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arXiv:physics/9911058v1 [physics.atm-clus] 24 Nov 1999Semiempirical charge distribution of clusters in the ion sputtering of metal Victor I. Matveev and Olga V. Karpova Heat Physics Department of Uzbek Academy of Sciences, 28 Katartal Str., 700135 Tashkent, Uzbekistan Abstract We propose the generalization of a known established empiri cally (Wahl W. and Wucher A. Nucl. Instrum. Meth. B 94, 36(1994)) power law, describing relative mass-spectra of neutral sputtered clusters, on th e cases of arbitrary clus- ter charges. The fluctuation mechanism of charge state forma tion of sputtering products in the form of large clusters with the number of atom sN≥5 is also proposed. The simple formula obtained by us has been shown a g ood agreement with the experimental data. PACS numbers: 79.20.*, 36.40*. 0Sputtering of solids under the ion bombardment is one of the m ain applied and fundamental problems which is importance in the many directions of contemporary science and technology. Consid erable tech- nological possibilities in the micro- and nanoelectronics , cosmic and thermonuclear technologies have stimulated increase of th e number of works devoted to application and basic investigations of sp uttering phe- nomenon (see, for example, recent reviews [1-3] and referen ces therein). The theoretical description and estimations of sputtering processes are rather difficult due to the multiparticle character of proble m both at the stage of ion penetration to solid and at the stage of formatio n sputtering products which consist of not only single target atoms but al so polyi- atomic particles, i.e., clusters. Presently, some perspec tives on carrying out of ”first principle” calculations are connected (see, al so, estimates [4,5]) with computer simulation by molecular dynamics meth ods. How- ever, such calculations are complicated in technical plan, especially in the case of increasing of the number of atoms in cluster and they a re difficult for performing excluding the authors of these calculations .. The maximal using of possibilities of empirically established sputter ing regularities is reasonable in this case. For cluster sputtering so-called p ower law for rel- ative neutral cluster yield which was discovered experimen tally (see for instance [6]) could be most important. According to this the norlmalized neutral cluster yield is described by the law Nξ, where Nis the number of atoms in the cluster and the parameter ξdepends on bombardment conditions and target type. One of most complex problems is a lso pro- cess of charge state formation of surface sputtering produc ts. Consid- erable number of experimental and theoretical works are dev oted (see, for example, review [7]) to the investigations of charge sta te formation of single atomic particles at the surface scattering or sput tering of metal surface. On other hand, the mechanism of charged structure f ormation 1of polyatomic particles had been less investigated both the oretically and experimentally. In this paper the generalization of well kn own empiri- cally established power law describing relative mass-spec tra of neutral clusters for cluster emission of arbitrary charges is offere d. The fluc- tuation mechanism of charge state formation of sputtering p roducts in form of large clusters with the number of atoms N≥5 is also proposed. Derived simple formulas are in a good accordance with the exp erimental data. We use the old conception according to which large clus ters are emitted as a whole agglomerate in the form of block of atoms (s ee also [8,9]) . We will consider the probability WNof events corresponding to correlated movement of N-atomic block as given. Let us determine the charged state of the block of N-atoms. For this purpose we wil l follow the statistical deriving of Saha-Lengmuir’s formula [10], and assume that with moving off of the cluster from the metal surface up to some distance (so-called critical distance) the exchange between the ele ctrons of metal conduction zone and electrons of cluster atoms is possible. When cluster moves away from the metal surface to the distance exceeding c ritical one, the electron exchange stop unadiabatically. Further b elow saying about cluster electrons, we will mean valence electrons onl y and corre- sponding aggregate of states we will call the cluster conduc tion zone. We will also assume that namely between the zones of metal and cluster the exchange is possible. Then average number of electrons nτon the energy electron level ετof cluster, according to the Fermi distribution, is defined by nτ={exp[(ετ−µ)/Θ] + 1 }−1,where Θ is temperature, µis the chemical potential. Let us denote via ∆n2τthe average of square de- viation numbers of occupation nτfrom the equilibrium nτ- values. Then ∆n2τ=(nτ−nτ)2=nτ(1−nτ) [11]. Obviously, the average number of electrons is Ne=/summationtext τnτ. Let the number of electron in cluster conduc- tion zone is Ne. Then, according to definition, the average of square 2deviation of number of electron in cluster conduction zone f rom average value is ∆N2e=(Ne−Ne)2=/summationtext τ∆n2τ.The cluster, having Neelectrons in conduction zone, will be electrically neutral, if Ne=Ne, where Ne is the average number of electrons in the cluster conduction zone which is equal to the number of atoms in N-atomic cluster multiplied to va- lencyγ(i.e., to the number of atomic electrons, yelding by neutral metal atom to the conduction zone). Thus, cluster charge is Qe= (Ne−Nγ)e, where eis electron charge. Further calculations with these formul ae require knowledge of the electronic structure of cluster an d generally speaking cannot be performed in general form. However, if to consider cluster size is large enough and electronic states are quasi -continuous, one can exchange summing over the electronic states on integ ration over the zone [11]. Therefore, for the temperatures less than the degeneration temperature, i.e. for µ/Θ≫1, one has ∆N2e≈21/2V m3/2 e π2¯h3√µΘ, where meis electron mass in conduction zone, Vis cluster volume and chemical potential of the degenerated Fermi gas with the num ber of particles Nein the cluster volume Vis [11] µ= (3π2)2 3¯h2 2me Ne V 2 3 . Thus, the average of square deviation of cluster charge from the equilib- rium value of Qe=(Ne−Nγ)e= 0, is1 (∆QN)2=e2∆N2e=e231 2 π4 3meΘ ¯h2 Ne V 1 3 V. (1) 1In principle, equality to zero of equilibrium cluster charg e follows from the assumption that Fermi levels in cluster and metal coincide. If it is not executed, a symmetry between positive and negative charged clusters will be observed and corresponding change s in following formulas can be easy made. 3Probabilities PN(Q) of values Qwe will determine by making use of standard formula for probability of fluctuations, i.e., PN(Q) =1 DNexp/braceleftbigg −1 2Q2 (∆QN)2/bracerightbigg , (2) where normalizing factor DNis defined by summing (2) over all possible values ( 2) Q= 0,±e,±2e, .... Thus, to obtain probability WQ Nof cluster emission with number of atoms Nand charge Qeone should multiply the probability of occurrence of events WNcorresponding to correlated moving of N-atomic agglomerate, on PN(Q): WQ N=WNPN(Q). (3) On other hand, according to experiment, neutral clusters ar e distributed by power law Nξ, and so W(Q=0) N=WNPN(Q= 0) = Nξ. (4) ThusWQ Ncan be written as follow WQ N=1 PN(Q= 0)NξPN(Q). (5) AsPN(Q= 0) = 1 /DN, then definitive expression for probability of N-atomic cluster emission and having charge Qwill have a form WQ N=Nξexp/braceleftbigg −1 2Q2 (∆QN)2/bracerightbigg , (6) where, according to equation ( 1), (∆QN)2=e231 2 π4 3meΘ ¯h2/parenleftBigg1 d/parenrightBigg2 3 γ1 3N , (7) where dis the number of atoms in the unit of cluster volume, i.e. conc en- tration (which we have accepted equal to the atomic target co ncentration for numerical calculations). 4Simplest characteristic of cluster charge distribution, c onsisting of given number of atoms N, is the ionization coefficient κQ Nwhich is equal to the ratio of number of clusters with charge Q/ne}ationslash= 0 and number of neutral clusters with the same number of atoms N. In our case ionization coefficient is κQ N=WQ N WQ=0 N=exp/braceleftbigg −1 2Q2 (∆QN)2/bracerightbigg . (8) Obviously, our consideration is not applicable for the sput tering of single atoms or small clusters. From comparison with the experimen tal data one can made a conclusion (see also [8,9]) on applicability o f the model beginning from the concrete number of cluster atoms ( N≥5). In exper- iment one measures, usually, the relative probabilities of the cluster yield with different number of atoms. Therefore, to compare theore tical data with the experiment ones, one should at first divide the proba bility (6) to the probability of cluster emission with ( 6) N= 5 (more exactly, we can choose any value N≥5, but it is more conveniently for us, when N= 5 ) , i.e. YQ N=WQ N/WQ 5. The experimental data will be same normalized. Farther, if it is necessary, one can pass to arbitrary conven ient system of units. The results of analysis of the general formulas and pe rformed nu- merical calculations and experimental data which are given in Figs. 1-3 allow to come to the following conclusions: a) The charge sta te changes by the variation of target temperature, moreover the ioniza tion coeffi- cients increase by increasing of the temperature; b) relati ve mass-spectra of the neutral clusters do not depend on target temperature, while rel- ative mass-spectra of charged clusters depend on it very str ongly, but by increasing of temperature they approach to mass-spectra of neutral clusters; c) the more cluster charge, the more seldom they ar e found; for example, the number of clusters with charge 2, as a rule, less than the number of clusters with charge 1; d) large clusters are ioniz ed in larger 5degree; e) tendency to saturation of ionization coefficients with growth of cluster dimension is an important peculiarity, qualitat ively the same behavior has been noted in experiments [12], that confirms th e conclu- sions about coincidence of the relative mass-spectra of cha rged clusters with neutral ones, when values of N are large (i.e., when N≫1) ). As it is well known, the experimental registration of the charg ed clusters is simpler technically than one of neutral clusters. Theref ore the data of measurements of charged clusters allow restoring of neut ral clusters distribution indirectly and experimental set up is simplifi ed very much. 6References 1. H.H. Andersen, K.Dan. Vidensk. Selsk. Mat. Fys. Medd. 43, 127(1993). 2. H.M. Urbassek and W.O. Hofer, K.Dan. Vidensk. Selsk. Mat. Fys. Medd. 43, 97(1993). 3. G. Betz and W. Wahl, International J. of Spectrometry and I on Pro- cesses. 140, 1(1994). 4. A. Wucher and B.Y. Garrison, J.Chem Phys. 105, 5999(1996). 5. Th.J. Colla, H.M. Urbassek, A. Wucher, C. Staudt, R. Heinr ich, B.J. Garrison, C. Dandachi and G. Betz Nucl. Instrum. Meth. (1998 ),B 143, 284(1998). 6. A. Wucher and W. Wahl, Nucl. Instrum. Meth. B 115 , 581(1996). 7. M.L. Yu, Topics of Applied Phys. Sputtering by Particle Bo mbard- ment III. Ed. by R. Behrisch and K. Wittmaack, Springer-Verl ag, (1991) p. 91-160. 8. V.I. Matveev, S.F. Belykh and I.V. Veryovkin, Zh. Tekh. Fi z.69, 64(1999), [Technical Physics, 44, 323(1999).]. 9. S.F. Belykh, V.I. Matveev, I.V. Veryovkin, A. Adriaens, F . Adams. Nucl. Instrum. Meth. B 155 , 409(1999). 10. Dobretsov L.N., Gomounov M.V. Emissional electronics, Moscow, Nauka, 1966. 11. L.D. Landau and E.M. Lifshitz, Statistical physics, Par t 1, Moscow, Nauka, 1964. 12. W. Wahl and A. Wucher, Nucl. Instrum. Meth. (1994), B 94, 36(1994). 13. S.F. Belykh, U.Kh. Rasulev, A.V. Samartsev and I.V. Very ovkin, Nucl. Instrum. Meth. B 136-138 , 773(1998). 7Figure captions: Fig.1. The dependence of coefficients of single and double ion ization of clusters from 5 and 10 Ta-atoms on target temperature Θ. Fig.2. The dependence of coefficients of single ionization on the num- ber of atoms in cluster of Ag: dotted line - our calculations at target temperature Θ = 500oK,•- experimental data from [12]. Fig.3. Relative yield Y1 Nof one charge cluster of Ta+1 Nin dependence on number Nof atoms in cluster under one-charged ion of Au−1(with the energy 6 keV) bombardment of tantalum at target and targe t tem- perature Θ = 2273oK: unbroken line - calculated values of Y1 N,•- experiment [13]. 8050010001500200025003000 TARGETTEMPERATURE00.10.20.30.40.50.6IONIZATIONCOEFFICIENTS k25k15k110 k2100510152025 N,CLUSTERSIZE1.·10-60.00010.011k1N,IONIZATIONCOEFFICIENT1.·10-60.00010.011 3 571015 N,CLUSTERSIZE0.0010.010.1110Y1N,CLUSTERYIELD3 571015 0.0010.010.1110

arXiv:physics/9911059v1 [physics.class-ph] 24 Nov 1999Dedicated to Irene Z. CLASSICAL TUNNELING AS THE RESULT OF RETARDATION IN CLASSICAL ELECTRODYNAMICS: NONRELATIVISTIC CASE Alexander A. Vlasov High Energy and Quantum Theory Department of Physics Moscow State University Moscow, 119899 Russia In nonrelativistic approximation one-dimensional motion of Sommerfeld sphere in the case of potential barrier is numerically investigate d. The effect of classi- cal tunneling is confirmed once more - Sommerfeld sphere over comes the barrier and finds itself in the forbidden, from classical point of vie w, area 03.50.De The problem of radiation reaction in classical electrodyna mics is still dis- cussed in the literature (for ex. see [1]). This problem can b e formulated in the following way: it is known that classical charged body movin g with acceleration must radiate. Thus there is back reaction of outgoing electr omagnetic waves. But what quantity feels this back reaction - pure mechanical mass of a charged body or an effective mass, constructed from the mechanical ma ss and energy of self electromagnetic field? Is this effective mass constant o n the trajectory of a moving body or a function of time? In another words, what is th e dynamics of a charged body due to radiation reaction? To answer these questions long time ago [2] was proposed by So mmerfeld model of sphere with uniform surface charge Qand mechanical mass m. In nonrelativistic approximation such sphere obeys the equat ion (see also [3,4,5,6]): m˙/vector v=/vectorFext+η[/vector v(t−2a/c)−/vector v(t)] (1) herea- radius of the sphere, η=Q2 3ca2, /vector v=d/vectorR/dt, /vectorR- coordinate of the center of the shell. One can find in the literature the opinion [1], that the equati on (1) has no unphysical solutions and ”free of the problems that have pla gued the theory for most of this century”. But the fact is that (as was shown in [7]) equation of motion fo r Sommerfeld model possesses some strange solution which can be interpre ted as ”classical tunneling” (see also [8,9] ). The physics of this effect is sim ple: due to retarda- 1tion the body ”understands” that there is the potential barr ier ”too late” and thus can fall through the barrier. Here we consider one-dimensional motion of the shell in more simple, then in [7], case - in nonrelativistic case for potential barrier, p roduced by homogeneous static electric field Ez, stretched in z- direction for 0 < z < L (like in plane condenser): Ez=  0, z < 0; E,0< z < L ; 0, L < z ; For dimensionless variables y=R/L, x =ct/L, a∗= 2a/L, taking for simplicity a∗= 1, the equation of motion of Sommerfeld sphere in nonrelati vistic approximation (1) with external force produced by Ez Fext=/integraldisplay d/vector rρ·Ez=EQ·f, where ρ=Qδ(|/vector r−/vectorR| −a)/4πa2, f=  0, y < −1/2; (2y+ 1)/2,−1/2< y < 1/2; (−2y+ 3)/2,1/2< y < 3/2; 0, 3/2< y; reads d2y dx2=k·/bracketleftbiggdy(x−1) dx−dy(x) dx/bracketrightbigg +λf (2) herek=2Q2 3mc2a, λ=LQE mc2, It is useful to compare solutions of (1) with classical point charge motion in the same field, governed by the following nonrelativistic equation without radiation force: d2y dx2=FE (3) here FE=λ  0, y < 0; 1,0< y < 1; 0,1< y; A.Dividing x-axis into unit intervals, one can find solutions of (2) on eac h interval in elementary functions (exponents) and then sew t hem together with appropriate boundary conditions (position of the center of the shell and its velocity must be continuous) thus constructing the solutio n of (2) on the whole 2x-axis. But for our goal it will be more effective to obtain solu tions of (2) through numerical calculations. Numerical calculations of eq. (2) show that there is the effec t of classical tunneling for Sommerfeld sphere. Indeed, classical point particle motion, governed by eq. (3 ), is simple: v2= 2λ+v2 0,0< y < 1 heredy dx=v,v0- initial velocity. Thus for given initial velocity for 2 |λ|> v2 0there is the turning point - i.e. classical particle cannot overcome the potential barrier. But for Sommerfeld sphere the result is different. Numerical results are on fig. (A.1-A.3) (vertical axis is vel ocitydy/dx , hor- izontal axis is coordinate y,−1/2< y < 3/2 - i.e. inside the barrier). Onfig. A.1 we can see the effect of tunneling for the following values of k andλ: k= 1, λ=−0.5. Velocities of the shell are v= 0.4, v= 0.6, v= 0.7 (- all give rebounce); v= 0.8 (and here is tunneling) and all of them are from the ”forbidden area” v≤/radicalbig 2|λ|= 1.0. Onfig. A.2 we can see the effect of tunneling for the following values of k andλ: k= 1, λ=−0.1. Velocities of the shell are: v= 0.12, v= 0.3 (rebounce); v= 0.4 (tunneling) and all of them are from the ”forbidden area” v≤/radicalbig 2|λ|= 0.4472.... Comparing fig. A.3 withfig. A.2 , we can see that the more greater the value of k(”more” retardation), the more stronger becomes the effect o f tunneling: onfig. A.3 :k= 10, λ=−0.1; velocities of the shell are the same as for fig. A.2: v= 0.12 (rebounce); v= 0.3, v= 0.4 (tunneling) Thus we see that the effect of classical tunneling exists not o nly for point- like particles, governed by Lorentz-Dirac equation [8], bu t also exists for charged bodies of finite size. REFERENCES 1. F.Rohrlich, Am.J.Phys., 65(11), 1051(1997). 2. A.Sommerfeld, Gottingen Nachrichten, 29 (1904), 363 (19 04), 201 (1905). 3. L.Page, Phys.Rev., 11, 377 (1918) 34. T.Erber, Fortschr. Phys., 9, 343 (1961) 5. P.Pearle in ”Electromagnetism”,ed. D.Tepliz, (Plenum, N.Y., 1982), p.211. 6. A.Yaghjian, ”Relativistic Dynamics of a Charged Sphere” . Lecture Notes in Physics, 11 (Springer-Verlag, Berlin, 1992). 7. Alexander A.Vlasov, physics/9905050. 8. F.Denef et al, Phys.Rev. E56, 3624 (1997); hep-th/960206 6. 9. Alexander A.Vlasov, Theoretical and Mathematical Physi cs, 109, n.3, 1608(1996). v=0.4v=0.6v=0.7v=0.8 -6.20e-1-4.78e-1-3.36e-1-1.94e-1-5.20e-29.00e-22.32e-13.74e-15.16e-16.58e-18.00e-1 -5.00e-1-3.00e-1-1.00e-11.00e-1 3.00e-1 5.00e-1 7.00e- 1 9.00e-1 1.10e0 1.30e0 1.50e0 Fig.A.1 4k=1,v=0.12k=1,v=0.3k=1,v=0.4 -3.00e-1-2.30e-1-1.60e-1-9.00e-2-2.00e-25.00e-21.20e-11.90e-12.60e-13.30e-14.00e-1 -5.00e-1-3.00e-1-1.00e-11.00e-1 3.00e-1 5.00e-1 7.00e- 1 9.00e-1 1.10e0 1.30e0 1.50e0 Fig.A.2 5k=10,v=0.12k=10,v=0.3k=10,v=0.4 -1.20e-1-6.80e-2-1.60e-23.60e-28.80e-21.40e-11.92e-12.44e-12.96e-13.48e-14.00e-1 -5.00e-1-3.00e-1-1.00e-11.00e-1 3.00e-1 5.00e-1 7.00e- 1 9.00e-1 1.10e0 1.30e0 1.50e0 Fig.A.3 6

(revised August, 2006)Symmetry Principles of the Unified Field Theory (a "Theory of Everything") John A. Gowan http://www.people.cornell.edu/pages/jag8/index.html There is nothing so valuable as a fresh perspective. Contents: Abstract Row one: Symmetric Energy States: Creation Event Light and Spacetime Noether's Theorem Particles, Leptoquarks Symmetry Breaking IVBs Row two: Particles - Raw Energy Conservation Mass Time and Entropy The Interval The Mechanism of Gravitation Quantum Mechanics and Gravitation Quarks and Leptons Quarks Fermions and Bosons Neutrinos Row three: Charges - Symmetry Conservation Electric Charge Gravitational Charge Color Charge Number Charge Row four: Fields - Symmetry Payments Photons Gravitons8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 1 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlNote (1): I recommend the reader consult the "preface" or "guide" to this paper, which may be found at and the " ".Quantum Radiance and Black Holes Gluons IVBs Summary Links And References ABSTRACT: The conceptual basis of the Unified Field Theory, as presented in these pages, can be briefly sketched as follows: "Noether's Theorem" states that in a multicomponent field such as the electromagnetic field (or the metric field of spacetime), where one finds a symmetry one finds an associated conservation law, and vice versa. In matter, light's symmetries are conserved by charge and spin; in spacetime, by inertial and gravitational forces. All forms of energy, including the conservation/entropy domain of spacetime, originate as light. During the "Big Bang", the asymmetric interaction of primordial, high energy light with the metric structure of spacetime produces matter; matter carries charges which are the symmetry (and entropy) debts of the light which created it. Charges produce forces which act to return the material system to its original symmetric state (light), paying matter's symmetry/ entropy debts. Repayment is exampled by matter-antimatter annihilation reactions, particle and proton decay, the nucleosynthetic pathway of stars, and Hawking's "quantum radiance" of black holes. Identifying the broken symmetries of light associated with each of the 4 forces of physics is the first step toward a conceptual unification of those forces. The charges of matter are the symmetry debts of light. Row 1 - and the "Big Bang" Symmetric Energy States "About the Papers: An Introduction" The Sun Archetype Note (2): The format of this paper ("Row 1", "Row 2", etc.) follows a which the reader should access and print out for ready reference. This table provides a convenient way to organize an extensive subject matter, and is furthermore part of a , which facilitates comparison and correlation with other "world systems". An introductory paper: provides a general summary of the topic. See also for another version of the primary table.4x4 table General System, or Fractal Model of the Universe "Synopsis of the Unification Theory: The System of Spacetime" " 4x4 Table of Conservation Law vs Forces" Note (3): The symmetries usually discussed in physics articles with regard to the four forces are highly technical and mathematical, often described as parameters or dimensions of an imaginary "phase space". The less technical but equally valid symmetries presented in this article and throughout this webpage are more general, with broad significance, relating common and recognizable features of the forces to common and recognizable features of the conservation laws. Note (4): In each of the four rows below I suggest a financial metaphor for the energetic process characteristic of the row, beginning with the assumption of a debt, followed by two contrasting payment modes, and ending with a full repayment. The intent is to help the reader gain an overview of and feeling for the unfolding energy budget of the Cosmos as outlined in this model, by reference to an analogous quantitative system with which we are all familiar. Incurring the energy, entropy, and symmetry debt - "opening the account" - symmetry-breaking during the Big Bang. Important concepts in Row 1 include the nature of light and its intrinsic motion, as gauged by "velocity c"; the establishment of the spacetime metric; "Noether's Theorem" and the conserved symmetries of Row 1:8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 2 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlThe Universe, our theory, and this exposition all begin with light - free electromagnetic energy - which is a perfectly symmetric energy form. Light is massless, carries no charges of any kind, produces no gravitational field, and has no time dimension in the ordinary sense. Light's intrinsic motion (gauged by "velocity c") is the entropy drive of free energy, and also the gauge of a "non-local" symmetry condition formally characterized by Einstein as light's zero "Interval". Light's zero "Interval" (the "Interval" is a mathematically invariant quantity of spacetime) defines light's symmetric energy state of "non-locality". light; the interaction of light with metric space to create the "particle sea"; and finally, the breaking of the symmetry of light, the spacetime metric, and matter-antimatter particle pairs by the asymmetric interactions of the weak force with matter vs antimatter. Symmetry-breaking results in the creation of isolated particles of matter - the atoms which form our material Universe. How the universe actually begins (for example, "inflationary" scenarios) is not considered in this account. I assume, however, that the initiating positive energy is effectively balanced by some type of negative energy (such as gravity). Furthermore, it is not unreasonable to suppose that our universe is but one of many (a member of the "multiverse"), whose basic physical constants are constrained by the "anthropic principle" (must allow the evolution of humans). Light and Spacetime Light is a 2-dimensional transverse wave whose intrinsic motion sweeps out a third spatial dimension. Lacking both a time dimension and one spatial dimension (in its direction of propagation), light's position in 3- dimensional space or 4-dimensional spacetime cannot be specified. Since both time and distance are meaningless to light, and yet light has intrinsic motion, light has in effect an infinite amount of time to go nowhere. Hence in its own reference frame (moving freely in the vacuum of spacetime at velocity c), light must be considered to be everywhere simultaneously. From this results the "non-local" (and therefore atemporal and acausal) symmetric energy state of light. "Non-locality" is the principle symmetry condition of massless free energy, and its chief distinction from massive, local, temporal, and causal bound energy. Several other symmetries are associated with light's non-local energy state, all of which require conservation (in accordance with "Noether's Theorem"). Light's "zero Interval" means that light is everywhere throughout its conservation domain simultaneously - a symmetry condition with respect to the distribution of light's energy in spacetime ("symmetry" refers to a condition of balance, sameness, or equality). It is due to this symmetry condition that we can circumnavigate the universe within a human lifetime - in a rocket ship moving at nearly velocity c. At exactly c it takes no time at all. The electromagnetic constant c is the universal "gauge" or regulator (in the sense of railroad track or wire gauges) for the "metric" of spacetime, the fixed relationship which establishes the equivalence of measurement within and between the dimensions: 300,000 km of space is metrically equivalent to 1 second of time. At c this equivalence is complete and time is suppressed to a locally implicit state (light has no time dimension). The suppression of the asymmetric time dimension (and time's asymmetric companions, mass, charge, and gravitation), and the equilibration of the 3 spatial dimensions, is the principle symmetry-keeping function of c. To think of c as a velocity, even as a "non-ordinary" velocity, is to miss the point: the physical significance of c is that it is the symmetry gauge and the primordial entropy drive of free electromagnetic energy in its metric, dimensional, or spatial expression. It is because of these "gauge" functions that c appears to us as an effectively "infinite" and invariant velocity. Another famous gauge function of c fixes the energetic equivalence of free to bound electromagnetic energy: E = mcc. c also functions as the gauge or messenger of causality. These various gauge functions indicate the primacy of light in our Universe - and the fundamental significance of Einstein's Theory of Special Relativity. In a universe of pure light, before the creation of matter, the metric is everywhere the same, as no gravitational fields are present to disturb its symmetry. The metric is a necessary condition of the spatial domain, indeed the very reason for its existence, as it is the regulatory mechanism which performs the conservation function of the domain (via "inertial" forces), controlling and coordinating the rate of expansion and cooling of space both 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 3 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlglobally and locally, regardless of the changing size of the expanding Universe. It is for this reason that a "non- local" metric gauge such as c is required - one whose regulatory influence can be everywhere simultaneously, irrespective of the physical extent (or expansion) of its domain. Both space and its metric are created by the intrinsic motion of light. Without the metric every photon could have a unique velocity; it is the metric which imposes the universal constant c upon them all. While we conceive of the metric as produced by light, the metric's origin is in the inherent conservation characteristic of light, including entropy and symmetry. (See: " ".)The Higgs Field vs the Spacetime Metric The entropy drive of light is expressed through its intrinsic motion, expanding and cooling the Universe, hence reducing the Cosmos' capacity for work. But it is light's intrinsic motion which also creates the conservation domain of spacetime and maintains its metric symmetry, suppressing time, etc. Therefore light and space are related through the first and second laws of thermodynamics, while c functions as both the entropy drive and the symmetry gauge of free energy. It is the function of entropy to create a dimensional conservation domain in which energy can be transformed, used, and yet conserved. Without entropy (the 2nd law of thermodynamics), the Universe could not spend its energy capital, since the 1st law of thermodynamics (energy conservation) would forbid any use of energy at all. The dimensions of spacetime are entropy domains, created by the intrinsic (entropic) dimensional motions of light (creating space), time (creating history), and gravitation (creating time and spacetime), as gauged by "c" (the intrinsic motion of light), "T" (the intrinsic motion of time), and "G" (the gravitational constant). The intrinsic motion of time is also primarily gauged by c as the temporal duration (measured by a clock) required by light to move a given distance. G is the entropy conversion gauge, fixing the volume of space which must be annihilated and converted to time per given mass. Gravitation converts the entropy drive of free energy (the intrinsic motion of light as gauged by velocity c) to the entropy drive of bound energy (the intrinsic motion of time as gauged by velocity T) and vice versa (as in stars). See: ; and " "."A Description of Gravitation" Spatial vs Temporal Entropy Our physical universe, including the conservation domain of spacetime, is wholly the product of a single form of energy - electromagnetic energy (the "monotheism" of physics). Light is the most primordial form of this energy, which we know because light has the greatest symmetry of any energy form, and provides the basic gauges, both metric and energetic, for either free or bound electromagnetic energy. Light is the only energy form which can produce its own conservation domain from its own nature (intrinsic motion c) - matter must produce its historic domain from preexisting space via the gravitational conversion of space to time. Finally, light is the form from which all other kinds of energy are made, and to which they all reduce and return. (See: .)"Entropy, Gravitation, and Thermodynamics" Noether's Theorem "Noether's Theorem" (Emmy Noether, 1918) states that in a multicomponent field (such as the electromagnetic field, or the metric field of spacetime), where one finds a symmetry, one will find an associated conservation law, and vice versa. Noether's Theorem is saying that in the conversion of light to matter (for example), not only must the raw energy of light be conserved in the mass and momentum of particles, but the symmetry of light must also be conserved - not only the quantity but the quality of energy must be conserved. Noether's Theorem seems to apply individually to both halves of the "frequency-wavelength" or " particle- wave" (or "electric-magnetic") duality that characterizes light. The "frequency" characteristic of light apparently corresponds to the particle-antiparticle expression of light's energy (light's raw energy and potential information content), which is conserved through the charges (and spin) of matter. The "wavelength" characteristic of light apparently corresponds to the metric field of inertial forces and intrinsic dimensional motion (the structural and entropic aspects of light), which are conserved through the gravitational and temporal attributes of matter. ("Spin" seems to be a wholly conserved intermediate or mixed state of charge and inertial force.) Hence symmetry-breaking we find Noether's Theorem expressed through: 1) the inertial forces of metric symmetry-keeping as gauged by "velocity c", suppressing the asymmetric time dimension; 2) through the electrical annihilation of particle-antiparticle pairs, suppressing the asymmetric appearance of any immobile before8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 4 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlin which Beauty corresponds to Symmetry and Truth corresponds to Conservation. bound energy form, whether matter or antimatter. symmetry-breaking (in the "Big Bang"), we find Noether's Theorem expressed through: 1) the metric fields of gravitation and time; 2) the conserved charges (and spin) of particles - which all work together (as in the Sun) to return asymmetric matter to its original form of symmetric light. The process (of symmetry conservation) drives to completion via Hawking's "quantum radiance" of black holes.After I think of Noether's theorem as the "Truth and Beauty" theorem, in reference to Keat's great poetic intuition: "... Beauty is truth, truth beauty, - that is all Ye know on earth, and all ye need to know" ("Ode on a Grecian Urn": John Keats,1819) The two common examples of Noether's Theorem enforced in Nature - charge (and spin) conservation among the particles, and gravitational and inertial forces in the spacetime metric - are the more enlightening because the former is an example of symmetry conservation and debt payment deferred indefinitely through time, while the latter is an example of raw energy conservation in which the debt must be paid immediately. Furthermore, in the case of inertial forces, we see the implication that gravitation will also fall under the conservation mantle of Noether's Theorem, via Einstein's "Equivalence Principle". This indication is borne out and verified by the discovery that gravitation is indeed a symmetry debt of light, responding to and conserving light's non-local spatial distribution, a symmetry broken by the immobile, undistributed concentrations of mass energy (E = mcc) represented by matter. Noether's theorem tells us why the forces of nature are busy converting matter to light: matter was created from light in the "Big Bang", but since light has greater symmetry than matter, it is to conserve light's symmetry that all the charges and forces of matter work to accomplish the return of bound energy to its original symmetric state. These charges produce forces which act to return the system of matter to light (free energy). Our Sun is an archetypical example of symmetry conservation in nature.The charges of matter are the symmetry debts of light. A program of unification is therefore clearly indicated by Noether's Theorem: identify the (broken) symmetries of light carried, represented, and conserved by the charges of matter. The actions of the forces produced by these charges should offer clues as to what these (broken) symmetries are. This will allow us to refer all the charges and forces of matter to their respective origins as specific symmetries of light, accomplishing our conceptual unification. Matter is but an asymmetric form of light, as time is an asymmetric form of space, and gravity is an asymmetric form of inertia. Charges and forces of matter act to return bound energy to its symmetric, original state of free energy. In the pages which follow, we will follow out this simple conceptual program of force unification, by identifying the broken symmetries of light represented by the conserved charges of matter - including gravity's "location" charge. While this is a conceptual rather than a quantitative unification, is is hoped that by framing the argument firmly within the constraints of the conservation laws, a route to a more formal, quantitative, mathematical unification will be indicated. Particles Matter consists of two types of massive particles, the elementary particles with no internal parts, called leptons, and composite particles with internal parts (quarks) called hadrons. Together they comprise atomic matter, the electron a member of the lepton family, and the nuclear particles (protons and neutrons) examples of the hadron family. Hadrons containing a quark-antiquark pair are known as mesons, while those containing 3 quarks are called baryons; no other quark combinations are thought to exist in nature - at least commonly (see: "The Year in Science" Jan. 2006 page 39). (See: ).Discover "The Particle Table" Together, light and metric spacetime have the capacity to produce particles, which are essentially a "packaging" of light's free energy. The mechanism by which the primordial transformation of free to bound electromagnetic energy occurs is still unknown, although actively investigated. We believe our universe began as an incredibly 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 5 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlhot, energy dense, and spatially tiny "singularity" (the standard "Big Bang" model - see Steven Weinberg's ). One can readily appreciate that a simple "packaging" mechanism for compactly storing the wave energy of light - which by its very nature (its intrinsic motion) takes up a lot of space - would be useful in the spatially cramped conditions of the initial moments of the Big Bang. In a purely pragmatic way this "packaging" concept accounts for the existence of particles and some of their salient features: the spectrum of identical elementary particles of various masses (the leptonic series), the heavier ones presumably more useful "packages" at earlier times and higher energy densities, and similarly, the spectrum of composite particles (baryons), which can store additional energy internally, as if they contained a set of compressible springs (the quarks). Finally, massive particles can store an unlimited quantity of energy as momentum, a feature of particular utility in the early universe, helping to avoid the "still birth" of a cosmic "black hole". (The conversion from a spatial (free energy) to a temporal (bound energy) entropy drive, preserving the Universe's capacity for work by storing energy as immobile, non-expanding mass (E = mcc), is perhaps an even better "reason" (from the "anthropic perspective") for the initial conversion of light to matter. See: .)" " The First Three Minutes "Spatial vs Temporal Entropy" I presume there is a mechanical or resonant relationship between the metric of spacetime and the structure of particles - the dimensional structure of spacetime is carried into, reflected in, or otherwise directly influences, the structure of particles. Light exists as a 2-dimensional energetic vibration of the metric structure of spacetime. Usually this vibration is simply transmitted by the metric grid at velocity c, the "inertial" symmetry condition imposed upon light by its conserving metric. However, it is also possible for this vibrational energy to become entangled in the metric and tie itself into higher dimensional "knots", which cannot be transmitted at c because they are no longer 2-dimensional. The mysterious is thought to play a central role in these entanglements, endowing the elementary particles with mass. (I think of the "Higgs" as the "sticky" component of the metric mesh, ensnaring free photons like the glue of a spider's web.) Such metric "knots" comprise particle-antiparticle pairs, and their energy, structure, and information content is derived from the mixture of metric spacetime and light's energy. The otherwise inexplicable existence of three energy families of both quarks and leptons is probably a consequence of the origin of particles as electromagnetic "knots" in the 3 spatial dimensions of the metric. The mathematical/geometric connection between energy, the metric, and the structure of particles is currently being investigated (in 10 or 11 dimensions!) by "string" theory (see Brian Greene's ). In this paper, however, I sketch much simpler ideas in the usual 4 dimensions."Higgs" boson "The Elegant Universe" It remains a mystery how the elementary leptons are related to the composite baryons, but it is plausible that this relationship is through an ancestral, heavy, leptonic particle (the "leptoquark"), which "fractured" under the high pressure of the Big Bang, and so could arrange its internal fractional charges in electrically neutral configurations - as in the neutron. This notion is based on the theory of "asymptotic freedom" (Politzer, Gross, Wilczek - ) - a symmetry principle which observes that as the quarks of a baryon are squeezed together, the strong force which binds them becomes weaker, affording the quarks more freedom of movement. If the quarks are squeezed together completely - as by the ambient pressure of the "Big Bang", the "X" Intermediate Vector Boson (IVB), or the gravitational pressure of a black hole - the color charge of the gluon field sums to zero (see Row 4, "Gluons", below), leaving a particle indistinguishable from a heavy lepton, the hypothetical "leptoquark". A "colorless" and electrically neutral leptoquark would therefore be susceptible to a typical weak force decay via a leptoquark neutrino and the "X" IVB, hypothetical particles we examine in the following section. (See: " ".)2004 Nobel Prize The Origin of Matter and Information Symmetry Breaking and the Weak Force (See: " ".)Leptons as Alternative Charge Carriers The Particle Table The leptonic elementary particles (charge-bearing particles with no internal parts or sub-units) function as alternative charge carriers for the hadrons (mass-bearing particles containing quarks). Without these alternative charge carriers (electrons carry electric charge, neutrinos carry number or "identity" charge), the massive hadrons would remain unmanifest, locked in symmetric particle-antiparticle pairs, forever annihilating and reforming. Hence we discover that in order to produce an asymmetric, isolated particle of matter from a symmetric particle-antiparticle hadron pair, we require: 1) the primary mass-carrying field (the quarks); 2) a 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 6 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlThe field vectors or force carriers of the weak force are known as Intermediate Vector Bosons, or IVB's. The IVBs include the W+, W-, and Z (neutral) particles. As a group, they are the most unusual particles known and the most difficult to understand (I also include in this group the hypothetical super-heavy "X" particle thought to be responsible for producing leptoquark and proton decay.) The charge carried or mediated by the IVBs is the "number" or "identity" charge of the weak force.secondary field of alternative charge carriers (the leptons - electrons, neutrinos, and their kin); 3) the secondary field must furthermore be asymmetric in its interaction with the primary field, such that its reactions with particles proceed at a different rate than its reactions with antiparticles; 4) interactions between the hadron and lepton field are brokered by a third quantized mediating field, the Intermediate Vector Bosons (IVBs) of the weak force, the W, Z, and X particles (in which the asymmetric principle is probably located); 5) a final requirement is that there must exist some fundamental basis of similarity between all three fields if they are to interact at all - they must be able to recognize and mesh with each other at the quantum level of charge. For example, the electrical charge of the proton must be exactly equal in magnitude to that of the positron, electron, or the IVBs. Obviously, the relationship between the hadrons and leptons must be intimate, and almost certainly they are related through ancestry, that is, one is derived from the other, both are derived from the metric, both are decay products of the leptoquark, etc. A complex arrangement, but nothing less will suffice to break the initial symmetry of free energy and the particle-antiparticle pairs it so abundantly produces. Free energy is flirting with the danger of manifestation in the ready creation of these virtual particle pairs, and in the end it pays the price, as flirts usually do. (See: ) ( ). "The "W" IVB and the Weak Force Mechanism" also available in HTML format IVBS - Quantum Process and Particle Transformation The weak force is the asymmetric and symmetry-breaking physical mechanism that produces elementary massive particles from light (more specifically, from light's particle-antiparticle form), and governs the creation, destruction, and transformation of elementary particles, both quarks and leptons. Only 3 massive leptonic elementary particles are known, the electron, muon, and tau, identical in all their properties other than mass and identity ("number") charge. This is the leptonic particle family, series, or spectrum. It is a quantized mass series, each member separated from the others by a large, discreet, and exact mass difference. I suspect the leptoquark is the 4th and heaviest member of this series, representing the primordial common ancestor of the baryons and leptons. It is the role of the IVBs to mediate or broker the transformation, creation, and destruction of the elementary leptons, and transformations of quark "flavors" in certain situations, notably in the decays of neutrons and heavy baryons ("hyperons"). The "Z" governs neutral weak force interactions in which particles simply scatter ("bounce") or swap identities; the super heavy "X" is hypothesized to govern proton and leptoquark decay. The actual weak force transformation mechanism is discussed below. (See also: )."The Weak Force: Identity or Number Charge" What is most remarkable about the IVBs is that they seem to be "metric" particles providing bridges between real particles and their counterparts in the "virtual particle sea" of the vacuum. The IVBs are not particles like the leptons and baryons which form stable matter; they are particles of interaction, present only when mediating a reaction, "virtual" particles usually known only by their effects, existing within the "Heisenberg Interval" for virtual reality, but real enough and producible as distinct, massive entities if the ambient energy density is sufficient. The IVBs are an especially complex example of nature's penchant for quantization, and like other quantum processes, are responsible for a good deal of head-scratching. I can think of two reasons why the process of particle transformation should be quantized: 1) quantized units are indefinitely reproducible without loss of information or precision (Nature's "digital" information coding); 2) how otherwise could the asymmetry of the weak force be built into its structure? The W particle (which is nowadays readily produced in accelerators) is approximately 90 times heavier than the proton, which explains the relative weakness of the weak force - there is a huge energy barrier to surmount before weak interactions can occur. However, this also raises the obvious question of what this massive particle 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 7 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.html "Down payment", "money up front", "pay now" - raw energy conservation. The major concepts of Row Two center on bound energy, mass, momentum, particles, time, gravitation, and inertial forces as raw energy debts, conserved states, or reactions occasioned by the conversion of light to matter in the Big Bang. The local, temporal, causal nature of matter vs the non-local, atemporal, and acausal nature of light is emphasized. The elementary particles of matter, the quarks and leptons, are examined. Einstein's most famous formula, E = mcc, expresses the notion that the energy stored in mass is enormous and somehow related to light through the electromagnetic gauge constant c. DeBroglie noted that Planck's formula for the energy of light E = h (where = the frequency of light, and h = Planck's constant) contained the same E; putting the two together, DeBroglie wrote h = mcc, expressing the conversion of free energy to its bound form (or vice versa). This equation states that all the energy of light is conserved in massive form in this transformation.is composed of - certainly not ordinary matter, the stuff of baryons and leptons. My guess is that this particle (and the IVBs generally) is nothing less than a piece of very compact spacetime metric, similar to the dense metric of the early moments of the Big Bang. The huge mass energy of the particle is the binding energy required to compress the metric, perhaps fold it, and secure it in the particular configuration that characterizes the W, Z, or X. Hence these particles are perhaps similar to the compacted, topological, multidimensional particles of "string" theory. The hypothetical "Higgs" boson may also be a "metric" particle. (See some IVB reaction examples listed in ). "The Particle Table" In the initial phase of particle creation, particle-antiparticle pairs, presumably of all types, are created but annihilate each other instantly, recreating the light energy from which they were made. So long as these pairs are created and annihilated in equal numbers, the symmetry of the light universe is maintained. But there is an inherent asymmetry in the way the weak force interacts with matter vs antimatter, with the consequence that even though particle pairs are created symmetrically, they do not decay symmetrically. Most probably these asymmetric decays occur in neutral leptoquarks, heavy analogs of the neutron. An excess of matter is produced in this process, breaking the symmetry of the particle-antiparticle pairs and the light universe, creating the matter comprising the Cosmos we see today. (See: .) It is the consequence of this broken symmetry of light, manifesting as massive particles, their quantized charges, and time and gravitation, that we will trace in the remaining rows of the model."The Formation of Matter and the Origin of Information" Row 2 - Particles - Conservation Raw Energy Row 2: Mass vv v We might think with some justification that energy conservation is satisfied by DeBroglie's equation and nothing more need be said. But this is just raw or total energy conservation, conservation of quantity, not quality. The conservation of the quality, or symmetry, of free energy has not been addressed by this formula, nor has the conservation of light's entropy. No massive particle can be created from free energy without engendering a symmetry (and entropy) debt and charge of some sort; if the free energy is simply absorbed by an existing massive system (for example, the absorption of a photon by the electron shell of an atom) without the creation of a new charged particle, then at least a gravitational charge will be recorded. Whenever we encounter the intrinsic dimensional motions of "velocity c" (light), "velocity T" (time), or "velocity G" (gravity), we are dealing with the entropy drives of free and bound energy in their primitive and primary forms. At its most basic level, the gravitational charge represents the transferal, conversion, and conservation of the entropy drive of one system to the entropy drive of another (in the case of gravity a symmetry debt is always combined with the entropy drive). Free energy cannot be transferred to bound energy (or vice versa) without also transferring, converting, or conserving the entropy drive of that energy; in massive systems, the intrinsic motion of time is the primordial entropy drive of the system. Time is created by the gravitational (or quantum mechanical) conversion of space and the drive of spatial entropy into time and the drive of historical entropy (see: ). Hence we must include time, the primordial entropy drive of bound energy, along with gravitation in Row Two, keeping in mind, however, that gravitation has in addition to its entropy conservation role a symmetry conservation role which links it to the charges and discussion of Row Three."Entropy, Gravitation, and Thermodynamics"8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 8 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlThe basic function of mass and momentum is apparently the compaction ("packaging") and storage of free energy (and the conversion of light to a bound energy form with a less destructive entropy drive), as touched upon in the discussion of Row One. Mass is bound electromagnetic energy, and it is asymmetric in many ways by comparison to the free electromagnetic energy (light) from which it is created. For this reason mass carries various charges, which are symmetry debts whose origins we have traced to the conservation of light's perfect symmetry (see Row 3). The most fundamental symmetry debt of mass is dimensional - mass is 4-dimensional, with no (net) intrinsic spatial motion, but with a time dimension which moves instead. Because time exists (among other reasons) to establish and control causality, the time dimension itself is necessarily one-way, hence asymmetric. Free energy, from which mass is formed, is a 2-dimensional transverse wave, whose intrinsic motion sweeps out a third spatial dimension. Four-dimensional massive matter or bound energy is local, temporal, and causal; two-dimensional massless light or free energy is non-local, atemporal, and acausal. Time is a dimensional asymmetry, or dimensional symmetry debt of mass; time is also the primordial expression of entropy in matter: the intrinsic motion of time is the entropy drive of bound energy and history. Gravitation creates the time dimension of matter by converting space into time, conserving in the process the spatial entropy drive (the intrinsic motion of light) of the free energy which originally created the matter. Essentially, gravitation converts the intrinsic motion of free energy (as gauged by "velocity c") into its entropic and metric equivalent, the intrinsic motion of matter's time dimension (as gauged by "velocity T"). The intrinsic motion of light creates space and the intrinsic motion of gravity creates time. The intrinsic motion of light is the spatial entropy drive of free energy, and the intrinsic motion of time is the historical entropy drive of bound energy. Space and the drive of spatial entropy (S) are gravitationally transformed into time and the drive of historical entropy (T), a transformation which can be symbolically represented in a "concept equation" as: -Gm(S) = (T) -Gm(S) - (T) = 0 (See: .) "A Description of Gravitation" Bound energy's most obvious asymmetry (matter's 4-dimensional energy state) is due to matter's lack of intrinsic spatial motion c, meaning bound energy is "local" and associated with temporal causality chains. The 4-dimensional energy state of matter gives bound energy a different inertial status than free energy, because light is 2-dimensional. The "Interval" of free energy = 0 and light produces no gravitational field; bound energy has a real, positive Interval (because of its time dimension and third spatial dimension) and a gravitational field. Both time and gravity are asymmetric dimensional attributes. I associate the gravitational charge ("location") with the entropy drive of bound energy (the intrinsic motion of time), and with the broken symmetry of the universally equitable distribution of light's energy throughout space (light's symmetric "non-local" energy state or "zero Interval"). Both local time and local gravity vary in intensity with the quantity and density of matter, demonstrating their association with the local character of bound energy, and with the significant dimensional parameters of the asymmetric spacetime distribution of matter's immobile energy content, especially matter's location, quantity, and concentration. Time and Entropy Note to Readers Concerning "Entropy": Unless the context indicates otherwise, when I refer to "entropy" in these papers (especially in such phrases as "space and spatial entropy" or "time and historical entropy"), I am referring to entropy in its most primordial or pure form, as the intrinsic motion of light "gauged" or regulated by "velocity c" (in the case of "spatial entropy"), or as the intrinsic motion of time "gauged" or regulated by "velocity T" (in the case of "historical entropy"). Of course, time is also ultimately "gauged" or regulated by "velocity c", since time is defined as the duration (measured by a clock) required by light to travel a given distance. (See: " "; and " ".)Spatial vs Temporal Entropy The Tetrahedron Model of Energy and Conservation Law8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 9 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlG is the connecting link or conversion force between c and T; gravity converts space and the drive of spatial entropy to time and the drive of historical entropy. Whereas the electromagnetic constant c is the gauge of the equivalency between space and time, the gravitational constant G is the gauge of the equivalency between space and time. A portion (-Gm) of the entropy-energy driving the spatial expansion of the Universe is gravitationally converted to the entropy-energy driving the historical expansion of the Universe. (See: " ".)The three primordial forms of entropy and their "gauges", "drives", or "intrinsic motions" c, G, T (the "intrinsic motions" of light, gravity, and time) are as follows (see also: " "): The Conversion of Space to Time c) Positive spatial entropy (the drive of spatial expansion, the intrinsic motion of light as "gauged" or regulated by "velocity c"); G) Negative spatial entropy (the drive of spatial contraction, the intrinsic motion of matter's gravitational field, as "gauged" or regulated by "velocity G"); T) Positive historical entropy (the drive of historical expansion, the intrinsic motion of matter's time dimension, as "gauged" or regulated by "velocity T"). metric entropic A Spacetime Map of the Universe The gravitational conversion of space and the drive of spatial entropy to time and the drive of historical entropy is physically demonstrated by black holes, and mathematically formulated in the Bekenstein-Hawking theory relating the surface area of a black hole to its entropy content. (See: " ".)The Half-Life of Proton Decay and the 'Heat Death' of the Cosmos (See: " ".)The Dimensions The Time Train The dimensions of spacetime are conservation/entropy domains, created by the entropic, "intrinsic" motions of free and bound electromagnetic energy (light and matter). These domains function as arenas of action, where energy in all its forms can be simultaneously used, transformed, and yet conserved. This is the major connection between the 1st and 2nd laws of thermodynamics. (See: "Entropy, Gravitation, and Thermodynamics".) Bound energy requires a time dimension both to establish and maintain causality, and to balance its energy accounts, because the energy contained in mass varies with its relative velocity, and velocity involves time. Light does not require this accommodation because light's absolute velocity is non-relative and invariant; light's energy varies not with velocity but with frequency. Time is one-way because raw energy conservation forces the continual updating of matter's energy accounts, from one instant to the next, protecting causality, the temporal sequence of cause and effect. The "local" character of matter requires a causal temporal linkage, whereas the "non-local" character of light does not. Causality itself requires the one-way character of time; energy conservation requires the presence and protection of causality and its associated temporal entropy drive in every system of bound energy. The intrinsic motion of time ("velocity T") is the primordial entropy drive of bound energy, causing the aging and decay of matter and information, and creating and expanding history, the conservation domain of information and matter's "causal matrix". History is the temporal analog of space: "intrinsic motion T" and "intrinsic motion c" are metric equivalents. The entropy drives T and c both produce analogous dimensional conservation domains for their energy types, history for information (matter's "causal matrix"), space for light. It is the energetic nature of light that requires a spatial entropic domain, whereas it is the causal nature of matter that requires an historic entropic domain. Gravitation (entropy drive "G") converts space into time and matter into light (as in the stars), producing the equilibrated joint dimensional conservation domain of historic spacetime, where both free and bound electromagnetic energy can interact and find their conservation needs satisfied. (See: " ".) A Spacetime Map of the Universe Time and gravitation always appear together, both engendered by mass. This one-way dimension and one-way 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 10 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlI begin this section by pasting in part of a paragraph (page 253) from my late father's book: "Trance, Art, Creativity", which presents a marvelous mathematical insight into the nature of the time dimension, as illuminated by Einstein's formula for the "Interval": (The book is linked to and can be accessed in its entirety (without charge) from my homepage).dimensional force are more than coincidentally connected; gravity and time induce each other in an endless loop, much as the electric and magnetic fields of light ceaselessly induce each other. The role of entropy (in its primordial form of intrinsic motion) is to provide a dimensional conservation domain for energy (free or bound) in which energy can be simultaneously used/transformed and conserved. Entropy allows us to use energy because it prevents us from abusing energy, as for example by the "perpetual motion" machine. Entropy says we cannot use the same energy twice to produce the same net work. Entropy is like a bodyguard protecting the 1st law of thermodynamics, energy conservation - or like the interest we must pay for the use of energy capital. Without entropy, the 1st law would forbid any use of energy at all, as there would be no safeguards against its abuse. (See: " ".) Spatial vs Temporal Entropy Entropy performs its primordial role via the intrinsic motions of light and time, which are effectively infinite velocities within their dimensional domains, ensuring the escape of radiant heat and opportunity beyond the reach of fast "space ship" or "time machine", establishing and protecting causality into the bargain. Similarly, gravity seals the dimensional borders of spacetime (against "wormholes") at the "event horizon" and central "singularity" of black holes. Energy conservation is protected through entropy. The dimensions of spacetime are entropy domains from which there is no escape and which allow no loopholes. This is the intimate connection between the 1st and 2nd laws of thermodynamics. Entropy is a necessary corollary of energy conservation, actually responsible for the creation of our dimensional experience of spacetime through the intrinsic (entropic) motions of light, time, and gravitation (the entropy drives or gauges c, T, G). (See: " ".)The Tetrahedron Model of Energy and Conservation Law The Interval "Analysis of this equation [the "Interval"] provides us with the proportion that time is to space as "i" (the square root of -1) is to 1. Now "i" multiplied by itself is -1, so that in a metaphoric sense we can say that the time dimension is "half" a space dimension. Curiously one finds this out intuitively. We have full intuition of the three spatial dimensions, but we cannot intuit the fourth dimension, so we experience it as "time." Furthermore this experience is not full; it is partial, for we are on a one way street indicated by "time's arrow" which allows us always to experience duration as getting later and later, but never the opposite." "Trance, Art, Creativity" The "Interval" is Einstein's mathematical formulation of a quantity of spacetime that is invariant for all observers regardless of their motion, uniform or accelerated. It is the analog of the Pythagorean theorem in 4 dimensions. The "Interval" of light is zero, which means light is "non-local". This is the fundamental symmetry condition of light. Light could not create its spacetime conservation domain, perform its entropy function, nor gauge its metric without the spatio-temporal symmetry of non-locality. But the Interval of mass, or bound energy, is always some positive quantity greater than zero, and this is because the time dimension is necessarily explicit for immobile, local mass, for reasons of entropy, causality, and energy conservation we have considered above. Conversely, because light is missing both the X and the T dimensional parameters, light's position in 4 dimensional spacetime cannot be specified. The basic function of Einstein's "Interval" is to rescue causality from the shifting perspectives of Einstein's relativistic reference frames. This all makes sense when we think about space filled only with light - in such a domain there is no purely spatial Interval because there is nothing to distinguish one place or point from another - all is uniform and indistinguishable spatial, metric, and energetic symmetry. But enter mass and with it its inevitable companions, time, charge, and gravitation (the asymmetric "gang of four"), and immediately we can distinguish a point or place - here is the particle - more importantly, here is the gravitational field pointing to the particle's location 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 11 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlfrom every other place in space (the influence of the field is universal in extent). But one more thing is needed to pin down this location as absolutely unique: because the universe is always moving, either expanding or contracting (due to the spatial entropy drive of light's intrinsic motion), the time dimension is also required to specify which of an endless succession of moving locations we are to consider. The positive Interval of mass represents a dimensional asymmetry because it is unique, distinguishable, and invariant for all observers. Light has no associated gravitational field because it has no "location". Being non- local, light cannot provide a center for a gravitational field, and an uncentered gravitational field constitutes a violation of energy conservation (because of producing "net" motion and energy). Consequently, freely moving light cannot and does not produce a gravitational field. Light's zero Interval is precisely the symmetry condition necessary to prevent the formation of an explicit time dimension and its associated gravitational field. Light could hardly function as the metric gauge of spacetime if it were itself plagued by a metric-warping "location" charge and gravitational field. Finally, light has no time dimension nor the gravitational field which could produce one. This is the basic conservation reason why the intrinsic motion of light - whatever its actual numerical value - must be the "velocity of non-locality", the symmetry gauge and entropy drive of free energy, the gauge of the metrical equivalence between time and space, effectively an infinite velocity within its spatial domain. Otherwise light would have a "location charge", a time dimension, and a gravitational field, and spacetime would immediately collapse into a black hole. (If light produced a gravitational field, the Universe would have been "still born" as a black hole; instead of a "Big Bang" there would have been a "Big Crunch". The fact that the scientific "establishment" believes that free light produces a gravitational field continues to be a major conceptual roadblock in the ongoing effort to unify gravitation with the other forces. This is a major, crucial, and (at least in principle) testable point of difference between my theory of the role of entropy, charge, and gravity within a unified theory and "establishment" physics.) In fact, the recently announced "acceleration" of the cosmic expansion of spacetime (see, for example, March, 2005, pages 32-39) provides observational evidence for this difference between the two theories. As mass is converted to light in stars and quasars, by quantum radiance and particle and proton decay (and by analogous conservation processes in "dark matter"), the total gravitational field of the Cosmos is reduced, resulting, over time, in the observed "acceleration". (See: " ".)Sky and Telescope Does Light Produce a Gravitational Field? In terms of conservation: in obedience to Noether's theorem, bound energy stores the symmetry of light as the conserved charges (and spin) of matter; in obedience to the first law of thermodynamics, bound energy stores the raw energy of light as the mass and momentum of matter; in obedience to the second law of thermodynamics, bound energy stores the entropy drive of light as the gravitational field and temporal entropy drive of matter. Gravitation and time induce each other endlessly. Thus entropy produces the dimensional conservation domains of free energy (space - through the intrinsic motion of light), of information and matter's "causal matrix" (history - through the intrinsic motion of time), and the compound domain of free and bound energy (historic spacetime, produced by the intrinsic motion of gravity, welding together space and time as gravity converts one into the other). This is the iron linkage between the first and second laws of thermodynamics. Noether's theorem is drawn into this "trinity" of natural law because velocity c is both the entropy drive and symmetry gauge of free energy and as a conservation consequence, gravitation is a symmetry as well as an entropy debt. (See: ). "The Double Conservation Role of Gravitation" The Mechanism of Gravitation Time and space are both implicit in the description of the motion of an electromagnetic wave: "frequency" (time) multiplied by "wavelength" (space) = c, the velocity of light. In the quantum-mechanical creation of a time "charge", when an electromagnetic wave collapses or becomes "knotted", it switches from the spatial or "wavelength" character of a moving wave to the temporal or "frequency" character of a particle or stationary wave - like a coin flipping from heads to tails. It is reasonable to call this temporal expression a "charge" because time is asymmetric; being one-way, time has the asymmetric or informational character of any other isolated charge of matter. Time differs from the other charges in that it is an "entropic charge" - a charge with intrinsic dimensional motion. The asymmetric time charge produces a specific "location" in the otherwise 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 12 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlsymmetric field of space - giving the massive particle it is associated with a positive "Interval", whereas the light from which the particle was produced had a "zero" Interval. (See: " ".) Gravity Diagram No. 2 This is the formal character of gravity's "location" charge - the positive Interval of bound energy breaks the non- local symmetry of the free energy which created it (light's Interval = zero). This non-local symmetry state produces the equitable distribution of light's energy throughout spacetime, a symmetry broken by the concentrated lump of immobile energy represented by bound energy's undistributed "rest mass". It is the distributional asymmetry of matter's energy content which is the origin of gravity's "location" charge. Demonstrating this point, the "location" or gravitational charge records the spacetime position, quantity, and concentration of the asymmetric energy distribution represented by any form of bound energy. Nor is gravity a passive signal: gravity will direct you to the center of this asymmetry by carrying you there bodily. Finally, gravity will attempt to repay the symmetry debt by converting bound to free energy in stars and via Hawking's "quantum radiance" of black holes. Time is the active principle of gravity's "location" charge, and time is unique among the charges of matter in that it is an entropic charge - a charge with intrinsic dimensional motion. Hence as soon as it is formed, time moves into the expanding historic domain of information, an entropy/conservation domain at right angles to all three spatial dimensions. However, because space and time are connected, when time moves it drags space after it. It is this spatial motion that we recognize as the gravitational field, but it is actually caused by the intrinsic motion of time dragging space along behind it. This is the "secondary" or macroscopic phase of the creation of a time charge by the action of a gravitational field, a continuous, cyclic process, in contrast to the quantum mechanical collapse of an electromagnetic wave, the microscopic one-time "primary" process which is crucial because it both "sets" and "gauges" the initial time charge, which the gravitational process simply copies and sustains (see: " "). The "primary process" reflects the entropy debt of gravitation, the "secondary process" reflects the symmetry debt, insofar as these can be usefully distinguished. The magnitude of G is determined by the energy difference between the symmetric spatial entropy drive of free energy (the intrinsic motion of light, as gauged by "velocity c") (S), vs the asymmetric historical entropy drive of bound energy (the intrinsic motion of time, as gauged by "velocity T") (T) - or equivalently, between implicit (S) and explicit (T) time: S - T = -G. The Gravity Diagram As space is dragged after the time charge, it is pulled symmetrically from all possible 3-dimensional spatial positions, (because time is connected equivalently to all spatial dimensions), and at the center of mass or at the locus of the time charge itself, space self-annihilates: +x cancels -x, +y cancels -y, and +z cancels -z, leaving behind, of course, a new residue of +t (the metric equivalent of the annihilated space), which cannot cancel because time being asymmetric, there is no -t. The new time charge exactly reproduces and replaces the old, which has moved down the one-way, one dimensional time line into the historic domain of causal information, and the cycle repeats - the new time charge moves down the time line dragging more space after it, space self- annihilates at the center of mass (as it tries to squeeze into the zero-dimensional beginning of the one- dimensional time line), producing a new time charge, etc., forever. Hence gravity and time induce each other, creating the continuous one-way flow of time, the continuous one-way flow of gravitation, and a spherically symmetric gravitational field with a local "center of mass" where the field self-annihilates or vanishes. The acceleration of gravity is produced by the application of a continuous force - the incessant march of time. All this is fully in accord (as any viable gravitational theory must be) with Einstein's "Equivalence Principle" relating inertial and gravitational mass and force, and accelerated vs gravitational reference frames. The one-way character of time is a necessary consequence of causality protection and energy conservation in bound energy systems, as noted before. The time flow produces information's expanding historic entropy/ conservation domain which is the analog of, and derived from, the expanding spatial entropy/conservation domain of light. The two are welded into historic spacetime by gravitation; historic spacetime is visible in our great telescopes as we look out into space and back into time. (See: .) History is the conservation domain of matter's expanding "causal matrix" of information. The reality of today depends upon the continuing reality of yesterday. The historic causal domain is necessary to uphold the reality of the "Universal Present Moment"; we are all immortal in history."A Spacetime Map of the Universe" Note that in the gravitational flow we have symmetric space "chasing" asymmetric time, exactly the reverse of the situation producing the intrinsic motion of light, where symmetric space "flees" the internal threat of manifestation posed by asymmetric time ("wavelength" flees "frequency" = c). (See: " The Conversion of Space 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 13 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlMass assumes quantized, specific, particulate form as the strong force quarks and hadrons, and the weak force leptons. Hadrons are defined as particles containing quarks; hence all hadrons carry "color" charge, the source of the strong force. Leptons contain no quarks and hence carry no color charge. Leptons carry lepton "number" or "identity" charge, the source of the weak force. The leptons are true elementary particles whereas the quarks are sub-elementary. Electrons are familiar examples of the heavy members of the lepton family (electron, muon, tau, and (?) leptoquark); neutrinos are (nearly) massless members of the lepton family (there is a separate and distinct neutrino for each heavy lepton). Protons and neutrons are familiar examples of the "hadron" family; they are further distinguished as members of the "baryon" class of hadrons, which are composed of 3 quarks. The only other hadrons are the mesons, which are composed of quark-antiquark pairs (see: ). In general, the baryons function as mass carriers, and the leptons and mesons function as alternative charge carriers, ".) This is just the difference between implicit and explicit time, or the negative spatial entropy of gravitation vs the positive spatial entropy of light. (See: ; .)to Time "Gravity Diagram No. 2" "Gravity Diagram No. 3" As magnetism is the invisible, "intrinsic", projective, "electro-motive" (electrically active) force of the loadstone, so gravity is the invisible, "intrinsic", projective, "inertio-motive" (dimensionally active) force of the ordinary rock. In the case of magnetism, we trace the force back to the moving electric charges of the atoms in the loadstone; in the case of gravity, we trace the force back to the moving temporal charges of bound energy in the rock. A moving electric charge creates a magnetic field; a moving temporal charge creates a gravitational field. In both cases the field is produced at right angles to the current. The relation is reciprocal as well: moving magnetic and spatial fields (gravity) create electric and temporal currents (time). This is the intuitive analogy between electromagnetism and gravitation which so intrigued Einstein. Quantum Mechanics and Gravitation Gravitation is both a symmetry debt and an entropy debt, unique among the charges and their forces. Gravity's double conservation role is due to the double gauge role of c, which gauges both the entropy drive and the non- local symmetry state of free energy. Gravity cannot conserve either gauge function of c without conserving both. This double nature is reflected in two different mechanisms, both of which convert space to time, one at the quantum level of charge - the entropy debt, and one at the macroscopic level of gravitational force - the symmetry debt. (See: .) "The Double Conservation Role of Gravity" The two mechanisms are distinct but both are part of the gravitational conversion of space to time, connecting the quantum-mechanical aspect of gravitational charge (the entropy debt) to the macroscopic aspect of gravitational flow (the symmetry debt). Both are linked by the entropy/symmetry gauge c and Noether's Theorem requiring the conservation of light's non-local symmetry. The gravitational charge, "location", is unique among charges in that its active principle is time. The gravitational charge is an "entropic" charge, a charge with intrinsic dimensional motion. It is the entropic nature of the gravitational charge which connects the quantum mechanical (particle-charge-time-entropy) and macroscopic (mass-location-time-symmetry) aspects of gravity. In turn, the double nature of the gravitational charge gives gravity a double conservation role: 1) conserving the entropy drive of free energy by converting the intrinsic motion of light to the entropy drive of bound energy - the intrinsic motion of time; 2) conserving the non-local symmetry of light (responsible for the the equitable distribution of light's energy throughout spacetime) by converting bound to free energy (as in our Sun). This duality extends backward in a conservation chain to the dual role of light's intrinsic motion, which is at once the symmetry gauge and the entropy drive of free energy. Gravity must conserve both roles of light's intrinsic motion if it conserves either one. (See also: " ".) Currents of Entropy and Symmetry The "graviton" or field vector of the gravitational charge is a quantum unit of temporal entropy, a quantum unit of time, the transformed, "flipped", or inverted spatial entropy drive or intrinsic motion of the photon (implicit vs explicit time = photon vs graviton = S/T vs T/S). Time is the active principle of gravity's "location" charge; time is the implicit entropy drive of free energy and the explicit entropy drive of bound energy; time is the connecting link between Quantum Mechanics and General Relativity. (See: " " and " ".)Gravity Diagram No 2 The Conversion of Space to Time Quarks and Leptons "The Particle Table"8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 14 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlbalancing charges in place of antiparticles. 3 Families of 4 Particles The quarks and the leptons each occur in "families" of three energy levels; the quark and lepton families appear to be paired in these 3 families as follows (a precisely corresponding set of antiparticles exists but is not shown): 1) up, down (u, d) quarks and the electron and electron neutrino (e, e); 2) strange, charm (s, c) quarks and the muon and muon neutrino ( , ); 3) bottom, top (b, t) quarks and the tau and tau neutrino ( , ).v uvu tvt There is no generally accepted explanation why there should be 3 energy levels of particles, or how the quarks and leptons are related. Ordinary matter (including stars) is composed of the 1st family only. It seems likely that the quarks and leptons are both derived from a high energy, primordial "ancestor" particle, the "leptoquark"; it is also likely that the 3 energy families of particles are somehow reflecting the 3-dimensional structure of space. (See: " "; and also: " ".) The Leptoquark Diagram The Hourglass Diagram Quarks In contrast to the "long-range" electrical and gravitational forces, which have an infinite range through spacetime, the strong force is a "short-range" force, an internal characteristic of nuclear matter. Quarks occur in only two kinds of particles: "baryons" composed of 3 quarks, and "mesons" composed of quark-antiquark pairs. Baryons are familiar to us as neutrons and protons, but there are many other 3 quark combinations possible using the heavier members of the quark family. In addition, every quark combination seems to have many possible energetic expressions, or resonances, just as electron orbits have many "excited" states. Typically, all excited states are exceedingly short-lived. Six quarks are known in three "energy families"; the quarks are named "up, down"; "charm, strange"; and "top, bottom". Ordinary matter consists only of the up, down quarks in their unexcited or "ground" state. All Quarks carry a 1/2 unit of strictly conserved "spin", and a partial "flavor" ("number" or "identity") charge; the latter is only partially conserved. The whole unit identity or number charge of the baryon is apparently strictly conserved, analogously to the strictly conserved number charges of the leptons. Quarks also carry partial electric charges (u, c, t quarks carry +2/3; d, s, b quarks carry -1/3) and their distinguishing charge, color. There are 3 color charges, red, green, yellow (not actually colors, just names of convenience) which are exchanged between quarks by the "gluon" field; each "gluon" is composed of a color-anticolor charge pair. One of the nine possible combinations of color-anticolor is doubly neutral ("green-antigreen"), leaving 8 effective members of the gluon field. The constant "round-robin" exchange of the (massless) gluons from one quark to another (at velocity c) is the strong force mechanism which binds the quarks together. The baryon is an incredible, miniature universe of structure, information, charge, and activity. A large compound atomic nucleus is a swarming "hive", a veritable metropolis of quantum mechanical action and force exchange, all quite beneath our notice, due to the short-range character of the strong force. The essential miracle of matter resides in the baron. Being composed of color-anticolor charges, the gluon field as a whole sums to zero, a crucial symmetry property known as "asymptotic freedom". Quarks are permanently confined by gluons to meson or baryon combinations; they never occur alone or in any other combinations in nature. Finally, only quark combinations which electrically sum to zero or unit (leptonic) electric charge, and neutral or "white" color charge, are allowed. Hence the quark-antiquark pairs composing mesons carry a single color and its corresponding anticolor (summing to "neutral" color), whereas in baryons the color charges of the gluon field pair with anticolors in all possible combinations (summing to "white" color). Quarks are sub-elementary particles, as they carry electric charges which are fractions of the unit electric charge of the leptons, the only truly elementary particles. When one considers the properties of a baron, it is hard to escape the impression that this is what a lepton would have to look like if it were somehow fractured into three parts. Since, by definition, you cannot "really" fracture an elementary particle, perhaps you could do so "virtually", provided the parts could never become "real", that is, separated, but remained forever united in 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 15 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlcombinations that sum to elementary leptonic charges. In this way, the fractured particle would still "look like" an elementary particle to the outside observer; nature is not above such tricks, as we have learned from the virtual particles and Heisenberg's "Uncertainty Principle". It seems probable that baryons are, in some sense, primordially "fractured" leptons. Such an origin (the "leptoquark") would go far toward explaining both the differences and the similarities of these two fundamental classes of particles. Fermions and Bosons Collectively, the hadrons and leptons, which comprise the material component of atomic matter (the nucleus, electron shell, and associated neutrinos), are known as "fermions". All fermions have a "spin", or quantized spin angular momentum, in 1/2 integer units of Planck's energy constant (1/2, 3/2, etc.); fermions obey the Pauli exclusion principle, which simply states that no two fermions can be in the same place at the same time, if all their quantum numbers are also the same. Fermions cannot pile up on top of one another indiscriminately; they keep their own counsel, which is why we get specific, discreet, crystalline atomic structure rather than goo. In contrast to the fermions is the class of energy forms known as "bosons", which includes the force carriers or field vectors of the 4 forces: the photons of electromagnetism (the quantum units of light), the gravitons of gravity, and the gluons of the strong force. As their name implies, the IVBs (Intermediate Vector Bosons) of the weak force have some characteristics of both classes, being very massive bosons. Together, the fermions and bosons comprise the particles and forces of matter. Bosons have whole integer spins (1, 2, etc.) and they can and do superimpose or pile up on one another. Thus a photon or graviton can have any energy because it can be composed of an indefinite number of superimposed quanta, whereas an electron has a single, specific rest energy and charge. The bosons all bear some relationship to light and the metric, their probable common origin. Thus we have the photon (ordinary massless light), the graviton (inverted light), the gluon (sticky light), and the IVBs (massive light). If we add the charges, or symmetry debts of matter (including spin and the entropic forms of intrinsic motion), to the fermions and bosons, we have a complete list of the fundamental (unexcited) energy states; spacetime is the dimensional conservation domain created by entropy (intrinsic motion) and occupied by free and bound forms of electromagnetic energy (light and matter). Symmetry-breaking creates fermions from light; fermions carry charges producing forces whose field vectors are bosons. All forces act to return the material system to the primordial symmetric state of free energy (light) from which it was created (Noether's Theorem). Thus massive leptons and quarks bear electric charges whose field vectors are photons; elementary particles bear identity charges whose field vectors are the IVBs; all quarks bear color charges whose field vectors are gluons; all forms of bound energy bear gravitational charges ("location" charge) whose field vectors are gravitons. Once again we have a natural dichotomy which invites our curiosity, experiment, and speculation. What is the relationship between the quarks and leptons? They seem made for each other - are they indeed made from each other - perhaps both arising from a common ancestor? I speculate that the ancestral particle of the quarks and leptons is the "leptoquark", the heaviest member of the leptonic elementary particle series. (See: " ".) The leptoquark is a lepton at very high (primordial) energy densities, when its quarks are compressed (by ambient pressure during the Big Bang) sufficiently that its color charge vanishes through the principle of "asymptotic freedom". (The gluon field, being composed entirely of color-anticolor charges, sums to zero when compressed to "leptonic size".) At lower energy densities, the quarks expand under their mutual quantum mechanical and electrical repulsion, causing the color charge to become explicit. The explicit color charge stabilizes the baryon, since neutrinos, which would otherwise cause its decay, do not carry color charge. Through the internal expansion of its 3 quarks, the leptoquark becomes a baryon, decaying eventually to the ground state proton, producing leptons and mesons (via the W) along the way, which function as alternative charge carriers for the electric and identity charges of quarks and other leptons. (See: " ".)The Leptoquark Diagram Introduction to the Weak Force Neutrinos The neutrinos remain mysterious particles and are actively being researched. Whether or not neutrinos actually have mass is still a question. If neutrinos have mass, why is it so small, and how do they escape carrying an 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 16 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.html "Mortgage", "pay later", "pay through time". Symmetry and charge conservation in obedience to Noether's theorem are the primary topics of Row 3. Each of the 4 forces is examined in terms of its fundamental charge and the broken symmetry of light which that charge represents. Quantized charges are conserved through time for payment at some future date. Gravitation pays the interest on this "mortgage" or symmetry debt by creating the time dimension, taking the necessary energy from the expansion of the Cosmos. Time is the relevant dimensional context in which concepts such as "the time deferred payment" or "cancellation of a conserved debt or charge" can have meaning. Charge (and spin) conservation is symmetry conservation; the forces generated by these charges are the demand for payment of the symmetry debt. Noether's theorem is the formal theory addressing the conservation of the symmetry of free electromagnetic energy (radiation, light). Charges are quantized to protect their values from inflation or deflation over time by entropy or relative motion in spacetime; otherwise, charge conservation would have little meaning. This is also the reason why matter must be separated and protected from the expansive or enervating effects of its entropy drive, time. Matter does not participate in the expansion of its causal information matrix, the historic domain of spacetime; matter maintains a tangential position with respect to history, existing only in the "universal present moment". electric charge, as do all other massive particles? Is there a 4th "leptoquark" neutrino? What is the smallest possible natural mass quanta? Are neutrinos composite or elementary particles? It is currently believed that neutrinos have a very small mass and "oscillate" between their several possible identities, just as the massive leptons, whose identity charges they carry, can change identities among themselves via reversible weak force decays, as mediated by the IVBs. (See: , Vol. 306, 26 Nov. 2004, page 1458.)Science Neutrinos were thought to be massless leptons with intrinsic motion c. They are now thought to have a tiny mass and to move very nearly at velocity c because they are so energetic when formed. Neutrinos are the explicit form of lepton number ("identity") charge, which is hidden or implicit in the massive leptons (and probably also in the massive baryons and leptoquark). Neutrinos, if they have any mass at all, are so light that they are apparently completely dominated by their deBroglie matter waves. Hence in the particle-wave spectrum of energy forms, neutrinos are more wave than particle. (See: " ".) deBroglie Matter Waves Each massive lepton (electron, muon, tau, and (perhaps) the hypothetical leptoquark) is associated with a specific neutrino, or number charge, which I refer to as an "Identity" charge to acknowledge the symmetry debt carried by the weak force. All photons are indistinguishable one from another, but the leptons do not share this "symmetry of anonymity". While all electrons are identical, they are distinct from the photon, and from the other elementary particles - the muon, tau, and leptoquark. Neutrinos are the hallmark of an elementary particle; they are telling us that there are only three or four; all else is a composite (or, as in the case of the quarks, a subunit). Due to Noether's Theorem, the conservation domain requires this identity asymmetry to be recognized and accounted for, but it is economical in its bookkeeping, concerning itself only with massive elementary particles. All neutrinos have left-handed spin, while all antineutrinos have right-handed spin, neatly distinguishing the leptonic series from its antimatter counterpart. Evidently these specific "identity" charges function to facilitate annihilation reactions between matter and antimatter, allowing the various particle species to identify their proper "anti-mates". Through the facilitation of timely annihilation reactions (within the Heisenberg limit for virtual reality), the identity charges make their contribution to conserving light's symmetry. Neutrinos are quanta of information keeping the symmetry records of spacetime concerning the identity and number of all massive elementary particles within its domain. Combined with the metrical warpage of gravitation, we see that spacetime contains an actual structural "knowledge" of the location, mass, and identity of every elementary particle. This startling fact informs us that spacetime is as scrupulous concerning symmetry conservation as it is concerning raw energy conservation. We have already noted that historical spacetime contains a complete causal record (in the form of information) of all past events. We are only beginning to appreciate the comprehensive meaning of the term "conservation domain". Row 3 - : The Symmetry Debts of LightCharges Row 3: Electric Charge The charges of matter are the symmetry debts of light.8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 17 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.html(See: " ".) The Time Train We do not ordinarily realize that the symmetry of energy is conserved as well as its total amount, but it has been known for a long time that this must be true. In a famous theorem, Emmy Noether proved mathematically that in a multicomponent field, such as the electromagnetic field (or the metric field of spacetime), wherever there is a symmetry one also finds an associated conservation law, and vice versa. This theorem has become the mathematical basis ("group theory") for modern efforts to unify the forces. In the model presented here, I trace the unity of the forces back to their origins as the conserved debts of light's broken symmetry. (See: ). " "Emmy Noether: A Tribute to her Life and Work Charges arise naturally from the process of symmetry breaking. When particle-antiparticle pairs are created from light, each member of the pair carries various charges which function to ensure instant and successful annihilation, reconstituting the light from which they were created. Since light itself carries no charges, it can only create particle pairs whose charges balance, cancel, or neutralize each other, summing to zero. The electric charge is prototypical of this effect. Initially, all massive elementary particles are created in particle-antiparticle pairs with equal but opposite electric charges (among others) summing to zero. Opposite electric charges attract each other powerfully, and at long range, allowing the particles to find each other anywhere in space and recombine, motivating an annihilation reaction which returns their bound energy to light, thereby conserving the symmetry of the free energy which created them. Since photons, or light quanta, are the field vectors (force carriers) of electric charge, we see light actively protecting its own symmetry in annihilation reactions, through the forces generated by electric charge. Finally, because the electrical annihilations of virtual particles are caused by photons traveling at velocity c, virtual particles are created and destroyed within the Heisenberg time limit imposed upon virtual reality. Virtual particles do not live long enough to exist in "real" time, and hence they also, like the light which created them, cannot produce a gravitational field. When one member of a particle-antiparticle pair is isolated, as by the asymmetric decay of matter-antimatter pairs during the Big Bang, the charges of that isolated particle, which were intended to motivate and facilitate an annihilation reaction with its antimatter partner, are simply "hung" in time. The isolated particle is one-half of a symmetric particle-antiparticle pair, one-half of light's symmetric particle form, and its uncanceled charges can therefore be fairly characterized as the "debts" or "remainders" of light's broken symmetry. While electric charge is always associated with mass, it is independent of the quantity of mass; the three leptonic particles (electron, muon, and tau), for example, have vastly different masses but carry the same electric charge. Electric charge is not associated with bosons which move with intrinsic motion c, such as the gluon, photon, or graviton. There is definitely a major, general asymmetry associated with the loss of light's intrinsic motion which electric charge is powerfully guarding against, and we would like to distinguish it from the asymmetry associated with the gravitational charge. The gravitational charge, however, is related to mass as well as to the loss of light's intrinsic motion. (Unlike their electric charges, the gravitational field (Gm) associated with the 3 elementary leptons cited above varies with each particle's mass.) The asymmetry I single out as the cause of electric charge is dimensional - light is 2-dimensional, mass is 4- dimensional. Light lacks the x, t dimensions of bound energy, as Einstein discovered. The jump from 2 to 4 dimensions in the conversion of light to particles (or bound to free energy) is a general loss of symmetry, since the 4th dimension inevitably includes time, which is an asymmetric, one-way dimension. It is this particular asymmetry, time, which electric charge protects against. Electric charge, through matter-antimatter annihilations, protects light's dimensional symmetry by preventing light from devolving into matter, gravitation, and the asymmetric time dimension which is matter's entropy drive and causal relation. Electric charge is not related to the quantity of mass because the dimensional asymmetry of time applies equally to all 4-dimensional massive forms, irrespective of magnitude. Like most symmetry debts, electric charge is a charge of "quality" not "quantity". Raw energy debts (mass, momentum) are "quantity" debts. Gravity is unusual in that it partakes of both, as gravity is both an entropy (quantity) and a symmetry (quality) debt of light - see below. Gravitational Charge8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 18 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlGravitation is a "spacetime" charge, at once the most common and familiar, but perhaps the most mysterious and intractable to explain. The symmetry debt associated with gravitation is "location", representing the broken spatio-temporal distributional symmetry of light's "non-local" character. When light is converted to mass, light loses its intrinsic motion and hence its non-local symmetric energy state. Whereas light (in its own reference frame) is everywhere simultaneously within its conservation domain (light's Interval = 0), mass has "intrinsic rest" and acquires a positive Interval. The distributional symmetry of light's energy within spacetime is therefore broken; mass is a concentrated lump of undistributed energy with a specific location in spacetime; this location is actually identified energetically in terms of both the quantity and concentration of bound energy by the warped metric produced by the gravitational field of mass. Whereas light is 2-dimensional, mass is 4-dimensional; the acquisition of the extra dimensions, especially time, identifies the spacetime coordinates and specific location of immobile mass-energy. Gravity identifies not only the coordinate position but the local severity of the broken symmetry of light's equitable energy distribution. As mentioned earlier, there are 3 "color" charges which are exchanged between quarks by the "gluon" field; gluons are composed of a color-anticolor charge pair. The constant "round-robin" exchange of the massless gluons (at velocity c) from one quark to another is the strong force mechanism which binds the quarks together. But the gravitational charge is unusual in that it is more than just a symmetry debt; unlike electric charge, color, or number, gravity is also the entropy debt of light. The gravitational force creates time and spacetime (bound energy shares spacetime with free energy as a compound conservation domain), converting space to time. Gravity and time induce each other: they are primordial expressions of entropy in matter. -Gm = the negentropic energy of mass, the energy associated with the time dimension of bound energy. The complexity of gravitation is due to the fact that its conservation function addresses both the first and second laws of thermodynamics (through time, causality, and entropy), as well as symmetry conservation (through the "location" charge and the positive Interval), simultaneously. The active principle of the gravitational "location" charge is time, which is both a symmetry (4-D location) and an entropy (intrinsic dimensional motion) debt. It is the entropic character of gravitation (time is an entropic charge) that causes gravitation to so aggressively and relentlessly pursue its symmetry conservation agenda (the conversion of bound to free energy, as in stars), unlike electric charge, for example, which is only a symmetry debt and is readily neutralized. (See: " ".)The Double Conservation Role of Gravitation Gravity is a collapsing spatial wave centered on a massive particle whose dynamic is supplied by the intrinsic motion of time, the entropy drive associated with the bound energy of the particle. The collapse of space produces a metrically equivalent temporal residue, whose entropic march into history collapses more space in an endless self-regenerating cycle. The temporal entropy drive thus supplied to matter is the conserved entropy drive of the free energy which originally created the particle - the transformed intrinsic motion of light. The temporal entropy drive of matter is not quenched until it succeeds in returning bound energy to its original free state, as seen in stars and via Hawking's "quantum radiance" of black holes, fulfilling the mandate of Noether's Theorem regarding the conservation of light's symmetric non-local energy state. This is the gravitational pathway of symmetry conservation, employing the engine of entropy. The electrical pathway is via chemistry and matter-antimatter annihilations, and the strong and weak force pathways are through particle fusion, fission, and proton decay - all with the same end, the conservation (restoration) of light's symmetric energy state. For a more complete discussion of the gravitational charge and its mechanism, see: and " "."Entropy, Gravitation, and Thermodynamics" A Description of Gravitation The Strong Force Color Charge Quarks are sub-elementary particles, as we know from their fractional electric charges which are either 1/3 or 2/3 of the unit charge carried by the truly elementary leptons such as the electron. Allowed quark combinations always sum to zero or unit leptonic values of electric charge: the proton is +1, the neutron 0, mesons are 0, +1 or -1. The symmetry which the strong force is protecting is this quantum unit of electric charge, the elementary leptonic charge, and whole unit charges generally. If quarks were not confined as they are, there would be no way to annihilate or even neutralize their partial electric charges, or other partial charges they may carry (such as color and identity). Symmetry could not be restored and conserved in such a case. The strong force protects 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 19 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlThe leptonic charge is known as "number" charge. I prefer to call it "identity" charge, a name which better reflects its reason for existence. Photons (individual light quanta) are indistinguishable and anonymous. They are all alike, and hence form a symmetry of identity which I call "anonymity". Elementary particles, on the other hand, are not all alike; they are distinguishable as to type. symmetry by confining these sub-elementary particles into whole quantum unit packages of charge which can be neutralized and/or annihilated by elementary (leptonic) unit anticharges. The strong force protects the quantum mechanical requirement of whole unit charge in the service of symmetry conservation. If one were to fracture an elementary particle into 3 parts, but require that when it became "real in time" it must retain its "virtual" leptonic character in terms of whole quantum units of charge, one would need a confining force with exactly the characteristics of the strong force as produced by the gluon field of the color charge. Earlier we noted that the ability to assume electrically neutral internal configurations (as in the neutron or neutral leptoquark) was the fundamental reason why the baryon must be a composite particle, if it is to break the symmetry of the primordial particle-antiparticle pairs. (See also: " ".)Proton Decay and the Heat Death of the Cosmos The two "particle forces", the strong and weak forces (the "short range" forces), form a symmetric-asymmetric force pair which is essential to the creation of matter. In this regard, they are provocatively similar to the two "spacetime" forces, electromagnetism and gravitation (the "long range" forces). (See: " ".)Diagram of the Spacetime and Particle Forces The Weak Force: Lepton "Number" or "Identity" Charge We know of three distinct elementary particles, comprising the leptonic spectrum or series: electron, muon, and tau, differing in their masses which increase from electron through muon to tau. Each has a specific neutrino associated with it, which functions as an alternative carrier of leptonic "number" ("identity") charge. (Neutrinos are the "bare" or "explicit" form of this charge, which is also carried in "hidden" or implicit form by the massive leptons). (See also: ). "The Weak Force: Identity or Number Charge" The leptonic series has the appearance of a mass quantum series - that is, these elementary particles are always created with a specific, discreet mass and no other; there are no elementary massive particles in the gaps between their mass "slots", much like the discreet gaps between the rungs of a ladder or the energy levels of atomic electron shells. The neutrino that is associated with each is evidently the hallmark of the truly elementary particle (the sub-elementary quarks have no associated neutrinos). It seems likely, however, that there is an undiscovered neutrino associated with the ancestral particle which gave rise to the quarks and baryons, which I assume to be the heaviest member of the leptonic series, the so-called "leptoquark". If we ever see proton decay, we would expect to see a leptoquark neutrino produced in the process. (The leptoquark neutrino is possibly the source of the "dark matter" or "missing mass" of the Universe - if neutrinos have mass at all.) The lepton "number" or "identity" charge evidently facilitates the annihilation process, identifies the several types of elementary particles, and by the handedness of neutrino spin neatly distinguishes matter particles from their antimatter counterparts (and so identifies suitable annihilation partners). Neutrinos also comprise a type of accounting system, recording the number and identity of elementary particles (or antiparticles) contained within the conservation domain of spacetime. Identity or number charge plays a special role in the creation of the material universe. We can characterize the light universe, before the creation of matter, with just 2 numbers representing its symmetric charge state: Interval = 0, and Number = 0. After the creation of matter, both symmetries are broken and become positive: Interval > 0, and Number > 0. (Electric charge is zero both before and after the creation of matter, while color is an internal property of baryons, also summing to zero). The positive Interval represents gravitation and time, the positive number charge represents the weak force identity charge and particles. The metric Universe, the Universe of the 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 20 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.html "Retiring the debt, closing the account" - symmetry restoration via the four forces. In Row 4 we list the various ways in which the 4 forces act through their conserved charges to fully repay the original energy, symmetry, and entropy debts incurred by the conversion of free to bound energy during the Big Bang. All energy, entropy, and symmetry debts are fully repaid by the conversion of bound to free energy, returning matter to its original form of light. The electrical symmetry debt can be repaid partially by neutralization, or wholly by annihilation, since unlike gravitation, electric charge is bipolar rather than monopolar (two-way rather than one-way). Whereas the gravitational symmetry debt can only be repaid by the conversion of mass to light, electric charge can be neutralized by its opposite matter charge, as well as annihilated by its antimatter charge. Electric charge acts to prevent the conversion of free to bound energy (as in the suppression of virtual particles via matter-antimatter annihilation reactions). Failing in this, it seems to have little further ability to restore symmetry, other than an eternal readiness to motivate an antimatter annihilation if the opportunity arises. Instead, electric charge contents itself with neutralizing opposite matter charges, confining them to small regions of spacetime which "pays down" its symmetry debt as far as it can. Conversely, gravitation does not act to prevent the formation of bound energy, but once matter is formed, seems to have a real "agenda" for its ultimate destruction - not "divide and conquer", but "collect and conquer". In this we discern the entropic character of gravitation, in contrast to the activity of any other charge or symmetry debt.dimensional conservation domains, responds to the positive number asymmetry by providing an asymmetric temporal entropy drive, an historic conservation domain for information and matter's causal matrix, and a compound conservation domain for both light and particles (spacetime), all through the quantum mechanical and gravitational conversion of space to time. The universe manifests through the identity charge, as identity provides a basis for the interaction between the symmetric quark field (the leptoquarks), the leptonic alternative charge carriers (the neutrinos), and the asymmetric field of the IVBs. It is through the identity charge that the IVBs recognize and separate leptoquark from antileptoquark, setting them upon separate and asymmetric decay pathways, breaking the symmetry of their particle-antiparticle pairs. Neutrinos are alternative carriers for identity charge, which allows this charge to be conserved or canceled without the presence of antiparticles (antileptoquarks) and their inevitable annihilation reactions. The fact that the fields interact electrically is not sufficient to break their primordial symmetry, since the electrical field is also perfectly symmetrical. It is for this reason that we feel the leptoquark neutrino must exist. For a more complete discussion, see: " ". The Formation of Matter and the Origin of Information Row 4 - : The Force Carriers as Symmetry Payments Field Vectors Row 4: Photons - The Electric Force The field vector (force carrier, charge carrier) of electric charge is the photon, the quantum unit of light and the electromagnetic force. In the annihilation of matter-antimatter particle pairs, we see the photon protecting its own symmetry. Electric charge is bipolar, consisting of opposite charges which attract each other powerfully over an infinite range of spacetime. The strength of this arrangement is that it permits matter-antimatter pairs to find each other, no matter how great their spatial separation. The weakness of this arrangement is that electric charges can neutralize as well as annihilate each other. It is therefore possible for a composite particle like the baryon to arrange the partial charges of its quarks to a neutral electrical configuration, as in the neutron. It is just such an arrangement that is exploited by the weak force to produce the asymmetric decays of neutral leptoquarks and create an excess of matter in the "Big Bang". Electrical neutrality is the fundamental reason why a composite particle (such as baryons composed of quarks) is necessary if matter is to be isolated from antimatter and the primordial symmetric energy state of the Cosmos. After the formation of matter, electric charge can do little to restore the symmetric state of energy because its force is quenched by its ability to neutralize itself. The net electric charge of the Cosmos is zero, both before and after the creation of matter. In chemical reactions, electric charge will drive toward the lowest bound energy state, but chemical releases of energy are insignificant compared with the total energy content of matter. Electrical annihilations of matter-antimatter particle pairs continuously suppresses the manifestation of particles from the 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 21 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlIf we are to believe Einstein, gravitons, the field vectors of gravitation, must connect directly to the dimensional structure of spacetime. This connection is attractive only, without a repulsive counterpart, as in electricity. The effect is to "warp" or "bend" spacetime, reducing the local gauge of the metric - the magnitude of the electromagnetic constant c. Time and space are affected in metrically equivalent terms. It may be difficult to imagine how anything could connect to something so intangible as a dimension, yet this is certainly the best explanation we have. And the dimensions are not so intangible when we encounter them through gravitational or inertial forces ("g" forces felt during acceleration); the intrinsic motion of time, the intrinsic motion of light, and gravitation itself can also be considered inertial forces in that they are all dimensional (metric) expressions of energy, entropy, or symmetry conservation. Like the other charges of matter, gravitation has a symmetry debt to pay, and like the other charges, if gravitation "vacuum", but the process is so effective we are normally quite unaware of this ongoing symmetry maintenance function performed by electric charge. Electric charge, however, in the form of the electron shell of atoms and the interplay of electric and magnetic forces, is instrumental in building a negentropic information pathway (with energy supplied mostly by gravitation) which culminates in biological systems and the rise of consciousness. In this, electric charge seems to be attempting to reconstruct the original connectivity of light, even if it cannot reconstruct its symmetry. The primordial system of light was not only a wholly symmetric, but also a wholly connected entity. Electric charge, whose field vector is the photon, can be thought of not only as a debt of light's dimensional symmetry, but also as a debt of light's dimensional connectivity, the holistic character of the primordial energy state. Hence electric charge seems to function as a "memory" of a preexisting state of connectivity and unity as well as symmetry. Similarly, we may see "beauty" as an emergent expression of symmetry conservation in the "Information Pathway" of biology. (See also: " ".) DeBroglie Matter Waves and the Evolution of Consciousness Although it cannot restore symmetry chemically, electric charge nevertheless attempts to reconstruct connectivity in material systems by means of a chemical (molecular) information pathway. For example, biology is nothing if not a web of interconnections, and through the evolution of conscious information systems, humans have not only become aware of the essential connectivity of the Cosmos, both intuitively and rationally, but are now engaged in the process of extending this physical web of connection between the planets of our solar system, and on into the galaxy. Significantly, through humanity, the biological Information Pathway has converged with the abiotic gravitational symmetry conservation pathway, converting bound to free energy through hydrogen fusion and the nucleosynthetic process. (See also: " and .The Information Pathway"; "Chardin: Prophet of the Information Age" Gravitons - Gravitation A dynamical view of gravitational action is allowed by Einstein's equations, via his own "Equivalence Principle". We are free to view a reference frame as either at rest in a static negative gravitational potential (as on the surface of the Earth) or as accelerated in spacetime by an equivalent positive motive force (as in a rocket ship). Hence we can view gravitation as the accelerated motion of spacetime itself, rather than as a static, "warped", or "curved" metric field. It seems to me this dynamic view offers a physically simpler way to visualize gravitational action, and is heuristically more fruitful, leading to other insights as well. The equivalence principle follows from the notion that we cannot distinguish between moving ourselves through spacetime (acceleration), or spacetime moving itself through us (gravitation). In the dynamic view, all objects fall with the same acceleration not because the static gravitational potential is the same but because they are all carried along in the same accelerated flow of spacetime. Similarly, velocity c and the local metric are reduced simply by the subtractive effect of the physical flow of spacetime; co-movers with the flow (free fall, orbit) are of course unaware of its motion - all the ordinary gravitational effects are as readily explained by one view as by the other. (See: " ".) Extending Einstein's Equivalence Principle "Quantum Radiance" and Black Holes8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 22 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlcannot pay off the debt outright, it will always move in that direction by at least "paying down" the debt as much as possible. Since an atom or a planet can have the same center of mass or "location", the gravitational concentration of massive particles reduces the scatter of individual "location" charges, confining them to as small a volume of spacetime as physically possible (the attractive principle of gravitation (-Gm), however, is simply the collapse of space caused by the intrinsic motion of time). (See also: " ".) If enough mass is accumulated, the fusion reactions of the nucleosynthetic pathway are initiated, converting a portion of the bound energy to light, a direct payment of the symmetry (and entropy) debt. However, nucleosynthesis can only go so far, as baryon number conservation prevents the great bulk of any stellar mass from converting to light. Nevertheless, gravitation drives on, collapsing the electron shells of atoms in "white dwarfs", and finally driving this "electron sea" into the protons, forming neutron stars, essentially gigantic atomic nuclei held together by gravitational forces. Still unsatisfied, if enough mass is present, gravitation collapses even nuclear matter to the singularity of a black hole, surely the most bizarre and fearsome object in the universe. In addition to its important role in confining quarks to elementary whole-quantum charge units, the strong force The Conversion of Space to Time Black holes can convert much more of the bound energy of atoms to radiation than nucleosynthesis, including extracting energy from the rotational energy of the hole, from the gravitational potential energy of highly accelerated particles (including any relativistic increase in mass), and even from the binding energy of nuclear particles, which the intense gravitational field of the hole replaces. Through such effects, up to 40% of the mass- equivalent energy of a particle can be converted to light as it falls into the event horizon. In the creation of a black hole, gravitation reaches its goal, for as Stephen Hawking has shown, through the principle of "quantum radiance" the total mass of a black hole will eventually be converted to light. The defining feature of a black hole is that the gravitational acceleration of spacetime reaches the equivalent of the intrinsic motion of light. As in the venerable saying, "the extremes meet": matter began as light with intrinsic motion c; matter ends by itself achieving intrinsic motion c through the gravitational acceleration of spacetime, a total reversal of the roles of intrinsic motion. But this full circle regenerates matter as light again, an amazing story of purposeful and relentless symmetry conservation which no one would believe if Einstein's and Hawking's mathematics were not there to prove it. Because the spatial entropy drive of light (intrinsic motion c) has greater symmetry than the one-way historical entropy drive of time (intrinsic motion T), Hawking's quantum radiance demonstrates that even the symmetry of entropy is conserved. It is symmetry conservation and the ultimate expression of Noether's theorem that drives the evaporation of black holes. The event horizon of a black hole is a temporal entropy surface (The Bekenstein- Hawking theorem), displacing space somewhat as a ship displaces water, providing physical proof of the gravitational conversion of space and the drive of spatial entropy to time and the drive of historical entropy. One reason we cannot see into a black hole is because there is no space to look into. The surface of the black hole is an expansion of the central dimensionless point which begins the time line, hugely enlarged; nevertheless, it remains spatially dimensionless. (See: " ".) A Description of Gravitation In thermodynamic terms, the conversion of light's entropy drive (light's intrinsic motion) to matter's entropy drive (time's intrinsic motion) reaches a limiting case in the black hole. Because at the Schwarzschild radius the inflow of space is already at velocity c, it is not physically possible to simply continue increasing the intensity of the field as matter is added to the hole. Therefore, the only accommodation possible for further mass inputs is to increase the size of the surface over which this maximum spatial flow is realized, resulting in the Hawking- Bekenstein theorem relating the entropy of a black hole to its surface area. Therefore black holes are somewhat larger than one might otherwise assume. Paradoxically, this effect does not reduce the critical density of the hole as it grows larger, because we are dealing with a time surface, not a spatial volume. Space is displaced by time, not by a competing spatial volume, which is where the ship displacement analogy fails. Since gravity is creating the time dimension for the mass of the hole, the constraint on the size of the entropy expression also applies to other related gravitational parameters. Hence the surface area of a black hole should be directly proportional not only to its entropy, but to its time dimension, its mass, and its total gravitational field energy as well. (See also: ). "Proton Decay and the 'Heat Death' of the Universe" Gluons - The Strong Force: Fusion and Proton Decay8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 23 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlcontains an important internal symmetry. The color charge of the strong force consists of three parts, designated (for convenience of reference only) red, green, and yellow. Each quark carries one color charge, which it swaps with its neighbors in a ceaseless round-robin exchange by means of an internal field of "gluons". Gluons are massless particles, moving at velocity c, the bosons or force carriers of the color charge and strong force. They have been compared to "sticky light". Each gluon is composed of a color-anticolor charge, in every combination, hence there should be nine of them, except one is doubly neutral ("green-antigreen"), leaving eight effective charge carriers. Because the gluon field is composed of color-anticolor charges, it sums overall to zero color, a crucial charge symmetry. The gluon field is internally confined to baryons, the class of particles containing 3 quarks. (There is enough "leakage" of the gluon field to allow gluon-sharing between nucleons (the color analog of a magnetic field), permitting the building of compound atomic nuclei). Physically squeezing the quarks together has the effect of summing up the gluon field, so that as quarks crowd together, the strong force relaxes and the quarks move more easily with respect to each other, an effect known as "asymptotic freedom" ( ). "In the limit", if the quarks are fully compressed, the color charge sums to zero and vanishes. This is the configuration of the leptoquark, and is the condition of "color symmetry" (color = 0) which is necessary for proton or leptoquark decay. Usually, quarks repel each other electrically and through other quantum mechanical forces (Pauli's "Exclusion Principle"); as quarks spread apart, the color force becomes explicit, limiting their expansion. Because the color charge is conserved, the weak force cannot cause baryon decay while the color charge is explicit (neutrinos do not carry color charge). But if for some reason the color charge should self-annihilate (as in the extreme pressures of the Big Bang, a black hole, or via the "X" IVB), the leptonic decay of a baryon can go forward. It is this effect that allows the weak force decays of electrically neutral leptoquark-antileptoquark pairs during the birth of the Cosmos. Politzer, Gross, and Wilczek: 2004 Nobel Prize for Physics "In the limit" the color charge vanishes. This limit probably translates physically to compacting the quarks of a baryon to "leptonic size"; in this condition, with no color charge present, a baryon is indistinguishable from a heavy lepton, reverting to its ancestral form, the "leptoquark". When fully compressed, the leptoquark is a lepton and the color charge is implicit; when the pressure is relieved, the color charge becomes explicit and the leptoquark becomes a baryon. As a lepton, the leptoquark must have an associated neutrino, but as a baryon, this neutrino cannot cancel the explicit color charge. Thus the baryon is stable against "proton decay" in its normal (expanded) state. Only when the quarks are fully compressed, vanishing the color charge, does the baryon return to its leptonic ancestral state, and proton decay becomes possible with the emission of a leptoquark neutrino. Achieving a condition of electrical neutrality is the fundamental reason why the elementary mass-carrier must be a composite particle whose constituent parts (the quarks) can assume an electrically neutral configuration (as in the neutron). This requirement in turn demands the creation of the gluon field and color charges to permanently control and confine these partial charges in combinations that sum to whole unit (leptonic) quantum numbers. The simplest way to create all these particles and fields is simply to split an elementary heavy lepton into three parts, demand (to satisfy symmetry conservation) that nevertheless it remain a "virtual" elementary particle in terms of its effective whole quantum unit (leptonic) charges, and the creation of the gluon field must follow of necessity. Presumably, all baryons have one and the same "number" charge, as all stem from the same leptoquark ancestor, and all must revert to this same high-energy form to decay, resulting in the extraordinary stability of the proton. Other than the hypothetical superheavy "X" IVB, it seems likely that only the gravitational pressures of a black hole can provide sufficient symmetric force to routinely cause proton decay. If this is so, then the interior of black holes may consist of nothing but gravitationally trapped light, a condition strangely reminiscent of the gluons or "sticky light" trapped within a baryon. (While a neutron star is like a gigantic gravitationally bound atomic nucleus, a black hole represents the next level of simplification, a gigantic gravitationally bound baryon.) Trapped light would solve the question of the infinite compressibility of matter at the central singularity, as there is no quantum-mechanical limit to the superposition of photons. (See also: " '")A Connection Between 'Inflation' and the 'Big Crunch Acting together, and energized by gravity, the strong and weak forces make common cause to restore light's symmetry through nuclear fission/fusion, resulting in the creation of the heavy elements in the nucleosynthetic pathway of stars. This pathway, however, is relatively short and ineffective, as only a small fraction of the energy 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 24 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.html(See also: " ".)stored in baryons can be released through nuclear fusion. The gravitational process goes to completion via Hawking's "quantum radiance" of black holes. Proton decay also completely converts nuclear mass to light, but the process is so rare that the proton, in human terms, is virtually eternal. We owe the stability of matter to the great strength of the strong force, the weakness of gravity, and the huge mass-energy barrier of the "W" and "X" IVBs. But the seeds of its own destruction are contained within the baryon, through the symmetry principle of "asymptotic freedom" and the self-annihilation of the color charge. (See: " .")The Half-Life of Proton Decay and the Heat Death of the Cosmos The Weak Force IVBs: Fission, Identity Charge Introduction to the Weak Force Because it is the weak force which breaks the symmetric state of energy in the Big Bang and brings the material Universe into existence, we might not expect this force to be particularly active in returning the material system to symmetry. Yet, the force that creates matter can also destroy matter, and it does so in several ways - through the decay of heavy particles to their ground state, through the fission of heavy compound nuclei (radioactivity), through contributions to fusion in the nucleosynthetic pathway of stars, and through the process of proton decay, for which it provides the annihilating identity charge (the leptoquark antineutrino) as well as the "X" IVB. (See: " ".) The Particle Table When we consider an elementary particle, such as the electron (e-), we often forget that this particle carries two charges, electric charge and identity (or "number") charge. The electric charge is indicated by the negative sign, the identity charge is indicated by the "e" (this charge is sometimes referred to as "flavor"). We say that identity charge is "hidden", or carried in implicit form, by the massive electron, but is revealed in its explicit, "bare", and nearly massless form as the electron neutrino. (Whether or not the neutrino is actually massless has little to do with its symmetry debt of "identity". Most charges are in fact carried by massive particles). Usually the identity charge is simply called lepton or baryon "number" charge (or even "flavor" charge), which obscures the true meaning of this charge. If "number charge" adequately described its function, then the number charge of the electron would also serve as the number charge of the muon and tau; but as we have discovered, there is a specific and distinct neutrino associated with each member of the elementary leptonic spectrum, so the charge is more accurately described as "identity". Moreover, we can readily assign "identity" as the plausible symmetry debt of light's "anonymity", with a sensible function to perform in annihilation reactions (facilitating the choice of the correct antimatter partner), arguments and contact with Noether's theorem which we cannot make for the generalized "number" charge. It is at first a curious fact, and then after reflection an obvious one, that the "identity" charge is the key to manifestation. It is identity that brings matter into existence as the principle or "cardinal" symmetry debt. But then, how could it be otherwise? Identity is the essence of asymmetry, the key ingredient of information that must be isolated from the symmetric field of energy if manifestation is to occur. (See also: .)"The Weak Force "W" Particle as the Bridge Between Symmetric (2-D) and Asymmetric (4-D) Reality" In addition to the mesons, the leptonic field of elementary particles functions as an alternative charge carrier, both for the symmetric, composite field of the quarks and hadrons, and for other leptons. The massive leptons function as alternative carriers of electric charge, the (nearly) massless neutrinos function as alternative carriers of identity charge, the mesons function as alternative carriers of quark flavor and color. Without these services, the symmetric quark field could not manifest, since in the absence of alternative carriers, quarks could only balance their charges with antiquarks, and they would remain forever locked in mutually annihilating particle-antiparticle pairs. Without neutrinos, the massive leptons would likewise remain locked in their particle-antiparticle pairs, themselves lacking an alternative carrier of identity. Hence it is that the neutrino, the least of all particles, becomes the "mouse which nibbles the lion's net", providing an alternative conserved carrier of identity charge, and through this service, unleashes the information potential of the Cosmos. (See: .)"The Weak Force: Identity or Number Charge" Just as we see the information pathway of the electromagnetic force evolving to reestablish the primordial 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 25 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlWe have asserted that light and metric space create matter in symmetric particle-antiparticle pairs, and that these, through the mechanism of mutually interlocking charges, annihilate each other to recreate the light which formed them. During the "Big Bang", the asymmetric mechanism of the weak force breaks the symmetry of the particle-antiparticle pairs, producing an excess of matter. We understand that the raw energy of light is stored (conserved) in the mass and momentum of the particles, and that the charges of matter, which appear to be gratuitous from the point of view of raw energy conservation, are in fact necessary from the viewpoint of symmetry conservation: not only the raw energy of light, but its symmetric state must be conserved (Noether's Theorem). This interaction occurs within the metric arena of spacetime, the entropic dimensional setting which houses and conserves the energy play. How is spacetime related to this play of light and particles? What is this play about?connective unity of light and emergent forms of symmetry (beauty) throughout material systems, so we also see through the rise of consciousness and the emergence of organisms with definite individuality and personality, the reemergence and exploration of weak force "identity" in the biological realm. (See also: .)"Chardin: Prophet of the Information Age" Summary There is another apex of the which involves the second law of thermodynamics, entropy. It is through entropy that we are able to complete the conservation linkage between the dimensional structure of space, light, and matter. The primordial entropy of light is expressed through its intrinsic motion, which creates not only space and its metric (the conservation domain of light), but the expansion and cooling of space as well. The primordial entropy of bound energy is expressed through the intrinsic motion of time (and gravity), creating historic spacetime, the conservation domain of matter's causal information matrix. The intrinsic motion of time causes the aging and decay of matter and information and the expansion and dilution of history. Time and gravity are therefore the conserved form of light's entropy (second thermodynamic law); mass and momentum are the conserved form of light's raw energy (first law, energy conservation); the charges of matter are the conserved form of light's various symmetries, and constitute the essential information which particles require, in the absence of antimatter, to return to their original symmetric state (Noether's Theorem). Tetrahedron Model of Light and Conservation Law Before "Big Bang" symmetry-breaking, in the absence of matter, Noether's Theorem is expressed through metric symmetry conservation and the suppression of virtual particles by matter-antimatter annihilations, all gauged by "velocity c". After symmetry breaking, in the presence of matter, Noether's Theorem is expressed through charge and spin conservation, gravitation and time, and the inertial forces of the spacetime metric. Entropy conservation allows the conversion of free energy to work; symmetry conservation allows the conversion of free energy to information; raw energy conservation allows the conversion of free energy to mass and momentum. These three conservation laws, acting in concert, allow (but do not cause) weak force symmetry-breaking and the conversion of light to our familiar material Universe. Time, causality, gravitation, and historic spacetime provide the connective linkages of matter's "causal matrix" and information field. History is the functional analog of space. The continued reality of historic spacetime and matter's "causal matrix" are necessary to uphold the continuing reality of the "Universal Present Moment" of material existence. Light's entropy drive (light's intrinsic spatial motion), creates, expands, and cools light's dimensional conservation domain, space; matter's entropy drive (the intrinsic motion of time) creates, expands, and dilutes information's dimensional conservation domain, history. Gravity, the entropy conversion force, welds space and the drive of spatial entropy (light's intrinsic motion) into time and the drive of historical entropy (time's intrinsic motion), creating historic spacetime, the joint entropy/conservation domain of free and bound energy. The first and second laws of thermodynamics are connected through the entropic creation of the dimensional conservation domains of light and matter. The function of entropy is to create a dimensional domain (space, history, historic spacetime) appropriate to its energy type (free or bound), in which energy can be transformed, used, and yet conserved. Gravity equilibrates the two entropy drives (by extracting time from space) so that interaction between them is possible, creating their joint dimensional conservation domain, historic spacetime. The metric equivalency between space and time is gauged (regulated) by the universal energy constant c; the 8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 26 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlentropic equivalency between space and time is gauged by G, the universal gravitational constant. G is related to c as entropy is related to energy. The magnitude of G is determined by the small energy difference between the symmetric spatial entropy drive (S) of light (the intrinsic motion of light, as gauged by "velocity c"), and the asymmetric historical entropy drive (T) of matter (the intrinsic motion of time, as gauged by "velocity T"): S - T = -G. This is equivalent to the small energy difference between implicit (S) and explicit (T) time. The gravitational conversion of space and the drive of spatial entropy (S) to time and the drive of historic entropy (T) can be symbolically represented by a "concept equation" as : -Gm(S) = (T) -Gm(S) - (T) = 0 The spatial entropy drive of free energy therefore funds the historical entropy drive of bound energy, and the expansion of the Cosmos must decelerate accordingly. (See: " ".) The Conversion of Space to Time The cosmic drama begins innocently enough with the entrance of pure light and light's creation, the spacetime metric. The interaction of light with the spacetime metric creates a cast of virtual particles in symmetric particle- antiparticle pairs: some heavy (hadrons), some light (leptons), some composite and complex, some elementary and simple, but all related and all derived from the interaction between light's energy and the metric structure of spacetime. They are costumed in various charges which allow them to alternate with blinding speed between their particle and wave forms, a counterpoint between manifest and unmanifest reality, a true magic show. But then a symmetry disaster strikes, and the plot literally thickens. Some of the heavy, composite particles of antimatter have reverted to their wave form via their neutrinos and the weak force "X" IVB, without annihilating their matter counterparts. Caught by surprise in an expanding Universe, the matter particles have no way of reverting to their wave form in the absence of their antimatter partners; they are trapped in the 4th dimension of explicit time, whereas before they existed in the virtual realm of two or three symmetric spatial dimensions. We recognize them now as baryons. In the rapidly expanding and cooling Universe, they are left in their asymmetric and massive forms, one half of light's particle form, with all their charges intact and exposed, charges which had previously functioned to unite them with their antimatter partners and return both to light. Like Hamlet's father, the baryons have been treacherously thrust into a new realm without the chance to absolve their "sins".These charges are the symmetry debts of light. Spacetime becomes the dimensional entropy/conservation stage upon which the play now unfolds, a negentropic arena provided by the energy of gravitation (energy borrowed, in turn, from the expansion of space). Gravitation creates the time dimension through its ceaseless annihilation of space; time and gravity endlessly induce each other. The argument of the play is this: can the particles, using their conserved symmetry charges, either individually or collectively revert to their symmetric wave form in the absence of their antimatter partners? Is one-half of the information contained in the original particle-antiparticle pair enough to accomplish this magical transformation? The answer is yes, but only in the additional dimension of time, and in two modes: a collective process (gravity) and an individual process (proton decay). Both will arrive at the same result, the complete transformation of the particle to light. In the meantime, as a sort of subplot, or "play within a play", an electromagnetic information pathway develops (through biology), which attempts to express or reconstitute in material systems its charge-memory of the symmetry and connective unity of its primordial state. The development of personal "identity" and the abstract information systems of humans reprise and recollect our physical origins, in religious, aesthetic, psychological, and rational terms, even including the fractal algorithm of the information pathway. (See: ). "The Information Ladder" As for the issue of "intelligent design", the recent concept of the "Multiverse" in service of the "Anthropic Principle" offers a completely satisfactory resolution of the problem of the "special balancing" or "exquisite adjustment" of our Universe's physical constants. According to this view, we naturally find ourselves inhabiting that special Universe, of perhaps infinitely many realized possibilities, in which the physical constants of Nature are so adjusted, by chance alone, as to favor the evolutionary development of our life form. But it could hardly be otherwise. We might as well be amazed at how perfectly our skin fits our body. While this is a completely rational explanation for the peculiar characteristics of our Universe, it actually says nothing at all regarding the existence of a "First Cause" or "Creator" - neither for nor against. Concerning the issue of evolution, it is simply a biological form of negative entropy driven by Natural Selection, as factual and impersonal as gravity or chemistry. (See: " ".) Newton and Darwin: the Evolution and Abundance of Life in the Universe8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 27 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlThe meaning of the biological information pathway that develops through time in the negentropic domain of gravitation, the significance of the human experience and the Universe, are separate topics which I address in other papers (see: and: ); " "; " "). See also my late father's book: in regard to the significance and meaning of the human experience."The Information Pathway"; "Chardin: Prophet of the Information Age" The Human Condition Is There Life After Death? "Trance, Art, Creativity" Links: (pdf file) (html file) (diagrams) Related articles on my Entropy, Gravitation, and Thermodynamics A Description of Gravity Spatial vs Temporal Entropy The Double Conservation Role of Gravity: Symmetry vs Entropy The Higgs Boson vs the Spacetime Metric DeBroglie Matter Waves and the Evolution of Consciousness The Conversion of Space to Time The Time Train Extending Einstein's Equivalence Principle Principles of the Unified Field Theory: A Tetrahedral Model A General Systems Approach to the Unified Field Theory Currents of Symmetry and Entropy The Formation of Matter and the Origin of Information The Information Pathway Chardin: Prophet of the Information Age Is There Life After Death? The Fractal Organization of Nature A Spacetime Map of the Universe Introduction to the Weak Force The Weak Force: Identity or Number Charge The Weak Force "W" Particle as the Bridge Between Symmetric (2-D) and Asymmetric (4-D) Reality The "W" IVB and the Weak Force Mechanism The "W" IVB and the Weak Force Mechanism Gravity A New Gravity Diagram The Gravity Diagram The Three Entropies: Intrinsic Motions of Gravity, Time, and Light Unified Diagram of the Four Forces Diagram of the Particle Forces Diagram of the Spacetime Forces The Particle Table The Tetrahedron Model of Light and Conservation Law The Interaction of 4 Conservation Laws with the 4 Forces of Physics Unified Field Table: Simple Form Unified Field Table: "Bare" Form About the Papers: An Introduction The Sun Archetype home page References Noether, E. . Brewer, J. W. and M. K. Smith, eds. M. Dekker, New York, , 180 + x pp. + 10 plates. Weinberg, S. . Bantam. , 177 + x pp.Emmy Noether: A Tribute to her Life and Work 1981 The First Three Minutes 19778/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 28 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.htmlCronin, J. W. CP Symmetry Violation: the Search for its Origin. , 212, 1221-8 (Nobel lecture). Hawking, S. W. Particle Creation by Black Holes. , 43 (3), 199-220. Green, B. . W.W. Norton & Co. , 448 + xiii pp. Bekenstein, J. D. Black Holes and Entropy. , , 7(8), 2333-46. Pierre Teilhard de Chardin: French: Editions du Seuil, Paris, 1955; English: Harper and Row, New York, 1959. D. J. Gross and F. Wilczek. 1973. Ultraviolet Behavior of Non-Abelian Gauge Theories. Phys. Rev. Lett. 30: 1343. H. D. Politzer. 1973. Phys. Rev. Lett. 30: 1346. Gowan, J. C. 1975. Science1981 Communications in Mathematical Physics 1975 The Elegant Universe 1999 Physical Review D 1973 The Phenomenon of Man. Gross, Politzer, Wilczek: 15 October 2004 vol. 306 page 400: "Laurels to Three Who Tamed Equations of Quark Theory."Science: "Trance, Art, Creativity"8/27/06 10:55 AM Symmetry Principles of the Unified Field Theory (a "Theory of Everything") Page 29 of 29 file:///Macintosh%20HD/Desktop%20Folder/TheoryOfEverything/appendix.html

arXiv:physics/9911061v1 [physics.gen-ph] 24 Nov 1999About Charge Density Wave for Electromagnetic Field-Drive Benoˆ ıt T. Guay† Qu´ ebec, November 24, 1999 Abstract To generate a propulsive force without propellant and ex- ternal couplings, it has been shown that two confined macro- scopic and time-varying charge density waves well separate d in space are needed. Here, some physical conditions will be proposed to support and maintain these particular collecti ve modes of charge distributions. I. INTRODUCTION Within the framework of classical electrodynamics, it has been shown [1] how an electromagnetic propulsive force and, in particular, an electric (conservative) propu l- sive force can be generated without propellent mass and external couplings by using two confined, time-varying, neutral and macroscopic charge density waves (CDW). These CDW own a same symmetry axis, are adequately separated in space and have a relative temporal phase- shift. This last one controls the propulsive force’s inten- sity. From far fields point of view, these CDW are able to induce an asymmetry into the space distribution of the far fields momentum variation rate along the symmetry axis. They can do that because the relative temporal phase-shift controls the space distribution of construc- tive and destructive interferences of far fields produced by the two CDW [1]. So, this relative temporal phase-shift controls the asymmetry. When this last one is created, an electromagnetic propulsive force along the symmetry axis is generated and applied on both CDW in a same direction. Such propulsive effect is impossible in statics because fields’ interferences can be produced only with time-varying fields. Because this propulsive force is gen- erated by a spatial asymmetry in the (electromagnetic) field, it is a propulsion driven by the electromagnetic field or more simply an electromagnetic field-drive (EFD). In our first paper [1] we have used the CDW concept in a theoretical way. Actually, nothing has been said about the material or the conductive fluid needed to sustain a neutral macroscopic charge density wave. The only thing we have mentioned was this CDW is a longitudi- nal (i.e. φdirection in cylindrical coordinates) charge oscillation mode, it has a wave number “n”, it oscillates at frequency ωand it is pinned (circular standing wave) inside a ring made with an electrical conductor. In this simplified model, we have used two identical planar fili- form rings with radii R’, placed in vacuum and separated by a distance D along the z axis. Planes of rings were per-pendicular to the z axis; the symmetry axis, the thrust axis. In a more realistic way, rings have a cross section Rosmaller than D and R’ according to section 4 in [1]. However, we have never mentioned that a relation must exist (dispersion relation) between n and ωand what is this relation. Furthermore, what are needed conditions to support and maintain a time-varying CDW able to create the desired propulsive effect? Is it possible to use solid rings? Metallic ones? Or what else? In this work, we would like to give preliminary and partial answers to some of those questions. II. A LONGITUDINAL PLASMA MODE A time-varying longitudinal CDW involves a time- varying longitudinal charge separation among opposite charges. In that case, there must be a restoring force among these charges and consequently, this creates a col- lective oscillation mode (i.e. longitudinal plasma mode) at plasma frequency ωp[2,3,4,5,6]. So, to sustain a large amplitude of charge separations in a neutral conductor or, more generally, in a conductive “fluid” and then sup- port and maintain sources of large electric fields, our fre- quency ωmust be close to (at least equal or greater than) the “resonant” frequency ωp. Thus, we will get an ap- propriate CDW (n /negationslash=0) if each neutral conductive fluid of our two rings is a neutral plasma. The other reason to use a ω > ω pis this. In a sense ωpcan be considered as a cut-off [7]. So, if ωis greater than this cut-off, fields created by one conductive fluid in a given ring will penetrate deeply inside the conduc- tive fluid of the other ring to create propulsive effect throughout the ring’s cross section for non-filiform rings (i.e. torus for instance). Actually, if ω < ω pfields gen- erated by one ring will remain near the surface of the other; they will be mostly reflected by this one and they will be nearly zero inside of it except to its surface. In such a case, the thrust’s amplitude will be limited and restricted to the rings’ surface. In addition, this will in- crease the probability of cold emission like in a metal (see below) because fields must be relatively strong (i.e. at least about 100kv) to get a good thrust [1]. So, things like that can reduce the propulsive effect. According to the model in [1], the value of ωmust be in the range of radio frequency or TV range. Consequently, our plasma must have a ωpin these ranges too. However, if n=0 there are no charge separations at all; we have only a uniform longitudinal current on each ring. In this last case, we don’t need a longitudinal plasma mode; a 1neutral conductive fluid with a ωpmuch larger than ωcan be used. But let’s remember this, if n=0 the propulsive force has no electric contribution (i.e. no conservative part), only a magnetic one (i.e. a dissipative part because radiative) and we know that this last contribution has a poor efficiency according to section 6 in [1]. For our purpose and for now, at least four limits or con- ditions must be considered in a neutral plasma. The first is related to the collision rate f. Our macroscopic time- varying CDW is a collective oscillation mode; a longitu- dinal plasma mode. Collisions break “coherence” among charges’ motions and then break the collective oscillation and, the CDW itself can be destroyed. To get collective oscillations, we must have f≪ωp∼(e2no/meǫo)1/2(SI units) [2,3,4,5,6]; meis the effective electron’s mass, e the electron’s charge, nothe electron density when n=0 (for electronic plasma with heavy positive ions as uniform background) and ǫois the vacuum electrical permittivity. We have to mention that fincreases with noand tem- perature (see below). The second limit is associated with the wave number or the wave length of the charge density in a conductive fluid. For us, this is related to “n”. There is an upper limit for this wave number. Above this limit, the CDW cannot oscillate; the damping (i.e. Landau damping [8]) is too strong and thus, CDW does not exist (it’s too “vis- cous”). In a nondegenerate conductive fluid, like an ion- ized gas with relatively small density of electrons and ions for instance, this upper wave number is the Debye wave number kDgiven by kD−1∼(Te/no)1/2cm [9,10] ( no in cm−3).Teis the electrons’ temperature (in Kelvin); a measure of their mean kinetic energy. An electron within kD−1cannot move easily (“viscous” area) but outside, it can. So, if the wave length of our CDW is larger than kD−1, the damping won’t exist or it will be weak or quite weak and then, this CDW will survive and will be able to oscillate. In a degenerate neutral conductive fluid like an electron gas in a solid metal at low temperature (i.e. low com- pared to the Fermi energy EF[11,12]), kDis replaced by the Fermi wave number kF[9]. In that case, the typical kinetic energy is EFnotTe. In our situation, we need a neutral plasma with ωp∼ 100 MHz (radio frequency as order of magnitude) and akDorF−1smaller than about 10−2cm. 10−2cm is a lower limit for the wave length of our CDW; a macro- scopic length scale for which our classical approach in [1] is certainly correct. With solid alkali metals like Li, Na, etc. or solid noble metals like Cu, Ag, Au, kF−1 respects the above condition. For example, solid copper (Cu) at room temperature ( ∼300K), kF−1∼10−8cm [13,14]. But the problem with solid metals like alkali (or noble) is their ωpbelong to ultraviolet frequency range (∼1015Hz) [15,16]. The reason for such a big value is a large no(∼1022cm−3) [16] and a very small effective mass of charge carrier (i.e. electron). So, solid metals can bee used only if n = 0 (i.e. uniform currents on rings) according to above discussion ( ω < ω p).For instance, if n = 0, we could use two metallic and solid torus (i.e. planar rings with cross section Rosmaller than D and R’ according to above), fixed apart with dis- tance D by some adequat isolators and placed in good vacuum at “room temperature”. However, one possible problem with metals is the cold emission [17]; when fields applied over metallic crystal become relatively strong, electrons (carriers) can be expelled outside the crystal by “quantum tunneling”. In that case, the charge’s mo- mentum of carriers won’t be given to the whole crystal along the thrust axis so, the momentum transfer effi- ciency will be diminished and then, the propulsion too. Furthermore, with metals we will have ω < ω pand, as mentioned above, this is limitative. With n /negationslash= 0, we need something else. For instance an ionized gas; a neutral conductive gas formed by electrons and ions with a smaller electron density: no∼108cm−3. In that case this neutral plasma has a ωpin the range that we want according to its expression given above. On the other hand, we want a relatively “cold” plasma because we wish to satisfy the condition f≪ωpand also because we want to avoid any complications about plasma confinement (“walls”). For example, let’s con- sider a temperature Tebetween 1000K to 10000K. Such values for Teandnogive us a kD−1∼10−3to 10−2 cm according to the above expression so, they give us a classical plasma (i.e. nondegenerate electrons gas where classical statistics can be applied) quite similar to the ionosphere’s one [18]. Actually, at 90km into ionosphere, collision rate is f∼106Hz and at 300km, f∼103Hz [19]. Such last values respect the preceding inequality between fandωp. This doesn’t mean ion species we need must be the same as the ionosphere’s ones. Best ion species we need is another issue. But it shows that such a kind of plasma exist. So, a priori, a neutral ionized gas with relatively “low”temperature, 103to 104K, and low elec- trons density, no∼108cm−3, (i.e. a cold plasma) could be a good candidate for our purpose when n /negationslash= 0. Let’s take an example to get an order of magnitude of the propulsive force when a cold plasma gas is un- der consideration. Let us consider a lithium gas with electrons density no∼108cm−3and electrons temper- ature Te∼5000K. According to above expressions, ωp ∼564MHz and kD−1∼7×10−3cm. By simplicity, let’s imagine all atoms of lithium are ionized such as Li → Li++ e−. Atomic weight of Li is about 6.9a.m.u. so lithium mass density is about: no×6.9×1.66×10−27kg∼ 10−18kg/cm3. (Of course, this doesn’t take into account the mass of confining “walls”). Mass of Li+is about 104times larger than the one of e−. So, ion Li+is at rest compared to e−; only electrons move at frequency ωalong φdirection. Now to get an order of magnitude of the propulsive force, we can use the Coulomb force expression. Coulomb force is one of main contributions (conservative part) to the thrust in [1]. So, in these con- ditions if we consider a small volume of 1cm3of charges on each ring (or torus), the force we can get between 2these small volumes if D = 0.1m (same order of magni- tude than the one used in [1]) is given approximately by (1cm3)2·/parenleftbig no2e2/4πǫoD2/parenrightbig ∼10−10N. This evaluation is amaximum one because it doesn’t take into account de- structive interferences among fields produced by positive and negative charges in a same CDW and applied over charges in the other CDW. The reason for such a small force is the relatively small value of no. If we increase no, condition kD−1≪10−2cm will be always satisfied but certainly not ωp∼100MHz. However, if we use an “ionic plasma” instead of an “elec- tronic one” as in the above example, we will have ωp∼ (e2no/miǫo)1/2andkD−1∼(Ti/no)1/2cm where nois now the ions density, Tithe ions temperature and mi, the ion mass. Consequently, if nois increased, we will keep ωpfixed if we take an appropriate ion mass milarger thanme. Let’s give an example. Let’s take Li + Cl →Li++ Cl−. Ion chlorine Cl− is about 5 times heavier than ion Li+. Here, mi≡ mLi= 11.4×10−27kg and Ti≡TLi. As before we take same temperature TLi∼5000K. Now to get the same plasma frequency; ωp∼564MHz, we must take no∼ 1.3×1012cm−3. In that case, kD−1∼6.2×10−5cm and (1cm3)2·/parenleftbig no2e2/4πǫoD2/parenrightbig ∼3.9×10−2N with the same D as before. Here, condition f≪ωpis still respected. We can evaluate fby using its expression [20,21] for an ideal gas (i.e. low density and pressure). One has f∼no¯vσi=no(8kBTi/πm i)1/2σi∼641Hz. kBis the Boltzmann’s constant, σi≡σLi∼4π(1˚A)2is the scat- tering cross section of lithium ion and ¯ v, its mean speed. Finally, mass density is no(6.9+35.4)×1.66×10−27kg∼ 9.1×10−14kg/cm3. So, as we can see, the choice of ion species is quite important. The neutral plasma gas must be ionized by some exter- nal source (at the beginning at least) but, because tem- perature is relatively small, after a specific time there are recombinations among electrons and ions (or ions-ions) and then a radiation (named secondary here) is emitted. The primary radiation is the one emitted by the longi- tudinal plasma oscillations of both CDW at frequency ω > ∼ωp. Other kinds of secondary radiations can also be emitted like breaking radiation (bremsstrahlung) [22,23] and spectral radiation coming from excited atoms (not ionized). Recombinations among charges imply that a third limit has to be considered in our neutral plasma. This limit is given by fr≪ωpwhere fris the recombina- tion rate between negative and positive charges. Clearly, this quantity depends on electrons (or ions) density no and electrons (or ions) temperature Te(orTi).frin- creases when temperature decreases because kinetic en- ergy of opposite charges (i.e. their thermal energy) be- comes smaller than their potential energy (i.e. mutual attraction). This is why temperature, on the other hand, cannot be too small.III. ANISOTROPIC CONDUCTIVE GAS According to the model given in [1], charges must be well confined along the z direction (i.e. the thrust direc- tion) and along the ρdirection in some restricted regions (i.e. “filiform” rings). So, some constraints have to ex- ist to maintain charges in these limited areas along those directions. These constraints have to ensure also the mo- mentum transfer from charges to confining “walls”, spe- cially along z. In that sense, the conductive fluid (or gas) must be strongly anisotropic; charges can move eas- ily along φbut should be nearly “at rest” along z and ρ directions. Now, to get an appropriate anisotropic conductive gas (ionic and cold plasma gas), the cross section’s radii Roof a ring (or torus) must be smaller or equal to kD−1so, the fourth limit is Ro< ∼kD−1∼6.2×10−5cm (using pre- ceding value of chlorine-lithium gas) so, a “micro-torus” with a relatively large radii R’. The reason is this. Any charges inside kD−1, around the heavier ion; the Cl− in our previous example, are in “viscous” area. This is true for Li+ions and for induced dipoles of the dielectric “walls” (see below). Consequently, with the above limit, any relative motions between Cl−and Li+along z and ρ are quite well limited and this is true also among Cl−and dipoles, induced by this ion, inside the internal surface of the dielectric walls along those directions. In addition, the wall of this micro-torus must be a good dielectric. The neutral ionized gas will fill the micro- torus. The dielectric wall must be transparent to primary and secondary radiations. This is obvious for primary fields according to above; fields must reach the gas. But it is also important for the secondary to maintain a fixed temperature and get and sustain an equilibrium between ionization and recombination. Furthermore, this dielec- tric wall must be able to support high mechanical stress and relatively high temperature. IV. CONCLUSION In this paper, a well confined neutral ionized gas at relatively low density and temperature (i.e. a nondegen- erated conductive gas; a “cold plasma”) is proposed as a substrate in which a CDW (n /negationslash= 0) can be sustained; the CDW needed to produce a conservative propulsive force, according to the model given in our first work. Up to now, cold plasma is the most appropriate mate- rial to meet conditions given in this paper. But, plasma stability, plasma confinement, momentum transfer from accelerated charges to the confining “walls” along the thrust axis, choice of best ion species and dispersion re- lation are certainly complicated issues to deal with in the near-term. In addition, the fourth condition is difficult to satisfy from a technological point of view now. On the other hand, as shown in [1], this model (i.e. rings and the specific charge and current density distributions 3used; the CDW) has a poor efficiency. For all of these reasons, modifications to this model (i.e. to charge dis- tributions) are needed to get a more efficient and realistic near-term EFD. †e-mail address: bguay@interlinx.qc.ca [1]Guay, B.T., Propulsion Without Propellent Mass; a Time-Varying Electromagnetic Field Effect , physics/9908048 [2]Jackson, J.D., Classical Electrodynamics ,second ed., Wiley and Sons, 1975, chapter 7, p. 288, chapter 10, p. 492. [3]Jordan, E.C. and Balman, K.G., Electromagnetic Waves and Radiating Systems ,second ed., Prentice- Hall, 1968, chapter 9, p. 293. [4]Lorrain, P. et Corson, D.R., Champs et ondes ´ electromagn´ etiques ,´ ed. Armand Colin, Paris, 1979, chapter 11, p. 506. (French version of: Electromag- netic Fields and Waves ,edited by W.H. Freemann and Company, U.S.A., 1962, 1970). [5]Ashcroft, N.W. and Mermin, N.D., Solid State Physics ,1st ed., Saunders College/HRW, 1976, chapter 1, p. 18. [6]Kittel, C., Physique de l’ ´Etat Solide ,5e ´ ed., Dunod, 1983, chapter 10, p. 289, (French version of: Introduc- tion to Solid State Physics ,Wiley and Sons, 1976. [7]ref. [3], chapter 9, p. 295. [8]ref. [2], chapter 10, pp. 495-496. [9]ref. [2], chapter 10, p. 494, 497. [10]ref. [3], chapter 17, pp. 697-698. [11]ref. [5]. chapter 8, pp. 141-142. [12]ref. [6], chapter 6, p. 156. [13]ref. [6], chapter 6, p. 152, table 1. [14]ref. [5], chapter 2, p. 38, table 2.1. [15]ref. [5], chapter 1, p. 18. [16]ref. [6], chapter 10, p. 291. [17]Yavorski, B. et Detlaf, A., Aide-M´ emoire de Physique ,3e ´ ed., Editions Mir, Moscou, 1975, 1984, pp. 440-441. [18]Plasma Science Report, Contents and Overview, 1995, Intro., see figure S.1, (http://www.nap.edu/readingroom/books /plasma/contents.html #intro). [19]ref. [3], chapter 17, p. 670. [20]Reif, F., Fundamentals of Statistical and Thermal Physics ,1st ed., McGraw-Hill, Inc., 1965, chapter 12 p. 490. [21]Reichl, L.E., A Modern Course in Statistical Physics ,1st ed., University of Texas Press, 1980, chapter 13 pp. 457-459. [22]ref. [17], p. 605. [23]ref. [2], chapter 15, pp. 708-715. 4

