xi: henceJshould be constant with probability 1, at least if the stationary SRB distributions can be reasonably suppose d to have some property of ergodicity with respect to space translations . §8. Entropy driven intermittency in reversible dissipation . A further argument for the equivalence conjectures in the ab ove electric conduction models can be related to the fluctuation theorem: the quantit yα(x) is also proportional to the phase space contraction rate σ(x) = (3N−1)α(x). Therefore, denoting in general with a subscript + the SRB average (or the time average) of an o bservable, the probability thatσ(x) deviates from its average σ+= (3N−1)α+can be studied as follows. If the number Nof particles is large the time scale τ0over which σ(Stx) evolves will be large compared to the microscopic evolution rates, becau seσt(x) is the sumof the ∼6Nrates of expansion and contraction of the ∼6Nphase space directions out of x (sometimes called the “ local Lyapunov exponents ”).4 4The exact number of exponents depends on how many constants o f motion the system has: for instance in the case of the conduction model (1) in §6 above the number of exponents is 6 N−1 because the kinetic energy is conserved and the system has no other (obvi ous) first integrals. Furthermore one of such exponents is 0 since every dynamical system in continuous ti me has one zero exponent (corresponding to the direction ˙ xof the flow). 2/febbraio/ 2008; 1:06 9Consider a large number mof time intervals I1,I2,...,I mof sizeτ0and letσjbe the (average) value of σ(Stx) fort∈Ij. Then the fraction of the j’s such that σj−σ+≃σ+p will be proportional to πτ0(p)≃eτ0ζ(p)(8.1)8.1 andζ(p)<ζ(1) ifp/\e}atio\slash= 1. Since we can expect that ζ(p) is proportional to Nwe see that the fraction of time intervals Ijin whichσj/\e}atio\slash=σ+will be exponentially small with N. For instance the fraction of time intervals in which σj≃ −σ+will be, by the fluctuation theorem e−(3N−1)α+τ0(8.2)8.2 In order that the above argument holds it is essential that Nis large to the point that we can think that the time scale τ0over which σ(Stx) varies is much larger than the microscopic scales: so that we can regard τ0large enough for the fluctuation theorem to apply. In this respect this is not really different from the pr eviously quoted argument in [Ru99b]. However the change of perspective gives further in formation. In fact we get the following picture: Nis large and for most of the time the (stationary) evolution uneventfully proceeds as if σ(Stx)≡σ+(thus justifying conjecture 1). Very rarely, however, it proceeds as if σ(Stx)/\e}atio\slash=σ+, for instance with σ(Stx) =−σ+: such “bursts of anomalous behavior ” occur very rarely. But when they occur “everything else goes the wrong way” because, as discussed in detail in [Ga99c ], while the phase space contraction is opposite to what it “should be” (in the averag e) then it also happens that all observables evolve following paths that are the time rev ersal of the expected paths , This is the content, see [Ga99c], of the following theorem which i s quite close (particularly if one examines its derivation) to the Machlup–Onsager theo ry of fluctuation patterns (note that, however, it does not require closeness to equili brium) Theorem (conditional reversibility theorem): If Fis an observable with even (or odd), for simplicity, time reversal parity and if τis large then the evolution or “fluctuation pattern”ϕ(t)and its time reversal Iϕ(t)≡ϕ(−t),t∈[−τ0/2,τ0/2], will be followed with equal likelihood if the first is conditioned to an entropy cre ation ratepand the second to the opposite −p. In other words systems with reversible dynamics can be equiv alent to systems with irreversible dynamics but they show “ intermittent behavior ” with intermittency lapses that become extremely rare very quickly as N→ ∞. Sometimes they can be really dramatic, as in the cases in which σ=−σ+: alas they are unobservable just for this reason and one can wonder (see §9 below) whether this is really of any interest. §9. Local fluctuations and observable intermittency. As a final comment upon the analysis of the equivalence of ense mbles attempted above we consider a very large system with volume Vand a small subsystem of volume V0which is large but not yet really macroscopic so that the number of p articles inV0is not too large, a nobler way to express the same notion is to say that we consider a “mesocopic” subsystem of our macroscopic system. Here it is quite important to specify the system because we wa nt to make use of as- pects of the equivalence conjectures that are model depende nt. Therefore we consider the conduction models (2) or (3) of §5: these are models in which dissipation occurs “homogeneously” throughout the system. In this case we can i magine to look at the part of the system in the box V0: ifj1,...,j N0are the particles which at a certain instant are insideV0and ˙xj=fj(x) are the equations of motion we can define 2/febbraio/ 2008; 1:06 10σV0(x) =N0/summationdisplay i=1∂xjkfjk(x) (9 .1)9.1 which is (by definition) the part of phase space contraction d ue to the particles in V0. Since the part of the system inside the microscopically larg e but macroscopically small V0can be regarded as a new dynamical system whose properties sh ould not be different from the ones of the full system enclosed in the full volume Vwe may expect that the subsystem inside V0is in a stationary state and the quantity σV0has the same fluctuation properties as σV,i.e. (1) /a\}bracketle{tσV0/a\}bracketri}ht+=V0σ+, /a\}bracketle{tσV/a\}bracketri}ht+=Vσ+ (2)πV0 τ(p) =eζ(p)τ V0, πV τ(p) =eζ(p)τ V(9.2)9.2 whereζ,σ+arethe same for V,V0andp=τ−1/integraltextτ/2 −τ/2σV0(Stx)//a\}bracketle{tσV0/a\}bracketri}htdtor respectively p=τ−1/integraltextτ/2 −τ/2σV(Stx)//a\}bracketle{tσV/a\}bracketri}htdt. HereσV0is naively defined as the contribution to σ coming from the particles in V0. In other words in large stationary systems with homogeneous reversible dissipation phase space contractions fluctuate in an extensive way , i.e. they are regulated by the same deviation function ζ(p) (volume independent). This is very similar to the well known property of equilibriu m density fluctuations in a gas of density ρ: ifV⊃V0are a very large volume Vin a yet larger container and V0 is a small but microscopically large ( i.e.mesoscopic) volume V0then the total numbers of particles in VandV0will beNandN0and the average numbers will be ρVandρV0 respectively. Then setting p= (N−ρV)/ρV, or, p = (N0−ρV0)/ρV0 (9.3)9.3 the probability that the variable phas a given value will be proportional to πV(p) =eζ(p)V, πV0(p) =eζ(p)V0(9.4)9.4 again with the same function ζ(p). This means that we can observe ζ(p)by performing fluctuations experiments in small boxes, ideally carved out of the large container, where the densit y fluctuations are not too rare. A “local fluctuation law” should hold more generall y in cases of models in which dissipation occurs homogeneously across the system, like the above considered conduction models. The intuitive picture for the above “local fluctuation relat ion” inspired (and was sub- stantiated) a mathematical model in which a local fluctuatio n relation can be proved as a theorem: it has een discussed in [Ga99c], see also below. Going back to the conduction model we see that the intermitte ncy phenomena discussed above can be actually observed by looking at the fluctuations of the contribution to phase space contraction due to a small subsystem. And such “ entropy driven ” intermittency will be model independent for models which a re equivalent in the sense of the previous sections provided th e models used are equivalent and one of them is reversible. An extreme case is provided by models (1)%(3), §7, for electric conduction (conjectured to be equivalent, see §7). In fact at first the model (3), the viscous thermostat, mig ht look uninteresting as, obviously, in this case σV(x) = 3Nν, σV0(x) = 3N0ν (9.5)9.5 2/febbraio/ 2008; 1:06 11andσV/Vhas no fluctuations. However the equivalence conjecture makes a statement about expectation values of the same observable : hence we should consider the quantity ˆ σV0(x) =E·JV0//summationtext j˙x2 jand we should expect that its statistics with respect to an eleme nt of the ensemble E′is the same as that of the same quantity with respect to the corre sponding elements of the ensembles E,˜E. Hence in particular the functions ζ(p) which control the large fluctuations ofσV(p) will verify ζ(−p) =ζ(p)−p/a\}bracketle{tˆσV0/a\}bracketri}ht+/V0=ζ(p)−3ρνp=ζ(p)−eEmJ + kBTp (9.6)9.6 where the first equality expresses the validity of a fluctuati on theorem type of relation due to the fact that the small system, by the equivalence conj ecture, should behave as a closed system; the second equality expresses a consequence of the equivalence conjecture between models (2) and (3) while the third is obtained by expr essing the current via Drude’s theory (again assuming the conjectures of equivale nce 1,2 of §7). §10. Fluids. The chaotic hypothesis was originally formulated to unders tand developed turbulence, [Ru78]: it is therefore interesting to revisit fluid motions theory. The incompressible Navier Stokes equation for a velocity fie lduin a periodic container Vof sideLcan be considered as an equation for the evolution in time of i ts Fourier coefficients ukwhere the “ mode”khas the form 2 πL−1nwithn/\e}atio\slash= 0andnan integer components vector.5Furthermore uk=u−kandk·uk≡0. Ifpis the pressure field andfa simple forcing we shall fix the ideas by considering f(x) =f esinkf·x wherekfis some prefixed mode and eis a unit vector orthogonal to kf. The Navier Stokes equation is then ˙u+u /tildewide·∂ /tildewideu=−∂p+f+ν∆u (10.1)10.1 and it is convenient to use dimensionless variables u0,p0,ϕ0,ξ,τ: so we define them as u(x,t) =fL2ν−1u0(L−1x,L−2νt), ξ =L−1x, τ=L−2νt p(x,t) =fLp0(L−1x,L−2νt), Rdef=fL3ν−2 f(x,t) =fϕ0(L−1x)(10.2)10.2 with max |ϕ0|= 1. The result, dropping the label 0 and calling again x,tthe new variablesξ,τ, is that the Navier Stokes equations become an equation for a divergenceless fieldudefined onV= [0,1]3, with periodic boundary conditions and equations ˙u+Ru /tildewide·∂ /tildewideu=−∂p+ϕ+ ∆u, ∂ ·u= 0 (10 .3)10.3 with max |ϕ|= 1. Equation (10.3) is our model of fluid motion, where Rplays the role of “forcing intensity” and the term ∆ urepresents the “thermostatting force”. As Rvaries the stationary distributions µRwhich describe the SRB statistics of the motions (10.3) defin e a set E of probability distributions which forms an “ensemble”. The mathematical theory of the Navier Stokes equations is fa r from being understood: however phenomenology establishes quite clearly a few key p oints. The main property 5The value n= 0is excluded because, having periodic boundary conditions, it is not restrictive to suppose that the space average of uvanishes (galilean invariance). The convention for the Fou rier transform that we use is u(x) =/summationtext kei k·xuk. 2/febbraio/ 2008; 1:06 12is that if (10.3) is written as an equation for the Fourier com ponents of uthen one can assume that uk≡0for|k|>K(R), for some finite K(R). Therefore the equation (10.3) should be thought of as a “trun cated equation” in mo- mentum space by identifying it with the equation obtained by projecting also u /tildewide·∂ /tildewideuon the same function space. Should one develop anxiety about the mathematical aspects o f the Navier Stokes equa- tion one should therefore think that an equally good model fo r a fluid is the mentioned truncation providedK(R) is chose large enough. The idea is that for K(R) =Rκ, withκlarger than a suitable κ0the results of the theory, i.e.the statistical properties of µRbecomeκ–independent for Rlarge. The simplest evaluation of κ0givesκ0= 9/4 as a consequence of the so called K41 theory of homogeneous turbulence, see [LL71]. If (10.3) is a good model for a fluid when Lis large then it provides us with an “ensemble” Eof SRB distributions (on the space of the velocity fields comp onentsukof dimension ∼8πK(R)3/3).6 We should expect, following the discussion of the statistic al mechanics cases, that there can be other “ensembles” ˜Ewhich are equivalent to E. HereRplays the role of the volume in non equilibrium statistical m echanics, so that R→ ∞will play the role of the thermodynamic limit, a limit in whic h the effective number of degrees of freedom, ∼4πR3κ/3, becomes infinite. The role of the local observables will be plaid by the (smooth) functions F(u) of the velocity fields uwhich depend onuonly via its Fourier components that have mode kwith|k|< Bfor someB: F(u) =F({uk}|k|≤B). We shall call Lthe space of such observables: examples can be obtained by se tting F(u) =|/integraltext ei k·xu(x)dx|2orF(u) =/integraltext ψ(x)·u(x)dxwhere the function has only a finte number of harmonics, ψ(x) =/integraltext/summationtext |k|<Bei k·xu(x)dx,etc. As in non equilibrium statistical mechanics we can expect th at the equations of motion themselves become part of the definition of the ensembles. Fo r instance one can imagine defining the ensemble ˜Eof the SRB distributions ˜ µVfor the equations ˙u+Ru /tildewide·∂ /tildewideu=−∂p+ψ+ν(u)∆u (10.4)10.4 called GNS equations in [Ga97a], or “gaussian Navier Stokes ” equations, where ν(u) is so defined that Ξ =/integraldisplay V(∂ /tildewideu)2dx/(2π)3=/summationdisplay kk2|uk|2(10.5)10.5 is exactly constant and equal to Ξ. The equations (10.4) are i nterpreted as above with the same momentum cut off K(R) =Rκ. An element ˜ µΞof˜Eand oneµRofE, SRB distributions for the two different dynamics (10.3) and (10.4), “correspond to each other” if Ξ =/integraldisplay µR(du)/parenleftbig/integraldisplay V(∂ /tildewideu)2dx/(2π)3/parenrightbigdef= Ξ R (10.6)10.6 whereµR∈ Eis the SRB distribution at Reynolds number Rfor the previous viscous Navier Stokes equation, (10.3), and we naturally conjectur e 6There are about 4 πK(R)3/3 vectors with integer components inside a sphere of radius K(R), thus the number of complex Fourier components with mode label |k|< K(R) would be 3 times as much, but the divergenceless condition leaves only 2 complex compone nts for ukalong the two unit vectors orthogonal to kand the reality condition further divides by 2 the number of “ free” components. 2/febbraio/ 2008; 1:06 13Conjecture 3 (equivalence GNS–NS): If R→ ∞then for all local observables F∈ Lit isµR(F) = ˜µΞR(F)if (10.6) holds. It is easy to check that the GNS model “viscosity” ν(u), having to be such that the quantity Ξ in (10.5) is exactly constant must be ν(u) =/integraltext V/parenleftbig ϕ·∆u−R∆u·(u /tildewide·∂ /tildewideu)/parenrightbig dx /integraltext V(∆u)2dx(10.7)10.7 and we see that while (10.3) is an irreversible equation the (10.4) is reversible , with time reversal symmetry given by Iu(x,t) =−u(x,t) (10 .8)10.8 as one can check. More generally one may wish to leave the “Kolmogorov paramet er”κas a free pa- rameter: in this case the SRB distributions will form an ense mble whose elements can be parameterized by R,κand the equivalence conjecture can be extended to this case yielding equivalence between µR,κand ˜µΞ,κ. This is of interest, particularly if one has numerical experiments in mind. Ifκ > κ 0then the value of κshould be irrelevant : but ifκ < κ 0the phenomenology will be different from the one of the Navier Stokes equation an d equivalence might still hold but one cannot expect either equation to have the proper ties that we expect for the usual Navier Stokes equations ( i.e.in this situation one would have to be careful in making statements based on common experience). If we takeκto be exactly equal to the value κ0= 9/4 (i.e.if we take the ultraviolet cut–off to be such that, according to the K41 theory, for large r values it is needlessly large and for lower values it is incorrectly low and shows a phenomenology which will depend on its actual value) then we may speculate that the “attracti ng set” is the full phase space (available compatibly with the constraint Ξ = Ξ R). Therefore the divergence of the equations of motion, which is given by a rather involved e xpression in which only the first term seems to dominate at large R, namely σ(u) = (/summationdisplay |k|<K(R)k2)ν(u)−/parenleftbig/integraldisplay V∆ϕ·∆udx/parenrightbig /parenleftbig/integraldisplay V[(∆u)2− −R∆u·(∆(u /tildewide·∂ /tildewideu))−R(∆u /tildewide)·(∆u)·(∂ /tildewideu)−R∆u·(∆∂ /tildewideu)u /tildewide+ +ν(u)∆u·∆2u]dx/parenrightbig //integraldisplay V(∆u)2dx(10.9)10.9 will verify the fluctuation theorem, i.e.the rate function ζ(p) for the average phase space contraction p=τ−1/integraltextτ/2 −τ/2σ(Stu)dt/τσ +will be such that ζ(−p) =ζ(p)−pσ+. If the chaotic hypothesis is valid together with the equival ence conjecture the validity of the fluctuation relation can be taken as a criterion for det erminingκ: it would be the lastκbefore which the fluctuation relation between ζ(p) andζ(−p) holds. However this conclusion can only be drawn if the attracting set in pha se space is the full ellipsoid Ξ = Ξ Rat least for K(R) =Rκ0. The latter property might not be realized: and in such case th e fluctuation theorem does not apply directly, although the equivalence conjectu res still hold. In fact one can try to extend the fluctuation theorem to cover reversible cas es in which the attracting set is smaller than the full phase space left available by the con straints. In such cases under 2/febbraio/ 2008; 1:06 14suitable geometric assumptions , [BG97] and the earlier work [BGG97], one can derive a relation like ζ(−p) =ζ(p)−pσ+ϑ, 0≤ϑ≤1 (10 .10)10.10 whereϑis a coefficient that can be related to the Lyapunov spectrum of the system, c.f.r.[BG97], [Ga97a]. In fact numerical work to check the theory p roposed in [Ga97a] is currently being performed (private communication by Rondo ni and Segre) with not too promising results which, optimistically, can be attribute d to the fact that the ultraviolet cut off is too small due to numerical litmitations: clearly th ere is more work to do here. The preliminary numerical results give, so far, the somewhat surprising linearity inpbut with a slope that, although of the correct order of magnitude , seems to have a value that does not match the theory within the error bounds. Coming back to the Navier Stokes equation we mention that we m ay imagine to write it as (10.3) butwith the different constraint U=/integraldisplay Vu2dx= const (10 .11)10.11 rather than (10.5). This case has been considered in [RS99] and the multiplier ν(u) is in this case ν(u) =/integraltext Vϕ·udx/integraltext Vu2dx, σ (u) = (3/summationdisplay |k|<K(R)|k|2−1)ν(u) (10 .12)10.12 and we can (almost) repeat the above considerations and equi valence conjectures. This constraint is a gaussian constraint that Uis constant obtained by imposing its constancy on the Euler evolution via Gauss’ principle with a suitable d efinition of the notion of “constraint effort” (this notion is not unique, see [Ga97a] f or another definition) and we do not discuss it here to avoid overlapping with §12 below. Thd intuitive motivation for the equivalence conjectures i s that for large Rthe phase space contraction σ(u) and the coefficient ν(u)7are “global quantities” and depend on the global properties of the system ( e.g.σ(u) is the sum of all the local Lyapunov exponents of the system whose number is O(K(R)3)): they will “therefore” vary over time more slowly than any time scale of the system and can be co nsidered constant. The argument is not very convincing in the case of the equatio ns with the constraint (10.11) because the σ(u) in (10.12) is proportional to/integraltext Vϕ·udxwhich clearly depends onlyon harmonics of uwithksmall, i.e.it is a “local observable”. Note that this does not apply to the GNS equations with the constrained vorticit y Ξ, (10.6) where the “main” contribution to σ(u), see (10.7), comes from the term proportional to Rwhich contains all harmonics. Therefore the result in [RS99] about the equi valence between the GNS equations, (10.4) with the constraint (10.5), and the equat ions with constraint (10.11) is interesting and puzzling: it might be an artifact of the smal lness of the cut off that one has to impose in order to have numerically feasible simulati ons. Finallyσ+(u)/σ+,i.e.essentiallyν(u)/ν+will fluctuate taking values sensibly differ- ent from their average value 1, at very rare intervals of time : but when such fluctuations will occurr one shall see “bursts” of anomalous behavior: i.e.the motion will be “ inter- mittent ” as in the case discussed in non equilibrium statistical mec hanics. 7Which in the case (10.9) are simply proportional and in the ca se of (10.4) they are related in a more involved way, see (10.8),(10.9), but which are still probab ly proportional to leading order as R→ ∞. 2/febbraio/ 2008; 1:06 1511. Entropy creation rate and entropy driven intermittency . Of course if Ris large the number of degrees of freedom is large and intermi ttency on the scale of the fluid container will not be observable due to i ts extreme unlikelyhood (expected and quantitatively predicted by the fluctuation t heorem). Therefore we look also here, in fluid motions, for a local fluctuation relation . Fluids seem particularly suitable for verifying such local fluctuation s relations because dissipation occurs homogeneously ,i.e.friction strength is translation invariant. This implies that we can regard a very small volume V0of the fluid as a system in itself (as always done in the derivation of the basic fluid equations ,e.g.see [Ga97b]) and we can expect that the phase space contraction due to such volum e elements is simply σ(u), given by (10.9) or (10.12) (“equivalently” because of our eq uivalence conjectures) with the integrals in the numerator and denominator being extend ed to the volume V0rather than to the whole box, and expressing (essentially by definit ion) the “local phase space contraction” σV0(u). Thenp=τ−1/integraltextτ/2 −τ/2σV0(Stu)//a\}bracketle{tσV0/a\}bracketri}ht+will have a rate function ζ(p) which will verify, under the same assumptions as in (10.10), a large deviation r elation as ζ(−p) =ζ(p)−p/a\}bracketle{tσV0/a\}bracketri}ht+ϑ (11.1)11.1 for someϑ: as mentioned the theoretical value of this slope ϑseems currently inaccessible to theory (as the theory proposed in [BG67], [Ga97a] may need substantial modifications, c.f.r.comment following (10.10)). The /a\}bracketle{tσV0/a\}bracketri}htandζ(p) will be proportional to V0:ζ(p) = V0ζ(p) with aV0–independent ζ(p). Note that ζ(p) depends also on R. The small volume element of the fluid will therefore be subjec t to rather frequent varia- tions: in spite of ζ(p)being proportional to V0, because now V0is not large. The conse- quent intermittency phenomena can therefore be observed. A nd as in §9 once the phase space contraction is intermittent all properties of the sys tem show the same behavior. And in fact intermittency in observations averaged ove a tim e spanτwill appear with a time frequency of the form eV0(ζ(p)−ζ(1))τ: the quantity pcan be interpreted as a measure of the “ strength of intermittency ” observable in easurements averaged over a time τ because as noted in §9 and in [Ga99b] the size of pcontrols the statistical properties of “most” other observables. Therefore the function ζ(p)(henceζ(p)) might be directly measurable and it should be rather directly related to the qu antities that one actually observes in intermittency experiments. And the difference ζ(p)−ζ(−p) can be tested for linearity in pas predicted by the analysis above. Note that in an extended system the volume Vis much larger than V0and we shall see “for sure” intermittency (for observables averaged ove r a timeτ) of strength pin a region of volume V0somewhere within a volume Wsuch that W V0eV0(ζ(p)−ζ(1))τ≃1 (11 .2)11.2 At this point it seems relevant to recall that it is rather hea tedly being debated whether the name of “ entropy creation rate ” that some authors (including the present one) give to the phase space contraction rate is justified or not, see [A n82]. The above properties not only propose the physical meaning of the quantity pand bring up the possibility of measuring its rate function ζ(p) in actual experiments but also provide a further justification of the name given to σas “entropy creation rate” and fuel for the debate thatinevitably the word entropy generates at each and every occurrence. 2/febbraio/ 2008; 1:06 16§12. Benard convection, intermittency and the Ciliberto–La roche experi- ment. A very interesting attempt at checking some of the above idea s has been made recently by Cilberto and Laroche in an experiment on real fluids which h as been performed with the aim of testing the relation (11.1) locally in a small volu me element, [CL98]. By “real” we mean here non numerical : a distinction that, however, has faded away together with the XX–th century but that some still cherish: the system is p hysically macroscopic (water in a container of a size of the order of a liter). This being a real experiment one has to stretch quite a bit the very primitive theory developed so far in order to interpret it and one has to add to t he chaotic hypothesis other assumptions that have been discussed in [BG97], [Ga97 a] in order to obtain the fluctuation relation (10.12) and its local couterpart (11.1 ). The experiment attempts at measuring a quantity that is even tually interpreted as the differenceζ(p)−ζ(−p), by observing the fluctuations of the product ϑuzwhereϑis the deviation of the temperature from the average temperature in a small volume element ∆ of water at a fixed position in a Couette flow and uzis the velocity in the zdirection of the water in the same volume element. The result of the experiment is in a way quite unexpected: it i s found that the function ζ(p) is rather irregular and lacking symmetry around p= 1:nevertheless the function ζ(p)−ζ(−p)seems to be strikingly linear . As discussed in [Ga97a], predicting the slope of the entropy creation rate would be difficult but if the equival ence conjecture considered above and discussed more in detail in [Ga97a] is correct then we should expect linearity ofζ(p)−ζ(−p). In the experiment of [CL98] the quantity ϑuzdid not appear to be the divergence of the phase space volume simply because there was no model prop osed for a theory of the experiment. Nevertheless Ciliberto–Laroche select the qu antity/integraltext ∆ϑuzdxon the basis of considerations on entropy and dissipation so that there i s hope that in a model of the flow this quantity can be related to the entropy creation rate discussed in §10,§11. Here we propose that a model for the fluid, that can be reasonab ly used, is Rayleigh’s model of convection, [Lo63], [LL71] and [Ga97b] sec. 5. An at tempt for a theory of the experiment could be the following. One supposes that the equations of motion of the system in the whole container (of linear size of the order of 30 cm) are written for the quantities t,x,z,ϑ,u in terms of the heightHof the container (assumed to be a horizontal infinite layer), of the temperature difference between top and bottom δTand in terms of the phenomenological “ friction constants ”ν,χof viscosity, dynamical thermal conductivity and of the the rmodynamic dilatation coefficient α. We suppose that the fluid is 3–dimensional but stratified, so that velocity and temperature fields do not depend on the coor dinatey, and gravity is directed along the z–axis:g=g e, e= (0,0,−1). The temperature deviation ϑis defined as the difference betwen the temperature T(x,y,z ) and the temperature that the fluid would have at height zin absence of convection, i.e.T0−zδT/H ifT0is the bottom temperature. In such conditions the equations, including the boundary co nditions (of fixed temper- ature at top and bottom and zero normal velocity at top and bot tom), the convection equations in the Rayleigh model, see [Lo63] eq. (17), (18) wh ere they are called the Saltzman equations , and [Ga97b] §1.5, become ∂·u= 0,/integraldisplay uxdx=/integraldisplay uydx= 0 ˙u+u /tildewide·∂ /tildewideu=ν∆u−αϑg−∂p′, (12.1)12.1 ˙ϑ+u /tildewide·∂ /tildewideϑ=χ∆ϑ+δT Huzϑ(0) = 0 =ϑ(H), u z(0) = 0 =uz(H), 2/febbraio/ 2008; 1:06 17The function p′is related to the pressure p: within the approximations it is p=p0− ρ0gz+p′. We shall impose for simplicity horizontal periodic bounda ry conditions in x,y so that the fluid can be considered in a finite container Vof sideafor somea>0 prefixed (which in theoriginal variable would correspond to a contai ner of horizntal size aH). It is useful to define the following adimensional quantities τ=tνH−2, ξ=xH−1, η=yH−1, ζ=zH−1, ϑ0=αϑ αδT, u0= (/radicalbig gHαδT )−1u R2=gH3αδT ν2, R Pr=ν κ(12.2)12.2 and one checks that the Rayleigh equations take the form ˙u+Ru /tildewide·∂ /tildewideu= ∆u−Rϑe−∂p, ˙ϑ+Ru /tildewide·∂ /tildewideϑ=R−1 Pr∆ϑ+Ruz, ∂·u= 0 uz(0) =uz(1) = 0, ϑ (0) =ϑ(1) = 0,/integraldisplay Vuxdx=/integraldisplay Vuydx= 0(12.3)12.3 where we again call t,x,y,z,u,ϑthe adimensional coordinates τ,ξ,η,ζ, u0,ϑ0in (12.2). The numbers R,R Prare respectively called the Reynolds and Prandtl numbers of the problem:RPr=∼6.7 for water while Ris a parameter that we can adjust, to some extent, from 0 up to a rather large value. According to the principle of equivalence stated in [Ga97a] here one could impose the constraints /integraldisplay V/parenleftbig u2+1 RPrϑ2/parenrightbig dx=C (12.4)12.4 on the “ frictionless equations ” (i.e.the ones without the terms with the laplacians) and determine the necessary forces via Gauss’ principle of mini mal effort, see footnote3 and [Ga96a], [Ga97a]. We use as effort functional of an acceleration field aand of a temperature variation field sthe quantity E(a,s)def=/parenleftbig (a+∂p−f),(−∆)−1(a+∂p−f)/parenrightbig + (12 .5)12.5 +/parenleftbig (s−ϕ),(−∆)−1(s−ϕ)/parenrightbig with fdef=−Rϑe, ϕdef=Ruz and require it to be minimal over the variations δ(x) ofa=d u dtandτ(x) ofs=dϑ dt with the constraints that for all xit is∂·δ= 0, besides those due to the boundary conditions. The result is ∂·u= 0 ˙u+Ru /tildewide·∂ /tildewideu=Rϑe −∂p′+τth ˙ϑ+Ru /tildewide·∂ /tildewideϑ=Ruz+λth ϑ(0) = 0 =ϑ(H),/integraldisplay Vuxdx=/integraldisplay Vuydx= 0(12.6)12.6 2/febbraio/ 2008; 1:06 18where the frictionless equations are modified by the thermostats forces τth,λth: the latter impose the nonholonomic constraint in (12.4) with th e effort functional defined by (12.5). Looking only at the bulk terms we see that the equatio ns obtained by imposing the con1straints via Gauss’ principle become the (12.3) wit h coefficients in front of the Laplace operators equal to νG,νGR−1 Pr, respectively, with the “gaussian multiplier” νG being an oddfunctions of u, see [Ga97a]: setting ˜CV(u,ϑ) =/integraltext V/parenleftbig (∂ /tildewideu)2+R−1 Pr(∂ϑ)2/parenrightbig dx one finds νG=˜CV(u,ϑ)−1R(1 +R−1 Pr)/integraldisplay Vuzϑ dx (12.7)12.7 And the equations become, finally ∂·u= 0 ˙u+Ru /tildewide·∂ /tildewideu=Rϑe −∂p′+νG∆u ˙ϑ+Ru /tildewide·∂ /tildewideϑ=Ruz+νG1 RPr∆ϑ ϑ(0) = 0 =ϑ(H),/integraldisplay uxdx=/integraldisplay uydx= 0(12.8)12.8 If one wants the equivalence between the ensembles of SRB dis tributions for the equation (12.8) and for (12.3) one has to tune, [Ga97a], the value of th e constantCin (12.4) so that the time average value /a\}bracketle{tνG/a\}bracketri}ht+ofνGis precisely the physical one: namely /a\}bracketle{tνG/a\}bracketri}ht= 1 by (12.3). This is (again) the same, in spirit, as fixing the te mperature in the canonical ensemble so that it agrees with the microcanonical temperat ure thus implying that the two ensembles give the same averages to the local observable s. The equations (12.8) are time reversible (unlike the (12.3) ) under the time reversal map: (u,ϑ) = (−u,ϑ) (12 .9)12.9 and they should be supposed, by the arguments in [Ga97a] and §10,11: “equivalent” to the irreversible ones (12.3), The (12.8) should therefore have a “divergence” σ(u,ϑ) whose fluctuation function ζ(p) verifies a linear fluctuation relation, i.e.ζ(p)−ζ(−p) should be linear in p. Note that the divergence of the above equations is proportional to νGif one supposes that the high momenta modes with |k|> K(R) =Rκwithκsuitable can be set equal to 0 so that the equation (12.8) becomes a system of finite differenti al equations for the Fourier components of u,ϑ. For instance the Lorenz’ equations, [Lo63] see also §17 of [Ga97b], reduced the number of Fourier components necessary to describe (12.3) to just thr ee components, thus turning it into a system of three differential equations. Proceeding in this way the divergence of the equations of mot ion can be computed as a sum of two integrals one of which proportional to νGin (12.7). If instead of integrating over the whole sample we integrate over a small region ∆, like in the experiment of [CL98], we can expect to see a fluctuation relation for the ent ropy creation rate if the fluctuation theorem holds locally ,i.e.for the entropy creation in a small region. As for the cases in §11 this is certainly not implied by the proof in [GC95]: howev er when the dissipation is homogeneous through the system , as it is the case in the Rayleigh model there is hope that the fluctuation relation holds local ly because “a small subsystem should be equivalent to a large one”. As noted in §9 the actual possibility of a local fluctuation theorem in systems with homogeneous dissipatio n has been shown in [Ga99c], after having been found through numerical simulations in [G P99], and this example was relevant because it gave us some justification to imagine tha t it might apply to the present situation as well. 2/febbraio/ 2008; 1:06 19The entropy creation is due to the term R/integraltext ∆uzϑdx/˜C∆(u,ϑ), where ∆ is the region where the measurements of [CL98] are performed, hence we hav e a proposal for the explanation of the remarkable experimental result. Unfort unately in the experiment [CL98] the contributions not explicitly proportional to Rto the entropy creation rates have not been measured nor has been the ˆCin (12.7) which also fluctuates (or might fluctuate). In any event they might be measurable by improvin g the same apparatus, so that one can check whether the above attempt to an explanatio n of the experiment is correct, or try to find out more about the theory in case it is no t right. If correct the above “theory” the experiment in [CL98] would be quite impor tant for the status of the chaotic hypothesis. §13. Conclusions. The chaotic hypothesis promises a point of view on non equili brium that has proved so far of some interest. Here we have exposed the basic ideas and attempted at drawing some consequences: admittedly the most interesting rely on rather phenomenological and heuristic grounds. They are summarized below. (1) The definition of nonequilibrium ensembles with the prop osal that out of equilibrium also the equation of motion should be considered as part of th e definition of ensemble. This is take into account that while in equilibrium the syste m is uniquely defined by its microscopic forces and constituents in non equilibrium it is not so . Systems must be put in contact with thermostats if we want them to become stat ionary after a transient time. And (for large systems) there may be several equivalen t ways of taking heat out of a system, i.e.several thermostats, without affecting the properties of st ationary state that is eventually reached by the system itself. (2) Equivalence of ensembles has the most striking aspect th at systems which evolve with equations that are very different may exhibit the same statis tical properties. In particular reversible evolutions might be equivalent to non reversibl e ones, thus making it possible to apply results that require reversibility, in particular the fluctuation relations, to cases in which it is not valid. (3) An interpretation of the quantity pthat intervenes in the fluctuation theorems in terms of an intermittency phenomenon and as a further quanti tative measure of it. (4) The possibility of applying the theory to strongly turbu lent motions was the origin of the Ruelle’s principle that evolved into the chaotic hypoth esis: therefore not surprisingly the ideas can be applied to fluid dynamics. We have discussed a possible approach. The approach leads again to a proposal for the theory of certain i ntermittency phenomena which appear quantitatively related to entropy creation flu ctuations. (5) The possibility of measurement of the rate ζ(p) leads to a a possible prediction of the spatial frequency of internittent events of strength por, as I prefer, with entropy creation ratep(see (4) above, (11.2) and §12). This seems testable in concrete experiments (both real and numerical). (6) We have used the results in (2)%(8) to hint at an interpret ation of the experiment by Ciliberto and Laroche on Benard convection in water. Although the theory is still at its beginning and it migh turn out to be not really of interest it seems that at this moment it is worth trying to tes t it both in its safest, c.f.r. §2%§8, and in its most daring, c.f.r. §9%§12, predictions. Acknowledgements: I am greatly indebted to Professor R. Newton for giving me the opportunity to collect the above thoughts which, collected together, go far beyond what I originally planned after he proposed to me to write this revi ew. Work partially supported by Rutgers University and by MPI through grant # ??????????. 2/febbraio/ 2008; 1:06 20References. [An82] Andrej, L.: The rate of entropy change in non–Hamiltonian systems , Physics Letters, 111A , 45–46, 1982. And Ideal gas in empty space , Nuovo Cimento, B69, 136– 144, 1982. 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H.A.: Resolution of Loschmidt’s para- dox: the origin of irreversible behavior in reversible atom istic dynamics , Physical Review Letters, 59, 10–13, 1987. [Lo63] Lorenz, E.: Deterministic non periodic flow , J. of the Athmospheric Sciences, 20, 130- 141, 1963. [LL71] Landau, L., Lifchitz, E.: M´ ecanique des fluides , MIR, Moscou, 1971. [RS99] Rondoni, L., Segre, E.: Fluctuations in two dimensional reversibly damped turbu- lence, Nonlinearity, 12, 1471–1487, 1999. [Ru76] Ruelle, D.: A measure associated with Axiom A attractors , American Journal of Mathematics, 98, 619–654, 1976. [Ru78] Ruelle, D.: Sensitive dependence on initial conditions and turbulent b ehavior of dynamical systems , Annals of the New York Academy of Sciences, 356, 408–416, 1978. This is the first place where the hypothesis analogous to the l ater chaotic hypothesis was formulated (for fluids): however the idea was exposed orally at least since the talks given to illustrate the technical work [Ru76], which appeared as a preprint and was submitted for publication in 1973 but was in print three years later. [Ru96] Ruelle, D.: Positivity of entropy production in nonequilibrium statis tical mechan- ics, Journal of Statistical Physics, 85, 1–25, 1996. And Ruelle, D.: Entropy production in nonequilibrium statistical mechanics , Communications in Mathematical Physics, 189, 365–371, 1997. [Ru99a] Ruelle, D.: Smooth dynamics and new theoretical ideas in non-equilibri um sta- tistical mechanics , Journal of Statistical Physics, 95, 393–468, 1999. [Ru99b] Ruelle, D.: A remark on the equivalence of isokinetic and isoenergetic t her- mostats in the thermodynamic limit , preprint IHES 1999, to appear in Journal of Statis- tical Physics. [Se87] Seitz, F.: The modern theory of solids , Dover, 1987 (reprint). Internet: Authors’ preprints downloadable (latest versio n) at: http://ipparco.roma1.infn.it (link)http://www.math.rutgers.edu/ ∼giovanni e-mail: giovanni.gallavotti@roma1.infn.it 2/febbraio/ 2008; 1:06 22

arXiv:physics/0001072v1 [physics.ao-ph] 31 Jan 2000Monodisperse approximation in the metastable phase decay V.Kurasov Victor.Kurasov@pobox.spbu.ru Abstract A new simple method for the first order phase transition kinet - ics is suggested. The metastable phase consumption can be im ag- ined in frames of the modisperse approximation for the distr ibution of the droplets sizes. In all situations of the metastable ph ase decay this approximation leads to negligible errors in the total n umber of droplets appeared in the system. An evident advantage of the pre- sented method is the possibility to investigate the situati on of the metastable phase decay on several sorts of heterogeneous ce nters. Attempts to give a theoretical description for the first orde r phase tran- sition appeared practically simultaneously with creation of the classical nu- cleation theory [1]. An idea to give the global description o f the first order phase transition was attractive and induced many publicati ons (for example, see Wakeshima [2], Segal’ [3]). But all of them were intended only to estimate the main characteristics of the phase transition. The time o f the cut-off of the nucleation clearly observed in experiments was adopted in these publica- tions without any proper justification. The first theoretica l description with explicit determination of time behavior of supersaturatio n was given in [4] where a homogeneous condensation was investigated. The met hod used in [4] was an iteration solution of integral equation of a subst ance balance. To give a global picture of phase transition one has to take in to account a presence of active centers of condensation. The iteration method can be spread on heterogeneous condensation on similar centers (s ee [5]), but for con- densation on several types of centers one can not calculate i terations with 1a proper accuracy (see also [5]). The system of condensation equations be- comes so complex that it can not be directly solved without si mplifications. As the result it would be rather attractive to suggest some si mple approxi- mations which can be used in the complex situation of the cond ensation on several types of centers. Certainly, this approximation ha s to be well based. Here we shall suggest a monodisperse approximation of the dr oplets size distribution to calculate the number of molecules in the liq uid phase. From the first point of view this approximation is strange - we have already at- tracted attention to the necessity to know the behavior of a s upersaturation which determines the form of the droplets size spectrum. But here we are going to show that with an appropriate choice of the cut-off (w hich can be also justified) one can give an adequate description of a nucl eation period. The monodisperse approximation presented here differs from the total monodisperse approximation used in [7] for description of t he intensive con- sumption of the metastable phase and can not be applied durin g a nucleation period. Here we use a special modification of the mentioned ap proximation which allows to describe the period of nucleation. This publication is intended to give the main idea of the mono disperse approximation which would be clear for experimenters. That ’s why we start from situations of homogeneous condensation and heterogen eous condensa- tion on similar centers which can be solved even without this approximation. Some technical details are excluded to give the most clear ve rsion (for exam- ple, a complete version of limit situations and monodispers e approximation in the intermediate situation is described in [6], the overl apping of the inter- mediate and limit situations is analysed in [8], the transit ion of the special monodisperse approximation to the total one is discussed in [9]). We use the physical model of nucleation kinetics described i n [5]. It is rather standard but to avoid misunderstanding we shall cons ider [5] as the base of references. 1 Homogeneous nucleation The condensation kinetics equation for the number Gof the molecules in the liquid phase can be written in a well known form [4], [5] G(z) =f/integraldisplayz 0dx(z−x)3exp(−ΓG(x)) (1) 2where parameter fis the amplitude value of the droplets sizes distribution Γ is some fixed positive parameter. One can analyse behavior of subintegral function gdefined by G(z) =/integraldisplayz 0g(z, x)dx (2) as a function of a size ρ=z−x,z. This function has the sense of the distribution of the number of molecules in droplets over the ir sizes ρ. In some ”moment” z( ort(z)) it can be presented in the following form •When ρ > z it is equal to zero (there are no droplets with such a big size) •When ρ <0 it is also equal to zero (there aren’t droplets with a negati ve size) •At the intermediate ρit grows rather quickly with a growth of ρ. It is easy to note that it grows faster than ρ3grows. Really, if one takes into account that supersaturation decr eases in time then we get g∼ρ3. But supersaturation falls in time and there aren’t so many droplets of the small size as of the big size. As the result one can see that the function gas the function of ρhas the sharp peak near ρ≈z. This property takes place under the arbitrary z(or t(z)). The sharp peak of gallows to use for gthe monodisperse approximation - a representation in the δ-like form with a coordinate corresponding to a position of the peak of function g, i.e. g∼δ(ρ=z) As the result one can state that the monodisperse approximat ion is based now. But it is necessary to determine the number of droplets i n this approx- imation. It would be wrong to include the total number of already appea red droplets in this peak. Really, in the spectrum of sizes there are many droplets with small sizes. One can not describe these droplets as cont aining the same substance as the droplets of a big size. It would be more corre ct to exclude 3them from the substance balance. So, it is necessary to cut off the small droplets. It can be done according to two recipes. The first recipe is the differential one. One can note that duri ng all times which don’t exceed the time of nucleation essentially the fu nction gnear maximum is close to gappr=fρ3 This approximation corresponds to the constant value of sup ersaturation. One can cut off this approximation at a half of amplitude value (i.e. at a levelfz3/2). Then one can get for the width ∆ diffzthe following expression ∆diffz= (1−2−1/3)z This cut off means that all droplets ρ < z−∆diffzare excluded from consideration and all droplets with ρ > z−∆diffzare taken into account in aδ-like peak. The second recipe is the integral one. One can integrate gapprand require that /integraldisplayz 0gappr(z, x)dx=Nz3 An integration gives /integraldisplayz 0gappr(z, x)dx=f/integraldisplayz 0(z−x)3dx=fz4 4 The width of spectrum is defined from condition that the numbe r of droplets has to be equal to the amplitude multiplied by the width of spe ctrum ∆ intz: N=f∆intz This gives the following expression ∆intz=z/4 One can see that ∆ diffzand ∆ intzpractically coincide. This shows the high selfconsistency of this approximation. The second rec ipe will be more convenient for concrete calculations. In fig.1 one can see the application of the monodisperse appro ximation in the homogeneous case. 4As the result one can say that all parameters of approximatio n are defined. Now it will be used to solve (1). Instead of (1) one can get G(z) =N(z/4)z3 where N(z/4) =fz/4 is the number of droplets formed until t(z/4). This leads to G(z) =fz4/4 which coincides with the resulting iteration in the iterati on method [4], [5]. It is known (see [4], [5]) that this expression is very accura te which shows the effectiveness of the monodisperse approximation. Here the c ut off of the tail of the sizes spectrum compensates the unsymmetry of the init ial spectrum. The main result of the nucleation process is the total number of the droplets which can be found as Ntot=f/integraldisplay∞ 0dxexp(−ΓG(x))dx or Ntot=f/integraldisplay∞ 0dxexp(−fΓz4/4) =f3/4Γ−1/4D where D=/integraldisplay∞ 0exp(−x4/4)dx= 1.28 The error of this expression is less than two percents (it is t he same as in the iteration method). 2 Heterogeneous condensation on similar cen- ters The condensation equations system can be written in the foll owing form [5] G(z) =f/integraldisplayz 0dx(z−x)3exp(−ΓG(x))θ(x) 5θ(z) = exp( −b/integraldisplayz 0exp(−ΓG(x))dx) with positive parameters f,b, Γ. An appearance of a new function θwhich is a relative number of free heterogeneous centers requires th e second equation. The first equation of the system is rather analogous to the hom ogeneous case. The subintegral function here is also sharp. A functio nθis a decreasing function of time according to the second equation of the syst em. Then the function gwhich is again determined by (2) is more sharp than in the homo - geneous case. As far as the supersaturation has to fall one ca n see that gis more sharp than gappr. It allows to use here the monodisperse approximation for all zort(z). As the result the monodisperse approximation is based for he terogeneous condensation. One needs here only the sharp peak of g(ρ) which can be easily seen. The successive application of the monodisperse approximat ion in the ho- mogeneous case shows that all droplets necessary for a metas table phase consumption at t(z) were formed until t(z/4). In the heterogeneous case the exhaustion of heterogeneous centers increases in time. So, all essential at t(z) droplets were formed before t(z/4). At the same time the presence of a long tail in the situation of a weak exhaustion of heterogeneous centers requires to cut off the s pectrum for the monodisperse approximation. As far as the long tail is essen tial in the situa- tion of a weak exhaustion one has to cut off the spectrum by the s ame recipe as in the situation of the homogeneous condensation: one has to exclude all droplets formed after z/4 which have the sizes ρ < z−∆intz=z−z/4. One can see in fig. 2 the monodisperse approximation in the sit uation of the heterogeneous condensation on similar centers. The for m of the spectrum in this situation is illustrated in fig. 3. So, the way to const ruct approxima- tion is known. Now one can turn to concrete calculations. The number of the droplets formed until t(z/4) has to be calculated as N(z/4) =f b(1−θ(z/4)) An approximation for Ghas the form G(z) =f b(1−θ(z/4))z3 6The total number of droplets can be determined as Ntot=f b(1−θ(∞)) or Ntot=f b(1−exp(−b/integraldisplay∞ 0exp(−ΓG(x))dx)) or Ntot=f b(1−exp(−b/integraldisplay∞ 0exp(−Γf b(1−θ(z/4))z3)dz)) or Ntot=f b(1−exp(−b/integraldisplay∞ 0exp(−Γf b(1−exp(−b/integraldisplayz/4 0exp(−ΓG(x))dx))z3)dz)) The last expression has a rather complicate form. It contain s several itera- tions in a hidden form which ensures the high accuracy. The last expression can be simplified. One of the possible rec ipes is the following. One can note that an expression for Gis necessary at Γ G∼1. Then zattains some values ∆ ζz. But until ∆ ζz/4 the value Γ Gis small and exp(ΓG(z)) is close to unity. This leads to simplification of last expr ession which can be written in the following form Ntot=f b(1−exp(−b/integraldisplay∞ 0exp(−Γf b(1−exp(−bz/4))z3)dz)) Then one can fulfil calculation according to the last formula . The relative error is less than two percents. Here it is a little bit greate r than in the homogeneous case because the form of initial spectrum is cha nged and there is no full compensation of the unsymmetry of spectrum and an e xclusion of the tail. The relative error in the situation of heterogeneo us condensation on similar centers is drawn in fig. 4. Now one can turn to explicit calculation of the integral in th e last expres- sion. After the appropriate renormalization the subintegr al function is more sharp than exp( −x3) and more smooth than exp( −x4). Both these functions have a sharp back front of spectrum. It allows to introduce th e characteristic scale ∆ zby equation f bΓ(1−exp(−b(∆z)/4))(∆z)3≈1 7Then/integraldisplay∞ 0exp(−Γf b(1−exp(−bz/4))z3)dz= ∆zA+B 2 where A=/integraldisplay∞ 0exp(−x3)dx= 0.89 B=/integraldisplay∞ 0exp(−x4)dx= 0.90 Now the calculation is reduced to some algebraic manipulati ons. The error of the last approximation is less than one percent. As the result Ntot=f b(1−exp(−b∆zA+B 2)) One can note that it is possible to formulate the recipe alrea dy in terms of ∆z. The long way is adopted here to give the most clear picture fo r the monodisperse approximation. 3 Nucleation on several types of heteroge- neous centers The main advantage of monodisperse approximation is the pos sibility to use it for the condensation on the several types of centers. The i teration proce- dure can not be applied in this case successfully. The result of calculations according to [5] shows this fact explicitly. The reason is th e existence of the cross influence of the different types of centers through vapo r consumption. In the condensation on similar heterogeneous centers in the situation of exhaustion the influence of this phenomena on the vapor consu mption isn’t important because in the situation of consumption the conve rging force of the heterogeneous centers exhaustion is extremely high. But in the situation with two types of heterogeneous centers the exhaustion of the firs t type centers can have a certain influence on a vapor consumption but the exh austion of the second type centers is weak and there is no converging for ce due to the weak exhaustion of the second type centers. This effect in very thin and it can not be taken into account in t he second iteration. But one can not calculate the third iteration ana lytically and this stops an application of iterations. Really, this phenomena isn’t evident from 8the first point of view but it exits and leads to the error of the second iteration in many times. The application of the monodisperse approximation is based on the sharp- ness of function g. This property takes place already in this situation. So, there are no objections to apply the monodisperse approxima tion here. Here we shall reproduce the same formulas but with the lower i ndexes which determine the sort of heterogeneous centers. The system of condensation equations can be written in the fo llowing form [6] Gi(z) =fi/integraldisplayz 0dx(z−x)3exp(−Γ/summationdisplay jGj(x))θi(x) θi(z) = exp( −bi/integraldisplayz 0exp(−Γ/summationdisplay jGj(x))dx) where the lower indexes denote the sorts of centers. This sys tem can be seen by the direct generalization of the one type case. The subintegral function in the substance balance equation s is also sharp. As far as all θiare the decreasing functions of arguments then the function g defined by (2) (with proper indexes) is sharper than without t he exhaustion of heterogeneous centers. So, due to the supersaturationde creasing gis more sharp than gappr. It allows here to use the monodisperse approximation for allzort(z). As the result one can see that the monodisperse approximatio n in this case is justified on the base of the sharpness of g(ρ). The same properties as in the previous case can be also seen he re. One has to cut off the spectrum at z/4. Here all justifications are absolutely same as in the previous section. The characteristic situation fo r the nucleation on two types of heterogeneous centers is drawn in fig.5. As the re sult the way to construct the monodisperse approximation is known. Now one can present calculations. The number of the droplets formed until t(z/4) on the centers of sort i has to be calculated as Ni(z/4) =fi bi(1−θi(z/4)) An approximation for Gican be now presented as Gi(z) =fi bi(1−θi(z/4))z3 9The total number of droplets is defined as Ni tot=fi bi(1−θi(∞)) or Ni tot=fi bi(1−exp(−bi/integraldisplay∞ 0exp(−Γ/summationdisplay jGj(x))dx)) or Ni tot=fi bi(1−exp(−bi/integraldisplay∞ 0exp(−Γ/summationdisplay jfj bj(1−θj(z/4))z3)dz)) or Ni tot=fi bi(1−exp(−bi/integraldisplay∞ 0exp(−Γ /summationdisplay jfj bj(1−exp(−bj/integraldisplayz/4 0exp(−Γ/summationdisplay kGk(x))dx))z3)dz)) Now one can simplify the last expression by the same way as in t he one type case. Expressions for Giare essential at Γ/summationtext jGj∼1. Then zis near ∆ ζz. Until ∆ ζz/4 the value Γ/summationtext jGjis small and exp(Γ/summationtext jGj(z)) is near unity. It leads to Ni tot=fi bi(1−exp(−bi/integraldisplay∞ 0exp(−Γ/summationdisplay jfj bj(1−exp(−bjz/4))z3)dz)) Now one can fulfil the calculations according the explicit fo rmula. The relative error of the last expression is less than five percen ts (here it increases slightly due to the complex form of the spectrums on different sorts. The relative error in the number of droplets is drawn in fig. 6. The calculation of the last integral is absolutely analogous to the previous se ction. The subin- tegral function after renormalization lies between exp( −x3) and exp( −x4). It allows to get the characteristic size ∆ zfrom Γ/summationdisplay jfj bj(1−exp(−bj(∆z)/4))(∆z)3≈1 Then /integraldisplay∞ 0exp(−Γ/summationdisplay jfj bj(1−exp(−bjz/4))z3)dz= ∆zA+B 2 10The relative error of the last expression is less than one per cent. As the result Ni tot=fi bi(1−exp(−bi∆zA+B 2)) The formula is similar to the final expression in the previous section. But parameters in the last formula have to be determined in anoth er manner. The physical sense of the last expression is the separate exh austion of heterogeneous centers. One sort of centers can influence on t he other sort only through a vapor consumption. This fact can be seen also i n the initial precise system of the condensation equations. References [1] Frenkel, J., Kinetic theory of liquids, Oxford Universi ty Press, New York, 1977 [2] Wakeshima H., Time lag in self nucleation, J.Chem. Phys. , 1954, v.22. N.9, p.1614-1615 [3] Segal’ R.B., The Journal of experimental and theoretica l physics (USSR), Vol. 21, Issue 7, p. 814 (1951) [4] F.M. Kuni, A.P.Grinin, A.S. Kabanov, Kolloidn. journ. ( USSR), v. 46, p. 440 (1984) [5] Kuni F.M., NOvojilova T. Yu., Terent’iev I.A. Teoretica l and matemat- ical physics, (USSR) v.60, p 276 [6] Kurasov V.B., Deponed in VINITI 2594B95 from 19.09.1995 , 28 p. [7] Kuni F.M., Kolloidn. journ. (USSR) vol. 46, p.674, p.902 , p.1120 (1984) [8] Kurasov V.B., Preprint cond-mat@xxx.lanl.gov get 0001 091 [9] Kurasov V.B., Preprint cond-mat@xxx.lanl.gov get 0001 104, 0001108, 0001112 [10] Kurasov V.B., Preprint cond-mat@xxx.lanl.gov get 000 1119 11[11] Kurasov V.B. Universality in kinetics of the first order phase transitions, SPb, 1997, 400 p. Kurasov V.B., Developement of the universality concep[tio n in the first order phase transitions, SPb, 1998, 125 p. 12................................................. .... ... .. .. .. .. .. . .. . . .. . . .. . . . . . . .. . . . . . . . . . . . . . . . .✻ ✲zfz3 ρg A............................................... ..... ... ... .. .. .. . .. . .. . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . .✻ ✲ Bzfz3 ρg ................................................. ... .. .. . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .✻ ✲zfz3 ρg C................................... . .. . . . . . . . . . . . . . . . . . . .✻ ✲zfz3 ρg D Fig.1 Monodisperse approximation in homogeneous condensation. Here one can see four pictures for different periods of time (or for differe nt values of z. One can introduce ∆zaccording to ΓG(∆z) = 1 and it will be the characteristic scale of the supersaturation fall. In part ”A” z= ∆z/2, in part ”B” z= ∆z, in part ”C” z= 3∆z/2, in part ”D” z= 2∆z. One can see that the spectrums in part ”A” and part ”B” are practically the same. It correspo nds to the property of the similarity of spectrums until the end of the n ucleation period. 13.............................................. ... ... .. .. .. . .. . .. . . .. . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . .✻ ✲zfz3 ρg A......................................... .... .. .. .. .. .. . .. . . .. . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .✻ ✲ Bzfz3 ρg ...................................... .. .. .. .. . .. . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .✻ ✲zfz3 ρg C............................. .. .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .✻ ✲zfz3 ρg D Fig.2 Monodisperse approximation in condensation on the similar centers. The value ∆zis the same as in the previous figure (i.e. it is determined wit hout the exhaustion of centers). Now ΓG(∆z)<1and∆zwill be the character- istic scale of the supersaturation fall in the situation wit hout exhaustion. In part ”A” z= ∆z/2, in part ”B” z= ∆z, in part ”C” z= 3∆z/2, in part ”D”z= 2∆z. One can see that the spectrums in part ”A” and part ”B” aren’t similar. Now all spectrums are more sharp than in the h omogeneous case. 14.................................................................... ............ ......... ....... ....... ..... ..... ..... .... .... ..... ... .... .... ... ... ... ... ... ... ... ... .. ... .. ... .. ... .. .. .. .. ... .. .. .. .. . .. .. .. .. .. . .. .. .. . .. . .. .. . .. . .. . .. . .. . . .. . .. . . .. . . .. . . . .. . . .. . . . . .. . . . . .. . . . . . .. . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . .. . . . .. . . .. . . .. . .. . .. . .. .. . .. .. .. .. .. .. .. ... ... .. .... .... ..... ............................................................................................................ .......... .... .... ... ... ... .. .. .. .. .. .. .. . .. .. . .. . .. . . ..................................... ............. ....... ..... .... ... ...... .... ....... ... .. ..... ... .. ... .. . .. .. ... . .. .. .. . .. . . ... . .. .. . . . .. . . .. . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. .. . . .... . . .. . .. .. .. ... .. .................................................................... ... ... .. .. .. ... . .. ... . . .✻ ✲zf ρdistributions Fig.3 One can see two curves which are going from ρ=zto the small sizes ρ. The lower curve corresponds to the real spectrum calculate d with account of heterogeneous centers exhaustion. The upper curve corre sponds to the condensation without exhaustion of heterogeneous centers which is the worst situation where there is converging force due to the centers exhaustion. Concrete situation drawn here corresponds to b= 2after renormalization (the values of parameters fandΓcan be canceled by appropriate renormal- ization). The value of zhere equals to 3∆z/2. The solid lines correspond to the precise numerical solutio n. The dashed lines correspond to application of monodisperse approxima tion. One can not separate the numerical solutions from the approximate ones except the slight deviation in the region of small ρ. As far as all (precise and approximate) solutions will go to zero there will be no deviations for z≫∆z(i.e. we stop at the worst moment). 15.................................. ... . .. . .. ................................................................................................... ........✻ ✲10.01 berror Fig.4 The relative error of approximate solution for the nucleati on on the sim- ilar heterogeneous centers. Here the values fandΓcan be canceled after renormalization and there remains only one parameter bwhich is the argu- ment of the function drawn here. It is clear that this functio n is small. All asymptotes can be checked analytically (see [9]). 16. . ... ....... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ............................................................................................ ... ... ... ... ... .. ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ............................................ .. ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .✻ ✲zf ρdistributions Fig.5 Characteristic behavior of size spectrums for nucleation o n two types of heterogeneous centers. One can cancel f1,Γby renormalization. One can putf2<1due to the choice of centers. Here b1= 2,f2= 1/2,b2= 1/2. The value zis taken as 2∆z(see fig.1). The are three curves here. The lower one corresponds to the spectrum of droplets formed on t he first type centers, the intermediate one corresponds to the droplets s ize spectrum for the second type centers, the upper one corresponds to the spe ctrum calculated without exhaustion of heterogeneous centers (the reasons a re the same as in fig.3). The solid lines are the numerical solutions, the dash ed lines are the approximate solutions. One can not see the difference for the lower curve. For the upper and intermediate curve one can see only very sli ght difference. All spectrums are renormalized to have one and the same ampli tude (which is marked by f). 17..................................................................................... ..................... . .................... . .................... . . ................... . . ................... . . . .................. . . . .................. . . . .................. . . ................... . . . .................. . . . .................

arXiv:physics/0001073v1 [physics.acc-ph] 31 Jan 2000SLAC–PUB–8358 January 2000 Formation of Patterns in Intense Hadron Beams. The Amplitude Equation Approach∗ Stephan I. Tzenov Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Abstract We study the longitudinal motion of beam particles under the action of a single resonator wave induced by the beam itself. Based on the metho d of multiple scales we derive a system of coupled amplitude equations for the slowly varying part of the longitudinal distribution function and for the r esonator wave envelope, corresponding to an arbitrary wave number. The equation gov erning the slow evolution of the voltage envelope is shown to be of Ginzburg– Landau type. Paper presented at: Second ICFA Advanced Accelerator Workshop on THE PHYSICS OF HIGH BRIGHTNESS BEAMS UCLA Faculty Center, Los Angeles November 9–12, 1999 ∗Work supported by Department of Energy contract DE–AC03–76 SF00515.1 Introduction So far, extensive work has been performed on the linear stabi lity analysis of collective motion in particle accelerators [1]. Nonlinear theories [2]–[7] o f wave interaction and formation of patterns and coherent structures in intense beams are howev er less prevalent, in part, due to the mathematical complexity of the subject, but also beca use of the commonly spread opinion that highly nonlinear regime is associated with poo r machine performance that is best to be avoided. Nevertheless, nonlinear wave interaction is a well observe d phenomenon [2], [8] in present machines, complete and self-consistent theory explaining the processes, leading to the forma- tion of self-organized structures mentioned above is far fr om being established. The present paper is aimed as an attempt in this direction. The problem addressed here (perhaps, the simplest one) is th e evolution of a beam in longitudinal direction under the influence of a resonator vo ltage induced by the beam itself. Linear theory is obviously unable to explain bunch (droplet ) formation and bunch breakoff (especially in the highly damped regime), phenomena that ha ve been observed by numerical simulations [2], [3], [7], but it should be considered as the first important step towards our final goal – nonlinear model of wave interaction developed in Section 3. It is well-known that within the framework of linear stabili ty analysis the solution of the original problem is represented as a superposition of plane waves with constant amplitudes, while the phases are determined by the spectrum of solutions to the dispersion equation. Moreover, the wave amplitudes are completely arbitrary and independent of the spatial and temporal variables. The effect of nonlinearities is to cause variation in the amplitudes in both space and time. We are interested in describing these va riations, since they govern the relatively slow process of formation of self-organized pat terns and coherent structures. The importance of the linear theory is embedded in the disper sion relation and the type of solutions it possesses. If the dispersion relation has no imaginary parts (no dissipation of energy occurs and no pumping from external energy sources is available) and its solutions, that is the wave frequency as a function of the wave number are all real, then the correspond- ing amplitude equations describing the evolution of the wav e envelopes will be of nonlinear Schr¨ odinger type. Another possibility arises for conserv ative systems when some of the roots of the dispersion equation appear in complex conjugate pair s. Then the amplitude equations can be shown to be of the so called AB–type [10]. For open syste ms (like the system studied here) the dispersion relation is in general a complex valued function of the wave frequency and wave number and therefore its solutions will be complex. It can be shown [10] that the equation governing the slow evolution of the wave amplit udes in this case will be the Ginzburg–Landau equation. Based on the renormalization group approach we have recentl y derived a Ginzburg– Landau equation for the amplitude of the resonator voltage i n the case of a coasting beam [5]. The derivation has been carried out under the assumptio n that the spatial evolution of the system is much slower compared to the temporal one. This r estriction has been removed here, and the present paper may be considered as an extension of [5]. Using the method of multiple scales we derive a set of coupled amplitude equations for 2the slowly varying part of the longitudinal distribution fu nction and for the intensity of a single resonator wave with an arbitrary wave number (and wav e frequency, specified as a solution to the linear dispersion equation). The equation g overning the evolution of the voltage envelope is shown to be of Ginzburg–Landau type. 2 Formulation of the Problem It is well-known that the longitudinal dynamics of an indivi dual beam particle is governed by the set of equations [9] dz1 dt=k0∆E ;d∆E dt=eωsVRF 2π(sinφ−sinφs) +eωs 2πV1, (2.1) where k0=−ηωs β2sEs(2.2) is the proportionality constant between the frequency devi ation of a non synchronous particle with respect to the frequency ωsof the synchronous one, and the energy deviation ∆ E= E−Es. The quantity k0also involves the phase slip coefficient η=αM−γ−2 s, where αMis the momentum compaction factor [9]. The variables z1=θ−ωst;φ=φs−hz1. (2.3) are the azimuthal displacement of the particle with respect to the synchronous one, and the phase of the RF field, respectively. Here VRFis the amplitude of the RF voltage and his the harmonic number. Apart from the RF field we assume that bea m motion is influenced by a resonator voltage V1due to a broad band impedance ∂2V1 ∂z2 1−2γ∂V1 ∂z1+ω2V1=2γeR ωs∂I1 ∂t, (2.4) where ω=ωr ωs;γ=ω 2Q;I1(θ;t) =/integraldisplay d∆E(ωs+k0∆E)f1(θ,∆E;t),(2.5) f1(θ,∆E;t) is the longitudinal distribution function, ωris the resonant frequency, Qis the quality factor of the resonator and Ris the resonator shunt impedance. It is convenient to pass to a new independent variable (“time ”)θand to the new dimen- sionless variables [2], [6]: 3τ=νsθ;z=z1√νs;u=1√νsk0∆E ωs, (2.6) f1(θ,∆E;t) =ρ0|k0| ωs√νsf(z, u;θ) ; V1=λ1V ;I1=ωsρ0I, (2.7) where ν2 s=ehk0VRFcosφs 2πωs;λ1= 2γ0eRωsρ0. (2.8) In the above expressions the quantity ρ0is the uniform beam density in the thermodynamic limit. The linearized equations of motion (2.1) and equatio n (2.4) in these variables read as: dz dτ=u ;du dτ=−z+λV, (2.9) ∂2V ∂z2−2γ0∂V ∂z+ω2 0V=−∂I ∂z;I(z;θ) =/integraldisplay du(1 +u√νs)f(z, u;θ),(2.10) where γ0=γ√νs;ω0=ω√νs;λ=e2Rγ0k0ρ0 πνs√νs. (2.11) We can now write the Vlasov equation for the longitudinal dis tribution function f(z, u;θ), which combined with the equation for the resonator voltage V(z;θ) ∂f ∂τ+u∂f ∂z−z∂f ∂u+λV∂f ∂u= 0, (2.12) ∂2V ∂z2−2γ0∂V ∂z+ω2 0V=−∂I ∂z, (2.13) I(z;θ) =/integraldisplay du(1 +u√νs)f(z, u;θ), (2.14) comprises the starting point for our subsequent analysis. 43 Derivation of the Amplitude Equations for a Coast- ing Beam In this Section we analyze the simplest case of a coasting bea m. The model equations (2.12) and (2.13) acquire the form [6] ∂f ∂θ+u∂f ∂z+λV∂f ∂u= 0, (3.1) ∂2V ∂z2−2γ∂V ∂z+ω2V=/integraldisplay du/parenleftigg∂f ∂θ−∂f ∂z/parenrightigg , (3.2) where the parameter λshould be calculated for νs= 1. In what follows it will be convenient to write the above equations more compactly as: /hatwideF/parenleftigg∂ ∂θ,∂ ∂z, u/parenrightigg f+λV∂f ∂u= 0, (3.3) /hatwideV/parenleftigg∂ ∂z, ω/parenrightigg V=/hatwideL/parenleftigg∂ ∂θ,∂ ∂z/parenrightigg /an}bracketle{tf/an}bracketri}ht, (3.4) where we have introduced the linear operators /hatwideF/parenleftigg∂ ∂θ,∂ ∂z, u/parenrightigg =∂ ∂θ+u∂ ∂z, (3.5) /hatwideV/parenleftigg∂ ∂z, ω/parenrightigg =∂2 ∂z2−2γ∂ ∂z+ω2, (3.6) /hatwideL/parenleftigg∂ ∂θ,∂ ∂z/parenrightigg =∂ ∂θ−∂ ∂z, (3.7) /an}bracketle{tG(z, u;θ)/an}bracketri}ht=/integraldisplay duG(z, u;θ). (3.8) To obtain the desired amplitude equation for nonlinear wave s we use the method of multiple scales [10], [11]. The key point of this approach is to introd uce slow temporal as well as spatial scales according to the relations: 5θ;T1=ǫθ;T2=ǫ2θ;. . .;Tn=ǫnθ;. . . (3.9) z;z1=ǫz;z2=ǫ2z;. . .;zn=ǫnz;. . . (3.10) where ǫis a formal small parameter. Next is to utilize the perturbat ion expansion of the longitudinal distribution function f, the resonator voltage V f=f0(u) +∞/summationdisplay k=1ǫkfk;V=∞/summationdisplay k=1ǫkVk, (3.11) and the operator expansions /hatwideF/parenleftigg∂ ∂θ+∞/summationdisplay k=1ǫk∂ ∂Tk,∂ ∂z+∞/summationdisplay k=1ǫk∂ ∂zk, u/parenrightigg = =/hatwideF/parenleftigg∂ ∂θ,∂ ∂z, u/parenrightigg +∞/summationdisplay k=1ǫk/hatwideF/parenleftigg∂ ∂Tk,∂ ∂zk, u/parenrightigg , (3.12) /hatwideL/parenleftigg∂ ∂θ+∞/summationdisplay k=1ǫk∂ ∂Tk,∂ ∂z+∞/summationdisplay k=1ǫk∂ ∂zk/parenrightigg = =/hatwideL/parenleftigg∂ ∂θ,∂ ∂z/parenrightigg +∞/summationdisplay k=1ǫk/hatwideL/parenleftigg∂ ∂Tk,∂ ∂zk/parenrightigg , (3.13) /hatwideV/parenleftigg∂ ∂z+∞/summationdisplay k=1ǫk∂ ∂zk/parenrightigg =/hatwideV+ǫ/hatwideVz∂ ∂z1+ǫ2 2/parenleftigg /hatwideVzz∂2 ∂z2 1+ 2/hatwideVz∂ ∂z2/parenrightigg +. . . (3.14) where/hatwideVzimplies differentiation with respect to ∂/∂z. Substituting them back into (3.3) and (3.4) we obtain the corresponding perturbation equatio ns order by order. It is worth noting that without loss of generality we can miss out the spa tial scale z2, because it can be transformed away by a simple change of the reference frame. F or the sake of saving space we will omit the explicit substitution and subsequent calcu lations and state the final result order by order. First order O(ǫ): /hatwideFf1+λV1∂f0 ∂u= 0, (3.15) 6/hatwideVV1=/hatwideL/an}bracketle{tf1/an}bracketri}ht. (3.16) Second order O(ǫ2): /hatwideFf2+λV2∂f0 ∂u=−/hatwideF1f1−λV1∂f1 ∂u, (3.17) /hatwideVV2=/hatwideL/an}bracketle{tf2/an}bracketri}ht+/hatwideL1/an}bracketle{tf1/an}bracketri}ht −/hatwideVz∂V1 ∂z1. (3.18) Third order O(ǫ3): /hatwideFf3+λV3∂f0 ∂u=−/hatwideF1f2−/hatwideF2f1−λV1∂f2 ∂u−λV2∂f1 ∂u, (3.19) /hatwideVV3=/hatwideL/an}bracketle{tf3/an}bracketri}ht+/hatwideL1/an}bracketle{tf2/an}bracketri}ht+/hatwideL2/an}bracketle{tf1/an}bracketri}ht −/hatwideVz∂V2 ∂z1−/hatwideVzz 2∂2V1 ∂z2 1, (3.20) where/hatwideFnand/hatwideLnare the corresponding operators, calculated for Tnandzn. In order to solve consistently the perturbation equations f or each order we need a unique equation for one of the unknowns; it is more convenient to hav e a sole equation for the distribution functions fnalone. This will prove later to be very efficient for the remova l of secular terms that appear in higher orders. By inspecting the above equations order by order one can catch their general form: /hatwideFfn+λVn∂f0 ∂u=αn;/hatwideVVn=/hatwideL/an}bracketle{tfn/an}bracketri}ht+βn, (3.21) where αnandβnare known functions, determined from previous orders. Elim inating Vnwe obtain: /hatwideV/hatwideFfn+λ∂f0 ∂u/hatwideL/an}bracketle{tfn/an}bracketri}ht=−λ∂f0 ∂uβn+/hatwideVαn. (3.22) Let us now proceed with solving the perturbation equations. The analysis of the first order equations (linearized equations) is quite standard, and fo r the one-wave solution we readily obtain: V1=E(zn;Tn)eiϕ+E∗(zn;Tn)e−iϕ∗, (3.23) 7f1=−λ∂f0 ∂u/bracketleftiggE(zn;Tn) /tildewideF(iΩ,−ik, u)eiϕ+E∗(zn;Tn) /tildewideF∗(iΩ,−ik, u)e−iϕ∗/bracketrightigg +F(zn, u;Tn), (3.24) with ϕ= Ωθ−kz, (3.25) where given the wave number k, the wave frequency Ω( k) is a solution to the dispersion equation: /tildewideD(k,Ω(k))≡0. (3.26) The dispersion function/tildewideD(k,Ω) is proportional to the dielectric permittivity of the bea m and is given by the expression /tildewideD(k,Ω) =/tildewideV(−ik) +λ/tildewideL(iΩ,−ik)/angbracketleftigg1 /tildewideF(iΩ,−ik, u)∂f0 ∂u/angbracketrightigg , (3.27) where /hatwideFeiϕ=/tildewideF(iΩ,−ik, u)eiϕ;/hatwideVeiϕ=/tildewideV(−ik)eiϕ;/hatwideLeiϕ=/tildewideL(iΩ,−ik)eiϕ.(3.28) Note that the wave frequency has the following symmetry prop erty: Ω∗(k) =−Ω(−k). (3.29) The functions E(zn;Tn) and F(zn, u;Tn) in equations (3.23) and (3.24) are the amplitude function we wish to determine. Clearly, these functions are constants with respect to the fast scales, but to this end they are allowed to be generic fun ctions of the slow ones. In order to specify the dependence of the amplitude function s on the slow scales, that is to derive the desired amplitude equations one need to go be yond the first order. The first step is to evaluate the right hand side of equation (3.22 ) corresponding to the second order with the already found solution (3.23) and (3.24) for t he first order. This yields terms (proportional to eiϕ) belonging to the kernel of the linear operator on the left ha nd side of equation (3.22), which consequently give rise to the so ca lled secular contributions to the perturbative solution. If the spectrum of solutions to t he dispersion equation (3.26) is complex (as is in our case), terms proportional to e−2Im(Ω)θappear on the right hand side of (3.22). Since, the imaginary part of the wave frequency we consider small, the factor e−2Im(Ω)θis slowly varying in θand we can replace it by e−2Im(Ω)Tn, where the slow temporal scaleTnis to be specified later. This in turn produces additional sec ular terms, which need 8to be taken care of as well. (Note that exactly for this purpos e we have chosen two amplitude functions at first order). The procedure to avoid secular ter ms is to impose certain conditions on the amplitudes E(zn;Tn) and F(zn, u;Tn), that guarantee exact cancellation of all terms proportional to eiϕand terms constant in the fast scales zandθ(containing e−2Im(Ω)Tn) on the right hand side of equation (3.22). One can easily check b y direct calculation that the above mentioned conditions read as: ∂/tildewideD ∂Ω∂E ∂T1−∂/tildewideD ∂k∂E ∂z1=−iλ/tildewideL/angbracketleftigg1 /tildewideF∂F ∂u/angbracketrightigg E, (3.30) /hatwideF1F+ 2λ2Im(Ω)∂ ∂u 1 /vextendsingle/vextendsingle/vextendsingle/tildewideF/vextendsingle/vextendsingle/vextendsingle2∂f0 ∂u |E|2e−2Im(Ω)Tn=−λ ω2∂f0 ∂u/hatwideL1/an}bracketle{tF/an}bracketri}ht. (3.31) Noting that the group velocity of the wave Ω g=dΩ/dkis given by ∂/tildewideD ∂k+∂/tildewideD ∂ΩdΩ dk= 0 = ⇒ Ωg=−∂/tildewideD ∂k/parenleftigg∂/tildewideD ∂Ω/parenrightigg−1 (3.32) we get ∂E ∂T1+ Ω g∂E ∂z1=−iλ/parenleftigg∂/tildewideD ∂Ω/parenrightigg−1 /tildewideL/angbracketleftigg1 /tildewideF∂F ∂u/angbracketrightigg E. (3.33) The above equations (3.31) and (3.33) are the amplitude equa tions to first order. Note that if Im(Ω) = 0 we could simply set Fequal to zero and then equation (3.33) would describe the symmetry properties of the original system (3. 1) and (3.2) with respect to a linear plane wave solution. However, we are interested in th e nonlinear interaction between waves (of increasing harmonicity) generated order by order , and as it can be easily seen the first nontrivial result taking into account this interactio n will come out at third order. To pursue this we need the explicit (non secular) second order s olutions for f2andV2. Solving the second order equation (3.22) with the remaining non secular part of the second order right hand side and then solving equation (3.18 ) with the already determined f2we find f2=SF(k,Ω, u)E2e2iϕ+c.c.+F2(zn, u;Tn), (3.34) V2=SV(k,Ω)E2e2iϕ+fVeiϕ+c.c.+GV(zn, Tn; [F]), (3.35) 9where c.c.denotes complex conjugation. Without loss of generality we can set the generic function F2(zn, u;Tn) equal to zero. Note that, in case Im(Ω) = 0 we could have set F= 0, as mentioned earlier, but we should keep the function F2nonzero in order to cancel third order secular terms depending on the slow scales only. Moreo ver, the functions SF,SV,fV and the functional GV([F]) of the amplitude Fare given by the following expressions: SF(k,Ω, u) =λ2 2/tildewideV(−2ik) /tildewideD(2k,2Ω)1 /tildewideF(iΩ,−ik, u)∂ ∂u/bracketleftigg1 /tildewideF(iΩ,−ik, u)∂f0 ∂u/bracketrightigg , (3.36) SV(k,Ω) =λ2/tildewideL(iΩ,−ik) /tildewideD(2k,2Ω)/angbracketleftigg1 /tildewideF(iΩ,−ik, u)∂ ∂u/bracketleftigg1 /tildewideF(iΩ,−ik, u)∂f0 ∂u/bracketrightigg/angbracketrightigg , (3.37) fV=i /tildewideV(−ik)/bracketleftigg iλ/tildewideI/hatwideL1E−/tildewideVk(−ik)∂E ∂z1/bracketrightigg , (3.38) GV(zn, Tn; [F]) =1 ω2/hatwideL1/an}bracketle{tF/an}bracketri}ht, (3.39) /tildewideI(k,Ω) =/angbracketleftigg1 /tildewideF(iΩ,−ik, u)∂f0 ∂u/angbracketrightigg , (3.40) where the k-index implies differentiation with respect to k. The last step consists in evaluating the right hand side of eq uation (3.22), corresponding to the third order with the already found first and second orde r solutions. Removal of secular terms in the slow scales leads us finally to the amplit ude equation for the function F(zn, u;Tn), that is ∂ ∂T2/parenleftigg ω2F+λ/an}bracketle{tF/an}bracketri}ht∂f0 ∂u/parenrightigg +2λγ ω2∂f0 ∂u∂ ∂z1/hatwideL1/an}bracketle{tF/an}bracketri}ht+λ∂F ∂u/hatwideL1/an}bracketle{tF/an}bracketri}ht= =λ2ω2/bracketleftigg∂ ∂u/parenleftigg1 /tildewideF∗∂f0 ∂u/parenrightigg fVE∗+∂ ∂u/parenleftigg1 /tildewideF∂f0 ∂u/parenrightigg f∗ VE/bracketrightigg e−2Im(Ω)T2. (3.41) Elimination of secular terms in the fast scales leads us to a g eneralized cubic Ginzburg– Landau type of equation for the amplitude E(zn, Tn): i∂/tildewideD ∂Ω∂E ∂T2=A∂2E ∂z2 1+λa∂ ∂z1{G([F])E}+λB|E|2Ee−2Im(Ω)T2− −λ2CGV([F])E+λ/tildewideLG([F])fV, (3.42) 10where the coefficients a(k),A(k),B(k) and C(k) are given by the expressions: a(k) =/tildewideVk/parenleftigg∂/tildewideD ∂Ω/parenrightigg−1 , (3.43) A(k) = 1 +/tildewideVk /tildewideV/bracketleftig/tildewideVk+iλ/tildewideI(1 + Ω g)/bracketrightig , (3.44) B(k) =/tildewideL/angbracketleftigg1 /tildewideF∂SF ∂u/angbracketrightigg −λ/tildewideLSV/angbracketleftigg1 /tildewideF∂ ∂u/parenleftigg1 /tildewideF∗∂f0 ∂u/parenrightigg/angbracketrightigg , (3.45) C(k) =/tildewideL/angbracketleftigg1 /tildewideF∂ ∂u/parenleftigg1 /tildewideF∂f0 ∂u/parenrightigg/angbracketrightigg , (3.46) and the functional G([F]) of the amplitude Fcan be written as G([F]) =/angbracketleftigg1 /tildewideF∂F ∂u/angbracketrightigg . (3.47) Equations (3.41) and (3.42) comprise the system of coupled a mplitude equations for the intensity of a resonator wave with a wave number kand the slowly varying part of the longitudinal distribution function. Note that the depe ndence on the temporal scale T1(involving derivatives with respect to T1) in equations (3.41) and (3.42) through the operator/hatwideL1and the function fVcan be eliminated in principle by using the first order equations (3.31) and (3.33). As a result one obtains a system of coupled second order partial differential equations for FandEwith respect to the variables T2andz1. 4 Concluding Remarks We have studied the longitudinal dynamics of particles movi ng in an accelerator under the action of a collective force due to a resonator voltage. For a sufficiently high beam density (relatively large value of the parameter λ) the nonlinear wave coupling, described by the nonlinear term in the Vlasov equation becomes important, an d has to be taken into account. This is manifested in a spatio-temporal modulation of the wa ve amplitudes in unison with the slow process of particle redistribution. As a result of this wave-particle interaction (coupling between resonator waves and particle distribution modes) c oherent, self-organized patterns can be formed in a wide range of relevant parameters. We have analyzed the slow evolution of the amplitude of a sing le resonator wave with an arbitrary wave number k(and wave frequency Ω( k) defined as a solution to the dispersion 11relation). Using the method of multiple scales a system of co upled amplitude equations for the resonator wave envelope and for the slowly varying part o f the longitudinal distribu- tion function has been derived. As expected, the equation fo r the resonator wave envelope is a generalized cubic Ginzburg–Landau (GCGE) equation. We argue that these ampli- tude equations govern the (relatively) slow process of form ation of coherent structures and establishment of wave-particle equilibrium. Acknowledgments The author wishes to thank Y. Oono and C. Bohn for careful read ing of the manuscript and for making valuable comments. This work was supported by the US Department of Energy, Office o f Basic Energy Sci- ences, under contract DE-AC03-76SF00515. References [1] A.W. Chao, Physics of Collective Beam Instabilities in High-Energy Ac celerators , Wiley, New York, 1993. [2] P.L. Colestock, L.K. Spentzouris and S.I. Tzenov, In Proc. International Symposium on Near Beam Physics , Fermilab, September 22–24, 1997, FNAL-Conf-98/166, 1998 , pp. 94–104. [3] A. Gerasimov, Phys. Rev. E 49, (1994), p. 2331. [4] S.I. Tzenov and P.L. Colestock, FNAL-Pub-98/258, 1998. [5] S.I. Tzenov, FNAL-Pub-98/275, 1998. [6] S.I. Tzenov, In Proc. Workshop on Instabilities of High Intensity Hadron Be ams in Rings , Upton, New York, June/July 1999, T. Roser and S.Y. Zhang eds. , AIP Conf. Proc. 496, 1999, pp. 351–360. [7] P.H. Stoltz and J.R. Cary, Physics of Plasmas ,7, (2000), p. 231. [8] L.K. Spentzouris, Ph.D. Thesis, Northwestern Universi ty, 1996. [9] H. Bruck, Accelerateurs Circulaires de Particules , Presses Universitaires, Paris, 1966. [10] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons and Nonlinear Wave Equations , Academic Press, London, 1982. [11] Lokenath Debnath, Nonlinear Partial Differential Equations for Scientists an d Engi- neers, Birkhauser, Boston, 1997. 12

arXiv:physics/0002001v1 [physics.bio-ph] 1 Feb 2000A Simple Explanation for Taxon Abundance Patterns Johan Chu and Christoph Adami W.K. Kellogg Radiation Laboratory 106-38 California Institute of Technology, Pasadena, CA 91125 Classification: Biological Sciences (Evolution), Physical Sciences (Appl ied Mathematics) Corresponding author: Dr. Chris Adami E-mail: adami@krl.caltech.edu Phone: (+) 626 395-4256 Fax: (+) 626 564-8708J. Chu and C. Adami 2 Abstract For taxonomic levels higher than species, the abundance dis tributions of number of subtaxa per taxon tend to approximate power laws, b ut often show strong deviationns from such a law. Previously, these devia tions were attributed to finite-time effects in a continuous time branching process at the generic level. Instead, we describe here a simple discrete branchin g process which generates the observed distributions and find that the distr ibution’s deviation from power-law form is not caused by disequilibration, but r ather that it is time-independent and determined by the evolutionary prope rties of the taxa of interest. Our model predicts—with no free parameters—th e rank-frequency distribution of number of families in fossil marine animal o rders obtained from the fossil record. We find that near power-law distributions are statistically almost inevitable for taxa higher than species. The branchi ng model also sheds light on species abundance patterns, as well as on links betw een evolutionary processes, self-organized criticality and fractals. Taxonomic abundance distributions have been studied since the pioneering work of Yule [1], who proposed a continuous time branching proces s model to explain the distributions at the generic level, and found that they w ere power laws in the limit of equilibrated populations. Deviations from the geo metric law were attributed to a finite-time effect, namely, to the fact that the populatio ns had not reached equilibrium. Much later, Burlando [2, 3] compiled data that appeared to corroborate the geometric nature of the distributions, even though clea r violations of the law are visible in his data also. In this paper, we present a model whi ch is based on a discrete branching process whose distributions are time-independe nt and where violations of the geometric form reflect specific environmental condition s and pressures that the assemblage under consideration was subject to during evolu tion. As such, it holds the promise that an analysis of taxonomic abundance distrib utions may reveal certain characteristics of ecological niches long after its inhabi tants have disappeared. The model described here is based on the simplest of branchin g processes, known in the mathematical literature as the Galton-Watson process. Consider an assemblage of taxa at one taxonomic level. This assemblage can be all the fa milies under a particular order, all the subspecies of a particular species, or any oth er group of taxa at the same taxonomic level that can be assumed to have suffered the same e volutionary pressures. We are interested in the shape of the rank-frequency distrib ution of this assemblage and the factors that influence it. We describe the model by explaining a specific example: the di stribution of the number of families within orders for a particular phylum. Th e adaptation of this model to different levels in the taxonomic hierarchy is obvio us. We can assume that the assemblage was founded by one order in the phylum and that this order consisted of one family which had one genus with one species. We further assume thatJ. Chu and C. Adami 3 new families in this order are created by means of mutation in individuals of extant families. This can be viewed as a process where existing fami lies can “replicate” and create new families of the same order, which we term daughters of the initial family. Of course, relatively rarely, mutations may lead to the crea tion of a new order, a new class, etc. We define a probability pifor a family to have idaughter families of the same order ( true daughters ). Thus, a family will have no true daughters with probability p0, one true daughter with probability p1, and so on. For the sake of simplicity, we initially assume that all families of this ph ylum share the same pi. We show later that variance in piamong different families does not significantly affect the results, in particular the shape of the distributi on. The branching process described above gives rise to an abundance distribution of f amilies within orders, and its probability distribution can be obtained from the La grange expansion of a nonlinear differential equation [4]. Using a simple iterati ve algorithm [5] in place of this Lagrange expansion procedure, we can calculate rank-f requency curves for many different sets of pi. It should be emphasized here that we are mostly concerned wi th the shape of this curve for n/lessorsimilar104, and not the asymptotic shape as n→ ∞, a limit that is not reached in nature. For different sets of pi, the theoretical curve can either be close to a power-law, a power law with an exponential tail or a purely exponential d istribution (Fig. 1). We show here that there is a global parameter that distinguis hes among these cases. Figure 1: Predicted abundance pattern P(n) (probability for a taxon to have nsub- taxa) of the branching model with different values of m. The curves have been individually rescaled.J. Chu and C. Adami 4 Indeed, the mean number of true daughters, i.e., the mean num ber of different families of the same order that each family gives rise to in the example above, m=∞/summationdisplay i=0i·pi (1) is a good indicator of the overall shape of the curve. Univers ally,m= 1 leads to a power law for the abundance distribution. The further mis away from 1, the further the curve diverges from a power-law and towards an ex ponential curve. The value of mfor a particular assemblage can be estimated from the fossil record also, allowing for a characterization of the evolutionary proces s with no free parameters. Indeed, if we assume that the number of families in this phylu m existing at one time is roughly constant, or varies slowly compared to the averag e rate of family creation (an assumption the fossil record seems to vindicate [6]), we find that mcan be related to the ratio Ro/Rfof the rates of creation of orders and families—by m= (1 +Ro Rf)−1(2) to leading order [5]. In general, we can not expect all the families within an order to share the same m. Interestingly, it turns out that even if the piandmdiffer widely between different families, the rank-frequency curve is identical to that obt ained by assuming a fixed m equal to the average of macross the families (Fig. 2), i.e., the variance of the piacross families appears to be completely immaterial to the shape of the distribution—only the average µ≡ /angbracketleftm/angbracketrightcounts. In Fig. 3, we show the abundance distribution of families wit hin orders for fossil marine animals [7], together with the prediction of our bran ching model. The theo- retical curve was obtained by assuming that the ratio Ro/Rfis approximated by the ratio of the total number of orders to the total number of fami lies Ro Rf∼=No Nf(3) and that both are very small compared to the rate of mutations . The prediction µ= 0.9(16) obtained from the branching process model by using (3) as the sole parameter fits the observed data remarkably well ( P= 0.12, Kolmogorov-Smirnov test, see inset in Fig. 3). Alternatively, we can use a best fit to determine the ratio Ro/Rfwithout resorting to (3), yielding Ro/Rf= 0.115(20) ( P= 0.44). Fitting abundance distributions to the branching model thus allows us to determine a ratio of parameters which reflect dynamics intrinsic to the taxon u nder consideration, andJ. Chu and C. Adami 5 Figure 2: Abundance patterns obtained from two sets of numer ical simulations of the branching model, each with µ=/angbracketleftm/angbracketright= 0.5.mwas chosen from a uniform probability distribution of width 1 for the runs represented by crosses, and from a distribution of width 0 .01 for those represented by circles. Simulations where mandpiare allowed to vary significantly and those where they are severely const ricted are impossible to distinguish if they share the same /angbracketleftm/angbracketright. the niche(s) it inhabits. Indeed, some taxa analyzed in Refs . [2, 3] are better fit with 0.5< µ < 0.75, pointing to conditions in which the rate of taxon formati on was much closer to the rate of subtaxon formation, indicating either a more “robust” genome or richer and more diverse niches. In general, however, Burlando’s data [2, 3] suggest that a wi de variety of taxonomic distributions are fit quite well by power laws ( µ= 1). This seems to imply that actual taxonomic abundance patterns from the fossil record are cha racterized by a relatively narrow range of µnear 1. This is likely within the model description advanced here. It is obvious that µcan not remain above 1 for significant time scales as this woul d lead to an infinite number of subtaxa for each taxon. What abou t lowµ? We propose that low values of µare not observed for large (and therefore statistically imp ortant) taxon assemblages for the following reasons. If µis very small, this implies either a small number of total individuals for this assemblage, or a v ery low rate of beneficial taxon-forming (or niche-filling) mutations. The former mig ht lead to this assemblage not being recognized at all in field observations. Either cas e will lead to an assemblage with too few taxons to be statistically tractable. Also, sin ce such an assemblage eitherJ. Chu and C. Adami 6 Figure 3: The abundance distribution of fossil marine anima l orders [7] (squares) and the predicted curve from the branching model (solid line ). The fossil data has been binned above n= 37 with a variable bin size [5]. The predicted curve was generated using Ro/Rf=No/Nf= 0.115, where NoandNfwere obtained directly from the fossil data. The inset shows Kolmogorov-Smirnov (K -S) significance levels Pobtained from comparison of the fossil data to several predi cted distributions with different values of Ro/Rf, which shows that the data is best fit by Ro/Rf= 0.135. The arrow points to our prediction Ro/Rf= 0.115 where P= 0.12. A Monte Carlo analysis shows that for a sample size of 626 (as we have h ere), the predicted Ro/Rf= 0.115 is within the 66% confidence interval of the best fit Ro/Rf= 0.135 (P= 0.44). The K-S tests were done after removal of the first point, w hich suffers from sampling uncertainties. contains a small number of individuals or is less suited for f urther adaptation or both, it would seem to be susceptible to early extinction. The branching model can—with appropriate care—also be appl ied to species- abundance distributions, even though these are more compli cated than those for higher taxonomic orders for several reasons. Among these ar e the effects of sexual re- production and the localized and variable effects of the envi ronment and other species on specific populations. Still, as the arguments for using a b ranching process model essentially rely on mutations which may produce lines of ind ividuals that displace oth- ers, species-abundance distributions may turn out notto be qualitatively as different from taxonomically higher-level rank-frequency distribu tions as is usually expected. Historically, species abundance distributions have been c haracterized using fre- quency histograms of the number of species in logarithmic ab undance classes. ForJ. Chu and C. Adami 7 Figure 4: The abundance distribution of fossil marine anima l orders in logarithmic abundance classes (the same data as Fig. 3). The histogram sh ows the number of orders in each abundance class (left scale), while the solid line depicts the number of families in each abundance class (right scale). Species ran k-abundance distributions where the highest abundance class present also has the highe st number of individuals (as in these data) are termed canonical lognormal [9]. many taxonomic assemblages, this was found to produce a hump ed distribution trun- cated on the left—a shape usually dubbed lognormal [8, 9, 10]. In fact, this distri- bution is not incompatible with the power-law type distribu tions described above. Indeed, plotting the fossil data of Fig. 3 in logarithmic abu ndance classes produces a lognormal (Fig. 4). For species, µis the mean number of children each individual of the species has. (Of course, for sexual species, µwould be half the mean number of children per individual.) In the present case, µless than 1 implies that extant species’ populations decrease on average, while µequal to 1 implies that average populations do not change. An extant species’ population can decline due to the introdu ction of competitors and/or the decrease of the size of the species’ ecological ni che. Let us examine the former more closely. If a competitor is introduced into a sat urated niche, all species currently occupying that niche would temporarily see a decr ease in their muntil a new equilibrium was obtained. If the new species is significantl y fitter than the previously existing species, it may eliminate the others. If the new spe cies is significantly less fit, then it may be the one eliminated. If the competitors are a bout as efficient as the species already present, then the outcome is less certai n. Indeed, it is analogous to a non-biased random walk with a possibility of ruin. The eff ects of introducing aJ. Chu and C. Adami 8 single competitor are transient. However, if new competito rs are introduced more or less periodically, then this would act to push mlower for all species in this niche and we would expect an abundance pattern closer to the exponenti al curve as opposed to the power-law than otherwise expected. We have examined t his in simulations of populations where new competitors were introduced into the population by means of neutral mutations—mutations leading to new species of the s ame fitness as extant species—and found that these are fit very well by the branchin g model. A higher rate of neutral mutations and thus of new competitors leads t o distributions closer to exponential. We have performed the same experiment in more s ophisticated systems of digital organisms (artificial life) [11, 12] and found the same result [5]. If no new competitors are introduced but the size of the niche is gradually reduced, we expect the same effect on mand on the abundance distributions. Whether it is possible to separate the effects of these two mechanisms in ec ological abundance patterns obtained from field data is an open question. An anal ysis of such data to examine these trends would certainly be very interesting. So far, we have sidestepped the difference between historica l and ecological distri- butions. For the fossil record, the historical distributio n we have modeled here should work well. For field observations where only currently livin g groups are considered, the nature of the death and extinction processes for each gro up will affect the abun- dance pattern. In our simulations and artificial-life exper iments, we have universally observed a strong correlation between the shapes of histori cal and ecological distribu- tions. We believe this correspondence will hold in natural d istributions as well when death rates are affected mainly by competition for resources . The model’s validity for different scenarios is an interesting question, which could be answered by comparison with more taxonomical data. Our branching process model allows us to reexamine the quest ion of whether any type of special dynamics—such as self-organized criticali ty [13] (SOC)—is at work in evolution [14, 15]. While showing that the statistics of t axon rank-frequency pat- terns in evolution are closely related to the avalanche size s in SOC sandpile models, the present model clearly shows that instead of a subsidiary relationship where evo- lutionary processes may be self-organized critical, the po wer-law behaviour of both evolutionary andsandpile distributions can be understood in terms of the mec hanics of a Galton-Watson branching process [5, 16]. The mechanics of this branching pro- cess are such that the branching trees are probabilistic fra ctal constructs. However, the underlying stochastic process responsible for the obse rved behaviour can be ex- plained simply in terms of a random walk [17]. For evolution, the propensity for near power-law behaviour is found to stem from a dynamical proces s in which µ≈1 is selected for and highly more likely to be observed than other values, while the “self- tuning” of the SOC models is seen to result from arbitrarily e nforcing conditions which would correspond to the limit Ro/Rf→0 and therefore m→1 [5].J. Chu and C. Adami 9 References [1] Yule, G. U. (1924) A mathematical theory of evolution. Proc. Roy. Soc. London Ser. B 213, 21-87. [2] Burlando, B. (1990) The fractal dimension of taxonomic s ystems. J. theor. Biol. 146, 99-114. [3] Burlando, B. (1993) The fractal geometry of evolution. J. theor. Biol. 163, 161- 172. [4] Harris, T. E. (1963) The Theory of Branching Processes . (Springer, Berlin; Prentice-Hall, Englewood Cliffs, N.J.). [5] Chu, J. & Adami, C. (1999) Critical and near-critical bra nching processes (sub- mitted). [6] Raup, D. M. (1985) Mathematical models of cladogenesis. Paleobiology ,11, 42- 52. [7] Sepkoski, J. J. (1992) A Compendium of Fossil Marine Animal Families , 2nd ed. (Milwaukee Public Museum; Milwaukee, WI; 1992) with emenda tions by J. J. Sepkoski based largely on The Fossil Record 2 , Benton, M. J., ed. (Chapman & Hall; New York; 1993). [8] Preston, F. W. (1948) The commonness, and rarity, of spec ies.Ecology 29, 255- 283. [9] Preston, F. W. (1962) The canonical distribution of comm onness and rarity. Ecology 43, 185-215, 410-432. [10] Sugihara, G. (1980) Minimal community structure: An ex planation of species abundance patterns. Am. Nat. 116, 770-787. [11] Adami, C. (1998) Introduction to Artificial Life . (Telos, Springer-Verlag, New York) [12] Chu, J. & Adami, C. (1997) Propagation of information in populations of self- replicating code, in Artificial Life V: Proceedings of the Fifth International Wo rk- shop on the Synthesis and Simulation of Living Systems , Langton, C. G. and Shimohara, K. eds., p. 462-469 (MIT Press, Cambridge, MA). [13] Bak, P., Tang, C. & Wiesenfeld, K. (1987) Self-organize d criticality—an expla- nation of 1/ fnoise. Phys. Rev. Lett. 59, 381-384. Self-organized criticality. Phys. Rev. A 38, 364-374 (1988).J. Chu and C. Adami 10 [14] Sneppen, K., Bak P., Flyvbjerg, H. & Jensen, M. H. (1995) Evolution as a self- organized critical phenomenon. Proc. Nat. Acad. Sci. USA 92, 5209-5213. [15] Adami, C. (1995) Self-organized criticality in living systems. Phys. Lett. A 203, 29-32. [16] Vespignani, A. & Zapperi, S. (1998) How self-organized criticality works: A unified mean-field picture. Phys. Rev. E 57, 6345-6362. [17] Spitzer, F. (1964) Principles of Random Walk . (Springer-Verlag, New York). Acknowledgments. We would like to thank J. J. Sepkoski for kindly sending us his amended data set of fossil marine animal families. Acces s to the Intel Paragon XP/S was provided by the Center of Advanced Computing Resear ch at the California Institute of Technology. This work was supported by a grant f rom the NSF. Correspondence and requests for materials should be addres sed to C.A. (e-mail: adami@krl.caltech.edu).

arXiv:physics/0002002v1 [physics.gen-ph] 1 Feb 2000The Multifractal Time and Irreversibility in Dynamic Syste ms L.Ya.Kobelev Department of Physics, Urals State University Lenina Ave., 51, Ekaterinburg 620083, Russia E-mail: leonid.kobelev@usu.ru The irreversibility of the equations of classical dynamics (the Hamilton equations and the Liouville equation ) in the space with multifractal time is demonstrat ed. The time is given on multifractal sets with fractional dimensions. The last depends on densities o f Lagrangians in a given time moment and in a given point of space. After transition to sets of time points with the integer dimension the obtained equations transfer in the known equations of class ical dynamics. Production of an entropy is not equally to zero in space with multifractal time, i.e. t he classical systems in this space are non-closed. 01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p. I. INTRODUCTION The dynamic equations of the physical theories are re- versible, it is well known. The kinetic equations of the statistical theory are irreversible. The irreversibility the Boltzmann’s statistical theory was in due time the main reason of non-recognition by Poincare of Boltzmann’s statistical theory. For the irreversibility introducing i n the physical dynamical equations (for example, in the equation of the Liouville) it is necessary to introduce the dissipation terms [1], [2] or the functionals of a micro- scopic entropy and time [3] ensuring realization of the second law of thermodynamics. In the first case irre- versibility in the dynamic equations arises as a sequence of mathematical approaches. Prigogin’s the point of view is consists in recognizing the primacy of irreversible pro- cesses and it seems intuitively more reasonable. Are more general, than mentioned, a methods of introducing of the irreversibility in the dynamical equations of physics ex- ist? Is it possible the reversibility of the equations of the dynamical theories to introduce as result of approximate transition from more rigorous the dynamical irreversible equations (these dynamic equations may be obtained as a result of generalization of the known equations) to the idealized and reversible, but approximate equations? The purpose of a note is to introduce one of a possible generalizations of the dynamical theories of physics real- ized by replacement of time with topological dimension equal unity on ”multifractal” time (for the first time it was introduced in [4]). In the mentioned theory the time is characterized in each time and space points by frac- tional (fractal) dimensions (FD) dt(r(t), t). The marked replacement dimension of time by fractional dimensions gives in the origin of irreversibility in the dynamical equa - tions of physics and the existence of irreversibility in our world may be interpreted by new reason. It is not contra- dicts (for FD is small differs from unity for weak physical fields on the Earth) an experimental data, and allows to receive a new interesting physical results. The equationsof the classical mechanics (the equations of the Hamilton - Liouville) are chosen for research as an typical example of dynamic systems. II. EQUATION OF A MECHANICS WITH TIME DEFINED ON MULTIFRACTAL SETS The method of generalization of the classical mechanics equations is founded on the new model of an approach to the problem of a nature of time [4].This model is consists in replacement of usual time by the time defined on a mul- tifractal subsets stof continuously set Mt(the measure carrier). The multifractal set Stconsists of subsets st, i.e. very small time intervals (in further named ”points”), which are also multifractal, and each of them is character- ized in turn by its global fractal dimension FD) dt(r(t), t) ( defined as box-dimension, [5], [6] and so on) that de- pends of a nature of sets st, and depends at coordinates and time (see [4], [7]). So each time subset is character- ized by its global fractal dimension dt(r(t), t) which char- acterize the scaling characteristics for this subset. The continuity dt(r(t), t) is supposed. The new approach to a nature of time is consists in the replacing the usual time points of time axe by selection for describing of the time’s intervals (the ”points” on a time axis consisting of sets st that defined on the measure carrier Mt) only the ”points” that characterized by sets st. The time axe (or, in other selections time plane or time volume Rn) is the carrier of measure of all the multifractal time subsets stdefined on it. The researching of the dynamic equations and physical quantities (in particular, the entropy) with time ”points” with fractal characteristics defined on multifrac - tal sets stwith FD dt(r(t), t), lead to irreversibility of all dynamic theories used in physics. It is stipulated by an openness of dynamic systems with multifractal time (role of a thermostat plays set Rn) that is appears in a time de- pendencies of all physical and mathematical (except for zero) objects. For describing of small changes of func- tions defined on multifractal time sets, it is impossible 1to apply ordinary or fractional (in sense of the Riemann - Liouville) derivatives and integrals, since to different time points there corresponds to different fractal dimen- sions. For describing of changes of such functions need’s introduction of generalized fractional derivatives and in - tegrals [4]. In this note the multifractal properties of the space sets srare not considered, since the irreversibility of the dynamic physical equations arises already at the using only of multifractal time (see also [7])and global FD of small time sets intervals st. III. GENERALIZED FRACTIONAL DERIVATIVES AND INTEGRALS ON MULTIFRACTAL SET STOF TIME POINTS It is necessary if we want to describe the dynamics of time-dependent functions determined on multifractal set Stto enter the functionals that extends the fractional derivatives and integrals of the Riemann - Liouville on the set Stwith FD dt(r(t), t)) that is different in each subsets st[4] ) Ddt +,tf(t) =/parenleftbiggd dt/parenrightbiggnt/integraldisplay adt′ f(t′) Γ(n−dt(t′))(t−t′)dt(t′)−n+1 (1) Ddt −,tf(t) = (−1)n× ×/parenleftbiggd dt/parenrightbiggnb/integraldisplay tdt′ f(t′) Γ(n−dt(t′))(t′−t)dt(t′)−n+1(2) where Γ-is Euler gamma -function, a < b ,aandbis stationary values selected on an axis (from −∞to∞), n−1≤dt< n,n={dt}+ 1,{dt}is an integer part of dt≥0,n= 0 for dt<0,dt=dt(r(t), t)-is the fractal dimensions (FD). The dependencies FD from time and space coordinates are defined by the Lagrangians’s densi- ties of a viewed problem [4], [7], [8]. The generalized frac- tional derivative (GFD) (1)-(2) coincide with fractional derivatives or fractional integrals of the Riemann - Liou- ville [9] in the case dt=const. Atdt=n+ε(t),ε→0 GFD are represented by usual derivatives and integrals [4]. The functions and integrals in (1)-(2) are considered as generalized functions given on the set of finitary func- tions [10]. The definitions the GFD (1)-(2) allow to de- scribe the dynamics of functions defined on multifractal sets and GFD substitute (for such functions) the usual or fractional differentiation and integration (GFD partially conserve the memory about of the last time events). IV. HAMILTON EQUATIONS The Hamilton equations for system from Nof clas- sical particles with identical masses m on the set withmultifractal time (i.e. time defined on multifractal set St) reads: Ddt +,tri=∂H ∂pi, Ddt −,tpi=−∂H ∂ri,pi=Ddt +,tri(3) H=/summationdisplay i=1..Np2 i 2m+1 2/summationdisplay i/negationslash=j=1..NV(|ri−rj|) (4) The equations (refeq3)-(ref4) differ from the classical Hamilton equations by replacement the derivatives with respect to time by GFD (refeq1)-(ref2) and coincide with the classical equations of a mechanics at dt= 1. The equations for arbitrary function Bdynamic variable p,r will look like Ddt +,tB=˜Ddt +,tB+∂H ∂pi∂B ∂ri−∂H ∂ri∂B ∂pi(5) The figure ˜Ddt +,tin (5) differs from Ddt +,tin (1) by replace- ment the complete derivative with respect to time ton a partial differential with respect to time t. Let’s show, that the Hamiltonian function Hin space with multifrac- tal time is not integral of a motion of the equation (5), i.e. does not convert a right part of (5) in zero. Substitution Hin (5) gives in Ddt +,tH=˜Ddt +,tH (6) From the (1) follows, that equation (6) is of the form (fordt(r(t), t) = 1 + ε(r(t), t),ε→0, when a simplifying assumption about lack at dt,εof explicit dependence fromtis valid) Ddt +,tH=˜Ddt +,tH=εH Γ(1 + ε)tdt(7) and is equal to zero when dt(r(t), t) = 1. So, in space with multifractal time, at classical system with a Hamil- tonian Hthat not depends explicitly at time (conser- vative systems) the GFD with respect to a total en- ergy depends on time and decreases with the increases of time, i.e. in the model of multifractal time the rigor- ously conservative classical systems does not exist. For differs dt(r(t), t) from unity by a little bit (that it is valid about it represents experimental data about time and results of [4] ) the changing of energy of system will be very small. Let’s consider the problem of change Hwith change of time in more general case ( ε=ε(r(t), t)). For this purpose we shall be restricted to a case, when FD of timedt(r(t), t) = 1 is not considerably differs from unity: dt= 1 + ε(r(t), t) = 1, |ε| ≪1. In this case GFD is represented as [4] (integral is calculated as the total of a principal value and residue in a singular point) ˜D1+ε +,tH≈∂ ∂tH∓1 2∂ ∂t[ε(r(t), t)H Γ(1 + ε(r(t), t))] + +εH Γ(1 + ε)tdt, ε > 0, d t<1 (8) 2˜D1+ε +,tH≈∂ ∂t[1 Γ(1−ε)H]±1 2∂ ∂t[ε(r(t), t)H Γ(1−ε)] + +εH Γ(1 + ε)tdt, ε > 0, d t>1 (9) The selection of signs (plus or minus) in (8)-(9) is deter- mined by sign of εand requirements of a regularization of integrals and selection of FD dt(greater or smaller unity). Let dt(r(t), t)<1. In this case from (8) follows (forHdo not containing explicit time dependence) Ddt +,tH=˜Ddt +,tH≈ ±1 2H∂ε(r(t), t) ∂t+εH Γ(1 + ε)tdt(10) Fort→ ∞ , the basic contribution in the (10) im- ports corrections proportional to velocity of change FD dt(r(t), t). The total energy conservative (in sense, that the Hamiltonian has not an explicit dependence at time ) systems now in space with multifractal time is not con- servative systems. It changes can be at t→ ∞ of any sign, and depend on a sign of derivatives with respect to time from the fractional correction to dimension of time ε. For ε= 0 the total energy of system is conserves and all relations coincide with known relations following from the dynamic equations of classical systems mechanics. V. LIOUVILLE EQUATION The equation of the Liouville for a N-partial distribu- tion function ρ(X, t) (X- are coordinate and impulses of particles of a 6 N-dimension phase space) is equivalent to the Hamilton equations for system from Nof classical particles and is invariant in relation to transformations ri→ri,pi→ −pi, t→ −t, i= 1,2,3, ..N (11) (equation is reversible). On set Stof multifractal time complete derivative ρ(X, t) will reads: Ddt +,tρ(X, t) =˜Ddt +,tρ(X, t) +∂H ∂pi∂ρ(X, t) ∂ri− −∂H ∂ri∂ρ(X, t) ∂pi=˜Ddt +,tρ(X, t)−Lρ(X, t) (12) pi=Ddt +,tri where Ddt +,tρis defined by (1), L-functional of the Liou- ville and differs from the functional of the known equa- tion of the Liouville by replacement of ordinary deriva- tives with respect to time on GFD. For a demonstration of the irreversibility of expression (12) to transformatio ns (11) we shall mark the following: the distribution func- tions ρ(X, t) and ρ(X0, t0) viewed in different moments t0andtare connected by the relation ρ(X, t)dX=ρ(X0, t0)dX0 (13) in which because of change of fractal dimension with a change time dX/ne}ationslash=dX0. Therefore ρ(X, t)/ne}ationslash=ρ(X0, t0)andρ(X, t) evolves with a time. Complete derivative Ddt +,tρ(X, t), in particular and in that connection, is not equal to zero. Intuitively it is clear, that derivative Ddt +,tρ(X, t) is determined by a functional from function ρ(X, t) equal to zero at dt= 1 + ε= 1. Let’s designate this functional describing change ρ(X, t) owing to inter- action with ”thermostat”, by ϕ=ϕ0(ρ, dt, t)ε(r(t), t). The role of the thermostat plays the set Mt(being the carrier of a measure of a subsets of time points stand belonging to one of spaces Rn), as was already marked. The appealing of the functional ϕis caused not to in- terior processes happening with change of energy inside system, but is determined by different properties of time setsstin different instant of time (change of dimension ofstwith a time changes). Complete derivative in this case will be equal to a functional ϕand (12) will reads as the equation ˜Ddt +,tρ(X, t) +∂H ∂pi∂ρ(X, t) ∂ri−∂H ∂ri∂ρ(X, t) ∂pi= =ϕ0(ρ, dt, t)ε(r(t), t) (14) The equation (14) is analogous of the Liouville equation of the classical systems with the time defined on multi- fractal sets. The analog of a collision integral in a right member (14) is stipulated by interaction with the carrier of a measure of multifractal set Mtand is equal’s to zero if sets of time stis substitutes by sets with topological dimension equal to unity. VI. PRODUCTION OF AN ENTROPY Let’s consider a classical system which is be found in an equilibrium state at the usual describing of the time.The production of an entropy in such system is equal to zero. Let’s consider the production of the en- tropy S=/integraltext ρ(X, t)ℓnρ(X, t)dXof same classical sys- tem defined on multifractal set of time points St(for dt= 1−ε <1): Ddt +,tS=/integraldisplay Ddt +,t[ρℓnρ]dX (15) Permissible, as well as earlier, that dt= 1 + ε(r(t), t), |ε≪1|. As ρ(X, t) has for the equilibrium system at dt/ne}ationslash= 1 the complete GFD which is non-equal zero, the right member (15) is not equal to zero. It means, that equilibrium systems does not exist in space with multi- fractal time. Really, as Ddt +,tS=/integraldisplay Ddt +,t[ρℓnρ]dX≈ εS tdt±1 2∂ ∂t(εS) +∂S ∂t/ne}ationslash= 0 (16) that (16) is an inequality and in a case∂S ∂t= 0. For ∂ε ∂t= 0 the production of the entropy is positive. For 3∂ε ∂t/ne}ationslash= 0,∂S ∂t= 0 the production of the entropy can have any sign (in particular, the entropy can be decreasing too). Let’s mark in that circumstance, that all new re- sults for behavior of the entropy are stipulated by the multifractality of the time and disappear after transition to time with topological dimension equal to unity. VII. ABOUT CONNECTION FD DTWITH LAGRANGIANS OF PHYSICAL FIELDS In the monograph [4] the following approximating con- nection of fractional dimension of time dt(r(t), t) with a Lagrangian density Lof all physical fields in a point r(t) in an instant t(see also [7], [8]) is obtained: dt(r(t), t) = 1 +/summationdisplay iβiLi(r(t), t) (17) where βiis dimensional numerical factors ensuring a zero dimension of products βiLi. In [4] it is shown, that for coinciding the results founded on the (17) with results of the theory general relativity (GR) it is necessary (for gravitational forces) to choose β=2 c2(c- speed of light). For correspondence with results of a quantum mechanics, for electric fields the βehas an order of magnitude βe= (2mc2)−1(m-is mass of particle or body creating the electric charge). The small differences FD from unity is satisfied by condition /summationdisplay iβiLi=ε≪1 (18) The connection εwith density of Lagrangians adds phys- ical sense GFD (more in detail about it see [7]) and renders concrete relations obtained in the previous para- graphs. VIII. CONCLUSIONS The present note is devoted to the appendix of idea of the multifractal time offered in [4] [8], for researching of the problem of an irreversibility in large classical sys- tems consisting from an identical objects. The following results, leading from this paper, on our sight, are essen- tial: 1. The equations describing behavior of conservative sys- tems (in usual time), are irreversible in space with mul- tifractal time; 2. The neglecting by the fractionality of dimension of time and transition in space with topological dimension of time equal to unity allows to receive the known re- versible equations of classical dynamics; 3. In space with the multifractal time there are no invari- able objects, since the GFD with respect to stationary values are not equal to zero. From the physical point of view it is the reflection of non-stationary of the Universewith multifractal time. Last statement corresponds to mathematical exposition of behavior of physical objects and not contradict the exposition of the Einstein type of Universe , in which, in connection with its expansion, there are no invariable objects; 4. The quantity of the fractional additional to topologi- cal dimension of time member εis determined by physical fields and depends on the density of energy that presents in the given moment in the given point of space . At the small densities of energy the corrections are very small. So, for the gravitational fields at distances more larger that gravitational radius (for example, for FD created by mass of the Earth on a surface of Earth) and for electric fields on atomic distances the value of εis equal ε∼10−8. Therefore the multifractal nature of time ( dt∼1+ε) does not contradicts an existing experimental data. [1] L.Boltzmann Wien. Der., 1872, v.66, p.275 [2] Yu.L.Klimontovich Statistical Theory of Open Systems1 (Kluwer, Dordrecht, 1995); Statistical Theory of Open Systems2 (Yanus,Moscow,1999) [3] Prigogin I., From existing to incipient , (Moscow: Science, 1985) [4] Kobelev L.Ya. Fractal theory of time and space (Ekater- inburg, Konross,1999)(in Russian); Kobelev L.Ya. The fractal theory of time and space ; Urals State University, Dep. in VINITI. 22.01.99, No.189-B99 [5] Hausdorf F., Math. Ann. 79 (1919), P.157-179 [6] Renyi A. Introduction to information theory, Appendix in: Probability theory (North Holland, Amsterdam, 1988) [7] Kobelev L.Ya. ”What Dimensions Do the Time and the Spase Have: Integer or Fractional?” LANL arXiv:physics/0001035 17Jan 2000 [8] Kobelev L.Ya. ”Can a Particle Velosity Exeed the Speed of Light in Empty Space?” LANL arXiv:gr-qc/0001042 15 Jan 2000 [9] S.G.Samko , A.A.Kilbas , O.I.Marichev, Fractional Inte- grals and Derivatives - Theory and Applications (Gordon and Breach, New York, 1993) [10] I.M.Gelfand, G.E.Shilov, Generalized Functions (Aca- demic Press, New York, 1964) 4

arXiv:physics/0002003v1 [physics.gen-ph] 1 Feb 2000. 1IS IT POSSIBLE TO TRANSFER AN INFORMATION WITH THE VELOCITIES EXCEEDING SPEED OF LIGHT IN EMPTY SPACE? L.Ya.Kobelev∗ Department of Physics, Urals State University Lenina Ave., 51, Ekaterinburg 620083, Russia abstract:On the base of the theory of time and space with the f ractional dimensions a pos- sibility for information transferring with any velocities is demonstrated. 01.30.Tt.05.45, 64. 60.A; 00.89.98.02.90. + p 1 Introduction In the theory of special relativity (SR) the maximal velocit y of any signal does not exceed speed of light in the empty space (the existence of optical ta xions does not break SR ) In the frame of multifractal theory of time and space [1] it is po ssible to construct the theory of almost inertial systems [2]. In this theory an arbitrary v elocities of moving particles are possible if the approximate independence of speed of light f rom the velocity of the light source and the approximate constancy of speed of light in vacuum are valid (the breaking the low of constancy speed of light are less than possibility of modern experiment and consist ∼10−10c, see [1]). Is the transfer of the information within the frame work of the theory [2]-[4] possible with any velocities? The difficulty of create a signal carrier of the information spreading with arbitrary large (practically infinitely large) velocity is not the main difficulty at the answer to this question. These signals can be, for example, a beams o f charged particles (protons, ionized atoms) accelerated up to velocities greater then th e speed of light (their energy must be more then energy E0103where E0=moc2) and then spontaneously accelerated at almost infinite quantity velocity. These beams may be the carriers o f the transferring information. The difficulty consists in the creating the receivers (detect ors) of the information recorded by beams (or single) faster than light particles. According to the theory [2]-[4] a particle with velocity v > c is spontaneously accelerated up to the velocity v=∞and practically ceases to interact with a surrounding medium. The purpose of this pape r is the attempt to analyze some opportunities of detection of such particles. If the proble m of detectors for registration of the faster than light particles will be decided, the problem of p ractically instantaneous transfer of the information at any distances is solved positively. ∗E-mail: leonid.kobelev@usu.ru 22 What physical effects are existing for detection of particles moving with velocity v > c? Let us suppose validity of the laws of the electrodynamics fo r the velocities v > c . After replacing β=/radicalbig 1−v2/c2byβ∗=4/radicalbig (1−v2/c2)2+ 4a2, (see designations ain [2]-[4]) the Lorentz’s transformation also may be used. In that case for t he moving electrical charged particle possessing velocity v≃ ∞ and energy E=√ 2E0, near to the device playing role of the detector, there are following effects can be probably use d for detection of the fact of transit of a particle: a) In the real physical world any of the physical quantities c an not be equal infinity, so we shall introduce for designation of maximal velocity of a particle designation vm(vmis the velocity of a faster than light particle for which the energy loss acco mpanying by increase of velocity is compensated by magnification of the energy gained from a me dium in which the beam of the particles flow by, i.e. the velocity of a particle becomes practically stationary value, for example be in thermodynamic equilibrium with the relic radi ation that gives the particle the velocity vm∼500c). There are an almost instantaneous impulses of electrical and magnetic fields from an electrical current formed by transit of the fas ter than light particle through the media. These impulses could be discovered by detectors that are capable to detect super short impulses electrical or magnetic fields; b)The kinetic energy of a faster than light particle at v > c looks like Ek≈√ 2E0c2/v2 m. The transfer of parts of this energy basically can be registered by high precision detector’s (counters of prompt particles for example based on use of an inner photo electric effect) in case when a faster than light particle has the collision at a proton, nuc lear or electron; c) In the lengthy detector filled by substances with large den sity (small free length of collisions for particles) will arise the multiple collisions of faster than light particles with atoms. It can gives energy transition from substances to a faster than lig ht particle and by that to decreasing of its velocity. The power transmission from medium to a part icle will gives in decreasing of temperature of medium and besides gives the radiation of Che renkov-Vavilov type (in an region of frequencies defined by number of collisions with atoms of t he substance of the detector); d) When the faster than light particle fly through the substan ce with many energy levels with negative temperatures as result may be lost of energy of subs tance without radiation and decreasing of negative temperature of active optical subst ance. Physical laws do not forbid all numbered methods of detection for ordinary particles with f aster than light velocities and their experimental realization (as well as many other method’s ar e based on an energy exchange of a faster than light particle with energy of medium) are pos sible. The numbered methods realization depends on the value of maximal velocity vm. 3 Are the particles with v > c and real mass exist in nature? Let us put the question: are the faster than light particles e xist in our world? When and where such particles can be discovered? As the one of consequences of the theory of fractal time (see [1]), the particles with velocities exceeding the velocity of light must have an energy exceeding their the rest energy Eoin 103times. Such particles may be borne for example by explosions of stars ( in that case it is possible to expect the appearance of the maximum in the spectrum of γ- quanta for the energies E0103or at the first moments of ”big bang” when temperatures of the early Universe exceed 1016K. If a neutrinos have the rest mass and its rest energy are smal l 3and have the order (or less) 1 ev.the neutrinos with faster then light velocities may be produ ced by stars, by nuclei explosions and in the reactions of thermo nuclear controlled syntheses. May some super civilization use the faster then light particles , if this civilization has the technology of receiving the beams of such particle, for record and trans lating information with the faster than light velocities ? In that case it is necessary to seek su ch particles by mentioned above (or similar) methods. 4 Conclusion On the basis of the above-stated treatment of possibilities of detection of the particles with the faster than light speed, it is possible to make a deduction: t he prohibitions for transfer and receiving of the information with faster than light speed ar e absent (if the theory [1]-[4] are valid). The question about an existence of the ordinary part icles (protons, electrons, neutrinos) with velocities faster than light and the real mass in nature ( that question was presented (and decided) for the first time in the paper [2] as one of the conseq uences of the theory of almost inertial systems that lays beyond of the special relativity and coincides with SR in the case of ideal inertial systems) is now unsolved. The search of taxio ns continue more than thirty years. I don’t mention about the optical taxions. The existence of t he optical taxions do not contra- dict the SR and apparently they are discovered. I think that o nly careful experimental search of the ordinary particles with the real mass (the faster than light particles) and experiments that may examine the fractal theory of time [1] may throw ligh t on this very interesting problem. We suggest to carry out the experiments for receiving by acce lerating the protons with en- ergies equal ∼1012ev.(that gives a protons the velocity equal the speed of light if the theory [1]-[2] are valid) , then to verify the predictions of the the ory that presented in this paper and papers [2]-[4] References [1] Kobelev L.Ya. Fractl theory of time and space /Edit.by Kobelev L.Ya., Ekaterinburg, Kon- ross,1999,p.136 ;LANL arXiv:physics/0001035 17 Jan 2000 [2] Kobelev L.Ya. Multifractality of time and special theory of relativity Urals State University, Dep. v VINITI 19.08.99, No.2677-B99 1. [3] Kobelev L.Ya. Can a particle moves with velosity exeeding the speed of ligh t in empty space? Urals State University,/Dep. v VINITY, 20.10.99, No.3129- B99;LANL arXiv:gr- qc/0001042 15 Jan 2000 [4] Kobelev L.Ya. Physical effects of a particles moving with velosity exeedin g the speed of light in empty space /Dep.v VINITY.20.10.99, No.3128-B99; LANL arXiv:gr-qc/0 001043 17 Jan 2000 4

arXiv:physics/0002005v1 [physics.atom-ph] 3 Feb 2000Bethe logarithms for the 11S,21S and 23S states of helium and helium-like ions Jonathan D. Baker1Robert C. Forrey2Malgorzata Jeziorska3John D. Morgan III4 1National Institute of Standards and Technology, Gaithersb urg, MD 20899 2Penn State University, Berks-Lehigh Valley College, Readi ng, PA 19610-6009 3Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland 4Department of Physics and Astronomy, University of Delawar e, Newark, DE 19716 (February 2, 2008) We have computed the Bethe logarithms for the 11S, 21S and 23S states of the helium atom to about seven figure-accuracy using a generalization of a meth od first developed by Charles Schwartz. We have also calculated the Bethe logarithms for the helium- like ions Li+, Be++, O6+and S14+ for all three states to study the 1 /Zbehavior of the results. The Bethe logarithm of H−was also calculated with somewhat less accuracy. The use of our B ethe logarithms for the excited states of neutral helium, instead of those from Goldman and D rake’s first-order 1 /Z-expansion, reduces by several orders of magnitude the discrepancies be tween the theoretically calculated and experimentally measured ionization potentials of these st ates. PACS numbers: 31.15.Ar, 31.30.Jv Ever since the invention of quantum mechanics, the helium at om has served as an important testing-ground for our understanding of fundamental physics. In 1929 Hylleraa s’ calculation of the binding energy of the non-relativisti c helium atom Hamiltonian showed that Schroedinger’s formul ation of quantum mechanics provided a quantitatively accurate description of not just two-body but three-body sy stems [1]. During the 1950’s, with the advent of fast digital computers, calculations by Kinoshita [2] and Peker is [3] of not only the non-relativistic binding energy but also of relativistic corrections of O( α2) Rydberg greatly improved the agreement between theory and experiment, and showed that the estimation of O( α3) Rydberg effects arising from quantum electrodynamics was i mportant for obtaining agreement between theory and experiment at the le vel of 1 part in 106or better. During the 1960’s and 1970’s the variational techniques employed by Pekeris on th e lowest states of singlet and triplet symmetry were extended to a wide range of excited states of the helium atom [ 4] [5]. During the 1980’s, with the advent of two- photon spectroscopy with counterpropagating laser beams, which can be used to eliminate the 1st-order Doppler shift due to the thermal motion of the atoms, it became possib le to measure the wavelengths for transitions between excited states of the helium atom with a precision of 1 part in 109or better [6]. Though numerous examples of excellent agreement between theory and experiment in a wide variety of contexts leave no reasonable doubt that quantum electrodynamics is the correct theory for describi ng the interactions of charged particles at low energies, the extraordinary accuracy recently achieved in high-prec ision measurements on the helium atom poses a challenge to theorists to develop computational techniques capable o f matching such accuracies. Since α3is of order 10−6, it is clear that the coefficient of the lowest-order QED correction s needs to be evaluated with a relative accuracy of 10−3 or better, and the effects of contributions with higher power s ofαmust also be estimated, to match the experimental accuracy of 1 part in 109or better. For a helium atom or helium-like ion of atomic number Z, the leading O( α3) Rydberg contribution to the Lamb shift is given by the expression [7] EL,2=8 3Zα3Ψ2 0(0)/bracketleftbigg 2 ln/parenleftbigg1 α/parenrightbigg −ln/parenleftbiggk0 Ry/parenrightbigg +19 30/bracketrightbigg Ry, where the so-called Bethe logarithm [8] is defined by an infini te and slowly-convergent sum over all bound and continuum eigenstates: ln(k0/Ry) =β D= /summationtext n| /angb∇acketleftΨn|p|Ψ0/angb∇acket∇ight |2(En−E0)ln|En−E0|/summationtext n| /angb∇acketleftΨn|p|Ψ0/angb∇acket∇ight |2(En−E0), Herepis the sum of single-particle momentum operators ( p=/summationtext ipi) and Ψ0is an eigenfunction with eigenvalue E0 of the Hamiltonian Hof the atom. For simplicity, we assume that His the nonrelativistic Hamiltonian of an atom with atomic number Z, with a point nucleus of infinite mass: H=T+V=/summationdisplay ip2 i/2−Z/summationdisplay i1/ri+/summationdisplay i>j1/rij,which has the important and useful property that it is unitar ily equivalent to the scaled Hamiltonian Z2 /summationdisplay ip2 i/2−/summationdisplay i1/ri+ (1/Z)/summationdisplay i>j1/rij , which after division by Z2tends to a well-defined limit as Z→ ∞. (The effects of the reduced mass µ=meMN/(me+ MN) due to the finiteness of the nuclear mass MNare subsequently included by scaling by appropriate powers ofµ/m e, and the negligible effect of the ‘mass-polarisation’ term M−1 N/summationtext i>jpi·pjon the Bethe logarithm is here ignored.) With the help of the closure relation/summationtext n|n/angb∇acket∇ight(En−E0)/angb∇acketleftn|=H−E0, the commutation relation ( H−E0)pΨ0=i(∇V)Ψ0, an integration by parts, and Gauss’ Law ( ∇2V= 4πZ/summationtext iδ(3)(ri)), the denominator Dis easily evaluated: D=/angb∇acketleftΨ0|p·(H−E0)p|Ψ0/angb∇acket∇ight= 2πZΨ2 0(0), but the logarithmic factor makes the numerator βmuch harder to evaluate. Even for a very simple one-electron system such as the hydrogen atom, βcannot be evaluated in closed form, though several rapidly c onvergent methods can be used to evaluate it to high accuracy [9], [10], [11], [1 2], and the Bethe logarithm of the electronic ground state of H+ 2was recently evaluated numerically [13]. For a two-electro n system such as the helium atom, whose unknown wavefunction Ψ 0must be represented by an expansion in a large basis set, the n umerical challenges are even more daunting. In the early 1960’s C. Schwartz recast the numerator as integ ral over the virtual photon energy k[14], β= lim K→∞/parenleftBig −K/angb∇acketleftΨ0|p·p|Ψ0/angb∇acket∇ight+Dln(K)+ /integraldisplayK 0kdk/angb∇acketleftΨ0|p·(H−E0+k)−1p|Ψ0/angb∇acket∇ight/parenrightBig , (1) and thereby replaced the insuperable difficulties associate d with accurately summing over an infinite number of bound and continuum eigenstates of Hwith the more tractable difficulty of numerically integratin g an accurate representation of the matrix element of the resolvent ( H−E0+k)−1for small, intermediate and large values of k. Whenkis very large, Schwartz found it sufficient to approximate the matrix element with a simple asymptotic formula. For smaller values ofk, the action of the resolvent is solved explicitly as the solu tion of a system of linear equations in a suitable basis withp-wave symmetry. For intermediate kthe convergence was greatly improved by including a single f unction which has the same leading-order asymptotic behavior as the true solution as k→ ∞. Despite growing problems with the numerical linear depende nce of his basis as the number of basis functions was increased, Schwartz was able to compute for the 11S ground state of the neutral helium atom a Bethe logarithm of 4.370(4) Rydbergs, which yielded a theoretical ionizati on potential for this state in agreement with the best experimental values available at that time, and which remai ned unsurpassed until very recently. The results presented in this letter were generated by an app roach very similar to that used by Schwartz, in which the integral in Eq. (1) is split into a low kregionβLand a high kregionβH. The counterterms in Eq. (1) are then brought inside the integral to cancel explicitly the diverg ent behavior at large k: β=βH+βL=/integraldisplay∞ 1dk k/angb∇acketleftΨ0|p·(H−E0)|ψH(k)/angb∇acket∇ight +/integraldisplay1 0kdk/parenleftbigg /angb∇acketleftΨ0|p|ψL(k)/angb∇acket∇ight −/angb∇acketleftΨ0|p·p|Ψ0/angb∇acket∇ight k/parenrightbigg , (2) whereψL(k) andψH(k) are solutions of the equations (H−E0+k)ψL(k) =pΨ0, (3) (H−E0+k)ψH(k) = (H−E0)pΨ0. (4) SinceHpossesses overall rotational symmetry, the solutions ψL(k) andψH(k) have a total angular momentum quantum number which can differ by only ±1 from that of Ψ0. In this work Ψ0hasS-symmetry, so ψL(k) andψH(k) haveP-symmetry. An elegant derivation of Eq. (2) can be found in the work of For rey and Hill [12], which examines Schwartz’s method from a fresh perspective and provides many useful computati onal techniques. We evaluate the two integrals in Eq. (2) numerically, using the procedure described by Forrey an d Hill, computing the matrix element of the resolvent ateach integration knot by solving variationally for ψLorψHin Eq. (3) and Eq. (4). When kis very large, we use the asymptotic approximation [14] /angb∇acketleftΨ0|p(H−E0)|ψH(k)/angb∇acket∇ight= 2ZD k/bracketleftbigg√ 2k−Zln(k) +C+D√ k+···/bracketrightbigg . (5) The constants CandDhave been computed in closed form only for the hydrogen atom; in this work they are estimated by extrapolating the values generated by the solution of Eq. (4) at successive integration knots. This equation was solved explicitly at each successive knot, running in the di rection of increasing k, until the relative difference between successive extrapolated estimates of Cwas roughly 1%. For larger kthe resulting asymptotic formula was used. For the helium ground state our estimates of CandDare 4.988(1) and -18.8(3) respectively, with the errors res ulting mainly from extrapolation uncertainty. These estimates ca n be compared with the value 5.18 computed by Schwartz [14] forCand the value -20 ±3 he assumed for D. The non-relativistic wavefunction Ψ0was computed variationally using our modification [10], [15 ] of the basis set first developed by Frankowski and Pekeris [4], which expl oits knowledge of the analytic structure of the true wavefunction at the 2- and 3-particle coalescences to impro ve the convergence of the variational trial function to the exact unknown wavefunction: Ψ0=/summationdisplay νcν(φν(s,t,u)±φν(s,−t,u)) φν(s,t,u) =sntlum(lns)je−as+ct wheres,t, anduare the Hylleraas coordinates defined by s=r1+r2,t=r2−r1andu=r12and the ±sign is chosen so that the product of Ψ0and the spin function is antisymmetric under exchange of the electrons. Our bases for representing ψL(k) andψH(k) include functions of four different types. The k-independent functions χ(1) ν=r1φν(s,t,u)±r2φν(s,−t,u) together with the single function χ(2)=pΨ0provide a good solution space for small k. For largekthe solution ψH(k) becomes concentrated in k-dependent regions of configuration space for which one electron is very close to the nucleus and the other electron i s much further away, so it is essential to use explicitly k-dependent basis functions. Of primary importance is the ‘S chwartz function’ χ(3), an approximate solution of Eq. (4) that reproduces the first two terms in the asymptotic expa nsion in Eq. (5): χ(3)=/parenleftBigg p1exp (−√ 2kr1)−1 r1Ψ0/parenrightBigg ±(r1↔r2). To help approximate that part of ψH(k) which is orthogonal to the ‘Schwartz function’, we also use a fourth set of functionsχ(4) ν, which are symmetrized sums of products of single-variable Laguerre functions Li(Rj)=Li(Rj)e−Rj/2 of the three perimetric coordinates R1=r1+r2−r12,R2=r1−r2+r12, andR3=−r1+r2+r12: χ(4) ν= (r1Lp(aR1)Lq(bR2)Lr(cR3))±(r1↔r2). Combinations of the exponential parameters a,b, andccan be chosen to reflect the strong ‘in-out’ correlation in ψH(k) for largek. For anykthe overlap matrix elements for these basis functions are ve ry small or zero far from the main diagonal, which enables us to avoid the severe problems with numerical linear dependence which prevented Schwartz from using a large basis of functions of the form of powers of r1,r2,r12times a highly asymmetrical exponential ofr1andr2. We seta=b+cto eliminate from χ(4) νany exponential r12-dependence, which would complicate the evaluation of matrix elements between these functions and t he other types of basis functions. Analytic considerations [12] suggest that the integrand is optimised if b≈(2k)1/4andc≈Z. We coarsely search the parameter space in the neighborhood of these values of bandcseeking to maximize the two integrands of Eq. (2) in accordan ce with the variational principle described in [12]. The calculation of βLwas fast and straightforward. In this case kwas small enough that there was no need to include explicit k-dependence in the basis. χ(3)was omitted altogether, and a single average value of the parameter bwas used in the χ(4) νfunctions, independent of the value of kat a particular integration knot. We solved for ψL(k)in a basis with 92 χ(1) νfunctions, the χ(2)function, and 120 χ(4) νfunctions. The parameters bandcwere varied to maximize the integrand. Changes in the integrand due to smal l variations in bandcwere used to assess convergence. TheβHintegral was computationally expensive, primarily becaus e including the ‘Schwartz function’ χ(3)requires evaluating algorithmically complicated matrix elements. Sinceχ(3)is intended primarily to accelerate the convergence for very large k, and since over half of the knots in our integration scheme co rrespond to k <40, we chose to omit χ(3)from the basis for knots below k≈40. At each node we solved for ψHin a basis consisting of 92 χ(1) νfunctions, theχ(2)function, the χ(3)function (for high k), and 220χ(4) νfunctions. We then recomputed the solution of Eq. (4) after first reducing the number Nofχ(4) νfunctions in the existing matrices to study convergence of t he integrand. A simple polynomial fit in the variable 1 /Nwas applied to the sequence of results with N= 220, 165, 120, and 84 to generate the values of the Bethe logarithms in this letter. T he error associated with the finiteness of the basis for ψH was taken as the entire difference between the extrapolated v alue and the value corresponding to N= 220. Other sources of numerical error arise from the numerical in tegration itself (for which there are good analytic error bounds [12]), and the finiteness of the basis used to approxim ateΨ0. The latter error is assumed to be comparable to the relative error in Din all cases. For neutral helium, independent runs with less accurate representations of Ψ0indicate that this estimate of this error is somewhat conser vative. The numerical integration was parametrized to keep the absolute error in βHandβLbelow 10−8. The results of independent calculations carried out for ne utral helium with a coarser mesh were consistent with the analytic error bound. The uncertainties assigned to the Bethe logarithms in this l etter are the sums of the uncertainty due to extrapolation ofψHand the uncertainty due to approximation of Ψ0in a finite basis. The uncertainty in ψLand the numerical integration error bounds are negligible by comparison. Our Bethe logarithms for the 11S, 21S, and 23S states are listed in Tables I, II, and III, respectively. Th e values of k0have been divided by Z2to illustrate their approach to the hydrogenic limit as Zbecomes large. Scaled values of the nonrelativistic binding energy Enr/Z2andD/Z4are also listed to provide some measure of the accuracy of Ψ0. Uncertainties in Dwere computed by comparison with highly accurate results fo r<δ(r1)>provided by Drake [16]. The exact hydrogenic limits of ln( k0/Ry) [11], [17] and of EnrandDare displayed in the bottom row of each table, labeled by 1 /∞(exact). Immediately above the bottom row, in the row labele d 1/∞, we list the hydrogenic values and the corresponding uncertainties computed using the met hod described in this letter with 1 /Z= 0 so that the 1/r12term is removed from the Hamiltonian. ForZ= 1 the Hamiltonian Hhas a single bound state of 11S symmetry. As Z→1 from above all the singly excited bound states of a two-electron ion disappear into th e continuum as the ‘outer’ electron moves infinitely far away. Hence as Z→1 from above, the energies and all other finite-range propert ies of the states should tend toward those for a single hydrogen atom in its ground state with Z= 1. The approach of the Bethe logarithms and other properties toward their hydrogenic values as Z→1 from above is visible in Tables II and III for the 21S and 23S states, respectively. We have fit our ionic results to the 1 /Zexpansion developed by Goldman and Drake [18] ln(k0/Ry) =C0+C1/Z+C2/Z2+··· C0= ln 2 + 2 ln Z+ ln(kH/Ry), (6) where ln(kH/Ry) is the weighted sum of the two hydrogenic Bethe logarithm s corresponding to the state. Table IV displays the results of a three parameter polynomial fit for C1,C2, andC3using data for Z= 4,8,and 16. The listed uncertainties come from the formal propagation of error thr ough the regression formula and do not include truncation errors from higher-order terms in the expansion. Our results for the 11S state and the 21S states of neutral helium are in complete agreement with the recent calculations of Korobov and Korobov [19]. The most accurate previous value of the Bethe logarithm of the 23S state came from Goldman and Drake’s 1st-order 1 /Zexpansion [17], [18]. A numerical comparison of results for neutral helium appears in Table V. Preliminary values of our Bethe logarithms for the 11S, 21S [20] and 23S states of helium were used in a recent comparison of theory and experiment by Drake and Martin [21] . The values in this letter make slight corrections to the theoretical ionization energies of the 21S and the 23S levels in that work, while the 11S state is unaffected. Modifying “ Bethe log cor. ” contribution in Drake and Martin’s Table II to include the v alues in this letter yields the theoretical results in Table VI, which are compared with res ults from several recent experiments [22], [23], [24], [25] , [26]. We are indebted to P.J. Mohr for helpful discussions related to this work and for his assistance in securing resources at NIST. All numerical results in this letter were generated in the fall of 1998 on either the NIST J40 IBM RS/6000SMP machine or on the IBM SP2, also at NIST.1We would also like to acknowledge R.N. Hill for contributing several useful ideas for setting up and performing the numer ical integration over virtual photon energy k. We thank G.W.F. Drake for helpful discussions at an earlier stage of t his work, for kindly providing us with unpublished data from his work on helium-like ions [27], and also for performi ng additional calculations to facilitate our estimation of the uncertainty in D. We also thank W.C. Martin for helpful discussions and Janin e Shertzer and Tony Scott for their assistance and advice with the evaluation of integ rals. We are also grateful to V.I. Korobov for keeping us informed of his calculation of the Bethe logarithms. Some co mputer runs with an earlier version of this program were performed on an RS/6000 system at the University of Washingt on kindly made available to us by W.P. Reinhardt, and also on the SP-2 system at the Cornell Theory Center. This work was supported by NSF grants PHY-8608155 and PHY-9215442 and by a NIST Precision Measurement Grant to J.D. Morgan at the University of Delaware, and by an NRC Postdoctoral Fellowship held by J.D. Baker at NIST. J.D. Morgan thanks the Institute for Theoretical Atomic and Molecular Physics at Harvard University, and its previous director, A. Dalgarno, for support in 1989-90 and 1992. J.D. Morgan and J.D. Baker thank D. Herschbach and t he members of his research group at Harvard University for their hospitality. which has greatly facili tated this work. They also thank the Institute for Nuclear Theory at the University of Washington for providing suppor t in the spring of 1993. J.D. Morgan is further indebted to C.J. Umrigar and M.P. Teter of the Cornell Theory Center fo r sabbatical support in 1995. [1] E.A. Hylleraas, Z. Phys. 54, 347 (1929). [2] T. Kinoshita, Phys. Rev. 105, 1490 (1957); ibid.115, 366 (1959). [3] C.L. Pekeris, Phys. Rev. 112, 1649 (1958); ibid.115, 1216 (1959); ibid.127, 509 (1962). [4] K. Frankowski and C.L. Pekeris, Phys. Rev. 146, 46 (1966); K. Frankowski, Phys. Rev. 160, 1 (1967). [5] Y. Accad, C.L. Pekeris, B. Schiff Phys. Rev. A 4, 516 (1971). [6] E. Giacobino and F. Biraben, J. Phys. B: At. Mol. Phys. 15, L385 (1982); L. Hlousek, S. A. Lee, and W. M. Fairbank, Jr., Phys. Rev. Lett. 50, 328 (1983); P. Juncar, H. G. Berry, R. Damaschini, and H. T. D uong, J. Phys. B: At. Mol. Phys. 16, 381 (1983); C. J. Sansonetti and W. C. Martin, Phys. Rev. A 29, 159 (1984); C. J. Sansonetti, J. D. Gillaspy, and C. L. Cromer, Phys. Rev. Lett. 65, 2539 (1990). [7] P.K. Kabir and E.E. Salpeter, Phys. Rev. 108, 1256 (1957). [8] H.A. Bethe, Phys. Rev. 72, 399 (1947). [9] S.P.Goldman, Phys.Rev. A, 30, 1219, (1984). [10] J.D. Baker, R.N. Hill, and J.D. Morgan III, “High Precis ion Calculation of Helium Atom Energy Levels”, in AIP Conference Proceedings 189,Relativistic, Quantum Electrodynamic, and Weak Interacti on Effects in Atoms (AIP, New York, 1989), 123; [11] G.W.F. Drake and R.A. Swainson, Phys. Rev. A 41, 1243 (1990); [12] R. C. Forrey and R. N. Hill, Ann. Phys. 226, 88 (1993). [13] R. Bukowski, B. Jeziorski, R. Moszy´ nski, and W. Kolos, Int. J. Quantum Chem. 42, 287 (1992). [14] C. Schwartz, Phys. Rev. 123, 1700 (1961). [15] D.E. Freund, B.D. Huxtable and J.D. Morgan III, Phys. Re v. A29, 980 (1984); J.D. Baker, D.E. Freund, R.N. Hill, and J.D. Morgan III, Phys. Rev. A 41, 1247-1273 (1990). [16] G.W.F. Drake, private communication. [17] G.W.F. Drake, “High Precision Calculations for Rydber g States of Helium”, in Long Range Casimir Forces: Theory and Recent Experiment on Atomic Systems (Plenum Press, New York, 1993), 163. [18] S. P. Goldman and G.W.F. Drake, J. Phys. B 16, L183 (1983); 17, L197 (1984). [19] V.I. Korobov and S.V. Korobov, Phys. Rev. A 59, 3394 (1999). [20] J.D. Baker, R.C. Forrey, J.D. Morgan III, R.N. Hill, M. J eziorska, J. Shertzer, Bull. Am. Phys. Soc., 38, 1127 (1993). [21] G.W.F. Drake and W.C. Martin, Can. J. Phys. 76, 597 (1998). [22] K.S.E. Eikema, W. Ubachs, W. Vassen, and W. Hogervorst, Phys. Rev. Lett. 76, 1216 (1996); K.S.E. Eikema, W. Ubachs, W. Vassen, and W. Hogervorst, Phys. Rev. A 55, 1866 (1997). 1Certain commercial equipment, instruments, or materials a re identified in this paper to foster understanding. Such ide nti- fication does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessa rily the best available for the purpose.[23] S.D. Bergeson, A. Balakrishnan, K.G.H. Baldwin, T.B. L ucatorto, J.P. Marangos, T.J. McIlrath, T.R. O’Brian, S.L. Rolston, C.J. Sansonetti, J. Wen, and N. Westbrook, Phys. Re v. Lett. 80, 3475 (1998). [24] W. Lichten, D. Shiner, and Z.-X. Zhou, Phys. Rev. A 43, 1663 (1991); ibid.45, 8295 (1992). [25] C.J. Sansonetti and J.D. Gillaspy, Phys. Rev. A 45, R1 (1992), and unpublished data (1996). [26] C. Dorrer, F. Nez, B. de Beauvoir, L. Julien and F. Birabe n, Phys. Rev. Lett. 78, 3658 (1997). [27] G.W.F. Drake, Can. J. Phys. 66, 586 (1988).TABLE I. ln( k0/(Z2Ry)), Enr/Z2andD/Z4for the 11S state. 1/Z ln(k0/(Z2Ry)) Enr/Z2(a.u.) D/Z4(a.u.) 1/1 2.992 97(5) -0.527 751 015 308 2.067 80(4) 1/2 2.983 864(2) -0.725 931 094 259 2.843 815 67(5) 1/3 2.982 624(2) -0.808 879 268 074 3.189 069 9(1) 1/4 2.982 503(1) -0.853 472 889 901 3.376 853 2(1) 1/8 2.982 948(2) -0.924 321 798 793 3.677 270 31(4) 1/16 2.983 448(1) -0.961 551 275 290 3.835 858 39(2) 1/∞ 2.984 128 6(7) -1.0 4.0 1/∞(exact) 2.984 128 556 -1.0 4.0 TABLE II. ln( k0/(Z2Ry)) , Enr/Z2andD/Z4for the 21S state. 1/Z ln(k0/(Z2Ry)) Enr/Z2(a.u.) D/Z4(a.u.) 1/1 (exact limit) 2.984 128 556 -0.5 2.0 1/2 2.980 115(1) -0.536 493 511 514 2.056 896 21(2) 1/3 2.976 362(2) -0.560 097 416 177 2.103 163 60(3) 1/4 2.973 976(1) -0.574 054 618 459 2.132 593 60(7) 1/8 2.969 797(4) -0.597 793 082 931 2.185 583(1) 1/16 2.967 459(3) -0.610 956 015 708 2.216 320 3(6) 1/∞ 2.964 977 7(4) -0.625 2.25 1/∞(exact) 2.964 977 593 -0.625 2.25 TABLE III. ln( k0/(Z2Ry)) , Enr/Z2andD/Z4for the 23S state. 1/Z ln(k0/(Z2Ry)) Enr/Z2(a.u.) D/Z4(a.u.) 1/1 (exact limit) 2.984 128 556 -0.5 2.0 1/2 2.977 742(1) -0.543 807 344 559 2.074 008 93(2) 1/3 2.973 852(1) -0.567 858 596 952 2.124 087 184(9) 1/4 2.971 735(1) -0.581 072 911 861 2.152 566 566(2) 1/8 2.968 414(2) -0.602 260 114 376 2.199 147 9(4) 1/16 2.966 705(1) -0.613 440 895 056 2.224 062 6(2) 1/∞ 2.964 977 6(2) -0.625 2.25 1/∞(exact) 2.964 977 593 -0.625 2.25 TABLE IV. Coefficients of the 1 /Zexpansion using a 3 parameter fit. The exact values of C1are due to Drake [17]. 1/Zcoeff 11S 21S 23S C1(exact) -0.0123 03(1) 0.040 771(1) 0.027 760(1) C1 -0.0123 2(5) 0.040 78(10) 0.027 73(5) C2 0.0228(8) -0.016(2) -0.001 0(6) C3 0.002(2) -0.011(6) -0.007(2) TABLE V. Comparison of ln( k0/Ry) for neutral helium. The uncertainty in the 1st-order 1 /Zexpansion due to uncalculated higher-order terms could not be readily estimated until mor e exact calculations were done. State 1st-order 1 /Zexpansion Schwartz Korobov This work 11S 4.364(?) 4.370(4) 4.370 157 9(5) 4.370 159(2) 21S 4.372(?) ——— 4.366 409 1(5) 4.366 409(1) 23S 4.365(?) ——— —————— 4.364 036(1)TABLE VI. Ionization potentials, in MHz, as described in [21 ], but with theoretical values corrected slightly by the res ults in this letter. a: from [22], b: from [23], c: an average of val ues from [24] and [25], d: from [26]. State This work Experiment Difference 11S 5945 204 226(91) 5945 204 238(45)a12(102) 5945 204 356(48)b130(103) 21S 960 332 040.9(25.0) 960 332 041.01(15)c0.1(25.0) 23S 1152 842 738.2(25.2) 1152 842 742.87(6)d4.7(25.2)

arXiv:physics/0002006v1 [physics.optics] 3 Feb 2000Spatiotemporally Localized Multidimensional Solitons in Self-Induced Transparency Media M. Blaauboer,a,bB.A. Malomed,aG. Kurizki,b aDepartment of Interdisciplinary Studies, Faculty of Engin eering, Tel Aviv University, Tel Aviv 69978, Israel bChemical Physics Department, Weizmann Institute of Scienc e, Rehovot 76100, Israel (December 9, 2013) ”Light bullets” are multi-dimensional solitons which are localized in both space and time. We show that such solitons exist in two- and three-dimensional self-induced-transpa rency media and that they are fully stable. Our approximate an- alytical calculation, backed and verified by direct numeric al simulations, yields the multi-dimensional generalizatio n of the one-dimensional Sine-Gordon soliton. PACS numbers: 42.50 Rh, 42.50 Si, 3.40 Kf physics/0002006 The concept of multi-dimensional solitons that are localized in both space and time, alias ”light bullets” (LBs), was pioneered by Silberberg [1], and has since then been investigated in various nonlinear optical media, with particular emphasis on the question of whether these solitons are stable or not. For a second-harmonic gener- ating medium, the existence of stable two- and three- dimensional (2D and 3D) solitons was predicted as early as in 1981 [2], followed by studies of their propagation and stability against collapse [3–6], and of analogous 3D quantum solitons [7]. In a nonlinear Schr¨ odinger model both stable and unstable LBs were found [8] and it was suggested that various models describing fluid flows yield stable 2D spatio-temporal solitons [9]. Recently, the first experimental observation of a quasi-2D ”bullet” in a 3D sample was reported in Ref. [10]. In this letter we predict a new, hitherto unexplored type of LBs, obtainable by 2D or 3D self-induced trans- parency (SIT). SIT involves the solitary propagation of an electromagnetic pulse in a near-resonant medium, irre- spective of the carrier-frequency detuning from resonance [11,12]. The SIT soliton in 1D near-resonant media [13] is exponentially localized and stable. In order to investigat e the existence of ”light bullets” in SIT, i.e. solitons that are localized in both space and time, one has to consider a 2D or 3D near-resonant medium. Here we present an approximate analytical solution of this problem, which is checked by and in very good agreement with direct numerical simulations. Our starting point are the two-dimensional SIT equa- tions in dimensionless form [14] −iExx+Ez− P= 0 (1a) Pτ− EW= 0 (1b) Wτ+1 2(E∗P+P∗E) = 0. (1c)HereEandPdenote the slowly-varying amplitudes of the electric field and polarization, respectively, Wis the inversion, zandxare respectively the longitudinal and transverse coordinates (in units of the effective ab- sorption length αeff), and τthe retarded time (in units of the input pulse duration τp). The Fresnel number F (F >0), which governs the transverse diffraction in 2D and 3D propagation, is incorporated in xand the detun- ing ∆Ω of the carrier frequency from the central atomic resonance frequency is absorbed in EandP[15]. We have neglected polarization dephasing and inversion de- cay, considering pulse durations that are much shorter than the corresponding relaxation times. Eqs. (1) are then compatible with the local constraint |P|2+W2= 1, which corresponds to conservation of the Bloch vector [14]. The first nontrivial question is to find a Lagrangian representation for these 2D equations, which is necessary for adequate understanding of the dynamics. To this end, we rewrite the equations in a different form, introducing the complex variable φdefined as follows [16] φ≡1 +W P=P∗ 1−W⇐⇒ P =2φ∗ φφ∗+ 1, W=φφ∗−1 φφ∗+ 1. (2) Eqs. (1b) and (1c) can then be expressed as a single equa- tion,φτ+(E/2)φ2+(1/2)E∗= 0. Next, we define a vari- ablefso that φ≡2fτ/(Ef). In terms of f, the previous equation becomes fττ−(Eτ/E)fτ+ (1/4)|E|2f= 0.This equation is equivalent to fτ=1 2Eg (3a) gτ=−1 2E∗f, (3b) withg≡f φ. Applying the same transformations to Eq. (1a) yields −iExx+Ez−2fg∗= 0. (4) The Lagrangian density corresponding to Eqs. (3) and (4) can now be found in an explicit form, L(x, τ) =1 4ExE∗ x+i 8(EE∗ z− EzE∗)−i 2(f∗gE −fg∗E∗) −i 2/parenleftBig f˙f∗−˙ff∗/parenrightBig −i 2(g˙g∗−˙gg∗). (5) 1Now we proceed to search for LB solutions. Before resorting to direct simulations, we obtain an analytical approximation of the solutions. The starting point for this approximation is the well-known soliton solution for 1D SIT (the Sine-Gordon soliton) [12,14,17] E(τ, z) =±2αsechΘ (6a) P(τ, z) =±2 sechΘ tanhΘ (6b) W(τ, z) = sech2Θ−tanh2Θ, (6c) with Θ( τ, z) =ατ−z α+ Θ0, and α, Θ0arbitrary real parameters. Equation (6a) is also called a 2 π-pulse, be- cause its area/integraltext∞ ∞E(τ, z)dτ=±2π. Returning to the 2D SIT equations, we notice by straightforward substitution into Eqs. (3) that a 2D so- lution with separated variables, in the form E(τ, z, x) = E1(τ, z)E2(x) (and similarly for fandg), does not exist. To look for less obvious solutions, we first split equa- tions (1) into their real and imaginary parts, writing E ≡ E 1+iE2andP ≡ P 1+iP2: E2xx+E1z− P1= 0 (7a) E1xx− E2z+P2= 0 (7b) P1τ− E1W= 0 (7c) P2τ− E2W= 0 (7d) Wτ+E1P1+E2P2= 0. (7e) In the absence of the x-dependence, these equations are invariant under the transformation ( E1,P1)↔(E2,P2). This suggests a 1D solution in which real and imaginary parts of the field and polarization are equal, E1=E2and P1=P2, and such that the total field and polarization reduce to the SG solution (6). Our central result is an approximate but quite accurate (see below) extension of this solution, applicable to the 2D SIT equations. In terms of the original physical variables it is given by E(τ, z, x) =±2α/radicalbig sechΘ 1sechΘ 2exp(−i∆Ωτ+iπ/4) (8a) P(τ, z, x) =±/radicalbig sechΘ 1sechΘ 2{(tanhΘ 1+ tanh Θ 2)2+ 1 4α2C4[(tanh Θ 1−tanh Θ 2)2− 2(sech2Θ1+ sech2Θ2)]2}1/2exp(−i∆Ωτ+iµ) (8b) W(τ, z, x) = [1−sechΘ 1sechΘ 2{(tanh Θ 1+ tanh Θ 2)2+ 1 4α2C4[(tanh Θ 1−tanh Θ 2)2− 2(sech2Θ1+ sech2Θ2)]2}]1/2, (8c) with Θ1=ατ−z α+ Θ0+Cx Θ2=ατ−z α+ Θ0−Cx, µ≡arctan( P2/P1). Hereα, Θ0andCare real constants. Equations (8) sat- isfy the two-dimensional SIT equations (7a) and (7b) andobey the normalization condition P2 1+P2 2+W2= 1. They reduce to the Sine-Gordon solution for C= 0. The accuracy to which Eqs. (8) satisfy Eqs. (7c)-(7e) is O(αC2), which requires that |α|C2≪1. This is the single approximation made. Numerical simulations dis- cussed later on verify that Eq. (8) indeed approximates the exact solution of Eq. (7) to a high accuracy. In ad- dition, we have checked that substitution of (8) into the Lagrangian (5) and varying the resulting expression with respect to the parameters αandCyields zero. This ”variational approach” is commonly used to obtain an approximate ”ansatz” solution to a set of partial differ- ential equations in Lagrangian representation [18]. Equa- tions (8) represents a light bullet , which decays both in space and time and is stable for all values of z. The latter follows directly from (8a) and also from the Vakhitov- Kolokolov stability criterion [19]. -3 0 3time -30030 x02 |E| -3 0 time FIG. 1. The electric field in the 2D ”light bullet”, |E|, as a function of time τ(in units of the input pulse duration τp) and transverse coordinate x(in units of the effective absorption length αeff) after propagating the distance z= 1000. Param- eters used correspond to α= 1,C= 0.1 and Θ 0= 1000. -30 -20 -10 0 10 20 30-30-20-100102030 2FIG. 2. Contourplot of Fig. 1 in the ( τ,x)-plane. Regions with lighter shading correspond to higher values of the elec tric field. Note the different time scale than that of Fig. 1. -3 0 3time -30030 x01 |P| -3 0 time FIG. 3. The polarization in the 2D ”bullet”, |P|, as a func- tion of time τand transverse coordinate x. Parameters used are the same as in Fig. 1. Figs. 1-3 show the electric field and polarization, gener- ated by direct numerical simulation of the 2D SIT equa- tions (1) at the point z= 1000, using (8) as an initial ansatz for z= 0. To a very good accuracy (with a de- viation <1%), they still coincide with the initial con- figuration and analytic prediction (8). The electric field has a typical shape of a 2D LB, localized in time and the transverse coordinate x, with an amplitude 2 αand a nearly sech-form cross-section in a plane in which two of the three coordinates τ,zandxare constant. The ratio C/αdetermines how fast the field decays in the transverse direction. For |C/α| ≪1 (then |C|<1, as |α|C2≪1), we have a relatively rapid decay in τand slow fall-off in the x-direction, as is seen in Fig. 1. In the opposite case, |C/α| ≫1, the field decays more slowly in time and faster in x. The polarization field has the shape of a double-peaked bullet. Its cross-section at constant x displays a minimum at Θmin≈0, where |P(Θmin)| ≈0, and maxima at Θ ±=±Arcosh(√ 2), where |P(Θ±)| ≈1. The field and polarization decay in a similar way, which is a characteristic property of SIT [14]. Also the inver- sion decays both in time and in x, but to a value of −1 instead of zero, corresponding to the atoms in the ground state at infinity. A numerical calculation of the field area atx= 0 yields/integraltext∞ −∞dτ|E(τ, z,0)|= 6.28±0.05≈2π, irrespective of z. By analogy with the SG soliton, one might thus name this a ”2 πbullet”. We have also numerically obtained axisymmetric stable LBs in a 3D SIT medium, see Fig. 4. The 3D medium is described by Eqs. (1) with the first one replaced by −i(Err+r−1Er) +Ez− P= 0, (9) where r≡/radicalbig x2+y2is the transverse radial coordinate. Searching for an analytic 3D bullet solution in the trans-verse plane proves to be difficult. However, in the limit of either large or small r, an approximate analytic so- lution may be found. For large r, it again takes the form (8), but now with Θ 1=ατ−z/α+ Θ0+Crand Θ2=ατ−z/α+ Θ0−Cr, where α, Θ0, and Care constants, |α|C2≪1, and it is implied r≫1/|C|. It is in sufficiently good agreement (deviations <5%) with results of simulation of the 3D equations, using this so- lution as an initial ansatz. Comparison of Figs. 1 and 4 shows that the 2D and 3D bullets have similar shapes, but the 3D one decays faster in the radial direction for small rthan the 2D bullet in its transverse direction. -2 0 2time-505 r02 |E| -2 0 2time FIG. 4. The electric field in the 3D ”light bullet”, |E|, as a function of time τand transverse radial variable rafter prop- agating the distance z= 1000. Parameters used correspond toα= 1,C= 0.1 and Θ 0= 1000. For constant τ, the 2D and 3D bullets are localized in both the propagation direction zand the transverse direction(s). One may also ask whether there exist SIT solitons which are traveling (plane) waves in zand lo- calized in x(andy). Using a symmetry argument, it is straightforward to prove that they do notexist. Starting from the SIT equations (1) (in 2D, the 3D case can be considered analogously) we adopt a plane-wave ansatz for EandP, changing variables as follows: x→√ kx(assum- ingk >0),E(τ, z, x)→ E(τ, x) exp(−ikz),P(τ, z, x)→ k−1P(τ, x) exp(−ikz), and W(τ, z, x)→k−1W(τ, x). The equations for the real and imaginary parts of the field then become E2xx− E2− P1= 0 (10a) E1xx− E1+P2= 0, (10b) with the equations for PτandWτgiven by (7c)-(7e). Us- ing the transformation ( E1,P1)↔(E2,P2), which leaves the last three equations invariant but changes the first two, one immediately finds that (10) only admits the trivial solution E1=E2=P1=P2= 0,W=−1. The observation of ”light bullets” in a SIT process re- quires high input power of the incident pulse and high 3density of the two-level atoms in the medium, in order to achieve pulse durations short compared to decoher- ence and loss times. These requirements are met e.g. for alkali gas media, with typical atomic densities of ∼1011atoms/cm3and relaxation times ∼50 ns [20], and for optical pulses generated by a laser with pulse duration τp<0.1 ns. In order to include transverse diffraction, the incident pulse should be of uniform trans- verse intensity and satisfy αeffd2/λ <1 [20], where λand dare its carrier wavelength and diameter respectively [20]. The parameter αin the solution (8), which deter- mines the amplitude of the bullet and its decay in time, corresponds to α∼κzτpvp[13], with κzthe wavevec- tor component along the propagation direction zandvp the velocity of the pulse in the medium, and can thus be controlled by the incident pulse duration and veloc- ity. The parameter C∼κxLx, where κxis the trans- verse component of the wavevector and Lxis the spa- tial transverse width of the pulse, is also controlled by the characteristics of the incident pulse and should sat- isfy the condition κzκ2 xLzL2 x≪1. For a homogeneous (atomic beam) absorber, the effective absorption length αeff∼104m−1and the Fresnel number Fcan range from 1 to 100 [20]. The bullets then decay on a time scale of t∼1−10τp∼10 ns and transverse length of x∼0.1−1 mm, which is well within experimental reach. In conclusion, we predict the existence of fully stable ”light bullets” in 2D and 3D self-induced transparency media. The prediction is based on an approximate analytical solution of the multi-dimensional SIT equa- tions and verified by direct numerical simulation of these PDE’s. Our results suggest an experiment aimed at de- tection of this ”bullet” in an SIT-medium and opens the road for analogous searches for ”light bullets” in other nonlinear optical processes, such as, e.g., stimulated Ra- man scattering, which is analogous to SIT. M.B. acknowledges support from the Israeli Council for Higher Education. Support from ISF, Minerva and EU (TMR) is acknowledged by G.K. [1] Y. Silberberg, Opt. Lett. 15, 1281 (1990). [2] A.A. Kanashov and A.M. Rubenchik, Physica D 4, 122 (1981). [3] K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993). [4] B.A. Malomed, P. Drummond, H.He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997). [5] D. Michalache, D Mazilu, B.A. Malomed, and L. Torner, Opt. Comm. 152, 265 (1998). [6] H. He and P. Drummond, Phys. Rev. E 58, 5025 (1998). [7] K.V. Kheruntsyan and P.D. Drummond, Phys. Rev A 58, 2488 (1998); ibid.R2676 (1998).[8] D.J. Frantzeskakis, K. Hizanidis, B.A. Malomed, and C. Polymilis, Phys. Lett. 248, 203 (1998). [9] G. Gottwald, R. Grinshaw, and B.A. Malomed, Phys. Lett.248, 208 (1998). [10] X. Liu, L.J. Qian, and F.W. Wise, Phys. Rev. Lett. 82, 4631 (1999). [11] S.L. McCall and E.L. Hahn, Phys. Rev. Lett. 18, 408 (1967). [12] G.L. Lamb, Phys. Rev. 43, 99 (1971). [13] See for a review on SIT-theory e.g. A.A. Maimistov, A.M. Basharov, S.O. Elyutin, and Yu. M. Sklyarov, Phys. Rep. 191, 1 (1990) and references in [17]. [14] A.C. Newell and J.V. Moloney, Nonlinear Optics , (Addison-Wesley, Redwood City, 1992). [15] Bringing the Fresnel number Fand detuning ∆Ω back explicitly into Eqs. (1) requires the transformations x→F−1/2x,E(τ,z, x)→ E(τ, z, x)exp( −i∆Ωτ) and P(τ, z, x)→ P(τ, z, x)exp( −i∆Ωτ). [16] G.L. Lamb, Phys. Rev. Lett. 31, 196 (1973); see also M.J. Ablowitz, D.J. Kaup, and A.C. Newell, J. Math. Phys. 15, 1852 (1974) and V.E. Zakharov and A.B. Shabat, Sov. Phys. JETP 34, 62 (1972). [17] G.P. Agrawal in: Contemporary Nonlinear Optics , edited by G.P. Agrawal and R.W. Boyd, (Academic Press, San Diego, 1992). [18] D. Anderson, Phys. Rev. A 27, 3135 (1983); D. Ander- son, M. Lisak, and T. Reichel, J. Opt. Soc. Am. 5, 207 (1988). [19] See e.g. Ref. [4] and references therein. [20] R.E. Slusher in: Progress in Optics Vol. 12, Editor E. Wolf, (North-Holland, Amsterdam, 1974). 4