THE LIGHT VELOCITY CASIMIR EFFECT Does the Velocity of Light in a Vacuum Increase When Propagating Between the Casimir Plates? Tom Ostoma and Mike Trushyk 48 O’HARA PLACE, Brampton, Ontario, L6Y 3R8 miket1@home.com Wednesday, November 24, 1999 ACKNOWLEGMENTS We wish to thank Paul Almond for originally pointing out this effect to us and for his review and comments on this work. We also want to thank Paul for the many interesting e-mail exchanges on the subject of space, time, light, matter, and CA theory. We thank R. Mongrain for many long discussions on the nature of quantum theory and space-time, and for his insight on the way photons propagate through the quantum vacuum.2ABSTRACT Our theory of quantum gravity called Electro-Magnetic Quantum Gravity (EMQG) depends heavily on an important property of the quantum vacuum; it’s ability to effect the velocity of photon propagation under two very special physical conditions. In the first case, photon propagation in the vacuum is altered when the electrically charged virtual particle density changes, as it does between the Casimir plates. In the second case, photon propagation in the vacuum is also altered when there is a coordinated acceleration given to the electrically charged virtual particles of the quantum vacuum, such as near a large mass like the earth (EMQG). These effects can be understood through the familiar process that photons partake when interacting with all electrically charged particles; ‘Photon Scattering’. Here we propose experiments that might be set up to detect the increase in the velocity of light in a vacuum in the laboratory frame for the first case, that is when photons travel between (and perpendicular to) the Casimir plates in vacuum. The Casimir plates are two closely spaced, conductive plates, where an attractive force is observed to exist between the plates called the ‘Casimir Force’. We propose that the velocity of light in a vacuum increases when propagating between the Casimir Plates, which are in a vacuum. We call this effect the ‘Light Velocity Casimir Effect’ or LVC effect. In the second case where light propagates upwards or downwards on the earth, the change in light velocity predicted by EMQG is associated with a corresponding curved 4D space-time in general relativity, where light velocity is taken as constant. We find that it is impossible to distinguish between these two conflicting views of light propagation in large gravitational fields by experimental means at this time. The LVC effect happens because the vacuum energy density in between the plates is lower than that outside the Casimir plates. The conductive plates disallow certain frequencies of electrically charged virtual particles to exist inside the plates, thus lowering the inside vacuum particle density, compared to the density outside the plates. The Casimir plates also disallow certain wavelengths of virtual photons as well, which is the basis for the calculation of the Casimir force first done by H.B. G. Casimir in 1948. The reduced (electrically charged) virtual particle density results in fewer photon scattering events inside the plates, which should increase the light velocity between the plates in a vacuum relative to the normal vacuum light speed (as measured with instruments in the laboratory frame). A similar effect, involving light velocity change, happens when light travels through two different real material densities; for example when light propagates from water to air, a process known as optical refraction. We also propose an experiment to demonstrate the Casimir refraction of light moving at a shallow angle that is nearly perpendicular to a series of unequally spaced Casimir plates, which cause a permanent shift in the direction of light propagation. Furthermore we propose a method to determine the index of refraction for light propagating from the ordinary vacuum to the less dense Casimir vacuum.3TABLE OF CONTENTS ABSTRACT ________________________________ ____________________________ 2 1. INTRODUCTION ________________________________ _____________________ 4 1.1 DEFINITION OF EINSTEIN CAUSALITY ________________________________ __5 1.2 INTRODUCTION TO THE QUANTUM VACUUM LIGHT SCATTERING ______ 5 2. THE VIRTUAL PARTICLES OF THE QUANTUM VACUUM ________________ 8 2.1 INTRODUCTION TO THE CASIMIR FORCE EFFECT ______________________ 9 2.2 EVIDENCE FOR THE EXISTENCE OF VIRTUAL PARTICLES (** Optional) __11 2.3 INTRODUCTION TO QUANTUM INERTIA THEORY (** Optional) __________ 12 3. LIGHT SCATTERING THEORY ________________________________ _______ 16 3.1 CLASSICAL SCATTERING OF PHOTONS IN REAL MATTER ______________ 16 3.2 QUANTUM FIELD THEORY OF PHOTON SCATTERING IN MATTER _____ 18 3.3 THE SCATTERING OF PHOTONS IN THE QUANTUM VACUUM ___________ 19 3.4 FIZEAU EFFECT: LIGHT VELOCITY IN A MOVING MEDIA (** Optional) ___21 3.5 LORENTZ SEMI-CLASSICAL PHOTON SCATTERING (** Optional) ________ 22 3.6 PHOTON SCATTERING IN THE ACCELERATED VACUUM (** Optional) ___23 4. NON-LOCALITY AND SUPERLUMINAL PHOTONIC TUNNELING ________ 24 5. THE PROPOSED CASIMIR LIGHT VELOCITY EXPERIMENTS ___________ 26 6. CONCLUSIONS ________________________________ _____________________ 29 7. REFERENCES ________________________________ ______________________ 30 8. FIGURE CAPTIONS ________________________________ _________________ 31 APPENDIX A: BRIEF REVIEW OF EMQG ________________________________ 3441. INTRODUCTION “The relativistic treatment of gravitation creates serious difficulties. I consider it probable that the principle of the constancy of the velocity of light in its customary version holds only for spaces with constant gravitational potential.” - Albert Einstein (in a letter to his friend Laub, August 10, 1911) The subject of this work might seem like scientific heresy to the reader. No doubt many of you will instantly reject this proposal on the grounds that it violates special relativity. Einstein’s 1905 prediction that the speed of light in a vacuum is an absolute constant for all inertial observers has now been well established theoretically and experimentally, although there may still be a few small cracks in the armor as witnessed by quantum non- locality and quantum tunneling effects (section 4). Special relativity has stood the test of time for 95 years without any evidence to the contrary, and has thus become a corner stone of modern theoretical physics. It would seem to be a career ending move for anyone to propose that the light velocity in a vacuum is anything but the accepted fixed value! Yet we nevertheless propose an experiment which we call the ‘Light Velocity Casimir’ experiment or the ‘LVC’ experiment, to prove that the velocity of light (front velocity) propagating in vacuum inside and perpendicular to two closely spaced, electrically conducting and transparent plates called the Casimir plates, will actually increase inside the plates compared to the light velocity in the normal vacuum as measured in the laboratory frame (figure 1). The Casimir plates are a pair of closely spaced, conductive plates, where an attractive force is observed to exist between the plates called the ‘Casimir Force’. H.B.G. Casimir theoretically predicted the existence of this force in 1948 (ref. 6). Recently the Casimir force has been verified experimentally for a plate spacing of about 1 micrometer, with an accuracy of about 5% (1996, ref. 7), by using an electromagnetic- based torsion pendulum. More recently U. Mohideen and A. Roy have made an even more precise measurement of the Casimir force in the 0.1 to 0.9 micrometer plate spacing to an accuracy of about 1% (1998, ref. 8) using the techniques of atomic force microscopy. We believe that there must exist a ‘Vacuum Casimir Index of Refraction’ called ‘n vac’ for light traveling from outside, and then through the Casimir plates. The Casimir vacuum index of refraction is defined as the ratio of the velocity of light in normal vacuum conditions (‘c‘ or 299,792.458 km/sec) divided by the light velocity measured perpendicular to the Casimir plates in vacuum (‘c c‘ or slightly greater than 299,792.458 km/sec). The vacuum Casimir index of refraction ‘n vac’ is given by: n vac = c / c c , which is slightly less than one for the Light Velocity Casimir (LVC) experiment. The process is comparable to the familiar index of refraction for light propagating from water to air, where the light velocity in air is greater than the light velocity in water. Figure #2 shows a possible experiment to observe the Casimir vacuum index of refraction by witnessing the deflection of light on a shallow angle propagating through a series of unequally spaced, Casimir plates (described in section 5).5Furthermore we have theoretical reason to believe that the index of refraction n vac will vary with the Casimir plate spacing ‘d’, just as the Casimir force varies with plate spacing. In the Casimir force effect the force varies as the inverse fourth power of the plate spacing. The light velocity dependence on plate spacing is unknown at this time. Before we go into details on the LVC experiment we explain precisely what we mean by the increase in light velocity in the vacuum. 1.1 DEFINITION OF EINSTEIN CAUSALITY Often in the literature we find statements made regarding Einstein causality, and that nothing propagates faster than the speed of light in the vacuum. Since light consists of photons, which have both particle and wave properties, care must be taken to ensure that velocity of light is properly defined. This is especially important when taking into account the quantum wave packet properties of photons. In L. Brillouin classic book titled ‘Wave Propagation and Group Velocity, 1960’, he identified five different definitions for the velocity of a finite-bandwidth pulse of electromagnetic radiation. We will be concerned only with the ‘front velocity’ of light, which is defined below. The proper definition for light velocity that we use here is the ‘front velocity’, which was given by Sommerfield and L. Brillouin and (ref. 33). Suppose that there is a light source at point x=0, which is switched on at the time t=0. Some distance ‘d’ away from the source at x, no effect can be detected that is coming from point x before the time ‘d/c’. The beginning of the signal is a discontinuity in the signal envelope (or in a higher derivative). The beginning of the signal is known as the front velocity and this may not exceed the velocity of light in a vacuum in order to fulfill the Einstein causality condition. We maintain that the front velocity for a beam of light traveling through the Casimir plates will exceed the velocity of light in the ordinary vacuum during transit through the Casimir plates. 1.2 INTRODUCTION TO THE QUANTU M VACUUM LIGHT SCATTERING How did we come to such a drastic conclusion regarding the increased velocity of light, in spite of the heavy body of scientific evidence to the contrary? This conclusion is partly based on our work on a new quantum theory of gravity called Electro-Magnetic Quantum Gravity (EMQG) ref. 1, which depends heavily on an important characteristic of the quantum vacuum; it’s ability to affect the velocity of photon propagation under certain special conditions through the familiar process called ‘Photon Scattering’. EMQG requires a photon scattering process, photon scattering in the accelerated quantum vacuum near the earth, in order to understand gravity and the Newtonian equivalence of inertial mass and gravitational mass on a quantum scale. According to EMQG the light velocity in a region where the virtual particle vacuum density is lower than in the normal vacuum, is greater than the light velocity in the normal vacuum. This occurs for an6observer in the laboratory frame using his clocks and rulers to measure the front velocity of light. According to the general principles of EMQG theory, the LVC effect occurs because: 1. The fundamental virtual matter particles (fermions) that make up the quantum vacuum (not the ZPF or virtual photons as in the Casimir force effect) are ultimately electrically charged at the lowest level, just as we believe for real ordinary matter. 2. Since at the lowest level the virtual particles of the vacuum are electrically charged, they will interact with light (photon particles) propagating through the vacuum. 3. According to quantum theory when a photon interacts with an electrically charged virtual particle, the propagation is delayed at each electrically charged virtual particle of the quantum vacuum, before the photon continues propagating. Why is there a photon delay ? There is a time delay during the absorption and subsequent re-emission of the photon by a given charged virtual particle. The uncertainty principle places a lower limit on this time delay, and forbids it from being zero. In other words, according to the uncertainty principle the time delay due to the absorption and re- emission time of the photon cannot be exactly equal to zero. 4. The time delay caused by the absorption and subsequent re-emission of the photon by a given electrically charged virtual particle results in a lower TOTAL AVERAGE velocity for the propagation of the photons on the macroscopic distance scale, as compared to the average velocity of the photon without the presence of any constraining Casimir plates. 5. In the case of the vacuum between the Casimir plates, the virtual fermion particle density in the vacuum is lower between the plates compared to that outside the plates. The Casimir plates prohibit certain wavelengths of the fermion wave function from existing (as it does for certain photon wavelengths in the Casimir force calculation). 6. Therefore, the TOTAL AVERAGE light velocity inside the plates must be greater than that outside the plates. Our review of the physics literature has not revealed any previous work on the time delay analysis of photon propagation through the ordinary quantum vacuum or any evidence to contradict our hypothesis of photon vacuum delay, presumably because of the precedent set by Einstein’s postulate of light speed constancy. Note: Suppose we can place a tiny observer A and his clocks and rulers (somehow) in between the Casimir plates. We would find for observer A that his measurement of the speed of light is the same as the conventional value! However for an observer B outside the plates in the laboratory frame, his measurement does show an increase in light velocity through the plates. Furthermore, the individual space and time measurements ‘dlab’ and ‘t lab’ made by observer B in the laboratory frame do not agree with the same measurements of ‘ dplate ’ and ‘t plate ’ made by our tiny observer A inside the Casimir plates. The general relativists could argue that the 4D space-time inside the Casimir plates is altered compared to outside, and that the light velocity is still an absolute constant in all cases! This argument results because of the crucial importance of light propagation to7the fundamental nature of space and time measurements, a theme that was first championed by Einstein. This same controversy rears it’s ugly head in gravity where we are forced to choose between the two experimentally indistinguishable views; 4D curved space-time in general relativity and variable light velocity proposal of EMQG. It turns out to be impossible to distinguish between curved 4D space-time in gravitational frames, and variations in light velocity in gravitational frames experimentally. More will be said on this important point in appendix A (section A10 and ref. 1). Another result from EMQG states that the velocity of light without the existence of any of the virtual particles of the quantum vacuum ought to be much greater than the observed average light velocity in the vacuum. The electrically charged virtual fermion particles of the quantum vacuum frequently scatter photons, which introduce many tiny delays for the photon propagation. This causes a great reduction in the total average light velocity in the vacuum that is populated by countless numbers of virtual, electrically charged, particles. In other words; the low level light velocity (between virtual particle scattering events) is much greater than the measured average light velocity after vacuum scattering in the normal vacuum . A similar effect is known to occur when light propagates through glass, where photons scatter with electrons in the glass molecules, which subsequently reduces the average light velocity through glass compared to the normal vacuum light velocity. Here we propose an experiment that might be able to detect this predicted increase in the velocity of light between the Casimir plates in vacuum. The experiment is designed to compare the velocity of light in the ordinary vacuum against the light velocity that is propagating in between the Casimir plates, also in a vacuum state (figure 1). A light source is split into two parallel paths by a beam splitter; one path is the reference vacuum path of light, and in the other path the light is allowed to propagate in vacuum perpendicular to two closely spaced, electrically conductive and transparent plates. The two paths are then recombined by a beam splitter, and routed to an interferometer. After the apparatus is calibrated (and if our prediction is correct), the phase of the interference pattern will indicate that light travels faster in the Casimir plate leg of the interferometer. Furthermore, it might be possible to measure the index of refraction for light traveling from the normal vacuum to the Casimir vacuum. As we have said the light velocity increases between the Casimir plates because the vacuum energy-density in between the plates is lower than that outside the plates. This difference in vacuum energy-density or ‘pressure’ is actually the cause of the attractive force between the plates, where the energy-density is greater outside. This implies that it might be possible to extract a virtually unlimited supply of energy the quantum vacuum, which is an active area of recent research (ref. 11). Before we go into detail on the proposed LVC experiments, we present a brief introduction to the virtual particles of the quantum vacuum that is crucial to the understanding of the LVC effect.82. THE VIRTUAL PARTICLES OF THE QUANTUM VACUUM Philosophers: “Nature abhors a vacuum.” In order to make a complete vacuum, one must remove all matter from an enclosure. However one would find that this is still not good enough. One must also lower the temperature inside the closure to absolute zero in order to remove all thermal electromagnetic radiation. Nernst correctly deduced in 1916 (ref. 32) that empty space is still not completely devoid of all radiation after this is done. He predicted that the vacuum is still permanently filled with an electromagnetic field propagating at the speed of light, called the zero-point fluctuations (or sometimes called by the generic name ‘vacuum fluctuations’). This result was later confirmed theoretically by the newly developed quantum field theory that was developed in the 1920’s and 30’s. Later with the development of QED (the quantum theory of electrons and photons), it was realized that all quantum fields should contribute to the vacuum state. This means that virtual electrons and positron particles should not be excluded from consideration. These particles possess mass and have multiples of half integer spin (such as the electron), and therefore belong to the generic class of particles known as fermions. We refer to virtual electrons and virtual anti-electrons (positron) particles as virtual fermions. We believe that ultimately all fermions can be broken down to a fundamental entity that is also electrically charged, as well as having half integer spin and mass (technically, mass-charge as described in Appendix A). According to modern quantum field theory, the perfect vacuum is teeming with activity as all types of quantum virtual particles (and virtual bosons or the force carrying particles) from the various quantum fields appear and disappear spontaneously. These particles are called ‘virtual’ particles because they result from quantum processes that generally have short lifetimes, and are mostly undetectable. One way to look at the existence of the quantum vacuum is to consider that quantum theory forbids the complete absence of propagating fields. This is in accordance with the famous Heisenberg uncertainty principle. In general, it is known that all the possible real particles types (for example electrons, quarks, etc.) will also be present in the quantum vacuum in their virtual particle form. In the QED vacuum, the quantum fermion vacuum is produced from the virtual particle pair creation and annihilation processes that create enormous numbers of virtual electron and virtual positron pairs. We also have in QED the creation of the zero-point-fluctuation (ZPF) of the vacuum consisting of the electromagnetic field or virtual photon particles. Indeed in the standard model, we also find in the vacuum every possible boson particle, such as the gluons, gravitons, etc., and also every possible fermion particle, such as virtual quarks, virtual neutrinos, etc.92.1 INTRODUCTION TO THE CASIMIR FORCE EFFECT The existence of virtual particles of the quantum vacuum reveals itself in the famous Casimir effect (ref. 6), which is an effect predicted theoretically by the Dutch scientist Hendrik Casimir in 1948. The Casimir effect refers to the tiny attractive force that occurs between two neutral metal plates suspended in a vacuum. He predicted theoretically that the force ‘F’ per unit area ‘A’ for plate separation D is given by: F/A = - π2 h c /(240 D4 ) Newton’s per square meter (Casimir Force ‘F’) (2.1) Casimir obtained this formula by calculating the sum of the quantum-mechanical zero- point energies of the normal modes of the electromagnetic field (virtual photons) between two conductive plates. The origin of this minute force can be traced to the disruption of the normal quantum vacuum virtual photon distribution between two nearby metallic plates as compared to the vacuum state outside the plates. Certain virtual photon wavelengths (and therefore energies) are forbidden to exist between the plates, because these waves do not ‘fit’ between the two plates (which are both at a relative classical electrical potential of zero). This creates a negative pressure due to the unequal energy distribution of virtual particles inside the plates as compared to those outside the plate region. The pressure imbalance can be visualized as causing the two plates to be drawn together by radiation pressure. Note: Even in the vacuum state, the virtual photon particles do carry energy and momentum while they exist. Although the Casimir effect has been attributed to the zero-point fluctuations (ZPF) in the EM field inside the plates, Schwinger showed in the late 70’s that the Casimir effect can also be derived in terms of his source theory (ref. 13), which has no explicit reference to the ZPF of the EM field between the plates. Recently Milonni and Shih have developed a theory of the Casimir force effect, which is totally within the framework of conventional QED (ref. 15). Therefore it seems that it is only a matter of taste whether we attribute the Casimir force effect to the ZPF fields or to the matter fields in vacuum (ref. 23). Recently Lamoreaux made accurate experimental measurements for the first time of the Casimir force existing between two gold-coated quartz surfaces that were spaced on the order of a micrometer apart (ref. 7). Lamoreaux found a force value of about 1 billionth of a Newton, agreeing with the Casimir theory to within an accuracy of about 5%. More recently, U. Mohideen and A. Roy have made an even more precise measurement in the 0.1 to 0.9 micrometer plate spacing to an accuracy of about 1% (1998, ref. 8). Therefore the experimental reality of this effect is beyond question. Can the vacuum state be disrupted by other physical processes besides the Casimir plates? One might ask what happens to the virtual particles of the quantum vacuum that are subjected to a large gravitational field like the earth? Since the quantum vacuum is composed of virtual fermions (as well as virtual bosons), the conclusion is inescapable: all10the virtual fermions possessing mass must be falling (accelerating) on the average towards the earth during their very brief lifetimes . This vacuum state is definitely different from the vacuum of far outer space away from gravitational fields. Yet to our knowledge, no previous authors have acknowledged the existence of this effect, or studied the physical consequences that result from this. It turns out that the free fall state of the virtual, electrically charged fermion particles of the vacuum is actually the root cause of 4D space-time curvature and also leads to a full understanding of the principle of equivalence. In EMQG (appendix A) we fully study the consequences of a falling quantum vacuum in quantum gravity, which does lead to new experimentally testable predictions. The physics of the Casimir force effect implies that the quantum vacuum contains an enormous reservoir of energy (ref. 11). Although in standard quantum field theory the number density of virtual particles is unlimited, some estimates place a high frequency cut- off at the plank scale which is estimated to be a density of 1090 particles per cubic meter (ref. 11)! Generally this energy-density is not available because the energy-density is uniform and it permeates everything. It’s like the situation in the deep ocean, where deep sea fishes easily tolerate the extreme pressures in the abyss, because the pressure inside and outside the fish’s body balance. If a human goes into these depths, a great difference in pressure must be maintained to support atmospheric pressure inside the human body. Some physicists are looking at ways in which this vast energy reservoir can be tapped (ref. 11) If the vacuum is capable of exerting a mechanical force between the two Casimir plates, might the vacuum’s effect be felt in a less exotic way? Most physicists believe that the answer is no. Yet there is a small number of physicist who believe otherwise. In 1994, R. Haisch, A. Rueda, and H. Puthoff (ref. 5) were the first to propose a theory of inertia (known here as HRP Inertia), where the quantum vacuum played a central role in Newtonian inertia. They suggested that inertia is due to the strictly local electrical force interactions of charged matter particles with the immediate background virtual particles of the quantum vacuum. We have built on their work and developed a theory of quantum gravity and quantum inertia based on their idea. According to EMQG, the quantum vacuum affects all masses that are in the state of acceleration. In the EMQG model, the force of inertia is actually caused by the resistance force to acceleration by the electrical force interactions between charged particles that make up a mass and the electrically charged virtual particles of the quantum vacuum. We call this quantum inertia, which plays a central role in our quantum theory of gravity that closely links inertia and gravity. We introduce this important concept in section 2.3 (this section can be omitted if desired, since it is not essential in order to understand the LVC experiments).112.2 EVIDENCE FOR THE EXISTENCE OF VIRTUAL PARTICLES (** Optional) There is other evidence for the existence of virtual particles besides the Casimir force effect. We present a very brief review of some theoretical and experimental evidence for the existence of the virtual particles of the quantum vacuum: (1) The extreme precision in the theoretical calculations of the hyper-fine structure of the energy levels of the hydrogen atom, and the anomalous magnetic moment of the electron and muon are both based on the existence of virtual particles in the framework of QED. These effects have been calculated in QED to a very high precision (approximately 10 decimal places), and these values have also been verified experimentally to an unprecedented accuracy. This indeed is a great achievement for QED, which is essentially a perturbation theory of the electromagnetic quantum vacuum. Indeed, this is one of physics greatest achievements. (2) Recently, vacuum polarization (the polarization of electron-positron pairs near a real electron particle) has been observed experimentally by a team of physicists led by David Koltick. Vacuum polarization causes a cloud of virtual particles to form around the electron in such a way as to produce an electron charge screening effect. This is because virtual positrons tend to migrate towards the real electron, and the virtual electrons tend to migrate away. A team of physicists fired high-energy particles at electrons, and found that the effect of this cloud of virtual particles was reduced the closer a particle penetrated towards the electron. They reported that the effect of the higher charge for the penetration of the electron cloud with energetic 58 giga-electron volt particles was equivalent to a fine structure constant of 1/129.6. This agreed well with their theoretical prediction of 128.5 of QED. This can be taken as verification of the vacuum polarization effect predicted by QED, and further evidence for the existence of the quantum vacuum. (3) The quantum vacuum explains why cooling alone will never freeze liquid helium. Unless pressure is applied, vacuum energy fluctuations prevent its atoms from getting close enough to trigger solidification. (4) For fluorescent strip lamps, the random energy fluctuations of the virtual particles of the quantum vacuum cause the atoms of mercury, which are in their exited state, to spontaneously emit photons by eventually knocking them out of their unstable energy orbital. In this way, spontaneous emission in an atom can be viewed as being directly caused by the state of the surrounding quantum vacuum. (5) In electronics, there is a limit as to how much a radio signal can be amplified. Random noise signals are always added to the original signal. This is due to the presence of the virtual particles of the quantum vacuum as the real radio photons from the transmitter propagate in space. The vacuum fluctuations add a random noise pattern to the signal by slightly modifying the energy of the propagating radio photons.12(6) Recent theoretical and experimental work done in the field of Cavity Quantum Electrodynamics suggests that the orbital electron transition time for excited atoms can be affected by the state of the virtual particles of the quantum vacuum immediately surrounding the excited atom in a cavity, where the size of the cavity modifies the spectrum of the virtual particles. In the weight of all this evidence, only a few physicists doubt the existence of the virtual particles of the quantum vacuum. Yet to us, it seems strange that the quantum vacuum should barely reveal it’s presence to us, and that we only know about it’s existence through rather obscure physical effects like the Casimir force effect and Davies-Unruh effect. This is especially odd considering that the observable particles of ordinary real matter in an average cubic meter of space in the universe constitute a minute fraction of the total population of virtual particles of the quantum vacuum at any given instant of time. Some estimates of the quantum vacuum particle density (ref. 11) place the vacuum particle numbers at about 1090 particles per cubic meter! Instead, we believe that the quantum vacuum plays a much more prominent role in physics. We maintain that the effects of the quantum vacuum are present in virtually all physical activity. In fact, Newton’s three laws of motion can be understood to originate directly from the effects of the quantum vacuum (Appendix A). Furthermore, the quantum vacuum plays an extremely important role in gravity, which is generally well understood by the physics community. In order not to distract the reader from the main theme of this paper, we have included a brief review of EMQG theory which summarizes the central role that light scattering in the accelerated quantum vacuum has in our quantum gravity theory; and for the principle of equivalence, inertia, and 4D space-time curvature. This can be found in Appendix A of this paper. A full account is given in reference 1. We provide an optional introduction to Quantum Inertia in the next section for those readers who are interested. 2.3 INTRODUCTION TO QUANTUM INERTIA THEORY (** Optional) Recently it has been proposed that Newtonian Inertia is strictly a quantum vacuum phenomenon! If this is true, then the existence of the quantum vacuum actually reveals it’s presence to us in all daily activities! Unlike the hard-to-measure Casimir force effect, the presence of the inertial force is universal and it’s presence prevails throughout all of physics. For example, the orbital motion of the earth around the sun is a balancing act between inertia force and gravitational force. If quantum inertia is true, every time you accelerate, you are witnessing a quantum vacuum effect! This is a far cry from an exotic and almost impossible measurement of the feeble Casimir force between two plates. In 1994, R. Haisch, A. Rueda, and H. Puthoff (ref. 5) were the first to propose a theory of inertia (known here as HRP Inertia), where the quantum vacuum played a central role in Newtonian inertia. They suggested that inertia is due to the strictly local electrical force13interactions of charged matter particles with the immediate background virtual particles of the quantum vacuum (in particular the virtual photons or ZPF as the authors called it). They found that inertia is caused by the magnetic component of the Lorentz force, which arises between what the author’s call the charged ‘parton’ particles in an accelerated reference frame interacting with the background quantum vacuum virtual particles. The sum of all these tiny forces in this process is the source of the resistance force opposing accelerated motion in Newton’s F=MA. The ‘parton’ is a term that Richard Feynman coined for the constituents of the nuclear particles such as the proton and neutron (now called quarks). We have found it necessary to make a small modification to HRP Inertia theory as a result of our investigation of the principle of equivalence. Our modified version of HRP inertia is called “Quantum Inertia” (or QI), and is described in detail in Appendix A. This theory also resolves the long outstanding problems and paradoxes of accelerated motion introduced by Mach’s principle, by suggesting that the vacuum particles themselves serve as Mach’s universal reference frame (for acceleration only), without violating the principle of relativity of constant velocity motion. In other words our universe offers no observable reference frame to gauge inertial frames (non-accelerated frames where Newton’s laws of inertia are valid), because the quantum vacuum offers no means to determine absolute constant velocity motion. However for accelerated motion, the quantum vacuum plays a very important role by offering a resistance to acceleration, which results in an inertial force opposing the acceleration of the mass. Thus the very existence of inertial force reveals the absolute value of the acceleration with respect to the net statistical average acceleration of the virtual particles of the quantum vacuum. If this is correct then Newton’s three famous laws of motion can be understood at the quantum level (ref. 20). There have been various clues to the importance the virtual particles of the quantum vacuum for the accelerated motion of real charged particles. One example is the so-called Davies-Unruh effect (ref. 18), where uniform and linearly accelerated charged particles in the vacuum are immersed in a heat bath, with a temperature proportional to acceleration (with the scale of the quantum heat effects being very low). However, the work of reference 5 is the first place we have clearly seen the identification of inertial forces as the direct consequence of the interactions of real matter particles with the quantum vacuum. It has also even been suggested that the virtual particles of the quantum vacuum are somehow involved in gravitational interactions. The prominent Russian physicist A. Sakharov proposed in 1968 (ref. 16) that Newtonian gravity could be interpreted as a van der Waals type of force induced by the electromagnetic fluctuations of the virtual particles of the quantum vacuum. Sakharov visualized ordinary neutral matter as a collection of electromagnetically, interacting polarizable particles made of charged point-mass sub- particles (partons). He associated the Newtonian gravitational field with the Van Der Waals force present in neutral matter, where the long-range radiation fields are generated by the parton ‘Zitterbewegung’. Sakharov went on to develop what he called the ‘metric elasticity’ concept, where space-time is somehow identified with the ‘hydrodynamic elasticity’ of the vacuum. However, he did not understand the important clues about the14quantum vacuum that are revealed by the equivalence principle, nor the role that the quantum vacuum played in inertia and Mach’s principle. We maintain that the quantum vacuum also make it’s presence felt in a very big way in all gravitational interactions (Appendix A) just as it does in inertia! There have been further hints that the quantum vacuum is involved in gravitational physics. In 1974 Hawkings (ref. 17) announced that black holes are not completely black. Black holes emit an outgoing thermal flux of radiation due to gravitational interactions of the black hole with the virtual particle pairs created in the quantum vacuum near the event horizon. At first sight the emission of thermal radiation from a black hole seems paradoxical, since nothing can escape from the event horizon. However the spontaneous creation of virtual particle and anti-particle pairs in the quantum vacuum near the event horizon can be used to explain this effect (ref. 18). Heuristically one can imagine that the virtual particle pairs (that are created with wavelength λ that is approximately equal to the size of the black hole) ‘tunnel’ out of the event horizon. For a virtual particle with a wavelength comparable to the size of the hole, strong tidal forces operate to prevent re- annihilation. One virtual particle escapes to infinity with positive energy to contribute to the Hawking radiation, while the corresponding antiparticle enters the black hole to be trapped forever by the deep gravitational potential. Thus the quantum vacuum is important in order to properly understand the Hawking radiation. As a result of all these and other considerations, we have developed a new approach to the unification of quantum theory with general relativity referred to as Electro-Magnetic Quantum Gravity or EMQG (ref. 1 and summary in appendix A). EMQG had its early origins in Cellular Automata (CA) theory (ref. 2,4,9 and 34), and on a theory of inertia proposed by R. Haisch, A. Rueda, and H. Puthoff (ref. 5). In EMQG, the quantum vacuum plays an extremely important, if not a central role, in both inertia and gravitation. It also plays a major role in the origin of 4D curved space-time curvature near gravitational sources. We maintain that anybody who believes in the existence of the virtual particles of the quantum vacuum and accepts the fact that many virtual particles carry mass (virtual fermions), will have no trouble in believing that the virtual particles of the vacuum are falling in the presence of a large gravitational mass like the earth during their brief lifetimes. We believe the existence of the downward accelerating virtual particles, under the action of a large gravitational field, turns out to be the missing link between inertia and gravity. It leads us directly to a full understanding of the principle of equivalence. Although the quantum vacuum has been studied in much detail in the past, to our knowledge no one has examined the direct consequences of a quantum vacuum in a state of free-fall near the earth. This concept is the central theme behind EMQG. Reference 14 offers an excellent introduction to the motion of matter in the presence of the quantum vacuum, and on the history of the discovery of the virtual particles of the quantum vacuum.15We propose that the virtual particles of the quantum vacuum can be viewed as kind of a transparent fluid medium, sort of like a kind of a 21th century “ether”. Unlike ordinary transparent fluids like water, the vacuum does not resist constant velocity motion. However the virtual particles of the quantum vacuum can be made to take on a coordinated (average) accelerated motion with respect to an observer in two different physical instances, and this has very important consequences for mass particles in the following two cases: (1) The quantum vacuum looks disturbed from the perspective of a mass that is being accelerated (by a rocket for example). Here the observer and his mass are the physical entities that are actually accelerating, and the quantum vacuum only appears to be accelerated in the reference frame of the observer. Here the vacuum acceleration is not actually real ! However, the quantum vacuum effects are very real, and are the root cause of inertial force. (2) The vacuum actually is disturbed by the presence of a near-by gravitational field of a large mass, like the earth. In this case, the coordinated vacuum particle (net average) acceleration with respect to the earth’s surface is real , and is caused by direct graviton exchanges between the earth and the individual virtual fermion particles of the quantum vacuum. Therefore, the vacuum fluid can be viewed as falling just as the Niagara Water Falls. However the vacuum fluid does not accumulate at the earth’s surface as a real liquid water fluid might, because the virtual particles are short lived and are constantly being replaced by new ones. These considerations imply that Newton’s principle of equivalence of gravitational mass and inertial mass can be understood to be caused by the virtual particles of the quantum vacuum. The inertial mass ‘m’ is defined in the formula F = ma, and has the same magnitude as the gravitational mass ‘m’ defined in F = GmM e/r2, which are two independent definitions for the same mass value. Newton equivalence results because: The quantum vacuum looks the same from the perspective of an accelerated mass ‘m’ on the floor of a rocket accelerated at 1g, as it does from the perspective of a stationary mass ‘m’ on the earth! In order to see how the Newtonian Equivalence principle of inertia and gravitational mass follows from the accelerated quantum vacuum effects, we only have to recall our Quantum Inertia principle: The cause of inertia is the electrical resistance force that appears between the electrically charged, real matter particles that constitute a mass, and the surrounding electrically charged, virtual fermions of the quantum vacuum, where there exists a state of relative acceleration between the real and virtual particle species. In other words, an accelerated mass feels the inertial force from the sum of the tiny electrical forces that originate from each electrically charged particle that make up a mass.16Similarly the gravitational mass of the same object, stationary on the earth’s surface, also feels the exact same sum of the tiny electrical forces that originate from each electrically charged particle that make up a mass, where now it’s the virtual particles of the quantum vacuum that accelerates downwards . What is the cause of the vacuum particle acceleration on the earth? According to EMQG, which is a quantum field theory of gravity, it is the graviton exchanges between the fermion particles of the earth and the virtual fermion particles of the quantum vacuum. These ideas are fully elaborated in EMQG theory in Appendix A (attached). We now review the basic notions of photon scattering in real matter and in the quantum vacuum. 3. LIGHT SCATTERING THEORY Since photon scattering is essential to our model of the predicted LVC effect (and in EMQG theory), we will examine the general principles of photon scattering in some detail. First we review the conventional physics of light scattering in a real media such as water or glass, including the concept of the index of refraction and Snell’s Law of refraction. We also introduce photon scattering when the real media is moving at a constant velocity, where the velocity of light varies in the moving media and known as the Fizeau effect. Next we introduce an accelerated medium for the real medium and examine how the photons scatter. This is important to understanding EMQG theory. Readers that are only interested in the LVC affect can skip sections 3.3, 3.5 and 3.6. We generalize these arguments to examine photon scattering with the electrically charged virtual particles of the quantum vacuum. 3.1 CLASSICAL SCATTERING OF PHOTONS IN REAL MATTER It is a well-known result of classical optics that light moves slower in glass than in air. Furthermore it is recognized that the velocity of light in air is slower than that of light’s vacuum velocity. This effect is described by the index of refraction ‘n’, which is the ratio of light velocities in the two different media. The Feynman Lectures on Physics gives one of the best accounts of the classical theory for the origin of the refractive index and the slowing of light through a transparent material like glass (ref. 42, chap. 31 contains the mathematical details). When light passes from a vacuum into glass, with an incident angle of θ0 it deflects and changes it’s direction and moves at a new angle θ1 , where the angles follow Snell’s law: n = sin θ0 / sin θ1 (3.11) This follows geometrically because the wave crests on both sides of the surface of the glass must have the same spacing, since they must travel together (ref. 42). The shortest distance between crests of the wave is the wavelength divided by the frequency. On the vacuum side of the glass surface it is λ0 = 2πc/ω, and on the other side it is given by λ =172πv/ω or 2πc / ωn since we define v=c/n. If we accept this, then Snell’s law follows geometrically (ref. 42). In some sense, the existence of the index of refraction in Snell’s law is confirmation of the change in light speed going from the vacuum to glass. Snell’s law does not tell us why we have a change in light velocity, nor does it give us any insight into the phenomena of dispersion and back scattering of light in refraction. A good classical account of the derivation of the index of refraction is given by Feynman himself in ref. 42. Feynman derives the index of refraction for a transparent medium by accepting that the total electric field in any physical circumstance can be represented by the sum of the fields from all charge sources, and by accepting that the field from a single charge is given by it’s acceleration evaluated with a retardation speed ‘c’ (the propagation speed of the exchanged photons). We only summarize the important points of his argument below, and the full details are available in reference 42: (1) The incoming source electromagnetic wave (light) consists of an oscillating electric and magnetic field. The glass consists of electrons bound elastically to the atoms, such that if a force is applied to an electron the displacement from its normal position will be proportional to the force. (2) The oscillating electric field of the light causes the electron to be driven in an oscillating motion, thus acting like a new radiator generating a new electromagnetic wave. This new wave is always delayed, or retarded in phase. These delays result from the time delay required for the bound electron to oscillate to full amplitude. Recall that the electron carries mass and therefore inertia. Therefore some time is required to move the electron. (3) The total resulting electromagnetic wave is the sum of the source electromagnetic wave plus the new phase-delayed electromagnetic wave, where the total resulting wave is phase-shifted. (4) The resulting phase delay of the electromagnetic wave is the root cause of the reduced velocity of light observed in the medium. Feynman goes on to derive the classic formula for the index of refraction for atoms with several different resonant frequency ωk which is given by: n = 1 + [q e2 / (2e0m)] Σk Nk / [ ωk2 - ω2 + iγkω] (3.12) where n is the index of refraction, q e is the electron charge, m is the electron mass, ω is the incoming light frequency, γk is the damping factor, and N k is the number of atoms per unit volume. This formula describes the index of refraction for many substances, and also describes the dispersion of light through the medium. Dispersion is the phenomenon where the index of refraction of a media varies with the frequency of the incoming light, and is the reason that a glass prism bends light more in the blue end than the red end of the spectrum.18If the medium consists of free, unbound electrons in the form of a gas such as in a plasma (or as the conduction electrons in a simple metal) then the index of refraction with the conditions γk << ω and ωk = 0 is given by (ref. 42): n ≈ 1 - [N k qe2 / (2e0m)] / ω2≈ {1 - [N k qe2 / (e0m)] / ω2 }1/2 (3.13) where we recall that (1-x)1/2 ≈ 1 - x/2 if x is much lees than 1. The quantity Ω = [ N k qe2 / (e0m) ] 1/2 is sometimes called the Plasma frequency Ω, where there is a transition to the transparent state at Ω = ω. 3.2 QUANTUM FIELD THEORY OF PHOTON SCATTERING IN MATTER Although the classical account of scattering predicts the experimentally confirmed results, the correct account must be a quantum mechanical account. To quote R. Feynman himself: “ … yes, but the world is quantum not classical dam-it ”. The propagation of light through a transparent medium is a very difficult subject in QED. It is impossible to compute the interaction of a collection of atoms with light exactly. In fact, it is impossible to treat even one atom’s interaction with light exactly in QED. However the interaction of a real atom with photons can be approximated by a simpler quantum system. Since in many cases only two atomic energy levels play a significant role in the interaction of the electromagnetic field with atoms, the atom can be represented by a quantum system with only two energy eigenstates. In the text book “Optical Coherence and Quantum Optics” a thorough treatment of the absorption and emission of photons in two-level atoms is given (ref. 43, Chap. 15, pg. 762). When a photon is absorbed, and later a new photon of the same frequency is re- emitted by an electron bound to an atom, there exists a time delay before the photon re- emission. The probabilities for emission and absorption of a photon is given as a function of time Δt for an atom frequency of ω0 and photon frequency of ω1 : Probability of Photon Absorption is: K [ sin (0.5( ω1 - ω0) Δt) / ( 0.5( ω1 - ω0)) ]2 Probability of Photon Emission is: M [ sin (0.5( ω1 - ω0) Δt) / ( 0.5( ω1 - ω0)) ]2 (3.21) (where K and M are complex expressions defined in ref. 43) The important point we want to make from eq. 3.21 is that the probability of absorption or emission depends on the length of time Δt, where the probability of the emission is zero, if the time Δt = 0. In other words according to QED, a finite time is required before re- emission of the photon. There are other factors that affect the probability, of course. For example, the closer the frequency of the photon matches the atomic frequency, the higher the probability of re-emission in some given time period. We maintain that these delays are the actual route cause of the index of refraction in a medium.19We believe that a similar thing happens when photons propagate through the quantum vacuum. Therefore, we want to address the effect of the virtual particles of the quantum vacuum on the propagation velocity of real (non-virtual) photons, a subject that is largely ignored in the physics literature. 3.3 THE SCATTERING OF PHOTONS IN THE QUANTUM VACUUM In section 3.1 we discussed photon scattering in a real matter medium and in a real negatively charged electron gas. The electron gas model is the closest model we have towards understanding the Casimir index of refraction of the quantum vacuum. However, there are several important differences between the charged electron gas medium and the electrically charged virtual fermion particles of the quantum vacuum as a medium. First, and most importantly, virtual particles do not carry any net average energy. Instead an individual virtual particle ‘borrows’ a small amount of energy during it’s brief existence, which is then paid back quickly in accordance to the uncertainty principle. It is because quantum mechanics forbids knowing the value of two complementary variables precisely (in this case energy and existence time) for a virtual particle that virtual particles are allowed to exist at all. Therefore unlike the electron gas, the vacuum is incapable of permanently absorbing light that propagates through it. Thus the quantum vacuum does not absorb any light over macroscopic distance scales. This statement seems trivial, but it is never-the-less important when considering the quantum vacuum as a medium. On microscopic scales real photons are absorbed and re- emitted by individual virtual particles, in accordance with QED. Photon energy is lost in some collisions and regained in others so that on the average the energy loss is zero. This is because during the brief existence time of a virtual fermion particle, the virtual particle does possess energy, which is paid back almost immediately. This quantum process happens an enormous number of times as light travels through macroscopic distance scales. The energy balances out to zero over sufficiently large distance scales. Furthermore unlike the electron gas, there can be no dispersion of light in the quantum vacuum. In other words all frequencies of electromagnetic radiation move at the same speed through the quantum vacuum in spite of the incredible numbers of virtual particle interactions that occur for any particular frequency of photon. Zero dispersion follows experimentally from many astronomical observations of distant supernova, where there is a dramatic change in light and electromagnetic radiation with time. Observations have been made of specific events in the light curves of supernovae light curves that range from the radio band frequencies to the X-ray / Gamma Ray frequencies. All the different frequencies are observed to arrive on the earth at the same time. With distances of thousands or millions of light years away, any discrepancy in the photon velocity of supernovae at different frequencies would be very apparent. For example with20the relatively nearby supernova 1987A (which exploded about 160,000 years ago in the Large Magellanic cloud) all the different frequencies of EM waves has reached us very much at the same time. If there had been a dispersion of only 0.01 m/sec in light velocity (i.e. 3 parts in 10-11) between two different frequencies, then the light of one frequency would arrive on the earth: 160000 x 365.25 x 24 x 60 x 60 x 10-2 / (3x10-8) =170 seconds or 2.8 minutes later! A result like this obviously disagrees with observations made of the spectrum of supernova 1987A. Spectra have been obtained for very distant supernovae up to a few billion light years away in other galaxies. One study places the maximum allowed dispersion to be on the order of 1 part in 10-21 (ref. xx). Thus we conclude that there is no dispersion of light in the vacuum. Is there a possibility for an index of refraction in the vacuum, as we have in an electron gas? Remember that an index of refraction requires two different media in which to compare the relative velocities of light. However the vacuum particle density must nearly uniform, with no transitions in density. Let us imagine a situation where somehow we have removed all the virtual particles in half of an empty box in vacuum, and the other half has the normal population of virtual particles in the normal quantum vacuum state. Would there be an index of refraction as light traveled from one side of the box to the other? This is a very important question because the validity of special relativity at the sub- microscopic distance scales comes into question here. You might think that if the vacuum has no energy, there should no effect on the propagation speed of photons. However we believe that the virtual particles in the quantum vacuum do indeed delay the progress of photons through electrically charged vacuum particle scattering effects. Thus we believe that photon scattering reduces the light velocity on the half of the box with electrically charged virtual particles. How can we justify this belief, in spite of the contradiction to special relativity? Special relativity is a classical theory, and was developed in the macroscopic domain of physics. It is almost impossible to measure light velocities over the extremely short distance scales that we are talking about. The electrically charged virtual particles in the quantum vacuum all have random velocities and move in random directions. They also have random energies ΔE during their brief life time Δt, which satisfies the uncertainty principle: ΔE Δt > h/(2 π). Imagine a real photon propagating in a straight path through the electrically charged virtual particles in a given direction. The real photon will encounter an equal number of virtual particles moving towards it as it does moving away from it. The end result is that the electrically charged quantum vacuum particles do not contribute anything different than the situation where all the virtual particles in the it’s path were at relative rest. Thus we can consider the vacuum as some sort of stationary crystal medium of virtual particles with a very high density, where each virtual particle is short-lived and constantly replaced (and carry no net average energy as discussed above).21The progress of the real photon is delayed as it travels through this quantum vacuum ‘crystal’, where it meets uncountable numbers of electrically charged virtual particles. Light travels through this with no absorption or dispersion. Based on our general arguments in the sections 3.1 to 3.4 above, we postulate that the photon is delayed as it travels through the quantum vacuum. We can definitely say that the uncertainty principle places a lower limit on the emission and absorption time delay, and forbids the time delay from being exactly equal to zero. Therefore we conclude that the electrically charged virtual particles of the quantum vacuum frequently absorb and re-emit the real photons moving through the vacuum by introducing small delays during absorption and subsequent re-emission of the photon, thus reducing the average propagation speed of the photons in the vacuum (compared to the light speed of photons between absorption/re-emission events). Our examination of the physics literature has not revealed any previous work on a quantum time delay analysis of photon propagation through the quantum vacuum, presumably because of the precedent set by Einstein’s postulate of light speed constancy in the vacuum under all circumstances. We will take the position that the delays due to photon scattering through the quantum vacuum are real. These delays reduce the much faster and absolutely fixed ‘low-level light velocity c l’ (defined as the photon velocity between vacuum particle scattering events) to the average observed light velocity ‘c’ in the vacuum (300,000 km/sec) that we observe in our actual experiments. Furthermore, we propose that the quantum vacuum introduces a sort of Vacuum Index of Refraction ‘n vac’ (compared to a vacuum without all virtual particles) such that c = c l / nvac. If this is true, what is the low-level light velocity? It is unknown at this time, but it must be significantly larger than 300,000 km/sec. In fact we believe that the vacuum index of refraction ‘n vac’ must be very large because of the high density of virtual particles in the vacuum. This concept is required in EMQG theory, and has become central to understanding the equivalence principle and 4D space-time curvature in accelerated frames and in gravitational fields (Appendix A). 3.4 FIZEAU EFFECT: LIGHT VELOCITY IN A MOVING MEDIA (** Optional) It also has been known for over a century that the velocity of light in a moving medium differs from its value in the same stationary medium. Fizeau demonstrated this experimentally in 1851 (ref. 41). For example, with a current of water (with refractive index of the medium of n=4/3) flowing with a velocity V of about 5 m/sec, the relative variation in the light velocity is 10-8 (which he measured by use of interferometry). Fresnel first derived the formula (ref. 41) in 1810 with his ether dragging theory. The resulting formula relates the longitudinal light velocity ‘v c’ moving in the same direction as a transparent medium of an index of refraction ‘n’ defined such that ‘c/n’ is the light velocity in the stationary medium, which is moving with velocity ‘V’ (with respect to the laboratory frame), where c is the velocity of light in the vacuum:22Fresnel Formula: v c = c/n + (1 – 1/n2) V (3.41) Why does the velocity of light vary in a moving (and non-moving) transparent medium? According to the principles of special relativity, the velocity of light is a constant in the vacuum with respect to all inertial observers. When Einstein proposed this postulate, he was not aware that the vacuum is not empty. However he was aware of Fresnel’s formula and derived it by the special relativistic velocity addition formula for parallel velocities (to first order). According to special relativity, the velocity of light relative to the proper frame of the transparent medium depends only on the medium. The velocity of light in the stationary medium is defined as ‘c/n’. Recall that velocities u and v add according to the formula: (u + v) / (1 + uv/c2) Therefore: vc = [ c/n + V ] / [ 1 + (c/n) (V)/c2 ] = (c/n + V) / ( 1 + V/(nc) ) ≈ c/n + (1 – 1/n2) V (3.42) The special relativistic approach to deriving the Fresnel formula does not say much about the actual quantum processes going on at the atomic level. At this scale, there are several explanations for the detailed scattering process in conventional physics. We investigate these different approaches in more detail below. 3.5 LORENTZ SEMI-CLASSICAL PHOTON SCATTERING (** Optional) The microscopic theory of the light propagation in matter was developed as a consequence of Lorentz’s non-relativistic, semi-classical electromagnetic theory. We will review and summarize this approach to photon scattering, which will not only prove useful for our analysis of the Fizeau effect, but has become the basis of the ‘Fizeau-like’ scattering of photons in the accelerated quantum vacuum near large gravitational fields (EMQG theory, Appendix A). To understand what happens in photon scattering inside a moving medium, imagine a simplified one-dimensional quantum model of the propagation of light in a refractive medium. The medium consisting of an idealized moving crystal of velocity ‘V’, which is composed of evenly spaced, point-like atoms of spacing ‘l’. When a photon traveling between atoms at a speed ‘c’ (vacuum light speed) encounters an atom, that atom absorbs it and another photon of the same wavelength is emitted after a time lag ‘ τ’. In the classical wave interpretation, the scattered photon is out of phase with the incident photon. We can thus consider the propagation of the photon through the crystal is a composite signal. As the photon propagates, part of the time it exists in the atom (technically, existing as an electron bound elastically to some atom), and part of the time as a photon propagating with the undisturbed low-level light velocity ‘c’. When the photon changes existence to being a bound electron, the velocity is ‘V’. From this, it can23be shown (ref. 41, an exercise in algebra and geometry) that the velocity of the composite signal ‘v c’ (ignoring atom recoil, which is shown to be negligible) is: vc = c [1 + (V τ/l) (1 - V/c)] / [1 + (c τ/l) (1 - V/c)] (3.51) If we set V=0, then v c = c / (1 + c τ/l) = c/n. Therefore, τ/l = (n – 1)/c. Inserting this in the above equations give: vc = [(c/n) + (1 – 1/n) V (1 - V/c)] /[1 - (1 – 1/n)(V/c)] ≈ c/n + (1 – 1/n2) V (to first order in V/c). (3.52) Again, this is Fresnel’s formula. Thus the simplified non-relativistic atomic model of the propagation of light through matter explains the Fresnel formula to the first order in V/c through the simple introduction of a scattering delay between photon absorption and subsequent re-emission. This analysis is based on a semi-classical approach. What does quantum theory say about this scattering process? The best theory we have to answer this question is QED. 3.6 PHOTON SCATTERIN G IN THE ACCELERATED VACUUM (** Optional) Anyone who believes in the existence of virtual fermion particles in the quantum vacuum that carry mass, will acknowledge the existence of a coordinated general downward acceleration of these virtual particles near any large gravitational field. In EMQG (Appendix A) gravitons from the real fermions on the earth exchange gravitons with the virtual fermions of the vacuum (which carry electric charge), causing a downward acceleration. The virtual particles of the quantum vacuum (now accelerated by a large mass) acts on light (and matter) in a similar manner as a stream of moving water acts on light in the Fizeau effect. How does this work mathematically? Again, it is impossible to compute the interaction of an accelerated collection of virtual particles of the quantum vacuum with light exactly. However, a simplified model can yield useful results. We will proceed using the semi-classical model proposed by Lorentz, above. We have defined the raw light velocity ‘c r’ (EMQG, ref. 1) as the photon velocity in between virtual particle scattering. Recall that raw light velocity is the shifting of the photon information pattern by one cell at every clock cycle on the CA, so that in fundamental units it is an absolute constant. Again, we assume that the photon delay between absorption and subsequent re-emission by a virtual particle is ‘ τ’, and the average distance between virtual particle scattering is ‘l’. The scattered light velocity v c(t) is now a function of time, because we assume that it is constantly varying as it moves downwards towards the surface in the same direction of the virtual particles. The virtual particles move according to: a =gt, where g = GM/R2. Therefore we can write the velocity of light after scattering with the accelerated quantum vacuum:24vc(t) = c r [1 + (gt τ/l) (1 - gt/c r)] / [1 + (c rτ/l) (1 - gt/c r)] (3.61) If we set the acceleration to zero, or gt = 0, then v c(t) = c r / (1 + c rτ/l) = c r/n. Therefore, τ/l = (n – 1)/c r. Inserting this in the above equation gives: vc(t) = [(c r/n) + (1 – 1/n) gt (1 - gt/c r)] / [1 - (1 – 1/n)(gt/c r)] ≈ cr/n + (1 – 1/n2) gt to first order in gt/c r. (3.62) Since the average distance between virtual charged particles is very small, the photons (which are always created at velocity c r) spend most of the time existing as some virtual charged particle undergoing downward acceleration. Because the electrically charged virtual particles of the quantum vacuum are falling in their brief existence, the photon effectively takes on the same downward acceleration as the virtual vacuum particles (as an average over macroscopic distances). In other words, because the index of refraction of the quantum vacuum ‘n’ is so large (relative to no vacuum particles), and because c = c r/n and we can write in equation 3.62: vc(t) = c r/n + (1 – 1/n2) gt = c + gt = c (1 + gt/c) if n >>1. (3.63) Similarly, for photons going against the flow (upwards): v c(t) = c (1 - gt/c) (3.64) This formula is used in EMQG for the variation of light velocity near a large gravitational field, and leads to the correct amount of general relativistic space-time curvature taking into account some additional assumptions as shown in Appendix A. It is this path Einstein followed 4. NON-LOCALITY AND SUPERLUMINAL PHOTONIC TUNNELING Is there any other evidence in physics for phenomena that potentially exhibit faster-than- light propagation? In quantum mechanics there definitely exist such phenomena: (1) The Quantum Non-Locality of quantum entangled particles, which apparently communicate each other’s quantum state faster-than-light . (2) Apparent faster-than-light tunneling of photons through a potential barrier, which is classically too large for the photon to penetrate. Both phenomena are well known and described in standard text books on quantum theory. A good account of both is given in an article titled “FASTER THAN LIGHT?”. In Scientific American, August 1993 by R. Chiao, P. Kwiat, A. Steinberg (ref. 26). Non- locality is an effect where two particles (or more) are causally connected or entangled (in other words where the two particle’s wave functions are dependent on each other), can influence each other apparently instantaneously no matter how far apart they are. A famous example of entanglement is the famous Einstein EPR proposal of 1935 (ref. 10),25and subsequent experimental verification by Aspect (ref. 12). It was J.S. Bell (ref. 28) that first derived a set of inequalities that Nature should obey if locality and reality were obeyed, and in which are violated by quantum mechanics. In some interpretations of quantum theory this appears to be contradiction of strict Einstein locality or causality, and therefore provides evidence for faster-than-light signaling. However to our knowledge no one has been able to devise a method of sending information faster than light from one location to another using quantum non-locality methods (excluding the claims by Nimtz, ref. 27. Quantum Tunneling is a phenomenon that is in some sense related to non-locality. In photon tunneling, a photon has a finite probability of moving through a barrier that it should not be able to pass through according to classical physics. What is remarkable about tunneling of photons (or for other types of quantum particles) is that when a measurement is made for the tunneling velocity, one finds that it is greater than the velocity of light in a vacuum. However some physicists maintain that it is not possible to talk about the photon actually having a definite velocity while it passes through the barrier. Indeed the act of assigning a definite time to the tunneling process has also been questioned (ref. 24). These arguments appeal to the probabilistic nature of the wave packet that describes the photon. The problem of defining the tunneling time of photon penetration through a barrier has a long history, which dates back to the 1920’s, when Hund first proposed the quantum mechanical “barrier penetration” phenomena. Quantum tunneling has important applications in electronics, and the very operation of the tunnel diode depends on the existence of the quantum tunneling phenomenon. An excellent account of quantum tunneling of photons is given by R. Y. Chiao, P. Kwiat, and A. Steinberg in the August 1993 (ref. 26) Scientific American magazine titled “Faster than Light?”. These authors claim that their “ Experiments in quantum optics show that two distant events can influence each other faster than any signal could have traveled between them .” They report that during several days of data collection (of more than one million photons tunneling through their barrier), that on average the tunneling photons arrived before the unimpeded photons. Their results imply that the average tunneling velocity for the photons is about 1.7 times that of light (ref. 25). What is even more bizarre is that the ‘tunneling velocity’ (which is a questionable concept) does not depend on the width of the barrier! R. Chiao et al. provide an explanation for faster-than-light tunneling that is not based on the concept of a tunneling velocity or a tunneling time. They point out that the photon’s quantum mechanical wave function of the tunneling photon is greatly reduced in amplitude in comparison to the unimpeded photon’s wave function. Recall that the amplitude of the wave function at a point represents the probability of finding a photon at that point. They point out that the center of the photon wave packet is the place of greatest probability of detection in the experiment. They claim that the wave front of the tunneling and unimpeded photons move together at the same rate, but that the tunneling photons arrive first because of the change in wave shape. They illustrate this elegantly with racing tortoises, where the two noses of the tortoises are locked in step, but the smaller one has a26narrower wave packet width and subsequently is detected first (illustration on bottom of page 55, ref. 26). Other experiments by G. Nimitz using microwave frequency photons instead of light, demonstrate that the microwave ‘tunneling velocity’ is 4.7 times light speed. Furthermore in order to illustrate that tunneling can convey information faster-than-light, they modulated the microwave beam with the audio track of the 1st movement of Mozart’s 40th symphony. They report that they were able to send this message through the barrier at 4.7 times light speed (ref. 27). The proposed LVC experiments borrow much of the same techniques used by R. Chiao et. al. to measure the increase in light velocity in quantum tunneling. The task is similar, that is to compare the arrival times of photons from two different photon paths that started at the same time, where the difference in arrival times is incredibly small and hard to detect. The speed of light is so great at laboratory distances that conventional electronics is tens of thousands of times too slow to measure the small differences in light arrival time. To solve this problem they used twin photon interferometers to measure the required time delays, a technique that we will borrow for our proposed experiments. 5. THE PROPOSED CASIMIR LIGHT VELOCITY EXPERIMENTS We propose experiments to look for the Light Velocity Casimir (LVC) effect and to measure the Casimir vacuum index of refraction. Figure 1 shows the conceptual block diagram of the first experiment to observe the increase in light velocity. Figure 3 gives a more detailed account, which we will describe later. Here we have two identical light paths that originated from a common light source such as a laser, where one light path travels straight through the vacuum unimpeded, and becomes our standard reference path for the light velocity in the vacuum. The other light path travels between two transparent electrically conducting, Casimir plates in a vacuum, which are closely spaced and have an adjustable plate spacing ‘d’. When the laser light source is switched on, the light path through the Casimir plates arrives at its detector first, where the arrival time becomes sooner with decreasing plate spacing. NOTE: Light must propagate perpendicular through the plates in the LVC experiment, because it is in this direction that vacuum density decreases. If light travels parallel through the Casimir plates (which would offer a longer path to increase light velocity), the vacuum density and light velocity along this direction are not changed . This illustrates that the vacuum process inside the Casimir plates is a dynamic effect. If the Casimir plates are visualized as being part of a rectangular enclosure and air is pumped out of the box, the velocity of light would increase compared to the velocity of light in air, no matter what direction light traveled in the box. If the two ends of the rectangular box are removed, then the inside and outside air pressure would balance. This is not so for the Casimir plates. Even though the two ends of the ‘Casimir’ box are removed, the Casimir plates maintain a decreased vacuum density, but only in the direction27perpendicular to the plates. This pressure change is maintained dynamically by the quantum vacuum process discussed in section 2.1. We claim that the front velocity of the light traveling through the Casimir plates will exceed the velocity of light in the ordinary vacuum, as defined in section 1.1. In order to enhance the magnitude of the light velocity increase, it is desirable to increase the optical path length for the laser light through the Casimir plates. This can be done by optically arranging a set of mirrors to direct the light back and forth several times through the Casimir plates (which must also be done in the reference path). In order to simplify the discussions, this arrangement is not shown in any of the diagrams, and the method used depends on the detailed experimental arrangement chosen. The two beams are then routed to two individual detectors (Figure 3). The detectors electronic outputs are fed to an electronic instrument that can accurately measure time delays between the two outputs. Although we have been unable to compute the value of the light velocity increase, we expect it to be very small. We believe that using an ordinary laser beam (which contains enormous numbers of photons) as a light source will not be effective in measuring the LVC effect. Instead we borrow some techniques from quantum optics and from the measurement of optical tunneling times to detect the LVC effect for single photons. The time that it takes light to transverse the Casimir plate spacing (assuming the plates are not there) with a 1 µm spacing is 33 fs (33x10-15 seconds)! Therefore we expect that the time to propagate through the plates to be smaller than this value. That means we would like to have perhaps 0.1 fs resolution in time. Currently the best photon detectors only have a picosecond-scale response time, which is not fast enough for this application. However a device called the ‘Hong-Ou-Mandel Interferometer’ has femtosecond-scale time resolution, which is ideal for this experiment. Figure 3 shows a conceptual Hong-Ou-Mandel interferometer-based setup that should be capable of measuring the Casimir light velocity increase for individual photons! It is very similar to the experimental arrangement to measure the quantum tunneling of individual photons through a barrier used by R. Chiao et al. (ref. 25 and 26). In order to make sure both photons start traveling at the same time through the apparatus we suggest the use of a ‘Spontaneous Parametric Down-Conversion’ crystal, which absorbs the incoming ultraviolet photon from an argon laser and emits two new photons simultaneously, and which are strongly correlated. The energies of the two photons equal the energy of the incoming photon. Furthermore the two photons are quantum mechanically entangled, which is beneficial to overcoming potential sources of errors in performing the experiment (ref. 25). The two photons reflect off the two mirrors and travel through two equal length paths and meet at the beam splitter. One path is the reference vacuum path, and the other is the Casimir plate path, where the plates are originally removed to calibrate the optical paths to null the interferometer (ref. 25). The advantage of using the Hong-Ou-Mandel Interferometer is that it results in a narrow null in the coincidence count rate as a function of the relative delay between the two photons, a destructive interference effect that was first observed by Hong, Ou, and28Mandel. The narrowness of the coincidence minimum combined with a good signal to noise ratio should provide a measurement of the relative delay between the two photons to a precision of ±0.2 fs (ref. 25). There are four possible outcomes at the beam splitter: both photons might pass through, both might be reflected from the beam splitter, both might go off to one side (one reflected, and one transmitted), or both may go off the other side. We are interested in the first two cases, where the two photons reach different detectors that result in coincidence detection . We adjust the optical lengths until coincidence detection disappears. This means that any deviation in the relative velocities of the two path’s results will become readily apparent because of the narrowness of the interference null. The detectors chosen for the tunneling experiment (ref. 25) are Geiger-mode silicon avalanche photo-diodes. Aslo, care must be taken not to choose a Casimir plate spacing which nulls out the photon (multiples of the photon frequency), since the Casimir plates are electrically conductive. The Vacuum Casimir Index of Refraction given by n vac = c / c c (which should be slightly less than one) where ‘c c’ is the light velocity between the Casimir plates. This can be measured in the experimental arrangement of figure 3, assuming the LVC effect is observed. In order to calculate n vac one must measure c c. Let ‘h’ be the optical path length of the reference leg of the interferometer. Let ‘d’ be the distance between the Casimir plates. Let ‘ δt’ be the time a photon takes to traverse the Casimir plates. Let ‘ Δt’ be the time a photon takes to traverse the plate spacing, with no plates present. The interferometer measures the time t diff = Δt - δt, or the decrease in the relative time of propagation of the photon through Casimir plate distance ‘d’. In order to calculate nvac: • Determine the optical length ‘h’ through the reference leg of the interferometer, which is also equal to the optical length of the Casimir plate leg of the interferometer. • Determine the time delay t diff = Δt - δt , which is also the time difference between the arrival of a photon first at detector 2 followed by the detection at detector 1. • Calculate Δt = d/c, and then calculate δt = Δt - tdiff . • It follows that the light velocity inside the Casimir plates is given by: c c = d / δt, which should be slightly less than ‘c’ in the reference leg. • Finally we have the Vacuum Casimir Index of Refraction: n vac = c / c c Figure 2 shows an alternate way to demonstrate the Light Velocity Casimir effect, based on the classical idea of light refraction. Conceptually we start with a single laser light source and direct the light at a shallow angle ‘t 0‘ that is close to the perpendicular of the two transparent, electrically conducting Casimir plates (see the bubble in figure 2). This must be so, because the refraction only happens in the nearly perpendicular direction of the Casimir plates (see the note at the beginning of section 5). The light deflects at the first plate by an angle t 1, such that n vac = sin(t 0) / sin(t 1). When the light exits the second plate it returns to the original direction because it goes to the normal vacuum state which is an increase in density. The end result is that the light beam is still in line with the incoming beam, but slightly shifted to the right. In these discussions we29ignore any refraction through the transparent plate material, which must be taken into account when performing the actual experiment. Because the spacing between the plates is quite small, the light bending effect would be very hard to detect with only a two Casimir plate experiment. In order to enhance the bending effect it is desirable to have a series of transparent and conductive Casimir plates, such that the spacing between the subsequent plates decreases with distance. Figure 2 shows an example arrangement of four such Casimir plates for illustration purposes. With this arrangement the exit angle of the light is permanently different from the original angle, because the vacuum density between the plates varies in different steps between each new pair of plates. This means that over a large distance the angle can be easily measured. In practice the resulting angle is going to be extremely small, because of the very slight differences in energy-density per pair of plates. However a small angle can translate to a large shift or deflection in the total light path, when the light is allowed to travel over a large distance in normal air. For example if the difference in angle of the incoming and outgoing light paths is Δθ = 0.001 degrees (which is only 3.6 seconds of arc), at a distance of h=100 meters the difference in position of the expected light beam is given by: h tan( Δθ) = 1.7 millimeters, which is a distance that can be easily measured. 6. CONCLUSIONS 1. We believe that the velocity of light (specifically the front velocity) propagating in vacuum inside (and perpendicular) two closely spaced, electrically conducting and transparent plates called the Casimir plates, will increase inside the plates compared to the light velocity in the normal vacuum, measured in the laboratory frame. This conclusion should be subjected to future experimental verification. We have proposed two such experiments; an interferometer, which is designed to resolve the difference in arrival time of individual photons, and another experiment to measure the refraction of light propagating at a shallow angle (nearly perpendicular) through a series of Casimir plates with gradually decreasing plate separation. 2. We believe that there must exist a ‘Vacuum Casimir Index of Refraction’ called ‘n vac’ for light traveling from outside, and through the Casimir plates in vacuum. The Casimir vacuum index of refraction is defined as the ratio of the velocity of light in normal vacuum conditions divided by the light velocity measured propagating inside the Casimir plates in vacuum ‘c c‘. The vacuum Casimir index of refraction ‘n vac’ is thus defined as: n vac = c / c c , which should be slightly less than one. 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BELL, Long Island City, New York, Physics 1, 195, (1964) (29) SPLITTING THE ELECTRON by B.Daviss, New Scientist, Jan. 31, 1998. (30) OPTICAL COHERENCE AND QUANTUM OPTICS by L. Mandel and E. Wolf., Cambridge (31) RELATIVITY: THE GENERAL THEORY by J.L. Synge, 1971, North-Holland, Amsterdam, p. IX. (32) VERH. DEUTSCH. PHYS. GES. 18, 83 , 1916, W. Nernst. (33) WAVE PROPAGATION AND GROUP VELOCITY by L. Brillouin, 1960, Academic Press. (34) DIGITAL MECHANICS: An Informational Process based on Reversible Cellular Automata by Edward Fredkin, Physica D 45 (1990) 254-270. (35) INTRODUCTION TO THE THEORY OF RELATIVITY by P.G. Bregmann, Chap. IV, pg.33. (36) UNIVERSITY PHYSICS by H. Benson, Wiley, Chap. 39, pg. 797 (37) ON THE “DERIVATION” OF EINSTEIN’S FIELD EQUATIONS by S. Chandrasekhar, AJP Volume 40, pg. 224 (1972). (38) ESSENTIAL RELATIVITY by W. Rindler, Chap. 1, pg. 10 (Rise and fall of Absolute Space). (39) GRAVITATION AND COSMOLOGY by S. Weinberg, Chap. 8, pg. 179. (40) VERIFICATION OF THE EQUIVALENCE OF GRAVITATIONAL AND INERTIAL MASS FOR THE NEUTRON by L. Koester, Physical Rev. D, Vol. 14, Num. 4, pg.907 (1976). (41) DOES THE FIZEAU EXPERIMENT REALLY TEST SPECIAL RELATIVITY by G. Clement, Am. J. Phys. 48(12), Dec. 1980. (42) THE FEYNMAN LECTURES ON PHYSICS by Feynman, Leighton, and Sands, Vol. 1, Chap. 31 The Origins of the Refractive Index. 8. FIGURE CAPTIONS The captions for the figures are shown below: Fig. 1 - Schematic Diagram of the Light Velocity Casimir Effect Fig. 2 - Experiment to measure the Refraction of Light through Multiple Casimir Plates Fig. 3 - An Interferometer Setup for the Light Velocity Casimir Experiment32 33 34APPENDIX A: BRIEF REVIEW OF EMQG35This appendix gives a very brief review of Electromagnetic Quantum Gravity (EMQG) and it’s connection to the quantum vacuum and the Casimir Light Velocity effect. The full paper can be found in reference A1. This review is intended to briefly summarize the essential ideas of EMQG and the central role that the quantum vacuum plays in EMQG. We have developed a new approach to the unification of quantum theory with general relativity referred to as Electro-Magnetic Quantum Gravity or EMQG (ref. 1). Figure A1 at the end of the appendix illustrates the relationship between EMQG and the rest of physical theory. EMQG has its origins in Cellular Automata (CA) theory (ref. 2,4,9 and 34), and is also based on the new theory of inertia that has been proposed by R. Haisch, A. Rueda, and H. Puthoff (ref. 5) known here as the HRP Inertia theory. These authors suggest that inertia is due to the strictly local force interactions of charged matter particles with their immediate background virtual particles of the quantum vacuum. They found that inertia is caused by the magnetic component of the Lorentz force, which arises between what the author’s call the charged ‘parton’ particles in an accelerated reference frame interacting with the background quantum vacuum virtual particles. The sum of all these tiny forces in this process is the source of the resistance force opposing accelerated motion in Newton’s F=MA. We have found it necessary to make a small modification of HRP Inertia theory as a result of our investigation of the principle of equivalence. The modified version of HRP inertia is called “Quantum Inertia” (or QI). In EMQG, a new elementary particle is required to fully understand inertia, gravitation, and the principle of equivalence (described in the next section). This theory also resolves the long outstanding problems and paradoxes of accelerated motion introduced by Mach’s principle, by suggesting that the vacuum particles themselves serve as Mach’s universal reference frame (for acceleration only), without violating the principle of relativity of constant velocity motion. In other words, our universe offers no observable reference frame to gauge inertial frames, because the quantum vacuum offers no means to determine absolute constant velocity motion. However for accelerated motion, the quantum vacuum plays a very important role by offering a resistance to acceleration, which results in an inertial force opposing the acceleration of the mass. Thus the very existence of inertial force reveals the absolute value of the acceleration with respect to the net statistical average acceleration of the virtual particles of the quantum vacuum. Reference 14 offers an excellent introduction to the motion of matter in the quantum vacuum, and on the history of the discovery of the virtual particles of the quantum vacuum. (A-1) EMQG and the Quantum Theory of Inertia EMQG theory presents a unified approach to Inertia, Gravity, the Principle of Equivalence, space-time Curvature, Gravitational Waves, and Mach’s Principle. These apparently different phenomena are the common results of the quantum interactions of the real (charged) matter particles (of a mass) with the surrounding virtual particles of the quantum vacuum through the exchange of two force particles: the photon and the graviton. Furthermore, the problem of the cosmological constant is solved automatically in the framework of EMQG. This new approach to quantum gravity is definitely non- geometric on the tiniest of distance scales (Plank Scales of distance and time). This is36because the large scale relativistic 4D space-time curvature is caused purely by the accelerated state of virtual particles of the quantum vacuum with respect to a mass, and their discrete interactions with real matter particles of a mass through the particle force exchange process. Because of this departure from a universe with fundamentally curved space-time, EMQG is a complete change in paradigm over conventional gravitational physics. This paper should be considered as a framework, or outline of a new approach to gravitational physics that will hopefully lead to a full theory of quantum gravity. We modified the HRP theory of Inertia (ref. 5) based on a detailed examination of the principle of equivalence. In EMQG, the modified version of inertia is known as “Quantum Inertia”, or QI. In EMQG, a new elementary particle is required to fully understand inertia, gravitation, and the principle of equivalence. All matter, including electrons and quarks, must be made of nature’s most fundamental mass unit or particle, which we call the ‘masseon’ particle. These particles contain one fixed, fundamental ‘quanta’ of both inertial and gravitational mass. The masseons also carry one basic, smallest unit or quanta of electrical charge as well, of which they can be either positive or negative. Masseons exist in the particle or in the anti-particle form (called anti-masseon), that can appear at random in the vacuum as virtual masseon/anti-masseon particle pairs of opposite electric charge and opposite ‘mass charge’. The earth consists of ordinary masseons (with no anti- masseons), of which there are equal numbers of positive and negative electric charge varieties. In HRP Inertia theory, the electrically charged ‘parton’ particles (that make up an inertial mass in an accelerated reference frame) interact with the background vacuum electromagnetic zero-point-field (or virtual photons) creating a resistance to acceleration called inertia. We have modified this slightly by postulating that the real masseons (that make up an accelerating mass) interacts with the surrounding, virtual masseons of the quantum vacuum, electromagnetically (although the details of this process are still not fully understood). The properties of the masseon particle and gravitons are developed later. (A-2) EMQG and the Quantum Origin of Newton’s Laws of Motion We are now in a position to understand the quantum nature of Newton’s classical laws of motion. According to the standard textbooks of physics, Newton’s three laws of laws of motion are: An object at rest will remain at rest and an object in motion will continue in motion with a constant velocity unless it experiences a net external force. The acceleration of an object is directly proportional to the resultant force acting on it and inversely proportional to its mass. Mathematically: ΣF = ma, where F and a are vectors. If two bodies interact, the force exerted on body 1 by body 2 is equal to and opposite the force exerted on body 2 by body 1. Mathematically: F 12 = -F 21. Newton’s first law explains what happens to a mass when the resultant of all external forces on it is zero. Newton’s second law explains what happens to a mass when there is a37nonzero resultant force acting on it. Newton’s third law tells us that forces always come in pairs. In other words, a single isolated force cannot exist. The force that body 1 exerts on body 2 is called the action force, and the force of body 2 on body 1 is called the reaction force. In the framework of EMQG theory, Newton’s first two laws are the direct consequence of the (electromagnetic) force interaction of the (charged) elementary particles of the mass interacting with the (charged) virtual particles of the quantum vacuum. Newton’s third law of motion is the direct consequence of the fact that all forces are the end result of a boson particle exchange process. (A-3) NEWTON’S FIRST LAW OF MOTION: In EMQG, the first law is a trivial result, which follows directly from the quantum principle of inertia (postulate #3, appendix A-11). First a mass is at relative rest with respect to an observer in deep space. If no external forces act on the mass, the (charged) elementary particles that make up the mass maintain a net acceleration of zero with respect to the (charged) virtual particles of the quantum vacuum through the electromagnetic force exchange process. This means that no change in velocity is possible (zero acceleration) and the mass remains at rest. Secondly, a mass has some given constant velocity with respect to an observer in deep space. If no external forces act on the mass, the (charged) elementary particles that make up the mass also maintain a net acceleration of zero with respect to the (charged) virtual particles of the quantum vacuum through the electromagnetic force exchange process. Again, no change in velocity is possible (zero acceleration) and the mass remains at the same constant velocity. (A-4) NEWTON’S SECOND LAW OF MOTION: In EMQG, the second law is the quantum theory of inertia discussed above. Basically the state of relative acceleration of the charged virtual particles of the quantum vacuum with respect to the charged particles of the mass is what is responsible for the inertial force. By this we mean that it is the tiny (electromagnetic) force contributed by each mass particle undergoing an acceleration ‘A’, with respect to the net statistical average of the virtual particles of the quantum vacuum, that results in the property of inertia possessed by all masses. The sum of all these tiny (electromagnetic) forces contributed from each charged particle of the mass (from the vacuum) is the source of the total inertial resistance force opposing accelerated motion in Newton’s F=MA. Therefore, inertial mass ‘M’ of a mass simply represents the total resistance to acceleration of all the mass particles. (A-5) NEWTON’S THIRD LAW OF MOTION: According to the boson force particle exchange paradigm (originated from QED) all forces (including gravity, as we shall see) result from particle exchanges. Therefore, the force that body 1 exerts on body 2 (called the action force), is the result of the emission of38force exchange particles from (the charged particles that make up) body 1, which are readily absorbed by (the charged particles that make up) body 2, resulting in a force acting on body 2. Similarly, the force of body 2 on body 1 (called the reaction force), is the result of the absorption of force exchange particles that are originating from (the charged particles that make up) body 2, and received by (the charged particles that make up) body 1, resulting in a force acting on body 1. An important property of charge is the ability to readily emit and absorb boson force exchange particles. Therefore, body 1 is both an emitter and an absorber of the force exchange particles. Similarly, body 2 is also both an emitter and an absorber of the force exchange particles. This is the reason that there is both an action and reaction force. For example, the contact forces (the mechanical forces that Newton was thinking of when he formulated this law) that results from a person pushing on a mass (and the reaction force from the mass pushing on the person) is really the exchange of photon particles from the charged electrons bound to the atoms of the person’s hand and the charged electrons bound to the atoms of the mass on the quantum level. Therefore, on the quantum level there is really is no contact here. The hand gets very close to the mass, but does not actually touch. The electrons in one’s hand exchange photons with the electrons in the mass. The force exchange process works both directions in equal numbers, because all the electrons in the hand and in the mass are electrically charged and therefore the exchange process gives forces that are equal and opposite in both directions. (A-6) Introduction to the Principle of Equivalence and EMQG Are virtual particle force exchange processes originating from the quantum vacuum also present for gravitational mass? The answer turns out to be a resounding yes! As we suggested, there is some evidence of the interplay between the virtual particles of the quantum vacuum and gravitational phenomena. In order to see how this impacts our understanding of the nature of gravitational mass, we found it necessary to perform a thorough investigation of Einstein’s Principle of Equivalence of inertial and gravitational mass in general relativity under the guidance of the new theory of quantum inertia. We have uncovered some theoretical evidence that the SEP may not hold for certain experiments. There are two basic theoretical problems with the SEP in regard to quantum gravity. First, if gravitons (the proposed force exchange particle) can be detected with some new form of a sensitive graviton detector, we would be able to distinguish between an accelerated reference frame and a gravitational frame with this detector. This is because accelerated frames would have virtually no graviton particles present, whereas a large gravitational field like the earth has enormous numbers of graviton particles associated with it. Secondly, theoretical considerations from several authors regarding the emission of electromagnetic waves from a uniformly accelerated charge, and the lack of radiation from the same charge subjected to a static gravitational field leads us to question the validity of the SEP for charged particles radiating electromagnetically. How does the WEP hold out in EMQG? The WEP has been tested to a phenomenal accuracy (ref 24.) in recent times. Yet in our current understanding of the WEP, we can39only specify the accuracy as to which the two different mass values (or types) have been shown experimentally to be equal inside an inertial or gravitational field. There exists no physical or mathematical proof that the WEP is precisely true. It is still only a postulate of general relativity. We have applied the recent work on quantum inertia (ref. 5) to the investigation of the weak principle of equivalence, and have found theoretical reasons to believe that the WEP is not precisely correct when measured in extremely accurate experiments. Imagine an experiment with two masses; one mass M 1 being very large in value, and the other mass M 2 is very small (M 1 >> M 2). These two masses are dropped simultaneously in a uniform gravitational field of 1g from a height ‘h’, and the same pair of masses are also dropped inside a rocket accelerating at 1g, at the same height ‘h’. We predict that there should be a minute deviation in arrival times on the surface of the earth (only) for the two masses, with the heavier mass arriving just slightly ahead of the smaller mass. This is due to a small deviation in the magnitude of the force of gravity on the mass pair (in favor of M 1) on the order of (N 1-N2)i * δ, where (N 1-N2) is the difference in the low level mass specified in terms of the difference in the number of masseon particles in the two masses (defined latter) times the single masseon mass ‘i’, and δ is the ratio of the gravitational to electromagnetic forces for a single (charged) masseon. This experiment is very difficult to perform on the earth, because δ is extremely small ( ≈10-40), and ΔN = (N 1- N2) cannot in practice be made sufficiently large in order to produce a measurable effect. However, inside the accelerated rocket, the arrival times are exactly identical for the same pair of masses. This, of course, violates the principle of equivalence, since the motion of the masses in the inertial frame is slightly different then in the gravitational frame. This imbalance is minute because of the dominance of the strong electromagnetic force, which is also acting on the masseons of the two masses from the virtual particles of the quantum vacuum. This acts to stabilize the fall rate, giving us nearly perfect equivalence. This conclusion is based on the discovery that the weak principle of equivalence results from lower level physical processes. Mass equivalence arises from the equivalence of the force generated between the net statistical average acceleration vectors of the charged matter particles inside a mass with respect to the immediate surrounding quantum vacuum virtual particles inside an accelerating rocket. This is almost exactly the same physical force occurring between the stationary (charged) matter particles and the immediate surrounding accelerating virtual particles of the same mass near the earth. It turns out that equivalence is not perfect in the presence of a large gravitational field like the earth. Equivalence breaks down due to an extremely minute force imbalance in favor of a larger mass dropped simultaneously with respect to a smaller mass. This force imbalance can be traced to the pure graviton exchange force component occurring in the gravitational field that is not present in the case of the identically dropped masses in an accelerated rocket. This imbalance contributes a minute amount of extra force for the larger mass compared to the smaller mass (due to many more gravitons exchanged between the larger mass as compared to the smaller mass), which might be detected in highly accurate measurements. In the case of the rocket, the equivalence of two different falling masses is perfect, since it is the floor of the rocket that accelerates up to meet the two masses simultaneously. Of course, the breakdown of the WEP also means the downfall of the SEP.40In EMQG, the gravitational interactions involve the same electromagnetic force interaction as found in inertia based on our QI theory. We also found that the weak principle of equivalence itself is a physical phenomenon originating from the hidden lower level quantum processes involving the quantum vacuum particles, graviton exchange particles, and photon exchange particles. In other words, gravitation is purely a quantum force particle exchange process, and is not based on low level fundamental 4D curved space-time geometry of the universe as believed in general relativity. The perceived 4D curvature is a manifestation of the dynamic state of the falling virtual particles of the quantum vacuum in accelerated frames, and gravitational frames. The only difference between the inertial and gravitation force is that gravity also involves graviton exchanges (between the earth and the quantum vacuum virtual particles, which become accelerated downwards), whereas inertia does not. Gravitons have been proposed in the past as the exchange particle for gravitational interactions in a quantum field theory of gravity without much success. The reason for the lack of success is that graviton exchange is not the only exchange process occurring in large-scale gravitational interactions; photon exchanges are also involved! It turns out that not only are there both graviton and photon exchange processes occurring simultaneously in large scale gravitational interactions such as on the earth, but that both exchange particles are almost identical in their fundamental nature (Of course, the strength of the two forces differs greatly). The equivalence of inertial and gravitational mass is ultimately traced down to the reversal of all the relative acceleration vectors of the charged particles of the accelerated mass with respect to the (net statistical) average acceleration of the quantum vacuum particles, that occurs when changing from inertial to gravitational frames. The inertial mass ‘M’ of an object with acceleration ‘a’ (in a rocket traveling in deep space, away from gravitational fields) results from the sum of all the tiny forces of the charged elementary particles that make up that mass with respect to the immediate quantum vacuum particles. This inertial force is in the opposite direction to the motion of the rocket. The (charged masseon) particles building up the mass in the rocket will have a net statistical average acceleration ‘a’ with respect to the local (charged masseon) virtual particles of the immediate quantum vacuum. A stationary gravitational mass resting on the earth’s surface has this same quantum process occurring as for the accelerated mass, but with the acceleration vectors reversed. What we mean by this is that under gravity, it is now the virtual particles of the quantum vacuum that carries the net statistical average acceleration ‘A’ downward. This downward virtual particle acceleration is caused by the graviton exchanges between the earth and the mass, where the mass is not accelerated with respect to the center of mass of the earth. (Note: On an individual basis, the velocity vectors of these quantum vacuum particles actually point in all directions, and also have random amplitudes. Furthermore, random accelerations occur due to force interactions between the virtual particles themselves. This is why we refer to the statistical nature of the acceleration.) We now see that the gravitational force of a stationary mass is also the same sum of the tiny forces that originate for a mass undergoing accelerated motion in a gravitational field from the virtual particles of the quantum vacuum according to Newton’s law ‘F = MA’. In other words, the same inertial force F=MA is also found hidden inside gravitational interactions of41masses! Mathematically, this fact can be seen in Newton’s laws of inertia and in Newton’s gravitational force law by slightly rearranging the formulas as follows: Fi = Mi (Ai) ... the inertial force F i opposes the acceleration A i of mass M i in the rocket, caused by the sum of the tiny forces from the virtual particles of the quantum vacuum. Fg= Mg (Ag) = Mg (GM e/r2) ... the gravitational force F g is the result of a kind of an inertial force given by ‘M g Ag’ where A g = GM e/r2 is now due to the sum of the tiny forces from the virtual particles of the quantum vacuum (now accelerating downwards). Since F i=Fg, and since the acceleration of gravity is chosen to be the same as the inertial acceleration, where the virtual particles now have: A g = A i = GM e/r2 , therefore M i=Mg , or the inertia mass is equal to the gravitational mass (M e is the mass of the earth). Here, Newton’s law of gravity is rearranged slightly to emphasis it’s form as a kind of an ‘inertial force’ of the form F=MA, where the acceleration (GM e/r2) is now the net statistical average downward acceleration of the quantum vacuum virtual particles near the vicinity of the earth. This derivation is not complete, unless we can provide a clear explanation as to why F i=Fg , which we know to be true from experimental observation. In EMQG, both of these forces are understood to arise from an almost identical quantum vacuum process! For accelerated masses, inertia is the force F i caused by the sum of all the tiny electromagnetic forces from each of the accelerated charged particles inside the mass; with respect to the non-accelerating surrounding virtual particles of the quantum vacuum. Under the influence of a gravitational field, the same force Fg exists as it does in inertia, but now the quantum vacuum particles are the ones undergoing the same acceleration A i (through graviton exchanges with the earth); the charged particles of the mass are stationary with respect to earth’s center. The same force arises, but the arrows of the acceleration vectors are reversed. To elaborate on this, imagine that you are in the reference frame of a stationary mass resting on the surface with respect to earth’s center. An average charged particle of this mass ‘sees’ the virtual particles of the quantum vacuum in the same state of acceleration, as does an average charged particle of an identical mass sitting on the floor of an accelerated rocket (1 g). In other words, the background quantum vacuum ‘looks’ exactly the same from both points of view (neglecting the very small imbalance caused by a very large number of gravitons interacting with the mass directly under gravity, this imbalance is swamped by the strength of the electromagnetic forces existing). These equations and methodology illustrate equivalence in a special case: that is between an accelerated mass M i and the same stationary gravitational mass M g. In EMQG, the weak equivalence principle of gravitational and inertial frames holds for many other scenarios such as for free falling masses, for masses that have considerable self gravity and energy (like the earth), for elementary particles, and for the propagation of light. However, equivalence is not perfect, and in some situations (for example, antimatter discussed in section 7.1) it simply does not hold at all!42An astute observer may question why all the virtual particles (electrons, quarks, etc., all having different masses) are accelerating downwards on the earth with the same acceleration. This definitely would be the case from the perspective of a mass being accelerated by a rocket (where the observer is accelerating). Since the masses of the different types of virtual particles are all different according to the standard model of particle physics, why are they all falling at the same rate? Since we are trying to derive the equivalence principle, we cannot invoke this principle to state that all virtual particles must be accelerating downward at the same rate. It turns out that the all quantum vacuum virtual particles are accelerating at the same rate because all particles with mass (virtual or not) are composed of combinations of a new fundamental “masseon” particle, which carries just one fixed quanta of mass. Therefore, all the elementary virtual masseon particles of the quantum vacuum are accelerated by the same amount. These masseons can bind together to form the familiar particles of the standard model, like virtual electrons, virtual positrons, virtual quarks, etc. Recalling that the masseon also carries electrical charge, we see that all the constituent masseons of the quantum vacuum particles fall to earth at same rate through the electromagnetic interaction (or photon exchange) process, no matter how the virtual masseons combine to give the familiar virtual particles. This process works like a microscopic principle of equivalence for falling virtual particles, with the same action occurring for virtual particles as for large falling masses. The properties of the masseon particle is elaborated in section 7 (the masseon may be the unification particle sought out by physicist, in which case it will have other properties to do with the other forces of nature). For now, note that the masseon also carries the fundamental unit of electric charge as well. This fundamental unit of electric charge turns out to be the source of inertia for all matter according to Quantum Inertia. By postulating the existence of the masseon particle (which is the fundamental unit of ‘mass charge’ as well as ‘electrical charge’) all the quantum vacuum virtual particles accelerate at the same rate with respect to an observer on the surface of the earth. We have postulated the existence of a fundamental “low level gravitational mass charge” of a particle, which results from the graviton particle exchange process similar to the process found for electrical charges. This ‘mass charge’ is not affected when a particle achieves relativistic velocities, so we can state that ‘low level mass charge’ is an absolute constant. For particles accelerated to relativistic speeds, a high relative velocity between the source of the force and the receiving mass affects the ordinary measurable inertial mass, as we have seen (in accordance to Einstein’s mass-velocity formula). (A-7) Summary of the Basic Mass Definitions in EMQG EMQG proposes three different mass definitions for an object: (1) INERTIAL MASS is the measurable mass defined in Newton’s force law F=MA. This is considered as the absolute mass in EMQG, because it results from force produced by the relative (statistical average) acceleration of the charged virtual particles of the quantum vacuum with respect to the charged particles that make up the inertial mass. The43virtual particles of the quantum vacuum become Newton’s absolute reference frame. In special relativity this mass is equivalent to the rest mass. (2) GRAVITATIONAL MASS is the measurable mass involved in the gravitational force as defined in Newton’s law F=GM 1M2/R2. This is what is measured on a weighing scale. This is also considered as absolute mass, and is almost exactly the same as inertial mass. (3) LOW LEVEL GRAVITATIONAL ‘MASS CHARGE’, which is the origin of the pure gravitational force, is defined as the force that results through the exchange of graviton particles between two (or more) quantum particles. This type of mass analogous to ‘electrical charge’, where photon particles are exchanged between electrically charged particles. Note: This force is very hard to measure because it is masked by the background quantum vacuum electromagnetic force interactions, which dominates over the graviton force processes. These three forms of mass are not necessarily equal! We have seen that the inertial mass is almost exactly the same as gravitational mass, but not perfectly equal. All quantum mass particles (fermions) have all three mass types defined above. Note that bosons (only photons and gravitons are considered here) have only the first two mass types. This means that photons and gravitons transfer momentum, and do react to the presence of inertial frames and to gravitational fields, but they do not emit or absorb gravitons. Gravitational fields effect photons, and this is linked to the concept of space-time curvature, described in detail later (section 9). It is important to realize that gravitational fields deflect photons (and gravitons), but not by force particle exchanges directly. Instead, it is due to a scattering process (described later). To summarize, both the photon and the graviton do not carry low level ‘mass charge’, even though they both carry inertial and gravitational mass. The graviton exchange particle, although responsible for a major part of the gravitational mass process, does not itself carry the property of ‘mass charge’. Contrast this with conventional physics, where the photon and the graviton both carry a non-zero mass given by M=E/C2. According to this reasoning, the photon and the graviton both carry mass (since they carry energy), and therefore both must have ‘mass charge’ and exchange gravitons. In other words, the graviton particle not only participates in the exchange process, it also undergoes further exchanges while it is being exchanged! This is the source of great difficulty for canonical quantum gravity theories, and causes all sorts of mathematical renormalization problems in the corresponding quantum field theory. Furthermore, in gravitational force interactions with photons, the strength of the force (which depends on the number of gravitons exchanged with photon) varies with the energy that the photon carries! In modern physics, we do not distinguish between inertial, gravitational, or low level ‘mass charge’. They are assumed to be equivalent, and given a generic name ‘mass’. In EMQG, the photon and graviton carry measurable inertial and gravitational mass, but neither particle carries the ‘low level mass charge’, and therefore do not participate in graviton exchanges.44We must emphasize that gravitons do not interact with each other through force exchanges in EMQG, just as photons do not interact with each other with force exchanges in QED. Imagine if gravitons did interact with other gravitons. One might ask how it is possible for a graviton particle (that always moves at the speed of light) to emit graviton particles that are also moving at the speed of light. For one thing, this violates the principles of special relativity theory. Imagine two gravitons moving in the same direction at the speed of light, which are separated by a distance d, with the leading graviton called ‘A’ and the lagging graviton called ‘B’. How can graviton ‘B’ emit another graviton (also moving at the speed of light) that becomes absorbed by graviton ‘A’ moving at the speed of light? As we have seen, these difficulties are resolved by realizing that there are actually three different types of mass. There is measurable inertial mass and measurable gravitational mass, and low level ‘mass charge’ that cannot be directly measured. Inertial and gravitational masses have already been discussed and arise from different physical circumstances, but in most cases give identical results. However, the ‘low level mass charge’ of a particle is defined simply as the force existing between two identical particles due to the exchange of graviton particles only, which are the vector bosons of the gravitational force. Low level mass charge is not directly measurable, because of the complications due to the electromagnetic forces that are present simultaneously from the virtual particles. It would be interesting to speculate what the universe might be like if there were no quantum vacuum virtual particles present. Bearing in mind that the graviton exchange process is almost identical to the photon exchange process, and bearing in mind the complete absence of the electromagnetic component in gravitational interactions, the universe would be a very strange place indeed. We would find that large masses would fall faster than smaller masses, just as a large positive electric charge would ‘fall’ faster than a small positive charge towards a very large negative charge. There would be no inertia as we know it, and basically no force would be required to accelerate or stop a large mass. (A-8) The Quantum Field Theory of the Masseon and Graviton Particles EMQG addresses gravitational force, inertia, and electromagnetic forces only, and the weak and strong nuclear forces are excluded from consideration. EMQG is based on the idea that all elementary matter particles must get their quantum mass numbers from combinations of just one fundamental matter (and corresponding anti-matter particle), which has just one fixed unit or quanta of mass that we call the ‘masseon’ particle. This fundamental particle generates a fixed flux of gravitons that are exchanged during gravitational interactions. The exchange process is not affected by the state of motion of the masseon (as you might expect from the special relativistic variation of mass with velocity). We also purpose that nature does not have two completely different long-range forces, for example gravity and electromagnetism. Instead we believe that there exists an almost perfect symmetry between the two forces, which is hidden from view because of the mixing of these two forces in all measurable gravitational interactions. In EMQG the graviton and photon exchange process are found to be essentially the same, except for the strength of the force coupling (and a minor difference in the treatment of positive and45negative masses discussed later). EMQG treats graviton exchanges by the same successful methods developed for the behavior of photons in QED. The dimensionless coupling constant that governs the graviton exchange process is what we call ‘ β‘ in close analogy with the dimensionless coupling constant ‘ α‘ in QED, where β ≈ 10-40 α. As we stated, EMQG requires the existence of a new fundamental matter particle called the ‘masseon’ (and a corresponding ‘anti-masseon’ particle), which are held together by a new unidentified strong force. Furthermore, EMQG requires that masseons and anti- masseons emit gravitons analogous with the electrons and anti-electrons (positrons) which emit photons in QED. Virtual masseons and anti-masseons are created in equal amounts in the quantum vacuum as virtual particle pairs. A masseon generates a fixed flux of graviton particles with wave functions that induce attraction when absorbed by another masseon or anti-masseon; and an anti-masseon generates a fixed flux of graviton particles with an opposite wave function (anti-gravitons) that induces repulsion when absorbed by another masseon or anti-masseon. A graviton is its own anti-particle, just as a photon is its own antiparticle. This process is similar to, but not identical to the photon exchange processes in QED for electrons of opposite charge. In QED, an electron produces a fixed flux of photon particles with wave functions that induces repulsion when absorbed by another electron, and induces attraction when absorbed by a positron. A positron produces a fixed flux of photon particles with wave functions that induces attraction when absorbed by another electron, and induces repulsion when absorbed by a positron. From this it can be seen that if two sufficiently large pieces of anti-matter can be fabricated which are both electrically neutral, they will be found to repel each other gravitationally! Thus anti-matter can actually be thought of as ‘negative’ mass (-M), and therefore negative energy. This grossly violates the equivalence principle. These subtle differences in the exchange process in QED and EMQG produce some interesting effects for gravitation that are not found in electromagnetism. For example, a large gravitational mass like the earth does not produce vacuum polarization of virtual particles from the point of view of ‘mass-charge’ (unlike electromagnetism). In gravitational fields, all the virtual masseon and anti-masseon particles of the vacuum have a net average statistical acceleration directed downwards towards a large mass. This produces a net downward accelerated flux of vacuum particles (acceleration vectors only) that effects other masses immersed in this flux. In contrast to this, an electrically charged object does produce vacuum polarization. For example, a negatively charged object will cause the positive and negative (electrically charged) virtual particles to accelerate towards and away, respectively from the negatively charged object. Therefore, there is no energy contribution to other real electrically charged test particles placed near the charged object from the vacuum particles, because the electrically charged vacuum particles contribute equal amounts of force from both the upward and downward directions. The individual electrical forces from the vacuum cancel out to zero.46In gravitational fields, the vacuum particles are responsible for the principle of equivalence, precisely because of the lack of vacuum polarization due to gravitational fields. Recall that ‘masseon’ particles of EMQG are equivalent to the ‘parton’ particle concept that was introduced by the authors of reference 5 concerning HRP Quantum Inertia. Recall that the masseons and anti-masseons also carry one quantum of electric charge of which there are two types; positive and negative charges. For example masseons come in positive and negative electric charge, and anti-masseons also come in positive and negative charge. A single charged masseon particle accelerating at 1g sees a certain fixed amount of inertial force generated by the virtual particles of the quantum vacuum. In a gravitational field of 1g, a single charged masseon particle on the surface of the earth sees the same quantum vacuum electromagnetic force. In other words, from the vantage point of a masseon particle that makes up the total mass, the virtual particles of the quantum vacuum look exactly the same from the point of view of motion and forces whether it is in an inertial reference frame or in a gravitational field. We propose a new universal constant “i” called the ‘inertion’, which is defined as the inertial force produced by the action of virtual particles on a single (real charged) masseon particle undergoing a relative acceleration of 1 g. This force is the lowest possible quanta of inertial mass. All other masses are fixed integer combinations of this number. This same constant ‘i’ is also the lowest possible quanta of gravitational force. The electric charge that is carried by the electron, positron, quark and anti-quark originate from combinations of masseons, which is the fundamental source of the electrical charge. This explains why a fixed charge relationship exists between the quarks and the leptons, which belong to different families in the standard model. For example, according to the standard model, 1 proton charge precisely equals 1 electron charge (opposite polarity), where the proton is made of 3 quarks. This precise equality arises from the fact that charged masseon particles are present in the internal structure of both the quarks and the electrons (and every other mass particle). The mathematical renormalization process is applied to particles to avoid infinities encountered in Quantum Field Theory (QFT) calculations. This is justified by postulating a high frequency cutoff of the vacuum processes in the summation of the Feynman diagrams. Recall that QED is formulated on the assumption that a perfect space-time continuum exists. In EMQG, a high frequency cutoff is essential because space is quantized as ‘cells’, specified by Cellular Automata (CA) theory. In CA theory there is quantization of space as cells. If particles are sufficiently close enough, they completely lose their identity as particles in CA theory, and QFT does not apply at this scale. Since graviton exchanges are almost identical to photon exchanges, we suspect that EMQG is also renormalizable as is QED, with a high frequency cutoff as well. This has not been proven yet. The reason that some current quantum gravity theories are not renormalizable boils down to the fact that the graviton is assumed to be the only boson involved in gravitational interactions. The graviton must therefore exhibit all the characteristics of the gravitational field, including space-time curvature.47In EMQG, the photon exchange and graviton exchange process is virtually identical in its basic nature, which shows the great symmetry between these two forces. As a byproduct of this, the quantum vacuum becomes ‘neutral’ in terms of gravitational ‘mass charge’, as the quantum vacuum is known to be neutral with respect to electric charge. This is due to an equal number of positive and negative electrical charged virtual particles and ‘gravitational charged’ virtual particles created in the quantum vacuum at any given time. This in turn is due to the symmetrical masseon and anti-masseon pair creation process. (EMQG does not resolve the problem of why the universe was created with an apparent imbalance of real ordinary matter and anti-matter mass particles.) This distortion of the acceleration vectors of the quantum vacuum ‘stream’ serves as an effective ‘electromagnetic guide’ for the motion of nearby test masses (themselves consisting of masseons) through space and time. This ‘electromagnetic guide’ concept replaces the 4D space-time geodesics (which is the path that light takes through curved 4D space-time) that guide light and matter in motion. Because the quantum vacuum virtual particle density is quite high, but not infinite (at least about 1090 particles/m3), the quantum vacuum acts as a very effective and energetic guide for the motion of light and matter. (A-9) Introduction to 4D Space-Time Curvature and EMQG The physicist A. Wheeler once said that: “space-time geometry ‘tells’ mass-energy how to move, and mass-energy ‘tells’ space-time geometry how to curve”. In EMQG, this statement must be somewhat revised on the quantum particle level to read: large mass- energy concentrations (consisting of quantum particles) exchanges gravitons with the immediate surrounding virtual particles of the quantum vacuum, causing a downward acceleration (of the net statistical average acceleration vectors) of the quantum vacuum particles. This downward acceleration of the virtual particles of the quantum vacuum ‘tells’ a nearby test mass (also consisting of real quantum particles) how to move electromagnetically, by the exchange of photons between the electrically charged, and falling virtual particles of the quantum vacuum and the electrical charged, real particles inside the test mass. This new view of gravity is totally based on the ideas of quantum field theory, and thus acknowledging the true particle nature of both matter and forces. It is also shows how nature is non-geometric when examined on the smallest of distance scales, where Riemann geometry is now replaced solely by the interactions of quantum particles existing on a kind of quantized 3D space and separate time on the CA. Since this downward accelerated stream of charged virtual particles also affects light or real photons and the motion of real matter (for example, matter making up a clock), the concept of space-time must be revised. For example, a light beam moving parallel to the surface of the earth is affected by the downward acceleration of charged virtual particles (electromagnetically), and moves in a curved path. Since light is at the foundation of the measurement process as Einstein showed in special relativity, the concept of space-time must also be affected near the earth by this accelerated ‘stream’ of virtual particles. Nothing escapes this ‘flow’, and one can imagine that not even a clock is expected to keep48the same time as it would in far space. As a result, a radically new picture of Einstein’s curved space-time concept arises from these considerations in EMQG. The variation of the value of the net statistical average (directional) acceleration vector of the quantum vacuum particles from point to point in space (with respect to the center of a massive object) guides the motion of nearby test masses and the motion of light through electromagnetic means. This process leads to the 4D space-time metric curvature concept of general relativity. With this new viewpoint, it is now easy to understand how one can switch between accelerated and gravitational reference frames. Gravity can be made to cancel out inside a free falling frame (technically at a point) above the earth because we are simply taking on the same net acceleration as the virtual particles at that point. In this scenario, the falling reference frame creates the same quantum vacuum particle background environment as found in an non-accelerated frame, far from all gravitational fields. As a result, light travels in perfectly straight lines when viewed by a falling observer, as specified by special relativity. Thus in the falling reference frame, a mass ‘feels’ no force or curvature as it would in empty space, and light travels in straight lines (defined as ‘flat’ space-time). Thus the mystery as to why different reference frames produce different space-time curvature is solved in EMQG. It is interesting that in an accelerated rocket 4D space-time curvature is also present, but now is caused by another mechanism; the accelerated motion of the floor of the rocket itself. In other words, the space-time curvature, manifesting itself as the path of curved light, is really caused by the accelerated motion of the observer! The observer (now in a state of acceleration with respect to the vacuum), ‘sees’ the accelerated virtual particle motion in his frame. Furthermore, the motion appears to him to be almost exactly the same as if he were in an equivalent gravitational field. This is why the space-time curvature appears the same in both a gravitational field and an equivalent accelerated frame. These differences between accelerated and gravitational frames imply that equivalence is not a basic element of reality, but merely a result of different physical processes, which happen to give the same results. In fact, equivalence is not perfect! According to EMQG, all metric theories of gravity, including general relativity, have a limited range of application. These theories are useful only when a sufficient mass is available to significantly distort the virtual particle motion surrounding the mass; and only where the electromagnetic interaction dominates over the graviton processes (or where the graviton flux is not too large). For precise calculation of gravitational force interactions of small masses, EMQG requires that the gravitational interaction be calculated by adding the specific Feynman diagrams for both photon and graviton exchanges. Thus, the use of the general relativistic Schwarzchild Metric for spherical bodies (even if modified by including the uncertainty principle) is totally useless for understanding the gravitational interactions of elementary particles. The whole concept of space-time ‘foam’ is incorrect according to EMQG, along with all the causality problems associated with this complex mathematical concept.49(A-10) Space-Time Curvature is a Pure Virtual Particle Quantum Vacuum Process 4D Minkowski curved space-time takes on a radically new meaning in EMQG, and is no longer a basic physical element of our reality. Instead, it is merely the result of quantum particle interactions alone. The curved space-time of general relativity arises strictly out of the interactions between the falling virtual particles of the quantum vacuum near a massive object and a nearby test mass. The effect of the falling quantum vacuum acts somewhat like a special kind of “Fizeau-Fluid” or media, that effects the propagation of light; and also effects clocks, rulers, and measuring instruments. Fizeau demonstrated in the middle 1850’s that moving water varies the velocity of light propagating through it. This effect was analyzed mathematically by Lorentz. He used his newly developed microscopic theory for the propagation of light in matter to study how photons move in a flowing stream of transparent fluid. He reasoned that photons would change velocity by frequent scattering with the molecules of the water, where the photons are absorbed and later remitted after a small time delay. This concept is discussed in detail in section 9.3. If Einstein himself had known about the existence of the quantum vacuum when he was developing general relativity theory, he may have deduced that space-time curvature was caused by the “accelerated quantum vacuum fluid”. He was aware of the work by Fizeau, but was unaware of the existence of the quantum vacuum. After all, Einstein certainly realized that clocks were not expected to keep time correctly when immersed in an accelerated stream of water! We show mathematically in this paper that the quantity of space-time curvature near a spherical object predicted by the Schwarzchild metric is identical to the value given by the ‘Fizeau-like’ scattering process in EMQG. In EMQG when we find an accelerated vacuum disturbance, there follows a corresponding space-time distortion (including the possibility of gravitational waves for dynamic accelerated disturbances). We have seen that both accelerated and gravitational frames qualify for the status of curved 4D space-time (although caused by different physical circumstances). We have found that in EMQG there exists two, separate but related space-time coordinate systems. First, there is the familiar global four dimensional relativistic space-time of Minkowski, as defined by our measuring instruments, and is designated by the x,y,z,t in Cartesian coordinates. The amount of 4D space-time curvature is influenced by accelerated frames and by gravitational frames, which is the cause of the accelerated state of the quantum vacuum. Secondly there is a kind of a quantized absolute space, and separate time as required by cellular automaton theory. Absolute space consists of an array of numbers or cells C(x,y,z) that changes state after every new clock operation Δt. C(x,y,z) acts like the absolute three dimensional pre-relativistic space, with a separate absolute time that acts to evolve the numerical state of the cellular automata. The CA space (and separate time) is not affected by any physical interactions or directly accessible through any measuring instruments, and currently remains a postulate of EMQG. Note that EMQG absolute space does not correspond to Newton’s idea of absolute space. Newton postulated the existence of50absolute space in his work on inertia. He realized that absolute space was required in order to resolve the puzzle of what reference frame nature uses to gauge accelerated motion. In EMQG, this reference frame is not the absolute quantized cell space (which is unobservable), but instead consists of the net average state of acceleration of the virtual particles of the quantum vacuum with respect to matter (particles). A very important consequence of the existence of absolute quantized space and quantized time (required by cellular automaton theory) is the fact that our universe must have a maximum speed limit! (A-11) THE BASIC POSTULATES OF EMQG Here is a summary of the basic postulates of EMQG. Reference 1 gives a much more complete description of the postulates and their consequences. Note that we do not include Einstein’s principle of equivalence as one of our basic postulates. This is because equivalence is not a fundamental principle. Instead equivalence is simply a consequence of quantum particle interactions. The basic postulates of EMQG are: POSTULATE #1: CELLULAR AUTOMATA The universe is a vast cellular automaton computation, which has an inherently quantized absolute 3D space consisting of ‘cells’, and absolute time. The numeric information in a cell changes state through the action of the numeric content of the immediate neighboring cells (26 neighbors) and the local mathematical rules, which are repeated for each and every cell. The action of absolute time (through clock cycles) synchronizes the state transition of all the cells. The number of ‘clock cycles’ elapsed between the change of the numeric state on the CA is a measure of the absolute time elapsed. The cells are interconnected (mathematically) to form a simple 3D geometric CA. Matter, forces, and motion are the end result of information changing in the cells as absolute time progresses. Gravity, motion, and any other physical process do not effect low-level absolute 3D space and absolute time in any way. Photons propagate in the simplest possible manner on the CA. Photons simply shift from cell to adjacent cell on each and every 'clock cycle' in a given direction. This rate represents the maximum speed that information can be moved during a CA ‘clock cycle’. The quantization scale is not known yet, but it must be much finer than the Plank Scale for distance and time. POSTULATE #2: GRAVITON-MASSEON PARTICLES The masseon is the most elementary form of matter (or anti-matter), and carries the lowest possible quanta of low level, gravitational ‘mass charge’. The masseon carries the lowest possible quanta of positive gravitational ‘mass charge’, where the low level gravitational ‘mass charge’ is defined as the (probability) fixed rate of emission of graviton particles in close analogy to electric charge in QED. Gravitational ‘mass charge’ is a fixed constant and analogous to the fixed electrical charge concept. Gravitational ‘mass charge’ is not governed by the ordinary physical laws of observable mass, which appear as ‘m’ in the various physical theories, including Einstein’s special relativity mass-velocity relationship: E=mc2 or m = m 0 (1 - v2/c2)-1/2. Masseons simultaneously carry a positive gravitational51‘mass charge’, and either a positive or negative electrical charge (defined exactly as in QED). Therefore we conclude that masseons also exchange photons with other masseon particles. Masseons are fermions with half integer spin, which behave according to the rules of quantum field theory. Gravitons (which are closely analogous to photons) have a spin of one (not spin two, as is commonly thought), and travel at the speed of light. Anti- masseons carry the lowest quanta of negative gravitational ‘mass charge’. Anti-masseons also carry either positive or negative electrical charge, with electrical charge being defined according to QED. An anti-masseon is always created with an ordinary masseon in a particle pair as required by quantum field theory (specifically, the Dirac equation). The anti-masseon is the negative energy solution of the Dirac equation for a fermion, where now the mass is taken to be ‘negative’ as well, in clear violation of the principle of equivalence. Another important property exhibited by the graviton particle is the principle of superposition . This property works the same way as for photons. The action of the gravitons originating from all sources acts to yield a net vector sum for the receiving particle. EMQG treats graviton exchanges by the same successful methods developed for the behavior of photons in QED. The dimensionless coupling constant that governs the graviton exchange process is what we call ‘ β‘ in close analogy with the dimensionless coupling constant ‘ α‘ in QED, where β ≈ 10-40 α. POSTULATE #3: QUANTUM THEORY OF INERTIA The property which Newton called the inertial mass of an object, is caused by the vacuum resistance to acceleration of all the individual, electrically charged masseon particles that make up the mass. This resistance force is caused by the electromagnetic force interaction (where the details of this process are unknown at this time) occurring between the electrically charged virtual masseon/anti-masseon particle pairs created in the surrounding quantum vacuum, and all the real masseons particles making up the accelerated mass. Therefore inertia originates in the photon exchanges with the electrically charged virtual masseon particles of the quantum vacuum. The total inertial force F i of a mass is simply the sum of all the little forces f p contributed by each of the individual masseons, where the sum is: F i = (Σ fp) = MA (Newton’s law of inertia). POSTULATE #4: PHOTON FIZEAU-LIKE SCATTERING IN THE VACUUM Photons have an absolute, fixed velocity resulting from its special motion on the CA, where photons simply shift from cell to adjacent cell on every CA ‘clock cycle’. This ‘low level’ photon velocity (measured in CA absolute space and time units) is much higher (by an unknown amount) than the observed light velocity of 300,000 km/sec. This is because a photon traveling in the vacuum (in an inertial frame) takes on a path through the quantum vacuum, that is the end result of a vast number of electromagnetic scattering processes with the surrounding electrically charged virtual particles. Each scattering process introduces a small random delay in the subsequent remission of the photon, and results in a cumulative reduction in the velocity of photon propagation. Real photons that travel near a large mass like the earth, take a path through the quantum vacuum that is the end result of a large number of electromagnetic scattering processes with the falling (statistical52average) electrically charged virtual particles of the quantum vacuum. The resulting path is one where the photons maintain a net statistical average acceleration of zero with respect to the electrically charged virtual particles of the quantum vacuum, through a process that is very similar to the Fizeau scattering of light through moving water. Through very frequent absorption and re-emission (which introduces a small delay) by the accelerated charged virtual particles of the quantum vacuum, the apparent light velocity assumes an accelerated value with respect to the center of mass in absolute CA space and time units. (Note: The light velocity is still an absolute constant when moving in between virtual particles, and is always created at this fixed constant velocity dictated by the CA rules ). The accelerated virtual particles of the quantum vacuum that appears in gravitational and accelerated reference frames can be viewed as a special Fizeau-like vacuum fluid. This fluid effects the motion of matter and light in the direction of the fluid acceleration, which is ultimately responsible for 4D space-time curvature. (A-12) Experimental Verification of EMQG Theory EMQG proposes several new experimental tests that give results that differ from the conventional general relativistic physics, and can thus be used to verify the theory. (1) EMQG opens up a new field of physics, which we call anti-matter gravitational physics. We propose that if two sufficiently large pieces of anti-matter are manufactured to allow measurement of the mutual gravitational interaction (with a torsion balance apparatus for example), then the gravitational force will be found to be repulsive! The force will be equal in magnitude to -GM2/r2 where M is the mass of each of the anti-matter masses, r is their mutual separation, and G is Newton’s gravitational constant. This is a gross violation of the principle of equivalence, since in this case, M i = - M g , instead of being strictly equal. Antimatter that is accelerated in far space has the same inertial mass ‘Mi’ as ordinary matter, but when interacting gravitationally with another antimatter mass it is repelled (M g). Note that the earth will attract bulk anti-matter because of the large abundance of gravitons originating from the earth of the type that induce attraction. This means that no violation of equivalence is expected for anti-matter dropped on the earth, where anti-matter falls normally. However, an antimatter earth will repel a nearby antimatter mass. Recent attempts at measuring earth’s gravitational force on anti-matter (for example on anti-protons) will not reveal any deviation from equivalence, according to EMQG. However, if there were two large identical masses of matter and anti-matter close to each other, there would be no gravitational force existing between them because of the balance of “positive and negative” masses, for example equal numbers of gravitons that induce attraction and repulsion. This gravitational system is considered gravitationally ‘neutral’ as is the quantum vacuum, which is also gravitationally neutral. (2) For an extremely large test mass and a very small test mass that is dropped simultaneously on the earth (in a vacuum), there will be an extremely small difference in the arrival time of the masses on the surface of the earth in slight violation of the principle of equivalence. This effect is on the order of ≈ ΔN x δ, where ΔN is the difference in the number of masseon particles in the two masses, and δ is the ratio of the gravitational to53electric forces for one masseon. This experiment is very difficult to perform on the earth, because δ is extremely small ( ≈10-40), and ΔN cannot be made sufficiently large. To achieve a difference of ΔN =1030 masseons particles between the small and large mass requires dropping a molecular-sized cluster and a large military tank simultaneously in the vacuum in order to give a measurable deviation. Note that for ordinary objects that might seem to have a large enough difference in mass (like dropping a feather and a tank), the difference in arrival time would be obscured by background interference, and possibly by quantum effects like the Heisenberg uncertainty principle which restrict the accuracy of arrival time measurements. (3) If gravitons can be detected by the invention of a graviton detector/counter in the far future, then there will be experimental proof for the violation of the strong principle of equivalence. The strong equivalence principle states that all the laws of physics are the same for an observer situated on the surface of the earth as it is for an accelerated observer at 1 g. The graviton detector will find a tremendous difference in the graviton count in these two cases. This is because gravitons are vastly more numerous here on the earth. Thus a detector can manufactured with an indicator that distinguishes between whether an observer is in an inertial frame or in a gravitational frame. This of course is a gross violation of the strong equivalence principle. (4) Since the gravitational mass of an object has a strong electrical force component, mass measurements near the earth might be disrupted experimentally by manipulating some of the electrically charged virtual particles of the nearby quantum vacuum through electromagnetic means. If a rapidly fluctuating magnetic field (or rotating magnetic field) is produced directly under a mass it might effect the instantaneous charged virtual particle spectrum and disrupt the tiny electrical forces for many of the masseons in the mass. This may reduce the measured gravitational (and inertial masses) of a test mass. In a sense this device would act like a primitive and weak “anti-gravity” machine. The virtual particles are constantly being “turned-over” in the vacuum at different rates, with the high frequency virtual particles (and therefore, the high-energy virtual particles) being replaced the quickest. If a magnetic field is made to fluctuate fast enough so that it does not allow the new electrically charged virtual particle pairs to replace the old and smooth out the disruption, the spectrum of the virtual particles in the vicinity may be altered. According to conventional physics, the energy density of virtual particles is infinite, which means that all frequencies of virtual particles are present. In EMQG there is an upper cut-off to the frequency, and therefore the highest energy according to the Plank’s law: E=h υ, where υ is the frequency that a virtual particle can have. We can state that the smallest wavelength that a virtual particle can have is about 10-35 meters, e.g. the plank wavelength (or a corresponding maximum Plank frequency of about 1043 hertz for very high velocity ( ≈c) virtual particles). Unfortunately for our “anti-gravity” device, it is technologically impossible to disrupt these highest frequencies. Recall that according to the uncertainty principle, the relationship between energy and time is: ΔE Δt > h/(2 π). This means that the high frequency end of the spectrum consists of virtual particles that “turns- over” the fastest. To give maximum disruption to a significant percentage of the high54frequency virtual particles require magnetic fluctuations on the order of at least 1020 cycles per seconds. Therefore only lower frequencies virtual particles of the vacuum can be practically affected in the future, and only small changes in the measured mass can be expected with today’s technology. As a result of this we conclude that the higher the frequency the greater the mass loss. Recent work on the Quantum Hall Effect by Laughlin on fractional electron charge suggests that, under the influence of a strong magnetic field, electrons might move in concert with swirling vortices created in the 2D electron gas. This leads to the possibility that this ‘whirlpool’ phenomenon also holds for the virtual particles of the quantum vacuum under the influence of a strongly fluctuating magnetic field. These high-speed whirlpools might disrupt the high frequency end of the distribution of electrically charged virtual particles into small pockets. Therefore, there might be a greater mass loss under these circumstances, an idea this is very speculative at this time. Experiments that report mass reduction associated with rapidly rotating superconducting magnets, which generate high frequency rotating magnetic fields are inconclusive at this time. Reference 6 gives an excellent and detailed review of the various experiments on reducing the gravitational force with superconducting magnets.55Figure #1 - BLOCK DIAGRAM OF RELATIONSHIP OF CA AND EMQG WITH PHYSICS* CELLULAR AUTOMATA PARADIGM The fastest known Parallel Computer Model. Here strict locality prevails, and there exists a maximum limiting speed for the transfer of information. There automatically exists an absolute, quantized 3D space in the form of 'cells', and quantized time. Quantum Field Theory and Quantum Electrodynamics (QED) All Forces result from Particle Exchanges. Dirac equ. predicts particle-antiparticle pair creation, with all charge types reversedQuantum Mechanics The links between this and Cellular Automata theory are not fully known.Special Relativity This theory follows as a direct consequence of Cellular Automata. * VIRTUAL PARTICLES OF THE VACUUM The existence of the 'Electrically-Charged' and 'Mass- Charged' Virtual Particles (Masseons) of the Vacuum. These are responsible for inertia. Their existence automatically resolves the Cosmological ConstantBOSON PARTICLE EXCHANGE PARADIGM * ElectroMagnetic Quantum Gravity (EMQG) Theory This theory is based on both Photon and Graviton exchanges occuring with the virtual particles. In Inertia, only Photon exchanges occur between matter particles and the Virtual Particles. In Gravitational Fields, this process still occurs with the addition of graviton exchanges with the vacuum particles. The Equivalence Principle is derived from this.Classical Electro- Magnetism General Relativity* QUANTUM INERTIA This is based on the Photon Exchanges between matter particles and Virtual Particles.Mach's Principle Deep connection with the vacuum.Newton's Laws of Motion Deep Connection with the quantum vacuum. A Finalized Quantum Gravity Theory Curved Riemann 4D Space-Time CurvatureGRAVITON PARTICLE Responsible for gravity. Principle of Equivalence * Newly Developed Theory56................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ............................................................................................................... ............................ ............................ ............................ ............................ ............................ ........................................ ............ ............................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... ................................................................................... .............................................................................................................. ............................ ............................ ............................ ............................ ............................ ........................................ ............ ............Acceleration of the Rocket is 1 g Figure 2A - Masses '2M' and 'M' at rest on the floor of the rocketFigure 2B - Masses '2M' and 'M' in free fall inside of a rocket Figure 2C - Masses '2M' and 'M' at rest on Earth's surfaceFigure 2D - Masses '2M' and 'M' in free fall above the Earth LEGEND: I = Relative downward acceleration (1g) of a virtual particle i = Relative downward acceleration (1g) of a real matter particle . = A real stationary matter particle (with respect to the earth's center)LEGEND : . = A virtual particle of the quantum vacuum (taken as the rest frame) = A real mass particle undergoing relative upward acceleration of 1g = A real matter particle at relative rest with respect to the vacuum Figure #2 -SCHEMATIC DIAGRAM OF THE PRINCIPLE OF EQUIVALENCE1 g 1 g . . . . i i i i . . i iIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIAcceleration of Box is 1 g Surface of the Earth where gravity produces a 1 g accelerationSNAPSHOT OF MASSES IN FREE FALLSNAPSHOT OF MASSES IN FREE FALL UNEQUAL MASSES AT REST ON SURFACEUNEQUAL MASSES AT REST ON THE FLOOR Equivalence Equivalence57

page: 1A twist in chiral interaction between biological helices A. A. Kornyshev Research Center "Jülich", D-52425 Jülich, Germany S. Leikin* Laboratory of Physical and Structural Biology, National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, MD 20892, USA Using an exact solution for the pair interaction potential, we show that long, rigid, chiral molecules with helical surface charge patterns have a preferential interaxial angle ~ RH L where L is the length of the molecules, R is the closest distance between their axes, and H is the helical pitch. Estimates based on this formula suggest a solution for the puzzle of small interaxial angles in α-helix bundles and in cholesteric phases of DNA. The existence of all living things depends on the molecular chirality of helices inside them. Chiral amino acids form chiral α-helices that self-assemble into structural domains of many proteins. Chiral DNA forms cholesteric liquid crystals right inside living cells [1]. Chiral interactions between biological helices present many puzzles [2]. In the cholesteric phase (Fig. 2), DNA molecules rotate by a fraction of a degree from layer to layer [1]. Their interaxial angle is much smaller than expected [3], and so is the angle between long α-helices in proteins and in membranes. This produces a macroscopic cholesteric pitch, ~0.4-5 µm in DNA [1] and even larger in cholesteric assemblies of α-helices in organic solvents [4]. One of the challenges of the physics of chiral macromolecules is to understand how the interaxial angle isencoded in intermolecular interactions [3]. Fig. 1 A simplified, heuristic model of interaction betweentwo long, net-neutral helical macromolecules in a nonpolarenvironment. On the left , the molecules are shown schematically by solid lines, their electric fields by light grayshading, and the overlap (interaction) area by dark grayshading. On the right , a small fragment of the overlap area is magnified in a side view. We assume that the molecularsurface charge pattern is composed of one negatively and onepositively charged, thin spiral lines shifted with respect toeach other by approximately one half of the pitch. To describe the position of the strands on each helix ( ν=1,2), we use the axial distance z ν between the positively charged strand on each molecule and the x-axis connecting the points of the closest approach between molecules (as shown for molecule 2on the right). At z 1=z2, the positively charged strand of one molecule opposes the negatively charged strand of the othermolecule creating a strong attraction. We assume that themolecules cross each other approximately in the middle.page: 2In the present paper, we suggest an explanation for the small angle puzzle. We start from a heuristic model of interaction between two long, net-neutral helices in an electrolyte-freeenvironment (Fig. 1). We present qualitative arguments followed by rigorous derivations based on an exact solution for the interaction potential. We apply the model to α-helix bundles in proteins and address the origin of the cholesteric pitch in DNA, extending our ideas tomultimolecular interaction in an electrolyte solution. Balance of forces . Consider two identical, right-handed, rigid, net-neutral helices (Fig. 1) immersed in an electrolyte-free medium and forming a small interaxial angle ψ (|sinψ|<<1). The length of each molecule, L, is much larger than its helical pitch, H. Each helix produces an electric field that decays as exp(- gr) with the distance r away from it (g=2π/H) [5]. The electric fields overlap over the length L eff(ψ) which rapidly varies with ψ (Fig. 1), e.g., Leff(ψ) diverges at ψ→0 for infinite helices. The energy of interaction between helices is Eint=u(ψ)Leff, where u(ψ) is the linear energy density. u(ψ) does not diverge at ψ→0 and it can be expanded at small ψ so that Eu R g u R g Leff int , , ...=++011616 1 6 yy .( 1 ) Here R is the closest approach distance between the molecular axes and u1(R,g)ψ is the energy density of chiral interaction that defines the direction of favorable twist [6]. The helices experience two torques, the “overlap torque” ( tu d L deff 1=− y) and the “chiral torque” ( td u d Leff 2=− y). If the helices are free to rotate around their axes, they will always select the most favorable alignment of their strands at which u<0 and the molecules attract each other (Fig. 1). Then, the overlap torque tends to reduce | ψ| to maximize the attraction. On the contrary, the chiral torque tends to increase | ψ|. The competition between these torques establishes a nonzero equilibrium interaxial angle. At small ψ, there are two distinct regions with different behavior of Leff(ψ): 1. (|ψ|>ψ*) Tips of the helices are separated by more than g-1. They contribute little to the interaction and Leff∝1/(g|sinψ|) (Fig. 1), i.e. LRg geff()sinψγ ψ=1 6.( 2 ) The coefficient γ(Rg) is derived below (see Eq. (10)). Within this region, tt uu12 0 1 1 ≈> >ψ 2 7 since |ψ|<<1. The overlap torque wins and it reduces | ψ|. 2. (|ψ|<<ψ*) Tips of the helices overlap and Leff stops following Eq.(2). Instead, it levels off at the value of the helix length, L. Since Leff has a maximum at ψ=0, dL deffy /c121 38→→ 00 and t1/t2<<1. The chiral torque wins in this region. The torques become equal at the crossover from the first to the second region. Exact evaluation of the equilibrium interaxial angle requires the exact Leff(ψ) in the crossover range. However, we can estimate that the crossover occurs at | ψ|~ψ*, where ψγ *=Rg Lg16(3)page: 3is obtained by extrapolating Eq. (2) to Leff(ψ)=L. The value of ψ* may serve as an upper estimate for the interaxial angle. According to Eq. (3), this angle is small for long molecules. Afterderiving Eqs. (1)-(3) rigorously, we will show that this may explain small interaxial angles in in vitro and in vivo helical aggregates. Rigorous derivation . The charge pattern on each molecule shown in Fig. 1 can be described in its own cylindrical coordinate frame with the long axis of the helix as the z-axis. The Fourier transform of the charge density in the molecular frame is given by σπδνν(,) ( ( )) e x p ( )qnZe g aingz q ngn=− − +0 211 16 ,( 4 ) where ~(,) (,)σπ φ σ φνπ νφqn d d z z e ein iqz=− −∞∞II21 0216 , ν(=1,2) labels the molecules, e0 is the elementary charge, Z is the number of elementary charges per helical pitch on each strand, and zν defines the alignment of each helix (Fig. 1). After substitution of Eq. (4), into the expression for Eint derived in [6], we find E kTZlg ngz mgz mwI nga I mga e ww wwBB nmnmmR g w ij nimj mn mnn nm nmmnm int ,~() , ,, ,,sincos ~()() () ~()~()~()~(),ψ πψ ψ ψψ ψψψ ≠=−− + ×+ + + +−+ =−∞ =+ =+∞ ∑02 1 112 12 21 21 21 222 16 49 49,( 5 ) where a is the radius of the helices, ε is the dielectric constant of the medium, lek TBB=02ε is the Bjerrum length, and ~()cos sin,wng mg mgnmψψ ψ=−. (6) Eq. (5) is valid at all ψ≠0, when the molecules are long enough so that their tips do not contribute much to the interaction. At ψ=0 (see Ref. [6]), E kTZLl g ng z z I nga K nRg BBn i niintcos ( )ψ π==− − = =+∞ ∑042 22 122 0 0 211616 1 6 (7) In Eqs. (5) and (7), In(x) and K0(x) are modified Bessel functions. At L→∞, Eqs. (5) and (7) are exact. They contain no approximations or assumptions, except for the choice of the surface charge distribution, Eq. (4). A similar result, can be obtained for any helical charge pattern [6]. Typically Rg>>1 [7], and the series in Eqs. (5) and (7) rapidly converge because of exp-nRg2 7. Therefore, the summation can be truncated after n,m=±1 so thatpage: 4E kTZlgI g a egz z egz zBBRg Rgintcos sin() sin cossin coscos ( ) sinsin sincos ( )ψ πψ ψψ ψ ψψ ψψ ψ≠≈− +   −% &K ’K +−   +( )K *K− −04 2122 21222 12 22 12 22 121 616 16 16 1616 16.( 8 ) By plotting Eint, one can easily find that the energy has a minimum at z1=z2 and ψ→0 [8]. Expanding Eint(ψ≠0) in small ψ at z1=z2 and comparing the result with Eq. (7), we recover Eqs. (1), (2), where uR g kTuRg kTZlg I g a K R g BBB012 22 12 04 ,,()16 1616 == −π,( 9 ) and gpRge KR gRg16=− 0(), (10) After substitution of Eq. (10) into Eq. (3) and using that Rg>>1, we find ψπ *≈=2Rg LRH L, (11) which is a remarkably simple combination of the only three length scales in the system. The energy gain upon the twist from ψ=0 to ψ~ψ* is E kTuR g L kTZl ae BBBgR a * *~,12 22 2 1 6 16 y p≈−−(12) where we used the asymptotic behavior of K0(Rg) and I1(ga) at large ga. Interaction between α-helices in proteins . Many proteins incorporate bundles of α- helices. The backbone of each α-helix contains a spiral of negatively charged carbonyl oxygens and a shifted spiral of positively charged amide hydrogens. In terms of the charge distribution, it resembles the heuristic model analyzed above where a≈2.3 Å, H≈5.5 Å, L/H≈3-10, R≈7-12 Å [9,10], Z≈1.7 (~0.5e0 per carbonyl or amide, ~3.5 groups per helical turn), and ε≈2 (lB≈300 Å). For such helices, we find ψ*~0.1-0.5 rad (7-30 deg). Of course, amino acid side chains impose packing constraints that affect interaxial angles and play a major role in determining the structure of α-helix bundles in proteins [9]. Still, electrostatic interaction between backbones of α-helices that do not have bulky side chains ( R≈7- 8 Å) is energetically significant ( E*~2-5 kBT) and it may be an important player as well [11]. For instance, steric interactions define a set of preferential interaxial angles rather than a single angle[9]. Electrostatics may then determine which angle from the set is most favorable. It may not be acoincidence that the average observed angle (~19 deg [9]) is in the middle of the range predictedfor chiral electrostatic interactions. Cholesteric pitch of DNA . Concentrated solutions of 500-Å-long DNA fragments in 10-page: 5300 mM salt form a cholesteric phase (see Fig. 2) at 32 Å< R<49 Å [12,13]. Such DNA fragments are short enough to behave as rigid rods [14] and long enough to have many helical turns, L/H≈15. Fig. 2 Alignment of DNA helices in the cholesteric phase. Left: A sketch of the cholesteric phase that consists of layers of parallel molecules. Each layer isslightly rotated with respect to the layer underneath.Right: Top view of a molecular layer showing most favorable alignments of molecules in the layer above(black rods): (1) when molecules are homogeneouslycharged and (2) when molecules have helical patterns offully balanced surface charges. Direct measurements of intermolecular forces demonstrated that the energetics of this cholesteric phase is determined primarily by electrostatic interactions [15]. The interactions areessentially pairwise since each DNA helix overlaps with only one molecule in the layer below it[16]. Furthermore, only nearest neighbor pairs contribute to the energy because of the rapid,exponential decay of the field. The net interaction between each two molecules can be viewed as a sum of two forces. The first one is a repulsion due to the fraction of DNA charge not balanced by bound orcondensed counterions (~20-25% of "naked" DNA charge [17]). This repulsion is the same as between homogeneously charged cylinders in an electrolyte solution [6]. It favors parallel ( ψ=0) alignment of helices in a multimolecular ensemble (Fig. 2) because this maximizesintermolecular separation and reduces the repulsion [2,18]. The second force is due to the compensated part of DNA charge. At optimal molecular alignment, it is an attraction between negatively charged phosphate strands on one molecule andpositively charged grooves on the opposing molecule [5,6,19]. The physics of this force is thesame as in our heuristic model (Fig. 1), but some details are different. Specifically, DNA is adouble-stranded helix and it has a more complicated surface charge pattern than in Fig. 1 [20]. In addition, electrolyte reduces the decay length of the electric field from g/c451 to gD2212+/c45k 27 , where κD-1 is the Debye length. These details are important for accurate quantitative predictions [6], but for order-of-magnitude estimates they can be neglected. From Eqs. (11), (12), where L≈500 Å, a≈10 Å, H≈34 Å, R~40 Å, |Z|≈40 [21], and lB≈7 Å (ε≈80), we find ψ*≈0.07 rad and E*≈2-5 kBT [20]. The competition between the first force favoring ψ=0 and the second force favoring |ψ|~ψ* determines whether DNA will form a cholesteric phase and the pitch of this phase [22]. Even without a many-body statistical theory, it is clear that the equilibrium interaxial angle should be smaller than ψ* and, therefore, the expected pitch should be larger than P*=2πR/ψ*≈0.4 µm. This is in full agreement with experimentally measured values that lie between 0.4 µm and ~5 µm [1,12,13]. Also in agreement with experiments [12,13], the pitch should not depend much on the salt concentration at 10-300 mM salt. Indeed, since g≈0.2 Å-1, Pg R LD * +2214 122 kp 27 1 6 is almost unaffected by the corresponding variation of κD from 0.03 to 0.2 Å-1. DNA molecules much longer than one persistence length also form a cholesteric phasepage: 6with about the same pitch as 500 Å fragments [1]. In this case, the theory is more complicated since our description of molecules as straight, rigid rods does not apply. However, assuming thateach molecule behaves as a collection of independent, one-persistence-length-long fragments[14], we can explain the observations. Thus, the macroscopic cholesteric pitch in DNA aggregates may have the following origin. When tips of one-persistence-length long DNA fragments are separated (in lateralprojection) by more than g -1, the attraction between negatively charged phosphate strands and positively charged grooves tends to reduce ψ. When the tips are separated by ≥kD/c451, the repulsion associated with the uncompensated charge of DNA also tends to reduce ψ. These two forces overpower the chiral torque, reducing the separation of the tips and resulting in the pitch ≥gR L LD2214 1228 +k p 27 1 6~~0.4 µm. A more detailed theory, based on the same ideas, also shows why the chiral torque may disappear at R≤32 Å (a possible solution for the puzzle of nematic-to-cholesteric transition at R=32 Å) and it demonstrates that the direction of the chiral torque can be reversed upon a change in counterion binding pattern or in separation [6]. In conclusion , let us emphasize that the goal of this letter is to illustrate on a conceptual level why chiral helical molecules may not twist more than a couple degrees in the cholesteric phase of DNA and in bundles of long α-helices. Of course, estimates are not a substitute for an accurate statistical theory that accounts for the pair potential between helices of finite length andfor thermal motion. They are intended only to pave the way. The agreement with experimentsindicates, however, that we may be on the right track. We thank V.A. Parsegian and D.C. Rau for useful discussions. This work was started within the 1998 program “Electrostatic Effects in Complex Fluids and Biophysics” at theInstitute for Theoretical Physics, University of California at Santa Barbara. The program wassupported by the NSF Grant No. PHY94-07194. In addition, AAK acknowledges the financialsupport of his visits to Bethesda by the National Institute of Child Health and HumanDevelopment, NIH which allowed to complete this project. References and Footnotes * To whom reprint requests should be addressed. LPSB/NICHD, Bldg. 12A, Rm. 2041, NIH, Bethesda, MD 20892, USA; e-mail leikin@helix.nih.gov; FAX 1-301-496-2172. [1] Yu.M. Yevdokimov, S.G. Skurdin, and V.I. Salyanov, Liq. Cryst., 3, 1443 (1988); F. Livolant, Physica A, 176, 117 (1991). [2] P.G. deGennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993), 2nd ed. [3] A.B. Harris, R.D. Kamien, and T.C. Lubensky, Phys. Rev. Lett., 78, 1476 (1997). See also a review by the same authors [e-Print # 9901174 available from http://xxx.lanl.gov, Condensed Matter (1999)]. [4] C. Robinson, Tetrahedron, 13, 219 (1961); F. Livolant, J. Physique (Paris), 47, 1605 (1986); D.B. Dupre and R.W. Duke, J. Chem. Phys., 63, 143 (1975). [5] A.A. Kornyshev and S. Leikin, J. Chem. Phys., 107, 3656 (1997); Erratum, ibid., 108, 7035 (1998). A.A. Kornyshev and S. Leikin, Proc. Natl. Acad. Sci. USA, 95, 13579 (1998). [6] A.A. Kornyshev and S. Leikin, submitted to Phys. Rev. E[7] The diameter (2 a) of most biological helices is comparable to their pitch (2 a/H~1). Since Rg≥2ag=4πa/H, we find that Rg≥2π and the approximation of Rg>>1 is almost always valid. [8] From Eq. (8) one can directly estimate the energetic cost of rotation of each helix around its axis at fixed height. Such rotation of helix ν by Δφν results in the change of zν by gzν. Both for α-helices and forpage: 7DNA the cost of rotation is much larger than kBT (except at the very edge of existence of the cholesteric phase of DNA, R≈50 Å). This, in the first approximation, thermal rotations can be neglected. [9] See, e.g., C. Chothia, M. Levitt, and D. Richardson, J. Mol. Biol., 145, 215 (1981); P.B. Harbury, P.S. Kim, and T. Alber, Nature, 371, 80 (1994). [10] For heuristic reasons, we use this crude approximation of the α-helix charge pattern. In principle, the formalism developed in [6] allows more precise analysis.[11] The contribution of van der Waals to the chiral energy calculated based on the formulas derived in [S.A. Issaenko, A.B. Harris, T.C. Lubensky, Phys. Rev. E, 60, 578 (1999)] gives EvdW *≈0.06 kBT at R=7 Å, i.e., it is much smaller than the contribution of electrostatic forces. [12] A. Leforestier and F. Livolant, Biophys. J., 65, 56 (1993); D.H. Van Winkle, M.W. Davidson, W.X. Chen, and R.L. Rill, Macromolecules, 23, 4140 (1990). [13] D. Durand, J. Doucet, and F. Livolant, J. Phys. II France, 2, 1769 (1992). [14] The persistence length of DNA is ~500-1000 Å depending on the ionic strength. [15] H.H. Strey, V.A. Parsegian, and R.P. Podgornik, Phys. Rev. E, 59, 999 (1999). [16] In the direction normal to the layer below, DNA traverses the distance L|sinψ|≈Lψ≤ 25 Å (Fig. 2). This is less than the interaxial separation 32 Å< R<49 Å [13] between the molecules in this layer. [17] G.S. Manning, Q. Rev. Biophys., 11, 179 (1978). [18] This interaction tends to put axes of two isolated molecules at ψ=±π/2 [S.L. Brenner and V.A. Parsegian, Biophys. J., 14, 327 (1974)]. However, in a multimolecular aggregate such alignment reduces the minimal distance between molecules at the same density of the phase (compared to hexagonal packing of parallel helices). As a result this homogeneous repulsion favors ψ=0. [19] A.A. Kornyshev, S. Leikin, Phys. Rev. Lett., 82, 4138 (1999). [20] In the absence of counterion chemisorption, one should speak of an excess negative charge onphosphate strands and an excess positive charge in grooves separating the strands, rather than of a fixedcharge pattern like in Fig. 1. Quantitatively, this excess may vary with the counterion nature, but mostmonovalent counterions do exhibit a preference to reside in grooves (see e.g. [V.N. Bartenev, E. I. Golovanov, K.A. Kapitonova, M.A. Mokulskii, L.I. Volkova, and I. Ya. Skuratovskii, J. Mol. Biol., 169, 217 (1983); X. Shui, L. McFail-Isom, G.G. Hu, L.D. Williams, Biochemistry, 37, 8341 (1998)]). Details of the counterion adsorption (condensation) pattern may have a strong effect on the absolute value of theenergy of chiral interactions, but not on the most energetically favorable angle between helices [6].[21] Two phosphate strands contribute 20 elementary charges per pitch. Their electrostatic images on thenonpolar dielectric core of DNA effectively double the charge to Z~40. [22] In 1:1 electrolytes, the first force is stronger than the second one resulting in net repulsion between molecules [19]. Still, the first, non-chiral force produces no torque at ψ=0 (as it follows from symmetry). The second, chiral force does produce a torque at ψ=0. In terms of their effect on the interaxial angle, the second force is dominant at | ψ|<<ψ ∗ while the first force may become important at | ψ|~ψ∗.

arXiv:physics/9911064v1 [physics.atom-ph] 25 Nov 1999EUROPHYSICS LETTERS Europhys. Lett. , (), pp. () Residual Symmetries in the Spectrum of Periodically Driven Alkali Rydberg States Andreas Krug1,2,3and Andreas Buchleitner1,3 1Max-Planck-Institut f¨ ur Physik komplexer Systeme, N¨ oth nitzer-Str. 38, D-01069 Dresden;2Max-Planck-Institut f¨ ur Quantenoptik, Hans-Kopfermann -Str. 1, D-85748 Garching b. M¨ unchen;3Sektion Physik der Ludwig-Maximilians-Universit¨ at M¨ un chen, Schellingstr. 4, D-80799 M¨ unchen. (received ; accepted ) PACS. 32.80Rm– Multiphoton ionization and excitation to hi ghly excited states (e.g., Rydberg states). PACS. 05.45+b – Theory and models of chaotic systems. PACS. 42.50Hz – Strong-field excitation of optical transiti ons in quantum systems; multi- photon processes; dynamic Stark shift. Abstract. – We identify a fundamental structure in the spectrum of micr owave driven alkali Rydberg states, which highlights the remnants of the Coulom b symmetry in the presence of a non-hydrogenic core. Core-induced corrections with respe ct to the hydrogen spectrum can be accounted for by a perturbative approach. Introduction. – The excitation and subsequent ionization of Rydberg state s of atomic hydrogen by microwave fields is one of the most prominent exam ples of the manifestation of classically nonlinear dynamics in a realistic physical s ystem [1]. Given a driving field frequency comparable to the classical Kepler frequency of t he unperturbed Rydberg electron, the electron’s classical trajectory goes chaotic for suffici ently large driving field amplitudes, finally leading to its ionization on a finite time scale [2]. Co rrespondingly, large ionization rates are observed in experiments on real (i.e., quantum) Ry dberg states of atomic hydrogen, in the appropriate parameter range [1, 3]. As a matter of fact, already before the onset of classically c haotic motion, i.e. at not too large driving field amplitudes, individual quantum eige nstates of the atom in the field exhibit energies and ionization rates which are determined only by the orbital parameters of the classical trajectory they are associated with [4]. Th ose orbits which are the least stable under the external perturbation (i.e., which turn ch aotic for the lowest values of the driving field amplitude, such as straight line orbits parall el to the field polarization axis for a linearly polarized drive) induce the largest ionization r ates for their associated eigenstates. Consequently, in this near-integrable regime of classical dynamics, it is possible to classify the eigenstates of the atom in the field through quantum numbe rs associated with the orbital parameters of unperturbed Kepler ellipses, i.e. with the an gular momentum and the Runge- Typeset using EURO-T EX2 EUROPHYSICS LETTERS Lenz vector. An adiabatic invariant governs the slow evolut ion of these parameters under external driving [4]. It should be noted, however, that a considerable part of expe rimental data has been accumulated in experiments on Rydberg states of alkali atom s rather than of atomic hydrogen [5, 6, 7, 8, 9, 10]. A priori, a classical-quantum correspond ence as briefly sketched above for atomic hydrogen cannot be established here, due to the absen ce of a well and uniquely defined classical Hamiltonian. In particular, the atomic core dest roys the symmetry characteristic for the hydrogen atom and the Runge-Lenz vector is no more a const ant of motion. Indeed, experimental data systematically suggest strongl y enhanced ionization rates of nonhydrogenic (i.e., low angular momentum) alkali Rydberg states as compared to atomic hydrogen [5, 6, 7, 9, 10], though they also exhibit qualitati vely similar features, e.g. of the dependence of the ionization yield on the principal quantum number of the atomic state the atoms are initially prepared in [9, 10]. On the other hand, a d irect comparison of available hydrogen and alkali data is somewhat questionable, since re levant experimental parameters such as the interaction time of the atom with the field are typi cally different for different experiments. Furthermore, a rigourous theoretical treatm ent of alkali atoms exposed to microwave fields was not accomplished until now. It is the purpose of the present letter to outline such a rigou rous treatment which allows for the first time for a direct comparison of hydrogen and alkali ionization dynamics unde rprecisely the same conditions, without adjustable parameters. First results of our numerical experiments directly address the above question of quantum-classical c orrespondence for periodically driven alkali atoms. Theory. – Let us start with the nonrelativistic Hamiltonian of a one- electron atom exposed to a linearly polarized microwave field of (constant) amplit udeFand frequency ω, in length gauge, employing the dipole approximation and atomic units : H(t) =p2 2+Vatom(r) +Fzcosωt, r > 0. (1) As this Hamiltonian is periodic in time, we can use the Floque t theorem [11] to find the eigenstates (“dressed states”) of the atom in the field. Afte r integration over the solid angle we have to solve the time-independent, radial eigenvalue eq uation /parenleftbigg −d2 dr2+ℓ(ℓ+ 1) r2+ 2Vatom(r)−2kω−2ε/parenrightbigg |Ψk ε,ℓ/angbracketright +FrA ℓ+1/parenleftBig |Ψk−1 ε,ℓ+1/angbracketright+|Ψk+1 ε,ℓ+1/angbracketright/parenrightBig +FrA ℓ/parenleftBig |Ψk−1 ε,ℓ−1/angbracketright+|Ψk+1 ε,ℓ−1/angbracketright/parenrightBig = 0, withAℓ=/radicalbigg ℓ2−m2 4ℓ2−1;ℓ= 0,1,2, . . .;k=−∞, . . .,+∞. (2) The additional quantum number kcounts the number of photons that are exchanged between the atom and the field, and εdenotes the quasi-energy of the dressed state |Ψε/angbracketright=/summationdisplay kexp(−ikωt)|Ψk ε/angbracketright=/summationdisplay k,ℓexp(−ikωt)Yℓ,m(θ, φ)|Ψk ε,ℓ/angbracketright/r, (3) withYℓ,m(θ, φ) the spherical harmonics. mdenotes the angular momentum projection on the field polarization axis and remains a good quantum number, du e to the rotational symmetry of our problem around the field axis. For all numerical results p resented hereafter, its value was fixed to m= 0. As immediately obvious from the nondiagonal part of eq. ( 2), the interaction with the linearly polarised microwave field conserves the ge neralised parity Π = ( −1)k+ℓ. ThisA. Krug and A. Buchleitner Residual Symmetries in the Spectr um of Periodically Driven Alkali Rydberg States 3 just expresses the angular momentum transfer associated wi th the absorption (emission) of a photon. As a unique one-particle potential Vatom(r) for alkali atoms is unknown, we use a variant [12] of R-matrix theory to describe the interaction of the ou ter electron with the atomic core. Configuration space is divided in two regions: In the interna l region, 0 < r≤a, the external field is negligible compared to the field created by the atomic core, and the details of the interaction are unknown. With the help of quantum defect the ory [13], the solution of eq. (2) atr=acan be written as a linear combination of regular and irregul ar Coulomb-functions sℓ,E(r) and cℓ,E(r), Fℓ,E(r) = cos( πδℓ)sℓ,E(r) + sin( πδℓ)cℓ,E(r), r=a, (4) where the δℓare the quantum defects [13] known from spectroscopic exper imental data [14]. In the outer region, r > a, the difference between the actual atomic potential Vatom(r) and the Coulomb potential −1/rcan be neglected. However, the operator d2/dr2is no more hermitian in the reduced range a < r < ∞. To overcome this problem, a surface term δ(r−a)(∂ ∂r+Cℓ) is added [12, 16] to the diagonal part of (2). The matching con dition between inner and outer region at r=ais incorporated in the constant Cℓby defining Cℓ= (Fℓ,ε+kω(r))(−1)∂ ∂rFℓ,ε+kω(r). (5) Note that the function Fℓ,E(r) in eq. (4) has to be evaluated at the energy ε+kωin (5), i.e. at different energies for different photon indices k. This generalizes the approach outlined in [12] to periodically driven systems. Finally, due to the continuum coupling induced by the extern al field, all atomic bound states turn into resonances with finite ionization rates Γ ǫ. In order to extract the latter together with the energies ǫof the atom in the field, we use the method of complex scaling [1 5, 17]. After this nonunitary transformation the Floquet Hamiltonian am ended by the core induced surface term (5) is represented by a complex symmetric matrix, with c omplex eigenvalues ε−iΓε/2. These are obtained by diagonalization of the complex eigenv alue problem in a real Sturmian basis, using an efficient implementation of the Lanczos algor ithm. Together with the associated eigenvectors they provide a complete description of our pro blem [15]. Results. – The described theoretical/numerical apparatus is now app lied to alkali atoms in a microwave field. Since we want to identify the core induced e ffects in the alkali problem as compared to the hydrogen spectrum, we use parameter values w hich have been employed in earlier work on microwave driven Rydberg states [4, 15] of hy drogen. To keep the comparison as transparent as possible, we focus on a microwave frequenc yω= 1.07171794 ×10−4a.u. which is nonresonant with the hydrogen level spacing in the v icinity of the atomic initial state with principal quantum number n0= 23. The field amplitude is fixed to F= 1.072×10−7a.u., slightly below the onset of appreciable (chaos-induced [2] ) ionization of atomic hydrogen [4]. This choice of parameters defines a near-integrable phase sp ace structure for the classical dynamics of driven hydrogen, with an unambiguous signature in the associated quantum energies emerging from the n0= 23 manifold. The black dots in fig. 1 illustrate the situatio n: The driving field lifts the angular momentum degeneracy of th e substates of the manifold, which reorganize according to their localization properti es in classical phase space [4]. Those states with maximum angular momentum and spherical symmetr y experience the strongest field induced (“ac-”) shift in energy, whereas those with max imum radial component of the Runge-Lenz vector and “ λ-symmetry” [4, 18, 19] remain essentially unaffected by the e xternal perturbation. Since the low angular momentum states are str ongly mixed by the field (to4 EUROPHYSICS LETTERS build states with λ-symmetry [18, 19]), a new (semiclassical) quantum number p[4] is used to label the n0substates of the manifold in the field. pis an integer ranging from 0 to n0−1, and simply counts the number of quanta enclosed by a semiclas sical contour integral along the equipotential curves of the adiabatic Hamiltonian whic h generates the slow evolution of angular momentum and Runge-Lenz vector of the classical K epler ellipse under external driving [4]. The associated eigenstates exhibit spherical symmetry for p= 0. . .9, and λ- symmetry for p= 10. . .22, respectively [4]. Note that low and high p-values correspond to negligible ionization rates of the atom in the field, due to th eclassical stability of the associated trajectories under external driving [4]. Actually, the λ-states with large p, which quantize a classical straight line orbit perpendicular to the field pol arization axis, with maximum modulus of the Runge-Lenz vector, display the smallest ionization r ates [4]. In the presence of a non-hydrogenic core, the Runge-Lenz vec tor is no more a conserved quantity and the λ-symmetry defining associated eigenstates of the field free a tom [18] is destroyed. Therefore, no symmetry argument is available to predict a similar (semiclassical) organization of the alkali energy levels under external dri ving, alike the one observed for atomic hydrogen [4]. Nonwithstanding, our results for lithium Rydberg states ex posed to precisely the same external perturbation as for the hydrogen results clearly s how that the symmetry properties of the driven Coulomb problem prevail even in the presence of the core. As evident from the open triangles in fig. 1 (a), the hydrogenic part of the lithiu m manifold exhibits globally the same (semiclassical) structure as the hydrogen levels. For low values of p(≃0. . .9) this is not surprising as the associated classical trajectories (larg e angular momenta) do not probe the atomic core [4]. However, for large p-values ( ≃10. . .20), the classical solution of the Coulomb problem does impinge on the nucleus and will certainly suffer scattering off the nonhydrogenic core. Yet, in the presence of the field, this scattering obvio usly mixes states of λtype only and does not affect the overall separation of the spectrum in sphe rical and λstates, as a remnant of the classical phase space structure of the driven Coulomb dy namics. Neither does the presence of the core appreciably affect the ionization rates of the dre ssed states, as obvious from fig. 1 (b). Only at p= 10 is there a local enhancement of the width (by approx. one o rder of magnitude), due to the near resonant coupling of the state to the nonhydrogenic eigenstate originating from |n= 41, ℓ= 0/angbracketright, via a six-photon transition (similarly, a very weak mutlip hoton coupling slightly enhances the width of the p= 12 state). In the near integrable regime of the classical Coulomb dynamics we are considering here it is pre cisely this kind of multiphoton resonances between nonhydrogenic (low ℓ, such that δℓ/negationslash= 0) states and hydrogenic manifolds which provides a channel for enhanced ionization as compare d to atomic hydrogen. Note that without such a near resonant coupling, the non-hydrogenic s tates of a given manifold tend to bemore stable than the hydrogenic ones, as they are highly isolated in the spectrum. As an example, for the same field parameters, the lithium n0= 23ℓ= 0 (δℓ=0= 0.399468) and ℓ= 1 (δℓ=1= 0.047263) [14] states exhibit ionization rates Γ ε∼10−15a.u.as small as the most stable substates of the hydrogenic manifold of fig. 1. A d etailed analysis of enhanced ionization via core-induced multiphoton resonances will b e provided elsewhere. Closer inspection of fig. 1 (a) shows additional structure in the alkali spectrum, on top of the globally hydrogen-like structure: for large values of p(≥11), the alkali levels are shifted with respect to the hydrogenic energies. These shifts can be recovered by diagonalization of the hydrogen problem within the restricted subspace spanne d by the hydrogenic levels of the alkali Rydberg manifold [19, 20, 21, 22]. In other words, the shifted energies are the solutions of the eigenvalue equation PHhydP|Φk0/angbracketright= (E+k0ω)|Φk0/angbracketright, (6)A. Krug and A. Buchleitner Residual Symmetries in the Spectr um of Periodically Driven Alkali Rydberg States 5 where Hhydis obtained from from (1) setting Vatom(r) =−1/r, r∈]0,∞[, and Pthe projector onto the hydrogenic subspace of the alkali manifold labeled by the principal quantum number n0and the photon number k0. Such a procedure is legitimate as long as the states emergin g from the nonhydrogenic part of the alkali manifold have vani shing overlap with the complete hydrogen manifold emanating from ( n0, k0). This condition is fulfilled for the driving field strength considered here. Solving (6) for Eis tantamount to finding the roots of det(Q1 Hhyd−(E+k0ω)Q) = 0, (7) withQ= 1−Pthe projector onto the orthogonal complement of the hydroge nic subspace for given ( n0, k0). Without loss of generality we choose k0= 0 hereafter. Consequently, for one single non-vanishing quantum defect δℓ0, (7) becomes /summationdisplay ε|/angbracketleftn0, ℓ0|Ψk0=0 ε/angbracketright|2 ε−E= 0, (8) where |n0, ℓ0/angbracketrightspans the orthogonal complement of the hydrogenic subspace of the alkali atom within the ( n0, k0= 0) manifold. Note that (7) or (8) have to be evaluated for diff erent values of the generalized parity Π, and that we have to solve (8) sepa rately for ℓ0= 0 and ℓ0= 1, in order to recover the level shifts observed for lithium in fig. 1 (the ℓ0= 2 and ℓ0= 3 states of lithium remain within the range of P, due to their negligible quantum defects δℓ=2= 0.002129 andδℓ=2=−0.000077 [14], at the given field strength). Fig. 2 (a) shows the result of the projection method, compared to the exact numerical result – the agreement is very good. Since the low pstates essentially exhibit spherical symmetry with large a ngular momentum projection, their overlap with |n0, ℓ0= 0(ℓ0= 1)/angbracketrightvanishes and their energies remain unshifted as compared to the hydrogen results. The scenario which we described for lithium also applies for the heavier alkali elements, as illustrated in figs. 2 (b) and (c). Here we plot the shifts of th e exact energies of sodium and rubidium with respect to the hydrogen levels, as they emerge from the n0= 23 manifold, for precisely the same field parameters as used for the lithium re sults. Since for these elements also the ℓ1= 2 (sodium) and the ℓ1= 3 (rubidium) states are separated from the hydrogenic manifold due to their large quantum defects, the range of Qin (7) is two-dimensional and the evaluation of the determinant yields the expression /summationdisplay ε|/angbracketleftn0, ℓ0|Ψk0=0 ε/angbracketright|2 ε−E/summationdisplay ε|/angbracketleftn0, ℓ1|Ψk0=0 ε/angbracketright|2 ε−E−/bracketleftBigg/summationdisplay ε/angbracketleftn0, ℓ0|Ψk0=0 ε/angbracketright/angbracketleftΨk0=0 ε|n0, ℓ1/angbracketright ε−E/bracketrightBigg2 = 0.(9) Again, the solution of (9) gives very good agreement with the numerical result. In addition, we note that the larger the dimension of the range of Q, the smaller the values of pfor which the alkali levels are shifted as compared to the hydrogen ene rgies. This is a consequence of the dominance of small ℓcomponents in large pstates and of large ℓcomponents in small p states, since the heavier the element the larger the ℓvalues affected by non-negligible quantum defects. Summary. – In conclusion, the energy levels of alkali Rydberg states e merging from the hy- drogenic n0-manifold clearly reflect the phase space structure of the mi crowave driven Coulomb problem, despite the presence of a symmetry breaking atomic core. Also the ionization rates of the atoms reflect the underlying classical phase space str ucture, with the exception of local enhancements due to multiphoton resonances with nonhydrog enic sublevels of other manifolds.6 EUROPHYSICS LETTERS We have checked that the observed structure is robust under c hanges of the driving field amplitude, up to values where adjacent n-manifolds start to overlap. *** We thank Dominique Delande and Ken Taylor for fruitful discu ssions and an introduction to the R-matrix approach of [12]. 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[19]Delande D. ,Th` ese d’ Etat , Universit´ e Pierre et Marie Curie, Paris 1988 . [20]Fabre C. et al.,J. Phys. B ,17(1984) 3217. [21]Braun P. A. ,J. Phys. B ,18(1985) 4187. [22]Penent F. et al.,Phys. Rev. A ,15(1988) 4707.A. Krug and A. Buchleitner Residual Symmetries in the Spectr um of Periodically Driven Alkali Rydberg States 7 0 5 10 15 20 Quantum Number p0123456Ionisation Rate [10−12 a.u.]0 5 10 15 20 −9.475−9.465−9.455−9.445Energy [10−4 a.u.](a) (b) 0 5 10 15 20 Quantum Number p0240123E alk −E hyd [10 −7a.u.]0 5 10 15 20 0123 (a) (b) (c) Fig. 1. – Energies (a) and ionisation rates (b) of Rydberg sta tes of lithium (triangles) and of atomic hydrogen (dots) exposed to a linearly polarized microwave fi eld of frequency ω= 1.07171794 ×10−4a.u. and amplitude F= 1.072×10−7a.u., for principal quantum number n0= 23 and angular momentum projection m= 0 on the field polarization axis. The lithium spectrum lacks two of the 23 substates of the manifold, due to the quantum defects δℓ=0= 0.399468 and δℓ=1= 0.047263 of the ℓ= 0 andℓ= 1 states, respectively. The quantum defects δℓ=2= 0.002129 and δℓ=3=−0.000077 are negligible compared to the field induced splitting of the n0= 23 manifold (field-free energy E23≃ −9.452×10−4a.u.). Both spectra almost coincide (in energy and ionisation ra te) even for larger values ( p≥10) of the (semiclassical [4]) quantum number p, despite the fact that the localization properties of the associated eigenstates (close to the plan e defined by the field polarization axis) originate in the dynamical symmetry of the −1/rCoulomb potential [18]. The latter is destroyed by the presence of a nonhydrogenic core in alkali atoms. The ion ization rate of the p= 10 state of lithium is locally enhanced by approx. one order of magnitude with re spect to the corresponding hydrogen eigenstate, due to a six-photon resonance with the |n= 41, ℓ= 0/angbracketrightstate. Fig. 2. – Shifts Ealk−Ehydof the energies Ealkof lithium (a, triangles), sodium (b, diamonds), and rubidium (c, squares) as compared to those, Ehyd, of the n0= 23 manifold of atomic hydrogen in a linearly polarized microwave field, with the same parameter s as in fig. 1. Quantum defects employed for the sodium results: δℓ=0= 1.347964, δℓ=1= 0.85538, δℓ=2= 0.015543, δℓ=3= 0.001453, and for rubidium: δℓ=0= 3.1311, δℓ=1= 2.6415, δℓ=2= 1.3472, δℓ=3= 0.016312 [14]. Consequently, three respectively four energy levels are missing in (b) and (c). T he nonvanishing shifts for large p≥9 values can be accounted for by projecting out the low ℓcomponents (i.e. the ones with core induced energy shifts large with respect to the field induced splitting of th en0= 23 manifold) of the n0-manifold, as indicated by the crosses, see eqs. (8) and (9). The agreement between this perturbative approach and the exact quantum results is always better than the average l evel spacing of the hydrogen manifold (dots in fig. 1), except for the relatively large discrepancy atp= 11, in (c). The latter is due to a multiphoton resonance between the alkali eigenstate and a n onhydrogenic (low ℓ) state.

Unconventional Logic Elements on the Base of Topologically Modulated Signals Guennadi A. Kouzaev, Igor V. Nazarov, Andrew V. Kalita Laser and Microwave Information Systems Dept., Moscow State Institute of Electronics and Mathematics 3/12 Bol. Trekhsvaytitelsky per. Moscow ,110028 Russia e-mail: kouzaev@mail.ru , g132uenf@hotmail.com , www.topolog.da.ru Key words: Topological computing, super-high-speed signal processing, quantum calculations ABSTRACT The paper presents new results in the field of super high-speed and multi-valued signal processing Writting digital information into spatial structures (topological charts) of electromagnetic field pulses. allows to use passive circuits for fulfillment several subpicosecond spatial logical operations. This is confirmed by analysis of several physical effects in solids and micron circuits, which influence on time delay of signals. A subpicosecond circuit for spatially modulated signal switching is considered. An analogy between electromagnetic mode physics and several aspects of quantum mechanics is studied. On this base a new digital multi-valued device for spatially modulated signal processing is suggested and modeled. A conclusion on possibility to design a new threedimensional architecture of super –density IC has been made.1. INTRODUCTION One from final stages of development of the electronic integrated circuits is the passage to three- dimensional quasicontinuum medium for processing electromagnetic signals having complicated spatially - temporarily forms. Indication of this tendency is the minimization of interelement distances in three-dimensional VLSI, diminution of sizes of the active elements and magnification of velocity of their work [1,2]. The signals in new super high-speed ICs become three-dimensional spatial objects which are capable to carry digital or analog information by the space structures of pulse fields. The tendency allows to use some analogies to optical space methods for processing the electromagnetic signals in electronic ICs [3-9]. The first results in this direction were obtained in outcome of study of space structures of fields in the microwave three-dimensional integrated circuits, which were suggested and designed for airspace engineering [10]. It was shown, that the structure of force lines of fields (topological chart) is capable to carry discrete information [3-5]. The spatial field characteristics are changed discretely during diffraction of modes or package of modes on passive discontinuities. The further researches have shown a possibility of design full series of the Boolean logic elements for microwave spatially - modulated signals [4,9], and micron sized subpicosecond passive components which are capable to switch digital signals to different layers of the integrated circuits or to work as binary matched space filters [11,12]. For design such circuits have appeared useful some analogies to optical methods of processing spatially modulated signals. The purpose of the paper is study of multi-valued nature of the recently (1992) suggested electromagnetic signals (digital images) and modeling new devices on this base. 2. HIGH DENSITY INTEGRATED CIRCUITS AS QUASI-OPTICAL DEVICES Let's define a likeness and difference of optical and electromagnetic signals for correct application of method of analogies to high-density IC [13]. The optical signal represents an impulse sinusoidal field bytemporal duration up to units femtosecond. The electromagnetic signal in electronic circuits can look like sinusoidal segments of microwave oscillations or to represent impulse, which form is close to a rectangle. Its band of temporal frequencies can take a spectrum from 0 up to units terahertz. The band of space frequencies of an optical signal, as a rule, is much wider than similar performance of an electromagnetic signal. Space information can be transmitted by a ray in open space or by light waveguide. The signal in electronic circuits on strip transmission lines is capable to be spatially- modulated only under existence of multimode condition, for example at use of coupled transmission lines (Fig. 1). Thus the magnitude of the signal space spectrum is determined by number of propagation modes or number of strip conductors. Dimensions of separate components of the electronic circuits are significant less than the least wavelength of the signal and they are elements of “near field zone”, where the quasistatic six-component electromagnetic fields prevails. The effects of superposition of electromagnetic fields with origin of a fractal space structure of potentials are characteristic of this space area of electromagnetic field. The signal processing in this area is produced by the discrete active and passive elements and a possibility of separate transformation of magnetic and electrical fields is realized. Behind of nanocircuits, the optical components are devices using the wave mechanism of interference and diffraction in the far zone. It is possible to name the circuits as “far field” components. The active processing optical signals is possible only by using special nonlinear wave mediums or optoelectronic devices [14]. Thus, the common electromagnetic nature of signals in optic and electronic circuits allows to apply similar methods of their spatial processing. But the possibilities of electronic circuits in relation to optical engineering can be essentially large at the expense of significant variety of element basis [12-15- 18].3. SPATIALLY MODULATED SIGNALS IN ELECTRONIC IC AS TOPOLOGICAL OBJECTS During development the electronic circuits of this type the fact of low dimensionality of space spectrums of electromagnetic signals in electronic circuits was taken into account. The best kind of the signal has been recognized a pulse with discrete modulation of its electromagnetic field structure [3,4,19]. Thus the information carrier is the topology or topological chart of a picture of force lines representing an ordered combination of basic elements of the picture: separatrixes, positions of field equilibrium. The topological chart is possible to name as some kind of a quantum of spatial information According to the nature of the topology, the characteristic is capable only to discrete modifications [3,10]. The theory of such signals and methods of their processing have been already considered in [3- 6,8,9]. Its basic sense consists that the field topology can be changed discretely during diffraction or interference of electromagnetic waves. The effects, as well known, are linear operations concerning amplitudes of signals, but not topology of fields [8,11]. Therefore, the part of discrete logic operations can be fulfilled by passive integrated structures like in optical engineering. Small inertia of the passive circuits allows to process the vector images (topological charts) of electromagnetic fields practically in real time. The techniques of the signal processing is possible to name as topological computing. 4. PHYSICAL BASEMENT FOR SUPER HIGH SPEED SIGNAL PROCESSING BY PASSIVE ELECTRONIC COMPONENTS The writing digital information to a space structure of electromagnetic fields allows to realize some high speed logic operations on the base of using diffraction effects and fast-response equilibrium processes in conductors having continuous energy spectrums [11,12]. These effects are divided into two basic groups (Table 1). The first from them are composed from phenomena, having a place in solid states. For realization subpicosecond operation it should pay special attention on time characteristic of using materials. The second group is stipulated by macroeffects, having electromagnetic nature. For example,the special attention should be given to transients distorting the forms of switched impulses. The method of equivalent circuits [22] which are taking into account parasitic reactivities of discontinuities of strip transmission lines, was applied to an evaluation of duration of these processes (Fig. 2). It has been shown, that parasitic reactivities of the elementar discontinuities are the cause of transients, which duration are in limits from 0.01 to 0.2 picosecond, if the sizes of strip transmission lines are about several microns. Thus, the system analysis of main effects in strip transmission lines and circuits (Table 1) allows to make a conclusion about a possibility of realization subpicosecond operations with spatially - modulated signals by passive components, which have dimensions about several microns. 5. BINARY CIRCUITRY FOR TOPOLOGICAL SIGNAL PROCESSING The basic concepts of switching spatially modulated signals (Fig.1), are constructed on the principle of matched space filtration of impulses, the structures of which fields are varied discretely [5]. On Fig. 3, a circuit of the resistive switch for spatially modulated field signals and its truth-table are represented. Input pulses of even or odd modes of coupled strip transmission lines are switched to different outputs due to using the mechanism of matched spatial filtration (Fig.3, b). The resistors in the circuit allow to ensure a short aperiodic condition of transients at switching the signals. Minimization of the difference of even and odd modes speeds is achived due to using symmetrical construction of the coupled strip transmission lines [5-9]. On Fig.4 the switched signals are represented. The duration of transients did not exceed a several tenth long of picosecond for rectangular entering signals. Essentially smaller distortions are appeared if signals of the Gaussian form are used, the duration of which front makes about 1 picosecond. On the functions the considered switch is equivalent to the current transistor switch containing several transistors. The comparative analysis (Fig. 5 ) of ohmic power losses at transistor gates and offered switch indicates an advantage last [5,6-9,23,24].The inclusion in the circuits for space signals processing the active elements (Fig. 6) allows to achieve new functional advantages [8,25]. For example, the logic processing of amplitudes of impulses can be conjugate with discrete matched space filtration. The given type of evolution of the logic circuits is capable to design active, controlled analogs of hologram devices on the electronic basis [8,12,20]. The developed circuits were experimentally studied. The first results were obtained in microwave range for switches of different types. [4,9,23,24]. Another experimental results were for the digital signals and circuits with clock frequency about several megahertz [11,16,18,21,25,26]. On the base of scaling method a conclusion made on possibility to design a resistive switch for switching topologically modulated signals with clock frequency several hundred gigahertz. The last results are touched to developing and application picosecond generator of topologically modulated signals, designed by G. Domashenko [13]. 6. MULTI- VALUED NATURE OF TOPOLOGICALLY MODULATED SIGNALS AND THEIR APPLICATION FOR MODELING QUANTUM LOGIC ELEMENTS One of the most perspective methods of signal processing is application of the quantum algorithm and quantum devices. The approach uses physical parallelism for effective signal computing [27,28]. From an algorithmic point of view, the quantum logic circuits differ from the classical Boolean devices only by possibility of realization special type of multi-valued logic [29]. Besides in the quantum elements the superposition of quantum states corresponds to a new logic level. The last feature allows, for instance, to solve the well known problem of exponential complexity of calculations [29]. Comparison of quantum mechanics and electromagnetic mode physics in waveguides has been shown existence of an anology between them from the point of formal algorithmic view. For this purpose it is enough to compare multi-level energy diagram of a quantum element and multi-level frequency diagram for modes of a waveguide. Each mode has own topology of electromagnetic field. The mode topology is some kind of quantum of space information for mode and may correspond to a logical level. Duringsuperposition of modes, the topology (topological chart) of the modes may be changed abruptly, creating a new logical level of information system, like in a quantum element [8]. Besides, the topologically modulated signals carry digital information by their magnitudes. It allows to realize special types of multi-valued logic signal processing. The first results in this field were obtained in [4,5,7,8,30]. On the base of this physics a hybrid logic devices were developed and studied: NOT, OR/AND logical circuits ( the state of the last may be changed thanks to variation of amplitudes of comparing topologically modulated signals). On the base of the hybrid logical circuits a microwave trigger was suggested for processing the multi-valued signals of this type [4]. In [8,13,28,29, 31] a multi-valued logic circuit was suggested and modeled (Fig. 6,7). The device allows to process information contained in magnitudes and topological charts of the field signals due to diodes on the logical outputs of above considered passive switch (Fig.6, b ). The full number of logical states of topologically modulated signals in coupled strip transmission line is four . So the circuits may be pertinent for 4-value logical circuitry. (Fig.6, b). The results of modeling the multi-valued switch are shown on Fig.7. The switch transients depend, practically, only on parasitic reactivities of the used microwave diodes. It is obviously, that amplitudes of output signals depend on polarities and topological charts of input signals. An advantage of the type of multi-valued signal is a possibility to realize the logical circuits without using an additional energy for supporting these logical levels opposite to well-known amplitude multi- valued techniques. So, the application analogies of mode physics to quantum mechanics allows to develop and design new type circuits having multi –logic and quasi-neural features. The circuitry opens a possibility of modeling quantum algorithm on the base well-known technology of IC producing.7. A VIEW ON PERSPECTIVE HIGH - DENSITY INTEGRATED CIRCUIT ARCHITECTURES The increase of density of IC dictates to use spatial methods of signal processing in electronics. On this way the most perspective approach is development new IC elements and physical algorithms, allowing to achieve a new grade of density and new quality in signal processing. One of these approaches is a topological computing, dealing with discretely modulated spatial signals. Due to that it is possible to develop super-high speed circuits, realize multi-valued and pseudoquantum devices and consider a future architecture of integrated electronic circuits as a threedimensional medium - new type of the artifical, (designed) electronic hologram [8,12,32,33]. 8. CONCLUSIONS The paper contains original and overviewed results in the field of new logical signals – topologically modulated images of electromagnetic fields, carrying information by their magnitudes and field structures. It has been shown multi-valued nature of the signals and existence of “electromagnetic field topological logic”. Part operations from this logical system may be similar to the quantum logic. The suggested digital devices are able to process digitally modulated signals with subpicosecond time delay (passive components). It has been shown a formal possibility to simulate quantum logic operation by threedimensional circuits for topologically modulated signals. The combined circuits for parallel processing magnitude and topological information are suggested as a base for multi-valued signal devices. The considered approach (topological computing) allows to develop new threedimensional integrated circuits as an information processing medium with new possibilities.REFERENCES 1. Ferry, D.K., Akers, L.A., Greenwich, E.W. Ultra Large Scale Integrated Microelectronics . Prentice Hall, 1988. 2. Nishikawa, K., and et al. Three-dimensional silicon MMICs operating up to K-band , IEEE Trans. Microwave Theory Tech, 46 (1998), 677-684. 3. Kouzaev, G.A. Analysis of solutions convergence for topological problems for 3D microwave circuits. In Proceedings of Conference on Technique, Theory, Mathematical Modeling and CAD for Super High Speed Information processing systems. Moscow. 1992, 2, 238-241. 4. Gvozdev, V.I., Kouzaev, G.A. Microwave trigger. Russian Federation Patent , No 504442, dated on May, 26, 1992. 5. Kouzaev, G.A., Nazarov, I.V. On theory of hybrid logic devices. Journal of Communications Technology and Electronics. 39 (1994),130-136. 6. Kouzaev, G.A. Topological pulse modulation of electromagnetic field and super high-speed logical circuits in microwave range. In Proceedings of the International URSI Symposium on Electromagnetic Theory . St.–Petersburg, Russia, 23-26 May 1995, 584-586. 7. Kouzaev, G.A. and Nazarov, I.V. Topological impulse modulation of field and hybrid logic devices. In Proceedings of the Conference and Exhibition on Microwave Technique and Satellite Communications , Sevastopol, Ukraine, Sept.17-24, 1994 , 4, 443-446 (in Russian). 8. Kouzaev, G.A. Information Properties of Electromagnetic Field Superpositions. J. of Communications Technology and Electronics. 40 (1995), 39-47. 9. Kouzaev, G.A. and Gvozdev, V.I. Topological pulse modulation of field and new microwave circuits design for superspeed operating computers. 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Electrooptic generation and detection of femtosecond electrical transients , IEEE J. Quantum Electron. 24 (1988),184-197. 15. Kouzaev G.A. An active VLSI hologram for super high-speed processing electromagnetic field signals. In Proceedings of the 3-d International Conference on Theory and Technique for Transmission, Reception and Processing Digital Information , Kharkov, Ukraine, 16-18 September 1997,135-136 (in Russian). 16. Kouzaev, G.A. Theoretical and experimental estimations of speed of switches for topologically modulated signals. J. of Communications Technology and Electronics . 43, (1999), 76-82. 17. Kouzaev, G.A. Nazarov, IV ,. Tcherny, V.V. Super broad band passive components for integrated circuits signal processing. Proceedings of SPIE . 3465, (1998), 483-490. 18. Kuzaev, G.A. Experimental study of the temporal characteristics of a switch for topologically modulated signals Journal of Technical Physics . 40 (1998), 573-575.19. 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Moscow State Institute of Electronics and Mathematics. Moscow.1997 (in Russian).¹ Physical effect Time or frequency evaluation of an effect 1. Limited mode velocity in microstrip transmission lines. Time delay of signal in a microstrip transmission line on the substrate with dielectric permittivity ε: ~ 33.3 √ε , fs/mm 2. Inertia of interaction of electromagnetic field with free charge in the region of low values of photon energy.Defined by efficient or free mass of charges 3. Maxwell relaxation time of charges in conductors: ~ 0.001 – 0.01 fs 4. Collective effects in the electronic plasma. Period of plasma frequency in the conductors: ~ 0,067 – 0,2 fs 5. Relaxation phenomenas in dielectric. Time constant of electronic polarization: Time constant of atomic polarization:~ 1 – 10 fs ~ 10 – 10000 fs 6. Minimal time of transition an electron from one energy level on the another in atom: ~ 1 – 10 fs 7. Typical theoretical time of electron relaxation in quantum nanoelements: 100-1000 fs 8. Electron- phonone interaction. Resonant frequency in conductors: ~ 10 THz 9. Transient-time effects on discontinuities of strip transmission lines of micron sizes. Typical duration time of transient process on discontinuities (Idealized Oliner model for discontinuities): ~ 80-150 fs 10. Excitation of higher modes on discontinuities of microstrip transmission lines in VLSI. Cut-off frequency of the first higher mode: ~ 10- 100 THz 11. Excitation of surface waves in micron microstrip transmission lines. Critical coupling frequency of the strip and surface modes: 10-100 THz 12. Limited mode velocity in microstrip transmission lines. Time delay of signal in a microstrip transmission line on the substrate with dielectric permittivity ε: ~ 33.3 √ε , fs/mm 13. Inertia of interaction of electromagnetic field with free charge in the region of low values of photon energy.Defined by efficient or free mass of charges 14. Maxwell relaxation time of charges in conductors: ~ 0.001 – 0.01 fs 15. Limited mode velocity in microstrip transmission lines. Time delay of signal in a microstrip transmission line on the substrate with dielectric permittivity ε: ~ 33.3 √ε , fs/mm 16. Inertia of interaction of electromagnetic field with free charge in the region of low values of photon energy.Defined by efficient or free mass of charges 17. Maxwell relaxation time of charges in conductors: ~ 0.001 – 0.01 fs 18. Collective effects in the electronic plasma. Period of plasma frequency in the conductors: ~ 0,067 – 0,2 fs 19. Relaxation phenomenas in dielectric. Time constant of electronic polarization: Time constant of atomic polarization:~ 1 – 10 fs ~ 10 – 10000 fs 20. Minimal time of transition an electron from one energy level on the another in atom: ~ 1 – 10 fs 21. Typical theoretical time of electron relaxation in quantum nanoelements: 100-1000 fsFigure legends Fig. 1. An example of topologically modulated field pulses in coupled microstrip transmission lines. Digital information may contain in pulse field structures -topological charts (two logic levels) and in their amplitudes (other two logical levels).Fig.2. Transients on steps of microstrip transmission lines. ε-dielectric permittivity of substrate, h=1.3 mkm - thikness of substrate, W 1,2 - width of strip conductors, T- duration period of transients. Curve numberW1 W2 ε T 1 1.0 1.5 3.5 0.0011 2 1.0 1.5 9.6 0.0021 3 0.5 1.0 3.5 0.0035 4 0.5 1.0 9.6 0.0047 5 0.3 1.0 3.5 0.0077 6 0.5 1.5 9.6 0.0118 7 0.5 1.5 3.5 0.0132a b Fig. 3. Binary switch for topologically modulated pulse field signals (a) and its truth-table (b): I - input of the signals (coupled strip transmission lines with characteristic impedance Ze and Zo), II - output of logical "1" ( a two conductor transmission line with characteristic impedance R II ), III - output of logical "0" (a strip transmission line with characteristic impedance R III.a b Fig. 4. Impulses of odd (a) and even (b) modes, transmitted the switch. R is parametrically varying constant. Rii =Rii=50 Omh, ε=3.5, h=3 mkm, Δl=1 mkm, w=1 mkm, s=1 mkm.Fig. 5.Time-energy performances of passive switches for topologically modulated field signals and their transistors analogs.a b Fig. 6. A switch for multi-valued signal processing (a). Signal information contains in structure and amplitudes of field pulses. I –input coupled strip transmission lines, II- output strip transmission lines, III-two conductor transmission. Truth table for the switch (b).a b c d Fig. 7. Transients on the switch for multi-valued signals (U, V; t, ps): (a) Input signal- negative even mode pulse. The dotted line - signal at the input I, utter - on output II ; (b). Input signal -positive even mode pulse. The dotted line - signal at the input, utter - on output II; (c) Input signal- negative odd mode pulse. The dotted line - signal at the input I, utter - on output III. (d) Input signal- positive odd mode pulse. The dotted line - signal at the input I, utter - on output III.