arXiv:physics/0002007v1 [physics.ed-ph] 3 Feb 2000Teaching the EPR–Paradox at High School ? Gesche Pospiech Institut for Didaktik der Physik Johann Wolfgang Goethe University Frankfurt February 2, 2008Abstract The discovery of quantum mechanics in the beginning of our ce ntury led to a revolution of physical world view. Modern experiments on th e border of the classical and the quantum regime made possible by new techni ques open better insight and understanding of the quantum world and have impa ct on new techno- logical development. Therefore it seems important that stu dents and even pupils at higher grades become acquainted with the principles of qu antum mechanics. A suitable way seems to be given by treatment of the EPR–gedan kenexperiment.0.1 Introduction The first question to be answered is: why should quantum theor y be taught at school at all? For choosing this topic there are the followin g three reasons: 1. Quantum theory is the fundamental theory of modern physic s. It plays a significant role in nearly all modern developments of physi cs. Many recent experiments and research in nanostructures with lar ge applicability in technology rely on quantum effects. 2. Quantum theory has important philosophical aspects. Man y people are highly interested in interpretation and understanding qua ntum theory as shows up in the many popular books about this subject. 3. Pupils at the age from 16+ on are searching for their place i n the world. They are trying to understand the world and are open for philo sophical hints that help them in building their own world view. But often this highly fascinating subject is avoided at scho ol because of math- ematical and conceptual difficulties. I therefore want to sho w a possible way to introduce quantum theory in a manner suitable for interes ted pupils. In this article I concentrate on the mathematical part because here lie some dif- ficulties. For an introduction into the philosophical aspec ts I have developed a dialogue between philosophers from different times - class ical antiquity (Par- menides), the Enlightenment (Kant) and from our century, pu blished elsewhere, ([Pos98]). The goal of the following is to clarify the main di fficulties in teaching the physical basis of quantum theory and how to keep them mino r. Speaking qualitatively about quantum theory in an adequate way is nearly im- possible in itself since all our concepts and terms have been developed along everyday experience. Hence our language is well suited to co mmunicate about concrete physical objects with well determined properties or about psycholog- ical issues. In my opinion this last property should be used i n dealing with the interfering and superposing objects of quantum theory t hat may have more similarities or associations with pyschological feelings than with concrete balls or waves occurring in classical physics. In every attempt to talk about quan- tum theory one has to be aware of this principal difficulty alre ady recognized by Bohr, Heisenberg, Pauli and others. In the complementary worlds of the quantum regime and the classical regime we only are at home in the classical regime. The other regime remains accessible only through so phisticated experi- ments - even if an experimentalist would call them easy and si mple.... One way out might be to talk in images - but soon one arrives at poor ana logies. Hence, in order to reach more than only a superficial knowledge at lea st some hints to the mathematical background of quantum theory must be given . On the other hand at least in the beginning some themes should be avoided t o facilitate the pupils the understanding of the peculiarities of quantum th eory. 10.2 What can be done without too many technicalities? It is nearly impossible to understand quantum theory withou t considering its mathematical structure. Nevertheless at school the mathem atical apparatus of quantum mechanics has to be abandoned for the major part. T he main ideas, however, can be presented quite easily with help of th e typical quantum phenomenon “spin” having no classical analogues. Experime nts with polarized photons may help in conveying the essentials. In the followi ng I describe the reasons for taking spin as the first subject in treating quant um theory in more detail. Furthermore I explain with the example of EPR–gedan ken experiment how to proceed. Treating the phenomenon “spin” right in the beginning has se veral advantages: •Spin lies at the heart of quantum theory. Its properties are u sed to explain the different statistics, the fine structure of spectra, the s plitting of spectra in external (electrical or magnetic) fields. Already a spin s ystem consisting of two particles, i.e. living in a four–dimensional Hilbert space can no longer be described classically. This proof is similar to th e proof of Bell inequality, ([Bau]). •The procedure to describe spin mainly by its structure is typ ical of quan- tum theory. Furthermore, the mathematics of spin is quite si mple, using mainly the well–known Pauli– matrices: σx=/parenleftbigg0 1 1 0/parenrightbigg , σ y=/parenleftbigg0i −i0/parenrightbigg , σ z=/parenleftbigg1 0 0−1/parenrightbigg . With the help of these quite simple looking matrices, acting on two– dimensional Hilbert space, the most essential mathematica l structures of quantum theory can be explained and interpreted, see table 1 . Some de- tails are explained in the next section. •The meaning of Heisenberg uncertainty relation can be expla ined as prin- cipal non–existence of fixed values for properties hence of t heir non– determination, [Pos99]. Therefore the danger that the unce rtainty is perceived as measurement mistakes can be drastically dimin ished. Some helpful constructions even can be visualized on the blackbo ard. •Spin is a phenomenon of special importance in modern experim ents reach- ing from Nuclear Magnetic Resonance used in medical applica tions to re- alizations of the Einstein–Podolsky–Rosen gedankenexper iment. Its treat- ment opens the way to a discussion of philosophical aspects o f quantum theory which quickly reaches the main points: the question o f reality and objectivity in nature treated on a mathematical and physica l foundation. 20.2.1 An Example As an example I show how the arguments of Einstein, Podolsky a nd Rosen in their famous paper [EPR35] can be used to show the power of the mathematical formalism and - even more important - how the mathematical co nstructions can be interpreted in this framework. This allows a bridge to be built from the mathematical structures over the physical phenomena an d connecting to a philosophical discussion. Instead of arguing very sophis ticated within the mathematical formalism the main goal should be to uncover th e main aspects of quantum theory and in this way to build a solid fundament fr om which the mathematics can be developed further, (see also table 1). The argumens of EPR can be developed the following way: Step 1: The mathematical tools In 1935, the year of the EPR-paper the mathematical framework has just been settled implying t he following main points: •Thestate of a quantum object is given by a state vector ψcontaining all the available information, i.e. a complete description of the physical properties of the quantum object. •Eachphysical quantity is given together with all the possible results of a measurement of that quantity and corresponding eigenstate s, i.e. all the states a quantum object can attain after a measurement. A mat hematical realization of this concept is given for instance by matrice s. •Arbitrary states can be expressed with aid of the eigenstate s of such a matrix resp. physical quantity. In a deviation from the original argument of EPR I would advis e taking the spin realized with the above–mentioned Pauli–matrices as a conc rete example. The students can compute the eigenvalus and eigenstates easily from the matrices. The possible measurement results (eigenvalues) are +1 and −1 together with the corresponding eigenstates. The first expriment showing thi s property directly has been the Stern–Gerlach–experiment. Furthermore the Pa uli–matrices fulfill the condition crucial for the next step of argument of EPR: th ey do not have any eigenstates in common. Hence there always are several po ssibilities to rep- resent the spin state of a quantum object, namely with respec t to the respective eigenstates of the different matrices correponding to the sp in di rections. The representation of an arbitrary spin state ψ(s) with respect to the eigenstates of σxwould be ψ(s) =c1/parenleftBigg1√ 21√ 2/parenrightBigg +c2/parenleftBigg1√ 2 −1√ 2/parenrightBigg 3Mathematical termPhysical Interpretation Example vector physical state ψ=/parenleftbigg 1 0/parenrightbigg operator physical quantity σx=/parenleftbigg0 1 1 0/parenrightbigg eigenvalues of an operatorpossible results of measurements +1,−1 eigenstates of an operator (nor- malized to 1)physical states with a fixed value for the physical quantity in question/parenleftBigg1√ 21√ 2/parenrightBigg ,/parenleftBigg1√ 2 −1√ 2/parenrightBigg vector addition superposition (no fixed value for the physical quantity in question) development into eigenstatesrepresentation of arbitrary physi- cal states with respect to the cor- responding physical quantity/parenleftbigg1 0/parenrightbigg = 1√ 2/bracketleftBigg/parenleftBigg1√ 21√ 2/parenrightBigg +/parenleftBigg1√ 2 −1√ 2/parenrightBigg/bracketrightBigg coefficients (squared) of developmentprobability of getting the corre- sponding measurement resultprobability of getting either +1 or −1:/parenleftBig 1√ 2/parenrightBig ∗ ∗2 =1 2 4and with respect to the eigenstates of σzthe same state ψwould look like: ψ(s) =k1/parenleftbigg 1 0/parenrightbigg +k2/parenleftbigg 0 1/parenrightbigg with different coefficients ( ci)/negationslash= (ki). (For a concrete example look at the table.) This fact perhaps does not matter too much since we al ways can change coordinates (here it would be the rotation of a coordinate sy stem by an angle of 45 degree). But here it means that the spin state of a given s ystem possesses two different representations belonging to the “same piece o f reality”(EPR). The most interesting thing happens in the next step! Step 2: The experimental setup Two quantum objects, e.g. photons, are brought into interaction or produced in a single process and hence become entangled i.e. they share a common “history”. After that the y are separated from each other without any further manipulation, let us say one is brought to the moon, the second stays on earth. Because of their common “history” they are described by one c ommon state ψ which is not just the addition of the states of the single phot ons. This con- sideration is central for the whole argument of EPR. The deve lopment of the entangled state of both photons into eigenstates with respe ct to eigenstates of σxis given by: ψ(s1,s2) =ψ1(s1)/parenleftbigg 1 1/parenrightbigg +ψ2(s1)/parenleftbigg 1 −1/parenrightbigg and with respect to the eigenstates of σz: ψ(s1,s2) =φ1(s1)/parenleftbigg 1 0/parenrightbigg +φ2(s1)/parenleftbigg 0 1/parenrightbigg The only difference to the representations above is that the c oefficients now depend ons1The meaning of these two representations is that photon 1 is de- scribed differently depending on the description chosen for photon 2 , namely ψi(s1) resp.φi(s1). This is called the entanglement of the two photons. There- fore I would prefer to call the whole system consisting out of these two photons rather a “diphoton” in order to emphasize that they build one whole (also see step 5 below). Step 3: Classical Assumptions Assuming a fixed objective reality and demanding that physics has to give a complete description of reality Einstein arrives at a contradiction to the predictions of quantum the ory. More precisely, Einstein assumes: 1. Separability Classical Physics only knows action between objects in dire ct contact with 5each other. With “object” in this sense I also denote e.g. fiel ds. Hence if two objects are separated in space, including intermedia ting fields, all future manipulations on them are absolutely independent fr om each other. We could summarize this in the sentence: Spatially separate d objects also are physically separated. This is an implicit assumption of EPR that is not spoken out directly, but is underlying the whole argumen t as can be seen in the last paragraph of the famous EPR-paper [EPR35]. “Separability” hence means that the respective descriptio ns of two spa- tially separated photons should be totally independent fro m each other. 2. Physical Reality Einstein defines a pragmatic criterion for reality: Every we ll determined physical quantity has to have a representation in the theory . The point herein lies in the question: Which properties are well deter mined? Einstein regarded every physical quantity that can be measured as wel l determined. But quantum theory deviates in so far from classical physics as not all (in principle) measurable quantites have well determined prop erties at the same instant. They only possess them as a potentiality . From this view point the different descriptions from above (s tep 2) should not occur in a “good” physical theory. Step 4: Quantum Theoretical Outcome We can get information about the photons only after a measurement. What can possibly happ en then? There are several possibilities (as an example): 1. The spin of photon 2 is measured in x-direction. At the same instant the spin state of photon 1 is ψ1(s1) orψ2(s1) according to the result of the measurement at photon 2. 2. The spin of photon 2 is measured in z-direction. At the same instant the spin state of photon 1 is φ1(s1) orφ2(s1) according to the result of the measurement performed on photon 2. That means that photon 1 immediately “knows” the kindof measurement done on photon 2 far away as well as its result . Einstein calls this a “spooky action at a distance”, which may not occur in classical physics. Step 5: Interpretation The behaviour of both entangled photons is strongly connected to each other, they behave in spite of the ir spatial sepa- ration as one single quantum object. Therefore I propose to c all these both a “diphoton” which suggests more clearly that there is only one common state of the whole system, and not an addition of states of separated p hotons. Further- more, the outcome of measurements demonstrates that we may n ot assume that photon 1 or photon 2 had fixed values for their spin directions before measure- ment. For this purpose one could use the comfortable Dirac–n otation for spin 6states e.g.: ψ(s1,s2) =|1,0/angbracketright − |0,1/angbracketrightfor an entangled spin state instead of the above used vector–notation. The Dirac–notation has the adv antage of showing only the relative directions of spins of both photons, which is the only proper ty that is fixed and well determined (in absence of manipulation s). The directions themselves are not determined, they only show up after a measurement. Fixed values of properties do not exist in general, they only emerg e in measurements. Once this essential point is grasped the way is open for appli cations. This access consequently avoids possible pitfalls which in general erschweren understanding quantum theory. 0.3 Which Themes to Avoid in a First Approach? From historical reasons, having their roots in the developm ent of quantum the- ory, most ways of teaching the concepts of quantum physics re fer to classical models. This “procedere” causes principal difficulties in un derstanding. There- fore every reference to classical concepts should be avoide d as far as possible. The most important points to avoid are: •Speaking about position and momentum, i.e. about trajector ies If the concepts of position and momentum — well–known from ev eryday experience — are used at the very beginning of a course in quan tum the- ory there exists the danger of transferring classical think ing to quantum theory, although everybody would say: clearly, in quantum t heory there are no trajectories. Conceptual difficulties arising from us e of the terms “ position” and “velocity” can be avoided in the simplest man ner if these fundamental classical concepts do not play any role in the be ginning of a course in quantum theory. Then any association of classical ideas might disappear and students might recognize the philosophical s ignificance of quantum theory far more easily. The most prominent example i s the fa- mous Heisenberg uncertainty relation for position and mome ntum which easily is misunderstood in a sense that the uncertainty simp ly relies on disturbance by measurement in the usual sense. Instead the u ncertainty relations are kind of measure for distinguishing classical behaviour from quantum behaviour in that they determine whether two physic al quanti- ties can attain fixed values at the same instant. If two physic al quantities can attain fixed values at the same instant the quantum object in question behaves “classically”, if not it displays quantum behaviou r as e.g. spin. Hence the role and the implications of the non–existence of fi xed values for some properties at the same instant - as expressed in unce rtainty re- lations - might not be fully appreciated in their revolution ary potential if one concentrates on “position” and “momentum”. Besides u ndesired analogies to Newtonian mechanics the corresponding operat ors for posi- tion and momentum and their eigenstates are mathematically far more 7difficult to handle than the 2 ×2–spin–matrices. •Speaking about particle–wave–dualism Waves and particles both are classical concepts, complemen tary to each other. One could illustrate their relation by looking at the same object from different sights. E.g. a cylinder standing upright appe ars completely different from the above (a circle) compared to a look from the side (a rect- angle). But this observation does not meet the essential poi nt in quantum theory. A first step to avoid analogy to classical phenomena would be t o use the term “quantum object” instead of wave or particle. Only afte r the quan- tum mechanical concepts are fixed there might be a careful use of those “classical” terms be allowed where unevitable. Perhaps the importance of using suitable terms may become clear with the example of the double– slit–experiment. If it is replaced by the so called Taylor–e xperiment in which photons display at the same time wave properties - they show in- terference - as well as particle properties - they arrive at d istinct points on the film, the necessity of changing concepts gets far more o bvious. •Speaking about spin as sort of spinning around One should not give an image of spin. Especially one may not th ink in terms of an electron spinning around. The quantum mechanica l spin is simply structure manifesting itself and its behaviour thro ugh experiments, especially in the Stern–Gerlach–experiment which may serv e as an intro- ductory experiment. As shown above the abstract structure o f spin can de introduced to a certain extent, depending on the mathematic al capabilities of students. Those three points are mentioned here because their avoidan ce breaks with the tradition of teaching and speaking about quantum theory . The preceding sections showed an alternative. 0.4 Conclusion and Perspectives The recent EPR–experiments are the starting point for all th e current develop- ments concerning the fundamentals of quantum theory as well as technological utopies in the area of quantum computing and teleportation. In addition it widely opens the door to philosophical discussions. In so fa r the EPR–gedanken experiment lies at the heart of quantum theory and its interp retation. The entrance to quantum mechanics with help of the phenomeno n spin quickly gives gifted or interested pupils a possibility of discussi ng the properties of quantum objects, the mathematical structures and the inter pretation of quan- tum mechanics on a technically very modest level but nonethe less quite precise. As the spin inevitably is a purely quantum mechanical phenom enon this opens 8via the EPR–gedanken–experiment a short way into the crucia l points of un- derstanding concepts as well as philosophical implication s of quantum theory and hence gives the possibility for people to revisit their v iew of nature, their Weltbild . I regard this an important contribution to general educati on. References [Bau] Hellmut Baumg”artel. Mathematische Grundlagen der Q uantenthe- orie. manuscript, http://www.math.uni-potsdam.de/mp1/vorlesun.htm. [EPR35] Albert Einstein, Boris Podolsky, and Nathan Rosen. Can quantum- mechanical description of physical reality be considered c omplete? Physical Review , 48:696–702, 1935. [Pos98] Gesche Pospiech. Vom Atommodell zur Quantenphysik und zur”uck. In Fachverband Didaktik der Physik Deutsche Physi kalische Gesellschaft, editor, Didaktik der Physik – Beitrge zur 62. Physikerta- gung Regensburg 1998 , pages 178–183, 1998. [Pos99] Gesche Pospiech. Spukhafte Fernwirkungen in der Qu antentheorie? Physik in der Schule , 37:56–59, 1999. 9

arXiv:physics/0002009v1 [physics.hist-ph] 3 Feb 2000The singular points of Einstein’s universe † by Marcel Brillouin (translation by S. Antoci∗) 1. Einstein’s four-dimensional Universe is determined by t he ten gµνof its ds2. In order to determine them it is not sufficient to know the six independent partial differential equations that they must fulfil; one needs also to know the conditions at the b oundaries of the Universe, which are necessary for specialising the integrals in view of given pr oblems. These boundary conditions are of two sorts. One deals with the far away state of the Universe, c ompletely outside the region where we wish to study the events; it is the one whose choice, still in d ispute, is translated into this question: is the Universe infinite? Is it finite, although without limit ? I do not bother with this here. The other one deals with the singular lines that correspond to wh at, from the experimental viewpoint, we call the attractive masses. In Newtonian gravitation the material point of mass mcorresponds to the point of the Euclidean space where the integral of Lapl ace’s equation becomes infinite like m/r, where ris (in the neighbourhood of this point) the distance from the material point to the point where one studies the Newtonian potential. It is this k ind of singularities, characteristic of matter, that I come to consider. We first remark that, in the present state of our experimental knowledge, nothing entitles us to suppose that singular points (in four dimensions) may exi st in the Universe. From the analytical viewpoint, this impossibility is evidently connected with the distinction that exists between one of the variables and the remaining three, which allows for the s ign changes of ds2, like in acoustics. It would be interesting to give precision to this remark. 2. Let us consider a static, permanent state, i.e.a state in which the gµνdepend only on three of the variables, x1,x2,x3, and are independent of that x4whose g44in essentially positive. The simplest singular line of Universe is the one which, in the se ction with the three dimensions x1,x2, x3that we call space, corresponds to an isotropic singular point . For a Universe that contains only one line of this kind Schwar zschild has integrated, in 1916, the differential equations that rule the gµν, and he obtained a ds2that, by changing Schwarzschild’s notations, I write under the form ds2=γc2dτ2−1 γdR2−(R+ 2m)2(dθ2+sin2θdϕ2), γ=R R+ 2m, c, velocity of light (universal constant); θ,ϕ, the spherical polar angles in space; R, a length such that the sphere, whose centre is at the singula r point in the (non Euclidean) space R,θ,ϕ, and which has Rfor co-ordinate radius, has a total surface equal to 4 π(R+ 2m)2, and 2π(R+ 2m) as circumference of the great circle. The function γis positive, and equal to 1 at a large distance; mis a positive constant. If Ris not zero, γis finite and nonvanishing; there is no singularity either of gµνor ofds2. †Le Journal de Physique et Le Radium 23, 43 (1923). ∗Dipartimento di Fisica “A. Volta”, Universit` a di Pavia, Vi a Bassi 6 - 27100 Pavia (Italy). 1But, if Ris zero, γis zero, and the coefficient of dR2is infinite: there is a pointlike singularity in this point in space, and a line singularity in the Universe . 3. One may wonder whether this singularity limits the Universe, and one must stop at R= 0 or, on the contrary, it only traverses the Universe, which shall continue on the other side, for R <0. In the discussions at the “Coll` ege de France”, in particula r in the ones held during the Easter of 1922, it has been generally argued as if R= 0 would mean a catastrophic region that one needs to cross in order to attain the true, singular limit, that one only reaches when γis infinite, with R=−2m.In my opinion, it is the first singularity, reached when R= 0,γ= 0 (m >0), the one that limits the Universe and that must not be crossed [ C. R.,175, (November 27th 1922)]. The reason for this is peremptory, although up to now I have ne glected to put it in evidence: forR <0,γ <0, in no way the ds2any longer corresponds to the problem one aimed at dealing with. In order to see it clearly, let us put anew the letters x1...x4whose physical meaning shall not be suggested by old habits. Schwarzschild’s ds2corresponds to the following analytical problem: thegµνdepend only on the single variable x1; two more variables, θandϕ, enter in the manner that corresponds to the spatial isotropy around one point, a nd one has: ds2=γc2dx2 4−1 γdx2 1−(x1+ 2m)2(dθ2+sin2θdϕ2), γ=x1 x1+ 2mm >0. The term γis positive either when x1>0and x 1+ 2m >0 or when x1<0and x 1+ 2m <0; one has truly solved the proposed problem, x1is a spatial variable and x4is a time variable. 4.Ifγis instead negative, −2m < x 1<0, the characters of length and of duration are exchanged between x1andx4; in fact now the term −(1/γ)dx2 1is positive, while the term γc2dx2 4 is negative. Let us make this character visible by substituting the notat iontfor−x1: ds2=2m−t tdt2−t 2m−tdx2 4−(2m−t)2(dθ2+sin2θdϕ2). This is a ds2which has no longer any relation with the static problem that one aimed at dealing with. Theds2forx1<0does not continue the one that is appropriate for x1>0. This discontinuity is by far sharper than all the ones that have be en encountered up to now in the problems of mathematical physics. The frontier x1= 0,R= 0, is really an insurmountable one. While discussing Schwarzschild’s integration one notices that an arbitrary factor C4could have been left in the term with dx4, and that this factor is taken equal to 1 in order that the Universe become Euclidean when x1is infinite. No similar condition can be imposed in the interv al −2m < x 1<0; but C4is a real constant, and it can only be taken with a positive val ue. In fact, if it were negative, the ds2would no longer have any meaning that refer to an Einstein’s u niverse. 5. The conclusion seems to me unescapable: the limit R= 0 is insurmountable; it embodies thematerial singularity. 2The distance rfrom this origin ( R= 0) to a point with co-ordinate radius R, calculated along a radius vector ( θ=const.,ϕ =const. ) is(1) r=/radicalbig R(R+ 2m) +mlnR+m+/radicalbig R(R+ 2m) m. The ratio between the circumference 2 π(R+ 2m) and the radius is everywhere larger than 2 π; in particular at the origin ( r= 0,R= 0) this ratio becomes infinite. The circumference of the gre at circle of the origin is 4 πm; the spherical surface of that point has the finite value 4 π(2m)2. It is this singularity that constitutes what physics calls the material point ; it is the factor mappearing in it that must be called mass. In view of this occurrence, the word material point is perhaps ill chosen. In fact, due to the finite extension of the spherical surface of the point, the va riations of θand of ϕtruly displace the extremity of the radius vector (of zero length) over this sur face as it would occur over any other sphere whose co-ordinate radius Rand whose radius rwere nonvanishing. Anyway, since nothing more pointlike can be found in Einstei n’s Universe, and since one really needs attaining a definition of the elementary material test body which, according to Einstein, follows a geodesic of the Universe to which it belongs, I will maintain this abridged nomenclature, material point , without forgetting its imperfection. January 1923. (1)Erratum. In the already cited note of the C. R. the coefficient −m/2of the logarithm is incorrect. 3

arXiv:physics/0002010v1 [physics.bio-ph] 3 Feb 2000A-Tract Induced DNA Bending is a Local Non-Electrostatic Eff ect Alexey K. Mazur Laboratoire de Biochimie Th´ eorique, CNRS UPR9080 Institut de Biologie Physico-Chimique 13, rue Pierre et Marie Curie, Paris,75005, France. FAX: +33[0]1.58.41.50.26 Email: alexey@ibpc.fr (March 18, 2008) The macroscopic curvature induced in double helical B- DNA by regularly repeated adenine tracts (A-tracts) is a long known, but still unexplained phenomenon. This effect plays a key role in DNA studies because it is unique in the amount and the variety of the available experimental inform a- tion and, therefore, is likely to serve as a gate to the unknow n general mechanisms of recognition and regulation of genome sequences. The dominating idea in the recent years was that, in general, macroscopic bends in DNA are caused by long range electrostatic repulsion between phosphate groups wh en some of them are neutralized by proximal external charges. In the case of A-tracts this may be specifically bound solvent counterions. Here we report about molecular dynamics sim- ulations where a correct static curvature in a DNA fragment with phased adenine tracts emerges spontaneously in condi- tions where any role of counterions or long range electrosta tic effects can be excluded. RESULTS AND DISCUSSION Although the macroscopic curvature of DNA induced by adenine-tracts (A-tracts) was discovered almost two decades ago1,2structural basis for this phenomenon re- mains unclear. A few models considered originally3,4,5 suggested that it is caused by intrinsic conformational preferences of certain sequences, but all these and simi- lar theories failed to explain experimental data obtained later.6Calculations show that the B-DNA duplex is mechanically anisotropic,7that bending towards minor grooves of some A-tracts is strongly facilitated,8and that the macroscopic curvature becomes energetically prefer- able once the characteristic A-tract structure is main- tained by freezing or imposing constraints.9,10,11How- ever, the static curvature never appears spontaneously in calculations unbiased a priori and these results leave all doors open for the possible physical origin of the ef- fect. In the recent years the main attention has been shifted to specific interactions between DNA and solvent counterions that can bend the double helix by specifically neutralizing some phosphate groups.12,13,14,15,16The pos- sibility of such mechanism is often evident in protein- DNA complexes, and it has also been demonstrated by direct chemical modification of a duplex DNA.14In the case of the free DNA in solution, however, the available experimental observations are controversial.16,17Molec- ular dynamics simulations of a B-DNA in an explicitcounterion shell could neither confirm nor disprove this hypothesis.18Here we report the first example where sta- ble static curvature emerges spontaneously in molecular dynamics simulations. Its direction is in striking agree- ment with expectations based upon experimental data. However, we use a minimal B-DNA model without coun- terions, which strongly suggests that they hardly play a key role in this effect. Figure 1 exhibits results of a 10 ns simulation of dy- namics of a 25-mer B-DNA fragment including three A-tracts separated by one helical turn. This sequence has been constructed after many preliminary tests with shorter sequence motives. Our general strategy came out from the following considerations. Although the A-tract sequences that induce the strongest bends are known from experiments, probably not all of them would work in simulations. There are natural limitations, such as the precision of the model, and, in addition, the limited duration of trajectories may be insufficient for some A- tracts to adopt their specific conformation. Also, we can study only short DNA fragments, therefore, it is prefer- able to place A-tracts at both ends in order to maximize the possible bend. There is, however, little experimental evidence of static curvature in short DNA fragments, and one may well expect the specific A-tract structure to be unstable near the ends. That is why we did not simply take the strongest experimental “benders”, but looked for sequence motives that in calculations readily adopt the characteristic local structure, with a narrow minor groove profile and high propeller twist, both in the middle and near the ends of the duplex. The complementary duplex AAAATAGGCTATTTTAGGCTATTTT has been con- structed by repeating and inverting one such motive. The upper trace in plate (a) shows the time depen- dence of rmsd from the canonical B-DNA model. It fluc- tuates below 4 ˚A sometimes falling down to 2 ˚A, which is very low for the double helix of this length indicating that all helical parameters are well within the range of the B- DNA family. The lower surface plot shows the time evo- lution of the minor DNA groove. The surface is formed by 75 ps time-averaged successive minor groove profiles, with that on the front face corresponding to the final DNA conformation. The groove width is evaluated by using space traces of C5’ atoms as described elsewhere19. Its value is given in angstr¨ oms and the corresponding canonical B-DNA level of 7.7 ˚A is marked by the straight dotted lines on the faces of the box. It is seen that the 1FIG. 1. (a) FIG. 1. (c) FIG. 1. (b) FIG. 1. Representative results from the first 10 ns MD sim- ulation of a 25-mer double helical DNA fragment. (a) The time variation of the heavy atom rmsd from the canonical B-DNA form and the evolution of the profile of the minor groove. (b) Dynamics of B I↔BIIbackbone transitions. (c) The time evolution of the optimal helical axis, which is a bes t fit axis of coaxial cylindrical surfaces passing through sug ar atoms. In all cases considered here it is very close to that produced by the Curves algorithm.30Two perpendicular pro- jections are shown with the corresponding views of the avera ge conformation during the last nanosecond shown on the right. (d) The time variation the bending angle and direction. The bending angle is measured between the two ends of the opti- mal helical axis. The bending direction is characterized by the angle between the X-projection plane in plate (c) and the xz plane of the local DNA coordinate frame constructed in the center of the duplex according to the Cambridge convention20. 2overall groove shape has established after 2 ns and re- mained stable later, with noticeable local fluctuations. In all A-tracts the groove strongly narrows towards 3’ ends and widens significantly at the boundaries. There are two less significant relative narrowings inside non A- tract sequences as well. Dynamics of B I↔BIIbackbone transitions are shown in plate (b). The B Iand B IIconformations are distin- guished by the values of two consecutive backbone tor- sions, εandζ. In a transition they change concertedly from (t,g−) to (g−,t). The difference ζ−εis, there- fore, positive in B Istate and negative in B II, and it is used in Fig. (d) as a monitoring indicator, with the cor- responding gray scale levels shown on the right. Each base pair step is characterized by a column consisting of two sub-columns, with the left sub-columns referring to the sequence written at the top in 5’-3’ direction from left to right. The right sub-columns refer to the comple- mentary sequence shown at the bottom. It is seen that, in A-tracts, the B IIconformation is preferably found in ApA steps and that B I↔BIItransitions in neighboring steps often occur concertedly so that along a single A- strand B Iand B IIconformations tend to alternate. The pattern of these transitions reveals rather slow dynamics and suggests that MD trajectories in the 10 ns time scale are still not long enough to sample all relevant conforma- tions. Note, for instance, a very stable B IIconformation in both strands at one of the GpG steps. Plate (c) shows the time evolution of the overall shape of the helical axis. The optimal curved axes of all DNA conformations saved during dynamics were rotated with the two ends fixed at the OZ axis to put the middle point at the OX axis. The axis is next characterized by two per- pendicular projections labeled X and Y. Any time section of the surfaces shown in the figure gives the correspond- ing axis projection averaged over a time window of 75 ps. The horizontal deviation is given in angstr¨ oms and, for clarity, its relative scale is two times increased with re- spect to the true DNA length. Shown on the right are two perpendicular views of the last one-nanosecond-average conformation. Its orientation is chosen to correspond ap- proximately that of the helical axis in the surface plots. It is seen that the molecule maintained a planar bent shape during a substantial part of the trajectory, and that at the end the bending plane was passing through the three A-tracts. The X-surface clearly shows an in- crease in bending during the second half of the trajec- tory. In the perpendicular Y-projection the helical axis is locally wound, but straight on average. The fluctuating pattern in Y-projection sometimes reveals two local max- ima between A-tracts, which corresponds to two indepen- dent bends with slightly divergent directions. One may note also that there were at least two relatively long pe- riods when the axis was almost straight, namely, around 3 ns and during the fifth nanosecond. At the same time, straightening of only one of the two bending points is a more frequent event observed several times in the surface plots.Finally, plate (d) shows the time fluctuations of the bending direction and angle. The bending direction is characterized by the angle between the X-projection plane in plate (c) and the xzplane of the local DNA co- ordinate frame constructed in the center of the duplex. According to the Cambridge convention20the local xdi- rection points to the major DNA groove along the short axis of the base-pair, while the local zaxis direction is ad- jacent to the optimal helicoidal axis. Thus, a zero angle between the two planes corresponds to the overall bend to the minor groove exactly at the central base pair. In both plots, short time scale fluctuations are smoothed by averaging with a window of 15 ps. The total angle mea- sured between the opposite axis ends fluctuates around 10-15◦in the least bent states and raises to average 40- 50◦during periods of strong bending. The maximal in- stantaneous bend of 58◦was observed at around 8 ns. The bending direction was much more stable during the last few nanoseconds, however, it fluctuated at a roughly constant value of 50◦starting from the second nanosecond. This value means that the center of the ob- served planar bend is shifted by approximately two steps from the middle base pair so that its preferred direction is to the minor groove at the two ATT triplets, which is well distinguished in plate (c) as well, and corresponds to the local minima in the minor groove profiles in plate (a). During the periods when the molecule straightened the bending direction strongly fluctuates. This effect is due to the fact that when the axis becomes straight the bending plane is not defined, which in our case appears when the central point of the curved axis passes close to the line between its ends. It is very interesting, however, that after the straightening, the bending is resumed in approximately the same direction. Figure 2 exhibits similar data for another 10 ns tra- jectory of the same DNA fragment, computed in order to check reproducibility of the results. A straight DNA conformation was taken from the initial phase of the pre- vious trajectory, energy minimized, and restarted with random initial velocities. It shows surprisingly similar results as regards the bending direction and dynamics in spite of a somewhat different minor groove profile and significantly different distribution of B Iand B IIconform- ers along the backbone. Note that in this case the heli- cal axis was initially S-shaped in X-projection, with one of the A-tracts exhibiting a completely opposite bend- ing direction. Fluctuations of the bending direction are reduced and are similar to the final part of the first tra- jectory, which apparently results from the additional re- equilibration. In this case the maximal instantaneous bend of 71◦was observed at around 4 ns. Comparison of traces in plates (a) and (d) in Figs. 1 and 2 clearly shows that large scale slow fluctuations of rmsd are caused by bending. The rmsd drops down to 2˚A when the duplex is straight and raises beyond 6 ˚A in strongly bent conformations. In both trajectories the molecule experienced many temporary transitions to straight conformations which usually are very short liv- 3FIG. 2. (a) FIG. 2. (c) FIG. 2. (b) FIG. 2. Representative results from the second 10 ns MD trajectory of the same DNA fragment. Notation as in Fig. 1. 4FIG. 3. A stereo snapshot of the system at around 8.5 ns of the second trajectory. AT base pairs are shown in red and GC base pairs in blue. ing. These observations suggest that the bent state is rel- atively more stable than the straight one and, therefore, the observed behavior corresponds to static curvature. In conformations averaged over successive one nanosecond intervals the overall bending angle is 35-45◦except for a few periods in the first trajectory. Figure 3 shows a snap- shot from around 8.5 ns of the second trajectory where the rmsd from the straight canonical B-DNA reached its maximum of 6.5 ˚A. The strong smooth bent towards the minor grooves of the three A-tracts is evident, with the overall bending angle around 61◦. All transformations exhibited in Figs. 1 and 2 are isoenergetic, with the total energy fluctuating around the same level established during the first nanosecond already, and the same is true for the average helicoidal parameters. Plates (b), however, indicate that there are much slower motions in the system, and this observation precludes any conclusions concerning the global stabil- ity of the observed conformations. Moreover, we have computed yet another trajectory for the same molecule starting from the canonical A-DNA form. During 10 ns it converged to a similarly good B-DNA structure with the same average total energy, but the bending pattern was not reproduced. It appears, therefore, that the conforma- tional space is divided into distinct domains, with tran- sitions between them probably occurring in much longer time scales. However, the very fact that the stable curva- ture in good agreement with experimental data emergesin trajectories starting from a featureless straight canon - ical B-DNA conformation strongly suggests that the true molecular mechanism of the A-tract induced bending is reproduced. Therefore, it cannot depend upon the com- ponents discarded in our calculations, notably, specific interactions with solvent counterions and long-range elec - trostatic effects. We are not yet ready to present a detailed molecu- lar mechanism responsible for the observed curvature be- cause even in this relatively small system it is difficult to distinguish the cause and the consequences. We believe, however, that all sorts of bending of the double helical DNA, including that produced by ligands and that due to intrinsic sequence effects, have its limited, but high flexi- bility as a common origin. Its own conformational energy has the global minimum in a straight form, but this min- imum is very broad and flat, and DNA responds by dis- tinguishable bending to even small perturbations. The results reported here prove that in the case of A-tracts these perturbations are produced by DNA-water interac- tions in the minor groove. Neither long range phosphate repulsion nor counterions are essential. The curvature is certainly connected with the specific A-tract structure and modulations of the minor groove width, but it does not seem to be strictly bound to them. In dynamics, conformations, both smoothly bent and kinked at the two insertions between the A-tracts, are observed period- ically. Note also, that the minor groove profile somewhat differs between the two trajectories and that it does not change when the molecule straightens. We strongly be- lieve, however, the experimental data already available will finally allow one to solve this problem by theoretical means, including the approach described here, and we continue these attempts. METHODS Molecular dynamics simulations have been performed with the internal coordinate method (ICMD)21,22includ- ing special technique for flexible sugar rings23. The so- called “minimal B-DNA” model was used24,25which con- sists of a double helix with the minor groove filled with explicit water. Unlike the more widely used models, it does not involve explicit counterions and damps long range electrostatic interactions in a semi-empirical way by using distance scaling of the electrostatic constant and reduction of phosphate charges. The DNA model was same as in earlier reports,24,25namely, all torsions were free as well as bond angles centered at sugar atoms, while other bonds and angles were fixed, and the bases held rigid. AMBER9426,27force field and atom parameters were used with TIP3P water28and no cut off schemes. With a time step of 10 fs, these simulation conditions require around 75 hours of cpu per nanosecond on a Pen- tium II-200 microprocessor. The initial conformations were prepared by vacuum en- 5ergy minimization starting from the fiber B-DNA model constructed from the published atom coordinates.29The subsequent hydration protocol to fill up the minor groove24normally adds around 16 water molecules per base pair. The heating and equilibration protocols were same as before24,25. During the runs, after every 200 ps, water positions were checked in order to identify those penetrating into the major groove and those completely separated. These molecules, if found, were removed and next re-introduced in simulations by putting them with zero velocities at random positions around the hydrated duplex, so that they could readily re-join the core sys- tem. This procedure assures stable conditions, notably, a constant number of molecules in the minor groove hydra- tion cloud and the absence of water in the major groove, which is necessary for fast sampling25. The interval of 200 ps between the checks is small enough to assure that on average less then one molecule is repositioned and, therefore, the perturbation introduced is considered neg- ligible. ACKNOWLEDGEMENTS I thank R. Lavery for useful discussions as well as crit- ical comments and suggestions to the paper. 1Marini, J. C., Levene, S. D., Crothers, D. M. & Englund, P. T., Proc. Natl. Acad. Sci. USA 79, 7664–7668 (1982). 2Wu, H.-M. & Crothers, D. M., Nature 308, 509–513 (1984). 3Trifonov, E. N. & Sussman, J. L., Proc. Natl. Acad. Sci. USA77, 3816–3820 (1980). 4Levene, S. D. & Crothers, D. M., J. Biomol. Struct. Dyn. 1, 429–435 (1983). 5Calladine, C. R., Drew, H. R. & McCall, M. J., J. Mol. Biol.201, 127–137 (1988). 6Crothers, D. M. & Shakked, Z., in Oxford Handbook of Nucleic Acid Structure , edited by Neidle, S. (Oxford Uni- versity Press, New York, 1999), pp. 455–470. 7Zhurkin, V. B., Lysov, Y. P. & Ivanov, V. I., Nucl. Acids Res.6, 1081–1096 (1979). 8Sanghani, S. R., Zakrzewska, K., Harvey, S. C. & Lavery, R.,Nucl. Acids Res. 24, 1632–1637 (1996). 9von Kitzing, E. & Diekmann, S., Eur. Biophys. J. 14, 13– 26 (1987). 10Chuprina, V. P. & Abagyan, R. A., J. Biomol. Struct. Dyn. 1, 121–138 (1988). 11Zhurkin, V. B., Ulyanov, N. B., Gorin, A. A. & Jernigan, R. L., Proc. Natl. Acad. Sci. USA 88, 7046–7050 (1991). 12Mirzabekov, A. D. & Rich, A., Proc. Natl. Acad. Sci. USA 76, 1118–1121 (1979). 13Levene, S. D., Wu, H.-M. & Crothers, D. M., Biochemistry 25, 3988–3995 (1986).14Strauss, J. K. & Maher, L. J., III, Science 266, 1829–1834 (1994). 15Travers, A., Nature Struct. Biol. 2, 264–265 (1995). 16McFail-Isom, L., Sines, C. C. & Williams, L. D., Curr. Opin. Struct. Biol. 9, 298–304 (1999). 17Chiu, T. K., Zaczor-Grzeskowiak, M. & Dickerson, R. E., J. Mol. Biol. 292, 589–608 (1999). 18Young, M. A. & Beveridge, D. L., J. Mol. Biol. 281, 675– 687 (1998). 19Mazur, A. K., J. Mol. Biol. 290, 373–377 (1999). 20Dickerson, R. E. et al.,J. Mol. Biol. 205, 787–791 (1989). 21Mazur, A. K. & Abagyan, R. A., J. Biomol. Struct. Dyn. 6, 815–832 (1989). 22Mazur, A. K., J. Comput. Chem. 18, 1354–1364 (1997). 23Mazur, A. K., J. Chem. Phys. 111, 1407–1414 (1999). 24Mazur, A. K., J. Am. Chem. Soc. 120, 10928–10937 (1998). 25Mazur, A. K., Preprint http: // xxx.lanl.gov/abs/ physics/9907028, (1999). 26Cornell, W. D. et al.,J. Am. Chem. Soc. 117, 5179–5197 (1995). 27Cheatham, T. E., III, Cieplak, P. & Kollman, P. A., J. Biomol. Struct. Dyn. 16, 845–862 (1999). 28Jorgensen, W. L., J. Am. Chem. Soc. 103, 335–340 (1981). 29Arnott, S. & Hukins, D. W. L., Biochem. Biophys. Res. Communs. 47, 1504–1509 (1972). 30Lavery, R. & Sklenar, H., J. Biomol. Struct. Dyn. 6, 63–91 (1988). APPENDIX This section contains comments from anonymous ref- erees of peer-review journals where the manuscript has been considered for publication, but rejected. A. Journal of Molecular Biology 1. First referee Dr. Mazur describes molecular dynamics simulations where a correct static curvature of DNA with phased A-tracts emerges spontaneously in conditions where any role of counterions or long range electrostatic effects can be excluded. I have several problems with this manuscript: 1) The observed curvature is dependent on the starting model. In fact the manuscript uses the phrase ‘stable static curvature’ incorrectly to describe what is probably a trapped metastable state. The observed curve is neither stable nor static. 2) The choice of DNA sequence seems to be biased toward that which gives an altered structure in simula- tions, ad is not that which gives the most pronounced bend in solution. I would suggest a comparison of (CAAAATTTTTG)n and (CTTTTAAAAG)n. 3) The result is not consistent with solution results. See for example: 6Prodin, F., Cocchione, S., Savino, M., & Tuffillaro, A. “Different Interactions of Spermine With a Curved and a Normal DNA Duplex - (Ca(4)T(4)G)(N) and (Ct(4)a(4)G)(N) - Gel -Electrophoresis and Circular- Dichroism Studies” (1992) Biochemistry International 27, 291-901. Brukner, l, Sucis, S., Dlakic, M., Savic, A., & Pon- gor, S. “Physiological concentrations of magnesium ions induces a strong macroscopic curvature in GGGCCC - containing DNA” (1994) J. Mol. Biol. 236, 26-32. Diekmann, S., & Wang, J. C. “On the sequence deter- minants and flexibility of the kinetoplast DNA fragment with abnormal gel electrophoretic mobilities” (1985) J. Mol. Biol. 186, 1-11. Llaudnon, C. H., & Griffith, J. D. “Cationic metals promote sequence-directed DNA bending” (1987) Bio- chemistry 26, 3759-3762. 4) The result is not consistent with other simulations. See for example: Feig, M., & Pettitt, B. M. “Sodium and Chlorine ions as part of the DNA solvation shell” (1999) Biophys. J. 77, 1769-81. 5) The results should be given by objective statistical descriptions rather than a series of spot examples, as in “sometimes reveals two independent bends”. 2. Second referee This manuscript describes the modeling of a 25-residue DNA duplex using molecular dynamics simulations. The DNA sequence in question contains 3 A/T tracts ar- ranged in-phase with the helix screw and thus is expected to manifest intrinsic bending. Unlike previous MD stud- ies of intrinsically bent DNA sequences, these calcula- tions omit explicit consideration of the role of counteri- ons. Because recent crystallographic studies of A-tract- like DNA sequence have attributed intrinsic bending to the localization of counterions in the minor groove, the present finding that intrinsic bending occurs in the ab- sence of explicit counterions is important for understand- ing the underlying basis of A-tract-dependent bending. Overall, the MD procedure appears sound and the cal- culations were carried out with obvious care and atten- tion to detail. There are two specific issues raised by this study that should be addressed in revision, however. 1. Although the sequence chosen for this study was based on a canonical, intrinsically-bent motif consisting of three A tracts, it is unclear to what extent intrin- sic bending has been experimentally shown for this par- ticular sequence. There are known sequence-context ef- fects that modulate A-tract-dependent bending and thus the author should refer the reader to data in the litera- ture or show experimentally that intrinsic bending of the expected magnitude occurs for this particular sequence. Moreover, one A tract is out-of-phase with respect to the others and it is therefore not clear how this contributesto the overall bend. The author is understandably con- cerned about end effect with short sequences; this prob- lem can be ameliorated by examining DNA fragments that constrain multiple copies of the chosen motif or by extending the ends of the motif with mixed-sequence DNA. 2. Notwithstanding the authors remark bout separat- ing the cause and the effects with respect to intrinsic bending some comments about the underlying mecha- nism of bending seem appropriate. It would be particu- larly useful to know whether average values of any specific conformational variables are unusual or whether strongly bent states are consistent with narrowing of the minor groove within A-tracts, for example. 7

arXiv:physics/0002011v1 [physics.atom-ph] 4 Feb 2000A classical Over Barrier Model to compute charge exchange be tween ions and one–optical–electron atoms Fabio Sattin∗ Consorzio RFX, Corso Stati Uniti 4, 35127 Padova, ITALY In this paper we study theoretically the process of electron capture between one–optical–electron atoms (e.g. hydrogenlike or alkali atoms) and ions at low-to -medium impact velocities ( v/ve≤ 1) working on a modification of an already developed classica l Over Barrier Model (OBM) [V. Ostrovsky, J. Phys. B: At. Mol. Opt. Phys. 283901 (1995)], which allows to give a semianalytical formula for the cross sections. The model is discussed and th en applied to a number of test cases including experimental data as well as data coming from othe r sophisticated numerical simulations. It is found that the accuracy of the model, with the suggested corrections and applied to quite different situations, is rather high. PACS numbers: 34.70+e, 34.10.+x I. INTRODUCTION The electron capture process in collisions of slow, highly c harged ions with neutral atoms and molecules is of great importance not only in basic atomic phy sics but also in applied fields such as fusion plasmas and astrophysics. The process under study ca n be written as: A+q+B→A(q−j)++Bj+. (1) Theoretical models are regularly developed and/or improve d to solve (1) from first principles for a variety of choices of target Aand the projectile B, and their predictions are compared with the results of ever more refined experiments. In principle, one could compute all the quantities of intere st by writing the time-dependent Schr¨ odinger equation for the system (1) and programming a c omputer to solve it. This task can be performed on present–days supercomputers for moderately c omplicated systems. Notwithstanding this, simple approximate models are still valuable: (i) the y allow to get analytical estimates which are easy to adapt to particular cases; (ii) allow to get physi cal insight on the features of the problem by looking at the analytical formulas; (iii) finally, they ca n be the only tools available when the complexity of the problem overcomes the capabilities of the computers. For this reason new models are being still developed [1–3]. The present author has presented in a recent paper [3] a study attempting to develop a more accurate OBM by adding some quantal features. The model so de veloped was therefore called a semi–classical OBM. Its results showed somewhat an improve ment with respect to other OBMs, but not a dramatic one. In this paper we aim to present an OBM for dealing with one of th e simplest processes (1): that between an ion and a target provided with a single active elec tron. Unlike the former one [3], this model is entirely developed within the framework of a classi cal model, previously studied in [1] (see also [4]), but with some important amendments and improveme nts which, as we shall see, allow a quite good accordance with experiments. The paper is organized as follows: a first version of the model is presented and discussed in section II. In section III we will test our model against a first test ca se. From the comparison a further improvement to the model is proposed (section IV) and tested against the same case, as well as other data in section V. It will be shown that predictions wit h this correction are in much better agreement. ∗E-mail: sattin@igi.pd.cnr.it 1II. THE MODEL: FIRST PICTURE We consider the standard scattering experiment and label T,P, anderespectively the target ion, the projectile and the electron. The system T+eis the initial neutral atom. Let rbe the electron vector relative to TandRthe internuclear vector between TandP. In the spirit of classical OBM models, all particles are considered as classical objects. Let us consider the plane Pcontaining all the three particles and use cylindrical pola r coordinates (ρ, z, φ ) to describe the position of the electron within this plane. We can arbitrarily choose to set the angle φ= 0, and assign the zaxis to the direction along the internuclear axis. The total energy of the electron is (atomic units will be used unless otherwise stated): E=p2 2+U=p2 2−Zt/radicalbig ρ2+z2−Zp/radicalbig ρ2+ (R−z)2. (2) ZpandZtare the effective charge of the projectile and of the target se en by the electron, respectively. Notice that we are considering hydrogenlike approximation s for both the target and the projectile. We assigne an effective charge Zt= 1 to the target and an effective quantum number nto label the binding energy of the electron: En=Z2 t/2n2= 1/2n2. As long as the electron is bound to T, we can also approximate Eas E(R) =−En−Zp R. (3) This expression is used throughout all calculations in (I); however, we notice that it is asimptotically correct as long as as R→ ∞. In the limit of small R, instead, E(R) must converge to a finite limit: E(R)→(Zp+ 1)2En (4) (united atom limit). For the moment we will assume that Ris sufficiently large so that eq . (3) holds, but later we will consider the limit (4), too. On the plane Pwe can draw a section of the equipotential surface U(z, ρ, R ) =−En−Zp R. (5) This represents the limit of the region classically allowed to the electron. When R→ ∞ this region is divided into two disconnected circles centered around ea ch of the two nuclei. Initial conditions determine which of the two regions actually the electron liv es in. As Rdiminishes there can be eventually an instant where the two regions become connecte d. In fig. 1 we give an example for this. In the spirit of OBMs it is the opening of the equipotential cu rve between PandTwhich leads to a leakage of electrons from one nucleus to another, and theref ore to charge exchange. We make here the no-return hypothesis: once crossed the barrier, the ele ctron does not return to the target. It is well justified if Zp>>1. As we shall see just below, this hypothesis has important c onsequences. It is easy to solve eq. (5) for Rby imposing a vanishing width of the opening ( ρm= 0); further- more, by imposing also that there be an unique solution for zin the range 0 < z < R : Rm=(1 +/radicalbig Zp)2−Zp En. (6) In the region of the opening the potential Uhas a saddle structure: along the internuclear axis it has a maximum at z=z0=R1/radicalbig Zp+ 1(7) while this is a minimum along the orthogonal direction. Charge exchange occurs provided the electron is able to cros s this potential barrier. Let NΩbe the fraction of trajectories which lead to electron loss at t he time t. It is clear from the discussion above that it must be function of the solid opening angle angl e Ω, whose projection on the plane is the±θmangle. The exact expression for NΩwill be given below. Further, be W(t) the probability for the electron to be still bound to the target, always at tim et. Its rate of change is given by dW(t) =−NΩdt2 TemW(t), (8) 2withTemthe period of the electron motion along its orbit. It is important to discuss the factor dt(2/Tem) since it is an important difference with (I), where just half of this value was used. The meaning of this factor is to account for the fraction of electrons which, within the time interval [ t, t+dt] reach and cross the potential saddle. In (I) it was guessed that it should be equal to dt/T em, on the basis of an uniform distribution of the classical pha ses of the electrons. However, let us read again what the rhs of eq . (8) does mean: it says that the probability of loss is given by the total number of available electrons within the loss cone ( W(t)×NΩ), multiplied by the fraction of electrons which reach the pote ntial saddle. However, on the basis of the no–return hypothesis, only outgoing electrons can cont ribute to this term: an electron which is within the loss cone and is returning to the target from the projectile is not allowed, it should already have been captured and therefore would not be in the s etW. It is clear, therefore, that the effective period is Tem/2, corresponding to the outgoing part of the trajectory. A simple integration yields the leakage probability Pl=P(+∞) = 1−W(+∞) = = 1−exp/parenleftbigg −2 Tem/integraldisplay+tm −tmNΩdt/parenrightbigg .(9) In order to actually integrate Eq. (9) we need to know the coll ision trajectory; an unperturbed straight line with bimpact parameter is assumed: R=/radicalbig b2+ (vt)2. (10) The extrema ±tmin the integral (9) are the maximal values of tat which charge exchange can occur. If we identify this instant with the birth of the openi ng, using eq. (6) and (10), we find tm=√R2m−b2 v. (11) At this point it is necessary to give an explicit expression f orNΩ. To this end, we will consider first the case of an electron with zero angular momentum ( l= 0), and then will extend to nonzero values. In absence of the projectile, the classical electron trajec tories, with zero angular momentum, are ellipses squeezed onto the target nucleus. We are thus consi dering an electron moving essentially in one dimension. Its hamiltonian can be written as p2 2−1 r=−En. (12) The electron has a turning point at rc=1 En. (13) Obviously the approaching of the projectile modifies these t rajectories. However, in order to make computations feasible, we make the following hypothesis: e lectron trajectories are considered as essentially unperturbed in the region between the target an d the saddle point. The only trajectories which are thus allowed to escape are those whose aphelia are d irected towards the opening within the solid angle whose projection on the Pplane is ±θm(see fig. 1) provided that the turning point of the electron is greater than the saddle-point distance: rc≥z0. The validity of these approximations can be questionable, particularly if we are studying the col lision with highly–charged ions, which could deeply affect the electron trajectory. We limit to obse rve that it is necessary in order to make analytical calculations. A posteriori , we shall check the amount of error introduced by such an approximation. The angular integration is now easily done, supposing a unif orm distribution for the directions of the electrons: NΩ=1 2(1−cosθm). (14) In order to give an expression for θmwe notice that cos θm=z0/(ρ2 m+z2 0)1/2, with ρmroot of E(R) =/parenleftBigg ρ2 m+R2 (/radicalbig Zp+ 1)2/parenrightBigg−1/2 +Zp/parenleftBigg ρ2 m+ZpR2 (/radicalbig Zp+ 1)2/parenrightBigg−1/2 . (15) 3It is easy to recognize that, in the right-hand side, the first term is the potential due to the electron– target interaction, and the second is the electron–project ile contribution. Eq. (15) cannot be solved analytically for ρmexcept for the particular case Zp= 1, for which case: ρ2 m=/parenleftbigg2 E(R)/parenrightbigg2 −/parenleftbiggR 2/parenrightbigg2 . (16) The form of E(R) function of Rcannot be given analytically, even though can be quite easil y computed numerically [6]. In order to deal with expressions amenable to algebraic manipulations, we do therefore the approximation: first of all, divide the sp ace in the two regions R < R u, R > R u, where Ruis the internuclear distance at which the energy given by eq. (3) becomes comparable with its united–atom form: En+Zp Ru= (Zp+ 1)2En→Ru=Zp (Zp+ 1)2−11 En. (17) We use then for E(R) the united–atom form for R < R u, and the asymptotic form otherwise: E(R) =En+Zp R, R > R u = (Zp+ 1)2En, R < R u(18) It is worthwhile explicitly rewriting eq. (16) for the two ca ses: ρ2 m=R2/parenleftbigg4 (EnR+ 1)2−1 4/parenrightbigg , R > R u =1 4/parenleftbigg1 E2n−R2/parenrightbigg , R < R u(19) and the corresponding expressions for NΩare: NΩ=1−cosθm 2=1 8(3−EnR), R > R u =1 2(1−EnR), R < R u.(20) Note that NΩ= 1/2 for R= 0. This is a check on the correctness of the model, since, for sym- metrical scattering at low velocity and small distances we e xpect the electrons to be equally shared between the two nuclei. When Zp>1 we have to consider two distinct limits: when R→ ∞ we know that eventually ρm→0 (eq. 6). It is reasonable therefore to expand (15) in series of powers of ρm/Rand, retaining only terms up to second order: ρ2 m≈2/radicalbig Zp/parenleftbig/radicalbig Zp+ 1/parenrightbig4R2/bracketleftbigg/parenleftBig/radicalbig Zp+ 1/parenrightBig2 −Zp−EnR/bracketrightbigg . (21) Consistently with the limit R→ ∞, we have used the large– Rexpression for E(R). The limit R→0 is quite delicate to deal with: a straightforward solution of eq. (15) would give ρm≈1 (Zp+ 1)En+O(R), (22) but calculating cos θmand eventually NΩfrom this expression gives wrong results: it is easy to work out the result NΩ= 1/2, R→0. This is wrong because, obviously, the limit NΩ→1, Zp→ ∞ must hold. The reason of the failure lies in the coupling of eq . (15) with the united–atom form for E(R): one can notice that the expression thus written is perfect ly simmetrical with respect to the interchange projectile–target. Because of this symmetry, electrons are forced to be equally shared between the two nuclei. This is good when dealing with symmet rical collisions, Zp=Zt= 1, and is actually an improvement with respect to (I), where eq. (21 ) was used even for small R’s and one recovered the erroneous value NΩ(R= 0) = 3 /8. But when Zp>1 the asymmetry must be retained in the equations. The only way we have to do this is to extend eq . (21) to small R, obtaining 1−cosθm≈/radicalbig Zp (/radicalbig Zp+ 1)2/bracketleftBig (/radicalbig Zp+ 1)2−Zp−EnR/bracketrightBig . (23) 4It is straightforward to evaluate eq. (23) in the limit Zp→ ∞, R→0, and find the sought result, 2. We notice that, from the numerical point of view, it is not a gr eat error using eq. (21) everywhere: the approximation it is based upon breaks down when Ris of the order of Ruor lesser, which is quite a small range with respect to all other lengths involve d when Zp>1, while even for the case Zp= 1 it is easy to recover (see equations below) that the relati ve error thus introduced on Plis ∆Pl/Pl= 1/24 for small b(and–obviously–it is exactly null for large b). Therefore, eq. (21) could be used safely in all situations. However, we think that the r igorous altough quite lengthy derivation given above was needed since it is not satisfactory working w ith a model which does not comply with the very basic requirements required by the symmetries of the problem at hand. We have now to take into account that the maximum escursion fo r the electron is finite. If we put rc=z0and use for z0,rcrespectively the expressions given by (7) and (13), we obtai n an equation which can be easily solved for R: R=R′ m= (/radicalbig Zp+ 1)rc. (24) TheR′ mthus computed is the maximum internuclear distance at which charge exchange is allowed under the present assumptions. Since R′m< R m(compare the previous result with that of eq. 6 ) we have to reduce accordingly the limits in the integration i n eq. (9): it must be performed between ±t′ m, with the definition of t′ mthe same as tmbut for the replacement Rm→R′ m. The result for the leakage probability is: Pl= 1−exp/parenleftbigg −2F(um) +GZ Tem/parenrightbigg , (25) where we have defined F(u) =/radicalbig Zp (/radicalbig Zp+ 1)2/bracketleftbigg/parenleftBig (/radicalbig Zp+ 1)2−Zp/parenrightBigb vu−/parenleftbiggEnb2 2v/parenrightbigg/parenleftBig u/radicalbig 1 +u2+ arcsinh( u)/parenrightBig/bracketrightbigg , GZ= (3F(uu)−2tu) (Zp= 1) = 0 ( Zp>1), um=vt′ m/b , uu=vtu/b , tu=√R2u−b2 v.(26) The period can be easily computed by Tem= 2/integraldisplay1/En 0dr p=√ 2/integraldisplay1/En 0dr/radicalBig 1 r−En= 2πn3(27) (this result could be found also in [5]). The cross section can be finally obtained after integrating o ver the impact parameter (this last integration must be done numerically): σ= 2π/integraldisplaybm 0bPl(b)db . (28) Again, we have used the fact that the range of interaction is fi nite: the maximum allowable impact parameter bmis set equal to R′ m. Finally, we consider the case when the angular momentum is di fferent from zero. Now, orbits are ellipses whose minor semiaxis has finite length. We can still write the hamiltonian as function of just (r, p): p2 2−1 r+L2 2r2=−En. (29) Lis the usual term: L2=l(l+ 1). The turning points are now r± c=1±√ 1−2EnL2 2En. (30) 5andR′ m= (/radicalbig Zp+ 1)r+ c. Now the fraction of trajectories entering the loss cone is mu ch more difficult to estimate. In principle, it can still be determined: it is equal to the frac tion of ellipses which have intersection with the opening. Actual computations can be rather cumbers ome. Thus, we use the following approximation, which holds for low angular momenta l << n (with nprincipal quantum number): ellipses are approximated as straight lines (as for the l= 0 case), but their turning point is correctly estimated using eq. (30). Note that also the period is modifie d: its correct expression is Tem=√ 2/integraldisplayr+ r−dr/radicalBig 1 r−En−l(l+1) 2r2. (31) III. A TEST CASE As a first test case we consider the inelastic scattering Na++ Na(28d ,29s). We investigate this sytem since: (i) it has been studied experimentally in [7]; ( ii) some numerical simulations using the Classical Trajectory Monte Carlo (CTMC) method have also be en done on it [8], allowing to have detailed informations about the capture probability Plfunction of the impact parameter, and not simply integrated cross sections; (iii) finally, it has been used as test case in (I), thus allowing to assess the relative quality of the fits. In fig. (2) we plot the normalized cross section ˜ σ=σ/n4versus the normalized impact velocity ˜v=vnfor both collisions nl= 28d and nl= 29s (solid line). The two curves are very close to each other, reflecting the fact that the two orbits have very simil ar properties: the energies of the two states differ by a very small amount, and in both cases EnL2<<1. The two curves show reversed with respect to experiment: σ(28d) it is greater than σ(29s). The reason is that the parameter rc is larger in the former case than in the latter. We can distinguish three regions: the first is at reduced velo city around 0.2, where a steep increase of cross section appears while going towards lower velociti es. Over–barrier models do not appear to fully account for this trend: they have a behaviour at low spe ed which is ruled approximately by the 1/vlaw, consequence of the straight-line impact trajectory ap proximation: it is well possible that this approximation too becomes unadequate in this regi on. The second region covers roughly the range 0.3 ÷1.0. Here the nl= 29s data are rather well simulated while the present model overestimates the data fo rnl= 28d. The bad agreement for nl= 28d was already clear to Ostrovsky which attributed it to a de ficiency of the model to modelize l- changing processes. It seems clear that neither our treatme nt of the angular momentum is sufficient to cure this defect. Finally, there is the region at ˜ v >1, where again the OBM, as it stands, is not able to correctly reproduce the data. The reason for this discrepancy can be tr aced back to the finite velocity of the electron: the classical electron velocity is ve= 1/n, so ˜vcan be given the meaning of the ratio between the projectile and the electron velocity. When ˜ v≥1 the projectile is less effective at collecting electrons in its outgoing part of the trajectory (i.e. when it has gone beyond the point of closest approach). In simple terms: an electron is slower th an the projectile; when it is left behind, it cannot any longer reach and cross the potential barrier. IV. CORRECTIONS TO THE MODEL This picture suggests a straightforward remedy: a term must be inserted in eq. (8) to account for the diminished capture efficiency. This is accomplished form ally through rewriting NΩ→w(t,˜v)NΩ, withw≤1. We have put into evidence that wcan in principle be function of time and of the impact velocity. The simplest correction is made by assuming a perf ect efficiency for ˜ v <1,w(t,˜v <1) = 1, while, for ˜ v >1, no electrons can be collected after that the distance of mi nimum approach has been reached: w+≡w(t >0,˜v >1) = 0. This can appear too strong an assumption, since those electrons which are by the same side of the projectile with re spect to the nucleus, and which are close to their turning point may still be captured. In fig. (2) we can compare the original data with those for w+= 0 (dashed line). The sharp variation of σat ˜v= 1 is obviously a consequence of the crude approximations done choosing wwhich has a step–like behaviour with v. To get further insight, we plot in fig. 3 the quantity bPl(b)versus bfor the collision Na++Na(28d). The impact velocity is ˜ v= 1. The symbols are the CTMC results of ref. [8]. Solid line is the model 6result for w+= 1; dotted line, the result for w+= 0; dashed line, an intermediate situation, with w+= 1/2. Striking features are, for all curves, the nearly perfect accordance of the value b≈3000 at which Pl= 0 (it is bmaccording to our definition). The behaviour at small b’s (Pl≈1/2) is well reproduced for w+= 1 while it is slightly underestimated by the two other curve s. On the other hands, only by setting w+= 0 it is possible to avoid the gross overestimate of Plnear its maximum. It is thus evident that the agreement is somewhat improved in the region ˜ v≈1 by letting w+= 0. However, the high–velocity behaviour is still missed by the model, which predicts a power– law behaviour σ∝v−1, while the actual exponent is higher. Within our picture, th is suggests that also the capture efficiency w−=w(t <0) must be a decreasing function of ˜ v. An accurate modelization of the processes which affect this term is difficu lt, and we were not able to provide it. However, some semi–qualitative arguments can be given. Let us review again the process of capture as described in section II and shown in fig. (1): if ˜ v >1, an electron at time tcan be in the loss cone and still not to be lost, since within a time span ∆ t≈ρm/vthe position of the loss cone has shifted of such an amount that only those electrons which wer e closer to the saddle point than a distance ve∆tcould be caught. The fraction of these electrons is ∆ t(2/Tem)≈ρm(2/vTem). This correction gives an additional 1 /vdependence, thus now σ≈1/v2. As an exercise, we try to fit experimental data using was a free parameter instead that a function to be determined by first principles. We choose one of the simp lest functional forms: w=1 +|β|m 1 +|˜v−β|m, (32) withβ, mfree parameters to be adjusted. This form gives the two corre ct limits: w→1,˜v→0, andw→0,˜v→ ∞. The parameter βis not really needed; it has been added to reach a better fit. Its meaning is that of a treshold velocity, at which the ca pture efficiency begins to diminish. In fig. (2) we plot the fit obtained with β= 0.2, m= 4 (dotted line): this is not meant to be the best fit, just a choice of parameters which gives a very good ag reement with data. We see that the suggested corrections are still not enough to give the right power–law, if one needs to go to some extent beyond the region ˜ v= 1. V. OTHER COMPARISONS A. Iodine - Cesium collisions We apply now our model to the process of electron capture Iq++ Cs→I(q−1)++ Cs+(33) withq= 6÷30. This scattering process has been studied experimentall y in [9]. It is particularly interesting to study in this context since it has revealed un tractable by a number of other OBM’s, including that of (I) (for a discussion and results, see [3]) . The impact energy is chosen equal to 1.5×ZpkeV: since it corresponds to ˜ v << 1, we can safely assume w= 1. The Cesium atom is in its ground state with the optical electron in a sstate. In fig. 4 we plot the experimental points together with our est imates. In this case the fit is excellent. It is important to notice that this agreement is e ntirely consequence of our choice of limiting integration to Rgiven by eq. (24): to understand this point, observe that bec ause of the very high charge of the projectile, the exponential term in e q. (25) is small ( F, by direct inspection, is increasing with Zp) and thus Pl≈1. The details of the model which are in Fare therefore of no relevance. The only surviving parameter, and that which det ermines σ, isR′ m. It can be checked by directly comparing our fig. 4 with fig. 1 of ref. [3], where resu lts from model (I) are shown, which differ from ours just in replacing eq. (24) with eq. (6). There , the disagreement is severe. B. Ion - Na( n= 3) collisions As a final test case we present the results for collisions H–Na (3s,3p). They are part of a set of experiments as well as numerical simulations involving als o other singly–charged ions: He, Ne, and Ar (see [11] and the references therein and in particular [12 ]; ref. [13] presents numerical calculations for the same system). In fig. 5 we plot the results of our model t ogether with those of ref. [11]. Again, we find that only by neglecting w+some accordance is found. The low–energy wing of the 7curve is strongly underestimated for Na(3s), while the agre ement is somewhat better for Na(3p). Again, the slope of σfor relative velocities higher than 1 could not be reproduce d. We do not show results for other ions: they can be found in fig. 3 of ref. [11]. What is important to note is that differencies of a factor two (and even larger fo r 3s states) appear between light (H+, He+) and heavy (Ne+, Ar+) ions which our model is unable to predict. We can reasonably conclude therefore: (i) that the present model is not satisfactory fo rv/ve<<1 (it was already pointed out in sec. IV) and for v/ve>1 ; (ii) the structure of the projectile must be incorporated into the model otherwise different ions with the same charge should ca use the same effect, at odds with experiments. As emphasized in [11,12] the energy defect ∆ Eof the process is a crucial parameter: captures to states with ∆ E≈0 are strongly preferred. Obviously, the value of ∆ Edepends on the energy levels structure of the recombining ion. VI. SUMMARY AND CONCLUSIONS We have developed in this paper a classical OBM for single cha rge exchange between ions and atoms. The accuracy of the model has been tested against thre e cases, with results going from moderate–to–good (sec. III and IV), excellent (sec. V.A), a nd poor–to–moderate (sec. V.B). As a rule of thumb, the model can be stated to be very well suited fo r collisions involving highly charged ions at low velocities. The model is based upon a previous work [1], and adds to it a num ber of features, which we go to recall and discuss: (i) the finite excursion from the nucle us permitted to the electrons; (ii) the redefinition of the fraction of lost electrons dt/T em→dt(2/Tem); (iii) a more accurate treatment of the small impact parameter region for symmetrical collisio ns; (iv) the explicit-altough still somewhat approximate-treatment of the capture from l >0 states; (v) a correction to the capture probability due finite impact velocity. Let us discuss briefly each of thes e points: Point (i) and (ii) contribute a major correction: in particu lar, (i) is essential to recover that excellent agreement found in section V.A, while (ii) accounts for the c orrect bPlbehaviour at small b’s (see fig. 2). Point (iii) is unimportant for actual computations, but cor rects an inconsistency of the model. Point (iv) has been studied in less detail, in part for the lac k of experimental data on which doing comparisons. Point (v): a good theoretical estimate of wshould be of the outmost importance for developing a really accurate model of collision at medium-to-high impac t velocity. In this paper we have just attempted a step towards this direction which, however, has allowed to recover definitely better results. Finally we recall from sec. V.B that the treatment of the proj ectile–or better the process of the electron-projectile binding–is an aspect which probably a waits for main improvements. We just observe that it is a shortcoming of all classical methods, th at they cannot easily deal with quantized energy levels. ACKNOWLEDGMENTS It is a pleasure to thank the staff at National Institute for Fu sion Science (Nagoya), and in particular Prof. H. Tawara and Dr. K. Hosaka for providing th e data of ref. [9]. [1] V.N. Ostrovsky, J. Phys. B: At. Mol. Opt. Phys. 28, 3901 (1995). [2] G. Ivanovski, R.K. Janev, and E.A. Solov’ev, J. Phys B: At . Mol. Opt. Phys. 28, 4799 (1995). [3] F. Sattin, e-print physics/0001008 (to be published in J ournal of Physics B). [4] H. Ryufuku, K. Sasaki and T. Watanabe, Phys. Rev. A 21, 745 (1980). [5] L.D. Landau and E.M. Lifshitz Quantum Mechanics (Oxford, Pergamon, 1977) Eq. (48.5). [6] F. Sattin, Comp. Phys. Commun. 105, 225 (1997). [7] S.B. Hansen, L.G. Gray, E. Horsdal-Petersen and K.B. Mac Adam, J. Phys. B: At. Mol. Opt. Phys. 24, L315 (1991). 8[8] J. Pascale, R.E. Olson and C.O. Reinhold, Phys. Rev. A 42, 5305 (1990). [9] K. Hosaka et al,Electron capture cross sections of low energy highly charge d ions in collisions with alkali atoms , poster presented at the International Seminar on Atomic Processes in Plasmas held in Toki (Japan, 1999). To be published as a report of the National Institute for Fusion Science (NIFS). [10] M. Kimura et al, J. Phys. B: At. Mol. Opt. Phys. 28, L643 (1995); K. Hosaka et al1997 Fus Eng Design 34-35 , 781 (1997); A. Hiroyuki et al, Fus. Eng. Design 34-35 , 785 (1997); K. Hosaka et al, Phys. Scr. T73, 273 (1997). [11] J.W. Thomsen et al, Z. Phys. D 37, 133 (1996). [12] F. Aumayr, G. Lakits and H. Winter, Z. Phys. D 6, 145 (1987). [13] A. Dubois, S.E. Nielsen and J.P. Hansen, J. Phys. B: At. M ol. Opt. Phys. 26, 705 (1993). 9FIG. 1. The enveloping curve shows a section of the equipoten tial surface U=E, i.e. it is the border of the region classically accessible to the electron. Ris the internuclear distance. The parameter ρmis the radius of the opening which joins the potential wells, θmthe opening angle from T;z0is the position of the potential’s saddle point. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6010203040 v n (a.u.)σ/n4 (a.u.)Na(29s) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.601020304050 v n (a.u.)σ/n4 (a.u.)Na(28d) FIG. 2. Cross section for charge exchange for Na+–Na(29s) (upper) and Na+–Na(28d) (lower) collisions. Symbols, experi- mental data (adapted from ref. 7); solid line, present model withw+= 1; dashed line, model with w+= 0; dotted line, model withwgiven by eq. (32). Note that the experimental results are not absolutely calibrated, the data shown here are calibrated using as reference the CTMC results at ˜ v= 1 and nl= 28d. 100 500 1000 1500 2000 2500 3000 35000100200300400500600700800 b (a.u.)b P(b) (a.u.) FIG. 3. Probability of electron capture multiplied by impac t parameter, Plb, for Na+–Na(28d) collision at ˜ v= 1. Squares, CTMC data (adapted from ref. 8); solid line, present model wi thw+= 1; dashed line, w+= 0.5; dotted line, w+= 0. 5 10 15 20 25 3040060080010001200140016001800 Zpσ [10−20 m2] FIG. 4. Cross section for charge exchange in I+q–Cs collisions. Circles, experimental data with 20% error b ar; solid line, present model (where we have set w≡1, since we are dealing with v/ve<<1). 110.4 0.6 0.8 1 1.2 1.4 1.6 1.80102030405060 v/veσ [10−20 m2]Ne(3s) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2020406080100 v/veσ [10−20 m2]Ne(3p) FIG. 5. Cross section for charge exchange in H+–Na(3s) (upper) and H+–Na(3p) (lower) collisions. Symbols, experimental data from ref. (9); lines, present model. 12

arXiv:physics/0002012v1 [physics.bio-ph] 4 Feb 2000Rupture of multiple parallel molecular bonds under dynamic loading Udo Seifert Max-Planck-Institut f¨ ur Kolloid- und Grenzfl¨ achenforsc hung, Am M¨ uhlenberg 2, 14476 Golm, Germany Abstract Biological adhesion often involves several pairs of specifi c receptor-ligand molecules. Using rate equations, we study theoretically th e rupture of such multiple parallel bonds under dynamic loading assisted by t hermal activation. For a simple generic type of cooperativity, both the rupture time and force exhibit several different scaling regimes. The dependence o f the rupture force on the number of bonds is predicted to be either linear, like a square root or logarithmic. PACS: 87.15 By, 82.20 Mj Introduction. Single molecule force spectroscopy has made it possible to m easure the binding strength of a pair of receptor-ligand (“lock-key”) molecules using vesicles [1], the atomic force apparatus [2–4], or optical tweezers [5] as tra nsducers. Thus, the essential constituents mediating biological adhesion have become ac cessible to quantitative physical experiments [6]. This experimental progress has fostered t heoretical studies of the rupture of such pairs under dynamic loading. Thermal activation bei ng a main contributing factor, Kramers-like descriptions of the rupture process with time -dependent potentials show that the rupture strength of such bonds depends on the loading rat e [7–9]. Such behavior has been found experimentally indeed [10,11]. While unspecific theoretical models of the rupture process reveal generic features, molecular dynamic studie s can address the details of the dynamics of the rupture of specific pairs [8,12]. Adhesive contact and the rupture thereof often involves not just one but several molecular pairs of the same or different species [13]. The equilibrium p roperties of the cooperative effects of such specific interactions are well studied both in theory [14–16] and in experiments [17–19]. Concerning the dynamics of rupture of such a contac t under loading, detailed models for specific problems such as the peeling of a membrane [20,21 ] or the rolling of leucocytes in shear flow [22] have been solved numerically to extract a cr itical tension or shear rate for rupture. However, it is inherently difficult to separate gene ric dynamical properties from specific ones using such intricate models. As an example for a generic property consider 1the following question: How does the time and force necessar y to break an adhesive contact under dynamic loading depend on the number of bonds initiall y present? The present study addresses this question within a simple mo del that extends work on the dynamic failure of a single bond to that of a whole patch in volving several bonds of the same type. Quite generally, two different limiting cases must be distinguished. If the load is primarily concentrated on one bond at a time with rela xation of the load when the first bonds fails and subsequent loading of the next one, the r upture process basically is a sequence of similar single molecule events. The N0bonds initially present then act in series . The rupture time will be ∼N0whereas the force will exhibit a saw tooth-like pattern with a peak given by the rupture strength of a single bond. Such a be havior has been found and modeled in the related case of unfolding of proteins with sev eral identical domains like titin [23]. The main purpose of the present paper is to analyze the o ther case where the load is distributed (almost) uniformly among several bonds such th at these bonds act in parallel . As more and more bonds rupture, the force on the remaining one s increases. This simple type of cooperativity leads to different scaling regimes for the rupture time and rupture force. xx v K Kt ppt FIG.1: Model geometry for the rupture of parallel bonds. Sym bols are explained in the main text. Model. We model the rupture geometry generically as shown in Fig.1. One partner of the bond (“receptor”) is confined to a substrate. The other (“ ligand”) is connected by a polymer to a transducer which is connected by an elastic elem ent to a sled being pulled at velocity v. For simplicity, we model both the elasticity of the transdu cer and the polymers as Hookean springs with zero rest length and spring constant sKtandKp, respectively. As long as a bond is intact the corresponding polymer is stretch ed to an extension xpwhich we assume to be the same for all intact bonds. The elongation o f the transducer from its resting position is xt. Force balance on the transducer becomes NK pxp=Ktxtwhere N is the number of intact bonds. Geometry dictates the time dep endence xp(t) +xt(t) =vt. From these two relations, we find the time-dependent force on an intact bond as Fb(t) =Kpxp(t) =KpKt N(t)Kp+Ktvt. (1) Following Bell [13], we assume that the main effect of such a fo rce is to introduce an instan- taneous, time-dependent dissociation rate k0(t) according to 2k0(t) =k0exp[Fb(t)xb/kBT], (2) where k0is the dissociation rate in the absence of a force. The quanti tyxbis of the order of the distance between the minimum of the binding potential an d the barrier and kBTis the product of Boltzmann’s constant and temperature. We are mainly interested in the case of a softtransducer defined as Kt<∼Kp. In this case, eq. (1) shows that the force on a bond is inversely propo rtional to the number of intact bonds for all N(t). Hence, when a bond ruptures, the force on the remaining one s increases. We now discuss two different cases, irreversible and reversi ble bonds. In the former case, a bond, once ruptured, cannot rebind. Reversible bonds have a non-zero rebinding rate. Irreversible bonds. Initially N(t= 0)≡N0bonds are present. The rate equation for their time-dependent decrease is ∂tN=−N(t)k0exp[Fb(t)xb/kBT]. (3) We scale time with the dissociation rate in equilibrium k0according to τ≡tk0.The rate equation in the case of a soft transducer then becomes ∂τN=−Nexp[(µτ/N ] (4) with the loading parameter µ≡Ktxbv/kBTk0. (5) This simple rate equation seems not to have an analytical sol ution. However, its scaling behavior can be extracted by the following analysis. With th e substitution u(τ)≡τ/None obtains ∂τu=u(1/τ+ 1) + u(exp[µu]−1) (6) For small τ,u(τ)≈τ/N0and the second term in (6) can therefore be neglected. The solution u1(τ) of the corresponding equation becomes u1(τ) =τeτ/N0and hence a purely exponential decay for the number of intact bonds, N(τ) =N0e−τ. This approximation breaks down for τ>∼τ1withτ1implicitly defined by max(1 /τ1,1) = exp[ µu1(τ1)]−1. (7) Forτ > τ 1, we can then ignore both the first term and the “-1” in the secon d term of (6). The corresponding equation ∂τu=uexp[µu] is solved by E(µu1)−E(µu) =τ−τ1 (8) where E(x)≡/integraltext∞ xdx′e−x′/x′is the exponential integral and u1≡u1(τ1) is the cross over value of the first solution at the matching point τ1. Hence the time necessary for complete rupture, τ∗, can be estimated by setting u(τ∗) =τ∗/N=∞which leads to τ∗=τ1+τ2=τ1+E(µu1). (9) 3Based on this approximative solution of (6), three sub-regi mes can be identified: (i)µ<∼1: In this case, the exponential decay holds till N(τ)≃1. Physically, the rupture is then complete. In this trivial regime, where the loading i s too small to affect the rupture process at all, the time required for rupture is τ∗∼lnN0. (10) Note that the same result could have been obtained by analyzi ng the mean time required for the irreversible decay of N0independent bonds under no force. (ii) 1<∼µ<∼N0: In this regime, the exponential decay persists till τ1∼ln(N0/µ). At this time the number of bonds has reached N(τ1)∼µ. The remaining bonds decay according to (8) which leads to an additional time τ2of order 1 which is small compared to τ1. Hence the whole rupture time in this regime is of order τ∗∼ln(N0/µ). (11) (iii)N0<∼µ. In this case, the exponential decay applies till τ1∼(N0/µ) ln(µ/N 0). According to (8) the remaining time τ2∼(N0/µ)/ln(µ/N 0) is smaller than τ1. Hence the total rupture time is τ∗∼(N0/µ) ln(µ/N 0). (12) Thus we find for small loading rates that the rupture time is lo garithmic in the number of bonds initially present whereas for large loading, this tim e becomes linear in N0. For fixed N0 and increasing µ, the rupture time first is independent of µ. It then decays logarithmically inµand finally becomes inversely proportional to µ. The force measured by the transducer is given by Ft≡N(t)Kpxp(t)≈(kBT/x b)µτ. (13) Thus, the total force experienced by the soft transducer is i ndependent of the number of intact bonds and increases linearly in time. The dimensionl ess rupture force f∗=µτ∗is thus given by f∗∼µlnN0 forµ<∼1, ∼µln(N0/µ) for 1<∼µ<∼N0, ∼N0ln(µ/N 0) for N0<∼µ. (14) in the three regimes, respectively. Reversible bonds. So far, we have neglected the possibility that broken bonds c an reform. Hence, rupture from a genuine equilibrium situatio n where bonds form, break, and rebind requires a refined description where we add a term for r ebinding. We assume that one species of the receptor/ligand couple is limited to a total n umber N1withN(t) molecules bound and N1−N(t) unbound whereas the other species is available in excess. T he rate equation becomes ∂tN=−N(t)k0exp[µτ/N (t)] +kf(N1−N(t)), (15) 4where we assume for simplicity that the rate kffor bond formation is not affected by the force. Without loading, the equilibrium number of bonds is Neq=γN1/(1 +γ) (16) where γ≡kf/k0. As loading starts, the number of bonds decreases from this e quilibrium value. With u(τ)≡τ/Nas before, we get ∂τu=u(1/τ+ 1 + γ−γN1u/τ) +u(exp[µu]−1) (17) Forµ= 0, this equation is solved by u0(τ)≡τ/N eq, (18) which corresponds to the stationary equilibrium distribut ion. The loading term becomes relevant at a time τ=τ1for which (exp[µu0(τ1)/N1]−1)u0∼∂τu0= 1/Neq. (19) Two cases must then be distinguished: (i) For µ<∼Neq,τ1∼(Neq/µ)1/2. Up to this time, the loading has not significantly af- fected the number of bonds. The remaining time till all bonds are ruptured can be estimated to be of the same order as τ1using (17). Hence, τ∗∼(Neq/µ)1/2. (20) In this case, the rupture time increases as a square root of th e equilibrium bonds present and decreases as a square root of the loading parameter. (ii) For Neq<∼µ,τ1∼(Neq/µ) ln(µ/N eq), with a remaining time of the same order. Hence in this case, we recover the irreversible result (12) with N0replaced by Neq. Since in both cases the rupture time τ∗∼τ1, we get easily for the rupture force f∗=µτ∗∼(µNeq)1/2forµ<∼Neq, ∼Neqln(µ/N eq) for Neq<∼µ. (21) Stiff transducer. So far, we have considered the case of a soft transducer ( Kt<∼Kp) for which the force on a bond depends on the number of bonds. Anoth er limiting case is a stiff transducer with Kt>∼N0KpandKt>∼NeqKpfor the case of irreversible and reversible rupture, respectively. According to eq. (1), the force on a b ond then is (almost) independent of the number of bonds. Hence, the rupture time is only weakly dependent on the number of bonds. An analysis of the corresponding rate equations al ong similar lines as above shows for the irreversible case two subregimes with τ∗∼lnN0for ¯µ<∼1, ∼ln ¯µ/¯µfor ¯µ>∼1 (22) with a loading parameter 5¯µ≡Kpxbv/kBTk0 (23) dominated by the polymeric stiffness. For the dimensionless maximal force experienced by the transducer during the rupture process one finds f∗∼¯µN0for ¯µ<∼1, ∼N0ln ¯µfor ¯µ>∼1 (24) in the two cases. Similarly, for a stiff transducer and reversible bonds, one gets τ∗∼(lnNeq)1/2/¯µ1/2for ¯µ<∼1, ∼ln ¯µ/¯µ for ¯µ>∼1 (25) and for the dimensionless maximal force experienced by the t ransducer f∗∼¯µ1/2Neqfor ¯µ<∼1, ∼Neqln ¯µfor ¯µ>∼1. (26) Finally, there is a crossover regime for N0,eqKp>∼Kt>∼Kp, where the pulling starts as in the soft case. As the number of intact bonds decreases toward s the value ˜N≡Kt/Kp, the denominator in (1) becomes dominated by Ktand the rupture process proceeds as for a stiff transducer. For the reversible case, it turns out that both the rupture time and the rupture force are dominated by the soft part. Hence the results (20,2 1) apply for all Kt<∼NeqKp. For the irreversible case, analysis of the crossover regime is slightly more invo lved. The different scaling regimes for rupture time and force are show n in Fig. 2 without explicit derivation. Concluding perspective. Based on an analysis of rate equations, the comprehensive sc aling analysis presented in this paper has revealed several differ ent regimes for the rupture time and force of parallel molecular bonds under dynamic loading . The most distinctive regime is presumably the square root dependence of rupture time and force (20,21) on loading rate and number of bonds derived for reversible bonds under small loading. Such a square root behavior on the loading rate is different from both the irreve rsible case and the dependence on loading rate for rupture of a single bond or bonds in series . An experimental result showing such an exponent could therefore be taken as a signat ure of breaking multiple parallel reversible bonds. Of course, it will be important t o work with a model system where the number or density of bonds of at least one partner ca n be controlled in order to extract the dependence of rupture time and force on this cruc ial quantity. An obvious theoretical refinement of the present model would be to include fluctuations of the rupture time for individual bonds. Other ramification s can include allowing lateral interactions between the bonds, combining the simplistic H ookean transducer with a mem- brane patch with its own elasticity, or modeling the rupture process more delicately than done here to name just a few possibilities. It will be interes ting to see how robust the scaling regimes derived in this paper will be under such modi fications which can effectively lead to scenarios somewhere between the present “in paralle l” case and the “in series” case described briefly in the introduction. Finally, it should be clear that in spite of – or rather 6because of – the progress made in understanding the single bo nd behavior, the cooperative effects of several bonds under dynamic loading deserve furth er attention both in theory and in experiment. Acknowledgments: I thank E. Sackmann for a stimulating discussion and J. Shill cock for a critical reading of the manuscript. ln(N /µ) (1/µ) ln µ ln N 1 1 Nµ µ− −− ln(N /µ) ln N 1 1 Nµ µ− −µµN /µ) ln (µ/N ) ( N ln(µ/N ) N ln µ µ N−w(a) (b) 0 0 0 0 0000 000 0 FIG.2: Dynamical phase diagram for (a) the dimensionless ru pture time τ∗and (b) the dimensionless rupture force f∗as a function of the two loading parameters µ(5) and ¯µ(23) in the case of irreversible rupture. In the region w, the rupture force is given by f∗∼µln(N0¯µ/µ). 7REFERENCES [1] E. Evans, D. Berk, and A. Leung, Biophys. J. 59, 838 (1991). [2] E.-L. Florin, V. T. Moy, and H. E. Gaub, Science 264, 415 (1994). [3] V. T. Moy, E.-L. Florin, and H. E. Gaub, Science 266, 257 (1994). [4] G. U. Lee, D. A. Kidwell, and R. J. Colton, Langmuir 10, 354 (1994). [5] T. Nishizaka, H. Miyata, H. Yshikawa, S. Ishiwata and K. K inosita, Nature 377, 251 (1995). [6] for a review, see: P. Bongrand, Rep. Prog. Phys. 62, 921 (1999). [7] E. Evans and K. Ritchie, Biophys. J. 72, 1541 (1997). [8] S. Izrailev et al., Biophys. J. 72, 1568 (1997). [9] J. Shillcock and U. Seifert, Phys. Rev. E 57, 7301 (1998). [10] R. Merkel, P. Nassoy, A. Leung, K. Ritchie and E. Evans, N ature397, 50 (1999). [11] D. A. Simson, M. Strigl, M. Hohenadl, and R. Merkel, Phys . Rev. Lett. 83, 652 (1999). [12] H. Grubm¨ uller, B. Heymann, and P. Tavan, Science 271, 997 (1996). [13] G.I. Bell, Science 200, 618 (1978). [14] G.I. Bell, M. Dembo, and P. Bongrand, Biophys. J. 45, 1051 (1984). [15] D. Zuckerman and R. Bruinsma, Phys. Rev. Lett. 74, 3900 (1995). [16] R. Lipowsky, Phys. Rev. Lett. 77, 1652 (1996). [17] D. A. Noppl-Simson and D. Needham, Biophys. J. 70, 1391 (1996). 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arXiv:physics/0002013v1 [physics.optics] 6 Feb 2000Dynamics of Atom-Mediated Photon-Photon Scattering II: Ex periment M. W. Mitchell, Cindy I. Hancox and R. Y. Chiao Department of Physics, University of California at Berkele y, Berkeley, CA 94720, USA (February 2, 2008) Temporal and angular correlations in atom-mediated photon -photon scattering are measured. Good agreement is found with the theory presented in Part I. I. INTRODUCTION As described in Part I [1], atom-mediated photon- photon scattering is the microscopic process underlying the optical Kerr nonlinearity in atomic media. The Kerr nonlinearity produces such effects as self-phase modula- tion, self-focusing and self-defocusing and four-wave mix - ing. In an atomic medium, resonanant nonlinearities can give rise to very large nonlinear optical effects, suggestin g the possibility of nonlinear optical interactions with onl y a few photons [2]. Unfortunately, the same resonances which could facilitate such experiments make them dif- ficult to analyze [3]. In Part I we showed theoretically that the photon-photon interaction is not intrinsically lossy, and can be fast on the time scale of atomic relax- ation. Here we describe an experiment to directly mea- sure the time-duration of the photon-photon interaction in a transparent medium. In the scattering experiment, two off-resonance laser beams collide in a rubidium vapor cell and scattering products are detected at right angles. The process of phase-matched resonance fluorescence in this geometry has been described as spontaneous four-wave mixing [4], a description which applies to our off-resonant excitation as well. This geometry has been of interest in quantum optics for generating phase-conjugate reflection [5]. Ele- gant experiments with a barium atomic beam [6] showed antibunching in multi-atom resonance fluorescence, but a separation of timescales was not possible since the detun- ing, linewidth and doppler width were all of comparable magnitude. II. SETUP A free-running 30mW diode laser at 780nm was tem- perature stabilized and actively locked to the point of minimum fluorescence between the hyperfine-split res- onances of the D2 line of rubidium. Saturation spec- troscopy features could be observed using this laser, indi- cating a linewidth δν <200 MHz. This linewidth is small compared with the detuning from the nearest absorp- tion line δν= 1.3 GHz . Direct observation of the laser output with a fast photodiode (3 dB rolloff at 9 GHz) showed no significant modulation in the frequency band 100 MHz – 2 GHz. The laser beam was was shaped bypassage through a single-mode polarization-maintaining fiber, collimated and passed through a scattering cell to a retro-reflection mirror. The beam within the cell was lin- early polarized in the vertical direction. The beam waist (at the retroreflection mirror) was 0 .026 cm ×0.023 cm (intensity FWHM, vertical ×horizontal). The center of the cell was 1.9 cm from the retroreflection mirror, thus within a Rayleigh range of the waist. With optimal align- ment, the laser could deliver 1.95 mW to the cell, giving a maximal Rabi frequency of Ω Rabi≈2×109s−1, signif- icantly less than the minimal detuning of δ= 2π×1.3 GHz = 8 ×109s−1. For this reason, we have neglected saturation of the transitions in the analysis. The retro-reflected beam returned through the fiber and was picked off by a beamsplitter. The single-mode fiber acted as a near-ideal spatial filter and the returned power through the fiber provides a quantitative measure of the mode fidelity on passing through the rubidium cell. With optimal alignment it was possible to achieve a mode fidelity (described below) of 36%. The cell, an evacuated cuvette filled with natural abun- dance rubidium vapor, was maintained at a temperature of 330 K to produce a density of about 1.6 ×1010cm−3. Irises near the cell limited the field of view of the detec- tors. Stray light reaching the detectors was negligible, as were the detectors’ dark count rates of <100 cps. With the aide of an auxiliary laser beam, two single- photon counting modules (SPCMs) were positioned to detect photons leaving the detection region in opposite directions. In particular, photons scattered at right- angles to the incident beams and in the direction per- pendicular to the drive beam polarization were observed. Each detector had a 500 µm diameter active area and a quantum efficiency of about 70%. The detectors were at a distance of 70 cm from the center of the cell. The effective position of one detector could be scanned in two dimensions by displacing the alignment mirrors with inchworm motors. A time-to-amplitude converter and multichannel analyzer were used to record the time-delay spectrum. The system time response was measured us- ing sub-picosecond pulses at 850nm as an impulse source. The response was well described by a Lorentzian of width 810 ps (FWHM). Optimal alignment of the laser beam to the input fiber coupler could not always be maintained against thermal drifts in the laboratory. This affected the power of the 1drive beams in the cell but not their alignment or beam shape. These were preserved by the mode-filtering of the fiber. Since the shape of the correlation function depends on beam shape and laser tuning but not on beam power, this reduction in drive power reduced the data rate but did not introduce errors into the correlation signal. A. Experimental Results The time-delay spectrum of a data run of 45 hours is shown in Fig. 2. The detectors were placed to col- lect back-to-back scattering products to maximize the photon-photon scattering signal. A Gaussian function P(tA−tB) fitted to the data has a contrast [ P(0)− P(∞)]/P(∞) of 0.046±0.008, a FWHM of 1 .3±0.3 ns, and a center of −0.07±0.11 ns. This center position is consistent with zero, as one would expect by the symme- try of the scattering process. For comparison, a reference spectrum is shown. This was taken under the same con- ditions but with one detector intentonally misaligned by much more than the angular width of the scattering sig- nal. The angular dependence of the scattering signal was in- vestigated by acquiring time-delay spectra as a function of detector position. To avoid drifts over the week-long acquision, the detector was scanned in a raster pattern, remaining on each point for 300 s before shifting to the next. Points were spaced at 1 mm intervals. Total live acquisition time was 9 hours per point. The aggregate time-spectrum from each location was fitted to a Gaus- sian function with fixed width and center determined from the data of Fig. 2. The position-dependent con- trast C(x,y) is shown in Fig. 3. A negative value for the contrast means that the best fit had a coincidence dip rather than a coincidence peak at zero time. These neg- ative values are not statistically significant. Fitted to a Gaussian function, C(x,y) has a peak of 0.044 ±.010 and angular widths (FWHM) of 1 .1±0.7 mrad and 3 .7±0.4 mrad in the horizontal and vertical directions, respec- tively. These angular widths are consistent with the expected coherence of scattering products [7]. Seen from the de- tector positions, the excitation beam is narrow in the vertical direction, with a Gaussian shape of beam waist wy= 0.009 cm, but is limited in the horizontal direction only by the apertures, of size ∆ z= 0.08 cm. Thus we expect angular widths of 0 .9 mrad and 3 .25 mrad, where the first describes diffraction of a Gaussian, the second diffraction from a hard aperture. III. COMPARISON TO THEORY The correlation signal predicted by the theory of Part I is shown in Fig. 4. The ideal contrast is 1.53 and the FWHM is 870 ps. The shape of the time correlationsis altered by experimental limitations. First, beam dis- tortion in passing through the cell windows reduces the photon-photon scattering signal. Second, finite detector response time and finite detector size act to disperse the signal. None of these effects alters the incoherent scat- tering background. Beam distortion is quantified by the fidelity factor in- troduced in Part I F≡4/vextendsingle/vextendsingle/integraltext d3xG(x)H(x)/vextendsingle/vextendsingle2 /bracketleftbig/integraltextd3x(|G(x)|2+|H(x)|2)/bracketrightbig2(1) The greatest contrast occurs when His the phase- conjugate, or time-reverse of G, i.e., when H(x) =G∗(x). In this situation F= 1. Under the approximation that the field envelopes obey the paraxial wave equations d dzG=i 2k∇2 ⊥G d dzH=−i 2k∇2 ⊥H, (2) Green’s theorem can be used to show that the volume integral is proportional to the mode-overlap integral /integraldisplay d3xG(x)H(x) = ∆ z/integraldisplay dxdyG (x)H(x), (3) where the last integration is taken at any fixed zand ∆zis the length of the interaction region. Similarly, the beam powers are invariant under propagation and the mode fidelity can be expressed entirely in terms of sur- face integrals as F= 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay dxdyG (x)H(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ×/bracketleftbigg/integraldisplay dxdy/parenleftbig |G(x)|2+|H(x)|2/parenrightbig/bracketrightbigg−2 . (4) The overlap of GandHalso determines the efficiency of coupling back into the fiber. This allows us to determine F. In terms of Pin, the power leaving the output fiber coupler and Pret, the power returned through the fiber after being retro-reflected, this is F=4 (1 +T2)2Pret ηTPin. (5) where η= 0.883 is the intensity transmission coefficient of the fiber and coupling lenses and T= 0.92 is the trans- mission coefficient for a single-pass through a cell win- dow. We find F= 0.36±0.03. The mode fidelity acts twice to reduce contrast, once as the drive beams enter the cell, and again on the photons leaving the cell. This beam distortion has no effect on the incoherent scattering background, thus the visibility is reduced by F2. The finite time response of the detector system acts to disperse the coincidence signal over a larger time window. This reduces the maximum contrast by a factor of 0.27 2and increases the temporal width to 1.62 ns. Similarly, the finite detector area reduces the maximum contrast by a factor of 0.81 and spreads the angular correlations by a small amount. The resulting coincidence signal is shown in Fig. 6. Fitted to a Gaussian, the final signal contrast is 0.042 ±0.007, where the uncertainty reflects the uncer- tainty in F. This is consistent with the observed contrast of 0.044 ±0.010. IV. CONCLUSION We have measured the temporal and angular correla- tions in photon-photon scattering mediated by atomic rubidium vapor. We found good agreement between ex- periment and the perturbative theory presented in Part I. The observed temporal correlations are of the order of one nanosecond, much faster than the system can relax by radiative processes. This is consistent with the predic- tion that the duration of the photon-photon interactionis determined by the inhomogeneous broadening of the vapor. [1] M. W. Mitchell and R. Y. Chiao, Submitted for publica- tion . [2] A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Physical Review Letters 79, 1467 (1997). [3] P. Grangier, D. Walls, and K. Gheri, Physical Review Let- ters81, 2833 (1998). [4] A. Heidmann and S. Reynaud, Journal of Modern Optics 34, 923 (1987). [5] A. Gaeta and R. Boyd, Physical Review Letters 60, 2618 (1988). [6] P. Grangier et al., Physical Review Letters 57, 687 (1986). [7] T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, Phys . Rev. A 52, R3429 (1995). Diode Laser Isolator Reference Rb CellIsolator Lightproof Detection EnclosureSPCM SPCMSM/PM fiber Cell EnclosureReflection Monitor He-Ne LaserInput Fiber Coupler Output Fiber Coupler FIG. 1. FIG. 1. Experiment schematic. 3700072007400760078008000 7,4007,6007,8008,0008,200 -10 -5 0 5 10Signal CountsReference Counts Time Lag (ns) FIG. 2. FIG. 2. Observed coincidence rates for right-angle p hoton-photon scattering. Circles show data acquired with detectors aligned to collect back-to-back scattering prod ucts. Squares show data acquired with detectors misaligned by 10 cm ≈0.14 radian. The solid line is a Gaussian function fit to the da ta. -2-1012-2-1012 -0.0100.010.020.030.040.05 Contrast Vertical Offset (mm)Horizontal Offset (mm) FIG. 3. FIG. 3. Signal contrast vs. detector displacement. A displacement of 1 mm corresponds to an angular deviation of 1.43 mrad. -1.5 -1-0.5 00.5 11.50.511.522.5 τB−τA |A| (a.u.)2 FIG. 4. FIG. 4. Coincidence rates by photon-photon scatteri ng theory: ideal case. Mirror Cell G HT FIG. 5. FIG. 5. Geometry for retro-reflection measurements. 4-1.5 -1-0.5 00.5 11.5 20.850.90.9511.051.11.15 τB−τA |A| (a.u.)2 FIG. 6. FIG. 6. Coincidence rates by photon-photon scatteri ng theory: adjusted for beam shape, finite detection time and detector area. 5

arXiv:physics/0002014v1 [physics.optics] 6 Feb 2000Dynamics of Atom-Mediated Photon-Photon Scattering I: The ory M. W. Mitchell and R. Y. Chiao Department of Physics, University of California at Berkele y, Berkeley, CA 94720, USA (February 2, 2008) The mediated photon-photon interaction due to the resonant Kerr nonlinearity in an inhomo- geneously broadened atomic vapor is considered. The time-s cale for photon-photon scattering is computed and found to be determined by the inhomogeneous bro adening and the magnitude of the momentum transfer. This time can be shorter than the atomic r elaxation time. Effects of atom statistics are included and the special case of small-angle scattering is considered. In the latter case the time-scale of the nonlinear response remains fast, even though the linear response slows as the inverse of the momentum transfer. I. INTRODUCTION Recently there has been experimental and theoretical interest in the nonlinear optics of confined light [1]. A medium possessing an optical Kerr nonlinearity and con- fined within a planar or cylindrical Fabry-Perot resonator gives rise to new nonlinear optical phenomena such as soliton filtering and bilateral symmetry breaking [2,3]. The classical nonlinear optics of this system is described by the Complex Ginzburg-Landau equation (CGLE) ∂E ∂t=ic 2n0k∇2 ⊥E+iωAn2 n0|E|2E+ic∆k n0E −Γ(E−Ed), (1) where Eis the electric field envelope, kis the longitudinal wavenumber, ω=ck/n 0is the field envelope angular fre- quency, Ais a mode overlap factor, ∆ kis the wavenum- ber mismatch from the linear-cavity response and Γ is the field amplitude decay rate. The classical dynamics of Eq. (1) describes the mean-field behavior of a system of interacting photons coherently coupled to an external reservoir. A photonic system of this sort is a versatile model system for condensed matter physics in reduced dimensions [4], as the parameters ∆ k,n2, Γ, and Edin Eq. (1) are subject to experimental control. In partic- ular, an atomic vapor can provide a strong Kerr nonlin- earity which is tunable both in strength and in sign. In this case the nonlinearity arises from the saturation of the linear refractive index, which is a strong function of the drive laser frequency near an absorption resonance. Some of the most interesting proposed experiments for this system, including generation of few-photon bound states [5], direct observation of the the Kosterlitz- Thouless transition in an optical system [4] and observa- tion of quantum corrections to the elementary excitation spectrum of a 1D photon gas [6,7] intrinsically involve photon correlations. For this reason, it is important to understand the microscopic (and not just mean-field) be- havior of photons in an optical Kerr medium. We specifi- cally consider saturation of the resonant electronic polar - ization of a Doppler-broadened atomic vapor, a mediumwhich has been proposed for quantum cavity nonlinear optics experiments and used to observe a nonlinear cavity mode [2]. Thus the system under consideration involves dispersion, loss, inhomogeneous broadening, and the con- tinuum of transverse modes in an extended resonator. Sophisticated techniques have been developed for treating mediated interactions among photons in non- linear media. One approach is to obtain an effective theory in which the quanta are excitations of coupled radiation-matter modes, by canonical quantization of the macroscopic field equations [8,9], or by direct attack on a microscopic Hamiltonian [10]. This approach has the advantage of generality and is suited to multi-mode problems, but has basic difficulties with loss and disper- sion near resonance [11–13]. Microscopic treatments in- clude Scully-Lamb type theory [14,15] and application of phase-space methods [16,17]. A strength of these tech- niques is their ability to handle relaxation and popula- tion changes. They are, however, cumbersome to apply to inhomogeneously broadened media and to multi-mode problems. In this paper we characterize the atom-mediated photon-photon interaction using an accurate microscopic model and perturbation calculations. This allows us to determine the time-scale of the mediated photon-photon interaction in the atomic vapor, despite the complexity of the medium. We find that the interaction is fast and not intrinsically lossy, even for small momentum trans- fer. Thus the medium is suitable for quantum optical experiments, including experiments using the NLFP as a model for the interacting Bose gas. II. SCATTERING CALCULATIONS The complete system is treated as the quantized elec- tromagnetic field interacting via the dipole interaction with an vapor of atoms of mass M. The perturbation calculations are performed in momentum space, as is nat- ural for thermodynamic description of the atomic vapor. This also makes simple the inclusion of atomic recoil ef- fects. The dipole interaction term is identified as the 1perturbation, so that the eigenstates of the unperturbed Hamiltonian are direct products of Fock states for each field. In the rotating wave approximation, the unper- turbed and perturbation Hamiltonians are H0=/summationdisplay k,α¯hcka† k,αak,α+/summationdisplay n,p(¯hωn+¯h2p2 2M)c† n,pcn,p(2) H′=−E(x)·d(x) =−/summationdisplay k,α/radicalbigg 2π¯hck V/summationdisplay n,m,piek,α·µnmc† n,p+kcm,pak,α +h.c. (3) where ak,αis the annihilation operator for a photon of momentum ¯ hkand polarization α,cn,pis the annihila- tion operator for an atom in internal state nwith center- of-mass momentum ¯ hp,Eis the quantized electric field anddis the atomic dipole field. Polarization plays only a very minor role in this discussion so polarization indices will be omitted from this point forward. The simplest mediated interaction is photon-photon scattering, which transfers momentum from one pho- ton to another by temporarily depositing this momen- tum in the medium. Specifically, photons with mo- menta k,lare consumed and photons with momenta k′≡k+q,l′≡l−qare produced. The lowest-order processes to do this are fourth order, so we look for rel- evant terms in H′H′H′H′. A parametric process, i.e., one which leaves the medium unchanged, sums coher- ently over all atoms which could be involved [18]. Due to this coherence, the rates of parametric processes scale as the square of N/V, the number density of atoms. In contrast, incoherent loss processes such as Rayleigh and Raman scattering scale as N/V. Thus for large atomic densities, a given photon is more likely to interact with another photon than it is to be lost from the system. In this sense, the interaction is not intrinsically lossy, as are some optical Kerr nonlinearities such as optical pumping or thermal blooming. The latter processes re- quire absorption of photons before there is any effect on other photons. For this reason, they are unsuitable for quantum optical experiments such as creation of a two- photon bound state. One parametric process, photon-photon scattering at a single atom, is described by the diagram of Fig. 1. The relevant terms in H′H′H′H′contain c† a,pcd,p+l′a† l′c† d,p+l′cc,p−qal ×c† c,p−qcb,p+ka† k′c† b,p+kca,pak (4) or permutations k′↔l′,k↔lfor a total of four terms. Herepis the initial atomic momentum and athrough d index the atomic states involved. With the assumption that no atoms are initially found in the upper states b andd, i.e., nb=nd= 0, this reduces tona,p(1±nc,p−q)a† l′ala† k′ak (5) where the nare number operators for the atomic modes and the upper and lower signs hold for Bose and Fermi gases, respectively. The difference for atoms of differ- ent statistics reflects the fact that the scattering process takes the atom through an intermediate momentum state which could be occupied. Occupation of this intermedi- ate state enhances the process for Bose gases but sup- presses it for Fermi gases. A thermal average of the relevant terms in H′H′H′H′ gives the thermally averaged effective perturbation ∝angb∇acketleftH′ eff∝angb∇acket∇ight=(2π)3 V/summationdisplay klk′l′Vl′k′lka† l′ala† k′ak (6) where Vl′k′lk≡/summationdisplay a/integraldisplay d3pveff(p, a, c)∝angb∇acketleftna,p∝angb∇acket∇ight/summationdisplay c(1± ∝angb∇acketleftnc,p−q∝angb∇acket∇ight), (7) veff(p, a, c) =v(1) eff+v(2) eff+v(3) eff+v(4) eff(8) v(1) eff=c2√ klk′l′ (2π)4¯h ×/summationdisplay bd(el′·µda)∗el·µdc(ek′·µbc)∗ek·µba ×/bracketleftBig R(1) 1R(1) 2R(1) 3/bracketrightBig−1 (9) and similar expressions obtain for v(2−4) eff. and ∝angb∇acketleftna,p∝angb∇acket∇ightis the average occupancy of the atomic state |a,p∝angb∇acket∇ight. The R(1) iare the resonance denominators R(1) 1=c(k+l−k′)−¯h M[p·l′+l′2/2]−ωda+iγd R(1) 2=c(k−k′)−¯h M[−p·q+q2/2]−ωca+iη R(1) 3=c(k)−¯h M[p·k+k2/2]−ωba+iγb. (10) Here ¯hωij≡¯h(ωi−ωj) is the energy difference between states iandj,γiis the inverse lifetime of state iandη is a vanishing positive quantity. Here and throughout, the process is understood to conserve photon momen- tum, but for clarity of presentation this is not explicitly indicated. As described in Appendix A, intensity correlation func- tions for photon-photon scattering products contain a Fourier transform of the scattering amplitudes P(xA, tA, xB, tB)∝/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay dδk′Vl′k′l0k0exp[icδk′τ−]/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (11) 2where δk′is the output photon energy shift, xA,Band tA,Bare detection positions and times, respectively, and τ−≡tB−xB/c−tA+xA/cis the difference in retarded times. This expression allows us to determine the time correlations for photon-photon scattering in a number of important cases. III. LARGE-ANGLE SCATTERING The simplest configuration to understand is that of counterpropagating input beams producing counterprop- agating output photons scattered at large angles. This is also the most convenient experimental geometry. A. One Atom Process Scattering amplitudes and rates for right-angle scatter- ing by the one-atom process are shown in Fig. 2 and Fig. 3, respectively. For the moment we ignore the statistical correction due to the na,pnc,p−qterm in Eq. (7), which will be considered separately. The the vapor is treated as a gas of two-level atoms. The parameters are the Doppler width δD≡k(kBT/M)1/2, where kBis Boltzmann’s con- stant, the radiative linewidth γb=AE/2 where AEis the Einstein A coefficient, and the detuning ∆ ≡ck−ωba, in the ratios γb= 0.01δD, ∆ = 2 πδD. The amplitude units are arbitrary, but do not vary between graphs. At this point it is important to note that the dura- tion of the correlation signal is much shorter than the coherence lifetime of an individual atom, approximately γ−1 b. In fact, the duration of the correlation signal is de- termined by the momentum distribution, a property of the medium as a whole. This can be explained in terms of the coherent summation of amplitudes for scattering processes occurring at different atoms. The process is coherent only when it is not possible, even in principle, to tell which atom participated. This clearly requires momentum conservation among the photons, but it also limits the duration of the atomic involvement. An atom acting as intermediary to transfer momentum qis dis- placed during the time in remains in the state cof Fig. 1. If this displacement is larger than the thermal deBroglie wavelength Λ it is possible, in principle, to determine which atom participated. This limits the duration of the coherent process to δτ∼ΛM/¯hq. B. Statistical Correction As noted above, the quantum statistics of the atoms in the vapor contribute a correction to the single-atom scattering amplitude. This correction (with the sign appropriate for Bose atoms) is shown in Fig. 4 for a gas with phase space density NΛ3/V= 1/2, whereΛ≡(MkBT/2π¯h2)1/2is the thermal deBroglie wave- length. Parameters are as for Fig. 2. C. Simultaneous scattering A second parametric process, simultaneous scattering, is described by the diagram of Fig. 5. The relevant terms inH′H′H′H′contain c† a,pcd,p+l′a† l′c† c,p−qcb,p+ka† k′ ×c† d,p+l′cc,p−qalc† b,p+kca,pak (12) or permutations k′↔l′,k↔lfor a total of four terms. Making the same assumption as before, this reduces to na,pnc,p−qa† l′a† k′alak. (13) This process corresponds to the absorption of each pho- ton by an atom before emission of either, and thus de- scribes a two-atom process and is of the same order in the atomic number density as the Fermi and Bose cor- rections to single-atom scattering. The kinematical and geometric factors of Eq. (7) and Eq. (9) are the same for this process, and the resonance denominators are R(2) 1=c(k+l−k′)−¯h M[p·l′+l′2/2]−ωda+iγd R(2) 2=c(k+l)−¯h M[p·k+k2/2 + (p−q)·l+l2/2] −ωba+iγb−ωdc+iγd R(2) 3=c(k)−¯h M[p·k+k2/2]−ωba+iγb. (14) Amplitudes for simultaneous scattering are shown in Fig. 6 for a gas with a phase space density of one half. Parameters are as for Fig. 2. D. Fermi and Bose Gases The statistical correction and two-atom scattering con- tributions add coherently, giving considerably different correlation functions for moderate degeneracy Bose vs. Fermi gases. This is illustrated in Fig. 7 and Fig. 8, which show the scattering rates vs. delay for Bose and Fermi gases with a phase space density of one half. Pa- rameters are as for Fig. 2. E. Ladder Process In atoms with a “ladder” level structure, in which three levels a–care ordered in energy ωc> ωb> ωaand con- nected by matrix elements µba, µcb∝negationslash= 0, µca= 0, an additional process described by the diagram of Fig. 9 is possible. The relevant terms in H′H′H′H′contain 3c† a,pcd,p+l′a† l′c† d,p+l′cc,p+k+la† k′ ×c† c,p+k+lcb,p+kalc† b,p+kca,pak (15) or permutations k′↔l′,k↔lfor a total of four terms. Making the same assumption as before, this reduces to na,pa† l′a† k′alak. (16) This process corresponds to the absorption of both pho- tons by an atom before emission of either, and thus de- scribes a one-atom process which is of the same order in the atomic number density as one-atom scattering. The kinematical and geometric factors of Eq. (7) Eq. (9) are the same for this process, and the resonance denomina- tors are R(3) 1=c(k+l−k′)−¯h M[p·l′+l′2/2]−ωda+iγd R(3) 2=c(k+l)−¯h M[p·(k+l) +|k+l|2/2] −ωca+iγc R(3) 3=c(k)−¯h M[p·k+k2/2]−ωba+iγb. (17) Right-angle scattering amplitudes for this process are shown in Fig. 10. Parameters are as for Fig. 2. F. Lorentz-model Behavior It is interesting to consider the case of a ladder atom with equal energy spacing ωcb=ωbaand matrix elements |µcb|2= 2|µba|2. In this case the states a–care equivalent to the lowest three levels of a harmonic oscillator, i.e., to a Lorentz model, and the medium is effectively linear for two-photon processes. The amplitudes for the one atom process of Eq. (4) and the ladder process of Eq. (15) partially cancel. The resulting signal is smaller and lacks oscillations, as show n in Fig. 11. Parameters are as for Fig. 2. G. Background Events In addition to the photon-photon scattering processes, Rayleigh scattering (and Raman scattering for more complicated atoms) will create an uncorrelated coinci- dence background. This background is calculated in Ap- pendix A. The coincidence signal, consisting of both the Lorentz-model atom photon-photon scattering sig- nal and the incoherent background is shown in Fig. 12. The peak coincidence rate (at δτ= 0) is approximately twice the background, accidental coincidence rate. In the limit of large detuning, it becomes exactly twice ac- cidental rate. This can be explained in analogy with the Hanbury-Brown-Twiss effect as follows: For the op- timal geometry the drive beams are conjugates of each other H(x) =G∗(x) and the detectors are in oppositedirections. The linear atoms act to create a random in- dex grating which scatters a chaotic but equal (up to phase conjugation) field to each detector. As expected for chaotic light [19], the fourth-order equal-time correlati on function is twice the product of second-order correlation functions. /angbracketleftbig E2(xA, t)E2(xB, t)/angbracketrightbig = 2/angbracketleftbig E2(xA, t)/angbracketrightbig/angbracketleftbig E2(xB, t)/angbracketrightbig .(18) IV. SMALL-ANGLE SCATTERING Thus far the discussion has involved only large-angle scattering. In the context of cavity nonlinear optics all fields are propagating nearly along the optical axis of the cavity so it is necessary to consider scattering processes for nearly co-propagating or nearly counter-propagating photons. As argued above, the temporal width of the correlation signal scales as 1 /q, the inverse of the mo- mentum transfer. This is shown in Fig. 13 and Fig. 14, which show rates for scattering photons from beams in thex–zplane into the the y–zplane. In all cases the beam directions are 0 .1 radian from the zaxis. The co- incidence distribution shows oscillations which die out on the time-scale of the inverse Doppler width, and a non-oscillating pedestal with a width determined by the momentum transfer q. The pedestal, however, does not correspond to the du- ration of the nonlinear process in this case. As above, by considering a ladder atom with the energy spacings and matrix elements of a harmonic oscillator we can isolate the linear optical behavior. As shown in Fig. 15 and Fig. 16, this behavior includes the pedestal, but not the oscillations, indicating that the nonlinear optical proce ss is still fast, with a time-scale on the order of the inverse Doppler width. V. LIMITATIONS ON SCATTERING ANGLE Due to the limited width of the atomic momentum distribution, the resonance denominator R(1) 2is small if the input and output photons are not of nearly the same energy. Since the complete process must conserve pho- ton momentum, input photons with net transverse mo- mentum in the output photon direction will scatter less strongly. The width of this resonance is very narrow: a net transverse momentum ky+ly∼k/radicalbig kBT/Mc2is suf- ficient that few atoms will be resonant. As/radicalbig kBT/Mc2 is typically of order 10−6in an atomic vapor, this would be a severe restriction on the transverse momentum con- tent of the beams in a cavity nonlinear optics experiment. However, as shown in Fig. 16, the narrow resonance as- sociated with R(1) 2contributes the linear response of the medium. The nonlinear response, which has the same 4resonance character as the “ladder” process, is not lim- ited in this way because R(3) 2does not depend upon the output photon energies. VI. OUTPUT POLARIZATION The polarization of the output photons depends on the structure of the atom and can produce polarization- entangled photons. For example, if the input photons are propagating in the ±zdirections and are xpolarized, the two absorption events in the above diagram change the z component of angular momentum by δm=±1. In order for the process to return the atom to its initial state, the two emission events must both produce δm=±1 or both δm= 0. For right angle scattering with the detectors in the±ydirections, the output photons must therefore be either both xor both zpolarized. If both polarizations are possible, the emitted photons are entangled in polar- ization, as well as in energy and in momentum. VII. CONCLUSION Time correlations in photon-photon scattering provide an indication of the time-scale over which the atomic medium is involved in the interaction among photons in a nonlinear medium. It is found that the time-scale is determined by the inhomogeneous broadening of the medium and the magnitude of the momentum transfer. For large-angle scattering, the time-scale of involvement isδτ∼ΛM/¯hq, while for small-angle scattering the time- scale is δτ∼ΛM/¯hk. As this time-scale is shorter than the atomic relaxation time, calculations which contain an adiabatic elimination of the atomic degrees of freedom necessarily overlook the fastest dynamics in this process. APPENDIX A: PHOTON CORRELATIONS 1. Detection Amplitudes Unlike a genuine two-body collision process, atom- mediated photon-photon scattering has a preferred ref- erence frame which is determined by the atomic mo- mentum distribution. To calculate the photon cor- relations we work in the “laboratory” frame and as- sume the momentum distribution is symmetric about zero. We consider scattering from two input beams with beam shapes G(x)≡V−1/2/summationtext kg(k)exp[ik·x] and H(x)≡V−1/2/summationtext lh(l)exp[il·x] which are normalized as/summationtext k|g(k)|2=/summationtext l|h(l)|2= 1. We further assume that the beams are derived from the same monochromatic source and are paraxial, i.e., that g(k) is only appreciable in some small neighborhood of the average beam direction k0, and similarly for h(l) around l0. The geometry is shown schematically in Fig. 17. For convenience, thebeams are assumed to each contain one photon, so that the initial state of the field is |φ(0)∝angb∇acket∇ight=A† GA† H|0∝angb∇acket∇ight (A1) where the creation operators A† G, A† HareA† G≡/summationtext kg(k)a† kandA† H≡/summationtext lh(l)a† l.Scaling of the result to multiple photons is obvious. We use Glauber photodetection theory to determine the rates at which scattering products arrive at two detec- torsAandBat space-time points ( xA, tA) and ( xB, tB), respectively. We compute the correlation function in the Heisenberg representation P(xA, tA,xB, tB) =|∝angb∇acketleft0|Φ(+) H(xB, tB)Φ(+) H(xA, tA)|φ(0)∝angb∇acket∇ightH|2(A2) where the photon field operator is Φ(+) H(x, t)≡V−1/2/summationdisplay k,αak,α(t)exp[ik·x]. (A3) This field operator is similar to the positive fre- quency part of the electric field and is chosen so that Φ(−)(x, t)Φ(+)(x, t) is Mandel’s photon-density operator [20]. To make use of perturbation theory, Eq. (A2) is more conveniently expressed in interaction representatio n as P(xA, tA,xB, tB) =|∝angb∇acketleft0|Φ(+) I(xB, tB)UI(tB, tA)Φ(+) I(xA, tA)|φ(tA)∝angb∇acket∇ightI|2 =|∝angb∇acketleft0|Φ(+) I(xB, tB)Φ(+) I(xA, tA)|φ(tA)∝angb∇acket∇ightI|2 ≡ |A(xA, tA,xB, tB)|2(A4) where UIis the interaction picture time-evolution oper- ator, the interaction picture field operator is Φ(+) I(x, t) =V−1/2/summationdisplay k,αak,αexp[i(k·x−ckt)] (A5) and in passing to the second line we have made the as- sumption that a detection at ( xA, tA) does not physically influence the behavior of photons at ( xB, tB) although there may be correlations. The the amplitude of joint detection is A(xA, tA,xB, tB) =(2π)3 V2¯h/summationdisplay k′l′exp[i(k′·xA−ck′tA)] ×exp[i(l′·xB−cl′tB)] ×/summationdisplay klg(k)h(l)Vl′k′lk ×1−exp[ic(k′+l′−k−l)tA] c(k′+l′−k−l) +iη(A6) 5Although Vl′k′lkdepends strongly upon the magni- tudes of the initial and final photon momenta through the resonance denominators of Eq. (10), it depends only weakly on their directions through the geometrical fac- tors of Eq. (9). This and the assumption of paraxial input beams justify the approximation /summationdisplay klg(k)h(l)Vl′k′lk ≈Vl′k′l0k0/summationdisplay klg(k)h(l)δk+l,k′+l′ =Vl′k′l0k0/integraldisplay d3xG(x)H(x)exp[−i(k′+l′)·x].(A7) We can similarly treat the output photons in the parax- ial approximation for the case that the detection points are far from the interaction region, i.e., that xA, xB≫x. Making these approximations and dropping unphysical portions of the solution propagating inward from the de- tectors toward the source region, we find A(xA, tA,xB, tB) =−i ¯hc/integraldisplay k′dk′l′Vl′k′l0k0 ×/integraldisplay d3xG(x)H(x) |xA−x||xB−x| ×exp[i(k′·(xA−x)−ck′tA)] ×exp[i(l′·(xB−x)−cl′tB)] ×θ(τA)θ(τB) (A8) where cτA,B≡ctA,B−xA,Bare retarded times. A final approximation ignores the slow variation of k′, l′relative to that of the resonant Vl′k′l0k0. Further, we define G′(x)≡G(x)exp[ik0·x],H′(x)≡H(x)exp[il0·x] andk′≡k′ 0+δk′wherek′ 0is the value of k′which max- imizes Vl′k′l0k0subject to momentum and energy conser- vation. This gives a simple expression for the correlation function A(xA, tA,xB, tB) =−ik′l′ ¯hcexp[−ic(k′ 0τA+l′ 0τB)] ×/integraldisplay dδk′Vl′k′l0k0exp[icδk′(τB−τA)] ×/integraldisplay d3xG′(x)H′(x) |xA−x||xB−x| ×exp[i(k0+l0−k′ 0−l′ 0)·x] ×θ(τA)θ(τB). (A9) This can be interpreted as consisting of a carrier wave, a Fourier transform of the scattering amplitude and a coherent integration of the contributions from different parts of the interaction region. The spatial integral en- forces phase matching in the photon-photon scattering process.2. Detection Rates The probability for a coincidence detection at two de- tectors of specified area and in two specified time inter- vals is P=/integraldisplay d2xAd2xBcdtAcdtB|A(xA, tA,xB, tB)|2,(A10) where the integral is over the detector surfaces (each as- sumed normal to the line from scattering region to de- tector) and over the relevant time intervals. This is more conveniently expressed in terms of a rate Wof coinci- dence detections in terms of the detector solid angles δΩA,δΩBand the difference in retarded arrival times τ−≡τB−τA W=c2x2 Ax2 B|A(xA, tA,xB, tB)|2δΩAδΩBdτ−.(A11) Coincidence rate is largest when the detectors are placed in the directions which satisfy the phase-matching condition. We assume that k+l=k′+l′= 0 and that the detectors are small compared to the source-detector distance, i.e., that δΩA,B≪1. Under these conditions, the rate of coincidence events reduces to Wscattering =(k′l′)2 ¯h2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay dδk′Vl′k′l0k0exp[icδk′τ−]/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay d3xG(x)H(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 δΩBδΩBdτ−.(A12) 3. Signal Contrast In addition to the photon-photon scattering signal, un- correlated Rayleigh and Raman scattering events give a background of accidental coincidences. The rate of scat- tering into a small solid angle δΩ is WBG=BδΩ/integraldisplay d3xnk (A13) where B≡/summationdisplay a,c/integraldisplay d3p∝angb∇acketleftna,p∝angb∇acket∇ight(1± ∝angb∇acketleftnc,p′∝angb∇acket∇ight)k4 fc (2π)3¯h2 ×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay b(ef·µbc)∗ei·µba ck+ωab−¯h M[p·k+k2/2] +iγb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (A14) andnkis the number density of photons propagating in thekdirection. In terms of the beam-shape functions for two colliding beams, the rate of accidental coincidences is 6Waccidental =B2/bracketleftbigg/integraldisplay d3x|G(x)|2+|H(x)|2/bracketrightbigg2 ×δΩAδΩBdτ−. (A15) The ratio of coincidences due to photon-photon scatter- ing to accidental background coincidences is thus Wscattering Waccidental=(k′l′)2 4¯h2F B2 ×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay dδk′Vl′k′l0k0exp[icδk′τ−]/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (A16) where Fis the mode fidelity factor F≡4/bracketleftbig/integraltext d3xG(x)H(x)/bracketrightbig2 /bracketleftbig/integraltext d3x(|G(x)|2+|H(x)|2)/bracketrightbig2. (A17) [1] J. Boyce and R. Chiao, Physical Review A 59, 3953 (1999). [2] J. Boyce, J. P. Torres, and R. Y. Chiao, Submitted for publication (1999). [3] J. P. Torres, J. Boyce, and R. Y. Chiao, Submitted for publication (1999). [4] R. Y. Chiao, I. H. Deutsch, J. C. Garrison, and E. M. Wright, in Frontiers in Nonlinear Optics, The Sergei Akhmanov Memorial Volume , edited by H. Walther, N.Koroteev, and M. O. Scully (Institute of Physics Pub- lishing, Bristol, 1992), pp. 151–182. [5] I. Deutsch, R. Chiao, and J. Garrison, Physical Review Letters 69, 3627 (1992). [6] E. H. Lieb and W. Liniger, Physical Review 130, 1605 (1963). [7] E. H. Lieb, Physical Review 130, 1616 (1963). [8] P. Drummond and S. Carter, Journal of the Optical So- ciety of America B 4, 1565 (1987). [9] S. Carter, P. Drummond, M. Reid, and R. Shelby, Phys- ical Review Letters 58, 1841 (1987). [10] P. Drummond and M. Hillery, Physical Review A 59, 691 (1999). [11] M. Hillery and L. Mlodinow, Physical Review A 30, 1860 (1984). [12] P. Drummond, Physical Review A 42, 6845 (1990). [13] I. Abram and E. Cohen, Physical Review A 44, 500 (1991). [14] M. O. Scully and W. E. Lamb, Jr., Physical Review 159, 208 (1967). [15] M. Sargent, III, D. Holm, and M. Zubairy, Physical Re- view A 31, 3112 (1985). [16] P. Drummond and D. Walls, Physical Review A 23, 2563 (1981). [17] W. H. Louisell, Quantum Statistical Properties of Radia- tion,Wiley series in pure and applied optics (John Wiley & Sons, New York, 1973). [18] A. Heidmann and S. Reynaud, Journal of Modern Optics 34, 923 (1987). [19] D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin, 1994). [20] L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, New York, 1995). a,pb,p+k a,pk l' = l -q c,p-qk' = k+q l d,p+l FIG. 1. FIG. 1. Photon-photon scattering at a single atom. -4-2 024 -0.4-0.20.2-4-2 024 -0.8-0.6-0.4-0.2Amplitude (a.u.)τB−τABτ −τA Real Imag. FIG. 2. FIG. 2. Right-angle scattering amplitude Avs. time delay for the single-atom process of Fig. 1. The time unit is δ−1 D. 7-4 -2 0 2 40.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 3. FIG. 3. Right-angle scattering rate |A|2vs. time delay for the single-atom process of Fig. 1. Time uni t isδ−1 D. -4-2 024 -0.1-0.050.050.1 -4-2 024 -0.2-0.15-0.1-0.05Amplitude (a.u.)τB−τABτ −τA Real Imag. FIG. 4. FIG. 4. Statistical correction to the one-atom scatt ering amplitude. The time unit is δ−1 D. a,pb,p+k a,p l' = l-qc,p-q ld,p+lk k' = k +q c,p-q FIG. 5. FIG. 5. Two-atom photon-photon scattering. -4-2 024 -0.04-0.020.020.04 -4-2 024 -0.06-0.04-0.020.020.04Amplitude (a.u.)τB−τABτ −τAReal Imag. FIG. 6. FIG. 6. Scattering rate |A|2vs. time delay for the two-atom process of Fig. 5. The time uni t isδ−1 D. -4 -2 0 2 40.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 7. FIG. 7. Scattering rate |A|2vs. time delay for a Bose gas of phase-space density 1/2. The t ime unit is δ−1 D. 8-4 -2 0 2 40.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 8. FIG. 8. Scattering rate |A|2vs. time delay for a Fermi gas of phase-space density 1/2. The time unit is δ−1 D. a,pb,p+k a,pk l' = l -q c,p+k+lk' = k+q l d,p+l' FIG. 9. FIG. 9. “Ladder” process in a three-level atom. -4-2 024 -0.20.20.4 -4-2 024 -0.4-0.20.20.4Amplitude (a.u.)τB−τAτB−τAReal Imag. FIG. 10. FIG. 10. Scattering rate |A|2vs. time delay for the “ladder” process of Fig. 9. The time uni t isδ−1 D. -4 -2 0 2 40.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 11. FIG. 11. Scattering rate |A|2vs. time delay for a Lorentz-model atomic medium. The time un it isδ−1 D. -4 -2 0 2 40.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 12. FIG. 12. Coincidence rate vs. time delay for a Lorent z-model atomic medium. The constant background is accidental coincidences due to independent Rayleigh scatt ering events. The time unit is δ−1 D. 9-10 -5 0 5 100.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 13. FIG. 13. Small angle scattering rate |A|2vs. time delay for nearly co-propagating photons. Time unit isδ−1 D. -10 -5 0 5 100.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 14. FIG. 14. Right-angle scattering rate |A|2vs. time delay for nearly counter-propagating photons. Tim e unit is δ−1 D. -10 -5 0 5 100.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 15. FIG. 15. Coincidence rate |A|2vs. time delay for nearly co-propagating photons in a linear medium. γb= 0.01δD, ∆ = 2 πδD. Time unit is δ−1 D. -10 -5 0 5 100.20.40.60.81 τB−τA |A| (a.u.)2 FIG. 16. FIG. 16. Coincidence rate |A|2vs. time delay for nearly counter-propagating photons line ar medium. γb= 0.01δD, ∆ = 2 πδD. Time unit is δ−1 D. 10H(x) G(x)x xADetector A xBDetector Bk0 l0 FIG. 17. FIG. 17. Geometry of collision process. 11

arXiv:physics/0002015v1 [physics.chem-ph] 7 Feb 2000Density Functional Study of adsorption of molecular hydrog en on graphene layers J. S. Arellano∗, L.M. Molina, A. Rubio and J.A. Alonso Departamento de F´ ısica Te´ orica, Universidad de Valladol id, 47011 Valladolid, Spain. (December 2, 2012) Keywords: Hydrogen adsorption, graphite, density functio nal theory. Abstract Density functional theory has been used to study the adsorpt ion of molecular H2on a graphene layer. Different adsorption sites on top of atom s, bonds and the center of carbon hexagons have been considered and co mpared. We conclude that the most stable configuration of H2is physisorbed above the center of an hexagon. Barriers for classical diffusion are, h owever, very small. I. INTRODUCTION The adsorption of hydrogen by different forms of carbon has be en studied by different groups.1–7Dillon et al1were the first to study the storage of molecular hydrogen by assemblies of single wall carbon nanotubes (SWCNT) and poro us activated carbon. They pointed out that the attractive potential of the walls of the pores makes it possible a high density storage. From temperature-programmed desorption experiments Dillon et al1con- cluded that those forms of carbon are promising candidates f or hydrogen storage, although the density of hydrogen is still low in order to meet the requi rements of the DOE Agency for novel hydrogen storage systems. More recently Levesque et al2, Yeet al6, and Liu et al 7also studied the adsorption of molecular hydrogen on SWCNT a t different temperatures and pressures. Chambers et al3have reported obtaining an extraordinary storage capacity by some graphite nanofibers but Wang and Johnson4have tried unsuccessfully to confirm the high storage capacity by graphite nanofibers (slit pores ) and SWCNT. Hynek et al5 investigated ten carbon sorbents but only one of them could a ugment the capacity of com- pressed hydrogen gas storage vessels. The improvement was m arginal at 190 K and 300 K but nonexistent at 80 K. The storage capacity of carbon nanot ubes and graphitic fibers has been enhanced by doping with lithium and other alkali elemen ts8. The alkali atoms seem to have a catalytic effect in dissociating the H2molecule and promoting atomic adsorption. An advantage is that the doped systems can operate at moderat e temperatures and ambient pressure. ∗On sabbatical leave from Area de F´ ısica, Divisi´ on de Cienc ias B´ asicas e Ingenier´ ıa, Universidad Aut´ onoma Metropolitana Azcapotzalco, Av. San Pablo 180, 0 2200 M´ exico D.F., M´ exico. 1Some of the authors cited above2,4,9have also performed computer simulations of the ad- sorption of molecular hydrogen inside, outside and in the in terstices of an array of SWCNT and in idealized carbon slit pores using model pair potentia ls to describe the interactions. Wang and Johnson4adopted the semiempirical pair potential of Silvera and Gol dman10 for the H2−H2interaction and the H2−Cinteraction was modelled by a potential derived by Crowell and Brown11by averaging an empirical Lennard-Jones H2−Cpotential over a graphite plane. In the simulations Wang and Johnson used a h ybrid path integral-Monte Carlo method. Johnson9also studied the influence of electrical charging of the tube s. Stan and Cole12performed calculations based on a sum of isotropic Lennard- Jones interactions between the molecule and the C atoms of the tube. They calcula ted the adsorption poten- tial of a hydrogen molecule, considered as a spherically sym metric entity, as a function of distance from the axis of a SWCNT, along radial lines upon the center of an hexagon of carbon atoms and upon a carbon atom respectively. Those simu lations give useful insight to interpret the results of the experiments. However the descr iption of the interaction between H2and the graphitic surfaces of the SWCNT or the slit pores in th ose works is too simple. Simplicity is a necessary requirement for massive simulati ons involving several hundred (or several thousand) H2molecules and an assembly of SWCNT of realistic size, but one can expect more realistic results if the interaction potential is derived from an ab initio calcu- lation. The adsorption of ”atomic” hydrogen on a planar grap hene sheet, that is a planar layer exfoliated from graphite, has been studied previousl y13,14. Bercu and Grecu13used a semiempirical molecular orbital LCAO treatment at the INDO (intermediate neglect of dif- ferential overlap) unrestricted Hartree-Fock level and Je loaica and Sidis14used the Density Functional formalism (DFT)15. In both works the description of the graphene layers was simplified by modelling this layer by a finite cluster C 24-H12, where the hydrogen atoms sat- urate the dangling bonds on the periphery of the planar clust er. But, as mentioned above, hydrogen is adsorbed in molecular form by graphitic surface s (SWCNT and slit pores), so in this work we study the interaction of an H2molecule with a planar graphene layer. Since the graphene layers interact weakly in bulk graphite, the in teraction of H2with a graphitic surface is a localized phenomenon restricted to the outermo st plane. For this reason our cal- culations have relevance for understanding the adsorption ofH2on the walls of slit pores in graphite, and also for the case of adsorption by SWCNT, since these differ from a graphene layer only in the curvature of the layer. II. THEORETICAL METHOD AND TESTS To calculate the interaction between H2and a planar graphene layer we use the ab initio fhi96md code, developed by Scheffler et al16. This code uses the DFT15to compute the electronic density and the total energy of the system, an d we have chosen the local density approximation (LDA) for exchange and correlation.17Only the two electrons of the H2molecule and the four external electrons (2 s22p2) of each carbon atom are explicitly included in the calculations, while the 1 s2core of carbon is replaced by a pseudopotential. For this purpose we use the nonlocal norm-conserving pseudo potentials of Hamann et al 18,19. Nonlocality in the pseudopotential is restricted to ℓ= 2, and we take as local part of the pseudopotential the scomponent. The code employs a supercell geometry and a basis of plane waves to expand the electronic wave functions.20 2First we have tested the method for pure graphite. By minimiz ation of the total energy with respect to the interatomic distances we obtained an in- plane C−Cbond length equal to 4.61 a.u. and a distance between planar graphitic layers o f 12.55 a.u. The corresponding experimental values21are 4.65 a.u. and 12.68 a.u. respectively. The small (1%) und er- estimation of bond lengths is characteristic of the LDA. Nex t we have studied an isolated graphene layer. Since the computer code uses a periodic supe rcell method, the cell axis has to be large in the z-direction to avoid the interaction betwe en graphene sheets in different cells. Table I gives the calculated energy of the graphene la yer as a function of the length c of the unit cell in the z direction, or in other words, as a func tion of the distance between parallel graphene layers. Results given for c = 20, 25, 30 and 35 a.u. show that the energy is well converged for those layer separations and that for c = 20 a.u. the error in the energy per atom is only about 1 in 105. A cutoff of 40 Ry was used in all the calculations. We also tested the method by calculating the energy of the H2molecule, that was placed at the center of a simple cubic supercell. The total energy obtaine d for a plane wave cut-off energy of 40 Ry and supercell lattice constants of 18 a.u. and 20 a.u. is the same, -2.247 Ry as well as the bondlength, 1.48 a.u. Notice that this bond length is s mall compared to the C−C bond length. Anticipating the geometry to be used in the stud y of the interaction between H2and graphene, another set of calculations were performed fo r the energy of H2by placing the molecule in the superlattice described above in the stud y of the graphene layer, but this time without graphene. Calculations for distances between the imaginary graphene planes ranging from 20 a.u. to 35 a.u. (the plane wave cut-off was agai n 40 Ry) gave energies for theH2molecule, identical to the energies obtained for the cubic s uperlattice geometry. III. INTERACTION BETWEEN H2AND THE GRAPHENE LAYER For the periodicity of the system we have selected a unit cell with eight carbon atoms and one hydrogen molecule (see Fig. 1). If we place a hydrogen molecule at any point of the cell, the distance from this molecule to other in the nearest cells is 9.224 a.u. This separation is large compared to the bond length of H2(1.480 a.u.), and we have verified that there is no interaction between two hydrogen molecules separated by th at distance. The interaction of theH2molecule with the graphene sheet has been studied by perform ing static calculations for two orientations of the axis of the molecule: axis perpen dicular to the graphene plane and axis parallel to that plane. Three possibles configurations , called A, B and C below, have been selected for the perpendicular approach of the molecul e to the plane: (A) upon one carbon atom, (B) upon the center of a carbon-carbon bond, and (C) upon the center of an hexagon of carbon atoms. On the other hand, for the parallel a pproach the molecule is placed upon the center of an hexagon of carbon atoms with the molecul ar axis perpendicular to two parallel sides of the hexagon, and this is called configurati on D. These four configurations are given in the bottom panel of Fig. 1. To obtain the interact ion energy curve for each of those four cases, the distance between the hydrogen molecul e and the graphene layer was varied while maintaining the relative configuration. In the se calculations the bond length of theH2was held fixed at 1.48 a.u., the bondlength of the free molecul e. This is expected to be valid in the relevant region of the interaction. This cons traint will, however, be relaxed in simulations described at the end of this section. Calcula tions were first performed in the parallel configuration (D) for a superlattice such that t he distance between graphene 3layers is 30 a.u. The plane wave cut-off was 40 Ry. The interact ion energy curve is plotted in Figure 2 and the curve has a minimum at 5.07 a.u. For separat ions larger than this value the energy rises fast and reaches its asymptotic value for 10 - 11 a.u. The energy at the maximum possible separation between the center of mass o f theH2molecule and the graphene plane for this superlattice, 15 a.u., was taken as t he zero of energy. The figure also gives the results of a similar calculation for a smaller supe rlattice, such that the distance between graphene layers is 20 a.u. The corresponding energy curve, referred to the same zero of energy as above, is practically indistinguishable from t he former curve. The calculations also show that for all practical purposes the energy curve ha s reached its asymptotic value for a distance of 10 a.u., that is the longest separation allo wed for the superlattice of 20 a.u. This indicates that calculations using the smaller superla ttice are enough for our purposes of studying the H2- graphene interaction. Then, the results of calculations c orresponding to configurations A, B, C and D for a superlattice of 20 a.u. are gi ven in Figure 3. The potential energy curves for the perpendicular approach (A, B, C) rapid ly merge with each other for largeH2- graphene separation, becoming indistinguishable from on e another beyond 6.5 a.u. Actually, curves A and B are very close in the whole range of separations although B is marginally more attractive. The common value of the ener gy of curves A, B and C at separation 10 a.u. is taken as zero of energy in Figure 3. Th e predicted equilibrium positions and the binding energies (depth of the minimum) of the different curves are given in Table II. The small magnitude of the binding energies, les s than 0.1 eV, shows that the system is in the physisorption regime. Comparison of the four curves reveals that the most favorable position for the H2molecule is physisorbed in a position above the center of a carbon hexagon, and that the parallel configuration is sl ightly more favorable than the perpendicular one. We have verified that different orientati ons of the molecular axis with respect to the underlying carbon hexagon in the parallel con figuration lead, in all cases, to the same curve D plotted in Fig. 3. The differences in binding e nergy shown in Table II are very small. For instance, configuration D and A only differ by 1 6 meV, and configuration D and C by 3 meV. Figures 4a and 4b give the electron density of the pure graphe ne layer in two parallel planes, 5 and 3 a.u. above the plane of the nuclei, respective ly. The former one is very close to the preferred distance of approach for the H2molecule in configuration D. First of all one can note that the values of the electron density in tha t plane are very small, of the order of 10−5e/(a.u.)3, so the plane is in the tail region of the electron density dis tribution. Nevertheless the densities clearly reveal the topography o f the graphene layer. Electron density contours on top of carbon atoms surround other conto urs representing the large hexagonal holes. Densities are larger in the other plane, cl oser to the plane of nuclei. In each plane the density is larger in the positions above carbo n atoms and lower above the hexagons. A plot that complements this view is given in Figur e 5, that gives the electron density in a plane perpendicular to the graphene layer throu gh a line containing two adjacent carbon atoms, labelled C1andC2in the figure. Then, points labelled M and X represent the midpoint of a carbon-carbon bond and the center of an hexa gon respectively. The most noticeable feature is the existence of depressions of elect ron density in the regions above the centers of carbon hexagons. These hollow regions are sep arated by regions of larger density that delineate the skeleton of carbon-carbon bonds . In this figure the density of the most external contour is ρ= 1.11×10−2e/(a.u.)3and the interval between contours 4∆ρ= 1.11×10−2e/(a.u.)3 These observations correlate with the features in Fig. 3, an d lead to the following interpretation of the potential energy curves. Each curve c an be seen as arising from two main contributions, one attractive and one repulsive. The a ttractive contribution is rather similar for all the configurations (notice the similarity of the potential energy curves beyond 6 a.u.) and is mainly due to exchange and correlation effects. Neglecting correlation for the purposes of simplicity, the exchange contribution to the to tal energy is given, in the LDA, by the functional ELDA x[ρ] =Cx/integraldisplay ρ(r)4 3d3r, (1) where Cxis a well known negative constant.15In the regime of weakly overlapping densi- ties, and assuming no density rearrangements due to the clos e-shell character of H2, the contribution of exchange to the interaction energy becomes ∆Ex=Cx[/integraldisplay [ρH2(r) +ρg(r)]4 3d3r−/integraldisplay ρH2(r)4 3d3r−/integraldisplay ρg(r)4 3d3r], (2) where ρgandρH2represent the tail densities of the graphene layer and H2molecule respec- tively. A net ”bonding” contribution arises from the nonlin earity of the exchange energy functional. On the other hand the sharp repulsive wall is due to the short-range repulsion between the close electronic shell of the H2molecule and the electron gas of the substrate. This contribution is very sensitive to the local electron de nsity sampled by the H2molecule in its approach to the graphene layer and explains the correl ation between the position and depth of the different minima in Fig. 3 and the features of the s ubstrate electron density in Figs. 4 and 5. Similar arguments explain the physisorption o f noble gas atoms on metallic surfaces22and the weak bonding interaction between noble gases.23At very large separation the interaction energy curves should approach the Van der Wa als interaction, that is not well described, however, by the LDA. An interesting point concerns the comparison of the minima o f the curves C and D of Fig. 3. That of curve D is deeper and occurs at a shorter H2- graphene separation. The reason is that the surfaces of constant electron density of the H2molecule have the shape of slightly prolate ellipsoids instead of simple spheres. Consequentl y, for a given distance dbetween the center of mass of H2and the graphene plane, the molecule with the perpendicular orientation (C) penetrates more deeply into the electronic cloud of the substrate than in the parallel orientation (D). In other words, the repulsive wall is reached earlier, that is for larger d, in the perpendicular configuration (C). If we consider an el ectronic density contour inH2with a value ρ= 0.018 e/(a .u.)3, then the two semiaxes have lengths of 2.07 and 1.71 a.u. respectively and the difference between these two lengt hs is 0.36 a.u. This value is in qualitative agrement with the difference between the H2- graphene separations for the two minima of curves C and D, which is 0.20 a.u. This shape effect is usually neglected in the phenomenological approaches, that treat H2simply as a spherical molecule. Figure 6 gives a plot of the charge density difference ρdiff(r) =ρtot(r)−(ρg(r) +ρH2(r)), (3) where ρtot(r) is the calculated density of the total system, that is the H2molecule physisorbed in orientation D at a distance of 5 a.u. above the graphene lay er, whereas ρg+ρH2is the 5simple superposition of the densities of the pure graphene l ayer and H2molecule placed also in orientation D, 5 a.u. above the graphene layer. That densi ty difference ρdiff(r) is given in the same plane, perpendicular to the graphene lay er, used in Fig. 5. ρdiff(r) has positive and negative regions. The positive region is th e area bound by the contour labelled P. This region has the shape of two lobes joined by a n arrow neck. Contour P has a value ρdiff= 2.36×10−5e/(a.u.)3andρdiffincreases in this positive region as we move towards inner contours in the lobes. The innermost co ntour shown has a value ρdiff= 2.87×10−4e/(a.u.)3.TheH2molecule sits above the neck, so the figure reveals that the repulsive interaction produced by the close electronic shell of H2pushes some charge from the region immediately below the molecule (the neck reg ion) to form the lobes of positive ρdiff(r).This displacement of electronic charge is nevertheless qua ntitatively very small. Notice that ρgtakes values between 1 .6×10−3and 4.1×10−3e/(a.u.)3in a plane 3 a.u. above the graphene layer, while ρdiffhas values of the order 10−5−10−4e/(a.u.)3 in the same plane. The smallness of ρdiffjustifies the argument given in eq. (2) for the attractive exchange-correlation contribution to the inte raction potential. The static calculations discussed above have been compleme nted with dynamical simula- tions in which the H2molecule was initially placed in different orientations at d istances of 4 - 6 a.u. from the graphene layer and was left to evolve under the influence of the forces on the H atoms. The H2bondlength was allowed to adjust in the process. The simulat ions confirm the results of the static calculations, in the sense that the H2molecules end up in positions above the center of an hexagon at the end of the simulations. T he binding energies and H2- graphene layer distances practically coincide with those in Table II. Marginally small differences in separation or binding energy are due to very sm all changes of the bondlength ofH2, always smaller than 0 .3%. The result of one of the simulations is worth to be men- tioned. A configuration intermediate between those labelle d C and D above was obtained: the center of mass of the molecule was 5.10 a.u. above the cent er of a carbon hexagon, with the molecular axis forming an angle of about 30◦with the graphene plane. The binding energy in this new configuration was only 1 meV larger than in t he parallel configuration D. In summary, the picture arising from the calculations is rat her clear. The H2molecules prefer the hollow sites above the centers of carbon hexagons where the background electron density is lower than in channels on top of the skeleton of car bon-carbon bonds. The exchange-correlation contribution provides the weak attr action responsible for physisorption, but the preferred distance of approach is determined by the r epulsive part of the interaction potential. That repulsive contribution is due to the close- shell electronic structure of H2. We have performed static calculations of the barrier for the di ffusion of a molecule, initially in the parallel configuration D at the preferred distance of 5.0 7 a.u. above the graphene plane, to an equivalent configuration D above an adjacent hexagon. T he initial configuration of the molecule, with its axis perpendicular to two parallel ca rbon-carbon bonds, can be seen in the bottom panel of Fig. 1. The molecule was then forced to f ollow a path across one of those bonds, allowing for the reorientation of the molecu lar axis at each step in order to minimize the energy of the system. Although the molecule b egins with the axis parallel to the graphene plane, the orientation of the axis changes as the molecule approaches the carbon-carbon bond. In fact, when the center of mass of the mo lecule is precisely above that bond, the molecular axis becomes perpendicular to the graph ene plane, that is the molecule adopts configuration B, as indicated also in Fig. 1. The energ y difference between this 6saddle configuration and the starting one gives a calculated diffusional barrier of 14 meV. A temperature of 163 K is enough to surpass this barrier. The conclusions from the calculations are, in our view, gene ral enough that one can make some extrapolations to the case of adsorption of H2by carbon nanotubes. When adsorption occurs on the outside wall of an isolated nanotube, the predi ctions of Fig. 3 will be valid, with a minor influence of the nanotube curvature. If the tubes form a parallel bundle and we consider the interstitial channels between tubes, the effec ts seen in Figure 3 will be smoothed out because of the addition of contributions of different gra phitic surfaces non in registry. Addition of these contributions will give rise to an interst itial channel with a potential energy nearly independent of z, if we call z the direction par allel to the tube axis. Finally, the same smoothing effect will occur in the inner channel of a t ube if the tube diameter is not large. In summary we predict very easy diffusion of the H2molecule in arrangements of parallel tubes along the direction parallel to the tube axis , both inside the tube cavity and in the interstitial channels. Another system that can be ana lysed based on the results of Fig. 3 is graphite intercalated with a small amount of H2. In this case the binding energy can be estimated as the sum of the binding energies with the tw o layers above and below the molecule. The layer-layer separation in pure graphite i s 12.55 a.u. and it is safe to assume that a small amount of intercalated H2will not modify that distance. If we further assume a simple stacking between layers with hexagons exact ly on top of hexagons, then the most stable configuration is for the molecule equidistan t from the two layers, suspended between the centers of two hexagons in the perpendicular con figuration C. Consequently the orientation of the molecule changes compared to the case of a dsorption on a single graphene layer. However, the layer stacking observed in graphite is m ore complex. By analysing the different configurations, parallel and perpendicular, cons istent with this stacking we have found that a perpendicular orientation of the molecule, sus pended between the center of an hexagon and the carbon atom below, is the most stable configur ation, although again we notice that the binding energies for different relevant confi gurations are very similar. The present adsorption results can be partialy compared wit h those of Stan and Cole.12 They considered the H2molecule as a spherically symmetric entity and calculated t he ad- sorption potential inside zigzag (13,0) nanotubes (radius = 9.62 a.u.) based on a sum of isotropic Lennard-Jones interactions between the molecul e and the carbon atoms of the tube. Our calculation and that of Stan and Cole agree in that t he smallest binding energy is obtained for the H2upon one carbon atom and the largest one for the H2upon the center of the hexagon of carbon atoms. However Stan and Cole do not di stinguish between parallel and perpendicular orientations because they considered an spherical molecule. Their Fig. 1 shows a binding energy about 0.079 eV for adsorption in front of the center of an hexagon of carbon atoms and that the equilibrium distance between th e molecule and the nanotube wall is 5.7 a.u. This distance is consistent but a little larg er than those reported in our Table II. On the other hand, the value 0.079 eV for the binding energy is also consistent with the binding energies in Table II. Notice, however, that the binding energy for a tube of larger radius, or for a planar graphene sheet, will be a littl e smaller because the curvature of the tube increases the number of nearest neighbor carbon a toms. In fact, Wang and Johnson4calculated an adsorption binding energy near 0.050 eV for mo lecular hydrogen in an idealized carbon slit pore with a pore width of 17.4 a.u. 7IV. CONCLUSIONS By performing DFT calculations we confirm that physisorptio n ofH2on graphitic layers is possible. The differences between the binding energies co rresponding to different positions (on top of carbon atoms, on top of carbon-carbon bonds, on top of hexagonal holes) are small, and the diffusional barriers are also small, so easy diffusion is expected at low temperature. The nonsphericity of the H2molecule has some influence on the preferred orientation of t he molecular axis with respect to the graphene plane. These sma ll effects associated to different positions and orientations of the physisorbed molecule are expected to average out inside carbon nanotubes or in the interstitial channels in paralle l arrays of carbon nanotubes. ACKNOWLEDGEMENTS Work supported by DGES(Grant PB95-0720-C02-01), Junta de C astilla y Le´ on (Grant VA28/99) and European Community (TMR Contract ERBFM RX-CT96-0062- DG12-MIHT). L.M.M. is greatful to DGES for a Predoctoral Gra nt. J.S.A. wishes to thank the hospitality of Universidad de Valladolid during his sab batical leave and grants given by Universidad Aut´ onoma Metropolitana Azcapotzalco and by I nstituto Polit´ ecnico Nacional (M´ exico). Finally we thank the referee for constructive su ggestions. 8REFERENCES 1A.C. Dillon, K.M. Jones, T.A. Bekkedahl, C.H. Kiang, D.S. Be thune and M.J. Heben, Nature (London) 386, 377 (1997). 2F. Darkrim and D. Levesque, J. Chem. Phys. 109, 4981 (1998); F. Darkrim, J. Vermesse, P. Malbrunot and D. Levesque, J. Chem. Phys. 110, 4020 (1999). 3A. Chambers, C. Park, R.T.K. Baker and N.M. Rodriguez, J. Phy s. Chem. 102, 4253 (1998). 4Q. Wang and J. K. Johnson, J. Chem. Phys. 110, 577 (1999); J. Phys. Chem. B 103, 4809 (1999). 5S. Hynek, W. Fuller and J. Bentley, Int. J. Hydrogen Energy 22, 601 (1997). 6Y. Ye, C.C. Ahn, C. Witham, B. Fultz, J. Liu, A.G. Rinzler, D. C olbert, K.A. Smith and R.E. Smalley, Appl. Phys. Lett. 74, 2307 (1999). 7C. Liu, Y.Y. Fan, M. Liu, H.T. Cong, H.M. Cheng and M.S. Dresse lhaus, Science 286, 1127 (1999). 8P. Chen, X. Wu, J. Lin and K.L. Tan, Science 285, 91 (1999). 9V.V. Simonyan, P. Diep and J.K. Johnson, J. Chem. Phys. 111, 9778 (1999). 10I.F. Silvera and V.V. Goldman, J. Chem. Phys. 69, 4209 (1978). 11A.D. Crowell and J.S. Brown, Surf. Sci. 123, 296 (1982). 12G. Stan and M.W. Cole, J. Low Temp. Phys. 110, 539 (1998). 13M.I. Bercu, V.V. Grecu, Romanian J. of Phys. 41, 371 (1996). 14L. Jeloaica and V. Sidis. Chem. Phys. Lett. 300, 157 (1999). 15W. Kohn and L.J. Sham, Phys. Rev 140, A1133 (1965); R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford Universit y Press, New York (1989). 16M. Bockstedte, A. Kley, J. Neugebauer and M. Scheffler. Comp. P hys. Commun. 107, 187 (1997). 17J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 18G.B. Bachelet, D.R. Hamann, and M. Schluter, Phys. Rev. B 26, 4199 (1982). 19D.R. Hamann, Phys. Rev. B 40, 2980 (1989). 20M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias and J.D. Joann opoulos, Rev. Mod. Phys. 64, 1045 (1992). 21M.S. Dresselhaus, G. Dresselhaus and P.C. Eklund, Science o f Fullerenes and Carbon Nanotubes, Academic Press, San Diego (1996). 22N.D. Lang, Phys. Rev. Lett. 46, 482 (1981). 23R.G. Gordon and V.S. Kim, J. Chem. Phys. 56, 3122 (1972). 9TABLES TABLE I. Calculated energy (Ry) of the graphene layer for sev eral layer-layer distances. The energies are calculated for a plane wave cut-off energy of 40 R y. Layer-layer distance (a.u.) Energy (per atom) 20 -11.4234 25 -11.4235 30 -11.4235 35 -11.4235 TABLE II. Binding energy (eV) and equilibrium distance (a.u .) for H2physisorbed on a graphene layer. A, B and C correspond to the configurations in which the molecular axis is perpendicular to the graphene plane and the molecule is on to p of: a carbon atom (A), the midpoint of a carbon-carbon bond (B), the center of an hexagon (C). In c onfiguration D the molecule is on top of the center of an hexagon with the molecular axis parall el to the graphene plane. A B C D Binding energy 0.070 0.072 0.083 0.086 Distance 5.50 5.49 5.25 5.07 10FIGURES FIG. 1. Top panel gives a fragment of the graphene layer showi ng the eight carbon atoms in the unit cell, represented by large spheres. Bottom panel sh ows the three adsorption configurations with the molecular axis perpendicular to the graphene plane . These have the H2molecule above a carbon atom (A), above the midpoint of a carbon-carbon bond (B) and above the center of an hexagon (C). Also shown is an adsorption configuration (D) wi th the molecular axis parallel to the graphene plane and the molecule above the center of an hexago n. FIG. 2. Comparison of potential energy curves for the parall el approach of H2to the graphene layer upon the center of an hexagon of carbon atoms. The curve s were obtained using supercells such that the graphene layers are separated by 30 a.u. (circl es) or 20 a.u. (crosses). FIG. 3. Potential energy curves for the aproach of H2to the graphene layer in four different configurations. The axis of the molecule is perpendicular (A , B, C) or parallel (D) to the graphene layer. In the former orientation the molecule is above a carb on atom (A), above the center of a C-C bond (B), and above the center of an hexagon (C). In the par allel orientation (D) the molecule is above the center of an hexagon. FIG. 4. (a) Contours of constant electron density ρof a pure graphene layer in a plane 5 a.u. above the plane of the carbon nuclei. ρ= 8.64×10−5e/(a.u.)3in the innermost contours above carbon atoms. ρ= 6.08×10−5e/(a.u.)3in the innermost contours above the large hexagonal holes. Densities decrease monotonously between those two contour s with an interval ∆ ρ= 0.18×10−5 e/(a.u.)3. (b) Contours in a plane 3 a.u. above the plane of carbon nucle i.ρ= 3.92×10−3e/(a.u.)3 in the innermost contours above carbon atoms. ρ= 1.72×10−3e/(a.u.)3in the innermost contours above the large hexagonal holes. Densities decrease monoto nously between those two contours with an interval ∆ ρ= 0.16×10−3e/(a.u.)3 FIG. 5. Contours of constant electron density of pure graphe ne in a plane perpendicular to the graphene layer, going through a line containing two adjacen t carbon atoms, labelled C1andC2. Symbols M and X indicate the mid-point of a carbon-carbon bon d and the center of an hexagon, respectively. The outermost contour plotted is ρ= 1.11×10−2e/(a.u.)3and the interval between contours ∆ ρ= 1.11×10−2e/(a.u.)3 FIG. 6. Charge density difference ρdiff=ρtot−ρg−ρH2forH2physisorbed 5 a.u. above the graphene layer. The plane of the plot and the symbols C1, C2, M and X are the same as in Fig. 5. Contour labelled P has a value 2 .36×10−5e/(a.u.)3and encloses the region of positive ρdiff. 11/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0 /1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 BCA D0.0 5.0 10.0 15.0 DISTANCE (a.u.) FROM THE HYDROGEN MOLECULE TO GRAPHENE PLANE−0.10−0.050.000.05POTENTIAL ENERGY (eV) Interlayer distance 30 a.u. Interlayer distance 20 a.u.4.0 5.0 6.0 7.0 8.0 9.0 10.0 DISTANCE (a.u.) FROM THE HYDROGEN MOLECULE TO THE GRAPHENE PLANE−0.10−0.08−0.06−0.04−0.020.000.020.04POTENTIAL ENERGY (eV)Configuration A Configuration B Configuration C Configuration D’PLTOUT’’PLTOUT’C1MC2 X C1M0123456789 a.u.C2 M C1 X C2 MP P C1+ + 0510 a.u.

arXiv:physics/0002016v1 [physics.bio-ph] 8 Feb 2000Formalizing the gene centered view of evolution Y. Bar-Yam New England Complex Systems Institute 24 Mt. Auburn St., Cambridge MA Department of Molecular and Cellular Biology Harvard University, Cambridge, MA Abstract A historical dispute in the conceptual underpinnings of evo lution is the validity of the gene centered view of evolution. [1, 2] We transcend this debate by formalizing the gene centered view as a dynami c version of the mean field approximation. This establishes the condit ions under which it is applicable and when it is not. In particular, it br eaks down for trait divergence which corresponds to symmetry breakin g in evolving populations. The gene centered view addresses a basic problem in the inter play of selection and heredity in sexually reproducing organisms. Sexual rep roduction disrupts the simplest view of evolution because the offspring of an org anism are often as different from the parent as organisms that it is competing against. In the gene centered view the genes serve as indivisible units that are preserved from generation to generation. In effect, different versions of th e gene, i.e. alleles, compete rather than organisms. It is helpful to explain this using the “row- ers analogy” introduced by Dawkins. [1] In this analogy boat s of mixed left- and right-handed rowers are filled from a common rower pool. B oats compete in heats and it is assumed that a speed advantage exists for bo ats with more same-handed rowers. The successful rowers are then reassig ned to the rower pool for the next round. Over time, a predominantly and then t otally single handed rower pool will result. Thus, the selection of boats s erves, in effect, to select rowers who therefore may be considered to be competin g against each other. Other perspectives on evolution distinguish betwee n vehicles of selection (the organisms) and replicators (the genes). However, a dir ect analysis of the gene centered view to reveal its domain of applicability has not yet been dis- cussed. The analysis provided here, including all the equat ions, is applicable quite generally, but for simplicity it will be explained in t erms of the rowers 1analogy.1 The formal question is: Under what conditions (assumptions ) can allelic (rower) competition serve as a surrogate for organism (boat ) competition in the simple view of evolution. Formalizing this question req uires identifying the conditions attributed to two steps in models of evolutio n, selection and reproduction. In the selection step, organisms (boats) are selected, while in the sexual reproduction step, new organisms are formed from the organisms that were selected. This is not fully discussed in the rowers mode l, but is implicit in the statement that victorious rowers are returned to the r ower pool to be placed into new teams. The two steps of reproduction and sele ction can be written quite generally as: {N(s;t)}=R[{N′(s;t−1)}] (1) {N′(s;t)}=D[{N(s;t)}] (2) The first equation describes reproduction. The number of offs pring N(s;t) hav- ing a particular genome sis written as a function of the reproducing organisms N′(s;t−1) from the previous generation. The second equation descri bes selec- tion. The reproducing population N′(s;t) is written as a function of the same generation at birth N(s;t). The brackets on the left indicate that each equa- tion represents a set of equations for each value of the genom e. The brackets within the functions indicate, for example, that each of the offspring populations depends on the entire parent population. To formalize the gene centered view, we introduce a dynamic f orm of what is known in physics as the mean field approximation. In the mea n field approx- imation the probability of appearance of a particular state of the system (i.e. a particular genome, s) is the product of probabilities of the components (i.e. genes, si) P(s1, . . ., s N) =/productdisplay P(si). (3) In the usual application of this approximation, it can be sho wn to be equivalent to allowing each of the components to be placed in an environm ent which is an average over the possible environments formed by the other c omponents of the system, hence the term “mean field approximation.” The key to applying this in the context of evolution is to consider carefully the effec t of the reproduction step, not just the selection step. In many models of evolution that are discussed in the literat ure, the offspring are constructed by random selection of surviving alleles (a panmictic popula- tion). In the rowers analogy the return of successful rowers to a common pool is the same approximation. This approximation eliminates c orrelations in the 1The rowers analogy may be considered a model of a single gene i n an n-ploid organism withnthe number of rowers, or a model of ngenes with two alleles per gene and each pair labeled correspondingly. The formal discussion applies to complete genomes i.e. to homolog and non-homolog genes. 2genome that result from the selection step and thus imposes E q. (3), the mean field approximation, on the reproduction step for the allele s of offspring. Even though it is not imposed on the selection step, inserting thi s approximation into the two step process allows us to write both of the equations i n Eq. (4) together as an effective one-step update P′(si;t) =˜D[{P′(si;t−1)}] (4) which describes the allele population change from one gener ation to the next of offspring at birth. Since this equation describes the behavi or of a single allele it corresponds to the gene centered view. There is still a difficulty pointed out by Lewontin. [2] The effe ctive fitness of each allele depends on the distribution of alleles in the p opulation. Thus, the fitness of an allele is coupled to the evolution of other al leles. This is apparent in Eq. (4) which, as indicated by the brackets, is a f unction of all the allele populations. It corresponds, as in other mean fiel d approximations, to placing an allele in an average environment formed from th e other alleles. For example, there is a difference of likelihood of victory (fi tness) between a right-handed rower in a predominantly left-handed populat ion, compared to a right-handed rower in a predominantly right-handed popul ation. Since the population changes over time, fitnesses are time dependent a nd therefore not uniquely defined. This problem with fitness assignment would not be present if each allele separately coded for an organism trait. While this is a partial violation of the simplest conceptual view of evolution, how ever, the applicability of a gene centered view can still be justified, as long as the co ntextual assignment of fitness is included. When the fitness of organism phenotype is dependent on the relative frequency of phenotypes in a population of orga nisms it is known as frequency dependent selection, which is a concept that is be ing applied to genes in this context. A more serious breakdown of the mean field approximation aris es from what is known in physics as symmetry breaking. This corresponds i n evolution to trait divergence of subpopulations. Such trait divergence arises when correla- tions in reproduction exist so that reproduction does not fo rce complete mixing of alleles. The correlations in reproduction do not have to b e trait related. For example, they can be due to spatial separation of organisms c ausing correla- tions in reproduction among nearby organisms. Models of spa tially distributed organisms are sometimes called models of spatially structu red environments. However, this terminology suggests that the environment it self is spatially vary- ing and it is important to emphasize that symmetry breaking / trait divergence can occur in environments that are uniform (hence the termin ology “symmetry breaking”). In the rowers model this has direct meaning in te rms of the ap- pearance of clusters of mostly left and mostly right handed r owers if they are not completely mixed when reintroduced and taken from the ro wer pool. Trait related correlations in sexual reproduction (assortive ma ting) caused by, e.g. 3sexual selection, would also have similar consequences. In either case, the gene centered view would not apply. Historically, the gene-centered view of evolution has been part of the discus- sion of attitudes toward altruism and group selection and re lated socio-political as well as biological concerns. [3] Our focus here is on the ma thematical appli- cability of the gene-centered view in different circumstanc es. While the formal discussion we present here may contribute to the socio-poli tical issues, we have chosen to focus here on mathematical concerns. The problem of understanding the mean-field approximation i n application to biology is, however, also relevant to the problem of group selection. In typ- ical models of group selection asexually (clonally) reprod ucing organisms have fecundities determined both by individual traits and group composition. The groups are assumed to be well defined, but periodically mixed . Similar to the gene-centered model, an assumption of random mixing is equi valent to a mean field theory. Sober and Wilson (1998) have used the term “the a veraging fal- lacy” to refer to the direct assignment of fitnesses to indivi duals. This captures the essential concept of the mean-field approximation. Howe ver, both the limi- tations of this approximation in some circumstances and its usefulness in others do not appear to be generally recognized. For example, it is n ot necessary for well defined groups to exist for a breakdown in the mean-field a pproximation to occur. Correlations in organism influences are sufficient. Moreover, stan- dard group-selection models rely upon averaging across gro ups with the same composition. For this case, where well defined groups exist a nd correlations in mixing satisfy averaging (mean-field) assumptions by group composition, equa- tions developed by Price2separate and identify both the mean field contribution to fitness and corrections due to correlations. These equati ons do not apply in more general circumstances when correlations exist in a net work of interactions and/or groups are not well defined, and/or averaging across g roups does not apply. It is also helpful to make a distinction between the ki nd of objection raised by Lewontin to the use of averaging, and the failure th at occurs due to correlations when the mean-field approximation does not app ly. In the former case, the assignment of fitnesses can be performed through th e effect of the en- vironment influencing the gene, in the latter case, an attemp t to assign fitnesses to a gene would correspond to inventing non-causal interact ions between genes. The mean field approximation is widely used in statistical ph ysics as a “ze- roth” order approximation to understanding the properties of systems. There are many cases where it provides important insight to some as pects of a system (e.g. the Ising model of magnets) and others where is essenti ally valid (conven- tional BCS superconductivity). The application of the mean field approximation to a problem involves assuming an element (or small part of th e system) can be treated in the average environment that it finds throughout t he system. This is equivalent to assuming that the probability distributio n of the states of the 2See discussion on pp. 73–74 of [3]. 4elements factor. Systematic strategies for improving the s tudy of systems be- yond the mean field approximation both analytically and thro ugh simulations allow the inclusions of correlations between element behav ior. An introduction to the mean-field approximation and a variety of application s can be found in Bar-Yam (1997). [4] In conclusion, the gene centered view can be applied directl y in popula- tions where sexual reproduction causes complete allelic mi xing, and only so long as effective fitnesses are understood to be relative to th e prevailing gene pool. However, structured populations (i.e. species with d emes—local mating neighborhoods) are unlikely to conform to the mean field appr oximation / gene centered view. Moreover, it does not apply to considering th e consequences of trait divergence, which can occur when such correlations in organism mating occur. These issues are important in understanding problem s that lie at scales traditionaly between the problems of population biology an d those of evolu- tionary theory: e.g. the understanding of ecological diver sity and sympatric speciation. [5] References [1] Dawkins, R. (1989) The Selfish Gene, 2nd ed. (Oxford Unive rsity Press, Oxford) p. 86. [2] Lewontin, R. in R. N. Brandon and R. M. Burian, eds. (1984) Genes, Organ- isms, Populations: Controversies Over the Units of Selecti on (MIT Press, Cambridge). [3] Sober, E. and Wilson, D. S. (1998) Unto Others: The evolut ion and psy- chology of unselfish behavior (Harvard University Press, Ca mbridge). [4] Bar-Yam, Y. (1997) Dynamics of Complex Systems (Perseus Press, Cam- bridge). [5] Sayama, H., Kaufman L. and Bar-Yam, Y. in preparation. 5