physics/9911066 25 Nov 1999mH/G0A/G0A/G1B4 c/G0A/G0A2/G1B1 c/G0A/G0A2/G2D1/G0A/G0Ah c, E/G0C/G0A/G0Amc21/G08/G08v2 0 2c2/G08/G083v4 0 8c4, 3Vm/G27mc2 8/G0A/G0A/G1B4c, (1)The Rest Mass of the Hydrogen Atom from First Principles Ernst Karl Kunst The rest mass of the hydrogen (H) atom in its ground state is calculated from1 first physical principles and elementary geometric considerations. Key Words: Equivalence of mass and time - masss of the hydrogen atom Previously has been shown [1] that rest mass “m” and relativistic mass m’ = m/G0B -0 where /G0B is the Lorentz factor (1 - v/c) based on composite velocity v [2] - must be0 0 022-1/2 of like origin (are equivalent) and hence the former seems to be generated by the movement of a fourth spatial dimension of matter relative to a fourth dimensional manifold R, in which our R-world is embedded. This implies that - apart from a4 3 numerical factor - the following must be valid where m presumably is the rest mass of the hydrogen atom in its ground state, /G1BH 4 fundamental length in R and /G1B in R, respectively, /G2D = /G1B/c quantum of time, h4 1 1 11 Planck’s constant and c velocity of light. It is proposed to calculate the mentioned numerical factor as follows. By development of energy E’ = E/G0B = E(1 - v/c) to powers of v/c one receives0 0 022-1/2 22 where E means rest energy. As is widely known does the first term mc of the right-2 hand side express the rest mass and the second term mv/2 the classical kinetic02 energy of the material particle under consideration. Thus, the third term 3mv/(8c)042 must be the energy, which is due alone to the relativistic expansion of the moving material particle [2]. Therefore, if rest mass is generated by the movement of /G1B relative4 to R at velocity c it must be valid 3mc/8 = /G1B/c or4 42 where V is volume and /G27 Newtonian density of mass. It is to expect that the volumem m V of the hydrogen atom in R attains the minimum value of volume unit 1, which is theH 3 volume of the tetraoid formed by the four points 1; 2; 3; 4 with the coordinates x, y,11 z;...;x, y, z:1444(1,2,3,4)/G0A/G0A1 1×2×3/G12/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G12/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13/G13x1y1z11 x2y2z21 x3y3z31 x4y4z41 6×33dx1 2×33dx2 2×33dx3 2×/G27m/G0A/G0A/G1B4 c. 6×3 8×33 2×VHdx4 c/G0A/G0A/G1B4 c VHdx4 c/G0A/G0AVH/G27H/G0A/G0AmH/G0A/G0A8/G1B4 933c/G0A/G0A8h 933c. mH/G0A/G0A1.673456×10/G0924g, M/G0A/G0Am/G08/G08mV/G0A/G0Am1/G08/G082G(R/G09R1) 3c,2 (2) (3) (4) (5)Thus, if furthermore V = dxdxdx, from (1) followsH 123 From the foregoing it is clear that m = V/G27 is a four dimensional object so that mustmm be valid /G27 = /G083dx/(2c). Thus (2) deliversm 43 or Equ. (3) delivers for the rest mass of the H-atom1 which value agrees nearly, but not exactly with the experimental value 1,673559 × 10-24 g [3] - g means gram. As has been shown before [4] does the gravitational field of a body contribute to its inertial as well as to ist gravitational mass, namely where M is the sum of the mass (of a body) and of the surrounding gravitational field;, m the mass of the gravitational field, G the gravitatonal constant and R the radialV distance from the center of the body. Therefore, the integrated mass of the sub- microscopic H-atom in our medioscopic world must be composite of the central rest1mVH mV/G0A/G0AmHRVH mRV, RVH RV/G0A/G0ARH R; mVH/G0A/G0AmHRH, MH/G0A/G0AmH1/G08/G08RH63 8/G0A/G0A8h 933c1/G08/G08a063 8,3 (6) (7)mass (3) of the atom and the surrounding (gravitational) field vacuum. The mass of the gravitational field cannot be calculated straightforwardly, because there is no means to determine a bounda ry of R. But if the mass of the field vacuum of the H-atom is1 compared with the mass of the gravitational field of a body of equal density but different mass, according to (5) the ratios and result, where m means mass of the body of comparison and m the mass of itsV gravitational field. If m = R = 1, implying m = 1, we receiveV V where R is the radius of the atom. Hence the global mass of the H-atom, includingH1 the mass of its gravitational field, according to (3) to (6) is given by where R = a is the first Bohrian radius. Calculation delivers 1,673559 × 10 g, whichH0-24 fits exactly the experimental result [3]. It seems that (7) denotes the exact value of the rest mass of the smallest possible, electrically neutral and durable piece of matter - as seen from our medioscopic level. Of course, the theory delivers no explanation yet of the masses of the elementary particles. Presumably those masses are due to a hitherto not yet understood layered structure of space-time on the sub-microscopic level. Dedicated to Barbara.4 References [1] Kunst, E. K.: On the Origin of Time, physics/9910024 [2] Kunst, E. K.: Is the Kinematics of Special Relativity incomplete? physics/9909059 [3] CODATA-Bulletin No. 11, Dec. 1973 [4] Kunst, E. K.: Do Gravitational Fields Have Mass? Or on the Natur of Dark Matter, physics/9911007

arXiv:physics/9911068v1 [physics.atom-ph] 25 Nov 1999Two-dimensional sideband Raman cooling and Zeeman state pr eparation in an optical lattice∗ A.V. Taichenachev, A.M. Tumaikin, and V.I. Yudin Novosibirsk State University, Pirogova 2, Novosibirsk 630 090, Russia L. Hollberg Time and Frequency Division, National Institute of Standar ds and Technology, 325 Broadway, MS 847-10, Boulder, CO 80303 (September 21, 2013) A method of sideband Raman cooling to the vibrational ground state of the m= 0 Zeeman sublevel in a far-detuned two-dimensional optical lattice is proposed. In our scheme, the Raman coupling between vibrational manifolds of the adjacent Zee man sublevels is shifted to the red side- band due to the ac Stark effect induced by a weak pump field. Thus , cooling and optical pumping tom= 0 is achieved by purely optical means with coplanar cw laser beams. The optical lattice and cooling parameters are estimated in the framework of simple theoretical models. An application of the transverse sideband cooling method to frequency standa rds is discussed. Coherent population trapping for the sideband Raman transitions between the deg enerate vibrational levels is predicted. PACS: 32.80.Pj, 42.50.Vk Laser-cooled atoms play a critical role in modern frequency standards such as atomic fountains [1]. As is well-known, Sisyphus-type cooling in optical molasses with polarizati on gradients results in atoms with temperatures correspond ing to tens of the single-photon recoil energies εr= (¯hk)2/2M(for example, T∼30εr/kB∼3µKin the case of Cs[2]). Even lower temperatures can be achieved by velocity-select ive methods [3–5]. These methods, however, require more complicated technical implementations [4]. Recently, Poul Jessen and co-workers [6] demonstrated an el egant and efficient method of cooling atoms to the vibrational ground state of a far-off-resonance two-dimens ional optical lattice. Their method is a variant of Raman sideband cooling [7] based on transitions between the vibra tional manifolds of adjacent Zeeman substates. A static magnetic field is used to tune the Zeeman levels so that Raman r esonance occurs on the red sideband and results in cooling. Two circularly polarized fields are then used to r ecycle the atoms for repetitive Raman cooling. The cooling operates in the Lamb-Dicke regime with cw laser beam s and does not require phase-locked lasers; a transverse temperature of about 950 nKwas achieved. Unfortunately, Jessen’s scheme is difficult to apply to frequ ency standards for several reasons. First, atoms are accumulated in the stretched m=Fsubstate of the F= 4 ground-state hyperfine level of Cs. For clock applications it would be necessary to transfer atoms from |F= 4, m= 4/an}bracketri}htto|F= 4, m= 0/an}bracketri}htwithout additional heating. In principle, this can be realized by the adiabatic passage tec hnique [8]. Second, in the cooling scheme of Ref. [6] a static magnetic field in the range 100 −300mGis used to produce the required energy shift of the Zeeman sub states and, consequently, additional shielding of the Ramsey region of the clock is necessary. Finally, and most critically, the geometry of the cooling scheme requires pumping and repumpi ng beams propagating orthogonal to the cooling plane and these would, when present, produce unwanted light shift s for atoms in the Ramsey region. Stimulated by the concepts and results from Jessen [6], we pr opose a new variant of transverse sideband cooling, that avoids problems mentioned above, while maintaining most of the attractive features. In the present scheme, only cw lasers lying in the cooling plane are used. The basic differen ce from the method of Ref. [6] is that the linearly polarized pumping field now plays a two-fold role, both providing optic al pumping back to the m= 0 magnetic sublevel and causing a uniform ac Stark shift that replaces the external m agnetic field induced Zeeman shift in [6]. The lattice and cooling parameters are studied in the framework of simpl e theoretical models. The optimal magnitudes of the Raman transition amplitude, the pumping field intensity, an d the detuning are found. These results are confirmed by numerical calculations for a more realistic model of the c ycling F→F′=Ftransition. Apart from these, we find that coherence between degenerate (or nearly degenerate) l ower vibrational levels can lead, under certain conditions , to significant changes in the cooling efficiency and cooling ti me. The proposed cooling method may also be useful for atom optic s as a high-brightness well-collimated source of atoms, or for general purposes of quantum-state control in a non-dissipative optical lattice. ∗Contribution of NIST, not subject to copyright 1The field configuration used for the optical lattice consists of three linearly polarized beams having equal amplitudes and propagating in the xy-plane with angles of 2 π/3 between each other (Fig. 1). The polarization vectors of th ese beams are tilted through a small angle φwith respect to the z-axis. This field can be written as E(r, t) =E0E(r)exp(−iωLt) + c.c E(r) =ez3/summationdisplay i=1exp(ikir) + tan( φ)3/summationdisplay i=1eiexp(ikir), (1) where kiand tan( φ)eiare respectively the wave vectors and the in-plane componen ts of the polarization of the i-th beam. All the beams have the same frequency, ωL, far-detuned to the red of the D2resonance line. As was shown in Ref. [9], if the detuning is much greater than t he hyperfine splitting of the excited state, then the optical potential for the ground state takes the form /hatwideUF=−2 3us|E(r)|2+i 3usg(F)[E(r)∗× E(r)]·/hatwideF. (2) Hereg(F) = [F(F+ 1) + J(J+ 1)−I(I+ 1)]/[F(F+ 1)] where F,JandIare respectively the total, electron and nuclear angular momenta of the ground state, and /hatwideFis the angular-momentum operator. The single-beam light shiftus, defined as in Ref. [9], is proportional to the single-beam li ght intensity Iand inversely proportional to the detuning ∆ = ωL−ωF,F′max:us=−AI/∆. (For the D2line of133Csthe constant A≈1.5εrGHz/ (mW·cm−2)). In the zeroth order with tan( φ)≪1, the field (1) is linearly polarized along ezeverywhere. The vector term in Eq. (2) vanishes, resulting in the isotropic optical potential /hatwideU(0)=−4 3us/bracketleftBigg 3 2+ cos(√ 3kx) + cos(√ 3kx−3ky 2) + cos(√ 3kx+ 3ky 2)/bracketrightBigg . (3) In other words, contrary to the field configuration of Ref. [6] , all the Zeeman sublevels have the same optical shift. For red detunings ∆ <0, the minima of the potential (3) form a lattice consisting o f ideal triangles with a side 2 λ/3 (one of them has the coordinates x=y= 0). In the general case, the atomic motion in a periodic potentia l leads to a band energy structure. However for potentials with a periodicity of the order of the light wavel ength λand with the depth much larger than the recoil energy εr(6usin the case under consideration), both the tunneling probab ility and the width are exponentially small for bands close to a potential minimum. Hence, instead of a la ttice and energy bands we can consider vibrational levels as arising from independent potential wells. The spectrum o f the lower levels can be defined, with good accuracy, from the harmonic expansion in the vicinity of the well’s bot tom: /hatwideU(0)≈us[−6 + 3k2(X2+Y2)], where XandYare the displacements from the minimum. This expansion corr esponds to a 2D isotropic harmonic oscillator with the frequency ¯ hωv=√12usεr. Due to the isotropy, the n-th energy level is n+ 1 times degenerate. If the energy separation between adjacent vibrational leve ls is much greater than the recoil energy, the characteristi c size of lower vibrational states is l=/radicalbig ¯h/Mω v≪λ. In this case we have strong localization, and the Lamb-Dick e regime holds. Raman transitions between vibrational levels of adjacent m agnetic substates are induced by the small in-plane component of the field (1). To first order of tan( φ), the vector part of Eqn. (3) gives the correction /hatwideU(1)=1 3usg(F)tan(φ)M(r)·/hatwideF, (4) where Mhas the components Mx= 2√ 3[cos(3 ky/2)sin(√ 3kx/2) + sin(√ 3kx)] and My= 6 sin(3 ky/2)cos(√ 3kx/2). Since this term conserves the symmetry of the main potential (3), each well in the lattice obeys the same conditions for the Raman transitions. For the lower vibrational levels we use a first-order approximation with respect to the displacements X, Y from the minimum /hatwideU(1)≈3usg(F)tan(φ)k(X/hatwideFx+Y/hatwideFy). The operator /hatwideU(1)has off-diagonal elements both for the vibrational and for th e magnetic quantum numbers, inducing transitions with the selection rules ∆ n=±1 and ∆ m=±1 (for a quantization axis along ez). In order of magnitude, 2the Raman transition rate between the lower vibrational lev els isUR=ustan(φ)kl. As was shown in Refs. [9,6], side- band cooling and coherent quantum-state control require th is rate to be much greater than the spontaneous scattering rate of lattice photons γs= 6Γus/∆, where Γ is the natural width. In our lattice UR/γs≈0.2 tan(φ)∆/Γ(εr/us)1/4. Two other important requirements for efficient Raman sideban d cooling are a spatially independent energy shift of the magnetic sublevels and optical pumping. To achieve thes e, we propose to use another optical field, known as the pump beam, linearly polarized along the z-axis, propagating in the cooling plane, and detuned by seve ral Γ to the blue of the F→F′′=Ftransition of the D1line [10] (Fig. 1). In this case the m= 0 sublevel is dark and unshifted, while the others undergo the light shifts δm=m2∆pΩ2 p Γ2/4 + ∆2p, where Ω pis the Rabi coupling for the |F, m=±1/an}bracketri}ht → |F′′=F, m′′=±1/an}bracketri}httransitions and ∆ pis the detuning of the pump field. With a proper choice of Ω pand ∆ p, the states |m= 0, n+ 1/an}bracketri}htand|m=±1, n/an}bracketri}htwill have the same energy, which leads to efficient transition between them due to the Ram an coupling. The cooling picture is completed by the optical pumping, which provides the relaxation from |m=±1, n/an}bracketri}htto|m= 0, n/an}bracketri}ht(see Fig. 2.a). The vibrational quantum number nis conserved in this process due to the fact that atoms are in t he Lamb-Dicke regime. It is worth that, contrary to Ref. [6], in our case several levels take pa rt simultaneously in the cooling process due to the isotropy of the potential /hatwideU(0)(3). If ωv≫URthe state |m= 0, n= 0/an}bracketri}htis approximately dark and the majority of the atoms are eventually pumped into this target state. Thus, the describ ed cooling method can be viewed as a version of dark-state cooling. It is seen that the cooling scheme in the case under c onsideration is somewhat different from that of Ref. [6]. To make sure that there are no real constraints and to estimat e the cooling parameters, we consider a simple theoretical model of the double Λ-system (see Fig. 2.b), whi ch allows an analytical treatment of the problem. We have found the steady-state solution of the generalized o ptical Bloch equations involving the light-induced and spontaneous transitions and the Raman coupling. We are inte rested in the limits ∆p≫Γ ;ωv≫Ur, (5) because in this case the light shift exceeds the field broaden ing and we can shift the states |2/an}bracketri}htand|5/an}bracketri}htinto degeneracy with negligible perturbation of state |1/an}bracketri}ht. Under the conditions (5), the solution leads to the followi ng: (i) the population of the target state |1/an}bracketri}htis maximal at exact resonance Ω2 p/∆p=ωv(see Fig. 3.a); (ii) on resonance, the total population of the states coupled with light is small, a nd equal to ( UR/ωv)2, the probability of the |1/an}bracketri}ht → |6/an}bracketri}htRaman transitions multiplied by a factor 4; (iii) the population o f the state |2/an}bracketri}htcontains two terms: π2= 1/2(UR/ωv)2+ 1/16(γp/ωv)2. The second term is determined by the ratio of the width impos ed by light, γp= ΓΩ2 p/∆2 p, to the vibrational frequency ωv. As a result, the target state population is close to unity: π1≈1−a(UR/ωv)2−b(Γ/∆p)2. (6) The coefficients are a= 3/2 and b= 1/16 in the case of the double Λ-system model. We now turn to an estimate of the cooling dynamics. Instead of looking for a temporal solution of the Bloch equations (d/dt)ρ=Lρfor atomic density matrix ρ, we find the statistically averaged transition time τ=/integraltext∞ 0(ρ(t)−ρ(∞))dt [11]. This matrix obeys the equations Lτ=ρ(∞)−ρ(0), where ρ(∞) is the steady-state solution and ρ(0) is the initial distribution (we set π1=π2=π5=π6= 1/4 and the other elements equal to zero at t= 0). The cooling rate can be associated with the inverse transition time for the |1/an}bracketri}htstateγcool=τ−1 1. As a function of the optical frequency shift, the cooling rate is a Lorentzian curve with a width ∼/radicalBig 1/4γ2p+ 7U2 R(see Fig. 3.a). Exactly on resonance, ∆p= Ω2 p/ωvand in the limits (5), γcooltakes the form γcool=αγpU2 R γ2p+βU2 R. (7) Calculations within the framework of the double Λ-system gi veα= 8 and β= 28. Such a dependence of the cooling rate on γpandURcan be understood qualitatively if we consider cooling as op tical pumping into the dark state. Obviously, under the conditions (5), the cooling rate is det ermined entirely by the optical pumping rate γpand the Raman transitions rate UR, because other parameters do not appear. If UR≫γp, an atom passes from |2/an}bracketri}htto|5/an}bracketri}htvery quickly, and the cooling rate is proportional to the rate of t he slowest process of repumping from |5/an}bracketri}htto|1/an}bracketri}ht. In the inverse limit UR≪γp, the slowest process is the transition |2/an}bracketri}ht → |5/an}bracketri}ht. The corresponding rate, however, is not equal toUR, but is suppressed by the factor UR/γp. That can be explained as the inhibition of quantum transiti ons due to continuous measurements on the final state |5/an}bracketri}ht(quantum Zeno effect [12]). The cooling rate γcoolas a function of 3Ωp(on resonance) is shown in Fig.3.b; γcoolachieves a maximum γmax cool=URα/(2√β) at the optimal Rabi coupling Ωopt p= (β)−1/4ωv/radicalbig Γ/UR. The above described laws for the target-state population an d for the cooling dynamics are confirmed by numerical calculations for a more realistic model of the F→F′=Fcycling transition with a limited number of vibrational levels of the 2D oscillator taken into account. The numerica l data are fitted by the formulae (6,7) very well. The fitting coefficients a,b,αandβdepend on the angular momentum Fand on the initial distribution among the vibrational levels. The results, corresponding to the three vibrationa l levels (with initially equal populations), are presented in Table 1. In principle, two factors limit the number of vibrational le vels which participate in efficient cooling: both the anharmonicity and the violation of the Lamb-Dicke regime be come appreciable for higher vibrational levels. The second factor is the more stringent limitation and gives the following estimate for the maximal vibrational number: n∗≈0.1 ¯hωv/εr. It should be noted that in the case of a symmetric field configur ation for 2D and 3D lattices a degeneracy of the vibrational energy structure occurs. For a 2D lattice (f or example the field configuration of Ref. [6] and our configuration) in the harmonic approximation, the n-th vibrational level contains n+ 1 sublevels {|m, n x+ny=n/an}bracketri}ht}. We find that the coherence induced between the degenerate or n ear-degenerate vibrational levels can play an important role, significantly changing the efficiency of the Raman coupl ing. Indeed, if we consider two degenerate levels, for example, |m=F, nx= 1, ny= 0/an}bracketri}htand|m=F, nx= 0, ny= 1/an}bracketri}htcoupled by Raman transitions with the unique state |m=F−1, nx=ny= 0/an}bracketri}ht, as in Ref. [6], we can see that there exists a superposition o f degenerate states uncoupled with|m=F−1, nx=ny= 0/an}bracketri}ht. Hence, part of the population will be trapped in this superp osition state, in an analogy with well-known coherent population trapping in th e Λ-scheme [13]. In the case of coupling between higher levels |m=F, nx+ny=n/an}bracketri}htand|m=F−1, nx+ny=n−1/an}bracketri}ht, there always exists a coherent superposition of the sublevels |m=F, nx+ny=n/an}bracketri}ht, for which the operator of the Raman transitions is equal to z ero, as it is for the light-induced Λ-chains [14]. However, it should not be forg otten that for higher vibrational levels the anharmonicity has to be taken into account and the degeneracy is partly viol ated. In the scheme under consideration, this unwanted coherence effect is avoided by the simultaneous Raman coupli ng of the two degenerate states |m= 0, nx+ny= 1/an}bracketri}ht with the two other states |m=±1, nx=ny= 0/an}bracketri}htwith different amplitudes, in such a way that the conditions f or trapping can not be satisfied. Note that coherence within the vibrational structure might be very useful for other purposes, for instance in quantum state preparation. In order to provide the cycling interaction of atoms with the pump field, repumping from the other hyperfine level is necessary. We propose to use another light beam tuned in re sonance with a F→F′=F+ 1 transition of the D2 line. This beam is linearly polarized along ezand runs in the xy-plane. For example, if the pumping field operates on the F= 4→F′′= 4 of the D1line of Cs, the repumping field is applied to the F= 3→F′= 4 transition of the D2line. To minimize effects of optical pumping on the other hype rfine level, the intensity of the repumping field can be chosen close to the saturation intensity. It is noteworth y that in our lattice the potentials for both hyperfine levels have the same spatial dependence and, consequently, the req uirement on the repump intensity is not so stringent as in Ref. [6]. Let us give numerical estimations for133Cs(Γ≈2π5MHz andεr/¯h≈2π2kHz). If we take the lattice beams detuning ∆ = −2π10GHz(from the F= 4→F′= 5 transition of the D2line) and intensity I= 500 mW/cm2, then the single-beam shift us≈75εr≈2π150kHz. The lattice has the depth 6 us= 450 εr≈2π900kHz, supporting approximately 15 bound bands with the energy separation ¯ hωv= 30εr≈2π60kHz. Under the tilt angle tan( φ)≈0.1, the Raman transition rate is UR≈0.1 ¯hωv, providing the figure of merit UR/γs≈12≫1. Let the pumping field be applied to the F= 4→F′′= 4 transition of the D1line and repumping field to the F= 3→F′= 4 transition of the D2line. The repumping field should be tuned to resonance and hav e an intensity ∼10mW/cm2to saturate the transitions from all Zeeman sublevels. The opt imal pumping field detuning ∆ p≈0.2 Γωv/UR≈2 Γ and intensity Ip≈8mW/cm2give the cooling rate γcool≈0.4UR≈2π2.2kHz. As a result, approximately 95% of the population of lower levels having vibrational numbers up to n∗≈0.1 ¯hωv/εr≈3 will be accumulated in the target state|F= 4, m= 0, n= 0/an}bracketri}htduring τ≈γ−1 cool≈10−4s. Concluding, we have proposed a new scheme for 2D Raman sideba nd cooling to the zero-point energy in a far-off- resonance optical lattice. The main distinguishing featur es of our proposals are the use of the pumping field to shift the Raman coupling to the red sideband and the accumulation o f cold atoms in the m= 0 Zeeman sublevel. An elementary theoretical consideration allowed us to state t he basic laws for the cooling efficiency and for the cooling dynamics. Our estimates for Csshow that as much as 95% of atoms can be accumulated in the |F= 4, m= 0, n= 0/an}bracketri}ht state within the millisecond time range. This corresponds t o a kinetic temperature of order of 100 nKafter adiabatic release from the lattice [15]. A non-dissipative optical la ttice can be effectively loaded through the four-stage proce ss, as has been demonstrated in Ref. [6]. Also, coherent populat ion trapping for the sideband Raman transitions between degenerate vibrational levels is predicted. 4ACKNOWLEDGMENTS The authors thank Dr. J. Kitching and Prof. P. Jessen for help ful discussions. This work was supported in part by the Russian Fund for Basic Research (Grant No. 98-02-17794) . AVT and VIYu acknowledge the hospitality of NIST, Boulder. [1] J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wine land, Phys. Rev. Lett., 54, 1000 (1985); M. A. Kasevich, E. Riis, S. Chu, and R. G. DeVoe, Phys. Rev. Lett., 63, 612 (1989); K. Gibble and S. Chu, Phys. Rev. Lett., 70, 1771 (1993). [2] C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, and S . Guellati, Europhys. Lett., 12, 683 (1990). [3] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett., 61(1988) 826. [4] M. Kasevich and S. Chu, Phys. Rev. Lett., 69, 1741 (1992). [5] J. Lawall, F. Bardou, B. Saubamea, K. Shimizu, M. Leduc, A . Aspect, and C. Cohen-Tannoudji, Phys. Rev. Lett., 73, 1915 (1994). [6] S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deuts ch, and P. S. Jessen, Phys. Rev. Lett., 80, 4149 (1998). [7] D. J. Heinzen and D. J. Wineland, Phys. Rev. A, 42, 2977 (1990); R. Ta¨ ıeb, R. Dum, J. I. Cirac, P. Marte, and P. Z oller, Phys. Rev. A, 49, 4876 (1994); H. Perrin, A. Kuhn, I. Bouchoule, and C. Salomo n, Europhys. Lett., 42, 395 (1998). [8] P. Pillet, C. Valentine, R.-L. Yuan, and J. Yu, Phys. Rev. A,48, 845 (1993). [9] I. H. Deutsch and P. S. Jessen, Phys. Rev. A, 57, 1972 (1997). [10] We propose to use the D1line in order to avoid any interference with the repumping an d lattice beams, which operate on theD2line. [11] This method is a variant of a statistical consideration of a dynamical system first introduced in L. S. Pontryagin, A. A. Andronov, and A. A. Witt, Zh. Eksp. Teor. Fiz., 3, 165 (1933). [12] W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wine land, Phys. Rev. A, 41, 2295 (1990). [13] E. Arimondo, in Progress in Optics , edited by E. Wolf (North-Holland, Amsterdam, 1996), V. XXX V, p.259. [14] V. S. Smirnov, A. M. Tumaikin, and V. I. Yudin, Sov. Phys. JETP, 69, 913, (1989). [15] A. Kastberg, W. D. Phillips, S. L. Rolston, R. J. C. Spree uw, and P. Jessen, Phys. Rev. Lett., 74, 1542 (1995). TABLE I. The fitting parameters for different transitions F→F′′=F. F a b α β 1 3.0 0.13 6.0 27 2 2.6 0.13 4.6 27 3 2.5 0.13 4.1 27 4 2.4 0.13 3.8 27 5k/B31 k/B32k/B33 k/B52 k/B50 /c68 /B70 /B44/c102/c102 /c102z xy PumpRepump k/B31 k/B32k/B33 k/B52 k/B50(a) (b) /B46/B22 /B34 /B33/B46/B27 /B35 /B34 /B33 /B32 /B46/B3d/B34 /B46/B3d/B33Lattice/B50/B75/B6d/B70/B52/B65/B70/B75/B6d/B70/c68Cs/B31/B33/B33 /B70 /B44120î D D/B32 /B31 FIG. 1. (a) Field geometry. The basic lattice is formed by thr ee coplanar linearly polarized along the z-axis beams. The small in-plane component of polarizations induces the Rama n coupling. The pumping and repumping beams run in the xy-plane and have ezlinear polarization. (b) Possible choice of the field detuni ngs in the case of133Cs. The lattice beams are far-tuned to the red side from the F= 4→F′ max= 5 transition of the D2line, the pumping field is tuned to the blue side from the F= 4→F′′= 4 transition of the D1line, and the repumping field is tuned in resonance with the F= 3→F′= 4 transition of the D2line. 6/B32 /B31 /B30/B6e /B32 /B31 /B30/B6e /B32 /B31 /B30/B6e/B2d1□□□□□□□□□□□□□□□□□□□□□□□□□□□□0 1 /c43 /c451 0+1.□.□. .□.□..□.□. .□.□. /B32 /B31/B36 /B35/B34 /B33(b)(a) /c87/B70 /c87/B70 /c87/B70 FIG. 2. (a) Scheme of sideband Raman cooling. The Raman trans itions are shown by angle arrow lines. The transitions induced by the pumping field (solid lines) and spontaneous tr ansitions (dashed lines) provide the back relaxation to the |m= 0, n= 0/angbracketrightstate. The Zeeman sublevels are shown optically shifted. (b ) Simple double Λ-system model. 70.00 0.01 0.02 0.03 0.04 0.050.00.20.40.60.81.0 103γcool/Γoptical shift (in Γ units)103γcool/Γ b)a) π1π1 0.10 0.15 0.20 0.25 0.300.00.20.40.60.81.0 Ωp/Γ FIG. 3. (a) The target state population π1and the cooling rate vs the optical shift in the case of the dou ble Λ-system model. The parameters ∆ p= 5Γ, ωv= 0.01 Γ and UR= 0.001 Γ. (b) The target state population π1and the cooling rate vs the Rabi frequency in the exact resonance. The parameters ωv= 0.01 Γ and UR= 0.001 Γ. 8

arXiv:physics/9911069v1 [physics.class-ph] 25 Nov 1999The Evolved-Vacuum Model of Redshifts Eugene I. Shtyrkov February 2, 2008 Abstract A new interpretation of cosmological redshifts is proposed to construct the evolved-vacuum model of this phenomenon.The physical v acuum was considered to be a real matter with time-dependent permitti vity and per- meability.Time variation of these parameters (∆ε ε=∆µ µ= 9·10−11per year) was shown on the base of Maxwell’s equations and Hubble ’s law to obey the exponential behavior causing step-by-step decr ease of light velocity with a rate of about 2.7 m/s over 100 years.Cosmolog ical aspects are discussed to explain some features of reality of the Univ erse evolution. 1 Introduction It was experimentally established early in this century tha t the fainter distant galaxies and quasars are, the larger their shift of spectral lines toward the red region is. Assuming a plausible faintness-distance functi on, Hubble [1] discovered that there is a linear dependence between a redshift and dist anceL Z(L) =λshift−λo λo=/parenleftbiggHo co/parenrightbigg L (1) In this relation, Z(L) is the relative spectral shift, λois the wavelength of a spectral line from a source in the laboratory which is at rest relative to the Earth, λshiftis the wavelength of the same line emitted by a quasar and meas ured by a terrestrial observer in the laboratory, Hois the Hubble constant and cois the free space velocity of light. Until now, such spectral ob servations yielding information about extremely distant objects were being the only experimental clues to support an understanding of the Universe. This is be cause only such gigantic distances (of about 1022km) and time intervals (109years) suffice to reveal the small changes that occur while light is traveli ng through space. Therefore, a correct interpretation of such spectral observations to ascertain the real origin of redshifts and bring the Universe to light is es pecially important. At present, there are several alternative models to explain the redshift phe- nomenon. Usually, this one is interpreted as a Doppler effect which finally 1implies recession of galaxies and expansion of the Universe (the ”Big Bang” model).There are some doubts, however, about this interpre tation .Really, there is no physical explanation for singularities due to infinite density of matter at the point of creation, and some recession velocities appear to be extremely large. For example, for quasar Q01442+101 (at Z= 3,3) the recession velocity derived from this model is about of 0 .9co. Moreover, some super-clusters of galaxies seem to be older than the age of the Universe derived from this model. It was experimentally established recently that very extended ob jects (the huge sheets of galaxies stretching more than 200 million l.y. (light-ye ars) across , 700 million l.y. long and 20 million l.y. thick ) are like the super-clust ers of galaxies mapped by Tully earlier [ ?] .Still larger objects ( ”Cosmic Ladder” ) stretching across a distance of about seven billion light-years have been disco vered [3]. Because the maximal velocities of any objects experimentally observed in astronomy to be not more than 500 km/s it takes about 150 billion years to form this structure – more than seven times the number of years since the Big Bang t o form the Universe [3]. Besides the Big-Bang there are also alternative approaches based on ideas of time evolution of matter/light parameters, either the fu ndamental physical constants (Plank’s constant, charge and mass of elementary particles, the elec- tromagnetic parameter co, and so on) or the electromagnetic characteristics of the physical vacuum. The variation of fundamental physical constants as a possib le origin of the red- shifts has been discussed widely since Dirac [4] put forward the idea of ”Great Numbers”. Recently, however, some of the atomic constants w as really shown to be constant in time.For instance,Potekhin and Varshalov ich [5] have studied the fine splitting of the doublet absorption lines in quasar’ s spectra. They ana- lyzed 1414 doublets (CIV, NV, CVI, MgII, AlIII, and SiIV) in t he wide range of redshifts (0 .2< Z < 3.7). Their statistical analysis reveals no statistically significant time variation of the fine structure constant α=e2 hcoon time scale of about ten billion years.We can conclude from here that, at le ast, such the param- eters as charge of electron e, Plank’s constant hand electromagnetic coefficient coshould be considered as constant ones and, hence, can have no influence on redshifts. As for variation of the mass of elementary partic les, this idea discussed by Arp [6] in the intrinsic-redshift model to prove that reds hifts are supposedly related to the age of the objects. This idea, however, do not o bey the redshift- distance relation (1) at constant Ho[7] . Moreover, constancy of the electron mass in time appears to follow from the results of [5] as well. Really,because the fine splitting of energy levels δEdepends on Ridberg’s constant R=2π2me4 coh3as well we should conclude that mis also constant on a very large time scale . There are also alternative models of redshifts which obey th e redshift-distance relation and based on an idea of gradual change of light param eters due to in- teraction between light and matter while the light is travel ing gigantic distances through space for a very long time. There are two candidate wa ys for such inter- 2action to cause redshifts: gradual energy loss by the photon due to absorption during propagation of light with a constant velocity (tired -light model,see,for instance, [8]) and propagation of light with the variable ve locity and without absorption in free space (variable-light-velocity models ). Tired-light mechanism, however, results in obvious contra dictions between quan- tum and classical description based on the Maxwell’s equati ons. In fact, assum- ing that energy of a single photon is gradually decreasing du e to absorption, we may conclude that volume energy density of N photon flow wit h this same frequency and,hence,its intensity are also decreasing one s. From the electrody- namics point of view, it means that electric field strength Eshould gradually be decreased while this wave is traveling through space . Qua ntum descrip- tion says about simultaneous decreasing of frequency at cha nging of photon en- ergy.However,it is not difficult to be convinced that no such a combination, i.e. the simultaneous decaying space-dependent functions E(x) and ω(x) , obeys this same wave equation with stationary boundary condition s which is cur- rently used in quantum electronics and physical optics to ad equately describe the propagation of light at constant velocity in any non-conductors,including vacuum (see Appendix) . The electromagnetic coefficient co, which bridges the electric and magnetic phenomena, has the dimension of speed , and has been historically identified with a constant free space light velocity. In any m edia, however, the light velocity depends on its permittivity εand permeability µas well. At present, vacuum has been experimentally establish ed to be not a void but it is some material medium with definite but not so far inve stigated features. It was really confirmed by observation of several vacuum effec ts, for instance, zero oscillations and polarization of vacuum, generating t he particles in vac- uum due to electromagnetic interaction. Therefore, it was r easonable to assume that this real matter-physical vacuum can possess internal friction due to its small but a real viscosity to result in variation of light-ma tter interaction. That is, vacuum can affect on the light wave because of certain resi stance. Because physical vacuum is a real material with real characteristic s the light velocity can be non-constant, since it depends both on coandε,µof the vacuum ,all of which could have been space/time-dependent functions, in p rinciple. This may be a reason for the redshifts observed. For example, the situ ation with space variable ε(x) and µ(x) at constant electromagnetic parameter was discussed in [9] where a wave equation with the term analogous to one for a d amped simple oscillator was derived from Maxwell’s equations. Solution of this equation leads to a gradual increase of a wavelength ( redshifts at constanc y of the frequency) and variable light velocity. A drawback of the last approach is what one need to consider the permittivity and permeability rather as param eters of interaction but not just the characteristics of vacuum as real matter. Th is inconvenience was overcame in [10] where this same result was obtained, but at constant per- mittivity/permeability and the space-dependent electrom agnetic parameter co. However, following the publication of the work [5] followed by conclusion about constancy of cothe last model [10] must be obviously abandoned . In the prese nt 3paper we will consider more promising variable light veloci ty model of redshifts which let us join electrodynamical approach with the cosmol ogical principle and empirical data to be available in order to explain some featu res of reality. 2 Evolved-vacuum model (EVM) of cosmolog- ical redshifts This model is based on classical electrodynamics with takin g the time-dependent permittivity and permeability into account [11]. Let us mak e only one assump- tion that, in compliance with the cosmological principle, t he variation of the physical vacuum parameters occurs simultaneously and iden tically at any point of the infinite evolving Universe. Then the permittivity and permeability of the physical vacuum at the moment when light is leaving a distant galaxy (one point of space) would be different from what it would be when this lig ht is reaching the Earth (the other point of the Universe) to be a reason for s hifts of spectral lines. Let us consider this point in more detail by writing Ma xwell’s equations for the plane-polarized monochromatic wave propagating al ong the OX-axis and ε(t) and µ(t) as the functions of time ∂E ∂x=−1 co∂B ∂t(2) ∂H ∂x=−1 co∂D ∂t B=µ(t)H D=ε(t)E Consideration of the light wave as a plane one here is due to qu asars removed on infinity are practically point sources of light with flat wa ve fronts near the Earth. The simplest way of analyzing the situation under the se conditions is to write down a wave equation for the induction wave Dinstead of the electric field strength E. One can argue that wave characteristics of induction have t he same phase behavior as for the electric field. However, solvi ng the induction wave equation and then making use of the material relations i n (2) to find the electric field strenth is much simpler. The wave equation for induction can be derived from a chain of substitutions ε∂ ∂x/parenleftbigg∂E ∂x/parenrightbigg =−ε co∂2B ∂x∂t=ε c2o∂ ∂t/parenleftbigg µ∂D ∂t/parenrightbigg drawn from Eqs.2 by means of taking a partial derivative of th e left part of the first equation with respect to xand using the second equation in (2). This leads 4to the wave equation for electric induction ∂2D ∂x2−εµ c2o/parenleftbigg∂2D ∂t2+1 µ∂µ ∂t∂D ∂t/parenrightbigg = 0 (3) with the boundary and initial conditions : atx= 0 and t=ts(tsis the start instant when the light left the remote source) the electric field strength in the wave zone is E(0, ts) =Eoexpıωots, where the amplitude and frequency are constant, and D(0, ts) =ε(ts)E(0, ts) One can see from (3) that the vacuum propagation velocity of t he induction wave is a time-dependent function c(t) =co/radicalbig ε(t)µ(t)(4) and this must be the same as the light velocity. Let us seek a so lution of Eq.3 as a quasi-periodic function with variable phase D=aexpıφ(x, t) (5) The induction is an electric field strength in a void,i.e. wit hout any matter (including physical vacuum) filling a space.Because there i s no field-matter in- teraction in the void the amplitude of iduction in (5) can be c onsidered as a constant. Differentiating (5), inserting into (3) and separ ating real and imagi- nary parts, we obtain two equations ∂φ ∂x=±1 c∂φ ∂t(6) ∂2φ ∂x2−q(t)∂2φ ∂t2−p(t)∂φ ∂t= 0 where q(t) =1 c2(t)andp(t) =q µ(t)dµ(t) dt. In order to admit time dependence for both permeability and permittivity, we should repeat th e same analysis for the magnetic induction wave equation. Following the differe ntiating of the left side of the second equation (2) with respect to xand making the necessary substitutions using the first one we obtain the same equation as (3) but with B in place of Dandεin place of µin the bracket in the third term. Obviously, the solution is formally the same as (6), but with εin place of µin the definition of p(t). Using this conclusion, and the definition of cin (4) , we obtain dc(t)/dt c(t)=−dµ(t)/dt µ(t)=−dε(t)/dt ε=Q (7) 5where Qis either an as-yet unknown function on time or a constant. Ta king it into account we can rewrite the Eqs. (6) as follows ∂φ ∂x=±1 c∂φ ∂t(8) ∂2φ ∂x2−1 c2∂2φ ∂t2−/parenleftbigg1 c3dc dt/parenrightbigg∂φ ∂t= 0 It is seen from (7) that the behavior of the light velocity is t he same as permeability and permittivity time behavior at any point of space (in compliance with the cosmological principle as well).This means that an observer at any concrete space point on the light path sees the wave as a perio dic function whose period depends on the light velocity at this epoch. In o ther words, the light frequency perceived by the observer depends on time al one. Thus, the right part of the first equation (8), which keeps the frequency (∂φ ∂t) and light velocity, depends only on time. Therefore, let us seek the phase of the l ight in the form φ(x, t) =ς(t)±η(x). Inserting this form into the first equation (8), we derive dη(x) dx=±1 cdς(t) dt=k (9) Since η(x) depends only on space and ς(t) depends only on time, the parameter kmust be constant. It follows from solving the Eq.9 that η(x) is a linear function ofx,that is η(x) =±kx+φo. It is easy to show that this form will obey the second of Eqs. (8) as well. Thus the parameter kis a spatial derivative of phase, or a spatial frequency,i.e. a well-known wave number k=2π λ.Thus we have for phase φ(x, t) =ς(t)±kx (10) Thus we come to a very important conclusion: the induction wa ve, and hence the light one, must travel in vacuum with conservation of wave length even when the parameters are time dependent. This wave length is d etermined by the initial and boundary conditions (at the point of a quasar location at the moment of start of the light wave ,i.e. when light is leaving t he quasar) λ(ts) =2π k=2πc(ts) ω(ts)=λshift=const (11) where ω(ts) =ωo- the frequency of the atomic transition in question, tsis the start instant, when light left the source, c(ts) - the velocity of light at the concrete epoch of the Universe evolution. The time dependent frequency ω(t) =∂φ/∂t ,then, can be inferred from (8), given c(t) , which can in turn be inferred from (7), given Q(t). To determine this, let us refer to the redshift- distance relation (1). The distance covered by light depend s ontsandto-the observation time (our epoch at the Earth) and can be written a s L(to, ts) =/integraldisplayto tsc(t)dt (12) 6Taking this and (11) into account, we can rewrite the relatio n (1) in the form Z(ts, to) =λ(ts) λo−1 =Ho co/integraldisplayto tsc(t)dt (13) where initial conditions for the wave lengths of light are (1 1) for a remote source (atts) and (14) for a terrestrial source (at our epoch ) λo=2πco ωo(14) In the relation (13) the light velocity co=c(to) and Ho, measured at our epoch to, should be taken the same for different remote objects observ ed. Hence, the start moment should be an integration variable. Inserting ( 11) and (14) into Eq.13, differentiating it with respect to ts, using (12) to infer that dL(to, ts)/dts= −c(ts) and replacing ts→t, we obtain a simple differential equation for the light velocity dc(t) dt=−Hoc(t) (15) This result is in accord with the Eqs.7 derived from the wave e quation with parameter Qset to minus the Hubble’s constant Ho.Solving these equations in the range ts< t < t owith our initial conditions, we obtain an exponential law of time variation of the light velocity, permittivity and pe rmeability: c(t) =c(ts)e−Ho(t−ts)(16) ε(t) =ε(ts)eHo(t−ts) µ(t) =µ(ts)eHo(t−ts) Using (16) in (9) with the initial condition (11) and k= 2π/λ(ts) , we obtain the time-dependent part of the phase for the time range of ts< t < t oas follows ς(t) =ωo Ho/bracketleftBig 1−e−Ho(t−ts)/bracketrightBig (17) Because the time derivative of this phase is the frequency of the light wave we obtain the same behavior for frequency as for light velocity at the same time range ω(t) =ωoe−Ho(t−ts)(18) Thus the induction wave which obeys the wave equation (3) has a constant am- plitude,the wavelength shifted initially and the variable frequency due to gradual time variation of the vacuum parameters equaled throughout the Universe. The behavior of the electric field strength can be derived from th e last material re- lation in (2) with taking into account (5),(16) and the initi al conditions in (3) as follows E(x, t) =Eoe−Ho(t−ts)expıφ(x, t) (19) 7where the amplitude of the electric field is seen to decline wi th time,and the phase is φ(x, t) =ωo Ho/bracketleftBig 1−e−Ho(t−ts)/bracketrightBig −kx (20) We may use expressions (18),(19),(20) for any source in depe ndence on situation. For instance, in order to compare the parameters of light arr ived on the Earth with ones measured for terrestrial source at the same instan tto(our epoch) we should put t=toin using of (16) and (18) . As a result we have the following. remote source The wave length for the light arrived from a galaxy ( g−label) isλg(to) =2πc(to) ωg(to) where the light velocity at our epoch is c(to) =co(from Ex.16 at t=to) and frequency this light perceived by an observer is ωg(to) =ωoexp[Ho(to−ts)] (from Ex.18 at t=to) terrestrial source For light from the terrestrial source ( t−label) at this same time is λt(to) =2πc(to) ωt(to)where frequency of this source is ωt(to) =ωo(from Ex.18 at t=to=ts,because there is no time interval between emitting and observing the terrestrial source wave) comparison Using this, we obtain the relationλg(to) λt(to)=ωo ωg(to)=eHoτowhere τo=to−ts. Thus, in compliance with experiment, there is the redshift λshift=λg(to) =λoeHoτo Although reproducing the conclusions of the tired-light mo del, namely, about simultaneous decreasing the electric field strength and fre quency, this model has a different physical interpretation. Instead of energy l oss due to absorp- tion at constant light velocity,this mechanism is based on gradual change of the vacuum parameters that results in declining of the electric field strength. The electromagnetic wave is gradually slowing down, with conse rvation of the ini- tially shifted wavelength λshift. The frequency perceived by observers at any point on the light path depends on the light velocity at the ob servation time. 3 Cosmological aspects The cosmological principle implies that the Eqs.16 derived for the interval τo= to−tscan be extrapolated from present observation time toto any future or past one.If we take our epoch as zero point on the time scale th e light velocity in the Exp.16 can be rewritten as follows c(t) =coe−Ho(t−to)(21) 8where t < t oserves to define history of the Universe before our epoch. For t > t o we have future of the Universe.The exponential dependence i mplies no partic- ular points or singularities on the time axis.That is, all of the variations of the Universe parameters have neither beginning nor end but occu r always and ev- erywhere, identically. Such variation is very small (for in stance, as follows from Ex.21 for light velocity at co= 3·108m/sit is about of 2.7 m/s for the interval of 100 years).But it is quite measurable with contemporary tec hniques. Recently, Montgomery and Dolphin [12] performed a statistical analys is of extensive ex- perimental data to argue that light velocity is variable in t ime. This analysis shows the measured value of light velocity to have decreased slightly over the past 250 years. Such behavior of the light velocity can permi t a steady-state cosmology with the boundless Universe that has always exist ed, and is homo- geneous on the very large scale. Making use of (12) with (16), we can find the distance rcovered by light for any moment of time after the start time wh en the light left the quasar r(t) =c(ts) Ho/bracketleftBig 1−e−Ho(t−ts)/bracketrightBig (22) Unlike the constant light-velocity model, this model says t hat the distance ap- proaches a certain limit in a certain interval of time τh=th−ts. At, the τh∼=(5÷6)/Hodistance reaches the limit Lh∼=c(ts) Ho This limit distance is due to total declining the electric fie ld strength (19) and can be interpreted as a spatial cosmological horizon for lig ht. If we take this horizon into account the photometric Olbers’ paradox [13] h as a natural expla- nation. Indeed, the light from a galaxy cluster cannot possi bly reach the Earth if the Earth is situated beyond the light horizon that is avai lable for this clus- ter.In other words, a terrestrial observer can see only some remote clusters the horizons of which are in excess of the look-back time τo=to−ts, i.e. for τo< τh . It is interesting that the earlier light has been emitted, t he larger its horizon is, because of larger light velocity at this moment of start. Hen ce the light horizon for quasars associated with the younger Universe is larger t han that for more recent ones. Using (16) and (22) at t=toand (1) where L=L(τo) =r(to) we obtain the relative shift Z(τo) =Ho coL(τo) =eHoτo−1 (23) It follows from evaluation in (23) that the maximal Z, being for the horizon (τh∼=(5÷6)/Ho) is in the range of about 150-500. However, no empirical Zmeasured up to now exceed 5. If we take a real declining of ligh t intensity into account this can be explained in the following way. In re ality, we have a 9spherical wave front from the point source (remote galaxy) i ntensity of which is as the inverse square of the distance. In fact, the relation ∇2V=1 r∂2(rV) ∂r2 is valid for any function V(r) where the radius of a spherical wave is r=/radicalbig x2+y2+z2(see, for instance, [14]). Therefore, we may place the produ ct (rD) instead of Dinto the Eq.3 without changing of it (at the same direc- tions of the light beam randx). Thus, we obtain the induction in a wave zone (r >> λ/ 2π) as a spherical wave and,hence, the electric field strength h as a form E(r, t) =Eo re−Ho(t−ts)exp [ıφ(r, t)] (24) where φ(x, t) is the phase in (20). The intensity of this wave is I(r, t) =c(t)ε(t) 4πEE∗=coεo 4π/parenleftbiggEo r/parenrightbigg2 e−2Ho(t−ts)(25) In order to estimate decreasing intensity with distance and look-back time let us use of (16) and (22) at t=towith inserting the parameters of our epoch c(to), ε(to) and L(τo) =r(to) into (25). Following the substitutions we have I(L, τo) =Io/parenleftbiggHo co·e−Hoτo eHoτo−1/parenrightbigg2 (26) where Io=c(ts)ε(ts)E2 o 4π=coεo)E2 o 4π. Let us compare this one with the intensity of this wave taken a t some previous point of optical path from the source I(r1, τ1) where r1=r(τ1) and τ1= t1−ts. The distance r(τ1) is chosen much shorter than L(τo) but long enough to consider the galaxy as a point source. With aim to compare w e derive the relation β=I(L, τo)/I(r1, τ1) β=/parenleftbiggHoτ1e−Hoτo eHoτo−1/parenrightbigg2 (27) where we have taken τ1<< H−1 ointo account at series expansion of the ex- ponent. According to current data, the Hubble’s constant is in the range of 60−140kms−1Mpc−1. Making use of Ho= 100 kms−1Mpc−1,i.e. 2,9·10−18s−1, andτ1= 5·106years (at r1= 100 dwhere dis a size of the typical galaxy equaled of 50kps) to estimate β(τo) and Z(τo) for different look-back times we obtain the following table where β= (τ1/τo)2for the situation with constant light velocity Table 10τo(look-back time) Z(redshift) β(at variable c)β(at constant c) H−1 o= 1.08·1010years 1.8 1.5·10−112·10−7 2H−1 o 6.8 1.2·10−135·10−8 3H−1 o 21 2.1·10−152.2·10−8 4H−1 o 60 3·10−171.2·10−8 5H−1 o 170 4.5·10−198·10−9 6H−1 o 480 7·10−215·10−9 It is seen from here that intensity of light is falling more ab ruptly in the case of the variable light velocity compared to the constant ligh t velocity situation. This appears to result in very strong restriction of the visi ble horizon to make measuring of Zmore than 6 practically impossible at sensitivity of up-to- day equipment. 4 Conclusion A new model of cosmological redshifts developed on the base o f classical electro- dynamics and experimental Hubble’s law is discussed in this paper.In distinction from the usual wave equation the modified one (3) obtained her e has the third term taking into consideration interaction the light with p hysical vacuum as a real matter.The light was concluded from solving this equat ion to propagate in vacuum with a constant wavelength shifted initially (11) an d variable velocity (16) caused by gradual changing of permittivity and permeab ility of physical vacuum (a relative rate of about 10−10per year).Actually,for very long time of travel of the light in space from a quasar to the Earth,the w avelength of a terrestrial source is being shifted due to evolution of the Universe resulting in fractional redshifts.For this same reason the frequency of the traveling light perceived by the observer on the Earth is a function of time (18 ) to be different from the frequency ωospecified by the energy transition which remains constant in time at any point of the Universe fo r any atom in question. In distinction from the tired-light model,decre asing of the amplitude of the electric field strenth (19) during the travel trough sp ace is due to not ab- sorption.From this EVM such a behavior is a result of time evo lution of physical vacuum. This model offers novel explanations not only for the redshift origin but also for several other observed features of reality,for ins tance, Olber’s paradox and limitation of Z. 115 Appendix Consider the usual wave equation [14] with stationary bound ary conditions ∂2E ∂x2−/parenleftbiggεµ c2o/parenrightbigg∂2E ∂t2= 0 atEo, ωo, co, εoandµ- constant andE(0, t) =Eoexpıωotto study the propagation of light at constant velocity in any non-conductors,including vacuum. In order to discuss t he tired-light model let us seek a solution of this equation as a quasi-periodic fu nction with variable phase and x-dependent amplitude E=b(x)expıφ(x, t) Inserting this into the equation and separating real and ima ginary parts we obtain two following equations b′′−b(φ′)2+b/parenleftbiggεµ c2o/parenrightbigg (˙φ)2= 0 2b′φ′+bφ′′= 0 where ˙φ=ωo=const. Taking φ′=k=2π λinto account the second one can be rewritten as follows 2b′λ−bλ′= 0 It follows from here that b/λ= 2b′/λ′.Because the left side is always positive one the x- derivative signs for bandλshould be this same.It means that either we have redshifts ( λ′>0) with increasing of the amplitude (that is absurd) or decreasing amplitude ( b′<0) and violetshift ( λ′<0,in no compliance with experiments). Thus the situation of tired-light model ( λ′>0 with b′<0) is not acceptable for the usual wave equation with stationary conditions. References [1] E.P.Hubble, Proc. Nat. Acad. Sci. 15, 168 (1929). [2] R.Brent Tully, Astrophysical Journal,303,25(1986). R.Brent Tully,J.R.Fisher,Atlas of Nearby Galaxies,(Camb ridge:Cambridge University Press,1987) 12[3] E.J.Lerner, The Big Bang never happened,(Simon & Schust er Ltd,London,1992). [4] P.A.M. Dirac, Nature 139,323 (1937). [5] A. Potekhin,D. Varshalovich, Astronomy and Astrophysi cs Supplement 104,89 (1994). [6] H. Arp,Progress in New Cosmologies,(Plenum Press, New Y ork,1,1993). [7] H. Arp,Quasars, Redshifts and Controversies,Interste llar Media,(Berkeley,1987). [8] L.De Broglie,Cahiers de Physique,16,425 (1962). [9] E.I.Shtyrkov,Gal.Electrodynamics,3,66 (1992). [10] E.I.Shtyrkov,Progress in New Cosmologies, Plenum Pre ss, New York,327 (1993). [11] E.I.Shtyrkov,Gal. Electrodynamics,8,3,57 (1997). [12] A.Montgomery,L.Dolphin,Gal.Electrodynamics 5,93 ( 1993). [13] H.W.M.Olbers,Edinburg New Philosophical Journal,1, 141 (1826). [14] M.Born,E.Wolf,Principles of optics,(Pergamon Press ,New York,1964) 13

arXiv:physics/9911070v1 [physics.atom-ph] 26 Nov 1999EXACT SOLUTION TO THE SCHR ¨ODINGER EQUATION FOR THE QUANTUM RIGID BODY ZHONG-QI MA Department of College Computer Education, Hu’nan Nor- mal University, Changsha 410081, The People’s Republic of China, and Institute of High Energy Physics, Beijing 100039 , The People’s Republic of China. The exact solution to the Schr¨ odinger equation for the rigi d body with the given angular momentum and parity is obtained. Since the qua ntum rigid body can be thought of as the simplest quantum three-body pro blem where the internal motion is frozen, this calculation method is a g ood starting point for solving the quantum three-body problems. Key words: quantum three-body problem, rigid body, Schr¨ od inger equation. 1. INTRODUCTION The three-body problem is a fundamental problem in quantum m echanics, which has not been well solved. The Faddeev equations [1] pro vide a method for solving exactly the quantum three-body problems. Howev er, only a few analytically solvable examples were found [2]. The accurat e direct solution of the three-body Schr¨ odinger equation with the separated center-of-mass motion has been sought based on different numerical methods, such as the finite difference [3], finite element [4], complex coordinate rotation [5], hy- perspherical coordinate [6-8], hyperspherical harmonic [ 9-11] methods, and a large number of works [12-16]. In those numerical methods, t hree rotational degrees of freedom are not separated completely from the int ernal ones. In this letter we present a method to separate completely the ro tational degrees of freedom and apply it to the quantum rigid body as an example . The plan of this letter is organized as follows. In Sec. 2 we sh all introduce our notations and briefly demonstrate how to separate the rot ational degrees of freedom from the internal ones in a quantum three-body pro blem. The exact solution to the Schr¨ odinger equation for the rigid bo dy with the given angular momentum and parity is obtained in Sec. 3. A short con clusion is given in sec. 4. 2. QUANTUM THREE-BODY PROBLEM Denote by rjand byMj,j= 1,2,3, the position vectors and the masses of three particles in a three-body problem, respectively. The relative masses are mj=Mj/M, whereMis the total mass, M=/summationtextMj. The Laplace operator 1in the three-body Schr¨ odinger equation is proportional to/summationtext3 j=1m−1 j△rj, where △rjis the Laplace operator with respect to the position vector rj. Introducing the Jacobi coordinate vectors xandyin the center-of-mass frame, x=−/radicalbiggm1 m2+m3r1,y=/radicalbiggm2m3 m2+m3(r2−r3). (1) we obtain the Laplace operator and the total angular momentu m operator Lby a direct replacement of variables: △=3/summationdisplay j=1m−1 j△rj=△x+△y, L=3/summationdisplay j=1−i¯hrj× ▽rj=Lx+Ly, Lx=−i¯hx× ▽x,Ly=−i¯hy× ▽y.(2) The three-body Schr¨ odinger equation with the separated ce nter-of-mass mo- tion becomes −/parenleftBig ¯h2/2M/parenrightBig {△x+△y}Ψ +VΨ =EΨ, (3) whereVis a pair potential, depending only upon the distance of each pair of particles. In the hyperspherical harmonic method [11], for example, tw o Jacobi coordinate vectors are expressed in their spherical coordi nate forms, x∼(ρcosω,θx,ϕx),y∼(ρsinω,θy,ϕy). (4) whereρis called the hyperradius and Ω( ω,θx,ϕx,θy,ϕy) are the five hyper- angular variables. The wave function is presented as a sum of products of a hyperradial function and the hyperspherical harmonic func tion, Ψℓm(x,y) =/summationdisplay K,ℓxℓyψK,ℓxℓy(ρ)Yℓm K,ℓxℓy(Ω). There is huge degeneracy of the hyperspherical basis, and th e matrix ele- ments of the potential have to be calculated between differen t hyperspherical harmonic states [10], because the interaction in the three- body problem is not hyperspherically symmetric. The quantum rigid body (top) can be thought of as the simplest quantum three-body problem where the internal motion is frozen. To s olve exactly the Schr¨ odinger equation for the rigid body is the first step for solving exactly the quantum three-body problems. Wigner first studied the ex act solution for the quantum rigid body (see P.214 in [17]) from the group t heory. He characterized the position of the rigid body by the three Eul er anglesα,β, γof the rotation which brings the rigid body from its normal po sition into the position in question, and obtained the exact solution fo r the quantum rigid body, which is nothing but the Wigner D-function. For the quantum 2three-body problems, as in the helium atom, he separated thr ee rotational degrees of freedom from three internal ones (see Eq. (19.18) in [17]): Ψℓm(r1,r2) =/summationdisplay νDℓ mν(α,β,γ )∗ψν(r1,r2,ω), (5) where r1andr2are the coordinate vectors of two electrons, ωis their an- gle, and the Wigner D-function form [17] has been replaced with the usual D-function form [18]. Wigner did not write the three-body Sch r¨ odinger equa- tion explicitly. As a matter of fact, the three-body Schr¨ od inger equation (3) becomes very complicated if one replaces two coordinates ve ctors of electrons with the Euler angles as well as r1,r2, andωfor the internal motion. On the other hand, Wigner’s idea, to separate the degrees of fre edom completely from the internal ones, is helpful to simplify the calculati on for the quan- tum three-body problem. Hsiang and Hsiang in their recent pa per [19] also presented the similar idea. In this letter we will develop th e idea of Wigner and obtain the exact solution of the Schr¨ odinger equation f or the rigid body without introducing the Euler angles directly. This calcul ation method is a good starting point for solving the quantum three-body prob lems [19,20]. The Schr¨ odinger equation (3) is spherically symmetric so t hat its solution can be factorized into a product of an eigenfunction of the an gular momen- tumLand a ”radial” function, which only depends upon three varia bles, invariant in the rotation of the system: ξ1=x·x, ξ 2=y·y, ξ 3=x·y. (6) For the quantum rigid body, the potential makes the internal motion frozen so that those variables ξjare constant. For a particle moving in a central field, the eigenfunction of the angular momentum is the spherical harmonic function Yℓ m(θ,ϕ). How to generalize the spherical harmonic function to the three-body problem w ithout intro- ducing the Euler angles directly? As is well known, Yℓ m(x) =rℓYℓ m(θ,ϕ), where (r,θ,ϕ) are the spherical coordinates for the position vector x, is a homogeneous polynomial of degree ℓwith respect to the components of x, which does not contain r2=x·xexplicitly. Yℓ m(x), called the harmonic polynomial in the literature, satisfies the Laplace equatio n as well as the eigen-equation for the angular momentum. In the three-body problem there are two Jacobi coordinate ve ctorsxand yin the center-of-mass frame. We shall construct the eigenfu nctions of the angular momentum as the homogeneous polynomials of degree ℓwith respect to the components of xandy, which do not contain ξjexplicitly. According to the theory of angular momentum [18], they are Yℓq Lm(x,y) =/summationdisplay µYq µ(x)Yℓ−q m−µ(y)/an}bracketle{tq,µ,ℓ −q,m−µ|q,ℓ−q,L,m /an}bracketri}ht, 0≤q≤ℓ,whenL=ℓ,and 1 ≤q≤ℓ−1,whenL=ℓ−1.(7) where /an}bracketle{tq,µ,ℓ −q,m−µ|q,ℓ−q,L,m /an}bracketri}htare the Clebsch-Gordan coefficients. The remained combinations with the angular momentum L<ℓ −1 contain 3the factors ξ3explicitly [20]. In other words, the eigenfunctions of the t otal angular momentum L2with the eigenvalue ℓ(ℓ+1), not containing the factors ξjexplicitly, are those homogeneous polynomials of degree ℓor degree (ℓ+1). Let us introduce a parameter λ= 0 or 1 to identify them: Y(ℓ+λ)q ℓm(x,y) =/summationdisplay µYq µ(x)Yℓ−q+λ m−µ(y) × /an}bracketle{tq,µ,ℓ −q+λ,m−µ|q,ℓ−q+λ,ℓ,m /an}bracketri}ht, λ= 0 and 1 , λ ≤q≤ℓ.(8) Y(ℓ+λ)q ℓm(x,y) is the common eigenfunction of L2,L3,L2 x,L2 y,△x,△y, △xy, and the parity with the eigenvalues ℓ(ℓ+ 1),m,q(q+ 1), (ℓ−q+ λ)(ℓ−q+λ+ 1), 0, 0, 0, and ( −1)ℓ+λ, respectively, where L2andL3are the total angular momentum operators, L2 xandL2 yare the ”partial” angular momentum operators [see Eq. (2)], △xand△yare the Laplace operators respectively with respect to the Jacobi coordinate vectors xandy, and △xy is defined as △xy=∂2 ∂x1∂y1+∂2 ∂x2∂y2+∂2 ∂x3∂y3. (9) Because of the conservation of the angular momentum and pari ty, the solution Ψ ℓmλ(x,y) of the Schr¨ odinger equation (3) can be expanded in terms of Y(ℓ+λ)q ℓm(x,y), where the conserved quantum numbers ℓ,mandλ are fixed. Since those equations are independent of m, we can calculate them by settingm=ℓ, where [18] Yℓq ℓℓ(x,y) = (−1)ℓ/braceleftBigg [(2q+ 1)!(2ℓ−2q+ 1)!]1/2 q!(ℓ−q)!2ℓ+2π/bracerightBigg (x1+ix2)q(y1+iy2)ℓ−q, Y(ℓ+1)q ℓℓ(x,y) = (−1)ℓ/braceleftbigg(2q+ 1)!(2ℓ−2q+ 3)! 2q(ℓ−q+ 1)(ℓ+ 1)/bracerightbigg1/2/braceleftBig (q−1)!(ℓ−q)!2ℓ+2π/bracerightBig−1 ×(x1+ix2)q−1(y1+iy2)ℓ−q{(x1+ix2)y3−x3(y1+iy2)}λ.(10) By substituting Ψ ℓℓλ(x,y) into Eq. (3), a system of the partial differen- tial equations for the coefficients can be obtained. The parti al differen- tial equations will be simplified if one changes the normaliz ation factor of Y(ℓ+λ)q ℓℓ(x,y), namely Y(ℓ+λ)q ℓℓ(x,y) in Eq. (11) is replaced by Qℓλ q(x,y), which is proportional to Y(ℓ+λ)q ℓℓ(x,y): Ψℓℓλ(x,y) =ℓ/summationdisplay q=λψℓλ q(ξ1,ξ2,ξ3)Qℓλ q(x,y), Qℓλ q(x,y) = {(q−λ)!(ℓ−q)!}−1(x1+ix2)q−λ(y1+iy2)ℓ−q × {(x1+ix2)y3−x3(y1+iy2)}λ λ= 0,1, λ ≤q≤ℓ.(11) 4The partial differential equations for the functions ψℓλ q(ξ1,ξ2,ξ3) are: −¯h2 2M/braceleftBigg △ψℓλ q+ 4q∂ψℓλ q ∂ξ1+ 4(ℓ−q+λ)∂ψℓλ q ∂ξ2+ 2(q−λ)∂ψℓλ q−1 ∂ξ3 +2(ℓ−q)∂ψℓλ q+1 ∂ξ3/bracerightBigg = (E−V)ψℓλ q, λ≤q≤ℓ, λ = 0,1.(12) This system of the partial differential equations was first ob tained by Hsiang and Hsiang [19]. It is a good starting point for solving the qu antum three- body problems [19,20]. 3. QUANTUM RIGID BODY For the quantum rigid body, the potential preserves the geom etrical form of the rigid body fixed. It can be replaced by the constraints: ξ1= const. ξ 2= const. ξ 3= const. (13) Therefore, the solution of the Schr¨ odinger equation for th e quantum rigid body can be expressed as Ψℓℓλ(x,y) =ℓ/summationdisplay q=λfℓλ qQℓλ q(x,y). (14) wherefℓλ qare constant. Recall that Qℓλ q(x,y) is the solution of the Laplace equation. Due to the constraints (13) some differential term s with respect to ξjin the Laplace equation should be removed so that the Laplace equation is violated, namely, the rigid body obtains an energy E. On the other hand, as a technique of calculation, we can calculate those d ifferential terms first where ξjare not constant, and then set the constraints (13). The contribution from those terms is nothing but the minus energ y−Eof the rigid body. In the calculation, we first separate the six Jacobi coordina tes [see Eq. (4)] into three rotational coordinates and three internal c oordinates. The lengths of xandyand their angle ωare rx=/radicalbig ξ1, r y=/radicalbig ξ2,cosω=ξ3//radicalbig ξ1ξ2. (15) Obviously, those three variables are also constant in the co nstraints (13). Assume that in the normal position of the rigid body the Jacob i coordinate vector xis along the Zaxis and yis located in the XZplane with a positive Xcomponent. A rotation R(α,β,γ ) brings the rigid body from its normal position into the position in question. The Euler angles α,β, andγdescribe the rotation of the rigid body. The definition for the Euler an gles are different from that of Wigner (see Eq. (7) and Ref. [17]) because xandyhere are 5the Jacobi coordinate vectors. To shorten the notations, we define cα= cosα, c β= cosβ, c γ= cosγ, cx= cosθx, c y= cosθy, C = cosω, sα= sinα, s β= sinβ, s γ= sinγ, sx= sinθx, s y= sinθy, S = sinω.(16) According to the definition, we have [18] R(α,β,γ ) = cαcβcγ−sαsγ−cαcβsγ−sαcγcαsβ sαcβcγ+cαsγ−sαcβsγ+cαcγsαsβ −sβcγ sβsγ cβ , x1+ix2=rxeiαsβ, y 1+iy2=ryeiα(cβcγS+sβC+isγS), x3=rxcβ, y 3=ry(−sβcγS+cβC).(17) Through the replacement of variables: (rx,θx,ϕx,ry,θy,ϕy)− → (rx,ry,ω,α,β,γ ), α=ϕx, β =θx, C=cxcy+sxsycos(ϕx−ϕy), cotγ=sxcy−cxsycos(ϕx−ϕy) sysin(ϕx−ϕy),(18) we obtain △x=1 rx∂2 ∂r2xrx+· · ·, △y=1 ry∂2 ∂r2yry+1 r2yS∂ ∂ωS∂ ∂ω+· · ·,(19) where the neglected terms are those differential terms only w ith respect to the rotational variables α,βandγ. Now, ¯h2 2M/braceleftBigg 1 rx∂2 ∂r2xrx+1 ry∂2 ∂r2yry+1 r2yS∂ ∂ωS∂ ∂ω/bracerightBigg Ψℓℓλ(x,y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ξj=const. =EΨℓℓλ(x,y)|ξj=const.,(20) where Ψ ℓℓλ(x,y) is given in Eq. (14). Through a direct calculation, we obtain 1 rx∂2 ∂r2xrxQℓλ q(x,y) =q(q+ 1) r2xQℓλ q(x,y), 1 ry∂2 ∂r2yryQℓλ q(x,y) =(ℓ−q+λ)(ℓ−q+λ+ 1) r2yQℓλ q(x,y), 1 r2yS∂ ∂ωS∂ ∂ωQℓλ q(x,y) =/braceleftbig(ℓ−q)/bracketleftbig(ℓ−q+ 2λ)cot2ω−1/bracketrightbig +λ/parenleftbigcot2ω−1/parenrightbig/bracerightbigQℓλ q(x,y)/r2 y −(q−λ+ 1)(2ℓ−2q+ 2λ−1)/parenleftbigC/S2/parenrightbigQℓλ q+1(x,y)/(rxry) + (q−λ+ 1)(q−λ+ 2)S−2Qℓλ q+2(x,y)/r2 x.(21) 6Therefore, the coefficients fℓλ qsatisfies a system of linear algebraic equa- tions with the equation number ( ℓ−λ+ 1): (2ME/¯h2)fℓλ q=/braceleftBig q(q+ 1)/r2 x+ (ℓ−q+λ)(ℓ−q+λ+ 1)/r2 y +/bracketleftbig(ℓ−q)(ℓ−q+ 2λ)cot2ω−(ℓ−q) +λ/parenleftbigcot2ω−1/parenrightbig/bracketrightbig/r2 y/bracerightBig fℓλ q −/braceleftbig(q−λ)(2ℓ−2q+ 2λ+ 1)C//parenleftbigS2rxry/parenrightbig/bracerightbigfℓλ q−1 +/braceleftbig(q−λ)(q−λ−1)//parenleftbigS2r2 x/parenrightbig/bracerightbigfℓλ q−2. (22) whererx,ryandωare constant. Due to the spherical symmetry, the energy level with the give n total angular momentum ℓis (2ℓ+ 1)-degeneracy (normal degeneracy). Further- more, since λ≤q≤ℓ, there are ( ℓ+1) sets of solutions with the parity ( −1)ℓ andℓsets of solutions with the parity ( −1)ℓ+1. This conclusion coincides with that by Wigner (see P. 218 in [17]). When ℓ= 0 we have the constant solution with zero energy and even parity. When ℓ= 1, we have one set of solutions Ψ ℓm1with the even parity and two sets of solutions Ψ ℓm0with the odd parity: Ψ111(x,y) = (x1+ix2)y3−x3(y1+iy2), E11= ¯h2//parenleftbigMr2 x/parenrightbig+ ¯h2//parenleftBig 2Mr2 ysin2ω/parenrightBig , Ψ(1) 110=x1+ix2, E(1) 10= ¯h2//parenleftbigMr2 x/parenrightbig, Ψ(2) 110=C S2rxry(x1+ix2) +/parenleftBigg 2 r2x−1 S2r2y/parenrightBigg (y1+iy2), E(2) 10= ¯h2//parenleftBig 2Mr2 ysin2ω/parenrightBig ,(23) It is similar to obtain the solutions with the higher orbital angular momen- tumℓ. 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Making sense of Physics1 in the first year of study Shirley Booth Åke Ingerman Centre for educational development Department of Physics (MiNa) Chalmers University of Technology Chalmers University of Technology Abstract We address the question "How do students make sense of Physics from the point of view of constituting physics knowledge?". A phenomenographic study is described as a resultof which we present six qualitatively different ways in which students experience the first year of Physics. The variation is analysed in terms of the structure of experience, the nature of knowledge and an ethical aspect related to the identification of authority. Threeof these ways of experiencing the first year are considered to be unproductive in terms of making sense of physics, while the other three support to an increasing degree the formation of a well-grounded physics knowledge object. Implications for practice areconsidered. 1. Aim Students beginning to study engineering physics are faced in their first year with a bewildering array of new subjects and teachers. Such is the case at Chalmers Universityof Technology in Göteborg, Sweden, where our study took place. At the heart of the four and a half year programme is a vision of the all-round engineer-physicist, and in 1993 a reform of the programme was initiated, aimed at enhancing students' problem-solving andcommunications skills. Such moves are currently common (Baillie, 1999) but in this case it was followed by a drastic drop-out over the two following years, which is the immediate reason for the current study. The aim of the overall study is to illuminate the factors surrounding the drop-out that followed the reform. It was observed that the existing programme had been compressed in order to accommodate the extra curriculum. A survey of students indicated a puzzlingsplit: that about one third found it to be a stimulating programme and the same number found it grinding. We set out to examine the idea that the large number of courses that go to make up the first year, taught by relatively isolated teachers, led to a fragmentation ofthe content for some of the students. In particular, we hypothesised that the programme was not experienced as a whole and that becoming a physicist had been relegated to the background, while coping with many disparate courses – predominantly mathematics –had come to the fore. A more pragmatic aim of the study was to increase the programme board's awareness of the complexity of the first year from the students' perspective, and of the results of theirdecisions on the conditions for learning. The research questions addressed by the study as a whole are • How do students make sense of Physics from the point of view of constituting physics knowledge? 1 By "Physics" with a capital P we denote the programme of Engineering Physics at Chalmers; by "physics" with a small p we mean the physical world and the world of physicists.• What factors in the programme and the student experience of the programme can be related to the students’ approaches to studying and learning? • What implications are there for the individual student, in their quest for making sense of Physics? A wider aim is to illuminate the oft expressed goal of educators in designing new programmes or reforming old ones, that they want to improve the integration of thestudents’ knowledge, in particular mathematics and its applications. We ask ourselves what the implementations of this goal might look like from the students’ points of view. This paper focuses on the first of these questions, and the implications for faculty. 2. Method The study was carried out with a predominantly phenomenographic approach (Marton, 1981; Marton & Booth, 1997). This implies that we were interested in variation in the ways in which the students experienced their first year of study with respect to its contentand structure, it being made up of some twelve distinct courses distributed across mathematics, physics and engineering subjects in rough ratio 2:1:1. Data was collected primarily through interviews. These were held with 20 students sometime in the second year of studies, selected to represent a cross-section of success among those who remained on the programme (and thus including some who had reached the verge of failure). The interview was directed towards exploring the variation of waysin which the students had experienced their first year, with learning the subjects both as individual courses and as an integrated whole. First it probed specifically into the ways in which the student saw the relations between the courses, both structurally andmeaningfully, and engaged them in a semi-structured discussion of factors surrounding their studies. The students were initially given an A3 sheeet of paper with all the course titles spread over it, and they were asked to join them up according to perceivedrelationship. The interviewer began by asking the students for the meaning of the lines they had drawn, giving them opportunity to discuss the relationships between the course contents freely. The interviews then continued by developing the relations they referred toand by taking up students’ overall experience of the programme, their reasons for choosing Physics, what kept them going, the way in which the whole was seen, and their approaches to studying. In line with the phenomenographic approach, the interviews are seen as forming a "pool of meaning" in which the variation in ways of experiencing the phenomena of interest are to be seen. By reading the interviews repeatedly, now as expressions of individualstudents, now as series of extracts related to specific issues, we delved more and more deeply into the meaning of "Physics" as seen by the students. Categories were formed and reformed; extracts from interviews were sought to support and give substance to thecategories; and logical and empirical links between categories were explored. The aim was to offer a hierarchy of empirically grounded and logically consistent categories of description which capture the essence of the whole experience and reveal the essentialvariational structure of that experience. 3. Results Throughout the interviews we took up aspects of ways in which the students experienced the objects of their studies – the content of courses, the relations between them, therelation between them and the future (studies and work), understanding and difficulties. We have also taken note of the ways in which students relate to knowledge, to others and to self, in a system which we identify with an ethical aspect of experience. From the data we have analysed an outcome space of six categories of description, forming a hierarchy of increasing sense-making. We introduce the term “knowledge fragment ” to indicate the way in which the students seem to experience what constitutes the courses. These we see being experienced as self-contained pieces, bearing meaning only in a local sense, neither perceived as legitimate or recognisable outside the immediate educational locality. We also use the term “knowledge object ”, related only partially to the notion as used by Entwistle & Marton (1994). While they mean the ideal visualisable whole “made up of a tightly integrated and structured interconnected ideas and data which together make up our own personal understandings ” attained after a good deal of intensive study, we refer rather to the whole that students are experiencing, whatever its character might be. Based on our interpretation of the empirical data we even draw a distinction between "study knowledge object" and "physics knowledge object",where in the first focus is on the process of study, (we could say Physics), the latter is focused on the meaning of study related to the physical world – a figure-ground relationship. There follows a short description of the six ways of experiencing the first year in terms of the study content, with example extracts from the interviews. Note that we are not categorising individual students, but are analysing the whole experience as it is told bythe set of students and illustrating by individual statements. 3.1. A. Courses are identified with the study situation Here the engineering physics programme has been experienced as a discrete set of courses, a means to the end of a degree. These are related to authority, i.e. teachers andtradition, and common features, such as the ways in which courses were organised. S9 2 indicates such a way of experiencing the first year, here, for example, relating Mechanics and Strength of Materials: S9 It was about moments and suchlike, what can I say, forces here and there, but maybe there wasn ’t such a big link, they share the subject a bit, yes I suppose so Later in the same interview the same student refers thus to Complex Analysis and Fourier Analysis, bringing out the common organisational feature: S9 Well, what can I say, maybe there ’s not such an enormous connection, but it feels as if they are the same, more that it is the same sort of organisation in the course, sort of, more than what they areabout. I really like those courses, no problems to hand in, no bonus points to chase after, but it ’s just a case of learning really, and being able to work the problems out. I Is it the teaching more than anything? S9 Yes, more than the content maybe. But well, I liked them a lot. We would dub this a “study knowledge object ”, but it is taken for granted rather than being a focal concern. 3.2. B. One course is seen as a prerequisite for another course Courses are now related to their content to the extent that a preordained, correct sequence of acquisition of knowledge fragments is assumed. A "red thread"3 is sought in terms of 2 The 20 students who were interviewed are identified S1 to S20. I refers to the interviewerneeds and demands. Authority for the thread – content and structure – is still the domain of teachers and tradition. S7 First Algebra and the maths courses, you can ’t take them away, just like RealB4, I didn ’t enjoy that but it still has to be there anyway. Then the rest of the courses, I think they have to be there, but whether or not you could change their order I don't know, maybe you could. I don ’t know how important EM is, if you could put it earlier or later, it seems to be important because we ’ve done so much of it so far Asked if he could see a whole in the set of courses he had taken, S7 refers to the lack of insight into the teacher ’s intentions: S7 Yes, sort of … I can see how they ’ve tried to build it up but I don ’t know if I see the aim of it, to be honest, I feel I ’m moving forward but I don ’t really know where I ’m trying to get to In discussing courses that had common content S6 refers to perceived shortcomings in authority: S6 … but it sometimes feels as though the teachers don ’t really know what we know and don ’t know. Have you done this before? And have you taken that up? Like in ENets for example, where they took up Laplace transformations, and they came up an awful lot in the exam, on the first exam inany case, Laplace transformations, but it hadn ’t, in Complex they hadn ’t had time to take it up properly, and then in ENets there was no time to do it properly either, there they assumed we had gone through it in Complex. That sort of thing. It ’s a bit as if, things run into one another a bit vaguely, the boundaries are unclear, between Control and ENets, for example Compared with the previous category, the emerging “knowledge object ” is related to study, now being focused on in trying to find the fit of the fragments. 3.3. C. One course is seen as being useful in other courses Courses now support one another, but they still are necessarily arranged in a specific order. Reference is made to the knowledge fragments that constitute the courses, which mesh into one another, course-to-course. I And the line between Mechanics and MatStrength? S3 It ’s more a question of MatStrength having a bit of Mechanics in it, … the course in Strength of Materials, I thought that went smoothly, I didn ’t get any links to any other subjects at all except to just Mechanics, it was mostly force, forces and other things of course, but … A number of different knowledge fragments, not necessarily from the same course, may build a specific technique or application, and the future usage of such applications comesinto the picture. S12 I think that whatever courses you choose you can never cover so much that it ’ll be exactly what you finally work with, there are little bits in each course you have use of and recognise. I don ’t think it ’s the details, as long as you ’re not going to do research, and I have no idea if I ’ll do that. Now the knowledge object starts to have features relating to physics, while study as the object still dominates. 3 A “red thread ” is a Swedish term for the logical structure that is either planned or apparent. It is a very common term among both students, who demand them, and teachers, who try to make them apparent, bothin individual courses and in programmes of courses. 4 Courses are referred to by abbreviated names in interviews Analysis of single variable RealA Electro-magnetic fields EMFields Analysis of several variables RealB Passive & active electric networks ENets Complex analysis Complex Mechanics Mechanics Vector analysis Vector Strength of materials MatStrength Fourier analysis Fourier Automatic control Control Linear algebra Linear Numerical analysis Numeric3.4. D. Courses are related through mutual illumination Here is to be found sense-making for the first time. Courses now lend meaning to each other and understanding in an earlier course can be found in a later course. There are now networks that mesh and unmesh, knowledge fragments might be grouped together indifferent ways and offer different perspectives. There is a dynamic in what is focal or non-focal, and thematic or non-thematic. The Physics that is constituted takes on a dynamic form and begins to resemble a "physics knowledge object" rather than a “study knowledge object ”. What is met in one course can illuminate or explain what is met elsewhere: S12 I see that [ENets] more as a lot of things you just have to accept, currents that go here and there in ENets, they get explained in EMFields. That ’s what I think is essential when you do the Physics programme, that you get these explanations and don ’t simply apply things, but you go a bit further When discussing sudden insights he had had, S3 says: S3 Yes, in Numeric as well, when you studied optimisation and other things that you could sort of deduce from the theory from algebra and linear spaces and things, that you could …, there it comes in, you saw that it was that you were working at without thinking of it, and that you ’d done it before in RealB as well, without knowing that you were projecting it on a subspace sort of? There I felt sort of Wow, when I did Linear anyway. S3 refers to his need to put abstractions into context in order to find “physical meaning ”: S3 The relation between Vector and EMFields was really good. I failed the exam when I took it then, in the last quarter, largely because I didn ’t feel any, sort of had no connection to what it ’s used for actually. We did take EMFields at the same time, but we didn ’t get so far that you could start to look around … there was sort of no … you learn a bit about vorticity and so on but it has no physical meaning before you ’ve done the EMFields course. But then when we had learned electro- magnetic theory, learned a whole load of Vector, then the parts of that course started to come together Being able to confirm abstract concepts in a practical context is referred to: S12 There (Electrical Measurements) we measure, in some of the labs, things we learned about in EMFields, phenomena with reflections and suchlike, and see that they do in fact exist, that ’s a sort of link maybe 3.5. E. Courses fit together into an adaptable whole The courses are seen as constituting parts of a whole, and the strict ordering structure of the educational programme knowledge content is broken apart. An internal dynamic enables a picture to develop which is different on different occasions, depending on whataspects are brought into focus. S1, speaking of courses where he has gained understanding: S1 It was sort of, you discovered that in some way, like in RealB, that you suddenly can simply transform a two-dimensional [double-]integral to a three-dimensional [triple-]integral at once. Now it feels much more obvious that it is so. It didn ’t then. To be able to see something in a different way, that you couldn ’t see before S10 describes with pleasure tying things together, achieving a “knowledge object ” in Marton & Entwistle ’s sense: S10 Well, when we did Complex, and got towards the end of it, you sort of began to see how a lot of it is related to what you studied in the first year, then, you sort of got to tie in lots of the maths courses you ’d taken earlier, you got a bird ’s eye perspective over the whole thing [as you came to the end] of Complex, so you started to feel now, now I see some sort of connections anyway. That was really cool.And S1 takes s further step in realising that what has been encapsulated in one course can be seen as a special case of a more general field of knowledge; the knowledge object is not only visualisable but reformable when needed: S1 It ’s quite a lot of application. In Control you draw upon examples from Mechanics when you are working out your systems. And in MatStrength it ’s actually a question of, you actually take your mechanics systems and make them very very small, so that they can ’t shear and bend. You ’re taking Mechanics into a new dimension, that ’s why you use deformable bodies there [in Strength of Materials] instead. Large bodies. That sounded good! 3.6. F. Courses in Physics come into physics The borders between courses are erased, a physics knowledge object is constituted, physics and the physics world are one with the knower. S12 I think you get a lot of ahah-sensations in the EMFields course, you get to understand a lot of things that before you simply accepted. It ’s really courses like that that are fun to take, you understand how a microwave oven works and suchlike What is met in courses is related to potential others in potential situations outside university S10 That ’s how it was in Control. There you had to tackle problems and sort of feel that, if we had a specific problem here, something technical that an engineer could come across, how would I solveit? And how good would my solution be? There really ought to be a lot of that, things that an employer wants. You should be able to come up with a solution and then judge your solution critically, and see if it is acceptable. That feels right somehow. 3.7. Summary of the provenance of the categories Categories arise from the pool of meaning provided by the set of interviews, and not from individual students. We can see, however, extracts from individual interviews that indicate one category or the other. In table 1 we summarise the provenance of the categories. A S2, S9, S18 B S2, S5, S6, S7, S8, S9, S10, S11, S15, S17, S18, S19 C S1, S3, S4, S7, S10, S11, S12, S13, S14, S16, S17, S20 D S1, S3, S4, S8, S11, S12, S13, S14, S16, S17, S20 E S1, S10, S12, S16 F S10, S12, S16 Table 1. Individual interviews indicate a range of categories 4. Discussion The empirical study has resulted in an six-tiered outcome space of ways in which students of Physics experience their first year of study, which is a hierarchy of sense-making. The first three (A, B, C) refer to courses as courses, knowledge fragments being thecomponents of the courses, isolated in A, building on one another in B, and meshing into one another in C. The second group of three (D, E, F) bring the meaning of the content into focus and ascribe different relationships between the content – mutual in D, multiple in E and finally extending outside the programme to physics phenomena in F. The similarity to studies of conceptions of learning is striking (Marton et al., 1993, S äljö, 1979), in that meaning, or sense-making, is a watershed between two groups of threecategories.We have introduced the notions of “study knowledge object ” and “physics knowledge object ” to distinguish between making sense of the study situation in one way or another, and making sense of physics. To varying degrees these two aspects of the knowledgeobject are present throughout the categories, but “study ” dominates the earlier categories and “physics ” becomes increasingly in focus in the latter categories. 4.1. The structure of the experience of the first year of Physics In Table 2 we have analysed the results according to the structure of experience (Marton & Booth, 1997). It is seen that the referential aspect indicates clearly the shift from no- meaning to meaning between C and D. The external horizon of the structural aspect of the ways of experiencing shifts gradually from an unproblematised studying at the university,here and now, through a refocusing on future study and the world outside the university, to finally embrace physics as a way of seeing the world outside the university. The internal horizon of the structural aspect – how the parts of the ways of experiencing are related to one another and to the whole – shifts in a more discrete sense. Isolated, or possibly grouped, fragments are all there are in A, the blocks taking on a linear preordained arrangement in B. In C, thanks to overlapping fragments, the preordainedlinear arrangement has branches and parallel paths as well, while in D the fragments are related more by meshing facilitated by understanding, thus giving freedom for realignment and restructuring. In E forms of knowledge are constituted of the fragmentsto be found in courses, which give new perspectives and ways of seeing, while in F these ways of seeing are directed outside current experience to an unknown future. Based on the analytical device of the phenomenographic structure of experience, we have extended the analysis to consider the nature of knowledge, drawing largely on the characteristics of the internal horizon of the structure of the ways of experiencing the first year of Physics. Further, we consider, following Perry ’s seminal work “Epistemological and ethical development in the college years ” (Perry, 1970/99) an ethical aspect of the experience, drawing largely on the referential aspect. Let us relate the categories to the individuals who were interviewed. If we look back to Table 1 we see that almost all students expressed experiencing the first year in more than one of these ways, and most expressed ways that fall above and below the "sense- making" watershed. That so many voiced C, even if mainly speaking of sense-making, ishardly surprising giving the design of the interview, based as it was on a chart of individual courses. Of the 20 interviewed, 8 students expressed ways of experiencing their first year of Physics only in the range A to C, which can be interpreted as their notbeing competent to see the first year in a sense-making way. What these also have in common is reference to the weight of studies and the effects it has had on them. S2, an ambitious student not content to get less than top grades and having chosen Physicsbecause it is reputed to be the toughest programme, says at the end of his interview: S2 Sometimes it feels as though there ’s much too much to do. You can understand that a lot drop out. And there are periods when you can never take time off, there ’re always things to do but you don ’t have time. Then it is easy to lose interest and go over to something else instead … when you get to exams you generally have to learn what you need to and it often feels that during the study quarters you are mostly behind and don't know anything. S6, less confident of her abilities relative to her peers, says: S6 Interest has been killed by the tempo.Structural External horizonAspect Internal horizonReferential aspect Nature of knowledgeEthical aspect AUniversity Courses, tasks, organisation,teachers, exams,Gaining a degree Isolated fragments,encapsulated incoursesAuthority with teachers."We need thedegree" BUniversity, future years of PhysicsCourses, red threads,Building up the programmeaccording to the teachers’ intentionsOrdered fragmentsAuthority with teachers."Knowledge is what they want us to find" CUniversity, future years of Physics,world of workCourses, red threads, overlapand applicationBuilding up the programmeaccording to theteachers’intentionsFitting fragments Authority with teachers"Knowledge isthere to be puttogether" DUniversity, future years of Physics, world of workKnowledge fragments, related by explanationtheoreticalreasoning andconfirmed by empirical evidenceGaining an understanding of the basics of theprogrammeMeshed and re- arrangeable fragmentsintegrated byunderstandingResponsibility shift towards self. "Knowledge isthere tounderstand" EUniversity, future years of Physics, world of workKnowledge forms that give ways of seeingGaining new ways of seeingKnowledge object formingResponsibility with self. "Knowledge is ways of seeing" FUniversity, future years of Physics,physicsphenomena, worldof workKnowledge forms that give ways ofseeing physicsGaining physics ways of seeingKnowledge object related to self andthe physics worldCommitment to physics apossibility."Knowledge is away of experiencing the world" Table 2. Analysis of the variation in ways of experiencing the first year of Physics, with respect to learning physics One extension to this work has to be to make contact with students who have actually dropped out and see how their ways of experience fit into and extend this picture. Another is to look at the results in case studies of individual students. 4.2. Ethical aspect of the experience of the first year of Physics The clear watershed between category C and D is further emphasised if an ethical aspect of the categories is taken into account. The different interpretations of "authority" implies different views of knowledge. By "authority" we mean where the responsibility lies for the structure and outcome of the first year of study. In the first group of categories(A,B,C) the authority clearly lies outside of the student, the responsibility and problem formulation privilege are mainly taken by the teachers and other persons "in power", not necessarily known to the student. Following their guidance, the student is guaranteed asuccessful outcome of the studies. In the second group of categories (D,E,F), theresponsibility is taken and agenda is set mainly by the student. Drawing upon the work by Perry (1999), this is very similar to his developmental scheme from the dualistic world of Authorities and Absolutes to the relative world of Commitment and Nuances. Parallel to the responsibility aspect, different "coping strategies" could be observed. Even though the same physical act might exist in both groups, e.g. solving old exams (with given solutions) close before the exam (popularly called "tentakit"~"examfix"), thecontext is very different. In the first group this is one of the acts done to guess what "they", i.e. the authorities, teachers, want, but in the second this is a opportunity to delve into more complex problems with a context possibly easier to relate to earlier knowledge.We see these strategies as ways of creating a confidence, an assurance, trying to take control over the situation as it is perceived and bring a sense of purpose to one ’s studies. This leads us to relate the dichotomous approach to study – deep approaches vs. surface approaches (Marton et al., 1984) – to the individual's perception of authority and the source of the sense of control and/or self-assurance. A student who perceives authority for knowledge lying outside himself will seek ways of satisfying that authority – finding the "red thread" that teachers have built their courses round, trying to build knowledge fragments into a coherent whole according to their plan by studying their exam solutions, by reading over and over their notes and text-books – a classic surface approach in which attention is paid to the tokens. A student who sees the authority lying partly at least with himself will focus on the meaning of and relationships between knowledge fragments using strategies of studying exam solutions to see the variation in ways the fragments canmesh to one another, reading notes and text-books to spy hitherto unremarked connections – the classic deep approach of seeking what the tokens signify. We intend to extend this research with a study specifically aimed at studying the ethical aspects of students' study, their experience of authority and ways of coping with the need for assurance. 4.4. Conditions for learning and implications for faculty The Physics programme is the major factor in creating the conditions for learning for these students. The curriculum, embodied as it is in courses and teaching, is the major contributor to the students learning physics, becoming engineering physicists in knowledge, language and culture. While this study is not able to say much aboutindividual courses and individual teachers, and their effects on the conditions for learning, one can conclude from it that the programme as a whole, and how it is organised and conducted, has a profound effect. Any programme that is organised as this one is, as a set of courses given by subject specialists, (and degree programmes mostly are) has to have as an overriding goal that the students come to see the subject matter as a related whole, and that this provides themwith ways of seeing and coping with an as yet unknown world. This issue has been argued cogently by Bowden and Marton (1998) This study has a clear aim, which is to lead to improvements in the study situation for Physics students by informing and influencing the teachers and the leaders of the programme. The vision of the programme is to produce all-round engineering physicists, capable of working in a wide field of engineering research, development and leadership.The goals of the programme are less clearly articulated. An oft-stated desire of programme leaders, not least Physics, is to encourage an integration of knowledge so that students come to an understanding of a whole from the parts that are presented inindividual courses, yet neither the goals of the programme nor the goals of individual courses take this line. And, as we see from this study, the desired integration is not a self- evident result, even when courses are arranged to offer different aspects of a particularphenomenon. A naive belief in a given structure, known to and enforced by external authority, works against integration on the large scale, as it works against deep approaches on the smallscale. There are examples of groups of teachers who try hard to build "red threads" into their programmes, but fail to ask "whose red thread?" The evidence from this study shows that a red thread can be experienced as a security line to be clung to rather than anintegration guide through the constituents of an emerging physics knowledge object. Where integration becomes possible is in Category D, when knowledge fragments are perceived to mesh and unmesh like Lego blocks, as appropriate for current purpose. The main aim as we see it should be to encourage and support the students to develop a commitment to Physics and physics. How, though, can teachers create a study situation that disfavours the early categories with their strategies of coping in order to bringdisjointed bits of knowledge into the pattern demanded by external authority, and favours later categories in which there is a commitment to understanding and making a coherent adaptable whole of the fragments through which new phenomena can be seen andintegrated to form a new whole? The least but first step is to create a new, and hitherto lacking, College of teachers which goes across department boundaries, and where the whole programme and integration fromthe students ’ perspective is the theme. This is in line with the recommendations of Bowden & Marton (1998) who propose academic teams for curriculum design, cemented by a system of quality assurance that gives both team and individual responsibilities. Thatthe teachers learn about one another's subjects and – above all – about their students' learning, and to relate this to a theoretical framework for learning, needs to be the goal of the new forum. If we see this in terms of knowledge objects, we can say that the teachersare thus engaged in building a knowledge object which they will offer to their students. Another, more focused, approach we can refer to is that offered by Alant et al. (Alant et al., 1999) Focusing on the observed tendency for students to spend time on quantitative,algorithmic aspects of their physics studies at the expense of exploring the qualitative aspects that lead to an understanding, they devised and studied the results of a teaching experiment. The teachers on the course adopted strategies that would foster a conceptualfocus, foster reflection on the nature of the discipline, promote reflection on the value and relevance of what was taught, foster student activity, and make metacognition explicit. The results showed that the students ’ approaches to learning physics and their conceptions of learning shifted dramatically away from the rote-memorising with which they generally entered higher education. In addition, linked to shifts in the students ’ conceptions of learning were shifts in the ways in which they conceived of the nature of science, moving from an ‘immutable ’ conception of science to a more ‘tentative ’ conception of science They also relate their results to Perry ’s work (Perry, 1970/99), pointing out the epistemological variation that they observed. A distinct difference between the study ofAlant et al. and the present study is that whereas they were looking at students taking close-knit single and extensive physics courses, the students we were looking at were meeting a large number of teachers in a large number of separate courses. The content oftheir strategies, however, could well make a substantial contribution to the activities of a suggested College of teachers.Conclusion The question we addressed was "How do students make sense of Physics from the point of view of constituting physics knowledge?". We have identified six qualitatively different ways in which students experience the first year of Physics and analysed thevariation in terms of the structure of experience, the nature of knowledge and an ethical aspect related to the identification of authority. Three of these ways of experiencing the first year are considered to be unproductive in terms of making sense of physics, whilethe other three support to an increasing degree the formation of a well-grounded physics knowledge object, where fragments from different courses are integrated through ways of seeing physics. The ethical aspects have potentially profound implications for the waysstudents take on their studies, related in some sense to the deep and surface approach dichotomy, and deserve further investigation. 6. References Alant, B., Linder, C., & Marshall, D. (1999). Metacognitive-linked developments arising from the design and teaching of conceptual physics . Paper presented at 8th European Conference for Research on Learning and Instruction, August 24-28, 1999, G öteborg, Sweden Baillie, C. (1998). Addressing first year issues in engineering education. European Journal of Engineering Education , 23, 4, 453-465 Bowden, J. & Marton, F. (1998). The University of Learning. Beyond quality and competence in higher education. London: Kogan Page. Entwistle, N. & Marton, F. (1994). Knowledge objects: understandings constituted through intense academic study. British Journal of Educational Psychology, 64 , 161-178. Marton, F: (1981) Phenomenography – describing conceptions of the world around us. Instructional Science , 10, 177-200 Marton, F., Beaty, E. & Dall'Alba G. (1993). Conceptions of learning. International Journal of Educational Research , 19, 277-300. Marton, F. & Booth, S. (1997). Learning and Awareness . Mahwah: Lawrence Erlbaum Ass. Marton, F., Hounsell, D. & Entwistle, N. (Eds.) (1984). The Experience of Learning . Edinburgh: Scottish Academic Press. Perry, W. (1970/99). Forms of ethical and intellectual development in the college years. San Francisco: Jossey-Bass Inc. Säljö, R. (1979). Learning in the learner's perspective. I. Some common-sense conceptions. Reports from the Department of Education, Göteborg University, No 76.

arXiv:physics/9911072v1 [physics.flu-dyn] 26 Nov 1999On acoustic cavitation of slightly subcritical bubbles Anthony Harkin†Ali Nadim‡Tasso J. Kaper† †Department of Mathematics, Boston University, Boston, MA 0 2215 ‡Department of Aerospace and Mechanical Engineering, Bosto n University, Boston, MA 02215 July 24, 1998 Abstract The classical Blake threshold indicates the onset of quasis tatic evolution leading to cavitation for gas bubbles in liquids. When the mean pressure in the liquid is re duced to a value below the vapor pressure, the Blake analysis identifies a critical radius which separa tes quasistatically stable bubbles from those which would cavitate. In this work, we analyze the cavitatio n threshold for radially symmetric bubbles whose radii are slightly less than the Blake critical radius , in the presence of time-periodic acoustic pressure fields. A distinguished limit equation is derived t hat predicts the threshold for cavitation for a wide range of liquid viscosities and forcing frequencies. T his equation also yields frequency-amplitude response curves. Moreover, for fixed liquid viscosity, our s tudy identifies the frequency that yields the minimal forcing amplitude sufficient to initiate cavitation . Numerical simulations of the full Rayleigh- Plesset equation confirm the accuracy of these predictions. Finally, the implications of these findings for acoustic pressure fields that consist of two frequencies wil l be discussed. PACS Numbers: Primary 43.25.Yw, Secondary 43.25.Ts, 47.52.+j, 43.25.Rq Keywords: acoustic cavitation, nonlinear oscillations of gas bubble s, dynamic cavitation threshold, periodic pressure fields, quasiperiodic pressure fields, period-dou bling.I Introduction The Blake threshold pressure is the standard measure of stat ic acoustic cavitation [2, 1]. Bubbles forced at pressures exceeding the Blake threshold grow quasistati cally without bound. This criterion is especially important for gas bubbles in liquids when surface tension is the dominant effect, such as submicron air bubbles in water, where the natural oscillation frequencie s are high. In contrast, when the acoustic pressure fields are not quasis tatic, bubbles generally evolve in highly nonlinear fashions [21, 9, 16, 17]. To begin with, the intrin sic oscillations of spherically symmetric bubbles in inviscid incompressible liquids are nonlinear [16]. The phase portrait of the Rayleigh-Plesset equation [26, 25, 4], consists of a large region of bounded, stable states c entered about the stable equilibrium radius. The natural oscillation frequencies of these states depend on t he initial bubble radius and its radial momentum, and this family of states limits on a state of infinite period, namely a homoclinic orbit in the phase space, which acts as a boundary outside of which lie initial conditi ons corresponding to unstable bubbles. Time- dependent acoustic pressure fields then interact nonlinear ly with both the periodic orbits and the homoclinic orbit. In particular, they can act to break the homoclinic or bit, permitting initially stable bubbles to leave the stable region and grow without bound. These interaction s have been studied from many points of view: experimentally, numerically, and analytically via p erturbation theory and techniques from dynamical systems. In [26], the transition between regular and chaotic oscilla tions, as well as the onset of rapid radial growth, is studied for spherical gas bubbles in time-dependent pres sure fields. There, Melnikov theory is applied to the periodically- and quasiperiodically-forced Rayleigh -Plesset equation for bubbles containing an isothermal gas. One of the principal findings is that, when the acoustic p ressure field is quasiperiodic in time with two or more frequencies, the transition to chaos and the thresho ld for rapid growth occur at lower amplitudes of the acoustic pressure field than in the case of single-freque ncy forcing. Their work was motivated in turn by that in [13], where Melnikov theory was used to study the time -dependent shape changes of gas bubbles in time-periodic axisymmetric strain fields. The work in [25] identifies a rich bifurcation superstructur e for radial oscillations for bubbles in time- periodic acoustic pressure fields. Techniques from perturb ation theory and dynamical systems are used to analyze resonant subharmonics, period-doubling bifurcat ion sequences, the disappearance of strange attrac- tors, and transient chaos in the Rayleigh-Plesset equation with small-amplitude liquid viscosity and isentropic gas. The analysis in [25] complements the experiments of [8] and the experiments and numerical simulations of [14, 15, 20]. Analyzing subharmonics, these works quanti fy the impact of increasing the amplitude of the acoustic pressure field on the frequency-response curves. Other works examining the threshold for acoustic cavitatio n in time-dependent pressure fields have fo- cused on the case of a step change in pressure. In [4], the resp onse of a gas bubble to such a step change in pressure is analyzed by numerical and Melnikov perturbatio n techniques to find a correlation between the cavitation pressure and the viscosity of the liquid. One of t he principal findings is that the cavitation pres- sure scales as the one-fifth power of the liquid viscosity. A g eneral method to compute the critical conditions for an instantaneous pressure step is also given in [7]. The r esults extend numerical simulations of [19] and 2experimental findings of [24], and apply for any value of the p olytropic gas exponent. The goal of the present article is to apply similar perturbat ion methods and techniques from the theory of nonlinear dynamical systems to refine the Blake cavitation t hreshold for isothermal bubbles whose radii are slightly smaller than the critical Blake radius and whose mo tions are not quasistatic. Specifically, we suppose these bubbles are subjected to time-periodic acoustic pres sure fields and, by reducing the Rayleigh-Plesset equations to a simpler distinguished limit equation, we obt ain the dynamic cavitation threshold for these subcritical bubbles. The paper is organized as follows. In the remainder of this se ction, the standard Blake cavitation threshold is briefly reviewed. This also allows us to identif y the critical radius which separates stable and unstable bubbles that are in equilibrium. In section II, the distinguished limit (or normal form) equation of motion for subcritical bubbles ( i.e., those whose radii is slightly smaller than the critical val ue) is obtained from the Rayleigh-Plesset equation. This necessitates ide ntifying the natural timescale of oscillation of such subcritical bubbles which happens to depend upon how cl ose they are to the critical size. We begin section III by defining a simple criterion for determining wh en cavitation has occurred. We then analyze the normal form equation and determine the cavitation thres hold for a specific value of the acoustic forcing frequency (at which the corresponding linear undamped syst em would resonate). This pressure threshold is then compared to numerical simulations of the full Rayleigh -Plesset equation and the good agreement found between the two is demonstrated. The self-consistency of th e distinguished limit equation is further discussed in that section. Section IV generalizes the results to inclu de arbitrary acoustic forcing frequencies. Acoustic forcing frequencies which facilitate cavitation using the least forcing pressure are determined. An unusual dependence of the threshold pressure on forcing frequency i s discovered and explained by analyzing the “slowly-varying” phase-plane of the dynamical system. At t he end of section IV, our choice of a cavitation criterion is discussed in the setting of a Melnikov analysis . In section V we extend the cavitation results to the case of an oscillating subcritical bubble that is driven simultaneously at two different frequencies. We recap the paper in section VI by highlighting the main result s and discussing their applicability. Lastly, we conclude the paper with an appendix which qualitatively dis cusses the relation of our results to some recent experimental findings. I.1 Blake threshold pressure To facilitate the development of subsequent sections we firs t briefly review the derivation of the Blake threshold [17]. At equilibrium, the pressure, pB, inside a spherical bubble of radius Ris related to the pressure, pL, of the outside liquid through the normal stress balance acr oss the surface: pB=pL+2σ R. (1) The pressure inside the bubble consists of gas pressure and v apor pressure, pB=pg+pv, where the vapor pressure pvis taken to be constant — pvdepends primarily on the temperature of the liquid — and the pressure of the gas is assumed to be given by the equation of st ate: pg=pg0/parenleftbiggR0 R/parenrightbigg3γ , (2) 3withγthe polytropic index of the gas. For isothermal conditions γ= 1, whereas for adiabatic ones, γis the ratio of constant-pressure to constant-volume heat capaci ties. At equilibrium, the bubble has radius R0, the gas has pressure pg0and the static pressure of the liquid is taken to be p∞ 0. Thus, the equilibrium pressure of the gas in the bubble is given by pg0=p∞ 0−pv+2σ R0. Upon substituting this result into (2) we get the following e xpression for the pressure of the gas inside the bubble as a function of the bubble radius: pg=/parenleftbigg p∞ 0−pv+2σ R0/parenrightbigg/parenleftbiggR0 R/parenrightbigg3γ . (3) Upon combining equations (1) and (3), we find pL=/parenleftbigg p∞ 0−pv+2σ R0/parenrightbigg/parenleftbiggR0 R/parenrightbigg3γ +pv−2σ R. (4) Equation (4) governs the change in the radius of a bubble in re sponse to quasistatic changes in the liquid pressure pL. More precisely, by “quasistatic” we mean that the liquid pr essure changes slowly and uniformly with inertial and viscous effects remaining negligible duri ng expansion or contraction of the bubble. For very small (sub-micron) bubbles, surface tension is the dominan t effect. Furthermore, typical acoustic forcing frequencies are much smaller than the resonance frequencie s of such tiny bubbles. In this case, the pressure in the liquid changes very slowly and uniformly compared to t he natural timescale of the bubble. For very small bubbles, the Peclet number for heat transfer w ithin the bubble — defined as R2 0ω/α, with ωthe bubble natural frequency (see subsection II.1) and αthe thermal diffusivity of the gas — is small, and due to the rapidity of thermal conduction over such small length scales, the bubble may be regarded as isothermal. We therefore let γ= 1 for an isothermal bubble and define ˜G=/parenleftbigg p∞ 0−pv+2σ R0/parenrightbigg R3 0. Then equation (4) becomes pL=pv+˜G R3−2σ R. (5) The right-hand side of this equation is plotted in figure 1 (so lid curve), which shows a minimum value at a critical radius labeled Rcrit. Obviously, if the liquid pressure is lowered to a value below the corresponding critical pressure pLcrit, no equilibrium radius exists. For values of pLwhich are above the critical value but below the vapor pressu repv, equation (5) yields two possible solutions for the radius R. Bubbles whose radii are less than the Blake radius, Rcrit, are stable to small disturbances, whereas bubbles with R > R critare unstable to small disturbances. The Blake radius itself can be obtained by finding the minimum of the right-hand side of (5) for R >0. This yields the critical Blake radius Rcrit=/parenleftBigg 3˜G 2σ/parenrightBigg1/2 , (6) at which the corresponding critical liquid pressure is pLcrit=pv−/parenleftbigg32σ3 27˜G/parenrightbigg1/2 . (7) 4By combining the last two equations, it is also possible to ex press the Blake radius in the form: Rcrit=4σ 3(pv−pLcrit), relating the critical bubble radius to the critical pressur e in the liquid. Bubbles whose radii are smaller than Rcritare quasistatically stable, while bigger ones are unstable . To obtain the standard Blake pressure we assume that pvcan be ignored and recall that surface tension dominates in the quasistatic regime which amounts to p∞ 0≪2σ/R0. Under these approximations, ˜G≈2σR2 0 and the Blake threshold pressure is conventionally defined a s pBlake ≡p∞ 0−pLcrit ≈p∞ 0+ 0.77σ R0. In the quasistatic regime where the Blake threshold is valid ,pBlakeis the amplitude of the low-frequency acoustic pressure beyond which acoustic forcing at higher p ressures is sure to cause cavitation. When the pressure changes felt by the bubble are no longer quasist atic, a more detailed analysis taking into consideration the bubble dynamics and acoustic forcing fre quency must be performed to determine the cavitation threshold. This is the type of analysis we undert ake in this contribution. II The distinguished limit equation II.1 Derivation To make progress analytically, we focus our attention on “su bcritical” bubbles whose radii are only slightly smaller than the Blake radius at a given liquid pressure belo w the vapor pressure. We thus define a small parameter ǫ >0 by ǫ= 2/bracketleftbigg 1−R0 Rcrit/bracketrightbigg , (8) which measures how close the equilibrium bubble radius R0is to the critical value Rcrit. The value of the mean pressure in the liquid, corresponding to the equilibri um radius R0, can also be found from equation (5) to be p∞ 0−pv=2σ 3R0[(1−ǫ 2)−2−3] =−4σ 3R0[1−1 2ǫ−3 8ǫ2+O(ǫ3)]. (9) The liquid pressure p∞ 0and the critical pressure pLcritdiffer only by an O(ǫ2) amount. It turns out that the characteristic time scale for the natur al response of such subcritical bubbles also depends on the small parameter ǫ. This timescale for small amplitude oscillations of a spher ical bubble is obtained by linearizing the isothermal, unforced Rayleigh -Plesset equation [21] ρ/bracketleftbigg R¨R+3 2˙R2/bracketrightbigg =/parenleftbigg p∞ 0−pv+2σ R0/parenrightbigg/parenleftbiggR0 R/parenrightbigg3 +pv−2σ R−p∞ 0, (10) where the density of the liquid is given by ρand viscosity has been neglected. Specifically, we substitu te R=R0(1 +x) into (10) and keep terms linear in xto get: ¨x+/bracketleftbigg4σ ρR3 0+3(p∞ 0−pv) ρR2 0/bracketrightbigg x= 0. (11) 5Solutions to (11), representing small amplitude oscillati ons about equilibrium, are therefore x=x0cos(ωt+φ) with the angular frequency given by ω=/bracketleftbigg4σ ρR3 0+3(p∞ 0−pv) ρR2 0/bracketrightbigg1/2 . (12) We now use ωto define a nondimensional time variable: τ=ωt. We are interested in analyzing stability for values of ( R0, p∞ 0) near ( Rcrit, pLcrit). Hence, upon recalling (6), (7) and (8), we see that: τ=/bracketleftbigg2σ ρR3 0/parenleftbigg 2/bracketleftbigg 1−R0 Rcrit/bracketrightbigg/parenrightbigg/bracketrightbigg1/2 t=/bracketleftbigg2σǫ ρR3 0/bracketrightbigg1/2 t . We note that as ǫtends to zero, the timescale for bubble oscillations (the re ciprocal of the factor multiplying tin the last equation) increases as ǫ−1/2. Having determined the proper scaling for the time variable f or slightly subcritical bubbles, we can now find the distinguished limit (or normal form) equation for su ch bubbles in a time-periodic pressure field. We start with the isothermal, viscous Rayleigh-Plesset equat ion [21]: ρ/bracketleftbigg R¨R+3 2˙R2/bracketrightbigg + 4µ˙R R=/parenleftbigg p∞ 0−pv+2σ R0/parenrightbigg/parenleftbiggR0 R/parenrightbigg3 +pv−2σ R−p∞ 0+pAsin(Ωt). (13) The amplitude and frequency of the applied acoustic forcing are given by pAand Ω, respectively, and µrepresents the viscosity of the fluid. Here, the far-field pre ssure in the liquid has been taken to be p∞ 0−pAsin(Ωt), with p∞ 0given by equation (9). Setting R(t) =R0(1 +ǫx(τ)), with ǫthe same small parameter introduced above, we obtain at order ǫ2(noting that all of the O(1) and O(ǫ) terms cancel): ¨x+ 2ζ˙x+x−x2=Asin(Ω∗τ), (14) where ζ=/parenleftbigg2µ2 ǫσρR 0/parenrightbigg1/2 , A=pAR0 2σǫ2,Ω∗= Ω/parenleftbiggρR3 0 2σǫ/parenrightbigg1/2 . (15) In equation (14), each overdot represents a derivative with respect to τ. It is implicit in the above scaling that ζ,A, and Ω∗are nondimensional and O(1) with respect to ǫ. To see that this is reasonable, consider an air bubble in water with ρ= 998 kg/m3,µ= 0.001 kg/m ·s,σ= 0.0725 N/m. If we specify ǫ= 0.1 and take a modest equilibrium radius of R0= 2×10−6m then ζ= 0.38. Our analysis of (14) in subsequent sections will concentrate pr imarily on values of ζin the range 0 ≤ζ≤0.4. The parameters Aand Ω∗are related to the forcing conditions, and their magnitudes can be made order unity by choosing appropriate forcing parameters pAand Ω. As an example, if we again choose R0to equal 2 microns, then Ω∗= (2.35×10−7s)Ω/√ǫ. Moreover, setting ǫ= 0.05 gives Ω∗= (1.05×10−6s)Ω. Hence, the dimensionless parameter Ω∗isO(1) when Ω is in the megahertz range, and this is precisely the frequency range we are interested in exploring. Similar ly, with R0andσchosen as above, we find A= (1.38×10−5m·s2/kg)pA/ǫ2and thereby we see that if ǫ= 0.1 then pAcan become on the order of 103Pa. More data will be presented later, in figure 9, showing typ ical forcing pressures. II.2 Interpretation In the laboratory one can create a subcritical bubble by subj ecting the liquid to a low-frequency transducer whose effect is to lower the ambient pressure below the vapor p ressure. Then a second transducer of high 6frequency (high relative to the slow transducer) will give r ise to the forcing term on the right hand side of (14). The low-frequency transducer periodically increa ses and decreases the pressure in the liquid (and shrinks and expands the bubble, which follows this pressure field quasistatically). When the peak negative pressure is reached (and the bubble has expanded to its maxim um size), we can imagine that state as the new equilibrium state, and at that point bring in the effects o f the sound from the second transducer. This second field can then possibly make the bubble, which had alre ady grown to some large size (but still smaller than the critical radius), become unstable. This would all h appen very fast compared to the time scale of the original slow transducer, so the pressure field contribu ted by the original transducer remains near its most negative value throughout. The stability response of t he bubble to the high frequency component of the pressure field is the subject of the rest of this work. III Acoustic forcing thresholds ( Ω∗= 1) The value of Ω∗= 1 corresponds to the forcing frequency at which the linear a nd undamped counterpart of (14) would resonate. We therefore choose this value of the forcing frequency as a starting point and perform a detailed analysis of the dynamics inherent in the d istinguished limit equation at this value of Ω∗. We caution, however, that, as with most forced, damped nonli near oscillators, the largest resonant response occurs away from the resonance frequency of the linear oscil lator. We use Ω∗= 1 mainly as a starting point for the analysis, and the dynamics observed for a range of oth er Ω∗values is reported in section IV. Some special cases of (14) can be readily analyzed when Ω∗= 1. In the absence of forcing, i.e., when A= 0, the phase portraits of (14) with ζ≥0 are shown in figure 2. With no damping (figure 2a), the phase plane has a saddle point at (1,0) and a center at (0,0). The lat ter represents the equilibrium radius of the bubble which, when infinitesimally perturbed, results in si mple harmonic oscillations of the bubble about that equilibrium. The saddle point at (1,0) represents the e ffects of the second nearby root of the equation (5) which is an unstable equilibrium radius. When damping is added (figure 2b), the saddle point remains a saddle, but the center at (0,0) becomes a stable spiral, att racting a well-defined region of the phase space towards itself. In the presence of weak forcing (small A) but with no damping ( ζ= 0), the behavior of (14) can be seen in a Poincar´ e section shown in figure 3. III.1 Phase plane criterion for acoustic cavitation To determine when a slightly subcritical bubble becomes uns table we choose a simple criterion based upon the phase portrait of the distinguished limit equation (14) . For a given ζ, there exists a threshold value, Aesc, ofAsuch that the trajectory through the origin (0,0) grows with out bound for A > A esc, whereas that trajectory stays bounded for A < A esc. A stable subcritical bubble becomes unstable as Aincreases past Aesc. Thus there is a stability curve in the ( A, ζ)-plane separating the regions of this parameter space for which the trajectory starting at the origin in the phase-pla ne either escapes to infinity or remains bounded. Numerically, many such threshold ζ, Aescpairs (represented by the open circles in figure 4) were found with Ω∗= 1. The data are seen empirically to be well fitted by a least-s quares straight line, given by Aesc= 1.356ζ+ 0.058. 7For practical experimental purposes a linear regression cu rve based upon our escape criterion should provide a useful cavitation threshold for the acoustic pres sure in the following dimensional form: pA>3.835ǫ3/2σ1/2µ ρ1/2R3/2 0+ 0.116ǫ2σ R0. (16) Here, ǫis given by equation (8) and is itself a function of the equili brium radius R0, surface tension σand the pressure differential p∞ 0−pv. III.2 Period doubling in the distinguished limit equation It so happens that the stability curve for the trajectory of t he origin can also be interpreted in terms of the period doubling route to chaos for the escape oscillator (14 ). In other words, the value of Aeschappens to be very near the limiting value at which the oscillations bec ome chaotic, just before getting unbounded. For a fixed value of ζ >0 and a small enough A, the trajectory of the origin will settle upon a stable limit cycle in the phase plane. As Ais increased gradually, the period of this stable limit cycl e undergoes a doubling cascade as shown in figure 5 for a fixed value of ζ= 0.35. The period doubling sequence will continue as A is increased until the trajectory of the origin eventually b ecomes chaotic, but still remains bounded. Finally, at a threshold value of Athe trajectory of the origin will escape to infinity. This is t he value of Athat is given by the open circles on the stability diagram (figure 4). A typical bifurcation diagram for the escape oscillator (14) with ζ >0 is shown in figure 6 in which ζ= 0.375. III.3 Robustness of the simple cavitation criterion In this subsection we justify defining a cavitation criterio n based upon the fate of a single initial condition. In all simulations with ζ >0, there is a large region of initial conditions whose fate (e scaping or staying bounded) is the same as that of the origin (figure 7). In fact, w hen the trajectory through the origin stays bounded it is clear from the simulations that the origin lies in the basin of attraction of a bounded, attracting periodic orbit, and points in a large region around it all lie in the basin of attraction of the same orbit. The trajectories through all points in that basin remain bounde d. Then, as the forcing amplitude is increased the attractor is observed to undergo a sequence of period-doubl ing bifurcations, and this sequence culminates at the forcing magnitude when the origin and other initial cond itions in a large region about it escape, because there is no longer a bounded attracting orbit in whose basin o f attraction they lie. III.4 Comparison with full Rayleigh-Plesset simulations The stability threshold predicted by equation (16) can be co mpared to data obtained from simulating the Rayleigh-Plesset equation (13) directly. For small values ofǫ, figure 8 shows the resulting good agreement. The following is a brief description of how the simulations w ere carried out. The material parameters used to produce figure 8 were ρ= 998 kg/m3,µ= 0.001 kg/m ·s, and σ= 0.0725 N/m. Four values of ǫwere chosen, ǫ= 0.01,0.05,0.1,and 0 .2. For each fixed value of ǫand for a selected set of values of ζranging from 0 to 0.4, the parameters R0,Rcrit, (pv−p∞ 0), and Ω were calculated successively using the formulae: 8R0= 2µ2/(ζ2ǫρσ),Rcrit=R0/(1−ǫ/2),pv−p∞ 0= (2σ/R0)[1−(1/3)/(1−ǫ/2)2], and Ω = [(2 σǫ)/(ρR3 0)]1/2. (Note that this succession of computations is done for each c hosen value of ζin each of the four plots.) Having obtained the dimensional parameters required for th e simulation of the full Rayleigh-Plesset equations corresponding to a given ( ǫ,ζ) pair, we used a bisection procedure to determine ARP esc, the threshold value of Aseparating bounded and unbounded bubble trajectories. The bisection procedure was initiated by choosing a value of Aclose to the linear regression line. For this choice of A, the dimensional pressure, pA, was calculated using the middle equation in (15). Then, the i nitial condition R(0) = R0and˙R(0) = 0 was integrated forward in time using an implicit [22] fourth-order Runge-Kutta scheme. The adaptive, impli cit scheme we used offers an accurate and stable means to integrat e the governing equations. The time steps are large in those intervals in which the bubble radius does n ot change rapidly, and they are extremely short for the intervals where ˙Ror¨Ris large (see for example figure 4.7 on page 309 of [17]). If the bubble radius remained bounded during the simulation, then the value of Awas increased slightly, a new pAwas calculated, and a new simulation was begun. If, on the other hand, the bubb le radius became unbounded during the simulation, then the value of Awas slightly decreased and a new simulation was initiated. C ontinuing with this bisection of A, the threshold value, ARP esc, where the bubble first becomes unstable was determined. The dimensional counterpart ( pA-versus- R0) to figure 8 is shown in figure 9 along with the dimensional stability curve given by equation (16). Note that for a given parameter ǫ, the relationship which defines the dimensionless damping parameter ζ,i.e.,R0= 2µ2/(ζ2ǫρσ), can be thought of as defining the bubble radius R0. That is, for a given liquid viscosity µand with all other physical parameters being constant, ζcan only change by varying the equilibrium radius R0. As such, the dimensionless A-versus- ζcurves can be put in terms of the dimensional pA-versus- R0curves, drawn in figure 9. To show the way in which the bubble radius actually becomes un bounded in the full Rayleigh-Plesset simulations, figure 10 provides the radius-versus-time plo ts for three typical simulations with the same value ofζ= 0.3, where time is nondimensional. In this case, ARP esc≈0.51. The top two curves are obtained for values of Aof 0.3 and 0.5, respectively. They show stable oscillations although a period-doubling can be seen to have occurred in going from one to the other. The bot tom figure corresponds to A= 0.53 and shows that the bubble radius is becoming unbounded. The corr esponding dimensional parameters for the Rayleigh-Plesset simulations are given in the figure captio n. III.5 Consistency of the distinguished limit equation In this subsection, we argue, a posteriori , that it is self-consistent to use the escape oscillator as t he dis- tinguished limit equation (14) for the full Rayleigh-Pless et (13), i.e., we show that the higher-order terms encountered during the change of variables from Rtoxmay be neglected in a consistent fashion. Recall that, in the derivation of the escape oscillator, all of the O(1) and O(ǫ) terms dropped out, and the ordinary differential equation (14) was obtained by equa ting the terms of O(ǫ2). The remaining terms are of O(ǫ3) and higher. To be precise, at O(ǫ3), we find on the left-hand side: x¨x−2ζx˙x+3 2˙x2, 9and on the right-hand side: −3 4x+ 2x2−20 3x3. Moreover, we note that, for i≥4, all terms of O(ǫi) on the left-hand side are of the form xi−2˙x, while all terms of O(ǫi) on the right-hand side are polynomials in x. We already know that, for trajectories of the escape oscilla tor that remain bounded, the xand ˙xvari- ables stay O(1). Hence, all of the higher-order terms remain higher-ord er for these trajectories. Next, for trajectories that eventually escape ( i.e., those whose x-coordinate exceeds some large cut-off at some finite time), we know that xand ˙xare bounded until that time and afterwards they grow without bound. The fact that then Ralso grows without bound for these trajectories (due to the c hange of variables that defines x) is consistent with the dynamics of the full Rayleigh-Pless et equation. Potential trouble could arise with trajectories for which xbecomes negative and large in magnitude, e.g., when x∼ −1/ǫthe coefficient of ¨ xvanishes. (This corresponds to small R.) A glance at the Poin car´ e map for the escape oscillator reveals, however, that trajec tories which have x∼ −1/ǫat some time τcan never have x(τ) and ˙ x(τ) ofO(1) simultaneously, for any τ. Hence, these trajectories are not in the regime of interest, neither for the escape oscillator nor for the fu ll Rayleigh-Plesset equation. This completes the argument that it is self-consistent to use the escape oscill ator for this study. IV Pressure thresholds for general Ω∗ IV.1 Stability curves for various Ω∗ Until now, we have only examined the case Ω∗= 1 in the distinguished limit equation (14). In this section we examine the dependence of the stability threshold on the a coustic frequency, Ω, for subcritical bubbles. Specifically, for frequencies Ω∗between 0.1 and 1.1, we performed numerical simulations of t he distinguished limit equation (14) to determine many ( ζ, Aesc) pairs. These pairs are plotted in figure 11, and the data points at each dimensionless frequency Ω∗are connected by straight lines (in contrast to the least squ ares fitting done in subsection III.1). As in subsection III.1, good agreement between the distingu ished limit equation threshold and the full Rayleigh-Plesset equation is observed for various values o f Ω∗; this can been seen in figure 12 which compares the two results at four different values of Ω∗given by 0.6, 0.7, 0.8 and 0.9, for a fixed value of ǫ= 0.05. Various features observed in figure 11, such as the flattening of these curves as Ω∗decreases, are explained in the next subsection. IV.2 Minimum forcing threshold Suppose that we wish to determine the driving frequency, for a given bubble with an equilibrium radius R0 and a critical radius Rcrit, so that the acoustic forcing amplitude necessary to make th e bubble unstable is minimized. This can be done by choosing in figure 11, the value of Ω∗for which the corresponding threshold curve is below all the others for a given ζ. The result of such a procedure is provided in figure 13 as foll ows. Figure 13(a) provides the frequency of harmonic forcing at a given value of the damping parameter ζfor 10which the required amplitude of the acoustic pressure field t o create cavitation is the smallest. Figure 13(b) shows the dimensionless minimum pressure amplitude Aesccorresponding to the value of Ω∗just presented. In figure 13(a), for ζbetween 0 and 0.225, the frequency curve is nearly a straight line and we fit a linear regression line to the data in that interval: Ω∗=−1.12ζ+0.90 for 0 ≤ζ≤0.225. Correspondingly, in figure 13(b), we see that the minimum pressure curve is also nearly s traight for the same interval of ζvalues. The least squares line fitting the data in figure 13(b) is A= 1.03ζ+ 0.02 for 0 ≤ζ≤0.225. When ζ= 0.225, there is a discontinuity in the frequency curve, as seen in figure 13(a). At the same value of ζ, the pressure curve levels off to A≈0.25. This can be explained by a brief analysis of the normal form equation. The key observation will be that, in the escap e oscillator with constant forcing ( i.e., constant right-hand side), there is a saddle-node bifurcation when t he magnitude of the forcing is 1/4. In order to carry out this brief analysis, we consider the cases ζ >0 and ζ= 0 separately, beginning with ζ >0. Forζ >0, the Poincar´ e map of the normal form equation (14) has an as ymptotically stable fixed point (a sink), which corresponds to an attracting periodic orbit for the full normal form equation. Now, during each period of the external forcing, the location of this per iodic orbit in the ( x,˙x)-plane changes. In fact, for the small values of Ω∗we are interested in here (Ω∗≤0.3 approximately), the change in location occurs slowly, and one can write down a perturbation expansion for i ts position in powers of the small parameter Ω∗. The coefficients at each order are functions of the slow time z≡Ω∗t. To leading order, i.e.,atO(1), the attracting periodic orbit is located at the point ( x(z),0), where x(z) is the smaller root of x−x2=Asin(z), namely x(z) =1 2−1 2/radicalbig 1−4Asin(z). Therefore, one sees directly that A= 1/4 is a critical value. In particular, if one considers any fixe d value of A <1/4, then the attracting periodic orbit exists for all time, an d the trajectory of our initial condition (0 ,0) will be always be attracted to it. (Note that the viscosity ζ≥0.225 is large enough so orbits are attracted to the stable periodic orbit at a fast er rate than the rate at which the periodic orbit’s position moves in the ( x,˙x)-plane due to the slow modulation.) However, for any fixed va lue of A >1/4, the function giving x(z) becomes complex after the slow time zreaches a critical value z∗(A), where Asin(z∗) = 1/4, and where we write z∗(A) since z∗depends on A. Moreover, x(z) remains complex in the interval ( z∗(A), π−z∗(A)) during which Asin(z)>1/4. Viewed in terms of the slowly-varying phase portrait, the slowly-moving sink merges with the slowly-mo ving saddle in a saddle-node bifurcation when z reaches z∗(A), and they disappear together for z∗(A)< z < π −z∗(A). Hence, the attracting periodic orbit no longer exists when zreaches z∗(A), and the trajectory that started at (0 ,0) — and that was spiraling in toward the slowly-moving attracting periodic orbit while zwas less than z∗(A) — escapes, because there is no longer any attractor to which it is drawn. For the sake of completeness in presenting this analysis, we note that when A= 1/4, then z∗(A) =π/2; hence, it is precisely near this lowest value of A, namely A= 1/4, that we find the threshold for the acoustic forcing amplitude, and the escape happens near the slow time z=π/2. Moreover, for values of A >1/4, 0< z∗(A)< π/2, and so the escape happens at an earlier time. Numerically, the minimal frequency Ω∗appears to be Ω∗≈0, where Ω∗= 0.01 is the lowest value for which we conducted simulations. Moreover, this also explai ns why, as we see from figure 11 already, the 11curves are flat with A≈0.25 for Ω∗≤0.3. This is the range of small values of Ω∗for which the above analysis applies. Next, having analyzed the regime in which ζ >0, we turn briefly to the case ζ= 0. For small values, Ω∗≤0.3, the curves in figure 11 remain flat near A= 0.25 all the way down to ζ= 0. The full normal form equation with ζ= 0 is a slowly-modulated Hamiltonian system. One can again u se the slowly-varying phase planes as a guide to the analysis (although the periodic orbi t is only neutrally stable when ζ= 0 and no longer attracting as above), and the saddle-node bifurcati on in the leading order problem at Asin(z∗) = 1/4 is the main phenomenon responsible for the observation that the threshold forcing amplitude is near 0.25. (We also note that for a detailed analysis of the trapped orbi ts, one needs adiabatic separatrix-crossing theory, see [3], for example, but we shall not need that here. ) Finally, and most importantly, simulations of the full Rayl eigh-Plesset equation confirm all the quan- titative features of this analysis of the normal form equati on. The open circles in figure 13 represent the numerically observed threshold forcing amplitudes, and th ese circles lie very close to the curves obtained as predictions from the normal form equation. We attribute thi s similarity to the fact that the phase portrait of the isothermal Rayleigh-Plesset equation has the same st ructure — stable and unstable equilibria, sep- aratrix bounding the stable oscillations, and a saddle-nod e bifurcation when the forcing amplitude exceeds the threshold — as the normal form equation (see [26]). IV.3 The dimensional form of the minimum forcing threshold Recall, that for ζbetween 0 and 0.225, we fit linear regression lines to portion s of figures 13(a) and (b). Specifically, for figure 13(a) we found that, for a particular choice of ζ, the frequency which yields the smallest value of Aesccan be expressed as: Ω∗=−1.12ζ+0.90 for 0 ≤ζ≤0.225. And for figure 13(b) we found that the stability boundary for the minimum forcing is given by, A=  1.03ζ+ 0.02 for 0 ≤ζ≤0.225 0.25 for 0 .225≤ζ≤0.4.(17) Using the definitions of ζ, Aand Ω∗as given by (15), the “optimal” acoustic frequency to cause c avitation of a subcritical bubble is given in dimensional form by Ω =−2.24µ ρR2 0+ 1.27ǫ1/2σ1/2 ρ1/2R3/2 0for 0 ≤/parenleftbigg2µ2 σǫρR 0/parenrightbigg1/2 ≤0.225. Correspondingly, the minimum acoustic pressure threshold is pA>  2.91ǫ3/2σ1/2µ ρ1/2R3/2 0+ 0.04ǫ2σ R0for 0 ≤/parenleftBig 2µ2 ǫσρR 0/parenrightBig1/2 ≤0.225 ǫ2σ 2R0for 0 .225≤/parenleftBig 2µ2 ǫσρR 0/parenrightBig1/2 ≤0.4. IV.4 A lower bound for Aescvia Melnikov analysis The distinguished limit equation (14) can be written as the p erturbed system ˙ x=f(x) + ˜ǫg(x, τ) 12where, x= (x, y),f(x, y) = ( y, x2−x) and g(x, y, τ ) =/parenleftbig 0,¯Asin(Ω∗τ)−2¯ζy/parenrightbig withA= ˜ǫ¯A,ζ= ˜ǫ¯ζ. When ˜ ǫ= 0 the system has a center at (0,0) and a saddle point at (1,0). The homoclinic orbit to the unperturbed saddle is given by γ0(τ) = (x(τ), y(τ)) where x(τ) =−(1/2) + (3 /2)tanh2(τ/2) and y(τ) = (3/2)tanh( τ/2)sech2(τ/2). Following [12], the Melnikov function takes the form M(τ0) =/integraldisplay∞ −∞f(γ0(τ))∧g(γ0(τ), τ+τ0)dτ =3 2¯A/integraldisplay∞ −∞sin[Ω∗(τ+τ0)] tanh/parenleftBigτ 2/parenrightBig sech2/parenleftBigτ 2/parenrightBig dτ −9 2¯ζ/integraldisplay∞ −∞tanh2/parenleftBigτ 2/parenrightBig sech4/parenleftBigτ 2/parenrightBig dτ. The first integral can be done with a residue calculation, and the second integral is evaluated in a straight- forward manner, resulting in: M(τ0) =−/bracketleftbigg6π(Ω∗)2cos(Ω∗τ0) sinh(πΩ∗)/bracketrightbigg ¯A−12 5¯ζ. The Melnikov function has simple zeros when ¯A >¯Ah.tan., where ¯Ah.tan.=/parenleftbigg2 sinh( πΩ∗) 5π(Ω∗)2/parenrightbigg ¯ζ. (18) Hence the stable and unstable manifolds of the perturbed sad dle point intersect transversely for all sufficiently small ˜ ǫ/negationslash= 0 when ¯A >¯Ah.tan.[12]. The resulting chaotic dynamics is evident in figure 5, f or example. Since homoclinic tangency must occur before the trajectory throu gh the origin can escape, ¯Ah.tan.may be viewed as a precursor to Aesc. Figure 14 demonstrates that, for small enough ˜ ǫ, equation (18) provides a lower bound for the stability curves seen in figure 11. The reason why Melnikov analysis yields a lower bound for the cavitation threshold relates to how deeply the stable and unstable manifolds of the saddle fixed p oint of the Poincare map for equation (14) penetrate into the region bounded by the separatrix in the A, ζ= 0 case. For sufficiently small values of ˜ ǫ, long segments of the perturbed local stable and unstable man ifolds will stay O(˜ǫ) close to the unperturbed homoclinic orbit. However, as ˜ ǫgrows (and one gets out of the regime in which the asymptotic M elnikov theory strictly applies), these local manifolds will penet rate more deeply into the region bounded by the separatrix in the A, ζ= 0 case. In fact, there is a sizable gap in the parameter space between the homoclinic tangency values and the escape values corresponding to our c avitation criteria, i.e.,when the trajectory through the origin grows without bound. There is a similar ga p when other initial conditions are chosen. The Melnikov function was also calculated in [27]. There, a d etailed analysis of escape from a cubic potential is described and the fractal basin boundaries and occurrence of homoclinic tangencies are given. We also refer the reader to [11] in which a closely related sec ond order, damped and driven oscillator with quadratic nonlinearity is studied using both homoclinic Me lnikov theory, as was done here, and subharmonic Melnikov theory. The existence of periodic orbits is demons trated there, and period doubling bifurcations of these periodic orbits are examined. Their equation arise s from the study of travelling waves in a forced, damped KdV equation. 13V Pressure fields with two fast frequencies In this section, we consider what happens to the cavitation t hreshold if two fast frequency components are present in the acoustic pressure field, and the slow trans ducer, which lowers the ambient pressure and whose effect is quasistatic, is also still present. In figure 1 5, we show the results from simulations with quasiperiodic pressure fields. These were obtained from sim ulations of (14) with the forcing replaced by (A/2)(sin(Ω∗ 1τ) + sin(Ω∗ 2τ)), and a wide range of values for Ω∗ 1and Ω∗ 2. For a fixed value of ζ= 0.25, the cavitation surface shown in the figure was plotted by computi ng the triples (Ω∗ 1,Ω∗ 2, Aesc). We note that in figure 15, the intersection of the cavitation s urface and the vertical plane given by Ω∗ 1= Ω∗ 2represents cavitation thresholds for acoustic forcing of t he form Asin(Ω∗τ) (i.e.,a single fast frequency component and a quasistatic component). Further more, we see that the global minimum of the cavitation surface lies along the line Ω∗ 1= Ω∗ 2. Hence, for A/2 as our particular choice of quasiperiodic forcing coefficient, the addition of a second fast frequency c omponent in the pressure field does not lower the cavitation threshold beyond that of the single fast freq uency case. VI Discussion A distinguished limit equation has been derived which is sui table for use in determining cavitation events of slightly subcritical bubbles. This “normal form” equation allows us to study cavitation thresholds for a range of acoustic forcing frequencies. For Ω∗= 1, we find an explicit expression for the cavitation thresho ld via linear regression, since the simulation data reveal an appr oximate linear dependence of the nondimensional threshold amplitude, A, on the nondimensional liquid viscosity, ζ. When converted to dimensional form, this linear expression translates into a nonlinear dependence, cf. equation (16), on the material parameters. In all of our simulations, the acoustic threshold amplitude co incides with the amplitudes at which the cascades of period-doubling subharmonics terminate. Particular attention has also been paid to calculating the f requency, Ω∗, at which a given subcritical bubble will most easily cavitate. Expression (17) for the co rresponding minimum threshold amplitude A grows linearly in ζforζ <0.225 until the critical amplitude A= 1/4 is reached, and the threshold amplitude stays constant at A≈1/4 for larger ζ. For these larger values of ζ >0.225, the “optimal” frequency is essentially zero, as we showed by doing a slowly-varying pha se portrait analysis and exploiting the fact that the normal form equation undergoes a saddle-node bifurcati on at A= 1/4 in which the entire region of bounded stable orbits vanishes. The full Rayleigh-Plesset equation undergoes a similar bifurcation at forcing amplitudes very near A= 1/4 for sufficiently small ǫ. Overall, the results from the normal form equation are in excellent agreement with those of the full Rayleigh-P lesset equation, and this may be attributed to the high level of similarity between the phase-space struct ures of both equations. In view of the findings in [26], we may draw an additional concl usion from the present work. In a certain sense, we have extended the finding of lowered transition amp litudes reported in [26] to the limiting case of one low frequency and one fast frequency. We find that if a lo w frequency transducer prepares a bubble to become slightly subcritical, then the presence of a high f requency transducer can lower the cavitation 14threshold of the bubble below the Blake threshold. Our results on the optimum forcing frequency and minimum pre ssure threshold to cavitate a subcritical bubble may also be useful in fine-tuning experimental work on single-bubble sonoluminescence (SBSL). In SBSL [10, 6, 23], a single bubble is acoustically forced to un dergo repeated cavitation/collapse cycles, in each of which a short-lived flash of light is produced. While t he process through which a collapsing bubble emits light is very complex and involves many nonlinear phen omena, the possibility of better control over cavitation and collapse, e.g., through the use of multiple-frequency forcing, can perhap s be investigated using the type of analysis presented in this paper. Acknowledgments — We are grateful to Professor S. Madanshetty for many helpfu l discussions. The authors would also like to thank the referees for their comme nts. This research was made possible by Group Infrastructure Grant DMS-9631755 from the National Scienc e Foundation. A.H. gratefully acknowledges financial support from the National Science Foundation via t his grant. T.K. gratefully acknowledges support from the Alfred P. Sloan Foundation in the form of a Sloan Rese arch Fellowship. Appendix: Coaxing experiments in acoustic microcavitatio n To illustrate one application of our results, we now briefly c onsider the experimental findings of [18] on so-called “coaxing” of acoustic microcavitation. In these experiments, smooth submicrometer spheres were added to clean water and were found to facilitate the nucleat ion of cavitation events ( i.e.reduce the cavitation threshold) when a high-frequency transducer (originally a imed as a detector) was turned on at a relatively low pressure amplitude. Specifically, the main cavitation t ransducer was operating at a frequency of 0.75 MHz, while the active detector had a frequency of 30 MHz. In a t ypical experiment, with 0.984-micron spheres added to clean water, the cavitation threshold in th e absence of the active detector was found to be about 15 bar peak negative. When the active detector was turn ed on, producing a minimum pressure of only 0.5 bar peak negative by itself, it caused the cavitation thr eshold of the main transducer to be reduced from 15 to 7 bar peak negative. The polystyrene latex spheres were observed under scanning electron microscopes and their surface was determined to be smooth to about 50 nano meters. It was thus thought that any gas pockets which were trapped on their surface due to incomplet e wetting and which served as nucleation sites for cavitation, were smaller in size than this length. In [18 ], it is conjectured that the extremely high fluid accelerations created by the high-frequency active detect or, coupled with the density mismatch between the gas and the liquid, caused these gas pockets to accumulate on the surface of the spheres and form much larger “gas caps” (on the order of the particle size), which t hen cavitated at the lower threshold. Here we attempt to provide an alternative explanation for the obser ved lowering of the threshold in the presence of the active detector. To effect our estimates, we shall use the same physical parame ters as earlier: µ= 0.001 kg/m ·s,ρ= 0.998 kg/m3andσ= 0.0725 N/m. We also ignore the vapor pressure of the liquid at ro om temperature. Note also that 1 bar=105N/m2and that the transducer frequencies fcited above are related to the radian frequencies 15Ω used earlier by Ω = 2 πf. Let us begin by estimating a typical size for the nucleation s ites which cavitate at pLcrit=−15 bar in the absence of the active detector. Upon using Blake’s classica l estimate of pLcrit=−0.77σ/R0, the equilibrium radius of the trapped air pockets is estimated to be R0= 3.7×10−8m or 37 nm. This size is consistent with the observation that the surfaces of the spheres were smooth to within 50 nm. We note that such a small cavitation nucleus cannot exist within the homogeneous liq uid itself since it would dissolve away extremely fast due to its overpressure resulting from surface tension . However, when trapped in a crevice or within the roughness on solid surfaces, it can be stabilized agains t dissolution with the aid of the meniscus shape which separates it from the liquid. The natural frequency of a 37 nm bubble (if it were spherical) found from equation (12) would be 385 MHz which is very large compar ed to forcing frequency of the cavitation transducer which is 0.75 MHz. Therefore, consistent with Bl ake’s classical criterion, the pressure changes in the liquid would appear quasistatic to the bubble and at such a small size, surface tension does dominate the bubble dynamics. The Blake critical radius Rcritwhich corresponds to this equilibrium radius R0of 37 nm can be calculated to be Rcrit= 64 nm. Let us now suppose that the cavitation transducer is operati ng at pL=−7 bar peak negative as in the experiments with the active transducer also turned on. Usin g equation (5), the final expanded radius of the bubble when the liquid pressure is quasistatically reduced to−7 bar is found to be 4 .2×10−8m or 42 nm. In other words, a bubble of original radius 37 nm at a liquid pr essure of 1 bar, grows to a maximum size of 42 nm when the liquid pressure is reduced to −7 bar. Its critical radius is still 64 nm, reached if the liqui d pressure were to be reduced further to −15 bar. At this point, since the pressure changes in the liquid due to the 0.75 MHz cavitation transducer are occurring slowly compared both with the natural timescale o f the bubble and the 30 MHz detector, let us take the mean pressure in the liquid to be the p∞ 0=−7 bar, and imagine the bubble size at this pressure to be its new equilibrium radius R0= 42 nm, with the critical radius still given by Rcrit= 64 nm. This bubble is now assumed to be forced by the 30 MHz transducer at a n acoustic pressure amplitude of 1 .5 bar (i.e.−0.5 bar peak negative). Using these values, the perturbation p arameter ǫis calculated from equation (8) to be ǫ= 0.69. This parameter is too big for the results of the asymptoti c theory to provide meaningful quantitative agreement; nevertheless, we proceed with the discussion to see if we can at least obtain the right order of magnitude for the pressure threshold. With the given physical parameters, and using the forcing pr essure of pA= 1.5 bar and Ω = 2 π×30×10−6 s−1, the parameters ζ,Aand Ω∗are calculated from equation (15) to be: ζ= 0.98,A= 0.09 and Ω∗= 0.16. Upon examining figure 13, at the relatively large damping par ameter ζ= 0.98 (beyond the range originally considered) the minimum forcing threshold would appear to c orrespond to the constant value of A= 0.25. Here we also note that this same forcing threshold is also obs erved with a range of small Ω∗, including Ω∗= 0.16, see figure 11. In other words, the predicted threshold pre ssure for the active detector to cause cavitation is A= 0.25 which corresponds roughly to pA= 4 bar, whereas in the experiments the threshold was seen to be A= 0.09 or pA= 1.5 bar. Despite the lack of quantitative agreement, the theor etical predictions and the experiments do show the same trends. Namely, in the ab sence of the 30-MHz detector, the pressure in the liquid had to be reduced to −15 bar for cavitation to occur. With the high-frequency tran sducer 16turned on, however, cavitation occurred at a minimum pressu re of−7−1.5 =−8.5 bar in the experiments and at −7−4 =−11 bar based on the theory. (We are adding the negative pressu re contribution from the two transducers to arrive at the final minimum pressure). Thu s, the presence of the second high-frequency transducer does reduce the pressure threshold for cavitati on in both cases. 17References [1] R.E. Apfel, Some New results on cavitation threshold pre diction and bubble dynamics, in Cavitation and Inhomogeneities in Underwater Acoustics , Springer Series in Electrophysics, v.4, W. Lauterborn (ed.), 79–83 (Springer, Berlin, 1980). [2] F.G. Blake, Technical Memo 12, Acoustics Research Labor atory, Harvard University, Cambridge, MA (1949). [3] J.R. Cary, D.F. Escande and J. Tennyson, Adiabatic invar iant change due to separatrix crossing, Phys.Rev. A 34, 4256–4275 (1986). [4] H.-C. Chang and L.-H. Chen, Growth of a gas bubble in a visc ous fluid Phys. Fluids 29, 3530–3536 (1986). [5] L.A. Crum, Acoustic cavitation thresholds in water, in Cavitation and Inhomogeneities in Underwater Acoustics , Springer Series in Electrophysics, v.4, W. Lauterborn (ed .), 84–87 (Springer, Berlin, 1980). [6] L.A. Crum, Sonoluminescence, Physics Today , 22–29 (1994). [7] C. Dugu´ e, D.H. Fruman, J.-Y. Billard and P. Cerrutti, Dy namic criterion for cavitation of bubbles, J. Fluids Engineering 114(2), 250–254 (1992). [8] R. Esche, Untersuchung der Schwingungskavitation in Fl uessigkeiten, Acustica 2, AB208–AB218 (1952). [9] Z.C. Feng and L.G. Leal, Nonlinear bubble dynamics, Ann. Rev. Fluid Mech. 29, 201–243 (1997). [10] D.F. Gaitan, L.A. Crum, C.C. Church and R.A. Roy, Sonolu minescence and bubble dynamics for a single, stable, cavitation bubble, J. Acoust. Soc. Am. ,91, 3166–3183 (1992). [11] R. Grimshaw and X. Tian, Periodic and chaotic behavior i n a reduction of the perturbed KdV equation, Proc. R. Soc. Lond. A 445, 1–21 (1994). [12] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcati ons of Vector Fields , (Applied Mathematical Sciences 42, Springer-Verlag, New York, 1993). [13] I.S. Kang and L.G. Leal, Bubble dynamics in time-period ic straining flows, J. Fluid Mech. 218, 41–69 (1990). [14] W. Lauterborn, Numerical investigation of nonlinear o scillations of gas bubbles in liquids, J. Acoust. Soc. Am. 59, 283–293 (1976). [15] W. Lauterborn and A. Koch, Holographic observation of p eriod-doubled and chaotic bubble oscillations in acoustic cavitation, Phys. Rev. A 35(4), 1974–1976 (1987). [16] L.G. Leal, Laminar Flow and Convective Transport Processes (Butterworth-Heinemann Series in Chem- ical Engineering, Boston, 1992). [17] T.G. Leighton, The Acoustic Bubble (Academic Press Inc., San Diego, 1994). 18[18] S.I. Madanshetty, A conceptual model for acoustic micr ocavitation, J. Acoust. Soc. Am. 98, 2681–2689 (1995). [19] Y. Matsumoto and A.E. Beylich, Influence of homogeneous condensation inside a small gas bubble on it pressure response, J. Fluids Engineering 107, 281–286 (1985). [20] U. Parlitz, V. Englisch, C. Scheffczyk, and W. Lauterbor n, Bifurcation structure of bubble oscillators, J. Acoust. Soc. Am. 88, 1061–1077 (1990). [21] M.S. Plesset and A. Prosperetti, Bubble dynamics and ca vitation, Ann. Rev. Fluid Mech. 9, 145–185 (1977). [22] W. Press, W. Vetterling, S. Teukolsky and B.Flannery, Numerical Recipes in C - The Art of Scientific Computing, 2nd ed. , (Cambridge University Press, 1992). [23] S.J. Putterman, Sonoluminescence: sound into light, Scientific American ,272, 46–51 (1995). [24] Y. Sato and A. Shima, The growth of bubbles in viscous inc ompressible liquids, Report of Inst. of High Speed Mechanics , number 40-319, 23–49 (1979). [25] P. Smereka, B. Birnir and S. Banerjee, Regular and chaot ic bubble oscillations in periodically driven pressure fields, Phys. Fluids 30, 3342–3350 (1987). [26] A.J. Szeri and L.G. Leal, The onset of chaotic oscillati ons and rapid growth of a spherical bubble at subcritical conditions in an incompressible liquid, Phys. Fluids A 3, 551–555 (1991). [27] J.M.T. Thompson, Chaotic phenomena triggering the esc ape from a potential well, Proc. R. Soc. Lond. A421, 195–225 (1989). 19Figure Captions Figure 1 : Pressure in the liquid, pL, versus bubble radius, R, as governed by equation (5). Figure 2 : Phase portraits for the distinguished limit equation (14) . In (a), A= 0 and ζ= 0. The fixed point (0,0) is a center. The fixed point (1,0) is a saddle. In (b ),A= 0 and ζ= 0.09. The fixed point (0,0) is a stable spiral. Figure 3 : Poincar´ e section showing the unstable manifold of the sad dle fixed point (0.999769375,-0.024007197) forA= 0.048 and ζ= 0. Asymptotically, the saddle point is located at a distanc eO(A) from (1,0), specif- ically (1 −A2/10 +O(A4),−A/2−(13/200)A3+O(A5)). Invariant tori are shown inside a portion of the unstable manifold. The center point has moved a large distan ce from (0 ,0) in this case due to the 1:1 resonance when Ω∗= 1. And we note for comparison that with nonresonant values o f Ω∗, Poincar´ e sections show that the center only moves an O(A) distance. For example, when Ω∗= 0.6,0.7,0.85, the centers are located approximately at the points (0.009,0.045),(0.009 ,0.066) and (0.029,0.163) respectively. Figure 4 : Escape parameters for the trajectory of the origin (0,0). F or values of A,ζabove the regression line the trajectory of the origin grows without bound. Below this line the trajectory of the origin remains bounded. Least squares fit: A= 1.356ζ+ 0.058. Figure 5 : Period doubling route to chaos in the distinguished limit e quation (14). For ζ= 0.35, the limit cycles undergo period doubling as Ais increased. Figure 6 : Bifurcation diagram ( ζ= 0.375). Plotted is ˙ xversus A. For each fixed value of A, the origin (0,0) is integrated numerically and the value of ˙ xis plotted every ∆ τ= 2π. Figure 7 : Bounded trajectories for the distinguished limit equatio n with ζ= 0.2,A= 0.3 and Ω∗= 1.0. The dark region is the set of initial conditions whose trajec tories remain bounded; it is the basin of attraction of the periodic orbit that exists in the period-doubling hie rarchy for this value of A. Figure 8 : Simulations of the full Rayleigh-Plesset equation for fou r different values of ǫ. Each open circle represents an ( A, ζ) pair at which the bubble first goes unstable. Superimposed i s the linear re- gression line obtained from the simple criterion based upon the distinguished limit equation. In (a)–(d), ǫ= 0.01,0.05,0.1,0.2, respectively.Figure 9 : Simulations of the full Rayleigh-Plesset equation for fou r different values of ǫ. Each open circle represents a ( pA, R0) pair at which the bubble first goes unstable. Superimposed i s the threshold curve (16) obtained from the simple criterion based upon the distinguished limit equation. In (a)–(d), ǫ= 0.01,0.05,0.1,0.2, respectively. Figure 10 : Radius versus time plots ( ǫ= 0.1, ζ= 0.3, A= 0.3,0.5,0.53). Dimensional parameters: R0= 3.0µm,Rcrit= 3.2µm,pv−p∞ 0= 29.7 kPa, Ω = 0 .7 MHz. Top:pA= 141 .6 Pa. Middle: pA= 236 .0 Pa. Bottom: pA= 250 .2 Pa. Figure 11 : Stability threshold curves for the origin trajectory of th e distinguished limit equation (14) for many different values of Ω∗. The values of Ω∗on the right hand border label the different threshold curves . Figure 12 : Simulations of the full Rayleigh-Plesset equation for fou r different values of Ω∗. In each plot ǫ= 0.05. Each open circle represents an A, ζpair at which the bubble first goes unstable. Superimposed is the threshold curve obtained from the distinguished limi t equation. In (a)–(d), Ω∗= 0.6,0.7,0.8,0.9, respectively. Figure 13 : In (a), the Ω∗that minimizes Aescis plotted versus ζ. In (b), the minimum value of Aesc corresponding to the value of Ω∗in (a) is plotted versus ζ. The open circles represent Rayleigh-Plesset calculations of the minimum threshold with ǫ= 0.05. Figure 14 : Comparisons of the stability curves of figure 11 with the Mel nikov analysis for two values of Ω∗. The dotted line is the stability curve obtained from the dist inguished limit equation. The solid straight line is equation (18). In (a) and (b), Ω∗= 0.9,1.1, respectively. Figure 15 : Stability threshold surface for the trajectory through th e origin obtained by integrating the distinguished limit equation with quasiperiodic forcing. This was obtained from simulations of (14) with the forcing replaced by ( A/2)(sin(Ω∗ 1τ) + sin(Ω∗ 2τ)), and for 0 <Ω∗ 1,Ω∗ 2<1.3. The value of ζwas fixed at ζ= 0.25. The points below the surface correspond to parameter val ues for which the trajectory of the origin remains bounded whereas those points above the surface are p arameters which lead to an escape trajectory for the origin. Qualitatively and quantitatively similar r esults were obtained for ζ= 0.15 and ζ= 0.35.PF#1091 , Harkin, Nadim and Kaper – Figure 1 Rp L p v Rcrit p LcritPF#1091 , Harkin, Nadim and Kaper – Figure 2 −1 −0.5 0 0.5 1 1.5 2 2.5−1.5−1−0.500.511.5 (a)−0.5 0 0.5 1 1.5−1.2−1−0.8−0.6−0.4−0.200.20.40.60.8 (b)PF#1091 , Harkin, Nadim and Kaper – Figure 3

arXiv:physics/9911073v1 [physics.optics] 27 Nov 1999Study of Polarized Electromagnetic Radiation from Spatially Correlated Sources Abhishek Agarwal1, Pankaj Jain2and Jagdish Rai Physics Department Indian Institute of Technology Kanpur, India 208016 Abstract We consider the effect of spatial correlations on sources of p olarized electromagnetic radiation. The sources, assumed to be monochromatic, are constructed out o f dipoles aligned along a line such that their orientation is correlated with their position. In one repre sentative example, the dipole orientations are prescribed by a generalized form of the standard von Mises di stribution for angular variables such that the azimuthal angle of dipoles is correlated with their posi tion. In another example the tip of the dipole vector traces a helix around the symmetry axis of the source, thereby modelling the DNA molecule. We study the polarization properties of the radiation emitted from such sources in the radiation zone. For certain ranges of the parameters we find a rather striking ang ular dependence of polarization. This may find useful applications in certain biological systems as we ll as in astrophysical sources. 1 Introduction In a series of interesting papers Wolf and collaborators [1, 2, 3, 4, 5, 6, 7] studied the spectrum of light from a spatially correlated sources and found, remarkably, that in general the spectrum does not remain invariant under propagation even through vacuum. In the present paper we investigate Polarization properties of a correlated dipole array. Just as we expect spectral shifts f or spatially correlated non-monochromatic sources, we expect nontrivial polarization effects if the correlated source emits polarized light. A simple model of such a source can be constructed by arrangin g a series of dipoles along a line with their orientations correlated with the position of the sour ce. The dipoles will be taken to be aligned along thezaxis and distributed as a gaussian exp[ −z2/2σ2]. The orientation of the dipole is characterized by the polar coordinates θp,φp, which are also assumed to be correlated with the position z. A simple correlated ansatz is given by exp [αcos(θp) +βzsin(φp)] N1(α)N2(βz)(1) whereαandβare parameters, N1(α) =πI0(α) andN2(βz) = 2πI0(βz) are normalization factors and I0is the Bessel function. The basic distribution function exp( αcos(θ−θ0)) used in the above ansatz is the well known von Mises distribution which for circular data is in ma ny ways the analoque of Gaussian distribution for linear data [8, 9, 10]. For α>0 this function peaks at θ=θ0. Making a Taylor expansion close to its peak we find a gaussian distribution to leading power in θ−θ0. The maximum likelihood estimators for the mean angleθ0and the width parameter αare given by, <sin(θ−θ0)>= 0 and<cos(θ−θ0)>=dlog(I0(α))/dα respectively. In prescribing the ansatz given in Eq. 1 we hav e assumed that the polar angle θpof the dipole orientation is uncorrelated with zand the distribution is peaked either at θp= 0 (π) forα >0 (<0). The azimuthal angle φpis correlated with zsuch that for β >0 andz >0(<0) the distribution peaks at φp=π/2(3π/2). We next calculate the electric field at very large distance fr om such a correlated source. The observation pointQis located at the position ( R,θ,φ ) (Fig 1) measured in terms of the spherical polar coordinate s and we assume that the spatial extent of the source σ<<R . The electric field from such a correlated source at large distances is given by, E=−ω2 c2Rp0ei(−ωt+Rω/c)/integraldisplay∞ −∞dzexp/parenleftbig −z2/2σ2/parenrightbig/integraldisplayπ 0dθp/integraldisplay2π 0dφp 1current address: Physics Department, University of Roches ter, Rochester, NY 2e-mail: pkjain@iitk.ac.in 1yz xORQ(R,θ,φ) p θ φ Figure 1: The correlated source consisting of an array of dip oles aligned along the zaxis. The observation pointQis at a distance Rwhich is much larger than the spatial extent of the source. ×exp(αcosθp+βzsinφp) 2π2I0(α)I0(βz)exp/parenleftBig iωzˆR·ˆz/c/parenrightBig ×(ˆp·ˆRˆR−ˆp) (2) where ˆpis a unit vector parallel to the dipole axis, p0is the strength of the dipole, ωis the frequency of light andI0denotes the Bessel function. Since we are interested in the r adiation zone we have dropped all terms higher order in z/R. The resulting field is ofcourse transverse i.e. /vectorE·ˆR= 0. We have also assumed that all the dipoles radiate at same frequency and are in phas e. The correlation of the source with position is measured by the parameter β. It is convenient to define scaled variable z=z/σ,λ=λ/σwhereλ= 2πω/c is the wavelength, and β=βσ. The integrations over θpandφpcan be performed analytically. We numerically integrate ov er zfor various values of position of the observation point, the parameterαwhich determines the width of the distribution of θpand for different value of the correlation parameter β. The observed polarization is computed by calculating the coherency matrix, given by J=/parenleftbigg/angbracketleftEθE∗ θ/angbracketright /angbracketleftEθE∗ φ/angbracketright /angbracketleftE∗ φEθ/angbracketright /angbracketleftEφE∗ φ/angbracketright/parenrightbigg (3) The state of polarization can be uniquely specified by the Sto kes’s parameters or equivalently the Poincare sphere variables [11]. The Stoke’s parameters and the Poinc are sphere variables are obtained in terms of the coherency matrix as: S0=J11+J22 (4) S1=J11−J22 (5) S2=J12+J21 (6) S3=i(J21−J12) (7) The parameter S0is proportional to the intensity of the beam. The Poincare sp here is charted by the angular variables 2χ, and 2ψ, which can be expressed as: S1=S0cos2χcos2ψ (8) 2S2=S0cos2χsin2ψ (9) S3=S0sin 2χ (10) The angleχ(−π/4≤χ≤π/4) measures of the ellipticity of the state of polarization a ndψ(0≤ψ<π ) measures alignment of the linear polarization. For example ,χ= 0 represents pure linear polarization and χ=π/4 pure right circular polarization. 2 Results and Discussion We first study the situation where β >0 andα >0. The result for several values of ( θ,φ) are given in figures 2,3 which show plots of the Poincare sphere variables 2χand 2ψ. The scaled wavelength λ=λ/σof the emitted radiation is taken to be equal to π, i.e. the effective size of the source σis of the order of the wavelength λ. The results show several interesting aspects. The ellipti city of the state of polarization shows significant dependence on the position of the observer. The a ngleχ= 0, i.e. the beam is purely linearly polarized, for the polar angle cos( θ) = 0,1 for all values of azimuthal angle φ. It deviates significantly from 0 as cos(θ) varies from 0 to 1. For sin( φ) = 0, 2χ=π/2 at some critical value θtas cos(θ) varies between 0 andπ/2, i.e. the state of polarization is purely right circular at θ=θt. For sin(φ)>0, 2χalso deviates significantly from 0 and displays a peak at some value of θ. The precise position of the peak is determined by the values of the correlation parameters αandβ. The alignment of linear polarization also shows some very in teresting aspects. For sin( φ) = 0, we find thatψis either 0 or πdepending on the value of θ. The transition occurs at the same critical value of θ where the angle χshows a peak. The state of polarization is purely linear with the electric field along the ˆθ for cos(θ) = 0 and then acquires a circular component for increasing va lues of cos(θ). At the transition point θ=θt, the polarization is purely circular. With further increas e in value of θthe state of polarization is elliptical with the linearly polarized component aligned a longˆφ. The transition point is clearly determined by the condition S1=J11−J22= 0. For other values of sin( φ) we findψ= 0 for cos θ= 0 and then deviates significantly from 0 as θ approached θt, finally levelling off as cos θapproches 1. The final value of ψat cosθ= 1 depends on the correlation parameters and sin φbut for a wide range of parameters 2 ψ>π/ 2. Hence the linear polarizations from sources of this type shows striking characteristic, i. e. that the polarization angle ψis either 0 or close toπ/2 depending on the angle at which it is viewed. For sinφ<0 the Poincare sphere polar angle 2 χis same as for sin φ>0, however the orientation of the linear polarization 2 ψlies between πand 2π, i.e. in the third and fourth quadrants of the equatorial pla ne on the Poincare sphere. For a particular value of φthe azimuthal angle ψ(φ) =−ψ(−φ). If we change the sign of αwe do not find any change in linear polarization angle ψhowever the value of χchanges sign, i.e. the state of polarization changes from ri ght elliptical to left elliptical. Change in sign of βalso leaves ψunchanged while changing the sign of χ. Changing the signs of both αandβproduces no change at all. In the case of the limiting situation where β= 0 we find, as expected, linear polarization is independent of the angular position, i.e. χ= 0 andψ= 0. This is true for any value of the parameter α, which determines the polar distribution of the dipole orientations. Hence we see that the effect disappears if either the effective size of the source σ= 0 or the correlation parameter β= 0. The effect also dissappears in the limit α→ ∞. In this limit the distribution of θpis simply a delta function peaked at 0 and hence our model redu ces to a series of dipoles aligned along the z-axis, which cannot giv e rise to any nontrivial structure. In the numerical calculations above we have taken the effective size of the sou rceσof the order of the wavelength λ. If the sizeσ<<λ , the effect is again negligible since the phase factor ωzˆR·ˆz/cin Eq. 2 is much smaller than one in this case. Hence we find that in order to obtain a nontrivial angular depe ndence of the state of polarization the size of the source, assumed to be coherent, has to be of the ord er of or larger than the wavelength as well as the correlation length 1 /β. 32.1 Transition angle From our results we see that there exists a critical value of t he polar angle θat which the state of linear polarization changes very rapidly. This is particularly tr ue if we set sin φ= 0 where we find that that the orientation of linear polarization ψsuddenly jumps from 0 (or π) toπ/2 at some critical value of the polar angleθ=θt. We study this case in a little more detail. The θandφcomponents of the total electric field is given by, Eθ=−ω2p0 c2Re−iω(t−R/c)√ 2σ2πe−σ2ω2cos2θ/2c2I1(α) I0(α)sinθ Eφ=iω2p0 c2Re−iω(t−R/c)2 sinhα απI0(α)A A=/integraldisplay∞ −∞dze−z2/2σ2sin(ωzcosθ/c)I1(βz) I0(βz) In this case the Stokes parameter S2= 0. Forβ≥(<)0,S3≥(<)0 and hence χ≥(<)0. The point where the polarization angle 2 ψjumps from 0 to πis determined by the condition S1= 0. This is clearly also the point where 2 χ=±π/2. Explicitly the condition to determine the critical value θtis, A2=σ2π3α2e−σ2ω2cos2θt/c2sin2θtI1(α)2 2 sinh2α. This can be used to determine θtas a function of α,β. The result for cos θtas a function of αis plotted in figure 4 for several different values of β. For any fixed value of the parameter β, the transition angle θt decreases from π/2 to 0 asαgoes from zero to infinity. This is expected since as αbecomes large the polar angle distribution of the dipole orientations, peaked alon g thezaxis, becomes very narrow and hence the resultant electric field is aligned along the zaxis for a large range of polar angle θ. Furthermore we find, as expected, that as βgoes to zero the transition angle also tends towards 0. 2.2 Two Dipole Model Further insight into the behavior of such sources can be gain ed by considering a model consisting of two dipoles/vector p1and/vector p2which are located at zand−zrespectively and are oriented such that their polar angles θ1=θ2=θpand the azimuthal angles φ1=−φ2=π/2. We will assume that θplies between 0 and π/2. The strength of the dipoles is p0and they radiate at frequency ω. The electric field at any point is then an addition of two vectors ˆ p1·ˆRˆR−ˆp1and ˆp2·ˆRˆR−ˆp2with phase difference of 2 ωzcosθ/c. The vector ˆp·ˆRˆR−ˆpat any point ( R,θ,φ ) is ofcourse simply the projection of the polarization vect or ˆpon the plane perpendicular to ˆRat that point. The Stokes parameters are easily calculated for this model a nd are given by S0=/parenleftbiggω2p0 c2R/parenrightbigg2/bracketleftbig 4 cos2(ωzcosθ/c)cos2θpsin2θ+ 4 sin2(ωzcosθ/c)sin2θp(cos2θsin2φ+ cos2φ)/bracketrightbig S1=/parenleftbiggω2p0 c2R/parenrightbigg2/bracketleftbig 4 cos2(ωzcosθ/c)cos2θpsin2θ+ 4 sin2(ωzcosθ/c)sin2θp(cos2θsin2φ−cos2φ)/bracketrightbig S2=/parenleftbiggω2p0 c2R/parenrightbigg2 8 sin2(ωzcosθ/c)sin2θpcosθsinφcosφ S3=/parenleftbiggω2p0 c2R/parenrightbigg2 8 cos(ωzcosθ/c)sin(ωzcosθ/c)cosθpsinθpsinθcosφ Several of the features seen in the model prescribed by Eq. 1 c an be verified analytically in this case. First of all we notice that as z→0,S2,S3→0 and the entire effect disappears. The same is true for θp= 0 orπ/2 i.e. if both the dipoles are aligned along a single axis. At si nφ= 0,θ=π/2 the wave is linearly polarized (χ= 0) withψ= 0. Asθdecreases from π/2 to 0,χ >0 and the wave has general elliptical polarization. 4At a certain value of the polar angle θ=θtthe wave is purely right circularly polarized. As θcrossesθt, the linearly polarized component jumps from 0 to π/2, i.e. 2ψchanges from 0 to π. The value of the polar angleθtat which the transition occurs is determined by tan(ωzcosθt/c) =±sinθt/tanθp From this equation we see that as z→0,θtis close to zero for a wide range of values of θp. Only when θp→π/2, a solution with θtsignificantly different from 0 can be found. In general, howev er, we can find a solution with any value of θtby appropriately adjusting zandθp. We can also analytically verify the results for different cas es discussed in the previous model. For example, asφ→ −φ,S2changes sign while the remaining Stokes parameters remain u nchanged. This implies that 2 χ remains unchanged while ψ→ −ψ. We can also study the analoque of changing the sign of αin the previous example while keeping βfixed. In this situation, i.e. α <0, the distribution of polar angle θppeaks atπ instead of 0. In the present example this is equivalent to θp→π−θp. In this case only S3changes sign. The implies that χ→ −χand 2ψremains unchanged, i.e. the right elliptical polarization goes to left elliptical. If we keepθpfixed and change φp→ −φp, which is equivalent to keeping αfixed and changing the sign of β in the earlier example, we again find that only S3changes sign, i.e. χ→ −χand 2ψis unchanged. 2.3 Helical Model We next study an interesting generalization of the model dis cussed above. Instead of the having the peak of theφpdistribution fixed to −π/2 forz <0 andπ/2 forz >0 we allow it to rotate in a helix circling around the z-axis. In this case we replace the φpdependence by exp[ β(φp−ξz)]. As z goes from negative to positive values, the peak of the distribution rotates clo ckwise around the z-axis forming an helix. This is a reasonable model of the structure of DNA molecule and hence has direct physical application. We study this in detail by fixing the azimuthal angle of the dipole orie ntationφp=ξzand the polar angle θpto some constant value, i.e. the φpandθpdistributions are both assumed to be delta functions. This a llows us to perform the zintegration in Eq. 1 analytically. The resulting state of po larization, described by Poincare sphere angles 2 χand 2ψare shown in Figs. 5-8 . In this model we can extract a simple ru le to determine the transition angle for the special case θp=π/2 andξ=nπwherenis an integer. We set sin φ= 0 for this calculation since it is only for this value that the pola rization becomes purely circular for some value of θ=θtand the linearly polarized component flips by π/2 at this point. A straightforward calculation shows that this transition angle θtis given by: cos2θt=nλ/2 Herenrepresents the number of πradians that are traversed by the tip of electric field vector along the helical path and λis the wavelength. In order to get at least one transition λ <2/n. In the special case under consideration there is atmost one transition. Howeve r in general the situation is more complicated and for certain values of θpandξ, more than one transitions are possible. Some representati ve examples are shown in Figs. 5-8. 3 Conclusions In this paper we have considered spatially correlated monoc hromatic sources. We find that at large distance the polarization of the wave shows dramatic dependence on th e angular position of the observer. For certain set of parameters the linearly polarized component shows a s udden jump by π/2. If the symmetry axis of the source is taken to be the z-axis, the polarization show s a sudden transition from being parallel to perpendicular to the symmetry axis of the source, as the po lar angle is changed from π/2 to 0. The sources considered in this paper are idealized since we have assumed coherence over the entire source. For small enough sources, such as the DNA molecule, this may a reasonable approximation. In the case of macroscopic sources, this assumption is in general not appl icable. However in certain situations some aspects of the behavior described in this paper may survive even for t hese cases. For example, we may consider a macroscopic source consisting of large number of structure s of the type considered in this paper. As long as there is some correlation between the orientation of these s tructures over large distances we expect that some aspects of the angular dependence of the polarization of the small structures will survive, even if there does 5not exist any coherent phase relationship over large distan ces. Hence the ideas discussed in this paper may also find interesting applications to macroscopic and astro physical sources. As an interesting example we consider astrophysical sources of radio waves. It is well kn own that the polarization angle of these sources is predominantly observed to be aligned either parallel or per pendicular to the source orientation axis [12]. This difference has generally been attributed to the existence of different physical mechanism for the generation of radio waves in these sources. Our study, however, indicates that this difference in observed polarization angle could also arise simply due to different angles of observatio n. Hence orientation effects must be considered before attributing different physical mechanisms for differ ences in observed polarizations of these sources. Acknowledgements: We thank John Ralston for very useful comments. 6References [1] E. Wolf, Optics Communication 62, 12 (1987). [2] E. Wolf, Nature 326, 26 (1987). [3] E. Wolf, Phys. Rev. Lett. 63, 2220 (1989). [4] E. Wolf and D. F. V. James, Correlation induced spectral changes , Rep. Prog. Phys. 59, 771 (1996). [5] D. F. V. James and E. Wolf, Opt. Comm. 138, 257 (1997). [6] D. F. V. James and E. Wolf, Opt. Lett. 145, 1 (1998). [7] A. Dogriu and E. Wolf, Opt. Lett. 23, 1340 (1998). [8] K. V. Mardia, Statistics of Directional Data (Academic Press, London, 1972). [9] E. Batschelet, Circular Statistics in Biology , (London: Academic Press, 1981). [10] N. I. Fisher, Statistics of Circular Data , (Cambridge, 1993). [11] M. Born and E. Wolf, Principles of Optics (1980), Pergamon Press. [12] J. N. Clark et al, Mon. Not. R. Astron. Soc. 190, 205 (1980). 7cos(θ)cos(θ) cos(θ)cos(θ)sin(φ) sin(φ)sin(φ) = .25 sin(φ) = .5 sin(φ) = .752χ 2χ2χ 2χsin(φ) = 0 0.2.4.6.80.2.4.6.801 0.2.4.6.80.2.4.6.801 0.511.52 0 .2 .4 .6 .80.511.52 0 .2 .4 .6 .8 Figure 2: The polar angle on the Poincare sphere 2 χ, which is a measure of the eccentricity of the ellipse traced by the electric field vector. For pure linear polariza tion 2χ= 0 and for pure right circular polarization 2χ=π/2. The 3-D plot shows 2 χas a function of cos θand sinψwhereθandφare the polar and azimuthal angles of the point of observation. The 2-D plots on the right show the corresponding slices of the 3-D plots for different values of sin φ. The upper and lower plots correspond to β= 1,α= 0.25 andβ=α= 1 respectively. 8cos(θ)cos(θ) cos(θ)cos(θ) sin(φ)sin(φ)sin(φ) = 0 sin(φ) = .25 sin(φ) = .5 sin(φ) = .752ψ2ψ 2ψ 2ψ0.2.4.6.80.2.4.6.80123 0.2.4.6.80.2.4.6.80123 0123 0 .2 .4 .6 .80123 0 .2 .4 .6 .8 Figure 3: The azimuthal angle on the Poincare sphere 2 ψ. This measures the orientation of the linearly polarized component of the wave. The 3-D plot shows 2 ψas a function of cos θand sinψwhereθand φare the polar and azimuthal angles of the point of observatio n. The 2-D plots on the right show the corresponding slices of the 3-D plots for different values of sinφ. The upper and lower plots correspond to β= 1,α= 0.25 andβ=α= 1 respectively. 9αβ = 0.5 β = 1β = 0.1 β = 0.75 β = 1.5 β = 2sin(φ) = 0cosθt 00.10.20.30.40.50.60.70.80.91 0 0.5 1 1.5 2 2.5 3 Figure 4: The critical value of the polar angle θat which the state of linear polarization shows a sudden transition for sin φ= 0 as a function of the parameters αandβwhich specify the distribution of the dipole orientations. For any given value of the parameters αandβ, electric field is parallel ( ψ= 0) tozaxis if the cosine of the observation polar angle cos θis less than cos θt. On the other hand electric field is perpendicular to thezaxis if cosθis greater than cos θt.2χ cos(θ)sin(φ) = 1sin(φ) = 0.75sin(φ) = 0.5sin(φ) = 0.25sin(φ) = 0 sin(φ) = 0.99 -1.6-1.4-1.2-1-0.8-0.6-0.4-0.20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: The polar angle on the Poincare sphere 2 χ(radians) for the helical model as a function of cos( θ) (λ= 0.2π,θp=π/2,ξ=π). 10cos(θ)2ψ(φ) = 1sin(φ) = 0.75 sin(φ) = 0.99sin(φ) = 0.5sin(φ) = 0.25sin(φ) = 0 sin 2.533.544.555.566.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 6: The azimuthal angle on the Poincare sphere 2 ψ(radians) for the helical model as a function of cos(θ) (λ= 0.2π,θp=π/2,ξ=π). cos(θ)sin(φ) = 0.052χsin(φ) = 0.75 sin(φ) = 1sin(φ) = 0.5sin(φ) = 0.25sin(φ) = 0 -2-1.5-1-0.500.511.52 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 7: The polar angle on the Poincare sphere 2 χ(radians) for the helical model as a function of cos( θ) (λ= 0.4π,θp=π/4,ξ=π). 11cos(θ)sin(φ) = 1sin(φ) = 0.75sin(φ) = 0.5sin(φ) = 0.25sin(φ) = 0.05sin(φ) = 02ψ 01234567 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 8: The azimuthal angle on the Poincare sphere 2 ψ(radians) for the helical model as a function of cos(θ) (λ= 0.4π,θp=π/4,ξ=π). 12

arXiv:physics/9911074v1 [physics.bio-ph] 29 Nov 1999Monte Carlo implementation of supercoiled double-strande d DNA Zhang Yang1,2, Zhou Haijun1,3and Ou-Yang Zhong-can1,4 1Institute of Theoretical Physics, Academia Sinica, P.O. Bo x 2735, Beijing 100080, China 2Institut f ¨ur theoretische Physik, FU Berlin, Arnimallee 14, 14195 Berl in, Germany 3The State Key Lab. of Scientific and Engineering Computing, B eijing 100080, China 4Center for Advanced Study, Tsinghua University, Beijing 10 0084, China Abstract Metropolis Monte Carlo simulation is used to investigate th e elasticity of torsionally stressed double-stranded DNA, in which twis t and supercoiling are incorporated as a natural result of base-stacking inter action and backbone bending constrained by hydrogen bonds formed between DNA co mplementary nucleotide bases. Three evident regimes are found in extens ion versus torsion and/or force versus extension plots: a low-force regime in w hich over- and un- derwound molecules behave similarly under stretching; an i ntermediate-force regime in which chirality appears for negatively and positi vely supercoiled DNA and extension of underwound molecule is insensitive to t he supercoil- ing degree of the polymer; and a large-force regime in which p lectonemic DNA is fully converted to extended DNA and supercoiled DNA be haves quite like a torsionless molecule. The striking coincidence betw een theoretic calcu- lations and recent experimental measurement of torsionall y stretched DNA [Strick et al., Science 271, 1835 (1996), Biophys. J. 74, 2016 (1998)] strongly suggests that the interplay between base-stacking interac tion and permanent hydrogen-bond constraint takes an important role in unders tanding the novel properties of elasticity of supercoiled DNA polymer. Introduction 0Recent years have witnessed a remarkably intense experimen tal and theoretical activity in searching for the elasticity of a single supercoiled DNA mol ecule (see, e.g. Strick et al., 1996, 1998; Fain et al., 1997; Vologodskii and Marko, 1997; Moroz a nd Nelson, 1997; Bouchiat and Mezard, 1998, Zhou et al., 1999). Within a cell, native do uble-stranded DNA (dsDNA) often exists as a twisted, and heavily coiled, closed circle . Differing amount of supercoiling, in addition to affecting the packing of DNA within cells, influ ences the activities of proteins that participate in processes — such as DNA replication and t ranscription — that require the untwisting of dsDNA (Wu et al., 1988). It is believed that changes in supercoiling can also promote changes in DNA secondary structure that influen ces the binding of proteins and other ligands (Morse and Simpson, 1988). In recent experiments (Strick et al., 1996, 1998) on single t orsionally constrained DNA molecule, it was found that the supercoiling remarkably infl uences the mechanical property of DNA molecules. When applied with relatively low stretchi ng force, a supercoiled molecule can reduce its torque by writhing, forming structures known as plectonemes. Therefore, the distance between two ends of the polymer decreases with incr easing supercoiling. But above a certain critical force fc, this dependence of extension on supercoiling disappears. More strikingly, the value of fcis significantly different for positively and negatively sup ercoiled DNA, i.e. fc∼0.8pN for underwound molecule and fc∼4.5pN for overwound ones. On the theoretical side, harmonic twist elasticity and bendin g energy according to the wormlike chain model have been used to understand supercoiling of DNA polymer (Fain et al., 1997; Vologodskii and Marko, 1997; Moroz and Nelson, 1997; Bouchi at and Mezard, 1998), and some qualitative mechanic features of plectonemic structu res of supercoiled DNA polymer have been described by the harmonic twist elasticity. But be cause of the chiral symmetry of harmonic twist elasticity, the asymmetry of elastic beha viors of supercoiled DNA can not be understood by this model, and especially the three obviou s mechanic regimes observed in experiment of supercoiling DNA (Strick et al., 1996, 1998) s till need better understanding. To understand the supercoiling property as well as the highl y extensibility of DNA, we have developed a more realistic model in which the double-st randed nature of DNA is taken 1into account explicitly (Zhou et al., 1999). The supercoili ng property of highly extended DNA was investigated analytically. Here, we aim at performi ng a thorough and systematic investigation into the property of supercoiled DNA by using Monte Carlo simulations based on this model. As we have known, the bending energy of DNA polymer is mainly a ssociated with the covalent bonding between neighboring atoms of DNA backbone (Nossel and Lecar, 1991). In our previous work (Zhou et al., 1999), van der Waals interact ions between adjacent basepairs was introduced and this helps to explain the highly cooperat ive extensibility of overstretched DNA (Cluzel et al., 1996; Smith et al., 1996). And it has been s hown that the short-range base-stacking interaction takes a significant role in deter mining the elastical property of DNA. Lennard-Jones type potential between adjacent basepa irs can be written as U(θ) =  ǫ[(cosθ0 cosθ)12−2(cosθ0 cosθ)6],forθ >0, ǫ[cos12θ0−2 cos6θ0],forθ≤0,(1) (see also Fig. 1). The folding angle θof the sugar-phosphate backbones around DNA central axis is associated with the steric distance rof adjacent basepairs by r=r0cosθ, where r0 is the backbone arclength between adjacent bases. The asymm etric potential related to positive and negative folding angle θin Fig. 1 ensures a native DNA to take a right-handed double-helix configuration with its equilibrium folding an gleθeq∼θ0. This double-helix structure is anticipated to be very stable since ǫ(∼14kBT) is much higher than thermal energy kBTaccording to the results of quantum chemical calculations ( Saenger, 1984). In case that DNA polymer is torsionally constrained, the bas epair folding angle will deviate from the equilibrium position θeq. However, if the stretching force is very small, the folding angle may deviate from θeqonly slightly. This is because of the following reason: As we can infer from Fig. 1, the base-stacking potential is ve ry sharp around θ0, and a relatively large force is needed to make θdeviate considerably from its equilibrium value. It is reasonable for us to anticipate that a supercoiled DNA u nder low stretching force will convert its excess or deficit linking number into positive or negative writhing of its central axis. Since the central axis is symmetric with respect to pos itive or negative writhing, the 2elastic response of DNA at this force regime will certainly b e symmetric with positive or negative degree of supercoiling. Only when the stretching f orce becomes large enough will the chirality of supercoiled DNA appear. In this regime, it b ecomes more and more difficult for the central axis to writhe to absorb linking number and an increasing portion of the linking number will be converted to twisting number of the ba ckbones, which will certainly changes the twisting manner of dsDNA. Since Eq. 1 shows that f or dsDNA untwisting is much easier than overtwisting, chiral behavior is anticipa ted to emerge. This opinion is consistent with the experimental result of Strick et al. (19 96). In this paper, we investigate the mechanical properties of s upercoiled DNA by numerical Monte Carlo method. Base-stacking van der Waals interactio ns between adjacent basepairs are incorporated by introducing the new degree of freedom, n amely the folding angle θ. A fundamental difference from the previous approaches (See, e.g., Vologodskii and Marko, 1997), which try to include the twist degrees of freedom by ad ding extra terms to the free energy, is that the twist and supercoiling are treated as the cooperative result of base-stacking and backbones bending constrainted from permanent basepai rs. The striking coincidence between theoretic calculations and experimental data of su percoiling DNA (Stick et al., 1996, 1998) indeed confirms this treatment. Model and method of calculation In the simulation, the double-stranded DNA molecule is mode led as a chain of discrete cylinders, or two discrete wormlike chains constrained by b asepairs of fixed length 2 R(Fig. 2). The conformation of DNA molecule of N straight cylinder s egments is specified by the space positions of vertices of its central axis, ri= (x(i), y(i), z(i)) in 3-D Cartesian coordinate system, and the folding angle of the sugar-phosphate backbo nes around the central axis, θi, i= 1,2,· · ·, N. Each segment is assigned the same amount of basepairs, nbp, so that the length of the ith segment satisfies ∆si=|ri−ri−1|= 0.34nbpcosθi /angbracketleftcosθ/angbracketright0, (2) 3where /angbracketleft· · ·/angbracketright 0means the thermal average for a relaxed DNA molecule. Moreov er, bearing in mind the experimental fact that there are about 10.5 basepai rs for each turn of a native double helix DNA and the average distance between the adjace nt basepairs is about d0= 0.34nm, we have set the basepair length as 2 R= (10.5d0/π)/angbracketlefttanθ/angbracketright0in our model. Metropolis Monte Carlo method (Metropolis et al., 1953) is u sed to simulate the equi- librium evolution procedure of torsionally stretched dsDN A molecule. At each step of the simulation procedure, a trial conformation of the chain is g enerated by a movement from the previous one. The starting configuration is chosen arbitrar ily (except that some topology and bound conditions should be satisfied, see below) and the a veraged results of equilibrium ensemble are independent of the initial choice after numero us movements. The probability of acceptance of the movement depends on the difference in ene rgy between the trial and the current conformations, according to the Boltzmann weig ht. When a trial movement is rejected, the current conformation should be counted once m ore. This procedure is repeated numerous times to obtain an ensemble of conformations that, in principle, is representative of the equilibrium distribution of DNA conformation. The DNA model As we have known, double strand DNA is formed by winding two po lynucleotide back- bones right-handedly around a common central axis. Between the backbones nucleotide basepairs are formed with the formation of hydrogen bonds be tween complementary bases. In our continuous model (Zhou et al., 1999), the embeddings o f two backbones are defined byr1(s) andr2(s′). The ribbon structure of DNA is enforced by having r2(s′) separated fromr1(s) by a distance 2 R, i.e.r2(s′) =r1(s) + 2Rb(s) where the hydrogen-bond-director unit vector b(s) points from r1(s) tor2(s′). As the result of the wormlike backbones, the bending energy of two backbones can be written as EB=κ 2/integraldisplayL 0[(dr2 1 ds2)2ds+ (dr2 2 ds′2)2ds′]. (3) The formation of basepairs leads to rigid constraints betwe en the two backbones and at the same time they hinder considerably the bending freedom o f DNA central axis because 4of the strong steric effect. In the assumption of permanent hy drogen bonds (Everaers et al., 1995; Liverpool et al., 1998; Zhou et al, 1999), |s′−s|= 0. The relative sliding of backbones is prohibited and the basepair orientation lie s perpendicular to the tangent vectors t1=dr1/dsandt2=dr2/dsof the two backbones and that of the central axis, t: b·t1=b·t2=b·t= 0. By defining the folding angle as half of the rotation angle from t2(s) tot1(s), i.e., the intersection angle between tangent vector of ba ckbones t1(2)and DNA central axis t, we have   t1= cos θt+ sinθb×t t2= cos θt−sinθb×t.(4) Therefore, the bending energy of the two backbones can be rew ritten as EB=/integraldisplayL 0[κ(dt ds)2+κ(dθ ds)2+κsin4θ R2]ds (5) where dsdenotes arc-length element of the backbones, Lthe total contour length of each backbone, and κis the persistence length of one DNA backbone. Bearing in min d that the pairing and stacking enthalpy of the bases significantly increase bending stiffness of polymer axis, the experimental value of persistent length o f dsDNA polymer is considerably larger than that of a DNA single strand (See, e.g. Smith et al. , 1996). To incorporate the steric effect and also considering the typical experimen t value of persistent length of dsDNA p= 53nm, the simpliest way is to substitute kin the first term of Eq. 5 with a phenomenological parameter κ∗= 53.0/2/angbracketleftcosθ/angbracketright0nmkBT(Zhou et al., 1999), hereafter this is assumed. Taking into account Eqs. 1 and 5, the total energy of dsDNA mol ecule with Nsegments in our discrete computational model is expressed as E=αN−1/summationdisplay i=1γ2 i+α′N−1/summationdisplay i=1(θi+1−θi)2+κ R2N/summationdisplay i=1∆sisin3θitanθi+Nbp/summationdisplay j=1U(θj)−fz(N),(6) where γiis the bending angle between the ( i−1)th and the ith segments (Fig. 2), Nbpthe total number of basepairs of DNA polymer, and z(N) is the total extension of the DNA central axis along the direction of the external force f(assumed in the z-direction). 5Since Kuhn statistical length of dsDNA polymer is associate d with its bending stiffness (the Kuhn length is twice as persistence length of dsDNA poly mer according to the wormlike chain model), one can decide bending rigidity parameter αof the discrete chain accordingly. Suppose that we take the Ndiscrete segments to simulate the behaviors of a dsDNA polym er ofnKuhn statistical length, the length of m(=N/n) segments should correspond to one Kuhn statistical length. Therefore, for any chosen value m, we can decide the bending rigidity parameter αin the way (see Appendix) m=1 +/angbracketleftcosγ/angbracketright 1− /angbracketleftcosγ/angbracketright, (7) where /angbracketleftcosγ/angbracketright=/integraltextπ 0cosγexp(−αγ2) sinγdγ/integraltextπ 0exp(−αγ2) sinγdγ. (8) In principle, the discrete DNA model becomes continuous onl y when mapproaches infinity. The CPU time needed for a simulation, however, incr eases approximately as N2= (nm)2. So it is necessary to choose a value of mthat is large enough to ensure reliable results but small enough to keep the computational time within reasonable bounds. Our calculation and also previous work (Vologoskii et al., 1 992) showed that simulated properties do not depend on mif it exceeds 8. Therefore, m= 8 was used in the current calculation, for which the bending constant α= 1.895kBT. Furthermore, we have chosen N= 160 in consideration of the feasible computer time. Since K uhn statistical length of dsDNA is taken as 106nm, the B-form length of the polymer in ou r simulation corresponds toLB= 2120nm or 6234 base-pairs. The constant α′in the second term of Eq. 6 should be associated with stiffness of the DNA single strand. As an cr ude approximation, we have taken here α′=α= 1.895kBT.1 The fourth term in Eq. 6 accounts for van der Waals interactio ns between adjacent basepairs (see Eq. 1). θ0(= 62◦) is related to the equilibrium distance between a DNA 1Our unpublished data show that, the amount of second term of E q. 6 is quite small compared with other four terms. And the result of simulation is not sen sitive to α′. 6dimer. The base-stacking intensity ǫis generally influenced by composition and sequence of nucleotide chains. For example, the solubility experime nts in biphasic systems show that stacking interactions between purine and pyrimidine bases follow the trend purine −purine >pyrimidine −purine >pyrimidine −pyrimidine . Since we do not distinguish the specific base-sequence of pur ine and pyrimidine in our DNA model, we take statistic average of stack energies as ǫ= 14kBT, according to the result of quantum chemical calculations (Saenger, 1984). To simulate the extension of the stretched DNA chain, we fixed one of its ends at original point in 3-D Cartesian system and applied a force fdirected along the zaxis to the second end, which corresponds to the fifth term of Eq. 6. Calculation of link number The number of times the two strands of DNA double helix are int erwound, i.e., the link number Lk, is a topologic invariant quantity for closed DNA molecule a nd also for linear DNA polymer in case that the orientations of two extremities of the linear polymer are fixed and any part of polymer is forbidden to go round the extre mities of the polymer. An unstressed B-DNA molecule has one right-handed twist per 3. 4nm along its length, i.e., Lk0=LB/3.4. Under some twist stress, the link number of DNA polymer may be different from its torsionally relexed value. In all case when ∆ Lk=Lk−Lk0/negationslash= 0, the DNA polymer is called “supercoiled” (Vologodskii and Cozzarelli, 1994 ). The relative difference in link number σ=Lk−Lk0 Lk0(9) signifies the degree of supercoiling which is independent up on the length of DNA polymer. The native DNA of organisms living at physiological environ ment are found always slightly underwound and its supercoiling degree is between −0.03 and −0.09 (Bauer, 1978; Volo- godskii and Cozzarelli, 1994), which is believed significan tly relevant in some fundamental biological processes (Wu et al., 1988; Morse and Simpson, 19 88). 7In addition to counting directly the number of times the two s trands are interwound, the link number of closed DNA circle can be conveniently calc ulated by White’s theorem (White 1969) Lk=Tw+Wr. (10) The twist Twis the number of times basepair twist around central axis and does not depend upon the configuration of molecule axis. The writhe Wrof molecule is a simple function of only the molecule axis vector r(s) (White, 1969; Fuller, 1971) Wr=1 4π/integraldisplay /integraldisplay dsds′∂sr(s)×∂s′r(s′)·[r(s)−r(s′)] |r(s)−r(s′)|3. (11) Wris scale invariant and dimensionless and changes sign under reflection or inversion of r, reflecting the cross product in the formula above. Therefore Wr= 0 if r(s) is planar or otherwise reflection symmetric. In order to control and measure experimentally the supercoi ling degree of linear DNA polymer, Strick et al. (1996, 1998) attached one end of DNA mo lecule to a glass cover slip by DIG-anti-DIG links and other end to a paramagnetic bead by biotin-streptavidin links. Bearing in mind the diameter of magnetic bead ( ≈4.5µm) is far beyond that of polymer, the anchoring points can be considered as on impenetrable wa lls and ∼16-µm-long DNA (Strick et al., 1996) in fact is prohibited to pass around the ends of the polymer. A magnetic field pointing in the plane of the microscope slide was applie d to fix the orientation of the bead. Therefore, by rotating the magnets and counting the ti me of turns, the link number Lkof the linear DNA molecule can be controlled and measured exp erimentally. In Monte Carlo calculations, we restrict the DNA chain by two impenetrable parallel walls crossing the chain ends which is to simulate the above mentio ned experimental equipment of the magnet bead and the microscope slide (see also the trea tment in Vologodskii and Marko, 1997). The walls are always parallel to xyplane in our Cartian coordinate system and thus perpendicular to the direction of the force applied to the chain ends. One way to calculate the link number Lkof DNA molecule in our Monte Carlo simulation is to use the White’s formula Eq. 10. However, the writhe Wris defined only for closed 8chain. In order to solve the problem, we add three long flat rib bons to the two ends of the DNA chain in each conformation during the simulation pro cedure. The axes of these ribbons are kept in the same planar and consist a closed circl e together with the linear DNA chain. Since there is no any twist in the added three flat ribbo ns, it is not difficult to verify from Fig. 3 that the number of times two strands interwind Lklin Fig. 3a is equal to the link number of new closed polymer Lkcin Fig. 3d. Therefore, we only calculate Lkof the closed chain in our simulations according to Eqs. 10 and 11. Quite similar to the model by Tan and Harvey (1989) in which th e twist of each base-pair of DNA chain is explicitly specified, the folding angle of bac kbones in each segments has been given in our model. So the twist can be directly calculat ed by Tw=1 2πRN/summationdisplay i=1∆sitanθi. (12) The writhe Wrof the new DNA circle can be calculated according to Eq. 11. Simulation procedure For any given force, equilibrium sets of conformations of DN A chain are constructed using the Metropolis MC procedure (Metropolis et al., 1953) . Three kinds of movements have been considered in our simulations (see Fig. 4). In the first type of movement, a random chosen segment is under twisted or overtwisted by an angle λ1. In other words, the folding angle θiof the chosen segment is modified into a new value θ′ i=θi+λ1. When θ′ iis beyond the setting interval [ −θm, θm] from one side, it will re-enter the interval from the opposite side accordi ng to the periodicity assumption. Although the geometric limit of folding angle of DNA backbon e isθm=π/2, we set θm= 85o here to avoid the possible divergency in numerical calculat ion of potential of Eq. 1. It should be mentioned that, this movement modifies not only the foldin g angle of the chosen segment but also the coordinates of all the behind vertices rj, j=i,· · ·, Nalong the length, since when the folding angle θiis changed we have also changed the length of the segment ∆ si according to Eq. 2. So we should translate all those segments behind this one to make the chain match up (Fig. 4a). 9In the second type of movement, an interval subchain contain ing arbitrary amount of segments will be rotated by an angle of λ2around the straight line connecting the vertices bounding the subchain (Fig. 4b). The third type of movement i nvolves a rotation of the subchain between a chosen vertices and the free end by an angl e ofλ3, around an axis with arbitrary orientation (Fig. 4c). All three types of movemen ts satisfy the basic require- ment of the Metropolis procedure of microscopic reversibil ity, i.e. the probability of trial conformation Bwhen current conformation is Amust be equal to the probability of trial conformation Awhen current conformation is B. All three types of movements change the configurations of DNA chain. But from the viewpoint of energy, their functions are quite different. Wh ile the first type of movement concerns mainly with modifying twist and stacking energy, t he second one changes only the bending energy and the third modifies both bending energy and extension of DNA chain. Each of them is performed in the probability of 1 /3. The value of λ1, λ2, λ3are uniformly distributed over interval ( −λ0 1, λ0 1),(−λ0 2, λ0 2) and ( −λ0 3, λ0 3) respectively, and λ0 1, λ0 2andλ0 3 are chosen to guarantee that about half of the trial moves of e ach type are accepted. The starting conformation of DNA chain is unknotted. But the configurations after numerous steps of movements may become knotted, which viola tes the topologic invariance of chain and is incorporeal. Especially, both ends of molecu le are anchored in the experiment and knots never occur. In order to avoid knotted configuratio n, we should check the knot status for each trial conformation. The most effective way to clarify the knot categories of DNA circle is to calculate its Jones polynomial (Jones, 1985 ), which is strictly topological invariant for knot categories. But the computational calcu lation of Jones polynomial is quite prolix at this moment. In our case that it is only necessary to distinguish between unknot and knot categories, the classical Alexander polynomial (A lexander, 1928; Conway, 1969) is enough to meet this requirement although it is of weaker to pological invariants and does not distinguish mirror images. For trivial knot, Alexander polynomial ∆( t) = 1; and ∆( t) is 10usually not equal to 1 for knotted chain.2Convenient algorithms for computer calculation of Alexander polynomial had been well built (see, e.g. Volog odskii et al. 1974; Harris and Harvey, 1999). We only calculate the value of ∆( −1) in our simulation. In case that the trial movement knots the chain, the energy of trial conforma tion is set to be infinite, i.e. it will be rejected. Another interaction considered in our simulation is the ste ric effect of polymer chain. Since the segment has finite volume, other segments cannot co me into its own space region. This interaction evidently swells the polymer (Doi and Edwa rds, 1986). To incorporate this exclude-volume effect into our simulation, for each tri al conformation, we calculate the distance of between any point on the axis of a segment and any p oint on the axis of another non-adjacent segment and check whether this distance is les s than the DNA diameter 2 R. If the minimum distance for any two chosen segments is less th an 2R, the energy of trial conformation is set infinite and the movement is rejected. During the evolution of DNA chain, the supercoiling degree σmay distribute around all the possible values. In order to avoid the waste of comput ation events, we bound the supercoiling σof DNA chain inside the experimental region (Strick et al., 1 996, 1998), i.e. −0.12≥σ≥0.12. When the torsion degree of trial conformation is beyond t he chosen range, we simply neglect the movement and reproduce a new tri al movement again. Result of Monte Carlo simulation To obtain equilibrium ensemble of DNA evolution, 107elementary displacements are produced for each chosen applied force f. The relative extension xand supercoiling degree σof each accepted conformation of DNA chain are calculated. W hen the trial movement is rejected, the current conformation is count up twice (see Me tropolis et al., 1953). 2Although there are nontrivial knots whose Alexander polyno mials equal unity, this case is very rare. One of the example for nontrivial knot with ∆( t) = 1 can be found in Vologodskii et al. (1974). 11In order to see the dependence of mechanics property of DNA up on supercoiling degree, the whole sample is partitioned into 15 subsamples accordin g to the value of the supercoiling degree σ. For each subsample, we calculate the averaged extension /angbracketleftxj/angbracketright=1 Nj/summationtextNj i=1zi(N) LB, j= 1,· · ·,15 (13) and the averaged torsion /angbracketleftσ/angbracketright=1 NjNj/summationdisplay i=1σi, j= 1,· · ·,15, (14) where Njis the number of movements supercoiling of which belong to jth subsample. We display the force versus relation extension for all posit ive and negative supercoiling in Fig. 5a and c respectively. As a comparison, the experimen tal data (Strick et al., 1998) are shown in Fig. 5b and d. In Fig. 6 is shown the averaged exten sion as a function of supercoiling degree for 3 typical applied forces. At low f orce, the extension in our MC simulation saturates at a value greater than zero because of the impenetrable walls which astrict the vertical coordinate of the free end always highe r than that of any other points of the DNA chain. The same effect of the impenetrable walls was fo und in earlier works (see Fig. 9 of the paper by Vologodskii and Marko, 1997). For conci seness, we did not show the points the relative extension of which is less than 0 .15 in Fig. 5 and 6. In spite of quantitative difference between Monte Carlo resu lts and experimental data, the qualitative coincidence is striking. Especially, thre e evident regimes exist in both exper- imental data and our Monte Carlo simulations: i).At a low force, the elastic behaviour of DNA is symmetrical un der positive or negative supercoiling. This is understandable, since the DNA torsio n is the cooperative result of hydrogen-bond constrained bending of DNA backbones and t he base-stacking in- teraction in our model. At very low force, the contribution f rom applied force and the thermodynamic fluctuation perturbate the folding angle θof basepair to derive very little from the equilibrium position θ0. Therefore, the DNA elasticity is achiral at this region (see the Introduction part of this paper). For a fixed applied force, the 12increasing torsion stress tends to produce plectonemic sta te which shorten the distance of two ends, therefore, the relative extension of linear DNA polymer. These features can be also understood by the traditional approaches with ha rmonic twist and bending elasticity (Vologodskii and Marko, 1997; Bouchiat and Meza rd, 1998). ii).At intermediate force, the folding angle of basepairs are pu lled slightly further away from equilibrium value θ0where van der Waals potential is not symmetric around θ0. So the chiral nature of elasticity of the DNA molecule appea rs. In negative supercoiling region, i.e. θ < θ 0, the contribution of applied force dominates that of potential because of the low plateaus of U(θ). So the extension is insensitive to negative supercoiling degree. On the other hand, the positi ve supercoiling still tends to contract the molecule. iii).At higher force, the contribution of the applied force to the energy dominates that of van der Waals potential in both over- and underwound DNA. The extension of DNA accesses to its B-form length. Therefore, the plectonemic D NA is fully converted to extended DNA, the writhe is essentially entirely converted to twist and the force- extension behaviour reverts to that of untwisted ( σ= 0) DNA as expected from a torsionless worm-like chain model (Smith et al., 1992; Mark o and Siggia, 1995; Zhou et al., 1999). Because of the effect of impenetrable wall, how ever, the extension of DNA molecule in our calculation is slightly higher than expe rimental data. In conclusion, the elasticity of supercoiled double-stran ded DNA is investigated by Monte Carlo simulations. In stead of introducing an extra twist en ergy term, twist and supercoiling are leaded into as a nature result of cooperative interplay b etween base-stacking interac- tion and sugar-phosphate backbones bending constrained by permanent hydrogen bonds. Without any adjustable parameter, the theoretic results on the correlations among DNA extension, supercoiling degree and applied force agree qua litatively to recent experimental data by Strick et al (1996, 1998). It should be mentioned that there is an up-limit of supercoil ing degree for extended 13DNA in current model, i.e. σmax∼0.14, which corresponds to θ= 90oof folding angle. In recent experiments, Allemand et al. (1998) twisted the pl asmid up to the range of −5< σ < 3. They found that at this “unrealistically high” supercoil ing, the curves of force versus extension for different σsplit again at higher stretch force ( >3pN). As argued by Allemand et al. (1998), in the extremely under- and overwoun d torsion stress, two new DNA forms, denatured-DNA and P-DNA with exposed bases, will appear. In fact, if the deviation of the angle which specifies DNA twist from its equi librium value exceeds some threshold, the corresponding torsional stress causes loca l distraction of the regular double helix structure (Vologodskii and Cozzarelli, 1994). So the emergence of these two striking forms is essentially associated with the broken processes o f some basepairs under super- highly torsional stress. In this case, the permanent hydrog en constrain will be violated and the configuration of base stacking interactions be varie d considerably. We hope, with incorporation of these effects at high supercoiling degree, our model should reproduce the novel elastic behaviour of DNA. This part of work is in progre ss. Acknowledgements Parts of the computer calculations of this work were perform ed in the Computer Cluster of Institut f¨ ur Theoretische Physik (FU-Berlin) and the State Key Lab. of S cientific and Engineering Computing (Beijing), which our thanks are due t o. One of authors (Z. Y.) would like to thank U. H. E. Hansmann, B. -L. Hao, L. -S. Liu and W. -M. Zheng for discussions and helps. Appendix: Kuhn Statistical Length of Discrete Chain Let us consider a discrete chain of Nsegments with each of length l0, the end-to-end vector of which is written as R≡l0N/summationdisplay i=1ti, (15) where ti=Ri−Ri−1 |Ri−Ri−1|. 14For chains with bending stiffness, e.g. the DNA model describ ed in Eq. 6, /angbracketleftti+k·ti/angbracketrightdoes not vanish for k/negationslash= 0.ti+kcan be expressed relatively to i+k−1’th segment as ti+k= cos γi+k−1ti+k−1+ sinγi+k−1ni+k−1, (16) where γi+k−1is the bending angle between i+k−1’th and i+k’th segments as defined in Eq. 6, and ni+k−1is the unit vector coplanar with ti+kandti+k−1but perpendicular to the latter. If the average of ti+kis taken with the rest of the chain (i.e., ti,ti+1,· · ·,ti+k−1) fixed, one obtains /angbracketleftti+k/angbracketrightti,ti+1,···,ti+k−1fixed=/angbracketleftcosγi+k−1/angbracketrightti+k−1, (17) since /angbracketleftni+k−1/angbracketrightti,ti+1,···,ti+k−1fixed= 0 according to Eq.6. Multiplying both sides of Eq. 17 by tiand taking the average over ti,ti+1,· · ·,ti+k−1, one has /angbracketleftti+k·ti/angbracketright=/angbracketleftcosγ/angbracketright/angbracketleftti+k−1·ti/angbracketright, (18) where /angbracketleftcosγ/angbracketrightis not specific to segments and given by Eq. 8. This recursion e quation, with the initial condition t2= 1, is solved by /angbracketleftti+k·ti/angbracketright=/angbracketleftcosγ/angbracketrightk. (19) Thus for large N,/angbracketleftR2/angbracketrightis given by /angbracketleftR2/angbracketright=l2 0N/summationdisplay i=1N/summationdisplay j=1/angbracketleftti·tj/angbracketright =l2 0(N+ 2N−1/summationdisplay i+1N−i/summationdisplay k=1/angbracketleftti·ti+k/angbracketright) ≃Nl2 01 +/angbracketleftcosγ/angbracketright 1− /angbracketleftcosγ/angbracketright Therefore, Kuhn statistical length of the discrete chain ca n be written as b≡/angbracketleftR2/angbracketright Rmax=l01 +/angbracketleftcosγ/angbracketright 1− /angbracketleftcosγ/angbracketright, (20) where Rmaxis the maximum length of the end-to-end vector. 15References Alexander, J. W. 1928. Topological invariants of knots and k nots. Trans. Amer. Math. Soc., 30:275-306. Allemand, J. F., D. Bensimon, R. Lavery, and V. Croquette. 19 98. Stretched and over- wound DNA forms a Pauling-like structure with exposed bases .Proc. Natl. Acad. Sci. USA 95:14152-14157. Bauer, W. R. 1978. 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A program for analyzing knots represented by polygonal paths. J. Comput. Chem. 20:813-818. Jones, V. F. R. 1985. A polynomial invariant for links via von Neumann algebras. Bull. Am. Math. Soc. 12:103-112. Liverpool, T. B., R. Golestanian and K. Kremer. 1998. Statis tical mechanics of double- stranded semiflexible polymers. Phys. Rev. Lett. 80:405-408. Marko., J. F., and E. D. Siggia. 1995. Strething DNA. Macromolecules 28:8759-8770. Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, and A. H. Teller. 1953. Equation of State Calculations by Fast Computing Machines, J. Chem. Phys. 21:1087-1092. Moroz, J. D., and P. Nelson. 1997. Torsional directed walks, entropic elasticity, and DNA twist stiffness. Proc. Natl. Acad. Sci. USA 94:14 418-14 422. Morse, R. H., and R. T. Simpson. 1988. DNA in the nucleosome. Cell. 54:285-287. Nossal, R. J., and H. Lecar, 1991. Molecular and Cell Biophys ics. Addison-Wesley Pub- lishing Company. Saenger, W., 1984. Principles of Nucleic Acid Structure. Sp ringer-Verlag, New York. Smith., S. B., L. Finzi, and C. Bustamante. 1992. Direct mech anical measurements of elasticity of single DNA molecules by using magnetic beads. Science. 258:1122-1126. Smith, S. B., Y. Cui, and C. Bustamante, 1996. Overstretchin g B-DNA: The elastic re- sponse of individual double-stranded and single-stranded DNA molecules. Science. 271:795-799. Strick, T. R., J. F. Allemand, D. Bensimon, and V. Croquette. 1996. The elasticity of a single supercoiled DNA molecule. Science. 271:1835-1837. 17Strick, T. R., J. F. Allemand, D. Bensimon, and V. Croquette. 1998. Behavior of super- coiled DNA. Biophys. J. 74:2016-2028. Tan, R. K. Z., and S. C. Harvey. 1989. Molecular mechanics mod el of supercoiled DNA. J. Mol. Biol. 205:573-591. Vologodskii, A. V., and N. R. Cozzarelli. 1994. Conformatio nal and thermodynamic properties of supercoiled DNA. Annu. Biophys. Biomol. Struct. 23:609-643. Vologodskii, A. V., A. V. Lukashin, M. D. Frank-Kamenetskii and V. V. Anshelevich. 1974. The knot problem in statistical mechanics of polymer c hains. Sov. Phys. JETP 39:1059-1063. Vologodskii, A. V., S. D. Levene, K. V. Klenin, M. Frank-Kame netskii, and N. R. Cozzarelli. 1992. Conformational and thermodynamic properties of supe rcoiled DNA. J. Mol. Biol. 227:1224-1243. Vologodskii, A. V., and J. F. Marko. 1997. Extension of torsi onally stressed DNA by external force. Biophys. J. 73:123-132. White, J. H. 1969. Self-linking and Gauss integral in higher dimensions. Am. J. Math. 91:693-728. Wu, J. H., S. Shyy, J. C. Wang, and L. F. Liu. 1988. Transcripti on generates positively and negatively supercoiled domains in the template. Cell.53:433-440. Zhou Haijun, Zhang Yang, Ouyang Zhongcan. 1999. Bending and Base-stacking Interac- tions in Double-stranded DNA. Phys. Rev. Lett. 82:4560-4563. 18FIGURES FIG. 1. The van der Waals interaction potential versus foldi ng angle of sugar-phosphate backbones around DNA molecule axis. FIG. 2. The configuration of discrete DNA chain in our model. FIG. 3. The schematic diagram to calculate link number in our simulations. (a). For a linear supercoiled DNA chain with one end attached to a microscope s lide and with another end attached to a magnetic bead, when the orientation of the bead is fixed an d the DNA chain is forbidden to go round the bead, the number of times for two strands to interwi nd each other, the linking number of the linear DNA ( Lkl), is a topological constant. (b). The DNA double helix is str etched to a fully extended form while the orientation of bead keeps unchanged . The link number of linear DNA chain is equal to the twist number, i.e. Lkl=Twl. (c). Three long flat ribbons are connected to the two ends of the linear twisted DNA of (b). The link number of th e new double helix circle is equal to that of linear DNA chain, i.e. Lkc=Twc=Twl=Lklsince the writhe of the rectangle loop is 0. (d). The DNA circle in (c) can be deformed into a new circle, one part of which has the same steric structure as the linear supercoiled DNA chain in (a). So by adding three straight ribbons, the link number of linear double helix DNA can be obtained by c alculating the link number of the new DNA circle, i.e. Lkl=Lkc=Tw+Wr. FIG. 4. Trial motions of the DNA chain during Monte Carlo simu lations. The current confor- mation of DNA central axis is shown by solid lines and the tria l conformation by dashed lines. (a). The folding angle in ith segment θiis changed into θi+λ1. All segments between ith vertex and the free end are translated by the distance of |∆si−∆s′ i|. (b). A portion of the chain is rotated by an angle of λ2around the axis connecting the two ends of rotated chain. (c) . The segments from a randomly chosen vertex to the free end are rotated by an angle λ3around an arbitrary orientation axis which passes the chosen vertex. 19FIG. 5. Force versus relative extension curves for negative ly (a,b) and positively (c,d) super- coiling DNA molecule. Left two plots (a) and (c) are the resul ts of our Monte Carlo simulation, and the horizontal bars of points denote the statistic error of relative extension in our simulations. Right two plots (b) and (d) are the experimental data (Strick et al., 1998). The solid curves serve as guides for the eye. FIG. 6. Relative extension versus supercoiling degree of DN A polymer for three typical stretch forces. Open points denote the experimental data (Strick et al., 1998) and solid points the results of our Monte Carlo simulation. The vertical bars of the solid points signify the statistic error of the simulations, and the horizontal ones denote the bin-wid th that we partition the phase space of supercoiling degree. The solid lines connect the solid poin ts to guide the eye. 20-20-15-10-50510 -80 -60 -40 -20 0 20 40 60 80e (cos12q0-2cos6q0), q£0 U(q)=í e [(cosq0/cosq)12-2(cosq0/cosq)6], q>0 q0=62o qU(q)10-210-1110Force (pN) s .112± .008 .097± .008 .080± .008 .063± .008 .048± .008 .032± .008 .017± .008 0.0± .008MC Result (a)s .11 .088 .066 .040 .026 .013 0.0Data by Strick et al (b) 10-210-1110 0 0.25 0.5 0.75 1 Relative ExtensionForce (pN) s - .112± .008 - .096± .008 - .080± .008 - .064± .008 - .048± .008 - .032± .008 - .016± .008 0.0± .008MC Result (c) 0 0.25 0.5 0.75 1s - .040 - .026 - .013 0.0Data by Strick et al (d) Relative Extension00.20.40.60.81 -0.1 -0.075-0.05-0.025 0 0.025 0.05 0.075 0.1Force 8.0 pN 1.0 pN 0.2 pN Supercoiling degree sRelative Extension

arXiv:physics/9911075v1 [physics.space-ph] 29 Nov 1999Enhanced Phase Space Diffusion due to Chaos in Relativistic Electron-Whistler Mode Wave Particle Interactions in Planetary Magnetospheres. W J Wykes∗, S C Chapman, G Rowlands Space and Astrophysics Group, University of Warwick, UK (September 2, 2013) Abstract The chaotic interaction between electrons and whistler mod e waves has been shown to provide a mechanism for enhanced diffusion in phase s pace. Pitch angle diffusion is relevant for the scattering of electrons i nto the loss cones, thus providing a source for auroral precipitating electron s. A single whistler mode wave propagating parallel to the background magnetic fi eld has reso- nance with the electrons but the process is not stochastic. T he presence of a second, oppositely directed whistler wave has been shown pr eviously to intro- duce stochasticity into the system, thus enhancing phase sp ace diffusion. Here we generalise previous work to include relativistic effects . The full relativistic Lorentz equations are solved numerically to permit applica tion to a more ex- tensive parameter space. We consider parameters scaled to i ntrinsic planetary magnetospheres, for electron populations with ’pancake’ v elocity distributions i.e. large anisotropies in velocity space. We show that the d iffusion is rapid, occuring on timescales of the order of tens of electron gyrop eriods, and is strongly sensitive to the wave amplitude, the wave frequenc y and the perpen- dicular velocity. Using Voyager 1 data we give an estimate of the whistler wave amplitude in the Io torus at Jupiter and show that the two whistler mechanism produces pitch angle diffusion of up to ±10◦from an initial pan- cake distribution, on millisecond timescales. Keywords : Relativistic, Chaos, Whistler, Pitch Angle Diffusion, Sub storms. Typeset using REVT EX ∗Email: wykes@astro.warwick.ac.uk Fax: +44 (0)1203 692016 1INTRODUCTION The electron-whistler interaction has been considered as p otential mechanism for pitch angle scattering in planetary magnetospheres. Gyroresona nce processes with near parallel propagating whister waves have been considered (e.g. [1], [ 2]), although the process that they considered is not stochastic and requires a spectrum of frequencies to efficiently scatter electrons into the loss cone [3]. Whistler waves are able to resonate with electrons over a bro ad energy range, from less than 100 keV to several MeV [4]. In particular the Hamiltonia n has been obtained for relativistic electrons interacting with a whistler mode wa ve of single ˆk, revealing underlying behaviour that is dynamically simple [5]. Stochasticity has been introduced by coupling the bounce mo tion of the trapped electrons with a single whistler [6], whilst the presence of a second, o ppositely directed whistler wave was shown from the non-relativistic equations of motion to i ntroduce stochasticity into the system and was demonstrated numerically for a wave frequ ency of half the electron gyrofrequency [7]. This mechanism has been shown to exist in self-consistent simulations [8]. In this paper we generalise the work in [7] to consider a range of wave frequencies below the gyrofrequency and include relativistic effects. We cons ider the efficiency of the mecha- nism in scattering electrons with a high anisotropy in veloc ity spaceV⊥>V/bardbli.e. a ’pancake’ distribution. Recent plasma density models have shown that anisotropic distributions are required to fit the observed whistler dispersions in the Jovi an magnetosphere [9]. We investi- gate the dependence of the degree of stochasticity of the sys tem (using Lyapunov exponents) on the wave amplitude, wave frequency and perpendicular vel ocity. EQUATIONS OF MOTION We consider a total magnetic field of the form B=B0+B+ ω+B− ω where B0=B0ˆ xis the background magnetic field and B+ ωandB− ωare the whistler waves propagating parallel and anti-parallel to the background fi eld respectively (for coordinate system see Figure 1). We assume that the background field line s are uniform, since, as we will see, the interaction is sufficiently fast so that changes in the background field experienced by the electrons are small, e.g., for electrons close to Jupi ter’s magnetic equator at 6 RJ, the field changes by less than 1% for an MeV electron travelling at 0.9cand interacting with the field for 1000 electron gyroperiods (0.1s). The wavefields B+ ωandB− ωare given by B+ ω=Bω[cos(kx−ωt)ˆ y−sin(kx−ωt)ˆ z] B− ω=Bω[cos(−kx−ωt+θ)ˆ y−sin(−kx−ωt+θ)ˆ z] withˆ xparallel to the background field and ˆ yandˆ zperpendicular. The wave frequency, ω, and wave number, k, are given by the whistler mode dispersion relation: 2k2c2 ω2= 1−ω2 pe ω(ω−Ωe)(1) whereωpeis the plasma oscillation frequency and Ω eis the electron gyrofrequency. Electrons travelling at the correct parallel velocity will experienc e a constant field and will interact strongly with it. This resonance velocity, vr=vrˆ xis given by the resonance condition: ω−k·vr=nΩe/γ (2) wherenis an integer, and γ= (1−v2/c2)−1/2is the relativisic factor . The corresponding electric field is obtained from Maxwell’s relation for plane propagating waves, kEω=ωˆk∧Bω and the dispersion relation (1). We write v=v/bardblˆ x+v⊥cosφˆ y+v⊥sinφˆ z, whereφ=φ(t) is the phase of the perpen- dicular velocity and define the phase angles ψ1=kx−ωt+φandψ2=−kx−ωt+φ+θ as the angles between the perpendicular velocity and B+ ωandB− ωrespectivley. We substitute these into the Lorentz force law to give the ful l equations of motion: dv/bardbl dt=bv⊥ γ/parenleftbigg 1−ωv/bardbl kc2/parenrightbigg sinψ1+bv⊥ γ/parenleftbigg 1 +ωv/bardbl kc2/parenrightbigg sinψ2 (3) dv⊥ dt=−b γ/parenleftBigg v/bardbl−ω k/parenleftBigg 1 +v2 ⊥ c2/parenrightBigg/parenrightBigg sinψ1 −b γ/parenleftBigg v/bardbl+ω k/parenleftBigg 1 +v2 ⊥ c2/parenrightBigg/parenrightBigg sinψ2 (4) dψ1 dt=kv/bardbl−ω+1 γ−b γv⊥/parenleftbigg v/bardbl−ω k/parenrightbigg cosψ1 −b γv⊥/parenleftbigg v/bardbl+ω k/parenrightbigg cosψ2 (5) dψ2 dt=dψ1 dt−2kv/bardbl (6) dγ dt=bωv⊥ kc2(sinψ1−sinψ2) (7) whereb=Bω/B0is wave amplitude scaled to the background field, and time and velocity have been rescaled with respect to the gyrofrequency, Ω e, and the phase velocity, vphase= w/k, respectively. Reduced Equations The full relativistic equations can be reduced in the limit o f small wave amplitudes. We introduce two variables Θ = ( ψ1−ψ2)/2 ands= (ψ1+ψ2)/2 which in the limit of small b are proportional to the distance along the background field ,x, and time, t. For small perturbations in v/bardblwe havev/bardbl/v⊥≈b<< 1 and then to first order in b, we have: d2Θ ds2=1 k2v2r/parenleftBiggd2ψ1 dt2/parenrightBigg (8) 3=W2(1−a) sinψ1+W2(1 +a) sinψ2 (9) ⇒d2Θ ds2= 2W2(cos Θ sins−asin Θ coss) (10) where W2=b γω(v⊥/v2 r) a=1 γωc2+vr v2 ⊥ Thus we have a double pendulum equation with variables ˙Θ =v/bardbl/vrand Θ =kx. Perturbations in Θ = kxare thus proportional to the wave amplitude, b, to 1 /γand to the ratio of perpendicular velocity to the square of the reso nance velocity. For relativistic velocities and for large anisotropy ( v⊥>>v /bardbl), the constant a<< 1. NUMERICAL RESULTS Figure 2 shows numerical solutions of the full equations of m otion. The plots are stro- boscopic surfaces of section [10] constructed from cut-pla nes wherex= (n+ 1/2)π/k, to sample the full electron phase space. The initial parallel v elocity,v/bardbl, was varied over the range [ −vr,vr], wherevris the resonance velocity, given by the resonance condition (2). All electrons were given the same initial perpendicular vel ocity,v⊥, withv⊥= 0.7c (v⊥/vr≈20), and phase angle, ψ(defined as the angle between the perpendicular velocity and the first whistler wave B+ ω, see Figure 1) to give a pancake velocity distribution with high initial pitch angles. For low wave amplitudes, Figure 2 a) the trajectories are ess entially regular and char- acterised by two sets of resonances. As the wave amplitude is increased in Figures 2 b) and 2 c) stochastic effects are introduced into the region bet ween the two resonances. For higher wave amplitudes, Figure 2 d), the system becomes glob ally stochastic with regular trajectories confined to KAM surfaces close to the resonance s. There is significant diffusion of electrons throughout phase space, i.e. electrons with lo w parallel velocities can diffuse through phase space to regions of higher parallel velocity a nd undergo a significant reduction in pitch angle. In Figures 2 b)–d) the stochastic regions are bounded with up per and lower parallel velocity limits. These corresponds to the first untrapped el ectron orbit, which is regular, and bounds the stochastic region occupied by the orbits of tr apped particles. Stochastic electrons that diffuse through phase space to this maximum (m inimum) parallel velocity will have lowest (highest) pitch angles. In Figure 3 we show a sequence of phase plots for increasing pe rpendicular velocity. The reduced equation (Equation 10) describes pendulum like beh aviour with oscillations in v/bardbl proportional to both the wave amplitude and the ratio v⊥/v2 r. The resonance condition (Equation 2) shows that vr=vr(γ(v⊥,v/bardbl),k,ω), therefore (for v/bardbl<<v ⊥)vrin addition to the total electron energy, E, is a functions of the perpendicular velocity. By varying v⊥only we can consider the dependence of the degree of stochasticit y onv⊥,Eandv⊥/v2 r. 4In Figures 3 a) and b) the perpendicular velocity increases t o relativistic velocities v⊥= 0.3–0.6c (Energy 0.02–0.1 MeV). From the resonance conditi on (2) we see that increasing v⊥increasesγand hence reduces vrand the separation between the two resonances. Hence increasingv⊥increases the ratio v⊥/v2 rand we see an increase in the stochasticity in the system as expected from the reduced equations. In Figure 3 c) where v⊥= 0.88c(E=0.4 MeV) the resonance condition (2) is satisfied for vr= 0, consistent with maximal stochasticity from the reduced equation (10) as v⊥/v2 r→ ∞. Increasingv⊥further causes the resonances to pass through the v/bardbl= 0 line and change sign. In Figure 3 d) and e) we have v⊥= 0.97–0.98c(E=1–1.25 MeV). The resonance velocity now increases with v⊥, therefore the ratio v⊥/v2 rdecreases with v⊥. The degree of stochasticity decreases, until the system is no longer stochastic again (F igure 3 f)) with v⊥= 0.99c (E=1.75 MeV). The dependence, and in particular, the presence of a peak in t he degree of stochasticity on the ratio v⊥/v2 ris a relativistic effect. For non relativistic velocities γis constant and the degree of stochasticity continually increases with v⊥[7]. Lyapunov Exponents Lyapunov exponents are used to quantify the degree of stocha sticity in the system. The Lyapunov exponents are calculated using the method describ ed in [11]. All six Lyapunov exponents were calculated over phase space and evolved to th eir asymptotic limit. The only significant Lyapunov exponent corresponds to spacial pertu rbations along the background field. For positive Lyapunov exponents, two trajectories that are initially close together will diverge exponentially in time. For negative or zero Lyapuno v exponents, two trajectories that are initially close together will remain close togethe r. Positive (negative) Lyapunov exponents correspond to stochastic (regular) trajectorie s in phase space. In the top panel of Figure 4 the Lyapunov exponents are shown f or the same initial conditions as Figure 2 d): electrons have phase angle, ψ= 0, and parallel velocity in the range [ −vr,vr]. Trajectories in the chaotic region of Figure 2 d) have posi tive Lyapunov exponent while the Lyapunov exponent of the regular traject ories close to the resonances is zero. In the middle panel of Figure 4 we plot the averaged Lyapunov e xponent for increasing perpendicular velocities. The Lyapunov exponents have a de pendence on the ratio v⊥/v2 r; the Lyapunov exponents increase with v⊥(v⊥/v2 rincreasing) until v⊥satisfies the resonance equation for vr= 0, (v⊥= 0.88c), andv⊥/v2 r→ ∞. Asv⊥increases further the Lyapunov exponents decrease ( v⊥/v2 rdecreasing). In the bottom panel of Figure 4 we plot the averaged Lyapunov e xponent as a function of the corrected wave frequency, ωc=ω/γΩe. The Lyapunov exponent, and hence the degree of stochasticity varies strongly with frequency and appear s to be enhanced when ωc=m/n, wherem,n= 2,3,4,...It then follows that close to these frequencies the process m ay be the most efficient in pitch angle scattering. 5ESTIMATION OF WHISTLER WAVE AMPLITUDES. We use the analysis of Voyager 1 data in [12] and [13] to estima te the whistler wave amplitude. The plasma wave instrument on Voyager 1 measures the electric field spectral density of the whistler waves over a set of frequency channel s of finite width ∆ ωm. To estimate the wave amplitude we consider two possibilities. The minimum amplitude estimate is obtained if we assume the wave amplitude is constant over t he bandwidth, ∆ ωm, of the measurement giving an estimate of order b=Bω/B0= 0.0005. This is too low to give significant stochastic diffusion, and the changes in pitch an gle are small, of the order of less than 1◦. The optimum amplitude estimate for this process is obtained if we assume the majority of the wave power comes from a finite waveband smaller than tha t of the instrument. We consider the case where wavepower is enhanced at frequencie s coincident with maxima in the Lyapunov exponents: in Figure 4 we see that the Lyapunov e xponent is increased whenωc=ω/γΩe=m/n. The enhancement in the Lyapunov exponent occurs over a narrow frequency range of order Ω e/100 (see Figure 4). If we assume the that the measured wavepower occurs in this bandwidth we obtain an electric wav e amplitude of the order of 0.02mVm−1corresponding to a magnetic wave amplitude of 1 .5nT, givingb= 0.005, which is well into strong stochastic diffusion regime (see Figure 2 ). We obtain a similar estimate using data from Ulysses at the Jo vian magnetopause [14], [15]. To uniquely determine whether or not this process is si gnificant, wave amplitude, rather than spectral density measurements are needed. In th is context it interesting to note that for the Earth, direct amplitude measurements [16] and t he extrema of spectral density measurements [17] yield whistler amplitudes sufficient for s tochasticity by this mechanism; whereas average spectral density measurements do not [18]. This is suggestive that the process will be active under certain conditions only. Pitch Angle Diffusion in the Io torus at Jupiter Using the estimated wave amplitude we can estimate the rate o f diffusion from an initial pancake distribution of electrons. For the Io torus at Jupit er we have a gyrofrequency Ωe= 53.2kHz, corresponding to a background field of 302 nT, and a plasma frequency ωpe= 355kHz(see [12] and [13]). In Figures 5 and 6 we show phase plots similar to Figure 2 excep t we now plot pitch angle against phase. The initial conditions are b= 0.005,ωc= 1/3,v⊥= 0.65 and E=150 keV, The phase plots are qualitatively similar to Figure 2 and share m any of the same features. Regular trajectories are confined to close to the resonance pitch ang leαr= arctan(v⊥0/vr) where v⊥0is the initial perpendicular velocity (0.65c) and vris the resonance velocity. Stochastic particles can diffuse throughout the stochastic region of ph ase space and electrons with the maximum parallel velocity in Figure 2 will have the minimum p itch angle. In Figure 6 diffusion in pitch angle is very fast. Pitch angle d iffusion of up to ±10◦occurs on timescales of the order of tens of gyroperiods. On this tim escale electrons at Jupiters magnetic equator (L=6) experience changes in the magnetic fi eld of less than 1%, therefore the approximation in the numerical solutions that the backg round magnetic field lines are uniform is valid. 6DISCUSSION We have shown that the electron-whistler interaction intro duces stochasticity and can allow electrons to diffuse in phase space on very fast timesca les. The degree of stochasticity depends on three parameters; the wave amplitude, b, the wave frequency, ω, and the ratio of the perpendicular velocity over the resonance velocity s quared,v⊥/v2 r, which in turn is a function of γ. The degree of stochasticity of the system increases with bot h the wave amplitude, b, and the ratio v⊥/v2 r(and hence γ). However the resonance velocity is dependent on γ and decreases as γincreases. There exists a critical relativistic factor γcsuch the resonance condition is satisfied with vr= 0. Therefore when γ <γ cthe degree of stochasticity increases withγand whenγ >γ cthe degree of stochasticity decreases with γ. We have shown the Lyapunov exponent appears to be enhanced wh en the wave amplitude ωc=ω/γΩe=m/nwherem,n= 2,3,4,...This is a completely new phenomena and arises purely from the interaction of the two whistler waves. Deriv ing this analytically will be an area of future research. The two whistler interaction may form part of the pitch angle scattering process along- side other mechanisms, in particular it will enhance the slo wer processes, such as bounce- resonance, that require an initial v/bardblto operate as it specifically scatters electrons with high perpendicular velocities and low or zero parallel velociti es. Because of the ambiguity in the data for the Jovian magnetosphere it is probable that the mec hanism is ’switched on’ during periods of intense whistler wave activity. We have seen that for a single wave frequency the stochastic r egion is bounded by the regular trajectories of untrapped electrons. For the simpl e two wave process considered here, we can define a maximum and minimum parallel velocity gi ven byv0±∆v, where v0is the mean parallel velocity and ∆ vis the change in parallel velocity (the width of the stochastic region). In terms of the pitch angle this corresp onds to a mean pitch angle α0and a change in pitch angle ∆ α. Henceα0and ∆αare uniquely determined by the parameters ω,k,v ⊥,B0,Bω,Ωeandωpe. It would be more realistic to consider the more complex situa tion of a wave packet con- sisting of many more than two whistlers, with a range of frequ encies. This more complex case is difficult to parameterise so we have as as initial study considered the simple two wave case. However it is straightforward to qualitatively p redict the effect of adding more wave modes to form a wave packet, if we consider adding a secon d pair of waves at a lower frequency. This would add a second pair of resonances with re sonance velocities of a higher magnitude, which would have the effect of destroying the regu lar trajectories bounding the stochastic region for the original waves, so that a new, larg er, stochastic region encompasses the new pair of resonances. The resulting stochastic diffusi on in pitch angle would increase and electrons would be scattered to lower pitch angles. A mor e detailed investigation of this effect is required to calculate the diffusion coefficient for th e wave packet, however we can anticipate that the timescale for diffusion will still scale with the electron gyroperiod. Acknowledgements W J Wykes and S C Chapman are funded by PPARC. 7REFERENCES [1] Kennel C F, and H E Petschek. Limit on stably trapped parti cle fluxes. J. Geophys. 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Physica D, 95, 35-49, 1996 [9] Crary F J, F Bagenal, J A Ansher, D A Gurnett, W S Kurth. Anis otropy and proton density in the Io plasma torus derived from whistler wave dis persions. J. Geophys. Res., 101, 2699-2706, 1996 [10] Tabor M. Chaos and intergrability in nonlinear dynamic s - an introduction. New York, Chichester: Wiley, 1989. [11] Benettin G, L Galgani, J M Strelcyn. Kolmogorov entropy and numerical experiments. Physica A, 14, 6, 1976. [12] Scarf F L, F V Coroniti, D A Gurnett, W S Kurth. Pitch-angl e diffusion by whistler mode waves near the Io plasma torus. J. Geophys. Res., 6, 8, 1979. [13] Kurth W S, B D Strayer, D A Gurnett, F L Scarf. A summary of w histlers Observed by Voyager 1 at Jupiter. Icarus, 61, 497-507, 1985. [14] Tsurutani B, et al. Plasma wave characteristics of the Jovian magnetopause bo undary layer: Relationship to the Jovian Aurora? J. Geophys. Res., 102, A3, 4751-4764, 1997. [15] Hobara Y, S Kanemaru, M Hayakawa. On estimating the ampl itude of Jovian whistlers observed by Voyager 1 and implications concerning lightnin g.J. Geophys. Res., 102, A4, 7115-7125, 1997. [16] Nagano I, S Yagitani, H Kojima, H Matsumoto. Analysis of Wave Normal and Poynting Vectors of the Chorus Emissions Observed by GEOTAIL. J. Geomag. Geoelectr., 48, 299-307, 1996 [17] Parrot M, C A Gaye. A statistical survey of ELF waves in ge ostationary orbit. GRL 23, 2463, 1994. [18] Tsurutani B, et al. A statistical study of ELF-VLF plasma waves at the magnetop ause. J. Geophys. Res., 94, 1270, 1989. 8FIGURES ✲✻ ✁✁✁✁✁✁✁✁✁ ✕

arXiv:physics/9911076v1 [physics.chem-ph] 29 Nov 1999Spin-Lattice Relaxation in Metal-Organic Platinum(II) Complexes H. H. H. Homeier1,2, J. Strasser, and H. Yersin3 Institut f¨ ur Physikalische und Theoretische Chemie, D-93040 Regensburg, Germany Abstract The dynamics of spin-lattice relaxation (slr) of metal-org anic Pt(II) compounds is studied. Often, such systems are characterized by pronou nced zero-field splittings (zfs) of the lowest-lying triplets. Previous expressions f or the Orbach slr process do not allow to treat such splitting patterns properly. We disc uss the behavior of a modified Orbach expression for a model system and present res ults of a fit of the temperature dependence of the spin-lattice relaxation rat e of Pt(2-thpy) 2based on the modified expression. Key words: Metal-organic Platinum(II) complexes, Shpol’skii matric es, Spin-lattice relaxation, Orbach process, Raman process, D irect process, Triplets, Zero-field splittings Transition metal complexes with organic chelate ligands an d their lowest ex- cited states are of potential use for solar energy conversio n [1–7]. Recently, the processes of spin-lattice relaxation and the decay behavio r of excited states have been studied experimentally for such systems in Shpol’ skii matrices. [8– 14] Of special importance are compounds with a Pt(II) centra l ion. Pt(II) systems exhibit many different types of low-lying excited tr iplets that in- clude metal-centered (MC) dd∗states [15,16], metal-to-ligand-charge-transfer (MLCT) states [17–19], intra-ligand-charge-transfer (IL CT) states [10,13,20], ligand-ligand′-charge-transfer (LL′CT) states [6,7], and ligand-centered (LC) 1Author for correspondence. Address: PD Dr. H. H. H. Homeier, Institut f¨ ur Phy- sikalische und Theoretische Chemie, Universit¨ at Regensb urg, D-93040 Regensburg, Germany. FAX: +49-941-943 4719. Email: Herbert.Homeier@n a-net.ornl.gov 2WWW: http://www.chemie.uni-regensburg.de/ ∼hoh05008 3Author for correspondence. Address: Prof. Dr. H. Yersin, In stitut f¨ ur Physikali- sche und Theoretische Chemie, Universit¨ at Regensburg, D- 93040 Regensburg, Ger- many. FAX: +49-941-943 4488. Email: Hartmut.Yersin@chemi e.uni-regensburg.de Preprint submitted to Chemical Physics Letters 23 July 2013Table 1 Electronic origins E [ cm−1] (lowest triplet sublevel of T 1, lowest site), zero-field splittings[ cm−1] (∆Eba: Energy difference between |b/angbracketrightand|a/angbracketright, ∆Ecb: Energy dif- ference between |c/angbracketrightand|b/angbracketright), spin-lattice relaxation times τslr[ns] at 1.2 K, and transition types for various Pt(II) complexes with organic ligands Complex E ∆ Eba∆Ecb τslrType Ref. Pt(2-thpy) 2a)17156 7 9 710 LC/MLCT [9,21,14] Pt(2-thpy)(CO)Cla)18012 0.055 3.8 3000 LC/MLCT [14,24] Pt(phpy) 2a)19571 6.9 25.1 390 LC/MLCT [14] Pt(3-thpy) 2a)18020 13 9 ≈25 LC/MLCT [25,26] [Pt(bpy) 2](ClO 4)2b)21237 <1 <1>50·103LC/MC [23] Pt(qol) 2a)15426 <1 <1>60·103ILCT [10,13] Pt(qtl) 2a)13158 <1 <1 >7000 ILCT [13] Pt(phpy)(CO)Cla)20916 <1 6.4 LC/MLCT [27] Pt(bhq) 2c)19814 11 28 LC/MLCT [28] Pt(phpz) 2a)22952 9 7 LC/MLCT [25] 2-thpy−: 2-(2-thienyl)pyridinate; phpy−: 2,2′-phenylpyridinate; 3-thpy−: 2- (3-thienyl)pyridinate; bpy: 2,2′-bipyridine; qol−: 8-quinolinolate; qtl−: 8- quinolienthiolate; bhq−: benzo[h]quinolinolate; phpz−: 2,2′-phenylpyrazinate. a)In n-octaneb)Neat materialc)In n-decane states with some MLCT and/or MC contribution [21–23]. In the following, we focus to Pt(II) systems with heterocyclic chelate ligands. As shown in Tab. 1, the low-lying triplets of these systems ar e characterized by a rather large variation of zero-field splittings (zfs) in the range from less than 0.1 cm−1to about 40 cm−1. The larger splittings are mainly due to spin-orbit coupling. For the same complex in different matri ces, the lowest triplet states are shifted in energy (in many cases in the ran ge of 200 – 400 cm−1). The corresponding optical spectra show rich vibrational structure that may be well resolved (about 2cm−1) by choosing appropriate matrices and by employing methods of emission and/or excitation line narro wing. At low temperatures (several Kelvin), the processes of spin -lattice relaxation occurring between the triplet sublevels |a/angbracketright,|b/angbracketright, and |c/angbracketrightare relatively slow with relaxation times as long as hundreds of nano-seconds and eve n up to many micro-seconds (See Tab. 1 and Refs. [9–14]) due to the low den sity of phonon states corresponding to such zfs patterns. To discuss these processes, we assume that the perturbation Vcaused by the phonons couples the electronic states of the chromophore es sentially linearly 2abc a b c Fig. 1. Processes of spin-lattice relaxation: a) Direct pro cess. b) Orbach process. c) Raman process. (e.g. see Ref. [29, p. 228]) V=V1/summationdisplay kǫk+. . . (1) where ǫkis the strain corresponding to the phonon mode with wave vect ork in the long wavelength limit. The matrix elements of V1are denoted by Vba= |/angbracketleftb|V1|a/angbracketright|and analogous expressions for VcaandVcb. The energy differences are ∆Ebabetween |b/angbracketrightand|a/angbracketright, ∆Ecbbetween |c/angbracketrightand|b/angbracketright, and ∆ Ecabetween |c/angbracketrightand |a/angbracketright. The usual notation β= 1/(kBT) for given temperature Tand Boltzmann constant kB, and the abbreviations Cba=C V2 ba(∆Eba)3and analogous ones forCcaandCcbare also used. Here, the parameter C= 3/(2π¯h4ρv5) is defined in terms of mass density ρand (average) velocity vof sound of the matrix. The (∆ Eba)3dependence of Cbashould be kept in mind. The following relaxation processes (see Fig. 1) occur: Direct process: The rate is given by [30, p. 541], [29, p. 229] k(direct ) a,b =kab+kba =Cbacoth(β∆Eba/2). (2) Here, kabandkbaare the rate constants for the up and down processes, re- spectively, given by the expressions kab=Cba1 exp(β∆Eba)−1, kba=Cbaexp(β∆Eba) exp(β∆Eba)−1. (3) 3Analogous expressions hold for the up and down rates kbc,kcb,kac, and kca. Orbach process: The rate for this process vanishes for T→0 K exponen- tially. It depends on the splitting pattern of the three invo lved states: If the energy separation ∆ Ebaof the two lower states |a/angbracketrightand|b/angbracketrightis much smaller than both the energy separations ∆ Ecaand ∆ Ecbto the upper state |c/angbracketright, then the well-known expression k(Orbach ) a,b =2CcbCca (Cca+Ccb)1 (eβ∆E−1)(4) holds approximately for low T. This original Orbach expression is derived under the assumption that the energy differences are given by ∆E= ∆Eca= ∆Ecb>0. For a more general zfs pattern, the rate is given by the low- tempe- rature approximation [31] k(Orbach ) a,b =kackcb+kbckca−kbckba kca+kcb−kba(5) with up and down rates as given in Eq. (3). The modified express ion (5) contains Eq. (4) as a limiting case (see Ref. [31]). Raman process: For low temperature, the rate may be approximated by k(Raman ) a,b =D Tn(6) with a constant Dandn= 5 for non-Kramers ions [32]. In the cases under study, this T5dependence fits the experimental observations [31] better t han theT7dependence observed in other systems. The relative importance of the various slr processes is larg ely dependent on the size of the zfs and the energy separations to further elec tronic states. For instance, in systems like Pt(qol) 2and Pt(qtl) 2with a very small total zfs (see Tab. 1) and no further electronic states in the vicinity of T 1, direct and Orbach processes are expected to be very small due to the ∆ E3dependence of these processes, and the Raman process is expected to dominate. Co mpare also Ref. [31]. The behavior of the above expressions is illustrated for a model system (with- out a Raman process) and with parameters ∆ Eba= ∆Ecb= 7cm−1,v= 2000 m/s,ρ= 1.1 g/cm3,Vbc= 10cm−1,Vac= 20cm−1,Vab= 3cm−1. In Fig. 2, the relative errors of the approximations for both direct and Or bach process, i.e., 4Error (%) i)iii)ii) v) iv) ±20020406080100 2345678 Temperature (K)| a >| b > | c > i) ii) iii) iv) Fig. 2. Relative errors of the relaxation rate expressions w ith respect to Eq. (7) as a function of temperature T. Plotted are the errors of k(direct ) a,b+k(Orbach ) a,bwith Eq. (2) for the direct process in combination with the original O rbach expression (4) for different values of ∆ E( i) ∆ E= ∆Ecb= 7 cm−1, ii) ∆ E= ∆Eca= 14 cm−1, iii) ∆E= (∆Eca+ ∆Ecb)/2 = 10 .5 cm−1, iv) ∆ E= ∆Efit= 5.4 cm−1) and v) with the modified expression (5). for the sum k(direct ) a,b +k(Orbach ) a,b as obtained using Eq. (2) in combination either with Eq. (4) or Eq. (5), respectively. The errors are calcula ted with respect to the exact rate k(Orbach +direct ) a,b =1 2/parenleftbigg kbc+kac+kcb+kca+kba+kab/parenrightbigg −1 2/parenleftbigg (kbc+kcb−kab−kca−kac+kba)2 + 4(kcbkca−kabkca−kbakcb+kbakab)/parenrightbigg1/2 (7) for the three-level system that is obtained from the rate equ ations [31]. Applying the original Orbach expression, i.e., using Eq. (4 ) in combination with (2) for the direct process, the prefactor 2 CcbCca/(Cca+Ccb) was com- puted from the model parameters, but different values of the p arameter ∆ E have been used: ∆ E= ∆Ecbcorresponds to using the minimum distance of 5state |c/angbracketrightto the states |a/angbracketrightand|b/angbracketright(curve i) in Fig. 2), ∆ E= ∆Ecacorresponds to using the maximum distance (curve ii)), and ∆ E= (∆Eca+ ∆Ecb)/2 cor- responds to using the mean distance (curve iii)). The value ∆ E= ∆Efit= 5.4cm−1is obtained by a least square fit of the exact data with one fit pa rame- ter ∆E(curve iv)), i.e., for the direct process and the prefactor o f Eq. (4), the exact expressions were used during the fit. Interestingly, ∆ Efitis less than any of the other differences of the energies. Alternatively, one could try to use the prefactor in Eq. (4) as an additional fit parameter. But then, one cannot hope to extract the model values of CcbandCcafrom such a fit. Finally, curve v) in Fig. 2 was obtained using the modified expression (5) in combi nation with Eq. (2) for the direct process. Clearly, the modified approach yi elds much reduced errors over a large temperature range. Thus, Orbach’s origi nal expression (4) that was designed for a different pattern of the energy levels cannot be applied to a pattern with ∆ Ecb≈∆Ebafor any reasonable choice of the parameter ∆E. We remark that similar results are also obtained for differen t choices of the parameters. For instance, for a value of vsmaller by a factor f, the same results for the absolute rates would be obtained, if all the m atrix elements ofV1are also chosen smaller by a factor f5/2, e.g., for v= 1500 m/s and Vbc= 4.87 cm−1,Vac= 9.74 cm−1,Vab= 1.46 cm−1. Moreover, fixing all the other parameters, any rescaling of the three matrix element s by an arbitrary common positive factor yields the same error curves since we are dealing with relative errors and, under this scaling, all up and down rates kab,kbaetc., and, hence, all slr rates in the model are multiplied by a common fa ctor. It is of interest to present an example of the application of t he above formalism to the spin-lattice relaxation observed for the lowest trip let of the Pt(2-thpy) 2 complex in an n-octane matrix. This compound is depicted in o f Fig. 3, and some properties are collected in Table 1. The experimental s pin-lattice relax- ation rate k(slr)is obtained from the measured emission decay rate of state |b/angbracketrightby subtraction of the corresponding triplet deactivation r ate to the ground state [31]. For the fit, we used Eq. (2) for the direct process, the modified expression (5) for the Orbach process, and Eq. (6) with n= 5 for the Raman process, i.e., for a T5low temperature dependence. As prefactor of the direct process, we used the low temperature limit of k(slr). The ratio of Cca/Ccbcan be obtained independently from time-resolved excitation s pectra [9,31]. Also, all energy separations ∆ Ebaand ∆ Ecbare available from highly resolved spec- tra [9,21,14,31]. Thus, as fit parameter, only the prefactor Dof the Raman process and the constant Ccaremain. For such a two-parameter fit as displayed in Fig. 3, the result is highly satisfactory. A three-parameter fit based on the original Orbach expressio n (4) using the pa- rameters D, ∆Eand the prefactor in Eq. (4) yields the value ∆ E= 11.4cm−1 (and a nearly doubled prefactor Dfor the Raman process in comparison to 61 2 3 4 5 6 7 8Raman(T□)5directOrbachPt(2-thpy)2direct□+ Orbach□+□Raman 012345678 Temperature□(K)Rate□of□slr□(10□□s )6 -1N SN SN SN SPt Fig. 3. Fit of the spin-lattice relaxation rate k(slr)as a function of temperature for Pt(2-thpy) 2in an n-octane matrix. Displayed are the contributions of th e direct process (Eq. (2)), the Orbach process (using the modified exp ression (5)), and the Raman T5process (Eq. (6)). the fit displayed in Fig. 3). A similar value for ∆ Ewas obtained in Ref. [9] by a somewhat different fitting procedure. Both these values are unphysical since they do not correspond to any of the observed energy differenc es (see Tab. 1). We remark that the present study was triggered by this difficul ty of using the original Orbach expression (4). This result shows, as further ones presented in [33,31], tha t the use of the modified expression (5) for the Orbach process is necessary f or a detailed un- derstanding of the dynamics of the spin-lattice relaxation for low-lying triplets of metal-organic transition metal compounds with their cha racteristic patterns of zero-field splitting. Thus, although the present study co ncentrated on Pt(II) compounds, the result should be applicable to a more general class of com- pounds, namely, the whole platinum metal group complexes (c ompare, e.g., the recent results [8,9] for [Ru(bpy) 3]2+). Financial support by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is gratefully acknowledged. 7References [1] J. S. Conolly (Ed.), Photochemical Conversion and Stora ge of Solar Energy, Academic Press, New York, 1981. [2] A. Harriman, M. A. West (Eds.), Photogeneration of Hydro gen, Academic Press, London, 1982. [3] G. Calzaferri (Ed.), Proceedings of the 10thInternational Conference on Photochemical Transformation and Storage of Solar Energy, volume 38 of Solar Energy Materials and Solar Cells, Interlaken, 1994. [4] B. O’Regan, M. Gr¨ atzel, Nature 353 (1991) 737. [5] A. Juris, V. Balzani, F. Barigelletti, S. Campagna, P. Be lser, A. von Zelewsky, Coord. Chem. Rev. 84 (1988) 85. [6] A. Vogler, H. Kunkely, J. Am. Chem. Soc. 103 (1981) 1559. [7] S. D. Cummings, R. Eisenberg, J. Am. Chem. Soc. 118 (1996) 1949. [8] H. Yersin, W. Humbs, J. Strasser, in: H. Yersin (Ed.), Ele ctronic and Vibronic Spectra of Transition Metal Complexes, Vol. II, volume 191 o f Topics in Current Chemistry, Springer-Verlag, Berlin, 1997, p. 153. [9] J. Schmidt, J. Strasser, H. Yersin, Inorg. Chem. 36 (1997 ) 3957. [10] D. Donges, J. K. Nagle, H. Yersin, Inorg. Chem. 36 (1997) 3040. [11] H. Yersin, J. Strasser, J. Luminescence 72-74 (1997) 46 2. [12] H. Yersin, D. Braun, Coord. Chem. Rev. 111 (1991) 39. [13] D. Donges, J. K. 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arXiv:physics/9911077v1 [physics.data-an] 30 Nov 1999Mixtures of Gaussian process priors∗ J¨ org C. Lemm Institut f¨ ur Theoretische Physik I, Universit¨ at M¨ unste r D–48149 M¨ unster, Germany E-mail: lemm@uni-muenster.de http://pauli.uni-muenster.de/∼lemm Publication No.: MS-TP1-99-5 Abstract Nonparametric Bayesian approaches based on Gaussian proce sses have recently become popular in the empirical learning community. They encompas s many classical methods of statistics, like Radial Basis Functions or various splin es, and are technically convenient because Gaussian integrals can be calculated analytically . Restricting to Gaussian processes, however, forbids for example the implemention of genuine no nconcave priors. Mixtures of Gaussian process priors, on the other hand, allow the flexibl e implementation of complex and situation specific, also nonconcave a priori information. This is essential for tasks with, compared to their complexity, a small number of availa ble training data. The paper concentrates on the formalism for Gaussian regression prob lems where prior mixture models provide a generalisation of classical quadratic, typicall y smoothness related, regularisation approaches being more flexible without having a much larger c omputational complexity. Contents 1 Introduction 1 2 The Bayesian model 2 3 Gaussian regression 3 4 Prior mixtures 4 4.1 General formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.2 Maximum a posteriori approximation . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.4 High and low temperature limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.5 Equal covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 A numerical example 9 6 Conclusions 9 1 Introduction The generalisation behaviour of statistical learning algo rithms relies essentially on the correctness of the implemented a priori information. While Gaussian processes and the related regu larisation approaches have, on one hand, the very important advantage o f being able to formulate a priori ∗This is an extended version of a contribution to the Ninth Int ernational Conference on Artificial Neural Networks (ICANN 99), 7–10 September 1999, Edinburgh, UK. 1information explicitly in terms of the function of interest (mainly in the form of smoothness priors which have a long tradition in density estimation and regression problems [18, 17, 5]) they implement, on the other hand, only simple concave prior dens ities corresponding to quadratic errors. Especially complex tasks would require typically m ore general prior densities. Choosing mixtures of Gaussian process priors combines the advantage of an explicit formulation of priors with the possibility of constructing general non-concave p rior densities. While mixtures of Gaussian processes are technically a rela tively straightforward extension of Gaussian processes, which turns out to be a computational advantage, practically they are much more flexible and are able to produce in principle, i.e., in the limit of infinite number of components, any arbitrary prior density. As example, consider an image completion task, where an imag e have to be completed, given a subset of pixels (‘training data’). Simply requirin g smoothness of grey level values would obviously not be sufficient if we expect, say, the image o f a face. In that case the prior density should reflect that a face has specific constituents ( e.g., eyes, mouth, nose) and relations (e.g., typical distances between eyes) which may appear in v arious variations (scaled, translated, deformed, varying lightening conditions). While ways how prior mixtures can be used in such situations h ave already been outlined in [6, 7, 8, 9, 10] this paper concentrates on the general formal ism and technical aspects of mixture models and aims in showing their computational feasibility . Sections 2–4 provide the necessary formulae while Section 5 exemplifies the approach for an imag e completion task. Finally, we remark that mixtures of Gaussian process priors do usually notresult in a (finite) mixture of Gaussians [3] for the function of interest. Indee d, in density estimation, for example, arbitrary densities not restricted to a (finite) mixture of G aussians can be produced by a mixture of Gaussian prior processes. 2 The Bayesian model Let us consider the following random variables: 1.x, representing (a vector of) independent, visible variables (‘measurement situations’), 2.y, being (a vector of) dependent, visible variables (‘measurement results’), and 3.h, being the hidden variables (‘possible states of Nature’). A Bayesian approach is based on two model inputs [1, 11, 4, 12] : 1. Alikelihood model p(y|x, h), describing the density of observing ygiven xandh. Regarded as function of h, for fixed yandx, the density p(y|x, h) is also known as the ( x–conditional) likelihood ofh. 2. Aprior model p(h|D0), specifying the a priori density of hgiven some a priori information denoted by D0(but before training data DThave been taken into account). Furthermore, to decompose a possibly complicated prior density into simpler components, we introduce continuous hyperparameters θanddiscrete hyperparameters j(extending the set of hidden variables to ˜h= (h, θ, j)), p(h|D0) =/integraldisplay dθ/summationdisplay jp(h, θ, j|D0). (1) In the following, the summation over jwill be treated exactly, while the θ–integral will be approximated. A Bayesian approach aims in calculating the predictive density for outcomes yin testsituations x p(y|x, D) =/integraldisplay dh p(y|x, h)p(h|D), (2) given data D={DT, D0}consisting of a priori dataD0and i.i.d. training data DT={(xi, yi)|1≤ i≤n}. The vector of all xi(yi) will be denoted xT(yT). Fig.1 shows a graphical representation of the considered probabilistic model. 2training data DT test data x1  · · · xn  x ? ? ? y1  · · · yn  y @@I

arXiv:physics/9911078v1 [physics.flu-dyn] 30 Nov 1999submitted manuscript, ESPCI 1 On theViscosityofEmulsions By Klaus Kroy †, Isabelle Capron,Madeleine Djabourov Physiqe Thermique, ESPCI.10, rue Vauquelin, Paris.France (2February 2008) Combiningdirectcomputationswith invariancearguments, Taylor’sconstitutiveequationforan emulsion can be extrapolated to high shear rates. We show tha t the resulting expression is con- sistent with the rigorous limits of small drop deformation a nd that it bears a strong similarity to ana prioriunrelated rheological quantity, namely the dynamic (frequ ency dependent) linear shear response.Moreprecisely,within a largeparameterre gionthenonlinearsteady–stateshear viscosity is obtainedfrom the real part of the complexdynam ic viscosity,while the first normal stress difference is obtained from its imaginary part. Our e xperiments with a droplet phase of a binary polymer solution (alginate/caseinate) can be inte rpreted by an emulsion analogy. They indicate that the predictedsimilarity rule generalizesto the case of moderatelyviscoelastic con- stituentsthatobeytheCox–Merzrule. 1. Introduction Apart from their technologicalimportance,emulsions have served as model systems accessi- ble to rigorous theoretical modeling. The study of emulsion s consisting of droplets of a liquid dispersed in another liquid has thus contributed substanti ally to our understanding of the rhe- ology of complex fluids. However, although major theoretica l achievements date back to the beginningof the 20thcentury,furtherprogressturnedout to be difficult. The mac roscopicrheo- logicalpropertiesofemulsionsaredeterminedbythereact ionoftheindividualdropstotheflow field, which in turn is modified by the presence of other drops. The mutual hydrodynamic in- teractionsofdropscomplicatessubstantiallythemathema ticaldescription.Moreover,depending onsystemparametersandflowtype,dropletsmaybreakunders teadyflowconditionsifacertain critical strain rate is exceeded.Rigorouscalculationsof the constitutive equationhave therefore concentratedonverydiluteemulsionsandonconditionswhe redropsareonlyweaklydeformed. Sometimes, however, it is desirable to have an approximate e xpression, which — though not rigorous—canserveforpracticalpurposesasa quantitativ edescriptionintheparameterregion beyond the ideal limits. As far as the dependence of the visco elastic properties of an emulsion on the volume fraction φof the dispersed phase is concerned, such an approximation h as been givenbyOldroyd(1953).Itisnotrigorousbeyondfirst order inφbutserveswell somepractical purposesevenat rather highvolumefractions.It seems notr easonableto lookfor a comparably simple approximation for the dependence of shear viscosity ηon shear rate ˙γthat covers the whole range of parameters, where all kinds of difficult break –up scenarios are known to occur. In the next section, we propose instead a less predictive exp ression which contains an average drop size R(that may change with shear rate) as a phenomenologicalpara meter. The latter has to be determinedindependentlyeither from theoryor experi ment.It turnsout, however,that for a substantialrangeof viscosity ratiosandshear rates, the expressionfor η(˙γ)is to a largeextent independent of morphology. For conditions, where drops do n ot break outside this region, we point out a similarity relation between this expressionand the frequencydependentviscoelastic † klaus@pmmh.espci.fr2 KlausKroyklaus@pmmh.espci.fr,IsabelleCapron,Madelei neDjabourov moduliG′(ω),G′′(ω),similartothe Cox–Merz ruleinpolymerphysics.Moreprecisely,weshow that in the limit of small drop deformation, the constitutiv e equation of an emulsion composed ofNewtonianconstituentsofequaldensitycanbeobtainedf romthefrequencydependentlinear response to leading order in capillary number Cand (reciprocal) viscosity ratio λ−1. And we argue that the identification is likely to represent a good ap proximation beyond this limit in a larger part of theC−λparameter plane. Experimentally, this similarity relatio n can be tested directly,withoutinterferenceoftheoreticalmodeling,b ycomparingtwoindependentsetsofrhe- ological data. In summary, our theoretical discussion prov okes two major empirical questions. (1) If drops break: does the expression for the viscosity der ived in Eq.(2.18) describe the data withRtheaveragedropsizeatagivenshearrate?(2)Ifdropsdonot breakbelowacertainchar- acteristiccapillarynumber C∗(λ): doestheproposedsimilarityrulehold?To whatextentdoes it generalize to non–Newtonianconstituents? In Section 3 we a ddress mainly the second question byexperimentswithaquasi–staticdropletphaseofamixtur eofmoderatelyviscoelasticpolymer solutions. 2. Theory 2.1.Taylor’sconstitutiveequationforemulsions A common way to characterize the rheological properties of c omplex fluids such as emulsions, suspensions, and polymer solutions, is by means of a constitutive equation or an equation of state that relates the components pij+pδijof thestress tensor to therate–of–strain tensor e ij. Thisrelationcanaccountforalltheinternalheterogeneit yandthecomplexityandinteractionsof the constituents if only the system may be represented as a ho mogeneousfluid on macroscopic scales. Theformof possibleconstitutiveequationsis rest rictedby general symmetry arguments , which provideguidelinesfor the constructionof phenomeno logicalexpressions(Oldroyd1950, 1958).Ontheotherhand,forspecialmodelsystemstheconst itutiveequationsmaybecalculated directlyat least forsomerestricted rangeof parameters.A n earlyexamplefora direct computa- tionoftheconstitutiveequationofacomplexfluidisEinste in’sformula η=ηc/parenleftbigg 1+5 2φ/parenrightbigg (2.1) for the shear viscosity η≡p12/e12of a dilute suspension(particlevolume fraction φ≪1). It is obtainedbysolvingStokes’equationforaninfinitehomogen eousfluidofviscosity ηccontaining a single solid sphere.For a sufficientlydilute suspension, the contributionsof differentparticles totheoverallviscosity ηcanbeaddedindependently,givinganeffectproportionalt oφ.Inclose analogy Taylor (1932) calculated ηfor a steadily sheared dilute suspension of droplets of an incompressibleliquidofviscosity ηd≡ληcinanotherincompressibleliquidofviscosity ηc.For weaklydeformeddropsheobtained η=ηc/parenleftbigg 1+φ5λ+2 2λ+2/parenrightbigg ≡ηT, (2.2) which we abbreviate by ηTin the following. This expression includes Einstein’s resu lt as the limiting case of a highly viscous droplet, λ→∞. As in Einstein’s calculation, interactions of the drops are neglected. The result is independent of surfac e tension σ, shear rate ˙γ, and drop radiusR; i.e., it is a mereconsequenceof the presenceof a certainam ountφof disperseddrops, regardless of drop size and deformation (as long as the latte r is small). Moreover, the dynamic (frequency dependent) linear response of an emulsion has be en calculated by Oldroyd (1953). Hisresultsarequotedinsection2.4below. Under steady flow conditions,drop deformationitself is pro portionalto the magnitudeof theOnthe ViscosityofEmulsions 3 rate–of–strain tensor eij. More precisely, for simple shear flow with constant shear ra te˙γ, the characteristicmeasureofdropdeformationforgiven λisthecapillarynumber C=ηcR˙γ σ, (2.3) alsointroducedbyTaylor(1934).Itappearsasdimensionle ssexpansionparameterinaperturba- tion series of the drop shape under shear. To derive Taylor’s Eq.(2.2)it is sufficient to represent the drops by their spherical equilibrium shape. Aiming to im prove the constitutive equation, Schowalter, Chaffey & Brenner (1968) took into account defo rmations of drops to first order inC. The refined analysis did not affect the off–diagonalelemen ts of the constitutive equation, i.e. Taylor’s Eq.(2.2)for the viscosity, but it gave the (un equal) normal stresses to order O(C˙γ). Another limit, where exact results can be obtained, is the li mit of large viscosity ratios λ→∞ (Frankel & Acrivos 1970; Rallison 1980). To clarify the phys ical significance of the different limits we want to give a brief qualitative description of the behavior of a suspended drop under shear,basedonworkbyOldroyd(1953)andRallison(1980). In a quiescent matrix fluid of viscosity ηc, a single weakly deformed drop relaxes exponen- tially into its spherical equilibrium shape; i.e., defining dimensionless deformation by D:= (a−b)/(a+b)withaandbthe major and minor axis of the elongated drop, one has for a small initialdeformation D0, D=D0e−t/τ1. (2.4) Thecharacteristic relaxationtime (Oldroyd1953) τ1=ηcR σ(2λ+3)(19λ+16) 40(λ+1)(2.5) alsocharacterizesthemacroscopic stressrelaxation inanunstrainedregionofadiluteemulsion. Atωτ1≃1 one observesthe characteristic relaxationmode in the fre quencydependentmoduli. The relaxation time diverges for λ/σ→∞since it takes longer for a weak surface tension to drivea viscousdropbacktoequilibrium.Whathappensifthe matrixissteadilyshearedatshear rate˙γ?For˙γτ1≪1,theflowinducedinthedropbytheexternaldrivingisweakc omparedtothe internal relaxationdynamicsand the equilibriumstate is o nlyslightly disturbed,i.e., the dropis onlyweaklydeformed.Similarly,forlargeviscosityratio λ,the elongationof thedropbecomes veryslowcomparedtovorticity,andhenceagainverysmalli nthesteadystate,evenif τ1˙γisnot small.Technically,thisisduetotheasymptoticproportio nalitytoλ−1oftheshearratewithinthe drop. In both limits of weak deformation, the time τ1also controls the orientation of the major axisofthedropwithrespecttothe flowaccordingto π 4−1 2arctan(τ1˙γ). (2.6) Eq.(2.4) and Eq.(2.6)both can be used to determine the surfa ce tension σfrom observationsof singledropsunderamicroscope.Inpassing,wenotethatthe classicalmethodbasedontheresult obtainedbyTaylor(1934)forthesteady–statedeformation ,canonlybeusedif λisnottoolarge, whereasEq.(2.4)andEq.(2.6)aremoregeneral. The exactcalculationsmentionedso far becamefeasible bec ause(andareapplicableif) devi- ations of the drop from its spherical equilibrium shape are s mall. On the other hand, if neither thecapillarynumber(or τ1˙γ)is smallnortheviscosityratioislarge,i.e., C>∼1 and λ<∼1, (2.7) dropscanbestronglydeformedbythesymmetricpartoftheflo wfield.Experimentswithsingle drops by Grace (1982) and others have shown that this eventua lly leads to drop break–up if λ4 KlausKroyklaus@pmmh.espci.fr,IsabelleCapron,Madelei neDjabourov /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1 λ11C FIGURE1. Schematic representation of the limits (hatched) where E qs.(2.13), (2.18) for the viscosity η of an emulsion give rigorous results. In the shaded region th ey predict ηto be practically independent of capillary number. The frame of the box is meant to comprise th e wholeC-λparameter plane from zero to infinity (small Reynolds number understood). The curved sol id line is a sketch of the break–up curve for steadily sheared isolated Newtonian drops according to Gra ce (1982). Dashed lines indicate schematically our viscositymeasurements (see Section3). is smaller than some critical viscosity ratio. No general ri gorous result for the viscosity of an emulsion is known in this regime, where arbitrary drop defor mationsand break–up may occur. Below we will also be interested in such cases, where the cond itions required for the rigorous calculationsarenotfulfilled. 2.2.Second–ordertheory For the following discussion we introduce some additional n otation. The rate–of–strain tensor eijand the vorticity tensor ωijare defined as symmetric and antisymmetricparts of the veloc ity gradient∂jvi. Inparticular,fora steadysimpleshearflow vi=˙γx2δi1,and ∂jvi=eij+ωij=˙γ(δi1δj2+δi2δj1)/2+˙γ(δi1δj2−δi2δj1)/2. (2.8) The components pijof the stress tensor in the shear plane ( i,j∈ {1,2}) as obtained for finite λ bySchowalter et al.(1968)read pij=2ηTeij−ηcφτ119λ+16 (2λ+3)(λ+1)Deij+ηcφτ1˙γ225λ2+41λ+4 14(2λ+3)(λ+1)2δij+O(˙γC2).(2.9) Asusual,thematerialderivativehasbeendefinedby Dcij:=∂t+vk∂kcij+ωikckj+ωjkcik, (2.10) wheresummationoverrepeatedindicesisimplied,andthefir sttwotermsvanishforsteadyshear flow. Note that sinceDeijis diagonal, Eq.(2.9) implies η=ηT+O(C2), and hence Eq.(2.2) remainsvalidtofirst orderin Caswementionedalready. Can we extrapolate the exact second order result Eq.(2.9) fo r the stress tensor to arbitrary C andλby using the constraints provided by general invariance arg uments? For example, since the shear stress has to change sign if the direction of the she ar strain is inverted whereas the normalstresses donot,the shear stress andthe normalstres ses have to beodd/evenfunctionsof ˙γ, respectively.From this observation we could have foresee n that Eq.(2.2) cannot be improved by calculatingthe nextorderin ˙γ, i.e., byconsideringdropletdeformationto lowest order. More important are Galilean invariance and invariance under tra nsformations to rotating coordinateOnthe ViscosityofEmulsions 5 frames, which give rise to the material derivative introduc edabove. Applying the operator (1+ τ1D)to Eq.(2.9) adds to the right hand side of the equation a term 2 ηTτ1Deijplus a term of orderO(˙γC2), sothat oneobtains(intheshearplane) pij+τ1Dpij=2ηT(eij+τ2Deij)+ηcφτ1˙γ225λ2+41λ+4 14(2λ+3)(λ+1)2δij+O(˙γC2).(2.11) Asanothershort–handnotationwehaveintroducedasecondc haracteristictime τ2,whichtothe presentlevelofaccuracyin φisgivenby τ2/τ1=1−φ19λ+16 (2λ+3)(2λ+2)+O(φ2). (2.12) It sets the time scale for strain relaxation in an unstressed region and was named retardation time by Oldroyd(1953).Frankel& Acrivos(1970)realized th atup to the partlyunknownterms of orderO(˙γC2)on the right–hand side, Eq.(2.11) belongs to a class of possi ble viscoelastic equationsofstatealreadydiscussedbyOldroyd(1958).Hen ce,setting O(˙γC2)≡0ontheright– handsideofEq.(2.11),wecandefinea(minimal)modelviscoe lasticfluidthatbehavesidentical totheemulsiondescribedbyEq.(2.9)forsmallshearrates. IncontrasttoEq.(2.11),thetruncated formulaforthe viscosity η=ηT1+τ1τ2˙γ2 1+(τ1˙γ)2=ηc 1+(τ1˙γ)2/bracketleftbigg 1+φ5λ+2 2λ+2+/parenleftbigg 1+φ5(λ−1) 2λ+3/parenrightbigg (τ1˙γ)2/bracketrightbigg (2.13) thus obtained has a manifestly non–perturbativeform. Howe ver, no phenomenologicalparame- ters had to be introduced. Note that Eq.(2.13) comprises bot h exactly known limits:C→0 for fixedλ, andλ→∞for arbitraryC. Obviously,Frankel & Acrivos (1970) have forgottena term −25ηcφeij/2 in their Eq.(3.6) for pijin the limit λ→∞for fixedC. If the latter is included, Eq.(2.13)isalso inaccordwith their O(λ−1)−analysis.Moreover,Eq.(2.13)hastheproperlim- iting behavior for λ=1,σ→0, i.e.C→∞, which is an extreme case of Eq.(2.7). Since we assume equal densities for the two phases, the two–phase flui d actually reduces to a one–phase fluid in this degenerate case, and the viscosity is simply ηc, independent of morphology. For illustration,therigorouslimitsofEq.(2.13)inthe C−λplanearedepictedgraphicallyinFig.1. FollowingGrace(1982),a qualitativebreak–upcurveforsi ngledropsundersteadyshearisalso sketched.Insummary,Eq.(2.13)iscorrectforarbitrary λifC→0,andforarbitraryCifλ→∞, and for small and largeCifλ=1. Therefore, one can expect that Eq.(2.13) works reasonabl y well within a large parameter range (small Reynolds number u nderstood). This is further sup- portedbytheobservationthattheerrormadeingoingfromEq .(2.9)toEq.(2.13)ratherconcerns theshapeofthedropletthanits extension (itconsistsintruncatinga perturbationseriesinshape parametrisation). The final result, though sensitive to the latter, is probably less sensitive to the former. Nevertheless, one would not be surprised to see devi ations from Eq.(2.13) when drops become extremely elongated. Finally, due to changes in morp hology by break–up and coales- cence, the avarage drop size Rmay change. Observe, however, that for most viscosity ratio s (λ notclosetounity),Eq.(2.13)ispracticallyindependento fcapillarynumber(andthusof R)when break–up might be expected according Grace (1982) and other s. As an analytic function that is physically knownto be boundedfrom aboveand from below (the latter at least by the viscosity of a stratified two–phase fluid depicted in Fig. 3), η(λ,C)has to have vanishing slope in the C−directionforlargeC. AccordingtoEq.(2.13), η(λ,C)isalmostindependentofCfor C≫C∗≈40(λ+1)√ 3(2λ+3)(19λ+16), (2.14) whereC∗is the turning point in the dilute limit, determined by τ1˙γ=1/√ 3. For finite volume fractions ˜τ1fromEq.(2.16)replaces τ1.Hence,forC≫C∗,Eq.(2.13)anditsextensiontohigher6 KlausKroyklaus@pmmh.espci.fr,IsabelleCapron,Madelei neDjabourov η ληη c c FIGURE2. Self–consistent mean–field descriptionof volume–fract ion effects indisordered emulsions and suspensions. The fluid surrounding a test droplet is assumed to have the viscosity ηcof the continuous phase/the viscosity ηof the whole emulsion, within/outside the “free–volume–sp here” of radius R/φ1/3. volume fractions derived below in Eq.(2.18) are practicall y independent of drop deformation andmorphology.Forfinitevolumefractionsaroughinterpol ationforC∗derivedfromEq.(2.18) is given by 2 .4/(5(1+φ)+4λ). In a large part of parameter space we thus expect Eqs.(2.13) , (2.18)to be applicabletomonotonicshear historieswith Rgivenbythe averageinitial radiusof thedroplets.Fornon–monotonicshearhistories,therecan ofcoursebehysteresiseffectsin η(˙γ) that result from morphologicalchanges for C≫C∗. These can only be avoided by substituting forRthe radiuscorrespondingtotheactualaveragedropsize at t heappliedshearrate ˙γ. 2.3.Finite volumefractions A general limitation of the equations discussed so far, is th e restriction to small volume frac- tions. Above, we have implicitly assumed that second order e ffects from drop interactions are small compared to second order effects from drop deformatio n. Any direct (coalescence) and indirect (hydrodynamic) interactions of droplets have bee n neglected in the derivation. Hydro- dynamic interactions can approximately be taken into accou nt by various types of cell models. RecentlyPalierne(1990)proposedaself–consistentmetho danalogoustotheClausius–Mossotti or Lorentz–sphere method of electrostatics. For the case of a disordered spatial distribution of dropshisresultsreducetothosealreadyobtainedbyOldroy d(1953).Oldroydartificiallydivides thevolumearoundadropletofviscosity ηd≡ληcintoaninterior“freevolume”withaviscosity ηcof the bare continuousphase and an exterior part with the vis cosityηof the whole emulsion (seeFig.2).Accordingtothisscheme,animprovedversiono fEq.(2.2)shouldbe(Oldroyd1953) ˜ηT=ηc5+3(ηT/ηc−1) 5−2(ηT/ηc−1). (2.15) This equationpredictsa largerviscosity than its truncati onto first orderin φ, Eq.(2.2).Both are shownasdot–dashedlinesinFig.3.Eq.(2.15)is qualitatively superiortoEq.(2.2).Wenote,how- ever,that the limit λ→∞deviatesin secondorderin φfromthe result obtainedfor suspensions by Batchelor & Green (1972). Eq.(2.15) and likewise all of th e following equations containing quantities ˜ηT,˜τ1,˜τ2are onlyrigorousto first orderin φ. Thesamereasoningastotheviscosityappliestothecharact eristictimes τ1andτ2whichnow read(Oldroyd1953) ˜τ1=ηcR σ[19λ+16][2λ+3−2φ(λ−1)] 40(λ+1)−8φ(5λ+2), (2.16) ˜τ2=ηcR σ[19λ+16][2λ+3+3φ(λ−1)] 40(λ+1)+12φ(5λ+2). (2.17)Onthe ViscosityofEmulsions 7 0 0.2 0.4 0.6 0.8 1 φ1234 ηstratified fluid Taylor extrapolation to large λC Choi/Schowalter FIGURE3. Comparison of different mixing rules for emulsions with v iscosity ratio λ=3 (chosen arbi- trarily). The dot–dashed and dashed straightlines pertain to dilute emulsions described by the extrap- olation formula Eq.(2.13), which reduces to Taylor’s formu la Eq.(2.2) for small capillary numbers. The corresponding curvedlines are obtained from Eq.(2.18) where interactions of the droplets are taken into account in a mean–field approximation. The curved dottedlines are the predictions of the cell model by Choi & Schowalter (1975) for small shear rates, Eq.(2.19). T he curved solidline represents the viscosity η=ηc/[φ+(1−φ)λ−1]of a two–phase stratifiedfluidand is alower bound for anyvisc ositymixing rule. Theirratio ˜τ2/˜τ1isstill givenbyEq.(2.12).Finally,Eq.(2.13)becomes η=˜ηT1+˜τ1˜τ2˙γ2 1+(˜τ1˙γ)2 =ηc 1+(˜τ1˙γ)2/parenleftbigg2λ+2+3φ(λ+2/5) 2λ+2−2φ(λ+2/5)+2λ+3+3φ(λ−1) 2λ+3−2φ(λ−1)(˜τ1˙γ)2/parenrightbigg ,(2.18) whichtoourknowledgehasnotbeengivenbefore,andisoneof ourmainresults.(Foragraphical representationseeFig.4.)Inthelimit τ1˙γ→0itreducestoEq.(2.15),whereasfor τ1˙γ→∞only the second term in parentheses contributes and the curveddashed lines in Fig. 3 are obtained. FromtheforegoingdiscussiononeshouldexpectEq.(2.18)t obeapplicablewithinalargerange ofshearrates,viscosities, andvolumefractions. Finally,wenotethatmorecumbersomeexpressionsfor ˜ηT,˜τ1and˜τ2havebeenderivedwithin adifferentcellmodelbyChoi&Schowalter(1975).Hereweon lyquotetheirexpressionfor ˜ηT, ˜ηT ηcC&S=1+φ2[(5λ)−5(λ−1)φ7/3] 4(λ+1)−5(5λ+2)φ+42λφ5/3−5(5λ−2)φ7/3+4(λ−1)φ10/3(2.19) whichisalsorepresentedgraphicallybythedottedlinesin Fig.3.SinceourdatafavorEq.(2.15) over Eq.(2.19), and similar observations have been made by o thers before (see Section 3), we will notpursuethisalternativeapproachfurtherin thepre sentcontribution. 2.4.A similarity rule Itisinterestingtoobservethatifmorphologyisconserved (dropsize Rindependentofshearrate) forC<∼C∗, our Eq.(2.18) for the nonlinear shear viscosity is closely related to the expressions for the frequency dependent complex viscosity η∗(ω)≡η′(ω)+iη′′(ω)of an emulsion of two8 KlausKroyklaus@pmmh.espci.fr,IsabelleCapron,Madelei neDjabourov incompressibleNewtonianliquidsasderivedbyOldroyd(19 53), η∗(ω) =˜ηT1+˜τ2iω 1+˜τ1iω,η′(ω) =˜ηT1+˜τ1˜τ2ω2 1+(˜τ1ω)2,η′′(ω) =˜ηT(˜τ2−˜τ1)ω 1+(˜τ1ω)2.(2.20) Forconvenience,wealsogivethecorrespondingviscoelast icshearmodulus G∗(ω)≡iωη∗(ω)≡ G′(ω)+iG′′(ω), G′(ω) =ω˜ηT(˜τ1−˜τ2)ω 1+(˜τ1ω)2,G′′(ω) =ω˜ηT1+˜τ1˜τ2ω2 1+(˜τ1ω)2. (2.21) Obviously, the shear–rate dependent viscosity of Eq.(2.18 ) is obtained from the real part of the complexfrequencydependentviscosity η∗(ω)bysubstituting ˙γforω, η(˙γ)≃η′(ω). (2.22) Inthesame way,the firstnormalstress difference p11−p22=−2˙γ·φηcτ1˙γ19λ+16 (2λ+3)(2λ+2)(2.23) fromEq.(2.9)isobtainedtoleadingorderin φand˙γfrom2ωη′′(ω), i.e., p11−p22≃2ωη′′(ω). (2.24) Revertingthelineofreasoningpursuedsofar,wecanconclu dethattoleadingorderin Cand/or λ−1the weak deformation limit of the constitutive equation of e mulsions is obtained from the linearviscoelasticspectra G′(ω),G′′(ω).Further,thisidentificationcanpossiblybeextended(at least approximately)intoregionsofthe C−λplanewherethecriticalcapillarynumberfordrop breakup is somewhat larger than C∗of Eq.(2.14). In hindsight, it is not surprising that in the caseofweaklydeformeddropsthefrequencydependentvisco sityandthesteadyshearviscosity arerelated.Notethatundersteadyshear,dropsundergoosc illatorydeformationsatafrequencey 2ω=˙γifobservedfromaco–rotatingframeturningwithvorticity atafrequency ω=˙γ/2.Ifwe take Eq.(2.24)seriously beyond the rigorously known limit , we obtain an interesting prediction for the first normal stress difference. In contrast to Eq.(2. 23), Eq.(2.24) implies that the first normal stress difference saturates at a finite value 40 φσ/R[2λ+3+2φ(1−λ)]2for high shear rates.Thus,althoughtheinitialslopeofthefirstnormalst ressdifferencewith ˙γincreaseswith λ, itslimit forlarge ˙γdecreaseswith λ. Finally,weremarkthatbasedonqualitativetheoreticalar guments,thesimilarityrelationcon- tained in Eq.(2.22) and Eq.(2.24) has recently been propose d also for polymer melts (Renardy 1997).Usually,inpolymerphysicsaslightlydifferentrel ationisconsidered;namelyasimilarity betweenη(˙γ)and|η∗(ω)|, alsoknownas Cox–Merzrule (Cox&Merz1958).Inourcase, since η′′/η′=G′/G′′=O(φ), wecanwrite η(˙γ)≃ |η∗(ω)|+O(φ2). (2.25) Under the conditions mentioned at the beginning of this sect ion, the usual Cox–Merz rule is fulfilledto first orderin φforanemulsion.Eqs.(2.22),(2.24)areinterestingfromth e theoretical point of view, because they suggest a similarity of two a prioryrather different quantities. The results of this section also can be of practical use, since th ey suggest that two differentmethods maybeappliedto measurea quantityofinterest. 2.5.Non–Newtonianconstituents Generalization of the above theoretical discussion to the c ase of non–Newtonianconstituents is not straightforward.Indeed,as Oldroyd(1953)already kne w,hislinear–responseresultsquoted in Eq.(2.20) and Eq.(2.21) are readily generalized to visco elastic constituents by replacing theOnthe ViscosityofEmulsions 9 -1 0 1lnλ-1 0 1lnC 0lnη -1 0 lnλ0 1lnC FIGURE4. Eq.(2.18) normalized to ηcas a function of viscosityratio λandcapillarynumberC.The volume fractionof the dispersedphase is chosen tobe φ=0.3. viscosities ηd,cin the expressionfor η∗(orG∗) by complexviscosities η∗ d,c(ω)(Palierne 1990). As a consequence, the decompositions of η∗andG∗in real and imaginary parts are no longer those of Eqs.(2.20) and (2.21), and η′,η′′,G′,G′′are given by more cumbersome expressions. For the steady–state viscosity, on the other hand, one has to deal with a non–homogeneousvis- cosity even within homogeneous regions of the emulsion, sin ce the strain rate itself is non– homogeneousandtheviscositiesarestrainratedependent. We donotattempttosolvethisprob- lem here, nor do we try to account for elasticity in the nonlin ear case. Yet, it is an intriguing question,whether the similarity rule Eq.(2.22)can be gene ralizedto the case of non-Newtonian constituents if the constituents themselves obey the Cox–M erz rule (what many polymer melts andsolutionsdo).Ifbothconstituentshavesimilarphasea nglesθ≡arctanG′′/G′thegeneralized viscosityratio λ∗≡η∗ d η∗c=|η∗ d| |η∗c|ei(θd−θc)(2.26) that entersthe expressionsfor η∗andG∗, transformsapproximatelyto ηd(˙γ)/ηc(˙γ)by applying the Cox–Merz rule. Therefore, in this particular example, E q.(2.18) supports the expectation that the generalization may work at least approximately.If , on the other hand, the phase angles of the constituents behave very differently, the answer is l ess obvious. This problem has been investigatedexperimentallyandis furtherdiscussedin Se ction3. Inanycase,thegeneralizationcanonlyworkiftherepresen tationoftheemulsionbyasimple shear–rate dependent viscosity ratio ηd(˙γ)/ηc(˙γ), with˙γthe external shear rate, is justified. In the remainder of this section we construct an argument that a llows us to estimate the effective shear–rate dependent viscosity ratio that should replace λin Eq.(2.18). We take into account the deviation of the strain rate from the externally imposed flow only within the drops, because outsidethedropsthediscrepancyisalwayssmall.Insidead rop,thestrainratecanbesmalleven for high external shear rates if the viscosity ratio λ=ηd/ηcis large. Since we are looking for an effective viscosity ¯ηd(˙γ)for the whole drop to replace the viscosity ηdat small shear rate, we replace the non–Newtonian drop of non–homogeneous visco sity by an effective pseudo– Newtoniandropofhomogeneousbutshear–ratedependentvis cosity.A possibleansatzfor ¯ηdis obtainedbyrequiringthat thetotal energydissipatedwith inthe dropremainsconstantuponthis substitution.Hence,wehave /integraldisplay dV pijgij=2¯ηd/integraldisplay dV¯gij¯gij, (2.27)10 KlausKroyklaus@pmmh.espci.fr,IsabelleCapron,Madelei neDjabourov wheregijand ¯gijdenotetherate–of–strainfieldsintherealdropandintheco rrespondingmodel drop of effective viscosity ¯ηd, respectively. They both depend on the position within the d rop, whereastheaveragestrainrate ˙γ2 eff≡2 V/integraldisplay dV¯gij¯gij (2.28) that enters the right–hand side of Eq.(2.27) does not. Since the strain field ¯ gijwithin a drop of homogeneous viscosity is unique for a given system in a giv en flow field, ¯ηditself can be expressed as a function ¯ηd(˙γeff)of the average strain rate. Here, we approximatethis functi onal dependence by the strain rate dependence ηd(˙γ)of the viscosity of the dispersed fluid. Further, neglecting drop deformation we calculate ˙γefffrom the velocity field within a spherical drop (Bartok&Mason1958)andobtain ¯λ≡¯ηd(˙γeff) ηc(˙γ)≈ηd/parenleftBig√ 2˙γ/(¯λ+1)/parenrightBig ηc(˙γ). (2.29) A different prefactor (√ 7 in place of√ 2) in the expression for the effective strain rate ˙γeffwas obtained by de Bruijn (1989) using instead of the average in E q.(2.28) the maximum norm of ¯gij. To obtain the correction to Eq.(2.18) due to Eq.(2.29) in th e case of non–Newtonian con- stituents, the implicit equation for ¯λhas to be solved for given functions ηc(˙γ)andηd(˙γ). For shear thinning constituents, Eq.(2.29) implies a tendency of Eq.(2.18) to overestimate η(˙γ)if˙γ andλare large. In the actual case of interest, for the constituen ts that were used in the exper- iments discussed in Section 3, the viscosity ratio λ(ηdandηctaken at the external shear rate ˙γ) varies almost by a factor of 10. However, the corrections di scussed in this Section only be- come important for high shear rates, where the constituents are shear thinning. In this regime, the viscosity ratio (viscosities taken at the external shea r rate) only varies between 1 /2 and 2, and hence the correctionsexpectedfrom Eq.(2.29)are at bes t marginallysignificant at the level of accuracy of both Eq.(2.29) and the present measurements. Therefore, a representation of the drops by pseudo–Newtoniandrops of homogeneousbut shear–r ate dependent viscosity is most probably not a problem for the measurements presented in the following section. The question as to a generalizationof the similarity rule Eq.(2.22)to no n–Newtonianconstituentsseemswell defined. 3. Experiment 3.1.Materialsandmethods Theexperimentalinvestigationdealswithaphaseseparate daqueoussolutioncontainingapolysac- charide (alginate) and a protein (caseinate). This type of s olutions are currently used in the foodindustry.Themethodsforcharacterizingthe individu alpolymersin solutionare in general known,especially whendealingwith non–gellingsolutions wherecompositionandtemperature are the only relevant parameters. The polymers are water sol uble. When the two biopolymers in solution are mixed, a miscibility region appears in the lo w concentrations range and phase separation at higher concentrations.The binodal and the ti e lines of the phase diagramcan then be established by measuring the composition of each phase at a fixed temperature. In general, therheologicalbehaviorofphaseseparatedsystemsisdiffi culttoinvestigate,andasuitablepro- cedure is not fully established. In some cases, two–phaseso lutionsmacroscopicallyseparate by gravity within a short period of time, but in some other cases (such as ours) they remain stable for hours or days without appearance of any visible interfac e. These “emulsion type” solutions have no added surfactant. Following approaches developed f or immiscible blends, one may try to characterizethe partially separatedsolutionas an effe ctiveemulsionif the coarseningis slowOnthe ViscosityofEmulsions 11 enough.In order to establish a comparisonbetween phase sep arated solutions and emulsions, it isnecessarytoknow •the volumefractionofthephases, •theirshear–ratedependentviscosities(flowcurves), •theirviscoelasticspectra, •the interfacialtension σbetweenthephases, •the averageradiusof Rthedrops Only the ratio R/σenters rheological equations. Knowledge of either Rorσallows the other quantitytobe inferredfromrheologicalmeasurements. A difficultywhenworkingwith phase separatedsolutions,as opposedto immiscible polymer melts, arises from the fact that each phase is itself a mixtur e (and not a pure liquid) and there- fore the rheologyof the phase dependson its particular comp osition.If one wishes to minimize the number of parameters, it is important to keep the composi tion of the phases constant upon changingthevolumefractions.Thiscanbeachievedbyworki ngalonga tielineofthephasedi- agram. And this is precisely the procedurethat we followed. The polymerswere first dissolved, thenalargequantityoftheternarymixturewasprepared(35 0ml)andwascentrifuged.Thetwo phases were then collected separately. Both pure phases wer e found to be viscoelastic and to exhibit shear thinningbehavior,which is especially prono uncedfor the alginate rich phase with η(10−1s−1)/η(103s−1)≈20, while the caseinate rich phase is almost Newtonian below 102 s−1. The viscosity of the alginate rich phase is higher than that of the caseinate rich phase for shearratesbelow2 ·102s−1andlowerforhighershearrates.We checkedthat bothphases obey theusualCox–Merzrulein thewholerangeofappliedshearra tes. By mixing various amounts of each phase, the volume fraction of the dispersed phase was variedbetween10%and30%whilethecompositionofeachphas ewaskeptconstant.Inpartic- ular, the temperature was kept constant and equal to the cent rifugation temperature in order to avoid redissolution of the constituents. To prepare the emu lsion, the required quantities of each phase were mixed in a vial and gently shaken. Then the mixture was poured on the plate of the rheometer (AR 1000 from TA Instruments fitted with a cone and p late geometry 6 cm/2◦) and a constant shear rate was applied. The apparent viscosity fo r a particular shear rate was then recorded versus time until it reached a stable value. By shea ring at a fixed shear rate, one may expect to create a steady size distribution of droplets, wit h a shear rate dependent average size. After each shear experiment a complete dynamic spectrum was performed. In this way, shear rates rangingbetween3 ·10−2s−1and 103s−1were applied.The analysisof each spectrumac- cording to Palierne (1990)allowed us to derive by curve fitti ng the average drop radius Rat the correspondingshearrate. More technical details about the experimental investigati on along with more experimental results will be presentedelsewhere.Here, we concentrateo n the analysisof those aspects of the rheologicalmeasurementspertinenttothe theoreticaldis cussioninSection2. 3.2.Resultsanddiscussion In this section we present our experimental observations an d address the questions posed at the end of the introduction. Before we present our own data we want to comment briefly on relateddatarecentlyobtainedforpolymermeltsbyGrizzut i&Buonocore(1998).Theseauthors measuredtheshear–ratedependentviscosity ofbinarypoly mermeltsandcomparedthemto the low volume fraction limit of Eq.(2.18), i.e. Eq.(2.13), and to (a truncated form of) results of Choi & Schowalter (1975). They reported much better agreeme nt with Eq.(2.13) than with the truncated series from Choi & Schowalter (1975).Comparison with the full expressionsof Choi & Schowalter (1975) would have made the disagreement even wo rse (cf. Fig 3). The average radiusRthat enters the equation, was determined independently for each shear rate applied. The constituents where moderately non–Newtonian polymer m elts, the viscosity ratio varying12 KlausKroyklaus@pmmh.espci.fr,IsabelleCapron,Madelei neDjabourov 0.01 0.1 1 10 100 1000110 /c104' (/c119) experimentalG''//c119=/c104'(/c119) and /c104(/c103') (Pa.s) /c119(rad/s), /c103'(s-1)/c104(/c103')experimental /c104(/c103')Eq.(2.18) FIGURE5. The nonlinear shear viscosity η(˙γ)(opaque squares) and the real part η′(ω)of the dynamic viscosity η∗(ω)(lines) of a droplet phase of a mixture of weakly viscoelasti c polymer solutions (algi- nate/caseinate). Also shown is Eq.(2.18) for the viscosity of an emulsion of Newtonian constituents eval- uated for the actually non–Newtonian viscosities of the con stituting phases with the drop size obtained from the spectra (open triangles). Due to the scatter inthe d ynamic viscosity at low frequencies there is an uncertainty inthe average drop size,resulting incorrespo nding errorbars (multiplepoints) forEq.(2.18). betweenλ≈0.3...3overtherangeofshearratesapplied.Hence,theseexperim ents,arelocated in the interesting parameter range, where Eqs.(2.18) and Eq .(2.13) for η(˙γ)are expected to be sensitivetodropdeformationandbreak–up.Surprisingly, theresultsshowthattheydescribethe data very well over the whole range of shear rates although on e would not necessarily expect average dropdeformationto be verysmall. Unfortunately,d ropsizes have not been reportedby theauthors,soconclusionsconcerningthelocationinthe C−λparameterplaneandthevalidity of the similarity rule Eq.(2.22)cannot be drawn. Also the qu estion, whetherEq.(2.18)holdsfor small viscosityratios λ≪0.3,cannotbeanswered. Ourownmeasurementswerelocatedinaboutthesame λ−range.Aswenotedinthepreceding section,onlytheratio R/σentersrheologicalequationsandknowledgeofeither Rorσallowsthe other quantity to be inferred from rheological measurement s. Ding & Pacek (1999) determined the interfacial tension of the alginate/caseinate system u sed in our experiments by observing drop relaxation under a microscope and analyzing the data ac cording to Section 2. They found σ≃10−5N/m. Using this, we obtained an average drop size R≃10−5m from the measured spectraG′(ω),G′′(ω)according to Palierne (1990) for the experiments reported i n Fig. 5. By the method based on Palierne (1990), we could not detect a dec rease in drop size with shear rate as expected from the phenomenological phase diagram fo r single Newtonian drops under shear as established by Grace (1982) and others. Thus, the li mit of high capillary numbers and moderate viscosity ratios (the region above the break–up cu rve) in Fig. 1 has been accessed experimentally.Correspondinglocationshavebeenindica tedqualitativelyinthefigurebydashed lines.Withthedropsizebeingconstant,onecantrytotestt heproposedsimilarityruleEq.(2.22). By identifying the axis for frequency ωand shear rate ˙γ, data for the real part η′(ω)of the frequencydependentdynamicviscosity η∗(ω)arecomparedtodatafortheshear–ratedependent viscosity η(˙γ)in Fig. 5. The emulsion containing 30% of the alginate rich ph ase and 70% of the caseinate rich phase has been prepared at room temperatu re as described in the precedingOnthe ViscosityofEmulsions 13 section. Steady shear rates rangingbetween 3 ·10−2s−1and 103s−1correspondingto capillary numbersC≈6·10−2...103havebeenapplied.Theshearviscosity(opaquesquares)isr eported in the figure for each of these individual measurements. The m ultiple data sets for η′(lines) taken each between two successive steady shear measurement s, superimpose fairly well; i.e. the spectra appear to be remarkably independent of the prece ding steady shear rate. The good coincidence of η′(ω)andη(˙γ)in Fig. 5 show that the data obey the proposed similarity rule Eq.(2.22)overa largerangeof shear rates. The agreementne ar˙γ≈2·102s−1is a consequence oftheproximitytothetriviallimit λ=1,C=∞.Nevertheless,thedataprovidestrongevidence that Eq.(2.22) is an excellent approximation for a large ran ge of viscosity ratios and capillary numbers.Similarresults(notshown)havebeenobtainedfor othervolumefractions.Comparison withEq.(2.18)representedbytheopentrianglesinFig.5,o ntheotherhand,islesssuccessfulat large shear rates, althoughit is still not too far off for a th eoretical curve without any adjustable parameter. A discrepancyhad to be expected as a consequence of the non–Newtoniancharacter oftheconstituentsatlargeshearrates,whichisdefinitely nottakenintoaccountinEq.(2.18).For the plotofEq.(2.18)in Fig. 5we merelysubstituted ηd(˙γ)/ηc(˙γ)takenat the externalshearrate ˙γforλ. The average drop size Rwas obtained from fitting the viscoelastic moduli. The scatt er in thedynamicviscositydata givesrise toanuncertaintyin R,whichisreflectedbythe multiple open trianglesat low shear rates. It seems that the similari ty rule Eq.(2.22)is more generalthan Eq.(2.18), i.e., it still holds for rather viscoelastic con stituents (that obey the usual Cox–Merz rule), where the latter fails. This relation certainly dese rves further investigation with different materialsandmethods. In summary, we have succeeded in establishing an analogy firs t between a partially phase– separated polymer solution and an emulsion, and further bet ween the viscoelastic spectrum of the system and its nonlinear shear viscosity even in the case of (moderately) non–Newtonian constituents. This work was supported by the European Community under cont ract no FAIR/CT97-3022. WethankP.DingandA.W.Pacek(UniversityofBirmingham)fo rmeasuringthesurfacetension andS. CosteuxandG. Haaghforhelpfuldiscussionsandsugge stions. REFERENCES BARTOK, W. & M ASON, S. G. 1958 Particlemotions insheared suspensions. J.ColloidSci. 13, 293. BATCHELOR ,G. K. & G REEN,J. T.1972Thedeterminationofthebulkstressinasuspensi onofspherical particles toorder c2.J. FluidMech. 56, 401. DEBRUIJN, R. A. 1989 Deformation and breakup of drops in simple shear fl ows. PhD thesis, TU Eind- hoven, The Netherlands. CHOI, S. J. & S CHOWALTER ,W. R. 1975 Rheological properties of nondilute suspension s of deformable particles. Phys.Fluids 18, 420. COX, W. P. & M ERZ, E. H. 1958 Correlation of dynamic and steady flow viscositie s.J. Polym. Sci. 28, 619. DING, P. & P ACEK, A. W. 1999 unpublished. FRANKEL,N. A. & A CRIVOS, A. 1970Theconstitutiveequationforadiluteemulsion. J.FluidMech. 44, 65. GRACE, H. P. 1982 Dispersion phenomena in high viscosity immiscib le fluid systems and application of static mixers as dispersiondevices insuch systems. Chem.Eng. Commun. 14, 225. GRIZZUTI, N. & B UONOCORE , G. 1998 The morphology-dependent rheological behavior of an immisci- ble model polymer blend. In Proceedings of the 5thEuropean Rheology Conference (ed. I. Emri & R.Cvelbar), Progress and Trends inRheology , vol.5, p.80. Darmstadt: Steinkopff. OLDROYD, J. G. 1950 Onthe formulationof rheological equations of st ate.Proc.Roy. Soc. A 200, 523. OLDROYD, J. G. 1953 The elastic and viscous properties of emulsions a nd suspensions. Proc. Roy. Soc. A 218, 122.14 KlausKroyklaus@pmmh.espci.fr,IsabelleCapron,Madelei neDjabourov OLDROYD, J. G. 1958 Non-newtonian effects in steady motion of some id ealized elasto-viscous liquids. Proc. Roy.Soc. A 245, 278. PALIERNE , J. F. 1990 Linear rheology of viscoelastic emulsions with i nterfacial tension. Rheol. Acta 29, 204. RALLISON , J. M. 1980 Note onthe time-dependent deformation of avisco us drop whichisalmost spheri- cal.J.FluidMech. 98, 625. RENARDY, M. 1997 Qualitative correlation between viscometric and l inear viscoelastic functions. J. Non- Newtonian FluidMech. 68, 133. SCHOWALTER ,W. R., C HAFFEY, C. E. & B RENNER,H. 1968Rheological behaviorofadiluteemulsion. J. Colloidand Interface Sci. 26, 152. TAYLOR,G. I.1932Theviscosityofafluidcontainingsmalldropsofa notherfluid. Proc.Roy.Soc.A 138, 41. TAYLOR, G. I. 1934 The formationof emulsions indefinable fields of flo w.Proc. Roy.Soc. A 146, 501.

arXiv:physics/9912001v1 [physics.gen-ph] 1 Dec 1999THE VARIANT PRINCIPLE N. T. Anh1 Institute for Nuclear Science and Technique, Hanoi, Vietnam. Abstract Based on the principle of causality, I advance a new principl e of variation and try to use it as the most general principle for research in to laws of nature. 1 INTRODUCTION Abstractly, the Nature can be examined as a system of states a nd actions. State is a general concept that defines existence, structure, organiz ation, and conservation of all matter’s systems, and that stipulates properties, inner re lationships of all things and phenomena. Action is a operation that manifests self-influe nce and inter-influence of states, that presents dynamic power and impulsion of motion and development. Generally, state is object on which actions do. Each state has its action . Self-action makes state conservable and developable. Action of one state on other fo rms interaction between them. Self-action and inter-action cause variation of stat e from one to other. That variation establishes a general law of motion. Following this way I advance a new principle – that is called V ariant Principle. Uti- lizing this principle as the most general principle I hope th at it is useful for research on a logically systematic method to review known laws and to pre dict unknown laws. And I believe that some of the readers of this article will find out that this principle explains naturally inner origin of variation, rules evolutionary pr ocesses of things, and perhaps they will be the ones to complete the quest for theories of the Universe. The article is organized as follows. In Section 2, I advance t he ideas and concepts for leading the equation of motion. That is just the foundation o f the variant principle. A phenomenon in physics is illustrated by this principle in Se ction 3. Conclusion is given in Section 4. 1Email: anhnt@vol.vnn.vn 1The Variant Principle – N. T. Anh 2 2 THE EQUATION OF MOTION In the Nature, any state and its action are constituent eleme nts of a subject that I call it actor, A= (A&/hatwideA), (I) where Ais state, and /hatwideA is its action operator. 1. For any system in which there is only one actor {A}, that actor is in self-action. This fact causes actor either to be conserved or to be varied by action of itself with respect to all its possible inner degrees of freedom. Co nservation makes actor invariant. But variation obeys a equation of motion, /hatwideAA= 0, (II) where action operator /hatwideA may include differentiation, integration, and/or other fo r- mal operations doing with respect to some degrees of freedom (such as space, time, and/or some variable), depending on actually physical prob lems, and Amay nat- urally be a state function describing some considered objec t. The value ‘0’ on the right hand side of Eq. (II) means that variation of actor approaches to stability – invariance, i.e. self-action is equal to zero when variatio n finishes. Solution of the equation of motion describes variant proces s ofactor.Actor varies and finally closes to a new actor, that is solution of the equation of motion when variation finishes. 2. For any system consisting of many actors{A1;A2;...}, each actor is in its self-action and actions from others. This fact causes each actor to be varied by actions of itself and others with respect to all its possible inner and outer de grees of freedom. This variation obeys a equation of motion, (/hatwideA1;/hatwideA2;...)(A1;A2;...) = 0, (III) where action operators /hatwideAiofactorAiare operations doing with respect to some de- grees of freedom, and states AiofactorAiare functions characterized by considered objects. The value ‘0’ on the right hand side of Eq. (III) mean s that actions are equal to zero when variations of actors finishes, i.e. variations of actors approaches to stability – invariance. In fact, Eq. (III) is an advanced f orm of Eq. (II). Solutions of the equations of motion of actors describe their variant processes. All Actor s vary and finally close to a new actorA, that is solution of the equations of motion when variations of actors finishes: A= [A1,A2,...], (IV) where actors are in the same dimension of interaction.The Variant Principle – N. T. Anh 3 * For a system consisting of many actors{A1;A2;...}, the whole system can be considered as a total actor which includes component actors, {A}={A1;A2;...}. (V) Thereby, actorAis in self-action, and it either self-conserves or self-var ies with respect to all its possible inner degrees of freedom. An d variation obeys a equation of motion (II). Hence, the variant principle is stated as follows: -In the Nature every actor varied by actions of itself and othe rs with respect to all possible degrees of freedom to come to some new actor is solut ion of the equation of motion that describes its variant process. Indeed, every variation is caused by action of actor onto state, variation is to escape from action, or in other words, state varies to be agreeable t o action. This fact means that under actions actor must vary anyway with respect to all possible degrees of free dom – transportation facilities to come to new actor, and that its speed of variation is dependent on power of action, which is manifested by conservation of actor. Eigenvalue of action is expressed as instrument to promote v ariation, as easiness of variation. Its value over some degree of freedom shows proba bility of variation following this direction. Anyactor which is done by some action must vary somehow over all possib le degrees of freedom to come to new actor which is no longer to be done by any action. That process shows continuous variation of actor from the beginning to closing. Therefore, this reality proves that variation is imperativ e to have its cause, to have its agent, and that property of variation obeys the equation of m otion. Thereby, from Eqs. (II) and (III), equation of motion can be b uilt for any physical law. Using these equations (II) and (III) for research into p hysics is considered in the next section. I hope that the readers will understand more pr ofoundly about the variant principle. 3 The Rule of Universe’s Evolution The simplest form of self-action is expansion of actor about some degree of freedom, eδx/hatwide∂xf(x) =f(x+δx). (1) Here is just the equation of motion for any quantity f(x), withxdegree of freedom, andδxinfinitesimal of x.The Variant Principle – N. T. Anh 4 Universe’s evolution is described as a law of causality [2] e ssentially based on just this expansion. The form of Eq. (1) is nothing but Taylor’s series . Derivatives of f(x) with respect toxis just variations of f(x) over the degree of freedom x. Eq. (1) has an important application in modelling the multip lication and the combi- nation of quanta. Callα,β,γ,... quanta. For each quantum there is a rule of multiplication as follows αn→e∂ααn=n/summationdisplay i=0Cn iαn−i= (α+ 1)n(2) wherenis order of combination, δα= 1, andCn iis binary coefficient. Using Eq. (2) I consider two stages in the process of the Unive rse’s evolution: doublet and triplet. For two interactive quanta the rule of multiplication reads αn,βn→1 2(eβ∂ααn+eα∂ββn) =n/summationdisplay i=0Cn iαn−iβi= (α+β)n. (3) And similar to three interactive quanta αn,βn,γn→1 3(e(β+γ)∂ααn+e(γ+α)∂ββn+e(α+β)∂γγn) =n/summationdisplay mm/summationdisplay iCn mCm iαn−mβm−iγi = (α+β+γ)n. (4) And so fourth. Eqs. (3) and (4) can be drawn as schemata. ... ··· ··· ··· ··· 2 1 1 0 /circlecopyrt 2 1 1 2⊗2= 3⊕1 1 2 1 2⊗2⊗2= 4⊕2⊕2 1 3 3 1 ... 1 4 6 4 1 ... ··· ··· ··· ··· ··· ··· ···(5)The Variant Principle – N. T. Anh 5 is the schema for Eq. (3), where 2 means two quanta αandβ. The numbers in the triangle is the binary coefficients which are called weights o f classes. For example, 2⊗2= 3⊕1=1 1 ——– 1 ——– 1. And similar to Eq. (4) it reads ... 1 3 1 1 0 /circlecopyrt 3 1 1 1 1 2 1 3⊗3= 6⊕3 2 2 1 1 3 3 1 3⊗3⊗3= 10⊕8⊕8⊕1 3 6 3 3 3 1 1 4 6 4 1 4 12 12 4 3⊗3⊗3⊗3 6 12 6 4 4 ... 1(6) where 3 means three quanta α,βandγ. The coefficients in the pyramid are weights of classes, 1 1 1 - - - - - - - - 3⊗3= 6⊕3 =1 1 1 1 1 1,The Variant Principle – N. T. Anh 6 1 - - - - - - - 3⊗3 = 1⊕8 = 1 1 1 2 1 1 1. It is easily to identify that the above schemata have the form s similar to the SU(2) and theSU(3) groups. This means that for nquanta there is a corresponding schema according to the SU(n) group, and the multiplication and the combination of the Un iverse conform to the SUgroup. This rule is studied further in Ref. [3]. 4 CONCLUSION The theory of causality [1] is very useful to understand abou t the cause of variation. The coexistence of two different actors causes a contradiction. The solution to contradiction makes contradiction varied. That variation is just one of ea ch actor inclining to come to a new actor. It means the difference and the contradiction of t wo actors have inclining towards zero. Indeed, every system comes to equilibrium, st ability. A some state which has any immanent contradiction must vary to reach a new one ha ving no contradiction. The variant principle deals with the law of variation of acto rs, describes only actors with their actions and states, not to mention the difference a nd even the contradiction in them. In insight the variant principle is more elementary an d easier to understand than the causal principle since everything is referred as actor e xisting in nature. Self-action and inter-action of actors onto their states cause the world to b e in motion and in variation. Although the variant principle gives a powerful fundamenta l for application to research into laws of nature, there is no rule arisen yet for formulizi ng self-action and inter-action operators. However, there are some ways to enter operators i n the equation of motion that I hope that in some next article this ways will be synthes ized to a standard rule. For instance, in the quantum electromagnetic dynamics the e quations of motion of the electron-positron and the electromagnetic field are: iγµ∂µψ(x) +mec /planckover2pi1ψ(x) +e /planckover2pi1γµAµ(x)ψ(x) = 0, /squareAµ+ieψ(x)γµψ(x) = 0. The first line is the equation of motion of electron, the first t erm corresponds to the variation of electron with respect to space-time, the secon d gives conservation of electron, and the third is action of the electromagnetic field onto elec tron. The second line can be rewritten as ∂νFνµ−Jµ= 0, that is nothing but the Maxwell equation, with Fνµ=∂νAµ−∂µAνthe electromag- netic field tenser, Aµthe 4-dimensional potential, Jµ=−ieψ(x)γµψ(x) the 4-dimensionalThe Variant Principle – N. T. Anh 7 current density, the first term corresponds to the variation of the electromagnetic field, the second is the external current density of the electromag netic field, (here the mass of photon is zero, so the mass term is not present). This example is easy to show that: – The variation done over some degree of freedom is expressed as derivation with respect to that degree of freedom. – The conservation of actor is written as a term of actor multi plied by a constant characterized by its conservation. – The influence of other actor on a actor is represented as a mul tiplication of two actors. – The external actor stands equally with its variation, when an external influence does on an actor as an external current, an external source, or an e xternal force. Acknowledgments We would like to thank Dr. D. M. Chi for useful discussions and valuable comments. The present article was supported in part by the Advanced Res earch Project on Nat- ural 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). [2] N. T. Anh, Causality: The Nature of Everything , (1991), (available in web site: www.mt-anh.com-us.com). [3] N. T. Anh, The Universe’s Evolution , (1999), (to be published).