arXiv:physics/0002017v1 [physics.plasm-ph] 8 Feb 2000“Single-cycle” ionization effects in laser-matter interac tion Enrique Conejero Jarque∗and Fulvio Cornolti† Dipartimento di Fisica, Universit´ a di Pisa, Piazza Torric elli 2, Pisa, Italy Andrea Macchi†and Hartmut Ruhl‡ Theoretical Quantum Electronics, Darmstadt University of Technology, Darmstadt, Germany Abstract We investigate numerically effects related to “single-cycl e” ionization of dense matter by an ultra-short laser pulse. The strongly non-adia batic response of electrons leads to generation of a MG steady magnetic field in laser-solid interaction. By using two-beam interference, it is possibl e to create periodic density structures able to trap light and to generate relati vistic ionization fronts. I. INTRODUCTION In the adiabatic field ionization regime, the ionization rat e grows sharply when the electric field approaches the barrier suppression (BS) limi t, i.e. when the laser intensity is high enough that the electron in the ground state is able to “c lassically” escape the atomic potential barrier. The ionization rate for such field streng th may become higher than the ∗On leave from Departamento de F´ ısica Aplicada, Universida d de Salamanca, Spain. E-mail address enrikecj@gugu.usal.es †Also INFM, sezione A, Universit´ a di Pisa, Italy ‡Present address Max-Born Institut, Max-Born str. 2a, 12489 Berlin, Germany 1laser frequency and a regime in which most of the ionization i s produced within a single laser half-cycle is achievable. Here we present a numerical study of some effects of ultrafast ionization in the interaction of a short laser pulse with an initially transparent dense me dium. First, we will discuss the generation of megagauss steady magnetic fields in the surfac e “skin” layer of “solid” targets, i.e. slabs of hydrogen atoms with a number density close to th at of a solid medium (Macchi et al. 1999). Second, we will describe effects related to the combination o f two-beam interference with ultrafast ionization. We will show how it is possible to take advantage of this feature to create a layered dielectric-conductor structure able to tr ap the electric field, as well as a relativistic ionization front (Conejero Jarque et al. 1999 ). II. GENERATION OF STEADY MAGNETIC FIELDS The generation of steady currents and magnetic fields by ultr afast ionization is due to the non-adiabatic nature of the response of initially bound electrons to a strongly ramping laser field. Using a following “simple-man’s” model (SMM), v ery similar to the SMM used in studies of above-threshold ionization and harmonic gene ration in atoms, it can be shown that a single electron subject to an external sinusoidal int ense field can acquire a steady velocity (Macchi et al. 1999) vst=vI−vqo/radicalBig 1−(ET/Eyo)2, (1) where vIis the ejection velocity of the electron, vqo=eEy0/mω,Eyois the maximum field amplitude and ETis the field amplitude at the instant of ionization, which wil l be close to the threshold field for barrier suppression (for hydrogen, ET≈0.146Eau= 7.45×108V cm−1, being Eau= 5.1×109V cm−1the atomic field unit). If most of the electrons in the medium are ionized at the same i nstant, as may happen with a pulse which sharply rises above ET, one gets a net steady current which in turn 2generates a magnetic field. To obtain a larger current one may think to “tune” appropriately ETandEy0. This is possible if the ionization is no longer correlated w ith the oscillating field, i.e., it is produced independently of the field itself, like in the case studied by Wilks et al. (1988), in which a steady magnetic field Bst≈Ey0can be obtained in a very dense medium. For intense lasers ( I≥1018W cm−2), such a magnetic field would get values exceeding 100 MG and could explain (Teychenn´ e et al. 1998) t he experimental observation of high transparency of thin foil solid targets to 30 fs, 3 ×1018W cm−2pulses (Giulietti et al., 1997). However, it is questionable whether this high ma gnetic field may be obtained with superintense laser pulses. In this case, in fact, the “sourc e pulse” itself ionizes the medium and thus this will impose a constraint on the phase mismatch b etween the field and the velocity of the electrons. We will show by numerical simulat ions that the steady magnetic field exists but has values around 1 MG, being therefore too we ak to allow enhanced laser propagation. A. PIC simulations First we review the results of 1D3V PIC simulations with field ionization included. We choose pulses with a “sin2” envelope and with a “square” envelope. For all the PIC runs, the laser frequency was ωL= 2×1015s−1, close to that of Nd and Ti:Sapphire lasers. The thickness of the target was 0 .09µm and the density was no= 6.7×1022cm−3(ωpo≃7ωL). For the ionization rate we used a semi-empirical formula obt ained from atomic physics calculations (Bauer and Mulser, 1999). The laser energy los s due to ionization is included introducing a phenomenological “polarization” current (R ae and Burnett 1992, Cornolti et al. 1998, Mulser et al. 1998). Fig.1 shows the spatial profiles of the magnetic field and the f ree electron density five cycles after the end of a five cycles long (∆ tL= 15 fs) pulse, for three different field intensities in the “sin2” shape case, and for the square profile case at the intermedia te intensity value. The steady field is generated at the beginning of the interact ion and is always much weaker 3than the laser field, even for the most intense case (correspo nding to an intensity of 3 .5× 1018W cm−2); its sign varies according to the phase of the laser half cyc le where most of the ionization occurs. The ionization at the left boundary is ne arly instantaneous; however, even if the target is only 0 .1λthick, it is not ionized over its whole thickness due to insta ntaneous screening, except for the maximum intensity case. The fact that the produced magnetic field is much less than exp ected may be attributed to the instantaneous screening of the EM wave due to the ultra fast ionization. In fact, it is too weak to affect self-consistently the refractive index an d as a consequence it cannot lead to magnetically induced transparency as hypothesized by Te ychenn´ e et al. (1998). B. Boltzmann simulations To yield a further insight into the magnetic field generated b y ultrafast ionization we look at the results of 1D and 2D Boltzmann simulations. This c orresponds to the “direct” numerical solution of the Boltzmann equation for the electr on distribution function fe= fe(x,v, t), over a phase space grid: ∂tfe+v· ∇fe−eE m·∂vfe=νI(E)na(x, t)g(v;E(x, t)). (2) Herenais the density of neutral atoms (supposed at rest for simplic ity) and νIis the ionization rate. The term g(v;E) gives the “instantaneous” distribution of the just ionize d electrons, which is supposed to be known from atomic physics . A semiclassical picture which allows to define and evaluate g(v;E) was given by Cornolti et al. (1998). With respect to PIC simulations, the Boltzmann approach has the d isadvantage of larger memory requirements, but the advantages of reduced numerical nois e and the possibility to take into account the full kinetic distribution of the ionized electr ons. We first look at 2D2V Boltzmann simulations. We take a 0 .25µm, 1016W cm−2laser pulse impinging on a solid hydrogen target with number densi ty 2×1023cm−3= 12.5nc, and thickness 0 .1µm. The time envelope of the laser pulse is Gaussian with a FWHM duration 4of 2 cycles. The laser spot is also Gaussian with a FWHM of 2 µm. Fig.2 (a) shows the magnetic field and the density contours after the end of the la ser pulse. The steady magnetic field has constant (negative) sign over its extension. Its ma ximum intensity is about 3 MG. Fig.2 (b) shows the electron current density jyat the same time of the right plot of fig.2 (a). Among the parameters of our simulations, the magnetic field a ppears to be most sensitive to the temporal profile of the laser pulse, achieving its maxi mum value for a square pulse with zero risetime. In Fig.3 (a) we show the results of a 1D Bol tzmann simulations for a square pulse with I= 1016W cm−2,λ= 0.25µm, and a target with ne/nc= 12.5. The current density is jy∼1022c.g.s. units and extends over a distance comparable to dp≃ 1.2×10−2µm. The maximum magnetic field is consistent with Ampere’s law , which gives Bst∼4πjydp/c≃5 MG. Assuming a density ne≃no= 2.2×1023cm−3for the electrons which are instantaneously ionized, one gets a steady veloci tyvst≃jy/ene≃108cm s−1. This value is lower than the ejection velocity for hydrogen vI≃2×108cm s−1. This suggests that effects such as screening, nonzero ionization time, and velocity statistics act to keep the steady current well below the values that one may estimate ac cording to the SMM, eq.(1). Both laser and target parameters where varied in simulation s in order to investigate the scaling of the magnetic field with them. As an example, Fig.3 ( b) shows the results of a simulation for a target of hydrogenic ions with density and t hickness identical to Fig.3 (a), but where we assumed a nuclear charge Z= 2 and scaled the atomic parameters accordingly tox→Zx,t→Z2t,ω→Z−2ω,E→Z−3E. In order to have the ionization threshold to be exceeded at the same instant, the laser pulse had the same e nvelope and frequency but the intensity was scaled by Z6. With respect to the Z= 1 case, we obtain a steady field withlower peak amplitude which assumes both positive and negative val ues. We also performed 2D Boltzmann simulations for a pulse obliq uely incident at 15oon the target. The preliminary results show that the magnetic fi eld is much lower in this case. Therefore it appears that the steady magnetic field is sensit ive to the interaction geometry. In any case, the oblique incidence results further confirm th e conclusion that no magnetic field capable to affect the transmission through the target is generated. 5III. OPTICAL MICROCAVITIES AND IONIZATION FRONTS A. The model In this section, we study effects related to two beam-interfe rence in one spatial dimension and for wavelengths in the infrared and optical range. In our numerical experiment, a one- dimensional interference pattern is generated via an appro priate “target manufacturing”: the idea is to place a reflecting mirror on the rear side of the t arget, the one opposite to the laser. Such a mirror might be easily produced by a metal lic coating on a glass or plastic target. Taking a laser pulse with peak intensity bet ween IT/2 and IT, being IT the “threshold” value for ionization, a plasma is produced i n the target bulk around the maxima of interference pattern produced by the incident wav e and the wave reflected at the rear mirror. Since in this regime we deal only with moderate laser intensi ties, we may use a simple one-dimensional fluid model based on continuity, current an d wave equations for an ionizing medium, originally proposed by Brunel (1990), modified by th e inclusion of the polarization current. More details about the model and its validity can be found in Cornolti et al. (1998) and Conejero Jarque et al. (1999). B. Generation of layered plasmas We first consider a target with thickness L= 2πλ, being λ= 0.8µm, and density no= 10nc. The laser pulse has a sin2-shaped envelope with a duration of 80 fs (30 cycles) and a peak intensity I= 1.8×1014W cm−2. The target parameters are chosen to simulate a thin foil solid slab and it is enough to take the density as lo w as 10 ncsince the maximum electron density always remains much lower than this value. The electron density vs. space and time is shown in Fig.4. A cl ear layered density pattern with a spatial periodicity close to λ/2 is produced along nearly all the slab. The layers of overdense plasma are produced near the maxima of the interfe rence pattern. These maxima 6appear at close times because of the effect of the smooth envel ope of the laser pulse. The resulting quasi-periodic structure of the refractive inde x has in principle some similarities with the widely studied semiconductor microcavities and ph otonic band-gap materials (see reviews by Burstein and Weisbuch (1993) and by Skolnick et al . (1998)). C. Optical microcavities Since the density in the plasma layers is overcritical, and t he layers are created in a time shorter than a laser halfcycle, the portions of the stan ding wave between adjacent intensity maxima may be “trapped” into the cavity formed by t he two neighboring layers. This trapping effect is best seen in the case of a CO 2pulse impinging over a gas target withL=λ= 10.6µm and no= 5nc≃5×1019cm−3. For this target, two plasma layers are produced around the positions x= 0.25λ, x= 0.75λ. Fig. 5 shows the map of the electric field at early (a) and later (b) times, showing the ge neration of the constructive interference pattern which yields the layered ionization ( a), and the subsequent trapping of the field which remains in the cavity at times longer than the i ncident pulse duration (b). The non-ionized regions between density layers clearly act as optical microcavities. Since the microcavity length is Lc≤λ/2, light must have an upshifted wavevector k′≥k in order to persist inside the cavity. This implies also upsh ift of the laser frequency with ω′ L≥ωLas seen in Fig.5(b). The upshift decreases the critical dens ity value for the trapped radiation and therefore wavelengths much shorter than λescape from the cavity. Due to the small fraction of light that tunnels out of the cavity one observes radiation emission from the target for a time much longer than the pulse duration . Both the frequency upshift and the pulse lengthening may provide experimental diagnos tics for microcavity generation. The lifetime of the cavities is ultimately limited by proces ses such as recombination, which however should appear on times much longer than the pulse dur ation of a few tens of femtoseconds that are considered here and are available in t he laboratory. 7D. Ionization fronts As already shown, in our model target ionization is produced around the maxima of the “standing” wave which is generated due to the reflection a t the rear mirror. However, since ionization is instantaneous on the laser period times cale, it is produced as soon as the wave reflected at the rear mirror travels backwards and bu ilds up the standing wave by interference. Therefore, a backward propagating ioniza tion front is generated, as seen in Fig.4. The density at the front exceeds the critical density . This feature is not obtained for a single pulse impinging on a dense target, since it undergoe s immediate self-reflection and penetrates only in the “skin”surface layer (Macchi et al. 19 99). An example of “overdense” ionization front is obtained in th e case of a CO 2square pulse 15 cycles long impinging over a target with ne= 4nc. The ne(x, t) contour plot is shown in Fig.6. The ionized layers merge into a more homogene ous distribution and a “continuous” ionization front appears. The merging appear s because the time- and space- modulated refractive index perturbs the reflected wave subs tantially, leading to broadening of interference maxima. The velocity of the front in Fig.6 is near to, or even exceeds at some times that of light. This is clearly not a physical “movi ng mirror” with a velocity greater than c, but a reflective surface which is created apparently with su ch velocity due to a space-time phase effect. ACKNOWLEDGMENTS We acknowledge the scientific contributions of D. Bauer and L . Plaja as well as their suggestions. Discussions with G. La Rocca, R. Colombelli, L . Roso, and V. Malyshev are also greatly acknowledged. This work has been supported by t he European Commission through the TMR networks SILASI, contract No. ERBFMRX-CT96 -0043, and GAUSEX, contract. No. ERBFMRX-CT96-0080. E.C.J. also acknowledge s support from the Junta de Castilla y Le´ on (under grant SA56/99). 8REFERENCES BAUER, D. 1997 Phys. Rev. A 55, 2180. BAUER, D. & MULSER, P. 1999 Phys. Rev. A 59, 569. BRUNEL, F. 1990 J. Opt. Soc. Am. B 7, 521. BURSTEIN, E. & WEISBUCH, C., eds. 1993 Confined Electrons and Photons. New Physics and Applications (NATO ASI Series B: Physics, vol.340, Plenum Press, New York , 1993). CONEJERO JARQUE, E., CORNOLTI, F. & MACCHI, A. 2000 J. Phys. B : At. Mol. and Opt. Phys. 33, 1. CORNOLTI, F., MACCHI, A. & CONEJERO JARQUE, E. 1998 in Superstrong Fields in Plasmas , Proceedings of the First International Conference (Varen na, Italy, 1997), edited by M. Lontano et al., AIP Conf. Proc. No. 426(AIP, New York, 1998), p.55. GIULIETTI, D., GIZZI, L.A., GIULIETTI, A., MACCHI, A., TEYC HENNE, D., CHESSA, P., ROUSSE, A., CHERIAUX, G., CHAMBARET, J.P. & DARPENTIGNY , G. 1997 Phys. Rev. Lett. 79, 3194. MACCHI, A., CONEJERO JARQUE, E., BAUER, D., CORNOLTI, F. & PL AJA, L. 1999 Phys. Rev. E 59, R36. MULSER, P., CORNOLTI, F. & BAUER, D. 1998 Phys. of Plasmas 5, 4466. RAE, S. C. & BURNETT, K. 1992 Phys. Rev. A 46, 1084. SKOLNICK, M. S., FISHER, T. A. & WHITTAKER D. M. 1998 Semicond . Sci. Technol. 13, 645. TEYCHENN ´E, D., GIULIETTI, D., GIULIETTI, A. & GIZZI, L. A. 1998 Phys. R ev.E58, R1245. WILKS, S. C., DAWSON, J. M. & MORI, W. B. 1988 Phys. Rev. Lett. 61, 337. 9FIGURES FIG. 1. Spatial distribution of magnetic field (left) and ele ctron charge density (right) five cycles after the end of the pulse, for “sin2” pulses of 0.1 a.u. (dotted line), 1 a.u. (dashed line), 10 a.u. (solid line) maximum amplitude and a “square” pulse o f 1 a.u. amplitude (dashed-dotted line). All the pulses are 5 cycles long. The electric field ato mic unit is Eau= 5.1×109V cm−1 (corresponding to I= 3.5×1016W cm−2). (a) (b) FIG. 2. Grayscale contours of the magnetic field Bz(a) and the current density jy(b) five laser cycles after the laser pulse end, for a 2D2V Boltzmann simula tion. The dashed line in (a) and (b) giveBz/Boandjy/jo, respectively, along x= 2µm. The parameters Bo= 27.7 MG, jo= 2.2×1022 c.g.s. units. The solid lines give neutral density contours . The dashed-dotted lines mark the critical density surface. Simulation parameters I= 1016W cm−2,λ= 0.25µm,ne/nc= 12.5. 10(a) (b) FIG. 3. Profiles of the steady magnetic field Bz(solid) and the current jy(dashed) in 1D Boltz- mann simulations. The parameters common to (a) and (b) are λ= 0.25µm and ne/nc= 12.5. In the case (a) the atomic parameters are those of an hydrogen like atom with Z= 1, and I= 1016W cm−2,Bo= 27.7 MG, jo= 2.2×1022c.g.s. units. In the case (b) Z= 2 and laser param- eters are scaled accordingly to x→Zx,t→Z2t,ω→Z−2ω,E→Z−3E;I= 6.4×1017W cm−2, Bo= 50.6 MG, jo= 7×1022c.g.s. units. FIG. 4. Grayscale contourplot of free electron density ne(x,t) for a “solid” hydrogen target with a reflecting “metal” layer on the rear face (thick solid l ine). The pulse parameters are I= 1.8×1014W cm−2,λ= 0.8µm, ∆tL= 30(2 π/ωL)≃80 fs (“sin2” envelope). The target parameters are L= 2πλ,no= 10nc= 1.1×1022cm−3. 11FIG. 5. Evolution of the electric field inside the plasma slab during the interaction with the incident pulse (a) and 80 cycles later (b). The pulse paramet ers are I= 1.8×1014W cm−2, λ= 10.6µm, ∆tL= 15(2 π/ωL)≃530 fs (“sin2” envelope). The target parameters are L=λ, no= 5nc= 5×1019cm−3. FIG. 6. Grayscale contourplot of ne(x,t) for a hydrogen “gaseous” target with a reflect- ing “metal” layer on the right boundary. The pulse has square envelope and parameters I= 1.8×1014W cm−2,λ= 10.6µm, ∆tL= 15(2 π/ωL)≃530 fs Target parameters are L= 2πλ, ne= 4nc= 4×1019cm−3. 12

arXiv:physics/0002018v1 [physics.flu-dyn] 9 Feb 2000Relationships between a roller and a dynamic pressure distr ibution in circular hydraulic jumps Kensuke Yokoi1,2and Feng Xiao3 1Division of Mathematics and Research Institute for Electro nic Science, Hokkaido University, Sapporo 060-0812, Japan 2Computational Science Division, RIKEN (The Institute of Ph ysical and Chemical Research), Wako 351-0198, Japan 3Department of Energy Sciences, Tokyo Institute of Technolo gy, Yokohama 226-8502, Japan (To appear in Phys. Rev. E, Vol.61, Feb. 2000) We investigated numerically the relation between a roller and the pressure distribution to clarify the dynamics of the roller in circular hydraulic jumps. We found that a roller which characterizes a type II jump is associated with two hig h pressure regions after the jump, while a type I jump (without the roller) is associated with only one high pressure region . Our numerical results show that building up an appropriate pressure field is essential for a roller. PACS numbers: 83.50.Lh, 47.15.Cb, 47.32.Ff, 83.20.Jp As can be easily observed in a kitchen sink, a circu- lar hydraulic jump is formed when a vertical liquid jet impinges on a horizontal surface. The schematic figure of the circular hydraulic jump can be shown as in Fig. 1. The phenomenon has been investigated by many re- searchers through various approaches [1–11]. In some experiments, the depth on the outside of the jump can be controlled by varying the height of a circular walld, as shown in Fig. 1. Experimental results show that a circular hydraulic jump has two kinds of steady states which can be reached by changing d[7]. When d is small or 0, a type I jump is formed, as shown in Fig. 1(a). On increasing dthe jump becomes steeper until a critical dcis reached. If dbecomes larger than dc, the liquid outside of the jump topples. Then another steady state, a type II jump, is formed as shown in Fig. 1(b). The eddy on the surface in a type II jump, a secondary circulation, is usually called a “roller.” The existence of a roller distinguishes the two types of jumps. The roller is a common and important feature for many hydraulic phenomena. Recent experiments [9,10] demon- strate that various regular polygonal jumps can develop from a circular jump by controlling the height of the outer circular wall, and that all those polygonal jumps are as- sociated with rollers. Rollers are also observed in the channel flows and are useful for dissipating the excess energy of high velocity flows, such as from sluice gates and spillways [3]. It is widely recognized that rollers play an important role in hydraulic engineering. However, theoretical studies concerning the formation and evolution of the roller in a hydraulic jump are lim- ited because of the largely deformed interface. Some the- oretical studies have been proposed using a hydrostatic assumption in the vertical direction. Some reasonable re- sults have been obtained for the flows of type I jumps [8].However, a type II jump appears to be beyond the regime that this vertical-assumption theoretical model is able to deal with. Numerical modeling has also been used to in- vestigate the circular hydraulic jump problem. Due to the difficulties in the numerical treatment of largely dis- torted interfacial flows, the free boundary of the liquid surface was treated as the fixed boundary of a prescribed shape [7]. In our previous work [11], numerical simulations on cir- cular hydraulic jumps were conducted using some newly developed numerical schemes for multi-fluid flows. We investigated the transition from a type I jump to a type II jump. Non-hydrostatic pressure distributions in the gravitational direction were observed in our simulations. In our studies, we call ‘dynamic pressure’ the net amount of the pressure resulting from extracting the hydrostatic pressure from the actual pressure. We found that the dynamic pressure around the jump, which has been ne- glected in most of the theoretical studies to date, is im- portant for the transition. In a type I jump, a steeper jump is always associated with a higher wall height ( [7] and Fig. 3). Thus, as dis increased, the curvature of the interface immediately after the jump becomes larger, then the surface tension is strengthened, because the sur- face tension is proportional to the curvature. In order to counteract this surface tension and keep the jump sur- face steady, a larger rise in pressure is required (Figs.4 (a,b)). If the wall height is increased over the critical dc, the reverse pressure gradient generated by the dynamic pressure becomes stronger than the flow from below and a transition occurs. In this Rapid Communication, we intend to clarify the relationship between the roller and the pressure field. The simulation results show that the single high dynamic pressure region in a type I jump becomes two regions af- ter the transition to a type II jump. These two high pressure regions are located along the jump slope around the outer edge Routand the inner edge Rinof the roller. This pressure distribution appears important to the flow separation at the outer edge of the roller and then essen- tial to the maintenance of a roller. The governing equations, including effects of gravity, viscosity and surface tension can be written as ∂ρ ∂t+ (u· ∇)ρ=−ρ∇ ·u, (1) 1∂u ∂t+ (u· ∇)u=−∇p ρ+g+µ ρ∆u+Fsv ρ, (2) ∂e ∂t+ (u· ∇)e=−p ρ∇ ·u, (3) where ρis the density, uthe velocity, p the pressure, gthe gravitational acceleration, µthe viscosity coefficient, Fsv the surface tension force, and ethe inner energy. Both the liquid and the gas are assumed to have an equation of state in the form of a polytropic gas, but with quite different sound speeds (large for the liquid phase). The numerical model is constructed based on the C- CUP (CIP-Combined, Unified Procedure) method [12], the level set method [13,14] and the CSF (Continuum Surface Force) model [15]. By using the C-CUP method to solve multi-fluid flows, we are able to deal with both the gas and the liquid phase in a unified framework, and explicit treatment of the free boundary and interfacial discontinuity is not needed. The interface between the liquid and the gas is tracked using the level set method with the CIP (Cubic Interpo- lated Propagation) method [16] as the advection solver. A density function φgenerated from the level set func- tion of the level set method by the Heaviside function can be set as φ= 1 for the liquid and φ= 0 for the air. The density function is then used to define the physical properties, such as sound speed and viscosity for different materials. The surface tension force is modeled as a body force Fsvcalculated by the gradient of the density function, Fsv=σκ∇φ, where σis the fluid surface tension coef- ficient and κthe local mean curvature. κis computed fromκ=−(∇ ·n),where nis the outgoing unit normal vector to the interface and is evaluated from the level set function [14]. An axis-symmetric model has been constructed to deal with the circular hydraulic jump. The configuration of the simulation model on an r-z plane is shown in Fig. 2. This calculation model is validated by comparing the computed results with the scaling relation [6,11]. Simulations were carried out with different heights of the outer circular wall d. The volume flux of the inflow is Q= 5.6 ml/s and the viscosity of the liquid is νl= 7.6× 10−6m2/s. The steady surface profiles for the various wall heights are shown in Fig. 3. The three lower profiles are type I jumps, and the two upper profiles are type II jumps. We observe that the jump becomes steeper as the wall height increases for a type I jump, while for a type II, the slope of the jump appears less steep than that of type I with a high wall hight. These are consistent with the experimental results [7]. The roller is usually a consequence of a steepened jump, while its occurrence always leads to the destruction of the steepness. The dynamic pressure distributions of the second, the third and the fourth profiles from the lowest were plotted in Figs. 4(a-c). For the cases of type I (Figs. 4(a,b)), a high pressure region (referred to hereafter as the primaryhigh pressure) dominating a wide region under the jump surface is observed. In a type II jump, two high pres- sure regions are developed around the inner side of the jump (referred to as the primary high pressure) and the outer side of the jump (refereed to as the secondary high pressure) as shown in Fig. 4(c). This pressure distribu- tion is essential for the roller. We observe that the high pressure on the outer side of the jump (the secondary high pressure) coincides with the separation point of the flow, as shown in Fig. 4(d). This secondary high pres- sure continuously provides a pressure gradient force to maintain the upper reverse flow for the roller. The sec- ondary high pressure is associated with the surface ten- sion. In the steady state of the type II, the liquid surface appears convexly curved around the secondary high pres- sure region or the outer edge of the roller (Figs. 4(c,d)). This feature of the free surface around the outer edge of the roller is also observed in experiment [7]. To coun- teract the surface tension caused by this curved surface the small curvature and keep a steady surface, the sec- ondary high pressure must be required. The reverse flow from the separation point moves down along the jump surface until it meets another high pressure (the primary high pressure) on the upstream side of the jump. The fluid motion is decelerated when it approaches the high pressure on the inner side of the jump. The direction of motion is then changed, and joins the main stream again around the confluent point Rin. We further simulated the disappearance process of a roller (the transition process from a type II jump to a type I jump) to study the details of the relation between the pressure field and the roller. We started from the steady state of a type II jump (the fourth profile from the lowest in Fig. 3). Its surface profile is shown as the topmost one in Fig. 5. The time of this initial state was sett= 0. We simulated until the steady state of a type I jump (the second profile in Fig. 3) was reached by lowing the wall height at t= 0. Fig. 5 displays the surface pro- files at different instants. The flow experienced a transi- tion from a type II jump to a type I jump. The evolution of the dynamic pressure field and the maximum value of the secondary high pressure are shown in Fig. 6. The initial pressure distribution is characterized by two high pressure regions and a roller as discussed above. As time increases, the secondary high pressure becomes weaker, and finally vanishes around 0.55 s. It appears that the re- duction of the secondary high pressure is associated with the decline in the curvature of the surface around the secondary high pressure. Meanwhile, the primary high pressure does not experience any significant change and finally becomes to the primary high pressure in the type I jump. In order to give a quantitative measure for the roller, we calculated the horizontal width of the roller as Rr out−Rr in. Fig. 7 shows the time evolution of the roller width ( Rr out−Rr in). With the secondary high pressure abating, the roller width decreased. This process was significantly enhanced after the secondary high pressure disappeared completely (from 0.55 s) because the pres- 2sure gradient becomes perfectly opposite to the reverse flow of the roller. Around 0.75 s, the roller disappeared. With the secondary high pressure and the roller having abated, the fluid eventually approached the steady state of a type I jump. From this study, we have made clear that the existence of the high dynamic pressure regions and a secondary high pressure region around the outer edge of a roller are essential to the maintenance of a roller. The establish- ment of the high pressure field is a result of the balance among various fluid stresses, and the surface tension ap- pears to play an important role. The secondary high pressure provides a driving force to generate a reverse current beneath the jump surface in a type II jump. We would like to thank S. Watanabe and K. Hansen for many discussions. We also acknowledge the support of C. W. Stern. Numerical computations for this work were partially carried out at the Computer Information Center, RIKEN and the Yukawa Institute for Theoretical Physics, Kyoto University. [1] Lord Rayleigh, Proc. Roy. Soc. London A 90, 324 (1914). [2] I. Tani, J. Phys. Soc. Jpn. 4, 212 (1949). [3] V.T. Chow, Open channel hydraulic, McGraw-Hill, New York, 1959. [4] E. J. Watson J. Fluid. Mech. 20, 481 (1964). [5] A.D.D. Craik et al., J. Fluid Mech. 112, 347 (1981). [6] T. Bohr, P. Dimon, and V. Putkaradze, J. Fluid. Mech. 254, 635 (1993). [7] T. Bohr et al., Physica B 228, 1 (1996). [8] T. Bohr, V. Putkaradze, and S. Watanabe, Phys. Rev. Lett.79, 1038 (1997). [9] C. Ellegaard et al., Nature 392, 767 (1998). [10] C. Ellegaard et al., Nonlinearity 12, 1 (1999). [11] K. Yokoi and F. Xiao, Phys. Lett. A 257, 153 (1999). [12] T. Yabe and P.Y. Wang, J. Phys. Soc. Jpn. 60, 2105 (1991). [13] S. Osher and J.A. Sethian, J. Comput. Phys. 79, 12 (1988). [14] M. Sussman, P. Smereka, and S. Osher, J. Comput. Phys. 114, 146 (1994). [15] J.U. Brackbill, D.B. Kothe, and C. Zemach, J. Comput. Phys.100, 335 (1992). [16] T. Yabe et al., Comput. Phys. Commun. 66, 233 (1991).rz Nozzle Jump d(a) (b)Roller Rout inR FIG. 1. Schematic figures of the circular hydraulic jump. The radius of the wall is much larger than the radius of the jump. The flow from the nozzle is constant. In this experi- ment, a high viscous liquid is used for controlling the insta - bility of flow pattern. (a) and (b) are called type I and type II, respectively. The points of the inside and outside of the roller are defined as ( Rr in, Rz in) and ( Rr out, Rz out). z 0rAirdLiquid FIG. 2. Schematic figure for the initial condition of the simulation. The dark part indicates the no-slip wall. The liquid is jetted from the lower left to the right direction. A Cartesian grid with ∆ r= ∆z= 0.1 mm is used. 02 6 12z [mm] r [mm] FIG. 3. Surface profiles for varying wall heights. Q= 5.6 ml/s and νl= 7.6×10−6m2/s are used. The maximum values of the dynamic pressure around the jump (the primary high pressure) in type I jumps were 1 .77, 3.99, and 8 .47 Pa from the lowest respectively. 3 7 6 5 4 3 2 1 0.8 0.5 60110 160102030 60110 160102030(a) (d) 8 7 6 5 4 3 2 1 70 1001020 70 1001020 (c)(b) 3 2 1 70 1001020 70 1001020 60110160102030 60110160102030 60110160102030 FIG. 4. Dynamic pressure (Pa) contours and the surface profiles around the jump of the second (a), the third (b), and the fourth (c) from the lowest in Fig. 3. (d) shows the streamline, the dynamic pressure contours, and the surface profile of the fourth profile. 02 6 12z[mm] r[mm] FIG. 5. Time evolution of the surface profile from the type II jump to the type I jump at 0 .294 s intervals. The topmost profile is the initial state. 8 7 6 5 4 3 2 1 0.8 0.5 306090120150102030 306090120150102030t=0.59s 9 8 7 6 5 4 3 2 1 306090120150102030 306090120150102030t=0.94s 8 7 6 5 4 3 2 1 0.8 0.5 306090120150102030 306090120150102030t=0.47s 8 7 6 5 4 3 2 1 0.8 0.5 306090120150102030 306090120150102030t=0.29s 0.80.911.11.2 0 0.5Pressure [Pa] Time [s] FIG. 6. Time evolution of the dynamic pressure distribu- tion and the maximum value of the secondary high pressure. Fig.4 (c) and (a) correspond to t= 0 and the final steady state. 036 00.20.40.6Width of roller [mm] Time [s] FIG. 7. Time evolution of the width of the roller. 4

arXiv:physics/0002019v1 [physics.gen-ph] 9 Feb 2000ON M THEORY, QUANTUM PARADOXES AND THE NEW RELATIVITY Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University Atlanta, GA. 30314 Alex Granik Department of Physics University of the Pacific Stockton, California 95211 January 2000 (Dedicated to the memory of Leonard Ainsworth, a true pionee r and a scientist ) Abstract Recently a New Relativity Principle has been proposed by one of the authors as the underlying physical and geometrical foundations of String and MTheory. It is explicitly shown that within the framework of the New Relativity Theory, some Quantum Mechanical Paradox es like the Einstein-Rosen Podolsky and the Black Hole Information Loss, are easily resolved. Such New R elativity Theory requires the introduction of an Infinite Dimensional Quantum Spacetime as has been shown rec ently by one of us. This can be viewed as just another way of looking at Feynman’s path integral formulati on of Quantum Mechanics. Instead of having an infinite dimensional funcional integral over allpaths, smooth, forwards and backwards in time, random and fractal, in a finite-dimensional spacetime, one has a fini te number of paths in an Infinite Dimensional Quantum Spacetime. We present a few-lines proof why there is no such a thing as an EPR Paradox in this New Relativity theory. The reason is notdue to a superluminal information speed but to a divergent information charge density. In the infinite dimensional lim it, due to the properties of gamma functions, the hypervolume enclosed by a D-dim hypersphere, of finite nonzero radius, shrinks to zero : to ahyperpoint , the infinite-dimensional analog af a point. For this reason, Information flows through the infinite-dimensional hypersurface of nonzero radius, but zero size, the hyperpoi nt, in an instant. In this fashion we imbue an abstract mathematical ”point” with a true physical meaning : it is an entity in infinite dimensions that has zero hypervolume at nonzero radius . A plausible resolution of the Information Loss Paradox in Black Holes is proposed. 1. Introduction : Historical background From the very beginning the relativity principle has been on e of the cornerstones of mechanics. In 1686 , in the opening pages of Principia , Newton wrote : ” I do not define time, space, place, and motion , as being well known to all. Only I must observe, that the common people conceive those quantities under no other notions but from the relations they bear to sensible objects ”. Based on this preamble, Newton introduced the absolute space, time, and motion which in his own words do not bear ” relation to anything external ” . It is clear that such absolute space, time, motion, are purel y metaphysical notions, stand outside the realm of physics, and serve the purpose of ” geometrization ” of phy sical phenomena. Thus at the foundations of Newtonian mechanics lie the above metaphyisical concepts. The very same Metaphysical concepts that so many members of the scientific community, by their own admiss ion, are unwilling to accept. Much later, Mach rather broadly , and then Einstein definitel y, set Physics on physical ground by defining measurements relative only to the physical phenome na, and not to the metaphysical entities. Instead of vaguely-defined ( if at all) metaphyiscal concepts of abso lute space, time, and motion, new ( rather narrow) but well defined physical concepts of physical measurements had been introduced. The old ”absolutes ” had been dismissed by Einstein’s Theory of Relativity. Measure ments required a universal standard which had been furnished by a physical quantity : the speed of light. Based on this historical perspective we introduce the New Re lativity Principle [1] that rest on the four postulates presented in the next section. In the last sectio ns an amazingly simple proof is presented why there 1is noEPR Paradox in such New Relativity Theory. In addition, a plausi ble resolution of the Information Loss in Black Holes is proposed. 2. The Program of the New Relativity Principle : The Demoliti on of Today’s Absolutes Recently one of the authors has proposed that a New Relativit y principle may be operating in Nature which could reveal important clues to find the origins of Mtheory [1]. We were forced to introduce this new Relativity principle, where all dimensions and signatu res of spacetime are on the same footing, to find a fully covariant formulation of the p-brane Quantum Mechanical Loop Wave equations. This New Rel ativity Principle, or the principle of Polydimensional Covariance as has been called by Pezzaglia, has also been crucial in the derivation of Papapetrou’s equations of moti on of a spinning particle in curved spaces that was a long standing problem which lasted almost 50 years [2]. A Clifford calculus was used where all the equations were written in terms of Clifford-valued multivec tor quantities; i.e one had to abandon the use of vectors and tensors and replace them by Clifford-algebra val ued quantities, matrices, for example . The New Relativity Theory [1] rests on four postulates : 1. The old Bootstrap Idea of Chew : Each p-brane is made of allthe others. To view a single p-brane as a fundamental identity is a meaningless concept. p-branes are defined only in relation to others. This is Mach’s principle once again. For this reason, one must inc lude all dimensions and signatures on the same footing. Pezzaglia [2] has called this the principle of Poly dimensional Covariance or Dimensional Democracy. The New Relativity theory reshuffles, for example, a string hi story for a 5-brane history; a 9-brane history for a membrane history; an 11-brane history for a history enc ompassing all other p-branes...and so forth. Point and extended instantons and tensionless p-branes are also inlcluded. The tensionless p-brane history excitations of the infinite dimensional Quantum Spacetime a re the ” photons ” in this New Relativity Theory. We honestly believe that MTheory does notstand for mystery, membrane, matrix, master, mother, murky, Moyal....it stands for Mach. The New Relativity theo ry is based on the ultimate Machian view of the Quantum Universe ( the Ultimate Machian ” Quantum Comput er ” ) : Relationships among entities are the only meaningful statements one can make . A perfect ex ample of this are : spin-networks, quantum- networks, quantum sets, cellular networks, p-Adic Physics,...etc. Since it is undesirable to run off the l etters of the alphabet, by keep adding letters like M, F, S... Theory, we gather courage to say that by abandoning the Egocentric Anthropomorphic view of the Universe and, in stead, embracing Mach’s view that everything in the Quantum Universe is interconnected, one reaches the e nd of the alphabet at Ztheory : Zstands for the ultimate Machian view of the Quantum Universe ( that dism isses the egocentric view of the Universe ) for a Zenthropic Quantum Universe. Who are weto say that we know everything that an electron , a photon, a quark really ” sees ” ? Do electrons, quarks....per form Feynman diagrams ? Has anyone seen a point ? 2. Laurent Nottale’s Scale-Relativity theory [3]. In the final analysis, Physics involves, and is about, measur ements. Physics isan experimental science. Physics deals with experiences. On the other hand to measure something one needs a standard of measure- ment to compare measurements with. It is essential , it is of p rime importance, to introduce resolutions in Physics. It is meaningless to say that the one has a field at x. Byxmeaning : specifying the value of the real number xto an infinite number of non-periodic decimal places. In math ematics we can infinitely increase the accuracy ( or degree of resolution) at will of an y real number by adding digits. However, in practice we cannot have such arbitrary accuracy provided by mathematical constructions. Even writing an infinite aperiodic decimal fraction would require an infinit e amount of memory. Therefore, in Physics, it is necessary to have a finite universal physical ”yardstick” wh ich would define the ultimate Physical Resolution. Nottale’s Scale Realtivity takes the Planck scale as such Un iversal physical standard of measurement that is invariant, by definition, under Scale Relativistic trans formations of resolutions, like the speed of light was in Einstein’s Relativity. p-adic numbers and p-adic Physics is a nice attempt to eliminate the problem of ha ving to specify a real number up to infinite digits [11]. The Planck scale is therefo re taken as that universal standard of measure invariant under Scale Relativistic transformations of resolutions . In the same vein that the speed of light was taken by Einstein as the maximum speed in Nature, the Plan ck scale is taken to be the minimum length. The speed of light allowed Einstein to embrace space with tim e, since space and time have different units. 2By the same token, to embrace all dimensions one needs a Unive rsal length scale in alldimensions : the Planck Scale. As the years pass by, more and more planets have been found con firming Nottale’s predictions within his framework of Scale-Relativity. Instead of being proper ly rewarded with increased curiosity and interest in his remarkable theory, he has been increasingly rewarded wi th insults and a suffocating censorship [12]. As the number of his planet confirmations increases, so does the number of insults increases and the censorship of his work is tightened further. Unfortunately, the New Rel ativity Theory will never be able to explain such odd phenomena. The Universal scale, in units of ¯ h=c= 1, in anydimensions, D >2 ( in two dimensions the Einstein- Hilbert action is a topological invariant) is : Λ =G1 D−2 D=G1 D−3 D−1=........... = 10−33cms. (1a) where GD, GD−1.....are the Newton gravitational coupling constants in differen t dimensions. In the same fashion that in Newtonian Physics one only can assign a defini te meaning to the ratio of masses ( it is meaningless to say that one has a value of mwithout a comparison to another mass ), in the New Relativity theory it only makes sense to write : D−3 D−2=lnGD−1 lnGD. (1b) In the New Relativity Theory it is meaningless to talk about s uch things as ” compactification ” , ” decompactification ” used in the literature that relates the Newton constants in different dimensions through the small radius of a compactified unseen dimension at low ene rgies. It as meaningless as saying that the velocities of the gas molecules in a room experience a dynami cal or spontaneous ” compactification ” to a fixed average value. Problems with the compactification picture of Superstring t heory from 10to4were already alarming signals that something could be wrong. Billions and billion s of possible four-dimensional phenomenological theories of the world were obtained : the so-called uniqueness of string theory went out the window when this was found. String theory wasn’t the problem, assuming a fixed dimensions was ! Witten already proved long ago that something might be inherently wrong with the co mpactification schemes, when he showed using Index Theory arguments, that the standard 11-dim Supegravity Kaluza-Klein compactifications of ordinary manifolds did not yield chiral fermions in 4dimensions. This problem was bypassed in the second string revolution by saying that orbifold compactification were fine because orbifolds are not really ordinary manifolds, so things were satisfactory after all. The New Re lativity Theory does not have to face these challenges. One has a truly infinite-dimensional Quantum Un iverse which suggests that Topological Field Theories could be the most natural candidates for a theory of the world. Since below the Planck scale there is no such thing as a distance; it is very likely that Topology should play a more important role. Conformal Field Theories and their Higher Conformal spin ex tensions are the ones to use in D= 2. In D= 2, one has induced gravity : W2, W3...W∞gravity as a result of integrating out the conformal matter field fluctuations. This replaces the topological invariant Einstein-Hilbert action. In D= 1 dimension there is only extrinsic curvature. One can view a one-dim loop as th e boundary of a two-dimensional surface. This allowed [7] to write down a String Representation of Qua ntum Loops from a covariantized phase space Schild action path integral. The effective action for the bou ndary, with induced extrinsic curvature terms was obtained, in addition to the Polyakov Bulk partition fun ction and the holographic boundary Eguchi wave functional as well. 3. Noncommutative C-spaces. One of the authors was forced to enlarge the naive no tion of commuting spacetime coordinates to fully covariantize the Quantum Me chanical Loop Equations for p-branes. One achieved that goal if one extended the notion of ordinary spa cetime vectors and tensors, to a Noncommutative C-space, or Clifford manifold, where all p-branes were unified in one single footing by using Clifford-a lgebra valued multivectors quantities ( matrices) instead of ordinary vectors and tens ors. In order to combine objects of different dimensionality one needs a length scale : the Planck scale. There was a one-to-one correspondence between a nested hierarchy of point, loop, 2-loop,3-loop ....... p- loop histories in Ddimensions encoded in terms of hyper-matrices and single li nes in Clifford Manifolds. This is roughly similiar to the aim of Penrose’s twistor progam. B y using Clifford-algebra valued multivectors , 3one could argue why it may be meaningless to say that the cosmo logical constant is a constant in its definition ! The so-called cosmological constant is observer-depende nt in this New Relativity Theory : it is just one of the many components of the Clifford multivectors. Due to Po lydimensional Covariance, only the norm of such multivector is truly an invariant. So using this simp le argument one of us was able to argue why it is meaningless to try to measure such constant, unless one is specifying what is the frame of reference one lives in ! The reader may say that the value of p=−1 was not included here. Point and Extended Instantons can also be treated very naturally in this framework [1]. The New Relativity Theory reshuffles, for example, a loop-history represented by the coordinates : xµ, σµν, Ain one frame of reference, to another history, in another frame of reference, represented by the loop-instan tonx′µ,(σµν)′, A′= 0. The xCMare the center of mass coordinates of the loop. Ais the areal-time spanned by the motion of the loop through sp acetime. σµν are the holographic coordinates of the loop. It can reshuffle a massive point history ( a line ) : xµ, τ/negationslash= 0 to a massive point-instanton : x′µ, τ′= 0 in another frame of reference. An so forth. 4. Quantum Spacetime must be treated from a Multivector-Mult iscale point of view. The use of Clifford- valued multivectors was explained above. The multi-scale o r resolution aspects are based on Nottale’s fractals and El Naschie’ s Cantorian-Fractal Quantum Spacetime view s that dimensions are resolution dependent concepts and not fixed notions [3,4] . Nottale, by abandoning the hypothesis of the differentiabli ty of spacetime , was led to three effects ( at least ) : (i). The number of geodesics becomes infinite. This forces upon u s to jump to a statistical fluid-like description. (ii)Each geodesic becomes a fractal curve of higher and higher fr actal dimensionality as the resolution of the ” physical apparatus ” becomes finer and fine r, asympotically approaching the minimum Planck scale resolution where the fractal dimensionality b ecomes infinite. This forces us to embed the fractal geodesics in an spacetime of infinite-Hausdorff dimensions. (iii). The symmetry dt→ −dtis broken by the non-differentiablity which leads to a two-valuedness character of the average velocity vector and wh ich is, in Nottale’s view, the underlying reason why the wave function in QM is complex . This is not the ultimate status of things. To be consistent an d to move forward along the path charted by Mach and Einstein, one cannot, and should not , accept this status quo as the ” end of the road ” in Physics. This reminds us of the status of things at the end of t he19century when ” two clouds ” were the only obstacles hovering over the horizon that prevented the ” end ” of Physics. In fact, one cannot but to feel compelled to say that from the beginning, a truly quantu m mechanical description of the world must start by abandoning the very notion of spacetime itself and other ” idols ” from our minds, as Finkesltein has pointed out [8]. This is precisely the goal of p-Adic Physics [11] to remove the notion of spacetime per se and replace it by objects and their relationships. A tr ulyCategorical view of the Universe. An extension of Einstein’s motion Relativity and Nottale’s Sc ale Relativity into a unfied Scale-Motion Realtivity was outlined briefly in [13]. Whatever the ”final” view of the w orld may be, it seems that it is wrong to assume that Quantum Spacetime has a fixed dimension. On the co ntrary, it may have uncountably-infinite dimensions as El Naschie has argued [4]. Taking this infinite -dimensional point of view allows us to eliminate the notion of a EPR , and possibly, Black-Hole Information Loss ”paradoxes ”. F or this reason we believe it ought to be investigated further. Dimensions are not fixed absolutes. They are resolution dependent concepts. Quantum Gravity is nota quantization of the spacetime coordinates, metric.....I f this were the case, one would have had quantized the spacetime coordinates long ago. In String Theory, from the two-dim world sheet point of view , the spacetime coordinates are not hing but a finite number of scalar fields whose quantization is essentially trivial by selecting the confo rmal or orthonormal gauge. The same arguments applies with the ( linearized ) spin two graviton. Quantum Gr avity it is something much deeper than the naive notion of coordinates and gravitons. It is something t hat doesn’t need any spacetime background nor metrics whatsoever. Morever, it involves something that di sposes of the ill-conceived notion of having a fixed dimension. The classical spacetime that we perceive with ou r senses is just a long distance averaging effect associated with a quantum network of processeses of a deeper underlying Quantum Universe. Einstein’s Gravity is an effective theory as suspected long ago. To merge Quantum Mechanics with Relativity it is necessary to enlarge the Einsteinian view of Relativity to a New Relativity Principle [1]. To proceed further one has to demolish the concept of dimension as an absolute , as an idol. To sum up what has been said so far : 4The New Relativity Theory forces upon us to take a radically d ifferent view of the Quantum World, an ultimate Machian/Zenthropic view, and to dismiss the con cepts of false absolutes (idols) of dimensions, spacetime, cosmological constant , from our classical mind s, as Finkelstein [8] has advocated. If a true evolution ( revolution) of Physics is to take place one has to embrace th e plausible extensions of Relativity as Finkelstein has insisted [8]. For those who believe that w e have reached the end of the road, the end of Physics, we feel that they are setting themselves for simila r surprises that Lord Kelvin experienced with the advent of Quantum Mechanics and Relativity. To this day , to the best of our knowledge, there is no satisfac tory definition yet of Quantum Field Theory. QFT today is being challenged by deeper concepts : No ncommutative Geometry, Quantum Groups, Hopf algebras, Monoidal Braided Categories, Braided QFT, e tc.... Relativity itself is hereby extended to a deeper meaning by the New Relativity Theory : Scale Relativ ity and Cantorian-Fractal Geometry. A Nested Hierarchy of Histories have replaced the old fashion ed concepts of spacetime events; vectors and tensors have been replaced by Clifford-multivectors; Riema nnian Geometry by Finsler Geometry and by Fractal-Cantorian, Non-Archimedean, p−Adic, Noncommutative and Nonassociative Geometries..... Recently we have proposed to even abandon the the idea of the c osmological constant as a constant. The so-called cosmological constant is nota constant in its definition ! It is observer-dependent withi n the framework of the New Relativity Theory [1]. Trying to estima te the absolute values of such a ” constant ” is like trying to detect absolute spacetime motion and to ver ify the existence of the ether ! Such ideas that the vacuum energy could be observer dependent orginated wit h discussions held in Trieste by one of us with Miguel Cardenas and Devashis Benarjee [9]. Even the notion o f the ”vacuum” per se ! Special Relativity demolished such framework of thinking. We believe that the N ew Relativity Theory will also replace the existence of such ill-conceived notions that spacetime has a fixed dimension and that the cosmological constant has an well defined absolute value in all frame of ref erences. The observed spacetime dimension of D= 4 is interpreted in this New Relativity Theory [4] as a resul t of anaveraging procedure over all the possible infinite values of Quantum Spacetime. I n a sense it is similar to what happens with the statistical distribution o f velocities of a gas. There is an average velocity ( average over all the infinite possible values of the statisti cal ensemble ) proportional to the Temperature. To assume that there is a spacetime compactification from D=11 toD=4 ( like it is assumed in mainstream Physics today ) is an incongrous assumption in this New Relat ivity Theory : it is like saying that there is a ” velocity compactification/decompactification ” from h igher/lower velocities to the average observed velocity in a gas. Problems with the compactification pictur e of Superstring theory from 10to4were already alarming signals when billions and billions of possible fou r-dimensional theories of the world were obtained : the so-called uniqueness of string theory went out the wind ow when this was found. String theory wasn’t the problem, assuming a fixed dimensions was ! The fact that th ere might be an Statistical approach to the Dimensions, and to Quantum Gravity per se, was already lurki ng behind the scenes long ago in the work of Hawking : Black Hole Thermodynamics ! 3. There is No Such Thing as an Einstein-Rosen-Podolski Para dox in the New Relativity We will present a few-lines proof why there is no EPR Paradox within the framework of the New Relativity Theory if one assumes that information flows in a s imilar fashion as ordinary charges in Electro- magnetism ; i.e information is to be thought of as a ” field ” [14 ]. Interestingly enough, this will be our only assumption. We are notimplying that there is such a thing as a ” fifth ” force in Nature found one morning in the closet of our homes after a bad night. We are just voicin g out what has been irrefutably proven over and over by experiments. Take an electron-positron pair colliding at the center Oof an infinite dimensional sphere, SDforD→ ∞, at a givent moment we call t= 0. After the collision a pair of two photons will travel in op posite directions imposed by energy-momentum conservation. An any given mome nt after the collision, we can locate those two photons at the surface of a multidimensional sphere of ra diusR=ct. The flux of information from the center of the sphere Oflowing from the moment of the e−/e+collision radially outwards through the hypersurface is : Φ =/contintegraldisplay /vectorJD.d/vectorSD−1=JDSD−1. (2) 5This is nothing but the usual Gauss Law in Electromagnetism. TheD-dimensional information-current, JD, points radially outwards from the center O. Due to hyper-spherical symmetry its magnitude only depend s on the radius R=ct. At each given point on the hypersurface, the current is poin ting radially outwards and has the same value of magnitude, JD(R), along all the points of the hyper-sphere, This is why one ca n pull out the current outside the integral. The hypersurface SD−1encloses inside a VDvolume given in terms of gamma functions. Similar considerations apply to the highe r-dimensional solid angle : VD=πD/2RD Γ(D+2 2). S D−1=dVD dR=RD−1ΩD−1.ΩD−1=1 RD−1dVD dR. (3) Therefore, the total information-flux is given by the usual G auss Law : Φ =JD(R).RD−1ΩD−1=JD(R).RD−1.1 RD−1dVD dR=JD(R).DπD/2RD−1 Γ(D+2 2). (4) Now we take the D→ ∞ limit and make use of Stirling’s asymptotic formula for the g amma function : ( Pictures drawn on a Mathematica package also verify explici tly the results below ) limD→∞Γ(D+ 2 2)∼√ 2π(D+ 2 2)D+2 2e−D+2 2. (5) By Radius R=ctone means radius in Planck scale units. We will set the Planck scale to 1. So by lnRin all of the formulae below we mean ln(R/Λ) Otherwise the units will not match up. AsD→ ∞ one can verify that in the asymptotic D=∞limit the numerator expression for the flux approaches : exp[lnD+D 2lnπ+ (D−1)lnR]∼exp[lnD+D 2lnπ+DlnR ]∼exp[lnD+Dlnπ +DlnR ] (6) whereas the denominator approaches : exp[D+ 2 2ln(D+ 2 2)−D+ 2 2]∼exp[DlnD−D]∼exp[(D−1)lnD]∼exp[DlnD ]. (7) Hence, the flux in the infinite Dlimit is : Φ =Jexp [lnD+Dlnπ +DlnR−DlnD ] =Jeα. (8a) To be precise, upon reinserting the Planck scale one has that the flux is given in Planck units as : Φ =J(Λ)D−1eα=Jeα×1D−1=Jeα. (8b) where αis : α=D(lnπ+lnR) + (1−D)lnD∼D[lnπ+lnR−lnD] (9) Forfinite times , in units of Λ = 1, R=ct/negationslash=∞the coefficient αgoes to negative infinity : α∼ −DlnD→ −∞ ⇒ eα→0 (10) So limD→∞JD(R).DπD/2RD−1 Γ(D+2 2)→J∞(R)×0 = Φ⇒J∞(R)→ ∞. (11) hence, in the D=∞limit, the current ( in Planck units, Λ = 1 ) blows up. This is notbecause there is a superluminal speed of information. It is because the hyper-volume, hyper -area elements, for finite values ofR, go to zeroin infinite dimensions!. Everything shrinks to a hyperpoint despite the fact that the radius 6isnotzero ! The hyperpoint is the infinite-dimensional version of a point in ordinary fin ite-dimensional spacetime. The current is as usual of the form : J=ρv. As the hyper-volume, hyper-area elements, for finite values of R, go to zero, the information charge density ρ, charge per unit hyper-volume, blows up !. The information charge density diverges at the hyperpoint. The information velocity vis constant and cannot exceed the speed of light. From the point of view of an infinite -dimensional observer, allthe points of the hypersurface are interconnected . There is no such thing as non-locality in Quantum Mechanics. This is an illusion due to the shrinking to zero( for finite radius) of the infinite-dimensional volume of the hypersphere, resulting from the asymptotic behaviour of th e gamma functions ! This corroborates Mach’s brilliant insight that everythin g is connected in the ( Quantum ) Universe. What happens here and now, affects everything in the Universe in an instant. Based on the recent teletrans- portation experiments of a single photon by several experim ental teams, this view of the Quantum Universe may lead an advance future generation of open minded scienti sts to achieve the ultimate communication system : instant exchange of information to anyplace in the U niverse by tapping into the infinite dimensions of Quantum Spacetime. An speculative application of this wo uld be to tele-transport a quantum copy of the human genome to other distant Planets in the Universe sui table for life. This would be a way out of the Galactic bounds we live in and an escape of the ultimate fate o f the earth : consumed by the Sun when it becomes a Red Giant. Spacetime travel in an instant will be much harder to achieve if by travel on means tele-transporting a quantum copy of ourselves to another point in the Universe. I n order to do that one has to be able to tele- transport our consciousness as well. We adscribe to Penrose ’s view that consciousness is a non-algorithmic process. This agrees with the Uncountably-infinite number o f dimensions of the Cantorian-Fractal Spacetime view of El Naschie [4]. It would be impossible for a Quantum Tu ring Machine ( a Quantum Computer) to quantum-process such vast of uncountably-infinite number o f quantum bits. Never, in our wildest dreams we could possibly count such large number of dimensions of th e Cantorian-Fractal Spacetime of El Naschie [4] . Such World is not a mere Mathematical abstraction : it is essential for Consciousness to emerge. It is desirable that The Theory of ” Everyhthing ” should include C onsciousnes. The Theory of Everything has to account for the existence of Conscious life and when, why, ho w, and for what it emerged from the Quantum Universe. A ”pointeless ” Universe is another one of those al arming signals that something is inherently incomplete with our view of the World. We believe that it is no t sufficient to dismiss these questions as ”meaningless metaphyiscs ”. Upon closer inspection of eq-(11), if one were to set J=finite ; this would imply that the information flux Φ = 0 so by Gauss Law there is no net information charge enclosed in the hypersphere. This is not correct for the following reason. Nottale’s Scale Relativity implies that it is not possible t o have zero measures with zeroresolutions. It is possible to have zero measures but with ( nonzero) Planc k scale resolutions. The e−/e+pair never goes beyond the minimum Plank scale resolution. The center Oof the hypersphere is not a physical point. It is a smeared fuzzy hypersphere of infinite dimensions but w ith a nonzero Planck scale radius. This is a reason why Noncommutative Geometry, Fuzzy Phyiscs, Quantu m Groups ....could be the right approaches to look at the world at small scales. Thus the information cha rge is distributed ” uniformly ” , in discrete bits of Planck hyper-area, in Planck units , over the outer ” s urface ” of the hyperball of Planck radius. There is noinside. Inside is meaningless notion below the Planck scale , this is why the information charge has to reside on the ” surface ”. It would not be so surprising i f this mechanism could be linked to the Bekenstein-Hawking entropy-area relationship of Black Ho les. The number of dimensions increases as one probes finer and fine rresolution -scales ( not to be confused with lengths, although they both have the same units). By res olutions one means the resolutions that a physical apparatus can resolve. Resolutions which are not t he same thing as the spacetime labels of a ” point ”, event ” like xµ. Resolutions that so far ( until Nottale) have been overlook ed in the description of Physics. As one approaches asymptotically the Planck scale resolution , the hypersphere of Planck scale radius becomes more and more ” visible” to us . To be able to rea ch this limiting ” threshold ” of resolutions in our physical apparatus, an infinite amount Energy is requi red as Nottale has argued. By the same token that it takes an infinite amount of energy to accelerate a mass ( nonzero rest mass) from rest to the speed of light, it takes an infinite energy to probe Planck scale-reso lutions. The final infinite-dimensional hypersphere, 7containing the information charge located at the ” origin ” O, shrinks to a hyperpoint of zero size , but finite Planck radius, The information charge density also diverges at the infinite-dimensional point : the hyperpoint of nonzero Planck radius. Exactly in the same way it did for hyperspheres of radius R=ctupon taking the infinite dimension limit. Concluding, the flux Φ is notzero. There is a net information charge enclosed by the hyper sphere with center Oand radius R=ct. Exactly the same argument occur if one asks the question : Wha t does one of the photons ” see ” ? It will ” see ” the other photon at a distance l= 2R= 2ct( in Planck units) move away at the speed of light. Due to the Doppler effect, the frecuency will be redshi fted so much that the photon will appear to be completely dark, with zero frecuency . For finite values of th e radius, l= 2R= 2ct, the hypersphere centered at one of the photons will again shrink to a zero size , to a hype rpoint, in the infinite Dimensional ( large D) limit. Therefore, when a Macroscopic Observer with a Physi cal Apparatus measures the polarization of one particular photon, it will transfer its information t o the other photon in an instant due to the fact that both of the photons have access to an extremely large num ber of dimensions in comparison to the macroscopic observers; i.e the photons truly live inside th e hyperpoint. For this reason, they are able to exchange information in an instant without actually having a superluminal speed of information ! It is the information charge density ( and information current J) that diverges once again at the hyperpoint, and not the information speed. It is only an illusion due to the shrin king to zero of the hypersphere in the infinite dimensional limit. Of course, the e−/e+pair does not attain such infinite energies to probe Planck sc ale resolutions, they come very close to each other but never reach the Planck scale . As they approach each other more and more dimensions become visible to them. Much more dimension s than the dimensions of the apparent one-dimensional world to a macroscopic observer looking at the line between the two emerging photons while performing his experiment. . Effectively, the number o f dimensions of the world visible to the e−/e+ pair, and the two emerging photons , is very high in compariso n to the apparent D= 1, D= 4 of the macrospcopic observers , that for all practical purpopses, one can take the infinite dimensional limit of the gamma functions. The diagrams explicitly show that the hype rvolume, hyperareas fall-off very rapidly to zero as Dmoves far away from the D= 4. It is not necessary to actually take the infinite dimensio nal limit too literate. A related textbook issue is the following : Imagine a rapid mo ving observer passing by ourselves during the night while we are gazing at the stars. Due to the Lorentz c ontraction the celestial sphere that he experiences will naturally shrink with respect to us. It shr inks, but does it appear flattened ? The answer is no. One can view the Lorentz transformations in spinor ter ms as a SL(2, C) Mobius transformation. Since the Mobius transformation maps circles to circles, the cele stial sphere will have shrunk in radius only but it willnotbe flattened. Similar analogy happens to the photon. What does a photon ” se e ” ? Since we have said earlier that one cannot for certain answer such questions. We can onl y follow what we know so far : Due to the infinite Lorentz contraction the celestial sphere will shri nk to a point. Doesn’t this contradict Nottale view that the Planck scale is the mimimal length ? The answer is no. Once again we have to take the variable dimensions of the Quantum World that a photon experiences. T he photon is a quantum entity. Nobody can deny this. The photon of a given energy E= ¯hωwill probe resolutions larger than the Planck scale. Rigorously speaking , we should write : E= ¯heff(k2)ω. In [1] we have shown that the New Relativity Theory demands an energy-dependent effective Planck consta nt so that [ x, p] =i¯heff(k2) to reproduce the full blown Quantum Spacetime Uncertainty Relations that ar e more general than the String Uncertainty Relations : we have included the effects of allextended objects [1]. It was shown rigorously why one cannot probe resolutions smaller than the Planck scale. As energy b egins to be pump-in, one cannot probe smaller scales. Spacetime actually starts to grow. It is possible th at a polymerization growth process of the Quantum Spacetime begins : an infinite chain of self similar branched polymers is triggered and baby universes branch off. The Quantum Universe might be an ever self-reproducing , self-recursive, self-iterated fractal process as Linde has suggested. Only at infinite energy will a photon be able to probe the Planc k scale. The celestial hypersphere that the photon ” sees ” has a radius of the order of the inverse phot on Energy, roughly, assuming it is a low energy photon, Energy and resolution are inversely correla ted at that level., not at higher enegy levels. Scale 8Relativity implies that the Compton wavelength and momentu m aredecoupled as one approaches Planck scales. It takes an infinite energy to probe the Planck scale. The Planck scale is the ultimate Ultaviolet Regulator. However, due to the effectively large number of di mensions that the photon has access to, despite the fact the hypersphere has a nonzero radius, the celestial hypersphere shrinks to zerosize consistent with the infinite Lorentz contraction ! . It is true that one has to c onstruct the full Scale-Motion Relativity [13] to be fully rigourous and consistent. We have presented a sol ution to the apparent paradox of how one can have a zero measure/size ( due to the infinite Lorentz contrac tion) with a nonzero resolution for a radius : Infinite ( large number of ) Dimensions is the key once again ! Therefore, in essence : By introducing the notion of hyperpoint in physics, which is forced upon us by the New Relativity Principle as a result of having a truly i nfinite dimensional Quantum World. we have imbued a mathematical point with a true physical meaning : it is an infinite-dimensional hypersphere, of zerosize but nonzero radius !. When t=∞then the coefficient αwillnolonger be negative infinity due to a cancellation between lnRandlnD: α∼D[lnπ+lnR−lnD] =D[lnπ+ln(ct)−lnD]∼Dlnπ⇒eα→ ∞. (12) In this case, one has the opposite result : the value of the inf ormation current JDatR=∞collapses to zero, as it should on physical grounds. The information field must vanish at infinity in anydimension, finite or infinite. As the photons move away from eachother, if one wa its an infinite amount of time to peform the EPR experiment, the photons will nolonger be correlated ! To sum up : The EPR Paradox only occurs to the one-dimensional beings ( o r finite-dimensional beings) living along the linear path ( around the linear path) of the photons who wi sh to perform the EPR gedanken experiment. From their finite-dimensional point of a view, QM appears to b enon-local : a superluminal transfer of information appears to take place. From the point of view of t he New Relativity Theory there is no paradox because Quantum Spacetime is truly infinite-dimensional. F or those Quantum-dimensional beings who were able to tap into the effectively ”infinite” number of dimensio ns of Quantum Spacetime at the very ” moments ” when the e−/e+pair collided, at a very small distance separation among the m, distance which cannot be smaller than the Planck scale as indicated by Nottale’s Sc ale Relativity, there is no such Paradox at all ! : the information current blows up because from their in finite-dimensional point of view, for finite values of the radius, the hypersphere has shrunk to a hyperpo int. The transfer of information to the two photons, about the spin and other quantum numbers of the e−/e+pair, occurs in an instant ! Every point in their universe is inter-connected as Mach argued long ago . Similar arguments apply to the two photons when a macroscopic observer measures the polarization of on e photon, the information is transfered to the other photon in an instant via an effectively ” infinite ” dimen sional ( relative to the macroscopic observers) Quantum Spacetime accesible to them. This should encourage us to view Feynman’s path integral for mulation of QM taking allposible paths in afinite dimensional spacetime, from the New Relativity Theory poin t of view : it is possible to have a finite number of paths in an Infinite Dimensional Quantum Spac etime. The main question is : Where does the Feynman statistical complex-weighting of the paths via theeiScomes from ? The partial answer was given by Ord [10] , Nottale [3] and othe rs : Since fractal paths have a dominant weight in the path integr al compared to the smooth ones, the latter have a zero measure compare to the former, roughly spe aking, Quantum Effects manifest or channel themselves via the fractality of spacetime. Although there are people who do not subscribe to this view. Fractal curves are continous but nowhere differentiable. Th is means that the derivatives are discrete - valued. The discrete jumps of the values of the tangents are ” quantized” in units of what has been called by mathematicians the ” Planck ” constant of a curve. In this f ashion the Feynamn eiSweighting factor is interpreted although , we must say that no rigorous proof of t his has been given as far as we know. Fractals and Scale Relativity are essential because as the r esolutions that a physical apparatus can resolve reach the mimimal Planck scale resolution ( resolut ions must not to be confused with statistical uncertainties nor with ordinary lengths) the number of frac tal dimensions blows up. For a newPhase space path integral derivation of Feynman’s particle propagator that is roughly based on these ideas that a fractal particle ” path ” can have a meaning in QM see [7]. 9The apparent superluminal information velocity happens in other aspects of Physics. There is a very simple analogy with superluminal jets in Astrophysics [5] . If one takes a flash light at a sufficiently large distance from a wall and rotates it very rapidly , the image on the wall can appear to move faster than light. However the image is nota truly physical object. The physical photons never move fas ter than light. The image is comprised of many different photons and not of a fixed p articular number of them . The maximum angular velocity of rotation of the flash light is bounded by S pecial Relativity : ωmax=c r(13) where ris the length of the flashlight. If the distance to the wall is Rthen the apparent velocity of the shadow is : v=cR r> c. (14) The ( unphysical object) shadow can move faster than light. O ne does not even have to go to such extremes of achieving the maximal angular speed for the flash light , on e can simply choose the wall far enough, and the flashlight sufficently bright, ( Rlarge enough ) so the image on the wall moves with a superlumin al velocity ωR > c . Exactly similar arguments occur with the phase velocity in wave propagation. The phase velocity can be greater that cbut the physical group velocity is always bounded by c. Taking the number of Dimensions to infinity, mimics this simple example of taking the distance to the wall far enough and rotating the flash light fast enough. Similar arguments can be taken with the so called Back Hole In formation loss Paradox. Since Quantum Spacetime is truly infinite dimensional, there is no such thi ng as an Information Loss. This information is stored in allthe infinite number of dimensions that are inaccesible to an o utside low energy observer. There is information radiated away and a remnant ” hidden ” in the in finite number of dimensions inaccesible to the outside observers. Black Hole evaporation stops at the minimal scale in Nature : the Planck scale, reaching a maximum temperature, Planck’s Temperature. Sca le Realtivity not only induces an effective value of the Planck’s constant : ¯ heff(k2) [1], it also affects the Boltzmann constant as well : kB(k2) so that :kB(k2)T= ¯heff(k2)ω. As one reaches the Planck scale, energy blows up but the temp erature reaches asymptotically the maximum Planck Temperature ( thermal Re lativity). One must have a standard of temperature to compare temperatures with. That maximum uni versal standard is the Planck Temperature whose definition in D= 4 is : TP=/radicalBigg ¯hc5 Gk2 B= 1.42×1032K. (15) Astrophysicits have been baffled by recent findings that there are unexplained extremely bright and unrelenting sources of energy. It is warranted to study thes e phenomena within the framework of the New Relativity Theory. To be able to ” see ” all the informatio n one has to tap into all the infinite number of dimensions of the Quantum Spacetime. To achieve th at one requires infinite amount of energy to probe the Planck scale resolutions according to the Scale Relativity Principle . At that scale (infinite) Dimensions, (infinite) Energy and (infinite) Information me rges into the ” Omega ” hyperpoint , the ” Trinity ” hyperpoint..... the ultimate infinite-dimension al point : At that scale, the ” Trinity ” hyperpoint, Dimensions, Energy and Information are indistinguishable from each other. More details will be given later. Acknowledgements We are indebted to E. Spallucci for a very constructive criti cal remarks. We thank G. Chapline. L.Nottale , W. Pezzaglia, M. El Naschie and D. Finkelstein fo r illuminating discussions. Finally many thanks to C. Handy and M. Handy for their assistance and encou ragement . References 1. C. Castro : ” The String Uncertainty Relations follow from the New Relativity Principle ” hep-th/0001023. ” ” Hints of a New Relativity Principle from p-brane Quantum Mechanics ” hep-th/9912113. ” Is Quantum Spacetime Infinite Dimensional ? hep-th/000113 4. 10”Towards the Search for the Origins of MTheory, Loop Quantum Mechanics and Bulk/Boundary Duality ........hep-th/9809102. 2. W. Pezzaglia : ” Dimensionally Democratic Calculus and Pr inciples of Polydimensional Physics ” gr-qc/9912025. 3. L. Nottale : Fractal Spacetime and Microphysics, Towards the Theory of Scale Relativity World Scientific 1992. L. Nottale : La Relativite dans Tous ses Etats. Hachette Lite rature. Paris. 1999. 4. M. El Naschie : Jour. Chaos, Solitons and Fractals vol 10 nos. 2-3 (1999) 567. 5. Phillip Morrison : Conversations held with Carlos Castro at MIT in 1980. 6. . C. Castro, A. Granik et al : In preparation. 7. S. Ansoldi, A. Aurilia and E. Spallucci : Eur. J. Physcs C 21(2000) 1-12. quant-ph/9910074. S.Ansoldi, C. Castro, E. Spallucci : Class. Quantum. Gravit y16(1999) 1833. 8. D. Finkelstein : ” Third Relativity ” Georgia Tech preprin t, January 2000. ” Emptiness and Relativity ” Georgia Tech preprint. Decembe r 1999. 9. D. Benarjee, M. Cardenas : Private Communication. 10. G. Ord : J. Chaos, Solitons and Fractals 10(2-3) (1999) 499. 11. M. Altaisky, B. Sidharth : Journal of Chaos, Solitons and Fractals vol 10 (2-3) (1999) 167. l. Brekke, P. Freund : Phys. Reports 231(1993) 1-66. V. Valdimorov, I. Volovich, E. Zelenov : p-adics in Mathematical Physics. World Scientific 1992. A. Khrennikov : Non Archimedean Analyis, Quantum Paradoxes , Dynamical Systems and Biological Models. Kluwer Publisng 1998. 12. L. Nottale : Private Communication 13. C. Castro : J. Chaos, Solitons and Fractals 10(2-3) (1999) 295. 14. M. El Naschie : On the Unification of the Fundamental Force s and Complex Time in theE(∞)Space. Jour. Chaos. Solitons and Fractals 11(2000) 1149-1162. 11

arXiv:physics/0002020v1 [physics.acc-ph] 9 Feb 2000SIMULATION OF LASER-COMPTON COOLING OF ELECTRON BEAMS∗ TOMOMI OHGAKI Lawrence Berkeley National Laboratory Berkeley, California 94720, USA We study a method of laser-Compton cooling of electron beams . Using a Monte Carlo code, we evaluate the effects of the laser-electron interact ion for transverse cooling. The optics with and without chromatic correction for the coolin g are examined. The laser- Compton cooling for JLC/NLC at E0= 2 GeV is considered. 1. Introduction A novel method of electron beam cooling for future linear col liders was proposed by V.Telnov.1During head-on collisions with laser photons, the transver se distri- bution of electron beams remains almost unchanged and also t he angular spread is almost constant. Because the Compton scattered photons fol low the initial electron trajectory with a small additional spread due to much lower e nergy of photons (a few eV) than the energy of electrons (several GeV). The emitt anceǫi=σiσ′ ire- mains almost unchanged ( i=x, y). At the same time, the electron energy decreases fromE0toEf. Thus the normalized emittances have decreased as follows ǫn=γǫ=ǫn0(Ef/E0) =ǫn0/C, (1) where ǫn0,ǫnare the initial and final normalized emittances, the factor o f the emittance reduction C=E0/Ef. The method of electron beam cooling allows further reduction of the transverse emittances after dampi ng rings or guns by 1-3 orders of magnitude.1 In this paper, we have evaluated the effects of the laser-Comp ton interaction for transverse cooling using the Monte Carlo code CAIN.2The simulation calculates the effects of the nonlinear Compton scattering between the l aser photons and the electrons during a multi-cooling stage. Next, we examine th e optics for cooling with and without chromatic correction. The laser-Compton c ooling for JLC/NLC3 atE0= 2 GeV is considered in section 4. A summary of conclusion is g iven in section 5. ∗This work was supported in part by the U.S. Department of Ener gy under Contract No. DE- AC03-76SF00098. 1Table 1. Parameters of the electron beams for laser-Compton cooling. The value in the parentheses is given by Telnov’s formulas. E0(GeV) Ef(GeV) C ǫ n,x/ǫn,y(m·rad) βx/βy(mm) σz(mm) δ(%) 2 0.2 10 7 .4×10−8/2.9×10−84/4 0.5 11 (9.8) 5 1 5 3 .0×10−6/3.0×10−60.1/0.1 0.2 19 (19) Table 2. Parameters of the laser beams for laser-Compton coo ling. The value in the parentheses is given by Telnov’s formulas. E0(GeV) λL(µm) x0 A(J) ξ R L,x/RL,y(mm) σL,z(mm) 2 0.5 0.076 300 (56) 2.1 (2.2) 0.3/0.3 1.25 5 0.5 0.19 20 (4) 1.5 (1.5) 0.1/0.1 0.4 2. Laser-Electron Interaction 2.1.Laser-Electron Interaction In this section, we describe the main parameters for laser-C ompton cooling of electron beams. A laser photon of energy ωL(wavelength λL) is backward-Compton scattered by an electron beam of energy E0in the interaction point (IP). The kinematics of Compton scattering is characterized by the di mensionless parameter1 x0≡4E0ωL m2ec4= 0.019E0[GeV] λL[µm], (2) where meis electron mass. The parameters of the electron and laser be ams for laser-Compton cooling are listed in Tables 1 and 2. The param eters of the electron beam with 2 GeV are given for JLC/NLC case in section 4. The par ameters of that with 5 GeV are used for simulation in the next subsection. The wavelength of laser is assumed to be 0.5 µm. The parameters of x0with the electron energies 2 GeV and 5 GeV are 0.076 and 0.19, respectively. The required laser flush energy with ZR≪lγ≃leis1 A= 25le[mm]λL[µm] E0[GeV](C−1) [J], (3) where ZR,lγ(∼2σL,z), and le(∼2σz) are the Rayleigh length of laser, and the bunch lengths of laser and electron beams. From this formula , the parameters of A with the electron energies 2 GeV and 5 GeV are 56 J and 4 J, respe ctively. The nonlinear parameter of laser field is1 ξ2= 4.3λ2 L[µm2] le[mm]E0[GeV](C−1). (4) In this study, for the electron energies 2 GeV and 5 GeV, the pa rameters of ξare 2.2 and 1.5, respectively. The rms energy of the electron beam after Compton scattering is1 σe=1 C2/bracketleftbig σ2 e0[GeV2] + 0.7x0(1 + 0 .45ξ)(C−1)E2 0[GeV2]/bracketrightbig1/2[GeV] ,(5)where the rms energy of the initial beam is σe0and the ratio of energy spread is defined as δ=σe/Ef. If the parameter ξorx0is larger, the energy spread after Compton scattering is increasing and it is the origin of the e mittance growth in the defocusing optics, reacceleration linac, and focusing opt ics. The energy spreads δ for the electron energies 2 GeV and 5 GeV are 9.8% and 19%, resp ectively. The equilibrium emittances due to Compton scattering are1 ǫni,min=7.2×10−10βi[mm] λL[µm](i=x,y) [m·rad], (6) where βiare the beta functions at IP. From this formula we can see that small beta gives small emittance. However the large change of the beta f unctions between the magnet and the IP causes the emittance growth. Taking no acco unt of the emittance growth, for the electron energies 2 GeV and 5 GeV, the equilib rium emittances are 5.8×10−9m·rad and 1 .4×10−10m·rad, respectively. The equilibrium emittances depended on ξin the case ξ2≫1 were calculated in Ref. 1. 2.2.Simulation of Laser-Electron Interaction For the simulation of laser-electron interaction, the elec tron beam is simply assumed to be a round beam in the case of E0= 5 GeV and C= 5. Taking no account of the emittance growth of optics, the one stage for c ooling consists two parts as follows: 1. The laser-Compton interaction between the electron and l aser beams. 2. The reacceleration of electrons in linac. In the first part, we simulated the interactions by the CAIN co de.2This simulation calculates the effects of the nonlinear Compton scattering b etween the laser photons and the electrons. We assume that the laser pulse interacts w ith the electron bunch in head-on collisions. The βxandβyat the IP are fixed to be 0.1 mm. The initial energy spread of the electron beams is 1%. The energy of laser pulse is 20 J. The difference of the pulse energy between the simulation and the formula depends on the transverse sizes of the electron beams at IP. The polariz ation of the electron and laser beams are Pe=1.0 and PL=1.0 (circular polarization), respectively. When thex0andξparameters are small, the spectrum of the scattered photons does not largely depend on the polarization combination. In order to accelerate the electron beams to 5 GeV for recovery of energy in the second part, we sim ply added the energy ∆ E= 5 GeV −Eavefor reacceleration, where Eaveis the average energy of the scattered electron beams after the laser-Compton inter action. Here we define the transverse, longitudinal, and 6D emittanc es in the simulation. Thex, y-transverse emittance is ǫn,i=/radicalBig σ2 iσ2 i′−(/angbracketlefti·i′/angbracketright − /angbracketlefti/angbracketright/angbracketlefti′/angbracketright)2(i=x,y), (7) where the symbol /angbracketleft /angbracketrightmeans to take an average of all particles in a bunch.The longitudinal emittance is ǫn,l=/radicalBig σ2zσ2pz−(/angbracketleftz·pz/angbracketright − /angbracketleftz/angbracketright/angbracketleftpz/angbracketright)2/(mec). (8) The 6D emittance is defined as ǫ6N=ǫn,x·ǫn,y·ǫn,l. (9) Figure 1 shows the longitudinal distribution of the electro ns after the first laser- Compton scattering. The average energy of the electron beam s is 1.0 GeV and the energy spread δis 0.19. The longitudinal distribution seems to be a boomera ng. If we assume a short Rayleigh length of laser pulse, the energy l oss of head and tail of beams is small. The number of the scattered photons per inc oming particle and the average of the photon energy at the first stage are 40 and 96 MeV (rms energy 140 MeV), respectively. 0 1 2 E□(GeV)/c180101□□0 0.6 0.4 0.2 01.0 0.8 Number□of□electrons/c215 /c215/c215 /c215/c215 /c215/c215 /c215/c215/c215/c215/c215 /c215/c215/c215 /c215/c215/c215/c215 /c215 /c215 /c215 /c215/c215/c215/c215 /c215/c215 /c215/c215 /c215/c215/c215/c215 /c215/c215 /c215/c215 /c215 /c215/c215/c215 /c215/c215 /c215 /c215/c215 /c215 /c215/c215/c215 /c215 /c215/c215 /c215 /c215 /c215/c215/c215/c215/c215/c215 /c215 /c215/c215/c215 /c215/c215/c215 /c215/c215 /c215/c215/c215 /c215/c215/c215 /c215/c215/c215/c215 /c215 /c215/c215/c215 /c215/c215/c215/c215 /c215/c215/c215 /c215/c215/c215/c215/c215/c215 /c215 /c215/c215/c215 /c215/c215 /c215/c215/c215 /c215 /c215/c215/c215/c215 /c215 /c215/c215 /c215/c215/c215 /c215/c215/c215/c215 /c215/c215/c215 /c215/c215/c215/c215 /c215/c215 /c215/c215/c215 /c215/c215 /c215 /c215/c215/c215 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The longitudinal distribution of the electrons. (a) The energy vs.z.(b) The energy distribution of the electrons. The bin size is 40 MeV. 01 2 3 4 5678 9 Stage10-810-7 Transverse□size (m)□□( )/c115ii=x,y /c115y/c115x Fig. 2. The transverse sizes of the electron beams.01 2 3 4 5678 9 StageAngle (rad)□□( )/c113ii=x,y10-3 10-510-4/c113y/c113x Fig. 3. The angles of the electron beams.The transverse sizes of the electron beams in the multi-stag e cooling are shown in Fig. 2. During collisions with the laser photons, the tran sverse distribution of the electrons remains almost unchanged. But they decrease w hen we focus them for the next laser-Compton interaction due to the lower norm alized emittance and the fixed β-function at IP ( σi=/radicalbig βiǫn,i/γ). The angles of the electron beams in the multi-stage cooling are shown in Fig. 3. As a result of rea cceleration, the angles of the electrons decrease. They increase when we focus them f or the next laser- Compton interaction. Finally the angles attain the average of Compton scattering angle and the effect of cooling saturates. Figure 4 shows the transverse emittances of the electron bea ms in the multi- stage cooling. From Eq.(6), ǫni,min= 1.4×10−10m·rad, and the simulation presents ǫni,min= 1.2×10−9m·rad. Figure 5 shows the longitudinal emittance of the electr on beams in the multi-stage cooling. Due to the increase of the e nergy spread of the electron beams from 1% to 19%, the longitudinal emittance ra pidly increases at the first stage. After the first stage, the normalized longitudin al emittance is stable. The 6D emittance of the electron beams in the multi-stage coo ling is shown in Fig. 6. The second cooling stage has the largest reduction fo r cooling. The 8th or 9th cooling stages have small reduction for cooling. The i nitial and final 6D emittances ǫ6Nare 1.5×10−13(m·rad)3and 1.2×10−19(m·rad)3, respectively. Figure 7 shows the polarization of the electron beams in the m ulti-stage cooling. The decrease of the polarization during the first stage is abo ut 0.06. The final polarization Peafter the multi-stage cooling is 0.54. 01 2 3 4 5678 9 Stage10-5 10-910-8 10-1010-710-6 Emittance (m□rad)□□( )/c101 /c215n i,i=x,y/c101n x, /c101n y, Fig. 4. The transverse emittances of the electron beams.01 2 3 4 5678 9 StageEmittance (m□rad)/c101 /c215n l,0.1 0.09 0.08 0.07 0.06 0.05 0.010.04 0.03 0.02 Fig. 5. The longitudinal emittance of the electron beams. 3. Optics Design for Laser-Compton Cooling 3.1.Optics without chromaticity correction There are three optical devices for the laser-Compton cooli ng of electron beams as follows: 1. The focus optics to the first IP.01 2 3 4 5678 9 Stage10-1810-16 10-2010-1410-12 Emittance (m□rad)/c101 /c2156□□N3 Fig. 6. The 6D emittance of the electron beams.01 2 3 4 5678 9 StagePolarization1.0 0.8 0.6 00.4 0.2 Fig. 7. The polarization of the electron beams. 2. The defocus optics from the first IP to the reacceleration l inac. 3. The focus optics from the linac to the next IP. Figure 8 shows schematics of the laser-Compton cooling of el ectron beams. The optics 1 is focusing the electron beams from a few meters of β-function to several millimeters in order to effectively interact them with the la ser beams. The optics 2 is defocusing them from several millimeters to a few meters for reacceleration of electron beams in linac. In a multi-stage cooling system, th e optics 3 is needed for cooling in the next stage. The problem for the focus and defoc us optical devices is the energy spread of electrons and the electron beams with a l arge energy spread are necessary to minimize or correct the chromatic aberrati ons avoiding emittance growth. Linac Defocus□lens Focus□lens Focus□lensIP IP 5□GeV 1GeV 5□GeV /c100=4% /c100=1% /c100=19%Laser□Beam Next□Stage Fig. 8. Schematics of the laser-Compton cooling of electron beams. In this subsection, we discuss the optics for laser-Compton cooling without chro- matic corrections. For the focus and defocus of the beams, we use the final doublet system which is similar to that of the final focus system of the future linear col- liders.3The pole-tip field of the final quadrupole BTis limited to 1.2 T and the pole-tip radius ais greater than 3 mm. The strength of the final quadrupole is κ=BT/(aBρ)≤120/E[GeV] [m−2], (10) where B,ρ, andEare the magnetic field, the radius of curvature and the energy of the electron beams. In our case, the electron energies in the optics 1, 2, and 3 are 5.0, 1.0, and 5.0 GeV, respectively and the limit of the stren gth of the quadrupolein laser cooling is much larger than that of the final quadrupo le of the future linear colliders. Due to the low energy beams in laser cooling, the s ynchrotron radiation from quadrupoles and bends is negligible. The difference of three optical devices is the amount of the en ergy spread of the beams. In the optics 1,2, and 3, the beams have one, several te ns, and a few % energy spread. In order to minimize the chromatic aberratio ns, we need to shorten the length between the final quadrupole and the IP. In this stu dy, the length from the face of the final quadrupole to the IP, lis assumed to be 2 cm. Here we estimated the emittance growth in the optics 2, because the chromatic e ffect in the optics 2 is the most serious. Figure 9 shows the defocus optics without c hromaticity correction for laser-Compton cooling by the MAD code.4The input file is attached to Ref. 5. The parameters of the electron beam for laser-Compton cooli ng at E0= 5 GeV and C= 5 are listed in Table 3. The initial βxandβyafter laser-Compton interaction are 20 mm and 4 mm, respectively. The final βxandβyare assumed to be 2 m and 1 m, respectively. The initial and final αx(y)with no energy spread δ= 0 are 0 in this optics. The strength κof the final quadrupole for the beam energy of 1 GeV from Eq. (10) is assumed to be 120 m−2. Table 3. Parameters of the electron beam for laser-Compton c ooling at E0= 5 GeV and C= 5 for the optics design. E0(GeV) ǫn,x/ǫn,y(m·rad) βx/βy(mm) σx/σy(m) σz(mm) 5 1 .06×10−6/1.6×10−820/4 3 .3×10−5/1.8×10−70.2 In our case, the chromatic functions ξxandξyare 18 and 148, respectively. The momentum dependence of the emittances in the defocus optics without chromaticity correction is shown in Fig. 10. In the paper,6the analytical study by thin-lens approximation has been studied for the focusing system, and here the transverse emittances are calculated by a particle-tracking simulati on. The 10000 particles are tracked for the transverse and longitudinal Gaussian di stribution by the MAD code. The relative energy spread δis changed from 0 to 0.4. Due to the larger chromaticity ξy, the emittance ǫyis rapidly increasing with the energy spread δ. If we set a limit of 200% for ∆ ǫi/ǫi(i=x, y), the permissible energy spread δxand δyare 0.11 and 0.012 which mean the momentum band widths ±22% and ±2.4%, respectively. The results are not sufficient for cooling at E0= 5 GeV and C= 5, because the beams through the defocusing optics have the ene rgy spread of several tens %. On the one hand, the optics can be useful as the optics 1 and 3 with the energy spread of a few %. 3.2.Optics with chromaticity correction The optics without chromaticity correction for the optics 2 does not work as we saw in the previous subsection. In this subsection we appl y the chromaticity correction for the optics 2. The lattice for cooling is desig ned referring to the final focus system of the future linear colliders by K. Oide.7The final doublet system0.0 .25 .50 .75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 s□(m)0.02.4.6.8.10.12.14.16.18.20. /c98□□ □ 0.020.40.60.80.100.120.140.160.W/c98x /c98y Wx Wy Fig. 9. The defocus optics without chromaticity correction for laser-Compton cooling. 0 0.1 0.4 0.2 0.3 /c100Emittance (m□rad)/c101 /c215y10-8 10-9 10-1210-1110-10Emittance (m□rad)/c101 /c215x10-8 10-9 10-10 0 0.1 0.4 0.2 0.3 /c100 Fig. 10. Momentum dependence of the emittances in the defocus optics without chromaticity correction. is the same lattice as the optics before subsection. The meth od of chromaticity correction uses one family of sextupole to correct for verti cal chromaticity and moreover we added two weak sextupoles in the lattice to corre ct for horizontal chromaticity. Figure 11 shows the defocus optics with chrom aticity correction for the laser-Compton cooling. The input file is attached to Ref. 5. The total length of the lattice is about 63 m. The momentum dependence of the emittances in the defocus opt ics with chro- maticity correction is shown in Fig. 12. The 10000 particles are tracked for the transverse and longitudinal Gaussian distribution by the M AD code. The relative energy spread δis changed from 0 to 0.06 with the conservation κ2θB, where κ2 andθBare the strength of the sextupole and the angle of the bending magnet. The initial βxandβyafter laser-Compton interaction are 20 mm and 4 mm, respecti vely. The final βxandβyare assumed to be 2 m and 1 m, respectively. The initial and finalαx(y) with no energy spread δ= 0 are 0 in this optics. After the chromaticity correction, the chromaticity functions ξxandξyare 9.3 and 1.6, respectively. If we set a limit of 200% for ∆ ǫi/ǫi(i=x, y), the permissible energy spread δxand δyare 0.040 and 0.023 which mean the momentum band widths ±8% and ±4.6%, respectively. By the comparison with the results of the opti cs without chromaticity correction at a limit of 200% for ∆ ǫi/ǫi(i=x, y), the ǫyof the optics with chro- maticity correction is about two times larger than that of th e one before subsection,but the ǫxof the optics with chromaticity correction is three times sm aller than that of the one before. The results are still not sufficient for cool ing with E0= 5 GeV andC= 5. These results emphasize the need to pursue further ideas for plasma lens.8 /c100E/ p 0c = 00.0 10. 20. 30. 40. 50. 60. 70. 80. s□(m)0.0100.200.300.400.500.600. /c98(m) 0.0.0005.0010.0015.0020.0025.0030.0035.0040.0045.0050 D (m)/c98x /c98y Dx Dy . s□(m) /c100E/ p 0c = 0 .0.020.40.60.80.100.120.140.160.180.200.W 0.0.2.4.6.81.01.21.41.6 /c109(rad/ 2 /c112) Wx Wy /c109x /c109y 0.0 10. 20. 30. 40. 50. 60. 70. 80. Fig. 11. The defocus optics without chromaticity correction for laser-Compton cooling.Emittance (m□rad)/c101 /c215y10-9 10-1210-1110-10 0 0.02 0.06 0.04/c100Emittance (m□rad)/c101 /c215x10-8 10-9 10-10 0 0.02 0.06 0.04/c100 Fig. 12. Momentum dependence of the emittances in the defocus optics with chromaticity correction. 4. Laser-Compton Cooling for JLC/NLC at E0= 2 GeV 4.1.Optics For the future linear colliders, the method of laser-Compto n cooling is effective to reduce the transverse emittances after damping rings. Wh ere can it be placed? There are two possibilities for JLC/NLC9as follows: 1. After the first bunch compressor (BC1) and before the pre-l inac.E0= 2 GeV andσz= 0.5 mm. 2. After the second bunch compressor (BC2) and before the mai n linac. E0= 10 GeV and σz= 0.1 mm. Case 2 needs a large energy for recovery after Compton scatte ring and we consider case 1 in this study. The parameters of the electron and laser beams for laser- Compton cooling for JLC/NLC at E0= 2 GeV and C= 10 are listed in Tables 1 and 2. The energy of laser pulse is 300 J. The simulation resul ts of the laser-electroninteraction by the CAIN code are summarized as follows. The e nergy spread of the electron beam is 11%. The decrease of the longitudinal polar ization of the electron beam is 0.038 ( Pe= 1.0, PL= 1.0). The number of the scattered photons per incoming particle and the average of the photon energy are 20 0 and 8.9 MeV (rms energy 19 MeV), respectively. Table 4. Parameters of the defocus optics for laser-Compton cooling for JLC/NLC at E0=2 GeV andC= 10. l Length of Q1 Field of Q1 Aperture Total length 5 mm 2 cm 1.2 Tesla 0.5 mm 7.4 cm The electron energy after Compton scattering in case 2 is 0.2 GeV and the strength of the final quadrupole from Eq. (10) is 600 m−2. Table 4 lists the param- eters of the defocusing optics for laser-Compton cooling fo r JLC/NLC at E0= 2 GeV and C= 10. The final βxandβyare assumed to be 1 m and 0.25 m, respec- tively. The chromaticity functions ξxandξyare 18 and 23, respectively. Using the MAD code, the emittance growth in the defocus optics are ∆ǫdefocus n,x =ǫn,x−ǫn,x0∼1.0ǫn,x0∼7.6×10−8[m·rad], (11) ∆ǫdefocus n,y =ǫn,y−ǫn,y0∼1.6ǫn,y0∼4.6×10−8[m·rad], (12) where the normalized emittances before and after the defocu s optics are ǫn,i0and ǫn,i(i=x, y), respectively. The emittance growth in the other two focus optics are negligible. 4.2.Reacceleration Linac In the reacceleration linac, there are two major sources of t he emittance increase9 as follows: 1. The emittance growth due to the misalignment of the quadru pole magnet and the energy spread. 2. The emittance growth due to the cavity misalignment. The emittance growth due to these sources in the reaccelerat ion linac (L-band linac) are formulated by K. Yokoya9 ∆ǫlinac n,x∼3.4×10−9[m·rad]∼0.045ǫn,x0, (13) ∆ǫlinac n,y∼3.4×10−9[m·rad]∼0.12ǫn,y0. (14) The final emittance growth and the final emittance with C= 10 are ∆ǫn,x∼7.9×10−8[m·rad]∼1.1ǫn,x0⇒ǫn,x∼0.21ǫn,x0, (15) ∆ǫn,y∼4.9×10−8[m·rad]∼1.7ǫn,y0⇒ǫn,y∼0.27ǫn,y0. (16)The total reduction factor of the 6D emittance of the laser-C ompton cooling for JLC/NLC at E0= 2 GeV is about 18. The decrease of the polarization of the electron beam is 0.038 due to the laser-Compton interaction . 5. Summary We have studied the method of laser-Compton cooling of elect ron beams. The ef- fects of the laser-Compton interaction for cooling have bee n evaluated by the Monte Carlo simulation. From the simulation in the multi-stage co oling, we presented that the low emittance beams with ǫ6N= 1.2×10−19(m·rad)3can be achieved in our beam parameters. We also examined the optics with and withou t chromatic correc- tion for cooling, but the optics are not sufficient for cooling due to the large energy spread of the electron beams. The laser-Compton cooling for JLC/NLC at E0= 2 GeV and C= 10 was considered. The total reduction factor of the 6D emittance o f the laser-Compton cooling is about 18. The decrease of the polarization of the e lectron beam is 0.038 due to the laser-Compton interaction. Acknowledgments We would like to thank Y. Nosochkov, K. Oide, T. Takahashi, V. Telnov, M. Xie, and K. Yokoya for useful comments and discussions. References 1. V. Telnov, Phys. Rev. Lett. 78, 4757 (1997); ibid.80, 2747 (1998); in Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Asp ects of Beam Physics , Monterey, CA, Jan 4-9, 1998, BUDKERINP-98-33 (1998). 2. P. Chen, T. Ohgaki, A. Spitkovsky, T. Takahashi, and K. Yok oya, Nucl. Instrum. and Methods Phys. Res. A 397, 458 (1997). 3. Zeroth-Order Design Report for the Next Linear Collider, LBNL-PUB-5424, SLAC- Report-474 (1996); JLC Design Study, KEK-Report-97-1 (199 7); Conceptual Design of a 500 GeV Electron Positron Linear Collider with Integrat ed X-Ray Laser Facility, DESY-97-048, ECFA-97-182 (1997). 4. H. Grote and F.C. Iselin, The MAD Program (Methodical Acce lerator Design) Version 8.19: User’s Reference Manual, CERN-SL-90-13-AP (1996). 5. T. Ohgaki, LBNL-44380 (1999). 6. B.W. Montague and F. Ruggiero, CLIC-NOTE-37 (1987). 7. K. Oide, Nucl. Instrum. Meth. Phys. Res. A 276, 427 (1989); in Proceedings of the DPF Summer Study on High Energy Physics in the 1990’s , Snowmass, CO, Jun 27-Jul 15, 1988, SLAC-PUB-4806 (1988); in Proceedings of the 1st Workshop on the Japan Linear Collider (JLC), Tsukuba, Japan, Oct 24-25, 1989, KEK-Preprint-89-1 90 (1989). 8. P. Chen, K. Oide, A.M. Sessler, and S.S. Yu, Phys. Rev. Lett .64, 1231 (1990); in Proceedings of the Fourteenth International Conference on High Energy Accelerators , Tsukuba, Aug 22-26, 1989, SLAC-PUB-5060 (1989). 9. K. Yokoya, in Proceedings of the International Symposium on New Visions i n Laser- Beam Interactions , Tokyo, Oct 11-15, 1999.