arXiv:physics/9912002v1 [physics.atom-ph] 1 Dec 1999High sensitivity two-photon spectroscopy in a dark optical trap, based on electron shelving. L. Khaykovich, N. Friedman, S. Baluschev, D. Fathi, and N. Da vidson Department of Physics of Complex Systems, Weizmann Institu te of Science, Rehovot 76100, Israel We propose a new spectroscopic method for measuring weak tra nsitions in cold and trapped atoms, which exploits the long interaction times and tight confinem ent offered by dark optical traps to- gether with an electron shelving technique to achieve extre mely high sensitivity. We demonstrate our scheme by measuring a 5 S1/2→5D5/2two-photon transition in cold Rb atoms trapped in a new single-beam dark optical trap, using an extremely weak p robe laser power of 25 µW. We were able to measure transitions with as small excitation rate as 0.09 sec−1. PACS number(s): 39.30.+w, 32.80.Pj, 32.80.Rm, 32.90.+a The strong suppression of Doppler and time-of-flight broadenings due to the ultra low temperatures, and the possibility to obtain very long interaction times are obvious advantages of using cold atoms for spec- troscopy. Convincing examples of such precision spectro- scopic measurements are cold atomic clocks [1]. For RF clock transitions long interaction time is usually obtaine d in an atomic fountain [2], while for optical metastable clock transitions free expanding atomic clouds are used [3]. Even longer interaction times can be obtained for cold atoms trapped in optical dipole traps [4]. To obtain long atomic coherence times, spontaneous scattering of pho- tons and energy level perturbations caused by the trap- ping laser are reduced by increasing the laser detuning from resonance [5]. To further reduce scattering, blue- detuned optical traps, where repulsive light forces con- fine atoms mostly in the dark (dark traps), have been developed, achieving atomic coherence of 7 s [6]. The wide use of dark traps was limited by relatively complex setups that require multiple laser beams or gravity as- sistance. Recent development of single-beam dark traps make them more attractive for precision spectroscopy [7], [8]. Dark traps have an additional advantage that makes them especially useful for the spectroscopic measure- ments of extremely weak optical transitions. While pre- serving long atomic coherence times those traps can provide large spring constants and tight confinement of trapped atoms [7] to ensure good spatial overlap even with a tightly focused excitation laser beam. Therefore the atoms can be exposed to a much higher intensity of the excitation laser, yielding a further increase in sensi- tivity for very weak transitions. In this letter we present a new and extremely sensitive method for measuring weak transitions with cold atoms in a far detuned single-beam dark trap using electron shelving spectroscopy [9]. Recently, a similar technique was adapted to demonstrate quantum-limited detection of narrow-linewidth transitions on a free expanding cold atomic cloud [10]. Our scheme is based on a Λ system. Atoms with two ground states (for example, two hyper-7 7 7 .9 n m7 7 7 .9 n m 5 S 1 /26 P 3 /2 6 P 1 /2F = 0F = 1 F = 2 F = 3 F = 4 F = 5 5 P 1 /2 5 P 3 /24 2 0 .3 n m 5 D 5 /2 5 D 3 /2 D etectio n B ea mR ep u m p in g B e am 7 8 0 .2 4 n m7 8 0 .2 3 n m F = 3 | g2 F = 2 | g1 |e FIG. 1. Energy levels of85Rb and the transitions between them which are involved in the experiment. Spectroscopy of the|g1/angbracketright → |e/angbracketrighttransition (5 S1/2F= 2→5D5/2F′in the case of85Rb) is performed. Atoms which undergo the transition are shelved in the level |g2/angbracketright(5S1/2F= 3 in85Rb), from which they are detected using a cycling transition (to 5 P3/2F= 4). fine levels) are stored in the trap in a level |g1/angbracketrightthat is coupled to the upper (excited) state, |e/angbracketright, by an extremely weak transition. An atom that undergoes the weak tran- sition, may be shelved by a spontaneous Raman transi- tion on the second ground level, |g2/angbracketright, that is uncoupled to the excited level by the weak transition. After wait- ing long enough, a significant fraction of the atoms will be shelved on this second level. Finally, the detection scheme benefits from the multiply excited fluorescence of a strong closed transition from |g2/angbracketright, that utilizes quan- tum amplification due to the electron shelving technique. We realized this scheme on a 5 S1/2→5D5/2two- photon transition in cold and trapped85Rb atoms (see Fig. 1 for the relevant energy levels) using extremely weak (25 µW) laser beam and we were able to measure transitions with an excitation rate as small as 0.09 s−1. Precision spectroscopy of the two-photon transition in Rb atoms was previously demonstrated in a hot vapor 1with much higher laser power [11] [12]. In cold Rb atoms this transition was measured either on free expanding atoms using a mode-locked laser [13] [14] or on atoms trapped in a doughnut mode magneto-optical trap [15]. In all those schemes the fluorescent 420 nm photons were used to detect the two-photon transition. Our spectroscopic measurement was made on cold 85Rb atoms trapped in a rotating-beam optical trap (ROBOT). The operation principles of the ROBOT are described elsewhere [16]. Briefly, a linearly polar- ized, tightly focused (16 µm 1/e2radius) Gaussian laser beam is rapidly (100 kHz) rotated by two perpendicular acousto-optic scanners, as seen in Fig. 2. This forms a dark volume which is completely surrounded by a time- averaged dipole potential walls. The wavelength of the trapping laser was 770 nm (10 nm above the D 2line) and its power was 380 mW. The initial radius of the rotation was optimized for efficient loading of the ROBOT from a magneto-optical trap (MOT). 700 ms of loading, 47 ms of compression and 3 ms of polarization gradient cooling produced a cloud of ∼3×108atoms, with a temperature of 9µK and a peak density of 1 .5×1011cm−3. On the last stage of the loading procedure, the atoms were opti- cally pumped into the F= 2 ground state by shutting off the repumping laser 1 ms before shutting off the MOT beams. After all laser beams were shut off (except for the ROBOT beam which was overlapping the center of the MOT), ∼3·105atoms were typically loaded into the trap, with temperature and density comparable with those of the MOT. Next, we adiabatically compressed the trap by reducing the radius of rotation of the trapping beam from 70 µm to 29 µm such that the atoms will match the waist of the two-photon laser, to further increase the efficiency of the transition. The size of the final cloud in the radial direction was measured by absorption imaging and the temperature of the atoms was measured by time of flight fluorescence imaging. From these measurements and using our precise characterization of the trapping potential [16], the parameters of the final cloud are: ra- dial size (1 /e2radius) of 19 µm, axial size of 750 µm, rms temperatures of 55 µK [9µK] in the radial [axial] direction, and a density of 7 ·1011atoms/cm−3. The 1 /e lifetime of atoms in the trap was measured to be 350 ms for both hyperfine ground-states and was limited by col- lisions with background atoms. We measured the spin relaxation time of the trapped atoms to be >1 s, by measuring spontaneous Raman scattering between the two ground state levels [17] [7]. The spectroscopy was performed with an external- cavity diode laser which was tuned to the 5 S1/2F= 3→ 5D5/2F′two-photon transition (777 .9 nm) and was split into two parts. The first part (10 mW) was used to fre- quency stabilize the laser using the 420 nm fluorescence signal from the two-photon excitation obtained from a 1300CRb vapor cell. The laser was focused into the cellM O T T r a p b e a m M O T a n d r e p u m p i n g b e a m s D e t e c t i o n b e a m T w o p h o t o n b e a mA O S A O Sλ 2 ( = 7 8 0 . 2 4 n m )λ ( = 7 7 7 . 9 n m )λ ( = 7 7 0 n m )λ P B SP B S FIG. 2. Schematic diagram of the experimental setup. Two acousto-optic scanners (AOS) rotate a 10 nm blue-detuned laser beam that produce the ROBOT trap. The two-photon beam and the detection beam are co-aligned with the elon- gated axis of the trap. 2to∼100µm 1/e2radius and reflected back to obtain Doppler-free spectra. We locked the laser to the atomic line either by Zeeman modulation technique [18] or di- rectly to the side of the line. From the locking signal we estimated the peak-to-peak frequency noise of the laser to be ∼3 MHz. The second part of the diode laser beam passed through an acousto-optic modulator that shifted the laser frequency toward two-photon resonance with the 5S1/2F= 2→5D5/2F′transition. The laser beam was then focused to a 26 µm(1/e2radius) spot size in the center of the vacuum chamber, in order to optimize the efficiency of the two-photon transition and was carefully aligned with the long (axial) axis of the ROBOT. We used a normalized detection scheme to measure the fraction of atoms transferred to F= 3 by the two-photon laser. To detect the total number of atoms in the trap we applied a strong 200 µs laser pulse, resonant with the 5S1/2F= 3 →5P3/2F= 4 closed transition together with the repumping laser and imaged the fluorescent sig- nal on photomultiplier tube (PMT). To measure only the F= 3 population we applied the detection pulse without the repumping laser. The F= 3 atoms were simultane- ously accelerated and Doppler-shifted from resonance by the radiation pressure of the detection beam within the first 100 µs of the pulse. Then we could detect the F= 2 atoms by switching on the repumping laser that pumped F= 2 population to the F= 3 state where atoms were measured by the second part of the detection pulse. This normalized detection scheme is insensitive to shot-to-sho t fluctuations in atom number as well as fluctuations of the detection laser frequency and intensity. After the adiabatic compression of the atoms in the ROBOT was completed, the two-photon laser on reso- nance with 5 S1/2F= 2→5D5/2F= 4 was applied for various time intervals and the resulting F= 3 normalized population fraction was detected. The results for a 170 µW two-photon laser are presented on Fig. 3. After 100 ms,∼85% of the atoms are pumped to the F= 3 state. This steady state population is less then 100% since spon- taneous Raman scattering from the trapping laser and from the two-photon laser (absorption of onephoton fol- lowed by spontaneous emission ) tend to equalize the pop- ulations of the two ground levels and therefore compete with the measured two-photon process. The characteris- tic 1/etime of the four-photon spontaneous Raman scat- tering process which is induced by the two-photon laser (5S1/2F= 2→5D5/2F′→6P3/2F′→5S1/2F= 3, see Fig. 1) is obtained from a fit to the data as τ4p= 25 ms. The corresponding (four-photon) rate is γ4p= 1/τ4p= 40 s−1. Using the theoretical value of the two-photon cross- section of σ= 0.57×10−18cm4/W [19], the exact branch- ing ratio (68%) for the two-photon excitation to decay to F= 3 [20], and our maximal excitation laser intensity of 16 W/cm2we calculate γ4p= 391 s−1, a factor of ∼10 larger than the measured rate. Using a measured value for the two-photon cross-section [21] yields a somewhat FIG. 3. F= 3 normalized population fraction as a func- tion of the interrogation time of the 170 µW two-photon laser tuned to resonance with the 5 S1/2F= 2→5D5/2F= 4 line (/squaresolid). The solid line is a fit of the measurements by the func- tionNF=3/Ntotal=A(1−e−t/τ4p), resulting A= 0.85 as the steady state population, and τ4p= 25 ms as the four-photon spontaneous Raman scattering time (see text). Spontaneous Raman scattering rate caused by trapping laser is also given (/trianglesolid). larger value of γ4p= 823 s−1. The main factor that reduced the measured excitation rate was the linewidth of the two-photon laser that was ∼6 times larger than the 300 kHz natural linewidth of the two-photon transition [12]. The inhomogeneous broadening due to Stark-shift was calculated for the com- pressed trapping potential to be ∼400 kHz, which is smaller than the laser linewidth , hence it does not con- tribute to the reduced excitation rate. An additional re- duction of the excitation rate may be caused by imperfect matching between the trapped atomic sample and the maximal intensity of the two-photon laser, so the overall agreement between the measured and the expected γ4p is reasonable. To measure the excitation spectrum of the 5 S1/2F= 2→5D5/2F′transition we scanned the frequency of the two-photon laser using the acousto-optic modulator. For each frequency point the whole experimental cycle was repeated, with 50 ms interrogation time of the two- photon laser. The F= 3 fraction of atoms as a function of the frequency of the two-photon laser is presented in Fig. 4a. A 1 .75 MHz linewidth (FWHM) of the atomic lines was determined by fitting the data with a multi- peak Gaussian function and is limited by the linewidth of the two-photon laser. This measurement agrees well with the frequency noise of the laser estimated from the locking signal. The distances between the lines obtained from this fit are 4 .48 MHz, 3 .76 MHz and 2 .76 MHz, and are in excellent agreement with previously reported values of 4 .50 MHz, 3 .79 MHz and 2 .74 MHz [12]. The 3FIG. 4. A: Frequency scan of the 5 S1/2F= 2→5D5/2F′= 4,3,2,1 line of the two-photon transition, after 50 ms exposure to a 170µW two-photon laser. The solid line is a fit to the data by a multi-peak Gaussian function (see text). B: The same frequency scan as in (A), after 500 ms exposure to a 25 µW two-photon laser. (The dashed curve connects the points and is given to guide the eye). height-ratio between the lines obtained from the fit are 1 : 0.86 : 0.47 : 0.21 for F′= 4,3,2,1 respectively. The expected values were calculated using the strength of the two-photon transitions [12] together with the two photon decay via the 6 P3/2level [20] to be 1 : 0 .85 : 0.4 : 0.1, in good agreement with the measured values, except for the weakest line. Note that although the two-photon tran- sition F= 2→F′= 0 is allowed, a two-photon decay with ∆ F= 3 is forbidden and therefore this line is not detected. Finally, we reduced the power of the two-photon laser to 25 µW, which reduced the transition rate by a factor of 46. Here, the interrogation time of the two-photon laser was 500 ms and the measured F= 3 population is shown in Fig. 4b. A spectrum similar to that taken with higher intensity is observed. A transition rate as small as0.09 s−1(for the F= 3→F′= 1 transition) is detected in this scan. The ”quantum rate amplification” due to electron shelving (the ratio between the measured γ4p transition rate and the rate of the one-photon transition used for detecting the F=3 population) is ∼107for this case. In conclusion, we demonstrate a new and extremely sensitive scheme to measure weak transitions using cold atoms. The key issues in our scheme are the long spin relaxation times combined with tight confinement of the atoms in a dark optical dipole trap, and the use of a shelving technique to enhance the signal to noise ratio. We demonstrated our scheme by measuring a two-photon transition 5 S1/2→5D5/2for85Rb atoms trapped in a far-detuned rotating beam dark trap using only 25 µW laser power. The huge quantum amplification due to elec- tron shelving increases the sensitivity of our scheme far beyond the photon shot noise and technical noise encoun- tered in the direct detection of two-photon induced fluo- rescence [11] [12] [13] [21]. Our measurements may be improved in several ways. Improvements of the lifetime and spin relaxation time of atoms in the trap will allow much longer observation times and enable detection of much weaker transitions. This can be done by increasing the trapping laser de- tuning, where even longer spin-relaxation times are ex- pected due to quantum interference between the two D lines [17]. Reduction of the linewidth of the two-photon laser will allow further improvements in the sensitivity of our scheme. It can also be combined with mode-locked laser spectroscopy [13] to obtain even larger sensitivitie s for a given time-average power of the laser. Finally, our technique can be applied for other weak (forbidden) tran- sitions such as optical clock transitions [3] [10] and parit y violating transitions where a much lower mixing with an allowed transition could be used. [1] K. Gibble and S. Chu, Metrologia 29, 201 (1992). [2] M. A. Kasevich, E. Riis, S. Chu, and R. G. DeVoe, Phys. Rev. Lett. 63, 612 (1989). [3] F. Ruschewitz, J. L. Peng, H. Hinderth¨ ur, N. Schaffrath, K. Sengstock, and W. Ertmer, Phys. Rev. Lett. 80, 3173 (1998). [4] S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57, 314 (1986). [5] J. D. Miller, R. A. Cline, and D. J. Heinzen, Phys. Rev. A47, R4567 (1993). [6] N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, Phys. Rev. Lett. 74, 1311 (1995). [7] R. Ozeri, L. Khaykovich, and N. Davidson, Phys. Rev. A 59, R1750 (1999). [8] N. Friedman, L. Khaykovich, R. Ozeri, and N. Davidson, 4OPN, Optics in 99, in press (1999). [9] See, e.g., W. Nagorney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56, 2797 (1986); D. J. Winland, J. C. Bergquist, W. M. Itano, and R. E. Drullinger, Opt. Lett. 5, 245 (1980). [10] T. Kurosu, G. Zinner, T. Trebst, and F. Riehle, Phys. Rev. A 58, R4275 (1998). [11] W. Zapka, M. D. Levenson, F. M. Schellenberg, A. C. Tam, and G. C. Bjorklund, Opt. Lett. 8, 27 (1983). [12] F. Nez, F. Biraben, R. Felder, Y. Millerioux, Optics Comm. 102, 432 (1993). [13] M. J. Snadden, A. S. Bell, E. Riis, and A. I. Ferguson, Optics Comm. 125, 70 (1996). [14] M. J. Snadden, R. B. M. Clarke, and E. Riis, Optics Comm. 152, 283 (1998). [15] M. J. Snadden, A. S. Bell, R. B. M. Clarke, E. Riis, and D. H. McIntyre, J. Opt. Soc. Am. B 14, 544 (1997). [16] L. Khaykovich, N. Friedman, R. Ozeri, and N. Davidson, Technical digest QELS’99, QPD11-1 (Baltimore, MD, 1999). [17] R. A. Cline, J. D. Miller, M. R. Matthews, and D. J. Heinzen, Opt. Lett. 19, 207 (1994). [18] T. Yabuzaki, T. Kawamura, and T. Ogawa, Abstr. 10th Int. Conf. Atomic Physics, p. 184, (Tokyo, 1986); A. Weis and S. Derler, Appl. Opt. 27, 2662 (1988). [19] M. Marinescu, V. Florescu, and D. Dalgarno, Phys. Rev. A49, 2714 (1994). [20] I. I. Sobelman, Atomic Spectra and Radiative Transitions (Springer-Verlag, Berlin, 1979). [21] C. L. A. Collins, K. D. Bonin, and M. A. Kadar-Kallen, Opt. Lett. 18, 1754 (1993). 5

arXiv:physics/9912003v1 [physics.chem-ph] 2 Dec 1999A simple scaling law between the total energy of a free atom and its atomic number W. T. Geng Department of Physics & Astronomy, Northwestern Universit y, Evanston, IL 60208 A simple, approximate relation is found between the total en ergy of a free atom and its atomic number: E≃ −Z2.411. The existence of this index is inherent in the Coulomb and ma ny-body nature of the electron-electron interaction in the atomic s ystem and cannot be fabricated from the existing fundamental physical constants. In a recent work on the calculation of the cohesive energy of elemental crystals, [1] we have calculated the to- tal energy for all atoms with Z ≤92. Our calculations were based on the density functional theory with the local den- sity approximation. [2] Parameterization of the exchange- correlation interaction is that of Hedin-Lundqvist. [3] Fo r the first time, we plotted the total energy ( E) versus the atomic number ( Z) curve (Fig. 1), in an attempt to gain some physical insight into the density functional theory. Surprisingly, it is found that this curve can be very well fitted by a scaling law E=−Zn, n= 2.411 To make the E∼Zrelation more illustrative, the n∼Zcurve is plotted in Fig. 2 (down triangles)). The power index nis almost constant (close to 2.41) for atoms with 4 ≤Z≤92. If there is no interaction between elec- trons, n= 3; and if there is only one electron outside this nucleus, n= 2. Since the electron-electron interaction increases the total energy (i.e., less negative), nshould meet 2 < n < 3. The existence of such a near-constant power index is astounding because, due to the complex- ity of the quantum many-body problem, it’s never been expected that the total energy of an atom other than hy- drogen should have so simple a relation with its atomic number. Exceptions occur in the cases of hydrgen, helium, and lithium. For hydrogen, Z= 1 and E=−0.976Ry, therefore, nhas no definite value. For helium and lithium, n=2.506 and 2.447, respectively, apparently larger than 2.41. Experimental data [4], which is non- relativistic and available up to argon, are denoted by triangles in Fig.2. Also listed are the calculated power index from Desclaux’s Hartree-Fock atomic total energy data. [5] Open circles represent non-relativistic treatme ntand solid circles denote relativistic treatment. It is seen that all these four groups of nhave values with very lim- ited diviation. It’s then concluded that the scaling law is not an outcome of the density functional theory, where both the exchange and correlation interactions are con- sidered, nor a result of the Hartree-Fock approximation, in which only the exchange interaction is counted. Al- though relativistic effects makea difference in the index n, the approximate scaling law holds for both cases. As ionization potentials show very strong effects of chemical periodicity, it is of much interest to see whether they exert a periodic effect on the atomic total energy too. We replot the n∼Zcurves in Fig.3, a higher res- olution graph. nshows apparent oscillatory behavior for atoms lighter than krypton. But for heavier atoms, it dis- plays monotonic character. This is due to the fact that the ionization potentials are so small as to be averaged out for the heavy atoms. It is worth noting that the solid circle denoting Desclaux’s relativistic francium falls ou t of the otherwise smooth curve. There must be an abrupt mistake, probably a typo, in the reported total atomic total energy of francium. From the comparisons between density functional theory and Hartree-Fock approximation, relativistic and non-relativistic treatments, we can conclude that the ex- istence of such a simple relation between the total energy of an atom and its atomic number is independent of the framework in which the calculations of atomic total en- ergy are carried out. It is inherent in the Coulomb and many-body nature of the atomic system. Apparently, this power index cannot be fabricated from the existent fundamental physical constants such as ¯ h,c,e, etc., and can only be built into a new many-body quantum theory. 1ACKNOWLEDGMENTS The author acknowledges helpful discussions with Professors Ding-Sheng Wang and A. J. Freeman. [1] W. T. Geng, Ph.D Thesis, Institute of Physics, Chinese Academy of Sciences, Beijing 1998. [2] P. Hohenberg and W. Kohn, Phys. Rev. 136, 864B (1964); W. Kohn and L. J. Sham, ibid.140, 1133A (1965). [3] L. Hedin and B. Lundqvist, J. Phys. C 4, 2064 (1971); U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972). [4] R. C. Weast, Ed., CRC Handbook of Chemistry and Physics, 58th ed., Chemical Rubber Co., Cleverland, 1977. [5] J. P. Desclaux, Atomic Data and Nuclear Data Tables 12, 311-406 (1973). FIG. 1. Atomic total energy (Ry) given by density func- tional theory. FIG. 2. Calculated power index nin relation E=−Zn. Down triangles are our results from density functional the- ory; open (non-relativistic) and solid (relativistic) cir cles are calculated from Desclaux’s Hartree-Fock data; triangles d e- note experimental data (non-relativistic). FIG. 3. A replot of Fig.2 with higher resolution. 20.0-10000.0-20000.0-30000.0-40000.0-50000.0-60000.0 0 20 40 60 80 100DFT Atomic Total Energy (Ry) Atomic Number Fig.10 20 40 60 80 1002.02.22.42.62.83.0 UHeFr KrPower Index n Atomic Number Z Fig.20 20 40 60 80 1002.382.402.422.442.462.482.502.52 UUHe Fr KrPower Index n Atomic Number Z Fig.3

arXiv:physics/9912005v1 [physics.data-an] 2 Dec 1999Bayesian Field Theory Nonparametric Approaches to Density Estimation, Regression, Classification, and Inverse Quantum Problems J¨ org C. Lemm∗ Institut f¨ ur Theoretische Physik I Universit¨ at M¨ unster Wilhelm–Klemm–Str.9 D–48149 M¨ unster, Germany Abstract Bayesian field theory denotes a nonparametric Bayesian appr oach for learning functions from observational data. Based on th e principles of Bayesian statistics, a particular Bayesian field theory i s defined by combining two models: a likelihood model, providing a proba bilistic description of the measurement process, and a prior model, p roviding the information necessary to generalize from training to no n–training data. The particular likelihood models discussed in the pap er are those of general density estimation, Gaussian regression, clust ering, classi- fication, and models specific for inverse quantum problems. B esides problem typical hard constraints, like normalization and p ositivity for probabilities, prior models have to implement all the sp ecific, and often vague, a priori knowledge available for a specific task. Nonpara- metric prior models discussed in the paper are Gaussian proc esses, mixtures of Gaussian processes, and non–quadratic potenti als. Prior models are made flexible by including hyperparameters. In pa rticular, ∗Email: lemm@uni-muenster.de, WWW: http://pauli.uni-mue nster.de/∼lemm/ 1the adaption of mean functions and covariance operators of G aussian process components is discussed in detail. Even if construc ted using Gaussian process building blocks, Bayesian field theories a re typically non–Gaussian and have thus to be solved numerically. Accord ing to in- creasing computational resources the class of non–Gaussia n Bayesian field theories of practical interest which are numerically f easible is steadily growing. Models which turn out to be computational ly too demanding can serve as starting point to construct easier to solve parametric approaches, using for example variational tech niques. Contents 1 Introduction 5 2 Bayesian framework 9 2.1 Basic model and notations . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Independent, dependent, and hidden variables . . . . . 9 2.1.2 Energies, free energies, and errors . . . . . . . . . . . . 11 2.1.3 Posterior and likelihood . . . . . . . . . . . . . . . . . 13 2.1.4 Predictive density . . . . . . . . . . . . . . . . . . . . . 15 2.1.5 Mutual information and learning . . . . . . . . . . . . 16 2.2 Bayesian decision theory . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Loss and risk . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Loss functions for approximation . . . . . . . . . . . . 21 2.2.3 General loss functions and unsupervised learning . . . 24 2.3 Maximum A Posteriori Approximation . . . . . . . . . . . . . 25 2.4 Normalization, positivity, and specific priors . . . . . . . . . . 28 2.5 Empirical risk minimization . . . . . . . . . . . . . . . . . . . 31 2.6 Interpretations of Occam’s razor . . . . . . . . . . . . . . . . . 33 2.7A priori information and a posteriori control . . . . . . . . . . 34 3 Gaussian prior factors 39 3.1 Gaussian prior factor for log–probabilities . . . . . . . . . . . 39 3.1.1 Lagrange multipliers: Error functional EL. . . . . . . 39 3.1.2 Normalization by parameterization: Error functiona lEg44 3.1.3 The Hessians HL,Hg. . . . . . . . . . . . . . . . . . . 45 3.2 Gaussian prior factor for probabilities . . . . . . . . . . . . . . 47 3.2.1 Lagrange multipliers: Error functional EP. . . . . . . 47 23.2.2 Normalization by parameterization: Error functiona lEz49 3.2.3 The Hessians HP,Hz. . . . . . . . . . . . . . . . . . . 50 3.3 General Gaussian prior factors . . . . . . . . . . . . . . . . . . 51 3.3.1 The general case . . . . . . . . . . . . . . . . . . . . . 51 3.3.2 Example: Square root of P. . . . . . . . . . . . . . . . 53 3.3.3 Example: Distribution functions . . . . . . . . . . . . . 54 3.4 Covariances and invariances . . . . . . . . . . . . . . . . . . . 55 3.4.1 Approximate invariance . . . . . . . . . . . . . . . . . 55 3.4.2 Approximate symmetries . . . . . . . . . . . . . . . . . 56 3.4.3 Example: Infinitesimal translations . . . . . . . . . . . 57 3.4.4 Example: Approximate periodicity . . . . . . . . . . . 58 3.5 Non–zero means . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6 Quadratic density estimation and empirical risk minimi zation 61 3.7 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.7.1 Gaussian regression . . . . . . . . . . . . . . . . . . . . 64 3.7.2 Exact predictive density . . . . . . . . . . . . . . . . . 71 3.7.3 Gaussian mixture regression (cluster regression) . . . . 74 3.7.4 Support vector machines and regression . . . . . . . . . 75 3.8 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.9 Inverse quantum mechanics . . . . . . . . . . . . . . . . . . . 77 4 Parameterizing likelihoods: Variational methods 81 4.1 General parameterizations . . . . . . . . . . . . . . . . . . . . 81 4.2 Gaussian priors for parameters . . . . . . . . . . . . . . . . . . 83 4.3 Linear trial spaces . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Mixture models . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Additive models . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.6 Product ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.7 Decision trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.8 Projection pursuit . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.9 Neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5 Parameterizing priors: Hyperparameters 93 5.1 Prior normalization . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2 Adapting prior means . . . . . . . . . . . . . . . . . . . . . . . 98 5.2.1 General considerations . . . . . . . . . . . . . . . . . . 98 5.2.2 Density estimation . . . . . . . . . . . . . . . . . . . . 98 5.2.3 Unrestricted variation . . . . . . . . . . . . . . . . . . 99 35.2.4 Regression . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 Adapting prior covariances . . . . . . . . . . . . . . . . . . . . 102 5.3.1 General case . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3.2 Automatic relevance detection . . . . . . . . . . . . . . 103 5.3.3 Local smoothness adaption . . . . . . . . . . . . . . . . 104 5.3.4 Local masses and gauge theories . . . . . . . . . . . . . 105 5.3.5 Invariant determinants . . . . . . . . . . . . . . . . . . 106 5.3.6 Regularization parameters . . . . . . . . . . . . . . . . 108 5.4 Exact posterior for hyperparameters . . . . . . . . . . . . . . 10 9 5.5 Integer hyperparameters . . . . . . . . . . . . . . . . . . . . . 114 5.6 Local hyperfields . . . . . . . . . . . . . . . . . . . . . . . . . 115 6 Non–Gaussian prior factors 121 6.1 Mixtures of Gaussian prior factors . . . . . . . . . . . . . . . . 12 1 6.2 Prior mixtures for density estimation . . . . . . . . . . . . . . 1 23 6.3 Prior mixtures for regression . . . . . . . . . . . . . . . . . . . 123 6.3.1 High and low temperature limits . . . . . . . . . . . . 125 6.3.2 Equal covariances . . . . . . . . . . . . . . . . . . . . . 127 6.3.3 Analytical solution of mixture models . . . . . . . . . . 128 6.4 Local mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.5 Non–quadratic potentials . . . . . . . . . . . . . . . . . . . . . 134 7 Iteration procedures: Learning 137 7.1 Numerical solution of stationarity equations . . . . . . . . . . 137 7.2 Learning matrices . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.1 Learning algorithms for density estimation . . . . . . . 1 40 7.2.2 Linearization and Newton algorithm . . . . . . . . . . 141 7.2.3 Massive relaxation . . . . . . . . . . . . . . . . . . . . 142 7.2.4 Gaussian relaxation . . . . . . . . . . . . . . . . . . . . 146 7.2.5 Inverting in subspaces . . . . . . . . . . . . . . . . . . 147 7.2.6 Boundary conditions . . . . . . . . . . . . . . . . . . . 148 7.3 Initial configurations and kernel methods . . . . . . . . . . . . 150 7.3.1 Truncated equations . . . . . . . . . . . . . . . . . . . 150 7.3.2 Kernels for L. . . . . . . . . . . . . . . . . . . . . . . 151 7.3.3 Kernels for P. . . . . . . . . . . . . . . . . . . . . . . 153 7.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4.1 Density estimation with Gaussian specific prior . . . . 1 54 7.4.2 Density estimation with Gaussian mixture prior . . . . 1 59 41 Introduction The last decade has seen a rapidly growing interest in learni ng from observa- tional data. Increasing computational resources enabled s uccessful applica- tions of empirical learning algorithms in various areas inc luding, for example, time series prediction, image reconstruction, speech reco gnition, computer to- mography, and inverse scattering and inverse spectral prob lems for quantum mechanical systems. Empirical learning, i.e., the problem of finding underly- ing general laws from observations, represents a typical in verse problem and is usually ill–posed in the sense of Hadamard [204, 205, 208, 137, 107, 210]. It is well known that a successful solution of such problems req uires additional a priori information. It is a priori information which controls the general- ization ability of a learning system by providing the link be tween available empirical “training” data and unknown outcome in future “te st” situations. We will focus mainly on nonparametric approaches, formulat ed directly in terms of the function values of interest. Parametric meth ods, on the other hand, impose typically implicit restrictions which are oft en extremely diffi- cult to relate to available a priori knowledge. Combined with a Bayesian framework [10, 14, 30, 136, 187, 161, 16, 64, 196, 97], a nonpa rametric ap- proach allows a very flexible and interpretable implementat ion of a priori information in form of stochastic processes. Nonparametri c Bayesian meth- ods can easily be adapted to different learning situations an d have there- fore been applied to a variety of empirical learning problem s, including re- gression, classification, density estimation and inverse q uantum problems [157, 220, 134, 133, 129, 206]. Technically, they are relate d to kernel and reg- ularization methods which often appear in the form of a rough ness penalty approach [205, 208, 177, 195, 141, 212, 83, 78, 108, 210]. Com putationally, working with stochastic processes, or discretized version s thereof, is more demanding than, for example, fitting a small number of parame ters. This holds especially for such applications where one cannot tak e full advantage of the convenient analytical features of Gaussian processe s. Nevertheless, it seems to be the right time to study nonparametric Bayesian ap proaches also for non–Gaussian problems as they become computationally f easible now at least for low dimensional systems and, even if not directly s olvable, they provide a well defined basis for further approximations. In this paper we will in particular study general density est imation prob- lems. Those include, as special cases, regression, classifi cation, and certain types of clustering. In density estimation the functions of interest are the 5probability densities p(y|x,h), of producing output (“data”) yunder con- ditionxand unknown state of Nature h. Considered as function of h, for fixedy,x, the function p(y|x,h) is also known as likelihood function and a Bayesian approach to density estimation is based on a probab ilistic model for likelihoods p(y|x,h). We will concentrate on situations where yandxare real variables, possibly multi–dimensional. In a nonparam etric approach, the variablehrepresents the whole likelihood function p(y|x,h). That means, h may be seen as the collection of the numbers 0 ≤p(y|x,h)≤1 for allxand ally. The dimension of his thus infinite, if the number of values which the variablesxand/orycan take is infinite. This is the case for real xand/ory. A learning problem with discrete yvariable is also called a classifica- tion problem . Restricting to Gaussian probabilities p(y|x,h) with fixed vari- ance leads to (Gaussian) regression problems . For regression problems the aim is to find an optimal regression function h(x). Similarly, adapting a mixture of Gaussians allows soft clustering of data points. Furthermore, extracting relevant features from the predictive density p(y|x,data) is the Bayesian analogon of unsupervised learning . Other special density estimation problems are, for example, inverse problems in quantum mechanics whereh represents a unknown potential to be determined from observ ational data [134, 133, 129, 206]. Special emphasis will be put on the expl icit and flexible implementation of a priori information using, for example, mixtures of Gaus- sian prior processes with adaptive, non–zero mean function s for the mixture components. Let us now shortly explain what is meant by the term “Bayesian Field Theory”: From a physicists point of view functions, like h(x,y) =p(y|x,h), depending on continuous variables xand/ory, are often called a ‘field’.1 Most times in this paper we will, as common in field theories in physics, not parameterize these fields and formulate the relevant probab ility densities or stochastic processes, like the prior p(h) or the posterior p(h|f), directly in terms of the field values h(x,y), e.g.,p(h|f) =p(h(x,y),x∈X,y∈Y|f). (In the parametric case, discussed in Chapter 4, we obtain a prob ability density p(h|f) =p(ξ|f) for fieldsh(x,y,ξ) parameterized by ξ.) The possibility to solve Gaussian integrals analytically m akes Gaussian processes, or (generalized) free fields in the language of ph ysicists, very at- 1We may also remark that for example statistical field theorie s, which encompass quan- tum mechanics and quantum field theory in their Euclidean for mulation, are technically similar to a nonparametric Bayesian approach [232, 96, 118] . 6tractive for nonparametric learning. Unfortunately, only the case of Gaussian regression is completely Gaussian. For general density est imation problems the likelihood terms are non–Gaussian, and even for Gaussia n priors addi- tional non–Gaussian restrictions have to be included to ens ure positivity and normalization of densities. Hence, in the general case, den sity estimation corresponds to a non–Gaussian, i.e., interacting field theo ry. As it is well known from physics, a continuum limit for non-Ga ussian the- ories, based on the definition of a renormalization procedur e, can be highly nontrivial to construct, if possible at all. We will in the fo llowing not dis- cuss such renormalization procedures but focus more on prac tical, numerical learning algorithms, obtained by discretizing the problem (typically, but not necessarily in coordinate space). This is similar, for exam ple, to what is done in lattice field theories. Gaussian problems live effectively in a space with dimension not larger than the number of training data. This is not the case for non– Gaussian problems. Hence, numerical implementations of learning al gorithms for non– Gaussian problems require to discretize the functions of in terest. This can be computationally challenging. For low dimensional problems, however, many non–Gaussian m odels are nowadays solvable on a standard PC. Examples include predic tions of one– dimensional time series or the reconstruction of two–dimen sional images. Higher dimensional problems require additional approxima tions, like projec- tions into lower dimensional subspaces or other variationa l approaches. In- deed, it seems that a most solvable high dimensional problem s live effectively in some low dimensional subspace. There are special situations in classification where positi vity and normal- ization constraints are fulfilled automatically. In that ca se, the calculations can still be performed in a space of dimension not larger than the number of training data. Contrasting Gaussian models, however the equations to be solved are then typically nonlinear. Summarizing, we will call a nonparametric Bayesian model to learn a function one or more continuous variables a Bayesian field theory , having especially in mind non–Gaussian models. A large variety of B ayesian field theories can be constructed by combining a specific likeliho od models with specific functional priors (see Tab. 1). The resulting flexib ility of nonpara- metric Bayesian approaches is probably their main advantag e. 7likelihood model prior model describes measurement process (Chap. 2) generalization behavior (Chap. 2) is determined by parameters (Chap. 3, 4) hyperparameters (Chap. 5) Examples include density estimation (Chap. 3) hard constraints (Chap. 2) regression (Chap. 3) Gaussian prior factors (Chap. 3) classification (Sect. 3) mixtures of Gaussians (Sect. 6) inverse quantum theory (Sect. 3) non–quadratic potentials (Sect. 6) Table 1: A Bayesian approach is based on the combination of tw o models, a likelihood model, describing the measurement process use d to obtain the training data, and a prior model, enabling generalization t o non–training data. Parameters of the prior model are commonly called hype rparameters. In “nonparametric” approaches the collection of all values of the likelihood function itself are considered as the parameters. A nonpara metric Bayesian approach for likelihoods depending on one or more real varia bles is in this paper called a Bayesian field theory. (Learning is treated in Chapter 7.) 8The paper is organized as follows: Chapter 2 summarizes the B ayesian framework as needed for the subsequent chapters. Basic nota tions are de- fined, an introduction to Bayesian decision theory is given, and the role of a priori information is discussed together with the basics of a Maxim um A Posteriori Approximation (MAP), and the specific constrai nts for density estimation problems. Gaussian prior processes, being the m ost commonly used prior processes in nonparametric statistics, are trea ted in Chapter 3. In combination with Gaussian prior models, this section als o introduces the likelihood models of density estimation, (Sections 3.1, 3. 2, 3.3) Gaussian re- gression and clustering (Section 3.7), classification (Sec tion 3.8), and inverse quantum problems (Section 3.9). Notice, however, that all t hese likelihood models can also be combined with the more elaborated prior mo dels dis- cussed in the following sections of the paper. Parametric ap proaches, useful if a numerical solution of a full nonparametric approach is n ot feasible, are the topic of Chapter 4. Hyperparameters, parameterizing pr ior processes and making them more flexible, are considered in Section 5. Tw o possibil- ities to go beyond Gaussian processes, mixture models and no n–quadratic potentials, are presented in Section 6. Chapter 7 focuses on learning algo- rithms, i.e., on methods to solve the stationarity equation s resulting from a Maximum A Posteriori Approximation. In this section one ca n also find numerical solutions of Bayesian field theoretical models fo r general density estimation. 2 Bayesian framework 2.1 Basic model and notations 2.1.1 Independent, dependent, and hidden variables Constructing theories means introducing concepts which ar e not directly ob- servable. They should, however, explain empirical findings and thus have to be related to observations. Hence, it is useful and common to distinguish observable (visible) from non–observable (hidden) variab les. Furthermore, it is often convenient to separate visible variables into de pendent variables, representing results of such measurements the theory is aim ing to explain, and independent variables, specifying the kind of measurem ents performed and not being subject of the theory. Hence, we will consider the following three groups of variab les 91. observable (visible) independent variables x, 2. observable (visible) dependent variables y, 3. not directly observable (hidden, latent) variables h. This characterization of variables translates to the follo wing factorization property, defining the model we will study, p(x,y,h ) =p(y|x,h)p(x)p(h). (1) In particular, we will be interested in scenarios where x= (x1,x2,···) and analogously y= (y1,y2,···) are decomposed into independent components, meaning that p(y|x,h) =/producttext ip(yi|xi,h) andp(x) =/producttext ip(xi) factorize. Then, p(x,y,h ) =/productdisplay ip(yi|xi,h)p(xi)p(h). (2) Fig.1 shows a graphical representation of the factorizatio n model (2) as a directed acyclic graph [172, 117, 99, 186]. The xiand/oryiitself can also be vectors. The interpretation will be as follows: Variables h∈Hrepresent possible states of (the model of) Nature , being the invisible conditions for dependent variablesy. The setHdefines the space of all possible states of Nature for the model under study. We assume that states hare not directly observable and all information about p(h) comes from observed variables (data) y,x. A given set of observed data results in a state of knowledge fnumerically represented by the posterior density p(h|f) over states of Nature. Independent variables x∈Xdescribe the visible conditions (measure- ment situation, measurement device) under which dependent variables (mea- surement results) yhave been observed (measured). According to Eq. (1) they are independent of h, i.e.,p(x|h) =p(x). The conditional density p(y|x,h) of the dependent variables yis also known as likelihood ofh(undery givenx). Vector–valued ycan be treated as a collection of one–dimensional y with the vector index being part of the xvariable, i.e., yα(x) =y(x,α) =y(˜x) with ˜x= (x,α). In the setting of empirical learning available knowledge is usually sep- arated into a finite number of training data D={(xi,yi)|1≤i≤n} ={(xD,yD) and, to make the problem well defined, additional a priori in- formationD0. For dataD∪D0we writep(h|f) =p(h|D,D 0). Hypotheses h 10x1x2xn y1y2yn h··· ···❄ ❄ ❄ ▼✍■ ✒ Figure 1: Directed acyclic graph for the factorization mode l (1). represent in this setting functions h(x,y) =p(y|x,h) of two (possibly multi- dimensional) variables y,x. In density estimation yis a continuous variable (the variable xmay be constant and thus be skipped), while in classification problemsytakes only discrete values. In regression problems on assum es p(y|x,h) to be Gaussian with fixed variance, so the function of intere st be- comes the regression function h(x) =/integraltextdyyp(y|x,h). 2.1.2 Energies, free energies, and errors Often it will turn out to be convenient to work with log–proba bilities, un- normalized probabilities, or energies, instead of probabi lities. For example, the posterior p(h|f) can be written as p(h|f) =eL(h|f)=Z(h|f) Z(H|f)=e−βE(h|f) Z(H|f) =e−β(E(h|f)−F(H|f))=e−βE(h|f)+c(H|f), (3) with (posterior) log–probability L(h|f) = lnp(h|f), (4) unnormalized (posterior) probabilities or partition sums Z(h|f), Z (H|f) =/integraldisplay dhZ(h|f), (5) (posterior) energy E(h|f) =−1 βlnZ(h|f) (6) 11and (posterior) free energy F(H|f) = −1 βlnZ(H|f) (7) =−1 βln/integraldisplay dhe−βE(h|f), (8) yielding Z(h|f) =e−βE(h|f), (9) Z(H|f) =/integraldisplay dhe−βE(h|f), (10) where/integraltextdhrepresent a (functional) integral, for example over variab les (func- tions)h(x,y) =p(y|x,h), and c(H|f) =−lnZ(H|f) =βF(H|f). (11) Note that for the sake of simplicity we did not include the β–dependency of the functions Z,F,cin the notation. A central topic will be the maximization of the posterior (se e Section 2.3) which corresponds to minimizing the posterior energy E(h|f). Because in the context of regularization theory and empirical risk m inimization, an optimalh∗is obtained by minimizing an error functional we will often a lso refer to the posterior energy E(h|f) as (regularized) error functional forh. (see Section 2.5). Let us take a closer look to the integral over model states h. The variables hrepresent the parameters describing the data generating pr obabilities or likelihoods p(y|x,h). In this paper we will mainly be interested in “nonpara- metric” approaches where the ( x,y,h )–dependent numbers p(y|x,h) itself are considered to be the primary degrees of freedom which “pa rameterize” the model states h. Then, the integral over his an integral over a set of real variables indexed by x,y, under additional positivity and normalization condition./integraldisplay dh→/integraldisplay/parenleftigg/productdisplay x,ydp(y|x,h)/parenrightigg . (12) Mathematical difficulties arise for the case of continuous x,ywherep(h|f) represents a stochastic process. and the integral over hbecomes a functional integral over (non–negative and normalized) functions p(y|x,h). For Gaus- sian processes such a continuum limit can be defined [46, 72, 2 12, 135] while 12the construction of continuum limits for non–Gaussian proc esses is highly non–trivial (See for instance [43, 33, 96, 232, 174, 217, 218 , 31, 182] for per- turbative approaches or [72] for non–perturbative φ4–theory.) In this paper we will take the numerical point of view where all functions a re considered to be finally discretized, so the h–integral is well–defined (“lattice regular- ization” [37, 189, 150]). Varying the parameter βgenerates an exponential family of densities which is frequently used in practice by (simulated or determ inistic) annealing techniques for minimizing free energies [106, 144, 185, 38, 1, 188, 226, 63, 227, 228]. In physics βis known as inverse temperature and plays the role of a Lagrange multiplier in the maximum entropy approach to stat istical physics. Inverse temperature βcan also be seen as an external field coupling to the energy. Thus, the free energy F(H|f) (orc(H|f)) is a generating function for the cumulants of the energy, meaning that cumulants of Ecan be obtained by taking derivatives of Fwith respect to β[60, 7, 11, 150]. For the sake of clarity, we have chosen to use the common notat ion for conditional probabilities also for energies and the other q uantities derived from them. The same conventions will also be used for other pr obabilities, so we will write for example for likelihoods p(y|x,h) =e−β′(E(y|x,h)−F(Y|x,h)), (13) fory∈Y. Temperatures may be different for prior and likelihood. Thu s, we may choose β′∝ne}ationslash=βin Eq. (13) and Eq. (3). 2.1.3 Posterior and likelihood Bayesian approaches require the calculation of posterior d ensities. Model stateshare commonly specified by giving the data generating probabi lities or likelihoods p(y|x,h). Posteriors are linked to likelihoods by Bayes’ theorem p(A|B) =p(B|A)p(A) p(B), (14) which follows at once from the definition of conditional prob abilities, i.e., p(A,B) =p(A|B)p(B) =p(B|A)p(A). Thus, one finds p(h|f) =p(D|h)p(h|D0) p(D|D0)=p(yD|xD,h)p(h|D0) p(yD|xD,D0)(15) 13=/producttext ip(xi,yi|h)p(h|D0)/integraltextdh/producttext ip(xi,yi|h)p(h|D0)=/producttext ip(yi|xi,h)p(h|D0)/integraltextdh/producttext ip(yi|xi,h)p(h|D0), (16) usingp(yD|xD,D0,h) =p(yD|xD,h) for the training data likelihood of hand p(h|D0,xi) =p(h|D0). The terms of Eq. (15) are in a Bayesian context often referred to as posterior =likelihood ∗prior evidence. (17) Eqs.(16) show that the posterior can be expressed equivalen tly by the joint likelihoods p(yi,xi|h) or conditional likelihoods p(yi|xi,h). When working with joint likelihoods, a distinction between yandxvariables is not neces- sary. In that case xcan be included in yand skipped from the notation. If, however, p(x) is already known or is not of interest working with condi- tional likelihoods is preferable. Eqs.(15,16) can be inter preted as updating (or learning) formula used to obtain a new posterior from a gi ven prior prob- ability if new data Darrive. In terms of energies Eq. (16) reads, p(h|f) =e−β/summationtext iE(yi|xi,h)−βE(h|D0) Z(YD|xD,h)Z(H|D0)/integraldisplay dhZ(YD|xD,h)Z(H|D0) e−β/summationtext iE(yi|xi,h)−βE(h|D0), (18) where the same temperature 1 /βhas been chosen for both energies and the normalization constants are Z(YD|xD,h) =/productdisplay i/integraldisplay dyie−βE(yi|xi,h), (19) Z(H|D0) =/integraldisplay dhe−βE(h|D0). (20) The predictive density we are interested in can be written as the ratio of two correlation functions under p0(h), p(y|x,f) =<p(y|x,h)>H|f (21) =<p(y|x,h)/producttext ip(yi|xi,h)>H|D0 </producttext ip(yi|xi,h)>H|D0, (22) =/integraltextdhp(y|x,h)e−βEcomb /integraltextdhe−βEcomb(23) where<···>H|D0denotes the expectation under the prior density p0(h) =p(h|D0) and the combined likelihood and prior energy Ecombcollects the 14h–dependent energy and free energy terms Ecomb=/summationdisplay iE(yi|xi,h) +E(h|D0)−F(YD|xD,h), (24) with F(YD|xD,h) =−1 βlnZ(YD|xD,h). (25) Going from Eq. (22) to Eq. (23) the normalization factor Z(H|D0) appearing in numerator and denominator has been canceled. We remark that for continuous xand/orythe likelihood energy term E(yi|xi,h) describes an ideal, non–realistic measurement because re alistic measurements cannot be arbitrarily sharp. Considering the functionp(·|·,h) as element of a Hilbert space its values may be written as scal ar product p(x|y,h) = (vxy, p(·|·,h) ) with a function vxybeing also an element in that Hilbert space. For continuous xand/orythis notation is only formal as vxy becomes unnormalizable. In practice a measurement of p(·|·,h) corresponds to a normalizable v˜x˜y=/integraltextdy/integraltextdxϑ(x,y)vxywhere the kernel ϑ(x,y) has to ensure normalizability. (Choosing normalizable v˜x˜yas coordinates the Hilbert space ofp(·|·,h) is also called a reproducing kernel Hilbert space [170, 104 , 105, 212, 135].) The data terms then become p(˜yi|˜xi,h) =/integraltextdy/integraltextdxϑi(x,y)p(y,x|h)/integraltextdyϑi(x,y)p(y,x|h). (26) The notation p(yi|xi,h) is understood as limit ϑ(x,y)→δ(x−xi)δ(y−yi) and means in practice that ϑ(x,y) is very sharply centered. We will assume that the discretization, finally necessary to do numerical c alculations, will implement such an averaging. 2.1.4 Predictive density Within a Bayesian approach predictions about (e.g., future ) events are based on the predictive probability density , being the expectation of probability for yfor given (test) situation x, training data Dand prior data D0 p(y|x,f) =/integraldisplay dhp(h|f)p(y|x,h) (27) =<p(y|x,h)>H|f. (28) 15ˆ =p(y|x,hi), hi∈H p(y|x,htrue)✛ p(y|x,f)F✒ Figure 2: The predictive density p(y|x,f) for a state of knowledge f= f(D,D 0) is in the convex hull spanned by the possible states of Natur ehi characterized by the likelihoods p(y|x,hi). During learning the actual pre- dictive density p(y|x,f) tends to move stochastically towards the extremal pointp(y|x,htrue) representing the “true” state of Nature. Here<···>H|fdenotes the expectation under the posterior p(h|f) = p(h|D,D 0), the state of knowledge fdepending on prior and training data. Successful applications of Bayesian approaches rely stron gly on an adequate choice of the model space Hand model likelihoods p(y|x,h). Note thatp(y|x,f) = =/summationtext ip(y|x,hi)p(hi|f) is in the convex cone spanned by the possible states of Nature h∈H, and typically not equal to one of thesep(y|x,h). The situation is illustrated in Fig. 2. During learning th e predictive density p(y|x,f) tends to approach the true p(y|x,h). Because the training data are random variables, this approach is sto chastic. (There exists an extensive literature analyzing the stochastic pr ocess of learning and generalization from a statistical mechanics perspective [ 57, 58, 59, 215, 222, 165]). 2.1.5 Mutual information and learning The aim of learning is to generalize the information obtaine d from training data to non–training situations. For such a generalization to be possible, there must exist a, at least partially known, relation betwe en the likelihoods 16p(yi|xi,h) for training and for non–training data. This relation is ty pically provided by a priori knowledge. One possibility to quantify the relation between two random variables y1andy2, representing for example training and non–training data, is to calculate their mutual information , defined as M(Y1,Y2) =/summationdisplay y1∈Y1,y2∈Y2p(y1,y2) lnp(y1,y2) p(y1)p(y2). (29) It is also instructive to express the mutual information in t erms of (average) information content or entropy, which, for a probability fu nctionp(y), is defined as H(Y) =−ln/summationdisplay y∈Yp(y) lnp(y). (30) We find M(Y1,Y2) =H(Y1) +H(Y2)−H(Y1,Y2), (31) meaning that the mutual information is the sum of the two indi vidual en- tropies diminished by the entropy common to both variables. To have a compact notation for a family of predictive densiti esp(yi|xi,f) we choose a vector x= (x1,x2,···) consisting of all possible values xiand corresponding vector y= (y1,y2,···), so we can write p(y|x,f) =p(y1,y2,···|x1,x2,···,f). (32) We now would like to characterize a state of knowledge fcorresponding to predictive density p(y|x,f) by its mutual information. Thus, we generalize the definition (29) from two random variables to a random vect orywith components yi, given vector xwith components xiand obtain the conditional mutual information M(Y|x,f) =/integraldisplay/parenleftigg/productdisplay idyi/parenrightigg p(y|x,f) lnp(y|x,f) /producttext jp(yj|xj,f), (33) or M(Y|x,f) =/parenleftbigg/integraldisplay dyiH(Yi|x,f)−H(Y|x,f)/parenrightbigg , (34) in terms of conditional entropies H(Y|x,f) =−/integraldisplay dyp(y|x,f) lnp(y|x,f). (35) 17In case not a fixed vector xis given, like for example x= (x1,x2,···), but a densityp(x), it is useful to average the conditional mutual informatio n and conditional entropy by including the integral/integraltextdxp(x) in the above formulae. It is clear from Eq. (33) that predictive densities which fac torize p(y|x,f) =/productdisplay ip(yi|xi,f), (36) have a mutual information of zero. Hence, such factorial states do not allow any generalization from training to non–training data. A sp ecial example are the possible states of Nature or pure states h, which factorize according to the definition of our model p(y|x,h) =/productdisplay ip(yi|xi,h). (37) Thus, pure states do not allow any further generalization. T his is consistent with the fact that pure states represent the natural endpoin ts of any learning process. It is interesting to see, however, that there are also other s tates for which the predictive density factorizes. Indeed, from Eq. (37) it follows that any (prior or posterior) probability p(h) which factorizes leads to a factorial state, p(h) =/productdisplay ip(h(xi))⇒p(y|x,f) =/productdisplay ip(yi|xi,f). (38) This means generalization, i.e., (non–local) learning, is impossible when starting from a factorial prior . A factorial prior provides a very clear reference for analyz ing the role of a–priori information in learning. In particular, with re spect to a prior factorial in local variables xi, learning may be decomposed into two steps, one increasing, the other lowering mutual information: 1. Starting from a factorial prior, new non–local data D0(typically called a priori information) produce a new non–factorial state with non–ze ro mutual information. 2.Local dataD(typically called training data) stochastically reduce th e mutual information. Hence, learning with local data corres ponds to a stochastic decay of mutual information . 18Pure states, i.e., the extremal points in the space of possib le predictive densities, do not have to be deterministic. Improving measu rement devices, stochastic pure states may be further decomposed into finer c omponents g, so that p(yi|xi,h) =/integraldisplay dgp(g)p(yi|xi,g). (39) Imposing a non–factorial prior p(g) on the new, finer hypotheses genables again non–local learning with local data, leading asymptot ically to one of the new pure states p(yi|xi,g). Let us exemplify the stochastic decay of mutual information by a simple numerical example. Because the mutual information require s the integration over allyivariables we choose a problem with only two of them, yaand ybcorresponding to two xvaluesxaandxb. We consider a model with four states of Nature hl, 1≤l≤4, with Gaussian likelihood p(y|x,h) = (√ 2πσ)−1exp (−(y−hi(x))2/(2σ2)) and local means hl(xj) =±1. Selecting a “true” state of Nature h, we sample 50 data points Di= (xi,yi) from the corresponding Gaussian likelihood using p(xa) =p(xb) = 0.5. Then, starting from a given, factorial or non–factorial, priorp(h|D0) we sequentially update the predictive density, p(y|x,f(Di+1,···,D0)) =4/summationdisplay l=1p(y|x,hl)p(hl|Di+1,···,D0), (40) by calculating the posterior p(hl|Di+1,···,D0) =p(yi+1|xi+1,hl)p(hj|Di···,D0) p(yi+1|xi+1,Di···,D0). (41) It is easily seen from Eq. (41) that factorial states remain f actorial under local data. Fig. 3 shows that indeed the mutual information decays rapid ly. Depend- ing on the training data, still the wrong hypothesis hlmay survive the decay of mutual information. Having arrived at a factorial state, further learning has to be local. That means, data points for xican then only influence the predictive density for the corresponding yiand do not allow generalization to the other yjwithj∝ne}ationslash=i. For a factorial prior p(hl) =p(hl(xa))p(hl(xb)) learning is thus local from the very beginning. Only very small numerical random fluctua tions of the mutual information occur, quickly eliminated by learning. Thus, the predic- tive density moves through a sequence of factorial states. 1910 20 30 40 500.20.40.60.81 10 20 30 40 500.0010.0020.0030.0040.0050.0060.007 10 20 30 40 500.20.40.60.81 10 20 30 40 500.0010.0020.0030.0040.0050.0060.007 10 20 30 40 500.20.40.60.81 10 20 30 40 50 -16-1.5 10 -16-1. 10 -17-5. 10 -175. 10 -161. 10 -161.5 10 -162. 10posterior mutual information (a) (b) (c) (d) (e) (f) Figure 3: The decay of mutual information during learning: M odel with 4 possible states hlrepresenting Gaussian likelihoods p(yi|xi,hl) with means ±1 for two different xivalues. Shown are posterior probabilities p(hl|f) (a,c,e, on the left hand side, the posterior of the true hlis shown by a thick line) and mutual information M(y) (b,d,f, on the right hand side) during learning 50 training data. ( a,b): The mutual information decays during learning and becomes quickly practically zero. ( c,d): For “unlucky” training data the wrong hypothesis hican dominate at the beginning. Nevertheless, the mutual information decays and the correct hypothesis has fin ally to be found through “local” learning. ( e,f): Starting with a factorial prior the mutual information is and remains zero, up to artificial numerical fl uctuations. For (e,f) the same random data have been used as for ( c,d). 202.2 Bayesian decision theory 2.2.1 Loss and risk InBayesian decision theory a setAof possible actions ais considered, to- gether with a function l(x,y,a ) describing the losslsuffered in situation xif yappears and action ais selected [14, 119, 172, 187]. The loss averaged over test datax,y, and possible states of Nature his known as expected risk , r(a,f) =/integraldisplay dxdyp (x)p(y|x,f)l(x,y,a ). (42) =<l(x,y,a )>X,Y|f (43) =<r(a,h)>H|f (44) where<···>X,Y|fdenotes the expectation under the joint predictive density p(x,y|f) =p(x)p(y|x,f) and r(a,h) =/integraldisplay dxdyp (x)p(y|x,h)l(x,y,a ). (45) The aim is to find an optimal action a∗ a∗= argmina∈Ar(a,f). (46) 2.2.2 Loss functions for approximation Log–loss: A typical loss function for density estimation problems is thelog– loss l(x,y,a ) =−b1(x) lnp(y|x,a) +b2(x,y) (47) with somea–independent b1(x)>0,b2(x,y) and actions adescribing proba- bility densities/integraldisplay dyp(y|x,a) = 1,∀x∈X,∀a∈A. (48) Choosingb2(x,y) =p(y|x,f) andb1(x) = 1 gives r(a,f) =/integraldisplay dxdyp (x)p(y|x,f) lnp(y|x,f) p(y|x,a)(49) =<lnp(y|x,f) p(y|x,a)>X,Y|f (50) =<KL(p(y|x,f), p(y|x,a))>X, (51) 21which shows that minimizing log–loss is equivalent to minim izing the (x– averaged) Kullback–Leibler entropy KL(p(y|x,f), p(y|x,a))[114, 115, 11, 41, 48]. While the paper will concentrate on log–loss we will also giv e a short summary of loss functions for regression problems . (See for example [14, 187] for details.) Regression problems are special density esti mation problems where the considered possible actions are restricted to y–independent func- tionsa(x). Squared–error loss: The most common loss function for regression prob- lems (see Sections 3.7, 3.7.2) is the squared–error loss. It reads for one– dimensional y l(x,y,a ) =b1(x) (y−a(x))2+b2(x,y), (52) with arbitrary b1(x)>0 andb2(x,y). In that case the optimal function a(x) is theregression function of the posterior which is the mean of the predictive density a∗(x) =/integraldisplay dyyp(y|x,f) =<y>Y|x,f. (53) This can be easily seen by writing (y−a(x))2=/parenleftig y−<y>Y|x,f+<y>Y|x,f−a(x)/parenrightig2(54) =/parenleftig y−<y>Y|x,f/parenrightig2+/parenleftig a(x)−<y>Y|x,f/parenrightig2 −2/parenleftig y−<y>Y|x,f/parenrightig/parenleftig a(x)−<y>Y|x,f/parenrightig2,(55) where the first term in (55) is independent of aand the last term vanishes after integration over yaccording to the definition of <y>Y|x,f. Hence, r(a,f) =/integraldisplay dxb1(x)p(x)/parenleftig a(x)−<y>Y|x,f/parenrightig2+ const. (56) This is minimized by a(x) =< y >Y|x,f. Notice that for Gaussian p(y|x,a) with fixed variance log–loss and squared-error loss are equi valent. For multi– dimensional yone–dimensional loss functions like Eq. (52) can be used whe n the component index of yis considered part of the x–variables. Alternatively, loss functions depending explicitly on multidimensional ycan be defined. For instance, a general quadratic loss function would be l(x,y,a ) =/summationdisplay k,k′(yk−ak)K(k,k′)(yk′−ak′(x)). (57) 22with symmetric, positive definite kernel K(k,k′). Absolute loss: For absolute loss l(x,y,a ) =b1(x)|y−a(x)|+b2(x,y), (58) with arbitrary b1(x)>0 andb2(x,y). The risk becomes r(a,f) =/integraldisplay dxb1(x)p(x)/integraldisplaya(x) −∞dy(a(x)−y)p(y|x,f) +/integraldisplay dxb1(x)p(x)/integraldisplay∞ a(x)dy(y−a(x))p(y|x,f) + const.(59) = 2/integraldisplay dxb1(x)p(x)/integraldisplaya(x) m(x)dy(a(x)−y)p(y|x,f) + const.′,(60) where the integrals have been rewritten as/integraltexta(x) −∞=/integraltextm(x) −∞+/integraltexta(x) m(x)and/integraltext∞ a(x)= /integraltextm(x) a(x)+/integraltext∞ m(x)introducing a median function m(x) which satisfies /integraldisplaym(x) −∞dyp(y|x,f) =1 2,∀x∈X, (61) so that a(x)/parenleftigg/integraldisplaym(x) −∞dyp(y|x,f)−/integraldisplay∞ m(x)dyp(y|x,f)/parenrightigg = 0,∀x∈X. (62) Thus the risk is minimized by any median function m(x). δ–loss and 0–1loss : Another possible loss function, typical for classifica- tion tasks (see Section 3.8), like for example image segment ation [141], is the δ–loss for continuous yor 0–1–loss for discrete y l(x,y,a ) =−b1(x)δ(y−a(x)) +b2(x,y), (63) with arbitrary b1(x)>0 andb2(x,y). Hereδdenotes the Dirac δ–functional for continuous yand the Kronecker δfor discrete y. Then r(a,f) =/integraldisplay dxb1(x)p(x)p(y=a(x)|x,f) + const., (64) so the optimal acorresponds to any mode function of the predictive density. For Gaussians mode and median are unique, and coincide with t he mean. 232.2.3 General loss functions and unsupervised learning Choosing actions ain specific situations often requires the use of specific loss functions. Such loss functions may for example contain additional terms measuring costs of choosing action anot related to approximation of the predictive density. Such costs can quantify aspects like th e simplicity, imple- mentability, production costs, sparsity, or understandab ility of action a. Furthermore, instead of approximating a whole density it of ten suffices to extract some of its features. like identifying clusters o f similary–values, finding independent components for multidimensional y, or mapping to an approximating density with lower dimensional x. This kind of exploratory data analysis is the Bayesian analogon to unsupervised learning methods . Such methods are on one hand often utilized as a preprocessin g step but are, on the other hand, also important to choose actions for s ituations where specific loss functions can be defined. From a Bayesian point of view general loss functions require in general an explicit two–step procedure [123]: 1. Calculate (an appr oximation of) the predictive density, and 2. Minimize the expectation of the l oss function under that (approximated) predictive density. (Empirical risk m inimization, on the other hand, minimizes the empirical average of the (possibl y regularized) loss function, see Section 2.5.) (For a related example see for in stance [130].) For a Bayesian version of cluster analysis, for example, par titioning a pre- dictive density obtained from empirical data into several c lusters, a possible loss function is l(x,y,a ) = (y−a(x,y))2, (65) with action a(x,y) being a mapping of yfor givenxto a finite number of cluster centers (prototypes). Another example of a cluster ing method based on the predictive density is Fukunaga’s valley seeking proc edure [56]. For multidimensional xa space of actions a(Pxx,y) can be chosen de- pending only on a (possibly adaptable) lower dimensional pr ojection of x. For multidimensional ywith components yiit is often useful to identify independent components. One may look, say, for a linear mapp ing ˜y= Myminimizing the correlations between different components o f the ‘source’ variables ˜yby minimizing the loss function l(y,y′,M) =/summationdisplay i∝negationslash=j˜yi˜y′ j, (66) with respect to Munder the joint predictive density for yandy′given 24x,x′,D,D 0. This includes a Bayesian version of blind source separatio n (e.g. applied to the so called cocktail party problem [12, 6]), ana logous to the treatment of Molgedey and Schuster [149]. Interesting proj ections of mul- tidimensional ycan for example be found by projection pursuit techniques [54, 95, 100, 195]. 2.3 Maximum A Posteriori Approximation In most applications the (usually very high or even formally infinite dimen- sional)h–integral over model states in Eq. (23) cannot be performed e xactly. The two most common methods used to calculate the hintegral approxi- mately are Monte Carlo integration [142, 84, 88, 184, 14, 65, 185, 18, 203, 221, 64, 157, 167, 158] and saddle point approximation [14, 40, 27, 159, 15, 232, 187, 64, 71, 123]. The latter approach will be studied in the following. For that purpose, we expand Ecombwith respect to haround some h∗ Ecomb(h) =e(∆h,∇)E(h)/vextendsingle/vextendsingle/vextendsingle h=h∗(67) =Ecomb(h∗) + (∆h,∇(h∗)) +1 2(∆h,H(h∗)∆h) +··· with ∆h= (h−h∗), gradient ∇(not acting on ∆ h), Hessian H, and round brackets ( ···,···) denoting scalar products. In case p(y|x,h) is parameterized independently for every x,ythe stateshrepresent a parameter set indexed byxandy, hence ∇(h∗) =δEcomb(h) δh(x,y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle h=h∗=δEcomb(p(y′|x′,h)) δp(y|x,h)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle h=h∗, (68) H(h∗) =δ2Ecomb(h) δh(x,y)δh(x′,y′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle h=h∗=δ2Ecomb(p(y′′|x′′,h)) δp(y|x,h)δp(y′|x′,h)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle h=h∗, (69) are functional derivatives [90, 98, 26, 32] (or partial deri vatives for discrete x,y) and for example (∆h,∇(h∗)) =/integraldisplay dxdy (h(x,y)−h∗(x,y))∇(h∗)(x,y). (70) Choosingh∗to be the location of a local minimum of Ecomp(h) the linear term in (67) vanishes. The second order term includes the Hes sian and corresponds to a Gaussian integral over hwhich could be solved analytically /integraldisplay dhe−β(∆h,H∆h)=πd 2β−d 2(detH)−1 2, (71) 25for ad–dimensional h–integral. However, using the same approximation for theh–integrals in numerator and denominator of Eq. (23), expand ing then alsop(y|x,h) aroundh∗, and restricting to the first ( h–independent) term p(y|x,h∗) of that expansion, the factor (71) cancels, even for infinit ed. (The result is the zero order term of an expansion of the predictiv e density in powers of 1/β. Higher order contributions can be calculated by using Wick ’s theorem [40, 27, 159, 232, 101, 150, 123].) The final approxim ative result for the predictive density is very simple and intuitive p(y|x,f)≈p(y|x,h∗), (72) with h∗= argminh∈HEcomb= argmaxh∈Hp(h|f)= argmaxh∈Hp(yD|xD,h)p(h|D0). (73) The saddle point (or Laplace) approximation is therefore al so called Maxi- mum A Posteriori Approximation (MAP). Notice that this can equivalently be seen as a saddle point approximation for the evidence of th e datayD p(yD|xD,D0) =/integraldisplay dhp(yD|xD,h)p(h|D0). (74) This equivalence is due to the assumption that p(y|x,h) is slowly varying at the stationary point and has not to be included in the saddle p oint approx- imation for the predictive density. For (functional) differ entiableEcombEq. (73) yields the stationarity equation, δEcomb(h) δh(x,y)= 0. (75) The functional Ecombincluding training and prior data (regularization, sta- bilizer) terms is also known as (regularized) error functional forh. In practice a saddle point approximation may be expected use ful if the posterior is peaked enough around a single maximum, or more g eneral, if the posterior is well approximated by a Gaussian centered at the maximum. For asymptotical results one would have to require /summationdisplay iE(yi|xi,h) =−1 β/summationdisplay iL(yi|xi,h), (76) to become β–independent for β→ ∞ withβbeing the same for the prior and data term. (See for example [36, 225]). If for example1 n/summationtext iL(yi|xi,h) 26converges for large number nof training data the low temperature limit 1/β→0 can be interpreted as large data limit n→ ∞, nEcomb=n/parenleftigg −1 n/summationdisplay iL(yi|xi,h) +1 nE(h|D0)/parenrightigg . (77) Notice, however, the factor 1 /nin front of the prior energy. For Gaussian p(y|x,h) temperature 1 /βcorresponds to variance σ2 1 σ2Ecomb=1 σ2/parenleftigg1 2/summationdisplay i(yi−h(xi))2+σ2E(h|D0)/parenrightigg . (78) For Gaussian prior this would require simultaneous scaling of data and prior variance. We should also remark that for continuous x,ythe stationary solution h∗ needs not to be a typical representative of the process p(h|f). A common example is a Gaussian stochastic process p(h|f) with prior energy E(h|D0) re- lated to some smoothness measure of hexpressed by derivatives of p(y|x,h). Then, even if the stationary h∗is smooth, this needs not to be the case for a typical hsampled according to p(h|f). For Brownian motion, for in- stance, a typical sample path is not even differentiable (but continuous) while the (stationary) mean path is smooth. Thus, for continuous v ariables only expressions like/integraltextdhe−βE(h)can be given an exact meaning as a Gaussian measure, defined by a given covariance with existing normali zation factor, but not the expressions dhandE(h) alone [46, 60, 212, 102, 78, 135]. Interestingly, the stationary h∗yielding maximal posterior p(h|f) is not only useful to obtain an approximation for the predictive de nsityp(y|x,f) but is also the optimal solution a∗for a Bayesian decision problem with log–loss and a∈A=H: Theorem: For a Bayesian decision problem with log–loss (47) argmina∈Hr(a,h) =h, (79) and analogously, argmina∈Fr(a,f) =f. (80) Proof: Jensen’s inequality states that /integraldisplay dyp(y)g(q(y))≥g(/integraldisplay dyp(y)q(y)), (81) 27for any convex function gand probability p(y)≥0 with/integraltextdyp(y) = 1. Thus, because the logarithm is concave −/integraldisplay dyp(y|x,h) lnp(y|x,a) p(y|x,h)≥ −ln/integraldisplay dyp(y|x,h)p(y|x,a) p(y|x,h)= 0 (82) ⇒ −/integraldisplay dyp(y|x,h) lnp(y|x,a)≥ −/integraldisplay dyp(y|x,h) lnp(y|x,h), (83) with equality for a=h. Hence r(a,h) = −/integraldisplay dx/integraldisplay dyp(x)p(y|x,h) (b1(x) lnp(y|x,a) +b2(x,y)) (84) =−/integraldisplay dxp(x)b1(x)/integraldisplay dyp(y|x,h) lnp(y|x,a) + const.(85) ≥ −/integraldisplay dxp(x)b1(x)/integraldisplay dyp(y|x,h) lnp(y|x,h) + const.(86) =r(h,h), (87) with equality for a=h. Fora∈Freplaceh∈Hbyf∈F. q.e.d. 2.4 Normalization, positivity, and specific priors Density estimation problems are characterized by their nor malization and positivity condition for p(y|x,h). Thus, the prior density p(h|D0) can only be non–zero for such hfor whichp(y|x,h) is positive and normalized over yfor allx. (Similarly, when solving for a distribution function, i.e ., the integral of a density, the positivity constraint is replace d by monotonicity and the normalization constraint by requiring the distribu tion function to be 1 on the right boundary.) While the positivity constraint is local with respect toxandy, the normalization constraint is nonlocal with respect to y. Thus, implementing a normalization constraint leads to no nlocal and in general non–Gaussian priors. For classification problems, having discrete yvalues (classes), the nor- malization constraint requires simply to sum over the differ ent classes and a Gaussian prior structure with respect to the x–dependency is not altered [219]. For general density estimation problems, however, i .e., for continu- ousy, the loss of the Gaussian structure with respect to yis more severe, because non–Gaussian functional integrals can in general n ot be performed analytically. On the other hand, solving the learning probl em numerically 28by discretizing the yandxvariables, the normalization term is typically not a severe complication. To be specific, consider a Maximum A Posteriori Approximatio n, mini- mizing βEcomb=−/summationdisplay iL(yi|xi,h) +βE(h|D0), (88) where the likelihood free energy F(YD|xD,h) is included, but not the prior free energy F(H|D0) which, being h–independent, is irrelevant for minimiza- tion with respect to h. The prior energy βE(h|D0) has to implement the positivity and normalization conditions ZX(x,h) =/integraldisplay dyip(yi|xi,h) = 1,∀xi∈Xi,∀h∈H (89) p(yi|xi,h)≥0,∀yi∈Yi,∀xi∈Xi,∀h∈H.(90) It is useful to isolate the normalization condition and posi tivity constraint defining the class of density estimation problems from the re st of the problem specific priors. Introducing the specific prior information ˜D0so thatD0= {˜D0,normalized,positive }, we have p(h|˜D0,norm.,pos.) =p(norm.,pos.|h)p(h|˜D0) p(norm.,pos.|˜D0), (91) with deterministic, ˜D0–independent p(norm.,pos.|h) =p(norm.,pos.|h,˜D0) (92) =p(norm.|h)p(pos.|h) =δ(ZX−1)/productdisplay xyΘ/parenleftig p(y|x,h)/parenrightig , (93) and step function Θ. ( The density p(norm.|h) is normalized over all pos- sible normalizations of p(y|x,h), i.e., over all possible values of ZX, and p(pos.|h) over all possible sign combinations.) The h–independent denomi- natorp(norm.,pos.|˜D0) can be skipped for error minimization with respect toh. We define the specific prior as p(h|˜D0)∝e−E(h|˜D0). (94) In Eq. (94) the specific prior appears as posterior of a h–generating pro- cess determined by the parameters ˜D0. We will call therefore Eq. (94) the 29posterior form of the specific prior. Alternatively, a specific prior can als o be inlikelihood form p(˜D0,h|norm.,pos.) =p(˜D0|h)p(h|norm.,pos.). (95) As the likelihood p(˜D0|h) is conditioned on hthis means that the normal- izationZ=/integraltextd˜D0e−E(˜D0|h)remains in general h–dependent and must be in- cluded when minimizing with respect to h. However, Gaussian specific priors withh–independent covariances have the special property that ac cording to Eq. (71) likelihood and posterior interpretation coincide . Indeed, represent- ing Gaussian specific prior data ˜D0by a mean function t˜D0and covariance K−1(analogous to standard training data in the case of Gaussian regression, see also Section 3.5) one finds due to the fact that the normali zation of a Gaussian is independent of the mean (for uniform (meta) prio rp(h)) p(h|˜D0) =e−1 2(h−t˜D0,K(h−t˜D0)) /integraltextdhe−1 2(h−t˜D0,K(h−t˜D0))(96) =p(t˜D0|h,K) =e−1 2(h−t˜D0,K(h−t˜D0)) /integraltextdte−1 2(h−t,K(h−t)). (97) Thus, for Gaussian p(t˜D0|h,K) withh–independent normalization the specific prior energy in likelihood form becomes analogous to Eq. (94 ) p(t˜D0|h,K)∝e−E(t˜D0|h,K), (98) and specific prior energies can be interpreted both ways. Similarly, the complete likelihood factorizes p(˜D0,norm.,pos.|h) =p(norm.,pos.|h)p(˜D0|h). (99) According to Eq. (93) positivity and normalization conditi ons are im- plemented by step and δ–functions. The positivity constraint is only active when there are locations with p(y|x,h) = 0. In all other cases the gradient has no component pointing into forbidden regions. Due to the combined effect of data, where p(y|x,h) has to be larger than zero by definition, and smoothness terms the positivity condition for p(y|x,h) is usually (but not always) fulfilled. Hence, if strict positivity is checked fo r the final solution then it is not necessary to include extra positivity terms in the error (see 30Section 3.2.1). For the sake of simplicity we will therefore not include posi- tivity terms explicitly in the following. In case a positivi ty constraint has to be included this can be done using Lagrange multipliers, or a lternatively, by writing the step functions in p(pos.|h)∝/producttext x,yΘ(p(y|x,h)) Θ(x−a) =/integraldisplay∞ adξ/integraldisplay∞ −∞dηeiη(ξ−x), (100) and solving the ξ–integral in saddle point approximation (See for example [57, 58, 59]). Including the normalization condition in the prior p0(h|D0) in form of a δ–functional results in a posterior probability p(h|f)=e/summationtext iLi(yi|xi,h)−E(h|˜D0)+˜c(H|˜D0)/productdisplay x∈Xδ/parenleftbigg/integraldisplay dyeL(y|x,h)−1/parenrightbigg (101) with constant ˜ c(H|˜D0) =−ln˜Z(h|˜D0) related to the normalization of the specific prior e−E(h|˜D0). Writing the δ–functional in its Fourier representation δ(x) =1 2π/integraldisplay∞ −∞dkeikx=1 2πi/integraldisplayi∞ −i∞dke−kx, (102) i.e., δ(/integraldisplay dyeL(y|x,h)−1) =1 2πi/integraldisplayi∞ −i∞dΛX(x)eΛX(x)(1−/integraltext dyeL(y|x,h)), (103) and performing a saddle point approximation with respect to ΛX(x) (which is exact in this case) yields P(h|f) =e/summationtext iLi(yi|xi,h)−E(h|˜D0)+˜c(H|˜D0)+/integraltext dxΛX(x)(1−/integraltext dyeL(y|x,h)). (104) This is equivalent to the Lagrange multiplier approach. Her e the stationary ΛX(x) is the Lagrange multiplier vector (or function) to be deter mined by the normalization conditions for p(y|x,h) =eL(y|x,h). Besides the Lagrange multiplier terms it is numerically sometimes useful to add a dditional terms to the log–posterior which vanish for normalized p(y|x,h). 2.5 Empirical risk minimization In the previous sections the error functionals we will try to minimize in the following have been given a Bayesian interpretation in term s of the log– posterior density. There is, however, an alternative justi fication of error 31functionals using the Frequentist approach of empirical risk minimization [208, 209, 210]. Common to both approaches is the aim to minimize the expected risk for actiona r(a,f) =/integraldisplay dxdyp (x,y|f(D,D0))l(x,y,a ). (105) The expected risk, however, cannot be calculated without kn owledge of the truep(x,y|f). In contrast to the Bayesian approach of modeling p(x,y|f) the Frequentist approach approximates the expected risk by theempirical risk E(a) = ˆr(a,f) =/summationdisplay il(xi,yi,a), (106) i.e., by replacing the unknown true probability by an observ able empirical probability. Here it is essential for obtaining asymptotic convergence results to assume that training data are sampled according to the tru ep(x,y|f) [208, 47, 179, 119, 210]. Notice that in contrast in a Bayesian appr oach the density p(xi) for training data Ddoes according to Eq. (16) not enter the formalism becauseDenters as conditional variable. For more detailed discussi on of the relation of quadratic error functionals with Gaussian p rocesses see for example [168, 170, 171, 104, 105, 141, 212, 135]. From that Frequentist point of view one is not restricted to l ogarithmic data terms as they arise from the posterior–related Bayesia n interpretation. However, like in the Bayesian approach, training data terms are not enough to make the minimization problem well defined. Indeed this is a t ypical inverse problem [208, 107, 210] which can, according to the classica l regularization approach [204, 205, 152], be treated by including additiona lregularization (stabilizer) terms in the loss function l. Those regularization terms, which correspond to the prior terms in a Bayesian approach, are thu s from the point of view of empirical risk minimization a technical too l to make the minimization problem well defined. Theempirical generalization error for a test or validation data set inde- pendent from the training data D, on the other hand, is measured using only the data terms of the error functional without regularizati on terms. In empir- ical risk minimization this empirical generalization erro r is used, for example, to determine adaptive (hyper–)parameters of regularizati on terms. A typi- cal example is a factor multiplying the regularization term s controlling the trade–off between data and regularization terms. Common tec hniques using the empirical generalization error to determine such param eters are cross– 32validation orbootstrap like techniques [153, 5, 214, 200, 201, 76, 35, 212, 49]. From a strict Bayesian point of view those parameters would h ave to be integrated out after defining an appropriate prior [14, 138, 140, 21]. 2.6 Interpretations of Occam’s razor The principle to prefer simple models over complex models an d to find an op- timal trade–off between data and complexity is often referre d to as Occam’s razor (William of Occam, 1285–1349). Regularization terms , penalizing for example non–smooth (“complex”) functions, can be seen as an implementa- tion of Occam’s razor. The related phenomena appearing in practical learning is ca lled over– fitting [208, 89, 21]. Indeed, when studying the generalizat ion behavior of trained models on a test set different from the training set, i t is often found that there is a optimal model complexity. Complex models can due to their higher flexibility achieve better performance on the traini ng data than sim- pler models. On a test set independent from the training set, however, they can perform poorer than simpler models. Notice, however, that the Bayesian interpretation of regul arization terms as (a priori ) information about Nature and the Frequentist interpretat ion as additional cost terms in the loss function are notequivalent. Complexity priors reflects the case where Nature is known to be simple whi le complex- ity costs express the wish for simple models without the assu mption of a simple Nature. Thus, while the practical procedure of minim izing an error functional with regularization terms appears to be identic al for empirical risk minimization and a Bayesian Maximum A Posteriori Approxima tion, the un- derlying interpretation for this procedure is different. In particular, because the Theorem in Section 2.3 holds only for log–loss, the case o f loss functions differing from log–loss requires from a Bayesian point of vie w to distinguish explicitly between model states hand actions a. Even in saddle point ap- proximation, this would result in a two step procedure, wher e in a first step the hypothesis h∗, with maximal posterior probability is determined, while the second step minimizes the risk for action a∈Aunder that hypothesis h∗[123]. 332.7A priori information and a posteriori control Learning is based on data, which includes training data as we ll asa pri- oridata. It is prior knowledge which, besides specifying the sp ace of local hypothesis, enables generalization by providing the neces sary link between measured training data and not yet measured or non–training data. The strength of this connection may be quantified by the mutual in formation of training and non–training data, as we did in Section 2.1.5. Often, the role of a priori information seems to be underestimated. There are theorems, for example, proving that asymptotically lea rning results be- come independent of a priori information if the number of training data goes to infinity. This, however,is correct only if the space of hyp otheseshis al- ready sufficiently restricted and if a priori information means knowledge in addition to that restriction. In particular, let us assume that the number of potential tes t situations x, is larger than the number of training data one is able to coll ect. As the number of actual training data has to be finite, this is always the case if xcan take an infinite number of values, for example if xis a continuous variable. The following arguments, however, are not restri cted to situations were one considers an infinite number of test situation, we ju st assume that their number is too large to be completely included in the tra ining data. If there are xvalues for which no training data are available, then learn- ing for such xmust refer to the mutual information of such test data and the available training data. Otherwise, training would be u seless for these test situations. This also means, that the generalization t o non–training situations can be arbitrarily modified by varying a priori information. To make this point very clear, consider the rather trivial si tuation of learning a deterministic function h(x) for axvariable which can take only two values x1andx2, from which only one can be measured. Thus, hav- ing measured for example h(x1) = 5 then “learning” h(x2) is not possible without linking it to h(x1). Such prior knowledge may have the form of a “smoothness” constraint, say |h(x1)−h(x2)| ≤2 which would allow a learning algorithm to “generalize” from the training data and obtain 3≤h(x2)≤7. Obviously, arbitrary results can be obtained for h(x2) by changing the prior knowledge. This exemplifies that generalization can be cons idered as a mere reformulation of available information, i.e., of training data and prior knowl- edge. Except for such a rearrangement of knowledge, a learni ng algorithm does not add any new information to the problem. (For a discus sion of the 34related “no–free-lunch” theorems see [223, 224].) Being extremely simple, this example nevertheless shows a s evere prob- lem. If the result of learning can be arbitrary modified by a priori informa- tion, then it is critical which prior knowledge is implement ed in the learning algorithm. This means, that prior knowledge needs an empiri cal foundation, just like standard training data have to be measured empiric ally. Otherwise, the result of learning cannot expected to be of any use. Indeed, the problem of appropriate a priori information is just the old induction problem, i.e., the problem of learning general la ws from a finite number of observations, as already been discussed by the anc ient Greek philosophers. Clearly, this is not a purely academic proble m, but is ex- tremely important for every system which depends on a succes sful control of its environment. Modern applications of learning algori thms, like speech recognition or image understanding, rely essentially on co rrecta priori in- formation. This holds especially for situations where only few training data are available, for example, because sampling is very costly . Empirical measurement of a priori information, however, seems to be impossible. The reason is that we must link every possible te st situation to the training data. We are not able to do this in practice if, as we assumed, the number of potential test situations is larger than the numbe r of measurements one is able to perform. Take as example again a deterministic learning problem like the one dis- cussed above. Then measuring a priori information might for example be done by measuring (e.g., bounds on) all differences h(x1)−h(xi). Thus, even if we take the deterministic structure of the problem fo r granted, the number of such differences is equal to the number of potential non–training situationsxiwe included in our model. Thus, measuring a priori information does not require fewer measurements than measuring directl y all potential non–training data. We are interested in situations where th is is impossible. Going to a probabilistic setting the problem remains the sam e. For exam- ple, even if we assume Gaussian hypotheses with fixed varianc e, measuring a complete mean function h(x), say for continuous x, is clearly impossible in practice. The same holds thus for a Gaussian process prior onh. Even this very specific prior requires the determination of a cova riance and a mean function (see Chapter 3). As in general empirical measurement of a priori information seems to be impossible, one might thus just try to guess some prior. One m ay think, for example, of some “natural” priors. Indeed, the term “ a priori ” goes back 35to Kant [103] who assumed certain knowledge to be necessaril y be given “ a priori” without reference to empirical verification. This means th at we are either only able to produce correct prior assumptions, for e xample because incorrect prior assumptions are “unthinkable”, or that one must typically be lucky to implement the right a priori information. But looking at the huge number of different prior assumptions which are usually possible (or “thinkable”), there seems no reason why one should be lucky. The question thus remains, how can prior assumptions get empirically ver ified. Also, one can ask whether there are “natural” priors in pract ical learning tasks. In Gaussian regression one might maybe consider a “na tural” prior to be a Gaussian process with constant mean function and smoo thness– related covariance. This may leave a single regularization parameter to be determined for example by cross–validation. Formally, one can always even use a zero mean function for the prior process by subtracting a base line or reference function. Thus does, however, not solve the pro blem of finding a correct prior, as now that reference function has to be know n to relate the results of learning to empirical measurements. In princ ipleanyfunction could be chosen as reference function. Such a reference func tion would for example enter a smoothness prior. Hence, there is no “natura l” constant function and from an abstract point of view no prior is more “n atural” than any other. Formulating a general law refers implicitly (and sometimes explcitly) to a “ceteris paribus” condition, i.e., the constraint that al l relevent variables, not explicitly mentioned in the law, are held constant. But a gain, verifying a “ceteris paribus” condition is part of an empirical measur ement of a priori information and by no means trivial. Trying to be cautious and use only weak or “uninformative” pr iors does also not solve the principal problem. One may hope that such p riors (which may be for example an improper constant prior for a one–dimen sional real variable) do not introduce a completely wrong bias, so that t he result of learning is essentially determined by the training data. Bu t, besides the problem to define what exactly an uninformative prior has to b e, such priors are in practice only useful if the set of possible hypothesis is already suffi- ciently restricted, so “the data can speak for themselves” [ 64]. Hence, the problem remains to find that priors which impose the necessar y restrictions, so that uninformative priors can be used. Hence, as measuring a priori information seems impossible and finding correct a priori information by pure luck seems very unlikely, it looks like a lso 36successful learning is impossible. It is a simple fact, howe ver, that learning can be successful. That means there must be a way to control a priori information empirically. Indeed, the problem of measuring a priori information may be artificial, arising from the introduction of a large number of potential test situations and correspondingly a large number of hidden variables h(representing what we call “Nature”) which are not all observable. In practice, the number of actual test situations is also always finite, just like the number of training data has to be. This means, that no tallpoten- tial test data but only the actual test data must be linked to t he training data. Thus, in practice it is only a finite number of relations which must be under control to allow successful generalization. (See a lso Vapnik’s dis- tinction between induction and transduction problems. [21 0]: In induction problems one tries to infer a whole function, in transductio n problems one is only interested in predictions for a few specific test situat ions.) This, however, opens a possibility to control a priori information em- pirically. Because we do not know which test situation will o ccur, such an empirical control cannot take place at the time of training. This means a priori information has to be implemented at the time of measuring th e test data. In other words, a priori information has to be implemented by the measurement process [123, 126]. Again, a simple example may clarify this point. Consider the prior in- formation, that a function his bounded, i.e., a≤h(x)≤b,∀x. A direct measurement of this prior assumption is practically not pos sible, as it would require to check every value h(x). An implementation within the measure- ment process is however trivial. One just has to use a measure ment device which is only able to to produce output in the range between aandb. This is a very realistic assumption and valid for all real measure ment devices. Values smaller than aand larger than bhave to be filtered out or actively projected into that range. In case we nevertheless find a valu e out of that range we either have to adjust the bounds or we exchange the “m alfunction- ing” measurement device with a proper one. Note, that this ra nge filter is only needed at the finite number of actual measurements. That means, a priori information can be implemented by a posteriori control at the time of testing. A realistic measurement device does not only produce bounde d output but shows also always input noise orinput averaging . A device with input noise has noise in the xvariable. That means if one intends to measure at 3720 40 60 80 1000.20.40.60.81 20 40 60 80 1000.20.40.60.81 Figure 4: The l.h.s. shows a bounded random function which do es not allow generalization from training to non–training data. Using a measurement device with input averaging (r.h.s.) or input noise the func tion becomes learnable. xithe device measures instead at xi+∆ with ∆ being a random variable. A typical example is translational noise, with ∆ being a, poss ibly multidimen- sional, Gaussian random variable with mean zero. Similarly , a device with input averaging returns a weighted average of results for di fferentxvalues instead of a sharp result. Bounded devices with translation al input noise, for example, will always measure smooth functions [120, 20, 123 ]. (See Fig. 4.) This may be an explanation for the success of smoothness prio rs. The last example shows, that to obtain adequate a priori information it can be helpful in practice to analyze the measurement proc ess for which learning is intended. The term “measurement process” does h ere not only refer to a specific device, e.g., a box on the table, but to the c ollection of all processes which lead to a measurement result. We may remark that measuring a measurement process is as diffic ult or impossible as a direct measurement of a priori information. What has to be ensured is the validity of the necessary restrictions dur ing a finite num- ber of actual measurements. This is nothing else than the imp lementation of a probabilistic rule producing ygiven the test situation and the training data. In other words, what has to be implemented is the predic tive density p(y|x,D). This predictive density indeed only depends on the actual test situation and the finite number of training data. (Still, the probability den- sity for a real ycannot strictly be empirically verified or controlled. We ma y take it here, for example, as an approximate statement about frequencies.) This shows the tautological character of learning, where me asuring a priori information means controling directly the corresponding p redictive density. 38Thea posteriori interpretation of a priori information can be related to a constructivistic point of view. The main idea of construct ivism can be characterized by a sentence of Vico (1710): Verum ipsum factum — the truth is the same as the made [211]. (For an introduction to co nstructivism see [216] and references therein, for constructive mathema tics see [22].) 3 Gaussian prior factors 3.1 Gaussian prior factor for log–probabilities 3.1.1 Lagrange multipliers: Error functional EL In this chapter we look at density estimation problems with G aussian prior factors. We begin with a discussion of functional priors whi ch are Gaussian in probabilities or in log–probabilities, and continue with g eneral Gaussian prior factors. Two section are devoted to the discussion of covari ances and means of Gaussian prior factors, as their adequate choice is essen tial for practical applications. After exploring some relations of Bayesian fi eld theory and empirical risk minimization, the last three sections intro duce the specific likelihood models of regression, classification, inverse q uantum theory. We begin a discussion of Gaussian prior factors in L. As Gaussian prior factors correspond to quadratic error (or energy) terms, co nsider an error functional with a quadratic regularizer in L (L,KL) =||L||2 K=1 2/integraldisplay dxdydx′dy′L(x,y)K(x,y;x′,y′)L(x′,y′),(107) writing for the sake of simplicity from now on L(x,y) for the log–probability L(y|x,h) = lnp(y|x,h). The operator Kis assumed symmetric and positive semi–definite and positive definite on some subspace. (We wil l understand positive semi–definite to include symmetry in the following .) For positive (semi) definite Kthe scalar product defines a (semi) norm by ||L||K=/radicalig (L,KL), (108) and a corresponding distance by ||L−L′||K. The quadratic error term (107) corresponds to a Gaussian factor of the prior density which h ave been called the specific prior p(h|˜D0) =p(L|˜D0) forL. In particular, we will consider 39here the posterior density p(h|f)=e/summationtext iLi(xi,yi)−1 2/integraltext dxdydx′dy′L(x,y)K(x,y;x′,y′)L(x′,y′)+/integraltext dxΛX(x)(1−/integraltext dyeL(x,y))+˜c,, (109) where prefactors like βare understood to be included in K. The constant ˜creferring to the specific prior is determined by the determin ant of Kac- cording to Eq. (71). Notice however that not only the likelih ood/summationtext iLibut also the complete prior is usually notGaussian due to the presence of the normalization conditions. (An exception is Gaussian regre ssion, see Section 3.7.) The posterior (109) corresponds to an error functiona l EL=βEcomb=−(L,N) +1 2(L,KL) + (eL−δ(y),ΛX), (110) withlikelihood vector (or function) L(x,y) =L(y|x,h), (111) data vector (function) N(x,y) =n/summationdisplay iδ(x−xi)δ(y−yi), (112) Lagrange multiplier vector (function) ΛX(x,y) = ΛX(x), (113) probability vector (function) eL(x,y) =eL(x,y)=P(x,y) =p(y|x,h), (114) and δ(y)(x,y) =δ(y). (115) According to Eq. (112) N/n=Pempis anempirical density function for the joint probability p(x,y|h). We end this subsection by defining some notations. While func tions of vectors (functions) and matrices (operators), like eL, will be understood element-wise, only multiplication is interpreted as matri x product. Element- wise multiplication is written with the help of diagonal mat rices. For that 40purpose we denote diagonal matrices corresponding to vecto rs by bold letters. For instance, the matrices (operators) I(x,y;x′,y′) =δ(x−x′)δ(y−y′), (116) L(x,y;x′,y′) =δ(x−x′)δ(y−y′)L(x,y), (117) P(x,y;x′,y′) =eL(x,y;x′,y′) (118) =δ(x−x′)δ(y−y′)P(x,y), (119) N(x,y;x′,y′) =δ(x−x′)δ(y−y′)N(x,y), (120) ΛX(x,y;x′,y′) =δ(x−x′)δ(y−y′)ΛX(x), (121) correspond to the vectors or functions, I(x,y) = 1, (122) and L=LI, P =PI, eL=eLI, N =NI,ΛX=ΛXI. (123) Being diagonal all these matrices commute with each other. Element-wise multiplication can now be expressed as (KL)(x′,y′,x,y) =/integraldisplay dx′′dy′′K(x′,y′,x′′,y′′)L(x′′,y′′,x,y) =/integraldisplay dx′′dy′′K(x′,y′,x′′,y′′)L(x,y)δ(x−x′′)δ(y−y′′) =K(x′,y′,x,y)L(x,y). (124) In general this is not equal to L(x′,y′)K(x′,y′,x,y). In contrast, the matrix product KLwith vector L (KL)(x′,y′) =/integraldisplay dxdyK(x′,y′,x,y)L(x,y), (125) does not depend on x,yanymore, while the tensor product or outer product, (K⊗L)(x′′,y′′,x,y,x′,y′) =K(x′′,y′′,x′,y′)L(x,y), (126) depends on additional x′′,y′′. Taking the variational derivative of (109) with respect to L(x,y) using δL(x′,y′) δL(x,y)=δ(x−x′)δ(y−y′) (127) 41and setting the gradient equal to zero yields the stationari ty equation 0 =N−KL−eLΛX. (128) Alternatively, we can write eLΛX=ΛXeL=PΛX. The Lagrange multiplier function Λ Xis determined by the normalization condition ZX(x) =/integraldisplay dyeL(x,y)= 1,∀x∈X, (129) which can also be written ZX=IXP=IXeL=IorZX=I, (130) in terms of normalization vector, ZX(x,y) =ZX(x), (131) normalization matrix, ZX(x,y;x′,y′) =δ(x−x′)δ(y−y′)ZX(x), (132) and identity on X, IX(x,y;x′,y′) =δ(x−x′). (133) Multiplication of a vector with IXcorresponds to y–integration. Being a non–diagonal matrix IXdoes in general not commute with diagonal matrices likeLorP. Note also that despite IXeL=IXeLI=II=Iin general IXP =IXeL∝ne}ationslash=I=ZX. According to the fact that IXandΛXcommute, i.e., IXΛX=ΛXIX⇔[ΛX,IX] = 0, (134) and the same holds for the diagonal matrices [ΛX,eL] = [ΛX,P] = 0, (135) it follows from the normalization condition IXP=Ithat IXPΛX=IXΛXP=ΛXIXP=ΛXI= ΛX, (136) i.e., 0 = (I−IXeL)ΛX= (I−IXP)ΛX. (137) 42For ΛX(x)∝ne}ationslash= 0 Eqs.(136,137) are equivalent to the normalization (129) . If there exist directions at the stationary point L∗in which the normalization of Pchanges, i.e., the normalization constraint is active, a Λ X(x)∝ne}ationslash= 0 restricts the gradient to the normalized subspace (Kuhn–Tucker condi tions [52, 17, 92, 178]). This will clearly be the case for the unrestricted var iations ofp(y,x) which we are considering here. Combining Λ X=IXPΛXfor ΛX(x)∝ne}ationslash= 0 with the stationarity equation (128) the Lagrange multiplier fu nction is obtained ΛX=IX(N−KL) =NX−(IXKL). (138) Here we introduced the vector NX=IXN, (139) with components NX(x,y) =NX(x) =/summationdisplay iδ(x−xi) =nx, (140) giving the number of data available for x. Thus, Eq. (138) reads in compo- nents ΛX(x) =/summationdisplay iδ(x−xi)−/integraldisplay dy′′dx′dy′K(x,y′′;x′,y′)L(x′,y′). (141) Inserting now this equation for Λ Xinto the stationarity equation (128) yields 0 =N−KL−eL(NX−IXKL) =/parenleftig I−eLIX/parenrightig (N−KL). (142) Eq. (142) possesses, besides normalized solutions we are lo oking for, also possibly unnormalized solutions fulfilling N=KLfor which Eq. (138) yields ΛX= 0. That happens because we used Eq. (136) which is also fulfil led for ΛX(x) = 0. Such a Λ X(x) = 0 does not play the role of a Lagrange multiplier. For parameterizations of Lwhere the normalization constraint is not necessarily active at a stationary point Λ X(x) = 0 can be possible for a normalized solution L∗. In that case normalization has to be checked. It is instructive to define TL=N−ΛXeL, (143) so the stationarity equation (128) acquires the form KL=TL, (144) 43which reads in components /integraldisplay dx′dy′K(x,y;x′,y′)L(x′,y′) =/summationdisplay iδ(x−xi)δ(y−yi)−ΛX(x)eL(x,y),(145) which is in general a non–linear equation because TLdepends on L. For existing (and not too ill–conditioned) K−1the form (144) suggest however an iterative solution of the stationarity equation accordi ng to Li+1=K−1TL(Li), (146) for discretized L, starting from an initial guess L0. Here the Lagrange multi- plier ΛXhas to be adapted so it fulfills condition (138) at the end of it eration. Iteration procedures will be discussed in detail in Section 7. 3.1.2 Normalization by parameterization: Error functiona lEg Referring to the discussion in Section 2.3 we show that Eq. (1 42) can alter- natively be obtained by ensuring normalization, instead of using Lagrange multipliers, explicitly by the parameterization L(x,y) =g(x,y)−ln/integraldisplay dy′eg(x,y′), L=g−lnZX, (147) and considering the functional Eg=−/parenleftig N, g−lnZX/parenrightig +1 2/parenleftig g−lnZX,K(g−lnZX)/parenrightig . (148) The stationary equation for g(x,y) obtained by setting the functional deriva- tiveδEg/δgto zero yields again Eq. (142). We check this, using δlnZX(x′) δg(x,y)=δ(x−x′)eL(x,y),δlnZX δg=IXeL=/parenleftig eLIX/parenrightigT, (149) and δL(x′,y′) δg(x,y)=δ(x−x′)δ(y−y′)−δ(x−x′)eL(x,y),δL δg=I−IXeL,(150) whereδL δgdenotes a matrix, and the superscriptTthe transpose of a matrix. We also note that despite IX=IT X IXeL∝ne}ationslash=eLIX= (IXeL)T, (151) 44is not symmetric because eLdepends on yand does not commute with the non–diagonal IX. Hence, we obtain the stationarity equation of functional Egwritten in terms of L(g) again Eq. (142) 0 =−/parenleftiggδL δg/parenrightiggTδEg δL=GL−eLΛX=/parenleftig I−eLIX/parenrightig (N−KL). (152) HereGL=N−KL=−δEg/δLis theL–gradient of −Eg. Referring to the discussion following Eq. (142) we note, however, that solvi ng forginstead forLno unnormalized solutions fulfilling N=KLare possible. In case lnZXis in the zero space of Kthe functional Egcorresponds to a Gaussian prior in galone. Alternatively, we may also directly consider a Gaussian prior in g ˜Eg=−/parenleftig N, g−lnZX/parenrightig +1 2/parenleftig g,Kg/parenrightig , (153) with stationarity equation 0 =N−Kg−eLNX. (154) Notice, that expressing the density estimation problem in t erms ofg, nonlo- cal normalization terms have not disappeared but are part of the likelihood term. As it is typical for density estimation problems, the s olutiongcan be calculated in X–data space, i.e., in the space defined by the xiof the training data. This still allows to use a Gaussian prior structure wit h respect to the x–dependency which is especially useful for classification p roblems [219]. 3.1.3 The Hessians H L, Hg The Hessian HLof−ELis defined as the matrix or operator of second deriva- tives HL(L)(x,y;x′y′) =δ2(−EL) δL(x,y)δL(x′,y′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle L. (155) For functional (110) and fixed Λ Xwe find the Hessian by taking the derivative of the gradient in (128) with respect to Lagain. This gives HL(L)(x,y;x′y′) =−K(x,y;x′y′)−δ(x−x′)δ(y−y′)ΛX(x)eL(x,y)(156) or HL=−K−ΛXeL. (157) 45The addition of the diagonal matrix ΛXeL=eLΛXcan result in a negative definite Heven if Khas zero modes. like in the case where Kis a differential operator with periodic boundary conditions. Note, however , that ΛXeLis diagonal and therefore symmetric, but not necessarily posi tive definite, be- cause ΛX(x) can be negative for some x. Depending on the sign of Λ X(x) the normalization condition ZX(x) = 1 for that xcan be replaced by the inequalityZX(x)≤1 orZX(x)≥1. Including the L–dependence of Λ Xand with δeL(x′,y′) δg(x,y)=δ(x−x′)δ(y−y′)eL(x,y)−δ(x−x′)eL(x,y)eL(x′,y′), (158) i.e., δeL δg=/parenleftig I−eLIX/parenrightig eL=eL−eLIXeL, (159) we find, written in terms of L, Hg(L)(x,y;x′,y′) =δ2(−Eg) δg(x,y)δg(x′,y′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle L =/integraldisplay dx′′dy′′/parenleftiggδ2(−Eg) δL(x,y)δL(x′′,y′′)δL(x′′,y′′) δg(x′,y′)+δ(−Eg) δL(x′′,y′′)δ2L(x′′,y′′) δg(x,y)δg(x′,y′)/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle L =−K(x,y;x′,y′)−eL(x′,y′)eL(x,y)/integraldisplay dy′′dy′′′K(x′,y′′;x,y′′′) +eL(x′,y′)/integraldisplay dy′′K(x′,y′′;x,y) +eL(x,y)/integraldisplay dy′′K(x′,y′;x,y′′) −δ(x−x′)δ(y−y′)eL(x,y)/parenleftbigg NX(x)−/integraldisplay dy′′(KL)(x,y′′)/parenrightbigg +δ(x−x′)eL(x,y)eL(x′,y′)/parenleftbigg NX(x)−/integraldisplay dy′′(KL)(x,y′′)/parenrightbigg .(160) The last term, diagonal in X, has dyadic structure in Y, and therefore for fixedxat most one non–zero eigenvalue. In matrix notation the Hess ian becomes Hg=−/parenleftig I−eLIX/parenrightig K/parenleftig I−IXeL/parenrightig −/parenleftig I−eLIX/parenrightig ΛXeL =−(I−PIX) [K(I−IXP) +ΛXP], (161) 46the second line written in terms of the probability matrix. T he expression is symmetric under x↔x′,y↔y′, as it must be for a Hessian and as can be verified using the symmetry of K=KTand the fact that ΛXandIX commute, i.e., [ ΛX,IX] = 0. Because functional Egis invariant under a shift transformation, g(x,y)→g′(x,y) +c(x), the Hessian has a space of zero modes with the dimension of X. Indeed, any y–independent function (which can have finite L1–norm only in finite Y–spaces) is a left eigenvector of/parenleftig I−eLIX/parenrightig with eigenvalue zero. The zero mode can be removed by pro- jecting out the zero modes and using where necessary instead of the inverse a pseudo inverse of H, for example obtained by singular value decomposi- tion, or by including additional conditions on glike for example boundary conditions. 3.2 Gaussian prior factor for probabilities 3.2.1 Lagrange multipliers: Error functional EP We writeP(x,y) =p(y|x,h) for the probability of yconditioned on xand h. We consider now a regularizing term which is quadratic in Pinstead of L. This corresponds to a factor within the posterior probabil ity (the specific prior) which is Gaussian with respect to P. p(h|f)=e/summationtext ilnPi(xi,yi)−1 2/integraltext dxdydx′dy′P(x,y)K(x,y;x′,y′)P(x′,y′)+/integraltext dxΛX(x)(1−/integraltext dyP(x,y))+˜c, (162) or written in terms of L= lnPfor comparison, p(h|f)=e/summationtext iLi(xi,yi)−1 2/integraltext dxdydx′dy′eL(x,y)K(x,y;x′,y′)eL(x′,y′)+/integraltext dxΛX(x)(1−/integraltext dyeL(x,y))+˜c. (163) Hence, the error functional is EP=βEcomb=−(lnP,N) +1 2(P,KP) + (P−δ(y),ΛX). (164) In particular, the choice K=λ 2I, i.e., λ 2(P, P) =λ 2||P||2, (165) can be interpreted as a smoothness prior with respect to the d istribution function of P(see Section 3.3). 47In functional (164) we have only implemented the normalizat ion condition forPby a Lagrange multiplier and not the positivity constraint. This is sufficient if P(x,y)>0 (i.e.,P(x,y) not equal zero) at the stationary point because then P(x,y)>0 holds also in some neighborhood and there are no components of the gradient pointing into regions with negat ive probabilities. In that case the positivity constraint is not active at the st ationarity point. A typical smoothness constraint, for example, together with positive probability at data points result in positive probabilities everywhere where not set to zero explicitly by boundary conditions. If, however, the st ationary point has locations with P(x,y) = 0 at non–boundary points, then the component of the gradient pointing in the region with negative probabi lities has to be projected out by introducing Lagrange parameters for each P(x,y). This may happen, for example, if the regularizer rewards oscilla tory behavior. The stationarity equation for EPis 0 =P−1N−KP−ΛX, (166) with the diagonal matrix P(x′,y′;x,y) =δ(x−x′)δ(y−y′)P(x,y), or multi- plied by P 0 =N−PKP−PΛX. (167) Probabilities P(x,y) are unequal zero at observed data points ( xi,yi) so P−1Nis well defined. Combining the normalization condition Eq. (136) for Λ X(x)∝ne}ationslash= 0 with Eq. (166) or (167) the Lagrange multiplier function Λ Xis found as ΛX=IX(N−PKP) =NX−IXPKP, (168) where IXPKP(x,y) =/integraldisplay dy′dx′′dy′′P(x,y′)K(x,y′;x′′,y′′)P(x′′,y′′). Eliminating Λ Xin Eq. (166) by using Eq. (168) gives finally 0 = (I−IXP)(P−1N−KP), (169) or for Eq. (167) 0 = (I−PIX)(N−PKP). (170) For similar reasons as has been discussed for Eq. (142) unnor malized solutions fulfillingN−PKPare possible. Defining TP=P−1N−ΛX=P−1N−NX−IXPKP, (171) 48the stationarity equation can be written analogously to Eq. (144) as KP=TP, (172) withTP=TP(P), suggesting for existing K−1an iteration Pi+1=K−1TP(Pi), (173) starting from some initial guess P0. 3.2.2 Normalization by parameterization: Error functiona lEz Again, normalization can also be ensured by parameterizati on ofPand solv- ing for unnormalized probabilities z, i.e., P(x,y) =z(x,y)/integraltextdyz(x,y), P =z ZX. (174) The corresponding functional reads Ez=−/parenleftbigg N,lnz ZX/parenrightbigg +1 2/parenleftbiggz ZX,Kz ZX/parenrightbigg . (175) We have δz δz=I,δZX δz=IX,δlnz δz=z−1= (ZXP)−1,δlnZX δz=Z−1 XIX, (176) with diagonal matrix zbuilt analogous to PandZX, and δP δz=δ(z/ZX) δz=Z−1 X(I−PIX),δlnP δz=Z−1 X/parenleftig P−1−IX/parenrightig ,(177) δZ−1 X δz=−Z−2 XIX,δP−1 δz=−P−2Z−1 X(I−PIX). (178) The diagonal matrices [ ZX,P] = 0 commute, as well as [ ZX,IX] = 0, but [P,IX]∝ne}ationslash= 0. Setting the gradient to zero and using (I−PIX)T= (I−IXP), (179) we find 0 =−/parenleftiggδP δz/parenrightiggTδEz δP 49=Z−1 X/bracketleftig/parenleftig P−1−IX/parenrightig N−(I−IXP)KP/bracketrightig =Z−1 X(I−IXP)/parenleftig P−1N−KP/parenrightig =Z−1 X(I−IXP)GP=Z−1 X(GP−ΛX) = (GP−ΛX)Z−1 X, (180) withP–gradientGP=P−1N−KP=−δEz/δPof−EzandGPthe cor- responding diagonal matrix. Multiplied by ZXthis gives the stationarity equation (172). 3.2.3 The Hessians H P, Hz We now calculate the Hessian of the functional −EP. For fixed Λ Xone finds the Hessian by differentiating again the gradient (166) of −EP HP(P)(x,y;x′y′) =−K(x′y′;x,y)−δ(x−x′)δ(y−y′)/summationdisplay iδ(x−xi)δ(y−yi) P2(x,y), (181) i.e., HP=−K−P−2N. (182) Here the diagonal matrix P−2Nis non–zero only at data points. Including the dependence of Λ XonPone obtains for the Hessian of −Ez in (175) by calculating the derivative of the gradient in (18 0) Hz(x,y;x′,y′) =−1 ZX(x)/bracketleftig K(x,y;x′,y′) −/integraldisplay dy′′/parenleftig p(x,y′′)K(x,y′′;x′,y′) +K(x,y;x′,y′′)p(x′,y′′)/parenrightig +/integraldisplay dy′′dy′′′p(x,y′′)K(x,y′′;x′,y′′′)p(x′,y′′′) +δ(x−x′)δ(y−y′)/summationdisplay iδ(x−xi)δ(y−yi) p2(x,y)−δ(x−x′)/summationdisplay iδ(x−xi) −δ(x−x′)/integraldisplay dx′′dy′′/parenleftig K(x,y;x′′,y′′)p(x′′,y′′) +p(x′′,y′′)K(x′′,y′′;x′,y′)/parenrightig + 2δ(x−x′)/integraldisplay dy′′dx′′′dy′′′p(x,y′′)K(x,y′′;x′′′,y′′′)p(x′′′,y′′′)/bracketrightig1 ZX(x′),(183) 50i.e., Hz=Z−1 X(I−IXP)/parenleftig −K−P−2N/parenrightig (I−PIX)Z−1 X −Z−1 X(IX(GP−ΛX) + (GP−ΛX)IX)Z−1 X, (184) =−Z−1 X/bracketleftig (I−IXP)K(I−PIX) +P−2N −IXP−1N−NP−1IX+IXNIX +IXGP+GPIX−2IXΛX/bracketrightig Z−1 X. (185) Here we used [ ΛX,IX] = 0. It follows from the normalization/integraltextdyp(x,y) = 1 that any y–independent function is right eigenvector of ( I−IXP) with zero eigenvalue. Because Λ X=IXPGPthis factor or its transpose is also contained in the second line of Eq. (184), which means that Hzhas a zero mode. Indeed, functional Ezis invariant under multiplication of zwith a y–independent factor. The zero modes can be projected out or r emoved by including additional conditions, e.g. by fixing one value of zfor everyx. 3.3 General Gaussian prior factors 3.3.1 The general case In the previous sections we studied priors consisting of a fa ctor (the specific prior) which was Gaussian with respect to PorL= lnPand additional normalization (and positivity) conditions. In this sectio n we consider the situation where the probability p(y|x,h) is expressed in terms of a function φ(x,y). That means, we assume a, possibly non–linear, operator P=P(φ) which maps the function φto a probability. We can then formulate a learning problem in terms of the function φ, meaning that φnow represents the hidden variables or unknown state of Nature h.2Consider the case of a specific prior which is Gaussian in φ, i.e., which has a log–probability quadratic in φ −1 2(φ,Kφ). (186) This means we are lead to error functionals of the form Eφ=−( lnP(φ), N) +1 2(φ,Kφ) + (P(φ),ΛX), (187) 2Besides φalso the hyperparameters discussed in Chapter 5 belong to th e hidden variables h. 51where we have skipped the φ–independent part of the Λ X–terms. In general cases also the positivity constraint has to be implemented. To express the functional derivative of functional (187) wi th respect to φwe define besides the diagonal matrix P=P(φ) the Jacobian, i.e., the matrix of derivatives P′=P′(φ) with matrix elements P′(x,y;x′,y′;φ) =δP(x′,y′;φ) δφ(x,y). (188) The matrix P′is diagonal for point–wise transformations, i.e., for P(x,y;φ) = P(φ(x,y) ). In such cases we use P′to denote the vector of diagonal elements ofP′. An example is the previously discussed transformation L= lnPfor whichP′=P. The stationarity equation for functional (187) becomes 0 =P′(φ)P−1(φ)N−Kφ−P′(φ)ΛX, (189) and for existing PP′−1=(P′P−1)−1(for nonexisting inverse see Section 4.1), 0 =N−PP′−1Kφ−PΛX. (190) From Eq. (190) the Lagrange multiplier function can be found by integration, using the normalization condition IXP=I, in the form IXPΛX= ΛXfor ΛX(x)∝ne}ationslash= 0. Thus, multiplying Eq. (190) by IXyields ΛX=IX/parenleftig N−PP′−1Kφ/parenrightig =NX−IXPP′−1Kφ. (191) ΛXis now eliminated by inserting Eq. (191) into Eq. (190) 0 = (I−PIX)/parenleftig N−PP′−1Kφ/parenrightig . (192) A simple iteration procedure, provided K−1exists, is suggested by writing Eq. (189) in the form Kφ=Tφ, φi+1=K−1Tφ(φi), (193) with Tφ(φ) =P′P−1N−P′ΛX. (194) Table 2 lists constraints to be implemented explicitly for s ome choices of φ. 52φ P(φ) constraints P(x,y)P=P norm positivity z(x,y)P=z//integraltextzdy — positivity L(x,y) = lnPP=eLnorm — g(x,y)P=eg//integraltextegdy — — Φ =/integraltextydy′PP=dΦ/dy boundary monotony Table 2: Constraints for specific choices of φ 3.3.2 Example: Square root of P We already discussed the cases φ= lnPwithP′=P=eL,P/P′= 1 and φ=PwithP′= 1,P/P′=P. The choice φ=√ Pyields the common L2–normalization condition over y 1 =/integraldisplay dyφ2(x,y),∀x∈X, (195) which is quadratic in φ, andP=φ2,P′= 2φ,P/P′=φ/2. For real φthe positivity condition P≥0 is automatically satisfied [77, 195]. Forφ=√ Pand a negative Laplacian inverse covariance K=−∆, one can relate the corresponding Gaussian prior to the Fisher information [34, 195, 191]. Consider, for example, a problem with fixed x(soxcan be skipped from the natotion and one can write P(y)) and ady–dimensional y. Then one has, assuming the necessary differentiability conditio ns and vanishing boundary terms, (φ,Kφ) =−(φ,∆φ) =/integraldisplay dydy/summationdisplay k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂φ ∂yk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (196) =dy/summationdisplay k/integraldisplaydy 4P(y)/parenleftigg∂P(y) ∂yk/parenrightigg2 =1 4dy/summationdisplay kIF k(0), (197) 53whereIF k(0) is the Fisher information, defined as IF k(y0) =/integraldisplay dy/vextendsingle/vextendsingle/vextendsingle∂P(y−y0) ∂y0/vextendsingle/vextendsingle/vextendsingle2 P(y−y0)=/integraldisplay dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂lnP(y−y0) ∂y0 k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 P(y−y0 k), (198) for the family P(· −y0) with location parameter vector y0. A connection to quantum mechanics can be found considering t he training data free case Eφ=1 2(φ,Kφ) + (ΛX, φ), (199) has the homogeneous stationarity equation Kφ=−2ΦΛX. (200) Forx–independent Λ Xthis is an eigenvalue equation. Examples include the quantum mechanical Schr¨ odinger equation where Kcorresponds to the system Hamiltonian and −2ΛX=(φ,Kφ) (φ, φ), (201) to its ground state energy. In quantum mechanics Eq. (201) is the basis for variational methods (see Section 4) to obtain approxima te solutions for ground state energies [50, 183, 24]. Similarly, one can take φ=/radicalig −(L−Lmax) forLbounded from above by Lmaxwith the normalization 1 =/integraldisplay dye−φ2(x,y)+Lmax,∀x∈X, (202) andP=e−φ2+Lmax,P′=−2φP,P/P′=−1/(2φ). 3.3.3 Example: Distribution functions Instead in terms of the probability density function, one ca n formulate the prior in terms of its integral, the distribution function. T he densityPis then recovered from the distribution function φby differentiation, P(φ) =dy/productdisplay k∂φ ∂yk=dy/productdisplay k∇ykφ=dy/circlemultiplydisplay kR−1 kφ.=R−1φ, (203) 54resulting in a non–diagonal P′. The inverse of the derivative operator R−1 is the integration operator R=/circlemultiplytextdy kRkPwith matrix elements R(x,y;x′,y′) =δ(x−x′)θ(y−y′), (204) i.e., Rk(x,y;x′,y′) =δ(x−x′)/productdisplay l∝negationslash=kδ(yl−y′ l)θ(yk−y′ k). (205) Thus, (203) corresponds to the transformation of ( x–conditioned) density functionsPin (x–conditioned) distribution functions φ=RP, i.e.,φ(x,y) =/integraltexty −∞P(x,y′)dy′. Because RTKRis (semi)–positive definite if Kis, a specific prior which is Gaussian in the distribution function φis also Gaussian in the densityP.P′becomes P′(x,y;x′,y′) =δ/parenleftig/producttextdy k∇yk′φ(x′,y′/parenrightig δφ(x,y)=δ(x−x′)dy/productdisplay kδ′(yk−y′ k).(206) Here the derivative of the δ–function is defined by formal partial integration /integraldisplay∞ −∞dy′f(y′)δ′(y−y′) =f(y′)δ(y′−y)|∞ −∞−f′(y). (207) Fixingφ(x,−∞) = 0 the variational derivative δ/(δφ(x,−∞)) is not needed. The normalization condition for Pbecomes for the distribution function φ= RPthe boundary condition φ(x,∞) = 1, ∀x∈X. The positivity condition forPcorresponds to the monotonicity condition φ(x,y)≥φ(x,y′),∀y≥y′, ∀x∈Xand toφ(x,−∞)≥0,∀x∈X. 3.4 Covariances and invariances 3.4.1 Approximate invariance Prior terms can often be related to the assumption of approxi mate invariances or approximate symmetries. A Laplacian smoothness functio nal, for exam- ple, measures the deviation from translational symmetry un der infinitesimal translations. Consider for example a linear mapping φ→˜φ=Sφ, (208) 55given by the operator S. To compare φwith˜φwe define a (semi–)distance defined by choosing a positive (semi–)definite KS, and use as error measure 1 2/parenleftig (φ−Sφ),KS(φ−Sφ)/parenrightig =1 2/parenleftig φ,Kφ/parenrightig . (209) Here K= (I−S)TKS(I−S) (210) is positive semi–definite if KSis. Conversely, every positive semi–definite K can be written K=WTWand is thus of form (210) with S=I−Wand KS=I. Including terms of the form of (210) in the error functional forcesφ to be similar to ˜φ. A special case are mappings leaving the norm invariant (φ, φ) = (Sφ,Sφ) = (φ,STSφ). (211) For realφand˜φi.e., (Sφ) = (Sφ)∗, this requires ST=S−1andS∗=S. Thus, in that case Shas to be an orthogonal matrix ∈O(N) and can be written S(θ) =eA=e/summationtext iθiAi=∞/summationdisplay k=01 k!/parenleftigg/summationdisplay iθiAi/parenrightiggk , (212) with antisymmetric A=−ATand real parameters θi. Selecting a set of (generators) Aithe matrices obtained be varying the parameters θiform a Lie group. Up to first order the expansion of the exponential f unction reads S≈1 +/summationtext iθiAi. Thus, we can define an error measure with respect to an infinitesimal symmetry by 1 2/summationdisplay i/parenleftiggφ−(1 +θiAi)φ θi,KSφ−(1 +θiAi)φ θi/parenrightigg =1 2(φ,/summationdisplay iAT iKSAiφ). (213) 3.4.2 Approximate symmetries Next we come to the special case of symmetries, i.e., invaria nce under un- der coordinate transformations. Symmetry transformation sSchange the arguments of a function φ. For example for the translation of a function φ(x)→˜φ(x) =Sφ(x) =φ(x−c). Therefore it is useful to see how Sacts on the arguments of a function. Denoting the (possibly impro per) eigen- vectors of the coordinate operator xwith eigenvalue xby (·, x) =|x), i.e., 56x|x) =x|x), function values can be expressed as scalar products, e.g. φ(x) = (x, φ) for a function in x, or, in two variables, φ(x,y) = (x⊗y, φ). (Note that in this ‘eigenvalue’ notation, frequently used by phys icists, for example 2|x)∝ne}ationslash=|2x).) Thus, we see that the action of Son some function h(x) is equivalent to the action of ST( =S−1if orthogonal) on |x) Sφ(x) = (x,Sφ) = (STx,φ), (214) or forφ(x,y) Sφ(x,y) =/parenleftig ST(x⊗y), φ/parenrightig . (215) Assuming S=SxSywe may also split the action of S, Sφ(x,y) =/parenleftig (ST xx)⊗y,Syφ/parenrightig . (216) An example from physics are vector fields where xandφ(·,y) form three dimensional vectors with yrepresenting a linear combination of component labels ofφ. Notice that S|x) does, for a general operator S, not have to be an eigen- vector of the coordinate operator xagain. Coordinate transformations, how- ever, are represented by operators S, which map coordinate eigenvectors |x) to other coordinate eigenvectors |σ(x)) (and not to arbitrary vectors being linear combinations of |x)). Hence, such coordinate transformations Sjust changes the argument xof a function φintoσ(x), i.e., Sφ(x) =φ(σ(x)), (217) withσ(x) a permutation or a one–to–one coordinate transformation. Thus, even for an arbitrary nonlinear coordinate transformation σthe correspond- ing operator Sin the space of φis linear. (This is one of the reasons why linear functional analysis is so useful.) A special case are linear coordinate transformations for wh ich we can writeφ(x)→˜φ(x) =Sφ(x) =φ(Sx), withS(in contrast to S) acting in the space ofx. An example of such Sare coordinate rotations which preserve the norm in x–space, analogously to Eq. (211) for φ, and form a Lie group S(θ) =e/summationtext iθiAiacting on coordinates, analogously to Eq. (212). 3.4.3 Example: Infinitesimal translations A Laplacian smoothness prior, for example, can be related to an approxi- mate symmetry under infinitesimal translations. Consider t he group of d– dimensional translations which is generated by the gradien t operator ∇. This 57can be verified by recalling the multidimensional Taylor for mula for expan- sion ofφatx S(θ)φ(x) =e/summationtext iθi∇iφ(x) =∞/summationdisplay k=0(/summationtext iθi∇i)k k!φ(x) =φ(x+θ). (218) Up to first order S≈1 +/summationtext iθi∆i. Hence, for infinitesimal translations, the error measure of Eq. (213) becomes 1 2/summationdisplay i/parenleftiggφ−(1 +θi∆i)φ θi,φ−(1 +θi∆i)φ θi/parenrightigg =1 2(φ,/summationdisplay i∇T i∇iφ)=−1 2(φ,∆φ). (219) assuming vanishing boundary terms and choosing KS=I. This is the clas- sical Laplacian smoothness term. 3.4.4 Example: Approximate periodicity As another example, lets us discuss the implementation of ap proximate pe- riodicity. To measure the deviation from exact periodicity let us define the difference operators ∇R θφ(x) =φ(x)−φ(x+θ), (220) ∇L θφ(x) =φ(x−θ)−φ(x). (221) For periodic boundary conditions ( ∇L θ)T=−∇R θ, where ( ∇L θ)Tdenotes the transpose of ∇L θ. Hence, the operator, ∆θ=∇L θ∇R θ=−(∇R θ)T∇R θ, (222) defined similarly to the Laplacian, is positive definite, and a possible error term, enforcing approximate periodicity with period θ, is 1 2(∇R(θ)φ,∇R(θ)φ) =−1 2(φ,∆θφ) =1 2/integraldisplay dx|φ(x)−φ(x+θ)|2.(223) As every periodic function with φ(x) =φ(x+θ) is in the null space of ∆ θ typically another error term has to be added to get a unique so lution of the stationarity equation. Choosing, for example, a Laplacian smoothness term, yields −1 2(φ,(∆ +λ∆θ)φ). (224) 58In caseθis not known, it can be treated as hyperparameter as discusse d in Section 5. Alternatively to an implementation by choosing a semi–posi tive definite operator Kwith symmetric functions in its null space, approximate sym me- tries can be implemented by giving explicitly a symmetric re ference function t(x). For example,1 2(φ−t,K(φ−t) ) witht(x) =t(x+θ). This possibility will be discussed in the next section. 3.5 Non–zero means A prior energy term (1 /2)(φ,Kφ) measures the squared K–distance of φto the zero function t≡0. Choosing a zero mean function for the prior process is calculationally convenient for Gaussian priors, but by n o means mandatory. In particular, a function φis in practice often measured relative to some non– trivial base line. Without further a priori information that base line can in principle be an arbitrary function. Choosing a zero mean fun ction that base line does not enter the formulae and remains hidden in the rea lization of the measurement process. On the the other hand, including expli citly a non– zero mean function t, playing the role of a function template (or reference, target, prototype, base line) and being technically relati vely straightforward, can be a very powerful tool. It allows, for example, to parame terizet(θ) by introducing hyperparameters (see Section 5) and to specify explicitly different maxima of multimodal functional priors (see Section 6. [123 , 124, 125, 126, 127]) All this cannot be done by referring to a single baselin e. Hence, in this section we consider error terms of the form 1 2/parenleftig φ−t,K(φ−t)/parenrightig . (225) Mean or template functions tallow an easy and straightforward implementa- tion of prior information in form of examples for φ. They are the continuous analogue of standard training data. The fact that template f unctionstare most times chosen equal to zero, and thus do not appear explic itly in the error functional, should not obscure the fact that they are o f key importance for any generalization. There are many situations where it c an be very valu- able to include non–zero prior means explicitly. Template f unctions for φcan for example result from learning done in the past for the same or for similar tasks. In particular, consider for example ˜φ(x) to be the output of an empiri- cal learning system (neural net, decision tree, nearest nei ghbor methods, ...) 59being the result of learning the same or a similar task. Such a ˜φ(x) would be a natural candidate for a template function t(x). Thus, we see that template functions could be used for example to allow transfer of knowledge between similar tasks or to include the results of earlier learning on the same task in case the original data are lost but the output of another lear ning system is still available. Including non–zero template functions generalizes functi onalEφof Eq. (187) to Eφ=−(lnP(φ), N) +1 2/parenleftig φ−t,K(φ−t)/parenrightig + (P(φ),ΛX) (226) =−(lnP(φ), N) +1 2(φ,Kφ)−(J, φ)+(P(φ),ΛX)+const.(227) In the language of physics J=Ktrepresents an external field coupling to φ(x,y), similar, for example, to a magnetic field. A non–zero field l eads to a non–zero expectation of φin the no–data case. The φ–independent constant stands for the term1 2(t,Kt), or1 2(J,K−1J) for invertible K, and can be skipped from the error/energy functional Eφ. The stationarity equation for an Eφwith non–zero template tcontains an inhomogeneous term Kt=J 0 =P′(φ)P−1(φ)N−P′(φ)ΛX−K(φ−t), (228) with, for invertible PP′−1and ΛX∝ne}ationslash= 0, ΛX=IX/parenleftig N−PP′−1K(φ−t)/parenrightig . (229) Notice that functional (226) can be rewritten as a functiona l with zero tem- platet≡0 in terms of/tildewideφ=φ−t. That is the reason why we have not included non–zero templates in the previous sections. For general no n–additive com- binations of squared distances of the form (225) non–zero te mplates cannot be removed from the functional as we will see in Section 6. Add itive combi- nations of squared error terms, on the other hand, can again b e written as one squared error term, using a generalized ‘bias–variance ’–decomposition 1 2N/summationdisplay j=1/parenleftig φ−tj,Kj(φ−tj)/parenrightig =1 2/parenleftig φ−t,K(φ−t)/parenrightig +Emin (230) withtemplate average t=K−1N/summationdisplay j=1Kjtj, (231) 60assuming the existence of the inverse of the operator K=N/summationdisplay j=1Kj. (232) andminimal energy/error Emin=N 2V(t1,···tN) =1 2N/summationdisplay j=1(tj,Kjtj)−(t,Kt), (233) which up to a factor N/2 represents a generalized template variance V. We end with the remark that adding error terms corresponds in it s probabilistic Bayesian interpretation to ANDing independent events. For example, if we wish to implement that φis likely to be smooth AND mirror symmetric, we may add two squared error terms, one related to smoothness an d another to mirror symmetry. According to (230) the result will be a sing le squared error term of form (225). Summarizing, we have seen that there are many potentially us eful ap- plications of non–zero template functions. Technically, h owever, non–zero template functions can be removed from the formalism by a sim ple substitu- tionφ′=φ−tif the error functional consists of an additive combination of quadratic prior terms. As most regularized error functiona ls used in practice have additive prior terms this is probably the reason that th ey are formulated fort≡0, meaning that non–zero templates functions (base lines) h ave to be treated by including a preprocessing step switching from φtoφ′. We will see in Section 6 that for general error functionals templates ca nnot be removed by a simple substitution and do enter the error functionals e xplicitly. 3.6 Quadratic density estimation and empirical risk minimization Interpreting an energy or error functional Eprobabilistically, i.e., assuming −βE+cto be the logarithm of a posterior probability under study, t he form of the training data term has to be −/summationtext ilnPi. Technically, however, it would be easier to replace that data term by one which is quadr atic in the probability Pof interest. Indeed, we have mentioned in Section 2.5 that such functiona ls can be justified within the framework of empirical risk minimizati on. From that 61Frequentist point of view an error functional E(P), is not derived from a log–posterior, but represents an empirical risk ˆ r(P,f) =/summationtext il(xi,yi,P), ap- proximating an expected risk r(P,f) for action a=P. This is possible under the assumption that training data are sampled accordi ng to the true p(x,y|f). In that interpretation one is therefore not restricted to a log–loss for training data but may as well choose for training data a qu adratic loss like1 2/parenleftig P−Pemp,KD(P−Pemp)/parenrightig , (234) choosing a reference density Pemp and a real symmetric positive (semi–)/- definite KD. Approximating a joint probability p(x,y|h) the reference density Pemp would have to be the joint empirical density Pjoint emp(x,y) =1 nn/summationdisplay iδ(x−xi)δ(y−yi), (235) i.e.,Pjoint emp=N/n, as obtained from the training data. Approximating con- ditional probabilities p(y|x,h) the reference Pemphas to be chosen as condi- tional empirical density, Pemp(x,y) =/summationtext iδ(x−xi)δ(y−yi) /summationtext iδ(x−xi)=N(x,y) nx, (236) or, defining the diagonal matrix NX(x,x′,y,y′) =δ(x−x′)δ(y−y′)NX(x) = δ(x−x′)δ(y−y′)/summationtext iδ(x−xi) Pemp=N−1 XN. (237) This, however, is only a valid expression if NX(x)∝ne}ationslash= 0, meaning that for all xat least one measured value has to be available. For xvariables with a large number of possible values, this cannot be assumed. For continuous x variables it is even impossible. Hence, approximating conditional empirical densities eit her non–data x– values must be excluded from the integration in (234) by usin g an operator KDcontaining the projector/summationtext x′∈xDδ(x−x′), orPempmust be defined also for such non–data x–values. For existing VX=IX1 =/integraltextdy1, a possible extension ˜PempofPempwould be to assume a uniform density for non–data xvalues, 62yielding ˜Pemp(x,y) =  /summationtext iδ(x−xi)δ(y−yi)/summationtext iδ(x−xi)for/summationtext iδ(x−xi)∝ne}ationslash= 0, 1/integraltext dy1for/summationtext iδ(x−xi) = 0.(238) This introduces a bias towards uniform probabilities, but h as the advantage to give a empirical density for all xand to fulfill the conditional normalization requirements. Instead of a quadratic term in P, one might consider a quadratic term in the log–probability L. The log–probability, however, is minus infinity at all non–data points ( x,y)∝ne}ationslash∈D. To work with a finite expression, one can choose smallǫ(y) and approximate Pempby Pǫ emp(x,y) =ǫ(y) +/summationtext iδ(x−xi)δ(y−yi)/integraltextdyǫ(y) +/summationtext iδ(x−xi), (239) provided/integraltextdyǫ(y) exists. For ǫ(y)∝ne}ationslash= 0 alsoPǫ emp(x,y)∝ne}ationslash= 0,∀xandLǫ emp= lnPǫ emp>−∞exists. A quadratic data term in Presults in an error functional ˜EP=1 2/parenleftig P−Pemp,KD(P−Pemp)/parenrightig +1 2(P,KP) + (P,ΛX), (240) skipping the constant part of the Λ X–terms. In (240) the empirical density Pempmay be replaced by ˜Pempof (238). Positive (semi–)definite operators KDhave a square root and can be written in the form RTR. One possibility, skipping for the sake of simplicity xin the following, is to choose as square root Rthe integration operator, i.e., R=/circlemultiplytext kRkandR(y,y′) =θ(y−y′). Thus,φ=RPtransforms the density functionPin the distribution function φ, and we have P=P(φ) =R−1φ. Here the inverse R−1is the differentiation operator/producttext k∇yk(with appropriate boundary condition) and/parenleftig RT/parenrightig−1R−1=−/producttext k∆kis the product of one– dimensional Laplacians ∆ k=∂2/∂y2 k. Adding for example a regularizing term (165)λ 2(P, P) gives ˜EP=1 2/parenleftig P−Pemp,RTR(P−Pemp)/parenrightig +λ 2(P, P) (241) =1 2/parenleftigg/parenleftig φ−φemp, φ−φemp/parenrightig −λ/parenleftig φ,/productdisplay k∆kφ/parenrightig/parenrightigg (242) 63=1 2m2/parenleftig φ,(−/productdisplay k∆k+m2I)φ/parenrightig −(φ,φemp) +1 2(φemp, φemp). (243) withm2=λ−1. Here the empirical distribution function φemp=RPempis given byφemp(y) =1 n/summationtext iθ(y−yi) (or, including the xvariable,φemp(x,y) = 1 NX(x)/summationtext x′∈xDδ(x−x′)θ(y−yi) forNX(x)∝ne}ationslash= 0 which could be extended to a linear ˜φ=R˜PempforNX(x) = 0). The stationarity equation yields φ=m2/parenleftigg −/productdisplay k∆k+m2I/parenrightigg−1 φemp. (244) Fordy= 1 (orφ=/producttext kφ) the operator becomes ( −∆ +m2I)−1which has the structure of a free massive propagator for a scalar field with massm2and is calculated below. As already mentioned the normalizatio n and positivity condition for Pappear forφas boundary and monotonicity conditions. For non–constant Pthe monotonicity condition has not to be implemented by Lagrange multipliers as the gradient at the stationary poin t has no compo- nents pointing into the forbidden area. (But the conditions nevertheless have to be checked.) Kernel methods of density estimation, like t he use of Parzen windows, can be founded on such quadratic regularization fu nctionals [208]. Indeed, the one–dimensional Eq. (244) is equivalent to the u se of Parzens kernel in density estimation [169, 156]. 3.7 Regression 3.7.1 Gaussian regression An important special case of density estimation leading to q uadratic data terms is regression for independent training data with Gaus sian likelihoods p(yi|xi,h) =1√ 2πσe−(yi−h(xi))2 2σ2, (245) with fixed, but possibly xi–dependent, variance σ2. In that case P(x,y) = p(yi|xi,h) is specified by φ=hand the logarithmic term/summationtext ilnPibecomes quadratic in the regression function h(xi), i.e., of the form (225). In an inter- pretation as empirical risk minimization quadratic error t erms corresponds to the choice of a squared error loss function l(x,y,a ) = (y−a(x))2for ac- tiona(x). Similarly, the technical analogon of Bayesian priors are additional (regularizing) cost terms. 64We have remarked in Section 2.3 that for continuous xmeasurement of h(x) has to be understood as measurement of a h(˜x) =/integraltextdxϑ(x)h(x) for sharply peaked ϑ(x). We assume here that the discretization of hused in numerical calculations takes care of that averaging. Diver gent quantities like δ–functionals, used here for convenience, will then not be pr esent. We now combine Gaussian data terms and a Gaussian (specific) p rior with prior operator K0(x,x′) and define for training data xi,yithe operator Ki(x,x′) =δ(x−xi)δ(x−x′), (246) and training data templates t=yi. We also allow a general prior template t0but remark that it is often chosen identically zero. Accordi ng to (230) the resulting functional can be written in the following forms, useful for different purposes, Eh=1 2n/summationdisplay i=1(h(xi)−yi)2+1 2(h−t0,K0(h−t0) )X (247) =1 2n/summationdisplay i=1(h−ti,Ki(h−ti) )X+1 2(h−t0,K0(h−t0) )X(248) =1 2(h−tD,KD(h−tD))X+1 2(h−t0,K0(h−t0))X+ED min(249) =1 2(h−t,K(h−t) )X+Emin, (250) with KD=n/summationdisplay i=1Ki, tD=K−1 Dn/summationdisplay i=1Kiti, (251) K=n/summationdisplay i=0Ki, t=K−1n/summationdisplay i=0Kiti, (252) andh–independent minimal errors, ED min=1 2/parenleftiggn/summationdisplay i=1(ti,Kiti)X+ (tD,KDtD)X/parenrightigg , (253) Emin=1 2/parenleftiggn/summationdisplay i=0(ti,Kiti)X+ (t,Kt)X/parenrightigg , (254) being proportional to the “generalized variances” VD= 2ED min/nandV= 2Emin/(n+ 1). The scalar product ( ·,·)Xstands forx–integration only, for 65the sake of simplicity however, we will skip the subscript Xin the following. The data operator KD KD(x,x′) =n/summationdisplay i=1δ(x−xi)δ(x−x′) =nxδ(x−x′), (255) contains for discrete xon its diagonal the number of measurements at x, nx=NX(x) =n/summationdisplay i=1δ(x−xi), (256) which is zero for xnot in the training data. As already mentioned for con- tinuousxa integration around a neighborhood of xiis required. K−1 Dis a short hand notation for the inverse within the space of train ing data K−1 D= (IDKDID)−1=δ(x−x′)/nx, (257) IDdenoting the projector into the space of training data ID=δ(x−x′)˜n/summationdisplay i=1δ(x−xi). (258) Notice that the sum is not over all ntraining points xibut only over the ˜n≤ndifferentxi. (Again for continuous xan integration around xiis required to ensure I2 D=ID). Hence, the data template tDbecomes the mean ofy–values measured at x tD(x) =1 nxnx/summationdisplay j=1 xj=xy(xj), (259) andtD(x) = 0 fornx= 0. Normalization of P(x,y) is not influenced by a change inh(x) so the Lagrange multiplier terms have been skipped. The stationarity equation is most easily obtained from (250 ), 0 =K(h−t). (260) It is linear and has on a space where K−1exists the unique solution h=t. (261) 66We remark that Kcan be invertible (and usually is so the learning problem is well defined) even if K0is not invertible. The inverse K−1, necessary to calculatet, is training data dependent and represents the covariance o pera- tor/matrix of a Gaussian posterior process. In many practic al cases, however, the prior covariance K−1 0(or in case of a null space a pseudo inverse of K0) is directly given or can be calculated. Then an inversion of a finite dimen- sional matrix in data space is sufficient to find the minimum of t he energy Eh[212, 71]. Invertible K 0: Let us assume first deal with the case of an invertible K0. It is the best to begin the stationarity equation as obtaine d from (248) or (249) 0 =n/summationdisplay i=1Ki(h−ti) +K0(h−t0) (262) =KD(h−tD) +K0(h−t0). (263) For existing K−1 0 h=t0+K−1 0KD(tD−h), (264) one can introduce a=KD(tD−h), (265) to obtain h=t0+K−1 0a. (266) Inserting Eq. (266) into Eq. (265) one finds an equation for a /parenleftig I+KDK−1 0/parenrightig a=KD(tD−t0). (267) Multiplying Eq. (267) from the left by the projector IDand using KDID=IDKD, a=IDa, tD=IDtD, (268) one obtains an equation in data space /parenleftig ID+KDK−1 0,DD/parenrightig a=KD(tD−t0,D), (269) where K−1 0,DD= (K−1 0)DD=IDK−1 0ID∝ne}ationslash= (K0,DD)−1, t0,D=IDt0. (270) 67Thus, a=CDDb, (271) where CDD=/parenleftig ID+KDK−1 0,DD/parenrightig−1, (272) and b=KD(tD−t0). (273) In components Eq. (271) reads, /summationdisplay l/parenleftig δkl+nxkK−1 0(xk,xl)/parenrightig a(xl) =nxk(tD(xk)−t0(xk)). (274) Having calculated athe solution his given by Eq. (266) h=t0+K−1 0CDDb=t0+K−1 0/parenleftig K−1 D+K−1 0,DD/parenrightig−1(tD−t0). (275) Eq. (275) can also be obtained directly from Eq. (261) and the definitions (252), without introducing the auxiliary variable a, using the decomposition K0t0=−KDt0+ (K0+KD)t0and K−1KD=K−1 0/parenleftig I+KDK−1 0/parenrightig−1KD=K−1 0∞/summationdisplay m=0/parenleftig −KDK−1 0/parenrightigmKD(276) =K−1 0∞/summationdisplay m=0/parenleftig −KDIDK−1 0ID/parenrightigmKD=K−1 0/parenleftig ID+KDK−1 0,DD/parenrightig−1KD.(277) K−1 0CDDis also known as equivalent kernel due to its relation to kern el smoothing techniques [194, 87, 83, 71]. Interestingly, Eq. (266) still holds for non–quadratic dat a terms of the formgD(h) with any differentiable function fulfilling g(h) =g(hD), wherehD =IDhis the restriction of hto data space. Hence, also the function of func- tional derivatives with respect to h(x) is restricted to data space, i.e., g′(hD) =g′ D(hD) withg′ D=IDg′andg′(h,x) =δg(h)/δh(x). For example, g(h) =/summationtextn i=1V(h(xi)−yi) withVa differentiable function. The finite dimensional vectorais then found by solving a nonlinear equation instead of a lin ear one [68, 70]. Furthermore, one can study vector fields, i.e., the case wher e, besides possiblyx, alsoy, and thus h(x), is a vector for given x. (Considering the variable indicating the vector components of yas part of the x–variable, this 68is a situation where a fixed number of one–dimensional y, corresponding to a subspace of Xwith fixed dimension, is always measured simultaneously.) I n that case the diagonal Kiof Eq. (246) can be replaced by a version with non– zero off–diagonal elements Kα,α′between the vector components αofy. This corresponds to a multi–dimensional Gaussian data generati ng probability p(yi|xi,h) =detKi1 2 (2π)k 2e−1 2/summationtext α,α′(yi,α−hα(xi))Ki,α,α′(xi)(yi,α′−hα′(xi)), (278) fork–dimensional vector yiwith components yi,α. Non-invertible K 0: For non–invertible K0one can solve for husing the Moore–Penrose inverse K# 0. Let us first recall some basic facts [53, 151, 13, 112]. A pseudo inverse of (a possibly non–square) Ais defined by the conditions A#AA#=A,AA#A=A#, (279) and becomes for real Athe unique Moore–Penrose inverse A#if (AA#)T=AA#,(A#A)T=A#A. (280) A linear equation Ax=b (281) is solvable if AA#b=b. (282) In that case the solution is x=A#b+x0=A#b+y−A#Ay, (283) wherex0=y−A#Ayis solution of the homogeneous equation Ax0= 0 and vectoryis arbitrary. Hence, x0can be expanded in an orthonormalized basis ψlof the null space of A x0=/summationdisplay lclψl. (284) For an Awhich can be diagonalized, i.e., A=M−1DMwith diagonal D, the Moore–Penrose inverse is A#=M−1D#M. Therefore AA#=A#A=I1=I−I0. (285) 69whereI0=/summationtext lψlψT lis the projector into the zero space of AandI1=I−I0 =M−1DD#M. Thus, the solvability condition Eq. (282) becomes I0b= 0, (286) or in terms of ψl (ψl, b) = 0,∀l, (287) meaning that the inhomogeneity bmust have no components within the zero space of A. Now we apply this to Eq. (263) where K0is diagonalizable because pos- itive semi definite. (In this case Mis an orthogonal matrix and the entries ofDare real and larger or equal to zero.) Hence, one obtains unde r the condition I0(K0t0+KD(tD−h)) = 0, (288) for Eq. (283) h=K# 0(K0t0+KD(tD−h)) +h0, (289) whereK0h0= 0 so that h0=/summationtext lclψlcan be expanded in an orthonormalized basisψlof the null space of K0, assumed here to be of finite dimension. To find an equation in data space define the vector a=KD(tD−h), (290) to get from Eqs.(288) and (289) 0 = (ψl,K0t0) + (ψl, a),∀l (291) h=K# 0(K0t0+a) +/summationdisplay lclψl. (292) These equations have to be solved for aand the coefficients cl. Inserting Eq. (292) into the definition (290) gives (I+KDK# 0)a=KDtD−KDI1t0−KD/summationdisplay lclψl, (293) usingK# 0K0=I1according to Eq. (285). Using a=IDathe solvability condition (288) becomes ˜n/summationdisplay i=1ψl(xi)a=−(ψl,K0t0),∀l, (294) 70the sum going over different xionly. Eq. (293) for aandclreads in data space, similar to Eq. (269), a=˜C˜b, (295) where ˜C−1=I+KDK# 0has been assumed invertible and ˜bis given by the right hand side of Eq. (293). Inserting into Eq. (292) the sol ution finally can be written h=I1t0+K# 0˜C˜b+/summationdisplay lclψl. (296) Again, general non–quadratic data terms g(hD) can be allowed. In that caseδg(hD)/δh(x) =g′(hD,x) = (IDg′)(hD,x) and Eq. (290) becomes the nonlinear equation a=g′(hD) =g′/parenleftig ID/parenleftig K# 0(K0t0+KD(tD−h)) +h0/parenrightig/parenrightig . (297) The solution(s) aof that equation have then to be inserted in Eq. (292). 3.7.2 Exact predictive density For Gaussian regression the predictive density under train ing dataDand priorD0can be found analytically without resorting to a saddle poin t ap- proximation. The predictive density is defined as the h-integral p(y|x,D,D 0) =/integraldisplay dhp(y|x,h)p(h|D,D 0) =/integraltextdhp(y|x,h)p(yD|xD,h)p(h|D0)/integraltextdhp(yD|xD,h)p(h|D0) =p(y,yD|x,xD,D0) p(yD|xD,D0). (298) Denoting training data values yibytisampled with covariance Kiconcen- trated onxiand analogously test data values y=yn+1bytn+1sampled with (co–)variance Kn+1, we have for 1 ≤i≤n+ 1 p(yi|xi,h) = det( Ki/2π)1 2e−1 2/parenleftig h−ti,Ki(h−ti)/parenrightig , (299) and p(h|D0) = det( K0/2π)1 2e−1 2/parenleftig h−t0,K0(h−t0)/parenrightig , (300) 71hence p(y|x,D,D 0) =/integraltextdhe−1 2/summationtextn+1 i=0/parenleftig h−ti,Ki(h−ti)/parenrightig +1 2/summationtextn+1 i=0ln deti(Ki/2π) /integraltextdhe−1 2/summationtextn i=0/parenleftig h−ti,Ki(h−ti)/parenrightig +1 2/summationtextn i=0ln deti(Ki/2π).(301) Here we have this time written explicitly det i(Ki/2π) for a determinant calcu- lated in that space where Kiis invertible. This is useful because for example in general det iKidetK0∝ne}ationslash= detiKiK0. Using the generalized ‘bias–variance’– decomposition (230) yields p(y|x,D,D 0) =/integraltextdhe−1 2/parenleftig h−t+,K+(h−t+)/parenrightig +n 2V++1 2/summationtextn+1 i=0ln deti(Ki/2π) /integraltextdhe−1 2/parenleftig h−t,K(h−t)/parenrightig +n 2V+1 2/summationtextn i=0ln deti(Ki/2π),(302) with t=K−1n/summationdisplay i=0Kiti,K=n/summationdisplay i=0Ki, (303) t+=K−1 +n+1/summationdisplay i=0Kiti,K+=n+1/summationdisplay i=0Ki, (304) V=1 nn/summationdisplay i=0/parenleftig ti,Kiti/parenrightig −/parenleftig t,K nt/parenrightig , (305) V+=1 nn+1/summationdisplay i=0/parenleftig ti,Kiti/parenrightig −/parenleftig t+,K+ nt+/parenrightig . (306) Now theh–integration can be performed p(y|x,D,D 0) =e−n 2V++1 2/summationtextn+1 i=0ln deti(Ki/2π)−1 2ln det(K+/2π) e−n 2V+1 2/summationtextn i=0lndeti(Ki/2π)−1 2lndet(K/2π)(307) Canceling common factors, writing again yfortn+1,KxforKn+1, detxfor detn+1, and using K+t+=Kt+Kxy, this becomes p(y|x,D,D 0) =e−1 2(y,Kyy)+(y,Kyt)+1 2(t,(KK−1 +K−K)t)+1 2ln detx(KxK−1 +K/2π). (308) Here we introduced Ky=KT y=Kx−KxK−1 +Kxand used that detK−1K+= det(I−K−1Kx) = detxK−1K+ (309) 72can be calculated in the space of test data x. This follows from K=K+−Kx and the equality det/parenleftigg 1−A0 B 1/parenrightigg = det(1 −A) (310) withA=IxK−1Kx,B= (I−Ix)K−1Kx, andIxdenoting the projector into the space of test data x. Finally Ky=Kx−KxK−1 +Kx=KxK−1 +K= (K−KK−1 +K), (311) yields the correct normalization of the predictive density p(y|x,D,D 0) =e−1 2/parenleftig y−¯y,Ky(y−¯y)/parenrightig +1 2lndetx(Ky/2π), (312) with mean and covariance ¯y=t=K−1n/summationdisplay i=0Kiti, (313) K−1 y=/parenleftig Kx−KxK−1 +Kx/parenrightig−1=K−1 x+IxK−1Ix. (314) It is useful to express the posterior covariance K−1by the prior covariance K−1 0. According to /parenleftigg 1 +A B 0 1/parenrightigg−1 =/parenleftigg (1 +A)−1−(1 +A)−1B 0 1/parenrightigg , (315) withA=KDK−1 0,DD,B=KDK−1 0,D¯D, and K−1 0,DD=IDK−1 0ID,K−1 0,D¯D= IDK−1 0I¯D,I¯D=I−IDwe find K−1=K−1 0/parenleftig I+KDK−1 0/parenrightig−1(316) =K−1 0/parenleftbigg/parenleftig ID+KDK−1 0,DD/parenrightig−1−/parenleftig ID+KDK−1 0,DD/parenrightig−1KDK−1 0,D¯D+I¯D/parenrightbigg . Notice that while K−1 D= (IDKDID)−1in general K−1 0,DD=IDK−1 0ID∝ne}ationslash= (IDK0ID)−1. This means for example that K−1 0has to be known to find K−1 0,DDand it is not enough to invert IDK0ID=K0,DD∝ne}ationslash= (K−1 0,DD)−1. In data space/parenleftig ID+KDK−1 0,DD/parenrightig−1=/parenleftig K−1 D+K−1 0,DD/parenrightig−1K−1 D, so Eq. (316) can be manipulated to give K−1=K−1 0/parenleftbigg I−ID/parenleftig K−1 D+K−1 0,DD/parenrightig−1IDK−1 0/parenrightbigg . (317) 73This allows now to express the predictive mean (313) and cova riance (314) by the prior covariance ¯y=t0+K−1 0/parenleftig K−1 D+K−1 0,DD/parenrightig−1(tD−t0), (318) K−1 y=Kx+K−1 0,xx−K−1 0,xD/parenleftig K−1 D+K−1 0,DD/parenrightig−1K−1 0,Dx. (319) Thus, for given prior covariance K−1 0both, ¯yandK−1 y, can be calculated by inverting the ˜ nטnmatrix/tildewiderK=/parenleftig K−1 0,DD+K−1 D/parenrightig−1. Comparison of Eqs.(318,319) with the maximum posterior sol utionh∗of Eq. (275) now shows that for Gaussian regression the exact pr edictive density p(y|x,D,D 0) and its maximum posterior approximation p(y|x,h∗) have the same mean t=/integraldisplay dyyp(y|x,D,D 0) =/integraldisplay dyyp(y|x,h∗). (320) The variances, however, differ by the term IxK−1Ix. According to the results of Section 2.2.2 the mean of the pred ictive density is the optimal choice under squared–error loss (52). For Gau ssian regression, therefore the optimal regression function a∗(x) is the same for squared–error loss in exact and in maximum posterior treatment and thus als o for log–loss (for Gaussian p(y|x,a) with fixed variance) a∗ MPA,log=a∗ exact,log=a∗ MPA,sq.=a∗ exact,sq.=h∗=t. (321) In case the space of possible p(y|x,a) is not restricted to Gaussian densi- ties with fixed variance, the variance of the optimal density under log–loss p(y|x,a∗ exact,log) =p(y|x,D,D 0) differs by IxK−1Ixfrom its maximum poste- rior approximation p(y|x,a∗ MPA,log) =p(y|x,h∗). 3.7.3 Gaussian mixture regression (cluster regression) Generalizing Gaussian regression the likelihoods may be mo deled by a mix- ture ofmGaussians p(y|x,h) =/summationtextm kp(k)e−β 2(y−hk(x))2 /integraltextdy/summationtextm kp(k)e−β 2(y−hk(x))2, (322) where the normalization factor is found as/summationtext kp(k)/parenleftigβ 2π/parenrightigm 2. Hence,his here specified by mixing coefficients p(k) and a vector of regression functions hk(x) 74specifying the x–dependent location of the kth cluster centroid of the mixture model. A simple prior for hk(x) is a smoothness prior diagonal in the cluster components. As any density p(y|x,h) can be approximated arbitrarily well by a mixture with large enough msuch cluster regression models allows to interpolate between Gaussian regression and more flexible d ensity estimation. The posterior density becomes for independent data p(h|D,D 0) =p(h|D0) p(yD|xD,D0)n/productdisplay i/summationtextm kp(k)e−β 2(yi−hk(xi))2 /summationtextm kp(k)/parenleftigβ 2π/parenrightigm 2. (323) Maximizing that posterior is — for fixed x, uniformp(k) andp(h|D0) — equivalent to the clustering approach of Rose, Gurewitz, an d Fox for squared distance costs [188]. 3.7.4 Support vector machines and regression Expanding the regression function h(x) in a basis of eigenfunctions Ψ kofK0 K0=/summationdisplay kλkΨkΨT k, h(x) =/summationdisplay knkΨk(x) (324) yields for functional (247) Eh=/summationdisplay i/parenleftigg/summationdisplay knkΨk(xi)−yi/parenrightigg2 +/summationdisplay kλk|nk|2. (325) Under the assumption of output noise for training data the da ta terms may for example be replaced by the logarithm of a mixture of Gauss ians. Such mixture functions with varying mean can develop flat regions where the error is insensitive (robust) to changes of h. Analogously, Gaussians with varying mean can be added to obtain errors which are flat compared to Ga ussians for large absolute errors. Similarly to such Gaussian mixtu res the mean– square error data term ( yi−h(xi))2may be replaced by an ǫ–insensitive error|yi−h(xi)|ǫ, which is zero for absolute errors smaller ǫand linear for larger absolute errors (see Fig.5). This results in a quadra tic programming problem and is equivalent to Vapnik’s support vector machin e [209, 69, 210, 198, 199, 44]. For a more detailed discussion of the relation between support vector machines and Gaussian processes see [213, 192]. 75Figure 5: Three robust error functions which are insensitiv e to small errors. Left: Logarithm of mixture with two Gaussians with equal var iance and different means. Middle: Logarithm of mixture with 11 Gaussi ans with equal variance and different means. Right: ǫ–insensitive error. 3.8 Classification In classification (or pattern recognition) tasks the indepe ndent visible vari- ableytakes discrete values (group, cluster or pattern labels) [1 4, 56, 21, 42]. We writey=kandp(y|x,h) =Pk(x,h), i.e.,/summationtext kPk(x,h) = 1. Having re- ceived classification data D={(xi,ki)|1≤i≤n}the density estimation error functional for a prior on function φ(with components φkandP= P(φ)) reads Ecl.=n/summationdisplay ilnPki(xi;φ) +1 2/parenleftig φ−t,K(φ−t)/parenrightig + (P(φ),ΛX). (326) In classification the scalar product corresponds to an integ ral overxand a summation over k, e.g., /parenleftig φ−t,K(φ−t)/parenrightig =/summationdisplay k,k′/integraldisplay dxdx′(φk(x)−tk(x))Kk,k′(x,x′)(φk′(x′)−tk′(x′)), (327) and (P,ΛX) =/integraltextdxΛX(x)/summationtext kPk(x). For zero–one loss l(x,k,a ) =δk,a(x)— a typical loss function for classifi- cation problems — the optimal decision (or Bayes classifier ) is given by the mode of the predictive density (see Section 2.2.2), i.e., a(x) = argmaxkp(k|x,D,D 0). (328) In saddle point approximation p(k|x,D,D 0)≈p(k|x,φ∗) whereφ∗minimiz- ingEcl.(φ) can be found by solving the stationarity equation (228). For the choice φk=Pkpositivity and normalization must be ensured. Forφ=LwithP=eLpositivity is automatically fulfilled but the Lagrange multiplier must be included to ensure normalization. 76likelihoodp(y|x,h) problem type of general form density estimation discretey classification Gaussian with fixed variance regression mixture of Gaussians clustering quantum mechanical likelihood inverse quantum mechanics Table 3: Special cases of density estimation Normalization is guaranteed by using unnormalized probabi litiesφk= zk,P=zk//summationtext lzl(for which positivity has to be checked) or shifted log– likelihoodsφk=gkwithgk=Lk+ln/summationtext leLl, i.e.,Pk=egk//summationtext legl. In that case the nonlocal normalization terms are part of the likelihood and no Lagrange multiplier has to be used [219]. The resulting equation can b e solved in the space defined by the X–data (see Eq. (154)). The restriction of φk=gkto linear functions φk(x) =wkx+bkyields log–linear models [143]. Recently a mean field theory for Gaussian Process classification has be en developed [164, 166]. Table 3 lists some special cases of density estimation. The l ast line of the table, referring to inverse quantum mechanics, will be disc ussed in the next section. 3.9 Inverse quantum mechanics Up to now we have formulated the learning problem in terms of a functionφ having a simple, e.g., pointwise, relation to P. Nonlocalities in the relation betweenφandPwas only due to the normalization condition, or, working with the distribution function, due to an integration. Inverse problems for quantum mechanical systems provide examples of more complicated, nonlocal relations between likelihoods p(y|x,h) =p(y|x,φ) and the hidden variables φ the theory is formulated in. To show the flexibility of Bayesi an Field Theory we will give in the following a short introduction to its appl ication to inverse 77quantum mechanics. A more detailed discussion of inverse qu antum problems including numerical applications can be found in [124, 134, 133, 129, 206]. The state of a quantum mechanical systems can be completely d escribed by giving its density operator ρ. The density operator of a specific system depends on its preparation and its Hamiltonian, governing t he time evolution of the system. The inverse problem of quantum mechanics cons ists in the reconstruction of ρfrom observational data. Typically, one studies systems with identical preparation but differing Hamiltonians. Con sider for example Hamiltonians have the form H=T+V, consisting of a kinetic energy part Tand a potential V. Assuming the kinetic energy to be fixed, the inverse problem is that of reconstructing the potential Vfrom measurements. A local potential V(y,y′) =V(y)δ(y−y′) is specified by a function V(y). Thus, for reconstructing a local potential it is the function V(y) which determines the likelihood p(y|x,h) =p(y|X,ρ) =p(y|X,V) =P(φ) and it is natural to formulate the prior in terms of the function φ=V. The possibilities of implementing prior information for Vare similar to those we discuss in this paper for general density estimation problems. It is the lik elihood model where inverse quantum mechanics differs from general densit y estimation. Measuring quantum systems the variable xcorresponds to an hermitian operator X. The possible outcome yof measurements are given by the eigen- values of X, i.e., X|y>=y|y>, (329) where |y>, with dual <y|, denotes the eigenfunction with eigenvalue y. (For the sake of simplicity we assume nondegenerate eigenvalues , the generaliza- tion to the degenerate case being straightforward.) Definin g the projector ΠX,y=|y><y | (330) the likelihood model of quantum mechanics is given by p(y|x,ρ) = Tr(Π X,yρ). (331) In the simplest case, where the system is in a pure state, say t he ground stateϕ0ofHfulfilling H|ϕ0>=E0|ϕ0>, (332) the density operator is ρ=ρ2=|ϕ0><ϕ 0|, (333) 78ρ general pure state |ψ><ψ | stationary pure state |ϕi(H)><ϕi(H)| ground state |ϕ0(H)|><ϕ 0(H)| time–dependent pure state |U(t,t0)ψ(t0)><U(t,t0)ψ(t0)| scattering limt→∞ t0→−∞|U(t,t0)ψ(t0)><U(t,t0)ψ(t0)| general mixture state/summationtext kp(k)|ψk><ψk| stationary mixture state/summationtext ip(i|H)|ϕi(H)><ϕi(H)| canonical ensemble (Tre−βH)−1e−βH Table 4: The most common examples of density operators for qu antum systems. In this Table ψdenotes an arbitrary pure state, ϕirepresents an eigenstate of Hamiltonian H. The unitary time evolution operator for a time–independent Hamiltonian His given by U=e−i(t−t0)H. In scattering one imposes typically additional specific boundary conditi ons on the initial and final states. and the likelihood (331) becomes p(y|x,h) =p(y|X,ρ) = Tr( |ϕ0><ϕ 0|y><y |) =|ϕ0(y)|2. (334) Other common choices for ρare shown in Table 4. In contrast to ideal measurements on classical systems, qua ntum mea- surements change the state of the system. Thus, in case one is interested in repeated measurements for the same ρ, that density operator has to be prepared before each measurement. For a stationary state at finite tempera- ture, for example, this can be achieved by waiting until the s ystem is again in thermal equilibrium. For a Maximum A Posteriori Approximation the functional der ivative of the likelihood is needed. Thus, for reconstructing a local p otential we have 79to calculate δV(y)p(y|X,V). (335) To be specific, let us assume we measure particle coordinates , meaning we have chosen Xto be the coordinate operator. For a system prepared to be in the ground state of the unknown H, we thus have to find, δV(y)|ϕ0(y)|2. (336) For that purpose, we take the functional derivative of Eq. (3 32), which yields (H−E0)|δV(y)ϕ0>= (δV(y)H−δV(y)E0)|ϕ0>. (337) Projecting from the left by <ϕ0|, using again Eq. (332) and the fact that for a local potential δV(y)H(y′,y′′) =δ(y−y′)δ(y′−y′′), shows that δV(y)E0=<ϕ0|δV(y)H|ϕ0>=|ϕ0(y)|2. (338) Choosing<ϕ0|δV(y)ϕ0>= 0 and inserting a complete basis of eigenfunctions |ϕj>ofH, we end up with δV(y)ϕ0(y′) =/summationdisplay j∝negationslash=01 E0−Eiϕj(y′)ϕ∗ j(y)ϕ0(y). (339) From this the functional derivative of the quantum mechanic al log–likelihood (336) corresponding to data point yican be obtained easily, δV(y)lnp(yi|X,V) = 2Re/parenleftig ϕ0(yi)−1δV(y)ϕ0(yi)/parenrightig . (340) The MAP equations for inverse quantum mechanics are obtaine d by including the functional derivatives of a prior terms for V. In particular, for a Gaussian prior with mean V0and inverse covariance KV, acting in the space of potential functionsV(y), its negative logarithm, i.e., its prior error functional , reads 1 2(V−V0,KV(V−V0)) + lnZV, (341) withZVbeing theV–independent constant normalizing the prior over V. Collecting likelihood and prior terms, the stationarity eq uation finally be- comes 0 =/summationdisplay iδV(y)lnp(yi|X,V) +KV(V−V0). (342) 80The Bayesian approach to inverse quantum problems is quite fl exible and can be used for many different learning scenarios and quantum systems. By adapting Eq. (340), it can deal with measurements of differen t observables, for example, coordinates, momenta, energies, and with othe r density oper- ators, describing, for example, time–dependent states or s ystems at finite temperature [134]. The treatment of bound state or scattering problems for quan tum many– body systems requires additional approximations. Common a re, for example, mean field methods, for bound state problems [50, 183, 24] as w ell as for scattering theory [73, 24, 131, 132, 121, 122, 207] Referrin g to such mean field methods inverse quantum problems can also be treated fo r many–body systems [133]. 4 Parameterizing likelihoods: Variational methods 4.1 General parameterizations Approximate solutions of the error minimization problem ar e obtained by restricting the search (trial) space for h(x,y) =φ(x,y) (orh(x) in regression). Functionsφwhich are in the considered search space are called trial functions . Solving a minimization problem in some restricted trial spa ce is also called a variational approach [90, 98, 26, 32, 24]. Clearly, minimal values obtained by minimization within a trial space can only be larger or equal than the true minimal value, and from two variational approximations tha t with smaller error is the better one. Alternatively, using parameterized functions φcan also implement the prior where φis known to have that specific parameterized form. (In cases whereφis only known to be approximately of a specific parameterized form, this should ideally be implemented using a prior with a param etrized tem- plate and the parameters be treated as hyperparameters as in Section 5.) The following discussion holds for both interpretations. Any parameterization φ=φ({ξl}) together with a range of allowed values for the parameter vector ξdefines a possible trial space. Hence we consider the error functional Eφ(ξ)=−( lnP(ξ), N) +1 2(φ(ξ),Kφ(ξ) ) + (P(ξ),ΛX), (343) 81forφdepending on parameters ξandp(ξ) =p(φ(ξ) ). In the special case of Gaussian regression this reads Eh(ξ)=1 2(h(ξ)−tD,KDh(ξ)−tD) +1 2(h(ξ),Kh(ξ) ). (344) Defining the matrix Φ′(l;x,y) =∂φ(x,y) ∂ξl(345) the stationarity equation for the functional (343) becomes 0 = Φ′P′P−1N−Φ′Kφ−Φ′P′ΛX. (346) Similarly, a parameterized functional Eφwith non–zero template tas in (226) would give 0 = Φ′P′P−1N−Φ′K(φ−t)−Φ′P′ΛX. (347) To have a convenient notation when solving for Λ Xwe introduce P′ ξ= Φ′(ξ)P′(φ), (348) i.e., P′ ξ(l;x,y) =∂P(x,y) ∂ξl=/integraldisplay dx′dy′∂φ(x′,y′) ∂ξlδP(x,y) δφ(x′,y′), (349) and Gφ(ξ)=P′ ξP−1N−Φ′Kφ, (350) to obtain for Eq. (346) P′ ξΛX=Gφ(ξ). (351) For a parameterization ξrestricting the space of possible Pthe matrix P′ ξis not square and cannot be inverted. Thus, let ( P′ ξ)#be the Moore–Penrose inverse of P′ ξ, i.e., (P′ ξ)#P′ ξ(P′ ξ)#=P′ ξ,P′ ξ(P′ ξ)#P′ ξ= (P′ ξ)#, (352) and symmetric ( P′ ξ)#P′ ξandP′ ξ(P′ ξ)#. A solution for Λ Xexists if P′ ξ(P′ ξ)#Gφ(ξ)=Gφ(ξ). (353) In that case the solution can be written ΛX= (P′ ξ)#Gφ(ξ)+VΛ−(P′ ξ)#P′ ξVΛ, (354) 82with arbitrary vector VΛand Λ0 X=VΛ−(P′ ξ)#P′ ξVΛ (355) from the right null space of P′ ξ, representing a solution of P′ ξΛ0 X= 0. (356) Inserting for Λ X(x)∝ne}ationslash= 0 Eq. (354) into the normalization condition Λ X= IXPΛXgives ΛX=IXP/parenleftig (P′ ξ)#Gφ(ξ)+VΛ−(P′ ξ)#P′ ξVΛ/parenrightig . (357) Substituting back in Eq. (346) Λ Xis eliminated yielding as stationarity equa- tion 0 =/parenleftig I−P′ ξIXP(P′ ξ)#/parenrightig Gφ(ξ)−P′ ξIXP/parenleftig VΛ−(P′ ξ)#P′ ξVΛ/parenrightig , (358) whereGφ(ξ)has to fulfill Eq. (353). Eq. (358) may be written in a form similar to Eq. (193) Kφ(ξ)(ξ) =Tφ(ξ) (359) with Tφ(ξ)(ξ) =P′ ξP−1N−P′ ξΛX, (360) but with Kφ(ξ)(ξ) = Φ′KΦ(ξ), (361) being in general a nonlinear operator. 4.2 Gaussian priors for parameters Up to now we assumed the prior to be given for a function φ(ξ)(x,y) de- pending on xandy. Instead of a prior in a function φ(ξ)(x,y) also a prior in another not ( x,y)–dependent function of the parameters ψ(ξ) can be given. A Gaussian prior in ψ(ξ) =Wψξbeing a linear function of ξ, results in a prior which is also Gaussian in the parameters ξ, giving a regularization term 1 2(ξ, WT ψKψWψξ) =1 2(ξ,Kξξ), (362) 83whereKξ=WT ψKψWψis not an operator in a space of functions φ(x,y) but a matrix in the space of parameters ξ. The results of Section 4.1 apply to this case provided the following replacement is made Φ′Kφ→Kξξ. (363) Similarly, a nonlinear ψrequires the replacement Φ′Kφ→Ψ′Kψψ, (364) where Ψ′(k,l) =∂ψl(ξ) ∂ξk. (365) Thus, in the general case where a Gaussian (specific) prior in φ(ξ) andψ(ξ) is given, Eφ(ξ),ψ(ξ)=−( lnP(ξ), N) + (P(ξ),ΛX) +1 2(φ(ξ),Kφ(ξ) ) +1 2(ψ(ξ),Kψψ(ξ) ), (366) or, including also non–zero template functions (means) t,tψforφandψas discussed in Section 3.5, Eφ(ξ),ψ(ξ)=−( lnP(ξ), N) + (P(ξ),ΛX) +1 2(φ(ξ)−t,K(φ(ξ)−t) ) +1 2(ψ(ξ)−tψ,Kψ(ψ(ξ)−tψ) ). (367) Theφandψ–terms of the energy can be interpreted as corresponding to a probability p(ξ|t,K,tψ,Kψ), (∝ne}ationslash=p(ξ|t,K)p(ξ|tψ,Kψ)), or, for example, top(tψ|ξ,Kψ)p(ξ|t,K) with one of the two terms term corresponding to a Gaussian likelihood with ξ–independent normalization. The stationarity equation becomes 0 = P′ ξP−1N−Φ′K(φ−t)−Ψ′Kψ(ψ−tψ)−P′ ξΛX (368) =Gφ,ψ−P′ ξΛX, (369) which defines Gφ,ψ, and for Λ X∝ne}ationslash= 0 ΛX=IXP/parenleftig (P′ ξ)#Gφ,ψ+ Λ0 X/parenrightig , (370) forP′ ξΛ0 X= 0. 84Variable Error Stationarity equation ΛX L(x,y)EL KL=N−eLΛX IX(N−KL) P(x,y)EP KP=P−1N−ΛX IX(N−PKP) φ=√ PE√ P Kφ= 2Φ−1N−2ΦΛX IX(N−1 2ΦKφ) φ(x,y)Eφ Kφ=P′P−1N−P′ΛX IX/parenleftig N−PP′−1Kφ/parenrightig ξEφ(ξ)Φ′Kφ=P′ ξP−1N−P′ ξΛXIXP/parenleftig (P′ ξ)#Gφ(ξ)+ Λ0 X/parenrightig ξEφ(ξ)ψ(ξ)Φ′K(φ−t) + Ψ′Kψ(ψ−tψ)IXP/parenleftig (P′ ξ)#Gφ,ψ+ Λ0 X/parenrightig =P′ ξP−1N−P′ ξΛX Table 5: Summary of stationarity equations. For notations, conditions and comments see Sections 3.1.1, 3.2.1, 3.3.2, 3.3.1, 4.1 and 4. 2. 4.3 Linear trial spaces Choosing a finite linear trial space is also called the Ritz me thod and is equivalent to solving a projected stationarity equation. H ere φ=/summationdisplay lclBl (371) is expanded in a basis Bl, not necessarily orthonormal, and truncated to terms with l<lmax. This gives for (187) ERitz=−( lnP(φ), N) +1 2/summationdisplay klckcl(Bk,KBl) + (P(φ),ΛX).(372) Solving for the coefficients cl,l<lmaxto minimize the error results according to Eq.[346) and Φ′(l;x,y) =Bl(x,y), (373) in 0 = (Bl,P′P−1N)−/summationdisplay kck(Bl,KBk)−(Bl,P′ΛX),∀l≤lmax,(374) corresponding to the lmax–dimensional equation KBc=NB(c)−ΛB(c), (375) 85with c(l) =cl, (376) KB(l,k) = (Bl,KBk), (377) NB(c)(l) = (Bl,P′(φ(c))P−1(φ(c))N), (378) ΛB(c)(l) = (Bl,P′(φ(c)) ΛX). (379) Thus, for an orthonormal basis BlEq. (375) corresponds to Eq. (189) pro- jected into the trial space by/summationtext lBT lBl. The so called linear models are obtained by the (very restrictive) choice φ(z) =1/summationdisplay l=0clBl=c0+/summationdisplay lclzl (380) withz= (x,y) andB0= 1 andBl=zl. Interactions, i.e., terms proportional to products of z–components like cmnzmzncan be included. Including all pos- sible interaction would correspond to a multidimensional T aylor expansion of the function φ(z). If the functions Bl(z) are also parametrized this leads to mixture models forφ. (See Section 4.4.) 4.4 Mixture models The function φ(z) can be approximated by a mixture model, i.e., by a linear combination of components functions φ(z) =/summationdisplay clBl(ξl,z), (381) with parameter vectors ξland constants cl(which could also be included into the vector ξl) to be adapted. The functions Bl(ξl,z) are often chosen to depend on one–dimensional combinations of the vectors ξlandz. For example they may depend on some distance ||ξl−z||(‘local or distance approaches’) or the projection of zinξl–direction, i.e.,/summationtext kξl,kzk(‘projection approaches’). (For projection approaches see also Sections 4.5, 4.8 and 4. 9). A typical example are Radial Basis Functions (RBF) using Gau ssian Bl(ξl,z) for which centers (and possibly covariances and also numbe r of com- ponents) can be adjusted. Other local methods include k–nearest neighbors methods (kNN) and learning vector quantizations (LVQ) and its variant s. (For a comparison see [146].) 864.5 Additive models Trial functions φmay be chosen as sum of simpler functions φleach depending only on part of the xandyvariables. More precisely, we consider functions φldepending on projections zl=I(z) lzof the vector z= (x,y) of allxand ycomponents. I(z) ldenotes an projector in the vector space of z(and not in the space of functions Φ( x,y)). Hence,φbecomes of the form φ(z) =/summationdisplay lφl(zl), (382) so only one–dimensional functions φlhave to be determined. Restricting the functions φlto a parameterized function space yields a “parameterized additive model” φ(z) =/summationdisplay lφl(ξ,zl), (383) which has to be solved for the parameters ξ. The model can also be gener- alized to a model “additive in parameters ξl” φ(z) =/summationdisplay lφl(ξl,x,y), (384) where the functions φl(ξl,x,y) are not restricted to one–dimensional functions depending only on projections zlon the coordinate axes. If the parameters ξl determine the component functions φlcompletely, this yields just the mixture models of Section 4.4. Another example is projection pursui t, discussed in Section 4.8), where a parameter vector ξlcorresponds to a projections ξl·z. In that case even for given ξlstill a one–dimensional function φl(ξl·z) has to be determined. An ansatz like (382) is made more flexible by including also in teractions φ(x,y) =/summationdisplay lφl(zl) +/summationdisplay klφkl(zk,zl) +/summationdisplay klmφklm(zk,zl,zm) +···.(385) The functions φkl···(zk,zl,···) can be chosen to depend on product terms like zl,izk,j, orzl,izk,jzm,n, wherezl,idenotes one–dimensional sub-variables of zl. In additive models in the narrower sense [202, 85, 86, 87] zlis a subset of x,ycomponents, i.e., zl⊆ {xi|1≤i≤dx} ∪ {yj|1≤j≤dy},dxdenoting the dimension of x,dythe dimension of y. In regression, for example, one takes usually the one–element subsets zl={xl}for 1≤l≤dx. 87In more general schemes the projections of zdo not have to be restricted to projections on the coordinates axes. In particular, the p rojections can be optimized too. For example, one–dimensional projections I(z) lz=w·zwith z,w∈X×Y(where ·denotes a scalar product in the space of zvariables) are used by ridge approximation schemes. They include for regression problems one–layer (and similarly multilayer) feedforward neural n etworks (see Section 4.9) projection pursuit regression (see Section 4.8) and hi nge functions [28]. For a detailed discussion of the regression case see [71]. The stationarity equation for Eφbecomes for the ansatz (382) 0 =P′ lP−1N−Kφ−P′ lΛX, (386) with P′ l(zl,z′) =δP(z′) δφl(zl). (387) Considering a density Pbeing also decomposed into components Pldeter- mined by the components φl P(z) =/summationdisplay lPl(φl(zl)), (388) the derivative (387) becomes P′ l(zl,z′ k) =δPl(z′ l) δφl(zl), (389) so that specifying an additive prior 1 2/summationdisplay kl(φk−tk,Kkl(φl−tl) ), (390) the stationary conditions are coupled equations for the com ponent functions φlwhich, because Pis diagonal, only contain integrations over zl–variables 0 =δPl δφlP−1N−/summationdisplay kKlk(φk−tk)−δPl δφlΛX. (391) For the parameterized approach (383) one finds 0 = Φ′ lP′ lP−1N−Φ′ lKφ−Φ′ lP′ lΛX, (392) with Φ′ l(k,zl) =∂φl(zl) ∂ξk. (393) For the ansatz (384) Φ′ l(k,z) would be restricted to a subset of ξk. 884.6 Product ansatz A product ansatz has the form φ(z) =/productdisplay lφl(zl), (394) wherezl=I(z) lzrepresents projections of the vector zconsisting of all x andycomponents. The ansatz can be made more flexible by using sum o f products φ(z) =/summationdisplay k/productdisplay lφk,l(zl). (395) The restriction of the trial space to product functions corr esponds to the Hartree approximation in physics. (In a Hartree–Fock appro ximation the product functions are antisymmetrized under coordinate ex change.) For additive K=/summationtext lKlwithKlacting only on φl, i.e., Kl=Kl⊗/parenleftig/circlemultiplytext l′∝negationslash=lIl′/parenrightig , with Ilthe projector into the space of functions φl=Ilφl, the quadratic regularization term becomes, assuming IlIl′=δl,l′, (φ,Kφ) =/summationdisplay l(φl,Klφl)/productdisplay l′∝negationslash=l(φl′, φl′). (396) ForK=/circlemultiplytext lKlwith a product structure with respect to φl (φ,Kφ) =/productdisplay l(φl,Klφl). (397) In both cases the prior term factorizes into lower dimension al contributions. 4.7 Decision trees Decision trees [29] implement functions which are piecewis e constant on rect- angular areas parallel to the coordinate axes zl. Such an approach can be written in tree structure with nodes only performing compar isons of the form x<a orx>a which allows a very effective hardware implementation. Such a piecewise constant approach can be written in the form φ(z) =/summationdisplay lcl/productdisplay kΘ(zν(l,k)−alk) (398) with step function Θ and zν(l,k)indicating the component of zwhich is com- pared with the reference value alk. While there are effective constructive 89methods to build trees the use of gradient–based minimizati on or maximiza- tion methods would require, for example, to replace the step function by a sigmoid. In particular, decision trees correspond to neura l networks at zero temperature, where sigmoids become step functions, and whi ch are restricted to weights vectors in coordinate directions (see Section 4. 9). An overview over different variants of decision trees togeth er with a com- parison with rule–based systems, neural networks (see Sect ion 4.9) techniques from applied statistics like linear discriminants, projec tion pursuit (see Sec- tion 4.8) and local methods like for example k-nearest neighbors methods (kNN), Radial Basis Functions (RBF), or learning vector quant ization (LVQ) is given in [146]. 4.8 Projection pursuit Projection pursuit models [55, 95, 45] are a generalization of additive models (382) (and a special case of models (384) additive in paramet ers) where the projections of z= (x,y) are also adapted φ(z) =ξ0+/summationdisplay lφl(ξ0,l+ξl·z). (399) For such a model one has to determine one–dimensional ‘ridge ’ functions φl together with projections defined by vectors ξland constants ξ0,ξ0,l. Adap- tive projections may also be used for product approaches φ(z) =/productdisplay lφl(ξ0,l+ξl·z). (400) Similarly,φmay be decomposed into functions depending on distances to adapted reference points (centers). That gives models of th e form φ(z) =/productdisplay lφl(||ξl−z||), (401) which require to adapt parameter vectors (centers) ξland distance functions φl. For high dimensional spaces the number of centers necessar y to cover a high dimensional space with fixed density grows exponential ly. Furthermore, as the volume of a high dimensional sphere tends to be concent rated near its surface, the tails become more important in higher dimen sions. Thus, typically, projection methods are better suited for high di mensional spaces than distance methods [195]. 904.9 Neural networks While in projection pursuit–like techniques the one–dimen sional ‘ridge’ func- tionsφlare adapted optimally, neural networks use ridge functions of a fixed sigmoidal form. The resulting lower flexibility following f rom fixing the ridge function is then compensated by iterating this parameteriz ation. This leads to multilayer neural networks. Multilayer neural networks have been become a popular tool f or regres- sion and classification problems [190, 116, 147, 89, 154, 215 , 21, 186, 8]. One-layer neural networks, also known as perceptrons, corr espond to the parameterization φ(z) =σ/parenleftigg/summationdisplay lwlzl−b/parenrightigg =σ(v), (402) with a sigmoidal function σ, parameters ξ=w, projection v=/summationtext lwlzl−b andzlsingle components of the variables x,y, i.e.,zl=xlfor 1≤l≤dx andzl=ylfordx+ 1≤l≤dx+dy. (For neural networks with Lorentzians instead of sigmoids see [67].) Typical choices for the sigmoid are σ(v) = tanh(βv) orσ(v) = 1/(1 + e−2βv). The parameter β, often called inverse temperature, controls the sharpness of the step of the sigmoid. In particular, the sigm oid functions become a sharp step in the limit β→ ∞, i.e., at zero temperature. In princi- ple the sigmoidal function σmay depend on further parameters which then — similar to projection pursuit discussed in Section 4.8 — wo uld also have to be included in the optimization process. The threshold or biasbcan be treated as weight if an additional input component is includ ed clamped to the value 1. A linear combination of perceptrons φ(x,y) =b+/summationdisplay lWlσ/parenleftigg/summationdisplay kwlkzk−bk/parenrightigg , (403) has the form of a projection pursuit approach (399) but with fi xedφl(v) = Wlσ(v). In multi–layer networks the parameterization (402) is casc aded, zk,i=σ/parenleftiggmi−1/summationdisplay l=1wkl,izl,i−1−bk,i)/parenrightigg =σ(vk,i), (404) 91withzk,irepresenting the output of the kth node (neuron) in layer iand vk,i=mi−1/summationdisplay l=1wkl,izl,i−1−bk,i, (405) being the input for that node. This yields, skipping the bias terms for sim- plicity φ(z,w) =σ mn−1/summationdisplay ln−1wln−1,nσ mn−2/summationdisplay ln−2wln−1ln−2,n−1···σ m0/summationdisplay l0wl1l0,1zl0,0 ···  , (406) beginning with an input layer with m0=dx+dynodes (plus possibly nodes to implement the bias) zl,0=zland going over intermediate layers with mi nodeszl,i, 0<i<n , 1≤l≤mito a single node output layer zn=φ(x,y). Commonly neural nets are used in regression and classificati on to param- eterize a function φ(x,y) =h(x) in functionals E=/summationdisplay i(yi−h(xi,w))2, (407) quadratic in hand without further regularization terms. In that case, reg u- larization has to be assured by using either 1. a neural netwo rk architecture which is restrictive enough, 2. by using early stopping like training procedures so the full flexibility of the network structure cannot compl etely develop and destroy generalization, where in both cases the optimal arc hitecture or al- gorithm can be determined for example by cross–validation o r bootstrap techniques [153, 5, 214, 200, 201, 76, 35, 212, 49], or 3. by av eraging over ensembles of networks [157]. In all these cases regularizat ion is implicit in the parameterization of the network. Alternatively, expli cit regularization or prior terms can be added to the functional. For regression or classification this is for example done in learning by hints [2, 3, 4] or curvature–driven smoothing with feedforward networks [19]. One may also remark that from a Frequentist point of view the q uadratic functional is not interpreted as posterior but as squared–e rror loss/summationtext i(yi− a(xi,w))2for actionsa(x) =a(x,w). According to Section 2.2.2 minimization of error functional (407) for data {(xi,yi)|1≤i≤n}sampled under the true densityp(x,y|f) yields therefore an empirical estimate for the regression function/integraltextdyyp(y|x,f). 92We consider here neural nets as parameterizations for densi ty estimation with prior (and normalization) terms explicitly included i n the functional Eφ. In particular, the stationarity equation for functional (3 43) becomes 0 = Φ′ wP′P−1N−Φ′ wKφ−Φ′ wP′ΛX, (408) with matrix of derivatives Φ′ w(k,l,i;x,y) =∂φ(x,y,w ) ∂wkl,i(409) =σ′(vn)/summationdisplay ln−1wln−1,nσ′(vln−1,n−1)/summationdisplay ln−2wln−1ln−2,n−1 ···/summationdisplay li+1wli+2li+1,i+2σ′(vli+1,i+1)wli+1k,i+1σ′(vli,i)zl,i−1, andσ′(v) =dσ(v)/dv. Whileφ(x,y,w ) is calculated by forward propagating z= (x,y) through the net defined by weight vector waccording to Eq. (406) the derivatives Φ′can efficiently be calculated by back–propagation according to Eq. (409). Notice that even for diagonal P′the derivatives are not needed only at data points but the prior and normalization term requ ire derivatives at allx,y. Thus, in practice terms like Φ′Kφhave to be calculated in a relatively poor discretization. Notice, however, that reg ularization is here not only due to the prior term but follows also from the restri ctions implicit in a chosen neural network architecture. In many practical c ases a relatively poor discretization of the prior term may thus be sufficient. Table 6 summarizes the discussed approaches. 5 Parameterizing priors: Hyperparameters 5.1 Prior normalization In Chapter 4. parameterization of φhave been studied. This section now discusses parameterizations of the prior density p(φ|D0). For Gaussian prior densities that means parameterization of mean and/or covar iance. The pa- rameters of the prior functional, which we will denote by θ, are in a Bayesian context also known as hyperparameters . Hyperparameters θcan be consid- ered as part of the hidden variables. In a full Bayesian approach the h–integral therefore has to be completed by an integral over the additional hidden variables θ. Analogously, the prior 93Ansatz Functional form to be optimized linear ansatz φ(z) =/summationtext lξlBl(z) ξl linear model φ(z) =ξ0+/summationtext lξlzl ξ0,ξl with interaction +/summationtext mnξmnzmzn+···ξmn,··· mixture model φ(z) =/summationtextξ0,lBl(ξl,z) ξ0,l,ξl additive model φ(z) =/summationtext lφl(zl) φl(zl) with interaction +/summationtext mnφmn(zmzn) +···φmn(zmzn),··· product ansatz φ(z) =/producttext lφl(zl) φl(zl) decision trees φ(z) =/summationtext lξl/producttext kΘ(zξlk−ξ0,lk)ξl,ξ0,lk,ξlk projection pursuit φ(z) =ξ0+/summationtext lφl(ξ0,l+/summationtext lξlzl)φl,ξ0,ξ0,l,ξl neural net (2 lay.) φ(z) =σ(/summationtext lξlσ(/summationtext kξlkzk))ξl,ξlk Table 6: Some possible parameterizations. 94densities can be supplemented by priors for θ, also be called hyperpriors , with corresponding energies Eθ. In saddle point approximation thus an additional stationar ity equation will appear, resulting from the derivative with respect to θ. The saddle point approximation of the θ–integration (in the case of uniform hyperprior p(θ) and with the h–integral being calculated exactly or by approximation) is also known as ML–II prior [14] or evidence framework [79, 80, 197, 138, 139, 140, 21]. There are some cases where it is convenient to let the likelih oodp(y|x,h) depend, besides on a function φ, on a few additional parameters. In regres- sion such a parameter can be the variance of the likelihood. A nother example is the inverse temperature βintroduced in Section 6.3, which, like φalso ap- pears in the prior. Such parameters may formally be added to t he “direct” hidden variables φyielding an enlarged ˜φ. As those “additional likelihood pa- rameters” are like other hyperparameters typically just re al numbers, and not functions like φ, they can often be treated analogously to hyperparameters. For example, they may also be determined by cross–validatio n (see below) or by a low dimensional integration. In contrast to pure prior p arameters, how- ever, the functional derivatives with respect to such “addi tional likelihood parameters” contain terms arising from the derivative of th e likelihood. Within the Frequentist interpretation of error minimizati on as empirical risk minimization hyperparameters θcan be determined by minimizing the empirical generalization error on a new set of test or validation data DTbeing independent from the training data D. Here the empirical generalization error is meant to be the pure data term ED(θ) =ED(φ∗(θ)) of the error functional for φ∗being the optimal φfor the full regularized Eφ(θ) atθand for given training data D. Elaborated techniques include cross–validation and bootstrap methods which have been mentioned in Sections 2.5 and 4.9. Within the Bayesian interpretation of error minimization a s posterior maximization the introduction of hyperparameters leads to a new difficulty. The problem arises from the fact that it is usually desirable to interpret the error termEθas prior energy for θ, meaning that p(θ) =e−Eθ Zθ, (410) with normalization Zθ=/integraldisplay dθe−Eθ, (411) 95represents the prior density for θ. Because the joint prior factor for φandθ is given by the product p(φ,θ) =p(φ|θ)p(θ), (412) one finds p(φ|θ) =e−E(φ|θ) Zφ(θ). (413) Hence, the φ–dependent part of the energy represents a conditional prior energy denoted here E(φ|θ). As this conditional normalization Zφ(θ) =/integraldisplay dφe−E(φ|θ), (414) is in general θ–dependent a normalization term EN(θ) = lnZφ(θ) (415) must therefore be included in the error functional when mini mizing with respect toθ. It is interesting to look what happens if p(φ,θ) of Eq. (410) is expressed in terms of joint energy E(φ,θ) as follows p(φ,θ) =e−E(φ,θ) Zφ,θ. (416) Then the joint normalization Zφ,θ=/integraldisplay dφdθe−E(φ,θ), (417) is independent of φandθand could be skipped from the functional. However, in that case the term Eθcannot easily be related to the prior p(θ). Notice especially, that this discussion also applies to the case where Eθ is assumed to be uniform so it does not have to appear explicit ly in the error functional. The two ways of expressing p(φ,θ) by a joint or conditional energy, respectively, are equivalent if the joint density f actorizes. In that case, however, θandφare independent, so θcannot be used to parameterize the density of φ. Numerically the need to calculate Zφ(θ) can be disastrous because nor- malization factors Zφ(θ) represent often an extremely high dimensional (func- tional) integral and are, in contrast to the normalization o fPovery, very difficult to calculate. 96There are, however, situations for which Zφ(θ) remainsθ–independent. Letp(φ,θ) stand for example for a Gaussian specific prior p(φ,θ|˜D0) (with the normalization condition factored out as in Eq. (91)). Th en, because the normalization of a Gaussian is independent of its mean, para meterizing the meant=t(θ) results in a θ–independent Zφ(θ). Besides their mean, Gaussian processes are characterized b y their covari- ance operators K−1. Because the normalization only depends on det Ka second possibility yielding θ–dependent Zφ(θ) are parameterized transfor- mations of the form K→OKO−1with orthogonal O=O(θ). Indeed, such transformations do not change the determinant det K. They are only non–trivial for multi–dimensional Gaussians. For general parameterizations of density estimation probl ems, however, the normalization term ln Zφ(θ) must be included. The only way to get rid of that normalization term would be to assume a compensating hyperprior p(θ)∝Zφ(θ), (418) resulting in an error term E(θ) =−lnZφ(θ) compensating EN(θ). Thus, in the general case we have to consider the functional Eθ,φ=−(lnP(φ), N) + (P(φ),ΛX) +Eφ(θ) +Eθ+ lnZφ(θ).(419) writingE(φ|θ) =EφandE(θ) =Eθ. The stationarity conditions have the form δEφ δφ=P′(φ)P−1(φ)N−P′(φ)ΛX, (420) ∂Eφ ∂θ=−Z′Z−1 φ(θ)−E′ θ, (421) with Z′(l,k) =δ(l−k)∂Zφ(θ) dθl, E′ θ(l) =∂Eθ ∂θl. (422) For compensating hyperprior Eθ=−lnZφ(θ) the right hand side of Eq. (421) vanishes. Finally, we want to remark that in case function evaluation o fp(φ,θ) is much cheaper than calculating the gradient (421), minimi zation methods not using the gradient should be considered, like for exampl e the downhill simplex method [181]. 975.2 Adapting prior means 5.2.1 General considerations A prior mean or template function trepresents a prototype, reference func- tion or base line for φ. It may be a typical expected pattern in time series prediction or a reference image in image reconstruction. Co nsider, for ex- ample, the task of completing an image φgiven some pixel values (training data). Expecting the image to be that of a face the template fu nctiontmay be chosen to be some prototypical image of a face. We have seen in Section 3.5 that a single template tcould be eliminated for Gaussian (specific) priors by solving for φ−tinstead for φ. Restricting, however, to only a single template may be a very bad choice. Indeed, faces for example a ppear on images in many variations, like in different scales, transla ted, rotated, var- ious illuminations, and other kinds of deformations. We may now describe such variations by a family of templates t(θ), the parameter θdescribing scaling, translations, rotations, and more general deform ations. Thus, we expect a function to be similar to only one of the templates t(θ) and want to implement a (soft, probabilistic) OR, approximating t(θ1) ORt(θ2) OR···. A (soft, probabilistic) AND of approximation conditions, o n the other hand, is implemented by adding error terms. For example, cla ssical error functionals where data and prior terms are added correspond to an approxi- mation of training data AND a priori data. Similar considerations apply for model selection . We could for example expectφto be well approximated by a neural network or a decision tree . In that caset(θ) spans for example a space of neural networks or decision tre es. Finally, let us emphasize again that the great advantage and practical feasi- bility of adaptive templates for regression problems comes from the fact that no additional normalization terms have to be added to the err or functional. 5.2.2 Density estimation The general case with adaptive means for Gaussian prior fact ors and hyper- parameter energy Eθyields an error functional Eθ,φ=−(lnP(φ), N)+1 2/parenleftig φ−t(θ),K(φ−t(θ))/parenrightig +(P(φ),ΛX)+Eθ.(423) Defining t′(l;x,y) =∂t(x,y;θ) ∂θl, (424) 98the stationarity equations of (423) obtained from the funct ional derivatives with respect to φand hyperparameters θbecome K(φ−t) =P′(φ)P−1(φ)N−P′(φ)ΛX, (425) t′K(φ−t) = −E′ θ. (426) Inserting Eq. (425) in Eq. (426) gives t′P′(φ)P−1(φ)N=t′P′(φ)ΛX−E′ θ. (427) If working with parameterized φ(ξ) extra prior terms Gaussian in some func- tionψ(ξ) can be included as discussed in Section 4.2. Then, analogou sly to templatestforφ, also parameter templates tψcan be made adaptive with hyperparameters θψ. Furthermore, prior terms EθandEθψfor the hyperpa- rametersθ,θψcan be added. Including such additional error terms yields Eθ,θψ,φ(ξ),ψ(ξ)=−(lnP(φ(ξ) ),N) + (P(φ(ξ) ),ΛX) +1 2/parenleftig φ(ξ)−t(θ),K(φ(ξ)−t(θ))/parenrightig +1 2/parenleftig ψ(ξ)−tψ(θψ),Kψ(ψ(ξ)−tψ(θψ))/parenrightig +Eθ+Eθψ, (428) and Eqs.(425) and (425) change to Φ′K(φ−t) + Ψ′Kψ(ψ−tψ) =P′ ξP−1N−P′ ξΛX, (429) t′K(φ−t) = −E′ θ, (430) t′ ψKψ(ψ−tψ) = −E′ θψ, (431) where t′ ψ,E′ θψ,E′ θ, denote derivatives with respect to the parameters θψ orθ, respectively. Parameterizing EθandEθψthe process of introducing hyperparameters can be iterated. 5.2.3 Unrestricted variation To get a first understanding of the approach (423) let us consi der the extreme example of completely unrestricted t–variations. In that case the template functiont(x,y) itself represents the hyperparameter. (Such function hyp er- parameters or hyperfields are also discussed in Sect. 5.6.) T hen,t′=Iand 99Eq. (426) gives K(φ−t) = 0 (which for invertible Kis solved uniquely by t=φ), resulting according to Eq. (229) in ΛX=NX. (432) The case of a completely free prior mean tis therefore equivalent to a situation without prior. Indeed, for invertible P′, projection of Eq. (427) into the x– data space by IDof Eq. (258) yields PD=Λ−1 X,DN, (433) where ΛX,D=IDΛXIDis invertible and PD=IDP. Thus for xifor which yiare available P(xi,yi) =N(xi,yi) NX(xi)(434) is concentrated on the data points. Comparing this with solu tions of Eq. (192) for fixed twe see that adaptive means tend to lower the influence of prior terms. 5.2.4 Regression Consider now the case of regression according to functional (247) with an adaptive template t0(θ). The system of stationarity equations for the regres- sion function h(x) (corresponding to φ(x,y)) andθbecomes K0(h−t0) =KD(tD−h), (435) t′ 0K0(h−t0) = 0. (436) It will also be useful to insert Eq. (435) in Eq. (436), yieldi ng 0 =t′ 0KD(h−tD). (437) For fixedtEq. (435) is solved by the template average t h=t= (K0+KD)−1(K0t0+KDtD), (438) so that Eq. (436) or Eq. (437), respectively, become 0 =t′ 0K0(t−t0), (439) 0 =t′ 0KD(t−tD). (440) 100It is now interesting to note that if we replace in Eq. (440) th e full template averagetbyt0we get 0 =t′ 0KD(t0−tD), (441) which is equivalent to the stationarity equation 0 =H′KD(h−tD), (442) (the derivative matrix H′being the analogue to Φ′forh) of an error functional ED,h(ξ)=1 2(h(ξ)−tD,KD(h(ξ)−tD) ) (443) without prior terms but with parameterized h(ξ), e.g., a neural network. The approximation h=t=t0can, for example, be interpreted as limit λ→ ∞, lim λ→∞h= lim λ→∞t=t0, (444) after replacing K0byλK0in Eq. (438). The setting h=t0can then be used as initial guess h0for an iterative solution for h. For existing K−1 0h =t0is also obtained after one iteration step of the iteration sc hemehi= t0+K−1 0KD(tD−hi−1) starting with initial guess h0=tD. For comparison with Eqs.(440,441,442) we give the stationa rity equations for parameters ξfor a parameterized regression functional including an ad- ditional prior term with hyperparameters Eθ,h(ξ)=1 2(h(ξ)−tD,KD(h(ξ)−tD) )+1 2(h(ξ)−t0(θ),K0(θ)(h(ξ)−t0(θ)) ), (445) which are 0 =H′KD(h−tD) +h′K0(h−t0). (446) Let us now compare the various regression functionals we hav e met up to now. The non–parameterized and regularized regression fun ctionalEh(247) implements prior information explicitly by a regularizati on term. A parameterized and regularized functional Eh(ξ)of the form (344) cor- responds to a functional of the form (445) for θfixed. It imposes restrictions on the regression function hin two ways, by chosing a specific parameteriza- tion and by including an explicit prior term. If the number of data is large enough, compared to the flexibility of the parameterization , the data term ofEh(ξ)alone can have a unique minimum. Then, at least technically, no 101additional prior term would be required. This corresponds t o the classical error minimization methods used typically for parametric a pproaches. Nev- ertheless, also in such situations the explicit prior term c an be useful if it implements useful prior knowledge over h. The regularized functional with prior– or hyperparameters Eθ,h(423) im- plements, compared to Eh, effectively weaker prior restrictions. The prior term corresponds to a soft restriction ofhto the space spanned by the pa- rameterized t(θ). In the limit where the parameterization of t(θ) is rich enough to allow t(θ∗) =h∗at the stationary point the prior term vanishes completely. The parameterized and regularized functional Eθ,h(ξ)(445), including prior parameters θ, implements prior information explicitly by a regular- ization term and implicitly by the parameterization of h(ξ). The explicit prior term vanishes if t(θ∗) =h(ξ∗) at the stationary point. The func- tional combines a hard restriction ofhwith respect to the space spanned by the parameterization h(ξ) and a soft restriction ofhwith respect to the space spanned by the parameterized t(θ). Finally, the parameterized and non–regularized functional ED,h(ξ)(443) implements prior information only implicitly by parameterizing h(ξ). In contrast to the functionals Eθ,hand Eθ,h(ξ)it implements only a hard restriction forh. The following table sum- marizes the discussion: Functional Eq. prior implemented Eh(247) explicitly Eh(ξ)(344) explicitly and implicitly Eθ,h(423) explicitly no prior for t(θ∗) =h∗ Eθ,h(ξ)(445) explicitly and implicitly no expl. prior for t(θ∗) =h(ξ∗) ED,h(ξ)(443) implicitly 5.3 Adapting prior covariances 5.3.1 General case Parameterizing covariances K−1is often desirable in practice. It includes for example adapting the trade–off between data and prior ter ms (i.e., the determination of the regularization factor), the selectio n between different symmetries, smoothness measures, or in the multidimension al situation the 102determination of directions with low variance. As far as the normalization depends on K(θ) one has to consider the error functional Eθ,φ=−(lnP(φ), N)+1 2/parenleftig φ−t,K(θ) (φ−t)/parenrightig +(P(φ),ΛX)+lnZφ(θ)+Eθ, (447) with Zφ(θ) = (2π)d 2(detK(θ))−1 2, (448) for ad–dimensional Gaussian specific prior, and stationarity equ ations K(φ−t) =P′(φ)P−1(φ)N−P′(φ)ΛX,(449) 1 2/parenleftig φ−t,∂K(θ) ∂θ(φ−t)/parenrightig = Tr/parenleftigg K−1(θ)∂K(θ) ∂θ/parenrightigg −E′ θ. (450) Here we used ∂ ∂θln detK=∂ ∂θTr lnK= Tr/parenleftigg K−1∂K ∂θ/parenrightigg . (451) In case of an unrestricted variation of the matrix elements o fKthe hyper- parameters become θl=θ(x,y;x′,y′) =K(x,y;x′,y′). Then, using ∂K(x,y;x′,y′) ∂θ(x′′,y′′;x′′′,y′′′)=δ(x−x′′)δ(y−y′′)δ(x′−x′′′)δ(y′−y′′′), (452) Eqs.(450) becomes the inhomogeneous equation 1 2(φ−t) (φ−t)T= Tr/parenleftigg K−1(θ)∂K(θ) ∂θ/parenrightigg −E′ θ. (453) We will in the sequel consider the two special cases where the determinant of the covariance is θ–independent so that the trace term vanishes, and where θis just a multiplicative factor for the specific prior energy , i.e., a so called regularization parameter. 5.3.2 Automatic relevance detection A useful application of hyperparameters is the identificati on of sensible di- rections within the space of xandyvariables. Consider the general case of a covariance, decomposed into components K0=/summationtext iθiKi. Treating the 103coefficient vector θ(with components θi) as hyperparameter with hyperprior p(θ) results in a prior energy (error) functional 1 2(φ−t,(−/summationdisplay iθiKi)(φ−t) )−lnp(θ) + lnZφ(θ). (454) Theθ–dependent normalization ln Zφ(θ) has to be included to obtain the correct stationarity condition for θ. The components Kican be the compo- nents of a negative Laplacian, for example, Ki=−∂2 xiorKi=−∂2 yi. In that case adapting the hyperparameters means searching for sens ible directions in the space of xoryvariables. This technique has been called Automatic Rel- evance Determination by MacKay and Neal [157]. The positivity constraint foracan be implemented explicitly, for example by using K0=/summationtext iθ2 iKior K0=/summationtext iexp(θi)Ki. 5.3.3 Local smoothness adaption Similarly, the regularization factor of a smoothness relat ed covariance op- erator may be adapted locally. Consider, for example, a prio r energy for φ(x,y) E(φ|θ) =1 2(φ−t,K(a,b)(φ−t) ), (455) with a Laplacian prior K(x,x′,y,y′;θ) =−mx/summationdisplay ieθx,i(x)δ(xi−x′ i)∂2 xi−my/summationdisplay ieθy,i(y)δ(y−y′ i)∂2 yi,(456) formx–dimensional vector xandmy–dimensional vector ydepending on func- tionsθx,i(x) andθy,i(y) (or more general θx,i(x,y) andθy,i(x,y)) collectively denoted by θ. Expressing the coefficient functions as exponentials exp( θx,i), exp(θy,i) is one possibility to enforce their positivity. Typically , one might impose a smoothness hyperprior on the functions θx,i(x) andθy,i(y), for ex- ample by using an energy functional E(φ,θ) +1 2mx/summationdisplay i(θx,i,Kθ,xθx,i) +1 2my/summationdisplay i(θy,i,Kθ,yθy,i) + lnZφ(θ),(457) with smoothness related Kθ,x,Kθ,y. The stationarity equation for a functions θx,i(x) reads 0 = ( Kθ,xθx,i)(x)−(φ(x,y)−t(x,y))/parenleftig ∂2 xi(φ(x,y)−t(x,y))/parenrightig eθx,i(x) +∂θx,i(x)lnZφ(θ). (458) 104The functions θx,i(x) andθy,i(y) are examples of function hyperparameters (see Sect. 5.6). 5.3.4 Local masses and gauge theories The Bayesian analog of a mass term in quantum field theory is a t erm propor- tional to the identity matrix Iin the inverse prior covariance K0. Consider, for example, K0=θ2I−∆, (459) withθreal (so that θ2≥0) representing a mass parameter. For large masses φtends to copy the template tlocally, and longer range effects of data points following from smoothness requirements become less import ant. Similarly to Sect. 5.3.3 a constant mass can be replaced by a mass functi onθ(x). This allows to adapt locally that interplay between “templa te copying” and smoothness related influence of training data. As hyperprio r, one may use a smoothness constraint on the mass function θ(x), e.g., 1 2(φ−t,M2(φ−t))−1 2(φ−t,∆(φ−t)) +λ(θ,Kθθ) + lnZφ(θ),(460) where Mdenotes the diagonal mass operator with diagonal elements θ(x). Functional hyperparameters like θ(x) represent, in the language of physi- cists, additional fields entering the problem (see also Sect . 5.6). There are similarities for example to gauge fields in physics. In parti cular, a gauge theory–like formalism can be constructed by decomposing θ(x) =/summationtext iθi(x), so that the inverse covariance K0=/summationdisplay i/parenleftig M2 i−∂2 i/parenrightig =/summationdisplay i(Mi+∂i)(Mi−∂i) =/summationdisplay iD† iDi, (461) can be expressed in terms of a “covariant derivative” Di=∂i+θi. Next, one may choose as hyperprior for θi(x) 1 2 mx/summationdisplay i(θi,−∆θi)−(mx/summationdisplay i∂xiθi,mx/summationdisplay j∂xjθj) =1 4mx/summationdisplay ijF2 ij (462) which can be expressed in terms of a “field strength tensor” (f or Abelian fields), Fij=∂iθj−∂jθi, (463) 105like, for example, the Maxwell tensor in quantum electrodyn amics. (To relate this, as in electrodynamics, to a local U(1) gauge symmetry φ→eiαφone can consider complex functions φ, with the restriction that their phase cannot be measured.) Notice, that, due to the interpretation of the prior as product p(φ|θ)p(θ), an additional θ–dependent normalization term ln Zφ(θ) enters the energy functional. Such a term is not present in quantum field theory, where one relates the prior functional directly to p(φ,θ), so the norm is independent ofφandθ. 5.3.5 Invariant determinants In this section we discuss parameterizations of the covaria nce of a Gaussian specific prior which leave the determinant invariant. In tha t case noθ– dependent normalization factors have to be included which a re usually very difficult to calculate. We have to keep in mind, however, that i n general a large freedom for K(θ) effectively diminishes the influence of the parameter- ized prior term. A determinant is, for example, invariant under general simi larity trans- formations, i.e., det ˜K= detKforK→˜K=OKO−1where Ocould be any element of the general linear group. Similarity transfo rmations do not change the eigenvalues, because from Kψ=λψfollows OKO−1Oψ=λOψ. Thus, if Kis positive definite also ˜Kis. The additional constraint that ˜K has to be real symmetric, ˜K=˜KT=˜K†, (464) requires Oto be real and orthogonal O−1=OT=O†. (465) Furthermore, as an overall factor of Odoes not change ˜Kone can restrict O to a special orthogonal group SO(N) with det O= 1. If Khas degenerate eigenvalues there exist orthogonal transformations with K=˜K. While in one dimension only the identity remains as transfor mation, the condition of an invariant determinant becomes less restric tive in higher di- mensions. Thus, especially for large dimension dofK(infinite for continuous x) there is a great freedom to adapt covariances without the ne ed to calcu- late normalization factors, for example to adapt the sensib le directions of a multivariate Gaussian. 106A positive definite Kcan be diagonalized by an orthogonal matrix O with det O= 1, i.e., K=ODOT. Parameterizing Othe specific prior term becomes 1 2/parenleftig φ−t,K(θ) (φ−t)/parenrightig =1 2/parenleftig φ−t,O(θ)DOT(θ) (φ−t)/parenrightig , (466) so the stationarity Eq. (450) reads /parenleftig φ−t,∂O ∂θDOT(φ−t)/parenrightig =−E′ θ. (467) Matrices OfromSO(N) include rotations and inversion. For a Gaussian specific prior with nondegenerate eigenvalues Eq. (467) all ows therefore to adapt the ‘sensible’ directions of the Gaussian. There are also transformations which can change eigenvalue s, but leave eigenvectors invariant. As example, consider a diagonal ma trixDwith di- agonal elements (and eigenvalues) λi∝ne}ationslash= 0, i.e., det D=/producttext iλi. Clearly, any permutation of the eigenvalues λileaves the determinant invariant and trans- forms a positive definite matrix into a positive definite matr ix. Furthermore, one may introduce continuous parameters θij>0 withi<j and transform D→˜Daccording to λi→˜λi=λiθij, λj→˜λj=λj θij, (468) which leaves the product λiλj=˜λi˜λjand therefore also the determinant invariant and transforms a positive definite matrix into a po sitive definite matrix. This can be done with every pair of eigenvalues defini ng a set of continuous parameters θijwithi < j (θijcan be completed to a symmetric matrix) leading to λi→˜λi=λi/producttext j>iθij/producttext j<iθji, (469) which also leaves the determinant invariant det˜D=/productdisplay i˜λi=/productdisplay i/parenleftigg λi/producttext j>iθij/producttext j<iθji/parenrightigg =/parenleftigg/productdisplay iλi/parenrightigg/producttext i/producttext j>iθij/producttext i/producttext j<iθji=/productdisplay iλi= detD. (470) A more general transformation with unique parameterizatio n byθi>0, i∝ne}ationslash=i∗, still leaving the eigenvectors unchanged, would be ˜λi=λiθi, i∝ne}ationslash=i∗;˜λi∗=λi∗/productdisplay i∝negationslash=i∗θ−1 i. (471) 107This techniques can be applied to a general positive definite Kafter diago- nalizing K=ODOT→˜K=O˜DOT⇒detK= det ˜K. (472) As example consider the transformations (469, 471) for whic h the specific prior term becomes 1 2/parenleftig φ−t,K(θ) (φ−t)/parenrightig =1 2/parenleftig φ−t,OD(θ)OT(φ−t)/parenrightig , (473) and stationarity Eq. (450) 1 2/parenleftig φ−t,O∂D ∂θOT(φ−t)/parenrightig =−E′ θ, (474) and for (469), with k<l, ∂D(i,j) ∂θkl=δ(i−j)/parenleftigg δ(k−i)λk/producttext l∝negationslash=n>kθkn/producttext n<kθnk+δ(l−i)λl/producttext n>lθln/producttext k∝negationslash=n<lθnl/parenrightigg ,(475) or, for (471), with k∝ne}ationslash=i∗, ∂D(i,j) ∂θk=δ(i−j)/parenleftigg δ(k−i)λk+δ(i−i∗)λi∗1 θk/producttext l∝negationslash=i∗θl/parenrightigg . (476) If, for example, Kis a translationally invariant operator it is diagonalized in a basis of plane waves. Then also ˜Kis translationally invariant, but its sensitivity to certain frequencies has changed. The opt imal sensitivity pattern is then determined by the given stationarity equati ons. 5.3.6 Regularization parameters Next we consider the example K(γ) =γK0whereθ≥0 has been denoted γ, representing a regularization parameter or an inverse temp erature variable for the specific prior. For a d–dimensional Gaussian integral the normalization factor becomes Zφ(γ) = (2π γ)d 2(detK0)−1/2. For positive (semi)definite Kthe dimensiondis given by the rank of Kunder a chosen discretization. Skipping constants results in a normalization energy EN(γ) =−d 2lnγ. With ∂K ∂γ=K0 (477) 108we obtain the stationarity equations γK0(φ−t) =P′(φ)P−1(φ)N−P′(φ)ΛX, (478) 1 2(φ−t,K0(φ−t)) =d 2γ−E′ γ. (479) For compensating hyperprior the right hand side of Eq. (479) vanishes, giving thus no stationary point for γ. Using however the condition γ≥0 one sees that for positive definite K0Eq. (478) is minimized for γ= 0 corresponding to the ‘prior–free’ case. For example, in the case of Gaussia n regression the solution would be the data template φ=h=tD. This is also known as “δ–catastrophe”. To get a nontrivial solution for γa noncompensating hy- perparameter energy Eγ=Eθmust be used so that ln Zφ+ENis nonuniform [14, 21]. The other limiting case is a vanishing E′ γfor which Eq. (479) becomes γ=d (φ−t,K0(φ−t)). (480) Forφ→tone sees that γ→ ∞. Moreover, in case P[t] represents a nor- malized probability, φ=tis also a solution of the first stationarity equation (478) in the limit γ→ ∞. Thus, for vanishing E′ γthe ‘data–free’ solution φ=tis a selfconsistent solution of the stationarity equations (478,479). Fig.6 shows a posterior surface for uniform and for compensa ting hyper- prior for a one–dimensional regression example. The Maximu m A Posteriori Approximation corresponds to the highest point of the joint posterior over γ,hin that figures. Alternatively one can treat the γ–integral by Monte– Carlo–methods [219]. Finally we remark that in the setting of empirical risk minim ization, due to the different interpretation of the error functional, regularization pa- rameters are usually determined by cross–validation or sim ilar techniques [153, 5, 214, 200, 201, 76, 35, 195, 212, 49, 78]. 5.4 Exact posterior for hyperparameters In the previous sections we have studied saddle point approx imations which lead us to maximize the joint posterior p(h,θ|D,D 0) simultaneously with 1092 4 6 8 10gamma -1012 hp 2 4 6 8gamma2 4 6 8 10gamma -1012 hp 2 4 6 8gamma Figure 6: Shown is the joint posterior density of handγ, i.e.,p(h,γ|D,D 0)∝ p(yD|h)p(h|γ,D 0)p(γ) for a zero–dimensional example of Gaussian regression with training data yD= 0 and prior data yD0= 1. L.h.s: For uniform prior p(γ)∝1 so that the joint posterior becomes p∝e−1 2h2−γ 2(h−1)2+1 2lnγ, having its maximum is at γ=∞,h= 1. R.h.s.: For compensating hyperprior p(γ)∝1/√γso thatp∝e−1 2h2−γ 2(h−1)2having its maximum is at γ= 0, h= 0. respect to the hidden variables handθ p(y|x,D,D 0) =p(yD|xD,D0)−1/integraldisplay dh/integraldisplay dθp(y|x,h)p(yD|xD,h)p(h|D0,θ)p(θ)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright ∝p(h,θ|D,D0),max w.r.t.θand h, (481) assuming for the maximization with respect to ha slowly varying p(y|x,h) at the stationary point. This simultaneous maximization with respect to both variab les is consis- tent with the usual asymptotic justification of a saddle poin t approximation. For example, for a function f(h,θ) of two (for example, one–dimensional) variablesh,θ /integraldisplay dhdθe−βf(h,θ)≈e−βf(h∗,θ∗)−1 2lndet(βH/2π)(482) for large enough β(and a unique maximum). Here f(h∗,θ∗) denotes the joint minimum and Hthe Hessian of fwith respect to handθ. Forθ–dependent determinant of the covariance and the usual definition of β, results in a functionfof the form f(h,θ) =E(h,θ) + (1/2β) lndet(βK(θ)/2π), where 110both terms are relevant for the minimization of fwith respect to θ. For largeβ, however, the second term becomes small compared to the first one. (Of course, there is the possibility that a saddle point appr oximation is not adequate for the θintegration. Also, we have seen that the condition of a positive definite covariance may lead to a solution for θon the boundary where the (unrestricted) stationarity equation is not fulfi lled.) Alternatively, one might think of performing the two integr als stepwise. This seems especially useful if one integral can be calculat ed analytically. Consider, for example /integraldisplay dhdθe−βf(h,θ)≈/integraldisplay dθe−βf(θ,h∗(θ))−1 2lndet(β 2π∂2f(h∗(θ)) ∂h2)(483) which would be exact for a Gaussian h–integral. One sees now that mini- mizing the complete negative exponent βf(θ,h∗) +1 2ln det(β(∂2f/∂h2)/2π) with respect to θis different from minimizing only fin (482), if the second derivative of fwith respect to hdepends on θ(which is not the case for a Gaussian θintegral). Again this additional term becomes negligible f or large enough β. Thus, at least asymptotically, this term may be altered or even be skipped, and differences in the results of the variant s of saddle point approximation will be expected to be small. Stepwise approaches like (483) can be used, for example to pe rform Gaus- sian integrations analytically, and lead to somewhat simpl er stationarity equations for θ–dependent covariances [219]. In particular, let us look at the case of Gaussian regression in a bit more detail. The following discussion, however, also applies to density estimation if, as in (483), the Gaussian first step integration is replac ed by a saddle point approximation including the normalization factor. (This r equires the calcu- lation of the determinant of the Hessian.) Consider the two s tep procedure for Gaussian regression p(y|x,D,D 0) =p(yD|xD,D0)−1/integraldisplay dθp(θ)/integraldisplay dhp(y|x,h)p(yD|xD,h)p(h|D0,θ) /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright exact/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright p(θ)p(y,yD|x,xD,D0,θ)∝p(y,θ|x,D,D 0)max w.r.t.θ, =/integraldisplay dθ p(θ|D,D 0)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright ∝exactp(y|x,D,D 0,θ)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright exact/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright p(y,θ|x,D,D 0),max w.r.t.θ(484) 111where in a first step p(y,yD|x,xD,D0,θ) can be calculated analytically and in a second step the θintegral is performed by Gaussian approximation around a stationary point. Instead of maximizing the joint posteri orp(h,θ|D,D 0) with respect to handθthis approach performs the h–integration analytically and maximizes p(y,θ|x,D,D 0) with respect to θ. The disadvantage of this approach is the y–, andx–dependency of the resulting solution. Thus, assuming a slowly varying p(y|x,D,D 0,θ) at the stationary point it appears simpler to maximize the h–marginalized posterior p(θ|D,D 0) =/integraltextdhp(h,θ|D,D 0), performing this h–integration exactly, p(y|x,D,D 0) =/integraldisplay dθ p(θ|D,D 0)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright exact/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright max w.r.t.θp(y|x,D,D 0,θ)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright exact. (485) Having found a maximum posterior solution θ∗the corresponding anaytical solution for p(y|x,D,D 0,θ∗) is then given by Eq. (312). The posterior density p(θ|D,D 0) can be obtained from the likelihood of θand a specified prior p(θ) p(θ|D,D 0) =p(yD|xD,D0,θ)p(θ) p(yD|xD,D0). (486) Hence, for Gaussian regression, the likelihood can be integ rated analyti- cally, analogously to Section 3.7.2, yielding [212, 220, 21 9], p(yD|xD,D0,θ) =/integraldisplay dhe−1 2/summationtextn i=0(h−ti,Ki(h−ti))+1 2/summationtextn i=0lndeti(Ki/2π) =e−1 2/summationtextn i=0(ti,Kiti)+1 2(t,Kt)+1 2lndetD(/tildewideK/2π) =e−1 2/parenleftig tD−t0,/tildewideK(tD−t0)/parenrightig +1 2ln detD/tildewideK−˜n 2ln(2π) =e−/tildewideE+1 2ln detD/tildewideK−˜n 2ln(2π), (487) where/tildewideE=1 2/parenleftig tD−t0,/tildewiderK(tD−t0)/parenrightig ,/tildewiderK= (K−1 D+K−1 0,DD(θ))−1=KD+ KDK−1KD, detDthe determinant in data space, and we used that from K−1 iKj=δijfori,j > 0 follows/summationtextn i=0(ti,Kiti) = (tD,KDtD) + (t0,K0t0) = (tD,Kt), with K=/summationtextn i=0Ki. We already mentioned in Section 2.3 that the Maximum A Posteriori Approximation (MAP) might also see n as sad- dle point approximation for the θ–likelihood p(yD|xD,D0θ), i.e., the ( θ– conditional) evidence of the data yD(see Eq.(74). Thus, in cases where 112the marginalization over h, necessary to obtain that evidence, cannot be per- formed analytically, but has to be done in saddle point appro ximation, we get the same results as for a MAP of the predictive density. Now we are able to compare the three resulting stationary equ ations forθ–dependent mean t0(θ), covariance K0(θ) and prior p(θ). Setting the derivative of the joint posterior p(h,θ|D,D 0) with respect to θto zero yields 0 =/parenleftigg∂t0 ∂θ,K0(t0−h)/parenrightigg +1 2/parenleftig h−t0,∂K0(θ) ∂θ(h−t0)/parenrightig −Tr/parenleftigg K−1 0∂K0 ∂θ/parenrightigg −1 p(θ)∂p(θ) ∂θ. (488) This equation which we have already discussed has to be solve d simulta- neously with the stationarity equation for h. While this approach is easily adapted to general density estimation problems, its difficul ty forθ–dependent covariance determinants lies in calculation of the derivat ive of the determi- nant of K0. Maximizing the h–marginalized posterior p(θ|D,D 0), on the other hand, only requires the calculation of the derivative of the determinant of the ˜nטnmatrix/tildewiderK 0 =/parenleftigg∂t0 ∂θ,/tildewiderK(t0−tD)/parenrightigg +1 2/parenleftigg (tD−t0),∂/tildewiderK ∂θ(tD−t0)/parenrightigg −Tr/parenleftigg /tildewiderK−1∂/tildewiderK ∂θ/parenrightigg −1 p(θ)∂p(θ) ∂θ. (489) Evaluated at the stationary h∗=t0+K−1 0/tildewiderK(tD−t0), the first term of Eq. (488), which does not contain derivatives of the covariance s, becomes equal to the first term of Eq. (489). The last terms of Eqs. (488) and (48 9) are always identical. Typically, the data–independent K0has a more regular structure than the data–dependent/tildewiderK. Thus, at least for one or two dimensional x, a straightforward numerical solution of Eq. (488) by discret izingxcan also be a good choice for Gaussian regression problems. Analogously, from Eq. (312) follows for maximizing p(y,θ|x,D,D 0) with respect toθ 0 =/parenleftigg∂t ∂θ,Ky(t−y)/parenrightigg +1 2/parenleftigg (y−t),∂Ky ∂θ(y−t)/parenrightigg −Tr/parenleftigg K−1 y∂Ky ∂θ/parenrightigg −1 p(θ|D,D 0)∂p(θ|D,D 0) ∂θ, (490) 113which isy–, andx–dependent. Such an approach may be considered if inter- ested only in specific test data x,y. We may remark that also in Gaussian regression the θ–integral may be quite different from a Gaussian integral, so a saddle point ap proximation does not necessarily have to give satisfactory results. In c ases one encoun- ters problems one can, for example, try variable transforma tions/integraltextf(θ)dθ=/integraltextdet(∂θ/∂θ′)f(θ(θ′))dθ′to obtain a more Gaussian shape of the integrand. Due to the presence of the Jacobian determinant, however, th e asymptotic interpretation of the corresponding saddle point approxim ation is different for the two integrals. The variablility of saddle point appr oximations results from the freedom to add terms which vanish asymtotically but remains finite in the nonasymptotic region. Similar effects are known in qua ntum many body theory (see for example [159], chapter 7.) Alternative ly, theθ–integral can be solved numerically by Monte Carlo methods[220, 219]. 5.5 Integer hyperparameters The hyperparameters θconsidered up to now have been real numbers, or vector of real numbers. Such hyperparameters can describe c ontinuous trans- formations, like the translation, rotation or scaling of te mplate functions and the scaling of covariance operators. For real θand differentiable posterior, stationarity conditions can be found by differentiating the posterior with respect toθ. Instead of a class of continuous transformations a finite num ber of al- ternative template functions or covariances may be given. F or example, an image to be reconstructed might be expected to show a digit be tween zero and nine, a letter from some alphabet, or the face of someone w ho is a mem- ber of known group of people. Similarly, a particular times s eries may be expected to be either in a high or in a low variance regime. In a ll these cases, there exist a finite number of classes iwhich could be represented by specific templatestior covariances Ki. Such “class” variables iare nothing else than hyperparameters θwith integer values. Binary parameters, for example, allow to select from two ref erence func- tions or two covariances that one which fits the data best. E.g ., fori= θ∈ {0,1}one can write t(θ) = (1 −θ)t1+θt2, (491) K(θ) = (1 −θ)K1+θK2. (492) 114For integer θthe integral/integraltextdθbecomes a sum/summationtext θ(we will also write some- times/summationtext iif integer and continuous hyperparameters occur), so that p rior, posterior, and predictive density have the form of a finite mixture with com- ponentsθ. For a moderate number of components one may be able to include all of the mixture components. Such prior mixture models will be studied in Section 6. If the number of mixture components is too large to include th em all explicitly, one again must restrict to some of them. One poss ibility is to select a random sample using Monte–Carlo methods. Alternat ively, one may search for the θ∗with maximal posterior. In contrast to typical optimizatio n problems for real variables, the corresponding integer opt imization problems are usually not very smooth with respect to θ(with smoothness defined in terms of differences instead of derivatives), and are theref ore often much harder to solve. There exists, however, a variety of deterministic and stoch astic integer op- timization algorithms, which may be combined with ensemble methods like genetic algorithms [91, 74, 39, 145, 113, 193, 148], and with homotopy meth- ods, like simulated annealing [106, 144, 185, 38, 1, 188, 226 , 63, 227, 228]. An- nealing methods are similar to (Markov chain) Monte–Carlo m ethods, which aim in sampling many points from a specific distribution (i.e ., for example at fixed temperature). For them it is important to have (nearl y) indepen- dent samples and the correct limiting distribution of the Ma rkov chain. For annealing methods the aim is to find the correct minimum (i.e. , the ground state having zero temperature) by smoothly changing the tem perature from a finite value to zero. For them it is less important to model th e distribution for nonzero temperatures exactly, but it is important to use an adequate cooling scheme for lowering the temperature. Instead of an integer optimization problem one may also try t o solve a similar problem for real θ. For example, the binary θ∈ {0,1}in Eqs. (491) and (492) may be extended to real θ∈[0,1]. By smoothly increasing an appropriate additional hyperprior p(θ) one can finally enforce again binary hyperparameters θ∈ {0,1}. 5.6 Local hyperfields Most, but not all hyperparameters θconsidered so far have been real or integer numbers orvectors with real or integer components θi. With the 115unrestricted template functions of Sect. 5.2.3 or the funct ions parameterizing the covariance in Sections 5.3.3 and 5.3.4, we have, however , also encountered function hyperparameters orhyperfields . In this section we will now discuss function hyperparameters in more detail. Functions can be seen as continuous vectors, the function va luesθ(u) being the (continuous) analogue of vector components θi. In numerical cal- culations, in particular, functions usually have to be disc retized, so functions stand for high dimensional vectors. Typical arguments of function hyperparameters are xand, for general density estimation, also yvariables. Such functions θ(x) orθ(x,y) will be called local hyperparameters orlocal hyperfields . Local hyperfields θ(x) can be used, for example, to adapt templates or covariances loca lly. (For general density estimation problems replace here and in the followi ngxby (x,y).) The price to be paid for the additional flexibility of functio n hyperparam- eters is a large number of additional degrees of freedom. Thi s can consid- erably complicate calculations and, requires a sufficient nu mber of training data and/or a sufficiently restrictive hyperprior to be able t o determine the hyperfield and not to make the prior useless. To introduce local hyperparameters θ(x) we express real symmetric, pos- itive (semi–) definite inverse covariances by square roots W,K=WTW=/integraltextdxWxWT xwhereWxrepresents the vector W(x,·). In components K(x,x′) =/integraldisplay dx′′WT(x,x′′)W(x′′,x′). (493) In terms of ‘filtered differences’ ω(x) =/integraltextdx′W(x,x′) (φ(x′)−t(x′)) the prior can be written p(φ)∝e−1 2/integraltext dx|ω(x)|2. (494) A local hyperparameter θ(x) may be introduced as follows p(φ|θ) =e−1 2/integraltext dx|ω(x;θ)|2−lnZφ(θ)=e−1 2/integraltext dx((1−θ(x))|ω1(x)|2+θ(x)|ω2(x)|2)−lnZφ(θ), (495) with ω(x;θ) = (1 −θ(x))ω1(x) +θ(x)ω2(x), (496) and, for instance, binary θ(x)∈ {0,1}. Local modifications of ω(x) can be constructed from variants of templates or covariances tx(θ) = (1 −θ(x))t1,x+θ(x)t2,x, (497) Wx(θ) = (1 −θ(x))W1,x+θ(x)W2,x, (498) 116the latter corresponding to K(θ) =/integraldisplay dxKx(θ) =/integraldisplay dx/bracketleftig (1−θ(x))W1,xWT 1,x+θ(x)W2,xWT 2,x/bracketrightig ,(499) where Kx(θ) =Wx(θ)WT x(θ). For real θ(x) in Eq. (496) additional terms θ2and (1 −θ(x))θ(x) would appear in Eq. (495) ). Notice, that also the unrestricted adaption of templates discussed in Sect. 5.2. 3 corresponds to the adaption of a real function θ(x). A realθvariable can be converted into a binary variable by replacin gθ in Eq. (496). for example by Bθ(x) = Θ(θ(x)−ϑ). (500) In case, however, the hyperprior is also formulated in terms ofBθ(x) this is completely equivalent to a binary formulation. Notice, that local templates tx(θ) for fixedxare still functions tx(x′;θ) of anotherx′variable. Indeed, to obtain ω(x), the function txis needed for allx′for which Whas nonzero entries. For a given θ(x) the corresponding effective template t(θ) and effective covariance K(θ) are, according to Eqs. (250,247), given by t(θ) =K(θ)−1/summationdisplay xKx(θ)tx(θ), (501) K(θ) =/parenleftigg/summationdisplay xKx(θ)/parenrightigg , (502) i.e., one may rewrite /summationdisplay x|ω(x,θ)|2= (φ−t,K(φ−t)) +/summationdisplay x(tx,Kxtx)−(t,Kt).(503) The MAP of Gaussian regression for a prior corresponding to ( 503) at optimal θ∗is therefore given by φ∗= (KD+K(θ∗))−1(KDtD+K(θ∗)t(θ∗)), according to Section 3.7. As example, consider the following prior energy, E(φ|θ) =1 2/parenleftig φ−t0(θ),(φ−t0(θ))/parenrightig +1 2/parenleftig φ,K0φ/parenrightig . (504) Because the covariance of the θ–dependent term is the identity, ( t0)x(x′;θ) is only needed for x=x′and we may thus directly write t0(θ) denoting a 117functiont0(x;θ). To get the effective prior template tforφ, however, both terms have to be combined, yielding E(φ|θ) =1 2/parenleftbigg/parenleftig φ−t(θ),K(φ−t(θ))/parenrightig +/parenleftig t0(θ),/parenleftig I−K−1/parenrightig t0(θ)/parenrightig/parenrightbigg ,(505) with effective template and effective inverse covariance t(θ) =K−1t0(θ),K=I+K0. (506) For differential operators Wthe effective t(θ) is thus a smoothed version of t0(θ). The extreme case would be to treat tandWitself as unrestricted hyper- parameters. Notice, however, that increasing flexibility t ends to lower the influence of the corresponding prior term. That means, using completely free templates and covariances without introducing additional restricting hyper- priors, just eliminates the corresponding prior term (see S ection 5.2.3). Hence, to restrict the flexibility, typically a smoothness h yperprior is im- posed to prevent highly oscillating functions θ(x). For example, to restrict the number of discontinuities for a one–dimensional x, one may include a factor like p(θ)∝e−κ 2/integraltext dxδ(1−Cθ(x)), (507) with constant κ, and Cθ(x) = Θ /parenleftigg∂θ ∂x/parenrightigg2 −ϑθ , (508) with threshold 0 ≤ϑθ<∞and step function Θ with Θ( x) = 0 forx≤0 and Θ(x) = 1 for 1 < x≤ ∞. In the binary case, where/parenleftig ∂θ ∂x/parenrightig2∈ {0,∞}, the term (508) counts the number of jumps. For real θ(x) an additional smoothness prior like ( θ,−∆θ) should be added in regions where it is defined (The space of φ–functions for which a smoothness prior ( φ−t,K(φ−t)) with discontinuous t(θ) is defined depends on the locations of the discontinuities. ) To enable differentiation the step function Θ could be replac ed by a sigmoidal function. The expression Cθof Eq. (508) can be generalized to Cθ(x) = Θ/parenleftig |ωθ(x)|2−ϑθ/parenrightig , (509) 118whereωθ(x) = (WθBθ)(x) andWθis some hyperprior operator, analogous to the operator Win the prior, acting on the function Bθdefined in Eq. (500). Discontinuous functions φcan be approximated by using discontinuous templatest(x;θ) or by eliminating matrix elements of the covariance which connect the two sides of the discontinuity. For example, con sider the discrete version of a negative Laplacian with periodic boundary cond itions, K=WTW= 2−1 0 0 0 −1 −1 2 −1 0 0 0 0−1 2 −1 0 0 0 0 −1 2 −1 0 0 0 0 −1 2 −1 −1 0 0 0 −1 2 , (510) and square root, W= 1−1 0 0 0 0 0 1 −1 0 0 0 0 0 1 −1 0 0 0 0 0 1 −1 0 0 0 0 0 1 −1 −1 0 0 0 0 1 . (511) The first three points can be disconnected from the last three points by setting W(3) and W(6) to zero, namely, W= 1−1 0 0 0 0 0 1 −10 0 0 0 0 0 0 0 0 0 0 0 1−1 0 0 0 0 0 1 −1 0 0 0 0 0 0 (512) so that the smoothness prior is ineffective between points fr om different re- gions, K=WTW= 1−1 0 0 0 0 −1 2 −10 0 0 0−1 1 0 0 0 0 0 0 1−1 0 0 0 0 −1 2 −1 0 0 0 0−1 1 . (513) 119In contrast to using discontinuous templates, training dat a are in this case required for both regions to determine the free constants wh ich correspond to the zero mode of the Laplacian. Non–Gaussian priors often provide an alternative to the use of function hyperparameters. Similarly to Eq. (508) one may define a B(x) directly in terms ofφ, B(x) = Θ/parenleftig |ω1(x)|2− |ω2(x)|2−ϑ/parenrightig , (514) like, for a negative Laplacian prior, B(x) = Θ /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(φ−t1) ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 −/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(φ−t2) ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 −ϑ . (515) Notice, that the functions ωi(x) andB(x) may be nonlocal with respect to φ(x), meaning they may depend on more than one φ(x) value. The thresh- oldϑcan be related to the prior expectations about ωi. A possible prior formulated in terms of Bcan be, p(φ)∝e−1 2/integraltext dx(|ω1(x)|2(1−B(x))+|ω2(x)|2B(x)−κ 2δ(1−C(x))), (516) with C(x) = Θ/parenleftig |(WBB)(x)|2−ϑB/parenrightig , (517) and some threshold ϑBand operator WB. Similarly to the introduction of hyperparameters, one can again treat B(x) formally as an independent function by including a term λ(B(x)−Θ (|ω1(x)|2− |ω2(x)|2−ϑ)) in the prior energy and taking the limit λ→ ∞. Eq. (516) looks similar to the combination of the prior (495) with the hyperprior (507), p(φ,θ)∝e−1 2/integraltext dx(|ω1(x)|2(1−Bθ(x))+|ω2(x)|2Bθ(x)−κ 2δ(1−Cθ(x))−lnZφ(θ)).(518) Notice, however, that the definition of Bθ(orCθ, respectively), is different from that of B(orC). If theωidiffer only in their templates, the normaliza- tion term can be skipped. Then, identifying Bθin (518) with a binary θand assumingϑ= 0,ϑθ=ϑB,Wθ=WB, the two equations are equivalent for θ= Θ (|ω1(x)|2− |ω2(x)|2). In the absence of hyperpriors, it is indeed easily seen that this is a selfconsistent solution for θ, givenφ. In general, however, there may be a trade off with the hyperprior, and another solut ion forθ, not 120selecting locally the smallest of the two prior contributio ns, might be better. Non–Gaussian priors will be discussed in Section 6.5. Hyperpriors, or analogous non–Gaussian prior terms, are fo r example useful to enforce specific global constraints for θ(x) orB(x). In images, for example, discontinuities are expected to form closed curve s. Hyperpriors, organizing discontinuities along lines or closed curves, a re thus important for image segmentation [65, 141, 61, 62, 221, 229]. 6 Non–Gaussian prior factors 6.1 Mixtures of Gaussian prior factors Complex, non–Gaussian prior factors, for example being mul timodal, may be constructed or approximated by using mixtures of simpler pr ior components. In particular, it is convenient to use as components or “buil ding blocks” Gaussian densities, as then many useful results obtained fo r Gaussian pro- cesses survive the generalization to mixture models [123, 1 24, 125, 126, 127]. We will therefore in the following discuss applications of m ixtures of Gaus- sian priors. Other implementations of non–Gaussian priors will be discussed in Section 6.5. In Section 5.1 we have seen that hyperparameters label compo nents of mixture densities. Thus, if jlabels the components of a mixture model, then jcan be seen as hyperparameter. In Section 5 we have treated th e corre- sponding hyperparameter integration completely in saddle point approxima- tion. In this section we will assume the hyperparameters jto be discrete and try to calculate the corresponding summation exactly. Hence, consider a discrete hyperparameter j, possibly in addition to con- tinuous hyperparameters θ. In contrast to the θ–integral we aim now in treating the analogous sum over jexactly, i.e., we want to study mixture models p(φ,θ|˜D0) =m/summationdisplay jp(φ,θ,j|˜D0) =m/summationdisplay jp(φ|˜D0,θ,j)p(θ,j). (519) In the following we concentrate on mixtures of Gaussian specific priors . No- tice that such models do notcorrespond to Gaussian mixture models for φ as they are often used in density estimation. Indeed, the for m ofφmay be completely unrestricted, it is only its prior or posterior d ensity which is mod- eled by a mixture. We also remark that a strict asymptotical j ustification of 121a saddle point approximation would require the introductio n of a parameter ˜βso thatp(φ,θ|˜D0)∝e˜βln/summationtext jpj. If the sum is reduced to a single term then ˜βcorresponds to β. We already discussed shortly in Section 5.2 that, in contras t to a product of probabilities or a sum of error terms implementing a proba bilistic AND of approximation conditions, a sum over jimplements a probabilistic OR. Those alternative approximation conditions will in the seq uel be represented by alternative templates tjand covariances Kj. A prior (or posterior) den- sity in form of a probabilistic OR means that the optimal solu tion does not necessarily have to approximate all but possibly only one of thetj(in a met- ric defined by Kj). For example, we may expect in an image reconstruction task blue or brown eyes whereas a mixture between blue and bro wn might not be as likely. Prior mixture models are potentially usefu l for 1. Ambiguous (prior) data. Alternative templates can for ex ample repre- sent different expected trends for a time series. 2. Model selection. Here templates represent alternative r eference models (e.g., different neural network architectures, decision tr ees) and deter- mining the optimal θcorresponds to training of such models. 3. Expert knowledge. Assume a priori knowledge to be formulated in terms of conjunctions and disjunctions of simple component s or build- ing blocks (for example verbally). E.g., an image of a face is expected to contain certain constituents (eyes, mouth, nose; AND) ap pearing in various possible variants (OR). Representing the simple compo- nents/building blocks by Gaussian priors centered around a typical example (e.g.,of an eye) results in Gaussian mixture models . This con- stitutes a possible interface between symbolic and statist ical methods. Such an application of prior mixture models has some similar ities with the quantification of “linguistic variables” by fuzzy metho ds [110, 111]. For a discussion of possible applications of prior mixture m odels see also [123, 124, 125, 126, 127]. An application of prior mixture mo dels to image completion can be found in [128]. 1226.2 Prior mixtures for density estimation The mixture approach (519) leads in general to non–convex er ror functionals. For Gaussian components Eq. (519) results in an error functi onal Eθ,φ=−(lnP(φ), N) + (P(φ),ΛX) −ln/summationdisplay je−(1 2(φ−tj(θ),Kj(θ)(φ−tj(θ)))+lnZφ(θ,j)+Eθ,j), (520) =−ln/summationdisplay je−Eφ,j−Eθ,j+cj, (521) where Eφ,j=−(lnP(φ), N)+(P(φ),ΛX)+1 2/parenleftig φ−tj(θ),Kj(θ) (φ−tj(θ))/parenrightig ,(522) and cj=−lnZφ(θ,j). (523) The stationarity equations for φandθ 0 =m/summationdisplay jδEφ,j δφe−Eφ,j−Eθ,j+cj, (524) 0 =m/summationdisplay j/parenleftigg∂Eφ,j ∂θ+∂Eθ,j ∂θ+Z′ jZ−1 φ(θ,j)/parenrightigg e−Eφ,j−Eθ,j+cj,(525) can also be written 0 =m/summationdisplay jδEφ,j δφp(φ,θ,j|˜D0), (526) 0 =m/summationdisplay j/parenleftigg∂Eφ,j ∂θ+∂Eθ,j ∂θ+Z′ jZ−1 φ(θ,j)/parenrightigg p(φ,θ,j|˜D0). (527) Analogous equations are obtained for parameterized φ(ξ). 6.3 Prior mixtures for regression For regression it is especially useful to introduce an inver se temperature mul- tiplying the terms depending on φ, i.e., likelihood and prior.3As in regression 3As also the likelihood term depends on βit may be considered part of a ˜φtogether regression function h(x). Due to its similarity to a regularization factor we have in cluded βin this chapter about hyperparameters. 123φis represented by the regression function h(x) the temperature–dependent error functional becomes Eθ,h=−lnm/summationdisplay je−βEh,j−Eθ,β,j+cj=−lnm/summationdisplay je−Ej+cj, (528) with Ej=ED+E0,j+Eθ,β,j, (529) ED=1 2(h−tD,KD(h−tD)), E 0,j=1 2(h−tj(θ),Kj(θ) (h−tj(θ))), (530) some hyperprior energy Eθ,β,j, and cj(θ,β) = −lnZh(θ,j,β) +n 2lnβ−β 2VD−c =1 2ln det ( Kj(θ)) +d+n 2lnβ−β 2VD (531) with some constant c. If we also maximize with respect to βwe have to include the ( h–independent) training data variance VD=/summationtextn iViwhereVi=/summationtextni ky(xk)2/ni−t2 D(xi) is the variance of the nitraining data at xi. In case everyxiappears only once VDvanishes. Notice that cjincludes a contribution from thendata points arising from the β–dependent normalization of the likelihood term. Writing the stationarity equation for the hyperparameter β separately, the corresponding three stationarity conditi ons are found as 0 =m/summationdisplay j/parenleftig KD(h−tD) +Kj(h−tj)/parenrightig e−βEh,j−Eθ,β,j+cj, (532) 0 =m/summationdisplay j/parenleftigg E′ h,j+E′ θ,β,j+ Tr/parenleftigg K−1 j∂Kj ∂θ/parenrightigg/parenrightigg e−βEh,j−Eθ,β,j+cj,(533) 0 =m/summationdisplay j/parenleftigg E0,j+∂Eθ,β,j ∂β+d+n 2β/parenrightigg e−βEh,j−Eθ,β,j+cj. (534) Asβis only a one–dimensional parameter and its density can be qu ite non– Gaussian it is probably most times more informative to solve for varying values ofβinstead to restrict to a single ‘optimal’ β∗. Eq. (532) can also be written h= KD+m/summationdisplay jajKj −1/parenleftigg KDtD+m/summationdisplay lajKjtj/parenrightigg , (535) 124with aj=p(j|h,θ,β,D 0) =e−Ej+cj /summationtextm ke−Ek+ck=e−βE0,j−Eθ,β,j+1 2lndetKj /summationtextm ke−βE0,k−Eθ,β,k+1 2lndetKk =p(h|j,θ,β,D 0)p(j|θ,β,D 0) p(h|θ,β,D 0)=p(h|j,θ,β,D 0)p(j,θ|β,D 0) p(h,θ|β,D 0),(536) being thus still a nonlinear equation for h. 6.3.1 High and low temperature limits It are the limits of large and small βwhich make the introduction of this additional parameter useful. The reason being that the high temperature limitβ→0 gives the convex case, and statistical mechanics provides us with high and low temperature expansions. Hence, we study the hig h temperature and low temperature limits of Eq. (535). In the high temperature limit β→0 the exponential factors ajbecome h–independent ajβ→0−→a0 j=e−Eθ,β,j+1 2ln detKj /summationtextm ke−Eθ,β,k+1 2lndetKk. (537) In case one chooses Eθ,β,j=Eβ,j+βEθone has to replace Eθ,β,jbyEβ,j. The high temperature solution becomes h=¯t (538) with (generalized) ‘complete template average’ ¯t= KD+m/summationdisplay ja0 jKj −1/parenleftigg KDtD+m/summationdisplay la0 jKjtj/parenrightigg . (539) Notice that ¯tcorresponds to the minimum of the quadratic functional E(β=∞)=/parenleftig h−tD,KD(h−tD)/parenrightig +m/summationdisplay ja0 j/parenleftig h−tj,Kj(h−tj)/parenrightig .(540) Thus, in the infinite temperature limit a combination of quad ratic priors by OR is effectively replaced by a combination by AND. 125In the low temperature limit β→ ∞ we have, assuming Eθ,β,j=Eβ+ Ej+βEθ, /summationdisplay je−β(E0,j+Eθ)−Eβ−Ej=e−β(E0,j∗+Eθ)−Eβ/summationdisplay je−β(E0,j−E0,j∗)−Ej(541) β→∞−→e−β(E0,j∗+Eθ)−Eβ−EjforE0,j∗<E0,j,∀j∝ne}ationslash=j∗, p(j∗)∝ne}ationslash= 0,,(542) meaning that ajβ→∞−→a∞ j=/braceleftigg 1 :j= argminjE0,j= argminjEh,j 0 :j∝ne}ationslash= argminjE0,j= argminjEh,j. (543) Henceforth, all (generalized) ‘component averages’ ¯tjbecome solutions h=¯tj, (544) with ¯tj= (KD+Kj)−1(KDtD+Kjtj), (545) provided the ¯tjfulfill the stability condition Eh,j(h=¯tj)<Eh,j′(h=¯tj),∀j′∝ne}ationslash=j, (546) i.e., Vj<1 2/parenleftig¯tj−¯tj′,(KD+Kj′) (¯tj−¯tj′)/parenrightig +Vj′,∀j′∝ne}ationslash=j, (547) where Vj=1 2/parenleftigg/parenleftig tD,KDtD/parenrightig +/parenleftig tj,Kjtj/parenrightig −/parenleftig¯tj,(KD+Kj)¯tj/parenrightig/parenrightigg . (548) That means single components become solutions at zero tempe rature 1/β in case their (generalized) ‘template variance’ Vj, measuring the discrepancy between data and prior term, is not too large. Eq. (535) for hcan also be expressed by the (potential) low temperature solutions ¯tj h= m/summationdisplay jaj(KD+Kj) −1m/summationdisplay jaj(KD+Kj)¯tj. (549) Summarizing, in the high temperature limit the stationarit y equation (532) becomes linear with a single solution being essential ly a (generalized) average of all template functions. In the low temperature li mit the sin- gle component solutions become stable provided their (gene ralized) variance corresponding to their minimal error is small enough. 1266.3.2 Equal covariances Especially interesting is the case of j–independent Kj(θ) =K0(θ) andθ– independent det K0(θ). In that case the often difficult to obtain determinants ofKjdo not have to be calculated. Forj–independent covariances the high temperature solution is according to Eqs.(539,545) a linear combination of the (potential) lo w temperature solutions ¯t=m/summationdisplay ja0 j¯tj. (550) It is worth to emphasize that, as the solution ¯tisnota mixture of the component templates tjbut of component solutions ¯tj, even poor choices for the template functions tjcan lead to good solutions, if enough data are available. That is indeed the reason why the most common choi cet0≡0 for a Gaussian prior can be successful. Eqs.(549) simplifies to h=/summationtextm j¯tje−βEh,j(h)−Eθ,β,j+cj /summationtextm je−βEh,j(h)−Eθ,β,j+cj=m/summationdisplay jaj¯tj=¯t+m/summationdisplay j(aj−a0 j)¯tj, (551) where ¯tj= (KD+K0)−1(KDtD+K0tj), (552) and (forj–independent d) aj=e−Ej /summationtext ke−Ek=e−βEh,j−Eθ,β,j /summationtext ke−βEh,k−Eθ,β,k=e−β 2aBja+dj /summationtext ke−β 2aBka+dk, (553) introducing vector awith components aj,m×mmatrices Bj(k,l) =/parenleftig¯tk−¯tj,(KD+K0) (¯tl−¯tj)/parenrightig (554) and constants dj=−βVj−Eθ,β,j, (555) withVjgiven in (548). Eq. (551) is still a nonlinear equation for h, it shows however that the solutions must be convex combination s of theh– independent ¯tj. Thus, it is sufficient to solve Eq. (553) for mmixture coeffi- cientsajinstead of Eq. (532) for the function h. 127The high temperature relation Eq. (537) becomes ajβ→0−→a0 j=e−Eθ,β,j /summationtextm ke−Eθ,β,k, (556) ora0 j= 1/mfor a hyperprior p(θ,β,j) uniform with respect to j. The low temperature relation Eq. (543) remains unchanged. Form= 2 Eq. (551) becomes h=2/summationdisplay jaj¯tj=¯t1+¯t2 2+ (a1−a2)¯t1−¯t2 2=¯t1+¯t2 2+ (tanh ∆)¯t1−¯t2 2,(557) with ( ¯t1+¯t2)/2 =¯tin caseEθ,β,jis uniform in jso thata0 j= 0.5, and ∆ =E2−E1 2=βEh,2−Eh,1 2+Eθ,β,2−Eθ,β,1 2 =−β 4a(B1−B2)a+d1−d2 2=β 4b(2a1−1) +d1−d2 2,(558) because the matrices Bjare in this case zero except B1(2,2) =B2(1,1) =b. The stationarity Eq. (553) can be solved graphically (see Fi gs.7, 8), the solution being given by the point where a1e−β 2ba2 1+d2= (1−a1)e−β 2b(1−a1)2+d1, or, alternatively, a1=1 2(tanh ∆ + 1) . (559) That equation is analogous to the celebrated mean field equat ion of the ferromagnet. We conclude that in the case of equal component covariances, in addition to the linear low–temperature equations, only a m−1–dimensional nonlinear equation has to be solved to determine the ‘mixing coefficient s’a1,···,am−1. 6.3.3 Analytical solution of mixture models For regression under a Gaussian mixture model the predictiv e density can be calculated analytically for fixed θ. This is done by expressing the predictive density in terms of the likelihood of θandj, marginalized over h p(y|x,D,D 0) =/summationdisplay j/integraldisplay dhdθp(θ,j)p(yD|xD,D0,θ,j) /summationtext j/integraltextdθp(θ,j)p(yD|xD,D0,θ,j)p(y|x,D,D 0,θ,j). (560) 1280.20.40.60.81a10.20.40.60.81a10.20.40.60.81a10.20.40.60.81a10.20.40.60.81a10.20.40.60.81a1 Figure 7: The solution of stationary equation Eq. (553) is gi ven by the point wherea1e−β 2ba2 1+d2= (1−a1)e−β 2b(1−a1)2+d1(upper row) or, equivalently, a1= 1 2(tanh∆ + 1) (lower row). Shown are, from left to right, a situa tion at high temperature and one stable solution ( β= 2), at a temperature ( β= 2.75) near the bifurcation, and at low temperature with two stable and one unstable solutionsβ= 4. The values of b= 2,d1=−0.2025βandd2=−0.3025βused for the plots correspond for example to the one–dimensional model of Fig.9 witht1= 1,t2=−1,tD= 0.1. Notice, however, that the shown relation is valid form= 2 at arbitrary dimension. (Here we concentrate on θ. The parameter βcan be treated analogously.) According to Eq. (487) the likelihood can be written p(yD|xD,D0,θ,j) =e−β/tildewideE0,j(θ)+1 2ln det(β 2π/tildewideKj(θ)), (561) with /tildewideE0,j(θ) =1 2(tD−tj(θ),/tildewiderKj(θ)(tD−tj(θ))) =Vj, (562) and/tildewiderKj(θ) = (K−1 D+K−1 j,DD(θ))−1being a ˜nטn–matrix in data space. The equality of Vjand/tildewideE0,jcan be seen using Kj−Kj(KD+Kj)−1Kj=KD− KD(KD+Kj,DD)−1KD=Kj,DD−Kj,DD(KD+K−1 j,DD)Kj,DD=/tildewiderK. For the predictive mean, being the optimal solution under squar ed–error loss and log–loss (restricted to Gaussian densities with fixed va riance) we find therefore ¯y(x) =/integraldisplay dyyp(y|x,D,D 0) =/summationdisplay j/integraldisplay dθbj(θ)¯tj(θ), (563) with, according to Eq. (318), ¯tj(θ) =tj+K−1 j/tildewiderKj(tD−tj), (564) 1290 0.25 0.5 0.75 1a1 01234 beta00.250.50.751 0 0.25 0.5 0.75 1a1 Figure 8: As in Fig.7 the plots of f1(a1) =a1andf2(a1) =1 2(tanh∆ + 1) are shown within the inverse temperature range 0 ≤β≤4. and mixture coefficients bj(θ) =p(θ,j|D) =p(θ,j)p(yD|xD,D0,θ,j) /summationtext j/integraltextdθp(θ,j)p(yD|xD,D0,θ,j) ∝e−β/tildewideEj(θ)−Eθ,j+1 2ln det(/tildewideKj(θ)), (565) which defines/tildewideEj=β/tildewideE0,j+Eθ,j. For solvable θ–integral the coefficients can therefore be obtained exactly. Ifbjis calculated in saddle point approximation at θ=θ∗it has the structure of ajin (536) with E0,jreplaced by/tildewideEjandKjby/tildewiderKj. (The inverse temperature βcould be treated analogously to θ. In that case Eθ,jwould have to be replaced by Eθ,β,j.) Calculating also the likelihood for j,θin Eq. (565) in saddle point ap- proximation, i.e., p(yD|xD,D0,θ∗,j)≈p(yD|xD,h∗)p(h∗|D0,θ∗,j), the terms p(yD|xD,h∗) in numerator and denominator cancel, so that, skipping D0and β, bj(θ∗) =p(h∗|j,θ∗)p(j,θ∗) p(h∗,θ∗)=aj(h∗,θ∗), (566) becomes equal to the aj(θ∗) in Eq. (536) at h=h∗. Eq. (565) yields as stationarity equation for θ, similarly to Eq. (489) 0 =/summationdisplay jbj/parenleftigg∂/tildewideEj ∂θ−Tr/parenleftigg /tildewiderK−1 j∂/tildewiderKj ∂θ/parenrightigg/parenrightigg (567) 1302 4 6 8 10beta -2-1012 hp 2 4 6 8beta2 4 6 8 10beta -2-1012 hp 2 4 6 8beta Figure 9: Shown is the joint posterior density of handβ, i.e.,p(h,β|D,D 0) ∝p(yD|h,β)p(h|β,D 0)p(β) for a zero–dimensional example of a Gaussian prior mixture model with training data yD= 0.1 and prior data yD0= ±1 and inverse temperature β. L.h.s.: For uniform prior (middle) p(β)∝ 1 with joint posterior p∝e−β 2h2+lnβ/parenleftig e−β 2(h−1)2+e−β 2(h+1)2/parenrightig the maximum appears at finite β. (Here no factor 1 /2 appears in front of ln βbecause normalization constants for prior and likelihood term have to be included.) R.h.s.: For compensating hyperprior p(β)∝1/√βwithp∝e−β 2h2−β 2(h−1)2+ e−β 2h2−β 2(h+1)2the maximum is at β= 0. =/summationdisplay jbj/parenleftigg/parenleftigg∂tj(θ) ∂θ,/tildewiderKj(θ)(tj(θ)−tD)/parenrightigg +1 2/parenleftigg (tD−tj(θ)),∂/tildewiderKj(θ) ∂θ(tD−tj(θ))/parenrightigg −Tr/parenleftigg /tildewiderK−1 j(θ)∂/tildewiderKj(θ) ∂θ/parenrightigg −1 p(θ,j)∂p(θ,j) ∂θ/parenrightigg . (568) For fixedθandj–independent covariances the high temperature solution is a mixture of component solutions weighted by their prior p robability ¯yβ→0−→/summationdisplay jp(j)¯tj=/summationdisplay ja0 j¯tj=¯t. (569) The low temperature solution becomes the component solutio n¯tjwith min- 1312 4 6 8 10beta -2-1012 hp 2 4 6 8beta2 4 6 8 10beta -2-1012 hp 2 4 6 8beta Figure 10: Same zero–dimensional prior mixture model for un iform hyper- prior onβas in Fig.9, but for varying data xd= 0.3 (left),xd= 0.5 (right). imal distance between data and prior template ¯yβ→∞−→¯tj∗withj∗= argminj(tD−tj,/tildewiderKj(tD−tj)). (570) Fig.11 compares the exact mixture coefficient b1with the dominant solution of the maximum posterior coefficient a1(see also [123]) which are related according to (553) aj=e−β 2aBja−/tildewideEj /summationtext ke−β 2aBka−/tildewideEk=bje−β 2aBja /summationtext kbke−β 2aBka. (571) 6.4 Local mixtures Global mixture components can be obtained by combining loca l mixture components. Predicting a time series, for example, one may a llow to switch locally (in time) between two or more possible regimes, each corresponding to a different local covariance or template. The problem which arises when combining local alternatives is the fact that the total number of mixture components grows exponenti ally in the number local components which have to be combined for a globa l mixture component. Consider a local prior mixture model, similar to Eq. (516), p(φ|θ) =e−/integraltext dx;|ω(x;θ(x))|2−lnZφ(θ)(572) 1322 4 6 8 10 12 14 0.50.60.70.80.9 Figure 11: Exact b1anda1(dashed) vs. βfor two mixture components with equal covariances and B1(2,2) =b= 2,/tildewideE1= 0.405,/tildewideE2= 0.605. whereθ(x) may be a binary or an integer variable. The local mixture var iable θ(x) labels local alternatives for filtered differences ω(x;θ(x)) which may differ in their templates t(x;θ(x)) and/or their local filters W(x;θ(x)). To avoid infinite products, we choose a discretized xvariable (which may include theyvariable for general density estimation problems), so that p(φ) =/summationdisplay θp(θ)e−/summationtext x|ω(x;θ(x))|2−lnZφ(θ), (573) where the sum/summationtext θis over all local integer variables θ(x), i.e., /summationdisplay θ=/summationdisplay θ(x1)···/summationdisplay θ(xl)= /productdisplay x/summationdisplay θ(x1) . (574) Only for factorizing hyperprior p(θ) =/producttext xp(θ(x)) the complete posterior factorizes p(φ) = /productdisplay x′/summationdisplay θ(x′) /productdisplay x/parenleftig p(θ(x))e−|ω(x;θ(x))|2−lnZφ(x,θ(x))/parenrightig =/productdisplay x/summationdisplay θ(x)/parenleftig p(θ(x))e−|ω(x;θ(x))|2−lnZφ(x,θ(x))/parenrightig , (575) because Zφ=/productdisplay x/summationdisplay θ(x)/parenleftig e−|ω(x;θ(x))|2/parenrightig =/productdisplay xZφ(x,θ(x)). (576) 133Under that condition the mixture coefficients aθof Eq. (536) can be ob- tained from the equations, local in θ(x), aθ=aθ(x1)···θ(xl)=p(θ|φ) =/productdisplay xaθ(x) (577) with aθ(x)=p(θ(x))e−|ω(x;θ(x))|2−lnZφ(x;θ(x)) /summationtext θ′(x)p(θ′(x))e−|ω(x;θ′(x))|2−lnZφ(x;θ′(x)). (578) For equal covariances this is a nonlinear equation within a s pace of dimension equal to the number of local components. For non–factorizin g hyperprior the equations for different θ(x) cannot be decoupled. 6.5 Non–quadratic potentials Solving learning problems numerically by discretizing the xandyvariables allows in principle to deal with arbitrary non–Gaussian pri ors. Compared to Gaussian priors, however, the resulting stationarity equa tions are intrinsically nonlinear. As a typical example let us formulate a prior in terms of nonli near and non–quadratic “potential” functions ψacting on “filtered differences” ω= W(φ−t), defined with respect to some positive (semi–)definite inve rse co- variance K=WTW. In particular, consider a prior factor of the following form p(φ) =e−/integraltext dxψ(ω(x))−lnZφ=e−E(φ) Zφ, (579) whereE(φ) =/integraltextdxψ(ω(x)). For general density estimation problems we understand xto stand for a pair ( x,y). Such priors are for example used for image restoration [65, 25, 155, 66, 231, 230]. For differentiable ψfunction the functional derivative with respect to φ(x) becomes δφ(x)p(φ) =−e−/integraltext dx′ψ(ω(x′))−lnZφ/integraldisplay dx′′ψ′(ω(x′′))W(x′′,x), (580) withψ′(s) =dψ(z)/dz, from which follows δφE(φ) =−δφlnp(φ) =WTψ′. (581) 134For nonlinear filters acting on φ−t,Win Eq. (579) must be replaced by ω′(x) =δφ(x)ω(x). Instead of one Wa “filter bank” Wαwith corresponding Kα,ωα, andψαmay be used, so that e−/summationtext α/integraltext dxψα(ωα(x))−lnZφ, (582) and δφE(φ) =/summationdisplay αWT αψ′ α. (583) The potential functions ψmay be fixed in advance for a given problem. Typical choices to allow discontinuities are symmetric “cu p” functions with minimum at zero and flat tails for which one large step is cheap er than many small ones [221]). Examples are shown in Fig. 12 (a,b). The cu sp in (b), where the derivative does not exist, requires special treat ment [230]. Such functions can also be interpreted in the sense of robust stat istics as flat tails reduce the sensitivity with respect to outliers [93, 94, 62, 23]. Inverted “cup” functions, like those shown in Fig. 12 (c), ha ve been ob- tained by optimizing a set of ψαwith respect to a sample of natural images [230]. (For statistics of natural images their relation to w avelet–like filters and sparse coding see also [162, 163].) While, for Wwhich are differential operators, cup functions promote smoothness, inverse cup functions can be used to implement s tructure. For suchWthe gradient algorithm for minimizing E(φ), φnew=φold−ηδφE(φold), (584) becomes in the continuum limit a nonlinear parabolic partia l differential equation, φτ=−/summationdisplay αWT αψ′ α(Wα(φ−t)). (585) Here a formal time variable τhave been introduced so that ( φnew−φold)/η→ φτ=dφ/dτ . For cup functions this equation is of diffusion type [160, 17 3], if also inverted cup functions are included the equation is of r eaction–diffusion type [230]. Such equations are known to generate a great vari ety of patterns. Alternatively to fixing ψin advance or, which is sometimes possible for low–dimensional discrete function spaces like images, to a pproximate ψby sampling from the prior distribution, one may also introduc e hyperparame- ters and adapt potentials ψ(θ) to the data. 135(a) -15 -10 -5 0 5 10 1500.20.40.60.81 (b) -15 -10 -5 0 5 10 1500.511.522.53 (c) -15 -10 -5 0 5 10 15-2.5-2-1.5-1-0.50 Figure 12: Non–quadratic potentials of the form ψ(x) =a(1.0−1/(1+(|x− x0|/b)γ)), [230]: “Diffusion terms”: (a) Winkler’s cup function [22 1] (a= 5, b= 10,γ= 0.7,x0= 0), (b) with cusp ( a= 1,b= 3,γ= 2,x0= 0), (c) “Reaction term” ( a=−4.8,b= 15,γ= 2.0x0= 0). 136For example, attempting to adapt a unrestricted function ψ(x) with hy- perpriorp(ψ) by Maximum A Posteriori Approximation one has to solve the stationarity condition 0 =δψ(s)lnp(φ,ψ) =δψ(s)lnp(φ|ψ) +δψ(s)lnp(ψ). (586) From δψ(s)p(φ|ψ) =−p(φ|ψ)/integraldisplay dxδ(s−ω(x))−1 Z2 φδψ(s)Zφ, (587) it follows −δψ(s)lnp(φ|ψ) =n(s)−<n(s)>, (588) with integer n(s) =/integraldisplay dxδ(s−ω(x)), (589) being the histogram of the filtered differences, and average h istogram <n(s)>=/integraldisplay dφp(φ|ψ)n(s). (590) The right hand side of Eq. (588) is zero at φ∗if, e.g.,p(φ|ψ) =δ(φ−φ∗), which is the case for ψ(ω(x;φ)) =β(ω(x;φ)−ω(x;φ∗))2in theβ→ ∞ limit. Introducing hyperparameters one has to keep in mind that the resulting additional flexibility must be balanced by the number of trai ning data and the hyperprior to be useful in practice. 7 Iteration procedures: Learning 7.1 Numerical solution of stationarity equations Due to the presence of the logarithmic data term −(lnP,N) and the normal- ization constraint in density estimation problems the stat ionary equations are in general nonlinear, even for Gaussian specific priors. An exception are Gaussian regression problems discussed in Section 3.7 for w hich−(lnP,N) becomes quadratic and the normalization constraint can be s kipped. How- ever, the nonlinearities arising from the data term −(lnP,N) are restricted to a finite number of training data points and for Gaussian spe cific priors one may expect them, like those arising from the normalization c onstraint, to be numerically not very harmful. Clearly, severe nonlinearit ies can appear for 137general non–Gaussian specific priors or general nonlinear p arameterizations P(ξ). As nonlinear equations the stationarity conditions have in general to be solved by iteration. In the context of empirical learning it eration procedures to minimize an error functional represent possible learning algorithms . In the previous sections we have encountered stationarity e quations 0 =δ(−Eφ) δφ=G(φ), (591) for error functionals Eφ, e.g.,φ=Lorφ=P, written in a form Kφ=T. (592) withφ–dependent T(and possibly K). For the stationarity Eqs. (144), (172), and (193) the operator Kis aφ–independent inverse covariance of a Gaussian specific prior. It has already been mentioned that for existi ng (and not too ill–conditioned) K−1(representing the covariance of the prior process) Eq. (592) suggests an iteration scheme φ(i+1)=K−1T(φ(i)), (593) for discretized φstarting from some initial guess φ(0). In general, like for the non–Gaussian specific priors discussed in Section 6, Kcan beφ–dependent. Eq. (359) shows that general nonlinear parameterizations P(ξ) lead to non- linear operators K. Clearly, if allowing φ–dependent T, the form (592) is no restriction of generality. One always can choose an arbitrary invertible ( and not too ill– conditioned) A, define TA=G(φ) +Aφ, (594) write a stationarity equation as Aφ=TA, (595) discretize and iterate with A−1. To obtain a numerical iteration scheme we will choose a linear, positive definite learning matrix A. The learning matrix may depend on φand may also change during iteration. To connect a stationarity equation given in form (592) to an a rbitrary iteration scheme with a learning matrix Awe define B=K−A,Bη=K−1 ηA, (596) 138i.e., we split Kinto two parts K=A+B=1 ηA+Bη, (597) where we introduced ηfor later convenience. Then we obtain from the sta- tionarity equation (592) φ=ηA−1(T−Bηφ). (598) To iterate we start by inserting an approximate solution φ(i)to the right– hand side and obtain a new φ(i+1)by calculating the left hand side. This can be written in one of the following equivalent forms φ(i+1)=ηA−1/parenleftig T(i)−Bηφ(i)/parenrightig (599) = (1−η)φ(i)+ηA−1/parenleftig T(i)−Bφ(i)/parenrightig (600) =φ(i)+ηA−1/parenleftig T(i)−Kφ(i)/parenrightig , (601) whereηplays the role of a learning rate or step width, and A−1=/parenleftig A(i)/parenrightig−1 may be iteration dependent. The update equations (599–601) can be written ∆φ(i)=ηA−1G(φ(i)), (602) with ∆φ(i)=φ(i+1)−φ(i). Eq. (601) does not require the calculation of B orBηso that instead of Adirectly A−1can be given without the need to calculate its inverse. For example operators approximatin gK−1and being easy to calculate may be good choices for A−1. For positive definite A(and thus also positive definite inverse) conver- gence can be guaranteed, at least theoretically. Indeed, mu ltiplying with (1/η)Aand projecting onto an infinitesimal dφgives 1 η(dφ,A∆φ) =/parenleftig dφ,δ(−Eφ) δφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle φ=φ(i)/parenrightig =d(−Eφ). (603) In an infinitesimal neighborhood of φ(i)where ∆φ(i)becomes equal to dφ in first order the left–hand side is for positive (semi) defini teAlarger (or equal) to zero. This shows that at least for ηsmall enough the posterior log–probability −Eφincreases i.e., the differential dEφis smaller or equal to zero and the value of the error functional Eφdecreases. 139Stationarity equation (128) for minimizing ELyields for (599,600,601), L(i+1)=ηA−1/parenleftigg N−Λ(i) XeL(i)−KL(i)+1 ηAL(i)/parenrightigg (604) = (1−η)L(i)+ηA−1/parenleftig N−Λ(i) XeL(i)−KL(i)+AL(i)/parenrightig (605) =L(i)+ηA−1/parenleftig N−Λ(i) XeL(i)−KL(i)/parenrightig . (606) The function Λ(i) Xis also unknown and is part of the variables we want to solve for. The normalization conditions provide the necessary ad ditional equations, and the matrix A−1can be extended to include the iteration procedure for ΛX. For example, we can insert the stationarity equation for Λ Xin (606) to get L(i+1)=L(i)+ηA−1/bracketleftig N−eL(i)(NX−IXKL)−KL(i)/bracketrightig . (607) If normalizing L(i)at each iteration this corresponds to an iteration procedur e forg=L+ lnZX. Similarly, for the functional EPwe have to solve (166) and obtain for (601), P(i+1)=P(i)+ηA−1/parenleftig T(i) P−KP(i)/parenrightig (608) =P(i)+ηA−1/parenleftig P(i)−1N−Λ(i) X−KP(i)/parenrightig (609) =P(i)+ηA−1/parenleftig P(i)−1N−NX−IXP(i)KP(i)−KP(i)/parenrightig .(610) Again, normalizing Pat each iteration this is equivalent to solving for z= ZXP, and the update procedure for Λ Xcan be varied. 7.2 Learning matrices 7.2.1 Learning algorithms for density estimation There exists a variety of well developed numerical methods f or unconstraint as well as for constraint optimization [175, 52, 81, 181, 82, 9, 17, 75, 178]. Popular examples include conjugate gradient, Newton, and q uasi–Newton methods, like the variable metric methods DFP (Davidon–Fle tcher–Powell) or BFGS (Broyden–Fletcher–Goldfarb–Shanno). All of them correspond to the choice of specific, often iterat ion dependent, learning matrices Adefining the learning algorithm. Possible simple choices 140are: A=I: Gradient (611) A=D: Jacobi (612) A=L+D: Gauss–Seidel (613) A=K: prior relaxation (614) where Istands for the identity operator, Dfor a diagonal matrix, e.g. the diagonal part of K, andLfor a lower triangular matrix, e.g. the lower tri- angular part of K. In case Krepresents the operator of the prior term in an error functional we will call iteration with K−1(corresponding to the co- variance of the prior process) prior relaxation . Forφ–independent KandT, η= 1 with invertible Kthe corresponding linear equation is solved by prior relaxation in one step. However, also linear equations are s olved by iteration if the size of Kis too large to be inverted. Because of I−1=Ithe gradient algorithm does not require inversion. On one hand, density estimation is a rather general problem r equiring the solution of constraint, inhomogeneous, nonlinear (int egro–)differential equations. On the other hand, density estimation problems a re, at least for Gaussian specific priors and non restricting parameteri zation, typically “nearly” linear and have only a relatively simple positivit y and normalization constraint. Furthermore, the inhomogeneities are commonl y restricted to a finite number of discrete training data points. Thus, we expe ct the inversion ofKto be the essential part of the solution for density estimati on problems. However, Kis not necessarily invertible or may be difficult to calculate . Also, inversion of Kis not exactly what is optimal and there are improved methods. Thus, we will discuss in the following basic optimi zation methods adapted especially to the situation of density estimation. 7.2.2 Linearization and Newton algorithm For linear equations Kφ=TwhereTandKare no functions of φa spectral radiusρ(M)<1 (the largest modulus of the eigenvalues) of the iteration matrix M=−ηA−1Bη= (1−η)I−ηA−1B=I−ηA−1K (615) would guarantee convergence of the iteration scheme. This i s easily seen by solving the linear equation by iteration according to (599) φ(i+1)=ηA−1T+Mφ(i)(616) 141=ηA−1T+ηMA−1T+M2φ(i−1)(617) =η∞/summationdisplay n=0MnA−1T. (618) A zero mode of K, for example a constant function for differential operators without boundary conditions, corresponds to an eigenvalue 1 ofMand would lead to divergence of the sequence φ(i). However, a nonlinear T(φ) orK(φ), like the nonlinear normalization constraint contained in T(φ), can then still lead to a unique solution. A convergence analysis for nonlinear equations can be done i n a linear approximation around a fixed point. Expanding the gradient a tφ∗ G(φ) =δ(−Eφ) δφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle φ∗+ (φ−φ∗)H(φ∗) +··· (619) shows that the factor of the linear term is the Hessian. Thus i n the vicinity ofφ∗the spectral radius of the iteration matrix M=I+ηA−1H, (620) determines the rate of convergence. The Newton algorithm us es the negative Hessian −Has learning matrix provided it exists and is positive definit e. Otherwise it must resort to other methods. In the linear appr oximation (i.e., for quadratic energy) the Newton algorithm A=−H: Newton (621) is optimal. We have already seen in Sections 3.1.3 and 3.2.3 t hat the inho- mogeneities generate in the Hessian in addition to Ka diagonal part which can remove zero modes of K. 7.2.3 Massive relaxation We now consider methods to construct a positive definite or at least invertible learning matrix. For example, far from a minimum the Hessian Hmay not be positive definite and like a differential operator Kwith zero modes, not even invertible. Massive relaxation can transform a non–in vertible or not positive definite operator A0, e.g.A0=KorA0=−H, into an invertible or positive definite operators: A=A0+m2I: Massive relaxation (622) 142A generalization would be to allow m=m(x,y). This is, for example, used in some realizations of Newton‘s method for minimization in regions where His not positive definite and a diagonal operator is added to −H, using for example a modified Cholesky factorization [17]. The mass ter m removes the zero modes of Kif−m2is not in the spectrum of A0and makes it positive definite ifm2is larger than the smallest eigenvalue of A0. Matrix elements (φ,(A0−zI)−1φ) of the resolvent A−1(z),z=−m2representing in this case a complex variable, have poles at discrete eigenvalues ofA0and a cut at the continuous spectrum as long as φhas a non–zero overlap with the corresponding eigenfunctions. Instead of multiples of the identity, also other operators may be added to remove zero modes. The Hessian HLin (157), for example, adds a x–dependent mass–like, but not necessarily positive definit e term to K. Similarly, for example HPin (182) has ( x,y)–dependent mass P−2Nrestricted to data points. While full relaxation is the massless limit m2→0 of massive relaxation, a gradient algorithm with η′can be obtained as infinite mass limit m2→ ∞ withη→ ∞ andm2/η= 1/η′. Constant functions are typical zero modes, i.e., eigenfunc tions with zero eigenvalue, for differential operators with periodic bound ary conditions. For instance for a common smoothness term −∆ (kinetic energy operator) as regularizing operator Kthe inverse of A=K+m2Ihas the form A−1(x′,y′;x,y) =1 −∆ +m2. (623) =/integraldisplay∞ −∞ddXkxddYky (2π)deikx(x−x′)+iky(y−y′) k2x+k2y+m2, (624) withd=dX+dY,dX= dim(X),dY= dim(Y). This Green‘s function or matrix element of the resolvent kernel for A0is analogous to the (Euclidean) propagator of a free scalar field with mass m, which is its two–point corre- lation function or matrix element of the covariance operato r. According to 1/x=/integraltext∞ 0dte−xtthe denominator can be brought into the exponent by intro- ducing an additional integral. Performing the resulting Ga ussian integration overk= (kx,ky) the inverse can be expressed as A−1(x′,y′;x,y;m) =md−2A−1(m(x−x′),m(y−y′); 1) = (2π)−d/2/parenleftiggm |x−x′|+|y−y′|/parenrightigg(d−2)/2 K(d−2)/2(m|x−x′|+m|y−y′|),(625) 143in terms of the modified Bessel functions Kν(x) which have the following integral representation Kν(2/radicalig βγ) =1 2/parenleftiggγ β/parenrightiggν 2/integraldisplay∞ 0dttν−1eβ t−γt. (626) Alternatively, the same result can be obtained by switching tod–dimensional spherical coordinates, expanding the exponential in ultra -spheric harmonic functions and performing the integration over the angle-va riables [109]. For the example d= 1 this corresponds to Parzens kernel used in density esti- mation or for d= 3 A−1(x′,y′;x,y) =1 4π|x−x′|+ 4π|y−y′|e−m|x−x′|−m|y−y′|. (627) The Green’s function for periodic, Neumann, or Dirichlet bo undary con- ditions can be expressed by sums over A−1(x′,y′;x,y) [72]. The lattice version of the Laplacian with lattice spacing areads ˆ∆f(n) =1 a2d/summationdisplay j[f(n−aj)−2f(n) +f(n+aj)], (628) writingajfor a vector in direction jand length a. Including a mass term we get as lattice approximation for A ˆA(nx,ny;mx,my) =−1 a2dX/summationdisplay i=1δny,my(δnx+ax i,mx−2δnx,mx+δnx−ax i,mx) −1 a2dY/summationdisplay j=1δnx,mx(δny+ay j,my−2δny,my+δny−ay j,my) +m2δnx,mxδny,my(629) Inserting the Fourier representation (102) of δ(x) gives ˆA(nx,ny;mx,my) =2d a2/integraldisplayπ −πddXkxddYky (2π)deikx(nx−mx)+iky(ny−my) × 1 +m2a2 2d−1 ddX/summationdisplay i=1coskx,i−1 ddY/summationdisplay j=1cosky,j , (630) 144withkx,i=kxax i, cosky,j= coskyay jand inverse ˆA−1(nx,ny;mx,my) =/integraldisplayπ −πddXkxddYky (2π)dˆA−1(kx,ky)eikx(nx−mx)+iky(ny−my) =a2 2d/integraldisplayπ −πddXkxddYky (2π)deikx(nx−mx)+iky(ny−my) 1+m2a2 2d−1 d/summationtextdX i=1coskx,i−1 d/summationtextdY j=1cosky,j.(631) (Form= 0 andd≤2 the integrand diverges for k→0 (infrared divergence). Subtracting formally the also infinite ˆA−1(0,0; 0,0) results in finite difference. For example in d= 1 one finds ˆA−1(ny;my)−ˆA−1(0; 0) = −(1/2)|ny−my| [96]. Using 1 /x=/integraltext∞ 0dte−xtone obtains [180] ˆA−1(kx,ky) =1 2/integraldisplay∞ 0dte−µt+a−2t/parenleftig/summationtextdX icoskx,i+/summationtextdY jcosky,j/parenrightig , (632) withµ=d/a2+m2/2. This allows to express the inverse ˆA−1in terms of the modified Bessel functions Iν(n) which have for integer argument nthe integral representation Iν(n) =1 π/integraldisplayπ 0dΘencos Θcos(νΘ). (633) One finds ˆA−1(nx,ny;mx,my) =1 2/integraldisplay∞ 0e−µtdX/productdisplay i=1K|nx,i−n′ x,i|(t/a2)dY/productdisplay j=1K|my,j−m′ y,j|(t/a2). (634) It might be interesting to remark that the matrix elements of the inverse learning matrix or free massive propagator on the lattice ˆA−1(x′,y′;x,y) can be given an interpretation in terms of (random) walks connec ting the two points (x′,y′) and (x,y) [51, 180]. For that purpose the lattice Laplacian is splitted into a diagonal and a nearest neighbor part −ˆ∆ =1 a2(2dI−W), (635) where the nearest neighbor matrix Whas matrix elements equal one for nearest neighbors and equal to zero otherwise. Thus, /parenleftig −ˆ∆ +m2/parenrightig−1=1 2µ/parenleftigg I−1 2µa2W/parenrightigg−1 =1 2µ∞/summationdisplay n=0/parenleftigg1 2µa2/parenrightiggn Wn,(636) 145can be written as geometric series. The matrix elements Wn(x′,y′;x,y) give the number of walks w[(x′,y′)→(x,y)] of length |w|=nconnecting the two points (x′,y′) and (x,y). Thus, one can write /parenleftig −ˆ∆ +m2/parenrightig−1(x′,y′;x,y) =1 2µ/summationdisplay w[(x′,y′)→(x,y)]/parenleftigg1 2µa2/parenrightigg|w| . (637) 7.2.4 Gaussian relaxation As Gaussian kernels are often used in density estimation and also in function approximation (e.g. for radial basis functions [176]) we co nsider the example A=∞/summationdisplay k=01 k!/parenleftiggM2 2˜σ2/parenrightiggk =eM2 2˜σ2: Gaussian (638) with positive semi–definite M2. The contribution for k= 0 corresponds to a mass term so for positive semi–definite MthisAis positive definite and therefore invertible with inverse A−1=e−M2 2˜σ2, (639) which is diagonal and Gaussian in M–representation. In the limit ˜ σ→ ∞ or for zero modes of Mthe Gaussian A−1becomes the identity I, corresponding to the gradient algorithm. Consider M2(x′,y′;x,y) =−δ(x−x′)δ(y−y′)∆ (640) where theδ–functions are usually skipped from the notation, and ∆ =∂2 ∂x2+∂2 ∂y2, denotes the Laplacian. The kernel of the inverse is diagonal in Fourier rep- resentation A(k′ x,k′ y;,kx,ky) =δ(kx−k′ x)δ(ky−k′ y)e−k2x+k2y 2˜σ2 (641) and non–diagonal, but also Gaussian in ( x,y)–representation A−1(x′,y′;x,y) =e−∆ 2˜σ2=/integraldisplaydkxdky (2π)de−k2x+k2y 2˜σ2+ikx(x−x′)+iky(y−y′)(642) 146=/parenleftigg˜σ√ 2π/parenrightiggd e−˜σ2((x−x′)2+(y−y′)2)=1 /parenleftig σ√ 2π/parenrightigde−(x−x′)2+(y−y′)2 2σ2, (643) withσ= 1/˜σandd=dX+dY,dX= dim(X),dY= dim(Y). 7.2.5 Inverting in subspaces Matrices considered as learning matrix have to be invertibl e. Non-invertible matrices can only be inverted in the subspace which is the com plement of its zero space. With respect to a symmetric Awe define the projector Q0= I−/summationtext iψT iψiinto its zero space (for the more general case of a normal A replaceψT iby the hermitian conjugate ψ† i) and its complement Q1=I−Q0=/summationtext iψT iψiwithψidenoting orthogonal eigenvectors with eigenvalues ai∝ne}ationslash= 0 of A, i.e.,Aψi=aiψi∝ne}ationslash= 0. Then, denoting projected sub-matrices by QiAQj =Aijwe have A00=A10=A01= 0, i.e., A=Q1AQ1=A11. (644) and in the update equation A∆φ(i)=ηG (645) onlyA11can be inverted. Writing Qjφ=φjfor a projected vector, the iteration scheme acquires the form ∆φ(i) 1=ηA−1 11G1, (646) 0 =ηG0. (647) For positive semi–definite Athe sub-matrix A11is positive definite. If the second equation is already fulfilled or its solution is postp oned to a later iteration step we have φ(i+1) 1 =φ(i) 1+ηA−1 11/parenleftig T(i) 1−K(i) 11φ(i) 1−K(i) 10φ(i) 0/parenrightig , (648) φ(i+1) 0 =φ(i) 0. (649) In case the projector Q0=I0is diagonal in the chosen representation the projected equation can directly be solved by skipping the co rresponding com- ponents. Otherwise one can use the Moore–Penrose inverse A#ofAto solve the projected equation ∆φ(i)=ηA#G. (650) 147Alternatively, an invertible operator ˜A00can be added to A11to obtain a complete iteration scheme with A−1=A−1 11+˜A−1 00 φ(i+1)=φ(i)+ηA−1 11/parenleftig T(i) 1−K(i) 11φ(i) 1−K(i) 10φ(i) 0/parenrightig +η˜A−1 00/parenleftig T(i) 0−K(i) 01φ(i) 1−K(i) 00φ(i) 0/parenrightig . (651) The choice A−1= (A11+I00)−1=A−1 11+I00, =A−1 11+Q0, for instance, results in a gradient algorithm on the zero space with additi onal coupling between the two subspaces. 7.2.6 Boundary conditions For a differential operator invertability can be achieved by adding an operator restricted to a subset B⊂X×Y(boundary). More general, we consider an projector QBon a space which we will call boundary and the projector on the interior QI=I−QB. We write QkKQl=Kklfork,l∈ {I,B}, and require KBI= 0. That means Kis not symmetric, but KIIcan be, and we have K= (I−QB)K+QBKQB=KII+KIB+KBB. (652) For such an Kan equation of the form Kφ=Tcan be decomposed into KBBφB=TB, (653) KIBφB+KIIφI=TI, (654) with projected φk=Qkφ,Tk=QkTso that φB=K−1 BBTB, (655) φI=K−1 II/parenleftig TI−KIBK−1 BBTB/parenrightig . (656) The boundary part is independent of the interior, however, t he interior can depend on the boundary. A basis can be chosen so that the proje ctor onto the boundary is diagonal, i.e., QB=IB=/summationdisplay j:(xj,yj)∈B(δ(xj)⊗δ(yj))⊗(δ(xj)⊗δ(yj))T. Eliminating the boundary results in an equation for the inte rior with adapted inhomogeneity. The special case KBB=IBB, i.e.,φB=TBon the boundary, is known as Dirichlet boundary conditions. 148Analogously, we may use a learning matrix Awith boundary, correspond- ing for example to a Kwith boundary conditions: A=AII+AIB+ABB: Boundary (657) A=AII+AIB+IBB: Dirichlet boundary (658) For linear ABBthe form (657) corresponds to general linear boundary condi - tions. One can, however, also allow nonlinear boundary cond itions. AIIcan be chosen symmetric, and therefore positive definite, and th e boundary of A can be changed during iteration. Solving A(φ(i+1)−φ(i)) =η(T(i)−K(i)φ(i)) gives on the boundary and for the interior φ(i+1) B=φi B+ηA−1 BB/parenleftig T(i) B−K(i) BBφ(i) B−K(i) BIφ(i) I/parenrightig , (659) φ(i+1) I=φi I+ηA−1 II/parenleftig T(i) I−K(i) IIφ(i) I−K(i) IBφ(i) B/parenrightig −A−1 IIAIB/parenleftig φ(i+1) B−φ(i) B/parenrightig , (660) For fulfilled boundary conditions with φ(i) B=/parenleftig K(i) BB/parenrightig−1T(i) BandK(i) BI= 0, or forηA−1 BB→0 so the boundary is not updated, the term φ(i+1) B−φ(i) Bvanishes. Otherwise, inserting the first in the second equation gives φ(i+1) I =φi I+ηA−1 II/parenleftig T(i) I−K(i) IIφ(i) I−K(i) IBφ(i) B/parenrightig (661) −ηA−1 IIAIBA−1 BB/parenleftig T(i) B−K(i) BBφ(i) B−K(i) BIφ(i) I/parenrightig . Even if Kis not defined with boundary conditions, an invertible Acan be obtained from Kby introducing a boundary for A. The updating process can then for example be restricted to the interior and the bou ndary changed during iteration. The following table summarizes the learning matrices we hav e discussed: Learning algorithm Learning matrix Gradient A=I Jacobi A=D Gauss–Seidel A=L+D Newton A=−H prior relaxation A=K massive relaxation A=A0+m2I linear boundary A=AII+AIB+ABB Dirichlet boundary A=AII+AIB+IBB Gaussian A=/summationtext∞ k=01 k!/parenleftig M2 2˜σ2/parenrightigk=eM2 2˜σ2 1497.3 Initial configurations and kernel methods 7.3.1 Truncated equations To solve the nonlinear Eq. (593) by iteration one has to begin with an ini- tial configuration φ(0). In principle any easy to use technique for density estimation could be chosen to construct starting guesses φ(0). One possibility to obtain initial guesses is to neglect some terms of the full stationarity equation and solve the resulting simpler (ide ally linear) equation first. The corresponding solution may be taken as initial gue ssφ(0)for solving the full equation. Typical error functionals for statistical learning proble ms include a term (L, N) consisting of a discrete sum over a finite number nof training data. For diagonal P′those contributions result (346) in nδ–peak contributions to the inhomogeneities Tof the stationarity equations, like/summationtext iδ(x−xi)δ(y−yi) in Eq. (144) or/summationtext iδ(x−xi)δ(y−yi)/P(x,y) in Eq. (172). To find an initial guess, one can now keep only that δ–peak contributions Tδarising from the training data and ignore the other, typically continuous pa rts ofT. For (144) and (172) this means setting Λ X= 0 and yields a truncated equation Kφ=P′P−1N=Tδ. (662) Hence,φcan for diagonal P′be written as a sum of nterms φ(x,y) =n/summationdisplay i=1C(x,y;xi,yi)P′(xi,yi) P(xi,yi), (663) withC=K−1, provided the inverse K−1exists. For ELthe resulting trun- cated equation is linear in L. ForEP, however, the truncated equations remains nonlinear. Having solved the truncated equation we restore the necessary constraints for φ, like normalization and positivity for Por nor- malization of the exponential for L. In general, a C∝ne}ationslash=K−1can be chosen. This is necessary if Kis not invertible and can also be useful if its inverse is difficult to calculate. One possible choice for the kernel is the inverse negative Hessi anC=−H−1 evaluated at some initial configuration φ(0)or an approximation of it. A simple possibility to construct an invertible operator fro m a noninvertible K would be to add a mass term C=/parenleftig K+m2 CI/parenrightig−1, (664) 150or to impose additional boundary conditions. Solving a truncated equation of the form (663) with Cmeans skipping the term −C(P′ΛX+ (K−C−1)φ) from the exact relation φ=CP′P−1N−C(P′ΛX+ (K−C−1)φ). (665) A kernel used to create an initial guess φ(0)will be called an initializing kernel . A similar possibility is to start with an “empirical solutio n” φ(0)=φemp, (666) whereφempis defined as a φwhich reproduces the conditional empirical densityPempof Eq. (236) obtained from the training data, i.e., Pemp=P(φemp). (667) In case, there are not data points for every x–value, a correctly normalized initial solution would for example be given by ˜Pempdefined in Eq. (238). If zero values of the empirical density correspond to infinite v alues forφ, like in the case φ=L, one can use Pǫ empas defined in Eq. (239), with small ǫ, to obtain an initial guess. Similarly to Eq. (663), it is often also useful to choose a (fo r example smoothing) kernel Cand use as initial guess φ(0)=Cφemp, (668) or a properly normalized version thereof. Alternatively, o ne may also let the (smoothing) operator Cdirectly act on Pempand use a corresponding φas initial guess, φ(0)= (φ)(−1)CPemp), (669) assuming an inverse ( φ)(−1)of the mapping P(φ) exists. We will now discuss the cases φ=Landφ=Pin some more detail. 7.3.2 Kernels for L ForELwe have the truncated equation L=CN. (670) 151Normalizing the exponential of the solution gives L(x,y) =n/summationdisplay iC(x,y;xi,yi)−ln/integraldisplay dy′e/summationtextn iC(x,y′;xi,yi), (671) or L=CN−lnIXeCN. (672) Notice that normalizing Laccording to Eq. (671) after each iteration the truncated equation (670) is equivalent to a one–step iterat ion with uniform P(0)=eL(0)according to L1=CN+CP(0)ΛX, (673) where only ( I−CK)Lis missing from the nontruncated equation (665), because the additional y–independent term CP(0)ΛXbecomes inessential if Lis normalized afterwards. Lets us consider as example the choice C=−H−1(φ(0)) for uniform initial L(0)=ccorresponding to a normalized PandKL(0)= 0 (e.g., a differential operator). Uniform L(0)means uniform P(0)= 1/vy, assuming that vy=/integraltextdy exists and, according to Eq. (138), Λ X=NXforKL(0)= 0. Thus, the Hessian (161) at L(0)) is found as H(L(0)) =−/parenleftigg I−IX vy/parenrightigg K/parenleftigg I−IX vy/parenrightigg −/parenleftigg I−IX vy/parenrightiggNX vy=−C−1,(674) which can be invertible due to the presence of the second term . Another possibility is to start with an approximate empiric al log–density, defined as Lǫ emp= lnPǫ emp, (675) withPǫ empgiven in Eq. (239). Analogously to Eq. (668), the empirical l og– density may for example also be smoothed and correctly norma lized again, resulting in an initial guess, L(0)=CLǫ emp−lnIXeCLǫ emp. (676) Similarly, one may let a kernel C, or its normalized version ˜Cdefined below in Eq. (680), act on Pempfirst and then take the logarithm L(0)= ln(˜CPǫ emp). (677) Because already ˜CPempis typically nonzero it is most times not necessary to work here with Pǫ emp. Like in the next section Pempmay be also be replaced by˜Pempas defined in Eq. (238). 1527.3.3 Kernels for P ForEPthe truncated equation P=CP−1N, (678) is still nonlinear in P. If we solve this equation approximately by a one– step iteration P1=C(P(0))−1Nstarting from a uniform initial P(0)and normalizing afterwards this corresponds for a single x–value to the classical kernel methods commonly used in density estimation. As norm alized density results P(x,y) =/summationtext iC(x,y;xi,yi)/integraltextdy′/summationtext iC(x,y′;xi,yi)=/summationdisplay i¯C(x,y;xi,yi), (679) i.e., P=N−1 K,XCN=¯CN, (680) with (data dependent) normalized kernel ¯C=N−1 C,XCandNC,Xthe diagonal matrix with diagonal elements IXCN. Again C=K−1or similar invertible choices can be used to obtain a starting guess for P. The form of the Hessian (182) suggests in particular to include a mass term on the dat a. It would be interesting to interpret Eq. (680) as stationari ty equation of a functional ˆEPcontaining the usual data term/summationtext ilnP(xi,yi). Therefore, to obtain the derivative P−1Nof this data term we multiply for existing ¯C−1 Eq. (680) by P−1¯C−1, whereP∝ne}ationslash= 0 at data points, to obtain /tildewideC−1P=P−1N, (681) with data dependent /tildewideC−1(x,y;x′,y′) =¯C−1(x,y;x′,y′) /summationtext i¯C(x,y;xi,yi). (682) Thus, Eq. (680) is the stationarity equation of the function al ˆEP=−(N,lnP) +1 2(P,/tildewideC−1P). (683) To study the dependence on the number nof training data for a given C consider a normalized kernel with/integraltextdyC(x,y;x′,y′) =λ,∀x,x′,y′. For such a kernel the denominator of ¯Cis equal tonλso we have ¯C=C nλ, P =CN nλ(684) 153Assuming that for large nthe empirical average (1 /n)/summationtext iC(x,y;xi,yi) in the denominator of/tildewideC−1becomesnindependent, e.g., converging to the true av- eragen/integraltextdx′dy′p(x′,y′)C(x,y;x′,y′), the regularizing term in functional (683) becomes proportional to n /tildewideC−1∝nλ2, (685) According to Eq. (77) this would allow to relate a saddle poin t approximation to a largen–limit. Again, a similar possibility is to start with the empirical d ensity ˜Pemp defined in Eq. (238). Analogously to Eq. (668), the empirical density can for example also be smoothed and correctly normalized again, so that P(0)=˜C˜Pemp. (686) with˜Cdefined in Eq. (680). Fig. 13 compares the initialization according to Eq. (679), where the smoothing operator ˜Cacts onN, with an initialization according to Eq. (686), where the smoothing operator ˜Cacts on the correctly normalized ˜Pemp. 7.4 Numerical examples 7.4.1 Density estimation with Gaussian specific prior In this section we look at some numerical examples and discus s implemen- tations of the nonparametric learning algorithms for densi ty estimation we have discussed in this paper. As example, consider a problem with a one–dimensional X–space and a one–dimensional Y–space, and a smoothness prior with inverse covariance K=λx(KX⊗1Y) +λy(1X⊗KY), (687) where KX=λ0IX−λ2∆x+λ4∆2 x−λ6∆3 x (688) KY=λ0IY−λ2∆y+λ4∆2 y−λ6∆3 y, (689) and Laplacian ∆x(x,x′) =δ′′(x−x′) =δ(x−x′)d2 dx2, (690) 154and analogously for ∆ y. Forλ2∝ne}ationslash= 0 =λ0=λ4=λ6this corresponds to the two Laplacian prior factors ∆ xforxand ∆yfory. (Notice that also for λx= λytheλ4– andλ6–terms do not include all terms of an iterated 2–dimensional Laplacian, like ∆2= (∆x+ ∆y)2or ∆4, as the mixed derivatives ∆ x∆yare missing.) We will now study nonparametric density estimation with pri or factors being Gaussian with respect to Las well as being Gaussian with respect to P. The error or energy functional for a Gaussian prior factor in Lis given by Eq. (109). The corresponding iteration procedure is L(i+1)=L(i)+ηA−1/parenleftig N−KL(i)−eL(i)/bracketleftig NX−IXKL(i)/bracketrightig/parenrightig . (691) Written explicitly for λ2= 1,λ0=λ4=λ6= 0 Eq. (691) reads, L(i+1)(x,y) =L(i)(x,y) +η/summationdisplay jA−1(x,y;xj,yj) (692) +η/integraldisplay dx′dy′A−1(x,y;x′,y′)/bracketleftiggd2 d(x′)2L(i)(x′,y′) +d2 d(y′)2L(i)(x′,y′) − /summationdisplay jδ(x′−xj)+/integraldisplay dy′′d2 d(x′)2L(i)(x′,y′′)+/integraldisplay dy′′d2 d(y′′)2L(i)(x′,y′′) eL(i)(x′,y′) . Here/integraldisplayyB yAdy′′d2 d(y′′)2L(i)(x′,y′′) =d d(y′′)L(i)(x′,y′′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleyB yA vanishes if the first derivatived dyL(i)(x,y) vanishes at the boundary or if periodic. Analogously, for error functional EP(164) the iteration procedure P(i+1)=P(i)+ηA−1/bracketleftig (P(i))−1N−NX−IXP(i)KP(i)−KP(i)/bracketrightig .(693) becomes for λ2= 1,λ0=λ4=λ6= 0 P(i+1)(x,y) =P(i)(x,y) +η/summationdisplay jA−1(x,y;xj,yj) P(i)(xj,yj)(694) +η/integraldisplay dx′dy′A−1(x,y;x′,y′)/bracketleftiggd2 d(x′)2P(i)(x′,y′) +d2 d(y′)2P(i)(x′,y′) 155− /summationdisplay jδ(x′−xj) +/integraldisplay dy′′P(i)(x′,y′′)d2P(i)(x′,y′′) d(x′)2 +/integraldisplay dy′′P(i)(x′,y′′)d2P(i)(x′,y′′) d(y′′)2/parenrightigg/bracketrightigg . Here/integraldisplayyB yAdy′′P(i)(x′,y′′)d2P(i)(x′,y′′) d(y′′)2= P(i)(x′,y′′)dP(i)(x′,y′′) d(y′′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleyB yA−/integraldisplayyB yAdy′′/parenleftiggdP(i)(x′,y′′) dy′′/parenrightigg2 , (695) where the first term vanishes for P(i)periodic or vanishing at the boundaries. (This has to be the case for id/dy to be hermitian.) We now study density estimation problems numerically. In pa rticular, we want to check the influence of the nonlinear normalization constraint. Furthermore, we want to compare models with Gaussian prior f actors forL with models with Gaussian prior factors for P. The following numerical calculations have been performed o n a mesh of dimension 10 ×15, i.e.,x∈[1,10] andy∈[1,15], with periodic boundary conditions on yand sometimes also in x. A variety of different iteration and initialization methods have been used. Figs. 14 – 17 summarize results for density estimation probl ems with only two data points, where differences in the effects of varying sm oothness priors are particularly easy to see. A density estimation with more data points can be found in Fig. 21. For Fig. 14 a Laplacian smoothness prior on Lhas been implemented. The solution has been obtained by iterating with the negative He ssian, as long as positive definite. Otherwise the gradient algorithm has b een used. One iteration step means one iteration according to Eq. (601) wi th the optimal η. Thus, each iteration step includes the optimization of ηby a line search algorithm. (For the figures the Mathematica function FindMi nimum has been used to optimize η.) As initial guess in Fig. 14 the kernel estimate L(0)= ln(˜C˜Pemp) has been employed, with ˜Cdefined in Eq. (680) and C= (K+m2 CI) with squared mass m2 C= 0.1. The fast drop–off of the energy ELwithin the first two iterations shows the quality of this initial guess. Indeed, this fast co nvergence seems to indicate that the problem is nearly linear, meaning that t he influence of 156the only nonlinear term in the stationarity equation, the no rmalization con- straint, is not too strong. Notice also, that the reconstruc ted regression shows the typical piecewise linear approximations well known fro m one–dimensional (normalization constraint free) regression problems with Laplacian prior. Fig. 15 shows a density estimation similar to Fig. 14, but for a Gaussian prior factor in Pand thus also with different λ2, different initialization, and slightly different iteration procedure. For Fig. 15 also a ke rnel estimate P(0) = (˜C˜Pemp) has been used as initial guess, again with ˜Cas defined in Eq. (680) andC= (K+m2 CI) but with squared mass m2 C= 1.0. The solution has been obtained by prior relaxation A=K+m2Iincluding a mass term with m2 = 1.0 to get for a Laplacian K=−∆ and periodic boundary conditions an invertible A. This iteration scheme does not require to calculate the Hes sian HPat each iteration step. Again the quality of the initial gues s (and the iteration scheme) is indicated by the fast drop–off of the ene rgyEPduring the first iteration. Because the range of P–values, being between zero and one, is smaller than that of L–values, being between minus infinity and zero, a larger Lapl a- cian smoothness factor λ2is needed for Fig. 15 to get similar results than for Fig. 14. In particular, such λ2values have been chosen for the two figures that the maximal values of the the two reconstructed probabi lity densities P turns out to be nearly equal. Because the logarithm particularly expands the distances b etween small probabilities one would expect a Gaussian prior for Lto be especially effective for small probabilities. Comparing Fig. 14 and Fig. 15 this e ffect can indeed be seen. The deep valleys appearing in the L–landscape of Fig. 15 show that small values of Lare not smoothed out as effectively as in Fig. 14. Notice, that therefore also the variance of the solution p(y|x,h) is much smaller for a Gaussian prior in Pat thosexwhich are in the training set. Fig. 16 resumes results for a model similar to that presented in Fig. 14, but with a ( −∆3)–prior replacing the Laplacian ( −∆)–prior. As all quadratic functions have zero third derivative such a prior favors, applied to L, quadratic log–likelihoods, corresponding to Gaussian pr obabilitiesP. In- deed, this is indicated by the striking difference between th e regression func- tions in Fig. 16 and in Fig. 14: The ( −∆3)–prior produces a much rounder regression function, especially at the xvalues which appear in the data. Note however, that in contrast to a pure Gaussian regression prob lem, in density estimation an additional non–quadratic normalization con straint is present. In Fig. 17 a similar prior has been applied, but this time bein g Gaussian 157inPinstead ofL. In contrast to a ( −∆3)–prior for L, a (−∆3)–prior for P implements a tendency to quadratic P. Similarly to the difference between Fig. 14 and Fig. 16, the regression function in Fig. 17 is also rounder than that in Fig. 15. Furthermore, smoothing in Fig. 17 is also les s effective for smaller probabilities than it is in Fig. 16. That is the same r esult we have found comparing the two priors for Lshown in Fig. 15 and Fig. 14. This leads to deeper valleys in the L–landscape and to a smaller variance especially at xwhich appear in the training data. Fig. 21 depicts the results of a density estimation based on m ore than two data points. In particular, fifty training data have been obtained by sampling with uniform p(x) from the “true” density Ptrue(x,y) =p(y|x,htrue) =1 2√ 2πσ0 e−(y−ha(x))2 2σ2 0+e−(y−hb(x))2 2σ2 0 ,(696) withσ0= 1.5,ha(x) = 125/18 + (5/9)x,hb(x) = 145/18−(5/9)x, shown in the top row of Fig. 18. The sampling process has been implemen ted using the transformation method (see for example [181]). The corresp onding empirical densityN/n(235) and conditional empirical density Pempof Eq. (236), in this case equal to the extended ˜Pempdefined in Eq. (238), can be found in Fig. 20. Fig. 21 shows the maximum posterior solution p(y|x,h∗) and its loga- rithm, the energy ELduring iteration, the regression function h(x) =/integraldisplay dyyp(y|x,htrue) =/integraldisplay dyyP true(x,y), (697) (as reference, the regression function for the true likelih oodp(y|x,htrue) is given in Fig. 19), the average training error (orempirical (conditional) log– loss) <−lnp(y|x,h)>D=−1 nn/summationdisplay i=1logp(yi|xi,h), (698) and the average test error (ortrue expectation of (conditional) log–loss ) for uniformp(x) <−lnp(y|x,h)>Ptrue=−/integraldisplay dydxp (x)p(y|x,htrue) lnp(y|x,h),(699) which is, up to a constant, equal to the expected Kullback–Le ibler distance between the actual solution and the true likelihood, KL/parenleftig p(x,y|htrue),p(y|x,h)/parenrightig =−/integraldisplay dydxp (x,y|htrue) lnp(y|x,h) p(y|x,htrue).(700) 158The test error measures the quality of the achieved solution . It has, in contrast to the energy and training error, of course not been available to the learning algorithm. The maximum posterior solution of Fig. 21 has been calculate d by mini- mizingELusing massive prior iteration with A=K+m2I, a squared mass m2= 0.01, and a (conditionally) normalized, constant L(0)as initial guess. Convergence has been fast, the regression function is simil ar to the true one (see Fig. 19). Fig. 22 compares some iteration procedures and initializat ion methods Clearly, all methods do what they should do, they decrease th e energy func- tional. Iterating with the negative Hessian yields the fast est convergence. Massive prior iteration is nearly as fast, even for uniform i nitialization, and does not require calculation of the Hessian at each iteratio n. Finally, the slowest iteration method, but the easiest to implement, is t he gradient algo- rithm. Looking at Fig. 22 one can distinguish data–oriented from pr ior–oriented initializations. We understand data–oriented initial gue sses to be those for which the training error is smaller at the beginning of the it eration than for the final solution and prior–oriented initial guesses to be t hose for which the opposite is true. For good initial guesses the difference is s mall. Clearly, the uniform initializations is prior–oriented, while an em pirical log–density ln(N/n+ǫ) and the shown kernel initializations are data–oriented. The case where the test error grows while the energy is decrea sing indi- cates a misspecified prior and is typical for overfitting. For example, in the fifth row of Fig. 22 the test error (and in this case also the ave rage train- ing error) grows again after having reached a minimum while t he energy is steadily decreasing. 7.4.2 Density estimation with Gaussian mixture prior Having seen Bayesian field theoretical models working for Ga ussian prior factors we will study in this section the slightly more compl ex prior mixture models. Prior mixture models are an especially useful tool f or implementing complex and unsharp prior knowledge. They may be used, for ex ample, to translate verbal statements of experts into quantitativ e prior densities [123, 124, 125, 126, 127], similar to the quantification of “l inguistic variables” by fuzzy methods [110, 111]. We will now study a prior mixture with Gaussian prior compone nts inL. 159Hence, consider the following energy functional with mixtu re prior EL=−ln/summationdisplay jpje−Ej=−(L,N) + (eL,ΛX)−ln/summationdisplay jpje−λE0,j(701) with mixture components Ej=−(L,N) +λE0,j+ (eL,ΛX). (702) We choose Gaussian component prior factors with equal covar iances but dif- fering means E0,j=1 2/parenleftig L−tj,K(L−tj)/parenrightig . (703) Hence, the stationarity equation for Functional (701) beco mes 0 =N−λK L−/summationdisplay jajtj −eLΛX, (704) with Lagrange multiplier function ΛX=NX−λIXK L−/summationdisplay jajtj , (705) and mixture coefficients aj=pje−λE0,j /summationtext kpke−λE0,k. (706) The parameter λplays here a similar role as the inverse temperature βfor prior mixtures in regression (see Sect. 6.3). In contrast to theβ–parameter in regression, however, the “low temperature” solutions fo rλ→ ∞ are the pure prior templates tj, and forλ→0 the prior factor is switched off. Typical numerical results of a prior mixture model with two m ixture components are presented in Figs. 23 – 28. Like for Fig. 21, th e true likelihood used for these calculations is given by Eq. (696) and shown in Fig. 18. The corresponding true regression function is thus that of Fig. 19. Also, the same training data have been used as for the model of Fig. 21 (Fig. 2 0). The two templates t1andt2which have been selected for the two prior mixture components are (Fig. 18) t1(x,y) =1 2√ 2πσt e−(y−µa)2 2σ2 t+e−(y−µb)2 2σ2 t , (707) t2(x,y) =1√ 2πσte−(y−µ2)2 2σ2 t, (708) 160withσt= 2,µa=µ2+ 25/9 = 10.27,µb=µ2−25/9 = 4.72, andµ2= 15/2. Both templates capture a bit of the structure of the true li kelihood, but not too much, so learning remains interesting. The avera ge test error of t1is equal to 2.56 and is thus lower than that of t2being equal to 2.90. The minimal possible average test error 2.23 is given by that of t he true solution Ptrue. A uniform P, being the effective template in the zero mean case of Fig. 21, has with 2.68 an average test error between the two templa test1andt2. Fig. 23 proves that convergence is fast for massive prior rel axation when starting from t1as initial guess L(0). Compared to Fig. 21 the solution is a bit smoother, and as template t1is a better reference than the uniform likelihood the final test error is slightly lower than for the zero–mean G aussian prior onL. Starting from L(0)=t2convergence is not much slower and the final solution is similar, the test error being in that particular case even lower (Fig. 24). Starting from a uniform L(0)the mixture model produces a solution very similar to that of Fig. 21 (Fig. 24). The effect of changing the λparameter of the prior mixture can be seen in Fig. 26 and Fig. 27. Larger λmeans a smoother solution and faster convergence when starting from a template likelihood (Fig. 26). Smaller λ results in a more rugged solution combined with a slower conv ergence. The test error in Fig. 27 already indicates overfitting. Prior mixture models tend to produce metastable and approxi mately sta- ble solutions. Fig. 28 presents an example where starting wi thL(0)=t2the learning algorithm seems to have produced a stable solution after a few it- erations. However, iterating long enough this decays into a solution with smaller distance to t1and with lower test error. Notice that this effect can be prevented by starting with another initialization, like for example with L(0)=t1or a similar initial guess. We have seen now that, and also how, learning algorithms for B ayesian field theoretical models can be implemented. In this paper, t he discussion of numerical aspects was focussed on general density estima tion problems. Other Bayesian field theoretical models, e.g., for regressi on and inverse quan- tum problems, have also been proved to be numerically feasib le. Specifically, prior mixture models for Gaussian regression are compared w ith so–called Landau–Ginzburg models in [123]. An application of prior mi xture mod- els to image completion, formulated as a Gaussian regressio n model, can be found in [128]. Furthermore, hyperparameter have been incl uded in numer- ical calculations in [124] and also in [128]. Finally, learn ing algorithms for inverse quantum problems are treated in [134] for inverse qu antum statistics, 161and, in combination with a mean field approach, in [133] for in verse quantum many–body theory. Time–dependent inverse quantum problem s will be the topic of [129]. In conclusion, we may say that many different Bayesian field th eoretical models have already been studied numerically and proved to b e computation- ally feasible. This also shows that such nonparametric Baye sian approaches are relatively easy to adapt to a variety of quite different le arning scenarios. 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Oxford: Oxford Science Publications. 1805 10 15246810 00.20.40.60.8 5 105 10 15246810 00.20.4 5 10 5 10 15246810 00.20.40.6 5 105 10 15246810 0.10.2 5 10 Figure 13: Comparison of initial guesses P(0)(x,y) for a case with two data points located at (3 ,3) and (7,12) within the intervals y∈[1,15] andx∈ [1,10] with periodic boundary conditions. First row: P(0)=˜CN. (The smoothing operator acts on the unnormalized N. The following conditional normalization changes the shape more drastically than in th e example shown in the second row.) Second row: P(0)=˜C˜Pemp. (The smoothing operator acts on the already conditionally normalized ˜Pemp.) The kernel ˜Cis given by Eq. (680) with C= (K+m2 CI),m2 C= 1.0, and a Kof the form of Eq. (687) withλ0=λ4=λ6= 0, andλ2= 0.1 (figures on the l.h.s.) or λ2= 1.0 (figures on the r.h.s.), respectively. 1812 4 6 8 10i1.91.9522.052.12.152.2Energy during iteration 2 4 6 8 10x2468101214Regression functionP 5 10 15y246810 x0.20.40.6 5 10 15yL 5 10 15y246810 x-4-3-2-1 5 10 15y Figure 14: Density estimation with 2 data points and a Gaussi an prior factor for the log–probability L. First row: Final PandL. Second row: The l.h.s. shows the energy EL(109) during iteration, the r.h.s. the regres- sion function h(x) =/integraltextdyyp(y|x,htrue) =/integraltextdyyP true(x,y). The dotted lines indicate the range of one standard deviation above and below the regression function (ignoring periodicity in x). The fast convergence shows that the problem is nearly linear. The asymmetry of the solution betw een thex– andy–direction is due to the normalization constraint, only req uired fory. (Laplacian smoothness prior Kas given in Eq. (687) with λx=λy= 1,λ0 = 0,λ2= 0.025,λ4=λ6= 0. Iteration with negative Hessian A=−Hif positive definite, otherwise with the gradient algorithm, i .e.,A=I. Initial- ization with L(0)= ln(˜C˜Pemp), i.e.,L(0)normalized to/integraltextdyeL= 1, with ˜Cof Eq. (680) and C= (K+m2 CI),m2 C= 0.1. Within each iteration step the optimal step width ηhas been found by a line search. Mesh with 10 points inx-direction and 15 points in y–direction, periodic boundary conditions in xandy. The 2 data points are (3 ,3) and (7,12).) 1822 4 6 8 10i22.22.42.6Energy during iteration 2 4 6 8 10x2468101214Regression functionP 5 10 15y246810 x00.20.40.6 5 10 15yL 5 10 15y246810 x-10-7.5-5-2.5 5 10 15y Figure 15: Density estimation with 2 data points, this time w ith a Gaussian prior factor for the probability P, minimizing the energy functional EP(164). To make the figure comparable with Fig. 14 the parameters have been chosen so that the maximum of the solution Pis the same in both figures (max P= 0.6). Notice, that compared to Fig. 14 the smoothness prior i s less effective for small probabilities. (Same data, mesh and periodic boun dary conditions as for Fig. 14. Laplacian smoothness prior Kas in Eq. (687) with λx=λy = 1,λ0= 0,λ2= 1,λ4=λ6= 0. Iterated using massive prior relaxation, i.e.,A=K+m2Iwithm2= 1.0. Initialization with P(0)=˜C˜Pemp, with ˜Cof Eq. (680) so P(0)is correctly normalized, and C= (K+m2 CI),m2 C= 1.0. Within each iteration step the optimal factor ηhas been found by a line search algorithm.) 1832 4 6 8 10i22.22.42.62.833.2Energy during iteration 2 4 6 8 10x2468101214Regression functionP 5 10 15y246810 x0.20.4 5 10 15yL 5 10 15y246810 x-4-3-2-1 5 10 15y Figure 16: Density estimation with a ( −∆3) Gaussian prior factor for the log–probability L. Such a prior favors probabilities of Gaussian shape. (Smoothness prior Kof the form of Eq. (687) with λx=λy= 1,λ0= 0,λ2 = 0,λ4= 0,λ6= 0.01. Same iteration procedure, initialization, data, me sh and periodic boundary conditions as for Fig. 14.) 1842 4 6 8 10i2.62.833.23.4Energy during iteration 2 4 6 8 10x2468101214Regression functionP 5 10 15y246810 x00.10.20.3 5 10 15yL 5 10 15y246810 x-8-6-4-2 5 10 15y Figure 17: Density estimation with a ( −∆3) Gaussian prior factor for the probability P. As the variation of Pis smaller than that of L, a smaller λ6 has been chosen than in Fig. 17. The Gaussian prior in Pis also relatively less effective for small probabilities than a comparable Gau ssian prior in L. (Smoothness prior Kof the form of Eq. (687) with λx=λy= 1,λ0= 0,λ2 = 0,λ4= 0,λ6= 0.1. Same iteration procedure, initialization, data, mes h and periodic boundary conditions as for Fig. 15.) 185True P 5 10 15y246810 x00.10.2 5 10 15yTrue L 5 10 15y246810 x-10-5 5 10y Template 1 (P) 5 10 15246810 00.0250.050.0750.1 5 10 15Template 1 (L) 5 10 15246810 -5-4-3 5 10 Template 2 (P) 5 10 15246810 00.050.10.15 5 10 15Template 2 (L) 5 10 15246810 -8-6-4-2 5 10 Figure 18: First row: True density Ptrue(l.h.s.) true log–density Ltrue= logPtrue(r.h.s.) used for Figs. 21–28. Second and third row: The two templatest1andt2of Figs. 23–28 for P(tP i, l.h.s.) or for L(tL i, r.h.s.), respectively, with tL i= logtP i. As reference for the following figures we give the expected test error/integraltextdydxp (x)p(y|x,htrue) lnp(y|x,h) under the true p(y|x,htrue) for uniform p(x). It is forhtrueequal to 2.23 for template t1equal to 2.56, for template t2equal 2.90 and for a uniform Pequal to 2.68. 1862 4 6 8 10x 24681012Regression function Figure 19: Regression function htrue(x) for the true density Ptrueof Fig. 18, defined as h(x) =/integraltextdyyp(y|x,htrue) =/integraltextdyyP true(x,y). The dotted lines indicate the range of one standard deviation above and below the regression function. Empirical density 0 5 10 15y 02.557.510 x00.010.020.030.04 0 5 10 15yConditional empirical density 0 5 10 15y 02.557.510 x00.250.50.751 0 5 10 15y Figure 20: L.h.s.: Empirical density N(x,y)/n=/summationtext iδ(x−xi)δ(y−yi)//summationtext i1. sampled from p(x,y|htrue) =p(y|x,htrue)p(x) with uniform p(x). R.h.s.: Corresponding conditional empirical density Pemp(x,y) = (N−1 XN)(x,y) =/summationtext iδ(x−xi)/summationtext iδ(y−yi)/summationtext i//summationtext iδ(x−xi). Both densities are obtained from the 50 data points used for Figs. 21–28. 187P 5 10 15y246810 x0.050.10.15 5 10 15yL 5 10 15y246810 x-4-3.5-3-2.5-2 5 10 15y 10 20 30 40i120122.5125127.5130132.5135Energy 2 4 6 8 10x4681012Regression function 10 20 30 40i2.22.32.42.52.62.7Av. training err. 10 20 30 40i 2.42.452.52.552.62.65Test error Figure 21: Density estimation with Gaussian prior factor fo r log–probability Lwith 50 data points shown in Fig. 20. Top row: Final solution P(x,y) =p(y|x,h) andL= logP. Second row: Energy EL(109) dur- ing iteration and final regression function. Bottom row: Ave rage train- ing error −(1/n)/summationtextn i=1logp(yi|xi,h) during iteration and average test error −/integraltextdydxp (x)p(y|x,htrue) lnp(y|x,h) for uniform p(x). (Parameters: Zero mean Gaussian smoothness prior with inverse covariance λK,λ= 0.5 and K of the form (687) with λx= 2,λy= 1,λ0= 0,λ2= 1,λ4=λ6= 0, massive prior iteration with A=K+m2Iand squared mass m2= 0.01. Initialized with normalized constant L. At each iteration step the factor ηhas been adapted by a line search algorithm. Mesh with 10 points in x-direction and 15 points in y–direction, periodic boundary conditions in y.) 18810 20 30 40i120122.5125127.5130132.5135Energy 10 20 30 40i2.22.32.42.52.62.7Av. training err. 10 20 30 40i 2.42.452.52.552.62.65Test error 1234567i120122.5125127.5130132.5135Energy 1234567i2.22.32.42.52.62.7Av. training err. 1234567i 2.452.52.552.62.65Test error 1 2 3 4 5i120125130135140Energy 1 2 3 4 5i2.112.122.132.142.152.162.17Av. training err. 1 2 3 4 5i 2.4152.422.4252.432.4352.442.4452.45Test error 10 20 30 40 50i120122.5125127.5130132.5135Energy 10 20 30 40 50i2.22.32.42.52.62.7Av. training err. 10 20 30 40 50i 2.452.52.552.62.65Test error 10 20 30 40 50i120125130135140Energy 10 20 30 40 50i2.112.122.132.142.152.162.17Av. training err. 10 20 30 40 50i2.382.42.422.44Test error 10 20 30 40 50i120130140150160Energy 10 20 30 40 50i1.9522.052.12.152.2Av. training err. 10 20 30 40 50i 2.452.52.552.62.65Test error Figure 22: Comparison of iteration schemes and initializat ion. First row: Massive prior iteration (with A=K+m2I,m2= 0.01) and uniform initial- ization. Second row: Hessian iteration ( A=−H) and uniform initialization. Third row: Hessian iteration and kernel initialization (wi thC=K+m2 CI, m2 C= 0.01 and normalized afterwards). Forth row: Gradient ( A=I) with uniform initialization. Fifth row: Gradient with kernel in itialization. Sixth row: Gradient with delta–peak initialization. (Initial Lequal to ln( N/n+ǫ), ǫ= 10−10, conditionally normalized. For N/nsee Fig. 20). Minimal num- ber of iterations 4, maximal number of iterations 50, iterat ion stopped if |L(i)−L(i−1)|<10−8. Energy functional and parameters as for Fig. 21. 189P 5 10 15y246810 x0.050.10.15 5 10 15yL 5 10 15y246810 x-4-3-2 5 10y 1 2 3 4i122124126128130132134Energy 2 4 6 8 10x4681012Regression function 1 2 3 4i2.32.42.52.6Av. training err. 1 2 3 4i2.442.462.482.52.522.54Test error Figure 23: Density estimation with a Gaussian mixture prior for log– probability Lwith 50 data points, Laplacian prior and the two template fun c- tions shown in Fig. 18. Top row: Final solution P(x,y) =p(y|x,h) andL= logP. Second row: Energy Energy EL(701) during iteration and final regres- sion function. Bottom row: Average training error -(1 /n)/summationtextn i=1logp(yi|xi,h) during iteration and average test error −/integraltextdydxp (x)p(y|x,htrue) lnp(y|x,h) for uniform p(x). (Two mixture components with λ= 0.5 and smoothness prior with K1=K2of the form (687) with λx= 2,λy= 1,λ0= 0,λ2 = 1,λ4=λ6= 0, massive prior iteration with A=K+m2Iand squared massm2= 0.01, initialized with L=t1. At each iteration step the factor ηhas been adapted by a line search algorithm. Mesh with lx= 10 points inx-direction and ly= 15 points in y–direction, n= 2 data points at (3 ,3), (7,12), periodic boundary conditions in y. Except for the inclusion of two mixture components parameters are equal to those for Fig. 21 . ) 190P 5 10 15y246810 x00.050.10.15 5 10 15yL 5 10 15y246810 x-5-4-3-2 5 10 15y 1 2 3 4i120130140150160170Energy 2 4 6 8 10x4681012Regression function 1 2 3 4i2.22.42.62.833.23.4Av. training err. 1 2 3 4i 2.42.52.62.72.82.9Test error Figure 24: Using a different starting point. (Same parameter s as for Fig. 23, but initialized with L=t2.) While the initial guess is worse then that of Fig. 23, the final solution is even slightly better. 191P 5 10 15y246810 x0.050.10.15 5 10 15yL 5 10 15y246810 x-3.5-3-2.5-2 5 10 15y 1 2 3 4 5i125130135140145Energy 2 4 6 8 10x4681012Regression function 1 2 3 4 5i2.22.32.42.52.62.7Av. training err. 1 2 3 4 5i2.42.452.52.552.62.65Test error Figure 25: Starting from a uniform initial guess. (Same as Fi g. 23, but initialized with uniform L.) The resulting solution is, compared to Figs. 23 and 24, a bit more wiggly, i.e., more data oriented. One recog nizes a slight “overfitting”, meaning that the test error increases while t he training error is decreasing. (Despite the increasing of the test error durin g iteration at this value ofλ, a better solution cannot necessarily be found by just chang ing λ–value. This situation can for example occur, if the initial guess is better then the implemented prior.) 192P 5 10 15y246810 x0.0250.050.0750.1 5 10 15yL 5 10 15y246810 x-4.5-4-3.5-3-2.5 5 10 15y 1 2 3 4i 128129130131132133134Energy 2 4 6 8 10x46810Regression function 1 2 3 4i2.52.552.62.65Av. training err. 1 2 3 4i 2.482.492.52.512.522.532.54Test error Figure 26: Large λ. (Same parameters as for Fig. 23, except for λ= 1.0.) Due to the larger smoothness constraint the averaged traini ng error is larger than in Fig. 23. The fact that also the test error is larger tha n in Fig. 23 indicates that the value of λis too large. Convergence, however, is very fast. 193P 5 10 15y246810 x00.10.20.3 5 10 15yL 5 10 15y246810 x-5-4-3-2-1 5 10y 24681012i105110115120125130135Energy 2 4 6 8 10x 24681012Regression function 24681012i 1.822.22.42.6Av. training err. 24681012i2.442.462.482.52.522.54Test error Figure 27: Overfitting due to too small λ. (Same parameters as for Fig. 23, except for λ= 0.1.) A small λallows the average training error to become quite small. However, the average test error grows al ready after two iterations. (Having found at some λ–value during iteration an increasing test error, it is often but not necessarily the case that a better s olution can be found by changing λ.) 194P 5 10 15y246810 x00.050.10.15 5 10 15yL 5 10 15y246810 x-5-4-3-2 5 10y 2 4 6 810i 130140150160170Energy 2 4 6 8 10x 4567891011Regression function 2 4 6 810i 00.20.40.60.81Mixing coefficients 2 4 6 810i 2.42.52.62.72.82.9Test error Figure 28: Example of an approximately stable solution. (Sa me parameters as for Fig. 23, except for λ= 1.2,m2= 0.5, and initialized with L=t2.) A nearly stable solution is obtained after two iterations, fo llowed by a plateau between iteration 2 and 6. A better solution is finally found w ith smaller distance to template t1. (The plateau gets elongated with growing mass m.) The figure on the l.h.s. in the bottom row shows the mixing coeffi cientsajof the components of the prior mixture model for the solution du ring iteration (a1, line anda2, dashed). 195