arXiv:physics/0002021v1 [physics.ed-ph] 11 Feb 2000Averages of static electric and magnetic fields over a spheri cal region: a derivation based on the mean-value theorem Ben Yu-Kuang Hu Department of Physics, 250 Buchtel Commons, University of A kron, Akron, OH 44325-4001. (February 2, 2008) 1The electromagnetic theory of dielectric and magnetic medi a deals with macroscopic electric and magnetic fields, because the microscopic details ofthese fields are usually experimentally irrelevant (with ce rtain notable exceptions such as scanning tunneling microscopy). The macroscopic fie lds are the average of the microscopic fields over a microscopically large but macroscopically small region.1,2This averaging region is often chosen to be spherical, denot ed here as E≡3 4πR3/integraldisplay Vdr′E(r′), (1) where Vis a sphere of radius Rcentered at r(with a similar definition for B).Ris a distance which is macroscopically small but nonetheless large enough to enclose many atoms. Th e macroscopic EandBfields obtained by averaging over a sphere exhibit properties which prove useful in certa in arguments and derivations. These properties are as follows:3,4 1. if all the sources of the E-field are outside the sphere, then Eis equal to the electric field at the center of the sphere, 2. if all the sources of the E-field are inside the sphere, E=−p/(4πǫ0R3), where pis the dipole moment of the sources with respect to the center of the sphere, 3. if all the sources of the B-field are outside the sphere, then Bis equal to the magnetic field at the center of the sphere, 4. if all the sources of the B-field are inside the sphere, B=µ0m/(2πR3), where mis the magnetic dipole moment of the sources. These results can be derived in a variety of ways. For example , Griffiths3derives properties 1 and 2 using a combination of results from Coulomb’s law and Gauss’ law, an d properties 3 and 4 by writing down the B-field in terms of the vector potential in the Coulomb gauge, and expli citly evaluating angular integrals.5The purpose of this note is to describe a relatively simple derivation of al l four results, based on the well-known mean-value theorem theorem (described in most textbooks on electromagnetic th eory): if a scalar potential Φ( r) satisfies Laplace’s equation in a sphere, then the average of Φ over the surface of the spher e is equal to Φ at the center of the sphere;6,7that is, if∇2Φ = 0 in a spherical region of radius r′centered at r, then Φ(r) =1 4π/contintegraldisplay dΩ Φ/parenleftBig r+r′ˆnΩ/parenrightBig , (2) where Ω is the solid angle relative to randˆnΩis the unit vector pointing in the direction of Ω. Taking the gradient ∇of both sides of Eq. (2) with respect to r, we immediately obtain E(r) =−∇Φ(r) =1 4π/contintegraldisplay dΩ/bracketleftBig −∇Φ/parenleftBig r+r′ˆnΩ/parenrightBig/bracketrightBig =1 4π/contintegraldisplay dΩE/parenleftBig r+r′ˆnΩ/parenrightBig ; (3) that is, if ∇ ·E= 0 inside a sphere, the average of Eover the surface of the sphere is equal to the Eat the center of sphere. Eq. (3) is the basis of the derivation of the four prop erties listed above. To simplify the notation, henceforth in this note it is assumed that Vis a sphere of radius Rand is centered at the origin 0. 1. Electric field with sources outside sphere When all the static charge sources are outside V,E=−∇Φ and ∇ ·E= 0 inside Vand hence Eq. (3) is valid. We shall now see that property 1 follows quite trivially as a spe cial case of Eq. (3) [and hence of Eq. (2)]. The average of EoverVcan be written as weighted integral of the average of Eover surfaces of spheres with radius r′< Rcentered at 0, E≡3 4πR3/integraldisplay Vdr′E(r′) =3 R3/integraldisplayR 0r′2dr′/bracketleftbigg1 4π/integraldisplay dΩE(r′ˆnΩ)/bracketrightbigg . (4) Using Eq. (3) in Eq. (4) immediately yields property 1, E=3 R3/integraldisplayR 0r′2dr′E(0) =E(0). (5) 22. Electric field with sources inside sphere For clarity we first prove property 2 for a single point charge inside the sphere. The general result can be inferred from the single point charge result using the superposition principle, but for completeness we generalize the proof for a continuous charge distribution. Utilizing the vector identity2,8 /integraldisplay V∇Φdr=/contintegraldisplay SΦda, (6) where Sis the surface of V, the average of the electric field over Vcentered at 0can be written as4 E=−3 4πR3/integraldisplay V∇Φ(r)dr=−3 4πR3/integraldisplay SΦda. (7) Eq. (7) implies the average Eis determined completely by the potential on the surface S. We now use a well-known result from the method of images solut ion for the potential of a point charge next to a conducting sphere: the potential on the surface Sfor a charge qatrinside the sphere is reproduced exactly by an image charge q′=Rq/d atr′= (R2/r)ˆroutside the sphere ( ˆris a unit vector). Eq. (7) therefore implies that the E for a point charge qatris exactly equal to that of an image point charge q′at (R2/r)ˆr. But since the image charge isoutside the sphere, we can use property 1 to determine E, E=Eimage(0) =−q′ˆr 4πǫ0|r′|2=−qr 4πǫ0R3=−p 4πǫ0R3, (8) where p=qris the dipole moment for a single point charge. Generalization to continuous charge distributions – Assume the charge distribution inside the spherical volum eVis ρ(r). A volume element dVatrinside Vcontains charge dq=ρ(r)dV. The image charge element outside the sphere which gives the same average electric field as dqisdq′=ρ′(r′)dV′=dq R/r atr′= (R2/r)ˆr. As in the discrete case, the contribution of dqto the average E-field in Vis equal to the electric field at the origin due to dq′, dE=dEimage(0) =−ρ′(r′)dV′ 4πǫ0r′2ˆr=−rρ(r)dV 4πǫ0R3. (9) The averaged electric field due to all charges in the sphere Vis therefore E=/integraldisplay VdE=−1 4πǫ0R3/integraldisplay VdVrρ(r) =−p 4πǫ0R3, (10) where p=/integraltext VdVrρ(r) is the dipole moment with respect to the origin of all the cha rges in V. 3. Magnetic field with sources outside the sphere When magnetic field sources are absent in V, both ∇ ·B= 0 and ∇ ×B= 0, and hence B=−∇φMwhere ∇2φM= 0 inside the sphere. Therefore, the same derivation for pro perty 1 holds here; that is, the average of the B-field over a sphere is equal to its value at the center of the sp here. 4. Magnetic field for sources inside the sphere – Using the vector potential description of the magnetic field ,B=∇ ×A, the average over the sphere Vcan be written as4 B=3 4πR3/integraldisplay V∇ ×A=−3 4πR3/contintegraldisplay SA×da. (11) The second equality in the above equation is a vector identit y.8This equation shows that, as in the case of the E-field and the scalar potential, the average B-field over any volume Vcan be computed from Aon the surface of V. We now consider the contribution to Bof current element J(r)dVinside V. We can do this by determining the image current element outside the sphere that exactly repro duces the vector potential due to J(r)dVon the surface of the sphere. We choose the Coulomb gauge 3Ai(r) =µ0 4π/integraldisplayJi(x) |r−x|dx, (12) where idenotes spatial component. Since Aiis related to Jiin the same way as the electric potential Φ is to the charge density ρ, the method of electrostatic images can also be used here to d etermine the image current element. The proof of property 4 proceeds analogously to that of prope rty 2. For J(r)dVinside the sphere V,Aon the surface Sis reproduced by an image current element J′(r′)dV′= (R/r)J(r)dV, where r′= (R2/r)ˆr. Since the image current is outside the sphere, we can use property 3. Thus, th eJ(r)dVcontribution to the average B-field in Vis equal to the B-field at the center due to J′(r′)dV′,9 dB=dBimage(0) =−µ0 4πJ′(r′)׈r r′2dV′=µ0 4πR3r×J(r)dV. (13) The contribution for the entire current distribution in Vis therefore B=µ0 4πR3/integraldisplay VdVr×J(r) =µ0m 2πR3, (14) where m=1 2/integraltext VdVr×J(r) is magnetic moment of a current distribution in V. Finally, note that similar arguments hold for charge distri butions which are constant along the z-direction, since the potentials for these satisfy Laplace’s equation in two d imensions, and the method of images is also applicable for line charges. Acknowledgement Useful correspondence with Prof. David Griffiths is grateful ly acknowledged. 1David J. Griffiths, Introduction to Electrodynamics, 3rd Ed. (Prentice-Hall, New Jersey, 1999). 2John David Jacskon, Classical Electrodynamics, 3rd Ed. (Wi ley, New York, 1999). 3Ref. 1, pgs. 156–157 and pg. 253. 4Ref. 2, pgs. 148–149 and pgs. 187–188. 5David J. Griffiths, Solutions manual to Introduction to Elect odynamics, 3rd Ed. (Prentice-Hall, New Jersey, 1999), pg. 108-109. 6Ref. 1, pg. 114. 7Ref. 2, Problem 1.10, pg. 52. 8Ref. 1, pg. 56. 9The fact that ∇′·J′(r′) may be non-zero is irrelevant, because we know from Jefimenk o’s equations (see Ref. 1, pg. 427–429; Ref. 2, pg. 246–248) that the B-field is dependent only on J and˙Jand not ˙ ρ. Retardation effects in Jefimenko’s equations can be ignored by assuming that the currents were turned on in finitely long ago. See also Ref. 1, Problem 7.55, pg. 339. 4

arXiv:physics/0002022v1 [physics.gen-ph] 14 Feb 20001 Bulk Viscous Cosmological Model with G and Lambda Variables Through Dimensional Analysis Jos´ e Antonio Belinch´ on Grupo Inter-Universitario de An´ alisis Dimensional Dept. F ´isica ETS Arquitectura UPM Av. Juan de Herrera 4 Madrid 28040 Espa˜ na Abstract —A model with flat FRW symmetries and GandΛ,vari- able is considered in such a way that the momentum-energy ten sor that describes the model is characterized by a bulk viscosit y parame- ter. For this tensor the conservation principle is taken int o account. In this paper it is showed how to apply the dimensional method in order to solve the outlined equations in a trivial way. Keywords — Odes, AD, FRW Cosmologies, variable canstant I.Introduction. Recently several models with FRW metric, where “con- stants” Gand Λ are considered as dependent functions on timethave been studied. For these models, whose energy- momentum tensor describes a perfect fluid, it was demon- strated that G∝tα, where αrepresents a certain positive constant that depends on the state equation imposed while Λ∝t−2is independent of the state equation (see [1],[2]). More recently this type of model was generalized by Arbab (see [3]) who considers a fluid with bulk viscosity (or sec- ond viscosity in the nomenclature of Landau (see [4])). The role played by the viscosity and the consequent dissipative mechanism in cosmology has been studied by many authors (see [5]).). In the models studied by Arbab constants Gand Λ are substituted by scalar functions that depend on time t. The state equation that governs the bulk viscosity is: ξ∝ξ0ργ where γis a certain indeterminate constant for the time being γ∈[0,1]. As we shall see, this problem is already solved, but our aim is to solve it through Dimensional Analysis. We mean to point out how an adequate use of this technique let us find the solution of the outlined equations in a trivial way, even pointing out that it is not necessary to impose the condition div(Tij) = 0.The paper is organized as follows: in the second section the model is showed, expounding the equations and showing the ingredients that compose the model. The third section is devoted to revise the solution, which is reached by means of standard techniques of ODEs integration (see in special [10]), while in the forth sectio n the dimensional technique will be developed in order to solve the model. This section is divided in two subsec- tions. In the first one, titled “ pretty simple method”, we point out how a naive use of D.A. brings us to find such E-mail: jabelinchon@alehop.comsolution in a trivial way. Several cases are also studied her e by Arbab I. Arbab (see [3]), while in the other subsection a finer dimensional technique is showed. That is why we call it “ not so simple method ”. This section is based on dimensional techniques (groups and symmetries, see [8]), in order to reduce the number of variables intervening in the expounded ODEs. They are so simplified that its inte- gration is immediate. We think that the technique showed here is so powerful that it shall be proved that imposing the condition div(Tij) = 0 is not necessary to impose in order to solve the equations. II.The model. This problem was posed by Arbab (see [3]). The equa- tions of the model are: Rij−1 2gijR−Λ(t)gij=8πG(t) c4Tij (1) and it is imposed that1: div(Tij) = 0 where Λ( t) represent (stand) the cosmological “constant”. The basic ingredients of the model are: 1. The line element defined by: ds2=−c2dt2+f2(t)/bracketleftbiggdr2 1−kr2+r2/parenleftbig dθ2+ sin2θdφ2/parenrightbig/bracketrightbigg (2) we only consider here the case k= 0. 2. The momentum-energy tensor defined by: Tij= (ρ+p∗)uiuj−pgij where ρis the energy density and p∗represents pressure [ρ] = [p∗]. The effect of viscosity is seen in: p∗=p−3ξH (3) where: pis the thermostatic pressure, H= (f′/f) and ξis the viscosity coefficient that follows the law: ξ=kγργ(4) 1we shall see that this condition it is not necessary to impose it2 where kγmakes the equation be homogeneous i.e. it is a constant with dimensions and where the constant γ∈[0,1]. Andpalso verifies the next state equation: p=ωρ ω =const. (5) where ω∈[0,1] (i.e. it is a pure number) so that the momentum-energy tensor verifies the so-called energy con- ditions. The field equations are: 2f′′ f+(f′)2 f2=−8πG(t) c2p∗+c2Λ(t) (6) 3(f′)2 f2=8πG(t) c2ρ+c2Λ(t) (7) deriving (7) and simplifying with (6) it yields ρ′+ 3(ω+ 1)ρH−9kγργH2+Λ′c4 8πG+ρG′ G= 0 (8) and at the moment we consider this other equation. div(Tij) = 0 ⇔ρ′+ 3(ρ+p∗)f′ f= 0 (9) if we develop the equation (9) we get: ρ′+ 3(ω+ 1)ρH−9kγργH2= 0 (10) III.Non Dimensional Method. In this section we will mainly follow Singh et al work (see [10]). If we take the equation (8) regrouped, we get: ρ′+ 3(ω+ 1)ρH−9kγργH2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright A1=−/bracketleftbigg ρG′ G+Λ′c4 8πG/bracketrightbigg /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright A2(11) if take into account the conservation principle ρ′+ 3(ω+ 1)ρH−9kγργH2= 0 (12) then we solve this equation by solving the equation A2 in (11), in such a way that the equation to be solved is now: /bracketleftbigg ρG′ G+Λ′c4 8πG/bracketrightbigg = 0 (13) this equation is tried to be solved like this (see [10]). It is defined Λ =3βH2 c2where βis a numerical constant, (hypothesis by Arbab (see [3]) as well as by Singh et al (see [10]), condition that as we shall see, it is not neces- sary to impose in the solution through D.A.) and from the equation (7) the following relationship is obtained: 8πGρ= 3(1 −β)H2. Hence if all the equalities are re- placed in the equation (13) it yields: 2 (1−β)H′ H=ρ′ ρ(14)which is easily integrated. H=C1ρ1/dd=2 (1−β)(15) we get to the equation (12) with all these results ρ′+ 3(ω+ 1)ρH−9kγργH2= 0 we arrive to the next equation: ρ′+ 3C1(ω+ 1)ρd+1 d−9C2 1kγρdγ+2 d= 0 (16) which has got a particular solution in the case γ=d−1 obtaining: ρ(t) =1 (a0t)d/ a0=/parenleftbig 3C1(ω+ 1)−9kγC2 1/parenrightbig d−1 and obtaining from it: f(t) =C2t1 (3(ω+ 1)−3kγC1)(1−γ) This is the most developed solution reached by Singh et al (see [10]) which is slightly different from the one by Arbab (see [3]). IV.Dimensional Method. We shall explore this section two dimensional methods. The first one, probably the simplest one, has the incon- venience of having to depend on Einstein criterion(see [7] and Barenblatt [6]), while the second one is more pow- erful and more elaborated. We shall finish showing an equation obtained without having to impose the condition div(Tij) = 0. A.Simple Method. The dimensional way followed in this section is probably the most basic and simplest one. On one hand we integrate independently the equation div(Tij) = 0 ⇔ρ′+ 3(ω+ 1)ρf′ f= 0 (17) not taking into account the term 9 kγργH2,since if we cal- culate its order of magnitude we verify that is very small ≈10−40following, then, an asymptotic method (or pertur- bative) but this must be justified from a physical and/or mathematical point of view .If we integrate the equation (17) it is obtained the well-known relationship: ρ=Aωf−3(ω+1)(18) from this equation it is obtained one of the dimensional con- stants of our problem: Aω,that has different dimensions and physical meaning depending on the state equation im- posed i.e. it depends on ω. The other dimensional constant considered has been obtained from the state equation (4) i.e.ξ=kγργ, such constant kγwill also have different di- mensions depending on the value γ, in such a way that the3 problem is reduced to the following set of quantities and constants M. M= (t, c, A ω, kγ, a) where its respective dimensional equations in regard to a baseB={L, M, T, θ }are (the base Bof this type of mod- els has been calculated in [9]): [t] =T[c] =LT−1[a] =L−1MT−2θ−4 [Aω] =L2+3ωMT−2[kγ] =Lγ−1M1−γT2n−1(19) where arepresents the radiation constant and it will be take into account when we consider the thermodynamics quantities. Having done these considerations our aim is, therefore to solve this model through D.A. The Pi-theorem will bring us to obtain two πdimensionless monomials; one of them will be the obtained in the case of a perfect fluid ([2]) and the other monomial will contain information on viscosity, showing in this way that this type of models are very gener- al, reproducing the results obtained in the case for perfect fluids. Since all solutions will depend on these two mono- mials we must take into account Barenblatt criterion if we mean to reach a satisfactory final solution coincident with the one obtained theoretically (see [3] and [10]). B.Solutions through D.A. We shall calculate through D.A. i.e. by applying Pi- Theorem variation of G(t) in function on t, energy density ρ(t),the radius of Universe f(t),the temperature θ(t), the entropy S(t),the entropy density s(t) and finally the varia- tion of the cosmological “constant” Λ( t).The dimensional method brings us to (see [6] and [9]): B.1Calculation of G (t) : G=G(t, c, A ω, kγ) where the dimensional equation of G regarding to the base Bis: [G] =L3M−1T−2. Under this circumstances, the application of Pi-Theorem brings us to obtain the following dimensionless monomials:  G t c A ωkγ L3 0 1 2 + 3 ω γ −1 M−1 0 0 1 1 −γ T−2 1 −1 −2 2γ−1  π1=t1+3ωc5+3ω GAωπ2=ct1+β A(γ−1)β ω kβ γ It is observed that the first monomial ( π1) is identical to the one obtained in the paper ([2]) for perfect fluids, while the second monomial contains information on flow viscosity2. These two monomials lead us to the following solution: G∝t1+3ωc5+3ω Aω·ϕ/parenleftigg ct1+β A(γ−1)β ω kβ γ/parenrightigg (20) 2these remarks, obviusly are valid for all the solutions obta ined bellowwhere ϕrepresent an unknown function (i.e. at the moment we have obtained a “partial” solution, in order to reach a more satisfactory solution we must take into account the Barenblatt criterion) and βis: β=1 3(ω+ 1)(γ−1) B.2Calculation of energy density ρ(t) ρ=ρ(t, c, A ω, kγ) regarding to the base B,the dimen- sional equation of the energy density is: [ ρ] =L−1MT−2 ρ∝Aω (ct)3(ω+1)·ϕ/parenleftigg ct1+β A(γ−1)β ω kβ γ/parenrightigg (21) B.3Calculation of radius of Universe f(t). f=f(t, c, A ω, kγ) where its dimensional equation is: [f] =L f∝ct·ϕ/parenleftigg ct1+β A(γ−1)β ω kβ γ/parenrightigg (22) B.4Calculation of temperature θ(t). θ=θ(t, c, A ω,a, kγ) being its dimensional equation: [ θ] = θ a1 4θ∝A1 4ω (ct)3 4(1+ω)·ϕ/parenleftigg ct1+β A(γ−1)β ω kβ γ/parenrightigg (23) B.5Calculation of entropy S(t). S=S(c, Aω,a, kγ, t) where [ S] =L2MT−2θ−1. S∝/parenleftig A3 ωa(tc)3(1−3ω)/parenrightig1 4·ϕ/parenleftigg ct1+β A(γ−1)β ω kβ γ/parenrightigg (24) B.6Entropy density s(t). s=s(t, c, A ω,a, kγ) where [ S] =L−1MT−2θ−1 s∝/parenleftbig A3 ωa/parenrightbig1 4 (ct)9 4(1+ω)·ϕ/parenleftigg ct1+β A(γ−1)β ω kβ γ/parenrightigg (25) B.7Calculation of cosmological “constant” Λ(t). Λ = Λ( t, c, A ω, kγ) being its dimensional equation [Λ] = L−2 Λ∝1 c2t2·ϕ/parenleftigg ct1+β A(γ−1)β ω kβ γ/parenrightigg (26) C.Different Cases. All the following cases that we go on to study now have been studied by Arbab (see [3]) confirming “¡!” his solution ([10]). In obtaining all solutions depending on two monomials we shall try to find a solution to the problem expounded by means of the Barenblatt criterion (for more details about the method used here see [6] and [9]).4 C.1γ= 1/2andω= 1/3,Radiation predominance. As we pointed out in the introduction the only mod- els topologically equivalent to the ones of classic FRW are those that follow the law ξ∝ξ0ρ1/2i.eγ= 1/2 for its vis- cous parameter. In this case we observe a Universe with radiation predominance ω= 1/3.In order to obtain a com- plete solution we shall take into account Barenblatt crite- rion since, we have obtained the solutions depending on an unknown function ϕ. In this case the substitution of the values of ωandγleads us to: G∝t2c6 Aω·ϕ/parenleftigg ct1/2 A1/4 ωk−1/2 γ/parenrightigg To get rid of the unknown function ϕwe apply Baren- blatt criterion, for this purpose we need to know the order of magnitude of each monomial3: π1=GAω t2c6≈10−10.59π2=ct1/2 A1/4 ωk−1/2 γ≈102.6 π1= (π2)mm=logπ1 logπ2 G∝t2c6 Aω/parenleftigg ct1/2 A1/4 ωk−1/2 γ/parenrightiggm / m≈ −4 G∝c2 k2γi.e. G ∝const. as we expected in having a model with γ= 1/2.We also obtain from this point that k2 γ=c2/G.Whit regard to the rest of quantities we operate identically finding without surprise that: ρ∝t−2f∝t1/2θ∝t−1/2S∝t0 s∝t2/3Λ∝const. As we see the model shows the same behavior in the prin- cipal quantities as in the classic FRW model with radiation predominance. Let see, for example, how fhas been calculated: Follow- ing the same steps as we have seen in the case of calcula- tions of Git is observed that: f∝ct·ϕ/parenleftigg ct1/2k1/2 γ A1/4 ω/parenrightigg (27) π1=f ct≈10−2.6π2=ct1/2k1/2 γ A1/4 ω≈102.6 f∝ct/parenleftigg ct1/2k1/2 γ A1/4 ω/parenrightiggm / m =−1 f∝/parenleftigg cA1/2 ω kγ/parenrightigg1 2 t1 2∝/parenleftbiggGAω c2/parenrightbigg1 4 t1 2 This solution coincides with the one obtained for a classic FRW model with radiation predominance. In any other cases kγas well as Aωwill have other values to calculate. 3see at the end of the text the table of numerical valuesC.2γ= 1/2andω= 0.Matter predominance A model with matter predominance ω= 0 y topologi- cally equivalent to a classic FRW. In this case we find the following relationships: Regarding to Gthe solution obtained is (after replacing values γandω): G∝tc5 Aω·ϕ/parenleftigg ct1/3 A1/3 ωk−2/3 γ/parenrightigg as we are working with a model described by matter instead of considering energy density we find more convenient to consider matter density which becomes a little dimensional readjustment in Aωwhich becomes [ Aω] =Min such a way that the solution pointed out above for Gis still the following law: G∝tc3 Aω·ϕ/parenleftigg ctk2 γ Aω/parenrightigg1/3 as in the previous case we apply Barenblatt criterion which brings us to: π1=tc3 GAω= 10−1.42π2=/parenleftigg ctk2 γ Aω/parenrightigg1/3 = 10−0.47 π1= (π2)m/m≈ −3 G∝c2 k2γi.e. G∝const. In regard to the rest of quantities if we operate as before, we get: ρ∝c2 Gt2f∝(MG)1 3t2/3Λ∝const. where we have used the equality k2 γ=c2/Gand we have identified Aωwith the total mass of Universe Mi.e. The same behavior has been obtained as in a FRW with matter predominance. Let see for instance how we calculate radius f: For this quantity the obtained solution is: f∝ct·ϕ/parenleftigg ctk2 γ Aω/parenrightigg1/3 Barenblatt criterion brings us to: π1=f ct= 100.5π2=/parenleftigg ctk2 γ Aω/parenrightigg1/3 = 10−0.47 π1= (π2)m/m≈ −1f∝(MG)1 3t2/35 C.3γ= 3/4andω= 1/3.An Universe with radiation predominance: In this case, as β=−1 we find the following solutions: G∝t2c6 Aω·ϕ/parenleftbiggckγ A1/4 ω/parenrightbigg as the unknown function ϕdoes not depend on twe can state fearlessly that ϕ/parenleftbiggckγ A1/4 ω/parenrightbigg =D=const. sincec, kγas well as Aωare constant through hypothesis, in such a way that G∝D′t2 where D′=Dc6/Aω.In this case we do not need to re- sort to Barenblatt criterion in order to obtain a definitive solution. In regard to the other quantities we obtain the following behaviors: ρ∝DAω (ct)4f∝Dct a1/4θ∝DA1/4 ω ct S∝D(A3 ωa)1/4s∝D(A3 ωa)1/4 (ct)3Λ∝D(ct)−2 In short, the obtained behaviors are: ρ∝t−4f∝t θ ∝t−1 S∝const. s ∝t−3Λ∝t−2 this case follows an identical behavior to the one obtained in a model described by a perfect fluid with Gand Λ vari- ables (see [1] and [2]) showing in this way the generality that we have obtained when considering a bulk viscous flu- id. C.4γ= 2/3andω= 0An Universe with matter predominance. . In this case also β=−1, finding the following relation- ships as in the previous case: G∝tc5 Aω·ϕ/parenleftbiggckγ A1/4 ω/parenrightbigg that in the previous case leads us to: G∝Dc5t Aω where D=ϕ/parenleftig ckγ A1/4 ω/parenrightig .Simplifying in the same way, without difficulty we reach: ρ∝t−3f∝tΛ∝t−2 These two last cases are identical to the ones studied in references (see [1] and [2]).V.Not so simple method. In this section we will combine dimensional techniques with standard techniques of ODEs integration. With the dimensional method, we go on to obtain dimensionless monomials, which will be replaced in the equations. Thus, the number of variables will be reduced in such a way that the resulting equation is integrable in a trivial way. We study two cases, the first in which we consider div(Tij) = 0, while in the other, as we shall see, such hypothesis is not needed. .5Considering the condition div(Tij) = 0.. In this case we shall pay attention to the equation: ρ′+ 3(ω+ 1)ρH−9kγργH2+ρG′ G+Λ′c4 8πG= 0 taking into account the relationship div(Tij) = 0 The fol- lowing equality is brought up: ρ′+ 3(ω+ 1)ρH−9kγργH2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright A1=−/bracketleftbigg ρG′ G+Λ′c4 8πG/bracketrightbigg /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright A2 The idea is the following: By using D.A. we obtain two π−monomials, which are replaced in the equation, achiev- ing a huge simplification of it. On the other hand we in- tegrate ( A1) and ( A2),solving completely in this way the problem, this time without Barenblatt. let see. The mono- mials obtained are: π1=ρk−1 1−γ γt−1 γ−1andπ2= Λc2t2i.e. ρ=ak1 1−γ γt1 γ−1 Λ =d c2t2 where aanddare numerical constants. In a generic way the solution is of the following form: ρ=ak1 1−γ γt1 γ−1if we define b=1 1−γthenρ=akb γt−bwhere a=const. ∈Rthen ρ′=−bakb γt−b−1(paying attention only to the term ( A1) of the equation) it yields: −bakb γt−b−1+ 3(ω+ 1)akb γt−bH−9kγ/parenleftbig akb γt−b/parenrightbigγH2= 0 (28) that simplifying it is reduced to: 9a(γ−1)(f′)2−3wt−1ff′+bt−2f2= 0 (29) f′=f t/bracketleftbigg1 6aγ−1/parenleftig w±(w2−4baγ−1)1 2/parenrightig/bracketrightbigg (30) where w= (ω+ 1),if it is defined D=/bracketleftbigg1 6aγ−1/parenleftig w±(w2−4baγ−1)1 2/parenrightig/bracketrightbigg (31) then, the solution has the following form: f=lBtD(32) where lis a certain numerical constant and Bis an integra- tion constant with dimensions, that can be identified with our result by making B=Aωkγ.6 Now we shall solve the other term of the equation (the A2). the equation (/bracketleftig ρG′ G+Λ′c4 8πG/bracketrightig = 0 (13)) can be solved in a trivial way if we follow the next results. If we replace the monomials π1=ρk−1 1−γγt−1 γ−1andπ2= Λc2t2in such equation the integration of it becomes trivial: ak1 1−γγt1 γ−1/parenleftbiggG′ G/parenrightbigg −dc2 4πGt3= 0 G′=dc2 a4πkbγtb−3=⇒G(t) =gdc2 a4πkbγtb−2(33) where a, dandg∈R(they are pure numbers). We can also observe that this integral needs not be solved since a more careful analysis about the number of π−monomials that we can obtain from the equation leads us to obtain a solution of the type: G=G(kγ, c, t) which brings us to: G(t) =gk−b γc2tb−2 This method, as we have seen, is more elaborated and the solution, therefore, finer though coincident with the previ - ous one. .6Case in which div(Tij) = 0is not considered. Let see now how we can tackle this problem from the D.A. point of view, without imposing the condition div(Tij) = 0. The base Bas before, is still B={L, M, T } while the fundamental set of quantities and constants this time is M={t, c, k γ}, with these data we can obtain the following monomials from the equation ρ′+ 3(ω+ 1)ρH−9kγργH2+ρG′ G+Λ′c4 8πG= 0 (34) considering that: ρ=ak1 1−γ γt1 γ−1 Λ =d c2t2(35) these two monomials are replaced into the equation, which is quite simplified: −bakb γt−b−1+ 3(ω+ 1)akb γt−bH−9kγ/parenleftbig akb γt−b/parenrightbigγH2+ +akb γt−bG′ G−dc2 4πGt3= 0 (36) simplifying this equation, it yields: −9a(γ−1)tH2+ 3wH−bt−1+G′ G−dc2 4πakbγtb−3 G= 0 (37) that along with the field equations (6) and (7) carry us to the next set of equations. For example we note that 3H2=a8πG c2kb γt−b+d t2that we replace into the equation that we are treating, re- sulting: −bt−1+ 3w/parenleftigg a8πkb γ 3c2Gt−b+d 3t2/parenrightigg1 2 − −9a(γ−1)/parenleftigg a8πkb γ 3c2Gt−b+d 3t2/parenrightigg t+G′ G−dc2 4πakbγtb−3 G= 0 that solving it results: G=gk−b γc2tb−2(38) where g∈Rrepresents a numerical constant. We final- ly observe that as in the previous section we could have taken into account the three monomials obtained from the equation i.e. ρ=akb γt−bΛ =d c2t2G=gc2tb−2 kbγ replacing them into the equation ρ′+ 3(ω+ 1)ρH−9kγργH2+ρG′ G+Λ′c4 8πG= 0 and calculate f,arriving at the same solution obtained in the above section i.e. f=lBtD We have proved that it is not necessary to impose the con- dition div(Tij) = 0 since it is obtained, in this case, the same solution as the one obtain imposing it. VI.Conclusions. We have studied a cosmological model described by a momentum-energy tensor characterized by a fluid with bulk viscosity, in which, furthermore, we have consid- ered the constants Gand Λ as functions depending on time i.e. as variables and we have imposed the condition div(Tij) = 0. We have proved how a suitable use of Dimen- sional Analysis enables us to find the solution of such model in a “trivial” way. With the “ Pretty simple method ”, we have obtained two π−monomials, one of them is the one obtained in the case for a perfect fluid ([2]) and the other monomial contains the information about viscosity, show- ing, in this way, that this type of models is very general being able to reproduce the result obtained in the case of a perfect fluid. In order to solve the problem we have taken into account Barenblatt criterion being able to arrive to ob - tain a complete solution of the problem. Standing out that our results coincide with the solutions obtained by Arbab I. Arbab [3]. We have shown too that with the “ not so simple method ” we arrive to solve the problem without ne- cessity of impose any condition. We believe, nevertheless, that the “ simple method ” can be more effective, since, we obtain more solutions with it or more complete solutions in the sense of finding in it solutions such as Λ ∝t−2as well as Λ = const. while the “ not so simple method ” the only solution that is obtained is Λ ∝t−2, but has the drawback of depending on Barenblatt criterion i.e. we depend on the always insecure numerical data.7 G c ρ fAω kγθ t ω= 1/3-10.17 8.47 -13.379 2690.62 13.5 0.436 20.25 ω= 0 -10.17 8.47 -26.397 2654 13.5 17 TABLE I The values refer to a logarithmic scale i.e. G≈10−10.17 etc... meassured in the International System {m, kg, s }. In the case ω= 0, ρrepresents mass density while in the case ω= 1/3represents energy density. References [1]A-M. M. Abdel-Rahman. Gen. Rel. Grav. 22,655, (1990). M. S. Bermann . Gen. Rel. Grav. 23, 465, (1991). Abdus- saltar and R. G. Vishwakarma. Class. Quan. Grav. 14, 945, (1997). [2]Belinch´ on, J.A . (physics/9811017) [3]Arbab I. Arbab. Gen. Rel. Grav. 29, 61-74, (1997). [4]Landau, L.D. and Lifshitz, E. M . Fluid Mechanics (Perga- mon, London 1976). [5]Weinberg,S. Gravitation and cosmology. (Wiley, N.Y. 1972) pp. 593-594. Murphy, G.L. Phys. Rev. D12, 4231, (1973). Padmanabhan, T, Chitre, S.M. Phys. Lett. A 120, 433, (1987). Barrow, J. D. Nuclear .Phys B310 , 743. (1988). [6]Barenblatt. Scaling, self-similarity and intermediate asymp- totics. Cambridge texts in applied mathematics N 14 1996 CUP . Palacios, J. Dimensional Analysis. Macmillan 1964 London. R. Kurth. Dimensional Analysis and Group Theory in Astro- physic. Pergamon 1972. [7]Einstein, A . Ann Phys. 35, 679-694,(1911) [8]Birkhoff, G. Hydrodynamics. Princeton U.P. 1960. Cari˜ nena, J. F. et at Adv. Electr. Elec. Phys. 72, 181, (1988) [9]Belinch´ on, J.A . (physics/9811016). [10]T. Singh, A. Beesham, W.S. Mbokazi, Gen. Rel. Grav. 30, 573, (1998).

arXiv:physics/0002023v1 [physics.optics] 14 Feb 2000Interfering with Interference: a Pancharatnam Phase Polar imeter Jeremy S. Heyl Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, California 91125 A simple variation of the traditional Young’s double slit ex periment can demonstrate several subtleties of interference with polarized light, includin g Berry and Pancharatnam’s phase. Since the position of the fringes depends on the polarization state of the light at the input, the apparatus can also be used to measure the light’s polarization without a quarter-wave plate or an accurate measurement of the light’s intensity. In principle this tec hnique can be used for any wavelength of photon as long as one can effectively polarize the incoming ra diation. I. INTRODUCTION Pancharatnam [1] explored how the phase of polarized light c hanges as the light passes through cycle of polarizations. He found that the phase increases by −Ω/2, where Ω is the solid angle that the geodesic path of polariz ations subtends on the Poincar´ e sphere. If the path does not consist of great circles, an additional dynamical phase will develop. Berry [2] developed corresponding theory for general quantum sys tems and re-derived Pancharatnam’s result [3]. A series of experiments have been performed which demonstra te the Pancharatnam phase [4–11] and the related geometric phase [12–14]. De Vito and Levrero [15] have criti cized many of these experiments because they use retarders which introduce a dynamical component to the phase. Berry & K lein [16] and Hariharan et al.[17,18] have performed a series of experiments using only polarizers and beam split ters to introduce and measure the geometric phase. In this paper, I describe a simple experiment using polarize rs and a double slit to demonstrate the Pancharatnam phase and use this phase to determine the polarization state of the incoming light. If the incoming light is linearly polarized and the polarizers are also linear, the resulting phase is limited to 0 or π. However, elliptically polarized light will result in intermediate values of the phase. This e xperiment uses a double slit rather than a half silvered mirror to split the beam, eliminating a possible source of dy namic phase; furthermore, with fewer and less complex optical elements this experiment may be performed over a wid e range of photon energies. Furthermore, since the optics are simple, one can analyze the experiment in terms of Maxwell’s equations, illustrating the connection between these equations and the Pancharatnam phase. II. EXPERIMENTAL APPARATUS Fig. 1 illustrates the experiment setup ( c.f.[19–21]). RP and FP denote rotatable and fixed linear polariz ers respectively. Each fixed linear polarizer covers one of the t wo slits and is oriented perpendicular to the other. Each rotatable polarizer is oriented at forty-five degrees to the polarizers to maximize throughput. Incidentally, if the fin al polarizer is removed, the interference pattern disappears according to the Frensel-Arago law [19]. The arrangement is similar to that used by Schmitzer, Klein & Dultz [11]. Instead of using a Babinet-Soleil compensator to vary the geometric phase. The phase depends o n the input and output polarizations of interferometer. If the two polarizers are aligned, the Pancharatnam phase be tween the two paths vanishes; and if they are orthogonal, the phase is π. 1RP RPFP FP Screenlaser beam double slit FIG. 1. Optical system of the Pancharatnam phase polarimete r The setup is identical to the standard physics demonstratio n of Young’s double-slit experiment with the exception that each slit is covered with a polarizing filter; consequen tly, both the construction and analysis of the experiment are amenable as a demonstration or student laboratory exper iment. Furthermore, the light follows the same spatial trajectory, independent of the position of the polarizers a nd the geometric phase observed. III. THE POINCAR ´E SPHERE Understanding how the apparatus works for a general polariz ation is most simply achieved by tracing the polar- ization of the light through the system along the Poincar´ e s phere. The initial polarization from the laser is unknown but it is depicted in the figures as left circular. The left pan el of Fig. 2 depicts the configuration for zero geometric phase. As the laser passes through the first polarizer, its po larization is projected onto the horizontal direction. Aft er passing through the two slits, the polarization is projecte d onto two orthogonal polarizations oriented at forty-five degrees to the horizontal. Finally, the last polarizer proj ects the polarization back onto the horizontal. One can form a closed loop by following the path along one leg and returnin g along the other leg ( c.f.[11]). This closed loop does not enclose any solid angle on the sphere. FIG. 2. The path the polarization follows on the Poincar´ e sp here for the input and output polarizers aligned in parallel (left) and orthogonally (right). 2The right panel of Fig. 2 illustrates the path of the polariza tion when the two rotatable polarizers are orthogonal. The polarization is now projected onto the vertical first, fo llowed by the two diagonal polarizations and the final horizontal projection. Constructing the closed loop as des cribed earlier yields an area of 2 πand a Pancharatnam phase of −πbetween the two slits. Since a constant phase difference of πis equivalent to −π, this implementation hides the fact that the area on the sphere is oriented and consequently the geometric phase may be positive or negative. A third configuration when the input polarizer is followed by a quarter-wave plate yieldin g right-hand circular polarized input to the interferomete r illustrates this point. The direction that the loop is trave rsed determines which slit is designated by φ1such that φ1−φ2=−Ω/2. The phase difference is given by π/2 if the final polarizer lies clockwise relative to polarizer behind the first slit, and −π/2 if the final polarizer lies counterclockwise. The converse result holds for left-hand polarized light. If the axis of polarizer behind left-hand slit (as one looks t oward the screen) lies clockwise of that of the final polarizer, one obtains the result that the fringes will shif t to the left (relative to their position for the input and out put polarizations being identical) if the light is left ellipti cally polarized and to the right if it is right elliptically p olarized. This configuration automatically includes the minus sign pr esent in Pancharatnam’s definition of the geometric phase. The upper and lower panels of Fig. 3 show the fringe pattern pr oduced by the apparatus in the configuration described above for the input and output polarizations being linear an d identical and linear and orthogonal. The middle panel shows the fringes for left-hand circularly polarized input light at the input with the input polarizer removed. FIG. 3. Schematic of the interference patterns for φ1−φ2= 0, π/2, πfor quasi-monochromatic light. IV. MEASURING THE INPUT POLARIZATION If the input polarizer is removed, the apparatus can be used t o measure the polarization of the light source. The procedure requires four measurements of the fringe positio ns: two for calibration and two to determine the geometric phases. 31. Locate the positions of the fringes with the input and outp ut polarizers parallel and midway between the polarizers at the slits. 2. Remove the input polarizer and compare the position of the fringes relative to the two previous measurements. Letα/(2π) be the ratio of the offset of the fringes in step (2) relative t o step (1) to the distance between the fringes in step (1). Also, note the direction that fringes sh ift – left for left-circular polarization and right for right-circular polarization. 3. Rotate either all of the polarizers by forty-five degrees o r rotate the light source by forty-five degrees and repeat steps (1) and (2) and denote the resulting ratio by β/(2π). The left panel of Fig. 4 depicts how the polarization evolves on the Poincar´ e sphere for the two configurations. It is straightforward to calculate the input polarization using spherical trigonometry by following the right panel of Fig. 4. sindgives the fraction of circular polarization and eis the angle between the long axis of the polarization ellips e and the vertical axis. Napier’s analogies yield a=−/braceleftbigg tan−1/bracketleftbiggsin1 2(α−β) sin1 2(α+β)/bracketrightbigg + tan−1/bracketleftbiggcos1 2(α−β) cos1 2(α+β)/bracketrightbigg/bracerightbigg (1) and the law of sines gives sind= sinasinβ. (2) Combining these two results yields tane 2= tan1 2(a−d)sin/parenleftBig π 4+β 2/parenrightBig sin/parenleftBig π 4−β 2/parenrightBig (3) α α α αβ ββ β β βα α B B= =π−β A A= =π−α c c= =π/2e eb ba aC C d d FIG. 4. Measuring and calculating the input polarization If the input polarization is linear, one finds that measureme nts of the fringes can only locate the polarization vector to within forty-five degrees. However, the contrast o f the interference pattern constrains the linear polarizat ion further, as well as the fractional polarization of the input light. 4V. CONCLUSIONS A variation of Young’s double-slit experiment provides a ex cellent and simple demonstration of the Pancharatnam phase for polarized light. Furthermore, the observed phase difference between the two slits is simply related to the polarization of the incoming light. The phase difference det ermines upon which great circle of the Poincar´ e sphere the polarization lies, and by performing the measurement af ter rotating the apparatus the input polarization can be determined precisely. The distinct advantage of the experiment is the simplicity o f the optics. Only linear polarizers and a double slit are required. Since the Pancharatnam phase is achromatic, the p rocedure may be performed for the wide range of photon energies where suitable materials are available. A compani on paper discusses the implementation of the experiment in X-rays and possible applications. ACKNOWLEDGMENTS I would like to thank Jackie Hewitt, Lior Burko and Eugene Chi ang for useful discussions and to acknowledge a Lee A. DuBridge Postdoctoral Scholarship. [1] S. Pancharatnam, Proc. Indian Acad. Sci 44, 247 (1956). [2] M. V. Berry, Proc. R. Soc. Lond. A 392, 84 (1984). [3] M. V. Berry, J. Mod. Opt. 34, 1401 (1987). [4] R. Bhandari, Phys. Lett. A. 133, 1 (1988). [5] R. Bhandari and J. Samuel, Phys. Rev. Lett. 60, 1211 (1988). [6] R. Simon, H. J. Kimble, and E. C. G. Sudarshan, Phys. Rev. L ett.61, 19 (1988). [7] R. Bhandari and T. Dasgupta, Phys. Lett. A. 143, 170 (1990). [8] R. Bhandari, Phys. Lett. A. 171, 262 (1992). [9] R. Bhandari, Phys. Lett. A. 171, 267 (1992). [10] R. Bhandari, Phys. Lett. A. 180, 21 (1993). [11] H. Schmitzer, S. Klein, and W. Dultz, Phys. Rev. Lett. 71, 1530 (1993). [12] R. Y. Chiao and W. S. Wu, Phys. Rev. Lett. 57, 933 (1986). [13] A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986). [14] R. Y. Chiao et al., Phys. Rev. Lett. 60, 1214 (1988). [15] E. De Vito and A. Levrero, J. Mod. Opt. 41, 2233 (1994). [16] M. V. Berry and S. Klein, J. Mod. Opt. 43, 165 (1996). [17] P. Hariharan, H. Ramachandran, K. A. Suresh, and J. Samu el, J. Mod. Opt. 44, 707 (1997). [18] P. Hariharan, S. Mujumdar, and H. Ramachandran, J. Mod. Opt.46, 1443 (1999). [19] E. Fortin, Am. Jour. Phys. 38, 917 (1970). [20] J. L. Hunt and G. Karl, Am. Jour. Phys. 38, 1249 (1970). [21] D. Pescetti, Am. Jour. Phys. 40, 735 (1972). 5

arXiv:physics/0002024v1 [physics.optics] 15 Feb 2000Soliton electro-optic effects in paraelectrics Eugenio DelRe and Mario Tamburrini Fondazione Ugo Bordoni, Via B. Castiglione 59, 00142 Roma, I taly, and INFM, Unita’ di Roma I, Italy Aharon J. Agranat Applied Physics Department, Hebrew University of Jerusale m, Jerusalem 91904, Israel (February 2, 2008) The combination of charge separation induced by the formati on of a single photorefractive screen- ing soliton and an applied external bias field in a paraelectr ic is shown to lead to a family of useful electro-optic guiding patterns and properties. Apart from their inherent interest as peculiar prod- ucts of nonlinearity, spatial solitons hold the promise of allowing viable optical steering in bulk environments [1] [2]. Photorefractive screening solitons differ from other known manifestations of spatial self-trapping for their peculiar ease of observation and versatility [3], and re- cent experiments in photorefractive strontium-barium- niobate (SBN) and potassium-niobate (KNbO 3) have demonstrated two conceptual applications of their guid- ing properties. In the first case, a tunable directional coupler was realized making use of two independent slab- solitons [4]; in the second, self-induced phase-matching was observed to enhance second-harmonic-generation [5]. Although results suggest a means of obtaining all-optical functionality, actual implementation is hampered by the generally slow nonlinear response [6], that can be ”ac- celerated” only at the expense of stringent intensity re- quirements [7]. In contrast, non-dynamic guiding struc- tures have been observed by fixing a screening soliton [8], or in relation to the observation of spontaneous self- trapping during a structural crystal phase-transition [9] . One possible method of obtaining acceptable dynam- ics is to make directly use of the electro-optic proper- ties of the ferroelectrics involved, in combination with the internal photorefractive space charge field deposited by the soliton. Since photorefractive charge-activation is wavelength dependent, one can induce charge sep- aration in soliton-like structures at one active wave- length (typically visible), and then read the electro- optic index modulation at a different, nonphotorefrac- tive, wavelength (typically infrared) [10] [11]. For non- centrosymmetric samples (such as the above mentioned crystals) that typically host screening soliton formation , the electro-optic index of refraction modulation is pro- portional to the static crystal polarization P, and thus to the electric field (linear electro-optic effect). For these, noelectro-optic modulation effects are possible: for whatever value of external constant electric field E ext, the original soliton supporting guiding pattern remains unchanged . Incentrosymmetrics , such as photorefrac- tive potassium-lithium-tantalate-niobate (KLTN), soli-tons are supported by the quadratic electro-optic effect [12] [13] [14] [15]. In this case, the ”nonlinear” combina- tion of the internal photorefractive field with an external electric field can give rise to new and useful soliton-based electro-optic phenomena, which we here study for the first time. The basic mechanism leading to screening soliton for- mation is the following: a highly diffracting optical beam ionizes impurities hosted in the lattice of an electro-opti c crystal. An externally applied electric field makes these mobile charges drift to less illuminated regions, forming a double layer that renders the resultant electric field in the illuminated region lower. For an appropriate electro- optic sample, this leads to a self-lensing and soliton prop- agation, when beam diffraction is exactly compensated. For slab solitons, i.e. those self-trapped beams that orig- inate from a beam that linearly diffracts only in one transverse dimension (x), for a given soliton intensity full-width-half-maximum (FWHM) ∆x, a given ratio be- tween the soliton peak intensity and the (generally arti- ficial) background illumination, u2 0=Ipeak/Ib(intensity ratio), solitons form for a particular value of applied ex- ternal biasing field E. The soliton-supporting electric field E is expressed by E=(V/L)(1+I(x)/I b)−1, where V is the external applied voltage, L is the distance between the crystal electrodes (thus E=V/L), and I(x) is the soli- ton optical intensity confined in the x transverse dimen- sion [12]. This electric field, a result of a complex non- linear light-matter interaction, is present even when the generating optical field is blocked, and the sample is illu- minated with a nonphotorefractively active light. Charge separation is smeared out only by slow recombination, as- sociated with dark conductivity, characterized by consid- erably long decay times. The nonphotorefractively active illumination, although not leading to any further evolu- tion in the internal charge field, will feel the index in- homogeneity due to the quadratic electro-optic response described by the relation ∆n =-(1/2) n3geffǫ2 0(ǫr-1)2E2, where n is the crystal index of refraction, geffis the ef- fective electro-optic coefficient for a given scalar configu- ration, ǫ0is the vacuum dielectric constant, and ǫris the 1relative dielectric constant. The actual electric field in the crystal is now E=(V/L)(1+I(x)/I b)−1-(V/L)+E ext, where E ext(in general /negationslash=E) is the externally applied electric field afterthe nonlinear processes have occurred (the ”read-out” field). The index pattern induced is ∆n=−∆n0/parenleftbigg1 1 +I(x)/Ib−1 +Eext V/L/parenrightbigg2 , (1) where ∆n 0=(1/2) n3geffǫ2 0(ǫr-1)2(V/L)2. In Fig.(1) we show two families of induced index patterns associ- ated with two solitons at different saturation levels. In Fig.(1a) a 7 µm FWHM soliton at λ=514 nm wavelength (∆n0≃5.4×10−4, for n=2.45) with an intensity ratio u2 0=4, leads to three characteristic pattern regimes: for η=Eext/(V/L)≃1, the soliton supporting potential is formed. For η≃0, an antiguiding hump appears, whereas for intermediate values of η, a twin-waveguide potential forms. Analogous results can be predicted for a strongly saturated regime shown in Fig.(1b), where a 11 µm soli- ton is formed for u2 0≃22. FIG. 1. Predicted electro-optic index patterns resulting from the soliton deposited space-charge field, for u0= 2 (a) andu0= 4.7 (b). Experiments are carried out with an apparatus that is well documented in literature [13] [14]. An en- larged TEM 00Gaussian beam from a CW Argon-ion laser operating at λ=514nm, is focused be means of an f=150mm cylindrical lens onto the input facet of an 3.7(x)×4.6(y)×2.4(z)mm sample of zero-cut paraelectric KLTN, at T=20◦C (with a critical temperature T c=11 ◦C), giving rise to an approximately one-dimensional x- polarized Gaussian beam of ∆x ∼=11µm (”soliton” beam), and the entire crystal is illuminated with a second, ho- mogeneous beam (”background” beam) from the same laser, polarized along the y axis. Both the focused and the plane-wave beams copropagate along the z-direction. The constant voltage V is applied along the crystal x direction, the crystal itself being doped with Vanadium and Copper impurities, and photorefractively active at the laser wavelength. Guiding patterns can be investi- gated either by illuminating the crystal with an infrared beam (as mentioned above), or simply by using the same soliton-forming wavelength, but at a lower intensity, sinc e photorefractive temporal dynamics are proportional to beam intensity. Here we use this read-out method, and inwhat follows all read/write experiments are at λ=514nm, with I read/Iwrite∼=20. By changing the value of the ap- plied readout voltage, V ext, we can explore the optical potential described by Eq.(1), through the variable η. Beam distribution is investigated by imaging the facets of the sample onto a CCD camera by means of a second lens placed after the sample (along the z direction). FIG. 2. Soliton formation: an input 11 µm beam (a) diffracts to 24 µm in linear propagation (V=0) (b) and self-traps for V exp=1.33 kV at T=20◦C, for u0≃4.7. In Fig.(2) the observation of a single photorefractive screening soliton is shown. The 11 µm soliton is observed with an intensity ratio u2 0∼=22 at V exp=1.33 kV, an- nulling linear diffraction to 24 µm . Soliton formation takes approximately 3 min, for an I peak≃1.8 kW/m2 (Ib≃80 W/m2), measured directly before the sample, thus meaning that erasure during readout would take, at the very least, about 1 hr (i.e. longer than the duration of any one of our experiments). Had we used an IR read-out beam, decay would be halted indefinetly. Given the sam- ple g eff=0.12m4C−2,ǫr≃9000, ∆ n0≃6.9×10−4, the expected value for soliton formation would be V th≃1.27 kV. FIG. 3. Output light distribution of the read-out beam. Forη=0-0.3 the beam is scattered. For η=0.45 the twin beam structure forms, whereas for η=1 the original guiding pattern emerges. In Fig.(3) we show the same region of the crystal in- vested by the less intense (but otherwise identical to the soliton generating) ”read” beam at various values of η. Forη=1 the output beam is identical to the soliton (apart 2from the actual intensity). For low values of η(η <0.4) the index pattern given by Eq.(1) is antiguiding, and the output beam is scattered and split into two diffracting beams (beam ”bursting”, see Fig.(1b)). As ηis increased, the defocusing is weakened and for η≈0.45 the sam- ple gives rise to a beam-splitting on the twin-waveguide structure formed by the two-hump potential, as shown in Fig.(1). The distance between the two beams is ≈20µm. As opposed to previous defocusing, in this case light is exciting a guided mode. FIG. 4. Electro-optic switching. The output light distribu - tion of the read beam (a) for η=0.45, the side-guided beam, when the crystal is shifted sideways of 10 µm (b), the output in the same condition, but with η=0.8 (c). Next we shift the crystal with respect to the optical beam in the x direction, so as to launch it directly into one of the twin-guides for intermediate values of η. For anη= 0.45, shifting the crystal by 10 µm, the beam is guided by the side hump, as shown in Fig.(4b). In this forward guiding condition, we change ηtoη=0.8. The potential commutes from a double-hump twin-waveguide to a single guiding pattern (see Fig.1). The optical beam is redirected as shown in Fig.(4c). It is therefore possible to realize, by means of the formation of a single photorefractive centrosymmetric screening soliton, three qualitatively different optical c ir- cuits: a single waveguide, a double waveguide beam- splitter, and an antiguiding beam-stopper. If the crys- tal is shifted so as to launch the guided beam into one of the twin-guides, it is possible to deviate the beam, maintaining its strong confinement, realizing an electro- optic switch. Had we used a longer sample, launching the beam in a twin-waveguide leads to a tunable directional coupler, as shown in Fig.(5). The observed phenomena represent an important step in the achievement of viable soliton based components in two major aspects. The first is that the observed phe- nomenology occurs with the formation of a single soliton, that is only used to deposit a pattern of charge displace- ment (a peculiar volume hologram), whereas switching from one regime to the other occurs only through the change of the applied electric field. Thus switching dy- namics are only limited by capacity charging times, as all other electro-optic devices. Secondly, whereas screen - ing soliton formation requires a constant applied external field, during read-out, the use of independent electrodes can allow the formation of composite circuitry in cascade, all from a single soliton. FIG. 5. Predicted evolution of an ≈7µm beam: top view of read-out in an 8mm sample for η=0.2 (beam deflection and diffraction) (left), one beat directional coupling for η=0.4 from right hump to left hump (center), and mode beating for η=0.8 (≈2mm mode beat). The work of E.D. and M.T. was carried out in the framework of an agreement between Fondazione Ugo Bor- doni and the Italian Communications Administration. Research carried out by A.J.A. is supported by a grant from the Ministry of Science of the State of Israel. [1] G.I. Stegeman and M. Segev, Science 286, 1518 (1999). [2] M. Segev and M. Stegeman, Phys. Today 51, 42 (1998). [3] B. Crosignani, P. Di Porto, M. Segev, G. Salamo, and A. Yariv, Riv. Nuovo Cimento 21, 1 (6) (1998). [4] S. Lan, E. DelRe, Z. Chen, M. Shih, and M. Segev, Opt. Lett.24, 475 (1999). [5] S. Lan, M. Shih, G. Mizell, J. A. Giordmaine, Z. Chen, C. Anastassiou, J. Martin, and M. Segev, Opt. Lett. 24, 1145 (1999). [6] L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The physics and applications of photorefractive materials (Clarendon Press, Oxford 1996). [7] K. Kos, G. Salamo, and M. Segev, Opt. Lett. 23, 1001 (1998). [8] M. Klotz, H. Meng, G. J. Salamo, M. Segev, and S. R. Montgomery, Opt. Lett. 24, 77 (1999). [9] E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, Phys. Rev. Lett. 83, 1954 (1999). [10] M. Shih, Z. Chen, M. Mitchell, M. Segev, H. Lee, R. S.Feigelson, and J. P. Wilde, J. Opt. Soc. Am. B 14, 3091 (1997). [11] For linear schemes based on screening see A. Bekker, A. Ped’el, N. K. Berger, M. Horowitz, and B. Fisher , Appl. Phys. Lett. 72, 3121 (1998) and Ph. Dittrich, G. Mon- temezzani, P. Bernasconi, and P. Gunter, Opt. Lett. 24, 1508 (1999). [12] M. Segev and A. J. Agranat, Opt. Lett. 22, 1299 (1997). [13] E. DelRe, B. Crosignani, M. Tamburrini, M. Segev, M. Mitchell, E. Refaeli, and A. J. Agranat, Opt. Lett. 23, 421 (1998). [14] E. DelRe, M. Tamburrini, M. Segev, E. Refaeli, and A. J. Agranat, Appl. Phys. Lett. 73, 16 (1998). [15] A.J. Agranat, R. Hofmeister, and A. Yariv, Opt. Lett. 17, 713 (1992). 3

arXiv:physics/0002025v1 [physics.ed-ph] 15 Feb 2000The oscillations in the lossy medium Korotchenko K.B. Chair of Theoretical and Experimental Physics, Tomsk Polytechnical University, Rossia Abstract The object of the work : to explore dependence mass point oscillatory motion parameters in the following cases: - without resistance (free oscillations); - the resistance force is proportional to the velocity vecto r; - the resistance force is proportional to the velocity squar ed. Used equipment : the work have doing on a personal computer. The os- cillatory motion simulation is carried out by the numerical solution of system of differential equations. This equations describe a motion of a particle under an elastic force action and exterior forces ( resistance force) with initial values and parameters being entered during the dialogue with the computer. 1 The theoretical part Let’s begin from the definition : If some physical quantity Funder specified physical conditions is described periodic oralmost-periodic function of time one can say that this physical quantity is in oscillatory pro cess or in oscilla- tions. As is known, a function F(t) is called periodic if F(t) =F(t+T).1At the oscillatory process the constant T= 2π/ωis called an oscillation period and the constant ωis called an oscillation frequency (circular or cyclic) . Obviously Tis a time interval by means of that the values of function F(t) are repeated. If the physical quantity is in oscillations described by the harmonic function of time (i.e. function sin( ωt) or cos( ωt)) the oscillations is called harmonic . Among all oscillatory processes the special interest is rep resented those which the man can observe directly without any devices. The most kn own oscillatory process having so remarkable property is the oscillatory motion . According to this, the oscillatory motion of a mass point we will call any its motion that the allphysical quantities describing motion are periodic (or almost-periodic ) functions of time. The major physical values describing a motion of a mass point are: - the radius vector of a particle /vector r(t), i.e. its coordinates (we shall remind the equation of a form /vector r=/vector r(t) is called the motion equation (orlaw); 1The definition almost-periodic functions will be introduce d later. 1- and the vector of the particle acceleration /vector a(t). If we take into account that the vectors of velocity and accel eration are defined uniquely by the radius vector /vector r(t) of a mass point, it is possible to formulate the following definition: any motion of a mass point at which the radius vector of a particle is a periodic (or almost periodic) function of time is called the oscillatory motion 1.1 Free simple harmonic motions The elementary oscillatory motion of a mass point is the harmonic oscillatory motion. Thus, according to the definition of harmonic oscill ations, we shall called by free simple harmonic motion such oscillatory motion, at which the radius vector of a particle is harmonic function of time. It means, the equation (the law) motion of a mass point that is in a free harmonic osci llatory motion, has a form /vector r(t) =/vector rosin(ωt+ϕo). (1) In eq.(1) the constant Φ = ωt+ϕois called by a phase of the oscillatory motion and its value at t= 0, i.e. ϕo, is called by an epoch angle accordingly. The constant /vector rois called by an amplitude of the oscillatory motion. From the equation (1) is obvious that the amplitude is the maximal val ue of radius vector the achievable at those point in time when sin( ωt+ϕo) = 1. Let’s note one important characteristic of the oscillatory motion described by the equation (1). The vector /vector rois aconstant vector , i.e. does not change neither in magnitude nor in the direction. Therefore the vec tor/vector r(t) can change onlyin magnitude (at the expense of function sin( ...)), but remains parallel to thesame line. It means that the harmonic oscillatory motion always has only onedegree of freedom . In other words, one coordinate is enough for describing of a harmonic oscillatory motion . For example, coordinates measured along an axisOX. So the vector equation (1) can always be replaced by one equa tion in the coordinate form x(t) =xosin(ωt+ϕo), (2) where xo=|/vector ro|is the module of the vector /vector ro. It is easy to see that the equation (1) is the solution of the differential equa- tion d2/vector r dt2+ω2/vector r= 0. (3) For this reason the differential equation (3) is called by the equation of free simple harmonic motions . So, one can say that free harmonic oscillatory motion of a mass point is any motion described by the equation of free simple harmonic motions (eq.(3)) Classical example of a free harmonic oscillatory motion is t he particle motion with the mass mdue to action of quasi-elastic force (i.e. simulative elast ic force) 2/vectorF=−k/vector r, where kis stiffness coefficient. To be convinced of it we shall describ e for such a mass point the dynamical equation (i.e. Newton’s s econd law) m/vector a=−k/vector r. (4) Taking into account that the acceleration is a second-order derivative of the particle radius vector, we shall obtain d2/vector r dt2+k m/vector r= 0. (5) Comparing the obtained equation with the equation of free si mple harmonic motions (3), we can see that the motion a mass point due to acti on of quasi- elastic force is really a free harmonic oscillatory motion. And the oscillation cyclic frequency of a mass point is equal ω=/radicalbigg k m. (6) 1.2 Damped oscillations In the previous section we have considered a free harmonic mo tion and were convinced that due to action of onlyelastic force the mass point makes just such motion. Let’s consider now motion a mass point due to action of quasi- elastic forces /vectorF=−k/vector rin medium under the action of resistance forces . Let, for example, the resistance force is proportional to a vector of the particle velocity /vectorFc=−b/vector v, where bis the resistance coefficient. Then the dynamical law (Newton ’s second law) for such the mass point will have a form m/vector a=/vectorF+/vectorFc=−k/vector r−b/vector v. (7) Taking into account that the velocity is a first-order deriva tive and that the acceleration is a second-order derivative of the particle r adius vector, we shall obtain d2/vector r dt2+b md/vector r dt+k m/vector r= 0. (8) It is easy to be convinced that the obtained equation coincid es with the equation of free simple harmonic motions only at absence of the resist ance forces (i.e. at b= 0 ). The solution of the equation (8) varies from the solutio n of the equation (3) as well. The eq.(3) is the equation of free simple harmoni c motions. So, the common solution of the equation (8) will have a form /vector r(t) =/vector roe−βtsin(ωt+ϕo), (9) where the following notation for parameters of an oscillatory motion de- scribed by the equation (8) are conventional damping factor −β=b 2m oscillation cyclic frequency of the free harmonic os- cillatory motion (i.e. at absence of the resistance forces) −ωo=/radicalbigg k m(10) oscillation cyclic frequency of the studied harmonic oscillatory motion −ω=/radicalbig ωo−β2 3Let’s note that in these notation the equation (8) will look l ike d2/vector r dt2+ 2βd/vector r dt+ωo/vector r= 0. (11) As well as in the case of free simple harmonic motions the osci llatory motion described by the equation (9) has only one degree of freedom . Hence, if to set the direction of constant vector /vector roparallelly to axis OXof a cartesian frame, the eq.(9) will have a form x(t) =xoe−βtsin(ωt+ϕo), (12) where, as well as in the equation (2), x is the length of a vecto r/vector ro. In fig.1 the qualitative view of the solution (12) is presente d. This figure demon- strate that the studied oscillatory motion represents osci llations with amplitude decreasing in time by exponential law (i.e. described by the function e−βt). Just for this reason an oscillatory motion described by the equat ion (11), named as the damped oscillatory motion . Accordingly, eq.(11) named as the equa- tion of damped oscillations . So the damped oscillatory motion of a mass point is any motion described by the equation of damped oscillations (i.e. eq.(11)) Let’s consider more in detail properties of the damped oscil latory motion. First of all it is obvious that in contrast to the free harmonic osci llatory motion the radius vector of the mass point in damped oscillations (i.e. expression (9) or (12)) is not periodic function of time /vector r(t)/negationslash=/vector r(t+T). Thus damped oscillations are not harmonic oscillations . According to the definition by H. Bohr (Danish mathematician ) the function f(t) satisfying the requirement |f(t+T)−f(t)|< ǫ, (13) where ǫis some positive number is named an almost-periodic function. Ac- cordingly, Tis named an almost-period such function. And the mean value of analmost-periodic function is always limitary lim T→∞1 TT/integraldisplay 0f(t)dt <∞. (14) It is easy to be convinced that for x(t) from expression (12) lim T→∞1 TT/integraldisplay 0x(t)dt= 0. (15) Moreover, it always is possible to select such positive numb erǫthat the absolute value of the difference |x(t+T)−x(t)|(where T= 2π/ω) will be less than this number. So the requirement (13) will be satisfied. 4Figure 1: Damped oscillation Figure 2: Aperiodic oscillati on Hence radius vector of the mass point making thedamped oscillations is an almost-periodic function with almost-period T. Let’s remind that according to (10) the damped oscillation c yclic frequency ωof mass point is equal to ω=/radicalbig ωo2−β2. (16) Obviously the quantity ωhas the meaning of oscillation frequency only in the caseωo2< β2. At ωo2> β2theωbecomes imaginary and, accordingly, the trigonometrical function sin( ωt) is transformed to the hyperbolic function sh(ωt). In this case the solution of the damped oscillations equat ion (11) be- comes /vector r(t) =/vector roe−βtsh(ωt+ϕo), (17) or in the coordinate notation x(t) =xoe−βtsh(ωt+ϕo), (18) Such a solution is neither aperiodic function noanalmost-periodic function . And, therefore, the motion described by the equation of damp ed oscillations at ωo2> β2isnotan oscillatory motion. This process is named as aperiodic oscillations . The diagram of a function x(t) for an aperiodic process (i.e. described by eq.(18) at ϕo= 0) is presented on fig.2. 2 The practices for simulation of physical pro- cesses Before simulation initiation of physical processes it is necessary to familiar- izewithblanket rules of operation with the digital computer and sim ple set of usual activities used at operation with the Borland software menu. In this laboratory work there is an opportunity to get the help information on its problem (and another) at any moment not quitting the program. To obtain the help information it is necessary to press the ke yF1. The set of practices performed by the student at the study of o scillatory motions are determined by the teacher and can vary over a wide range. 5Let’s consider practices the realization of which is necessary for understand- ing of features of the mass point oscillatory motion at prese nce (and absence) of resistance forces. According to the object of the laboratory work there should be two such practices. 2.1 Free simple harmonic motions In this practice state problem to study an oscillatory motio n of a mass point at absence of resistance force. Namely: - to make sure that a trajectory of a mass point is the harmonic function; - to find out how the mass point trajectory varies by change of t he following parameters: - particle mass mand stiffness coefficient kin expression for elastic force (eq. (4)) - initial kinematic parameters of a motion: the mass point co ordi- nates of the origin x(0) = xosin(ϕo) and its initial velocity v(0) = xoωcos(ϕo) (they are those parameters, you can change by changing value of the oscillations epoch angle ϕo); The practice consists of the following items: 2.1.1 After you entered in the menu and selected necessary laborat ory work (i.e. ”The oscillatory motion”) the title page of this work arises. Press Enter then the mainmenu with titles of all practices will arise. By keys ↑and ↓it is necessary to select practice ”Simple harmonic Motions ” and press Enter . 2.1.2 You will pass in the firstdialog box ”Parameters of the system”. In this box you must set the particle mass and stiffness coefficient in SI un its2and write down those values to table 1. Let’s note that the ending of input in all dialog boxes is poss ible by two paths: - by pressing the key Enter ; - by activation of the dialog box button Ok(with the help of the device Mouse ) 2.1.3 In the following dialog box (according to its title ”Epoch an gle”) you should choose an epoch angle ϕoof a mass point oscillatory motion2. Then write down this value to table 1 and press Enter . 2In the line of the context-sensitive help (bottom line of the display) range of values is indicated within the bounds of all parameters numerical v alues, you can change. 62.1.4 You will see the diagram representing a trajectory of a free h armonic oscillatory motion of a mass point with parameters chosen by you3. 2.1.5 You will pass at the next dialog box ”Change of mass”. Enter an other value of the particle mass (in comparison with the value entered in item 2.1.2 i.e. in the dialog box ”Parameters of the system” ). You will see twodiagrams corresponding to different values of the particle mass with u nchangeable others parameters. By pressing any key (according to the message in the bottom line) you will return to the same dialog box again. Iterate the desc ribed activities for one more value of mass. As a result you will see three diagrams corresponding tothreevalues of the particle mass is in a simple harmonic motions3. 2.1.6 You will pass at the next dialog box ”Change of K”. Enter anoth er value of the stiffness coefficient k(in comparison with value entered in item 2.1.2 i.e. in the dialog box ”Parameters of the system” ). You will see twodiagrams corre- sponding to different values of the stiffness coefficient with u nchangeable others parameters. By pressing any key (according to the message in the bottom line) you will return to the same dialog box again. Iterate the desc ribed activities for one more stiffness coefficient. In result you will see three diagrams corre- sponding to threevalues of the stiffness coefficient of the quasi-elastic force (by due to action of this force the mass point is in a simple harmon ic motions)3. 2.1.7 You will pass at the next dialog box ”Change of epoch angle”. E nter another value of the epoch angle (in comparison with value entered in item 2.1.3 i.e. in the dialog box ”Epoch angle” ). You will see twodiagrams corresponding to different values of the epoch angle with unchangeable others parameters. By pressing any key (according to the message in the bottom line ) you will return to the same dialog box again. Iterate the described activiti es for one more value of epoch angle. In result you will see three diagrams corresponding to three values of the epoch angle of the simple harmonic motions3. So the first practice is ended and you will be returned to the main menu. Let’s note that after you exit out of the first practice you can notenter there once again . Therefore, if you do not accept results of this practice and you want to iterate it you should start the program once again. 2.2 Damped oscillations In this practice state problem to study an oscillatory motio n of a mass point with the resistance force is proportional to a vector of velo city. Namely: 3If the obtained diagrams satisfy the object of the practice ( in your opinion) then sketch these diagrams in yours writing-book and press any ke y (according to the message in the bottom line). 7- to make sure that the trajectory of a mass point is the non har monic function, but almost-periodic function; - to find out how the mass point trajectory varies by change of t he following parameters: - particle mass m, stiffness coefficient kin expression for the elastic force (eq. (4)) and the resistance coefficient bfor the resistance force (eq. (7)) - initial kinematic parameters of a motion: the mass point co ordi- nates of the origin x(0) = xosin(ϕo) and its initial velocity v(0) = xoωcos(ϕo) (they are those parameters you can change by changing value of the oscillations epoch angle ϕo); The practice consists of the following items: 2.2.1 After you exited out of the first practice you will see the main menu with titles of all practices again. By keys ↑and ↓it is necessary to select the practice ”Damped oscillations” and press Enter . 2.2.2 You will pass in the firstdialog box ”Parameters of the system”. In this box you must set the particle mass, stiffness coefficient and resis tance coefficient in SI units2and write down those values to table 2. 2.2.3 In the following dialog box (according to its title ”Epoch an gle”) you should choose an epoch angle ϕoof mass point oscillatory motion2. Then write down this value to table 2 and press Enter . 2.2.4 You will see the diagram representing a trajectory of a dampe d oscillatory mo- tion of a mass point with parameters chosen by you3. 2.2.5 You will pass at the next dialog box ”Change of the parameters ”. Here you can change values of two parameters: the particle mass mand resistance coefficient b. In contrast to the first practice, you can have this box as much as long . Because after each new diagram (for the next pair of parameters m and b ) you will be returned here . However, as well as in the first practice, at the display draw no more three diagrams. Therefore we recommend to act as follows: - at first draw three diagrams with different values of the particle mass m and the constant resistance coefficient b3; 8- then draw three diagrams with different values of the resistance coeffi- cientband the constant particle mass m3; - then select the such underload resistance coefficient bmin(with the con- stant particle mass m) for which the particle motion will become aperiodic; write down values mandbminto table 2; - iterate operations of the previous item for another two values of the particle mass m; - then enter pairwise obtained values of mass mand coefficient bmin(from table 2) so to obtain all three aperiodic motions on one picture (i.e. in one frame) and sketch these diagrams in your writing-book; So the second practice is ended. In order to return to the mainmenu (if you have the dialog box ”Change of parameters”) is necessary to p ress the key Esc (or make active the dialog box button Exit by the device Mouse ). 3 Return4 3.1 Contents of the return The return should include the following items: 1.Object of work. 2.Summary theoretical part. 3.Practices: -Free simple harmonic motions. T A B L E 1 Diagrams x=x(t) with different mfor all three trajectories in one frame. Diagrams x=x(t) with different kfor all three trajectories in one frame. Diagrams x=x(t) with different ϕofor all three trajectories in one frame. -Damped oscillations. T A B L E 2 Diagrams x=x(t) with different m for all three trajectories in one frame. Diagrams x=x(t) with different b for all three trajectories in one frame. Diagrams x=x(t) with different bminfor all three trajectories in one frame. 4Title page of the return at the laboratory work on physical processes si mulation one should draw up on the same rules that the title page of the r eturn at the laboratory work is done in chair T&EPh experimental laboratories. 94.Conclusion 3.2 Design of the tables. The tables used in the return should be designed by following ways. T A B L E 1 Change mChange kChange ϕo m, k, ϕo, T A B L E 2 Change mChange bChange bmin m, b, bmin, k, ϕo, 4 Conclusion Let’s mark, that this paper is written on the basis of the prev ious works [1] car- ried out on chair T&EPh under the author leadership (or direc t participation). References [1] Korotchenko K.B., Sivov U.A. [2] 10

arXiv:physics/0002026v1 [physics.atom-ph] 15 Feb 2000Precision Measurement of the Lifetime of the 3 d2D5/2state in40Ca+ P. A. Barton, C. J. S. Donald, D. M. Lucas, D. A. Stevens, A. M. S teane, D. N. Stacey Centre for Quantum Computation, Department of Atomic and La ser Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, England. 10 February, 2000 Abstract We report a measurement of the lifetime of the 3 d2D5/2 metastable level in40Ca+, using quantum jumps of a sin- gle cold calcium ion in a linear Paul trap. The 4 s2S1/2– 3d2D5/2transition is significant for single-ion optical fre- quency standards, astrophysical references, and tests of atomic structure calculations. We obtain τ= 1.168± 0.007 s from observation of nearly 64 ,000 quantum jumps during ∼32 hours. Our result is more precise and sig- nificantly larger than previous measurements. Experi- ments carried out to quantity systematic effects included a study of a previously unremarked source of systematic error, namely excitation by the broad background of ra- diation emitted by a semiconductor diode laser. Com- bining our result with atomic structure calculations yield s 1.20±0.01 s for the lifetime of 3 d2D3/2. We also use quantum jump observations to demonstrate photon anti- bunching, and to estimate background pressure and heat- ing rates in the ion trap. 1 Introduction In this paper we present a measurement of the natural lifetime τof the 3 d2D5/2metastable level in singly-ionised calcium, using quantum jumps of a single cold calcium ion in a linear Paul trap. The 3 D5/2level is of interest as the source of an optical frequency standard at 729 nm with 1 /τ∼1 Hz natural linewidth [1], as a means of testing atomic structure calculations [2, 3, 4, 5, 6, 7, 8], and as a diagnostic in astrophysics [9, 3]. In addition, our experiments offer insights into the diagnostics on the performance of the Paul trap, and highlight the need to take into account the spectral properties of semiconductor diode laser emission when such devices are used in atomic physics experiments. Our result is τ= 1.168±0.007 s, and is shown together with other recent measurements and theoretical predic- tions in figure 1. Our result is higher (a longer lifetime) than all the previous measurements, and differs by sev- eral standard deviations from most of them. It is possible that this discrepancy is at least partly due to a previously unrecognised source of systematic error, namely the pres-ence of light of wavelength in the vicinity of 854 nm in the beam produced by a semiconductor diode laser emitting predominantly at 866 nm. This is discussed in section 4.1. Precise knowledge of atomic structure for atoms or ions with a single electron outside closed shells is currently in demand for the analysis of atomic physics tests of electro- weak theory, especially measurements of parity violation in Cs [10]. There has been a long-standing discrepancy at the 2% level between measured and theoretically predicted rates for electric dipole transitions [11]. Up until now mea - surements of the metastable lifetimes (electric quadrupol e transition rates) in Ca+have not been sufficiently precise to provide an independent test of ab initio calculations at this level of precision. Our measurement precision is 0.6%. Furthermore the electric quadrupole transition rate is harder to calculate accurately (recent calculated value s have a 15% spread), so our result is of particular interest to atomic structure theory. The paper is organized as follows. First we discuss, in section 2, the central features of our experimental method to measure τ. Details of the apparatus are provided in section 3. This is a completely new apparatus which has not been described elsewhere, so we give a reasonably full description. Section 4 presents our study of systematic effects in the experiment. These include collisions with the background gas, off-resonant excitation of the 854 nm 3 D5/2–4P3/2 transition and of the 850 nm 3 D3/2–4P1/2transition, heat- ing of the trapped ion, and noise in the fluorescence sig- nal. Section 5 contains a clean demonstration of photon anti-bunching, using the random telegraph method, and section 6 presents the final accurate measurements of τ. 2 Experimental method The experimental method was identical, in principle, to that adopted by Block et al. [12]; preliminary experi- ments were carried out somewhat differently (see section 5). A single ion of40Ca+is trapped and laser-cooled to around 1 mK. The transitions of interest are shown in figure 2. Laser beams at 397 nm and 866 nm continu- ously illuminate the ion, and the fluorescence at 397 nm is detected by a photomultiplier. The photon count signal 1is accumulated for periods of duration tb= 10.01 ms (of which 2.002 ms is dead time), and logged. In our studies of systematic effects, and for our demonstration of photon antibunching, tbwas set at 22 .022 ms. A laser at 850 nm drives the 3 D3/2–4P3/2transition. The most probable de- cay route from 4 P3/2is to the 4 S1/2ground state; alter- natively, the ion can return to 3 D3/2. However, about 1 decay in 18 occurs to 3 D5/2, the metastable “shelving” level of interest. At this point the fluorescence abruptly disappears and the observed photon count signal falls to a background level. A shutter on the 850 nm laser beam remains open for 100 ms before it is closed, which gives ample time for shelving. Between 5 and 10 ms after the shutter has closed, we begin one “observation”, i.e., we start to record the photomultiplier count signal (see figure 3a). We keep observing the photon count, in the 10 ms bins, until it abruptly increases to a level above a thresh- old. This is set between the background level and the level observed when the ion fluoresces continuously. The signature for the end of a dark period is taken to be ten consecutive bins above threshold. We record the number of 10 ms bins in the observed ‘dark’ period. The 100 ms period of fluoresence also serves to allow the 397 nm and 866 nm lasers to cool the ion. After this we re-open the shutter on the 850 nm laser. This process is repeated for long periods of time (1 to 8 hours), the laser intensities being also monitored and the frequencies servo-controlled . Subsequent analysis of the large collection of dark times consists primarily of gathering them into a histogram, and fitting the expected exponential distribution, in order to derive the decay rate from the shelved state (see figure 4). Note that we do not measure the length of the dark period from when the ion is first shelved. This is not necessary, since the probability of decay is independent of how long the ion has been in the metastable state. This gives us time to block the 850 nm light, in order to prevent both off-resonant excitation of the ion by this light, and the possibility of missed quantum jumps should this light rapidly (in less than 5 ms) re-shelve a decayed ion. The data from a given run were analysed as follows. The raw data consist of a series of counts indicating the aver- age fluorescence level in each bin of duration tb. A thresh- old is set, typically at (2 Sdark+Sbright)/3, where Sdarkis the mean count observed during dark periods, and Sbright the mean count observed during fluorescing periods. This setting is chosen because bright periods have more noise than dark periods. The number of consecutive bins below threshold is a single dark-time measurement xi. The xi are expected to be distributed according to the exponen- tial decay law, with Poissonian statistics describing the departures from the mean. It is appropriate to use a Pois- sonian fitting method, rather than least squares, because of the small numbers involved in part of the distribution (at large t). Ifn(ti) is the number of xiequal to ti/tbthenthe cost function is −ln/bracketleftBiggm/productdisplay i=0P(n(ti))/bracketrightBigg =m/summationdisplay i=0/bracketleftbig Ae−γti+ ln (n(ti)!) −n(ti)lnA+n(ti)γti] (1) where mis the number of bins, and Aandγare two fitted parameters (obtained by minimising the cost function); they are the amplitude and decay rate in the assumed ex- ponential decay Aexp(−γt) of population of D5/2. The residuals shown in figure 4 indicate that our data are well fitted by an exponential function. Only dark times of du- ration less than 5 s are included in the fit, since our data collection procedure misses some dark times longer than this. We found that the statistical error in the fitted pa- rameters was consistent with the expected√ N/N value, where Nis the number of xiin the whole data set. 3 Apparatus We use a linear radio-frequency (r.f.) Paul trap, combined with an all-diode laser system, to isolate and cool a single ion of40Ca+. The electrodes are made from stainless steel rods and mounted on two supports made from machinable ceramic (Macor); an end view is shown in figure 5. The radial r.f. electrodes, of diameter 1.2mm, are centred at the corners of a square of side 2 .6mm. The d.c. endcap electrodes, of diameter 1.0mm are centred on the z-axis and positioned 7.2mm apart. In addition to the electrodes which comprise the trap, there are a further four 1.6mm diameter electrodes positioned in a similar configuration to that of the r.f. electrodes, but centred at the corners of a 8.4mm square. These electrodes allow potentials to be added to compensate for stray electric fields in the trap- ping region. The complete experimental apparatus is shown schematically in figure 6, to which we refer in the remain- der of this section. The trap electrodes lie at the centre of a hexagonal stainless steel vacuum chamber. A high- voltage a.c. source RFsupplies a drive voltage of frequency 6.2MHz and peak-to-peak amplitude 135V for the radial electrodes, while high-voltage d.c. supplies DCprovide the voltages for the endcaps (95V in the present experiments) and compensation electrodes (typically around 60V are applied to the upper two compensation electrodes, while the lower two are grounded). The central chamber is pumped by a 25l/s ion pump IP and a 30l/s getter pump GP. An ion gauge IGon the oppo- site side of the main chamber monitors the pressure; this is below 2 ×10−11torr, the limit of the gauge’s sensitivity. To produce calcium ions in the trapping region, we use a cal- cium oven cand electron gun e. The oven is a thin-walled stainless steel tube filled with calcium granules, closed by crimping at each end and with a small hole at the centre pointing towards the trap region. The oven is heated by passing a 6A current along its length, which produces a 2beam of calcium atoms. The electron gun consists of a tungsten filament, also resistively heated, enclosed withi n a grounded stainless steel “grid” with respect to which it is negatively biased by 50V. To load the trap, the oven and electron gun are heated for a few minutes, the latter being left on for 10s after the oven has been turned off. We capture a small cloud of approximately 10 calcium ions using this procedure. This cloud is reduced to a single ion by applying to one of the endcaps a low-voltage “tickle” oscillation, close to the axial resonance frequency of the trap, expelling ions until only one remains in the trap. Violet light at 397nm is generated by frequency- doubling 794nm light from a master-slave diode laser sys- tem. A grating-stabilized master diode 794M, with a linewidth below 1MHz, is locked to a stabilized low-finesse reference cavity RCand used to inject a slave diode 794S. Light from the slave is frequency-doubled by a 10mm long Brewster-cut lithium triborate crystal LBOin an external enhancement cavity; H¨ ansch-Couillaud polarization anal - ysis of light reflected by the cavity provides a feedback sig- nal for a piezo-mounted mirror PZTused to lock the cavity length to the fundamental light. The measured character- istics of the doubling cavity are: finesse 130, enhancement 43, mode-matching efficiency 94%, input power 90mW at 794nm, output power 0.50mW at 397nm. Correcting for losses at the exit face of the crystal and the output cou- pler gives an internal crystal efficiency γ= 54(5) µW/W2, some 20% greater than previously reported values [13, 14] and about 75% of the theoretical optimum efficiency calcu- lated using a Boyd-Kleinman analysis [15] and an effective non-linear coefficient for LBO of deff= 0.855pm/V [16]. The 397nm beam passes through a λ/2 waveplate and polarizing beam splitter (PBS) cube to provide intensity control, and a lens focuses it to a spot size of 200 ×30µm (measured with a CCD camera) at the centre of the trap. The beam power used is typically 0.2mW. A grating-stabilized diode laser 866provides repump- ing light at 866nm whose intensity is adjusted by a λ/2 waveplate and PBS cube. A heated iodine bromide vapour cell IBrprovides an absolute frequency reference— unfortunately not suitable for reliable locking—and a triple-pass acousto-optic modulator AOM shifts the laser light some 650MHz into resonance with the 3 D3/2→ 4P1/2calcium transition. Spontaneous light near 854nm emitted by the laser diode (see section 4.1.2) is rejected by a diffraction grating DGand aperture A. The transmitted 866nm light is superimposed on the 397nm beam using a PBS cube and focused onto the trap. The spot size used in the final lifetime measurements was 250 ×130µm; the maximum beam power was 2 mW. The light at 850nm to shelve the ion is provided by a third grating-stabilized laser diode 850situated on a sep- arate optical table, and is directed into the ion trap via a polarization-preserving monomode optical fibre. The in- tensity is controlled by a λ/2 waveplate before the optical isolator. A mechanical shutter Sbefore the fibre allows complete extinction of this light. The spot size at thetrap is 450 ×450µm and the maximum power 0 .5mW. For clarity in figure 6 the detection optics are shown in the plane of the diagram: they are actually vertically above the ion trap. A wide-aperture compound lens gath- ers fluorescence emitted by the trapped ion and images it onto an aperture to reject scattered light; further lenses re-image the light, via a violet filter, onto a photomul- tiplier PMT connected to a gated photon counter. The net collection efficiency of the detection system, includ- ing the 16% quantum efficiency of the PMT at 397nm, is approximately 0 .12%. The peak photon count rate above background for a single cooled ion is typically 32kHz. A personal computer PCis used for data acquisition and control of the experiment; in particular it provides timing, logs PMT count data, controls the shutter S, and eliminates long-term drift of the 866nm laser by locking it to the fluorescence signal from the trapped ion at the end of each 20s acquisition period. 4 Searches for systematic effects We now consider effects which alter the measured shelving periods from those appropriate to an unperturbed ion sub- ject only to spontaneous decay. These are of two distinct types: those which alter the shelving periods themselves, because the ion is perturbed in some way, and those which cause systematic error in the process of measurement. We consider the two in turn. The significant difference be- tween our final result and previous work calls for a detailed discussion here. 4.1 Perturbations to the ion When the ion is fluorescing, it is cycling between the levels 42S1/2,42P1/2and 32D3/2. We denote this system of levels by Λ. During the shelving period, the ion is subject to radiation from two of the lasers (397 nm and 866 nm), to thermal radiation, and to the fields associated with the trap. There may also be collisions with the background gas. Any of these perturbations can transfer the ion to the Λ system. We consider them in turn. 4.1.1 Electric field The ion experiences a static electric field because of im- perfect compensation. The most significant effect on the internal state of the ion is to mix the 3 D5/2and 4 P3/2 levels, so that the measured lifetime is shortened by the presence of the induced 4 P3/2–4S1/2strong electric dipole transition. In fact, using the known matrix elements, one finds that the effect is negligible; the induced transition probability is 9 .0×10−14E2s−1, so that a field of 300 kV m−1(three orders of magnitude larger than the typical compensating fields used) would be necessary to produce a 1% reduction in the lifetime. We note, however, that 3there is another effect associated with an imperfectly com- pensated field, that of heating during the shelving period; we consider this in section 4.2.1. 4.1.2 Laser radiation From the standard theory of atom-light interaction, one finds that transitions from a lower level (1) to a higher level (2) in the multi-level ion can be stimulated by radiation of intensity Iand angular frequency ωLat a rate R12 (averaged over all Zeeman components) given by R12=2J2+ 1 2J1+ 1π2c3 ¯hω3 12A21I cg(ωL−ω12) (2) where J1andJ2are the total angular momenta of the levels, ω12is the atomic resonance angular frequency, and g(ωL−ω12) is the normalised lineshape function. In our case we may assume Lorentzian lineshapes, so that g(ωL−ω12) =Γ/(2π) (ωL−ω12)2+ Γ2/4(3) with the linewidth Γ = 1 /τ2determined by the sum of all the decay processes from the upper level. Since Block et al. [12] reported a significant dependence of the shelving time on the intensity of the repumping laser, we first consider excitation from 3 D5/2to 4P3/2by light at 866 nm. The lifetime τ2of the 4 P3/2level has been measured to be 6 .924±0.019 ns [11]. We use the value A21= (7.7±0.3)×106s−1fromab initio [7] and semiempirical [17] atomic structure calculations. The 4% uncertainty is our own estimate, based on the variation among the published calculations, and on the fact that the calculations produce other electric dipole matrix element s in agreement with experiment to better than this level of precision. In any case a 10% error in the value of A21 would have a negligible influence on our final result. To excite the 3 D5/2–4P3/2transition on resonance would require radiation of wavelength 854 nm. From equation (2), we find that with 866 nm light the rate is 9.9×10−5s−1/mW mm−2. The probability that the ion will subsequently decay back to 3 D5/2is the branching ra- tiob=A21τ2= 0.053, so the rate at which it will be trans- ferred to the Λ system is (1 −b)R12= 9.4×10−5s−1/mW mm−2. In our experiments, it is convenient to choose the 866 nm intensity such that the 3 D3/2→4P1/2transition is saturated. This requires I866≫0.08 mW mm−2if the laser frequency is set on resonance (taking this A21coeffi- cient 8 .4×106s−1[7, 17]), but to avoid the need to control this frequency precisely a much higher intensity was used, typically 1 .5 mW in a spot size of 250 ×130µm, or 30 mW mm−2. This results in a contribution to the depopu- lation rate of the 3 D5/2level of around 0.3% of that due to spontaneous emission to the ground level. This is far smaller than the experimental result of Block et al. However, in our own preliminary work we also found that there was a significant dependence of the shelvingtime on the intensity of the repumper laser, of the same order of magnitude as that reported by Block et al., and some 200 times larger than the theoretical value given above. This suggested that the 866 nm laser was emitting some radiation much closer in wavelength to 854 nm which was primarily responsible for the shortening of the appar- ent lifetime of the 3 D5/2level. This laser is a semicon- ductor diode device, operated with an extended cavity by use of a Littrow-mounted diffraction grating. Without the grating, the laser would operate close to 854 nm. Since any laser produces spontaneous emission over its gain profile, as well as stimulated emission at the lasing wavelength, there was in our preliminary work radiation incident on the ion in the vicinity of 854 nm. One might expect the intensity of this radiation to be greatly reduced because of the long (3m) beam path. In fact, this is not the case, be- cause the emitting region in the laser is small (dimensions of order microns) so the spontaneous component is well collimated; like the coherent component, it is transported to the trap with little loss. We therefore investigated the spectrum of the light from the laser by means of a diffraction grating, and found it to contain a broad background from 840 nm to 870 nm. When the total laser power near the ion trap was 2 mW, the power in the broad component was 8 µW, and the spot sizes were similar. This implies a mean spectral density around 854 nm sufficient to cause de-shelving rates con- siderably higher than those we observed; we ascribe the lower observed rates to the structure of the background. This was not resolved in our investigation, but is expected on the basis of other experimental work [18, 19] to include a long series of spikes separated by the longitudinal mode spacing of the diode (50 GHz). These occur because the gain is so high in these devices that there is some am- plification right across the gain profile; evidence that this occurs in our laser is provided by the high degree of po- larization (90%) of the background. It is thus reasonable to conclude that the 3 D5/2–4P3/2atomic resonance falls between peaks in the 50 GHz spaced comb. To study the effects of this background radiation we reduced its intensity at the position of the ion. This was done in two stages: first, we reduced it by a factor of 25 us- ing an interference filter centred on 866 nm. The observed de-shelving rate fell from 1 .85±0.06 s−1to 0.87±0.02 s−1, the two measurements being taken at the same 866 nm intensity. This provided convincing evidence that the de-shelving was indeed caused by the postulated mecha- nism. We therefore replaced the filter with a diffraction grating and iris aperture, arranged to pass light at 866 nm. The power transmitted by the system in the vicinity of 854 nm was then reduced compared with the unfiltered laser by three and a half orders of magnitude (for a given in- tensity at 866 nm), and the remaining light was scattered, thus increasing the illuminated area in the vicinity of the ion by a factor measured to be 40, making a net intensity reduction 8 ×10−6. Under these conditions transitions stimulated by this radiation become much less probable 4than those due to the 866 nm light itself. The observed de-shelving rate was 0 .858±0.007 s−1. De-shelving rates γat various settings of the intensity Iof the 866 nm laser beam as measured in these various experiments are shown in figure 7a. A straight line of the form γ=γ0+αIfitted to the points for which the grating and iris system were in place gives γ0= 0.857±0.016 s−1,α= (1.5±6)×10−3 s−1/mW mm−2. The value of γ0is consistent with our fi- nal more accurate measurements, given in section 6 below, while the slope is consistent with zero (and with the very small theoretical value for de-shelving by 866 nm radiation given above). The intercept with unfiltered light is greater than that given by our final data because our method of varying the laser intensity did not alter the unpolarized component of the background radiation. Spontaneous emission from the repumper laser appears also to be likely to account for the observations of Block et al.It is possible that it was present in previous work on the 3D5/2lifetime, and unaccounted for, explaining the lower values obtained by all earlier workers. This does not immediately apply to Block et al., however, because their measurement involved an extrapolation to zero laser intensity using neutral density filters. The 866 nm laser might also generate radiation near 854 nm if it went multimode owing to a degradation of the alignment of its own external cavity. If intermittent and at a low level, this effect could occur undetected. How- ever, multimode operation was found in practice to have an all-or-nothing character: if it occurred at all then it wa s obvious from a large increase in the noise of the fluores- cence signals, and in such a case the data were discarded. The grating and iris system were in place for our final data sets. Figure 7b shows our final rate measurements plotted against the intensity of the repumper laser. The observations are consistent with the theoretical value for the slight dependence on 866 nm intensity. Our final result for the lifetime was obtained with a slope fixed at the theoretical value (see section 6). The radiation at 397 nm is obtained by frequency- doubling and so does not contain any significant back- ground light. There is no transition from 3 D5/2near enough to this wavelength to cause significant de-shelving. A check for this or some other (unidentified) effect was nevertheless carried out, where we changed the power of this light by a factor 2, and we observed, to ten percent precision, no effect on the deshelving rate. 4.1.3 Thermal radiation The rate of de-shelving is (1 −b)B12ρ(ω21), where ρ(ω) is the energy density per unit frequency interval in the thermal radiation, and B12=g2π2c3A21/(g1¯hω3 21). In a thermal cavity we obtain the rate (1−b)2J2+ 1 2J1+ 1A21e−¯hω21/kBT(4)for ¯hω21≫kBT. For the 854 nm transition in a room- temperature cavity, the rate is of order 10−18s−1so is negligible. Non-negligible rates can be obtained, however , when we consider the radiation produced by room lights or the filament of an ion gauge. The spectral energy density is reduced from the value in equation (4) by a geometrical factor approximately equal to S/(4πr2) where Sis the area of the hot filament and ris its distance from the ion, if the ion is within line of sight of the filament. For example, taking T= 1700 K, S= 20 mm2,r= 30 cm, we obtain a rate ∼5×10−3s−1. In our vacuum system, although the ion gauge is at this distance from the trapped ion, it is not in line of sight, so we expect the rate of this process to be well below this value. 4.1.4 Collisional effects The ion in the shelved level can undergo a collision with an atom of the background gas. Either or both of two processes may then occur: the ion may gain a significant amount of kinetic energy, and it may be transferred to an- other state. This is a source of error since in both cases the apparent shelving time — the interval during which fluorescence is not observed — will be affected. To inves- tigate the nature and frequency of collisional effects we monitored the fluorescence from the ion for 8 hours with the 397 nm and 866 nm radiation present but with no laser operating at 850 nm to take the ion to the 4 P3/2 level. The diffraction grating and aperture were in place. During this period we observed abrupt disappearance of the fluorescence (within the resolution of the 22 ms bins) on 17 occasions. It reappeared after times of the order of a second, with 6 “dark periods” as short as a few tens of milliseconds. The reappearance was generally abrupt, but in 5 of the longer periods it was more gradual, occurring over several bins. One non-collisional effect which can lead to loss of fluo- rescence in this test is shelving in the 3 D5/2level. Shelv- ing caused by the 866 nm radiation exciting the 3 D3/2to 4P3/2transition is negligible, because the spontaneous de- cay to 3 D5/2has such a low branching ratio (the excitation rate can be calculated from the branching ratio band the 3D3/2→4P3/2coefficient A21= 0.91×106s−1[7, 17]). In contrast, it is not possible to rule out excitation by the spontaneously emitted light from the 866 nm laser at 850 nm because the rate depends on the spectral distribution of the spontaneous emission at 850 nm; if a peak happened to be close to the frequency of the 850 nm transition the process could be responsible for a significant number of the observed events. However, we would then expect an abrupt reappearance of the fluorescence on a time scale of the order of a second, and the data then suggest that the upper limit for events of this type is around eight. These could alternatively be caused by fine-structure changing collisions; the ion in the 3 D3/2level can be transferred to 3D5/2by a long range collision which may not transfer sig- nificant kinetic energy. At our working pressure of order 510−11mbar we would expect about 8 such events on the basis of an approximate value of the collisional mixing rate which has been determined for conditions similar to ours [20, 21]. Our observations thus provide a rough upper limit on the background gas pressure in our system directly at the location of the ion, to confirm our ion gauge reading. We note that the rates for D5/2→D3/2andD3/2→D5/2 are approximately equal at room temperature, therefore the present experiment does give an indication of the de- shelving rate due to fine-structure changing collisions in ourD5/2lifetime measurements. The other events are more obviously characteristic of collisions. In particular, the more gradual reappearance of the fluorescence after a long dark period is likely to be associated with the ion being cooled again after acquiring a significant amount of kinetic energy. On the basis of our measured pressure and reasonable estimates of cross- sections, we do not expect more than two or three colli- sions of this type. Some of the very short dark periods are likely to be much smaller perturbations by relatively dis- tant collisions. If such a perturbation were to occur during the lifetime measurement itself, fluorescence would not be observable for a period whatever state the ion was in after the collision. Fortunately, for the purposes of estimating the uncertainty in the lifetime measurement, detailed in- terpretation of the events is not necessary. An upper limit to the error introduced can be found by assuming that all 17 events are due to collisions, giving a rate of 6 ×10−4 s−1, and that such a collision occurring in the experimen- tal runs themselves while the ion was in the shelved level would have delayed (or hastened) the reappearance of flu- orescence by an average of order 1 second. The net con- tribution to the measured rate has thus an upper limit of ±6×10−4s−1which is an order of magnitude below the statistical error. 4.2 Systematic effects in the measure- ment process Both the fluorescence and the background signal on which it is superposed are subject to fluctuations. As well as the random error this introduces into the measurement, be- cause the instant at which fluorescence resumes is subject to statistical uncertainty, there are systematic effects. F or example, a large fluctuation in the background can sug- gest that the ion has decayed while it is still shelved; our data analysis procedure had to be developed and tested to minimise errors due to such effects. Further, the ion is not cooled during the shelved period, and if there is significant heating the fluorescence may be reduced for a time after the decay. 4.2.1 Heating during the shelved periods In our preliminary experiments, we found that when an electric field was applied in the vertical direction the flu- orescence reappeared only gradually after shelving, overperiods of 10–50 ms. We ascribe this to heating. When the d.c. electric field in the trap is not zero, the ion ex- periences the r.f. driving field, which is much more noisy than the d.c. field, and so the ion motion heats during the shelved periods when it is not laser cooled. We calculate that if the ion heats up to room temperature, it would take the lasers approximately 50 ms to cool the motion down again and thus for the fluorescence to reappear. Ev- idence supporting this interpretation is given by the fact that the non-abrupt reappearance of fluorescence was cor- related to the duration of the dark period, being more likely for longer dark periods. We therefore took care to ensure this phenomenon was not present in the runs used for our final data set. This was done by nulling the vertical field carefully before each run. In our linear trap geometry the r.f. field is 2- dimensional. To null the vertical field we adjusted the voltage on one of the d.c. field compensation electrodes, so as to minimise the linewidth of the ion fluorescence as a function of 397 nm laser frequency (at lowered 397 nm laser power). The 397 nm laser beam used for shelving measurements enters along the direction (√ 3,0,1) (where thexzplane is horizontal); this is sensitive to horizontal micromotion and hence the vertical field. We were able to null this field to ±3 V/m, and most runs were carried out with nulling to ±10 V/m. We measured the remain- ing heating rate, when this field was as well compensated as we could make it, by blocking the cooling laser beams (397 nm and 866 nm) for long periods, and looking for a non-abrupt return of the fluorescence when the beams were unblocked. No delay was observed (the limit of sen- sitivity being the bin size of 22 ms) unless the cooling lasers were blocked for more than 10 minutes, some 600 times longer than the shelving times occurring during the measurements themselves. The consequent systematic er- ror in the lifetime can thus be safely neglected; heating is likely to be relatively slow when the ion is first shelved, but even assuming that the delay of fluorescence is linear with shelving time the error is only of order 30 µs. We note that there were no soft edges in the runs used for our final data set. We found that a horizontal field gave a much smaller heating effect, and indeed some of our final data points were taken in the presence of a horizontal field of order 200 V/m. However, for our final experimental run, a further beam was introduced along (0 ,−2,−1) to allow horizontal field compensation; this beam was blocked once the com- pensation was optimised. Although not important for the present lifetime measurements, accurate compensation is necessary for our future work in which the ion is to be cooled below the Doppler limit. 4.2.2 Tests of the reliability of the data analysis We consider three possible sources of systematic error in the data analysis in turn. First, it is straightforward to show analytically that approximating the continuous ex- 6ponential distribution of dark times as a histogram causes no systematic errors in the fitted value of τ. Second, nu- merically simulated data was used to show that varying the threshold level used in the data analysis, did not sys- tematically change the fitted value of τby more than 0 .1 ms. Our real data has a further property: intensity fluc- tuations. The τvalue deduced from a real data set was found to vary as a function of threshold by amounts of or- der±0.5 ms. We estimate this may introduce a systematic error of order 0 .5 ms. Third, repeated fits to numerically simulated data sets were used to show that any systematic error arising purely from the exponential fitting procedure was less than 0 .25 ms. They also permitted us to verify the uncertainty (one standard deviation) in the fitted value of τ, derived from the cost function surface. These effects give a total uncertainty of ±0.6 ms. 5 Demonstration of photon anti- bunching The quantum jump method provides a convenient means to demonstrate photon antibunching [22], that is, genera- tion of a light field whose second order coherence g(2)(t) falls below 1 as t→0. The second-order coherence is the normalised autocorrelation function of the intensity. A classical definition of intensity leads to g(2)(t)< g(2)(0) andg(2)(0)>1, whereas a definition in terms of quantum electric field operators allows all values [22]. Therefore o b- servation of a g(2)(0) value below 1 is a direct signature of the quantum nature of the radiation field. Quantum jump observations yield g(2)easily, since whenever a de-shelving jump is observed, we may deduce, with close to 100% re- liability, that one photon at 729 nm has been emitted by the ion [23]. Therefore we deduce the presence of a 729 nm radiation field which has g(2)(t) =/angbracketleftn(t)n(0)/angbracketright/¯n2, where n(t) and n(0) are the numbers of jumps observed in two equal time intervals separated by a time t. This quantity is plotted in figure 8 for a typical data set in our experi- ment. For this study, we did not use the method described in section 2, in which the 850 nm light was blocked when the ion was shelved. Instead, this light was left perma- nently on, and the fluorescence monitored continuously. We therefore obtained the well-known ‘random telegraph’ signal, figure 3b. Since we detect very close to all the 729 nm photons, we can be confident the second order co- herence does not vary at time-scales too short for us to detect. Therefore the antibunching signature g(2)(0)<1 is clearly demonstrated. We can understand the complete form of g(2)(t) by solv- ing the rate equations for the populations of the ion’s lev- els. All the relevant processes are fast except for two, namely spontaneous decay from D5/2to Λ and excitation from Λ to D5/2, so the problem reduces to a two-level sys- tem. If we take t= 0 to be the centre of a short time interval (one bin) during which a jump occurred, so that n(0) = 1, then the probability for the atom to emit a729 nm photon at time tis proportional to the population ofD5/2att, given that it was zero at t= 0. We thus obtain g(2)(t) = 1−e−(R+γ)t(5) where Ris the excitation rate, γthe decay rate. This curve is plotted on figure 8 with no free parameters ( Ris taken as the inverse of the mean duration of bright periods in the same data set). The agreement between data and theory is evidence that we have a good understanding of our experimental situation. 6 Result and discussion Our final data set consisted of four 8-hour runs, and two 2-hour runs, all using the experimental method of section 2 (850 nm laser blocked during dark periods). The de- shelving rates observed in these runs, as obtained by the analysis described above, are shown in figure 7b. The size of each error bar is equal to the statistical uncertainty emerging from the analysis. The straight line through the data is a single-parameter weighted least-squares fit. The line is given the theoretically expected slope, and the best fit intercept is found to be γ0= 0.856±0.004 s−1. The de-shelving rate γobtained in the experiment may be written γ=1 τ+/summationdisplay iγi (6) where τis the natural lifetime we wish to measure, and theγiare due to other processes which contribute to the measured rate, chiefly laser excitation of transitions in th e ion and collisional processes. The effect of off-resonant excitation by 866nm radiation is already accounted for in our fitted intercept γ0. However, uncertainty ( ±30%) in the light intensity at the ion makes this accounting im- precise, leading to a further systematic error in γ0at the level of 0.001s−1. This and other contributions to the systematic error, from collisional processes and remainin g background light from the 866nm laser, are shown in Ta- ble 1. The statistical error from the data fitting dominates. The systematic effects are independent of one another, so we add their errors in quadrature. Adding this total sys- tematic error linearly to the statistical error, the value w e obtain for the natural lifetime of the D5/2level is τ= 1.168±0.007 s (7) We have already noted the significant difference between this result and earlier measurements. Here we compare it with theory. Our result differs from all the reported ab initio calculations by amounts large compared with our 0.6 % experimental uncertainty. The relative difference (τtheor .−τmeas.)/τmeas.is−10% for a recent calculation based on the Brueckner approximation [7], +5 .6% for rel- ativistic many-body perturbation theory [4], and −2.6% for multi-configurational Hartree Fock (MCHF) calcula- tions [5, 6]. Of these calculations, those which produced 7effect 1 σerrors (10−3s−1) statistical systematic data fitting ±4 ±0.4 866nm intensity uncertainty (30%) ±1 A21uncertainty (4%) ±0.1 collisional processes ±0.6 background from 866nm laser ∼+0.01 total ±4 ±1.2 Table 1: Statistical and systematic contributions to the error in the measured de-shelving rate γ. The systematic errors are added in quadrature, then their total is added linearly to the statistical error, to give a total uncertain ty of 5.2×10−3s−1. See also section 4. the smallest discrepancy with our measured Dstate life- time produced the largest discrepancy with the measured Pstate lifetimes obtained by Jin and Church [11]. A semi- empirical calculation based on MCHF and core polariza- tion [5], and a calculation using related methods [6], give a value close to our measured result (relative difference −0.7%). The most natural interpretation of these obser- vations is that the ab initio calculations of τcarried out so far have a precision of at best a few percent, and success at calculating electric dipole matrix elements does not guar- antee the same degree of success with other parameters such as electric quadrupole matrix elements. The ratio between the lifetimes of D3/2andD5/2is much less sensitive to imperfections in the calculations, and is given primarily by the frequency factor ω5, which leads to a ratio 1 .022. The factor emerging in recent calcu- lations of the two lifetimes is 1 .0335 [7], 1 .0283 [4], 1 .0175 [5, 6]. We take the standard deviation of these results to indicate the theoretical uncertainty on their mean, giving τ3/2/τ5/2= 1.026±0.007. Combining this with our mea- surement of τ5/2givesτ3/2= 1.20±0.01 s for the lifetime of the 3 d2D3/2level in40Ca+. 7 Conclusion We have described a new linear ion trap apparatus and its use to measure the lifetime of the D5/2level in40Ca+by quantum jump measurements on a single trapped ion. Our result is more precise than previous measurements, and significantly larger. We believe this discrepancy is mostly explained by previously unrecognised systematic errors, which tend to make the lifetime appear shorter than it in fact is. Our measurement provides a new precise test of ab initio atomic structure calculations; it is in only moderate agreement with currently reported calculations. We have discussed the spectral distribution of light emitted by a semiconductor diode laser operated in an extended cavity, emphasizing that in experiments where unwanted excitation of allowed atomic transitions is to be avoided, it is necessary to take into account the weak broad component in the spectrum, which extends manynanometres away from the main lasing wavelength. This is significant in many experiments in atomic physics, es- pecially coherent atom optics, ‘dark’ optical lattices, fr e- quency standards and quantum information processing. Finally, we have discussed various diagnostics on the performance of the ion trap, made possible by careful analysis of long periods of operation of the trap. These include an upper limit for the pressure in the system, and the heating rate in the trap. We demonstrated photon anti-bunching, finding it in agreement with theoretical ex- pectations. Our uncertainty is dominated by statistics, so it would be possible to obtain significantly higher precision by accu - mulating more data, though for single-ion work this would require integration for several weeks. To go further, one would need better knowledge of the laser intensities and collisional processes. 8 Acknowledgements We would like to acknowledge helpful correspondence with G. Werth, and a preprint of [12]. L. Favre, E. Hodby, M. McDonnell and J. P. Stacey contributed to development of the apparatus, D. T. Smith designed the high voltage r.f. supply, and we are grateful for general technical assis - tance from G. Quelch. We thank R. Blatt and the Inns- bruck group (especially F. Schmidt-Kaler, C. Roos, M. Schulz) for many helpful discussions and guidance, and similarly R. C. Thompson and D. N. Segal. This work was supported by EPSRC (GR/L95373), the Royal So- ciety, Riverland Starlab and Oxford University (B(RE) 9772). References [1] F. Plumelle, M. Desaintfuscien, and M. Houssin, IEEE Transactions on Instrumentation and Measure- ment42, 462 (1993). [2] M. A. Ali and Y. Kim, Phys. Rev. A 38, 3992 (1988). [3] C. J. Zeippen, Astron. Astrophys. 229, 248 (1990). [4] C. Guet and W. R. Johnson, Phys. Rev. A 44, 1531 (1991). [5] N. Vaeck, M. Godefroid, and C. Froese-Fischer, Phys. Rev. A 46, 3704 (1992). [6] T. Brage, C. Froese-Fischer, N. Vaeck, M. Godefroid, and A. Hibbert, Physica Scripta 48, 533 (1993). [7] S. S. Liaw, Phys. Rev. A 51, R1723 (1995). [8] E. Biemont and C. J. Zeippen, Comments on At. Mol. Phys.33, 29 (1996). [9] D. E. Osterbrock, Astrophysics of Gaseous Nebulae (Freeman, San Francisco, 1974). 8[10] S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. 82, 2484 (1999). [11] J. Jin and D. A. Church, Phys. Rev. Lett. 70, 3213 (1993). [12] M. Block, O. Rehm, P. Seibert, and G. Werth, The European Physical Journal D 7, 461 (1999). [13] S. Bourzeix, M. D. Plimmer, F. Nez, L. Julien, and F. Biraben, Optics Communications 99, 89 (1993). [14] C. S. Adams and A. I. Ferguson, Opt. Comm. 90, 89 (1992). [15] G. D. Boyd and D. A. Kleinman, Journal of Applied Physics 39, 3597 (1968). [16] Technical report, Casix Inc. (private communication) . [17] C. E. Theodosiou, Phys. Rev. A 39, 4880 (1989). [18] J. C. Camparo, Contemporary Physics 26, 443 (1985). [19] M. de Labachelerie and P. Cerez, Optics Communi- cations 55, 174 (1985). [20] F. Arbes, T. Gudjons, F. Kurth, G. Werth, F. Marin, and M. Inguscio, Zeitschrift fur Physik D 25, 295 (1993). [21] F. Arbes, M. Benzing, T. Gudjons, F. Kurth, and G. Werth, Zeitschrift fur Physik D 31, 27 (1994). [22] R. Loudon, The Quantum Theory of Light (Oxford University Press, 1995). [23] J. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 1699 (1986). [24] M. Knoop, M. Vedel, and F. Vedel, Phys. Rev. A 52, 3763 (1995). [25] T. Gudjons, B. Hilbert, P. Seibert, and G. Werth, Europhysics Letters 33, 595 (1996). [26] G. Ritter and U. Eichmann, Journal of Physics B - Atomic Molecular and Optical Physics 30, L141 (1997). [27] J. Lidberg, A. Al-Khalili, L.-O. Norlin, P. Royen, X. Tordoir, and S. Mannervik, Journal of Physics B 32, 757 (1999). 9Arbes et al. 1994 [21] 1054(61) • Knoop et al. 1995 [24] 994(38) • Gudjons et al. 1996 [25] 1064(17) • Ritter et al. 1997 [26] 969(21) • Lidberg et al. 1999 [27] 1090(50) • Block et al. 1999 [12] 1100(18) • This work 1168(7) • Ali & Kim 1988 [2] 950 • Zeippen 1990 [3] 1053 • Guet & Johnson 1991 [4] 1236 • Vaeck et al. 1992 [5] 1140 • Brage et al. 1993 [6] 1163 • Liaw 1995 [7] 1045 • Biemont & Zeippen 1996 [8] 1070 • 950 1000 1050 1100 1150 1200 1250 τ(ms) Figure 1: Values of τexperimentally measured (points with error bars) and theor etically calculated ab initio (circles). The first two columns give the reference, the third gives τin milliseconds. Earlier measured values are not shown since they have considerably larger experimental uncertai nty. 4S1/23D3/23D5/24P1/2 866□nm854□nm 850□nm 397□nm4P3/2 729□nm393□nm Figure 2: Low-lying energy levels of40Ca+and associated transitions. Lasers at 397 nm, 866 nm and 850 n m drive the corresponding transitions in the experiments. The othe r transitions are significant to the interpretation of the measured signals and to systematic effects. 102 3 4 5 6 7 8 9 10100200300400500 Time (s) Counts per 20ms0100200300 (b)(a) Counts per 8ms Figure 3: Observed fluorescence signals. The vertical axis i s the number of counts given by the photomultiplier during one counting bin, the horizontal axis is time. The horizonta l dashed lines show the threshold settings for the data analysis. (a) Shutter method, in which 850 nm laser is blocke d during dark periods. The vertical lines indicate shutter opening and closing events. (b) Random telegraph method, in which 850 nm laser is permanently on. (a) was used for accurate τmeasurements, (b) for g(2)measurements and various systematic studies. 110 1000 2000 3000 4000 5000-4-2024 Time (ms)Residuals020406080100120140 Frequency Figure 4: Typical data set from a single 8-hour run. The figure shows a histogram of the measured dark period durations xi, and the fitted exponential curve produced by our analysis pr ocedure (see text). The residuals are shown on an expanded scale, in the form (data −fit)/√(fit). In this example the analysis gave A= 117 .7±1.5, γ= 0.853±0.009. 12/X31/X2E/X36 /X31/X2E/X32/X32/X2E/X36 /X38/X2E/X34 /X31/X2E/X30/X32/X38/XB0/X30/X2E/X32/X30/X20/X73/X74/X72/X61/X64 Figure 5: End view of the linear ion trap electrode arrangeme nt, to scale, showing the a.c. trap electrodes (solid black circles) and d.c. compenstation electrodes (grey circles) . Dimensions are in mm. The position of the d.c. endcaps is also indicated (open circle); the separation between the tw o opposing endcaps is 7 .2 mm. The solid angle in which fluorescence is collected by the imaging system is 0 .2 steradians. /X4C/X42/X4F/X50/X5A/X54 /X49/X42/X72/X6C/X2F/X34 /X6C/X2F/X32 /X65 /X52/X46 /X44/X43 /X63 /X49/X47/X47/X50 /X49/X50 /X50/X4D/X54/X6C/X2F/X32 /X53 /X37/X39/X34/X4D/X37/X39/X34/X53 /X52/X43 /X38/X35/X30/X38/X36/X36 /X41/X44/X47 /X41/X4F/X4D/X36/X35/X30/X4D/X48/X7A /X70/X68/X6F/X74/X6F/X6E /X63/X6F/X75/X6E/X74/X65/X72/X50/X43/X49/X2F/X4F/X6C/X2F/X32 /X6C/X2F/X32/X6C/X2F/X32 Figure 6: Main features of the optical set-up; see section 3 f or key. 130 5 10 15 20 25 30 350.800.820.840.860.880.90Rate (s-1) 866nm laser intensity (mW/mm2)0.81.21.62.02.4 (b)with interference filterunfiltered laser light with diffraction grating Rate (s-1) 0 10 20 30 40 50 60 70 80 90(a) 0 10 20 30 40 50 60 70 80 900.81.21.62.02.4 Figure 7: Measured de-shelving rates. (a) Measurements usi ng 1 to 2 hour runs of the random telegraph method, at various 866 nm beam intensities, to test for intensity dep endence of the rate. The lines are 2-parameter straight line fits to the data, with fitted intercept and slope. With the diffraction grating and aperture in place, the slope is consistent with zero and with the small theoretically expec ted value (see text). (b) Final data set using the shutter method. Points taken at the same intensity are shown horizon tally offset for clarity. The error bars are the statistical uncertainties emerging from the analysis of each run. The li ne is a single-parameter least-squares fit; it is given the theoretically expected slope, and the best fit intercept is o btained. 140 1000 2000 3000 4000 5000 60000.00.20.40.60.81.01.21.4 g(2)(t) Time t (ms) Figure 8: Second-order coherence of the 729 nm radiation emi tted by the ion, as deduced from quantum jump observations. The data points give the number of times a jump was observed (therefore a 729 nm photon emitted) at time tafter a chosen jump, normalised to the mean jump rate, accumu lated in several data sets. The line is the theoretical expectation, described in the text. 15

arXiv:physics/0002027v1 [physics.optics] 15 Feb 2000Femtosecond soliton amplification in nonlinear dispersive traps and soliton dispersion management Vladimir N. Serkinaand Akira Hasegawab aBenemerita Universidad Autonoma de Puebla, Instituto de Ci encias Apdo Postal 502, 72001 Puebla, Pue., Mexico aGeneral Physics Institute, Russian Academy of Science, Vavilova 38, 117942 Moscow aemail address: vserkin@hotmail.com bKochi University of Technology and NTT Science and Core Tech nology ATR BLDQ., 2-2 Hikaridai Seikacho Sorakugun Kyoto, Japan 619-0288 (15 February 2000) The nonlinear pulse propagation in an optical fibers with var ying parameters is investigated. The capture of moving in the frequency domain femtosecond color ed soliton by a dispersive trap formed in an amplifying fiber makes it possible to accumulate an additi onal energy and to reduce significantly the soliton pulse duration. Nonlinear dynamics of the chirp ed soliton pulses in the dispersion managed systems is also investigated. The methodology deve loped does provide a systematic way to generate infinite “ocean” of the chirped soliton solution s of the nonlinear Schr¨ odinger equation (NSE) with varying coefficients. Keywords and PACS numbers: Femtosecond solitons amplification, dispersion managemen t 42.65 Tg, 42.81 Dp I. INTRODUCTION In 1973 Hasegawa and Tappert [1] showed theoretically that a n optical pulse in a dielectric fibers forms an envelope solitons, and in 1980 Mollenauer, Stolen and Gordon [2] demo nstrated the effect experimentally. This discovery is significant in its application to optical communications . Today the optical soliton is regarded as an important alternative for the next generation of high speed telecommu nication systems. The theory of NSE solitons was developed for the first time in 1 971 by Zakharov and Shabad [3]. The concept of the soliton involves a large number of interesting problems in a pplied mathematics since it is an exact analytical solution of a nonlinear partial differential equations. The theory of optical solitons described by the nonlinear Schr¨ odinger equation has produced perfect agreement between theory and experiment [4]. In this paper we present mathematical description of solita ry waves propagation in a nonlinear dispersive medium with varying parameters. The soliton spectral tunneling effect was theoretically pre dicted in [5]. This is characterized in the spectral domain by the passage of a femtosecond soliton through a potential b arrier-like spectral inhomogeneity of the group velocity dispersion (GVD), including the forbidden band of a positiv e GVD. It is interesting to draw an analogy with quantum mechanics where the solitons are considered to exhibit part icle-like behavior. The soliton spectral tunneling effect also can be considered as an example of the dynamic dispersio n soliton management technique. In the first part of the paper we will concentrate on the problem of femtosecon d solitons amplification. We will show that spectral inhomogeneity of GVD allows one to capture a soliton in a sort of spectral trap and to accumulate an additional energy during the process of the soliton amplification. In th e second part we will consider the problem of the short soliton pulse propagation in the nonlinear fiber with static non-uniform inhomogeneity of GVD. The methodology developed does provide a systematic way to generate infinite “ocean” of the chirped soliton solutions of NSE model with varying coefficients. II. FEMTOSECOND SOLITON AMPLIFICATION It is well known that due to the Raman self-scattering effect [ 6] (called soliton self-frequency shift [7]) the central femtosecond soliton frequency shifts to the red spectral re gion and so-called colored solitons are generated. This effe ct decreases significantly the efficiency of resonant amplificat ion of femtosecond solitons. The mathematical model we consider based on the modified NSE including the effects of mol ecular vibrations and soliton amplification processes (see details in [8]): 1i∂ψ ∂z=1 2∂2ψ ∂τ2+iσ∂3Ψ ∂τ3+ (1 −β)|ψ|2ψ+βQψ+G 2P (1) µ2∂2Q ∂t2+ 2µδ∂Q ∂t+Q=|ψ|2,and, γ a∂P ∂τ+P(1 +iγa∆Ω) =iψ, (2) As numerical experiments showed the GVD inhomogeneity as a p otential well allows one to capture a soliton in a sort of spectral trap. Figure 1 shows the nonlinear dynamics of the soliton spectral trapping effect in the spectral domain. As soliton approaches the well, it does not slow down but speeds up, and then, after it has got into the well, the soliton is trapped. There exists a long time of soliton tr apping in internal region of the well .This effect opens a controlled possibility to increase the energy of a soliton . As follows from our computer simulations the capture of moving in the frequency space femtosecond colored soliton b y a dispersive trap formed in an amplifying optical fiber makes it possible to accumulate an additional energy in the s oliton dispersive trap and to reduce significantly the soliton pulse duration. III. DISPERSION MANAGEMENT: CHIRPED SOLITONS Let us consider the propagation of a nonlinear pulse in the an omalous (or normal) group velocity dispersion fiber of lengthZ1.The complex amplitude qof the light wave in a fiber with variable parameters D2(Z) ,N2(Z) and Γ(Z) is described by the nonlinear Schrodinger equation i∂q ∂Z+1 2D2(Z)∂2q ∂T2+N2(Z)|q|2q=iΓ(Z)q (3) Theorem 1 . Consider the NSE (3) with varying dispersion, nonlinearit y and gain. Suppose that Wronskian W[N 2,D2] of the functions N 2(Z) and D 2(Z) is nonvanishing, thus two functions N 2(Z) and D 2(Z) are linearly inde- pendent. There are then infinite number of solutions of Eq. (3 ) in the form of Eq.4 q(Z,T) =/radicalBigg D2(Z) N2(Z)P(Z)Q[P(Z)·T] exp iP(Z) 2T2+iZ/integraldisplay 0K(Z′)dZ′  (4) where function Qdescribes fundamental functional form of bright Q= sech(P(Z)T) or darkQ= th(P(Z)T) NSE solitons and the real functions P(Z), D 2(Z), N2(Z) and Γ(Z) are determined by the following nonlinear system of equations : 1 P2(Z)∂P(Z) ∂Z+D2(Z) = 0 ; −1 2D2(Z)P(Z) +W[N2(Z),D2(Z)] 2D2(Z)N2(Z)= Γ(Z) (5) Theorem 2 . Consider the NSE (3) with varying dispersion, nonlinearit y and gain. Suppose that Wronskian W[N 2,D2] of the functions N 2(Z) and D 2(Z) is vanishing, thus two functions N 2(Z) and D 2(Z) are linearly dependent. There are then infinite number of solutions of Eq. (3) of the fo llowing form Eq. 6 q(Z,T) =C P(Z)Q[P(Z)·T] exp iP(Z) 2T2+iZ/integraldisplay 0K(Z′)dZ′  (6) where function Qdescribes the fundamental form of bright (or dark) NSE solit on and the real functions P(Z), D 2(Z), N2(Z) and Γ(Z) are determined by the following nonlinear system of equati ons : D2(Z) =−1 P2(Z)∂P(Z) ∂Z; Γ(Z) =1 21 P∂P(Z) ∂Z;N2(Z) =D2(Z)/C2(7) The function P(Z) is required only to be once-differentiable , but otherwise arbitrary function, there is no restriction s. To prove Theorems 1 and 2 we first construct a stationary local ized solution of Eq. (3) by introducing Kumar- Hasegawa’s quasi-soliton concept [9–11] through 2q(Z,T) =/radicalBigg D2(Z) N2(Z)P(Z)Q[P(Z)·T] exp iP(Z) 2T2+iZ/integraldisplay 0K(Z′)dZ′  (8) whereD2(Z),N2(Z),P(Z) andK(Z) are the real functions of Z.Substituting expression (8) into Eq. (3) and separating real and imaginary parts we obtain the system of t wo equations 1 2sign(D2)∂2Q ∂S2+Q3+/parenleftbigg E−S2 2·Ω2(Z)/parenrightbigg Q= 0 (9) ∂P ∂ZQ+P∂Q ∂S∂S ∂Z+1 21 D2(Z)∂D2 ∂ZPQ−1 21 N2(Z)∂N2 ∂ZPQ+1 2D2P2Q+D2P2T∂Q ∂S∂S ∂T= ΓPQ (10) Where S(Z,T) =P(Z)T;∂S ∂Z=T∂P ∂Z;∂S ∂T=P(Z) (11) In Eq. (9) the parameters Eand Ω are ’the energy’ and ’frequency’ of ordinary quantum me chanical harmonic ocsillator Ω2(Z) =D−1 2(Z) P2(Z)/parenleftbigg1 P2(Z)∂P ∂Z+D2(Z)/parenrightbigg ;E(Z) =−K(Z)/P2(Z)/D2(Z) (12) Eq. (9) represents the nonlinear Schrodinger equation for t he harmonic ocsillator. As must be in the case of Hamil- tonian system Eq. (9) may be written in the form δH δQ∗= 0 (13) H=/integraldisplay/bracketleftBigg 1 2sign(D2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂Q ∂X/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +1 2α|Q|4+/parenleftbigg E−X2 2·Ω2(Z)/parenrightbigg |Q|2/bracketrightBigg dX (14) The derivative in (13) is functional derivative. For the firs t time this equation was solved numerically by Kumar and Hasegawa in [9] and gave rise a new concept of quasi-solitons [10,11]. Now we make the important assumption about the solution of Eq. (9). Let us consider the complete nonlinear regime when Eq. (9) re presents the ideal NLS equation, i.e. we will allow Ω(Z)≡0 , then from (12) follows that 1 P2(Z)∂P(Z) ∂Z+D2(Z) = 0 (15) We now look for a solution of Eq. (10) which satisfies the condi tion (15). Substituting the expression (15) and relations (11) into Eq. (10) we obtain −1 2D2(Z)P(Z) +1 21 D2(Z)∂D2(Z) ∂Z−1 21 N2(Z)∂N2(Z) ∂Z= Γ(Z) (16) Using notation W{N2,D2}=N2∂D2(Z) ∂Z−D2∂N2(Z) ∂Z one can obtain the soliton solution of Eq. 3 in the form of the c hirped solitons Eqs. 4-5 and Eqs. 6-7.. Consequently, we have found the infinite ”ocean” of solutions. The methodol ogy developed does provide a systematic way of new and new chirped soliton solutions generation. 3IV. DIFFERENT REGIMES OF SOLITON MANAGEMENT Lemma 1 : Soliton GVD management. Consider the NSE (3) with constant nonlinear coefficient N 2=const and with varying along Z-coordinate GVD parameter. Suppose tha t dispersion management function is known arbitrary analytical function :D 2(Z)=Φ(Z) . The function Φ( Z) is required only to be once-differentiable and once integra ble, but otherwise arbitrary function, there is no restrictions . There are then infinite number of solutions of Eq. (3) of the form of the chirped dispersion managed dark and bright so litons Eq. 4, where the main functions P and Γ are given by D2(Z) = Φ(Z) ;P(Z) =−1/bracketleftbig C−/integraltext Φ(Z)dZ/bracketrightbig (17) Γ(Z) =1 2Φ(Z)/bracketleftbig C−/integraltext Φ(Z)dZ/bracketrightbig+1 21 Φ(Z)∂Φ(Z) ∂Z(18) Lemma 2: Soliton intensity management. Consider the NSE (3) with con stant nonlinear coefficient N 2=const and with varying along Z-coordinate the dispersion and gain. Su ppose that intensity of the soliton pulse is determined by the known management function: D 2(Z)P2(Z)=Θ(Z),where the function Θ( Z) is required only to be once-differentiable and once integrable, but otherwise arbitrary function, the re is no restrictions. There are then infinite number of solutions of Eq. (3) of the form of the chirped dispersion man aged dark and bright solitons Eq. 4 with parameters given by D2(Z)P2(Z) = Θ(Z) ;P(Z) =−/integraldisplay Θ(Z)dZ+C;D2(Z) =Θ(Z) /bracketleftbig C−/integraltext Θ(Z)dZ/bracketrightbig2 Γ(Z) =1 2Θ(Z)/bracketleftbig C−/integraltext Θ(Z)dZ/bracketrightbig+1 21 Θ(Z)∂Θ(Z) ∂Z(19) Lemma 3: Soliton pulse duration management: optimal soliton compre ssion. Consider the NSE (3) with constant nonlinear coefficient N 2=const and with varying along Z-coordinate the dispersion and gain coefficients. Suppose that pulse duration of a soliton is determined by the known an alytical function: P(Z)=Υ( Z), where the function Υ( Z) is required only to be two-differentiable , but otherwise arb itrary function, there is no restrictions. There are then infinite number of solutions of Eq. (3) of the form of the chirp ed dispersion managed dark and bright solitons Eq. 4 with the main parameters given by D2(Z) =−1 Υ2(Z)∂Υ(Z) ∂Z; Γ(Z) =1 2/parenleftbigg∂Υ(Z) ∂Z/parenrightbigg−1∂ ∂Z/parenleftbigg1 Υ(Z)∂Υ(Z) ∂Z/parenrightbigg (20) Lemma 4 : Soliton amplification management: optimal soliton compre ssion. Consider the NSE (3) with constant nonlinear coefficient N 2=const and with varying along Z-coordinate the dispersion and gain coefficients. Suppose that the gain coefficient is determined by the known control fu nction: Γ(Z)=Λ( Z),where the function Λ( Z) is required only to be once integrable , but otherwise arbitrary functio n, there is no restrictions. There are then infinite number of solutions of Eq. (3) of the form of the chirped dispersion m anaged dark and bright solitons of the Eq. 4 where |P(Z)|= exp/bracketleftbigg/integraldisplay exp/parenleftbigg/integraldisplay 2Λ(Z′′)dZ′′/parenrightbigg dZ′/bracketrightbigg (21) |D2(Z)|=exp/parenleftbig/integraltext2Λ(Z)dZ/parenrightbig exp/bracketleftbig/integraltext exp/parenleftbig/integraltext 2Λ(Z′′)dZ′′/parenrightbig dZ′/bracketrightbig (22) Lemma 5: Combined dispersion and nonlinear soliton management. Con sider the NSE (3) with varying nonlinear coefficient N 2(Z) and with varying along Z-coordinate the dispersion and gai n coefficients too. Suppose that Wronskian W[N 2,D2] is vanishing, or that the functions N 2(Z) and D 2(Z) are linearly dependent. Suppose also that the function D2(Z) is determined by the initial control function D 2(Z)=Ξ(Z),where the function Ξ( Z) is required only to be once integrable, but otherwise arbitrary function, there is no r estrictions. There are then infinite number of solutions of Eq. (3) of the form of the chirped dispersion managed dark and bright solitons of the Eq. 6 where 4P(Z) =−1/bracketleftbig C−/integraltext Ξ(Z)dZ/bracketrightbig;N2(Z) =D2(Z)/C2(23) Γ(Z) =1 2Ξ(Z)/bracketleftbig C−/integraltext Ξ(Z)dZ/bracketrightbig (24) The analytical solutions for the different regimes of the mai n soliton parameters management (intensity, pulse duratio n, amplification or absorption ) in the case of W[N 2,D2]=0 can be obtained by using theorem 2. Let us consider some examples. The case of Γ( Z)≡0 and N 2(Z)=N 2(0) corresponds to the problem of ideal GVD soliton management. The soliton solution in this case is: q(Z,T) =−ηN−1/2 2(0)exp(C 2Z) sech [ηTexp(CZ)] (25) exp/bracketleftbigg −iT2C 2exp(CZ)−i1 2η2Zexp(CZ)/bracketrightbigg (26) q(Z,T) =ηN−1/2 2(0)exp(C 2Z) th [ηTexp(CZ] (27) exp/bracketleftbigg iT2C 2exp(CZ)−iη2Zexp(CZ)/bracketrightbigg (28) HereTandZare ordinary variables and Cis arbitrary constant. If we use the expressions D 2(Z)=constant and N2=N2(0) then we obtain the following solutions of Eq. (3) in the fo rm of hyperbolically growing ideal bright and dark solitons (for the first time reported in [12,13] q(Z,T) =−χ N−1/2 2(0) (1−2Γ(0)Z)sech/bracketleftbiggχT (1−2Γ(0)Z)/bracketrightbigg exp/bracketleftbigg −iT2Γ(0) (1−2Γ(0)Z)−iχ2Z 2(1−2Γ(0)Z)/bracketrightbigg (29) q(Z,T) =χ N−1/2 2(0) (1−2Γ(0)Z)th/bracketleftbiggχT (1−2Γ(0)Z)/bracketrightbigg exp/bracketleftbigg iT2Γ(0) (1−2Γ(0)Z)−iχ2Z (1−2Γ(0)Z)/bracketrightbigg (30) In the case of Γ( Z)≡G0and N 2=N2(0) the solution of Eq. 3 is given by: Q(P(Z)T) =ηN−1/2 2(0) sech [ηP(Z)T] (31) Q(P(Z)T) =ηN−1/2 2(0) th [ηP(Z)T] (32) P(Z) =−P(0)exp(1 2G0(exp(2G0Z)−1)) (33) D2(Z) =D2(0)exp(2G0Z−1 2G0(exp(2G0Z)−1)) (34) When GVD is a hyperbolically decreasing function of Z D2(Z) =1 1 +βZ(35) then from Lemma 1 follows the explicit soliton solution in th e form of Eq. 4 P(Z) =−1 1−1 βln(1 +βZ)(36) 5Γ(Z) =1 2(1 +βZ)/bracketleftBigg 1−ln(1 +βZ) 1−1 βln(1 +βZ)/bracketrightBigg (37) Let us consider the soliton intensity management problem. C hirped soliton pulse of Eq. 3 with the constant intensity can be obtained by using Lemma 2 P(Z) =−CZ−1;D2(Z) =C/(1 +CZ)2; Γ(Z) =−C/2/(1 +CZ) (38) Let us consider some periodical chirped soliton solutions o f Eq. 3. Suppose that the soliton intensity varies periodica lly as D2(Z)P2(Z) = Θ(Z) = 1 +δsin2nZ (39) Then soliton solution in the case of n=2 is determined by Eq. 4 with parameters: D2(Z) = Θ(Z)/P2(Z);P(Z) =C−/bracketleftbigg Z+δ/parenleftbigg3Z 8−sin2Z 4+sin 4Z 32/parenrightbigg/bracketrightbigg (40) Γ(Z) =1 2/parenleftbig 1 +δsin4Z/parenrightbig C−/bracketleftbig Z+δ/parenleftbig3Z 8−sin 2Z 4+sin 4Z 32/parenrightbig/bracketrightbig+1 22 sin 2Zsin2Z/parenleftbig 1 +δsin4Z/parenrightbig (41) Let us consider some periodical solutions of Eq. 3 in the case of the linearly dependent parameters of the media. The simplest solution of Eq. 3 in the form of Eq. 6 is: P(Z) = Υ(Z) =−/parenleftbig 1 +δsin2Z/parenrightbig ;N2(Z) =D2(Z) =δsin 2Z /parenleftbig 1 +δsin2Z/parenrightbig2; (42) Γ(Z) =δ 2sin2Z/parenleftbig 1 +δsin2Z/parenrightbig (43) The next periodical soliton solution is given by D2(Z) =N2(Z) = cosZ;P(Z) =−1 (C−sinZ); Γ(Z) =cosZ 2 (C−sinZ)(44) The main soliton features of the solutions given by theorem 1 and theorem 2 were investigated by using direct computer simulations. We have investigated the interactio n dynamics of particle-like solutions obtained, their soli ton- like character was calculated with the accuracy as high as 10−9. We also have investigated the influence of high-order effects on the dynamics of dispersion and amplification manag ement. As follows from numerical investigations elastic character of chirped solitons interacting does not depend o n a number of interacting solitons and their phases. Figure 2 shows the computer simulation dynamics of three hyperbolic ally growing solitons Eq. 29. NSE solution with periodic dispersion coefficient is shown in Figure 3. Here the dispersi on management function is D2(Z) = 1 +δsin2(Z) (45) and the soliton solution is given by Eqs. 17-18. In Figure 3 pa rameters C=200 and δ=−0.9.Figure 4 represents the two dispersion managed solitons interaction in the case of e qual phases and in Figure 5 the interaction dynamics of two solitons is shown in the case of opposite phases. Figure 6 shows the intensity managed solitons dynamics of the form presented by Eq. 38. Figures 7-9 show the nonlinear prop agation and interaction of the dispersion and nonlinear managed solitons of Eqs. 42-43. The main parameters in compu ter simulations were C=200; δ=±0.9. Figure 10 illustrates the dynamics of the fission of the bound states of two hyperbolically growing solitons Eqs. 29-30 produced by self-induced Raman scattering effect given by Eqs.2-3. Th is remarkable fact also emphasize the full soliton features of solutions discussed. They not only interact elastically but they can form the bound states and these bound states split under perturbations. The possibility to find the pleth ora of soliton solutions in the case of strong dispersion management is reported in the recent paper of Zakharov and Ma nakov [14]. 6[1] A. Hasegawa, F. Tappert, “Transmission of stationary no nlinear optical pulses in dispersive dielectrical fibers”, Appl. Phys. Lett., v. 23, pp. 142-144, 1973. [2] L.F. Mollenauer, R.G. Stolen, J.P.Gordon, “Experiment al observation of picosecond pulse narrowing and solitons i n optical fibers”, Phys. Rev. Lett., v. 45, pp. 1095-1098, 1980. [3] V.E. Zakharov and A.B. Shabat, “Exact theory of two-dime nsional self-focusing and one-dimensional self-modulati on of waves in nonlinear media”, Zh. Eksp. Teor. Fiz., v. 36, pp.61 -71, 1971. [4]Optical solitons - theory and experiment , ed. by J.R. Taylor, Cambridge Univ. Press, 1992 [5] V.N. Serkin, V.A. Vysloukh and J.R. Taylor, “Soliton spe ctral tunneling effect”, Electron. Lett., v. 29, pp. 12-13, 1 993. [6] E.M. Dianov, A.Ya. Karasik, P.V. Mamyshev, A.M. Prokhor ov, V.N. Serkin, M.F. Stel’makh and A.A. Fomichev, “Stim- ulated Raman conversion of multisoliton pulses in quartz op tical fibers”, JETP Lett., v. 41, pp. 294-297, 1985. [7] F.M. Mitschke and L.F. Mollenauer, “Discovery of the sol iton self-frequency shift”, Opt. Lett., v. 11, pp. 659-661, 1986. [8] V.V. Afanasjev, V.N. Serkin and V.A. Vysloukh, “Amplific ation and compression of femtosecond optical solitons in ac tive fibers”, Sov. Lightwave Commun., v. 2, pp. 35-58, 1992. [9] S. Kumar, A. Hasegawa, “Quasi-soliton propagation in di spersion managed optical fibers”, Opt. Lett., v. 22, pp. 372- 374, 1997. [10] Akira Hasegawa, “Quasi-soliton for ultra-high speed c ommunications”, Physica D, v. 123, pp. 267-270, 1998. [11] Yuji Kodama, “Nonlinear pulse propagation in dispersi on managed system”, Physica D, V. 123, pp. 255-266, 1998. [12] John D. Moores, “Nonlinear compression of chirped soli tary waves with and without phase modulation”, Opt. Lett., v . 21, pp. 555-557, 1996. [13] V.Y. Khasilev, “Optimal control of all-optical commun ication soliton systems”, SPIE Proceedings, v. 2919, pp. 17 7-188, 1996. [14] V.E.Zakharov and S.V. Manakov, “On propagation of shor t pulses in strong dispersion managed optical lines”, JETP Lett., v.70, pp. 578-582, 1999. FIG. 1. Femtosecond soliton spectral trapping effect. FIG. 2. Mutual interaction of three hyperbolically growing chirped solitons of Eq. 29 in the case of equal amplitudes and phases. FIG. 3. Evolution of the chirped dispersion managed solitar y wave of Eqs. 17 and 18 as a function of the propagation distance. Dispersion managed function is periodic of the fo rm Eq. 45. Input conditions : C=200 and δ=-0.9. FIG. 4. Two dispersion managed solitons of Eqs. 17-18 and 45 i nteraction for the case of equal phases. Input conditions: C=200 and δ=-0.9. FIG. 5. Two dispersion managed solitons of Eqs. 17-18 and 45 i nteraction for the case of equal phases. Input conditions: C=200 and δ=-0.9. FIG. 6. Two intensity managed solitons of Eq. 38 interaction for the case of zero initial group velocities. FIG. 7. Evolution of the chirped solitaty wave of Eqs. 39-41 a s a function of the propagation distance. Input conditions: δ=0.9 and group velocity V0=10. FIG. 8. Evolution of the chirped solitary wave of Eqs. 39-41 f or the case: δ=-0.8 and group velocity V0=2.0. FIG. 9. Soliton dispersion trapping effect in the presence of the linearly dependence between the nonlinearity and GVD parameters. FIG. 10. Decay of high-order hyperbolically growing solito ns in the presence of third-order dispersion and Raman self-scattering effects. 7This figure "F01.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F02.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F03.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F04.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F05.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F06.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F07.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F08.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F09.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1This figure "F10.JPG" is available in "JPG" format from: http://arXiv.org/ps/physics/0002027v1

- 1 -A Connection Between Gravitation and Electromagnetism Douglas M. Snyder Los Angeles, California1 It is argued that there is a connection between the fundamental forces of electromagnetism and gravitation. This connection occurs because of: 1) thefundamental significance of the finite and invariant velocity of light in inertial reference frames in the special theory, and 2) the reliance of the general theory of relativity upon the special theory of relativity locally in spacetime. 2 The connection between the fundamental forces of electromagnetism and gravitationfollows immediately from these two points. Because this connection has not been acknowledged, a brief review is provided of: 1) the role of the finite and invariant velocity of light in inertial reference frames in the special theory, and 2) certain fundamental concepts of the general theory, including its reliance onthe special theory locally. 1. The Velocity of Light in the Special Theory It is known that the reason that the Lorentz transformation equations of special relativity differ from the Galilean transformation equations underlying Newtonian mechanics is due to the finite value for the invariant velocity of light in any inertial reference frame (i.e., a spatial coordinate system attached to a physical body and for which Newton's first law of motion holds) that is used todevelop simultaneity and time in general in any inertial reference frame. Other results obtained in the special theory of relativity depend on the Lorentz transformation equations, as opposed to the Galilean transformation equations, and are thus dependent on the finite and invariant velocity of light. (1,2,3) For an inertial reference frame W' with one spatial dimension, x', in uniform translational motion relative to an inertial reference frame W (with onespatial dimension, x) with velocity v along the x and x' axes in the direction of increasing values of x and x', the Lorentz transformation equations may be stated as: x' = (x - vt)/(1 - v 2/c2)1/2 (1) and t' = [t - (v/c2)x]/(1 - v2/c2)1/2 . (2)Gravitation and Electromagnetism - 2 -c is the finite and invariant value for the velocity of light in inertial reference frames. c is found in equations 1 and 2 because of the fundamental role of the velocity of light in the development of simultaneity and time in general in inertial reference frames in the special theory.(1) If an existent with an arbitrarily great velocity (also represented by c) were used instead of light to develop simultaneity in an inertial reference frame, c would be arbitrarily large and equations 1 and 2 would become: x’ = x - vt (3) and t' = t . (4) Equations 3 and 4 are Galilean transformation equations for W and W' for the circumstances described. As an example of the significance of the finite and invariant velocity of light in the special theory, it can be noted that the Lorentz transformation equations underlie the derivation of the force law in the special theory as well asthe transformation of force in inertial reference frames in uniform translational motion relative to one another. (4,5) These transformation of force equations in the special theory allow for the invariance of Maxwell's equations of electromagnetism in inertial reference frames in uniform translational motion relative to one another.(6) In fact, one can use the Lorentz transformation equations to transform the field differentials in Maxwell's laws of electromagnetism for two inertial reference frames in uniform translational motion relative to one another while holding that these laws should be invariant.Doing this, one finds that the transformations for the electric and magnetic components of the field indicated are exactly those that one finds by application of the transformation equations for force in the special theory to the electric and magnetic components of the electromagnetic field. It should be remembered that one of Maxwell's most fundamental results was the identification of electromagnetic radiation with light because themeasured velocity of light in vacuum is in accord with the velocity of electromagnetic radiation in vacuum derived using Maxwell's equations. It should be emphasized that this identification of light with electromagnetic radiation through Maxwell's equations provides the theoretical foundation for the empirical evidence that the velocity of light in vacuum is finite and invariantand thus does not depend on the velocity of the emitting body relative to the observer at rest in an inertial reference frame.Gravitation and Electromagnetism - 3 -2. The Dependence of the General Theory on the Special Theory 2.1 The Principle of Equivalence It is known that in the general theory of relativity gravitation corresponds to spacetime curvature. Spacetime curvature may be developed byconsidering an accelerating reference frame or pattern of such reference frames as a pattern of local Lorentz frames moving at slightly different uniform, translational velocities relative to one another. The accelerating reference frame, or frames, in general corresponds to an inertial reference frame experiencing a gravitational field because the descriptions of physical phenomena in these circumstances are equivalent. In particular, the descriptions of the motion of physical bodies when described from a uniformly accelerating reference frame or an inertial reference frame experiencing a gravitational field of uniform intensity are equivalent. (7,8,9) Thus, a uniformly accelerating reference frame serves as a bridge between the gravitation-free inertial reference frame from which the motion of the uniformly ac celerating reference frame is judged and the inertial reference frame experiencing a gravitational field of uniform intensity that is no different for the description of physical phenomena to the uniformly accelerating reference frame. The inertial reference frame experiencing the gravitational field of uniform intensity is equivalent to the gravitation-free inertial reference frame as concerns the description of physical phenomena. The principle of equivalence provides the theoretical foundation for the basic equality of inertial and gravitational mass and allows for the in general equivalence between accelerating reference frames and inertial reference frames experiencing a gravitational field.(9,10,11) In The Meaning of Relativity, Einstein stated the principle of equivalence in the following way and indicated how it provides the theoretical underpinning for the essential equality of inertial and gravitational mass: Let now K be an inertial system. Masses which are sufficiently far from each other and from other bodies are then, with respectto K, free from acceleration. We shall also refer these masses to a system of co-ordinates K', uniformly accelerated with respect to K. Relatively to K' all the masses have equal and para llel accelerations; with respect to K’ they behave just as if a gravitational field were present and K' were unaccelerated. Overlooking for the present the question as to the “cause” ofGravitation and Electromagnetism - 4 -such a gravitational field...there is nothing to prevent our conceiving this gravitational field as real, that is, the conception that K' is “at rest” and a gravitational field is present we may consider as equivalent to the conception that only K is an “allowable” system of co-ordinates and no gravitational field is present. The assumption of the complete physical equivalence of the systems of coordinates, K and K', we call the “principle of equivalence;” this principle is evidently intimately connectedwith the law of the equality between the inert and the gravitational mass, and signifies an extension of the principle of relativity to co-ordinate systems which are in non-uniform motion relatively to each other. In fact, through this conception we arrive at the unity of the nature of inertia and gravitation. Foraccording to our way of looking at it, the same masses may appear to be either under the action of inertia alone (with respect to K) or under the combined action of inertia and gravitation (with respect to K'). (12) 2.2 The Special Theory Holds Locally As noted, the derivation of the spacetime curvature associated with an accelerating reference frame may be based on the notion that an accelerating reference frame is composed of a pattern of local Lorentz frames (essentially, local, inertial reference frames). These local Lorentz frames are considered to have varying uniform, translational velocities, and the Lorentz transformation equations of special relativity thus hold for these local reference frames. In outlining the essentials of the general theory of relativity, Einstein discussed an example of how an observer in an accelerating reference frame would be affected in his measurements of temporal duration and how his or hermeasurements could be accounted for in terms of a local Lorentz frame. He wrote: Let us consider a space-time domain in which no gravitational field exists relative to a reference-body K whose state of motion has been suitably chosen. K is then a Galileian reference-body as regards the domain considered, and the results of the special theory of relativity hold relative to K. Let us suppose the same domain referred to a second body of reference K', which is rotating uniformly with respect to K. In order to fix our ideas,Gravitation and Electromagnetism - 5 -we shall imagine K' to be in the form of a plane circular disc, which rotates uniformly in its own plane about its centre. An observer who is sitting eccentrically on the disc K' is sensible of a force which acts outwards in a radial direction, and which would be interpreted as an effect of inertial (centrifugal force) byan observer who was at rest with respect to the original reference-body K. But the observer on the disc may regard his disc as a reference-body which is “at rest”; on the basis of the general principle of relativity he is justified in doing this. This force acting on himself, and in fact on all other bodies which areat rest relative to the disc, he regards as the effect of a gravitational field. But since the observer believes in the generaltheory of rel ativity, this does not disturb him: he is quite in the right when he believes that a general law of gravitation can be formulated -a law which not only explains the motion of the stars correctly, but also the field of force experienced by himself. The observer performs experiments on his circular disc with clocks and measuring-rods. In doing so, it is his intention to arrive at exact definitions for the signification of time- and space-data with reference to the circular disc K', these definitions being based on his observations. What will be his experience in this enterprise? To start with, he places one of two identically constructed clocks at the centre of the circular disc, and the other on the edgeof the disc, so that they are at rest relative to it. We now ask ourselves whether both clocks go at the same rate from the standpoint of the non-rotating Galileian reference-body K. As judged from this body, the clock at the centre of the disc has no velocity, whereas the clock at the edge of the disc is in motion relative to K in consequence of the rotation....the latter clock goes at a rate permanently slower than that of the clock at the centre of the circular disc, i.e., as observed from K. It is obvious that the same effect would be noted by an observer whom we will imagine sitting alongside his clock at the centre of the circular disc. Thus on our circular disc, or, to make the case more general, in every gravitational field, a clock will goGravitation and Electromagnetism - 6 -more quickly or less quickly, according to the position in which the clock is situated (at rest).(13) What is the general relation for the durations of some occurrence in two inertial reference frames moving in a uniform translational manner relative to one another? Consider the inertial reference frames W and W' discussed earlier. If this occurrence maintains its position in W', then the relation betweenthe durations of this occurrence in W and W' is given by: Dt = Dt'/(1 - v 2/c2)1/2 . (5) Spacetime curvature may then be found for an accelerating frame, or an associated inertial reference frame in a gravitational field, essentially by derivingthe pattern of special relativistic results that hold for the local Lorentz frames and that compose the global reference frame. (14,15,16) Specifically, the curvature of the spacetime continuum can be developed through the use of a field tensor, gik. The use of this tensor allows for local Lorentz frames while also allowing that these local Lorentz frames do not have to be identical in terms of their spacetime characteristics. Thus, there is the possibility of a pattern of different,local Lorentz frames, the result of which is a curved spacetime continuum. Theparticular coordinate scheme that one applies to the spacetime continuum is arbitrary as long as it is Gaussian in nature. g ik are certain functions of the Gaussian coordinate scheme that transform for a continuous coordinate transformation. gik has a symmetrical property (i.e., gik = gki) that holds in general for a Gaussian coordinate scheme applied to the spacetime continuum. gik must be able to account for global, gravitation-free inertial reference frames. In the case of such inertial reference frames, gik reduces to a particular form such that the resulting metric is that characteristic of a Lorentz reference frame. The general covariant field law that applies to gik of different representations of the spacetime continuum is given by a set of differential equations in the form of another symmetrical tensor, Rik.(17,18) Misner, Thorne, and Wheeler summed up the relation of the general and special theories in writing, “General relativity is built on special relativity”.(19) In elaborating on this statement, the authors wrote: A tourist in a powered interplanetary rocket feels “gravity.” Can a physicist by local effects convince him that this “gravity” is bogus? Never, says Einstein's principle of the local equivalence of gravity and accelerations. But then the physicist will make no errors if he deludes himself into treating trueGravitation and Electromagnetism - 7 -gravity as a local illusion caused by acceleration. Under this delusion, he barges ahead and solves gravitational problems by using special relativity: if he is clever enough to divide every problem into a network of local questions, each solvable under such a delusion, then he can work out all influences of any gravitational field. Only three basic principles are invoked: special relativity physics, the equivalence principle, and the localnature of physics. They are simple and clear. To apply them, however, imposes a double task: (1) take spacetime apart into locally flat pieces (where the principles are valid), and (2) put these pieces together again into a comprehensible picture. Toundertake this dissection and reconstitution, to see curveddynamic spacetime inescapably take form, and to see the consequences for physics: that is general relativity. (19) 3. The General Principle of Relativity The principle of equivalence led Einstein to maintain that all frames of reference should be equivalent for the expression of physical law. The conceptual basis of space-time structure in the general theory has been noted and has been shown to be dependent upon its description locally using the special theory. With these aspects of the general theory in mind, the general principle of relativity can be more precisely stated as: All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.(20) Einstein wrote in his original paper proposing the general theory: In the general theory of relativity, space and time cannot be defined in such a way that differences of the spatial co-ordinatescan be directly measured by the unit measuring-rod, or differences in the time co-ordinate by a standard clock. The method hitherto employed [in the special theory as well as Newtonian mechanics adhering to Euclidean geometry] forlaying co-ordinates into the space-time continuum in a definite manner thus breaks down, and there seems to be no other way which would allow us to adapt systems of co-ordinates to the four-dimensional universe so that we might expect from their application a particularly simple formulation of the laws ofGravitation and Electromagnetism - 8 -nature. So there is nothing for it but to regard all imaginable systems of co-ordinates, on principle, as equally suitable for the description of nature. This comes to requiring that: - The general laws of nature are to be expressed by equations which hold good for all systems of coordinates, that is, are co- variant with respect to any substitutions whatever (generally co- variant).(21) The use of arbitrary Gaussian coordinate schemes, and their accompanying metrical coefficients gik, to represent the spacetime continuum allows for the development of the general formulation of physical law that applies to these coordinate schemes. As noted, gik transform for a continuous coordinate transformation. Rik is a formulation of gravitational field law for gik. 4. Light and the General Theory An important result of relativity theory, specifically the special theory, is that light has properties associated with mass. This is a consequence of the general equivalence of mass and energy in the special theory. The general equation relating mass and energy is E = mc2 , (6) where E is the energy equivalent of the relativistic mass of a physical entity, m is this relativistic mass, and c is the finite and invariant velocity of light in inertial reference frames. In the special theory, the kinetic energy of a physical existent can be expressed by the equation K = mc2 - m0c2 , (7) where K is the kinetic energy, m is the relativistic mass of the physical existent, m0 is the rest mass of this existent, and c is the finite and invariant velocity of light in inertial reference frames. Another form for the kinetic energy is K = m0c2[1/(1 - v2/c2)1/2 - 1] , (8) where v is the uniform, translational velocity of the inertial reference frames relative to one another. The relation between the relativistic mass of a physical existent, m, and its rest mass, m0, is given by m = m0/(1 - v2/c2)1/2 , (9) and the momentum is given byGravitation and Electromagnetism - 9 -p = [m0/(1 - v2/c2)1/2]v . (10) Combining equations 8 and 10 by removing v results in (K + m0c2)2 = (pc)2 + (m0c2)2 . (11) Given E = K + m0c2 , (12) E2 = (pc)2 + (m0c2)2 . (22)(13) Light has no rest mass in the special theory as it is never at rest according to the empirically verified postulate of the special theory regarding the invariant andfinite velocity of light in inertial reference frames. But light does have momentum, p. For light, E = pc (14) in the special theory, a relation found in classical electromagnetic theory. Light therefore has effective mass. In sensitive enough conditions, the influence of a gravitational field on light should thus be detectable. Using the general theory, Einstein predicted that a light ray would be deflected toward the sun in accordance with the following equation m = 1.7 sec. of arc/ D (15) where m is the angle of deflection and D is the distance of the light ray from the center of the sun. Experiments conducted in 1919 confirmed Einstein's prediction concerning the influence of a gravitational field on the motion of light.(23,24) 4.1 The Velocity of Light: Curvilinear Globally, Finite and Invariant Locally Einstein distinguished the global velocity of light in the special theory from the global velocity of light in the general theory: With respect to the Galileian reference body K, a ray of light is transmitted rectilinearly with the velocity c [in accord with the postulate of the special theory]. It can be easily shown that the path of the same ray is no longer a straight line when we consider it with reference to the [uniformly] accelerated chest (reference body K') [relative to K]. From this we conclude, that,Gravitation and Electromagnetism - 10 -in general, rays of light are propagated curvilinearly in gravitational fields .(25) This is characteristic of the distinction usually made between the general and special theories concerning the motion of light. For example, in his original paper on the general theory, Einstein wrote: It will be seen from these reflexions that in pursuing the general theory of relativity we shall be led to a theory of gravitation,since we are able to “produce” a gravitational field merely by changing the system of co-ordinates. It will also be obvious thatthe principle of the constancy of the velocity of light in vacuomust be modified, since we easily recognize that the path of a ray of light with respect to K' [a uniformly accelerating reference frame] must be in general curvilinear, if with respect to K [a gravitation-free inertial reference frame relative to which K' is uniformly accelerating] light is propagated in a straight line with a definite constant velocity. (26) Usually, in discussions of the general theory, there is little emphasis placed on the finite and invariant velocity of light locally. Yet, Einstein did mention this last characteristic of light in the general theory in The Meaning of Relativity where he wrote: In the general theory of relativity also the velocity of light is everywhere the same, relatively to a local inertial system. This velocity is unity in our natural measure of time.(27) Also, in his original paper on the general theory, Einstein implied in a footnote that the velocity of light locally is invariant and finite: We must choose the acceleration of the infinitely small (“local”) system of co-ordinates so that no gravitational field occurs; this is possible for an infinitely small region. Let X1, X2, X3, be the co-ordinates of space, and X4, the appertaining co-ordinate of time measured in the appropriate unit.* If a rigid rod is imagined to be given as the unit measure, the co-ordinates, with a given orientation of the sy stem of co-ordinates, have a direct physical meaning in the sense of the special theory of relativity. * The unit of time is to be chosen so that the velocity of light in vacuo as measured in the “local” system of co-ordinates is to beGravitation and Electromagnetism - 11 -equal to unity.(28) 5. Light and Gravitation If the spacetime continuum is locally Lorentzian, and the velocity of light is therefore locally invariant and finite, then the locally invariant and finite velocity of light provides a connection between the fundamental forces of gravitation and electromagnetism. The invariant and finite velocity of light is fundamental to the special theory. The special theory is essential to the development of the spacetime continuum cha racteristic of the general theory. Therefore, the locally invariant and finite velocity of light is fundamental to thegeneral theory. Contrast this conception of the motion of light with that for the motion of light globally in the general theory. Here, the generally curvilinear motion of light is dependent on its having mass, as first indicated in relativity theory in thespecial theory. Light, like any other physical entity, must hold to the principle of equivalence. Thus, its curvilinear motion in an accelerating reference frame, which can be equivalently considered an inertial reference frame experiencing a gravitational field, can be interpreted as the effect of the gravitational field on thelight passing through it. But these interesting considerations regarding the motion of light globally do not alter the fundamental dependence of gravitation in the general theory on the finite and invariant velocity of light locally. In fact the predictions in the general theory concerning the motion of light in a gravitational field, which have been empirically supported, depend on the velocity of light being finite and invariant locally. From a global view, in the general theory, the velocity of light is usually not invariant in reference frames. This velocity varies as spacetime is usually acurved continuum. Nonetheless, locally, the general theory relies fundamentally on the special theory. And the special theory relies fundamentally on the finite and invariant velocity of light in inertial reference frames. Why has the dependence of the general theory on the finite and invariant velocity of light locally not been widely acknowledged? It may well be that it isdifficult to maintain simultaneously that globally light travels in curved spacetime and that locally it has a finite and invariant velocity. This, though, is the case. When the dependence of the general theory on the velocity of light locally is acknowledged, and light is not considered simply another physical entity that is affected by the general theory (as happens when light is consideredGravitation and Electromagnetism - 12 -globally and bent by a gravitational field), the connection between the fundamental forces of gravity and electromagnetism is evident. 6. Conclusion It has been extremely difficult to integrate gravitation with the other fundamental forces of nature. It is clear, though, that gravitation depends on a fundamental characteristic of light in the special theory, namely the finite and invariant velocity of light. It is the case that this finite and invariant velocity of light holds only in local Lorentz frames, unless of course one is concerned withflat spacetime. But in principle , the velocity of light is finite and invariant locally, even if on a global level in the general theory it cannot be said to generally have an invariant velocity. The reason this dependence has not been recognized is likely due to the focus on the very interesting results in the general theory concerning the velocity of light globally. This velocity stands in such contrast to the finite and invariant velocity in the special theory that it seems a contradiction that light could have a curvilinear velocity globally and yet have a finite and invariantvelocity locally. These are the circumstances, though, and the finite and invariant velocity of light locally in the general theory provides the theoretical basis for connecting the fundamental forces of gravitation as developed in the general theory and electromagnetism that is elegantly discussed in the special theory. Endnotes 1Email: dsnyder@earthlink.net 2Some authors, including Einstein, maintain that in the special theory, it is the velocity of light that is finite and invariant in vacuum. Other authors believe it is more accurate to write that the speed of light is finite and invariant in vacuum. I believe that velocity is the better term to use. It is true that light can move in more than a single direction, thus implying that its velocity is not invariant. But, the essence of the use of the term velocity as regards light in thespecial theory is that for a given ray of light in inertial reference frames in uniform translational motion relative to one another, its speed and direction in vacuum do not change. References 1. A. Einstein, "On the electrodynamics of moving bodies," in H. Lorentz, A. Einstein, H. Minkowski, and H. Weyl (editors), The Principle of Relativity, aGravitation and Electromagnetism - 13 -Collection of Original Memoirs on the Special and General Theories of Relativity (Dover, 1952, originally 1905), p. 53. 2. A. Einstein, Relativity, the Special and the General Theory (Bonanza, 1961, originally 1917), p. 44. 3. R. Resnick, Introduction to Special Relativity . (John Wiley & Sons, 1961), p.79 4. Ibid., p. 119.5. W. G. Rosser, Contemporary Physics 1, 453, (1960). 6. Ref. 3, p. 163.7. Ref. 2, p. 66.8. A. Einstein, "The foundation of the general theory of relativity," in H. Lorentz, A. Einstein, H. Minkowski, and H. Weyl (editors), The Principle of Relativity, a Collection of Original Memoirs on the Special and General Theories of Relativity (Dover, 1952, originally 1916), p. 113. 9. A. Einstein, The Meaning of Relativity (Princeton University Press, 1956, originally 1922), p. 57. 10. Ref. 2, p. 66.11. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics: Mainly Electromagnetism and Matter , (Addison-Wesley, 1964), Vol. 2, p. 42-8. 12. Ref. 9, p. 57.13. Ref. 2, p. 79.14. Ref. 8, p. 115.15. M. Born, Einstein's Theory of Relativity . (Dover, 1965, originally 1924), p. 336. 16. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, 1973), p. 164. 17. Ref. 15, p. 324.18. A. Einstein, Scientific American , 182(4), 13 (1950). 19. Ref. 16, p. 164.20. Ref. 2, p. 97.21. Ref. 8, p. 117.22. Ref. 3, P. 123.23. Ref. 2, p. 126.24. A. Pais, A., 'Subtle is the Lord....': The Science and the Life of Albert Einstein (Oxford University Press, 1982), p. 303. 25. Ref. 2, p. 75.26. Ref. 8, P. 114.27. Ref. 9, p. 92. 28. Ref. 8, p. 118.