arXiv:physics/9912006v1 [physics.gen-ph] 3 Dec 1999physics THE UNIVERSE’S EVOLUTION N. T. Anh1 2 Institute of Nuclear Science and Technique, Hanoi, Vietnam Abstract Based on a new theory of causality [1] and its development to t he theory of the Universe [2], we show, in this paper, new ideas for building a theory of everything. 1Mail Address: No.D27, 25B1 Cat Linh, Hanoi, Vietnam. 2E-Mail Address: anhnt@vol.vnn.vn 11 Introduction The discovery of natural world around us is an indispensable activity of mankind. And looking for a single theory that can explain every phenomeno n and every process is a good dream of scientists and especially physicists. Nowadays, p hysicists have been trying to find a single theory that unifies four familiar interactions, and they hope that it develops the theory of everything. A theory they believe to be the theo ry of everything is called the superstring theory. Notably, it is necessary to underst and that the superstring theory gives us description which only can unite four familiar forc es into a single framework. Of course, whether it is really called the theory of everythi ng or not since there are many unknown interactions (besides four familiar interact ions) absenting in the theory. Moreover, the theory of everything must give us a correct sol ution in every phenomenon and process (in all universe’s dimension, in all energy leve l, in all universe’s status and so fourth). The theory of everything is necessarily a theory of Creation, that is, it must necessarily explain everything from the origin of the Unive rse down to the lilies of the field. A theory of everything is also a theory of everyday. Thu s, this theory, when fully completed, will be able to explain the existence of every phe nomenon, the variation of every process, and many others. However, there is another way on which we can reach the theory of everything. That is to find a single law that implies all known laws and, therefo re, predicts unknown laws. The presence of this law has in every phenomenon, process and thing in nature. It is really to be the ultimate goal of all knowledge, the theory to end all theories, the ultimate answer to all questions. The present article is the first one of a series that we would li ke to say about the law and the theory of causality as well as its implementation for building a theory of the Universe. We hope that some of the readers of this article wil l find out that the law of causality is just the law of all laws, the theory of causality is just the single theory of everything, and perhaps they will be the ones to complete the quest for the Theory of the Universe. The article is organized as follows. In Section 2 we introduc e the ideas and concepts for leading the equation of causality [1]. In Section 3, as th e main part of the article, we attempt to simulate briefly the process of the Universe’s evo lution [2]. The conclusions and prospects are given in Section 4. 2 The Equation of Causality We can always conceive that the Universe is in unification. An d a surefire fact is that the Universe’s unification is only in a general intrinsic rel ationship which is nothing but the relationship of causality, and the unification only mani fests itself in that causal form. Then, a question is put on what is the ultimate cause of everyt hing? On further reflection, we find out that there exists an ultimate cause - that is the diff erence. 2Truly, there would not exist anything if there were not the di fference. If there were no difference, this world did not exist. And there is a fact tha t since the difference is the ultimate cause, it is the cause of itself, in other words, it i s also the effect of itself. The difference causes the difference, the difference is the coroll ary generated by the difference. In another way, we can imagine abstractly that the Nature is a set of positive actions3 and negative actions4. Then, what does the Nature act positively on? and what does t he Nature act negatively on? The answer to these questions give s us a law. That is, what do not have any intrinsic contradiction is acted positively on, what do have some intrinsic contradiction is acted negatively on. Both the positive act ion (in front of a process) and the negative action (in back of the process) have a final goal t hat is to reach and to end at a new action. Thus, we have started to come to a theory, axiom of which is the difference, object of which is actions5[1]. Consider two actions, we obtain a definition that coexistenc e of two actions which reject mutually generates contradiction. That is represen ted as follows: M=/braceleftbiggA/\egatio\slash=A−Action K1 A=A−Action K2. This means the higher the power of mutual rejection between t wo actions K1andK2 is the more severe the contradiction Mwill be. And the power of mutual rejection of two actions is estimated from the degree of difference. A contrad iction which is solved means that the difference of two actions diminishes to zero. Herein , two actions K1andK2all vary to reach and to end at a new action K3. The change, and one kind of which - the variation, is generate d by contradiction. In exact words, the variation is the manifestation of contra diction solving. The more severe contradiction becomes the more urgent need of solvin g out of contradiction will be, and hence the more violent the change, the variation of the st ate, i.e. of contradiction will become. Call the violence, or the quickness of the varia tion of contradiction Q, the contradiction state is M, the above principle can be represented as follows: Q=K(M)M where K(M)is means to solve the contradiction M.K(M)can be a function of the contra- diction state. It represents the degree of easiness to escap e the contradiction state. If the contradiction is characterized by quantities x, y, z, ... , these quantities themselves will be facilities to transport the contradiction, degrees of free dom over which the contradiction is solved. Hence, the degree of easiness is valued as the deri vative of the contradiction with respect to its degree of freedom K(M)∼|M′(x, y, z, ... )|. 3Positive action means ’affirmation’. 4Negative action means ”negative’. 5Action here is a general concept of anything, it may be a funct ion, a generator, an operator, or even a force, an interaction, a field, ect. depending on each consi dered subject. 3Thus, Q=a|M′|M where the coefficient agenerates from choosing the dimension. Advance a quantity T, inverse of Q, to be stagnancy of contradiction solving. The sum of stagnancy in the process of contradiction solving fro mM0toM0−∆Mwe call the time is generated by this variation ∆t≈−T+ (T+ ∆T) 2∆M. Thus, lim∆T→0,∆M→0,∆t→0∆M ∆t=dM dt=−1 T=−a|M′|M. Therefrom, we obtain the equation of causality, dM dt=−a|M′(x, y, z, ... )|M(x, y, z, ... ). (1) Truthly, the difference is the origin of all, but it has the mea ning in direct relationship, in direct comparison. Some state which has any intrinsic con tradiction must vary to reach a new one having no intrinsic contradiction, or exactly, hav ing infinitesimal contradiction. The greater the value of the contradiction derivative with r espect to some degree of freedom is, the better the ’scent’ for way out in that degree o f freedom will be, the greater the strength of the solved contradiction over that degree of freedom will be. It is easy to see that equation of causality (1) is represente d as a ’classical’ form. It can be developed to more general form in which the time is cons idered as a new degree of freedom. However, Eq. (1) looks like familiar equations, an d we will use it for applying to concrete problems. Though the law of causality (1) is abst ract its concrete form in each problem is very clear. And in the next Section we show the process of the Universe’s evolution which lays the foundation for building a theory of the Universe. 3 The Process of The Universe’s Evolution 3.1 The general mechanism To survey clearly the evolution of the Universe, we firstly re view four important concepts: time, space, matter, and motion. About the time. Can the time exist independently, if it is sep arated from space, matter, and motion? Evidently, no. If the time were separated from mo tion, the conception of it would have no meaning. The time cannot self-exist, it is the e ffect of motion. No motion, no time. About the motion. The motion also would not self-exist if it w ere separated from matter and space. 4And about the matter. The matter also cannot self-exist with out space. It exists owing to not only itself but also the coexistence of the space surrounding it. In essence, the matter is nothing but just some space with intrinsic rela tionship different from familiar space we see around us. Imagine that all are vanished: matter, space,..., and in gen eral, every difference is vanished. Then, there exists only one. It is homogeneous and limitless everywhere. It can self-exist. It is the first element. In this unique there i s nothing, but there exists the ’Nothing’. The Nothing is the origin of all, the cause of all, since it has the first difference. In Section 2, we have said the axiom of the theory of causality . That is the difference. Imagine that if the present Universe has many differences, th e first state of the Universe will be the state which has fewest differences. It is logical t o show that the first Universe’s state is the Nothing, and the transformation chain ”differen ce - contradiction - solving” is the expansion of the Universe. A remarkable consensus has been developing recently around what is called ”quantum cosmology”, which proposes a beautiful synthesis of seemin gly hostile viewpoints. In the beginning it was Nothing. No space, no matter or energy. But a ccording to the quantum principle, even Nothing was unstable. Nothing began to deca y, i.e. it began to ’boil’, with billions of tiny bubbles forming and expanding rapidly. Eac h bubble became an expanding sub-universe6. Sub-universes can literally spring into existence as a qua ntum fluctuation of Nothing. Resonances of vacuum fluctuations create first el ements of matter. In Ref. [2] we show the elementary equation of Evolution eΣaM/hatwide∂M=eΣ(−)∆t. ∂M, (2) and the conservation relation of quanta /summationdisplay j,...,kn/summationdisplay i=0(−)i n!Cn iMj...M k= 0, (3) where nis total of quanta, iis quantum number generated by each step of expansion of the Universe, Cn iis binary coefficient. It is easy to realize that Eq. (2) is also a form of equation of c ausality (1). But Eq. (2) gives us an important application in modelling the multipli cation and the combination of quanta. There are two objects from Eq. (2) we can use to study: one is actions, the other is quanta. Studying actions gives us laws, equations, repre sentations in each considered field. And studying quanta gives us models, classifications, arrangements of quanta. To describe the evolution of the Universe, it is better for us to investigate quanta. 6Our universe is actually part of a much larger ”multiverse” o f sub-universes. Our sub-universe may co-exist with other sub-universes, but our sub-universe ma y be one of the few compatible with life. This would answer the age-old question of why the physics constan ts of the universe fall in a narrow band compatible with the formation of life. If the universal cons tants were changed slightly, then life would have been impossible. 5Callα, β, γ, ... quanta. For each quantum there is a rule of multiplication as follows αn→e∂ααn=n/summationdisplay i=0Cn iαn−i= (α+ 1)n(4) where nis order of combination. Although Eq. (4) is obtained from Eq . (2) in considering for quanta, it can be found meaningly using the evolution pri nciple shown in Ref. [2]. Eq. (4) itself represents the evolution of the Universe. 3.2 Examples for the doublet and the triplet Using Eq. (4) we consider two stages in the process of the Univ erse’s evolution: doublet and triplet. For two interactive quanta the rule of multiplication reads αn, βn→1 2(eβ∂ααn+eα∂ββn) =n/summationdisplay i=0Cn iαn−iβi= (α+β)n. (5) And similar to three interactive quanta αn, βn, γn→1 3(e(β+γ)∂ααn+e(γ+α)∂ββn+e(α+β)∂γγn) =n/summationdisplay mm/summationdisplay iCn mCm iαn−mβm−iγi = (α+β+γ)n. (6) And so fourth. Eqs. 5 and 6 can be drawn as schemata. ... · · · · · · · · · · · · 2 1 1 0 /circlecopyrt 2 1 1 2⊗2= 3⊕1 1 2 1 2⊗2⊗2= 4⊕2⊕2 1 3 3 1 ... 1 4 6 4 1 ... 1 5 10 10 5 1 ... · · · · · · · · · · · · · · · · · · · · ·(7) 6is the schema for Eq. (5), where 2 means two quanta αandβ. The numbers in the triangle is the binary coefficients which give us weights of cl asses. For example, 2⊗2= 3⊕1=1 1 ——– 1 ——– 1. And similar to Eq. (6) we have ... 1 3 1 1 0 /circlecopyrt 3 1 1 1 1 2 1 3⊗3= 6⊕3 2 2 1 1 3 3 1 3⊗3⊗3= 10⊕8⊕8⊕1 3 6 3 3 3 1 1 4 6 4 1 4 12 12 4 3⊗3⊗3⊗3 6 12 6 4 4 ... 1(8) where 3 means three quanta α,βandγ. The coefficients in the pyramid give us weights of classes, 1 1 1 3⊗3= 6⊕3 =1 1 1 1 1 1, 1 3⊗3 = 1 ⊕8 = 1 1 1 2 1 1 1. 7It is easily to identify that the above schemata have the form s similar to the SU(2) and the SU(3) groups. This means that for nquanta we have a corresponding schema according to the SU(n) group, and the multiplication and the combination of the Un iverse conform to the SUgroup. And from these schemata we can draw periodic diagrams of the Universe’s quanta. For simplification, we show below the periodic diagram of the two quanta’s multipli- cation made of the schema (7). Remodel (7) with regard to the l evel splitting we have a new diagram, [] 1][ 2][][ 5][][][ 1]... 1[ 2]][ 2][[ 5]][][ 5][][[ 14]][][][ 14][][][[ 1]]... \ 1][ /... 1[[ 3]]][ 3][][ 3][[[ 9]]][][ 9][][][ 9][][[[ ... \ 28][][][][ /... 1]]] 1]][... 1][[ 1[[[ 4]]]][ 4]][][ 4][][[ 4][[[[ ... \ 14]][][][ 14][][][[ /... ... 1 ]]][... \1][][/... 1][[[ ... ... 5 ]]][][ 5][][][ 5][][[[ ... ... \ 20][][][][ /... ... 1 ]]]][ 1]][][... 1][][[ 1][[[[... ... 6]][][][ 6][][][[... ... 1 ]]][][... \1][][][/... 1][][[[ ... ... 7][][][][... ... Arrange this diagram in the order of the levels we obtain the s o-called periodic diagram 8• I • /up∼lope /down∼lope •↼•⇀• II • /up∼lope /down∼lope • − • − • III /up∼lope • /down∼lope •↼•/up∼lope•/down∼lope•⇀• • − • − • IV /up∼lope • /down∼lope • − • /up∼lope•/down∼lope• − • /up∼lope • − • − • /down∼lope V /up∼lope /up∼lope • /down∼lope /down∼lope •↼•/up∼lope• −/up∼lope•/down∼lope− •/down∼lope•⇀• • − • /up∼lope− • − /down∼lope• − • /up∼lope • − − • − − • /down∼lope VI /up∼lope /up∼lope • /down∼lope /down∼lope •↼•/up∼lope− • − /up∼lope•/down∼lope− • − /down∼lope•⇀• • − − • /up∼lope− • − /down∼lope• − − • /up∼lope • − − • − − • /down∼lope VII /up∼lope /up∼lope • /down∼lope /down∼lope •↼− •/up∼lope• − • /up∼lope•/down∼lope• − • /down∼lope• −⇀• /up∼lope /up∼lope /down∼lope /down∼lope, which is nothing but the Mendeleev periodic table built in th e energy levels. The pictures Fig.1 and Fig.2 have a very special significance besides the periodic law. They give us a model of the evolution in the pine-tree and the s piral from simplex to complex, from low-level to high-level. 4 Conclusions and Prospects There is a truth that everybody knows: the nature is difficult t o understand for us when it has not been discovered yet, but it is really beautiful whe n we understand it. This is science, where the ultimate worth of one’s ideas is that th ey lead to a genuine under- standing of nature. And an idea or a theory not only represent s daily phenomena but also makes predictions that survive comparison with observ ation and experiment based on fundamental principles and laws that underlie the univer se. By the present article, we can confirm an existence of an ultimate principle or an ultima te law from which others could be found out. We realize that the most important principle of nature is tha t all observable prop- erties of things are about relationships. The difference has meaning in direct relation- 9ship. Actions are in interaction in mutual relationships. C ontradictions are generated in mutual-rejection relationships. Transformation, change , or motion, variation, or exactly contradiction solving, does experience of relationships. Even space and time must be spoken about in terms of relationships. There is no such a thi ng as space independent of that which exists in it and no such thing as time apart from cha nge. These mean that the universe is in unification, and this unification is create d by relationships of causality. Relationships of causality give us an ultimate law which is c alled the law of causality. Following the logical source of the law of causality, we open up limitless horizons of a view of the universe. The Universe was born from Nothing, and its evolution created beautiful worlds of numerous form of things whose structure and complexity can be self- organized. We understand that there are natural processes, easily comprehensible, by which organization can arise naturally and spontaneously, without any need for a maker outside of the system. That is confirmed in the present articl e. Although the results we obtained in this article is similar t o ones that modern physics discovered, we open to the possibility that the answers to ma ny of the questions we have about why phenomena, things, the elementary particles, or t he fundamental forces are as they are and not otherwise, and why the nature created beauti ful worlds in the way we see not otherwise. Moreover, we have the expectation to answ er the greatest questions: ”Where does the universe come from?” or ”What is the evolutio n, the self-organization, the variety, or the fate of the universe?” or ”Where does the m atter come from and where is the missing matter?”. In the present article’s view of the universe, everything is from to nothing, everything may be smooth at the beginning but does not stay smooth foreve r, because today our universe is very inhomogeneous. So the universe was not perf ectly homogeneous either when it began or shortly after it began but, rather, it was sli ghtly inhomogeneous. It had small regions where the density of matter was slightly hi gher than average and other regions where it was slightly lower than average. They are re ally tiny. Yet tiny as they are to begin with, these inhomogeneities are very important because they are the seeds from which particles, star clusters, galaxies and, eventua lly, human beings, will grow in the way that their structure must be formed systematicall y from within by natural processes of self-organization such as periodic, multipli cative, combinative, evolutive, and etc. principles. Our universe has a variety of mysteries to discover. But we ca nnot say everything in a day. Many and very many beautiful worlds are in future of our discovery. This article is only the first one we would like to open up a first view of the un iverse. The first is the key idea behind evolution of the universe from nothing, the s econd the idea behind the principle of causality. These themes are only essential for understanding what happened, is happening, and will happen in the universe. Of course, this does not mean that theories will be discovere d, based on the principle of causality, are proven to be right; only observation and ex periment can, in the end, tell us that. But a definite fact that we enter the 21st century with new ideas and wide horizons, with much to do and everything to talk about. 10Acknowledgments We would like to thank Dr. D. M. Chi for useful discussions and valuable comments. The present article was supported in part by the Advanced Res earch Project on Nat- ural Sciences of the MT&A Center. References [1] D. M. Chi, The Equation of Causality , (1979). [2] N. T. Anh, Causality: The Nature of Everything , (1991). 11Figure 1: The pine-tree form of the periodic law Figure 2: The spiral form of the periodic law 12

arXiv:physics/9912007v1 [physics.gen-ph] 3 Dec 1999Physics/9912007 THE EQUATION OF CAUSALITY D. M. Chi1 Center for MT&Anh, Hanoi, Vietnam. (1979) Abstract We research the natural causality of the Universe. We find tha t the equation of causality provides very good results on physics. That is our first endeavour and success in describing a quantitative expression of the law of causalit y. Hence, our theoretical point suggests ideas to build other laws including the law of the Un iverse’s evolution. 1Mail Address: No. 13A, Doi Can Street, Hanoi, Vietnam. 1The Equation of Causality – D. M. Chi 2 1 Introduction The motivation for our theoretical study of problem of causa lity comes from three sources. The first is due to physical interest: what is the cause of all? The second is from a story happened about non-Euclidean geometry. And the last one is o ur review of the four basic concepts: time, space, matter and motion. 1.0.1 Cause of all If the World is in unification, then it must be unified by connec tions of causality, and the unification is to be indicated only in that sense. According to that spirit, contingency, if there is really so mething by chance, is only product of indispensably. Since the World is united in connections of causality, nothi ng of the World exists outside them, we can divide the World into two systems: Acomprises all of what are called causes, and Ball of what are called effects. Eliminating from two systems all alike elements, we thus hav e the following possibili- ties: 1. Both AandBare empty, i.e. there are not pure cause and pure effect. In oth er words, the World have no beginning and no end. 2.Ais not empty, but Bis. Thus, there is an existence of a pure cause. The World have a beginning but no end. 3.Ais empty, but Bis not. There are no pure cause but a pure effect. The World have no beginning but an end. 4. Both AandBare not empty. The World have both a beginning and an end. And only one of the four above possibilities corresponds wit h the reality. Which possibility is it and what is the fact dependent on? If the World is assumed as a unity system comprising causes an d effects, any effect must be a direct result of causes which have generated it, and these causes also had been effects, direct results of other causes before, etc. – there i s no effect without cause. A mystery motivation always hurries man to search for causes of every phenomenon and everything. Idealistic ideology believes that an absol ute ideation, a supreme spirit, or a Creator, a God,... is the supreme cause, the cause of all. Ma terialistic ideology thinks that matter is the origin of all, the first one of all. That actu ality is in contradiction. If it is honesty to exist a supreme cause, then one must be the d ifference! Indeed, if there were no existence of difference, there would not be any existence of anything, including idealistic ideology with its ideation , spirit and materialistic ideology with its material facilities. Briefly, If there were no differ ence, this World did not exist.The Equation of Causality – D. M. Chi 3 But, if the difference is the supreme cause, namely the cause o f all causes, then it must be the cause of itself, or in other words, it also must be the eff ect of itself. We have recognized the existence of difference, it means that we have tacitly recognized its relative conservation: indeed, you could not be idealis t if you now are materialist; anything, as long as it still is itself, then cannot be anythi ng else! 1.1 A story happened in geometry Let us return an old story: a matter of argument about the axio matics of Euclid’s geom- etry. Still by the only mystery motivation people always thirst fo r searching out “the supreme cause”. The goal here is humbler, it is restrained in geometry, and the first to realize that was Euclid. Euclid showed in his Elements how geometry could be deduced from a few defini- tions, axioms, and postulates. These assumptions for the mo st part dealt with the most fundamental properties of points, lines, and figures. His fir st four assumptions has been easily to be accepted since they seem seft-evident, but the fi fth, the so-called Euclidean postulate, incited everybody to suspect its essence: “this postulate is complicated and less evident”. For twenty centuries geometers tried to purify Euclid’s sys tem by proving that the fifth postulate is a logical consequence of his other assumpt ions. Today we know that this is impossible. Euclid was right, there is no logical inconsi stency in a geometry without the fifth postulate, and if we want it we will have to put it in at the beginning rather than prove it at the end. And the struggle to prove the fifth postula te as a theorem ultimately gave birth to a new geometry – non-Euclidean geometry. Without exception, their efforts only succeeded in replacin g the fifth postulate with some other equivalent postulate, which might or might not se em more self-evident, but which in any case could not be proved from Euclid’s other post ulates either. By that way they affirmed that this problem had solved, Euclid’ s postulate was just an axiom, because the opposite supposition led to non-Euclide an geometry without immanent contradiction. But... whether such a conclusion was accommodating? While everybody was joyful because it seemed that everythin g was arranged all right and the proposed goal had been carried out: minimized quanti ty of geometric axioms and purified them, whimsically, a new axiom was intruded unde rhand into: Lobachevski’s axiom – this axiom and Euclid’s fifth excluded mutually! Nobody got to know clearly and profoundly how this contradic tion meant. But contra- diction is still contradiction, it brought about many argum ents and violent opponencies, even grudges. Afterwards, since Beltrami had proved correctness of Lobac hevski’s geometry on pseu-The Equation of Causality – D. M. Chi 4 dosphere – an infinite two-space of constant negative curvat ure in which all of Euclid’s assumptions are satisfied except the fifth postulate, the sit uation was made less tense. If non-Euclidean geometers, from the outset, since setting to build their geometry, de- clared to readers that objects of new geometry were not Eucli dean plane surface but pseu- dosphere, not Euclidean straight line but line of pseudosph ere, maybe nobody doubted and opposed at all! What a pity ! or it was not a pity that nothing happened such a th ing? But an actual regret was: the whole of problem was not what was brought out and solved on stage but what - its consequence - happened on backs tage. Because, even if non-Euclidean geometry was right absolute ly anywhere, it meant: with the same objects of geometry – Euclidean plane surface a nd straight line – among them, nevertheless, there might be coexistence of two forms of mutually excludible rela- tionships which were conveyed in Euclid’s axioms and Lobach evski’s axioms. It was possible to allege something and other as a reason for f orcing everybody to accept this disagreeableness, but that fact was not faithfu l. Here, causal single-valuedness was broken; here, relative conservation of difference was co nfused white and black; there was a danger that one thing was other and vice versa. The usual way to “prove” that a system of mathematical postul ates is self-consistent is to construct a model that satisfies the postulates out of so me other system whose consistency is unquestioned. Axiomatic method used broadly in mathematics is clear to bri ng much conveniences, but this method is only good when causal single-valuedness i s ensured, when you always pay attention on order not to take real and physical sense awa y from considered subjects. Brought out forms of relationships of objects as axioms and d efied objects - real owners of relationships, it is quite possible that at a most unexpec ted causal single-valuedness is broken and contradiction develops. Because what we unify together is: objects are former ones, t heir relationships are corollaries formed by their coexistence, but is not on the co ntrary. If we have a system of objects and we desire to search for all po ssible relationships among them by logically arguing method, perhaps at first and a t least we have to know intrinsic relationships of objects. Intrinsic relationships control nature of objects, in turn nature of objects directs pos- sible relationships among them and, assuredly, among them t here may not be coexistence of mutually excludable relationships. Intrinsic relationship, according to the way of philosophe r’s speaking, is spontaneous- ness of things. Science today is in search of spontaneity of t hings in two directions: more extensive and more elementary. Now return the story, as we already stated, the same objects t hemselve of Euclid’s geometry had two forms of mutually excludable relationship s, how is this understood?The Equation of Causality – D. M. Chi 5 It is only possible that Euclid’s axiomatics is not complete d yet with the meaning that: comprehension of geometrical objects is not perfecte d yet. Euclid himself had ever put in definitions of his geometrical objects, but modern mat hematicians have criticized that they are “puzzled” and “heavily intuitive”. According to them, primary objects of geometry are indefinable and are merely called points, lines , and sufaces, etc. only for historic reason. But, geometrical objects have other names: “zero”-, “one”- , “two”-, and “three”- dimensional spaces (“zero”-dimesional space, thai is poin t, added by the author to com- plete a set). We can ask that, could the objects self-exist independently ? If could, why would they relate together? Following logical course of fact, we realize that conceptio ns of objects are developed from experience which is gained by practical activities of m ankind in nature, but which is not innate and available by itself in our head. (Therefore , we should not consider them apart from intuition, should not dispossess of ability to im agine them, how reasonless that is!). Acknowledging at deeper level, we can perceive that no all of geometrical objects may exist independently, but any n-dimensional space is intersection of two other spaces with dimension higher one ( n+ 1). Thus, it seems that we have definitions: point is intersectio n of two lines; line is intersection of two surfaces; surface is intersection of tw o volumes, and volume... of what is it intersection? However, in a geometry, by human imaginable capability, the y are evident to be inde- pendent objects, and for convenience, we call them spatial e ntities. Simplest geometrical objects are homogeneous entities. Th ey are elements, speaking simply, in which as transferring with respect to all their po ssible degrees of freedom, it is quite impossible to find out any inner difference. Objects of Euclid’s geometry are a part of a system of homogen eous entities. If we build an axiomatics only for this part, it is clear that this a xiomatics is not generalized. An axiomatics used for homogeneous spaces is just one for sph erical surface2. Euclid’s geometry is only a limited case of this generalized geometry . For spherical surface, that is homogeneous surface in gener al, there exists a following postulate: any two non-coincident “straight” lines (“stra ight” line is homogeneous line dividing the surface that contains it into two equal halves) always intersect mutualy at two points and these two points divide into two halves of each line. It is possible to express further: any two points on a homogen eous surface belong to only a sole “straight” line also on that surface if they do not divide this line into two equal halves. 2The surface of a sphere is a two-dimensional space of constan t positive curvature.The Equation of Causality – D. M. Chi 6 Applying this postulate for Euclidean plane surface as a lim ited case, we realize imme- diately that it is just the purport of the first Euclidean axio m: through two given points it is possible to draw only a sole straight line. Indeed, any t wo points in an investigated scope of Euclidean plane surface belong to only a sole “strai ght” line since they do not divide this line that contains it into two equal halves. So we can say that the mode of stating the fifth Euclidean postu late was inaccurate from the outset, because any two “straight” lines on a given h omogeneous surface al- ways intersect mutually at two points and divide into two hal ves of each other. In any sufficiently small region of the surface it would be possible t o find either only one their intersectional point and the other at infinity or no point - th ey are at infinities. In this case these two “straight” line are regarded to be parallel ap parently with each other. Equivalent stating the fifth postulate, after correcting in the sense of above comment, is quite possible to be proved as a theorem. There is a very important property of spatial entities that: any spatial entity is pos- sible to be contained only in other spatial entity with the sa me dimension and the same curvature, or with higher dimension but no higher curvature . This seems to be awfully evident: two circles with different c urvatures are impossible to be contained in each other; a spherical surface with any cu rvature is impossible to contain a circle with lower curvature... Similarly, two spaces with different curvatures are impossi ble to contain in each other. Curvature, here, is correspondent to any quantity characte rized by inner relationship of investigated object. 1.2 Contradiction generated based on difference is dynamic p ower of all In essence, the Nature is a system of positive actions and neg ative actions. What has the Nature thus positive actions on and negative act ions on? Those secrets are explored and discovered by science more an d more and in searching, if not counting its dynamic source, logical argument plays a great role. But what we call logic is true not a string of positive actions and negative actions with all orders? Because thought is only a phenomenon of the Nature, the law of positive actions and negative actions of thought is also the law of positive actio ns and negative actions of the Nature. In other words, the law of actions of the Nature is refl ected and presented in the law of actions of thought. This law is that: what without immanent contradiction is in p ositive action by itself, what with immanent contadiction is in negative action by its elf. Positive action (if looking after the process) and negative action (if looking back uponThe Equation of Causality – D. M. Chi 7 the process) both have an ultimate target which is coming to a nd closing with a new action. Let us take a class of similarly meaning concepts such as: hav ing, existence, con- servation, and positive action. In opposition to them, anot her class includes: nothing, non-existence, non-conservation, and negative action. They belong among the most general and basic concepts, becau se in any phenomenon of the Nature: sensation, thinking, motion, and variation, etc. there are always their presences. But it turned out to be that the powers of two classes of concep ts are not equivalent to each other (and that is really a lucky thing!). Let us now establish a following action, called Aaction : “Having all, existing all, conserving all, and acting posit ively on all.” And an another, called Baction , has opposite purport: “Nothing at all, non-existing all, non-conserving all, and acting negatively on all.” Acting positively Aaction is acting negatively Baction, and vice versa. Baction says that: – ‘Nothing at all’, i.e. not having Baction itself. – ‘Non-existing all’, i.e. not existing Baction itself. – ‘Non-conserving all’, thus Baction itself is not conserved. – ‘Acting negatively on all’, this is acting negatively on Baction itself. Briefly, Baction contains an immanent contradiction. It acts negatively on i tself. Self-acting negatively on, Baction auto-acts positively on Aaction . It means that: there is not existence of absolute nihility or absolute emptiness , and therefore, the World was born! AndAaction acts on all, including itself and Baction , butBaction self-acts negatively on itself, so Aaction has not immanent contradiction. Thus, in the sphere of Aaction all what do not self-act negatively on then self-act positively on.The Equation of Causality – D. M. Chi 8 1.3 What is the most elementary? There are four very important concepts of knowledge that: time,space,matter , and motion . They are different from each other, but is it true that they are equal to each other and they can co-exist independently? Let us start from time. Is it an entity? Could it exist independently apart from space, matter , andmotion ? Evidently not! Just isolated timeout of motion , the conception of it would be no longer here, time would be dead. And the concept ion of motion has higher independence than time’s. Sotimeis not the first. It could not self-exist, it is only consequen ce of remainders. Motion is not the first either. It could not self-exist apart from matter andspace. In fact, motion is only a manifestation of relationship betwee nmatter andspace. Thus, one of two remainders, matter andspace, which is the most elementary? which is the former? or they are equal to each other and were born by o ne more elementary other? Perhaps setting such a question is unnecessary, beca use just as timeandmotion , matter could not exist apart from space. For instance, a concrete manifestation of matter, it exists not only because of itself but also because of simul taneous existence of space which surrounds it (and contains it) so that it is still itsel f. Clearly, matter is also in spatial category and it is anything else if not the s pace with inner relationships different from those of usual space that we know?! But now, according to the property of spatial entities raise d in the previous subsection, this fact is contradiction: two same dimensional spaces of d ifferent curvatures (inner relationships) are impossible to contain in each other! Thus, either we are wrong: it is evident to place coincidenta lly two circles of different radii in each other or the Nature is wrong: different spaces ca n place in each other, defying contradiction. And contradiction generated by this reason is power of motio n, motion to escape from contradiction. Thus, we may to say that matter is spatial entity of some curva ture. But, where were spatial entities born from? and how can they e xist? or are they products of higher dimensional spaces?, so what is about hig her dimensional spaces? Let us imagine that all vanish, including matter, space, ... and as a whole all possible differences. Then, what is left? Nothing at all! But that is a unique remainder! Clearly, this unique remainder is limitless and homogeneou s “everywhere”. Otherwise,The Equation of Causality – D. M. Chi 9 it will violate our requirement. We now require the next: even the unique remainder vanishes, too. What will remain, then? Not hardly, we indentify immedeately that substitute which replace it is just itself! Therefore, we call it absolute space. The absolute space can vanish in itself, in other words, acti ng negatively it leads to acting positively itself. That means, the absolute space ca n self-exist not depending on any other. It is the former element. It is the “supreme cause”, too. Because, contrary to anyone’ s will, it still contains a difference. Indeed, in the unique there is not anything, but still is the N othing! Nothing is contained in Having, Nothing creates Having. Having, but No thing at all! Here, negative action is also positive action, Nothing is al so Having, and vice versa. Immanent contradiction of this state is infinitely great. Express mathematically, the absolute space has zero curvat ure. In this space there exist points of infinite curvatures. This difference is infini tely great and, therefore, con- tradiction generated is infinitely great, too. The Nature did not want to exist in such a contradiction state . It had self-looked for a way to solve, and the consequence was that the Nature was bor n. Thus, anew the familiar vague truth is that: “matter is not bo rn naturally (from nihility), not vanished naturally (in nihility), it always is in motion and transformation from one form to other. Nowadays, it is necessary to be affirmed again that: “matter is just created from nothing, but this is not motiveless. The fo rce that makes it generate is also the power makes it exist, move, and transform. 2 Representation of contradiction under quantitative formula: Equation of causality Any contradiction is originated by coexistence of two mutua lly rejectable actions. That is represented as follows: M=/braceleftbiggA/negationslash=A −Action K1 A=A −Action K2. Clearly, the more severe contradiction Mwill be if the higher power of mutual rejection between two actions K1andK2is. And power of mutual rejection of two actions is estimated only from the degree of difference of those two acti ons. A contradiction which is solved means that difference of two a ctions diminishes to zero. Herein, two actions K1andK2all vary to reach and to end at a new action K3.The Equation of Causality – D. M. Chi 10 Thus, what are the differences [ K1−K3] and [ K2−K3] dependent on? Obviously, these differences are dependent on conservation capacities of actions K1andK2. The higher conservation capacity of any action is, the lower diff erence between it and the last action is. And then, in turn what is conservation capacity of any action dependent on? There are two elements: It is dependent on immanent contradiction of action, the gre ater its immanent con- tradiction is the lower its conservation capacity is. It is dependent on new contradiction generated by variation of action. The greater this contradiction is the more variation of action is resist ed and, therefore, the higher its contradiction capacity is. Variation, and one kind of which - motion, is generated by con tradiction. More exactly, motion is a manifestation of solution to contradiction. The more severe contradiction becomes the more urgent need of solution to contradiction wi ll be, and hence the more violent motion, variation of state, i.e. of contradiction w ill become. Call the violence, or the quickness of variation of contradiction Q, the contradiction state is M, the above principle can be represented as follows: Q=K(M)M. We call it equation of causality, where K(M)is means of solution to contradiction. On simplest level, K(M)can be a function of contradiction state. Actually, it repre sents easiness of escape from contradiction of state. If contradiction is characterized by quantities x, y, z, ... , these quantities themselves will be facilities of transport of contradiction, degrees o f freedom over which contradiction is solved. Hence, easiness is valued as the derivative of con tradiction with respect to its degree of freedom. The greater derivative value of contradiction with respect to any degree of freedom is, the higher way-out “scenting” capability in this direct ion of state becomes, the more “amount of contradiction” escaped with respect to this degr ee of freedom is. Thus, K(M)∼|M′(x, y, z, ... )|. And we have Q=a|M′(x, y, z, ... )|M(x, y, z, ... ), where the coefficient ais generated only by choosing system of units of quantities. We said that difference is the origin of all, but difference its elf has no meaning. The so-called meaning is generated in direct relationship, in d irect comparison. The Nature cannot feel difference through “distance”. A some state which has any immanent contradiction must vary t o reach a new one having no intrinsic contradiction, or exactly, having infin itesimal contradiction.The Equation of Causality – D. M. Chi 11 That process is one-way, going continuously through all val ues of contradiction, from the beginning value to closing one. We thus have endeavored to convince that motion (variation) is imperative to have its cause and property of motion obeys the equation of causal ity. Then, must invariation, i.e. conservation be evident without any cause? It is possib le to say that: any state has only two probabilities – either conservable or variable, an d more exactly, all are conserved but if that conservation causes a contradiction, then it mus t let variation have place to escape contradiction and this variation obeys the equation of causality. If this theoretical point is true, our work is only that: lear ning manner of comprehen- sion, estimating exactly and completely contradiction of s tate, and describing it in the equation of causality, at that time we will have any law of var iation. But is such enough for our terminal perception about the Natu re, about people them- selves with own thought power, to explain wonders, which alw ays surprise generations: why can the Nature self-perceive itself, through its produc t - people?! 3 Using the causal principle in some concrete and simplest phenomena Advance a quantity T, inverse of Q, to be stagnancy of solution to contradiction. Thus, T=1 aM′M The sum of stagnancy in the process of solution to contradict ion from M0toM0−∆M called the time is generated by this variation (∆ t). Figure 1: Variation of contradiction From the above definition and Figure 1, we identify that ∆t≈−T+ (T+ ∆T) 2∆M. Thus, ∆M ∆t≈ −2 2T+ ∆T. We have lim∆T→0,∆M→0,∆t→0∆M ∆t=dM dt=−1 T=−a|M′|M. Therefrom, we obtain a new form of the equation of causality, dM dt=−a|M′(x, y, z, ... )|M(x, y, z, ... ).The Equation of Causality – D. M. Chi 12 Thus, if we consent to the time as an independent quantity and contradiction as a time-dependent one, speed of escape from contradiction wi th respect to the time is proportional to magnitude of contradiction and means of sol ution. In the case that contradiction is characterized by itself, n amely M=M(M), we have M=M0e−a(t−t0), where M0is the contradiction at the time t=t0. 3.1 Thermotransfer principle Supposing that inn a some distance of an one-dimensional spa ce we have a distribution of a some quantity L. If the distribution has immanent difference, i.e. immanent c ontradiction, it will self- vary to reach a new state with lowest immanent contradiction . That variation obeys the equation of causality, dM dt=−a|M′|M. For convenience, we spread this distribution out on the xaxis and take a some point to be an origin of coordinates. Because the distribution is one of a some quantity L, all its values at points in the space of distribution must have equidimension (homogeneity). An d the immanent difference of distribution is just the difference of degree. At two points x1andx2, the quantity Lobtains two values L1andL2, respectively. Due to the difference of degree, there is only a way of estimati on: taking the difference (L2−L1). But two points x1andx2only ‘feel’ the difference from each other in direct connecti on, contradiction may appear or may not only in that direct conne ction: at a boundary of two neighbouring points x1andx2,Lquantity obtains simultaneously two values L1andL2; two these actions act negatively on each other and magnitude of contradiction depends on the difference ( L1−L2). Therefore, in order for the difference ( L1−L2) to be the yield of direct connection between two points x1andx2, we must let, for example, x2tend infinitely to x1(but not coincide with it). Whereat, the immanent contradiction at infinitesimal neigh bourhood of x1will be valued as the limit of the ratio:L1−L2 x1−x2, asx2→x1, i.e. the derivative value of Lover the space of distribution at x1. From the presented problems, we have M=dL dx=∂L ∂x.The Equation of Causality – D. M. Chi 13 Substituting the value of Min the equation of causality: ∂ ∂t∂L ∂x=−a∂L ∂x. (1) The immanent contradiction at each point is solved as Eq. (1) . That makes the distribution vary. We will seek for the law of this variation. The immanent contradiction at neighbourhood of xis Mx,t=∂L ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle x,t. Later a time interval ∆ t, this contradiction is decreased to the value Mx,t+∆t=∂L ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle x,t+∆t. Thus, it seems that this variation has compressed a some amou nt of values of Lfrom higher valued points to lower ones, making ‘a flowing current ’ of values of Lthrough x (Figure 2). Figure 2: The law of variation for Lquantity Clearly, the magnitude of ‘the flowing current’, i.e. the amo unt of values of Lflows through xin the time interval ∆ t, is ℑx=∂ ∂t∂L ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle x∆t=−a∂L ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle x∆t. Similarly, at the point x+ ∆xwe have ℑx+∆x=−a∂L ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle x+∆x∆t. In this example, the current ℑxmakes values of Lat points in the interval ∆ xincrease, andℑx+∆xmakes them decrease. The consequence is that the increment ∆ Lthe interval ∆xobtains is ∆L|∆t=a∆t/parenleftbigg∂L ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle x+∆x−∂L ∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle x/parenrightbigg =a∆t∂2L ∂x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle x≤ξ≤x+∆x∆x.The Equation of Causality – D. M. Chi 14 The average density value ∆Lat each point in the interval ∆ xwill be ∆L/vextendsingle/vextendsingle ∆t∼=a∆t∂2L ∂x2/vextendsingle/vextendsingle/vextendsingle ξ∆x ∆x. The exact value reaches at the limit ∆ x→0, ∆L|x,∆t= lim ∆x→0∆L/vextendsingle/vextendsingle ∆t=a∆t∂2L ∂x2/vextendsingle/vextendsingle/vextendsingle/vextendsingle x. Thus lim ∆t→0∆L ∆t/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=a∂2L ∂x2, or ∂L ∂t=a∂2L ∂x2. (2) The time-variational speed of Lat neighbourhood of any point of the distribution is proportional to the second derivative over the space of dist ribution of this quantity right at that point. And as was known, Eq. (2) is just diffusion equation (heat-tra nsfer equation) that had been sought on experimental basis. On the other hand, the corollary of the above reasoning manne r has announced to us the conservation of values of the quantity Lin the whole distribution, although values of this quantity at each separate point may vary, whenever valu e at any point decreases a some amount, then value at its some neighbouring point incre ases right the same amount. If the space of distribution is limitless, then along with in crease of time the mean value of distribution will decrease gradually to zero. 3.2 Gyroscope The conservation of angular momentum vectors may be regarde d as the conservation of two components: direction and magnitude. If in a system the d irective conservation is not violated but the magnitude conservation of vectors is vi olated, this system must vary by some way so that the whole system will have a sole angular mo mentum vector. And in the case where the conservation not only of magnitude but a lso of direction are both violated, solution to contradiction of state depends on the form of articulation. We now consider the case, in which the gravitational and cent rifugal components may be negligible (Figure 3). Figure 3: Gyroscope with only angle degree of freedomThe Equation of Causality – D. M. Chi 15 For simplicity, we admit that there is a motor to maintain a co nstant angular velocity ω of system. Thus, we are only interested in the contradiction generated by violation of the directive conservation kω0. The action K1– the conservation of k− →ω0, say that: variational speed of the vector direction k− →ω0equals zero. But the action K2– the conservation of − →ω, say that: the direction k− →ω0must be varied with the angular velocity ωcosα. Thus, in macroscope, the difference [ K1−K2] =ωcosαis the origin of that contra- diction, and the contradiction is proportional to this diffe rence. M∼ωcosα, M=|k− →ω0×− →ω|=kω0ωcosα. The taken proportionality factor kω0(still in macroanalysis) is based on an argument: ifω0equals zero, the vector direction k− →ω0will not exist certainly, and therefore the problem of contradiction generated by its directive conser vation will not be invented. Taking the value of Minto the equation of causality, we obtain ∂M ∂t=−ak2ω2 0ωsinαcosα. Here, we have calcuted M′=M′ α. From the equation we identify that if α= 0, then the escaping speed of contradiction state will equal zero. The derivative of contradiction with respect to the time is ∂ ∂t(kω0ωcosα) =−ak2ω2 0ωsinαcosα, or∂α ∂t=akω0ωcosα, α /negationslash= 0. (3) The variation of αcauses a new contradiction, this contradiction is proporti onal to value of ∂α/∂t , therefore there is not motional conservation over the comp onent α. And thus the escaping speed in Eq. (3) is also just the instantane ous velocity of the axis of rotation plane surface over α. The time, so that the angle between the axis of rotation plane surface (i.e. the direction of the vector k− →ω0) and the horizonal direction varies from the value +0 to α, will be t=1 2akω0ωln1 + sin α 1−sinα/vextendsingle/vextendsingle/vextendsingle/vextendsingleα +0. 3.3 Buffer zone of finite space Supposing that there is a finite space [ A] with intrinsic structure satisfying the invariance for the principle of causality. This space is in the absolute space [ O]. At the boundary of these two space there exists a contradiction caused by difference between them.The Equation of Causality – D. M. Chi 16 Because both of the spaces conserve themselves, contradict ion is only possible to be solved by forming a buffer zone (i.e. field), owing to which diff erence becomes lesser and more harmonic. The structure of the buffer zone must have a som e form so that the level of harmonicity reaches to a greatest value, i.e. immanent co ntradiction at each point in the field has lowest possible value. It is clear that the farther from the center of the space [ A] it is the more the property of [A] diminishes. In other words, the [ A]-surrounding buffer zone (field) has also the property of [ A] and this property is a function of r– i.e. the distance from the center of the space [ A] to considered point in the field (Figure 4). Figure 4: The structure of the buffer field of a finite space From the problems presented and if the notation of the buffer z one is T, we will have T[A]=g(r)[A] r, where g(r) is an unknown function of ralone, characterized for the intrinsic harmonicity of the field. If in the field zone T[A]there is a space [ B] and this space does not disturb considerably the field T[A], whereat the difference between [ B] and T[A]forces [ B] to move in the field T[A]to approach to position where the difference between [ B] and T[A]has lowest value (here, we have admitted that the space [ B] has also self-conservation capability). This contradiction of state is proportional to the difference ([ B]−T[A]). If we detect a factor cto use for ‘translating language’ from the property of [ B] into the property of [ A], then the contradiction may be expressed as follows M=f./parenleftbigg c[B]−g(r)[A] r/parenrightbigg , where fis proportionality factor. And the law of motion of the space [B] in the field T[A] is sought by the equation of causality, ∂M ∂t=−a|M′|M =−af2[A]/vextendsingle/vextendsingle/vextendsingle/vextendsingleg(r)−rg′(r) r2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg c[B]−g(r)[A] r/parenrightbigg . Here, the transfer quantity (degree of freedom) of contradi ction is r. Because the motion of the space [ B] must happen simultaneously over all directions which have centripetal components, therefore the resultan t escaping velocity of the stateThe Equation of Causality – D. M. Chi 17 – i.e. the resultant velocity of the space [ B] in the field T[A]must be estimated as the integral of the escaping speed over all directions which hav e centripetal components. ∂M ∂t=−a[A]4πf2π/2/integraldisplay 0/vextendsingle/vextendsingle/vextendsingle/vextendsingleg(r)−rg′(r) r2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg c[B]−g(r)[A] r/parenrightbigg cos2ϕ dϕ =−aπ2f2[A]/vextendsingle/vextendsingle/vextendsingle/vextendsingleg(r)−rg′(r) r2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg c[B]−g(r)[A] r/parenrightbigg . Expanding the left side hand, we obtain f[A]/parenleftbiggg(r) r/parenrightbigg∂r ∂t=−af2π2[A]/vextendsingle/vextendsingle/vextendsingle/vextendsingleg(r)−rg′(r) r2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg c[B]−g(r)[A] r/parenrightbigg , or ∂r ∂t=−afπ2g(r)−rg′(r) r2/parenleftbigg c[B]−g(r)[A] r/parenrightbigg . Notice here that g(r) is a function of ralone. If proving that the variation of ras well as the conservation of ∂r/∂t causes a new contradiction proportional to right ∂r/∂t, then the escaping speed obtained is just the instantaneous velocity of [ B] in the field T[A].

arXiv:physics/9912008v1 [physics.gen-ph] 3 Dec 1999Physics/9912008 1CAUSALITY The Nature of Everything N. T. Anh∗† Department of Theoretical Physics, Hanoi National Univers ity, High College of Physics, Institute of Theoretical Physics, Hanoi , Vietnam. (1991) Abstract We pursue research leading towards the nature of causality i n the universe. We establish the equation of the universe’s evolution from the universe-sta te function and its series expansion, in which causes and effects connect together to construct a li nked chain of causality. And the equation of causality [1] is rederived. Therefrom, our theo ry informs the progress of the universe and, simulaneously, a law that presents everywhere. Furthe rmore, we lay some foundations for a new mathematical phrasing of the movement of physics. That is the life-like mathematics - a mathematics full of life. ∗Present address: Institute for Nuclear Science and Technique, Hanoi, Vietna m. †Email: anhnt@vol.vnn.vn 2CAUSALITY: The Nature of Everything – N. T. Anh 3 1 INTRODUCTION To pursue the idea of the previous article [1] a question aris es that if the law of causality is the most general law, then about what will it inform us in the whole of t he universe, in the birth, expansion and conclusion of the universe, as well as in some concrete ph enomenon and process in the universe? The transformation chain “ Difference →Contradiction →Solving of contradiction ” is also the expansion chain of the universe. The expansion chain of the u niverse starts from the absolute infinite and homogeneous space. In the absolute space there is a Nothi ng, but itself is the one, is the unique: the content is zero, the form is one. Thus as acting negativel y all the absolute space acts negatively just its unique existence. Therefrom the immanent contradi ction of this state of the universe is infinitely great, and the universe self-solves by expanding into infinite series of smaller contradictions and solving them. The consequence is that our world was born. In order to describe quantitatively all processes of the bir th, expansion and conclusion of the universe, we establish a universal equation from the univer se-state function in Sec. 2. And the equation of causality is anew devised exactly in Sec. 3. Our c onclusion is given in Sec. 4. In Appendices A and B we present a few foundations for functor an d life-like mathematics. 2 UNIVERSAL EQUATION 1!At the debut of each expansion of the universe, the f! function characterizing for the early state of the universe is equal gradually to zero everywhere and for ming the absolute vacuum-state function f!( ). f! =f!( ). Continuously, in this expansion, the vacuum state of the uni verse becomes the absolute space state but still is always equal to zero everywhere, because i n the vacuum of the universe there contain infinitely many absolute spaces f!( ) = f!(K), where f!(K) is called absolute space-state function of the set of absol ute spaces in the universe’s vacuum, with Kbeing absolute space set K={...k...}. Therefore, the absolute space-state function satisfies the following condition /braceleftbigg f!(K) = 0,∀k∈K/integraltext f!(K)dk= 1,(1.1)CAUSALITY: The Nature of Everything – N. T. Anh 4 where the notation “0” is the “nothing” in the universe’s vac uum, and “1” symbolizes the unique existence of the universe’s vacuum. f!(K) is equal to zero everywhere, but its integral with respect t o space is unique, or the whole of f!(K) over space is unique. (Right in whole integral region, f!(K) still is equal to zero - The absolute vacuum has nothing, but has uniquely that “nothing”.) f!(K) is limitless and homogeneous everywhere, is determined in every neighborhood of point spaces κiin a set of absolute point spaces K(κi∈ K={...κ...}), is continuous with ki=κi, and exists the limit lim ki→κif!(K) =f!(K).f!(K) is continuously differentiable on the set of point spaces K, in which all direct and reverse partial derivatives to all o rders exist and are continuous due to the existence of limits, respectively. 2!The state functions of the absolute vacuum as well as of the ab solute space determine the states of the homogeneous state vacuum and the homogeneous s pace, in which as transferring with respect to all degrees of freedom there cannot be any arbitra ry immanent difference. But, the absolute vacuum is a unique and homogeneous state in which there is nothing. As self-acting negatively, it leads to self-acting positivel y itself and vice versa. In the unique state there contains nothing but still exists a “nothingness”! For that reason, in overall the absolute vacuum still contains a difference: between the “nihility” and the unique state of this “nihility”. That difference is infinitely great, and therefore the immanent contradicti on of this state is also infinitely great. It is easy to be identified this contradiction in the relation (1 .1). So the self-solving is performed by expanding f!(K) into series in the whole of space,  ...... · · · A(0)(k)/summationtext hA(+1) h(k)· · · · · ·/summationtext hA(−1) h(k)A(0)(k)· · · ...... = ...... · · · A(0)(k)/summationtext hA(+1) h(k)/vextendsingle/vextendsingle/vextendsingle κ· · · · · ·/summationtext hA(−1) h(k)/vextendsingle/vextendsingle/vextendsingle κA(0)(k) · · · ......  × ...... · · · 1/summationtext h/integraltextk− h κ− hdκ− h· · · · · ·/summationtext h/integraltextk+ h κ+ hdκ+ h 1 · · · ...... , or in the general form, f!(K) =f!(K)|KSK K (1.2)CAUSALITY: The Nature of Everything – N. T. Anh 5 under the space reflection, where f!(K),f!(K)|K, and SK Kcontain infinitely many variable elements. Conversely, if there is a linear transfer variation from Kspace to Kspace, then f!(K)|K=f!(K)SK K, where SK Kis different from SK Kby permuting of superior and inferior limits of integrals. For that reason, if doing an action f!(K)|KontoSK K, then SK KSK K=SK KSK K=I (1.3) withIunit. 3!Now, we return to consider Eq. (1.2). If instead of Kwe let ( K− K), then f!(K) becomes f!(K− K) =f!(K− K)|(K−K)=0S(K−K) (K−K)=0, (1.4) And if we define/integraltextki κidκi=Mi, then SK K=M, namely  ...... · · · 1/summationtext h/integraltextk− h κ− hdκ− h· · · · · ·/summationtext h/integraltextk+ h κ+ hdκ+ h 1 · · · ...... = ...... · · · 1/summationtext hM− h −1!· · · · · ·/summationtext hM+ h +1!1· · · ...... . Thus, Eq. (1.4) will be f!(M) =f!(M)|M=0M (1.5) where each element equation has the form as a functor A(M) =eΣM/hatwideAA(M)|M=0. From calculating SK K, we are convenient to consider the condition (1.3). After we define/integraltextκi kidκi= −/integraltextki κidκi=−Mi, then SK K=M−1, namely  ...... · · · 1/summationtext h/integraltextκ− h k− hdκ− h· · · · · ·/summationtext h/integraltextκ+ h k+ hdκ+ h 1 · · · ...... = ...... · · · 1/summationtext h−M− h −1!· · · · · ·/summationtext h−M+ h +1!1 · · · ...... .CAUSALITY: The Nature of Everything – N. T. Anh 6 Thus, in order for MM−1=M−1M=Ithen M+M−=M−M+= 0, (1.6) where M+andM−are some contradictions which belong to the two reflective wo rlds, and notice here the formula n/summationdisplay i=0(−)i (n−i)!i!=1 n!n/summationdisplay i=0(−)i/parenleftbiggn i/parenrightbigg = 0, with meaning that the sum of generation and annihilation qua ntities is constant (conservation of quanta). 4!From the second in the condition (1.1) of the absolute space- state function, we consider in the set of point-spaces K, 1 =/integraldisplay κ•f!(K)dκ=f!(K)A(0)(k)/vextendsingle/vextendsingle κ, or can write 1 =f!(K− K)A(0)(k−κ)/vextendsingle/vextendsingle (k−κ)=0,i.e. 1 =f!(M)A(0)(M)/vextendsingle/vextendsingle M=0. (1.7) And consider in an interval of integration from the set of poi nt-spaces Kto the set of absolute spaces K 1 =/integraldisplayk κf!(K)dκ=f!(K− K)[k−κ](0),i.e. 1 =f!(M)M(0). (1.8) Following the life-like mathematics, let (1.7) and (1.8) si nk into together we obtain an equivalent relation A(0)(M)/vextendsingle/vextendsingle M=0=M(0). (1.9) After we differentiate partially (in direct and reverse orie ntations) to all orders with respect to components in the two elements A(0)(M)/vextendsingle/vextendsingle M=0andM(0), respectively, then (1.9) may be written asCAUSALITY: The Nature of Everything – N. T. Anh 7 the form  ...... · · · A(0)(M)/vextendsingle/vextendsingle M=0/summationtext hA(+1) h(M)/vextendsingle/vextendsingle/vextendsingle M=0· · · · · ·/summationtext hA(−1) h(M)/vextendsingle/vextendsingle/vextendsingle M=0A(0)(M)/vextendsingle/vextendsingle M=0· · · ...... = ...... · · · M(0)/summationtext ha+ hM(+1) h· · · · · ·/summationtext ha− hM(−1) h M(0)· · · ...... . Namely, f!(M)|M=0=M′, (1.10) where the factors aare degree-of-freedom transfer coefficients. Therefrom, we have the following equation, from Eq. (1.5), f!(M) =M′M. (1.11) This equation is named universal equation, where each eleme nt is of the form A(M) =eΣaM/hatwiderMM. The two spaces kiandκiin the respective sets KandKare different from each other, but the κispace lies in the kispace. Id est, these two spaces differ mutually but they conce rn directly with each other through limiting of integral sum, so they annihil ate mutually generating the contradiction between them [ki−κi] =Mi, where “ −” is signed annihilation operation between two spaces kiandκi. The dimension of contra- diction depends on the dimension of spaces kiandκiwhich contribute to create that contradiction. Thus, (1.9) means that A(0)(k)/vextendsingle/vextendsingle κat the point space κof the set Kis just the contradiction [k−κ] =Mbetween the absolute space kof the set Kand the point space κof the set Kcontained inK. For the reason that A(0)(k)/vextendsingle/vextendsingle κreflects right the contradiction between the κspace for which this function characterizes and the kspace, characterized by A(k) that is rather conservative than A(k)|κ(in immanent contradiction). Hence, partial derivatives of A(k)|κto all orders with respect to all possible quantities in poin t spaces κiof the set Kare just partial derivatives of the contradiction Mto all orders with respect to all possible degrees of freedom in spaces (i.e. with respect to degrees of transfer of contradiction), withCAUSALITY: The Nature of Everything – N. T. Anh 8 abeing the degree-of-freedom transfer coefficients. And then , the derivatives M′of contradiction with respect to contradiction-transfer facilities become the expedients for solving of contradiction /hatwiderM. InM, the unit elements in the main diagonal form a reflecting mirr or demarcating the two worlds. When an element is generated in one world, a some elem ent in the other is simultaneously annihilated, and conversely. (they are conserved of degree ). The two reflecting worlds seem to be alike but if an arbitrary element of the one unites some arbit rary element of the other, they both will annihilate mutually to zero, Eq. (1.6). The annihilati on to zero from two some elements of the two reflecting worlds is to satisfy the condition (1.3). Each element of one world will become a some element of the oth er only after it has passed the reflecting mirror, the reflecting mirror has annihilated all natures of old world in this element so that it can receive new natures when it enter into a new world. 5!Let us now generalize all above concepts. If the universe com prises a set of actions, thereat we can regard that Kis set of actions which exist in all absolute spaces and Kis the set of actions which exist in all point spaces. f!( ) is the state function of the zero action, f!(K) is the action function of absolute spaces, and f!(K)|Kis the action function of point spaces. They satisfy the condition (1.1), with “0” being “zero action”, and “1” being the action of its unique existence. A some contradiction, thereat, will be defined as the simulta neous coexistence of two mutual annihilation actions, Mi= [ki−κi], k i∈Kandκi∈ K, and satisfies Eq. (1.6). From the condition of the uniqueness of the zero action, we ob tain the universal equation (1.11) where M′contains elements as derivatives of contradictions with re spect to all possible degrees of freedom. On the other hand, they act onto contradictions m aking contradictions to vary with respect to some degrees of freedom – i.e. making contradicti ons to be solved over some facilities for transfer of contradiction, so derivatives of contradictio ns with respect to all degrees of freedom are just contradiction-solving expedients over all contradic tion-transfer facilities. The universal equation manifests the birth, evolution and c onclusion of the universe, beginning from the absolute vacuum and then its expanding out the whole of the universe to infinitely many repeat period chains. The universal equation expresses jus t solving an infinitely great immanent contradiction of the absolute vacuum state by dividing into infinitely many small contradictions to solve them.CAUSALITY: The Nature of Everything – N. T. Anh 9 In the absolute vacuum there are infinitely many point spaces , solving contradictions between the vacuum and point spaces makes point spaces concentrate a nd crystallized together (accretion) forming early matters. They combine together, and continuo us so, gradually forming the world in a unity entity of the nature. If speaking in the geometrical language, contradictions be tween the zero curvature of the absolute vacuum (or of a sphere with an infinitely large radius) and infi nitely high curvatures of point spaces (or of spheres with infinitely small radii) must vary to reach sta tes with lowest immanent contradictions, i.e. curvatures must vary to reach and conclude some curvatu res (following tendency). The curvature of the absolute vacuum continuously conserves and is equal t o zero, so these contradictions always are solved with respect to the tendency of decreasing curvature s of point spaces to zero (i.e. increasing their radii to infinity) so as to coincide with the curvature o f the absolute vacuum. As curvatures of point spaces decrease just as point spaces concentrate and c rystallize together to form new spaces with lower curvatures (i.e. with larger radii). Concreting that at each point in the absolute vacuum there is an appearance of infinitely many centripetal flows (inflow). Isophasic flows force to appear an impulse at this point, immediately, that impulse causes an immanent contradiction and the contr adiction is solved directly by emitting backward flows (outflow) to decrease strength of inward flows. Then after that, emitter is weak gradually and a new cycle will begin, etc. In the absolute vacuum such infinitely many point spaces crea te a global motion of the universe generating infinitely many thermal currents everywhere in t he universe. The universe becomes full of life. Motion of infinitely many point spaces leads to the form ation of groups separated by isophasic current points. (Maybe the opposite phase groups would crea te the positive and negative extrema or mutually symmetrical extrema. And points which have inte rmediary phases would disperse to form different groups then move gradually to one of extrema). Each group creates one proper wave of flows giving rise to the universe to “breathe” lively. In the universal equation, the sign “=” has the meaning that: at the debut of each expansion pro- cess, the equation performs generally in the forward direct ion (rightwards), after that the equation is in relative equilibrium when there are new contradiction s generated and there also are contra- dictions solved, finally the equation performs generally in the backward direction (leftwards) when contradictions are solved more and more – at that time the exp ansion series converges uniformly to f!(K) =f!( ) = f!. Each numerical value under contradictions is determined by reflecting them onto the quantity 0! of the absolute vacuum, and as contradictions vary the numer ical values also alter after. The state function thus expands into chains in the reversible process , contradictions are generated gradually,CAUSALITY: The Nature of Everything – N. T. Anh 10 and quantities also gradually appear more and more. In eleme nt equations, when expanding sums there will be chains of terms, each chain is a period, later ch ain is in higher development degree than sooner chain. In a few real concrete cases, the expansion of the equation re quires strictly to determine superior limits of sums – i.e. to obey the rule of filling “reservoirs” o f the universe. If the equation expands from small contradictions to great contradictions, then wh en a some contradiction which corresponds with a superior limit index of a sum is greater than other cont radiction which corresponds with an inferior limit index of a successive sum, the smaller contra diction will be expanded foremost. This is similar to the expansion with respect to levels of energy m inimization of elements in the periodic system. Thus, all phenomena in the universe seem to proceed in an acco mplished order and the whole universe continuously obeys a law. A some term of series does not really open out yet but it has denoted a progress orientation of future expansion. About application, above all the universal equation is used for deriving directly the equation of causality, after that for building equations of many quanti ty sets in each scope of a researched space and for researching constructions and systems of worlds in t he universe. 3 EQUATION OF CAUSALITY We now research the reciprocal reflection from the space worl d or the world of actions to the time world. The time world stands of a fundamental background or fi eld, onto which other worlds all reflect as a whole. Therefore the time world is relatively ind ependent, and other worlds seem to be timeworld-dependent. In the time world the process of varia tion and transformation also is performed and formed from the functions f! andf!( ) and expanded into series, finally reaches the homogeneou s state and closes a reflection period. As the world of actions reflects on the time world, the set of ac tionsKis dependent on the set of times T={...⊤...}which exist in absolute spaces, and Kis dependent on T={...τ...}in point spaces. Thereby, f!( ) reflects on time forming the zero-action function of the a bsolute vacuum in the non-time state, f!(K) becomes the function f!(K(T)); and f!(K)|Kgrows f!(K(T))/vextendsingle/vextendsingle K(T). Similarly, in the time world the function f!(K(T)) also is expanded into series by linear transfor- mation from the time TtoT, f!(K(T)) =f!(K(T))/vextendsingle/vextendsingle K(T)ST T, (2.1)CAUSALITY: The Nature of Everything – N. T. Anh 11 or may be written (in the time inflection)  ...... ..A(0)(k(⊤))/summationtext hA(+1) ⊤h(k(⊤)).. ../summationtext hA(−1) ⊤h(k(⊤))A(0)(k(⊤)).. ...... = ...... .. A(0)(k(⊤))/summationtext hA(+1) ⊤h(k(⊤))/vextendsingle/vextendsingle/vextendsingle κ(τ).. ../summationtext hA(−1) ⊤h(k(⊤))/vextendsingle/vextendsingle/vextendsingle κ(τ)A(0)(k(⊤)) .. ......  × ...... ... 1/summationtext h/integraltext⊤− h τ− hdt− h... .../summationtext h/integraltext⊤+ h τ+ hdt+ h 1 ... ...... . If instead of K(T)we write [ K(T)− K(T)], and after putting/integraltext⊤i τidti= ∆ti, ST T= Υ, withST TST T= ΥΥ−1=Iwhere ∆t+∆t−= ∆t−∆t+= 0, (2.2) or  ...... ... 1/summationtext h/integraltext⊤− h τ− hdt− h... .../summationtext h/integraltext⊤+ h τ+ hdt+ h 1 ... ...... = ...... ... 1/summationtext h∆t− h −1!... .../summationtext h∆t+ h +1!1 ... ...... , then Eq. (2.1) will be f!(M(Υ)) =f!(M(Υ))/vextendsingle/vextendsingle M(Υ)=0Υ, (2.3) where each element has the functor form, A(M(t)) =eΣ∆t/hatwide· AA(M(t))/vextendsingle/vextendsingle M(t)=0. From the unique condition of the zero action, similar to the p revious section, we obtain A(0)(M(t))/vextendsingle/vextendsingle M(t)=0=M(0) (t)(2.4)CAUSALITY: The Nature of Everything – N. T. Anh 12 in the time world. After direct and reverse partial differentiating to all orde rs both of the side hand of this element with respect to time, respectively, we obtain f!(M(Υ))/vextendsingle/vextendsingle M(Υ)=0=· M. (2.5) And we can express it under the form  ...... .. A(0)(M(t))/summationtext hA(+1) th(M(t))/vextendsingle/vextendsingle/vextendsingle M(t)=0.. ../summationtext hA(−1) th(M(t))/vextendsingle/vextendsingle/vextendsingle M(t)=0A(0)(M(t)) .. ...... = ...... .. M(0)/summationtext h(−)+1M(+1) th.. ../summationtext h(−)−1M(−1) thM(0).. ......  where coefficients ( −) arise from the reason that contradiction varies inversely as time. Thus, we have f!(M(Υ)) =· MΥ. (2.6) This equation is named time-world equation, where each elem ent is of the form A(M(t)) =eΣ(−)∆t/hatwider· MM(t), with the contradiction between the action k(⊤)of the set K(T)and the action κ(τ)of the set K(T) being Mand also dependent on the time t:M(t)in the set M(Υ). Therefore, all-order partial derivatives of the function A(0)(k(⊤))/vextendsingle/vextendsingle κ(τ)with respect to all times in point spaces are also just all-order partial derivatives of contradiction with respect to all times, with coefficients ( −) generated by reflecting from the world of actions to the time world. Therefrom, the derivative· Mof contradiction with respect to time becomes just the contr adiction-variation rapidity (violence)/hatwider· M. Conforming to the life-like mathematics, after sinking the universal equation into the time-world equation we obtain a general universal equation, f!(M(Υ)) =M′M=· MΥ. (2.7) Elements of the universal equation unite respective elemen ts of the time-world equation, where each functor element has the form A(M(t)) =eΣaM(t)/hatwiderMM(t)=eΣ(−)∆t/hatwider· MM(t).CAUSALITY: The Nature of Everything – N. T. Anh 13 Consider an arbitrary contradiction in M, we obtain aM′M=−· M∆t, (2.8) withM′∈ M′and· M∈· M. From this equation, we identify that ais not only to be a degree-of-freedom transfer coefficient but also to be a world transfer coefficient. In this case it is fr om the space world to the time world. Supposing that ( M′M) reflects from the space world to the time world, (M′M)k∼(M′M)t. If considering ∆ tas a non-variable contradiction or a varied but invariable c ontradiction, then as (M′M)t=−· M∆t, the equation aM′M=−· M∆tis evident due to the result of reflection M′M∼−· M∆t. And therefore aare generated to transfer worlds. When taking ∆ tintoathen we will obtain the equation of causality, aM′M=−· M. (2.9) In the case the contradiction is characterized by itself, na melyM=M(M), then M=M0e−a(t−t0), where M0is the contradiction at the time t=t0. We next consider a case that if ∆ t= [⊤−τ] = 0 – i.e. ⊤=τ, the time in the universe is identical in all spaces – thereat the time includes homogeneousness: i t has an identical value everywhere at an arbitrary point time of the universe, in the universe ther e is no deflection of time at one place and other, at one space and other. According to this significance, the time is a relatively inde pendent world, so whenever there is a time deflection then this deflection will must be solved to re ach a lowest possible deflection – i.e. following the tendency ∆ t= [⊤ −τ]→0. The time world stands of a background or a field, in which every thing, every system, every phenomenon and every state, etc. all reflect, and through whi ch to “know” differences in their motive process, to “know” differences between them and other s and to find ways and tendencies for solving.CAUSALITY: The Nature of Everything – N. T. Anh 14 Thus, if ∆ t= [⊤ −τ] = 0, in the time world there is no contradiction, then Υ will b e equal to the unit I. Therefrom, f!(M(Υ)) =· MI. And after soaking the universal equation into it, f!(M(Υ)) =M′M=· MI, with each functor element, A(M(t)) =eΣaM(t)/hatwiderMM(t)=M(t). (2.10) 4 CONCLUSION We have answered almost very difficult and mysterious questio ns of the universe. Why do everything, every phenomenon, every system, every state, every process and so on all perform and exhibit as what we know but not perform and exhibit otherwise? Because, in ve ry deep of things, phenomena and processes there are different elements to annihilate mutual ly appearing contradictions between them. Elements annihilate mutually leading gradually to do not an nihilate any more. This requirement makes them appear expedients for solving of contradiction a nd facilities for transfer of contradiction in the tendency of decreasing gradually contradictions bet ween mutual annihilation elements. And owing to variation of these contradictions as well as of diffe rent elements in things, phenomena and processes, in the universe there is an appearance of laws, in which everything, phenomenon, process and so on all perform and exhibit just as that we see in the natu re, and anywhere in the universe. Then, why do only different elements play an important role fo r performance and exhibition of everything, phenomenon and process but do not any other elem ents? Because the difference is the first axiom of the universe. If there were no existence of diffe rence, then in the universe everywhere all would be identical, and therefore none of anything would exist. For that reason, only different elements are the first elements and play an important role for exhibition and variation of things, phenomena and processes in the universe. If so, then how is th e universe in variation and in motion? Variation processes in the universe could not arise from con densed matter and then create a big-bang generating particles, material bodies, etc. and m aking the universe expand... Because the outset state of the universe must be a state without immanent difference, or more exactly, with a least quantity of immanent differences. If matter were condensed, then it would comprise infinitely many different elements, therefore condensed state is impossibl e to be the basic state. And if condensed matter is identical, then in the universe there will not be th e existence of a big-bang but matter willCAUSALITY: The Nature of Everything – N. T. Anh 15 must be varied in the expansion phase of the universal equati on. In reality, in the early universe infinitely many point spaces create infinitely many “big-ban gs” which do not explode out but burst into center of each point. Therefore, the universe varies and transforms in the proces s f!→f!( ) = 0 →f!(K)→ ∞ → (1)→f!. In the debut, f! is equal gradually to zero everywhere forming itself and f!( ), thereby the time world also is gradually formed and contributes to crystalli ze the space world. The process from f! to infinity is performed as in the universal equation. After t hat, because every contradiction solved all reach a state without difference – i.e. everywhere, every thing are all identical, and finally lead to a homogeneous state. At that time the process from infinity to (1) is performed. The end is the process of f!-zing everywhere, closing a motion period. Briefly, the whole universe is a unity entity in causal connec tions, all phenomena seem to progress on intrinsic order following the universal equation and the whole universe continually obeys a law – the law of causality. From observations in the microcosm as w ell as in the macrocosm science always discovers deterministic “disorders” of the universe which are controlled by a some “intelligence” – due to contradiction. Appendices A. Functor We identify that a function f(K) expanded into a Taylor series has the following form (withd dK/vextendsingle/vextendsingle k≡/hatwidef) f(K) =e(K−k)/hatwideff(k), or f(K) =e/integraltextK k/hatwidef(κ)dκf(k). This function is called functor. It is clear that the functor may be sought from the equation of causality /hatwideff=∂f ∂K.CAUSALITY: The Nature of Everything – N. T. Anh 16 For the multiplet series the functor f(K1, K2, ...) has the form as f(K1, K2, ...) =eΣi(Ki−ki)/hatwidefif(k1, k2, ...), or as a functional f(K(z)) =e/integraltextdz δK (z)/hatwidef(k,z)f(k(z)). This function also may be sought from the equation of causali ty /hatwideff(K) =δf(K) δK(z). Briefly, an arbitrary function can expand into the Taylor con vergent series then also can be written as a functor and it satisfies the equation of causalit y. In reality, every function can expand into the Taylor series if we recognize the existence of all ze ro-valued, finite-valued and infinite-valued derivatives. At that time, in the series there is no remainde r and the series has infinite order number. Thus, every phenomenon can be described in the form of functo r, and variation and motion (i.e. every process) of phenomena then can be expressed in the equa tion of causality. B. Life-like Mathematics The life-like mathematics is a new mathematical phrasing, i n which objects and quantities, etc. always are in motion. The life-like mathematics describes r elationships of equation-of-causality plu- rality and “living” of the universal equation, etc. Therewi thal, the life-like mathematics comprises not only pictures of the nature but also abstract quantities with copious proper lifetimes reflecting mutually in a unified common lifetime. 1! In the space of actions Kthere exists a set of free actions K={K1, K2, ..., K n, ...}. a. If actions have not any common thing and they are alike, then they are independent mutually K1, K2, ..., K n, ...≡K1, K2, ..., K n, ... b. If actions have one element or many common elements, alike e lements between them, then they connect mutually by annihilation operations “ −”. If those actions are different from each other, then they annihilate mutually generating contradictions:CAUSALITY: The Nature of Everything – N. T. Anh 17 1.Ki−Ki− · · · − Ki≡Ki. 2. If actions annihilate directly, then K1−K2− · · · − Kn=M, withnbeing less than number of actions in the space K. 2! In the operator space M′there exists a set of free action operators: M′={M′ 1, M′ 2, ..., M′ n, ...} so that: a. In the case where contradiction is characterized by itself M=M(M), then M′ (M)≡M(0)= 1. b. As operators act simultaneously onto contradiction: M′ i+M′ i+· · ·+M′ i≡M′ i, M′ 1+M′ 2+· · ·+M′ n≡M′, withnbeing less than number of operators in the space M′. IfM′ 2=M′ −1, then M′ 1+M′ 2=M′ 1+M′ −1= (0), (because they eliminate mutually.) c. As operators act onto contradiction in order: M′ iM′ i...≡M′′··· i, M′ 2M′ 1...≡M′ 2M′ 1..., M′ 2M′ 1...=M′ −1M′ 1= (0) if M′ 2=M′ −1, (because M′ 1andM′ −1eliminate mutually.) 3! Selection Rule.CAUSALITY: The Nature of Everything – N. T. Anh 18 a. As the two spaces of actions K1andK2sink into each other: each action in one space of actions searches and selects actions among appropriate actions in o ther space in order to unite mutu- ally when between them there are common elements (if not, the y have mutually independent tendency) and then forming contradictions in a new space M: |K1− K2|=M. Writing so means that it has eliminated actions which do not c ontribute to create contradictions. b. As the operator space M′sink into the space M: each some contradiction Miin the space M searches and selects necessary operators (e.g. M′ i) among appropriate operators in the operator space M′so that those contradictions are solved with respect to tran sfer facilities (degrees of freedom) which chosen operators have contained (if operato rs are not necessary, contradictions will not choose them). These make the space Mhave contradiction parts to be solved, that solving obeys the equation of causality and contradictions decrease gradually to reach a lowest contradiction state. aM′ iMi=−· Mi. Finally, the space Mreaches and concludes at a new space of actions K3. c. As the two operator spaces M′ 1andM′ 2sink into each other: then operators act onto each other obeying the rule M′ 2M′ 1=M′ 1M′ 2, where M′ 1∈ M′ 1andM′ 2∈ M′ 2. d. As the two spaces ( M′M)1and (M′M)2sink into each other: then the two spaces M1and M2sink into each other, and simultaneously the spaces M′ 1andM′ 2sink into the spaces M1 andM2. 4! Action Group. There exist actions interacting together under the annihil ation opearation ⊖and action operators under the successive action operation ·and simultaneous action operation ⊕. They obey the following laws:CAUSALITY: The Nature of Everything – N. T. Anh 19 a. In the space of actions {K}: there exist actions K+and anti-actions K−; [K±⊖K∓] = 0 ,(existence of anti-actions) [0⊖K±] = K±,(existence of unite 0) [K± i⊖K± j] = K± k∈ {K},(algebra) [K± i⊖K± j] = [ K± j⊖K± i],(permutation) [[K± i⊖K± j]⊖K± k] = [K± i⊖[K± j⊖K± k]] = [[K± k⊖K± i]⊖K± j]. b. In the space of action operators {/hatwiderM}: there exist action operators M+′and anti-action operators M−′; M+′·M−′=M−′·M+′= 1,(existence of anti-action operators) M+′⊕M−′=M−′⊕M+′= 1, M′·M′ (M)=M′ (M)·M′=M′⊕M′ (M)=M′ (M)⊕M′=M′, (existence of unite M′ (M)=M(0)= 1) M′⊕M′=M′;M′ i⊕M′ j=M′ j⊕M′ i=M′ k∈ {/hatwiderM}; M′·M′=M′′;M′ iM′ j=αjiM′ jM′ i, α ij=α−1 ji. Briefly, there are many more actions then many more such kinds of contradictions and derivatives of contradictions and also the same for the equation of causa lity. Depending on concrete problem, the annihilation operation may be a vectorial product, or may be a derivative limit, etc. and the degrees of freedom may be a rotation angle, may be coordinates, or also may be the time, etc. and similar to the world transfer coefficient may be a definite value depending upon reflecting from one world to other in that conc rete problem. Acknowledgments We would like to thank Dr. D. M. Chi for useful discussions and valuable comments. References [1] D. M. Chi, The Equation of Causality , (Vietnamese, 1979), see http://www.mt-anh.com-us.com.

arXiv:physics/9912009v1 [physics.flu-dyn] 3 Dec 1999Spinning jets J. Eggers1and M. P. Brenner2 1Universit¨ at Gesamthochschule Essen, Fachbereich Physik , 45117 Essen, Germany 2Department of Mathematics, MIT, Cambridge, MA 02139 A fluid jet with a finite angular velocity is subject to centrip etal forces in addition to surface tension forces. At fixed angular momentum, centripetal forc es become large when the radius of the jet goes to zero. We study the possible importance of this obs ervation for the pinching of a jet within a slender jet model. A linear stability analysis shows the mo del to break down at low viscosities. Numerical simulations indicate that angular momentum is ex pelled from the pinch region so fast that it becomes asymptotically irrelevant in the limit of th e neck radius going to zero. I. INTRODUCTION A fluid jet emanating from a nozzle will become unstable and br eak up due to surface tension. Some 30 years ago, a series of papers [1–4] investigated the modifications Rayl eigh’s classical analysis would undergo if the jet performe d a solid-body rotation. Such a rotation is easily imparted by spinning the nozzle at the appropriate frequency. The somewhat surprising result of the linear analysis is that th e rotation always destabilizes the jet, a wavenumber kbeing unstable if 0<kr 0<(1 +L−1)1/2, (1) with L=γ/(ρΩ2r3 0). (2) Hereγis the surface tension, ρthe density, Ω the angular frequency, and r0the unperturbed jet radius. Note that Ω appears in the denominator, so no rotation corresponds to L→ ∞, for which the stability boundary 0 <kr 0<1 found by Plateau is recovered. The theoretical growth rates were f ound to be in reasonable agreement with experiment [4] and growth of disturbances for kr0larger than 1 was confirmed. Recently it was pointed out by Nagel [5] that rotation might h ave an even more dramatic effect for the highly nonlinear motion near the point where the neck radius goes to zero. Assume for the sake of the argument that a cylinder of fluid of length wpinches uniformly, i.e. it retains its cylindrical shape. T hen the total angular momentum is M=π 2ρΩr4 0w, (3) and the volume V=πr2 0wis constrained to remain constant as r0goes to zero. The total interface pressure corre- sponding to the outward centripetal force is found to be pc=ρr2 0Ω2/2, and thus pc= 2M2/(V2ρr2 0). (4) Asr0goes to zero, this outward pressure will dominate the surface tension pressure γ/r0, raising the possibility that rotation is a singular perturbation for pinching: an arbitr arily small amount of angular momentum, caused by a symmetry breaking, could modify the breaking of a jet. However, a jet does not pinch uniformly, but rather in a highl y localized fashion [6]. If the above argument is applied to pinching, it must correspond to a rapidly spinning thread of fluid surrounded by almost stationary fluid. Frictional forces represented by the viscosity of the fluid will lead to a diffusive transport of angular momentum out of the pinch region, thus reducing its effect. Determining which effect do minates requires a fully nonlinear calculation, including effects of surface tension, viscosity, inertia, and centrip etal forces. In the spirit of earlier work for Ω = 0 [7,8] we will derive a one dimensional model, which only takes into ac count the leading order dependence of the velocity field on the radial variable. This will be done in the next section, together with a comparison of the linear growth rates between the model and the full Navier-Stokes calculation. I n the third section we analyze the nonlinear behavior. First we investigate possible scaling solutions of the mode l equations, then we compare with numerical simulations. In the final section, we present some tentative conclusions. 1II. THE MODEL In our derivation of the slender jet model we closely follow [ 7]. The Navier-Stokes equation for an incompressible fluid of viscosity νread in cylindrical coordinates: ∂tvr+vr∂rvr+vz∂zvr−v2 φ/r=−∂rp/ρ+ν(∂2 rvr+∂2 zvr+∂rvr/r−vr/r2), ∂tvz+vr∂rvz+vz∂zvz=−∂zp/ρ+ν(∂2 rvz+∂2 zvz+∂rvz/r), ∂tvφ+vr∂rvφ+vz∂zvφ+vrvφ/r=ν(∂2 rvφ+∂2 zvφ+∂rvφ/r−vφ/r2), with the incompressibility condition ∂rvr+∂zvz+vr/r= 0. Here we have assumed that the velocity field does not depend on the angleφ. Exploiting incompressibility, vzandvr can be expanded in a power series in r: vz(z,r) =v0(z) +v2(z)r2+... (5) vr(z,r) =−v′ 0(z) 2r−v′ 2(z) 4r3−.... Here and in the following a prime refers to differentiation wi th respect to z. The crucial trick to make an expansion in rwork in the presence of rotation is to rewrite vφ(z,r) in terms of the angular momentum per unit length ℓ(z) of the corresponding solid body rotation: vφ(z,r) =2ℓ(z) πρh4(z)r+br3+.... (6) Hereh(z) is the local thread radius, hence no overturning of the profi le is allowed. Just as without rotation, the equation of motion for h(z,t) follows from mass conservation based on the leading order e xpression for vz: ∂th+v0h′=−v′ 0h/2. (7) Finally, the pressure is expanded according to p(z,r) =p0(z) +p2(z)r2+.... (8) Plugging this into the equation of motion for vr, to leading order in rone finds the balance p2=2ℓ2 π2ρh8, (9) while the leading balance for the vz-equation remains ∂tv0+v0v′ 0=−p′ 0/ρ+ν(4v2+v′′ 0). (10) Lastly, the vφ-equation leads to ∂tℓ+ℓv′ 0+ 4ℓv0h′/h+v0h4(ℓ/h4)′=νh4(4πρb+ (ℓ/h4)′′) (11) to leading order. Equations (9)-(11) contain the unknown functions p0,v2, andbwhich need to be determined from the boundary conditions. The normal stress balance nσn=γκgives p0+p2h2=γκ−v′ 0, whereκis the sum of the principal curvatures. As in the case without rotation, the tangential stress balance nσt= 0 gives 2−3v′ 0h′−v′′ 0h/2 + 2v2h= 0 fortpointing in the axial direction, but a new condition πρhb=h′(ℓ/h4)′ fortpointing in the azimuthal direction. Putting this together , one is left with a closed equation for h,v0, andℓ: ∂th+vh′=−v′h/2 (12) ∂tv+vv′=−γ ρκ′+2 ρ2π2(ℓ2/h6)′+ 3ν(v′h2)′/h2 ∂tl+ (vl)′=ν(h4(ℓ/h4)′)′, where we have dropped the subscript on v0. The same equations were derived independently by Horvath a nd Huber [10]. The most obvious way to test this model is to compare with the k nown results for the stability of the full Navier- Stokes equation. To that end we linearize (12) about a soluti on with constant radius r0and rotation rate Ω: h(z,t) =r0(1 +ǫeωtcos(kz)) (13) v(z,t) =−2ǫω keωtsin(kz) ℓ(z,t) =π 2ρΩr4 0(1 +ǫeωtαcos(kz)). Eliminating α, this leads to the equation ¯ω3+4¯k2 Re¯ω2+¯k2 2(−1 +¯k2+L−1+ 6¯k2/Re2)¯ω+¯k4 2Re(−1 +¯k2−L−1) = 0, (14) where ¯k=kr0and ¯ω=ω(ρr3 0)1/2/γ1/2are dimensionless. We have introduced the Reynolds number Re= (γr0)1/2/(ρ1/2ν), based on a balance of capillary and viscous forces. Note th at this convention differs from that of [4]. Putting ¯ ω= 0 one reproduces the exact stability boundaries (1). Howev er one can see that the inviscid limit Re→ ∞ is a very singular one, in disagreement with the full solutio n. Namely, for this limit one finds the three branches ¯ω2 1/2=¯k2 2(1−¯k2−L−1),¯ω3=Re−1¯k2(1−¯k2+L−1) ¯k2−1 +L−1. (15) Thus ¯ω3is the only unstable mode in the range 1 −L−1<¯k2<1+L−1, but goes to zero when the viscosity becomes small. The reason for this behavior, which is not found in the solution of the full equations, lies in the appearance of a very thin boundary layer for small viscosities [3]. Namely , Rayleigh’s stability criterion for a rotating fluid implie s that the interior of the fluid is stabilized . This forces any disturbance to be confined to a boundary laye r of thickness δ=ω 2Ωk near the surface of the jet, and δbecomes very small for ¯k≈1. But this additional length scale is not captured by our slender jet expansion. Only for high viscosities is the b oundary layer smoothed out sufficiently, and from (14) one finds the dispersion relation ¯ω=Re 6(1−¯k2+L−1), (16) which is consistent with the full theory in the limit of small ¯k. III. NONLINEAR BEHAVIOR Our main interest lies of course in the behavior close to pinc h-off. Close to the singularity, one expects the motion to become independent of initial conditions, so it is natura l to write the equations of motion in units of the material 3parameters of the fluid alone. In addition to the known [9] uni ts of length and time, ℓν=ν2ρ/γandtν=ν3ρ2/γ2, one finds an angular momentum scale ℓ0=ν5ρ2/γ2. Note that this scale is only about 1 .9·10−14g cm/s for water, corresponding to a frequency of 2 ·10−11s−1for a 1 mm jet, so even the smallest amount of rotation will be p otentially relevant. Rewriting the equations of motion (12) in units of ℓν,tν, andℓ0, one can effectively put ρ=ν=γ= 1, leading to a universal form of the equations, independent of the type of fluid. In addition, one can look for similarity solutions [11] of th e form h(z,t) =t′α1φ(z′/t′β) (17) v(z,t) =t′α2ψ(z′/t′β) ℓ(z,t) =t′α3χ(z′/t′β), wheret′=t0−tandz′=z−z0are the temporal and spatial distances from the singularity wherehgoes to zero. We have assumed that everything has been written in units of t he natural scales ℓν,tν, andℓ0. By plugging (17) into the equations of motion, and looking for solutions that bala nce thet′-dependence, one finds a unique set of values for the exponents: α1= 1, α 2=−1/2, α 3= 5/2, β= 1/2. In addition, one obtains a set of three ordinary differential equations for the similarity functions φ,ψ, andχ. So far we have not been able to find consistent solutions to these equ ations, which match onto a solution which is static on the time scale t′of the singular motion. This is a necessary requirement sinc e the fluid will not be able to follow the singular motion as one moves away from the pinch point. This negative result is consistent with simulations of the e quations for a variety of initial conditions. To avoid spurious boundary effects, we considered a solution of (12) w ith periodic boundary conditions in the interval [ −1,1] and an additional symmetry around the origin. This ensures t hat the total angular momentum is conserved exactly. We took the fluid to be initially at rest and the surface to be hinit(z) =r0(1 + 0.3 cos(2πx)), (18) withr0= 0.1. The angular momentum was distributed uniformly with the i nitial value ℓinit, corresponding to L=π2 4γρr5 0 ℓ2 init. Figures 1 and 2 show a numerical simulation of (12) with Re= 4.5 andL= 0.25 using a numerical code very similar to the one described in [12]. Written in units of the i ntrinsic angular momentum scale, ℓinit/ℓ0= 6·103, so ℓis potentially relevant. The thread pinches on either side o f the minimum, pushing fluid into the center. As seen in the profiles of ℓ, the angular momentum is expelled very rapidly from the regi ons wherehgoes to zero and also concentrates in the center. This is confirmed by a plot of the m inimum ofℓversus the minimum of h. On the basis of the similarity solution (17), a power law ℓmin∝h5/2 minis to be expected. Instead, Fig. 3 shows that ℓmindecays more rapidly, making angular momentum asymptotically irreleva nt. Indeed, a comparison of the similarity function φas found from the present simulation shows perfect agreement w ith the scaling theory in the absence of rotation [11]. The behavior of ℓminshould in principal be derivable from the linear equation fo rℓwith known time-dependent functions h(z,t) andv(z,t). Unfortunately, ℓmindoes not seem to follow a simple scaling law except below h= 3·10−4, where the power is close to 3.13. It is not clear how to extract this p ower analytically. One might think that by increasing the angular momentum the s ystem would cross over to a different behavior. To test this, the initial angular momentum was doubled to give L= 0.0625. AtL= 0.5 centripetal and surface tension forces are balanced, so decreasing Lsignificantly below this value will cause rotation to be impo rtant initially. Indeed, instead of pinching down immediately, the fluid is first pulle d into a narrow disc, while the radius of the surrounding fluid remains constant, cf. Fig. 4. Eventually this outward m otion stops, as surface tension and centripetal forces reach an equilibrium. Only then does the fluid pinch down at th e edge of the disc. The behavior close to the pinch point is however exactly the same as for smaller angular mome ntum. As a word of caution, one must add that our model is certainly not valid at the edges of the disk, where sl opes become very large. In fact, the very sharp gradients encountered in this region may be due to the fact that the fluid really wants to develop plumes . As is observed for low viscosity [12], the viscous terms prevent the interface fro m overturning, but at the cost of producing unrealistically sharp gradients. 40.0 0.2 0.4 0.6 0.8 1.0 z0.000.050.100.150.200.250.30 h(z,t) FIG. 1. The height profile in a numerical simulation with Re= 4.5 and L= 0.25. Shown are the initial condition, and the times when the minimum has reached hmin= 10−1.5,10−2, and 10−5, at the end of the computation. 0.0 0.2 0.4 0.6 0.8 1.0 z0.0000.0200.0400.060 l(z,t) FIG. 2. The angular momentum profiles ℓ(z, t) corresponding to Fig. 1. 5−5.0 −4.0 −3.0 −2.0 −1.0 log10(hmin)−15.0−10.0−5.00.0 log10(lmin)5/2 FIG. 3. The minimum value of the angular momentum as function of the minimum height. It is found that ℓmindecreases faster than h5/2 min, which would exactly balance surface tension and centripet al forces. 0.0 0.2 0.4 0.6 0.8 1.0 z0.000.100.200.300.40 h(z,t) FIG. 4. A numerical simulation with twice the angular moment um of Fig. 1. The height profiles are shown in time intervals of 0.05 and at the end of the simulation. Centripetal forces d raw the fluid out into a disc. 6IV. CONCLUSIONS The present investigation is only a first step towards the und erstanding of the role of rotation in droplet pinching. A major challenge lies in finding a description valid at low vi scosities. This can perhaps be done by incorporating the boundary layer structure near the surface into the slender j et approximation. The relevance of this lies in the fact that angular momentum is potentially more important at low v iscosities, when there is less frictional transport out of the pinch region. In fact it can be shown that the inviscid v ersion of (12) does not describe breakup at all, since centripetal forces will always dominate asymptotically. T his result is of course only of limited use since the model equations are definitely flawed in that regime. In addition, there remains the possibility that a region in p arameter space exists where angular momentum modifies breakup even at finite viscosities. We cannot make a definite s tatement since the additional variable makes it hard to scan parameter space completely. Finally, spinning jets have not received much attention in terms of experiments probing the non-linear regime. The discs found at high spinn ing rates (cf. Fig. 4) are a tantalizing new feature, and to our knowledge have not been found experimentally. The low est value of Lreported in [4] is 0.43, which is even larger than the value of Fig. 1. However, 0.0625 would easily be reachable by increasing the jet radius. ACKNOWLEDGMENTS The authors are indebted to Sid Nagel for pointing out this pr oblem and for stimulating discussions. J.E. thanks Howard Stone for his hospitality, which he has shown in so man y ways, and for stimulating discussions. J.E. is also grateful to Leo Kadanoff and the Department of Mathemati cs at the University of Chicago, where this paper was written, for making this another enjoyable summer. M.B. acknowledges support from the NSF Division of Mathematical Sciences, and the A.P. Sloan foundation. J.E. was supported by the Deutsche Forschungsgemeinschaft through SFB237. [1] L. M. Hocking, Mathematika 7, 1 (1960). [2] J. Gillis and B. Kaufman, Q. J. appl. Math 19, 301 (1961). [3] T. J. Pedley, J. Fluid Mech. 30, 127 (1967) [4] D. F. Rutland and G. J. Jameson, Chem. Engin. Sc. 25, 1301 (1970). [5] S. R. Nagel, private communication (1998). [6] J. Eggers, Rev. Mod. Phys. 69, 865 (1997). [7] J. Eggers and T. F. Dupont, J. Fluid Mech. 262, 205 (1994). [8] S. E. Bechtel, M. G. Forest, and K. J. Lin, SAACM 2, 59 (1992). [9] D. H. Peregrine, G. Shoker, and A. Symon, J. Fluid Mech. 212, 25 (1990). [10] G. Huber, private communication (1999). [11] J. Eggers, Phys. Rev. Lett. 71, 3458 (1993). [12] M. P. Brenner et al., Phys. Fluids 9, 1573 (1997). 7

arXiv:physics/9912010v1 [physics.class-ph] 3 Dec 1999ELECTRODYNAMIC FORCES IN ELASTIC MATTER S. ANTOCI AND L. MIHICH Abstract. A macroscopic theory for the dynamics of elastic, isotropic matter in presence of electromagnetic fields is proposed her e. We avail of Gordon’s general relativistic derivation of Abraham’s e lectromagnetic energy tensor as starting point. The necessary description of the elas- tic and of the inertial behaviour of matter is provided throu gh a four- dimensional generalisation of Hooke’s law, made possible b y the intro- duction of a four-dimensional “displacement” vector. As in timated by Nordstr¨ om, the physical origin of electrostriction and of magnetostric- tion is attributed to the change in the constitutive equatio n of elec- tromagnetism caused by the deformation of matter. The part o f the electromagnetic Lagrangian that depends on that deformati on is given explicitly for the case of an isotropic medium and the result ing expres- sion of the electrostrictive force is derived, thus showing how more real- istic equations of motion for matter subjected to electroma gnetic fields can be constructed. 1.Introduction According to a widespread belief cultivated by present-day physicists, gen- eral relativity exerts its sovereign power in the heavens, w here it supposedly rules tremendous astrophysical processes and awesome cosm ological scenar- ios, but it has essentially nothing to say about more down to e arth issues like the physics of ordinary matter, as it shows up in terrest rial laboratories. This way of thinking does not conform to the hopes expressed b y Bernhard Riemann in his celebrated Habilitationsschrift [1]. While commenting upon the possible applications to the physical space of his new ge ometrical ideas, he wrote: Die Fragen ¨ uber das Unmeßbargroße sind f¨ ur die Natur- erkl¨ arung m¨ usige Fragen. Anders verh¨ alt es sich aber mit den Fragen ¨ uber das Unmeßbarkleine1. The latter is presently supposed to be the exclusive hunting ground for quan- tum physics, whose workings occur at their best in the flat spa ce of Newton. The classical field theories, in particular classical elect romagnetism, are be- lieved to have accomplished their midwife task a long time ag o. Although Key words and phrases. Classical field theory. General relativity. Electromagnet ism. Nuovo Cimento, in press. 1The questions about the infinitely great are for the interpre tation of nature useless questions. But this is not the case with the questions about t he infinitely small. 12 S. ANTOCI AND L. MIHICH they are still revered, since they act as cornerstones for th e applied sci- ences, and also provide the very foundation on which quantum mechanics and quantum field theory do stand, no fundamental insights ar e generally expected from their further frequentation. This mind habit cannot subsist without a strenuous act of faith in the final nature of the pres ent-day reduc- tionist programme: since for all practical purposes we have eventually at- tained the right microscopic laws, getting from there the ri ght macroscopic physics should be just a matter of deduction by calculation ( for the ever growing army of computer addicts, a sheer problem of computi ng power). Given time and endurance, we should be able to account for all the observed phenomena just by starting from our very simple microscopic laws! It is not here the place for deciding how much this bold faith i n the capa- bilities of today’s reductionist approach be strenghtened by its undeniable successes, and how much it depend on having tackled just the s ort of prob- lems that are most suited to such a method. However, when conf ronted with the end results of many reductionist efforts, the obdura te classicist cannot help frowning in puzzlement. While he expects to meet with macro- scopic laws derived from the underlying microscopic postul ates by a pure exercise of logic, the everyday’s practice confronts him wi th a much less palatable food. At best he is presented with rather particul ar examples usually worked out from the sacred principles through the su rreptitious ad- dition of a host of subsidiary assumptions. In the worst case s he is forced to contemplate and believe sequels of colourful plots and diag rams, generated by some computing device in some arcane way that he is simply i mpotent at producing again. In the intention of their proponents, both the “analytic” instances and their numerical surrogates should provide ty pical examples of some supposedly general behaviour, really stemming from th e basic tenets of the theory, and in many a case this lucky occurrence may wel l have oc- curred, since “God watches over applied mathematicians” [2 ]. Nevertheless, the longing of the classicist for macroscopic laws of clear c onceptual ancestry that do encompass in surveyable form a large class of phenome na remains sadly disattended. He is led to remind of the pre-quantum era , when both the reductionist approaches and the macroscopic ones were b elieved to be equally important tools for the advancement of physical kno wledge2, and to wonder whether relegating the macroscopic field approach to the “phe- nomenological” dustbin was really a wise move. Before being removed from center stage by more modern approaches, the macroscopic fiel d theory lived long enough for developing, in the hands of great natural phi losophers and mathematicians, theoretical tools of a very wide scope that , if still remem- bered and cultivated, would be recognized to be very useful t oday. 2It is remarkable how the otherwise daring Minkowski kept a ca utious attitude with respect to Lorentz’ atomistic theory of electricity both in his fundamental paper [3] of 1908 and also in his “Nachlass” notes [4], posthumously edit ed and published by Max Born.ELECTRODYNAMIC FORCES IN ELASTIC MATTER 3 2.Electrodynamic forces in material media One of the clearest instances in which recourse to macroscop ic field theory proves to be a quite helpful research tool occurs when one tri es to describe the electromagnetic forces in material media. Since the tim e of Lorentz this has been a very challenging task; reductionist approac hes starting from classical mechanics and from vacuum electrodynamics, for r easons clearly spelled out e. g.by Ott [5], end up in a disappointing gamut of possibilities also when the program of a rigorous special relativistic der ivation is tena- ciously adhered to [6], [7]. To our knowledge, a derivation o f the macroscopic forces exerted by the electromagnetic field on a material med ium performed by availing of quantum electrodynamics as the underlying mi croscopic the- ory, that should be de rigueur in the reductionist programme, has never been undertaken3. Happily enough, the theoretical advance in the methods for p roducing the stress energy momentum tensor of non gravitational fields oc curred with the onset of general relativity theory [11], [12] have allowed W . Gordon to find, through a clever reduction to the vacuum case of the latter th eory [13], a clear-cut argument for determining the electromagnetic f orces in matter that is homogeneous and isotropic in its local rest frame. We shall recall Gordon’s result in the next Section, since extending his out come to the case of an elastic medium is just the scope of the present paper. 3.Gordon’s reductio ad vacuum of the constitutive equation of electromagnetism We adopt hereafter Gordon’s conventions [13] and assume tha t the metric tensor gikcan be locally brought to the diagonal form gik=ηik≡diag(1,1,1,−1) (3.1) at a given event through the appropriate transformation of c oordinates. According to the established convention [14] let the electr ic displacement and the magnetic field be represented by the antisymmetric, c ontravariant tensor density Hik, while the electric field and the magnetic induction are accounted for by the skew, covariant tensor Fik. With these geometrical objects we define the four-vectors: Fi=Fikuk, Hi=Hikuk, (3.2) where uiis the four-velocity of matter. In general relativity a line ar electro- magnetic medium can be told to be homogeneous and isotropic i n its rest frame if its constitutive equations can be written as µHik=Fik+ (ǫµ−1)(uiFk−ukFi), (3.3) 3By availing once more of the midwife abilities of classical fi eld theory, the converse has instead been attempted: some forms of “phenomenologica l” classical electrodynamics in matter has been subjected to some quantisation process [8 ], [9], [10], thereby producing diverse brands of “phenomenological” photons.4 S. ANTOCI AND L. MIHICH where the numbers ǫandµaccount for the dielectric constant and for the magnetic permeability of the medium [13]. This equation pro vides the con- stitutive relation in the standard form: Hik=1 2XikmnFmn, (3.4) valid for linear media [15]. Gordon observed that equation ( 3.3) can be rewritten as µHik=/bracketleftbig gir−(ǫµ−1)uiur/bracketrightbig/bracketleftbig gks−(ǫµ−1)ukus/bracketrightbig Frs. (3.5) By introducing the “effective metric tensor” σik=gik−(ǫµ−1)uiuk, (3.6) the constitutive equation takes the form µHik=√gσirσksFrs, (3.7) where g≡ −det(gik). The inverse of σikis σik=gik+/parenleftbig 1−1 ǫµ/parenrightbig uiuk, (3.8) and one easily finds [13] that σ=g ǫµ, (3.9) where σ≡ −det(σik). Therefore the constitutive equation can be eventually written as Hik=/radicalbiggǫ µ√σσirσksFrs. (3.10) 4.Gordon’s derivation of Abraham’s energy tensor This result is the basis of Gordon’s argument: since, apart f rom the con- stant factor/radicalbig ǫ/µ, equation (3.10) is the constitutive equation for electro- magnetism in a general relativistic vacuum whose metric be σik, the La- grangian density that we shall use for deriving the laws of th e field is: L=1 4/radicalbiggǫ µ√σF(i)(k)Fik−siϕi, (4.1) wheresiis the four-current density, while ϕiis the potential four-vector that defines the electric field and the magnetic induction: Fik≡ϕk,i−ϕi,k. (4.2) We have adopted the convention of enclosing within round bra ckets the indices that are either moved with σikandσik, or generated by performing the Hamiltonian derivative with respect to the mentioned te nsors. The position (4.2) is equivalent to asking the satisfaction of t he homogeneous set of Maxwell’s equations F[ik,m]= 0, (4.3)ELECTRODYNAMIC FORCES IN ELASTIC MATTER 5 while equating to zero the variation of the action integral/integraltext LdΩ with respect toϕientails the fulfilment of the inhomogeneous Maxwell’s set Hik ,k=si. (4.4) In our general relativistic framework, we can avail of the re sults found by Hilbert and Klein [11], [12] for determining the energy tens or of the electro- magnetic field. If the metric tensor of our pseudo-Riemannia n space-time wereσik, Hilbert’s method would provide the electromagnetic energ y tensor by executing the Hamiltonian derivative of the Lagrangian d ensity Lwith respect to that metric: δL≡1 2T(i)(k)δσik, (4.5) and we would get the mixed tensor density T(k) (i)=FirHkr−1 4δk iFrsHrs, (4.6) which is just the general relativistic form of the energy ten sor density pro- posed by Minkowski in his fundamental work [3]. But gik, notσik, is the true metric that accounts for the structure of space-time and, th rough Einstein’s equations, defines its overall energy tensor. Therefore the partial contribu- tion to that energy tensor coming from the electromagnetic fi eld must be obtained by calculating the Hamiltonian derivative of Lwith respect to gik. After some algebra [13] one easily gets the electromagnetic energy tensor: Tk i=FirHkr−1 4δk iFrsHrs−(ǫµ−1)Ωiuk, (4.7) where Ωi≡ −/parenleftbig Ti kuk+uiTmnumun/parenrightbig (4.8) is Minkowski’s “Ruh-Strahl” [3]. Since Ωiui≡0, substituting (4.7) into (4.8) yields: Ωi=FmHim−FmHmui=ukFm/parenleftbig Hikum+Hkmui+Hmiuk/parenrightbig , (4.9) and one eventually recognizes that Tikis the general relativistic extension of Abraham’s tensor [16] for a medium that is homogeneous and isotropic when looked at in its local rest frame. The four-force densit y exerted by the electromagnetic field on the medium shall be given by (minus) the covariant divergence of that energy tensor density: fi=−Tk i;k, (4.10) where the semicolon stands for the covariant differentiatio n done by using the Christoffel symbols built with the metric gik. Abraham’s energy tensor is an impressive theoretical outcome, that could hardly hav e been antici- pated on the basis of heuristic arguments. Quite remarkably , the so called Abraham’s force density, that stems from the four-divergen ce of that tensor, has found experimental confirmation in some delicate experi ences performed by G. B. Walker et al. [17], [18]. Despite this, Abraham’s ren dering of the6 S. ANTOCI AND L. MIHICH electrodynamic forces is not realistic enough, for it does n ot cope with the long known phenomena of electrostriction and of magnetostr iction. We need to find its generalization, and we shall start from consideri ng the case of lin- ear elastic media, to which Hooke’s law applies. This task wo uld be made formally easier if one could avail of a relativistic reformu lation of the linear theory of elasticity; the next Section will achieve this goa l through a four- dimensional formulation of Hooke’s law [19] that happens to be rather well suited to our scopes. 5.A four-dimensional formulation of Hooke’s law By availing of Cartesian coordinates and of the three-dimen sional tensor formalism, that was just invented to cope with its far-reach ing consequences, Hooke’s law “ut tensio sic vis” [20] can be written as Θλµ=1 2Cλµρσ(ξρ,σ+ξσ,ρ), (5.1) where Θλµis the three-dimensional tensor that defines the stresses ar ising in matter due to its displacement, given by the three-vector ξρ, from a supposedly relaxed condition, and Cλµρσis the constitutive tensor whose build depends on the material features and on the symmetry pr operties of the elastic medium. It seems natural to wonder whether this v enerable formula can admit of not merely a redressing, but of a true gen eralization to the four-dimensions of the general relativistic spaceti me. From a formal standpoint, the extension is obvious: one introduces a four -vector field ξi, that should represent a four-dimensional “displacement”, and builds the “deformation” tensor Sik=1 2(ξi;k+ξk;i). (5.2) A four-dimensional “stiffness” tensor density Ciklmis then introduced; it will be symmetric in both the first pair and the second pair of i ndices, since it will be used for producing a “stress-momentum-energy” te nsor density Tik=CiklmSlm, (5.3) through the four-dimensional generalization of equation ( 5.1). It has been found [19] that this generalization can be physically meani ngful, since it al- lows one to encompass both inertia and elasticity in a sort of four-dimensional elasticity. Let us consider a coordinate system such that, a t a given event, equation (3.1) holds, while the Christoffel symbols are vani shing and the components of the four-velocity of matter are: u1=u2=u3= 0, u4= 1. (5.4) We imagine that in such a coordinate system we are able to meas ure, at the chosen event, the three components of the (supposedly sm all) spatial displacement of matter from its relaxed condition, and that we adopt these three numbers as the values taken by ξρin that coordinate system, whileELECTRODYNAMIC FORCES IN ELASTIC MATTER 7 the reading of some clock ticking the proper time and travell ing with the medium will provide the value of the “temporal displacement ”ξ4in the same coordinate system. By applying this procedure to all th e events of the manifold where matter is present and by reducing the collect ed data to a common, arbitrary coordinate system, we can define the vecto r field ξi(xk). From such a field we shall require that, when matter is not subj ected to ordinary strain and is looked at in a local rest frame belongi ng to the ones defined above, it will exhibit a “deformation tensor” Siksuch that its only nonzero component will be S44=ξ4,4=−1. This requirement is met if we define the four-velocity of matter through the equation ξi ;kuk=ui. (5.5) The latter definition holds provided that det(ξi ;k−δi k) = 0, (5.6) and this shall be one equation that the field ξimust satisfy; in this way the number of independent components of ξiwill be reduced to three4. A four-dimensional “stiffness” tensor Ciklmpossibly endowed with physical meaning can be built as follows. We assume that in a locally Mi nkowskian rest frame the only nonvanishing components of Ciklmare:Cλνστ, with the meaning of ordinary elastic moduli, and C4444=−ρ, (5.7) where ρmeasures the rest density of matter. But of course we need defi ning the four-dimensional “stiffness” tensor in an arbitrary co- ordinate system. The task can be easily accomplished if the unstrained matter is isotropic when looked at in a locally Minkowskian rest frame, and this i s just the oc- currence that we have already studied from the electromagne tic standpoint in Sections 3 and 4. Let us define the auxiliary metric γik=gik+uiuk; (5.8) then the part of Ciklmaccounting for the ordinary elasticity of the isotropic medium can be written as [21] Ciklm el.=−λγikγlm−µ(γilγkm+γimγkl), (5.9) where λandµare assumed to be constants. The part of Ciklmthat accounts for the inertia of matter shall read instead Ciklm in.=−ρuiukulum. (5.10) 4The fulfilment of equation (5.5) is only a necessary, not a suffi cient condition for the field ξito take up the tentative meaning that was envisaged above. Th e physical interpretation of the field ξican only be assessed a posteriori from the solutions of the field equations.8 S. ANTOCI AND L. MIHICH The elastic part Tik el.of the energy tensor is orthogonal to the four-velocity, as it should be [22]; thanks to equation (5.5) it reduces to Tik el.=Ciklm el.Slm=−λ(gik+uiuk)(ξm ;m−1) −µ[ξi;k+ξk;i+ul(uiξl;k+ukξl;i)], (5.11) while, again thanks to equation (5.5), the inertial part of t he energy tensor proves to be effectively so, since Tik in.=Ciklm in.Slm=ρuiuk. (5.12) The energy tensor defined by summing the contributions (5.11 ) and (5.12) encompasses both the inertial and the elastic energy tensor of an isotropic medium; when the macroscopic electromagnetic field is vanis hing it should represent the overall energy tensor, whose covariant diver gence must vanish according to Einstein’s equations [11], [12]: Tik ;k= 0. (5.13) Imposing the latter condition allows one to write the equati ons of motion for isotropic matter subjected to elastic strain [22]. We sh ow this outcome in the limiting case when the metric is everywhere flat and the four-velocity of matter is such that uρcan be dealt with as a first order infinitesimal quantity, while u4differs from unity at most for a second order infinitesimal quantity. Also the spatial components ξρof the displacement vector and their derivatives are supposed to be infinitesimal at first or der. An easy calculation [19] then shows that equation (5.6) is satisfied to the required first order, and that equations (5.13) reduce to the three equ ations of motion: ρξν ,4,4=λξρ,ν ,ρ+µ(ξν,ρ+ξρ,ν),ρ, (5.14) and to the conservation equation {ρu4uk},k= 0, (5.15) i. e., to the required order, they come to coincide with the well kn own equations of the classical theory of elasticity for an isotr opic medium. 6.Electrostriction and magnetostriction in isotropic matte r Having provided that portion of the equations of motion of ma tter that stems from the inertial and from the elastic part of the energ y tensor, we can go back to the other side of our problem: finding to what change s the elec- trodynamic forces predicted in isotropic matter by Gordon’ s theory must be subjected in order to account for electrostriction and for m agnetostriction. Driven by a suggestion found in the quoted paper by Nordstr¨ o m [15], we at- tribute the physical origin of the electrostrictive and of t he magnetostrictive forces to the changes that the constitutive relation (3.4) u ndergoes when matter is strained in some way. If one desires to represent ex plicitly the effect of a small spatial deformation on the constitutive rel ation of electro- magnetism, one can replace (3.4) with a new equation, writte n in terms of the new tensor density Yikpqmn, that can be chosen to be antisymmetricELECTRODYNAMIC FORCES IN ELASTIC MATTER 9 with respect to the first pair and to the last pair of indices, s ymmetric with respect to the second pair. This tensor density allows one to rewrite the constitutive relation as follows: Hik=1 2YikpqmnSpqFmn, (6.1) where Fikis defined by (4.2) and Sikis given by (5.2). For the intended application to isotropic matter it is convenient to split th e equation writ- ten above in two terms, one concerning the unstrained medium , that has already been examined in Sections 3 and 4, and one dealing wit h the spatial deformation proper. Due to equation (5.5) one finds 1 2upuq(ξp;q+ξq;p) =upuqξp ;q=upup=−1, (6.2) and the part (3.5) of the constitutive equation valid for the isotropic un- strained medium can be rewritten as: Hik (u.)=−√g µ/bracketleftbig gim−(ǫµ−1)uium/bracketrightbig/bracketleftbig gkn−(ǫµ−1)ukun/bracketrightbig upuqSpqFmn.(6.3) For producing the part of the constitutive equation that dea ls with the effects of the spatial deformation proper, we recall that an a rbitrary de- formation will bring the medium, which is now supposed to be i sotropic when at rest and unstrained, into a generic anisotropic cond ition. When the magnetoelectric effect is disregarded5, the electromagnetic properties of an anisotropic medium can be summarised, as shown e. g. by Sch¨ opf [24], by assigning two symmetric four-tensors ζik=ζkiandκik=κki, whose fourth row and column vanish in a coordinate system in which matter h appens to be locally at rest. This property finds tensorial expression in the equations ζikuk= 0, κikuk= 0; (6.4) ζikhas the rˆ ole of dielectric tensor, while κikacts as inverse magnetic per- meability tensor. Let ηiklmbe the Ricci-Levi Civita symbol in contravariant form, while ηiklmis its covariant counterpart. Then the generally covariant expression of the constitutive equation for the anisotropi c medium reads [24]: Hik=√g/bracketleftbig (uiζkm−ukζim)un−1 2ηikrsurκscudηcdmn/bracketrightbig Fmn. (6.5) We shall avail of this equation to account for Hik (s.), i. e. for the part of Hik produced, for a given Fmn, by the presence of ordinary strain in the otherwise isotropic medium. The tensors ζikandκikwill now be given a new meaning: they represent henceforth only the changes in the dielectri c properties and in the inverse magnetic permeability produced by the presen ce of strain. If the medium, as supposed, is thought to be isotropic when at rest and 5Such an effect indeed exists [23], but it is sufficiently rara avis to be neglected in the present context.10 S. ANTOCI AND L. MIHICH in the unstrained state, the dependence of ζikand of κikonSpqwill mimic the dependence on the four-dimensional deformation tensor exhibited by the elastic stress in an isotropic medium. One shall in fact writ e: ζkm=/bracketleftbig α1γkmγpq+α2(γkpγmq+γkqγmp)/bracketrightbig Spq, (6.6) where the constants α1andα2specify the electrostrictive behaviour of the isotropic medium. In the same way one is led to pose: κkm=/bracketleftbig β1γkmγpq+β2(γkpγmq+γkqγmp)/bracketrightbig Spq (6.7) to account for the magnetostrictive behaviour; β1andβ2are again the ap- propriate magnetostrictive constants for the isotropic me dium. By availing of the definitions (6.6) and (6.7) one eventually writes Hik (s.)=√g/bracketleftbig (uiζkm−ukζim)un−1 2ηikrsurκscudηcdmn/bracketrightbig Fmn (6.8) for the part of Hikdue to the ordinary strain. The overall Hikis: Hik=Hik (u.)+Hik (s.), (6.9) and the two addenda at the right-hand side of this equation ar e the right- hand sides of equations (6.3) and (6.8); therefore the overa ll constitutive relation has just the form intimated by equation (6.1) for a g eneral medium. As we did when electrostriction and magnetostriction were n eglected, we assume again that the Lagrangian density Lfor the electromagnetic field in presence of the four-current density sishall read: L=1 4HikFik−siϕi, (6.10) where ϕiis the four-vector potential, while Hikhas the new definition (6.9). Maxwell’s equations (4.3) and (4.4) are then obtained in jus t the same way as it occurred with the Lagrangian density (4.1). Like Hik, alsoLcan be split into an “unstrained” part L(u.), given by equation (4.1), and a term stemming from strain, that will be called L(s.). The Hamiltonian differenti- ation of L(u.)with respect to the metric gikproduces the general relativistic version of Abraham’s energy tensor, as we know from Section 4 . For clear- ness, we will rewrite it here as /parenleftbig Tik/parenrightbig (u.)=√g/bracketleftbig Fi rHkr (u.)−1 4gikFrsHrs (u.)−(ǫµ−1)Ωiuk/bracketrightbig , (6.11) where Ωinow reads: Ωi=ukFm/parenleftbig Hik (u.)um+Hkm (u.)ui+Hmi (u.)uk/parenrightbig . (6.12) Let us now deal with the explicit form of L(s.). For simplicity we shall do so when only electrostriction is present, i. e.when κik= 0. In this case one writes: L(s.)=1 4Hik (s.)Fik=1 4√g/bracketleftbig (uiζkm−ukζim)un/bracketrightbig FmnFik, (6.13)ELECTRODYNAMIC FORCES IN ELASTIC MATTER 11 where ζkmis defined by (6.6). Due to the antisymmetry of Fik,L(s.)can be rewritten as L(s.)=1 2√guiunζkmFmnFik=1 2√guiun/bracketleftbig α1γkmγpq +α2(γkpγmq+γkqγmp)/bracketrightbig SpqFmnFik. (6.14) Thanks to equation (5.5) one finds from (6.6): ζkm=α1(gkm+ukum)(ξs ;s−1) +α2/bracketleftbig ξk;m+ξm;k+us(ukξs;m+umξs;k)/bracketrightbig ,(6.15) hence: L(s.)=−1 2√gζkmFkFm=1 2√guiun/braceleftbigg α1(gkm+ukum)(ξs ;s−1) +2α2/bracketleftbig ξk;m+usukξs;m/bracketrightbig/bracerightbigg FikFmn. (6.16) But of course the expression uiukFikidentically vanishes; therefore the pre- vious equation reduces to: L(s.)=1 2√guaun/bracketleftbig α1gbm(ξs ;s−1) + 2 α2gsmξb ;s/bracketrightbig FabFmn. (6.17) In our path towards the equations of motion of elastic matter subjected to electrodynamic forces we are now confronted with two option s. We could attempt evaluating the Hamiltonian derivative of L(s.)with respect to gik, then add the resulting tensor density to the overall energy t ensor density Tik, of which we already know the inertial term from (5.12) , the e lastic part from (5.11), and the “unstrained” electromagnetic compone nt (6.11). The vanishing divergence of Tikwould then provide the equations of motion for the fields ξiandρ, once the appropriate substitutions have been made, in keeping with the definition (5.5) of the four-velocity ui. This program meets however with a certain difficulty: it requires assessing the m etric content of ξi ;kthrough extra assumptions of arbitrary character. An alternative way is however offered. One can determine dire ctly, with- out extra hypotheses, the contribution to the generalized f orce density fi(s.) stemming from electrostriction through the Euler-Lagrang e procedure: fi(s.)=∂L(s.) ∂ξi−∂ ∂xk/parenleftbig∂L(s.) ∂ξi ,k/parenrightbig . (6.18) If the metric gikis everywhere given by equation (3.1), and the velocity is so small that uρcan be dealt with as a first order infinitesimal quantity, whil e u4differs from unity only for second order terms, the Lagrangia n density (6.17) comes to read: L(s.)=1 2α1F4σF4σξρ ,ρ−α2F4ρF4σξρ,σ, (6.19)12 S. ANTOCI AND L. MIHICH and the nonzero components of its Hamiltonian derivative wi th respect to ξiare: fρ(s.)=δL(s.) δξρ=−∂ ∂xν/parenleftbigg∂L(s.) ∂ξρ,ν/parenrightbigg =−1 2α1/parenleftbig F4σF4σ),ρ−α2/parenleftbig F4ρF4ν/parenrightbig ,ν. (6.20) In the case of fluid matter α2is vanishing, and the expression of the force density given by this equation agrees with the one predicted long ago by Helmholtz with arguments about the energy of an electrostat ic system [25], and vindicated by experiments [26], [27] performed much lat er. 7.Conclusive remarks The results of the previous Sections can be availed of in seve ral ways. The full general relativistic treatment would require a simult aneous solution of Einstein’s equations, of Maxwell’s equations and of the equ ations Tik ;k= 0 fulfilled by the overall energy tensor, thereby determining in a consistent way the metric gik, the four-potential ϕi, the “displacement” four-vector ξi, and ρ. This approach is presently extra vires , due to our ignorance of the part ofTikstemming from L(s.), for the reason mentioned in the previous Sec- tion. Achievements of lesser consistency are instead at han d, like solving the equations for the electromagnetic field and for the material field described byξiandρwith a given background metric, or finding the motion of elast ic matter with a fixed metric, while the electromagnetic field is evaluated as if electrostriction were absent. Obviously enough, the cal culations become trivial when the metric is everywhere given by (3.1), while t he motion of matter occurs with non relativistic speed. In the present pa per we have required that matter be isotropic when at rest and unstraine d, but this lim- itation was just chosen for providing a simple example. The t heory can be extended without effort to cristalline matter exhibiting di fferent symmetry properties, for which reliable experimental data have star ted accumulating in recent years.ELECTRODYNAMIC FORCES IN ELASTIC MATTER 13 References [1] B. Riemann: G¨ ott. Abhand. 13, (1868) 1. [2] C. Truesdell: An idiot’s fugitive essays on science (Springer-Verlag, New York 1987), p. 625. [3] H. Minkowski: G¨ ott. Nachr., Math.-Phys. Klasse (1908) 53. [4] M. Born: Math. Ann. 68, (1910) 526. [5] H. Ott: Ann. Phys. (Leipzig) 11, (1952) 33. [6] S. R. De Groot and L. G. Suttorp: Physica 37, (1967) 284, 297. [7] S. R. De Groot and L. G. Suttorp: Physica 39, (1968) 28, 41, 61, 77, 84. [8] J. M. Jauch and K. M. Watson: Phys. Rev. 74, (1948) 950, 1485. [9] K. Nagy: Acta Phys. Hung. 5, (1955) 95. [10] I. Brevik and B. Lautrup: Mat. Fys. Medd. Dan. Vid. Selsk .38, (1970) 1. [11] D. Hilbert: G¨ ott. Nachr., Math.-Phys. Klasse (1915) 3 95. [12] F. Klein: G¨ ott. Nachr., Math.-Phys. Klasse (1917) 469 . [13] W. Gordon: Ann. Phys. (Leipzig) 72, (1923) 421. [14] E. J. Post: Formal Structure of Electromagnetics. (North-Holland Publishing Com- pany, Amsterdam 1962). [15] G. Nordstr¨ om: Soc. Scient. Fenn., Comm. Phys.-Math. 1.33, (1923). [16] M. Abraham: Rend. Circ. Matem. Palermo 28, (1909) 1. [17] G. B. Walker and D. G. Lahoz: Nature 253, (1975) 339. [18] G. B. Walker, D. G. Lahoz and G. Walker: Can. J. Phys. 53, (1975) 2577. [19] S. Antoci and L. Mihich: Nuovo Cimento 114B , (1999) 873. [20] R. Hooke: Lectures de Potentia restitutiva, or of Spring , (Martyn, London, 1678). [21] Y. Choquet-Bruhat and L. Lamoureux-Brousse: C. R. Acad. Sc. Paris A 276, (1973) 1317. [22] C. Cattaneo: C. R. Acad. Sc. Paris A 272, (1971) 1421. [23] D. N. Astrov: J. Exptl. Theoret. Phys. (U. S. S. R.) 40, (1961) 1035. [24] H.-G. Sch¨ opf: Ann. Phys. (Leipzig) 13, (1964) 41. [25] H. Helmholtz: Ann. d. Phys. u. Chem. 13, (1881) 385. [26] H. Goetz and W. Zahn: Zeitschr. f. Phys. 151, (1958) 202. [27] W. Zahn: Zeitschr. f. Phys. 166, (1962) 275. Dipartimento di Fisica “A. Volta” and I. N. F. M., Via Bassi 6, Pavia, Italy E-mail address :Antoci@fisav.unipv.it

arXiv:physics/9912011v1 [physics.optics] 3 Dec 1999Marcatili’s Lossless Tapers and Bends: an Apparent Paradox and its Solution Antonio-D. Capobianco1, Stefano Corrias1, Stefano Curtarolo1,2and Carlo G. Someda1,3 1DEI, Universit` a di Padova, Via Gradenigo 6/A 35131 Padova, Italy. 2Dept. of Materials Science and Engineering, MIT, Cambridge , MA 02139, USA 3corresponding author e-mail: someda@dei.unipd.it Proceedings of Jordan International Electrical and Electr onic Engineering Conference, JIEEEC’98, April 27-29, 1998, Amman, Jordan I. ABSTRACT. Numerical results based on an extended BPM algorithm indicate that, in Marcatili’s lossless ta- pers and bends, through-flowing waves are dras- tically different from standing waves. The source of this surprising behavior is inherent in Maxwell’s equations. Indeed, if the magnetic field is correctly derived from the electric one, and the Poynting vector is calculated, then the analytical results are reconciled with the numerical ones. Similar con- siderations are shown to apply to Gaussian beams in free space. II. INTRODUCTION. In 1985, Marcatili [1] infringed a historical taboo, by showing that lossless tapers and bends in dielectric waveguides can be conceived and de- signed, at least on paper. The key feature shared by all the infinity of structures which obey Mar- catili’s recipe, is the fact that the phase fronts of the guided modes which propagate in them, are closed surfaces. As well known, phase fronts which extend to infinity in one direction orthogonal to that of propagation do entail radiation loss, but closed fronts can avoid this problem. However, shortly after the first recipe [1], it was pointed out [2] that that recipe could generate some inconsis- tencies. In fact, a traveling wave with a closed phase front is either exploding from a point (or a line, or a surface), or collapsing into such a set. In a lossless medium where there are no sources, this is untenable. On the other hand, it was also pointed out in [2] that a standing wave with closed constant-amplitude surfaces is physically meaning- ful. Therefore, propagation of a through-flowing wave through any of Marcatili’s lossless tapers or bends has to be described in this way: the incom- ing wave must be decomposed as the sum of two standing waves, of opposite parity with respect to a suitable symmetry surface. The output wave was then to be found as the sum of the values taken by the two standing waves at the other end of thedevice. Another point raised in [2] was that very similar remarks apply to Gaussian beams in free space. Later on, the literature showed that interest in this problem was not so high, for a long time. Re- cently, though, we observed several symptoms of a renewed interest in low-loss [3–5] and lossless [6,7] tapers or bends. This induced us to try to go be- yond the results of [2], and to clarify further the difference between through-flowing and standing waves in Marcatili’s tapers. The new results reported in this paper can be summarized as follows. In Section III, we show that the numerical analysis (based on an extended BPM algorithm) of Marcatili’s tapers reconfirms that indeed through-flowing waves are drastically different from standing ones. The latter ones match very well the analytical predictions of the original recipe [1], but through-flowing waves have open wave fronts, which do not entail any phys- ical paradox. In Section IV, we provide an ana- lytical discussion of why, in contrast to what oc- curs with plane waves in a homogeneous medium and with guided modes in longitudinally uniform waveguides, through-flowing waves are so differ- ent from standing ones. We show that this is a rather straightforward consequence of Maxwell’s equations. From this we will draw the conclu- sion that a through-flowing wave propagating in one of Marcatili’s tapers is never strictly lossless. Nonetheless, our numerical results reconfirm that the recipes given in [1] do yield extremely low ra- diation losses. Finally, we address briefly the case of Gaussian beams in free space, and explain why they be- have essentially in the same way as the devices we discussed above. In fact, Maxwell’s equations show that in general the phase fronts of the mag- netic field in a Gaussian beam are not the same as the phase fronts of the electric field. Therefore, the Poynting vector is not trivially proportional to the square of the electric field. Consequently, a through-flowing beam, resulting from two super- imposed standing waves of opposite parities, can be surprisingly different from the parent waves.III. NUMERICAL RESULTS. The geometry of Marcatili’s tapers can eventu- ally be very complicated (e.g., see [8]). For our tests, however, we chose a simple shape, to avoid the danger that geometrical features could hide the basic physics we were trying to clarify. The re- sults reported here refer to a single-mode taper whose graded-index core region is delimited by the two branches of a hyperbola (labeled A and A’ in Figs. 1 and 2), and has a mirror symmetry with respect to its waist. This is a “superlinear” taper, according to the terminology of [1], with an index distribution (see again [1]) n=/braceleftbigg n0/radicalbig 1 + 2∆ /(cosh2η−sin2θ)θ1< θ < θ 2 n0 θ1> θ > θ 2 (1) where ηandϑare the elliptical coordinates, in the plane of Figs. 1 and 2. Fig. 1 refers to a stand- ing wave of even symmetry with respect to the waist plane, Fig. 2 to a standing wave of odd sym- metry. The closed lines are constant-amplitude plots. We see that they are essentially elliptical, so they agree very well with the predictions of [1]. FIG. 1. Constant-amplitude plot of a standing wave of even symmetry (with respect to the waist plane) in a superlinear Marcatili’s taper. As mentioned briefly in the Introduction, these results were generated using an extended BPM, which deserves a short description. In fact, it is well known that the standard BPM codes are suit- able to track only traveling waves, as they neglect backward waves. Our code (using a Pade’s op- erator of order (5,5)) also generates a traveling wave, and its direction of propagation is inverted whenever the wave reaches one of the taper ends. In order to generate single-mode standing waves, each reflection should take place on a surface whose shape matches exactly that of the wave front. Thisis very difficult to implement numerically, but the problem can be circumvented, letting each reflec- tion take place on a phase-conjugation flat mirror. Our code calculates then, at each point in the ta- per, the sum of the forward and backward fields, and stops when the difference between two itera- tions is below a given threshold. FIG. 2. Constant-amplitude plot of a standing wave of odd symmetry (with respect to the waist plane) in a superlinear Marcatili’s taper. Figs. 3 and 4 refer to a through-flowing wave. The almost horizontal dark lines in Fig. 3 are its phase fronts. They are drastically different from those predicted by the analytical theory in [1], which are exemplified in the same figure as a set of confocal ellipses. Note that the through-flowing wave has been studied numerically in two ways. One was simply to launch a suitable transverse field distribution, and track it down the taper with a standard BPM code. The other one was to cal- culate the linear combination (with coefficients 1 andj) of the even and odd standing waves shown in Figs. 1 and 2. The results obtained in these two ways were indistinguishable one from the other. This proves that indeed through-flowing waves are drastically different from standing ones. In par- ticular, as we said in the Introduction, they keep clear from any paradox connected with energy con- servation. Fig. 4 shows a field amplitude contour plot for the same through-flowing wave as in Fig. 3. It indicates that propagation through the taper is in- deed almost adiabatic. Therefore, as anticipated in the Introduction, insertion losses of Marcatili’s tapers are very low (at least as long as the length to width ratio is not too small), although they are not strictly zero. As a typical example, for a total taper length of 2 .5µm, a waist width of 0 .55µmand an initial-final width of 1 .65µm, BPM calculations yield that the lost power fraction is 1 .4×10−4. A typical plot of power vs. distance along a taperwith these features is shown in Fig. 5. FIG. 3. Phase fronts of a through-flowing wave in the same superlinear taper as in Figs. 1 and 2. IV. THEORETICAL DISCUSSION. For the sake of clarity, let us restrict ourselves to the case of two-dimensional tapers, like those of the previous section, where the geometry and the index distribution are independent of the zcoordinate, orthogonal to the plane of the figures. However, our conclusions will apply to 3-D structure also. The index distributions found in the corner- stone paper [1] are such that the TE modes (elec- tric field parallel to z) satisfy rigorously a wave equation which can be solved by separation of vari- ables. Obviously, the same equation is satisfied rig- orously by the transverse component of the mag- netic field. However, in general two solutions of these two wave equations which are identical, except for a proportionality constant, do not satisfy Maxwell’s equations in that structure. This is very easy to show, for example, for the case which was called “linear taper” in [1], namely, a wedged-shape re- gion with a suitable index distribution, where a guided mode propagates in the radial direction. The claim [1] that the dependence of Ezon the radial coordinate is expressed by a Hankel func- tion of imaginary order iν, related to other fea- tures of the taper, is perfectly legitimate. How- ever, one cannot extrapolate from it that the same is true for the magnetic field. In fact, calculat- ing the curl of the electric field we find that the azimuthal component of the magnetic field is pro- portional to the first derivative of the Hankel func- tion, which is never proportional to the function itself. The same is true for the Mathieu function of the fourth kind, which satisfy the wave equation in the coordinate system which fits the superlinear taper of the previous Section. This entails a dras-tic difference with plane waves, and with guided modes in uniform waveguides, where the deriva- tive of the exponential function that describes the propagation of the electric field is proportional to the function itself. In the cases at hand, the con- cept of wave impedance becomes ill-grounded. In fact, the electric field and the transverse magnetic field have identical dependencies on the transverse coordinate, so that their ratio is constant over each wavefront, but they are different functions of the longitudinal coordinate, as if the ‘wave impedance’ were not constant at all along the wave path. This indicates why it is very risky, in the case at hand, to make general claims on the Poynting vector start- ing from the spatial distribution of only the electric field. To strengthen our point, let us prove explic- itly that it is not self-consistent to claim that a purely traveling TE wave, whose radial dependence is expressed by a Hankel function of imaginary or- der,Hiν, can propagate along a linear taper. As we just said, for such a wave Ezis proportional toHiν,Hψis proportional to H′ iν, so the radial component of the Poynting vector is proportional toHiν(H′ iν)∗. In a purely traveling wave there is no reactive power in the direction of propagation. Combining with what we just said, it is easy to see that this would imply |Hiν|2=constant along the (radial) direction of propagation, a requirement that is not satisfied by Hankel functions. (Note, once more, that it is satisfied by exponential func- tions). Therefore, anywave along a linear taper whose radial dependence is expressed as a Hankel function must be at least a partially standing wave. A through-flowing wave, if it exists, must behave in a different way. FIG. 4. Field-amplitude contour plot, in the same superlinear taper as in the previous figures. Finally, let us address briefly the case of Gaus- sian beams in free space. It was pointed out in [2] that they behave essentially in the same way as the devices we discussed above. There is still some- thing to add to the discussion of [2]. Assume thatthe electric field of an electromagnetic wave has the classical features of a TEM 00Gaussian beam (see, e.g., [9]). Then, Maxwell’s equations show that the phase fronts of the magnetic field are not the same as those of the electric field, neither on the waist plane nor far from it. Hence, the Poynting vector is not trivially proportional to the square of the elec- tric field. This entails the presence of a reactive power (never accounted for in the classical class- room explanations of Gaussian beams), and an ac- tive power flow which is not always along the lines orthogonal to the electric field phase fronts. Once again, a through-flowing beam, resulting from two superimposed standing waves of opposite parities, is different from the parent waves, and the differ- ence is maximum on the symmetry plane, i.e. at the beam waist. Due to time and space limits, the details of this discussion must be left out of this presentation, and will be published elsewhere [10]. FIG. 5. Power vs. distance, in a superlinear taper of the shape shown in the previous figures, whose pa- rameters are specified in the text. V. CONCLUSION. We tried to shed new light on an old problem, namely, whether the idea of a guided mode trav- eling without any loss through a dielectric taper can be sustained without running into any physical paradox. Our numerical results, obtained with an extended BPM technique, have fully reconfirmed what was stated in [2]: in Marcatili’s tapers, stand- ing waves have the basic features outlined in [1], but through-flowing waves do not. This prevents them from running into a paradox, but on the other hand entails some loss, although very small indeed. Next, we have provided an explanation for the unexpected and puzzling result, a drastic difference between standing and through-flowing waves in the same structures. The source of these “surprise” is within Maxwell’s equations. It was pointed out in [2] that some of the prob- lems discussed here with reference to Marcatili’stapers apply to Gaussian beams in free space as well. [1] E.A.J. Marcatili, “Dielectric tapers with curved axes and no loss”, IEEE J. Quant. Electron. , vol. 21, pp. 307-314, Apr. 1985. [2] E.A.J. Marcatili and C.G. Someda, “Gaussian beams are fundamentally different from free-space modes”, IEEE J. Quant. Electron. , vol. 231, pp. 164-167, Feb. 1987. [3] O. Mtomi, K. Kasaya and H. Miyazawa, “Design of a single-mode tapered waveguide for low-loss chip-to-fiber coupling”, IEEE J. Quant. Electron. , vol. 30, pp. 1787-1793, Aug. 1994. [4] I. Mansour and C.G. Someda, “Numerical opti- mization procedure for low-loss sharp bends in MgO co-doped Ti−LiNbO 3waveguides”, IEEE Photon. Technol. Lett. , vol 7, pp. 81-83, Jan. 1995. [5] C. Vassallo, “Analysis of tapered mode transform- ers for semiconductor optical amplifiers”, Optical and Quantum Electron. , vol. 26, pp. 235-248, 1994. [6] M.-L. Wu, P.-L. Fan, J.-M. Hsu and C.-T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping”, IEEE J. Lightwave Technol. , vol. 14, pp. 2604-2614, Nov. 1996. [7] C.-T. Lee, M.-L. Wu, L.-G. Sheu, P.-L. Fan and J.- M. Hsu, “Design and analysis of completely adia- batic tapered waveguides by conformal mapping“, IEEE J. Lightwave Technol. , vol. 15, pp. 403-410, Feb. 1997. [8] J.I. Sakai and E.A.J. Marcatili, “Lossless dielectric tapers with three-dimensional geometry”, IEEE J. Lightwave Technol. , vol. 9, pp. 386-393, Mar. 1991. [9] C.G. Someda, “Electromagnetic Waves”, Chap- man & Hall, London, 1998, pp. 165-171. [10] A. D. Capobianco, M. Midrio and C. G. Someda, “TE and TM Gaussian beams in a homogeneous medium”, to be published.