An Essay on the Relationship between the Mind and the Physical WorldAn Essay on the Relationship between the Mind and the Physical World Douglas M. Snyder Tailor Press Los Angeles, CaliforniaCopyright 1997 by Douglas M. Snyder All rights reserved Cover: Fishing by Ruth C. Snyder Tailor Press P.O. Box 18310 Beverly Hills, California 90209-4310 ISBN 0-9653689-1-2 Library of Congress Catalog Card Number: 97-90196 Printed in the United States of AmericaFor Shoshana- vii -Table of Contents PREFACE...........................................................................ix 1 - HERE WE GO!......................................................................1 2 - FUNNY THINGS ABOUT LIGHT......................................................2 3 - A S LIGHT DIGRESSION ..............................................................8 4 - CONSERVATION LAWS AND WHAT?.................................................9 5 - IS GRAVITY A PROBLEM?..........................................................13 6 - QUANTUM MECHANICS - NO, IT CAN’T BE!.....................................18 7 - OH NO! TEMPERATURE TOO?.....................................................23 8 - OH, I’VE JUST FALLEN DOWN, OR IS THAT UP?..................................29 9 - A F INAL WORD....................................................................32 SOURCES FOR THE QUOTES.........................................................34 SOME OTHER SOURCES............................................................34- ix -Preface I’ve endeavored in this essay to make my points simply. If you would like a more technical and detailed discussion of the points that are to follow, you might look at my book, The Mind and the Physical World: A Psychologist’s Exploration of Modern Physical Theory . It also is published by Tailor Press. I would like to thank Dr. Arthur Huffman for thought-provoking discussions and for clarifying various elements of physical theory. I also wish to thank Donald Musco and Dr. Jeffrey Simon for their incisive editorial comments.- 1 -Chapter 1 Here We Go! We are all used to looking around in the world and considering that we are observing things that are really there, just as they appear, whether or not weare looking at them. We think that things in the world are not affected by our thinking about them. We think that the world was there, just as it appears to usnow, before we lived and will continue to be there after we are no longer alive. We know that our bodily actions occur in the physical world, but for the most part we don’t think our minds can directly affect the physical world, outside of our intentions, perhaps, that are expressed in our bodies’ actions. Instead, webelieve that things in the physical world determine how we experience them. When we hear something, for example, it is due to vibrations emanating from something in the physical world that travel through the air until they reach our ears. Contrary to our view of the relationship, or really lack of it, between the mind and the physical world, modern physical theory indicates that the mind directly affects the physical world. The mind affects both the structure of the physical world and what happens in it. You probably don’t believe me. But I think you will when we’re done. For now, please bear with me.- 2 -Chapter 2 Funny Things About Light Light is very strange, and it is also very important. More important than you think. I know you think it is important because you know that we use light to see. But there is more. Light also is at the root of time, and this is what weneed to explore. Imagine yourself driving down the highway at 50 miles per hour. And imagine further that a car passes you that is going in the same direction as you and that its speed is 65 miles per hour. If I ask you how fast this other car is going relative to your car, your answer will be 15 miles per hour. You would be right. The reason you said the other car is going 15 miles per hour relative to you is because your own velocity and the other car’s velocity relative to you need to add up to the velocity of the other car relative to the road. Now try this. Imagine that this other car is actually a ray of light. Relative to the road, the ray of light has a velocity of c (about 186,000 miles per second). If I ask you what is the velocity of the light ray relative to you, I bet I know what you will say. It is c - 50 miles per hour. You’ve subtracted your velocity relative to the road from the velocity of the light ray relative to the road. You know what? This is not what happens. If you are not sitting down, I think you should because it is very hard to believe that the velocity of light relative to you is also c. I will state this again because it is so surprising. The velocity of light relative to you is the same as it is relative to the road. Why thisis the case, no one knows. That this is the case is the basis for Einstein’s special theory of relativity. When physicists first realized the velocity of light was the same in situations like I mentioned, they could not believe it. And theydid everything they could do to make things add up. Einstein was a young man when all this was going on, and he had a different idea. He said that we should accept the constant velocity of light in situations like I mentioned. Moreover, he said we should make it the basis for time! Einstein followed through on his plan, and he came to very odd conclusions. They all hinged on one basic result. It has to do with what we call “the same time.” While you’re driving down the road, let’s say you suddenly see two explosions off to the side of the car, one behind you and one in front of you, at the same time. The explosions in front of you and behind you are the sameFunny Things - 3 -distance from you. I realize this is not an everyday occurrence, but for the sake of argument let’s just say it happens. Now if I asked you if a man standing on the side of the road whom you pass in your car just when the explosions occur sees the explosions at the same time, I think I know what you’ll say. “Yes, theother man sees the explosions at the same time.” But according to special relativity, there is another answer. Einstein decided to define time in situations like the car or the man at the side of the road in terms of the motion of light. Remember that light has the same velocity in all these situations. Because of Einstein’s definition of time, we cannot say that the man at the side of the road will also see the explosions at the same time as you driving down the road. Why? Because we no longer have the “adding up of velocities” when the velocity of light is involved. It is the adding up of velocities that keeps the time in your car the same as the time for the man at the side of the road. Without this adding up of velocities, we have lost the glue for the situations that keeps the observers in them seeing certain things the same. Like how long something is, or how long it takes for something to occur. When we work things out using the motion of light to define time in different situations like I’ve noted, we find that “the same time” in one situation is not “the same time” in the other situation. We discover that the length of an object along a certain direction in one situation is not the length of the object in the othersituation. Also, how long something takes in one situation is not how long it takes when viewed from another situation. So, for example, the man on the side of the road and you in the car come up with different lengths for a stick on the side of road that is facing the direction in which you are traveling. How long it takes for a lizard to crawl from one end of the stick to the other is not thesame for the man on the side of the road and you in the car. I know all of this sounds odd, but this is what experiments tell us is the way things work. Einstein’s decision to define time in terms of the motion of light was correct. Now I want to concentrate on this last thing, how long something takes. We use clocks to determine how long something takes. That’s not a surprise toyou. Einstein essentially relied on clocks which use the motion of light back and forth to determine how long something takes in situations like our cars and the side of the road that are moving relative to one another. Another very fine physicist, Feynman, pointed out something very interesting. If instead of usinglight clocks, we used clocks of some other kind, even our ordinary wind-up clocks at home, we would find that they work the same way as the light clocks.Chapter 2 - 4 -Why is this so interesting? The light clocks don’t have to be anywhere around the non-light clocks and the non-light clocks work the same way as the light clocks. Remember that light clocks exhibit the constant velocity of light that I discussed earlier, a property that our ordinary clocks don’t rely on because their functioning is not based on the motion of light. So why do the non-light clockswork the same way? There is no reason in the physical world why they have to. Feynman said they have to because if they don’t, people on the side of the road and in the car won’t see the things happening in the world according to thesame rules. Now you can imagine what a dilemma this would pose. Feynman’s conclusion is quite unusual. A physicist says that different kinds of things in the world work the same way in our different situations for no other reason than the need for the rules to work the same in them! What is the nature of the rules? The first thing someone thinks is that the rules are part of the physical world itself. But, if this is the case, why don’t we see some mechanism in the physical world for the light clocks to affect the non-light clocks? There is no mechanism in the physical world. So the rules are not part of the physical world itself, even though they are the basis for how the physical world works. We must go a step further in trying to discover where the rules are. Science is concerned with observation. There is no doubt about that. So we have to find some other area open to observation where we can determine the nature of the rules. Now remember, the rule that things have to work the same way on the side of the road or in the car is a fundamental principle of the theory of special relativity. We need to return to an earlier point. Remember I stated that the length of an object along a certain direction in one situation is not the length of theobject in the other situation and that how long something takes in one situation is not how long it takes when viewed from another situation. Which situation is the one where an object is longer and which situation is the one where the object is shorter? Which situation is the one where the duration of how long something takes is longer and which situation is the one where the duration of how long something takes is shorter? That’s right. I didn’t say. That’s because it’s both. Both? Yes, both. It depends on in which situation we first define “the same time” and which second. Our situations like the car and the road are moving relative to one another. Neither situation can claim to be absolutely at rest. We saw that this was the case in that the rules for how thingswork in the physical world need to be the same in our different situations. It’sFunny Things - 5 -your choice where you want objects to be shorter and where you want things to take more time. The situation where you first define “the same time” will find moving objects shorter along a certain direction and require a longer period forthings to happen. Consider a ruler. A moving ruler is shorter than an identically constructed ruler that is at rest and that is used to measure the length of the “moving” ruler. In a situation where the “moving” ruler is at rest, this ruler caninstead be used to measure the formerly “at rest” ruler which is now “moving.” The ruler that is now “at rest” will find the ruler that is now “moving” to be shorter. It sounds like a tongue twister because the whole scenario is reciprocal. We just have to decide which way we want to do our reasoning. Itcomes down to: In which situation is “the same time” defined first in our reasoning? “But,” you might say, “an object is an object and a clock is a clock. Even if things are the way they are described here, we still have something concrete, something the nature of which you can’t think away.” True enough itseems, but I can demonstrate that this concrete object or clock doesn’t seem so concrete even in a situation where it is at rest. For this next point, I will rely initially on Einstein’s own argument. Instead of automobiles going down a road, think of a railroad car going down atrack. Lightning flashes hit both ends of the track where the ends of the railroad car are and move toward each other. For an observer standing on the embankment, the lightning flashes travel equal distances toward each other before they meet. This observer concludes the lightning flashes occurred at thesame time. But what about an observer at the middle of the railroad car? If we rely on the time of the person on the embankment, which we are doing, the train moves toward one lightning flash and away from one lightning flash. Youknow they won’t meet the observer at the middle of the railroad car. This means these lightning flashes did not strike the points of the track corresponding to the ends of the railroad car at the same time for the observer on the railroad car. This last statement doesn’t sound right because, according to what I stated earlier, light should have the same velocity for the observer on the railroad car that it has for the observer on the side of the embankment. This issue needs further clarification. At this point though, how do we establish “thesame time” in the railroad car? For the light ray that has “slowed” because itChapter 2 - 6 -moves in the same direction as the train, we can find a light ray to meet it at the middle of the railroad car. But it has to leave the other end of the car where the “fast” one left later than the “fast” one. What does this mean for the length of the railroad car? It means that the length of the car is longer when we define “the same time” for it after defining “the same time” for the embankment. If we had gone through this whole sequence of things beginning with “the same time”for the car, it is the distance on the embankment that would have been longer and the length of the railroad car would have been what the distance on the embankment was when “the same time” was first defined for the embankment. Why did I present this argument now? To point out that the length of a physical object at rest in some situation depends on whether “the same time” is defined first or second when comparing length in this situation to another one moving at a constant velocity relative to it. A similar scenario exists for clocks that are used to measure the durations of occurrences. So you cannot even say, “The concrete nature of a physical object provides something unaffected in its nature by the thought of the person considering the situations.” It is the argument on “the same time” in the two situations that is essential to the physical results. Without this argument, the light flashes would meet both observers, the man on the railroad car and the man on the embankment, midpoint in their respective situations (in which they are at rest) because of the constant velocity of light. Think a little more about these light flashes used to establish “the same time.” How could we be dealing with different light flashes for the observer onthe embankment and the observer on the train? Each observer can use these flashes to establish “the same time” first in their respective situations before “thesame time” is defined in the other situation. Yet when light flashes are used to define “the same time” second in their respective situations, it looks like different light flashes are used than when “the same time” is defined first. I think you can see how the nature of the light flashes depends on the thought of the person thinking about time in the two situations that are moving relative to one another at a constant velocity. What about the light and non-light clocks? Science and observation. What else can be observed if not the physical world itself? Our thought processes. These are also involved in the nature of time in special relativity: 1) as an observer in one of our situations, and 2) as a theoretician workingFunny Things - 7 -through the arguments in the special theory, in particular the determination of “the same time” in our situations. There will be more support for this conclusion shortly.- 8 -Chapter 3 A Slight Digression It’s not easy figuring out the order in which to present the topics in this essay. In fact, knowing how to make logical arguments is never really easy. Itrequires the ability to make sense of things, to find order in them. People varyin their ability to find order in things, including those in the physical world. This ability to order things is distributed in a manner that resembles a curve that will be discussed in a later chapter on statistical mechanics. It is called the normal curve. In science, this order that people find in the physical world is anchored in theories, logical arguments, made about the world that are then subject to experimental test. The arguments I have presented so far, and will continue to present, are concerned with the thinker thinking about the physical world or the observer observing the physical world. They are not concerned with the physical world as if it were a peach pie sitting right in front of your eyes waiting to be eaten. They are about the ordering process itself. That’s why the points I am making have not been considered very much before. Physicists, though experts on the physical world, are not trained today to think about the process of thinking. They are trained to investigate phenomena in the physical world that are kind of like our peach pie. They probably don’t want to devour most of the things they study, but generally they are objects for them. Really, they are objects for their thought and perception. The reason I can make the points in this essay is that I was trained differently. I was trained as a psychologist and thus to think about the thinker who is thinking about things. And after this training, I became interested in physical theory. The resulting perspective is one where I look at the physical world in terms of the theory constructed by physicists but, in addition, I look at the theorizing itself and the observational activities of physicists and others watching the physical world.- 9 -Chapter 4 Conservation Laws and What? You’ve heard of conservation laws, how certain things don’t change in a group of physical objects that are interacting with each other if nothing from the outside disturbs them. The most well-known is the conservation of energy. It says that the energy of this group of objects doesn’t change. For example, consider hanging two polished steel balls on a string from a piece of wood and pull one back and then let it go so it hits the other ball square in the middle. Theother ball will move outward and then swing back and hit the ball that first hit it. Now this can go on for a while, especially if there is no air present around the balls. What’s going on? Energy is being conserved. That’s why the balls keep hitting each other. The conservation of energy has a very important feature. It depends on physical law not changing over time. One might assume that the laws governing the physical world could change as time elapses, but the fact that they don’t explains why the balls continue to hit each other for a very long time. Is there anything else to consider? There is. Measuring time requires a scale, just like measuring space requires a reference frame. What is meant by a reference frame? So far we’ve discussed rulers, distance, the length of an object. Actually, what we have been discussing is a reference frame. It is just a coordinate system that allows you to measure the position of things in space. In your automobile, you have such a coordinate system and the man at the side of the road has such a coordinate system. Just like you move relative to the man on the side of the road, so your reference frame moves as well. A reference frame needs something physical associated with it, but it is up to us todetermine what that something is. To show that a reference frame really doesn’t depend on any physical object external to you, imagine that a rock is at rest relative to the man on the side of the road and the rock is considered the physical anchor to which the man’s frame of reference is attached. The reference frame can be considered animagined set of giant antennae stretching up-down, in-out, and side-to-side, theantennae in each direction divided into pieces of the same length. Now imagine that the rock is set in motion. Maybe someone comes along, roles it over a fewtimes, and the rock keeps going. If the man wants to measure the distance of things from each other, what does he do? He would do the same thing he did before the rock started rolling. He needs to pick a new rock, or some otherChapter 4 - 10 -object, to attach his giant mental antennae to. That’s all. To put it differently, a particular rock has nothing to do in an essential way with his reference frame. Rather, the reference frame depends on him. If the reference frame was not at rest relative to the observer, he would not know what’s what concerning the motion of objects. The observer is at rest in his reference frame. That’s just another way of saying that the observer’s reference frame is at rest relative to him. That the observer is at rest in his reference frame is borne out by experience, and we willcome back to this point in a minute. There’s another factor to be considered that supports a connection between a reference frame and an observer. The relativity principles in physics maintain that there is no preferred reference frame that is absolutely at rest. In Newtonian mechanics, for example, physicalphenomena are described the same by observers in either of two situations moving at constant velocity relative to one another in a straight line (like the railroad car and the embankment). It shouldn’t make a difference for physical description in our situations in special relativity as well. (This is not to say that where “the same time” is first defined is unimportant. Rather, once the situations are defined and used in the arguments concerning length and time in the same way, for example, physical description in these situations should be equivalent.) You may ask me what I mean when I say an observer is at rest in a reference frame? How can an observer riding in an automobile feel himself to be at rest? There is no problem with the observer on the side of the road because anyone standing on the side of a road feels like he is at rest. But wait, isn’t this man also moving? Isn’t the observer on the side of the road moving with the earth as it rotates on its axis and as it moves around the sun? Actually, the earth is moving pretty fast. Surely, if a person were in a car moving as fast as the earth is moving, he would feel it. What is going on here? Now think. When you are riding in a car and you close your eyes, what do you experience? You feel a certain vibration in your body. Since you’ve ridden in a car before, you conclude that this bumpiness means you’re riding along a road. If you hadn’t ridden in a car before, you just wouldn’t know what the bumpy feeling meant. Let’s say the bumpiness is completely absent because you’re riding on a very smooth road in a luxury car. If the car was moving in a straight line, you would feel that you were at rest. That’s right, at rest. Just think of yourself in an airplane when you’re in the air. You don’t feel like you’re moving. In fact, it may amaze you when you disembark from your first flight inConservation Laws - 11 -a place distant from where you began that you actually got there without more of a feeling of motion. Imagine how the astronauts felt on their trip to the moon. What is responsible for the observer’s being at rest in his reference frame? A physical cause cannot be identified. We choose a physical object to attach our reference frame to that is at rest relative to us. A psychological phenomenon is at work here, but that shouldn’t be too surprising because what’s a reference frame anyway? It’s really a cognitive construction that we use in living in and studying the physical world, such as saying the distance between two points is one foot. There is nothing that says that a particular mental construction has to be used in constructing a reference frame. I could have said the distance between the two points is two feet and no one could havetold me it’s not. This point is so basic to our description of the physical world that any other results indicating that there are different ways to describe the motion of physical objects should not be too surprising. These other results depend on the flexibility in constructing a reference frame. Just like a reference frame, the time scale we use needs to be associated with something in the physical world. We can determine what that something is. It could be a clock with minute, hour, and second hands, or it could be some other physical process in the world. Even a person’s getting older could be the basis for a time scale. Now we can return to physical law remaining thesame over time. In testing whether physical law doesn’t change over time, youprobably thought the only kind of test would be to let the clock tick to see if thisis the case. If you think about it though, you could just shift your time scale forward or backward any way you wish. There really isn’t any difference in watching our clock or shifting the time scale mentally. Physical law wouldn’t change. This may seem silly, but I assure you it is not. It’s the simplest and most elementary things we need to be careful about. Perhaps it will seem a bit less silly if we consider space. We have already observed that changing the time of an occurrence doesn’t change physical law. A similar thing happens in space. Let’s say we change the position of an object from one point to another point along a straight line. We see that physical law doesn’t change. Besides moving the physical object itself in space, we can change the position of the ruler itself, for example, that acts as our straight line in the physical world. We can also shift the straight line mentally in relation to the physical object.Chapter 4 - 12 -We also could rotate the position of an object in space from one point to another, and we would find that physical law is no different at either of these points. This rotation can be accomplished, for example, by rotating the object itself or by mentally rotating the axes while leaving the object untouched. So you see certain conservation laws (those associated with physical law not changing over time, with the change in an object’s position along a straight line,and the rotation of an object from one position to another) are associated with our thinking. Certain physical and mental actions are equivalent with regard to the conservation of particular physical quantities. For the record, these quantities are energy, linear momentum, and angular momentum.- 13 -Chapter 5 Is Gravity a Problem? It is surprising Einstein himself didn’t say that the mind is linked to the physical world. He surely saw what Feynman saw, namely that the need to maintain the integrity of the special theory was the basis for various physical results. He even began discussing “the same time” in his first paper on special relativity in a non-quantitative psychological way. Einstein, though, didn’t think to make this connection, even though he knew that in general relativity and quantum theory a mental tie to the physical world is also plausible, even more plausible than in special relativity. Let’s take a look at general relativity. Einstein thought there was no reason the laws of physics should not hold in all reference frames. I have to confess something. The reference frames with regard to special relativity have included factors like gravity and friction. Really, though, it’s best to imagine all these previous situations occurring deep in outer space far away from any large bodies. It’s strange to think of automobiles and railroad cars in outer space, so instead think of space ships. In outer space far away from any large bodies, there is no gravitational influence on the space ships and there is no friction. There, when something moves in a straight line and with a certain velocity, it just keeps moving in a straight line with that velocity unless an external force impacts it. When something is stopped, it remains stopped in the absence of an external force. When you push something, the object accelerates in the direction of the force exerted on it in a way that depends on the mass of the object. These ways in which objects behave are called Newton’s laws of motion. Special relativity encompasses these laws and goes beyond them. General relativity encompasses special relativity and goes beyond it. We need to back up, as I’ve begun to do, and think about Newton’s laws of motion a bit more. This will allow us to see how Einstein developed general relativity and show us how the mind is linked to the physical world in this theory. After developing special relativity, Einstein said the laws of physics should apply to reference frames where there is gravity or that are accelerating as well as to those deep in space far from large objects where Newton’s laws ofmotion hold. Why could he say this? He noticed that there was nothing other than an object’s acceleration in these reference frames in outer space, as well as Newton’s laws of motion, that is the same in them. That these laws hold in them depends on the acceleration being the same. Einstein had some unusualChapter 5 - 14 -thoughts about the acceleration. He deduced that the acceleration feels the same as gravity for an observer in an accelerating reference frame deep in space. He also figured out that an observer accelerating due to gravity would think he wasin a reference frame like one deep in outer space. I should tell you about these situations. Let’s start with the second one first. Think of a man in an elevator in a skyscraper. Suddenly, the elevator cable snaps and the elevator falls toward theground. The man experiences the physical world as if he is in outer space. Forthis man, objects in the elevator act in accordance with Newton's laws of motion and do not appear to be under the influence of gravity. If he lets go of aball without pushing or pulling it, for example, it will stay even with him, just as it would in outer space. Let’s see how Einstein (1938/1966) put it. “If the observer pushes a body in any direction, up or down for instance, it always moves uniformly so long as it does not collide with the ceiling or the floor of the elevator. Briefly speaking, the laws of classical mechanics are valid for the observer inside the elevator [who does not experience gravity]” (p. 215). Let’s consider the first situation of a man in a windowless room in deep space. The room is towed by a rope attached to one end so that the room accelerates at a uniform rate. What does the man inside the room experience? He feels as if he is in a gravitational field. For example, if the man in the room hangs a rope from the ceiling and attaches an object to the bottom, the rope hangs toward the floor. What does the man in the room think? He thinks, “The suspended body experiences a downward force in the gravitational field, and this is neutralised by the tension of the rope” (Einstein, 1917/1961, p. 68). What does an observer outside the room deep in space think? He thinks, “The rope must perforce take part in the accelerated motion of the chest [the room], and it transmits this motion to the body” (Einstein, 1917/1961, p. 68). Einstein saw that an accelerating reference frame, the accelerating room, was equivalent to one of these reference frames deep in outer space that obeyed Newton’s laws of motion, but with a slight twist. The slight twist is that such aframe would experience a gravitational field. This is really not a problem because a gravitational field exerts a force on objects and this force affectsobjects the same way that I indicated objects are affected by forces deep in space, namely in accordance with Newton’s laws of motion. So Einstein couldseriously consider that the laws of physics should hold for all reference frames. That these different reference frames are capable of describing the sameGravity - 15 -physical phenomenon is the heart of the principle of equivalence in general relativity. Also, Einstein had a way to develop a system to measure space and time, really space-time, in accelerating reference frames and reference frames experiencing gravitational fields. What Einstein did was to break up a uniformly accelerating reference frame into a sequence of very tiny reference frames. Then each tiny reference frame is like a reference frame in special relativity. Two of these tiny reference frames move at a constant velocity relative to one another. We’ve seen how we can develop the space-time relationships between two inertial reference frames moving at a constant velocity relative to one another. Einstein could develop a space-time metric by considering these tiny reference frames in relation to one another. Some other well-known physicists commented on the importance of special relativity to general relativity. They wrote: General relativity is built on special relativity. (Misner, Thorne, & Wheeler, 1973, p. 164) Elaborating on this statement, they said: A tourist in a powered interplanetary rocket feels “gravity.” Can a physicist by local effects convince him that this “gravity” is bogus? Never, says Einstein’s principle of the local [over a small area] equivalence of gravity and accelerations. But thenthe physicist will make no errors if he deludes [italics added] himself into treating true gravity as a local illusion caused by acceleration. Under this delusion, he barges ahead and solves gravitational problems by using special relativity: if he is clever enough to divide every problem into a network of local questions, each solvable under such a delusion, then he can work out all influences of any gravitational field. Only three basic principles are invoked: special relativity physics, the equivalence principle, and the local nature of physics. They are simple and clear. To apply them, however, imposes a double task: (1) take space-time apart into locally flat pieces (where the principles [of the special theory] are valid), and (2) put these pieces together again into a comprehensible picture. To undertake this dissection and reconstitution, to see curvedChapter 5 - 16 -dynamic space-time inescapably take form, and to see the consequences for physics: that is general relativity. (p. 164) Einstein proceeded to develop physical law so that it would also encompass accelerating reference frames and reference frames experiencing gravitational fields. He also knew that he could provide a physical basis for anobject’s gravitational mass being essentially equal to the resistance of this object to acceleration by a force. This equality is important for everyone, especially Galileo who found that the acceleration of objects in the earth’s gravitational field was the same, regardless of the type of object or its gravitational mass. That the man in the room accelerating in deep space could describe the physicalworld as well as someone outside watching the room accelerating requires that this equality hold. The equivalence of their descriptions forms the foundation for the general principle of relativity and the across-the-board application of the laws of physics in different types of reference frames. Here’s the punch line, which you may have already deduced. The difference in the description of the physical world between the accelerating observer in deep space and an observer who is watching the room the man is in accelerate is due to both observers being at rest in different reference frames. Remember a reference frame is a cognitive construction that an individual employs in observing the physical world. We have seen in special relativity the importance of an observer’s being at rest in a certain kind of reference frame. Einstein extended this idea of an observer’s being at rest to accelerating reference frames as well. The man in the falling elevator and the man in the spaceship feel themselves fundamentally to be at rest. It’s all the “extras” that make people think they are moving. Einstein (1917/1961) discussed a passenger on a train where the brakes are suddenly applied and the passenger experiences a powerful “jerk forwards” (p. 62). Einstein said: It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognises in this the non-uniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a “real” acceleration (retardation) of the carriage. He might also interpret his experience thus: “My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational field which is directed forwards and which is variable with respect to time. Under theGravity - 17 -influence of this field, the embankment together with the earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continuously reduced (pp. 69-70). Really, the passenger in the railway car has to see himself at rest or else we will again be faced with privileged reference frames. The observer’s being at rest in his reference frame in the special theory of relativity is psychological in nature. His being at rest in his reference frame in the general theory of relativity is psychological in nature too. Remember that the general theory relies on the special theory for measuring lengths and durations. So the psychological parts of the special theory, notably how we decide on “the same time” in reference frames moving at a constant velocity relative to one another, hold for the general theory as well. A final note: Gravitational mass itself, terra firma, may be simply a matter of reference frames. Maybe this shouldn’t be so shocking after Einstein showed that mass is really equivalent to energy, and it is energy that is conserved, not mass as we usually think about it. But that’s a lot different than saying “reference frames” because reference frames clearly depend on the mind.- 18 -Chapter 6 Quantum Mechanics - No, It Can’t Be! Imagine that you are seated in front of a big screen. On the screen, shapes gradually take form, dot by dot, kind of like a painting by Seurat. If you watch the screen over a long period of time, you see a form take shape. It looks like a range of hills where all the hills are skinny and separated by deep valleys. In the middle of the screen is the highest hill and as you move to either side, the hills gradually become smaller. The left and right sides of the hills on the screen look the same. The higher the location on a hill, the more dots appear at that location. Now the other picture that you see takes shape dot by dot is just one wide hill, perfectly balanced between left and right of the middle. It looks like this.Quantum Mechanics - 19 -To be honest with you, these two pictures are not actually what an observer would see. But they portray accurately essential elements of the physical situation that an observer would indeed see. You notice a thin wall. This wall and our screen line up perfectly, the wall in front of the screen. The wall has two slits in it, each one a little to one side of the middle. Now there’s also a light just behind the wall and smack in the middle between the two holes. Looking down, one sees the wall, the screen, and the light line up this way. You notice the following. If the light is off for a while, we see the skinny hills. If the light is on for a while, we see one wide hill. In each case, we start with a blank screen. Now since you are a very careful investigator, you notice another thing that may be important. When the light is on, there are sudden light flashes to the left and right of the light that are associated with the dots appearing on the screen. The flashes to the left happen by the left slit and the flashes to the right happen by the right slit. Each flash to the left or right of the light is associated with a subsequent dot appearing on the screen. The flashes occur one at a time, and so the flashes and subsequent dots on the screen can be tracked. The question naturally is, what is responsible for the different pictures that appear on the screen, point by point? You probably think it’s the light being turned on or off, or more specifically the light flashes that occur only when the light is on. That’s a pretty good guess, one that most physicists have made. But think about the following experiment. Move the light to one side, say the left side, so that you only get light flashes from one slit in the wall.Chapter 6 - 20 -Now which pattern do you think you’ll get? Maybe you’ll say a mixture of skinny hills and one wide hill. What you get is one perfectly balanced wide hill like the one that occurred when the light was in the middle between both holes. So then, like any good investigator, you ask yourself what can cause the one wide hill pattern that is not present when we get the many skinny hills pattern. Is it the light flashes at both slits? No. It’s the light flashes at one slit? Well, track the dots that are not associated with light flashes, the dots that occur when there are no flashes. They form a certain pattern, another kind of wide hill. You see that there are dots at places where when the light is turned off there aren’t many dots at all. (Remember when the light is turned off, we get the many skinny hills pattern.) What is going on? Something is happening at the slit away from the light where there are no light flashes. It can’t be the light flashes at the slit where the light is. What else? That’s a good question. First let me give you one other piece of information.Quantum Mechanics - 21 -Take a scenario where the light is located near the left slit and you know how long a specific flash has to take place. If it takes place in that time, okay. If not, it will not take place at all. Let’s say the light is turned off before the time has elapsed for a specific flash, even just before the time is up, and a flash has not occurred. What do you think happens? We get the skinny hills pattern if this is repeated many times. Why? The reason is our knowledge has not changed. We haven’t seen a flash, and we haven’t allowed the full time for the possibility of a flash, so we are unable to deduce any associations between specific light flashes at the wall with the two slits, or their absence, and specificdots at the screen that subsequently appear. Knowledge is what’s behind our seeing one wide hill when the light is moved near one slit. In this scenario, there are no light flashes near one of the slits. The fact that there are light flashes at the other slit that can be associated with dots on the screen indicates that the other dots on the screen can be associated with their not being light flashes at the other, unilluminated slit. Alternatively, if there was a light near the unilluminated slit, there would be light flashes at this slit that could be associated with these other dots. By extension, knowledge is also behind our seeing one wide hill when the light is in the middle between both slits and we get light flashes near both. We have determined the common factor in the different scenarios: knowledge. Also, the shape of the pattern of dots associated with no light flashes in the scenario where the light is located near one slit is similar to the shape of the pattern of dots associated with light flashes from the other slit when the light source is between both slits and is on all the time. That these shapes are similar supports the thesis that physical interaction is not responsible for a specific pattern of dots at the screen that are associated with light flashes near slits in thewall. Further, the one wide hill pattern is the summation of patterns of dots at the screen like these shapes. These light flashes and dots could involve a variety of small particles, but let’s say for example they are electrons. A light flash by a slit in the wall occurs when an electron passes through a particular slit and is struck by light from the light source, and a dot occurs when the electron strikes our screen. Soyou could say we see one wide hill when we know through which slit electronspass. In the absence of this knowledge, we see many skinny hills. On to something else. I should tell you briefly about Schrödinger’s cat. You might have heard about the cat that is neither alive nor dead in a certainChapter 6 - 22 -situation, but is kind of both until an observation of it is made. Which state characterizes the cat is tied to the observer’s recognition that the cat is either alive or dead, no matter how far away the cat is from the observer. This meanseven across the universe. Even then, with the observer’s observation, the state of the cat changes immediately, and it is either alive or dead. That’s it. There’splenty of experimental evidence to support these points. I could give you many fancy deductions regarding Schrödinger’s cat, but in the end it comes down to what I just told you. That doesn’t make it less remarkable. It makes it more remarkable because of its basic nature. I hope all of this information has not upset you. You know, you look a little red. Maybe you should take your temperature to be sure you don’t have a fever. But, uh oh, as I will explain, temperature is tied to the mind. Can a person rely on anything anymore?- 23 -Chapter 7 Oh No! Temperature Too? In order to demonstrate a link between temperature and the mind, we first have to discuss some concepts from statistics. Consider 10 pennies that are equally weighted and flipped in a way that does not favor either heads up ortails up. Also, allow that each coin flip does not affect any other coin flip. In this situation there is no reason why a head should come up rather than a tail. So the most logical thing to decide is that head up or tail up for each penny is equally likely. If we want to figure out how many ways the pennies can land heads up and heads down, our task is relatively easy. For example, for the combination of one head up and 9 tails up, there are 10 ways the pennies can achieve this combination. One arrangement of one head up and nine tails up is: Each of the 10 ways is equally likely to occur when the pennies are flipped. Now consider five heads up and five tails up. There are many more ways that the pennies can achieve this combination than they can for one head up and nine tails up. Just think that instead of shifting the head up penny over one place at a time, there are five heads up pennies that can be shifted throughout the 10 pennies. In fact, the equal number of heads up pennies and tails up pennies provides the greatest number of ways the pennies can be arranged of the different possible combinations of heads up and tails up. This last point holds for any even number of pennies. The next drawing is a graph showing the distribution of heads up pennies when we are dealing with 100 pennies instead of 10 pennies. The left edge shows the relative number of ways 0 pennies with heads up can be obtained. The right edge shows the relative number of ways 100 pennies with heads up can be obtained. The middle shows the relative number of ways 50 pennies with heads up can be obtained. This graph yields a curve that looks like something called the normal curve. Since all sequences of pennies are equally likely, this graph also tells you the likelihood that the pennies will be ina particular combination of heads up and tails up.HHTTTTTTTTTChapter 7 - 24 -0 50 100 heads up pennies Now imagine that we have a very large even number of pennies in one group. And imagine that we have another group of pennies, say 10, in another group. These groups are in separate containers, like so. The pennies are always able to flip from heads up to tails up or from tails up to heads up. When in the air, pennies from one group cannot fall into the other group. Imagine that the pennies in the containers are separated from all other physical phenomena and that the pennies in each container can change from sequence to sequence through flipping. The pennies are not stuck in one sequence. If no other conditions are placed on the groups of pennies, what is the most likely combination of heads up and tails up that will be obtained at anytime when both groups of pennies are considered? It is 5 heads up and 5 tails up in the group of 10 pennies and 1/2 of the pennies in the other group heads up and 1/2 of the pennies tails up.10 penniesa very large number of penniesOh No! - 25 -Now we need to add the following conditions. Imagine that instead of being physically separated, the containers with the pennies touch each other andthat the walls of the containers can exchange energy in the form of heat. Allow that a penny with head up is associated with a different amount of heat than a penny with tail up and that both groups of pennies share a certain fixed amount of heat. As noted, this heat can be exchanged between the two groups of pennies through the walls of the containers, but the total amount does not change. Then in each group of pennies different combinations of heads up andtails up pennies are associated with different amounts of heat, but the total energy of both groups remains constant. Now we have the interesting question of what is the most likely combinations of heads up and tails up in each of the two groups of pennies? The answer is that the heat is in large part distributed so that the combinations of heads up and tails up pennies for the two groups yield the highest number ofways the pennies can be arranged in different sequences, given the constraint that the heat energy of the two systems together is constant. Why is this the case? Only one reason. Each sequence of heads up pennies and tails up pennies in a group is equally likely when a group of pennies is considered in isolation, like we did at the beginning of the chapter. The heat energies of the groups of pennies are associated with these sequences. About now, you might ask yourself why is he going to all this trouble to discuss these things? First, this scenario is behind the equilibrium of temperature that two physical systems tend toward when they are brought into the kind of contact I just discussed. A good approximation of this is when you10 penniesa very large number of penniesChapter 7 - 26 -put a thermometer in your mouth to take your temperature. The thermometer tends toward the same temperature as you. The thermometer is like the small system of pennies, and you are like the large system of pennies. Compared to you, the thermometer has a very limited number of possible sequences of its parts, and so the overall number of sequences for you and the thermometer is largely determined by you. As you can see, physical systems in statistical mechanics are made up of things like the atoms or molecules that comprise you and the thermometer, for example. The pennies, though, are a good way to discuss the statistical concepts. Instead of heads up and tails up, we would be concerned with velocities, for example, of atoms and molecules. Consider the following problem. In the physical world, we only get one sequence of pennies at a time. Correct? Hard to argue with that one. But then how do we get results concerning things like temperature that involve lots of sequences, specifically all the possible sequences for all the possible combinations? Physicists call this last group a representative ensemble of systems like the actual physical system. “But,” you might ask, “where did this concern with all of the sequences come from?” Certain physicists said that we can consider that the sequences occur quickly over time, one at a time, and the length of a measurement of temperature, for example, takes into account the passage of these systems through various sequences. In fact, that is what physicists first thought and that seems at first to be the most natural explanation. A physicist by the name of Tolman made it very clear, though, that this attempted solution does not work, and it is accepted today that the different sequences need not occur one ata time. This attempted solution doesn’t work because we are concerned with the application of statistical methods to physical processes, and it is the integrityof the statistical methods that is of primary concern. When the different sequences occur is not of great significance. The important thing is that they are all available due to the flexibility in the physical systems involved, and thus they all contribute to various quantities of the physical systems like temperature. Oh, this isn’t supposed to be possible because only one sequence can exist at a time. Another physicist, Kittel, wrote that the different sequences exist at one time and that the representative ensemble of sequences is an intellectual construction. What does this mean? It means that people think it. “What,” you say, “people think it?” Yes, that’s right. The physical worldOh No! - 27 -indicates that their thinking is part of what lies behind the temperature of a physical system, including your body. It also lies behind the conduction of nerve impulses in your body as well the operation of the battery in your car. Remember the equal likelihood of the different sequences. Why did I focus on this? Wouldn’t it have been easier to state some law about how the pennies would land? Maybe they were flipped a certain way that resulted in their landing in a certain sequence. Then physical law could tell us why the pennies land the way they do. This is an interesting argument, and when you look at each individual penny, one can see how it lawfully flies in the air. When we consider the pennies as a group, though, we are dealing with statistics, not law. (I have to admit, it really is ideal to think of individual pennies with a perfect weight distribution and perfectly flipped, but it really is statistics that we are dealing with.) What I mean is that we are dealing with the possible ways the pennies can fall into a sequence. This is a lot different than how individual pennies fly. Knowing how each penny flies doesn’t help a lot because one would have to know how the pennies had such widely different initial conditions when they were flipped. At some point, a statistical assumption must be made. Physical law only deals with the pennies once we know their initial conditions. Why make the assumption that the various possible sequences of pennies with heads up and pennies with tails up are equally likely? Now, this is important. It is the only reasonable assumption that can be made if physical law does not apply. I would like to take credit for this last statement, but I can’t. Remember Tolman, the physicist. He made it. Again, we’re back to a mental-sounding concept: assumption. We’re back to thinking about the physical world that leads to correct predictions aboutmeasurements, measurements for which there is no reasonable physical explanation. We come to this conclusion because physical law cannot be the basis for these predictions. You may have noticed that the discussion in this last section has really been about probability. What the physical world has furnished us is a situationto let probability work. The probabilistic considerations that have been discussed could have involved non-physical things just as well as physical things. It wouldn’t have made any difference regarding the predictions. For example, it may interest you that intelligence as measured by IQ tests is distributed in about the same way as our group of 100 pennies. That’s right. ItChapter 7 - 28 -is approximately normally distributed. I mentioned this earlier when I wrote about finding order in the world. So probabilistic considerations apply to IQ test results as well. And you know what probabilities are about. They are about knowledge of what will happen. Maybe I should have told you at the beginning of this chapter on statistical mechanics that probabilities are really concerned with knowledge about what will happen and that statistical mechanics is really about probabilities. But I thought you wouldn’t really take it very seriously. Now I think you do. To show you how good the results of statistical mechanics are, here is a quote from Einstein on thermodynamics. Statistical mechanics provides a deep theoretical explanation of thermodynamics. It [classical thermodynamics] is the only physical theory of universal content concerning which I am convinced that, within the framework of applicability of its basic concepts, it will neverbe overthrown. (Einstein, 1949/1969, p. 33) It may well be that of all the areas of physical theory statistical mechanics shows most clearly that non-physical processes are involved in the functioning of the physical world.- 29 -Chapter 8 Oh, I’ve Just Fallen Down, or Is That Up? Now, if your head is spinning, I really can’t blame you. The involvement and affect of mind in and on the physical world are so basic that they are a bit discomforting. In fact if you felt that up is down and down is up,I would understand. But I would understand for an additional reason besides your feeling a bit unbalanced at the moment. Maybe what’s coming will seem like bad news to you. Your feeling that up is down and down is up is not far from the way the world is. This is where the discipline of psychology really enters the picture. Research from psychology indicates that what we think is upand down in our visual experience and in the physical world is dependent on various factors pertaining to the observer. It seems that most important are the coincidence of visual and tactual (touch) sensations. Other factors involve kinesthetic stimuli (our own sensations of our muscles) as well as gravity. Butthese latter factors seem to be less significant. Thus, if the coincidence of visual and tactual sensations is rearranged from what normally occurs, we againget a sense of up and down in visual experience that feels as natural as the first did and which the observer does not readily distinguish from the first unless prompted to do so. Competency in visually guided action returns. I should explain this research a bit more. At the very end of the 1800s, a psychologist by the name of Stratton wore an apparatus on his head that rotated incoming light to one of his eyes 180 o. The other eye was covered. On the days he wore this contraption and went about many of his normal activities, he progressively adapted to inversion of the incoming light along the lines of the results I mentioned. Other psychologists using different optical devices found similar results. In the case where a psychologist thought his own findings did not support these results, many of his results indeed were in accord with them. A very prominent psychologist by the name of Boring thought they were too. Let’s take a look at one implication of this research. Consider an observer A, who isn’t wearing any optical contraption and who looks at an arrow in the world that is pointing up. The retinal image of the arrow, the pattern of light on the retina that forms an arrow, is inverted. It points down. That’s how the eye usually works.Chapter 8 - 30 -retinal imageobserver A world Consider another observer, B, who wears an apparatus like Stratton’s and whose vision has adapted. If he looks at the arrow in the world that is pointing up for observer A, the retinal image of the arrow points up and observer B also sees the arrow pointing up. Now this same retinal image for B can also occur for an observer like A (call him A ¢) who does not wear an optical apparatus. When it does, the retinal image is associated with an arrow in the world pointing down for A ¢. Thus, the arrow in the world for B points in the opposite direction to that which would be found by A ¢ in the world for the same retinal image. Although I don’t think it’s a substantive argument, someone might say, “Well, B is only wearing an optical contraption. Inversion of incoming lightretinal imageobserver B world retinal imageobserver A' worldOh, I’ve Just Fallen - 31 -only affects B’s experience because there’s a contraption involved. It really doesn’t affect the physical world itself.” Well, inversion of incoming light really does affect the physical world because a contraption is just a contraption,a physical instrument involved in the display of light on the observer’s retina. It doesn’t stop the observer’s adaptation to the altered pattern of incoming light on the retina. The orientation of objects and phenomena in the physical world depends in part on the observer . That includes, for example, the orientation of something called the spin of an electron along a particular spatial axis. Instead of being built the way we are, we could have been built with an optical apparatus around our heads. Then we all naturally would think of up and downwhile wearing this contraption on our heads. There is something else I can say if you don’t accept my point yet. What if someone’s retinas were simply rotated 180 o in his eyes? No nerves are severed. Just the retinas are rotated. Then there is no contraption added, and we would still get the same results. Please forgive me for thinking such a thing, but sometimes it takes a strong statement to make a point. Up can be down and down can be up in the physical world. An electron’s spin down along a spatial axis for one observer, for example, can be spin up along this same axis for another observer wearing an apparatus like Stratton’s and whose vision has adapted. Whether up is down or down is up depends on how incoming light enters an observer’s eyes and whose spatial structure we are considering.- 32 -Chapter 9 A Final Word Physical world, mental world. They are separate and not separate. Both are needed to explain the wonder of our experience of living in the physical world. There are certain important advantages to the results of this study. Einstein (1954) wrote, “the eternal mystery of the world is its comprehensibility” (p. 292). Our comprehension of the world is no longer so mysterious. We don’t have this chasm between the physical world and our experience of it. The mind is directly tied to it. Also, you know this chasm thatwe have supposed exists between the physical world and the mind has contributed to a great sense of meaningless and isolation on the part of people. In the midst of the great cosmos, here we are, apparently insignificant to its functioning. Well, this is not the case. Each of us significantly affects the structure and functioning of the world. Without us, there would be no reference frame. What would the motion of objects be without a reference frame, whatever physical theory we are considering? The reference frame is essential to the invariance of physical lawunderlying the conservation laws that have been discussed. Without us, there would be no special relativity, no general relativity, no quantum mechanics, andno statistical mechanics. There would not be an observer at rest in a reference frame, or if you like, a reference frame at rest relative to an observer. There wouldn’t be the relativity of simultaneity in special relativity or the principle ofequivalence in general relativity. Because we wouldn’t have these basic results,we wouldn’t have the other results of these theories. I don’t even think we would have gravitational mass. We wouldn’t have measurements in the physical world corresponding to our different pictures that we saw in quantum mechanics, namely the many skinny hills or the one wide hill. With regard to statistical mechanics, what would probabilistic knowledge mean without someone for whom the probabilities meant something? What would temperature and other characteristics of physical phenomena be that depend on probabilistic knowledge? How would the representative ensemble exist? I didn’t mention this to you, but you might have guessed. Those dots that make up the hills we talked about in quantum mechanics are also, to a significant degree, governed by probability. There is also the spatial directionality of the physical world that is dependent on us, a result that comes from psychological research. To sum up, without us, the physical world is difficult to imagine.A Final Word - 33 -Now there is a big challenge before us. We need to find out more about our relationship to the physical world. We need to really look at our theories inphysics and psychology, and perhaps elsewhere, to see if there are ways that these theories are relevant to domains that they were not developed for. We have to broaden our viewpoint and not say physics is only about the physical world and psychology is only about the mind and expressions of mind found inbodily action. We have to accept the fact that the boundaries of physics and psychology broadly overlap. We have to do experiments that rely on methodology and results from both psychology and physics. Experiments will answer our questions. With this perspective, I am confident that significant progress can be made in understanding our relationship to the physical world.- 34 -Sources for the Quotes Einstein, A. (1954). Ideas and Opinions . New York: Crown. Einstein, A. (1961). Relativity, the Special and the General Theory (R. Lawson, Trans.). New York: Bonanza. (Original work published 1917) Einstein, A., and Infeld, L. (1966). The Evolution of Physics from Early Concepts to Relativity and Quanta . New York: Simon and Schuster. (Original work published 1938) Einstein, A. (1969). Autobiographical Notes. In P. A. Schilpp (Ed.), Albert Einstein: Philosopher-scientist (Vol. 1) (pp. 1-94). La Salle, Illinois: Open Court. (Original work published 1949) Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation . San Francisco: W. H. Freeman. Some Other Sources Cook, R. J. (1990). Quantum Jumps. In E. Wolf (Ed.), Progress in Optics (Vol. 28) (pp. 361-416). Amsterdam: North-Holland. Dolezal, H. (1982). Living in a World Transformed . New York: Academic Press. Einstein, A. (1952). On the Electrodynamics of Moving Bodies. In H. Lorentz, A. Einstein, H. Minkowski, and H. Weyl (Eds.), The Principle of Relativity, a Collection of Original Memoirs on the Special and General Theories of Relativity (pp. 35-65) (W. Perrett and G. B. Jeffrey, Trans.). New York: Dover. (Original work published 1905) Einstein, A. (1952). The Foundation of the General Theory of Relativity. In H. Lorentz, A. Einstein, H. Minkowski, and H. Weyl (Eds.), The Principle of Relativity, a Collection of Original Memoirs on the Special and General Theories of Relativity (pp. 109-164) (W. Perrett and G. B. Jeffrey, Trans.). New York: Dover. (Original work published 1916) Einstein, A. (1956). The Meaning of Relativity (5th ed.) (E. Adams, Trans.). Princeton, New Jersey: Princeton University Press. (Original work published 1922) Eisberg, R., and Resnick, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). New York: Wiley. (Original work published 1974)Sources - 35 -Epstein, P. (1945). The Reality Problem in Quantum Mechanics. American Journal of Physics , 13, 127-136. Erismann, T., and Kohler, I. (1953). Upright Vision Through Inverting Spectacles [Film]. University Park, Pennsylvania: PCR: Films and Video in the Behavioral Sciences. Erismann, T., and Kohler, I. (1958). Living in a Reversed World [Film]. University Park, Pennsylvania: PCR: Films and Video in the Behavioral Sciences. Ewert, P. W. (1930). A Study of the Effect of Inverted Retinal Stimulation upon Spatially Coordinated Behavior. Genetic Psychology Monographs: Child Behavior, Animal Behavior, and Comparative Psychology , 7, 177-363. Feynman, R. P., Leighton, R. B., and Sands, M. (1963). The Feynman Lectures on Physics: Mainly Mechanics, Radiation, and Heat (Vol. 1). Reading, Mass.: Addison-Wesley. Feynman, P. R., Leighton, R. B., and Sands, M. (1965). The Feynman Lectures on Physics: Quantum Mechanics (Vol. 3). Reading, Massachusetts: Addison-Wesley. Kittel, C. (1969). Thermal Physics . New York: John Wiley & Sons. Kittel, C., and Kroemer, H. (1980). Thermal Physics (2nd ed.). New York: W. H. Freeman. (Original work published 1969) Liboff, R. (1993). Introductory Quantum Mechanics (2nd ed.). Reading, Massachusetts: Addison-Wesley. Newton, I. (1962). Sir Issac Newton's Mathematical Principles of Natural Philosophy and His System of the World (Vols. 1-2) (A. Motte and F. Cajori, Trans.). Berkeley: University of California. (Original work published 1686) Pronko, N. H., and Snyder, F. W. (1951). Vision with Spatial Inversion [Film]. University Park, Pennsylvania: PCR: Films and Video in the Behavioral Sciences. Renninger, M. (1960). Messungen ohne Störung des Meßobjekts [Observations Without Disturbing the Object]. Zeitschrift für Physik, 158, 417-421.Sources - 36 -Snyder, D. M. (1983). On the Nature of Relationships Involving the Observer and the Observed Phenomenon in Psychology and Physics. The Journal of Mind and Behavior , 4, 389-400. Snyder, D. M. (1986). Light as an Expression of Mental Activity. The Journal of Mind and Behavior , 7, 567-584. Snyder, D. M. (1989). The Inclusion in Modern Physical Theory of a Link between Cognitive-interpretive Activity and the Structure and Course of the Physical World. The Journal of Mind and Behavior , 10, 153-172. Snyder, D. M. (1990). On the Relation between Psychology and Physics. The Journal of Mind and Behavior , 11, 1-17. Snyder, D. M. (1992). Being at Rest. The Journal of Mind and Behavior, 13, 157-161. Snyder, D. M. (1992). Quantum Mechanics and the Involvement of Mind in the Physical World: A Response to Garrison. The Journal of Mind and Behavior, 13, 247-257. Snyder, D. M. (1993). Quantum Mechanics Is Probabilistic in Nature. The Journal of Mind and Behavior , 14, 145-153. Snyder, D. M. (1994). On the Arbitrary Choice Regarding Which Inertial Reference Frame is “Stationary” and Which Is “Moving” in the Special Theoryof Relativity. Physics Essays , 7, 297-334. Snyder, D. M. (1995). On the Quantum Mechanical Wave Function as a Link between Cognition and the Physical World: A Role for Psychology. The Journal of Mind and Behavior , 16, 151-179. Snyder, D. M. (1996). The Mind and the Physical World: A Psychologist’s Exploration of Modern Physical Theory . Los Angeles: Tailor Press. Snyder, D. M. (1996). On the Nature of the Change in the Wave Function in a Measurement in Quantum Mechanics. Los Alamos National Laboratory E-PrintPhysics Archive (WWW address: http://xxx.lanl.gov/abs/quant-ph/9601006). Snyder, F. W., and Pronko, N. H. (1952). Vision with Spatial Inversion . Wichita, Kansas: University of Wichita Press. Stratton, G. M. (1896). Some Preliminary Experiments on Vision without Inversion of the Retinal Image. The Psychological Review , 3, 611-617.Sources - 37 -Stratton, G. M. (1897). Vision without Inversion of the Retinal Image - 1. The Psychological Review , 4, 341-360. Stratton, G. M. (1897). Vision without Inversion of the Retinal Image - 2. The Psychological Review , 4, 463-481. Stratton, G. M. (1899). The Spatial Harmony of Touch and Sight. Mind: A Quarterly Review , 8, 492-505. Tolman, R. C. (1938). 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arXiv:physics/0002030v1 [physics.atom-ph] 16 Feb 2000Phase transition in the ground state of a particle in a double -well potential and a magnetic field H. Kunz1, J. Chapuis1, A. Jory2 1Institute of Theoretical Physics, Swiss Federal Institute of Technology Lausanne, CH 1015 Lausanne EPFL 2Department of Mathematics, Swiss Federal Institute of Tech nology Lausanne, CH 1015 Lausanne EPFL (February 9, 2008) We analyse the ground state of a particle in a double-well pot ential, with a cylindrical symmetry, and in the presence of a magnetic field. We find that the azimuth al quantum number mtakes the values m= 0,1,2. . .when we increase the magnetic field. At critical values of the magnetic field, the ground state is twice degenerate. The magnetisation sho ws an oscillatory behaviour and jumps at critical values of the magnetic field. This phase transiti on could be seen in the condensate of a dilute gas of charged bosons confined by a double-well potent ial. I. INTRODUCTION It is a well-known fact1that in the absence of a mag- netic field the ground state of bosons is non degener- ate, and therefore has the symmetry of the hamiltonian. Mathematically this result from the fact that the kernel of the operator e−tHis positive. This last property no more holds in the presence of a magnetic field so that de- generacy of the ground state may be expected, as well as symmetry breaking in it. One-body systems already may show this phenomenon. Indeed Lavine and O’Carrol2 proved the existence of spherically symmetric potentials for which, in the presence of a magnetic field, the ground state has a non-vanishing value for the zcomponent of angular momentum, so that the rotational symmetry is broken. Further examples were provided by Avron, Herbst and Simon3,4. On the opposite side, these last authors were able to prove that for the hydrogen atom the symmetry is not broken, as well as in the case where the poten- tial is monotonically increasing with the distance. These authors, however, mainly concerned with problems of atomic physics, did not discuss the degeneracy and the physical significance of it. On the other hand, two of us analysing the problem of a particle confined to a disc or an annulus in the pres- ence of a magnetic field found that the ground state was degenerate in the case of an annulus and for a disc with Neumann boundary conditions (with Dirichtlet boundary conditions in the disc case the degeneracy disappears.) The degeneracy appears each time the magnetic field reaches a critical value and the magnetisation jumps at these critical values, which form a discrete set5. Motivated by these results we consider in this arti- cle a class of systems for which similar phenomena oc- cur. Namely we analyse the ground state of a particle in three dimensions moving in a double-well type poten- tial, cylindrically symmetric, and submitted to a constant magnetic field in the zdirection.We find that the ground state has an azimuthal mo- mentum ¯hmtaking increasing values m= 0,1,2,...when we increase the magnetic field B. At critical values of B (Bm) the ground state is twice degenerate between the m and them+ 1 state. Moreover the magnetisation jumps at these critical values and shows in general an oscilla- tory behaviour reminiscent of the well known de Haas von Halphen oscillations in solid state physics. We show that this phenomenon can be understood by an analysis of the minima of the potential energy, fixing however the angular momentum to its quantised value ¯hm. In the two-dimensional case we can use the WKB method and obtain bounds on the energy in order to estimate the critical fields. But in general, we had to compute numerically the energies and compare them to estimates based on trial wave functions. The agreement is quite good in general. Concerning possible experimental verifications of these effects, which require basically to have a potential which has a minimum sufficiently far from the origin, we could think of two cases. The first one would be in some molecules where proton dynamics could be described by such an effective potential. The second one, more thrilling, would be the case of charged bosons undertak- ing a Bose-Einstein condensation. Our results suggest that in this case, the bosons would undertake a phase transition in their condensate, when we apply an in- creasing magnetic field. This phase transition would manifest itself by appearance of oscillations in the mag- netisation , which would jump at certain critical values of the magnetic field. II. THE MODEL We will consider the case of a particle of mass µ, charge q, in a potential Vwith a cylindrical symmetry, submit- ted to a magnetic field /tildewideBin thezdirection. We do not consider the effect of the spin of the particle. We choose 1for a unit of energy V0, and length r0, both being charac- teristic of the potential. The dimensionless hamiltonian reads ifr=/radicalbig x2+y2 (iǫ/vector∇ −/vectorA)2+V(r,z) (1) where ǫ=¯h r0√2µV0(2) measures the importance of the quantum effects and the vector potential in the symmetric gauge is given by /vectorA=/parenleftbig−By 2,Bx 2,0/parenrightbig (3) B=q cr0√2µV0/tildewideBbeing the dimensionless magnetic field. Thanks to the cylindrical symmetry, we can replace thezcomponent of the angular momentum Lzby its eigenvalueǫmso that the reduced hamiltonian reads Hm=−ǫ2/bracketleftbigg1 r∂ ∂rr∂ ∂r+∂2 ∂z2/bracketrightbigg +/parenleftbiggǫm r−rB 2/parenrightbigg2 +V(r,z) (4) The ground state energy of this hamiltonian and the cor- responding eigenfunction will be denoted Emandψm. It remains to specify V. We will basically consider a double-well potential of the form: V(r,z) =r4+z4−2(r2+z2) +vr2z2(5) withvsatisfyingv≥ −2, so thatVis bounded from below. Ifvis equal to 0 we can decouple the motion in thezdirection form the one in the plane perpendicular to the magnetic field. This is what we will call the two- dimensional case . Ifv= 2, we have in three dimensions a potential with spherical symmetry. We have chosen this double-well form because if we had taken the simple well V=r4+z4+ 2(r2+z2) +vr2z2 withv≥0 it follows from3that the ground state is not degenerate and corresponds to m= 0. A physical quantity of interest is the magnetisation in the ground state M=−∂E ∂B(6) in unitsq cr0/radicalBig V0 2µ We will denote by emthe ground state energy of the hamiltonian hm=−ǫ2/bracketleftbigg1 r∂ ∂rr∂ ∂r+∂2 ∂z2/bracketrightbigg +Vm(r,z) (7) with Vm=(ǫm)2 r2+B2 4r2+V (8)and by Em=em−ǫmB (9) the ground state energy of Hmgiven in (4), so that the real ground state energy is given by E= inf m≥0Em (10) since obviously negative mgive a larger energy. Finally we will use the following useful scaling property of the energy em em(ǫ,λ,v) =s2em/parenleftBigǫ s3/2,λ s,v/parenrightBig ∀s≥0 (11) where λ=B2 4−2 (12) is the parameter multiplying r2in the potential. Equa- tion (11) follows simply from the scaling transformation :r2→sr2andz2→sz2. This relation shows that we have effectively a two parameter dependence of the en- ergyemin general and a one parameter dependence in the two dimensional case. The choice s=|λ|ors=m1 3(m≥1) shows that large magnetic field or large angular momenta correspond to the semi-classical limit. In fact we shall see that in the classical limit ǫ→0 ground state with m/negationslash= 0 are favoured inducing ground state degeneracies at some val- ues of the magnetic field. It thus appears that the ten- dency to have a ground state with the same symmetry as the hamiltonian and therefore non degenerate is an effect due to quantum mechanics. III. THE CLASSICAL LIMIT One can gain some qualitative understanding of the problem by looking at the classical limit of it. This means that we neglect the quantum kinetic energy and define the ground state energy as E= inf m≥0inf (r,z)[Vm(r,z)−ǫmB] (13) where Vm=(ǫm)2 r2+r4+z4+B2 4r2−2r2+vr2z2(14) and consider that mis an integer. Two cases need to be considered separately: |v|<2 andv≥2. If|v|<2 we denote by xmandtmrespectively the values of r2andz2which minimise the potential Vm, and we find tm= 1−vxm 2(15) /parenleftbigg 2−v2 2/parenrightbigg xm+/parenleftbigg v−2 +B2 4/parenrightbigg =(ǫm)2 x2m 2On the other hand, considering for a while mas a con- tinuous variable, the absolute minimum of Vm−ǫmBis given by ǫˆm=B 2xˆm (16) From (15) this gives an absolute minimum of Vm−ǫmB given by xˆm=tˆm=1 1 +v 2(17) and therefore ǫˆm=B 21 1 +v 2(18) In considering the variable mas a continuous one we have treated the problem purely classically and the cor- responding ”ground state” energy is Ecl=−2 1 +v 2(19) We know that mis a discrete variable but for consistency we must consider ǫas a small number. Then if mdes- ignates the integer part of ˆ m, we have ˆm=m+θand if 0≤θ <1 2, the ground state has the quantum number m, whereas if1 2<θ≤1 it hasm+ 1. From this analysis we conclude that if Bm−1< B < Bmwhere Bm=ǫ(1 +v 2)(2m+ 1) (20) the ground state has the quantum number m. Hence we see that by increasing the magnetic field, we find in in- creasing order the values of m= 0,1,2,...and an infinite set ofcritical values of the magnetic field exist ,Bm for which the ground state is twice degenerate , being bothmandm+ 1. This picture is entirely confirmed by the numerical re- sults in the quantum case. It is also quite interesting to look at the magnetisation. In the state whose quantum number ism, we have Mm=ǫm−B 2xm (21) so that using (15) Mm= [ǫm−B 21 1 +v 2][1−v 2 1−v 2+B2 4] (22) whenBm−1<B <B m. This shows that the magnetisation has an ”oscillatory” type of behaviour reminiscent of the familiar de Haas von Halphen one in solid state physics and that the magneti- sation jumps at the critical values of the magnetic field, the jump being given by ∆Mm=ǫ1−v 2 1−v 2+B2 4(23)Once again this general behaviour is reproduced by the numerical results in the quantum case and the spac- ing between the values of the critical field is rather well represented by formula (20) when m≥1. In the two- dimensional case, i.e.v= 0 and neglecting the trivial zdependence, we can proceed further and look at a re- ally semi-classical approximation namely WKB, for the ground state energy /integraldisplayr+ r−dr/radicalbig em−Vm(r) =ǫπ 2(24) where Vm(r) =(ǫm)2 r2+r4+/parenleftbig −2 +B2 4/parenrightbig r2(25) and the ground state energy is Em=em−ǫmB. In fact this WKB approximation will give the best an- alytical results, apart from the variational estimates for the energy, which give unfortunately only exact upper bounds on the energy. When the potential has spherical symmetry v= 2, quantum effects are much more important and the clas- sical analysis gives only that the ground state has m= 0 ifB <2ǫ, is degenerate between m= 0 andm= 1 when 2ǫ≤B <4ǫ, has possibly m= 0,1,2 for 4ǫ≤B <6ǫ and so on. This only suggests that we have again the increasing sequence of m, when we increase the magnetic field and that critical values appear near 2 ǫm. Whenv > 2, we find that m= 0 is the ground state except when B= 2ǫm, where it is degenerate be- tweenmand 0. We may note however that the classical ground state correspond to points ( r= 0,z=±1) in configuration space for m= 0, whereas it corresponds to two circles ( r=ǫ 2B,z=±/radicalbig1−ǫ 2B) form= 1 and 2ǫ < B < 4ǫ, so that the wave function can be more spread in the m= 1 state than the in the m= 0 state, and that the kinetic energy of the m= 1 state is lower, favouring the m= 1 state. Hence we should expect, at least whenǫis small, a ground state with m= 0 for small fields and a ground state with m= 1, when 2 ǫ<B < 4ǫ. A similar argument can be given for the higher values of m. Finally, it is worth noticing that if we had taken a simple well type potential V(r,z) =r4+z4+ 2(r2+z2) +vr2z2(26) the classical analysis gives a ground state with m= 0, at least when v≥ −1. This is a correct result when v≥0 at the quantum level. 3IV. NUMERICAL RESULTS AND VARIATIONAL BOUNDS It is quite useful to undertake a numerical analysis of this problem. We have used a finite element method, choosing for the basis a product of two triangles func- tions. We discuss separately the two-dimensional prob- lem and the three dimensional ones. A. Two dimensions We first give pictures of the ground state energy for two typical values of ǫ, a small (ǫ= 0.03) and a large one (ǫ= 0.5) as a function of the magnetic field B. (figure 1). The cusps at the critical values of Bindicate a jump of the corresponding magnetisation. 0 0.1 0.2 0.3 0.4−2−1.5−1−0.500.511.510−2 BM 0 2 4 6 8−0.5−0.4−0.3−0.2−0.10 BM0 0.1 0.2 0.3 0.4−9.41−9.405−9.4−9.395−9.3910−1 ε = 0.03 BEo 0 2 4 6 8−0.500.511.522.533.5ε = 0.5 BEo FIG. 1. Energie and magnetisation dependence of Bfor ǫ= 0.03 and ǫ= 0.5 This last quantity shows first a diamagnetic behaviour at small field, but then a paramagnetic - diamagnetic oscillation at least when ǫ<∼0.3. Beyond this value the magnetisation is entirely negative (figure 1 bottom right). We can also note that when Bbecomes large the mag- netisation tends to −ǫ, its value in the Landau regime. The results clearly indicate that we go progressively from the states with m= 0,1,2...by increasing the magnetic field and that the magnetisation jumps at the critical values. The effect is more pronounced in the clas- sical regime. All these results are in qualitative argument with the classical picture presented before and the agree- ment is even quantitative when ǫ= 0.03 for example.123456789100.0270.02750.0280.02850.0290.02950.03∆M msimulation estimation FIG. 2. Comparison of the jumps given by simulation and the estimation The jumps of the magnetisation given by formula (22) are reproduced (figure 2) with a precision of less than 1 percent when ǫ= 0.03, and the spacing between the critical values of the magnetic field Bm+1−Bm ǫ= 2 + ∆ m (27) is given by ∆ m≤0.04 ifm≥1 andǫ= 0.1. ∆ mde- creases when mincreases in agreement with the scaling relationBm= (2m+1)ǫ, so that the simple classical for- mula reproduces rather well the results. By contrast, the jump between the m= 0 and the m= 1 state is largely of quantum mechanical origin, as well as the precise values of the critical fields. 0 1 2 3 4 500.10.20.30.40.50.60.70.80.91 Bε0 1 2 3 4 5 6 > 6 FIG. 3. Mvalues of the ground state depending on Bandǫ Figure 3 describes the various regions in the ǫ−Bplan. We can note that even when ǫ >0.25 a linear relation exists between Bmandǫ, as in the classical regime, which is a bit surprising. 400.5 11.5 2−1−0.500.5 rEm = 0 B = 0.34 ε = 0.2 00.5 11.5 200.511.5ψ0E0 = −0.61474 r00.5 11.5 2−1−0.500.5 rEm = 1 B = 0.34 ε = 0.2 00.5 11.5 200.511.5ψ1E0 = −0.61474 r FIG. 4. Potential and eigenfunction of the ground state for m= 0 (left) and m= 1 (right) with Vm(− −) = orbital kinetic energy ( − · −) + double-well ( − −) and energy levels ( · · ·) It is also interesting to look at the eigenfunctions when the magnetic field reaches its critical value. In figure 4 we give pictures of them at the critical value between the statem= 0 andm= 1 when ǫ= 0.2. We see that their maxima are located very near the minimum of the potential. Finally we compare the results with two theoretical estimates: first of all the WKB one, and a variational one. This last estimate is based on the following two parameters trial wave function ψm=rme−αr2−β(r−1)2(28) The variational upper bound on the energy can be ex- pressed in terms of Weber cylindrical functions, but we directly computed the corresponding integrals. Deg.Simul. WKBδ%Variat.δ% 0-10.0313 0.0314 0.230.0317 1.16 1-20.0944 0.0942 -0.15 0.0946 0.23 2-30.1573 0.1571 -0.13 0.1574 0.09 3-40.2201 0.2198 -0.13 0.2203 0.06 4-50.2830 0.2826 -0.12 0.2830 0.02 5-60.3457 0.3453 -0.12 0.3458 0.02 6-70.4085 0.4080 -0.12 0.4085 0.00 TABLE I. Magnetic field Bmat the seven first degeneracies withǫ= 0.03Deg. Simul. WKBδ%Variat.δ% 0-1-0.9405 -0.9401 0.66-0.9403 0.26 1-2-0.9404 -0.9400 0.65-0.9403 0.27 2-3-0.9403 -0.9399 0.64-0.9401 0.28 3-4-0.9401 -0.9397 0.63-0.9399 0.29 4-5-0.9399 -0.9395 0.61-0.9397 0.30 5-6-0.9396 -0.9392 0.59-0.9394 0.30 6-7-0.9392 -0.9389 0.56-0.9390 0.30 TABLE II. Energies Emat the seven first degeneracies withǫ= 0.03 Tables I,II,III and IV give a comparison of the results for two values of the parameter ǫ, and for the critical fields. Excellent agreement is found for the variational method (maximal error of the order of 2 % when ǫ= 0.5). WKB works quite well when ǫis small (ǫ= 0.03) as expected, but even better on the energies when ǫ= 0.5 and the error does not exceed 1%. Deg.Simul. WKBδ%Variat.δ% 0-11.538 1.661 7.951.508 -1.98 1-22.747 2.811 2.332.743 -0.15 2-33.842 3.882 1.063.842 0.02 3-44.891 4.919 0.564.894 0.05 4-55.920 5.940 0.345.924 0.07 5-66.941 6.954 0.186.943 0.02 6-77.953 7.964 0.127.956 0.02 TABLE III. Magnetic field Bmat the seven first degenera- cies with ǫ= 0.5 Deg.Simul. WKBδ%Variat.δ% 0-10.220 0.232 0.97 0.227 0.55 1-20.685 0.686 0.04 0.690 0.25 2-31.159 1.159 -0.02 1.163 0.16 3-41.639 1.638 -0.03 1.642 0.12 4-52.122 2.122 0.00 2.125 0.12 5-62.609 2.608 -0.02 2.612 0.07 6-73.098 3.098 0.00 3.101 0.07 TABLE IV. Energies Emat the seven first degeneracies withǫ= 0.5 5B. Three dimensions For the spherically symmetric potential ( v= 2), figure 5 gives the ground energies a well as the corresponding magnetisation for two different values of ǫ: 0.03, 0.5. 0 0.1 0.2 0.3 0.4−2.5−2−1.5−1−0.500.5110−2 BM 0 2 4 6 8−0.6−0.5−0.4−0.3−0.2−0.10 BM0 0.1 0.2 0.3 0.4−9.41−9.4−9.39−9.38−9.37−9.3610−1 ε = 0.03 BEo 0 2 4 6 800.511.522.533.54ε = 0.5 BEo FIG. 5. Energie and magnetisation dependence of Bfor ǫ= 0.03 and ǫ= 0.5 Once again we see that the values of min the ground state increases with B, and that the magnetisation jumps at critical values Bmof the magnetic field, where the ground state is doubly degenerate. These results are in qualitative agreement with the classical analysis. Figure 6 summaries the results in the ǫ-Bplane. Notice that in this 01234567891000.10.20.30.40.50.60.70.80.91 Bε0 1 2 3 4 5 6 > 6 FIG. 6. Mvalues of the ground state depending on Bandǫ case, when ǫ≥0.1 already the relation between Bm andǫis no more linear. On the other hand the spacingbetween the critical values of Bpredicted by the crude classical estimate: ∆Bm=Bm+1−Bm∼=2ǫ (29) is satisfied with a precision of 25% at m= 1 and becomes more accurate when mincreases, at least in the range ǫ≤0.1. Our best variational estimate for the energy was made with a three parameter trial wave function ψα,β,ζ=rme−αr2−β(√ r2+z2−ζ)2(30) Table V gives the values of the critical field Bmand Table VI the corresponding ground state energies, when ǫ= 0.05 estimated by the variational method and computed with the simulation. Simulation Variational Deg.BmEmBmδB%EmδE% 0-10.1180 0.1206 2.17-0.8986 -0.8982 0.38 1-20.2381 0.2310 -2.94 -0.8966 -0.8966 -0.01 2-30.3549 0.3509 -1.15 -0.8946 -0.8946 -0.06 3-40.4686 0.4616 -1.49 -0.8925 -0.8925 -0.00 4-50.5829 0.5785 -0.74 -0.8901 -0.8901 -0.01 5-60.6961 0.6905 -0.80 -0.8876 -0.8876 -0.00 TABLE V. Magnetic field Bmand energies Emat the six first degeneracies at ǫ= 0.05 Simulation Variational Deg.BmEmBmδB%EmδE% 0-12.7576 0.9415 2.6225 -4.89 0.8959 -2.34 1-24.2493 1.6345 4.0912 -3.72 1.5675 -2.54 2-35.6746 2.3190 5.4972 -3.12 2.2363 -2.49 3-47.0961 3.0126 7.0275 -0.96 2.9845 -0.69 4-58.5025 3.7055 8.2415 -3.07 3.5720 -2.83 5-69.7537 4.3248 9.6016 -1.55 4.2412 -1.57 TABLE VI. Magnetic field Bmand energies Emat the six first degeneracies at ǫ= 0.5 6Obviously there is a very good agreement, since the largest error for Bmis less than 2% and for Emless than 0.7%. Table VI gives the same but for ǫ= 0.5. Again we see a good agreement (error less than 5%). When ǫincreases we found that αincreases and βdecreases as well as ζand our trial wave function becomes less accurate, because the double-well nature of the potential is less important compared to the kinetic energy. 0 0.2 0.4−101234 Bvε = 0.02 0 0.5 1−101234 Bvε = 0.05 0 1 2−101234 Bvε = 0.1 0 2 4−101234 Bvε = 0.2 0246−101234 Bvε = 0.3 0 5 10−101234 Bvε = 0.5 FIG. 7. Mvalues of the ground state depending on Band vwith increasing ǫ Figure 7 describes the situation in the v-Bplane for m= 0,1,...,10 and different values of ǫ. We notice that whenvis less than 2 and ǫis not too large ( ǫ≤0.2), the situation is similar to the one already discussed, but that there is an abrupt change at v= 2 whenǫis small in agreement with the classical analysis. However when ǫ>0.2 the ground state m= 0 is definitely favoured as vincreases. 00.5 11.5 2−1.65−1.6−1.55−1.5−1.45−1.4−1.35−1.3 BEv = 0 00.5 11.5 2−0.85−0.8−0.75−0.7−0.65−0.6−0.55−0.5 BEv = 2 00.5 11.5 2−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4 BEv = 3 FIG. 8. Comparison for different vof the energy for m= 0,1,2,3,4,5 depending on Bwithǫ= 0.1 Figure 8 shows the energies for the first five mvalues computed with three different v: two-dimensional ( v= 0), spherical potential ( v= 2), andv= 3. We can see a new crossing between the m= 0 and the other mlevels whenvbecomes larger than 2, although this does not concern the ground state.V. BOUNDS ON THE CRITICAL FIELD IN THE TWO DIMENSIONAL CASE One might desire to get rigorous upper and lower bounds on the critical fields. One possible approach would consist in getting upper and lower bounds on the ground state energies Em. Whereas we have seen that one can obtain very good variational upper bounds, it is rather difficult to get good lower ones. In order to test these results, we analysed only the two-dimensional case. First we want to obtain conditions under which m= 0 is the ground state. Using the inequality l2 x+x2≥x2−l2 a2x+2l2 a(31) valid for any xandapositive, we deduce that e0[λ]≥2(ǫm)2 a+e0/bracketleftbig λ−/parenleftbigǫm a/parenrightbig2/bracketrightbig (32) On the other hand e0[λ]−e0/bracketleftbig λ−/parenleftbigǫm a/parenrightbig2/bracketrightbig =/integraldisplayλ λ−(ǫm a)2dλ′/angb∇acketleftr2/angb∇acket∇ight0(λ′) ≤/parenleftBigǫm a/parenrightBig2 /angb∇acketleftr2/angb∇acket∇ight0/bracketleftbig λ−/parenleftbigǫm a/parenrightbig2/bracketrightbig (33) since/angb∇acketleftr2/angb∇acket∇ight0(λ) is decreasing in λ. But /vextendsingle/vextendsingle/angb∇acketleftr2/angb∇acket∇ight0[λ]−λ 2/vextendsingle/vextendsingle≤/bracketleftBig e0[λ] +/parenleftbiggλ 2/parenrightbigg2/bracketrightBig1 2(34) The scaling relation and the fact that e0is increasing in ǫimply that whenλ 2≤ −1 e0[λ] +/parenleftbiggλ 2/parenrightbigg2 ≤/parenleftbiggλ 2/parenrightbigg2 (e0[−2] + 1) (35) Taking now asuch thatǫm a≥B 2(m≥1) we get combin- ing these inequalities that E0≤Em ∀m≥1 (36) if we can find t>B 2such that t2/braceleftbigg 1 +1 2/parenleftBig t2−B2 4/parenrightBig/bracerightbigg δ−2ǫ/parenleftBig t−B 2/parenrightBig <0 (37) whereδ= 1 +/radicalbig eo[−2] + 1 In the estimate for δwe can use our best variational upper bound. Inequality (37) will be satisfied if Bis less than some value B0, so that in this range m= 0 is the ground state. In order to see when m/negationslash= 0 is a ground state, we use the following trial wave function ψ(r) for a state with angular momentum m′. ψ(r) =rm′−mψm(r)m′≥m (38) 7whereψm(r) is the exact ground state wave function for the state with angular momentum m. An integration by parts shows that /integraldisplay∞ 0drr/bracketleftbig ψ′ m2r2(m′−m)+ 2(m′−m)r2(m′−m)−1ψ′ mψm/bracketrightbig =−/integraldisplay∞ 0drr2(m′−m)ψm(rψm)′(39) Therefore if we use the fact that ǫ2 r(rψ′ m)′= [Vm(r)−em]ψm (40) We see that /integraldisplay∞ 0drr/bracketleftbig ǫ2ψ′2+Vm′(r)ψ2/bracketrightbig =em/integraldisplay∞ 0drrψ2 +ǫ22m′(m′−m)/integraldisplay∞ 0drr2(m′−m)−1ψ2 m(41) and we conclude that em′≤em+ǫ22m′(m′−m)/angb∇acketleftr2(m′−m−1)/angb∇acket∇ightm /angb∇acketleftr2(m′−m)/angb∇acket∇ightm(42) In particular e1≤e0+ 2ǫ21 /angb∇acketleftr2/angb∇acket∇ight0(43) If we have a lower bound con/angb∇acketleftr2/angb∇acket∇ight0then we see that E1< E0 (44) if B >2ǫ c(45) We can use for the lower bound cthe one given in equa- tion (34) c=λ 2−/radicalbigg e0[λ] + (λ 2)2 (46) which is satisfactory when Bis not too large, but which becomes negative for large B. We can repair this by using the fact3that iffis an increasing function of r, its expectation value in the ground state is lowered by adding to the potential a new increasing potential. We can find a useful comparison potential W=a1r2+a2r4+a3r6≥V (47) which has a ground state wave function of the form ψ=eb1r2−b2r4b2>0 (48) so that /angb∇acketleftr2/angb∇acket∇ightWcan be computed explicitly for this poten- tial and we can take c=/angb∇acketleftr2/angb∇acket∇ightWin equation (45), which gives a more satisfactory result for large B.In any case we see that the state m= 1 if favoured over the statem= 0 ifBis larger than some value, and by continuity there must exist a field for which both states have equal energy. But in order to prove that the ground state ism= 1 whenBis in some range requires to show thatEm>E1∀m≥2. For this purpose let us consider mas a continuous parameter. Then ∂Em ∂m= 2ǫ2m/angb∇acketleft1 r2/angb∇acket∇ightm−ǫB (49) If we can show that∂Em ∂m≥0 for allm≥1, then we will have shown that Em>E1. Whenm≥1 we have /angb∇acketleft1 r2/angb∇acket∇ightm≥1 /angb∇acketleftr2/angb∇acket∇ightm(50) and (ǫm)2 /angb∇acketleftr2/angb∇acket∇ightm+/angb∇acketleftr2/angb∇acket∇ight2 m+λ/angb∇acketleftr2/angb∇acket∇ightm≤em (51) In order to get a variational bound on emwe can use the trial wave function ψ=rme−ar2, which gives em≤m+ 1 m+ 2Vm+2(xm+2) (52) wherexmis the value of xwhich minimises Vm(x) =(ǫm)2 x+x2+λx (53) Noting that equation (51) implies that /angb∇acketleftr2/angb∇acket∇ightm≤xm+/radicalbig em+Vm(xm) (54) one can see by combining equations (49), (50), (52) and (54) thatEm≥E1for allm≥2 if B2 8<1 1 +c2(55) with c2=1 ǫ2|λ|/bracketleftBig x1+/radicalbig V1+2m(x1+2m)−V1(x1)/bracketrightBig2 (56) which implies that Bshould be less than some value. We give in the table VII some numerical values for the bounds obtained by these methods. ǫE0<E1B0−1E1<E mB1−2E1<E0 0.01 0.0 - 0.005 0.011 0.0 - 0.026 0.030 0.022 - 0.05 0.0 - 0.024 0.054 0.0 - 0.124 0.163 0.146 - 0.10.0 - 0.047 0.121 0.0 - 0.221 0.359 0.366 - 0.20.0 - 0.088 0.340 0.0 - 0.364 0.878 1.788 - 0.50.0 - 0.189 1.610 0.0 - 0.609 2.745 2.834 - 1.00.0 - 0.310 3.686 0.0 - 0.826 5.846 4.277 - 2.00.0 - 0.469 7.816 0.0 - 1.066 11.910 8.141 - TABLE VII. Results of the inequalities on the energies and values Bm 8They show that whereas the range of values of Bfor whichE0< E1andE1< E0is reasonably estimated forǫ<∼0.1, there is no range of values of Bfor which our bounds show that m= 1 is the ground state except whenǫis very small (0.01) But in this range WKB works perfectly well. Obviously we have too poorly estimated the effect of the kinetic energy and that of the centrifugal barrier. Numerical computations for example show that the replacement of /angb∇acketleft1 r2/angb∇acket∇ight1by1 /angbracketleftr2/angbracketright1is not appropriate when ǫorBare too large. In conclusion, even in two dimensions improved rigor- ous bounds on the critical values of the magnetic field are needed, and the WKB method for which we have no estimate of the error gives the best analytic results. VI. CONCLUSION It could be of course quite interesting to see an ex- perimental verification of these surprising effects of the magnetic field. Even though we have found them in the case of a double-well, we think that the details of the potential do not matter too much. What is needed is a potential whose minimum is taken sufficiently far from the origin. We have thought of two possible fields where one could observe such effects. The first one is molecular physics where often the dynamics of electrons or protons is mod- elled by the motion of a quantum particle in a double-well (although admittedly often a one-dimensional one.) If we consider the case of the electron in the rotationally sym- metric double-well, the smallest value of the critical field where them= 1 andm= 0 states are degenerate is about 15 Tesla if we take for the depth of the potential 1 eV and for the distance to the origin of the minimum 2˚A. For protons the situation is more favourable since a field of 5 Tesla can create a degeneracy when the depth is kept to 1 eV and the minimum is at a distance of 1 .5 ˚A. Obviously a more detailed investigation is needed if one wants to see these unusual effects (like a change from diamagnietism to paramagnetism) in molecules. The other field is that of Bose-Einstein condensates of very cold atoms, which recently has made spectac- ular progress. If we consider free charged bosons in a magnetic field and in a potential V(r r0) one can show that there is a Bose-Einstein condensation in the ground state in three dimensions, in the limit r0going to infinity, for all potentials which have a quadratic dependence of rnear the origin. Our result supports therefore that free charged bosons in their condensate would show a phase transition when one varies the magnetic field. This tran- sition would manifest itself by jumps of the magnetisa- tion at some critical values of the magnetic field. The phenomenon would probably persist in a dilute gas of charged bosons in a neutralising background. It is how- ever probably quite difficult to create such a jellium in the laboratory and this remains a challenging task.VII. ACKNOWLEDGEMENTS We thank Ph. Martin and N. Datta for some use- ful discussions on the Bose-Einstein condensation in the presence of a magnetic field. 1Reed and Simon, Methods of Modern Mathematical Physics , Vol IV Chapter XIII.12, Academic Press, (1970) 2R. Lavine, M. O’Carrol, Journal of Mathematical Physics, 18, 1908, (1977) 3J. E. Avron, I. W. Herbst, B. Simon, Communications in Mathematical Physics, 79, 529, (1981) 4J. E. Avron, I. W. Herbst, B. Simon, Duke Mathematical Journal, 45, 847, (1978) 5Alessandro Jori, Queues de Lifschitz magn´ etiques , Th` ese No. 1813, EPFL, (1998) 9