arXiv:physics/9912012v1 [physics.bio-ph] 4 Dec 1999Dynamic fitness landscapes in the quasispecies model Claus O. Wilke∗, Christopher Ronnewinkel†and Thomas Martinetz‡ Institut f¨ ur Neuro- und Bioinformatik Medizinische Universit¨ at zu L¨ ubeck Seelandstr. 1a, 23569 L¨ ubeck, Germany March 17, 2008 Abstract The quasispecies model is studied for the special case of ext ernally vary- ing replication rates. Most emphasis is laid on periodic tim e dependencies, but other cases are considered as well. For periodic time dep endencies, the behavior of the evolving system can be determined analytica lly in several limiting cases. With that knowledge, the qualitative phase diagram in a given time-periodic fitness landscape can be predicted with out almost any calculations. Several example landscapes are analyzed in d etail in order to demonstrate the validity of this approach. For other, non-p eriodic time de- pendencies, it is also possible to obtain results in some of t he limiting cases, so that there can be made predictions as well. Finally, the re lationship between the results from the infinite population limit and th e actual finite population dynamics is discussed. ∗CO Wilke is on the leave to Caltech, Pasadena, CA. Email: clau s.wilke@gmx.net †C Ronnewinkel’s current postal address is: Institut f¨ ur Ne uroinformatik, Ruhr-Universit¨ at Bochum 44780 Bochum, Germany. Email: ronne@neuroinformat ik.ruhr-uni-bochum.de ‡Email: martinetz@informatik.mu-luebeck.de 1Contents 1 Introduction 3 2 Time-dependent replication rates 5 3 Periodic fitness landscapes 8 3.1 Differential equation formalism . . . . . . . . . . . . . . . . . . . . 8 3.1.1 Neumann series for X. . . . . . . . . . . . . . . . . . . . . 9 3.1.2 Exact solutions for R= 0 and R= 0.5 . . . . . . . . . . . . 12 3.1.3 Schematic phase diagrams . . . . . . . . . . . . . . . . . . . 14 3.2 Discrete approximation . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Example landscapes . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.1 One oscillating peak . . . . . . . . . . . . . . . . . . . . . . 17 3.3.2 Two oscillating peaks . . . . . . . . . . . . . . . . . . . . . . 21 3.3.3 Two oscillating peaks with flat average landscape . . . . . . 24 4 Aperiodic or stochastic fitness landscapes 27 5 Finite Populations 31 5.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.1.1 Loss of the master sequence . . . . . . . . . . . . . . . . . . 33 5.1.2 Persistency . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 A finite population on a simple periodic fitness landscape . . . . . . 36 5.2.1 The probability to skip one period . . . . . . . . . . . . . . 40 6 Conclusions 46 A High-frequency expansion of X (t)for a landscape with two alter- nating master sequences 48 21 Introduction Eigen’s quasispecies model [7] has been the basis of a vivid b ranch of molecular evolution theory ever since it has been put forward almost 30 years ago [31, 14, 12, 13, 8, 29, 17, 25, 15, 9, 28, 10, 20, 2, 36, 19, 3]. Its two mai n statements, the formation of a quasispecies made up of several molecular species with well defined concentrations, and the existence of an error thresh old above which all information is lost because of accumulating erroneous muta tions, have since then been observed in a large number of experimental as well as the oretical studies (see, e.g., [6] for the formation of a quasispecies in the RNA of the Qβphage, [1] for the observation of an error threshold in a system of self-replic ating computer programs, and, generally, the reviews [9, 10, 3] and the references the rein). Recently, a new aspect of the quasispecies model has been brought into co nsideration that was almost completely absent in previous works, namely the a spect of a dynamic fitness landscape [36, 19]. With the notion “dynamic fitness l andscape”, we mean all situations in which the replication and/or decay rates o f the molecules change over time. In the present work, we are only interested in situ ations where these changes occur as an external influence for the evolving syste m, and where there is no feedback from the system to the dynamics of the fitness land scape. Dynamic fitness landscapes of that kind are important, since almost a ny biological system is subject to external changes in the form of, e.g., daytime/ni ghttime, seasons, long- term climatic changes, geographic changes due to tectonic m ovements, to name just a few. The main problem one encounters when dealing with dynamic la ndscapes is the difficulty to find a correct generalization of the quasispecie s concept. In the origi- nal work of Eigen, the quasispecies is the equilibrium distr ibution of the different molecular species. It is reached if the system is left undist urbed for a sufficiently long time. Since in a dynamic landscape, the system is being d isturbed by the landscape itself, the concept of a quasispecies is meaningl ess in the general case. However, there are special cases in which a meaningful quasi species can be de- fined. If, for example, the landscape changes on a much slower time scale than what the system needs to reach the equilibrium, then the syst em is virtually in equilibrium all the time, and the concentrations at time tare determined from the landscape present at that time. Generally, certain symmetr ies in the dynamics of the landscape can allow for the definition of a quasispecies. One example we treat in this paper in detail is the case of time-periodic landscap es, which offer a natural quasispecies definition. An early study of dynamic landscapes has been done by Jones [1 2, 13]. How- ever, he has considered only cases in which all replication r ates change by a com- mon factor. Therefore, his approach excludes, among other c ases, in particular 3all situations in which the order of the molecules’ replicat ion rates changes over time, i.e., in which e.g. one of the faster replicating molec ules becomes one of the slower replicating molecules and vice versa. The more recen t work on dynamic fitness landscapes allow for such changes. Wilke et al. [35, 36] have developed a framework that allows to define and to calculate numericall y a quasispecies in time-periodic landscapes. Independent of them, Nilsson an d Snoad [19] have stud- ied the particular example of a stochastically jumping peak in an otherwise flat landscape. This work has been generalized and refined by Ronn ewinkel et al. [22], who could also define a meaningful quasispecies for a determi nistic version of the jumping peak landscape and related landscapes. Finally, in the related field of genetic algorithms, there exist also some theoretical stud ies of dynamic fitness landscapes. Let us mention two of them. First of all, there is the work of Schmitt et al. [27, 26]. These authors derive results for finite population s (note that most results for the quasispecies model are only valid in the infin ite population limit) in a relatively broad class of dynamic landscapes. However, th ey can only treat land- scapes in which the fitnesses get scaled, so that the same rest riction applies here that applied to Jone’s work. The order of the fitnesses must ne ver change. Second of all, Rowe [24, 23] has studied genetic algorithms with tim e-periodic landscapes. However, his approach has the caveat that it is tightly conne cted to the discrete time used in genetic algorithms, and that the dimension of th e transition matrices grows in proportion to the period length Tof the oscillation. This makes it hard to derive analytical results, and in addition to that, it ren ders landscapes with largeTinaccessible to numerical calculations. The remainder of this article is structured as follows. We be gin our discussion in Section 2 with a brief summary of the general aspects of dynam ic fitness landscapes in the quasispecies equation. In Section 3, we will develop t he main subject of this work, a general theory of time-periodic fitness landscapes. The theoretical part thereof is presented in Section 3.1, in which we demonstrate how a time-dependent quasispecies can be defined by means of the monodromy matrix, and how this monodromy matrix can be expanded in terms of the oscillation period T. In Section 3.2, we present an alternative approximation formu la for the monodromy matrix that is more suitable for numerical calculations, an d in Section 3.3, we compare, for several example landscapes, the results obtai ned from that formula with the general theory developed in Section 3.1. The restri ction of a time-periodic fitness landscape is weakened in Section 4, where we discuss t he implications of our findings for other, non-periodic fitness landscapes. In that section, we are going to see that the qualitative results of the study of Nilsson and S noad [19] are a direct consequence of the general theory for dynamic fitness landsc apes. Since our work is based on Eigen’s deterministic approach with differentia l equations, all results 4presented up to the end of Section 4 are only valid for infinite population sizes. In order to address this shortcoming, in Section 5 we give a br ief introduction into the problems involved when dealing with finite populati ons. In Section 5.1, some simulation results are shown, demonstrating the relat ionship between the results from the infinite population limit and the actual fini te population dynamics. Finally, an approximative analytical description of a finit e population evolving on a simple periodic landscape is developed in Section 5.2. We c lose this paper with some conclusions in Section 6. 2 Time-dependent replication rates The quasispecies model describes the evolution of self-rep licating macromolecules. It assumes that there exists only a finite number of different m olecular species, and that each species iis present in high abundance, such that it suffices to record only the concentrations of the species, xi(t). The reaction dynamics is thought of taking place in a well-stirred reactor, with some constan t out-flux of molecules E(t), such that the total concentration of molecules/summationtext ixi(t) remains constant for allt. The self-replication process may fail, leading to erroneo usly copied offspring. This is being described by a mutation matrix Qijthat gives the probability with which an offspring molecule of type iis generated from a parent j. Often, it is assumed that the molecules are RNA sequences, consisting of a string of letters A, G, C, U, or even simpler, the molecules are represented as b itstrings. In that case, one regularly makes the additional assumption that th e replication process copies the string letter by letter, and that therefore the pr obability of a wrongly copied letter is independent of the letter’s position in the string, and also of the type of the letter. In connection with that, it is useful to in troduce the error rate per letter, R. In case we conceive the molecules of bitstrings of fixed leng thl, the mutation matrix Qijthen takes on the form Qij= (1−R)l/parenleftbiggR 1−R/parenrightbiggd(i,j) , (1) where d(i, j) represents the Hamming distance between two sequences of t ypei andj. All our examples in later sections are based on that assumpt ion. Our general results, however, do not depend on this assumption. In vector notation, i.e. x= (x1, x2, . . .), the basic quasispecies equation reads ˙x(t) = [W(t)−E(t)1]x(t). (2) Here,1stands for the identity matrix, and W(t) is given by W(t) =Q(t)A(t)−D(t), (3) 5where the diagonal matrix A(t) contains the replication coefficients, the diagonal matrix D(t) contains the decay constants, and the matrix Q(t) is the above intro- duced mutation matrix. In the most general case, all these th ree matrices can be time dependent. The average excess production can be expres sed in terms of the matrices A(t) andD(t) as E(t) =et·[A(t)x(t)−D(t)x(t)], (4) where etis a vector containing only entries of 1s, i.e. et= (1, . . .,1). Because of the x(t) dependence of E(t), Eq. (2) is nonlinear. However, as in the case of constant W[31, 14], the introduction of new variables of the form y(t) = exp/parenleftbigg/integraldisplayt 0E(τ)dτ/parenrightbigg x(t) (5) removes this nonlinearity. The resulting equation reads ˙y(t) =W(t)y(t), (6) and the concentrations can be obtained from y(t) via x(t) =y(t) et·y(t). (7) Note that if all decay constants are equal at all times, i.e. D(t) = diag( D(t), . . ., D (t)), with a single scalar function D(t), then an extended transformation y(t) = exp/parenleftbigg/integraldisplayt 0[E(τ) +D(t)]dτ/parenrightbigg x(t) (8) leads to the even simpler equation ˙y(t) =Q(t)A(t)y(t). (9) The concentration vector x(t) can again be obtained from Eq. (7). The linearized quasispecies model, Eq. (6), has been studie d in great detail for constant W[9, 10]. Since the quasispecies is the equilibrium distribu tion of the molecular concentrations, the main question in that con text has been the prediction of the system’s behavior for t→ ∞. As a linear differential equation, Eq. (6) displays exponential growth [exponential damping d oes not occur because of the sign in front of the integral in Eq. (5)]. That growth ma y in principle be accompanied by exponentially amplified/damped oscillat ions. Of course, an equilibrium can only be defined if there are either no oscilla tions at all, or all 6oscillations die out for t→ ∞. Fortunately, this is typically the case. First of all, for symmetric Q, the whole spectrum of Wis real [25], because Wcan be transformed into a symmetric matrix by means of a similarity transformation, W=QA−D→A1/2WA−1/2=A1/2QA1/2−D. (10) For non-symmetric Q, we can apply the Frobenius-Perron theorem if the decay rates satisfy (D)ii<(QA)ii for all i. (11) The Frobenius-Perron theorem guarantees a real largest eig envalue. Consequently, we have at most exponentially damped oscillations as long as we obey (11). In addition to that, the Frobenius-Perron theorem states that the eigenvector corre- sponding to this largest eigenvalue has only strictly posit ive entries, and hence, that this eigenvector can be interpreted as a vector of chemi cal concentrations if normalized appropriately. Now consider the case of a full time dependency. In that case, we can map the quasispecies model onto a linear system with a symmetric mat rix˜W(t) ifQ(t) is symmetric for all t. This can be seen by introducing z(t) =A1/2(t)y(t). (12) Differentiation yields ˙z(t) =˜W(t)z(t) (13) with ˜W(t) =A1/2(t)Q(t)A1/2(t)−D(t) +/bracketleftbiggd dtA1/2(t)/bracketrightbigg A−1/2(t). (14) Nevertheless, we cannot write down a solution for Eq. (13) fr om the knowledge of the eigensystem of ˜W(t) if˜W(t) has an arbitrary time dependency. Therefore, the symmetric quasispecies equation (13) does not help us in solving Eq. (6). As a consequence, we have to focus on limiting cases for which gen eral statements can be made. The two most important limiting cases are very fast c hanges in W(t) on the one hand, and very slow changes in W(t) on the other hand. We begin with the case of very slow changes. For the rest of this work, we wil l assume that W(t) has a real spectrum for all t. From Eq. (13), we know that this covers at least all cases for which Q(t) is symmetric. To be on the safe side, we also assume that (11) is satisfied for all t. In that way, the Perron eigenvector of W(t) can always be interpreted as a vector of chemical concentrations. 7For every time t0, we can define a relaxation time τR(t0) =1 λ0(t0)−λ1(t0), (15) where λ0(t)0andλ1(t0) are the largest and the second largest eigenvalue of W(t0), respectively. The time τR(t0) gives an estimate on how long a linear system with matrix W(t0) needs to settle into equilibrium. Therefore, if the change s inW(t) happen on a timescale much longer than τR(t), the system is virtually in equilib- rium at any given point in time. Hence, for large enough t, the quasispecies will be given by the Perron eigenvector of W(t). Strictly speaking, this is only true if there is always some overlap between the largest eigenvecto r ofW(t) and the one ofW(t+dt), but in all but some very pathological cases we can assume th is to be the case. The situation of fast changes in W(t) is somewhat more difficult, because, as we are going to see later on, we have to define a suitable averag e over W(t) in order to make a general statement. Therefore, we postpone th at situation for a moment. A detailed discussion of fast changes will be given f or the particular case of periodic fitness landscapes in the next section, and later on, we will discuss fast changing landscapes in general. 3 Periodic fitness landscapes 3.1 Differential equation formalism In this section, we are going to study periodic time dependen cies in W(t), for which we can demonstrate several general statements. If the changes in W(t) are periodic, i.e., if there exists a Tsuch that W(t+T) =W(t) for all t, (16) then Eq. (6) turns into a system of linear differential equati ons with periodic coefficients. Several theorems are known for such systems [37 ]. Most notably, if Y(t, t0) is the fundamental matrix, such that every solution to Eq. ( 6) can be written in the form y(t) =Y(t, t0)y(t0), (17) then we can define a so-called monodromy matrix X(t0), X(t0) =Y(t0+T, t0), (18) 8which simplifies Eq. (17) to y(t) =Y(t0+φ, t0)Xm(t0)y(t0) =Xm(t0+φ)Y(t0+φ, t0)y(t0), (19) for the decomposition t=mT+φ+t0with the phase φ < T . In particular, we have y(φ+mT) =Xm(φ)y(φ), (20) so that for every phase φ, we have a well defined asymptotic solution, given by the eigenvector to the largest eigenvalue of X(φ). In other words, periodic fitness landscapes allow the definition of a quasispecies, much in th e same way as static fitness landscapes do. However, this quasispecies is time-d ependent, and the time- dependency is periodic with period T. 3.1.1 Neumann series for X We can derive a formal expansion in Tfor the monodromy matrix. This formal expansion is similar in spirit to the Neumann series which gi ves a formal solution to an integral equation, and it is based on the Picard-Lindel¨ o f iteration for differential equations. As the first step, we have to rewrite Eq. (6) in the f orm of an integral equation, i.e. y(t0+τ) =y(t0) +/integraldisplayτ 0W(t0+τ1)y(t0+τ1)dτ1. (21) Our goal is to solve this equation for y(t0+τ) by iteration. Our initial solution is y0(t0+τ) =y(t0), (22) which we insert into Eq. (21). As a result, we obtain the 1st or der approximation y1(t0+τ) =y(t0) +/integraldisplayτ 0W(t0+τ1)y(t0)dτ1. (23) Further iteration yields y2(t0+τ) =y(t0) +/integraldisplayτ 0W(t0+τ1)y(t0)dτ1 +/integraldisplayτ 0W(t0+τ1)/integraldisplayτ1 0W(t0+τ2)y(t0)dτ1dτ2,(24) 9and so on. Now we define W0(t0, τ) = 1, (25) W1(t0, τ) =1 τ/integraldisplayτ 0W(t0+τ1)dτ1, (26) and, in general Wk(t0, τ) =1 τk/integraldisplayτ 0W(t0+τ1)/integraldisplayτ1 0W(t0+τ2)· · ·/integraldisplayτk−1 0W(t0+τk)dτ1dτ2· · ·dτk, (27) and obtain the formal solution y(t0+τ) =∞/summationdisplay k=0τkWk(t0, τ)y(t0). (28) For suitably small τ, the infinite sum on the right-hand side is guaranteed to converge. When we compare this equation for τ=Tto the definition of the monodromy matrix Eq. (18), we find that [introducing Wk(t0) :=Wk(t0, T)] X(t0) =∞/summationdisplay k=0TkWk(t0). (29) In particular, since W1(t0) is identical to the time-average over W(t), regardless oft0, we have the high-frequency expansion X(t0) =1+TW+O(T2), (30) with W=1 T/integraldisplayT 0W(t)dt . (31) Equation (30) reveals that for very high frequency oscillat ions, the system behaves as being in a static landscape. That static landscape is give n by the dynamic landscape’s average over one oscillation period. The radius of convergence of the expansion Eq. (29) can be est imated as follows. Since all entries of W(t) are positive, we have for the tensor Wiν1Wν1ν2· · ·Wνk−1j(t0) :=1 Tk/integraldisplayT 0Wiν1(t0+τ1)/integraldisplayτ1 0Wν1ν2(t0+τ2) · · ·/integraldisplayτk−1 0Wνk−1j(t0+τk)dτ1dτ2· · ·dτk(32) 10the estimate Wiν1Wν1ν2· · ·Wνk−1j(t0)≤1 Tk/integraldisplayT 0Wiν1(t0+τ)dτ/integraldisplayT 0Wν1ν2(t0+τ)dτ · · ·/integraldisplayT 0Wνk−1j(t0+τ)dτ , (33) from which follows /parenleftbig Wk(t0)/parenrightbig ij≤/parenleftBig Wk/parenrightBig ij. (34) The matrix norm induced by the sum norm /⌊ard⌊l(y1, y2, . . ., y n)/⌊ard⌊l1=/summationdisplay i|yi| (35) is the column-sum norm /vextenddouble/vextenddoubleW/vextenddouble/vextenddouble 1= max j/braceleftBigg/summationdisplay i|Wij|/bracerightBigg . (36) With that norm, we can with the aid of Eq. (34) estimate /vextenddouble/vextenddoubleWk(t0)/vextenddouble/vextenddouble 1≤/vextenddouble/vextenddouble/vextenddoubleWk/vextenddouble/vextenddouble/vextenddouble 1≤/vextenddouble/vextenddoubleW/vextenddouble/vextenddoublek 1. (37) Hence, the expansion Eq. (29) converges certainly for those Tthat satisfy T/vextenddouble/vextenddoubleW/vextenddouble/vextenddouble 1<1. (38) Since all entries in Ware positive, we have further /vextenddouble/vextenddoubleW/vextenddouble/vextenddouble 1= max j/braceleftBigg/summationdisplay i|AjQij−Djδij|/bracerightBigg = max j/braceleftbig Aj−Dj/bracerightbig , (39) where the bar in AjandDjindicates that these quantities represent averages over one oscillation period. The second equality holds because o f (11) and because of/summationtext iQij= 1. Without loss of generality, we assume that the maximum is given by A0−D0. Then, Eq. (38) is satisfied for T <1 A0−D0. (40) 11It is interesting to compare this expression to the relaxati on time of the time- averaged fitness landscape, τR. To 0ths order, the principal eigenvalue of Wis given by W00. The second largest eigenvalue is to the same order given by t he second largest diagonal element of W, which we assume to be W11without loss of generality. Hence, the relaxation time is approximately gi ven by τR=1 W00−W11>1 W00≥1 A0−D0, (41) which is generally larger than the radius of convergence of E q. (29). In particular, if the largest and the second largest eigenvalue of Wlie close together, the relaxation time may be much larger than the largest oscillation period f or which the expansion is feasible. This restricts the usability of Eq. (29) to cons iderably high frequency oscillations in the landscape. The interesting regime in wh ich the changes in the landscape happen on a time scale comparable to the relaxatio n time of the system can unfortunately not be studied from Eq. (29). 3.1.2 Exact solutions for R= 0andR= 0.5 The two extreme cases R= 0 (no replication errors) and R= 0.5 (random offspring sequences) allow for an exact analytic treatment. The secon d case is identical to the situation in static landscapes, and therefore we will me ntion it only briefly. At the point of stochastic replication R= 0.5, the population dynamics becomes independent of the details of the landscape. As a consequenc e, temporal changes in the landscape must become less important as Rapproaches R= 0.5. However, this is not very surprising, since in most cases, an error rat e close to 0.5 implies that the population has already passed the error threshold, which in turn implies that it does not feel the changes in the landscape any more. The case of R= 0, on the other hand, is more complex than the corresponding case in a static landscape. Since the matrix Qbecomes the identity matrix for R= 0, Eq. (6) reduces to ˙y(t) = [A(t)−D(t)]y(t). (42) The matrices A(t) andD(t) are diagonal by definition, and hence, a solution to Eq. (42) is given by y(t) = exp/parenleftbigg/integraldisplayt t0[A(t′)−D(t′)]dt′/parenrightbigg y(t0). (43) When we compare this expression to Eqs. (17) and (18), we find Y(t, t0) = exp/parenleftbigg/integraldisplayt t0[A(t′)−D(t′)]dt′/parenrightbigg , (44) 12and, in particular, X(φ) = exp/parenleftbigg/integraldisplayφ+T φ[A(t′)−D(t′)]dt′/parenrightbigg . (45) The integral in the second expression is taken over a complet e oscillation period, and hence, it is independent of φ. Thus, we find for arbitrary φ X(φ) = exp( W) for R= 0. (46) With a vanishing error rate, the monodromy matrix becomes th e exponential of the time-average over W(t). Since the exponential function only affects the eigenval- ues, but not the eigenvectors, of a matrix, the quasispecies is given by the principal eigenvector of W, irrespective of the length of the oscillation period T. In other words, under the absence of mutations will the sequence iwith the highest average value of Ai(t)−Di(t) take over the whole population after a suitable amount of time, provided it existed already in the population at the be ginning of the process. By continuity, this property must extend to very small but po sitive error rates R. So, similar to the case of R= 0.5, the temporal changes in the landscape loose their importance when Rapproaches 0. There is, however, a caveat to the above argument. In case the largest eigen- value of Wis degenerate, temporal changes in the landscape may contin ue to be of importance for R= 0. A degeneracy of the largest eigenvalue of Wis possible, because the Frobenius-Perron theorem applies only to posit ive error rates. For degenerate quasispecies, the initial condition y(t0) determines the composition of the asymptotic population. In this context, let us consider the general solution for periodic fitness landscapes, Eq. (19). We have y(t) =Xm(φ)y(t0+φ) (47) with y(t0+φ) =Y(t0+φ, t0)y(t0). (48) So even if Xbecomes independent of φforR= 0, this need not be the case for y(t0+φ), because of Eq. (48). If the largest eigenvalue of Wis degenerate, these variations in y(t0+φ) will remain visible for arbitrarily large times t. Hence, we will see oscillations among the different quasispecies which cor respond to the largest eigenvalue. Clearly, this effect is the more pronounced the l arger the oscillation period T. 133.1.3 Schematic phase diagrams The results of the previous two subsections allow us to ident ify the general prop- erties of the quasispecies model with a periodic fitness land scape at the borders of the parameter space. We have to consider only the two paramet ers error rate R and oscillation period T, since all other parameters (replication rates, decay rate s, details of the matrix Q) do not influence the above results. In Fig. 1, we have summarized our findings. Along the abscissa runs the oscilla tion period. For very fast oscillations, the evolving population sees only the ti me-averaged landscape. For very slow oscillations, on the other hand, the populatio n is able to settle into an equilibrium much faster than the changes in the landscape occur. Hence, the population sees a quasistatic landscape. Along the ordinat e, we have displayed the error rate. For the error rate, we have disregarded the re gion above R= 0.5, in which anti-correlations between parent and offspring seq uences are present. For R= 0.5, all sequences have random offspring, and hence, all sequen ces replicate equally well. Therefore, for this error rate, the landscape becomes effectively flat. On the other side, for R= 0, we have again the time-averaged landscape. However, for large T, the fact that we see the average landscape does not mean that the concentration variables are asymptotically constant. Deg eneracies in the largest eigenvalue may cause a remaining time dependency due to osci llations between superposed quasispecies. The exact form of these oscillati ons is dependent on the initial condition y(0). For small T, the oscillations disappear, because the ratio of newly created sequences during one oscillation period an d remaining sequences from the previous oscillation period decays with T[Eq. (30)]. From the above observations, we can derive generic phase dia grams for periodic fitness landscapes. There are two main possibilities. The fit ness landscape may average to a landscape that has a distinct quasispecies, or i t may average to a flat landscape. These two cases are illustrated in Fig. 2. Note th at the diagrams are meant to illustrate the qualitative form and position of the different phases. In their exact appearance, they may differ substantially from t he exact phase diagram of a particular landscape. If a landscape averages to one with a distinct quasispecies, then for every oscillation period Tand every phase of the oscillation φ, we have a unique error threshold R∗(T, φ). For small T, the error threshold converges towards the one of the average fitness landscape, R∗ av, irrespective of the phase φ. For larger T, the error threshold oscillates between R∗ lo= min φR∗(T, φ) and R∗ hi= max φR∗(T, φ). In the limit of an infinitely large oscillation period, R∗ hiconverges towards R∗ max, which is the largest error threshold of all the (static) land scapes W(φ). Similarly, R∗ loconverges towards R∗ minin that limit, where R∗ minis accordingly defined as the smallest error threshold of all landscapes W(φ). For a fixed oscillation period T, 14oscillation period T0.5 0 error rate R smallT largeT(degeneracies cause fluctuations for large T)time-averagetime-averagedisorder quasi-static Figure 1: The appearance of a periodic fitness landscape at th e border regions of the parameter space. and a fixed error rate RwithR∗ lo< R < R∗ hi, we have necessarily R > R∗(T, φ) for some phases φ, andR < R∗(T, φ) for the rest of the oscillation period. As a result, a quasispecies will form whenever R > R∗(T, φ), but it will disappear again as soon asR < R∗(T, φ). This phenomenon has for the first time been observed in [36] , and there, the region of the parameter space in which it can be found has been called the temporarily ordered phase . In this phase, whether we observe order or disorder depends on the particular moment in time at which we study the system. In correspondence to that, we will call a phase “ordered” onl y if order can be seen for the whole oscillation period, and we will call a phase “di sordered” if during the whole oscillation period no order can be seen. The relati onship between the ordered phase, the disordered phase, and the temporarily or dered phase for the first type of landscapes is displayed in Fig. 2a). Compare als o the phase diagram of the oscillating Swetina-Schuster landscape in Fig. 3. In a landscape that averages to a flat one, on the other hand, th e disordered phase must extend over the whole range of Rfor sufficiently small T. Order can be observed only above a certain Tmin. However, slightly above that Tmin, no order will be found for error rates Rother than intermediate ones, since for R= 0 the landscape averages again to a flat one. Hence, what we will obs erve is an ordered or temporarily ordered phase restricted from above and from below. Instead of a 15oscillation period T0.5 0 error rate R largeT smallT¯W=non-flat landscape a)disordered temp. ordered ordered oscillation period T0.5 0 error rate R largeT smallT¯W=flat landscape b)temp. ordereddisordered ordered Figure 2: The two possible phase diagrams of a periodic lands cape. If W(t) averages to a non-flat landscape, there will typically be a lo wer error threshold, below which we always find order, and a higher error threshold , above which the system is always in a disordered state. If W(t) averages to a flat landscape, however, the disordered phase extends to the whole range of Rfor sufficiently small T. unique error threshold R∗(T, φ), we have for every phase φa lower threshold that marks the transition from disorder to order, and a higher thr eshold that marks the transition back to disorder. For longer oscillation per iods, the fluctuations in the degenerate quasispecies become important for R= 0, and this fact allows the ordered regime to extend to much smaller values of R. Hence, the lower disordered phase will fade out for T→ ∞. A typical phase diagram for this type of landscapes is displayed in Fig. 2b). 3.2 Discrete approximation The differential equation formalism we have used so far allow s for an elegant dis- cussion of the system’s general properties. However, if we w ant to obtain numerical solutions, this formalism does not help us very much, becaus e we do not have a general expression for the fundamental matrix Y(t, t0) from Eq. (17), nor for the monodromy matrix X(t0) from Eq. (18). Therefore, for our numerical treatment we will move over to the discretized quasispecies equation, y(t+ ∆t) = [∆ tW(t) +1]y(t). (49) In the case of constant W, the quasispecies obtained from that equation is identical to the one of Eq. (6), and it is also identical to the one of the e quation y(t+ 1) = Wy(t). (50) 16Equation (50) has been studied by Demetrius et al.[4], and has been employed by Leuth¨ ausser [15] for her mapping of the quasispecies model onto the Ising model. In the general time-dependent case, however, the additiona l factor ∆ tand the identity matrix 1of Eq. (49) are important, and cannot be left out. The analogu e of the fundamental matrix for Eq. (49) reads Y(t0+k∆t, t0) =T/braceleftBiggk−1/productdisplay ν=0[∆tW(t0+ν∆t) +1]/bracerightBigg , (51) where T {·}indicates that the matrix product has to be evaluated with th e proper time ordering [35]. Similarly, the analogue of the monodrom y matrix becomes X(t0) =Y(t0+T, t0) =T/braceleftBiggn−1/productdisplay ν=0[∆tW(t0+ν∆t) +1]/bracerightBigg , (52) where we have assumed that Tis an integral multiple of ∆ t, and have set n=T/∆t. The influence of the size of ∆ ton the quality of the approximation has been investigated in [35]. A more in-depths discussion of the rel ationship between the continuous and the discrete quasispecies model can also be f ound in [3]. 3.3 Example landscapes For the rest of this section, we are going to have a look at seve ral example land- scapes, in order to illustrate the implications of our gener al theory. In all cases considered, we represent the molecules as bitstrings of fixe d length l. Moreover, we assume that a single bit is copied erroneously with rate R, irrespective of the bit’s type and of its position in the string. 3.3.1 One oscillating peak In the previous works on the quasispecies model with periodi c fitness landscapes [35, 36], most emphasis has been laid on landscapes with a single o scillating sharp peak. As a generalization of the work of Swetina and Schuster [29], the master sequence has been given a replication rate A0(t)≫A, where Ais the replication rate of all other sequences. The replication rate A0(t) has been expressed as A0(t) =A0,statexp[ǫf(t)], (53) with a T-periodic function f(t). The parameter ǫallows a smooth crossover from a static landscape to one with considerable dynamics, and th e exponential assures 17thatA0(t) is always positive. In order not to duplicate work, we will n ot repeat the results of [35, 36] here. In short, it has been found that t he behavior at the border regions of the parameter space is indeed as it is depic ted in Fig. 1, and that a phase diagram of the form of Fig. 2a) correctly describ es the relationship of order and disorder in an oscillating Swetina-Schuster la ndscape. Here, our aim is to show that the phase borders in such a phase diagram can, f or an oscillating Swetina-Schuster landscape, be calculated approximately . For static landscapes with a single peak, the assumption of a vanishing mu- tational backflow into the master sequence allows to derive a n approximate ex- pression for the error threshold [16, 9, 10]. A similar formu la can be developed to calculate the error threshold as a function of time in a lan dscape with a single oscillating peak. But before we turn towards the dynamic lan dscape, we are going to rederive the expression for the master’s concentration x0in a static landscape, based on the neglect of mutational backflow. The expression w e are going to find is slightly more general than the one that was previously giv en, and it will be of use for the periodic fitness landscape as well. The 0th component of the quasispecies equation (2) becomes, after neglecting the mutational backflow, ˙x0(t) =W00x0(t)−E(t)x0(t). (54) The average excess production E(t) can be expressed in terms of x(t) andWas E(t) =/summationdisplay i,jWijxj(t). (55) With that expression, the solution of Eq. (54) requires the k nowledge of the sta- tionary mutant concentrations xj, which are usually unknown. To circumvent this problem, we make the somewhat extreme assumption that all mu tant concentra- tions are equal. Although this assumption, which is equival ent to the assumption of equal excess productions Eiin the usual calculation without mutational back- flow, will generally not be true, it works fine for Swetina-Sch uster type landscapes. With this additional assumption, Eq. (55) becomes E(t) =/summationdisplay i/bracketleftBigg/summationdisplay j>0Wij1−x0(t) N−1+Wi0x0(t)/bracketrightBigg , (56) where Nis the number of different sequences in the system. When we ins ert this into Eq. (54) and solve for the steady state, we find x0=W00−1 N−1/summationtext i/summationtext j>0Wij/summationtext iWi0−1 N−1/summationtext i/summationtext j>0Wij. (57) 18The expressions involving sums over matrix elements in Eq. ( 57) can be identified with the excess production of the master, E0=/summationdisplay iWi0 (58) and with the average excess production without the master, E−0=1 N−1/summationdisplay i/summationdisplay j>0Wij, (59) ifWhas the standard form QA−D. Therefore, Eq. (57) corresponds to the often quoted result x0=W00−E−0 E0−E−0. (60) However, Eq. (57) is more general in that it can be used even if Wis not given as QA−D. Our idea here is to insert the monodromy matrix into Eq. (57) i n order to obtain an approximation for x0in the case of periodic landscapes. But why can we expect this to work? After all, Eq. (57) has been derived fr om an equation with continuous time, Eq. (54), whereas the monodromy matri x advances the system in discrete time steps, as can bee seen in Eq. (20). The important point is here that we are only interested in the asymptotic state, w hich is given by the normalized Perron vector of the monodromy matrix, whether w e use discrete or continuous time. Therefore, we are free to calculate the asy mptotic state in a periodic landscape for a given phase φfrom ˙y(t) =X(φ)y(t), (61) even if this equation does not have a direct physical meaning for finite times. The asymptotic molecular concentrations are then given by the l imitt→ ∞ of x(t) =y(t) e·y(t). (62) From differentiating Eq. (62) and inserting Eq. (61), we obta in ˙x(t) =X(φ)x(t)−x(t)/parenleftbig e·[X(φ)x(t)]/parenrightbig . (63) When we neglect the backflow onto the master sequence, the 0th s component of that equation becomes identical to Eqs. (54) and (55), but wi th the matrix X(φ) 19R∗ lo(T)R∗ hi(T) ordered phase R∗ minR∗ avR∗ maxdisordered phase temporarily ordered phase 100 oscillation period T10 1 0.10.4 0.3 0.2error rate R 0.1 0.0 Figure 3: The phase diagram of an oscillating Swetina-Schus ter landscape [ A0(t) = e2.4exp(2 sin ωt)], numerically calculated from Eq. (57). instead of W. This shows that we may indeed use Eq. (57) as an approximatio n for the asymptotic concentration of x0. Of course, since we have neglected mutational backflow, this approximation works only for landscapes in wh ich a single sequence has a significant advantage over all others. But this restric tion does similarly apply to the static case. Numerically, we have found that Eq. (57) w orks well for a single oscillating peak, and that it breaks down in other cases as ex pected. With the aid of Eq. (57), we are now in the position that we can c alculate the phase diagram of the oscillating Swetina-Schuster landsca pe. When we insert the monodromy matrix X(φ) into Eq. (57), we are able to obtain (numerically) the error rate at which x0vanishes, R∗(T, φ). From that expression, we can calculate R∗ loandR∗ hi. The results of the corresponding, numerically extensive c alculations are shown in Fig. 3, together with R∗ av,R∗ max, and R∗ min, which have also been determined from Eq. (57). We find that both R∗ loandR∗ hiapproach R∗ avforT→0, as predicted by our general theory. For T→ ∞ ,R∗ higrows quickly to the level of R∗ max, but a slight discrepancy between the two values remains. It has i ts origin in the vast complexity of the numerical calculations involved for largeT. We can only approximate the monodromy matrix by means of Eq. (52), and we need ever more factors ∆ tW(t0+ν∆t) +1for large T. The discrepancy between R∗ loandR∗ min, on the other hand, has a different origin. The main cause here i s the fact that the relaxation into equilibrium is generally slower for smalle r error rates. Therefore, R∗ loneeds a much larger Tto reach R∗ minthan it is the case with R∗ hiandR∗ max. 203.3.2 Two oscillating peaks A single oscillating peak provides some initial insights in to dynamic fitness land- scapes. It is more interesting, however, to study situation s in which several se- quences obtain the highest replication rate in different pha ses of the oscillation period. The simplest such case is a landscape in which two seq uences become in turn the master sequence. Here, we will assume that the two are located at opposite corners of the boolean hypercube, i.e., that they a re given by a certain sequence and its inverse. In that way, it is possible to group sequences into error classes according to their Hamming distance to one of the two possible master sequences. As an example, we are going to study a landscape wi th the replication coefficients A0(t) =A0,statexp(ǫsinωt), (64a) Al(t) =A0,statexp(−ǫsinωt), (64b) Ai(t) = 1 for 0 < i < l . (64c) The subscripts in the replication coefficients stand for the H amming distance to the sequence 000 · · ·0. For single peak landscapes, it is instructive to characteri ze the state of the system at time tby the value of the order parameter ms(t) =1 ll/summationdisplay i=0xi(t)[l−2i], (65) where xi(t) is the cumulative concentration of all sequences of Hammin g distance ito the master sequence [15, 30]. If the master sequence makes up the whole population, we have ms(t) = 1. A completely disordered population, on the other hand, yields ms(t) = 0. In principle, ms(t) can also be used for a landscape with two alternating master sequences if they are each other’s in verse. In that case, the Hamming distance has to be measured with respect to one of the two master sequences. If the population consists only of sequences of t he type of the other master sequence, we have ms(t) =−1. However, there is a small problem with degenerate landscapes, in which the two peaks have the same r eplication rate. In such landscapes, the sequence distribution becomes symm etric with respect to the two peaks, i.e., x0=xl,x1=xl−1, and so on. Then, ms(t) becomes zero because of this symmetry, although the population may be in a n ordered state. To distinguish between the case of true disorder and the case of an ordered, but 21symmetrical population, we introduce the additional order parameters m+ s(t) =1 l⌊(l−1)/2⌋/summationdisplay i=0xi(t)[l−2i], (66) and m− s(t) =1 ll/summationdisplay i=l−⌊(l−1)/2⌋xi(t)[l−2i]. (67) Here, ⌊x⌋stands for the largest integer smaller than or equal to x. The quantity m+ s(t) is always positive, m− s(t) is always negative, and further- more, we have ms(t) =m+ s(t) +m− s(t). (68) If the population is uniformly distributed over the whole se quence space, we have m+ s(t) =−m− s(t) =1 l2l⌊(l−1)/2⌋/summationdisplay i=0/parenleftbiggl i/parenrightbigg (l−2i). (69) This expression goes to 0 for l→ ∞. If, on the other hand, only the two peaks are populated, each with half of the total population, we find m+ s(t) =−m− s(t) =1 2. (70) In the case that either m+ s(t) orm− s(t) equal to zero, the population is centered about the respective other peak. In the following, when it is important to distinguish betwee n true disordered populations and symmetric populations, we will use m+ s(t) and m− s(t). When the situation is non-ambiguous, we will use ms(t) alone, in order to improve the clarity of our plots. In Fig. 4, we have displayed m+ s(t),m− s(t) and ms(t) for the quasispecies in a fitness landscape of the type defined in Eq. (64). For a large os cillation period, T= 100, the quasispecies is at every point in time clearly cent ered around a single peak. The switch from one peak to the other happens ver y fast. When the landscape oscillates with a higher frequency, the trans ition time uses up a larger proportion of the total oscillation period. This mak es the transition from one peak to the other appear softer in the plots for smaller os cillation periods. For extremely small oscillations, the system sees the avera ge fitness landscape, 22/CC /BP /BD/BC/BC /CC /BP /BD/CC /BP /BC /BM /BD /CC /BP /BC /BM /BC/BD /B9/BD/BA/BC/B9/BC/BA/BI/B9/BC/BA/BE /BC/BA/BE/BC/BA/BI/BD/BA/BC/BC/BA/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC /B9/BD/BA/BC/B9/BC/BA/BI/B9/BC/BA/BE /BC/BA/BE/BC/BA/BI/BD/BA/BC/BC/BA/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC/B9/BD/BA/BC/B9/BC/BA/BI/B9/BC/BA/BE /BC/BA/BE/BC/BA/BI/BD/BA/BC/BC/BA/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC /B9/BD/BA/BC/B9/BC/BA/BI/B9/BC/BA/BE /BC/BA/BE/BC/BA/BI/BD/BA/BC/BC/BA/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC/D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D4/CW/CP/D7/CT Ꜷ /D4/CW/CP/D7/CT Ꜷ/D4/CW/CP/D7/CT Ꜷ /D4/CW/CP/D7/CT Ꜷ /D1 /B7/D7 /B4 Ꜷ /B5/D1/D7 /B4 Ꜷ /B5/D1 /A0/D7 /B4 Ꜷ /B5 Figure 4: Order parameters ms(t),m+ s(t),m− s(t) as a function of the oscillation phase φ= (tmodT)/Tin a landscape with two alternating peaks. The upper dashed line represents m+ s(t), the lower dashed line represents m− s(t), and the solid line represents ms(t). The sequence length is l= 10, and we have used R= 0.05 andn=T/∆t= 100 in all four examples. The parameters of the fitness lands cape areA0,stat=e2.4,ǫ= 2. which is a degenerate landscape with two peaks of equal heigh t. As noted above, the quasispecies becomes symmetric in such a landscape. In t he lower right plot of Fig. 4, for T= 0.01, we can identify this limiting behavior. Both m+ s(t) and m− s(t) are nearly constant over the whole oscillation period with an absolute value close to 0.5. The deviation from 0.5 stems from the finite valu e of the error rate, R= 0.05 in this example. We observe further that ms(t) lies very close to zero, thus wrongly indicating a disordered state. Note that the ab solute value of m± s(t) for a uniformly spread population lies for the parameters of this example at 0.12 according to Eq. (69). The observations from the landscape with two oscillating pe aks have to be interpreted in the light of the results of Schuster and Sweti na on static landscapes 23with two peaks [28]. They have found that if the peaks are far a way in Hamming distance (which is the case here), a quasispecies is general ly unable to occupy both peaks at the same time, unless they are of exactly the same hei ght and with the same neighborhood.1For two peaks with different heights, the quasispecies will for small Rgenerally form around the higher peak. For larger R, however, the quasispecies moves to the lower peak if this one has a higher m utational backflow from mutants, which is the case, for example, if the second pe ak is broader than the first one. The transition from the higher peak to the lower one with increasing R is very sharp, and can be considered as a phase transition. In a dynamic landscape with relatively slow changes, the quasispecies therefore s witches the peak quickly when the higher peak becomes the lower one and vice versa. The exact time at which the switch occurs depends of course on the error rate. The lower the error rate, the longer does the population rema in centered around the previously higher peak until it actually moves on to the n ew higher peak. Therefore, if we look at the system at a fixed phase, and change the error rate, the quasispecies does, for certain phases φ, undergo a transition similar to the one found in [28] for static landscapes. This is illustrated in F ig. 5, where we display the order parameter msas a function of the error rate R. At the beginning of the oscillation period, for φ= 0, the quasispecies is, for all error rates Rbelow the error threshold, dominated by the peak corresponding to ms=−1. This must be the case, as the replication coefficients of the two peaks inte rsect at φ= 0, so up to this point the quasispecies has not had a chance to build up around the other peak. For phases shortly after φ= 0, the quasispecies gains weight around the other peak, starting from the error threshold on downwards. Forφ= 0.15, for example, we observe a relatively sharp transition from the p eak corresponding to ms=−1 to the peak corresponding to ms+ 1 at R≈0.05. The transition then moves quickly towards R= 0, until the peak corresponding to ms= 1 dominates the quasispecies for all R. Forφ= 0.5, the replication coefficients intersect again, and the quasispecies is exactly the inverse of the one for φ= 0. 3.3.3 Two oscillating peaks with flat average landscape In Sec. 3.1.3, we have predicted a special phase diagram for l andscapes whose time average is completely flat. A particular realization of such a landscape is obtained if all replication coefficients are either set to a co nstant aor to a function a+bisin(ωt+δi), with arbitrary δiandbi< a. In comparison to the previous subsection, here we choose again a landscape in which a seque nce and its inverse 1This is only true for infinite populations, however. For finit e populations, one of the two peaks will always get lost eventually due to sampling fluctua tions. 24/B9/BD/BA/BC/B9/BC/BA/BI/B9/BC/BA/BE /BC/BA/BE/BC/BA/BI/BD/BA/BC/BC/BA/BC /BC/BA/BD /BC/BA/BE /BC/BA/BF /BC/BA/BG /BC/BA/BH/CT/D6/D6/D3/D6 /D6/CP/D8/CT /CA/D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D1 /D7 Ꜷ /BP /BC /BM /BC/BCꜶ /BP /BC /BM /BD/BCꜶ /BP /BC /BM /BD/BHꜶ /BP /BC /BM /BH/BC Figure 5: The order parameter msas a function of the error rate Rfor various oscillation phases φ= (tmodT)/T. The fitness landscape is identical to the one of Fig. 4, and the oscillation period is T= 100. Note that for φ= 0.10, the error threshold seems to have moved towards lower R, which is not the case. What we have instead is a symmetric population, as explained on page 22. A plot of m+ sor m− sreveals this immediately. However, we have not displayed su ch a plot here in order to enhance the clarity of this figure. are alternating, while all others remain constant. We set th e replication rates to A0(t) = 1 −bsinωt , (71a) Al(t) = 1 + bsinωt , (71b) Ai(t) = 1 for 0 < i < l . (71c) The order parameter of the quasispecies in such a landscape i s displayed in Fig. 6 as a function of Rfor various oscillation periods T. What is immediately apparent from the plot is the existence of a lower error thres hold in addition to the normal upper error threshold. This is in perfect agreeme nt with the phase diagram in Fig. 2b), which predicts such a lower error thresh old for landscapes with a flat average. With decreasing length Tof the oscillation period, the two thresholds approximate each other, reducing the region in w hich order can be seen. ForT= 20, the order parameter does not even reach the value ms= 0.1 anymore, and for T= 10, it would be indistinguishable from the R-axis in this plot. To be able to study the region of small Tin more detail, we have done an expansion of Xin terms of Tas given in Eq. (29), up to second order. The 25T= 30T= 40T= 90 T= 200.81.0 10−610−510−410−310−210−1order parameter ms error rate R0.6 0.4 0.2 0.0 Figure 6: The order parameter msas a function of Rin a landscape with two alternating peaks that average to a flat landscape [Eq. (71)] . In this case, ms= 0 corresponds always to true disorder, and therefore, we have refrained from display- ingm+ sandm− sin addition to ms, in order to enhance the clarity of the plot. The other parameters were l= 10, b= 9/10,φ= 0 and n=T/∆t= 100. corresponding integrals can be taken relatively easy for th is particular landscape. The details of the calculation are given in Appendix A. By com paring the results from this expansion with the results from the discrete appro ximation Eq. (52), this serves also as a test of the validity of Eq. (29). In Fig. 7, we have displayed the order parameter msobtained from the expan- sion of Xin terms of Tand from the discrete approximation of Xas a function of the phase φfor four different oscillation periods T. First of all, the order parameter clearly flattens out for T→0 (note that the ordinates are scaled differently in the four plots, which may obsfucate this fact on first glance). However, since the T2term in the expansion gives a time-dependent contribution for arbitrarily small T[Eq. (128)], we cannot define the transition point to complete disorder with rigor. But this is nothing ne w. The same applies to the standard error threshold in a static landscape. Analy tically, the order parameter never reaches zero for a finite string length land for R <0.5. This is related to the fact that the error transition is a surface tra nsition with complete wetting [30]. Since the surface is finite, the order paramete r indicating this surface transition remains always finite. Hence, the exact transiti on point can only be determined from the corresponding transition in the bulk. F or our purposes here, it suffices to note that for a finite oscillation period, here ab outT= 10, the order 26/A0 /BF /A2 /BD/BC /A0 /BI /A0 /BE /A2 /BD/BC /A0 /BI /A0 /BD /A2 /BD/BC /A0 /BI /BC /A2 /BD/BC /A0 /BI /BD /A2 /BD/BC /A0 /BI /BE /A2 /BD/BC /A0 /BI /BF /A2 /BD/BC /A0 /BI/BC/BA/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC /CC /BP /BC /BM /BC/BD/A0 /BF /A2 /BD/BC /A0 /BH /A0 /BE /A2 /BD/BC /A0 /BH /A0 /BD /A2 /BD/BC /A0 /BH /BC /A2 /BD/BC /A0 /BH /BD /A2 /BD/BC /A0 /BH /BE /A2 /BD/BC /A0 /BH /BF /A2 /BD/BC /A0 /BH/BC/BA/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC /CC /BP /BC /BM /BD/A0 /BF /A2 /BD/BC /A0 /BG /A0 /BE /A2 /BD/BC /A0 /BG /A0 /BD /A2 /BD/BC /A0 /BG /BC /A2 /BD/BC /A0 /BG /BD /A2 /BD/BC /A0 /BG /BE /A2 /BD/BC /A0 /BG /BF /A2 /BD/BC /A0 /BG/BC/BA/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC /CC /BP /BD/A0 /BG /A2 /BD/BC /A0 /BF /A0 /BE /A2 /BD/BC /A0 /BF /BC /A2 /BD/BC /A0 /BF /BE /A2 /BD/BC /A0 /BF /BG /A2 /BD/BC /A0 /BF/BC/BA/BC /BC/BA/BE /BC/BA/BG /BC/BA/BI /BC/BA/BK /BD/BA/BC /CC /BP /BD/BC/D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D1 /D7/D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D1 /D7 /D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D1 /D7/D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D1 /D7 /D4/CW/CP/D7/CT Ꜷ /D4/CW/CP/D7/CT Ꜷ/D4/CW/CP/D7/CT Ꜷ /D4/CW/CP/D7/CT Ꜷ Figure 7: The order parameter msfor a landscape in which a sequence and its inverse become alternatingly the master sequence. The erro r rate is R= 0.01. The solid lines give stem from the discrete approximation, the d otted lines stem from the expansion in terms of T, Eq. (29), evaluated up to second order. Clearly, the expansion Eq. (29) is only of use for relatively short oscill ation periods. parameter is almost zero for all replication rates R. This demonstrates that the phase diagram Fig. 2b) is indeed correct. Second of all, we observe that the expansion in terms of Tbreaks down for T larger than ≈1. This agrees well with our estimate for the radius of conver gence of the expansion given in Eq. (40), which guarantees converg ence only for T <1 in the present case. 4 Aperiodic or stochastic fitness landscapes Periodic fitness landscapes can be treated rather elegantly . We have been able to define a meaningful quasispecies, as well as we have been ab le to determine the general dynamics in the border regions of the parameter s pace. It would be desirable to obtain similar results for arbitrary dynamic l andscapes. After all, an aperiodic or stochastic change is much more realistic tha n an exactly periodic change. However, the definition of a time-dependent quasisp ecies is tightly con- nected to periodic fitness landscapes. For arbitrary change s, it does not make sense 27to speak of an asymptotic state. Regardless of that, we can de rive some results for the border regions of the parameter space. In Section 3.1, we derived the formal solution to Eq. (6), y(t0+τ) =∞/summationdisplay k=0τkWk(t0, τ)y(t0). (72) To first order in τ, the formal solution reads y(t0+τ) =y(t0) +τW1(t0, τ)y(t0). (73) Obviously, the composition of the sequence distribution ch anges very little over the interval [ t0, t0+τ] if the condition τ/vextenddouble/vextenddoubleW1(t0, τ)/vextenddouble/vextenddouble 1≪1 (74) is satisfied. This observation allows us to establish a gener al result for quickly changing fitness landscapes. If the landscape changes in suc h a way that for every interval of length τbeginning at time t0, the average W1(t0, τ) =1 τ/integraldisplayτ 0W(t0+τ1)dτ1 (75) is approximately the same for every t0, and the condition /⌊ard⌊lW1(t0, τ)/⌊ard⌊l ≪ 1/τ holds, then the system develops a quasispecies given by the n ormalized principal eigenvector of the average matrix W1(t0, τ). With “approximately the same” we mean that for two times t0andt1, the components of the averaged matrices satisfy /vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBig W1(t0, τ)/parenrightBig ij−/parenleftBig W1(t1, τ)/parenrightBig ij/vextendsingle/vextendsingle/vextendsingle/vextendsingle< ǫfor all i,j,t0,t1, (76) with a suitably small ǫ. In other words, if the fitness landscape changes very fast, but in stationary way, then the evolving population sees onl y the time-averaged fitness landscape. For the special case of R= 0, we can, as in Eq. (43), write the solution for the quasispecies equation as y(t) = exp/parenleftbigg/integraldisplayt t0[A(t′)−D(t′)]dt′/parenrightbigg y(t0). (77) Unlike in the case of a periodic landscape, however, this doe s not tell us the general behavior at R= 0, apart from the fact that for fast changes, the system sees the average fitness landscape. But we knew that already from Eq. ( 73). If we have the 280.5 0 error rate R time-averagedisorder fast slow quality of changes Figure 8: The appearance of a stochastic fitness landscape at the border regions of the parameter space. situation of a stochastic landscape with long time correlat ions, on the other hand, it is hard to make general statements. The reason for this is t hat from long time correlations, we cannot generally deduce that the system mu st be in a quasistatic state. It may be the case if, for example, the landscape chang es only rarely, but then drastically. On the other hand, one can easily come up wi th landscapes that are in a constant flux, and still display long time correl ations. Hence, there exists no direct equivalent to the large oscillation period case of periodic fitness landscapes for general stochastic landscapes. Neverthele ss, we can draw a diagram similar to Fig. 1, where on the x-axis we use the qualitative description “slow” and “fast” changes. Under “fast”, we subsume everything tha t satisfies the above stated conditions under which the system sees the average fit ness landscape, and under “slow” we subsume everything else, assuming that a par ameter exists that allows a smooth transition from the “fast” regime to the “slo w” regime. Then, the analogue to Fig. 1 is Fig. 8. Although this figure contains considerably less information than Fig. 1, the implications for actual landsc apes are more or less the same. Most real landscapes will have a regime that can be asso ciated with slow changes, and hence, we will typically observe phase diagram s of the type of either Fig. 2a) or b). 29As an example, consider the work of Nilsson and Snoad [19], an d its subsequent extension by Ronnewinkel et al. [22]. Nilsson and Snoad have studied a landscape in which a single peak performs a random walk through the sequ ence space. The peak jumps to a random neighboring position of hamming dista nce 1 whenever a time interval of length τhas elapsed. Ronnewinkel et al.have studied a very similar fitness landscape, but they have mainly been interested in de terministic movements of the peak that allow for the formal definition of a quasispec ies, similarly to the situation of periodic fitness landscapes in Section 3.1. Ron newinkel et al. could verify the results of Nilsson and Snoad on more fundamental t heoretical grounds. The parameter τin the jumping peak landscape determines whether the change s happen on a short or on a long time scale. If τis very large, the landscape is static most of the time, and the population has enough time to settle into equilibrium before the peak jumps to a new position. If τis very small, on the other hand, the peak has moved away long before the population has had the tim e to form a stable quasispecies. Nilsson and Snoad have found that, in additio n to the common error threshold at which the mutation rate becomes too high to allo w for quasispecies formation, another error threshold can be found at which the mutation rate be- comes too low to allow the population to adapt to the changing landscape. The region of the lower disordered phase grows with decreasing τ, until the lower and the higher error threshold coincide and no selection can tak e place anymore. This is clear from an intuitive point of view. The faster the peak m oves, the higher must the error rate be in order to allow the population to track the peak. Once the error rate needed to track the peak exceeds the highest error rate f or which selection is possible, everything breaks down and the population does no t feel any selective pressure any more. Nilsson and Snoad concluded therefore th at “dynamic land- scapes have strong constraints on evolvability”. However, this conclusion is not so straightforward if we reconsider their landscape from the v iewpoint of the general theory developed here. As we have pointed out several times s o far, the authorita- tive fitness landscape in the region of fast changes is the tim e averaged landscape. Thus, selection does not break down because of a fast changin g landscape itself, but it breaks down due to the neutrality of the time-averaged landscape in this particular case. If there was a region in the sequence space i n which the peak would assume a higher level than in the remaining sequence space, o r if the peak’s move- ments were confined to a small portion of the sequence space, w e would clearly see selection in these particular regions. This suggests the vi ewpoint that the time- averaged landscape gives the “regions of robustness” in the landscape, the regions in which even fast changes in the landscape do not destroy the quasispecies. 305 Finite Populations In the previous sections, we have been studying infinite popu lations exclusively. However, the huge genotype spaces that are generated even by moderately long sequences (there are already 1030different sequences of length 100, for example), will be almost empty for any realistic finite population. Whe n most of the possible sequences are not present in the population, the concentrat ion variables become useless, and the outcome of the differential equation formal ism may be completely different from the actual behavior of the population. For sta tic fitness landscapes, the effects of a finite population size are reasonably well und erstood. If the fitness landscape is very simple (a single peak landscape), the popu lation is reasonably well described by finite stochastic sampling from the infinit e population concen- trations. Moreover, the error threshold generally moves to wards smaller Rwith decreasing population size [20]. In a multi peak landscape, the finite population localizes relatively fast around one peak, and there it rema ins, with a dynamics similar to that in a single peak landscape. In the rare case th at a mutant discovers a higher peak, the population moves over to that peak, where i t remains again. The main difference between a finite and an infinite population on a landscape with many peaks is given by the fact that the infinite populati on will always build a quasispecies around the highest peak, whereas the finite po pulation may get stuck on a suboptimal peak. Above the error threshold, a finit e population starts to drift through the genotype space, irrespective of the lan dscape. A finite population on a dynamic landscape will of course show a similar behav- ior, but in addition to that, other effects come into play that are tightly connected to the dynamics of the landscape. The most important differen ce between static and dynamic landscapes is the possible existence of a tempor arily ordered phase in the latter case, and there we should expect the major new dy namic effects. In the infinite population limit, the temporarily ordered ph ase generates an alternating pattern of a fully developed quasispecies and a homogeneous sequence distribution. What changes if a finite population evolves in that phase? At those points in time when a quasispecies is developed, the finite po pulation’s sequence concentrations are given by stochastic sampling from the in finite population result, similarly to static landscapes. As soon as the quasispecies breaks down (and this may happen earlier than the infinite population equations pr edict, because of the error threshold’s shift to a lower error rate for a finite popu lation), the population starts to disperse over the landscape. Because of that, the p opulation may loose track of the peak it was centered about previously. Therefor e, when it enters again a time interval in which order should be seen, the population may not be able to form a quasispecies, thus effectively staying in the disorde red regime, or it may form a quasispecies at a different peak. In that way, the tempo rarily ordered phase 31can open up a third possibility for a population to leave a loc al peak, in addition to the escape via neutral paths or to entropy-barrier crossi ng, which are present exclusively in static landscapes [32]. 5.1 Numerical results The numerical results presented below have been obtained fr om a genetic algo- rithm with Nsequences per generation. We have used the following mutati on and selection scheme in order to stay as closely as possible with the Eigen model: 1. To all sequences iin time step t, we assign a probability to be selected and mutated, pi,mutate(t) =Ai(t)/summationtext i[1/∆t+Ai(t)−Di(t)]ni(t), (78) and a probability to be selected but not mutated, pi,select(t) =1/∆t−Di(t)/summationtext i[1/∆t+Ai(t)−Di(t)]ni(t). (79) Here, ∆ tis the length of one time step, and ni(t) is the number of sequences of type i. 2. From the set of probabilities {pi,mutate(t), pi,select(t)}, we choose Nsequences at random. These Nsequences are going to form the population in time stept+ ∆t. A sequence jthat is determined to be mutated is subsequently converted into sequence iaccording to the mutation matrix Qij. Note that we assume generally Di(t)<1 ∆tfor all i,t, (80) so that pi,select(t) defined in Eq. (79) is always positive. For an infinite population, the above described genetic algo rithm evolves ac- cording to the equation x(t+ ∆t) =G/parenleftbig x(t), t/parenrightbig , (81) where x(t) is the vector of concentrations at time t, and G(x, t) is the operator that maps a population at time tonto a population at time t+ 1, G(x, t) =[∆tW(t) +1]x et·/parenleftbig [∆tA(t)−∆tD(t) +1]x/parenrightbig. (82) 32Since we can replace the non-linear operator G(x, t) with a linear operator ˜G(y, t), ˜G(y, t) = [∆ tW(t) +1]y, (83) in Eq. (81), if we recover the true concentrations xvia x(t) =y(t) et·y(t), (84) we have a direct correspondence between the genetic algorit hm for an infinite population and the discrete quasispecies model, as can be se en by comparing Eq. (83) with Eq. (49). This implies in particular that for pe riodic landscapes, the expression for the monodromy matrix X(t0), Eq. (52), is exact. There is no approximation involved. For a finite population, it is still the operator G(x, t) that determines the dynamics. However, the deterministic description Eq. (81) has to be replaced by a probabilistic one, namely Wright-Fisher or multinomial s ampling. If Gi(x, t) denotes the iths component of the concentration vector in the next time st ep, the probability that a population x1= (m1, m2, . . .)/N,/summationtext imi=N, produces a population x2= (n1, n2, . . .)/N,/summationtext ini=N, in the next time step, is given by P(x1→x2, t) =N!/productdisplay iGi(x1, t)ni ni!. (85) A proof that the stochastic process described by Eq. (85) doe s indeed converge to the deterministic process Eq. (81) in the limit N→ ∞ has been given by van Nimwegen et al. [33]. 5.1.1 Loss of the master sequence Our first example of a finite population in a dynamic fitness lan dscape demon- strates what happens if in the temporarily ordered phase the master sequence is lost due to sampling fluctuations. In Fig. 9, we have presente d a run of a finite population consisting of N= 1000 sequences of length l= 15, initialized ran- domly at t= 0, in an oscillating Swetina-Schuster landscape. For a com parison, we have also plotted the theoretical result for an infinite po pulation. The infinite population is always in an ordered state, the order paramete rmsnever takes on values smaller than 0.2. Nevertheless, the finite populatio n is likely to loose the master sequence whenever the order parameter of the infinite population reaches its minimum, since the error threshold is shifted towards lo wer error rates for fi- nite populations. In our example run, the master sequence wa s lost at the end of 330.00.20.40.60.81.0 0 200 400 600 800 1000order parameter ms time tinfinite population finite population Figure 9: A single run of a population of N= 1000 sequences in the oscillating Swetina-Schuster landscape. The sequences had length l= 15. The other param- eters were A0(t) =e2.4exp(2 sin ωt),Ai= 1 for i >0,R= 0.06,T= 100, ∆ t= 1. The dashed line indicates the theoretical result for an infin ite population. the first oscillation period, but it was rediscovered shortl y afterwards, so that the population could follow the infinite population dynamics fo r most of the second oscillation period as well. Right after a loss of the master s equence, the probability to rediscover the master has its highest value, because the p opulation is still cen- tered around the master sequence. Once the population has ha d the time to drift away from the position of the master sequence, the probabili ty of a rediscovery drops rapidly. This is what happened at the end of the second o scillation period. The population completely lost track of the master sequence , and it took the pop- ulation more than 4 oscillation periods to rediscover it. Th is is the main difference between a finite and an infinite population in the temporarily ordered phase. For an infinite population, the interval of disorder has the same well defined length in each oscillation period, whereas for a finite population, once the population has entered the disordered state, it may take a long time unti l an ordered state is reached again. In fact, for the case of a single peak in a ver y large sequence space and a small population, the peak may effectively be lost forever once it has disappeared from the population. This can be seen as a dynamic version of Muller’s ratchet [18] . A trait whose advantageous influence on the overall fitness of an individua l is reduced at some point (it is not necessary that the trait becomes completely neutral or even dele- terious) may get lost from the population due to sampling fluc tuations. If then at 34-1.0-0.6-0.20.20.61.0 0 500 1000 1500 2000 time torder parameter ms Figure 10: A single run of a population of N= 1000 sequences in a landscape as given in Eq. (64). All parameters were identical to the setup of Fig. 9. The dashed line again indicates the theoretical result for an infinite p opulation. a later stage this trait becomes again very advantageous, it is not available to the population anymore, until it is rediscovered independentl y. However, a rediscovery may be very unlikely. 5.1.2 Persistency A second aspect of a finite population in a dynamic landscape i s persistency. This means, a finite population may not be able to follow the change s in the landscape, although the infinite population limit predicts this. An exa mple of that effect is given in Fig. 10. There, we have two alternating peaks at oppo site corners of the boolean hypercube, as given by Eq. (64). Note that the peaks’ minimal height is relatively small, but still larger than the rest of the lands cape’s height. In fact, all parameters are identical to the situation shown in Fig. 9, so that this figure can be seen as an example of the dynamics around one of the peaks in Fig. 10. The infinite population result in Fig. 10 predicts that the popul ation should move on to the other peak whenever this peak becomes the higher one. H owever, the finite population does not follow this scheme. It stays localized a round one of the two peaks for a long time. A finite population does not, unlike an i nfinite population, occupy all possible points in the sequence space at the same t ime. Therefore, if a peak grows at a distance too far from the currently occupied p eak, no sequence in the population is there to exploit the advantage, and hence t he new opportunity 35goes undetected. Only if the population looses track of the fi rst peak, which is possible because of the temporarily ordered phase, it can di scover the second peak during its random drift. In the run of Fig. 10, this has happen ed two times. The first time, the population had discovered the alternative pe ak at the end of the drift, and the second time, it had again rediscovered this sa me peak. The situation of a finite population in a dynamic landscape wi th several growing and shrinking peaks can be compared to its situation in a rugg ed, but static landscape. In the latter case, once the population has reach ed a local optimum it remains there, unless a rare mutation opens the possibility to move to a new, higher peak. The same applies to the dynamic situation. But in addit ion, the fluctuations and oscillations of the fitness values destabilize the popul ation on local optima, and allow it to continue its search for other local optima. If the landscape’s dynamics is such that the population, by following the local optima, mov es into regions of low average fitness (observed e.g. in [35]), the landscape might be called “deceptive”, and in the opposite case, it might be called “well-behaved”. 5.2 A finite population on a simple periodic fitness land- scape In the above examples, we saw that the time it takes until the m aster is rediscov- ered, once it has been lost in the temporarily ordered phase, may be much larger than the period length of the landscape. Hence, for several p eriods, the popula- tion does not follow the infinite population results, but rem ains in a disordered state. It would be desirable to have an analytic description of this behavior, and, in particular, to have an estimate of the probability with wh ich a complete period is skipped, i.e., with which the master sequence is missed fo r a whole oscillation period. Unfortunately, the continuous time dependency of t he master sequence’s replication rate employed in Sec. 5.1, A0(t) =A0,statexp(ǫsinωt), (86) renders the corresponding calculations very complicated. Therefore, in order not to get too distracted by technical details in the calculatio n, we study in this section a simplified fitness landscape that displays a temporarily or dered phase similar to Fig. 9, but that is much easier to handle analytically. For a fi tness landscape such as Eq. (86), we can—for sufficiently high error rate R—divide the oscillation period into two intervals. During the first interval I1, of length T1, the population is in an ordered state provided that the master sequence is presen t in the population, and during the second interval I2, of length T2, the population is in a disordered state, even if the master sequence is present. The beginning of the first interval 36need not coincide with the beginning of the oscillation peri od, but after a suitable shift of the time origin, this is always the case. Note that fo r a finite population, the second interval is larger than predicted by the infinite p opulation limit, and it may exist even if the infinite population limit predicts a len gthT2= 0, because the error threshold is shifted towards smaller error rates for fi nite populations [20, 34]. This can be seen clearly in Fig. 9, where the infinite populati on limit predicts T2= 0, but the master is lost anyway because of sampling fluctuat ions. Our approximation here is to keep the fitness landscape const ant during the intervals I1andI2. During the interval I1, we let the master replicate with rate A0≫1, while all other sequences replicate with A= 1. During the second interval on the other hand, the fitness landscape becomes flat. Then, al l sequences replicate withA= 1. We continue to study the discrete process and set ∆ t= 1, so that T1andT2give the number of time steps spent in each interval. In summa ry, the replication rate A0(t) satisfies A0(t) =/braceleftbigg a:φ≤T1 1: else .(87) In order to get expressions that can be easily treated even fo r a finite popula- tion, we use the error tail approximation introduced in [20] . In that approximation, the state of the system is fully described by the concentrati on of the master se- quence. All other sequences are assumed to be uniformly spre ad over the remaining genotype space. This approximation underestimates the mut ational backflow into the master sequence, and hence it underestimates the concen tration of the mas- ter itself, but this small deviation can be accepted in the li ght of the enormous simplifications in the calculations. Before we have a look at the finite population dynamics, let us quickly study the infinite population limit. We express the state of the sys tem at time tby a vector x(t) = (x0(t), x1(t))t, where x0(t) gives the concentration of the master sequence, and x1(t) = 1−x0(t) gives the total concentration of all other sequences. The generation operator G(x, t) maps the population at time tinto the population at time t+ 1, i.e., x(t+ 1) = G/parenleftbig x(t), t/parenrightbig . (88) Here, G(x, t) is given by G(x, t) =[QA(t) +1]x A0(t)x0+x1+ 1. (89) Qis the 2 ×2 matrix Q=/parenleftBigg (1−R)l 1−(1−R)l 2l−1 1−(1−R)l1−1−(1−R)l 2l−1/parenrightBigg , (90) 37andA(t) = diag( A0(t),1). The linear operator ˜G(t) =QA(t) +1describes the evolution of the variables y(t), y(t+ 1) = ˜G(t)y(t), (91) which map into the original variables via x(t) =y(t) et·y(t),et= (1,1). (92) Hence, the eigensystem of ˜Gfully describes the time evolution of x(t). For the eigenvalues of ˜G, we find λ0,1=1 2/bracketleftbigg ˜G00+˜G11±/radicalBig (˜G00−˜G11)2+ 4˜G01˜G10/bracketrightbigg , (93) where the plus sign corresponds to the index 0, and the minus s ign corresponds to the index 1. The eigenvectors are φ0,1=1 1 +ξ±(1, ξ±)t, (94) with ξ±=˜G00−˜G11 2˜G01±1 ˜G01/radicalbigg 1 4(˜G00−˜G11)2+˜G01˜G10. (95) Of course, the eigenvalues and the eigenvectors are differen t for the two intervals I1andI2. For the first interval, inserting the explicit expressions of˜Gijinto Eqs. (93)–(95) does not lead to a substantial simplification of the expressions, so we leave this out here. For the second interval, however, we fi nd for the eigenvalues λ(2) 0= 2, (96a) λ(2) 1= 2−1−(1−R)l 1−2−l, (96b) and for the eigenvectors φ(2) 0= (2−l,1−2−l)t, (97a) φ(2) 1= (1,−1)t. (97b) The superscript (2) indicates that these results are only va lid for the interval I2. From the above expressions, we get a simple formula for the ev olution of the master’s concentration during the interval I2. Let the interval start at time t, and 38let the concentration of the master at that moment in time be x0(t). Then we find ntime steps later x0(t+n) =α0φ(2) 0+α1/parenleftBig λ(2) 1/λ(2) 0/parenrightBign φ(2) 1 α0(et·φ(2) 0) +α1/parenleftBig λ(2) 1/λ(2) 0/parenrightBign (et·φ(2) 1), (98) where α0andα1have to be chosen such that x0(t) =α0φ(2) 0+α1φ(2) 1. (99) After solving Eq. (99) for α0andα1and inserting everything back into Eq. (98), we end up with x0(t+n) = 2−l+/bracketleftbig x0(t)−2−l/bracketrightbig/parenleftbigg 1−1−(1−R)l 2(1−2−l)/parenrightbiggn . (100) This formula is sufficiently close to the solution obtained fr om diagonalization of the full 2l×2lmatrix Qin a flat landscape, and can be considered a good approximation to the actual infinite population dynamics [2 1]. In principle, a similar formula can be derived for the interval I1, but again, the expressions become very complicated, and do not lead to any new insight, so we lea ve this out here. Equation (100) demonstrates that a macroscopic proportion of the master se- quence that may have built up during the interval I1quickly decays to the expected concentration in a flat landscape, 2−l. Now we address finite populations. We assume the duration of t he interval I1 is long enough so that the quasispecies can form. The asympto tic concentration of the master sequence can then be calculated from a birth and death process as done in [20]. The alternative diffusion approximation used i n [34] is of no use here because it allows only replication rates A0of the form A0= 1 + ǫwith a small ǫ[11]. In [20], the probabilities pkto find the master sequence ktimes in the asymptotic distribution are given by pk=˜pk/summationtextN i=0˜piwith ˜pk=µ+ k−1 µ− k˜pk−1and ˜p0= 1. (101) The probabilities µ+ iandµ− iread here µ+ i=N−i N/parenleftbigg/bracketleftBig ˜G(1) 00−1/bracketrightBigi N+˜G(1) 01N−i N/parenrightbigg (102) and µ− i=i N/parenleftbigg ˜G(1) 10i N+/bracketleftBig ˜G(1) 11−1/bracketrightBigN−i N/parenrightbigg . (103) 39The expected asymptotic concentration becomes x0(∞) =1 NN/summationdisplay k=0k pk. (104) Unfortunately, there exists no analytic expression for x0(∞). However, its value is easily computed numerically. By our above assumption on t he length of the interval I1, we can suppose that at the end of I1, the concentration of x0is given byx0(∞). During the interval I2, the concentration of the master will then decay. 5.2.1 The probability to skip one period If at the end of the interval I2the master sequence has been lost because of sampling fluctuations, and if in addition to that the correla tions in the population have decayed so far that we can assume maximum entropy, what i s the probability that the master sequence is rediscovered in the following in terval I1? The process of rediscovering the master consists of two steps. The maste r sequence has to be generated through mutation, and then it has to be fixated in the population, i.e., it must not get lost again due to sampling fluctuations. First of all, we calculate the probability Pmissthat the master is not generated in one time step. This corresponds to the probability that the multinomial sa mpling of the operator G(1)(x) maps a population x= (0,1)tinto itself. Hence, we have Pmiss=N!1/productdisplay i=0G(1) i(x)ni ni! =/parenleftbiggQ11+ 1 2/parenrightbiggN =/bracketleftbigg 1−1−(1−R)l 2l+1−2/bracketrightbiggN . (105) G(1) i(x) stands for the iths component of the outcome of G(1)(x). The probability that the master sequence gets fixated needs m ore work. Let π(x, t) denote the probability that the master sequence has reache d its asymptotic concentration at time t, given that it had the initial concentration xat time t= 0. The asymptotic concentration is given by x0(∞) defined in Eq. (104). Then, the probability π(x, t) satisfies to second order the backward Fokker-Planck equat ion ∂π(x, t) ∂t=/an}⌊ra⌋ketle{tdx0/an}⌊ra⌋ketri}ht∂π(x, t) ∂x+/an}⌊ra⌋ketle{t(dx0)2/an}⌊ra⌋ketri}ht 2∂2π(x, t) ∂x2. (106) The moments /an}⌊ra⌋ketle{tdx0/an}⌊ra⌋ketri}htand/an}⌊ra⌋ketle{t(dx0)2/an}⌊ra⌋ketri}htcan be calculated similarly to the calculations 40in [33], and we find /an}⌊ra⌋ketle{tdx0/an}⌊ra⌋ketri}ht=/parenleftbigg1 2λ(1) 0−1/parenrightbigg x0=:γx0, (107) /an}⌊ra⌋ketle{t(dx0)2/an}⌊ra⌋ketri}ht=x0(1−x0) N. (108) The solution to Eq. (106) for t→ ∞ is then obtained as in [33], and we find π∞:=π/parenleftbigg1 N,∞/parenrightbigg =1−/parenleftbig 1−1 N/parenrightbig2Nγ+1 1−(1−x0(∞))2Nγ+1(109) ≈1−e−2γ. (110) As the initial concentration of x0, we have used 1 /N, since it is—for the param- eter settings we are interested in—extremely unlikely that more than one master sequence is generated in one time step. The approximation in the second line is only valid for large population sizes. It generally underes timates the true value of π∞. Note that the expression for π∞given in Eq. (109) reaches the value 1 for the (relatively large) error rate Rclose to the error threshold for which x0(∞) = 1/N. Naively, one would assume that π∞decays with increasing error rate, since mutations increase the risk that good traits are lost, and in deed the approximate expression in Eq. (110) decays with increasing error rate. H owever, since π∞is the probability that the master sequence reaches its equilibri um concentration, and the equilibrium concentration vanishes close to the error t hreshold, π∞must rise to 1 at the error threshold. We have done some measurements with a finite population to tes t the validity of Eq. (109). For a number of runs, we have initialized the pop ulation at random, but with exactly one instance of the master sequence, and hav e counted how often the master’s concentration reached x0(∞) and how often it reached 0. The results of these measurements are shown in Fig. 11. Clearly, numeric al and analytical results are in good agreement. Finally, we need an estimate of the time τit takes from the time step the master sequence is discovered to the time step in which the equilibr ium concentration is reached for the first time. We follow again the calculations i n [33], and assume that the process of fixation can be treated in the infinite popu lation limit. From Eq. (89), we obtain for the change in the variable x0(t) during one time step in the interval I1 x0(t+ 1)−x0(t) =−(a−1)x0(t)2+ (Q00a−Q01−1)x0(t) +Q01 (a−1)x0(t) + 2.(111) 41A0= 4A0= 10 A0= 2 N= 500, l= 150.81.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 error rate R0.00.6 0.4 0.2 0.0fixation probability π∞ Figure 11: The fixation probability π∞as a function of the error rate Rfor three different heights of the peak. The solid lines stem from the an alytic expression Eq. (109), and the dotted lines stem from measurements on a fin ite population consisting of N= 500 sequences. We approximate this with a differential equation, dx0(t) dt≈x0(t+ 1)−x0(t), (112) which we can solve for tas a function of x0, and obtain t=b+ 4 z/parenleftbigg Atanhb−2sx0 z−Atanhb−2s/N z/parenrightbigg −1 2ln−sx2 0+bx0+Q01 −s/N2+b/N+Q01, (113) with s=a−1, (114) b=Q00a−Q01−1, (115) and z=/radicalbig 4sQ01+b2. (116) Therefore, for the estimated time it takes until the master s equence gets fixated we will use in the following τ=t/parenleftbig x0(∞)/parenrightbig , (117) 42withx0(∞) given in Eq. (104). Hence, we can now calculate the probability that the populat ion skips a whole period, i.e., that it does not find and fixate the master during one interval I1. The probability that the master sequence has concentration zer o at the beginning of the interval I1is (1−1/2l)N. Therefore, the probability that the master sequence is not fixated in the first time step reads 1−/bracketleftBigg 1−/parenleftbigg 1−1 2l/parenrightbiggN/bracketrightBigg π∞. (118) The probability that the master sequence does not get found a nd subsequently fixated in a subsequent time step is given by 1−(1−Pmiss)π∞. (119) Now, if the master sequence is found, it will roughly take the timeτgiven in Eq. (117) until the equilibrium concentration is reached. T herefore, if the master sequence is not found during the first T1−τtime steps, it normally will not reach the equilibrium concentration anymore in that period . Therefore, in order to calculate the probability Pskip(T1) that the whole interval I1is skipped, we have to consider only the first T1−τtime steps of I1. In case that T1< τ, we have Pskip(T1)≈1. We have only approximate equality because τis the average time until fixation occurs. In rare cases, the fixation may happen m uch faster. Of the T1−τtime steps, the first one is different because in that time step we do not know whether the master sequence is present or not, w hereas for the remaining T1−τ−1 time steps, we may assume that the master sequence is not present if fixation has not occurred. Therefore, we find Pskip(T1) =/parenleftBigg 1−/bracketleftBigg 1−/parenleftbigg 1−1 2l/parenrightbiggN/bracketrightBigg π∞/parenrightBigg [1−(1−Pmiss)π∞]T1−τ−1(120) ≈1−/bracketleftBig 1−/parenleftbig 1−1 2l/parenrightbigN/bracketrightBig π∞ 1−(1−Pmiss)π∞exp [−(T1−τ) (1−Pmiss)π∞].(121) Figure 12 shows a comparison between this result and numeric al measure- ments. The measurements were taken by letting a randomly ini tialized population evolve in a flat landscape for 100 generations, and then recor ding the time it took the population to find and fixate a peak that was switched on in g eneration 101. We observe that the analytic expression for Pskip(T1) predicts the right order of magnitude and the right functional dependency on T1, but that it generally un- derestimates the exact value. Since Eq. (120) contains thre e quantities for which 43N= 100 N= 400 N= 1000 0.010.11 0 100 200 300 400 500 600 length of on-period T1probability Pskip(T1) measured analytic analytic with num. π∞error rate R= 0.05 N= 100 N= 400 N= 1000 0.010.11 0 100 200 300 400 500 600 length of on-period T1probability Pskip(T1) measured analytic analytic with num. π∞error rate R= 0.08 Figure 12: The probability Pskip(T1) that the population skips a whole period without fixating the master sequence, as a function of the len gth of the interval I1, for several different settings of NandR. The string length is l= 15. 44we have only approximative expressions, namely Pmiss,π∞, andτ, at first it is not clear from where these discrepancies arise. However, a syst ematic check quickly reveals the main cause of the discrepancies. First of all, no te that τmerely shifts the curve to the right. Since the measured and the analytic cu rves reach the value 1 at very much the same positions in Fig. 12, we can assume that τ, as given by Eq. (117), is accurate enough for our purposes here. Now cons ider the quantity π∞. In Fig. 11, we saw that our expression for π∞generally gives a good estimate of the true value, but that there are some deviations. To chec k whether these deviations are responsible for the discrepancies visible i n Fig. 12, we have addi- tionally displayed Pskip(T1) with π∞determined from measurements. We find that the usage of the true value of π∞enhances the quality of Eq. (120), in particular for larger error rates. For small error rates, however, it do es not help much. More- over, the analytic expression is generally getting worse fo r smaller error rates. As a conclusion, the main problems arise from the expression fo rPmiss, Eq. (105). We have derived Pmissunder a maximum entropy assumption, i.e., we have assumed that all mutants are distributed homogeneously over the seq uence space. Under this assumption, the probability to find the master is exactl y the same in every time step. But in reality, the population collapses very rap idly, even in a neu- tral landscape, and then moves about as a cluster whose radiu s is determined by the error rate. This introduces very long range time correla tions in a population evolving in a flat landscape [5]. In particular for small erro r rates, the cluster is very small, and this can increase the probability Pmisssubstantially. Note that this effect corresponds to the underestimation of epoch duration s that van Nimwegen et al. found in their analysis of the Royal Road genetic algorithm [ 33]. An exact treatment of this effect would probably have to be done along t he lines of [5]. Unfortunately, we cannot simply use their expressions here , because of the term +1present in our definition of the operator G(x, t) [Eq. (89)]. In order to check the hypothesis that the violation of the max imum entropy condition causes the main discrepancies shown in Fig. 12, we did some additional measurements in which we dispersed the population “by hand” over the complete sequence space except the master in every time step in which t he master sequence was not discovered. With this setup, we found a very good agre ement between the numerical and the analytical results. Since it is the population’s collapse into a small cluster th at causes the de- viations between Eq. (120) and the measured Pskip(T1), it is clear that the true Pskip(T1) must always be larger than predicted by Eq. (120). Therefor e, we can use that equation as a lower bound on the true value. We notice that Pskip(T1) decays exponentially. This means that the probability 45to find the peak in one oscillation period, Pfind(T1) = 1 −Pskip(T1), (122) approaches 1 for large T1. This is due to the fact that the peak will certainly be rediscovered if only we wait long enough. However, the model we are studying here is that of a peak that gets switched on and off alternatingly, a nd for which each “on”-period is of fixed length T1. In that case, the probability to rediscover the peak within one oscillation period can be extremely small, a s we are going to see now.Pskip(T1) decays with a rate of (1 −Pmiss)π∞. We can neglect π∞here, as it is of the order of one. Then, the decay rate is for fixed Nand large lapproximately given by 1−Pmiss≈N1−(1−R)l 2l+1, (123) i.e., it decays as 2−l. This implies in turn that already for string lengths of 50–6 0 (which can be considered a rough lower bound for typical DNA s equence lengths) and moderate NandR, we have Pfind(T1)≈0 for moderate T1. Hence, in many cases it is extremely unlikely that the peak is rediscovered at all. The above conclusion is of course tightly connected to the fa ct that we have studied a landscape with a single advantageous sequence. In the other extreme of a mount Fujiama landscape, in which the population can sen se the peak from every position in the sequence space, the conclusions would look differently. Note, however, that neither the single sharp peak landscape nor th e mount Fujiama landscape are realistic landscapes. In a realistic, high-d imensional rugged land- scape, it is probably valid to assume that local optima, once they are lost from the population, are never rediscovered. In such situations, dy namic fitness landscapes can induce the loss of a local optimum, and thus, they can acce lerate Muller’s ratchet[18] like effects. 6 Conclusions In this paper, we have been able to derive several very genera l results on landscapes with periodic time dependency. First of all, a quasispecies can be defined by means of the monodromy matrix. This means that after a sufficiently l ong time, the state of the system depends only on the phase φ= (tmodT)/Tof the oscillation, but it does not depend on the absolute time tany more. Therefore, in periodic fitness landscapes, the quasispecies is not a fixed mixture of sequen ce concentrations. Instead, it is a T-periodic function of mixtures of sequence concentrations . We 46have given an expansion of the monodromy matrix in terms of th e oscillation period T, which leads to an extremely simple description of the syste m for very high oscillation frequencies. Namely—if we assume the muta tion matrix remains constant all the time—the time-averaged fitness landscape c ompletely determines the behavior of the system; the system becomes indistinguis hable from one in a static landscape. This leads to the important conclusion th at selection never ceases to exist, no matter how fast the landscape changes. The only e xception to this rule is generated by dynamic landscapes that have a complete ly flat average. In that case, the system behaves for very fast changes as being i n a flat landscape, which is indistinguishable from the behavior of a system abo ve the error threshold. Therefore, if the average landscape is flat, selection will b reak down if the changes occur with a frequency higher than some critical frequency ω∗= 2π/T∗. For very slow changes, on the other hand, the system is virtually in equilibrium all the time. This leads generally to a time dependent error thre shold R∗(t). For mutation rates Rsuch that min tR∗(t)< R < max tR∗(t), the system is below the error threshold for some times t, and it is past the error threshold for other times. We have dubbed this region of the parameter space the t emporarily ordered phase, as we see alternating patterns of order and disorder i n that phase (in the infinite population limit). We found these general consider ations to be in complete agreement with all example landscapes that we studied. For the case of non-periodic landscapes, we have argued that the main conclu- sions remain valid, even if our mathematical formalism is no t generally applicable in that case. Fast changes in the landscape will average out, whereas slow changes lead to a quasistatic adaption of the quasispecies to the cur rent landscape. With these concepts, it has been possible to give an explanation f or the occurrence of a lower error threshold in the work of Nilsson and Snoad [19]. While the molecular concentrations become T-periodic for t→ ∞ in the in- finite population limit, this is not necessarily the case whe n we consider finite populations. In the temporarily ordered phase, after a popu lation has made the transition to the disordered state, it is not said that it tra nsitions back to order the same moment the infinite population would. Rather the opposi te is the case. Once the population has lost the ordered state, it is often hard fo r it to return there. From a very simple analytical model, we have found that the pr obability that the ordered state is not rediscovered in one oscillation period decays exponentially in the length of the interval in which order is possible at all. T he decay constant, however, is extremely small for large l, and therefore the rediscovery can become very unlikely. In more complex landscapes, this can lead to a n acceleration of Muller’s ratchet. Throughout this paper, we have assumed that mutations arise in the copy pro- 47cess. An equally valid assumption is that of mutations arisi ng on a per-unit-time basis (cosmic ray mutations), as opposed to the per generati on basis implied by copy mutations. With the latter assumption, one has to study the parallel muta- tion and selection equations [3] instead of Eigen’s equatio ns. Since these equations can be linearized in the same way as the quasispecies equatio ns, the formalism we developed applies also for these equations. The only diffe rence between the two types of equations is that in the parallel case, the mutat ion matrix Qand the replication matrix Aare added, whereas in the quasispecies case they are multiplied. In future work, it should be tried to obtain an improved under standing of the properties of the monodromy matrix. In particular, an expan sion of that matrix in the error rate Rwould certainly be valuable. A High-frequency expansion of X (t)for a land- scape with two alternating master sequences With Eq. (29), we have given an expansion of the monodromy mat rix for periodic landscapes, X(t0), in terms of the period length T. Here, we want to calculate the expansion explicitly up to second order for an example la ndscape. In that landscape, there are two sequences (without loss of general ization, we assume them to be i= 0 and i= 1) that become alternatingly the master sequence. The replication rates are A0(t) =a+bsin(ωt), (124a) A1(t) =c+dsin(ωt), (124b) Ai(t) = 1 for all i >1. (124c) The decay rates are set to zero. With vanishing decay rates, t he matrix W(t) reduces to QA(t), and as a consequence, we can write the nths average Wk(t) as /parenleftbig Wk(t)/parenrightbig ij=/summationdisplay ν1/summationdisplay ν2· · ·/summationdisplay νk−1Qiν1Qν1ν2· · ·Qνk−1jAν1,ν2,...νk−1,j(t) (125) with the generalized replication coefficients Aν1,ν2,...νk−1,j(t) =1 Tk/integraldisplayT 0Aiν1(t0+τ1)/integraldisplayτ1 0Aν2(t0+τ2) · · ·/integraldisplayτk−2 0Aνk−1(t0+τk−1)/integraldisplayτk−1 0Aj(t0+τk)dτ1dτ2· · ·dτk.(126) 48For the landscape given in Eq. (124), the first order tensor of the generalized replication coefficients has three independent elements, wh ich are (assuming i >1) A0(t) =a , (127a) A1(t) =c , (127b) Ai(t) = 1. (127c) The second order tensor has already 9 independent entries. A fter some algebra, we obtain (assuming again i >1) A00(t) =a2 2, (128a) A01(t) =ac 2+ad−bc 2πcos(ωt), (128b) A0i(t) =a 2−b 2πcos(ωt), (128c) A10(t) =ac 2−ad−bc 2πcos(ωt), (128d) A11(t) =c2 2, (128e) A1i(t) =c 2−d 2πcos(ωt), (128f) Ai0(t) =a 2+b 2πcos(ωt), (128g) Ai1(t) =c 2+d 2πcos(ωt), (128h) Aii(t) =1 2. (128i) In principle, the generalized replication coefficients Aν1,ν2,...νk−1,j(t) can be calcu- lated to arbitrary order for the landscape given in Eq. (124) . However, the third order tensor has already 27 independent entries, and with ev ery higher order, the number of independent entries triples. References [1] Chris Adami and C. Titus Brown. Evolutionary learning in the 2D Artificial Life system ’Avida’. In Rodney A. 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arXiv:physics/9912013v1 [physics.atom-ph] 4 Dec 1999Atomic collisions and sonoluminescence Leszek Motyka1and Mariusz Sadzikowski2 1) Institute of Physics, Jagiellonian University, Reymont a 4, 30-059 Krak´ ow, Poland 2) Institute of Nuclear Physics, Radzikowskiego 152, 31-34 2 Krak´ ow, Poland (November, 1999) We consider inelastic collisions between atoms of different kinds as a potential source of photons in the sonoluminescence phenom ena. We esti- mate the total energy emitted in one flash and the shape of the s pectrum and find a rough agreement between the results of our calculat ion and the experimental data. We conclude that the atomic collisio ns might be a candidate for the light-emitting mechanism for sonolumin escence and discuss the implications. PACS numbers: 34.50.-s, 32.80.Cy, 78.60.Mq The sonoluminescence is a mechanism of the conversion of sou nd energy into the energy of light emitted in a picosecond flash. This phenomeno n was discovered a long time ago (the emission of light from the water was observed in [1]) but only recently has attracted much attention because it becomes possible to tra p a single bubble of a gas mixture in a sound field [2] (this is so called Single Bubble So noluminescence — SBSL or just SL). The detailed review of the topic can be found in [3 ]. The mechanism of sonoluminescence has not been definitely understood so far. There are several possible explanations, each of them possessing its own difficulties. L et us briefly review some of the models. One of the most promising mechanism is the Bremsstrahlung ra diation from ionised regions [4,5]. These regions are created by shock waves form ed during the collapse of the bubble. This model succeeds, in particular, in prediction o f the total radiation of energy per flash. However, the presence of the plasma would imply the sensitivity of this light- emitting mechanism to the external magnetic field [6], which is not observed. Another possible explanation is given by the model of blackbody radi ation [7]. The spectrum of the SL is well fitted by the blackbody radiation spectrum of temperature 25000 K (in water of temperature 22oC [8]). However, the time dependence of SL spectrum is independent of the color of the emitted light, which contrad icts the blackbody radiation model expectations [9]. Other models refer to purely quantu m effects. These are: the “dynamical Casimir effect” suggested by Schwinger [10] and t he Unruh effect elaborated in more details by Eberlein [11]. Unfortunately, the dynami cal Casimir effect gives ambiguous predictions depending on the renormalization pr ocedure that one chooses, while the Unruh effect gives the power of radiation too low by s ome orders of magnitude. 1Yet another model describes the SL radiation as a radiation e mitted from molecular collisions [12]. This model properly predicts the shape of t he spectrum, however it tends to overestimate the magnitude of the emitted energy if one us es the realistic values of the radiating volume and the emission time. Besides, recent developments suggest that due to sonochemical reactions mainly noble gases are left in side the bubble [13], which if true, may be troublesome for the mechanism of SL. Since non e of the mechanisms is fully satisfactory we may, in search for an alternative cand idate, address the question if the inelastic collisions between atoms may be the source o f light in sonoluminescence . Thus we consider in this article the exclusive reaction A1+A2→A1+A2+γ of photon production in which the atoms remain in their groun d state. There are interesting features concerning these reactions which enc ourage to study them in more detail: (i) The featureless spectrum of SL suggests that the emissio n of light from excited atoms do not play a crucial role in the phenomenon. (ii) The spectrum of photons emitted in atomic collisions ca n behave similarly to that of SL. Indeed, the emission of photons with a wavelength grea ter than the size of the atom is suppressed due to the mutual cancellations between t he contributions of the nucleus and the electronic cloud (c.f. the Rayleigh scatter ing process). The short wavelength photons are absent because of the cut-off on the en ergy available in the collision. In the case of SL from an air bubble in water the analysis of the scattering of the oxygen or nitrogen atoms on the atoms of argon should be pe rformed. The exact calculation for this process is extremely involved thus, us ing a simplified model, we estimate the order of magnitude of the cross-section and the shape of the spectrum. Hence, let us start from the process of scattering of a hydrog en atom in the Coulomb field of an infinitely heavy source, accompanied by emission o f light. After obtaining the cross-section we shall argue that this elementary proce ss has similar features to the atomic collision and we shall use the results to estimate the number of photons in the SL flash. We employ the standard non-relativistic hamiltonian whose leading part H0con- tains the operators of the kinetic energy and the electron-p roton interaction. The perturbation H1in the Coulomb gauge, after neglecting A2terms, take the following form: H1(re,rp) =−eA(re)pe me+eA(rp)pp mp−Ze2 re+Ze2 rp, (1) where re(rp) and pe(pp) are the position and momentum of the electron (proton), the source of the potential is situated in the origin and carries the electric charge of Ze, Ak(r) =εkeikr√ Va+ ε,kis the vector potential of the emitted plain wave with moment umk and the polarisation vector given by εk. The matrix element of the creation operator in the box is given by /angbracketleftγ(ε,k)|a+ ε,k|0/angbracketright=/radicalBig 2π k. The eigenstates of H0are given in the position space by the wave functions Ψ P,n(R,r) = 1/√ Vexp(−iP R)ψn(r) with P being the atom momentum, ψn(r) — the electron-proton wave function (for a bound 2or continuum state n) in the centre-of-mass frame and r=re−rp,R=αrp+βre, α=mp/(me+mp),β=me/(me+mp). For future use, we also introduce the mass of the atom MH=me+mp. We shall calculate the amplitude Mof the transition: ΨP,1S→ΨP′,1Sγ(k) in the second order of perturbative expansion which gives t he first non-vanishing contribution. In this approximation the amp litude reads M=/summationdisplay m/angbracketleftΨP′,1Sγ(k)|H1|m/angbracketright/angbracketleftm|H1|ΨP,1S/angbracketright EP,1S−Em+iǫ, (2) wheremruns over all the possible intermediate states. There are tw o groups of the vir- tual states depending on the photon content. The first one con tains only the electron- proton states moving with the total momentum P′′, which may be either virtual bound states or virtual continuum states. We shall denote the memb ers of this group by |ΨP′′,n/angbracketright. The other group includes the states |ΨP′′,n;γ(k)/angbracketrightbuilt of a photon of energy kand momentum kaccompanying states |ΨP′′,n/angbracketrightin the electron-proton sector. It is straightforward to observe, that the sum (2) may be expanded into two parts corre- sponding to final (FSR) and initial state (ISR) photon radiat ion amplitudes. ¿From the kinematics of the process and the conservation of moment um it is easy to find that the energy denominators take the following form: DFSR≃E1S−En+kand DISR≃E1S−En−kwhere, making the approximations, we have employed the hier - archyk≪P∼P′≪MH. We have checked, that the contributions to the amplitude Mof the intermediate states in which the electron and proton form a 1 Sstate are negligible, due to a2 0k2 suppression of the photon emission matrix elements. On the o ther hand, the spectrum of hydrogen atom has a large gap between the ground state and t he first excited state and the excited bound states lie close to each other. It also m ay be proven that the intermediate continuum states with energies En>|E1S|give a subleading contribution to the amplitude. It seems therefore quite natural to approx imate the energy difference E1S−Enby a characteristic energy, say E1S−En≃ −∆E=−|E1S|. Of course, this approximation works only for ksignificantly smaller than the ∆ E. Now, we can employ the completness relation for the Ψ P,nstates in the electron-proton sector and perform the sum over P,nto obtain1 M ≃/angbracketleftΨP′,1Sγ(k)|HSHE|ΨP,1S/angbracketright −∆E−k+/angbracketleftΨP′,1Sγ(k)|HEHS|ΨP,1S/angbracketright −∆E+k, (3) whereHSandHEare the components of H1responsible for the atom Coulomb scat- tering and for the photon emission respectively. After perf orming the necessary inte- grations and taking into account the relation εkk= 0 we get the following expression for the amplitude: 1This approach is in the spirit of the classical approximatio n applied in the calculation of van der Waals potential [14]. 3M ≃ −/radicalBigg 2π kV34πZe3 (P′+k−P)2× /braceleftBiggk ∆E2−k2εkq/bracketleftBiggβ mef(βq+k) +β mpf(βq) +α mef(αq) +α mpf(αq−k)/bracketrightBigg + 2∆E ∆E2−k2εk(P+P′)1 MH[f(αq)−f(αq+k) +f(βq)−f(βq+k)]/bracerightbigg (4) where q=P′−Pandf(l) = (1 +a2 0l2/4)−2is the electric form-factor of the ground state of the hydrogen atom of the Bohr radius given by a0. As expected, there occur cancellations between the form-factors, and the amplitude vanishes for vanishing k. We also notice that the terms proportional to εk(P+P′) are suppressed by a small factor a0kas compared to term containing εkqthus we retain only the later. Furthermore we may drop the 1 /mpterms. A more subtle problem arises when comparing the relative imp ortance of the con- tributionsβ mef(βq+k) andα mef(αq). The answer seems to depend on the details of the process in the physically relevant parameter space. For example, if the energy of the collision falls between 5 and 10 eV then the first term domi nates fork>3 eV while fork <3 eV the contribution of the other term is more important givi ng rise to an additional “red” peak of spectral density. However, if we ta ke an atom with the mass of 16MHand the Bohr radius a0(modelling the oxygen atom) this peak would move to energiesk<0.5 eV i.e. to the experimentally inaccesible region. Besides , theα mef(αq) term is very sensitive to the details of the charge form-fact or of the scattering atom. On the other hand, theβ mef(βq+k) part is universal i.e very weakly dependent on the details of the atom structure. In what follows, we shall f ocus only on this universal contribution of the amplitude which is relevant for the shap e of the spectrum in its largekpart. By this neglection we underestimate the number of the e mitted photons. Thus we get the estimate M=−/radicalBigg 2π kV34πZe3 MHk ∆E2−k2εkq q2(5) ¿From this amplitude, after performing the standard integr ations over the phase space, one gets the differential cross-section for the photon emiss ion from a collision at the energyE: dσ dk=8Z2e6 3MHEk3 (∆E2−k2)2log/parenleftBiggP+¯P P−¯P/parenrightBigg , (6) whereP=√2MHE,¯P=/radicalBig 2MH(E−k). For the purpose of the numerical calcula- tions, we take ∆ E= 13.6 eV. 4Let us point out, that the obtained cross-section correspon ds to a subprocess in which the nucleus scatters off the Coulomb field and the electr on cloud radiates a pho- ton when recombining around the scattered nucleus. The othe r contributions to the amplitude were shown to be subleading and neglected. The spe ctral density resulting from this cross-section rises as k4for smallki.e faster than this observed in the SL experiments. For kcloser to the kinematical cut-off, the initial k4dependece is mod- iffied and one gets approximately quadratic behavior over wid e range ofkwhich is still steeper that it follows from the experimental data. However , let us remind, that in our picture, the SL spectrum would be given by a convolution of th e spectra of individual collisions with the (unknown) distribution of the collisio n energy, so the two spectra may differ from each other. After integration of the differential cross-section (6) ove r the energy, we get the total cross-section σtot(E) growing with the collision energy Eapproximately as Eν withν≃3.5. For the choice of parameters relevant for the hydrogen ato m,Z= 1 and a typical collision energy of 7 eV (the choice is suggested by the observed cut-off on photon spectrum) we obtain σ0=σtot(7 eV) = 1.3·10−31m2. Taking this number and the parameters implied by the known facts concerning the bub ble dynamics we may, after some modifications, make a crude estimate of the number Nfof photons produced in one flash. We assume, that the bubble is filled with a mixture of atoms of a noble gas and atoms of other element (oxygen or nitrogen)2with atomic numbers ZNandZO, massesMNandMOand concentrations nNandnOcorrespondingly. The predictions for collisions of an atom of noble gas with an atom of, say, oxy gen may be formulated on the basis of the previous calculation. Namely, for the mom entum transfers relevant for SL the condition a2 0q2≫1 is fulfilled and the charge form-factors for the electrons suppress their contribution to the scattering amplitude — t he electrons cannot absorb momenta much larger than their average momentum in the atom. Then the nucleus of the noble atom acts as a bare source of the Coulomb field. Furth ermore, the amplitude of the electromagnetic radiation from the noble atom is smal ler then from the oxygen. It is caused by the smaller polarizability of the former atom which is governed by the relevant ionisation energy. Thus the cancellations which o ccur due to the destructive interferece do not reduce the cross-section substantially in contrast with the case of a collision of two objects of the same kind. This feature was re flected in our model by neglecting the radiation from the source of the Coulomb field . Of course, to complete the analogy, we should use in the formulae the reduced mass of the two atoms instead the mass of the hydrogen atom and include the modification of c harges byZNfor the source of the field and ZOfor the scattering atom. The charge ZOenters in fourth power since it contribute both to scattering and the radiation. Th erefore we approximate Nf by the following expression: Nf≃4πR3 s 3vτMH(MO+MN) MOMNZ2 NZ4 Oσ0nNnO, (7) 2At the temperatures and densities predicted by a shock-wave model the diatomic molecules are dissociated. 5whereRsis the radius of the hot gas region, τ= 100 ps is the light emission time [9] andv≃104m/s denotes the relative velocity of colliding atoms. Let us focus on the argon ( ZN= 18)–oxygen ( ZO= 8) process, which may be relevant for the SL of air bubble in water. As a first guess we take for Rsthe minimal radius of the bubble i.e. about 0.5 µm and assume for the concentrations nO+nN≃600n0in accordance with the measured compression factor [3], with n0= 2.7·1025m−3being the concentration of the ideal gas in the normal conditions. We obtain Nf≃5·105 fornO=nN, in a reasonable agreement with the experiment. In the alter native shock- wave description of gas dynamics lower values of the radius a re prefered, however the increase in concentration caused by the additional compres sion may easily compensate the decrease of the reaction volume. In fact, the true concen tration in the radiating region is not measured, and the models provide us only with am bigous predictions. The limitting value of the concentration is ∼1/(8a3 0) = 1030m−3thus there is still some room here and the constraints imposed on the cross-sect ion (7) by the experiment are not very stringent. It is also very probable, that the pro posed mechanism is not responsible for the emission of all photons but rather suppl ements the list of processes studied in [5]. In this case, including it could, in particul ar, improve the shape of the spectrum obtained in [5]. We can also take into account that, as follows from [13], the S L bubble is filled mainly with a noble gas, and may contain only some admixture o f oxygen (which may be continously provided from water) by modifying the ratio nO:nN, keepingnN+nO fixed. Thus for 10% of the oxygen in the gas mixture the number o f photons in one flash drops to about 2 ·105. In conclusion, we propose a novel light-emitting mechanism in sonoluminescence, in which the photons are radiated from atoms disturbed by col lisions. The number of produced photons has the proper order of magnitude, howev er the resulting photon spectrum comes out somewhat to steep. The spectral density i s featureless and univer- sal. Our model of SL requires the gas temperature to be approx imately 30 000 K, and the concentration of the order of 500 −1000n0. The bubble should contain atoms of two different gases: a noble gas and a gas with smaller ionisat ion energy e.g. oxygen, however one of the gases may appear at much smaller concentra tion than the other. The emitting process is insensitive to external magnetic fie ld and does not require the presence of molecules which are expected to dissociate in th e gas temperature sug- gested by the photon spectrum. Although the model we have ela borated is based on the simplified assumptions we expect that the obtained predi ctions estimate correctly the order of magnitude of the cross-section. Therefore this gives the strong motiva- tion to further study of the subject which would probably req uire the use of numerical analysis if one intends to cover all the details of the proces s. 6ACKNOWLEDGEMENTS We are very grateful to Professors W. Czy˙ z and K. Zalewski fo r useful comments and to L. Hadasz and B. Ziaja for helpful discussions. This re search was supported in part by the Polish State Committee for Scientific Research (K BN) grants 2P 03B 084 14 and 2P 03B 086 14. [1] N. Mainesco and J. J. Trillat, C. R. Acad. Sci. 196(1933) 858; H. Frenzel and H. Shultes, Zeit. Phys. Chem. 27B(1934) 421; [2] D. F. Gaitan, Ph. D. thesis, Univ. of Missisipi, 1990; D. F . Gaitan, L. A. Crum, C. C. Church and R. A. Roy, J. Acoust. Soc. Am. 91(1992) 3166. [3] B. P. Barber et al., Phys. Rep. 281(1997) 67. [4] C. C. Wu and P. H. Roberts, Phys. Rev. Lett. 70(1993) 3424. [5] S. Hilgenfeldt, S. Grossmann and D. Lohse, Nature 398(1999) 402. [6] J. B. Young, T. Schmiedel and W. Kang, Phys. Rev. Lett. 77(1996) 4816. [7] B. Noltingk and E. Neppiras, Proc. Roy. Soc. B63(1950) 674; R. L¨ ofstedt, B. P. Barber and S. J. Putterman, Phys. Fluids A5(1993) 2911. [8] R. Hiller, S. J. Putterman and B. P. Barber, Phys. Rev. Let t.69(1992) 1182. [9] B. Gompag et al., Phys. Rev. Lett. 79(1997) 1405; R. A. Hiller, S. J. Putterman and K. R. Weninger, Phys. Rev. Lett. 80(1998) 1090. [10] J. Schwinger, Proc. Natl. Acad. Sci. U.S.A. 90(1993) 958, 2105, 4505, 7285; 91(1994) 6473. [11] C. Eberlein, Phys. Rev. Lett. 76(1996) 3842. [12] L. Frommhold, A.A. Atchley, Phys. Rev. Lett. 73(1994) 2883. [13] S. Hilgenfeldt, D. Lohse and M. P. Brenner, Phys. Fluids 8(1996) 2808; D. Lohse et al., Phys. Rev. Lett. 78(1997) 1359; T. J. Matula and L. A. Crum, Phys. Rev. Lett. 80 (1998) 865; J. A. Ketterling and R. E. Apfel, Phys. Rev. Lett. 81(1998) 4991. [14] A. Uns¨ old, Z. Physik 43(1927) 563. 7

- 1 -Negative Observations in Quantum Mechanics Douglas M. Snyder Quantum mechanics is fundamentally a theory concerned with knowledge of the physical world. It is not fundamentally concerned with describing the functioning of the physical world independent of the observing, thinking person, as Newtonian mechanics is generally considered to be (Snyder, 1990, 1992). Chief among the reasons for the thesis that cognition and the physical world are linked in quantum mechanics is that all knowledge concerning physical existents is developed using their associated wave functions, and the wave functions provide only probabilistic knowledge regarding the physical world (Liboff, 1993). There is no physical world in quantum mechanics that is assumed to function independently of the observer who uses quantum mechanics to develop predictions and who makes observations that have consistently been found to support these predictions. Also significant is the immediate change in the quantum mechanical wave function associated with a physical existent that generally occurs throughout space upon measurement of the physical existent. This change in the wave function is not limited by the velocity limitation of the special theory of relativity for physical existents -the velocity of light in vacuum. Another relevant feature of quantum mechanics is the complex number nature of the wave function associated with a physical existent that is the basis for deriving whatever information can be known concerning the existent (Eisberg & Resnick, 1974/1985). A complex function is one that has both mathematically imaginary and real components. The physical world is traditionally described by mathematically real numbers, giving rise to Eisberg and Resnick’s (1974/1985) comment that “we should not attempt to give to wave functions [in quantum mechanics] a physical existence in the same sense that water waves have a physical existence” (p. 147). Nonetheless, the particular demonstration concerning the phenomenon of interference to be discussed in the next section is remarkable. Examining interference will spotlight the wave-particle duality in quantum mechanics, the key feature of this duality being that physical existents sometimes show particle-like characteristics and sometimes show wave-like characteristics. Wave functions exhibiting interference are based on the sum of two or more elementary wave functions. In contrast, where interference does notNegative Observations - 2 -characterize some physical phenomenon, this phenomenon is described by a wave function that consists of only one of these elementary wave functions. Feynman’s Two-Hole Gedankenexperiments Generally the change in the wave function that often occurs in measurement in quantum mechanics has been ascribed to the unavoidable physical interaction between the measuring instrument and the physical entity measured. Indeed, Bohr (1935) maintained that this unavoidable interaction was responsible for the uncertainty principle, more specifically the inability to simultaneously measure observable quantities described by non-commutingHermitian operators (e.g., the position and momentum of a particle). The following series of gedankenexperiments in this section will show that this interaction is not necessary to effect a change in the wave function. The series of gedankenexperiments indicates that knowledge plays a significant role in thechange in the wave function that often occurs in measurement (Snyder, 1996a, 1996b). Gedankenexperiment 1 Feynman, Leighton, and Sands (1965) explained that the distribution of electrons passing through a wall with two suitably arranged holes to a backstopwhere the positions of the electrons are detected exhibits interference (Figure 1).Electrons at the backstop may be detected with a Geiger counter or an electron multiplier. Feynman et al. explained that this interference is characteristic of wave phenomena and that the distribution of electrons at the backstop indicates that each of the electrons acts like a wave as it passes through the wall with two holes. It should be noted that when the electrons are detected in this gedankenexperiment, they are detected as discrete entities, a characteristic of particles, or in Feynman et al.’s terminology, “lumps” (p. 1-5). In Figure 1, the absence of lines indicating possible paths for the electrons to take from the electron source to the backstop is not an oversight. An electron is not taking one or the other of the paths. Instead, the wave function associated with each electron after it passes through the holes is the sum of two more elementary wave functions, with each of these wave functionsexperiencing diffraction at one or the other of the holes. Epstein (1945) emphasized that when the quantum mechanical wave of some physical entity such as an electron exhibits interference, it is interference generated only in thewave function characterizing the individual entity.Negative Observations - 3 -cross section of backstop with detectordistribution pattern along backstop electron impacting backstopcross section of wall with holes hole A hole Bwave function associated with projected electron Two-hole gedankenexperiment in which the distribution of electrons reflects interference in the wave functions of electrons. (Gedankenexperiment 1)Figure 1electron gun emitting electronsNegative Observations - 4 -The diffraction patterns resulting from the waves of the electrons passing through the two holes would at different spatial points along a backstopbehind the hole exhibit constructive or destructive interference. At some pointsalong the backstop, the waves from each hole sum (i.e., constructively interfere), and at other points along the backstop, the waves from each hole subtract (i.e., destructively interfere). The distribution of electrons at the backstop is given by the absolute square of the combined waves at different locations along the backstop, similar to the characteristic of a classical wave whose intensity at a particular location is proportional to the square of its amplitude. Because the electrons are detected as discrete entities, like particles, at the backstop, it takes many electrons to determine the intensity of the quantum wave that describes each of the electrons and that is reflected in the distribution of the electrons against the backstop. Gedankenexperiment 2 Feynman et al. further explained that if one were to implement a procedure in which it could be determined through which hole the electron passed, the interference pattern is destroyed and the resulting distribution of theelectrons resembles that of classical particles passing through the two holes in an important way. Feynman et al. relied on a strong light source behind the wall and between the two holes that illuminates an electron as it travels through either hole (Figure 2). Note the significant difference between the distribution patterns in Figures 1 and 2. In Figure 2, the path from the electron’s detection by the light to the backstop is indicated, but it is important to emphasize that this path is inferred only after the electron has reached the backstop. A measurement of the positionof the electron with the use of the light source introduces an uncertainty in its momentum. Only when the electron is detected at the backstop can one infer thepath the electron traveled from the hole it went through to the backstop. It is notsomething one can know before the electron strikes the backstop. In Feynman et al.’s gedankenexperiment using the light source, the distribution of electrons passing through both holes would be similar to that found if classical particles were sent through an analogous experimental arrangement in an important way. Specifically, as in the case of classical particles, this distribution of electrons at the backstop is the simple summation of the distribution patterns for electrons passing through one or the other of the holes. Figure 3 shows the distribution patterns of electrons passing throughNegative Observations - 5 -distribution pattern along backstop light sourcecross section of backstop with detectorcross section of wall with holes wave function associated with projected electron electron gun emitting electronselectron illuminated at hole A at time t and detected at backstop 1 electron illuminated at hole B at time t (t „t ) and detected at backstop122hole A hole B Two-hole gedankenexperiment with strong light source. (Gedankenexperiment 2)Figure 2Negative Observations - 6 -distribution of electrons from hole A light source distribution of electrons from hole Bcross section of backstop with detectorcross section of wall with holes wave function associated with projected electron electron gun emitting electronselectron illuminated at hole A at time t and detected at backstop1 electron illuminated at hole B at time t (t „t ) and detected at backstop122hole A hole B Two-hole gedankenexperiment with strong light source in which the distribution of electrons from each hole is shown.Figure 3Negative Observations - 7 -hole A and electrons passing through hole B in Gedankenexperiment 2. These distribution patterns are identical to those that would occur if only one or the other of the holes were open at a particular time. An inspection of Figure 3 shows that summing the distribution patterns for the electrons passing through hole A and those passing through hole B results in the overall distribution of electrons found in Gedankenexperiment 2. The Uncertainty Principle Feynman et al.’s gedankenexperiments are themselves very interesting in that they illustrate certain apparently incongruent characteristics of microscopic physical existents, namely particle-like and wave-like features. Feynman et al. discussed their gedankenexperiments in terms of Heisenberg’s uncertainty principle. Feynman et al. wrote: He [Heisenberg] proposed as a general principle, his uncertainty principle, which we can state in terms of our experiment as follows: “It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern.” If an apparatus is capable of determining which hole the electron goes through, it cannot be so delicate that it does not disturb the pattern in an essential way. (p. 1-9) Note that Feynman et al. implied in their description of the uncertainty principle that there is an unavoidable interaction between the measuring instrument (in their gedankenexperiment, the strong light source emitting photons) and the physical entity measured. Feynman et al. also wrote concerning Gedanken- experiment 2: the jolt given to the electron when the photon is scattered by it is such as to change the electron’s motion enough so that if it might have gone to where P12 [the electron distribution] was at a maximum [in Gedankenexperiment 1] it will instead land where P12 was at a minimum; that is why we no longer see the wavy interference effects. (p. 1-8) In determining through which hole an electron passes, Feynman et al., like most physicists, maintained that the electrons are unavoidably disturbed by the photons from the light source and it is this disturbance by the photons that destroys the interference pattern. Indeed, in a survey of a number of the textbooks of quantum mechanics, it is interesting that each author, in line withNegative Observations - 8 -Feynman and Bohr, allowed a central role in the change in the wave function that occurs in a measurement to a physical interaction between the physical existent measured and some physical measuring apparatus. The authors of these textbooks are Dicke and Witke (1960), Eisberg and Resnick (1974/1985), Gasiorowicz (1974), Goswami (1992), Liboff (1993), Merzbacher (1961/1970), and Messiah (1962/1965). It is important to note explicitly that some causative factor is necessary to account for the very different distributions of the electrons in Figures 1 and 2. Feynman et al. maintained that the physical interaction between the electronsand photons from the light source is this factor. Gedankenexperiment 3 Feynman et al.’s gedankenexperiments indicate that in quantum mechanics the act of taking a measurement in principle is linked to, and often affects, the physical world which is being measured. The nature of taking a measurement in quantum mechanics can be explored further by considering a certain variation of Feynman et al.’s second gedankenexperiment (Epstein, 1945; Renninger, 1960). 8 The results of this exploration are even more surprising than those presented by Feynman et al. in their gedanken- experiments. Empirical work on electron shelving that supports the next gedankenexperiment has been conducted by Nagourney, Sandberg, and Dehmelt (1986), Bergquist, Hulet, Itano, and Wineland (1986), and by Sauter, Neuhauser, Blatt, and Toschek (1986). This work has been summarized by Cook (1990).9 8 Epstein (1945) presented the essence of Gedankenexperiment 3 using the passage of photons through an interferometer. Renninger (1960) also discussed a gedankenexperiment in an articleentitled "Observations without Disturbing the Object" in which the essence of Gedankenexperiment 3 is presented. 9 In electron shelving, an ion is placed into a superposition of two quantum states. In each of these states, an electron of the ion is in one or the other of two energy levels. The transition to one of the quantum states occurs very quickly and the transition to the other state occurs very slowly. If the ion is repeatedly placed in the superposition of states after it transitions toone or the other of the superposed states, one finds the atomic electron in general transitions very frequently between the superposed quantum states and the quantum state characterized by the very quick transition. The photons emitted in these frequently occurring transitions to thequantum state characterized by the very quick transition are associated with resonance fluorescence of the ion. The absence of resonance fluorescence means that the ion has transitioned into the quantum state that occurs infrequently. Cook (1990) has pointed out that in the work of Dehmelt and his colleagues on electron shelving involving the Ba + ion, the resonance fluorescence of a single ion is of sufficientNegative Observations - 9 -In a similar arrangement to that found in Gedankenexperiment 2, one can determine which of the two holes an electron went through on its way to thebackstop by using a light that is placed near only one of the holes and which illuminates only the hole it is placed by (Figure 4). Illuminating only one of theholes yields a distribution of the electrons similar to that which one would expect if the light were placed between the holes, as in Feynman et al.’s second gedankenexperiment. The distribution is similar to the sum of the distributions of electrons that one would expect if only one or the other of the holes were open at a particular time. Moreover, when an observer knows that electrons have passed through the unilluminated hole because they were not seen to pass through the illuminated hole, the distribution of these electrons through the unilluminated hole resembles the distribution of electrons passing through the illuminated hole (Figure 5). Consider also the point that if: 1) the light is turned off before sufficient time has passed allowing the observer to conclude that an electron could not have passed through the illuminated hole, and 2) an electron has not been observed at the illuminated hole, the distribution of many such electrons passing through the wall is determined by an interference pattern that is the sumof diffraction patterns of the waves of the electrons passing through the two holes similar to that found in Gedankenexperiment 1 (Epstein, 1945; Renninger, 1960). Discussion of the Gedankenexperiments The immediate question is how are the results in Gedankenexperiment 3 possible given Feynman et al.’s thesis that physical interaction between the light source and electron is necessary to destroy the interference? Where the light illuminates only hole A, electrons passing through hole B do not interact with photons from the light source and yet interference is destroyed in the same manner as if the light source illuminated both holes A and B. In addition, the distribution of electrons passing through hole B at the backstop indicates that there has been a change in the description of these electrons, even though no physical interaction has occurred between these electrons and photons from the light source. intensity to be detectable by the dark-adapted eye alone, and the making of a negative observation, to be discussed shortly, is thus not dependent on any measuring device external tothe observer.Negative Observations - 10 -distribution pattern along backstop light sourcecross section of backstop with detectorcross section of wall with holes electron illuminated at hole A and detected at backstopwave function associated with projected electron electron gun emitting electrons electron not illuminated at hole A by the time it could have been detected there and electron subsequently is detected at backstophole A hole B Two-hole gedankenexperiment with strong light source illuminating only one hole. (Gedankenexperiment 3)Figure 4Negative Observations - 11 -distribution of electrons from hole B light source electron not illuminated at hole A by the time it could have been detected there and electron subsequently is detected at backstopcross section of backstop with detectorcross section of wall with holes wave function associated with projected electron electron gun emitting electronshole A hole B Two-hole gedankenexperiment with strong light source illuminating only one hole in which the distribution of electrons from unilluminated hole is shown.Figure 5Negative Observations - 12 -Epstein (1945) maintained that these kinds of different effects on the physical world in quantum mechanics that cannot be ascribed to physical causesare associated with “ mental certainty ” (p. 134) on the part of an observer as to which of the possible alternatives for a physical existent occurs. Indeed, the factor responsible for the change in the wave function for an electron headed forholes A and B, and which is not illuminated at hole A, is knowledge by the observer as to whether there is sufficient time for an electron to pass through the“illuminated” hole. To borrow a term used by Renninger (1960), when the timehas elapsed in which the electron could be illuminated at hole A, and it is not illuminated, the observer makes a “negative” (p. 418) observation. The common factor associated with the electron’s passage through the wall in a manner resembling that found for classical-like particles in Gedanken- experiments 2 and 3 is the observing, thinking individual’s knowledge as to whether an electron passed through a particular hole. The physical interaction between photons from the light source and electrons passing through either hole 1 or hole 2 is not a common factor. It should be remembered that some causative factor is implied by the very different electron distributions in Gedankenexperiments 1 and 2. It is reasonable to conclude that knowledge by the observer regarding the particular path of the electron through the wall is a factor in the change in the distribution of the electrons in Gedankenexperiment 1to that found for electrons in Gedankenexperiments 2 and 3. It might be argued that in Gedankenexperiment 3 a non-human recording instrument might record whether or not an electron passed through the illuminated hole in the time allowed, apparently obviating the need for a human observer. But, as has been shown, a non-human recording instrument is not necessary to obtain the results in Gedankenexperiment 3. And yet even ifa non-human instrument is used, ultimately a person is involved to read the results who could still be responsible for the obtained results. Furthermore, one would still have to explain the destruction of the interference affecting the distribution of the electrons at the backstop without relying on a physical interaction between the electrons and some other physical existent. Without ultimately relying on a human observer, this would be difficult to accomplish when the non-human recording instrument presumably relies on physical interactions for its functioning. It should also be emphasized that the change in the wave function for an electron passing through the unilluminated hole in Gedankenexperiment 3 provides the general case concerning what is necessary for the change in a waveNegative Observations - 13 -function to occur in a measurement of the physical existent with which it is associated. It was shown clearly in the extension of Feynman et al.’s gedankenexperiments that the change in the wave function of an electron or other physical existent is not due fundamentally to a physical cause. Instead, the change in the wave function is linked to the knowledge attained by the observer of the circumstances affecting the physical existent measured. There is one other point to be emphasized. The change in the wave function discussed in Gedankenexperiment 3 serves only to capture the role of knowledge in negative observation. That is, one need not even present a discussion of the wave function to attain the result that knowledge is a factor in the change in the electron distribution in Gedankenexperiment 1 to the electron distribution in Gedankenexperiments 2 and 3. This result depends only on the analysis of experimental results concerning the electron distributions in these three gedankenexperiments. The Schrödinger Cat Gedankenexperiment The nature of the change in the wave function that generally occurs in a measurement will now be discussed in more detail in terms of a gedanken- experiment proposed in 1935 by Schrödinger. In his gedankenexperiment, Schrödinger focused on the immediate change in the wave function that occurs upon observation of a measuring apparatus that records the value of a quantum mechanical quantity. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit ofradioactive substance, so small, that perhaps in the course of onehour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask ofhydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The Y-function of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.Negative Observations - 14 -It is typical of these cases [of which the foregoing example is one] that an indeterminancy originally restricted to the atomic domain becomes transformed into macroscopic indeterminancy,which can then be resolved by direct observation. (Schrödinger 1935/1983, p. 157) How does the gedankenexperiment indicate that the nature of the wave function as a link between cognition and the physical world is warranted? It does so in terms of the features of the quantum mechanical wave function cited earlier, one being that there is no source of information concerning the physicalworld in quantum mechanics other than the probabilistic predictions that yield knowledge of the physical world, predictions that have been supported by empirical test. The second is that these probabilities in general change immediately throughout space upon observation of a quantity of the physical existent that is described by the wave function which is the basis for the probabilistic predictions. Importantly, the velocity limitation of the special theory precludes a physical existent from mediating this change in the wave function. Note that Schrödinger does not specify how close the observer needs to be to the cat to resolve the indeterminancy. The observer can, in principle, be atany distance from the cat, even across the universe, and initiate this immediate change in the wave function, so long as the observer makes an observation regarding whether the cat is alive. Indeed, the observer does not even have to observe the cat directly but can rely on another observer who has observed the cat and who tells the former observer the result of his observation. In a related vein, Schrödinger did not explicitly discuss the role and significance of the person as observer in the measurement process in quantum mechanics. Physicists often use the term “observation” ambiguously. Changing the latter part of Schrödinger’s quote to indicate that the concern specifically is with a person making the observation does not lessen the statement’s validity: It is typical of these cases [of which the foregoing example is one] that an indeterminancy originally restricted to the atomic domain becomes transformed into macroscopic indeterminancy,which can then be resolved by direct [human] observation. Thus, in a circumstance where the observer is specified to be a person, the change in the wave function is tied explicitly to the perception by the humanNegative Observations - 15 -observer of the cat. This point is not limited to those circumstances where a human observer is explicitly specified. This point holds in the general case where a non-human macroscopic measuring instrument intervenes between aquantum mechanical entity and a human observer. It is a human observer whoultimately records the result of any observation. In the cat gedankenexperiment,for example, the cat acts as a macroscopic measuring instrument and comes to be characterized by the same probabilities as the microscopic physicalphenomenon (i.e., the radioactive substance) until a human observer makes hisown observation of the cat regarding its being alive or dead. It should be remembered that the Schrödinger cat gedankenexperiment portrays the special case where a macroscopic measuring instrument is used to make a measurement. As has been shown, Gedankenexperiment 3 discussed above provides the general case concerning what is necessary for the change in a wave function to occur in a measurement of the physical existent with which itis associated. There it was also shown that the change in the wave function is linked to the knowledge attained by the observer of the circumstances affecting the physical existent measured and that the change in this wave function is not due fundamentally to a physical cause. Knowledge and the Measurement of the Spin Component of Electrons Along a Spatial Axis It has been shown in gedankenexperiments using the two-hole interference scenario of Feynman, Leighton, and Sands that physical interaction is not necessary to effect the change in the wave function that generally occurs in measurement in quantum mechanics. Instead, the general case is that knowledge is linked to the change in the wave function. Another demonstrationof this point follows. The models for gedankenexperiments employing electrons (spin one-half particles) presented now are found in Feynman, Leighton, and Sands’s (1965) chapter on spin-one particles in their Lectures on Physics. Similar to the earlier gedankenexperiments, these gedanken- experiments also employ negative observation. But in contrast to the earlier gedankenexperiments, readily quantifiable results of the negative observations are developed. In addition, the significance of knowledge to the change of the wave function is emphasized because a concurrent physical interaction to thenegative observation between the existent measured and the measuring instrument is shown to be incapable of effecting the change in the wave function.Negative Observations - 16 -Basic Features of the Experimental Design Consider the case of a device like a Stern-Gerlach type apparatus (device A) which has an inhomogeneous magnetic field where the field direction and thedirection of the gradient are the same, for example along the z axis (Figure 6). An electron can pass along one of two paths as it moves through the apparatus. 10 This is due to the quantization of the spin angular momentum of the electron, more specifically the quantization of the spin component along anyspatial axis into two possible values. Initially, let an electron be in a state such that the probabilities of its going through either of the paths are equal. Which of the two possible paths anelectron has passed through depends on whether the electron’s spin component along the axis of the inhomogeneous magnetic field of the device is either in, or against, the direction of the magnetic field and its gradient. Given the initial probabilities, one-half of the electrons exiting from device A will be observed tohave spin up (i.e., in the direction of the magnetic field and gradient of device A), and one-half of the electrons exiting device A will be observed to have spin down (i.e., opposite to the direction of the magnetic field and gradient of device A). If, after an observation is made, the electron is now put through another Stern-Gerlach type device (device C), identical in construction to the first and oriented in the same direction, the electron will exit along the same path that it exited from in the first machine. In order to do this, the electron must first be brought back to its original direction of motion. This is accomplished through the use of another Stern-Gerlach type device (device B), the spatial orientation of which is up-down and right-left reversed with respect to the first device. In device B, the magnetic field and thegradient are in the opposite direction along the same spatial axis to that found for device A. The placement of these two devices is shown in Figure 7, with devices A and B right next to each other. 11 10 An electron is a member of a class of particles known as fermions. The spin component of a fermion along any spatial axis has two possible values when it is measured: +1/2 (h/2p) (spin up along this axis) and -1/2 (h/2p) (spin down along this axis). The results of the gedankenexperiment hold for fermions in general. 11 Note that no pathways are shown in Figure 7 for the electrons traveling through device AB. This is because quantum mechanics provides the correct description of the electrons, and it indicates that an electron does not travel over one or the other of the paths until an observationof the electron is made regarding which path it traveled. Instead, the wave function associated with an electron indicates that the probability is 1/2 that it will have spin up along the z axis and the probability is 1/2 that it will have spin down along the z axis when its spinNegative Observations - 17 -For electrons traveling toward Device A: P = 1/2z spin up P = 1/2z spin down direction of magnetic field and gradientspin up Device Ay axisz axis electrons traveling toward Stern-Gerlach apparatus direction of magnetic field and gradientz axis spin downDevice Ay axiselectrons traveling toward Stern-Gerlach apparatusor1/2 of the electronsexit device C with spin up using detector and human observer 1/2 of the electronsexit device C with spin down using detector and human observerElectrons passing through a Stern Gerlach device.Figure 6 component along this axis is measured. In devices like AB in other gedankenexperiments where both paths are open, the lack of path lines will similarly indicate a lack of knowledge regarding which path electrons take in going through the device.Negative Observations - 18 - or z axis Device C Device ABdirection of magnetic field and gradientz axis Device C Device ABelectrons traveling toward Stern-Gerlach apparatuses y axis electrons traveling toward Stern-Gerlach apparatuses y axis direction of magnetic field and gradientspin up spin down Electrons passing through a series of Stern-Gerlach devices oriented along the same spatial axis z.Figure 7For electrons traveling toward Device A: P = 1/2z spin up P = 1/2z spin downFor electrons traveling toward Device C: P = 1/2z spin up P = 1/2z spin down A B BANegative Observations - 19 -Two Gedankenexperiments Consider the following gedankenexperiments that adhere to quantum mechanical principles and that are supported by empirical evidence. They show that it is an individual’s knowledge of the physical world that is tied to the functioning of the physical world itself. Gedankenexperiment 4 Allow that device AB has a block inserted in it as portrayed in Figure 8. Then device AB allows only electrons with a spin up component along the z axis to exit it. Electrons with a spin down component along this axis are blocked from exiting. Allow that R electrons exit the device with a spin up component. Next to device AB a second device, DE, is placed that is identical in construction. D is the Stern-Gerlach-like device closest to B. The device DE is tilted around the y axis relative to device AB. aR electrons exit device DE with spin up (where 0 < a < 1). (Spin up here is relative to the z' axis and is in the direction of the magnetic field and gradient of device D.) Next to device DEis device C in the same spatial orientation as device A of AB and its magnetic field and gradient in the same direction along the z axis as device A. A block is inserted into device C that precludes electrons with spin down from exiting it. baR electrons exit device C with spin up (where 0 < b < 1). (Spin up here is relative to the z axis.) (Figure 9 displays the number of electrons exiting the various devices in this and succeeding gedankenexperiments.) Gedankenexperiment 5 The experimental arrangement is the same as that in Gedanken- experiment 4, except that no block is inserted in device DE (Figure 10). The numbers of electrons coming out of each device are as follows: (1) R electrons exit device AB with spin up along the z axis; (2) R electrons exit device DE; and (3) R electrons exit device C with spin up along the z axis. Discussion of Gedankenexperiments 4 and 5 How can one account for the results of Gedankenexperiments 4 and 5? An observer finds that R electrons exit device C in Gedankenexperiment 5, in accordance with the expectation that the spin components of the electrons alongthe z axis remain unaffected by the passage of the electrons through device DE. It appears that device DE, which has no block, has no effect on the spin components along the z axis of the electrons passing through it. R electrons exit device A with spin up along the z axis and R electrons exit device C withNegative Observations - 20 -z' axis direction of magnetic field and gradientz axis z axis baR electronsDevice AB Device DE Device C R electrons exit Device AB with spin up along z axis exit Device C with spin up along z axisy axis electrons traveling toward Stern-Gerlachapparatusesz axis A series of Stern-Gerlach devices where only electrons with spin up along the z axis pass through device AB, only electrons with spin up along the z' axis pass through device DE, and only electrons with spin up along the z axis pass through device C. (Gedankenexperiment 4)Figure 8aR electrons exit Device DE with spin up along z' axisA BD ENegative Observations - 21 - Specifications and results for Gedankenexperiments 4 through 7.Figure 9Negative Observations - 22 - R electronsz' axis direction of magnetic field and gradientz axis R electrons R electronsz axis Device AB Device DE Device C exit Device AB with spin up along z axis exit Device C with spin up along z axiselectrons traveling toward Stern-Gerlach apparatusesy axis exit Device DEz axis A series of Stern-Gerlach devices where only electrons with spin up along the z axis pass through device AB, paths in device DE for electrons with spin up or spin down along the z' axis are both open, and only electrons with spin up along the z axis pass through device C. (Gedankenexperiment 5)Figure 10A B DENegative Observations - 23 -spin up along the z axis. All electrons pass through device DE. But Gedanken- experiment 4 does not provide a similar result. A similar result would be that aR electrons would exit device C in Gedankenexperiment 4, not baR electrons. That is, the spin components of the electrons along the z axis would essentially remain unaffected by the passage of the electrons through device DE in Gedankenexperiment 4, just as device DE in Gedankenexperiment 5 does not appear to affect the spin components of electrons along the z axis. How is it that baR electrons exit from device C in Gedankenexperiment 4 instead of aR electrons? It is reasonable to conclude that something unusual is happening to the electrons in their passage through device DE in Gedankenexperiment 4, particularly in view of the results of Gedankenexperiment 5. Somehow the spincomponents of the electrons along the z axis are affected by their passage through device DE in Gedankenexperiment 4 while device DE in Gedanken- experiment 5 does not affect the spin components of electrons along the z axis. A comparison of Gedankenexperiments 4 and 5 indicates that the only physical feature of the measuring apparatus that can possibly be responsible forthe change in the component of the spin angular momentum along the z axis of the electron is the block that is inserted in device DE in Gedankenexperiment 4.Other than this one difference, the measuring apparatuses in Gedanken- experiments 4 and 5 are identical. The Block in Device DE The experimental consequences resulting from the presence or absence of the block in device DE in Gedankenexperiments 4 and 5 concern whether oneor both paths are open in device DE. Significantly, it is electrons traveling along the unblocked path in Gedankenexperiment 1 that exhibit the unusual behavior regarding the frequency of electrons exiting device C . Thus, the nature of the effect of the influence of the block on the electrons is indeedunusual from a conventional standpoint, a standpoint that would expect the change in spin components along the z axis of the electrons that travel along the unblocked path to somehow be changed by a physical interaction with the block. This physical interaction, though, is not possible. The scenario involving a block is thus in essence a negative observation. A negative observation occurs where an observation is made by deducing that a particular physical event must have occurred because another physical event did not occurwith subsequent consequences for the functioning of the physical world stemming from the change in knowledge. Physical interaction as the basis for the consequences in the physical world is ruled out. Remember that the spinNegative Observations - 24 -components of the electrons along the z axis traveling through device DE are affected by the change in knowledge, as evidenced by baR electrons exiting device C in Gedankenexperiment 4 instead of aR electrons. As previously noted, empirical work on electron shelving that supports the existence of negative observation has been conducted by Nagourney, Sandberg, and Dehmelt (1986), Bergquist, Hulet, Itano, and Wineland (1986), and by Sauter, Neuhauser, Blatt, and Toschek (1986). A Variation of the Gedankenexperiments Two other gedankenexperiments similar to Gedankenexperiments 4 and 5 will provide an even more remarkable demonstration that an individual’s knowledge of the physical world is tied to the functioning of the physical world itself. Gedankenexperiment 6 The experimental arrangement is the same as that in Gedanken- experiment 4, except that the blocks are inserted in devices DE and C such that spin up electrons along z' and z, respectively, cannot exit these devices and spin down electrons are allowed to proceed unimpeded (Figure 11). The numbers ofelectrons coming out of each device are as follows: (1) R electrons exit device AB with spin up along the z axis; (2) vR electrons exit device DE with spin down along the z' axis; and (3) uvR electrons exit device C with spin down along the z axis. Gedankenexperiment 7 The experimental arrangement is the same as that in Gedanken- experiment 6, except that device DE has both paths open (Figure 12). The numbers of electrons coming out of each device are as follows: (1) R electrons exit device AB with spin up along the z axis; (2) R electrons exit device DE; and (3) 0 electrons exit device C with spin down along the z axis. Discussion of Gedankenexperiments 6 and 7 The result in Gedankenexperiment 6 is remarkable. How is it that electrons with spin down along the axis of the magnetic field of the measuring device A, oriented in a particular direction along z, are found exiting device C, in which the axis of its magnetic field and its gradient are also oriented in the same direction along z? No electron with spin down along the z axis exits device AB. This result is particularly unusual when in Gedankenexperiment 7,Negative Observations - 25 - R electronsz axisz' axis direction of magnetic field and gradientz axis vR electrons uvR electronsDevice AB Device DE Device C exit Device AB with spin up along z axisexit Device AR with spin down along z' axis exit Device C with spin down along z axiselectrons traveling toward Stern-Gerlachapparatusesy axisz axis A series of Stern-Gerlach devices where only electrons with spin up along the z axis pass through device AB, only electrons with spin down along the z' axis pass through device DE, and only electrons with spin down along the z axis pass through device C. (Gedankenexperiment 6)Figure 11E DA BNegative Observations - 26 - R electronsz axisz' axis direction of magnetic field and gradientz axis R electrons 0 electronsDevice AB Device DE Device C exit Device AB with spin up along z axis exit Device C with spin down along z axiselectrons traveling toward Stern-Gerlach apparatusesy axis exit Device ABz axis A series of Stern-Gerlach devices where only electrons with spin up along the z axis pass through device AB, paths in device DE for electrons with spin up or spin down along the z' axis are open, and only electrons with spin down along the z axis pass through device C. (Gedankenexperiment 7)Figure 12E D BANegative Observations - 27 -using the same device DE, modified only by the removal of the block that prevents electrons with a spin up component along z' (the axis of the magnetic field in DE) to pass, there are no electrons that exit device C with spin down along the axis of its magnetic field, which has the same spatial orientation as themagnetic field of A along the z axis. In Gedankenexperiment 7, it appears as if the spin components of the electrons along the z axis were not affected by their passage through device DE, which has both paths open and which thus allowed all electrons to pass through. As reflected in the behavior of the electrons that pass through device C, the spin components of the electrons along the z axis in Gedankenexperiment 6 are affected by device DE, specifically by the insertion of the block in this device that prevents electrons with spin up components along the z' axis from exiting device DE. Again, no electrons with spin down along this axis were found to exit device AB. The electrons traveling along the unblocked path in device DE in Gedankenexperiment 6 exhibit this unusual behavior regarding the frequency of electrons exiting device C . No physical interaction between the block in device DE and any electron traveling along the unblocked path isresponsible for the frequency of electrons exiting device C. In Gedanken- experiment 6, a negative observation at device DE has resulted in electrons exiting device C with spin down along the z axis whereas in the absence of a negative observation, in Gedankenexperiment 7, no electrons exit device C withspin down along the z axis. Interference The difference in the observer’s knowledge of the spin components of electrons along an axis, and the difference in the spin components of the electrons themselves, in the pairs of gedankenexperiments that have been presented (i.e., Gedankenexperiments 4 and 5, and 6 and 7) reflect the presenceor absence of interference in the wave functions associated with each of the electrons. For example, in terms of the formalism, in Gedankenexperiment 4 the probability amplitude a 1 for an electron exiting device AB with spin up (AB+) and exiting device C with spin up ( C+) is given by a1 = <C+|DE+> <DE+|AB+> . (21) The probability of these events is derived by taking the absolute square of this probability amplitude, | a1|2. In contrast, in Gedankenexperiment 5, the probability amplitude d for an electron exiting device AB with spin up ( AB+) and exiting device C with spin up ( C+) is given byNegative Observations - 28 -d = <C+|DE-> <DE-|AB+> + <C+|DE+> <DE+|AB+> . (22) When the absolute square of the probability amplitude d is calculated to yield the probability that an electron exiting device AB with spin up will exit device C with spin up, it is evident that there will be two terms representing interference. These terms are (<C+|DE-> <DE-|AB+>)* (<C+|DE+> <DE+|AB+>) (23a) and (<C+|DE+> <DE+|AB+>)* (<C+|DE-> <DE-|AB+>) . (23b) It is these terms that distinguish | d|2, where there is interference, from S|ai|2 where one knows which path the electron took through device DE and there is no interference S|ai|2 = |a 1|2 + |a2|2 (24) or S|a i|2 = |<C+|DE+> <DE+|AB+>|2 + |<C+|DE-> <DE-|AB+>|2 . (25) It is important to emphasize that it is not the presence or absence of the block in device DE that interacts with electrons that is responsible for the presence or absence of interference in Gedankenexperiments 4 and 6. It is theact of knowing the value of the spin component of the electron along the z' axis that is responsible. The block in device DE in Gedankenexperiment 4 and the block in device DE in Gedankenexperiment 6 serve as bases for negative observations. Another Indication of the Importance of Knowledge in Measurement in Quantum Mechanics There is one more feature of the gedankenexperiments discussed in this paper that supports the theses that: 1) the macroscopic nature of a physical apparatus used for a measuring instrument is not central to making a measurement in quantum mechanics; 2) knowledge is central to making such measurements; and 3) the role of the block in device DE in Gedanken- experiments 4 and 6 is to provide information. Gedankenexperiments 4through 7 demonstrate the interesting point that the magnetic field of device DE itself is not sufficient to induce the change in the wave function that device DE which has a block along one path does for electrons traveling along the unblocked path and with which the block does not physically interact. UnlessNegative Observations - 29 -there is some way in the physical set up of device DE to determine the spin component of the electron along an axis z' (as is done in Gedankenexperiments 4 and 6 by the block in device DE), there is no change in the wave function of the electron concerning its spin components. In Gedankenexperiment 5 where there is no possibility in the physical set up that is device DE to know the spin component of the electron in device DE (because a block is not inserted along either the “spin up” or the “spin down” path), device DE does not affect the spin components along the z axis of the electrons as they travel through. That is, the number of electrons exiting devices C and AB are exactly the same. Also, in gedankenexperiment 7, no electrons with spin down along the z axis exit device C and only electrons with spin up along the z axis exit device AB. Without a block in device DE in Gedankenexperiment 7, there is no change in the wave function of an electron as regards its spin components. This is equivalent to saying that there has been no measurement of the spin component along the z' axis of the electron. To quote Feynman et al. (1965) regarding their filtering experiments with spin-one particles similar in principle to Gedankenexperiments 4 and 6: The past information [concerning spin along the z axis after exiting the first device] is not lost by the separation into...beams [in the second device], but by the blocking masks that are put in [the second device] (p. 5-9). In conclusion, if an interaction between a macroscopic physical apparatus and the existent to be measured were responsible for a change in the wave function of the physical existent measured, why, if a magnetic field by itself is unable to effect this change in the wave function for electrons, is the insertion of a block able to effect this change for electrons traveling through the unblocked path? When device DE does not contain a block along one of the paths, electrons traveling along what is the unblocked path in Gedankenexperiment 4 or the unblocked path in Gedankenexperiment 6 do notundergo any change in their wave function. The role of the block in Gedankenexperiments 4 and 6 is to provide information to a human observerconcerning electrons traveling along the unblocked path. With regard to these electrons, the role of the block in the measurement of their spin components along the z' axis does not depend on a physical interaction between them and the block.Negative Observations - 30 -The Time of a Measurement The question is often asked concerning quantum mechanics how can an observer finding out about a measurement that has presumably been made some time earlier be linked to the measurement itself? In terms of Gedanken- experiment 4, for example, if a human observer finds out about the electrons passing through the devices AB, DE, and C only after the electrons exit device C, how can this observer be considered responsible in some way for a measurement that was presumably made at device DE because of the inclusion of the block in that device? That is, a negative observation seems to be made only after the electrons exit device C, even though the block in device DE made the information available earlier (i.e., as soon as the time elapsed in which an electron passing through device DE could reach the block at the end of D). The analysis underlying the question presumes that some form of physical interaction occurring within a temporal framework provides the basis for measurement in quantum mechanics even though it clearly does not. In Gedankenexperiment 4, this presumed physical interaction does not occur in device DE. Measurement in quantum mechanics is fundamentally concerned with the development of knowledge. The course of physical interactions over time is not the central factor in the development of this knowledge. It is knowledge that is primary and within this knowledge, the functioning of the physical world, including the course of physical interactions over time, occurs. As has been discussed, there are other indications for this view concerning the importance of knowledge in quantum mechanics. Knowledge ofthe physical world is developed using wave functions, and wave functions provide only probabilistic knowledge. The quantum mechanical wave function associated with a physical existent generally changes immediately throughout space upon measurement of the physical existent. This change in the wave function is not limited by the velocity limitation of the special theory of relativityfor physical existents, the velocity of light in vacuum. There is the complex number nature of the wave function from which information concerning the physical world is derived. The Effect of Measurement on the Past One other point provides support for the central significance of knowledge in measurement in quantum mechanics. In Gedankenexperiments 4and 6, the presence of the block, or more accurately the knowledge that results from the presence of the block, at the exit of device D affects the electronsNegative Observations - 31 -traveling along the unblocked path in device D from their entry into device D for two reasons: 1. If the block is removed prior to the end of the time over which an electron could traverse device D along the blocked path, interference would not be destroyed and thenumber of electrons exiting device C inGedankenexperiment 4 (i.e., with spin up along the z axis), for example, is the same as the number of electrons exiting device AB. 2. With the block in place and the time elapsed over which an electron could have reached the block in device D, theinterference that was supposed to characterize the electron in its passage through device D did not occur as the electron could have traveled along only the unblocked path. If a detector had been set up along any part of the path in device D containing the block prior to the electron’shaving reached the end of device D where the block is situated, the electron would not have been detected along the path containing the block. A negative observation that the block allows for by providing information to an observer is thus seen to affect one’s knowledge of the past as well as the past itself, in the present case indicating that the electron has traveled down a particular path in device D as opposed to being characterized by a wavefunction demonstrating interference and not having traveled one pathexclusively. References Bergquist, J. C., Hulet, R. G., Itano, W. M., and Wineland, D. J. (1986). Observation of quantum jumps in a single atom. Physical Review Letters , 57, 1699-1702. Bohr, N. (1935). Can quantum-mechanical description of nature be considered complete? Physical Review , 49, 1804-1807. Cook, R. J. (1990). Quantum jumps. In E. Wolf (Ed.), Progress in Optics (Vol. 28) (pp. 361-416). Amsterdam: North-Holland. Dicke, R. H., and Wittke, J. P. (1960). Introduction to quantum mechanics . Reading, Massachusetts: Addison-Wesley. Eisberg, R., and Resnick, R. (1985). Quantum physics of atoms, molecules, solids, nuclei and particles (2nd ed.). New York: Wiley. (Original work published 1974) Epstein, P. (1945). The reality problem in quantum mechanics. American Journal of Physics , 13, 127-136.Negative Observations - 32 -Feynman, P. R., Leighton, R. B., and Sands, M. (1965). The Feynman lectures on physics: Quantum mechanics (Vol. 3). Reading, Massachusetts: Addison-Wesley. Gasiorowicz, S. (1974). Quantum physics . New York: John Wiley. Goswami, A. (1992). Quantum mechanics . Dubuque, Iowa: Wm. C. Brown. Liboff, R. (1993). Introductory quantum mechanics (2nd ed.). Reading, Massachusetts: Addison-Wesley. Mermin, N. D. (1985, April). Is the moon there when nobody looks? Reality and the quantum theory. Physics Today , 38-47. Merzbacher, E. (1970). Quantum mechanics (2nd. ed.). New York: John Wiley. (Original work published 1961) Messiah, A. (1965). Quantum mechanics (2nd ed.) (Vol. 1) (G. Tremmer, Trans.). Amsterdam: North-Holland. (Original work published 1962) Nagourney, W., Sandberg, J., and Dehmelt, H. (1986). Shelved optical electron amplifier: observation of quantum jumps. Physical Review Letters , 56, 2797-2799. Renninger, M. (1960). Messungen ohne Störung des Meßobjekts [Observations without disturbing the object]. Zeitschrift für Physik , 158, 417-421. Sauter, T., Neuhauser, W., Blatt, R. and Toschek, P. E. (1986). Observation of quantum jumps. Physical Review Letters , 57, 1696-1698. Schrödinger, E. (1983). The present situation in quantum mechanics. In J. A. Wheeler and W. H. Zurek, Quantum theory and measurement (pp. 152-167) (J. Trimmer, Trans.). Princeton, New Jersey: Princeton University Press. (Original work published 1935) Snyder, D. M. (1990). On the relation between psychology and physics. The Journal of Mind and Behavior , 11, 1-17. 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. (1996a). Cognition and the physical world in quantum mechanics. Paper presented at the annual convention of the Western Psychological Association, San Jose,California. Snyder, D. M. (1996b). On the nature of the change in the wave function in a measurement in quantum mechanics. Los Alamos National Laboratory E-Print Physics Archive (WWWaddress: http://xxx.lanl.gov/abs/quant-ph/9601006). Wigner, E. (1983). Remarks on the mind-body question. In J. A. Wheeler and W. H. Zurek, Quantum theory and measurement (pp. 168-181). Princeton, New Jersey: Princeton University Press. (Original work published 1961)

arXiv:physics/9912016v1 [physics.chem-ph] 6 Dec 1999Effect of Subphase Ca++Ions on the Viscoelastic Properties of Langmuir Monolayers R. S. Ghaskadvi, Sharon Carr, and Michael Dennin Department of Physics and Astronomy University of California at Irvine Irvine, CA 92697-4575. (February 2, 2008) Abstract It is known that the presence of cations like Ca++or Pb++in the water subphase alters the pressure-area isotherms for fatty acid monolayers. The corresponding lattice constant changes have been studied u sing x-ray diffrac- tion. Reflection-absorption spectroscopy has been used to p robe the chemical composition of the film. We report on the first measurements of the time evolution of the shear viscosity of arachidic acid monolaye rs in the presence of Ca++ions in the subphase. We find that the introduction of Ca++ions to the water subphase results in an increase of the film’s visc osity by at least three orders of magnitude. This increase occurs in three dis tinct stages. First, there is a rapid change in the viscosity of up to one order of ma gnitude. This is followed by two periods, with very different time constant s, of a relatively slow increase in the viscosity over the next 10 or more hours. The correspond- ing time constants for this rise decrease as either the subph ase pH or Ca++ concentration is increased. 68.10.Et,68.18,46.35.+z Typeset using REVT EX 1I. INTRODUCTION Over the last ten years there has been a renewed interest in th e study of Langmuir monolayers [1], due to the development and application of a n umber of powerful tools like x-ray diffraction [2,3], Brewster Angle microscopy (BAM) [4 ,5], and fluorescence microscopy [6–8]. Langmuir monolayers are monomolecular films at the ai r-water interface formed by amphiphilic molecules. Typically, these molecules have a l ong hydrophobic chain oriented away from the water surface and a polar, hydrophilic headgro up that interacts with the components of the aqueous subphase. Langmuir monolayers se rve as an excellent model for biological membranes and for surfactant stabilizers th at are added to foams. Also, they are the starting point for Langmuir-Blodgett depositions w here a solid substrate is passed through the Langmuir monolayers, transferring one or more l ayers of the molecules. For all three of these applications, understanding the interactio n between Langmuir monolayers and ions in the subphase is important for two reasons. First, the ions are often naturally present in these systems, either as biologically relevant chemical s or as contaminants. Second, the ions provide a mechanism for controlling the mechanical pro perties of the films, which is especially important in applications related to foams and L angmuir-Blodgett depositions. A number of techniques, including pressure-area isotherms [9–11], reflection-absorption spectrometry [12], and x-ray diffraction [13], have been use d to study the effects of divalent cations on the monolayer structure. These studies have high lighted the important role played by pH, especially for fatty acid monolayers, in modifying th e effects of divalent cations on the structure of the monolayer. The equilibrium phase behavior of fatty acids on a pure water, or low pH subphase, have been extensively studied [14]. Ther e is a generally applicable phase diagram that consists of both “tilted” and “untilted” phase s. A tilted phase is one in which the monolayer tails are tilted with respect to the surface no rmal. Generally, the untilted phases occur at higher pressures. One of the main effects of th e calcium ions, as the pH is increased, is to lower the transition pressure between the v arious phases [13]. Ultimately, at very high pH, the tilted phases no longer appear to exist. T his lowering of the transition pressure is often referred to as a “stiffening” of the monolay er. A common feature of these studies is that no long term variations in the monolayer prop erties were measured. This is reasonable if chemical equilibrium with the ions in solutio n is reached relatively rapidly. Despite the evidence from pressure-area isotherms that the cations cause a stiffening of the monolayer, there has been minimal efforts to measure effec ts of cations on the viscoelastic properties of the monolayer [15,16]. In this paper, we repor t on a series of measurements of the viscoelastic properties of arachidic acid monolayers i n the presence of Ca++. We have looked at the effect of pH and Ca++concentration on the time evolution of three properties of the monolayer: the isotherms; the viscosity ( η); and the complex shear modulus G. Our isotherm results at t = 1 hr are consistent with previous meas urements of fatty acids and divalent cations [13]. However, we have found a slow change i n the viscoelastic properties of the monolayer over a long time period. This behavior suggest s interesting kinetics for the chemical reaction between the arachidic acid and the Ca++. 2II. EXPERIMENTAL DETAILS The viscoelastic properties were measured using a two-dime nsional Couette viscometer that is described in detail elsewhere [17]. A schematic of th e apparatus is given in Fig. 1. A circular barrier made of twelve individual teflon fingers is i mmersed into water in a circular trough. A circular knife-edge torsion pendulum (rotor) han gs by a wire so that it just touches the water surface in the center of the trough. A stati onary teflon disk is placed in the water just under the pendulum. The disk has the same dia meter as the knife-edge pendulum. A Langmuir monolayer is made at the annular air-wa ter interface between the barrier and the rotor knife-edge. The barrier can be compres sed or expanded to control the monolayer pressure and rotated to generate a two dimensi onal Taylor-Couette flow. The angular position of the rotor can be measured by means of a pick-up coil attached to the rotor. This is used to measure the torque generated by flow in the monolayer on the inner rotor. The torque provides a measurement of the monola yer viscosity. In addition, an external torque can be applied to the rotor by manipulatin g an external magnetic field. This allows for both oscillatory measurements of the linear shear response of the monolayer and measurements of stress relaxation curves for monolayer s. The apparatus is also equipped with a Brewster Angle Microsc ope (BAM) for observation of the domain structure of the film. The BAM image measures the relative reflectivity of p- polarized light incident on the monolayer at the Brewster an gle for pure water. Variations of reflectivity of the monolayer correspond to changes in the or ientation of the tilted molecules from domain to domain. To study the effect of cations, it is imperative to start with w ater that has minimal ionic content. We achieved this by passing de-ionized water through a Millipore filter to obtain water with resistivity in excess of 18 MΩ /cm. The concentration of Ca++was set by adding CaCl 2.2H2O to the purified water. Most of the experiments used a 0.65 mM C a++ concentration so that the results would be comparable with R ef. [13]. The arachidic acid monolayer was made from a chloroform solu tion. The solution was placed on the aqueous subphase with a microsyringe and allow ed to relax for about 20 minutes to facilitate the evaporation of the solvent. Then i t was compressed to the pressure of 9 dyne/cm. All the data presented here were taken at 22◦C. At this temperature and pressure, the monolayer is in the L 2phase. One hour after the solution was placed on the subphase, the equilibrium angle, θ1, of the rotor was measured. The outer barrier was set into rotation to generate a Couette flow. The Couette flow caus es a torque τon the rotor displacing it to a new equilibrium position θ2such that τ=κ(θ2−θ1), where κis the torsion constant of the wire. After rotating the barrier for about 5 m inutes to achieve equilibrium, θ2 was measured, and then the rotation was stopped. This series of experiments was repeated every hour. The isotherms as well as the complex shear modulu s were measured separately. The barrier rotation rate was 0.0237 rad/sec. With Rinner= 3.81 cm and Rbarrier= 6.5 cm, this corresponds to a shear rate of 0 .057s−1. G was measured at ω= 0.251 rad/sec. III. RESULTS Figure 2 shows the time evolution of viscosity (measured by t he Couette flow method and henceforth referred to as η) for different concentrations of Ca++in the subphase. All of 3the measurements were done at pH 5.5. There are two points of n ote. One, the higher the concentration, the higher the rate of viscosity rise. Secon dly, the rise in viscosity is in three parts. There is an initial jump of about one order of magnitud e within the first hour. The next two periods are separated by η= 1 g/s, below and above which the viscosity clearly rises with different slopes on the semilog plot. This indicat es there are three time constants associated with the increase. As the first data point is taken after one hour of making the film, we cannot comment about the time constant for the viscos ity rise in the first stage, except that the upper limit for τ1is about 0.5 hour. It should be noted that both the time constants decrease with increasing concentration. For the film with 0.65 mM concentration of Ca++,τ2= 1.76 hour and τ3= 5.43 hr, where τ2andτ3are the time constants for the second and the third stage respectively. The time constants for the rest of the data are given in the figure caption. The three different stages of viscosity rise are also obvious in Fig. 3 which depicts the dependence of this rise on the subphase pH. Note that below pH = 4, the viscosity is small and almost constant. This is consistent with other studies w here the isotherms were seen to remain unchanged for about the same pH. As the pH values are in creased, both the initial jump in viscosity and the later rates of rise increase. The results of the oscillatory experiment are plotted in the Fig. 4. The complex shear modulus G (=G′+ iG′′) is known to depend on the strain amplitude for some Langmuir monolayers [18]. Here G′is the elastic component of the shear modulus and G′′is the viscous component. For a linear viscoelastic fluid, the relation bet ween G′′and the viscosity, η, is given by G′′=ωη, where ωis the oscillation frequency. In this case we found G′′to be weakly and G′to be strongly dependent on the strain amplitude. The depend ence was qualitatively the same as in Ref. [18] i.e., G was constant at small amplitud es and decreased for higher amplitudes. To ensure linear response, we measured G at a sma ll constant strain amplitude of about 10−3. As with η, G′′displays two distinct periods of increase after the first hou r. However, G′rises monotonically with time. Figure 5 shows the variation of the arachidic acid monolayer isotherm as a function of time for pH = 5.5. It must be noted that the isotherms were meas ured separately from the viscosity. There might be some differences in the rate of Calc ium attachment arising from the fact that there was no rotation of the monolayer or genera tion of circular flow in the subphase. But, the effect is bound to be minimal for two reason s. First, the flow during viscosity measurements only occurred for roughly 10% of the data run. Second, there was no turbulence during the flow, so the rate of mixing in the subpha se would not be substantially modified. Furthermore, the qualitative behavior of G′′was found to be the same when measured with the rotation as it was when measured without ro tation. This confirmed that the rotation had minimal effect on the Ca++binding rate. Figure 5 also shows the position of the kink in the isotherm that corresponds to the 2ndorder phase transition for arachidic acid monolayer without Ca++in the subphase (horizontal dashed line). The presence of Ca++does alter the isotherm in the first hour. There is a lowering o f the pressure at which the 2ndorder transition occurs by about 4 dyne/cm. This is consiste nt with the isotherms published in the literature [13]. From the X-ray data it is kn own that this change is due to the bound calcium changing the head group interactions so th at the molecules come closer together. However, after this initial drop, there is a slow c hange in the isotherm. This change corresponds to a decrease in the transition pressure by about 0.3 dyne/cm/hour. 4The kink also appears to become more rounded with time. Howev er, we believe that the apparent rounding is due to the high viscosity of the film and i s not a real effect. IV. DISCUSSION In summary, we find that there are many effects of Ca++ions on the arachidic acid monolayer. In the first hour, the isotherm shifts downwards i n pressure by about 4 dyne/cm. Over the next ten hours, it changes by about 3 dyne/cm. These d rops are accompanied by changes in viscosities, measured by either the rotating bar rier method or the oscillating rotor method. One can interpret the viscosity rising during the fir st hour as a direct result of the change in the head group interactions. This is consistent wi th the pH data. It is known that sufficiently low pH suppresses the binding of divalent ions to the monolayer [12,13,19,20], and we observe no viscosity increase at pH 3.4 and below. The slow rise associated with the late time evolution of the v iscosity is surprising. The 3 dyne/cm drop in the transition pressure in this period sugg ests a very slow rate of Cal- cium ionically binding to the carboxylate. For octadecanoi c acid monolayers, IR reflection- absorption studies [12] have shown that near pH = 6, the Ca++does not bind to all the molecules but that some undissociated acid molecules remai n in the film. The increase in the viscosity, taken together with the slow change in the iso therm suggest that the same is true for the arachidic acid and that these remaining acid m olecules slowly bind with the Ca++ions with a time constant of a few hours. This is supported by t he fact that the time constants τ2andτ3both decrease with increasing subphase Ca++concentration. The steady rise of G′seen in Fig. 4 is consistent with this picture; however, the e xistence of a single time constant needs to be explained. The presence of two different time constants, namely τ2andτ3, is also puzzling. If we accept that Ca++continues binding to the monolayer, then two broad possibil ities emerge: a. that the rate of the binding changes abruptly and this chan ge is reflected in the viscous response or b. that the rate of binding does not change but the rate of visc osity rise with respect to bound site concentration varies after reaching a critical v alue. At this point, it is difficult to say which of these pictures is m ore accurate, but both are interesting. If the first case is correct, it suggests intere sting long-term kinetics associated with the chemical reaction mechanism that undergo abrupt ch anges. If the latter reason is correct, it suggests an interesting interplay between th e microscopic structure of the monolayer and the macroscopic viscosity. One possible mechanism for the abrupt change in the evolutio n of the viscosity is the contribution of the line tension between domains in the mono layer to the viscosity. It is known from foams and other complex fluids that line tension (o r surface tension in three dimensions) can substantially alter the macroscopic visco sity of a fluid. The L 2phase of arachidic acid consists of a random domain structure. Fried enberg, et al. [21] report that for docosanoic acid monolayers in the L 2phase, domains stretched by an extensional flow do not relax back to their original shape. This indicates that t he line tension in the absence of Ca++is nearly zero. Similar behavior is observed for our samples of arachidic acid. However, with Ca++ions in the subphase, our BAM images show evidence of domain r elaxation. Presumably, line tension between domains will be dominated by Ca++absorption at the 5domain boundaries. If this saturates, the rate of change of t he viscosity would be altered. We are currently undertaking detailed studies of this behav ior to probe the impact of the line tension to the overall viscoelastic response of the mon olayer and the effect of Ca++ions on the line tension. ACKNOWLEDGMENTS Acknowledgment is made to the donors of The Petroleum Resear ch Fund, administered by the ACS, for partial support of this research. Also, we wou ld like to thank Doug Tobias and Charles Knobler for helpful conversations. 6REFERENCES [1] For reviews of Langmuir Monolayers, see H. Mohwald, Annu . Rev. Phys. Chem. 41, 441 (1990); H. M. McConnell, ibid.42, 171 (1991). [2] P. Dutta, J.B. Peng, B. Lin, J. B. Ketterson, M. Prakash, P . Georgopoulos, and S. Ehrlich, Phys. Rev. Lett. 58, 2228 (1987). [3] K. Kjaer, J. Als-Nielsen, C. A. Helm, L. A. Laxhuber, and H . M¨ ohwald, Phys. Rev. Lett.58, 2224 (1987). [4] S. H´ enon and J. Meunier, Rev. Sci. Instrum. 62, 936 (1991). [5] D. H¨ onig and D. M¨ obius, J. Phys. Chem. 95, 4590 (1991). [6] D. K. Schwartz and C. M. Knobler, J. Phys. Chem 97, 8849 (1993). [7] S. Riv` ere, S. H´ enon, J. Meunier, D. K. Schwartz, M. W. Ts ao, and C. M. Knobler, J. Chem. Phys. 101, 10045 (1994). [8] B. G. Moore, C. M. Knobler, S. Akamatsu, and F. Rondelez, J . Phys. Chem. 94, 4588 (1990). [9] K. Miyano, B. M. Abraham, S. Q. Xu, and J. B. Ketterson, J. C hem. Phys. 77, 2190 (1982). [10] E. Pezron, P. M. Claesson, J. M. Berg, and D. Vollhardt, J . Colloid. Interface Sci. 138, 245 (1990). [11] S. Bettarni, F. Bonosi, G. Gabrielli, and G. Martini, La ngmuir 7, 611 (1991). [12] A. Gericke and H. H¨ uhnerfuss, Thin Solid Films 245, 74 (1994). [13] M. C. Shih, T. M. Bohanon, J. M. Mikrut, P. Zschack, and P. Dutta, J. Chem. Phys. 96, 1556 (1992). [14] For a review of phase transitions in monolayers, see C. M . Knobler and C. Desai, Annu. Rev. Phys. Chem. 43, 207 (1992). [15] M. R. Buhaenko, J. W. Goodwin, and R. M. Richardson, Thin Solid Films 159, 171 (1988). [16] M. Yazdanian, H. Yu, and G. Zografi, Langmuir 6, 1093 (1990). [17] R. S. Ghaskadvi and M. Dennin, Rev. Sci. Instrum. 69, 3568 (1998). [18] R. S. Ghaskadvi, P. Dutta and J. B. Ketterson, Phys. Rev. E54-2, 1770 (1996). [19] J. M. Bloch and W. Yun, Phys. Rev. A 41, 844 (1990). [20] D. J. Ahn and E. I. Franses, J. Chem. Phys. 95, 8486 (1991). [21] M. C. Friedenberg, G. G. Fuller, C. W. Frank and C. R. Robe rtson, Langmuir 12, 1594 (1996). 7FIGURES RotorTorsion wire FilmRotor BarrierFingers (2 shown) Figure 1: R. S. Ghaskadvi and Michael Dennin, J. Chem. Phys. FIG. 1. Schematic drawing of the apparatus. 8024681012141618200.010.11 Ca++ concentration 0.001 mM 0.015 mM 0.040 mM 0.650 mM Figure 2: R. S. Ghaskadvi & Michael Dennin, J. Chem. Phys. time (hr)Viscosity (g/s) FIG. 2. The viscosity of the arachidic acid monolayer as a fun ction of time at 22◦C. The different curves correspond to different concentrations of C a++ions at pH 5.5. The solid lines corresponds to the least square fits to the equation y=Aex/τ. Fit values 0.001 mM : A=0.0043 g/s,τ2=12.77 hr; 0.015 mM : A=0.0077 g/s, τ2=4.10 hr; 0.04 mM : A=0.0162 g/s, τ2=2.76 hr, τ3=8.36 hr; 0.65 mM : A=0.0197 g/s, τ2=1.76 hr, τ3=5.43 hr. 90 2 4 6 8 100.010.11 Figure 3: R. S. Ghaskadvi & Michael Dennin, J. Chem. Phys. time (hr)Viscosity (g/s) pH = 2.6 pH = 3.4 pH = 4.1 pH = 6.1 FIG. 3. The viscosity of the arachidic acid monolayer as a fun ction of time at 22◦C, Π = 9 dyne/cm. The different curves correspond to different pH va lues of the subphase. The concentration of the Ca++ions is fixed (0.65 mM). 1000 02 04 06 08 10 120.1110 Figure 4: R. S. Ghaskadvi & Michael Dennin, J. Chem. Phys. time (hr)G' & G" (dyne/cm) G' G" FIG. 4. G′and G′′of the arachidic acid monolayer as a function of time at 22◦C, Π = 9 dyne/cm. The dotted lines are guides to the eye. 11192021222324252627280102030405060 ° Figure 5: R. S. Ghaskadvi & Michael Dennin, J. Chem. Phys.Pressure (dyne/cm) Area/molecule (A2) FIG. 5. Isotherms of arachidic acid monolayer at 22◦C, subphase pH = 5.5, and subphase Ca++concentration of 0.65 mM. The x-axis reading is accurate for the first isotherm only since the rest are shifted for the sake of clarity. The isotherms ar e taken one hour apart. The dotted line is drawn to guide the eye along the kink position. The das hed line represents the pressure at which the kink occurs for the monolayer without Ca++in the subphase. 12

arXiv:physics/9912017v1 [physics.ed-ph] 7 Dec 1999An elementary quantum mechanics calculation for the Casimir effect in one dimension Attila Farkas Institute of Condensed Matter Research, Timi¸ soara, Str. Tˆ arnava 1, RO-1900 Timi¸ soara, Romania Nistor Nicolaevici Technical University of Timi¸ soara, Department of Physics , P-t ¸a Horat ¸iu 1, RO-1900 Timi¸ soara, Romania February 2, 2008 Abstract We obtain the Casimir effect for the massless scalar field in on e dimension based on the analogy between the quantum field and t he continuum limit of an infinite set of coupled harmonical osci llators. 1 Introduction A well known fact in quantum mechanics is that, even though th e classical system admits a zero minimal energy, this does not generally hold for its quantum counterpart. The typical example is the1 2¯hωvalue for the non- relativistic harmonic linear oscillator, where ¯ his the Planck constant and ω its proper frequency. More generally, if the system behaves as a collection of such oscillators, the minimal (or zero point) energy is E0=¯h 2/summationdisplay nωn, (1) 1where the sum extends over all proper frequencies ωn. As often pointed out in quantum field theory textbooks1,2, non-interacting quantized fields can be pictured this way, in the limit of an infinite spatial density of oscillators. In particular, for the scalar field the analogy with a set of coup led oscillators can be constructed in a precise manner1, as we shall also sketch below. We shall use here the oscillator model to obtain the Casimir effe ct for the massless field, in the case of one spatial dimension. The calculation i s a simple exercise in non-relativistic quantum mechanics. What is usually refered to as the Casimir effect3is the attraction force be- tween two conducting parallel uncharged plates in vacuum. T he phenomenon counts as a direct evidence for the zero point energy of the qu antized elec- tromagnetic field: assuming the plates are perfect conducto rs, the energy to area ratio reads1(cis the speed of light and Lis the plates separation) E0 A=−π2¯hc 720L3, (2) from which the attraction force can be readily derived. Qual itatively, the L dependence in E0is naturally understood as originating in that displayed by the proper frequencies of the field between the plates. Actually, by summing over frequencies as in eq. (1) one obtai ns a di- vergent energy. This is a common situation in quantum field th eory, being remedied by what is called renormalization: one basically s ubtracts a di- vergent quantity to render the result finite, with the justifi cation that only energy differences are relevanta. Unfortunately, computational methods used to handle infinities to enforce this operationbpresent themselves, rather generally, as a piece of technicality with no intuitive supp ort; for the unac- customed reader, they might very well leave the impression t hat the result is just a mathematical artifact. The oscillator analogy com es to provide a context to do the calculations within a physically transpar ent picture, with no extra mathematical input required. aIn the assumption of neglecting gravitational phaenomena, see e.g. Ref 4. bi.e. regularization methods. An example follows next parag raph. 22 Quantum field theory calculation We briefly review first the field theoretical approach. Consid er the uncharged massless scalar field in one dimension −∞< x <∞, /parenleftBigg1 c2∂2 ∂t2−∂2 ∂x2/parenrightBigg ϕ(x, t) = 0, (3) subjected to the conditions ϕ(0, t) =ϕ(L, t) = 0 (4) for some positive L. We are interested in the zero point energy as a function ofL. We shall focus on the field in the “box” 0 < x < L . It is intuitively clear that the result for the exterior regions follows by mak ingL→ ∞. Note that by eqs. (4) the field in the box is causally disconnected f rom that in the exterior regions, paralleling thus the situation for the el ectromagnetic field in the previous chapter. Eqs. (3) and (4) define the proper frequencies as ωn=nπ L, n= 1,2, . . .∞, (5) obviously making E0a divergent quantity. A convenient way5to deal with this is by introducing the damping factors ωn→ωnexp(−λωn/c), λ > 0, (6) and to consider E0=E0(L, λ) in the limit λ→0. Performing the sum one obtains E0(L, λ) =π¯hc 8L/parenleftBigg cth2πλ 2L−1/parenrightBigg . (7) Using the expansion cthz=1 z+z 3+O(z3), (8) one finds E0(L, λ) =¯hc 2πλ2L−π¯hc 24L+O/parenleftBiggλ L/parenrightBigg . (9) 3Now, it is immediate to see that the λ−2term can be assigned to an infinite energy density corresponding to the case L→ ∞. The simple but essential observation is that, when considering also the energy of the exterior regions, the divergences add to an L-independent quantity, which makes them me- chanically irrelevant. Renormalization amounts to ignore them. Thus one can set E0(L) =−π 24¯hc L, (10) which stands as the analogous result of eq. (2). 3 Quantum mechanics calculation Consider the one dimensional system of an infinite number of c oupled oscil- lators described by the Hamiltonian (all notations are conv entional) H=/summationdisplay kp2 k 2m+/summationdisplay kk 2(xk+1−xk)2. (11) xkmeasures the displacement of the kth oscillator from its equilibrium po- sition, supposed equally spaced from the neighbored ones by distance a. Canonical commutations assure that the Heisenberg operato rs xk(t) =ei ¯hHtxke−i ¯hHt(12) obey the classical equation md2xk(t) dt2−k(xk+1(t) +xk−1(t)−2xk(t)) = 0 . (13) Let us consider the parameters mandkscaled such that a2m k=1 c2. (14) As familiar from wave propagation theory in elastic media, e q. (13) becomes the d’Alembert equation (3) with the correspondence xk(t)→ϕ(ka, t), (15) 4and letting a→0.xk,pmcommutations can be also shown to trans- late into the equal-time field variables commutations requi red by canonical quantization1. One can thus identify the quantum field with the continuum limit of the quantum mechanical system. Our interest lies in the oscillator analogy when taking into account con- ditions (4). It is transparent from eq. (15) that they formal ly amount to set inH x0=xN= 0, p 0=pN= 0, (16) withNsome natural number. In other words, the 0th and the Nth oscillator are supposed fixed. As in the precedent paragraph, we shall ca lculate the zero point energy of the oscillators in the “box” 1 ≤k≤N−1. The first step is to decouple the oscillators by diagonalizin g the quadrat- ical form in coordinates in eq. (11). Equivalently, one need s the eigenvalues λnof the N−1 dimensional square matrix Vkmwith elements Vk,k= 2, V k,k+1=Vk,k−1=−1, (17) and zero in rest. One easily checks they are λn= 4 sin2nπ N, n= 1,2, . . .N−1, (18) withλncorresponding to the (unnormalized) eigenvectors xn,k= sinnk N. It follows E0(N, a) =¯hc aN−1/summationdisplay n=1sinnπ 2N. (19) To make connection with the continuous picture, we assign to the system the length L=aN (20) measuring the distance between the fixed oscillators, and el iminate Nin favour of aandLin eq. (19). After summing the series one obtains E0(L, a) =¯hc 2a/parenleftbigg ctgπa 4L−1/parenrightbigg . (21) 5With an expansion similar to eq. (8) ctgz=1 z−z 3+O(z3), (22) it follows for a≪L E0(L, a) =/parenleftBigg2¯hcL πa2−¯hc 2a/parenrightBigg −π 24¯hc L+O/parenleftbigga L/parenrightbigg . (23) The result is essentially the same with that in eq. (9). The aindependent term reproduces the renormalized value (10). An identical c omment applies to the a→0 diverging terms. Note that the L→ ∞ energy density can be equally obtained by making N→ ∞ in eq. (19) and evaluating the sum as an integral. Physically put, this corresponds to an infinite cr ystal with vibration modes characterized by a continuous quasimomentum in the Br illouin zone 0≤k <π a, (24) and dispersion relation ω(k) =2c asinka 2. (25) Note also that the second term, with no correspondent in eq. ( 9), can be absorbed into the first one with an irrelevant readjustment o f the box length L→L−πa 4. 4 Quantum field vs oscillator model: quanti- tative comparison and a speculation Let us define for a >0 the subtracted energy ES 0(L, a) as the difference between E0(L, a) and the paranthesis in eq. (23), so that lim a→0ES 0(L, a) =E0(L). (26) One may ask when the oscillator model provides a good approxi mation for the quantum field, in the sense that ES 0(L, a) E0(L)=−3/braceleftBigg/parenleftbigg4L πa/parenrightbigg ctg/parenleftbiggπa 4L/parenrightbigg −/parenleftbigg4L πa/parenrightbigg2/bracerightBigg (27) 6is close to unity. Note that by eq. (20) expression above is a f unction of Nonly. The corresponding dependence is plotted in Fig.1. One sees, quite surprisingly, that already a number of around twenty oscill ators suffices to assure a relative difference smaller than 10−4. More precisely, one has that the curve assymptotically approaches zero as π2 2401 N2. (28) We end with a bit of speculation. Suppose there exists some pr ivileged scalel(say, the Plank scale) which imposes a universal bound for le ngths measurements, and consider the oscillator system with the s pacing given by l. The indeterminacy in Lwill cause an indeterminacy in energy (we assume L≫l) ∆ES 0 ES 0∼∆E0 E0∼l L. (29) On the other hand, the assymptotic expression (28) implies ES 0−E0 E0∼/parenleftBiggl L/parenrightBigg2 . (30) We are led thus to the conclusion that, as far as Casimir effect measurements are considered, one could not distinguish between the “real ” quantum field and its oscillator model. References [1] C. Itzykson and J.B. Zuber, Quantum Field Theory , chap. 3, (Mc-Graw Hill, 1980). [2] I.J.R. Aitchinson and A.J.G. Hey, Gauge Theories in Particle Physics , chap. 4, (Adam Hilger, 1989). [3] H.B.G. Casimir, Proc. K. Ned. Akad. Wet., vol. 51, 793 (19 48). For a recent review, see S.K. Lamoreaux, Am. J. Phys. 67(10), pp. 850-861 (1999). 7[4] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation , pp. 426-428 (Freeman, San Francisco, 1973). [5] N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space , chap. 4, (Cambridge University Press, Cambridge, 1982). 8Fig.1. E0SE0 Relative difference between andas a function of the N-1 oscillators in the box.

arXiv:physics/9912018v1 [physics.plasm-ph] 8 Dec 1999Stopping of ion beam in a temperature anisotropic magnetize d plasma H. B. Nersisyan,∗M. Walter, and G. Zwicknagel Institut f¨ ur Theoretische Physik II, Universit¨ at Erlangen, D - 91058 Erlangen, Germany (Dated: November 3, 2012) Abstract Using the dielectric theory for a weakly coupled plasma we in vestigate the stopping power of the ion in a temperature anisotropic magnetized electron pl asma. The analysis is based on the assumption that the energy variation of the ion is much less t han its kinetic energy. The obtained general expression for stopping power is analyzed for weak a nd strong magnetic fields (i.e., for the electron cyclotron frequency less than and greater than the plasma frequency), and for low and high ion velocities. It is found that the friction coefficient contains, in addition to the usual velocity independent friction coefficient, an anomulous term which di verges logarithmically as the projectile velocity approaches zero. The physical origin of this anomu lous term is the coupling between the cyclotron motion of the electrons and the long-wave length, low-frequency fluctuations produced by the projectile ion. PACS numbers: 34.50.BW, 52.35.-g, 52.40.Mj ∗Permanent address: Division of Theoretical Physics, Insti tute of Radiophysics and Electronics, 1 Alikhanian Brothers Str., Ashtarak -2, 378410, Armenia 1I. INTRODUCTION Energy loss of the ions in a plasma has been a topic of great int erest due to its considerable importance for the study of basic interactions of the charge d particles in real media. Recent applications are electron cooling of heavy ion beams [1, 2, 3 ] and energy transfer for inertial confinement fusion (ICF) (see [4] for an overview). Electron cooling is realized by mixing the ion beam periodically with a cold electron beam of the sam e average velocity. The interaction length is normally about a few meters and the ele ctron beam is guided by a magnetic field parallel to its direction of motion. The cooli ng of the ion beam may then be viewed as an energy loss in the common rest frame of both bea ms. Similar questions arise in heavy-ion-induced ICF. There a frozen hydrogen pel let is heated and compressed by stopping of ion beams in the surrounding converter. In this c ase the electrons of the solid state converter are acting like a plasma and absorb the incom ing energy. In the electron cooling process the velocity distribution o f the electron beam is highly anisotropic because of the acceleration from the cathode to the cooling section. It can be described by a Maxwell distribution with two different tempe ratures, a longitudinal T/bardbland a transversal T⊥[1, 2, 3]. Furthermore, an external, longitudinal magnetic field is needed to guide the electrons from the cathode to and through the ele ctron cooler and to stabilize the anisotropic velocity distribution by suppressing the t ransverse-longitudinal relaxation. In the present paper we are interested in the influences of the magnetic field and the temperature anisotropy on the ion beam stopping power. Since the early 1960’s several theoretical calculations of the stopping power in a mag- netized plasma have been presented [5, 6, 7, 8, 9, 10, 11, 12, 1 3, 14]. Stopping of a fast test particle moving with velocity Vmuch higher than the electron thermal velocity vth was studied in Refs. [5, 6, 8]. Energy loss of a charged partic le moving with arbitrary velocity was studied in Ref. [7]. The expression obtained th ere for the Coulomb logarithm, Λ = ln(λD/ρ⊥) (whereλDis the Debye length and ρ⊥is the impact parameter for scat- tering for an angle ϑ=π/2), corresponds to the classical description of collisions . In the quantum-mechanical case, the Coulomb logarithm is Λ = ln( λD/λB), whereλBis the de Broglie wavelength of plasma electrons [15]. In Ref. [10], the expressions were derived describing the st opping power of a charged particle in Maxwellian plasma placed in a classically stron g (but not quantizing) magnetic 2field (λB≪ac≪λD,whereacis the electron Larmor radius), under the conditions when scattering must be described quantum mechanically. Calcul ations were carried out for slow test particles whose velocities satisfy the conditions ( m/m i)1/3vth< V≪vth, wheremiis the mass of the plasma ions and mis the electron mass. In the recent paper [11] the stopping power in the magnetized plasma has been inves- tigated for high-velocity light particles taking into acco unt the Larmor rotation of a test projectile in a magnetic field. It has been shown that the stop ping power can exhibit an oscillatory dependence on the magnetic field and that it is mu ch greater than in the case without magnetic field. More attention has been paid on the stopping power in a strong ly magnetized plasma for ions which move along the magnetic field [11, 12, 13]. Both uncorrelated [11, 13] and correlated [12] situations have been discussed. These investigations have concentrated on the stopping pow er in temperature isotropic plasma. Extensions to nonlinear effects of ion stopping and t emperature anisotropy have been done recently by particle-in-cell (PIC) computer simu lation [14], where the case T/bardbl≪ T⊥has been investigated which is interesting for electron coo ling process. Here, in the framework of dielectric theory, we will focus on the stoppin g power at arbitrary temperature anisotropy T⊥/T/bardbl. The paper is organized as follows. We start in Sec. II, with so lving the linearized Vlasov- Poisson equations by means of Fourier transformation. This provides the general form of the linearized potential generated in a temperature anisotrop ic magnetized Maxwellian plasma by a projectile ion from which the stopping power is deduced. In the next Sec. III, is dedicated to apply our results to nonm agnetized plasma. Calcula- tions are carried out for small projectile velocities at arb itrary temperature anisotropy and arbitrary direction of ion motion with respect to the anisot ropy axis. Then we turn to the effect of a weak magnetic field on the stoppin g power in Sec. IV, while we concentrate on the influence of a strong magnetic fiel d in Sec. V. In contrast with the papers [11, 13] we consider an ion motion in arbitrary dir ection. As the last issue we investigate in Sec. VI the stopping power for small projectile veloc- ities at arbitrary magnetic field and temperature anisotrop y. The friction coefficient there contains an anomalous term which increases logarithmicall y when the projectile velocity approaches to zero. 3The achieved results are finally summarized and discussed in Sec. VII. II. DIELECTRIC THEORY For the temperature anisotropic plasma with two different te mperatures T/bardbl,T⊥of the electrons we define an average temperature T=1 3T/bardbl+2 3T⊥. Within the dielectric theory the electron plasma is described as a continuous, polarizable fl uid (medium), which is represented by the phase-space density of the electrons f(r,v,t). Here, only a mean-field interaction between the electrons is considered and hard collisions are neglected and the evolution of the distribution function f(r,v,t) is determined by the Vlasov-Poisson equation is valid for weakly coupled plasmas where the number of electrons in the D ebye sphere ND= 4πn0λ3 D≫ 1 is very large. Here n0is the electron density, λD= (kBT/4πn0e2)1/2is an averaged Debye length. In the following, we consider a nonrelativistic projectile ion with charge Zeand with a velocity Vthat moves in a magnetized temperature anisotropic plasma a t an angleϑwith respect to the magnetic field B0. The axis defined by B0also coincides with the degree of freedom with temperature T/bardbl. We assume that the energy variation of the ion is much smaller than its kinetic energy. The strength of the couplin g betweeen an ion moving with velocityVand the electron plasma is given by the coupling parameter Z=|Z| ND[1 +V2/v2 th]3/2. (1) Herevth= (kBT/m)1/2is the average thermal velocity of an electron. The derivati on of Eq. (1) is discussed in detail in Ref. [16]. The parameter Zcharacterizes the ion-target coupling, where Z ≪ 1 corresponds to weak, almost linear coupling and Z>∼1 to strong, nonlinear coupling. For a sufficiently small perturbation ( Z ≪1) the linearized Vlasov equation of the plasma may be written as ∂f1 ∂t+v∂f1 ∂r−ωc[v×b]∂f1 ∂v=−e m∂ϕ ∂r∂f0 ∂v, (2) wheref=f0+f1and the self-consistent electrostatic potential ϕis determined by the Poisson equation 4▽2ϕ=−4πZeδ(r−Vt) + 4πe/integraldisplay dvf1(r,v,t). (3) Thebis the unit vector parallel to B0,−eandωc=eB0/mcare the charge and Larmor frequency of plasma electrons respectively, f0is the unperturbed distribution function of plasma electrons, which in the case of temperature anisotro pic, homogeneous electron plasma is given by two Maxwellians for the longitudinal and transve rsal degrees of freedom f0(v/bardbl,v⊥) =n0 (2π)3/2v2 th⊥vth/bardblexp/parenleftBigg −v2 ⊥ 2v2 th⊥/parenrightBigg exp −v2 /bardbl 2v2 th/bardbl , (4) where /angbracketleftv2 /bardbl/angbracketright=v2 th/bardbl=kBT/bardbl/m, /angbracketleftv2 ⊥/angbracketright= 2v2 th⊥= 2kBT⊥/m. By solving Eqs. (2) and (3) in space-time Fourier components , we obtain the electrostatic potential ϕ(r,t) =Ze 2π2/integraldisplay dkexp [ik(r−Vt)] k2ε(k,kV), (5) which provides the dynamical response of the temperature an isotropic plasma to the motion of the projectile ion in the presence of the external magneti c field. Here ǫ(k,ω) is the dielectric function of a temperature anisotropic, magneti zed plasma which is given by ε(k,ω) = 1 +1 k2λ2 D/bardbl[G(s) +iF(s)] (6) = 1 +1 k2λ2 D/bardbl  1 +is√ 2∞/integraldisplay 0dtexp/bracketleftBig ist√ 2−X(t)/bracketrightBig +kvth/bardbl√ 2 ωcsin2α(1−τ)∞/integraldisplay 0dtsin/parenleftBiggωct√ 2 kvth/bardbl/parenrightBigg exp/bracketleftBig ist√ 2−X(t)/bracketrightBig   with X(t) =t2cos2α+k2a2 c⊥sin2α/bracketleftBigg 1−cos/parenleftBiggωct√ 2 kvth/bardbl/parenrightBigg/bracketrightBigg , (7) whereλD/bardbl=vth/bardbl/ωp,ωpis the plasma frequency, s=ω/kv th/bardbl,τ=T⊥/T/bardbl,ac=vth⊥/ωcand αis the angle between the wave vector kand the magnetic field. As shown in Appendix A, Eqs. (6) and (7) are identical with the Bessel function rep- resentation of ε(k,ω) derived e.g. by Ichimaru [17]. Eqs. (6) and (7) are, however , more convenient when studying the weak and strong magnetic field l imits in Secs. IV and V. 5The stopping power Sof an ion is defined as the energy loss of the ion in a unit length due to interactions with the plasma electrons. From Eq. (5) it is straightforward to calculate the electric field E=−▽ϕ, and the stopping force acting on the ion. Then, the stopping power of the projectile ion becomes S=−dE dl=Ze∂ ∂rϕ(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=Vt(8) =2Z2e2λ2 D/bardbl π2kmax/integraldisplay 0k3dk1/integraldisplay 0dµπ/integraldisplay 0dϕcos ΘF(s) [k2λ2 D/bardbl+G(s)]2+F2(s), whereµ= cosαwas the angle between kandB0, Θ is the angle between kandV,s= k·V/kvth/bardbl= (V/vth/bardbl) cosΘ, cos Θ = µcosϑ−√1−µ2sinϑcosϕ, andϑis the angle between VandB0. In Eq. (8) we introduced a cutoff parameter kmax= 1/rmin(whererminis the effective minimum impact parameter) in order to avoid the log arithmic divergence at large k. This divergence corresponds to the incapability of the lin earized Vlasov theory to treat close encounters between the projectile ion and the plasma e lectrons properly. For rmin we thus use the effective minimum impact parameter of classic al binary Coulomb collisions rmin=Ze2/mv2 rfor relative velocities vr≃(V2+v2 th)1/2, which is often called the “distance of closest approach.” Hence kmax=1 rmin=m(V2+v2 th) Ze2. (9) A two temperature description of an electron plasma is valid only when the ion beam- plasma interaction time is less than the relaxation time bet ween the two temperatures, T/bardbland T⊥. For an estimate we will briefly consider the field-free case, because the external magnetic field suppresses the relaxation between the transversal and longitudinal temperatures during the time of flight of the ion beam through plasma. The problem of a temperature relaxation in a temperature ani sotropic plasma with and without of an external magnetic field was considered by Ichim aru [17]. Within the dominant- term approximation the relaxation time ∆ τrelfor the plasma without magnetic field is given by 1 ∆τrel=8 15/radicalbiggπ mn0e4 (kBTeff)3/2ln Λ c, (10) 6where ln Λ c= ln(ND) is the Coulomb logarithm and the effective electron tempera tureTeff is defined through 1 T3/2 eff=15 21/integraldisplay 0µ2(1−µ2)dµ [µ2T/bardbl+ (1−µ2)T⊥]3/2(11) =5√ 3 12T3/2(1 + 2τ)3/2 (τ−1)2 τ+ 2/radicalBig |τ−1|p0(τ)−3 , p0(τ) =  ln1+√1−τ√τ, τ < 1 arctan√τ−1, τ > 1. (12) The relaxation time calculated from Eq. (11) are of the order of 10−6s, 0.5×10−5s and 10−3s for averaged temperatures T= 10−2eV,T= 0.1eV andT= 1eV, respectively, for anisotropies τ≃0.01−100. The interaction time (for instance, for ICF or for elect ron cooling) is about 10−7−10−8s. Therefore, ion beam-plasma interaction time can be very small compared to the plasma relaxation time. III. STOPPING POWER IN PLASMA WITHOUT MAGNETIC FIELD Let us analyse expression (8) in the case when a projectile io n moves in a temperature anisotropic plasma without magnetic field. The plasma diele ctric function from Eqs. (6) and (7) now takes the form ε(k,ω) = 1 +1 k2λ2 D/bardbl1 A2W/parenleftbiggs A/parenrightbigg . (13) HereA= (µ2+τ(1−µ2))1/2andW(s) =g0(s) +if0(s) is the plasma dispersion function [18], g0(s) = 1−s√ 2Di/parenleftBiggs√ 2/parenrightBigg ;f0(s) =/radicalbiggπ 2sexp/parenleftBigg −s2 2/parenrightBigg , (14) where Di(s) = exp( −s2)s/integraldisplay 0dtexp(t2) (15) 7is the Dawson integral [18] which has for large arguments sthe asymptotic Di(s)≃1/2s+ 1/4s3. Substituting Eq. (13) into Eq. (8) and performing the k-integration we obtain S0=Z2e2 2π2λ2 D/bardbl1/integraldisplay 0dµπ/integraldisplay 0dϕcos Θ A2Q0/parenleftBiggv vth/bardblcos Θ A, ξ/bardblA/parenrightBigg , (16) whereξ/bardbl=kmaxλD/bardbland Q0(x,ξ) =f0(x) lnf2 0(x) + [ξ2+g0(x)]2 f2 0(x) +g2 0(x)(17) +2g0(x)/bracketleftBigg arctang0(x) f0(x)−arctanξ2+g0(x) f0(x)/bracketrightBigg . In the case of temperature isotropic plasma ( T⊥=T/bardbl≡T,andτ= 1)A= 1 and Eq. (16) coincides with the result of e.g. Ref. [19] S0=Z2e2 2πλ2 Dv2 th V2V/vth/integraldisplay 0dµµQ 0(µ,ξ), (18) wherevth=vth/bardbl=vth⊥,λD=vth/ωp, andξ=kmaxλD. When a projectile ion moves slowly through a plasma, the elec trons have much time to experience the ion attractive potential. They are accelera ted towards the ion, but when they reach its trajectory the ion has already moved forward a little bit. Hence, we expect an increased density of electrons at some place in the trail o f the ion. This negative charge density pulls back the positive ion and gives rise to the stop ping. This drag force is of particular interest for the electron cooling process. In th e limit of small velocities S≃R·V. This looks like the friction law of a viscous fluid, and accord inglyRis called the friction coefficient. However, in the case of an ideal plasma it should b e noted that this law does not depend on the plasma viscosity and is not a consequence of electron-electron collisions which are neglected in the Vlasov equation. The Taylor expansion of Eq. (16) for small V(V≪vth) yields the friction law S0=Z2/parenleftBig e2/λ2 D/parenrightBig 3√ 2πV vthψ(ξ)/bracketleftBig I1(τ) +I2(τ) sin2ϑ/bracketrightBig , (19) whereξ=kmaxλD= (1 +V2/v2 th)/Z ≃1/Z, 8I1(τ) =3 ψ(ξ)/parenleftbigg2τ+ 1 3/parenrightbigg3/21/integraldisplay 0dµµ2ψ(ξ/bardblA(µ)) A3(µ), (20) I2(τ) =3 2ψ(ξ)/parenleftbigg2τ+ 1 3/parenrightbigg3/21/integraldisplay 0dµ(1−3µ2)ψ(ξ/bardblA(µ)) A3(µ), (21) and the function ψis ψ(ξ) = ln(1 +ξ2)−ξ2 1 +ξ2. (22) In the case of temperature isotropic plasma ( τ= 1) we have I1= 1 andI2= 0. Then the Eq. (19) becomes the usual friction law in an isotropic pl asma [19]. For the strongly temperature anisotropic case, when τ≪1 (T⊥≪T/bardbl) we haveξ/bardbl≃√ 3/Zand I1≃ −√ 3 6ψ(ξ)/bracketleftBig Li2(1 +ξ2 /bardbl) + ln(1 +ξ2 /bardbl)/bracketrightBig , (23) I2≃√ 3 12ψ(ξ)/bracketleftBig ξ2 /bardbl+ 2 ln(1 +ξ2 /bardbl) + 3Li2(1 +ξ2 /bardbl)/bracketrightBig . (24) Here the functions I1andI2do not depend on τ, andLi2(x) is the dilogarithm function [20]. Note that Z ≪ 1 and therefore ξ≫1,ξ/bardbl≫1 in Eqs. (23) and (24). The Coulomb logarithms in Eqs. (23) and (24) are then the leading terms an d I1≃√ 3 6ln1 Z≪I2≃√ 3 8Z21 ln(1/Z). (25) The normalized friction coefficient (Eq. (19)) is thus domina ted by the second term and increases with increasing ϑ. In the opposite case, τ≫1 (T⊥≫T/bardbl), the evoluation of Eqs. (20) and (21) yields I1≃π√ 6 3ψ(ξ) /radicalBigg 1 +3 2ξ2−1−2 ln1 +/radicalBig 1 +3 2ξ2 2 , (26) I2≃π√ 6 6ψ(ξ) 1 +1/radicalBig 1 +3 2ξ2−2/radicalBigg 1 +3 2ξ2+ 6 ln1 +/radicalBig 1 +3 2ξ2 2 , (27) and I1≃ −I2≃π 2Zln(1/Z). (28) 9ThenI1+I2sin2ϑ≃I1cos2ϑand the normalized friction coefficient decreases with incre asing ofϑin this case. In Fig. 1 the normalized friction coefficient I1+I2sin2ϑis plotted as a function of temperature anisotropy τforϑ= 0 (solid line), ϑ=π/6 (dotted line), ϑ=π/3 (dashed line),ϑ=π/2 (dot-dashed line) and for fixed plasma density and average t emperature (Z= 0.2). Fig. 1 shows an enhancement of the friction coefficient whe n the ion moves along the direction with low temperature. This effect can be easily explained in a binary collision picture. Let us consider the particular case of strongly ani sotropic plasma T⊥≫T/bardbl. In this case the plasma electrons move mostly in the direction acros s to the anisotropy axis. For ϑ≃π/2 the projectile ion moves along the plasma electrons therma l fluctuation direction and effective impact parameter for electron-ion collision i s very small. Then the friction coefficient decreases. For ϑ≃0 the projectile ion moves across to the direction of plasma electrons thermal fluctuation. Therefore, the impact param eter for electron-ion collisions increases which rises the friction coefficient. For arbitrary projectile velocities we evaluated Eq. (16) n umerically. In Figs. 2 and 3 the stopping power is plotted for strongly temperature ani sotropic plasmas ( τ= 10−2 andτ= 102in Figs. (2) and (3) respectively) with n0= 108cm−3,T= 0.1eV and for four values of ϑ;ϑ= 0 (dotted line), ϑ=π/6 (dashed line), ϑ=π/3 (long-dashed line), ϑ=π/2 (dot-dashed line). The solid lines are plotted for tempera ture isotropic plasma with T=T= 0.1eV. The general behaviour of the stopping power for two anis otropy parameters τis characterized by an increase by comparision with the isot ropic case. At ϑ≃π/2 and τ= 10−2(Fig. (2)) the ion moves in direction accross to the longitud inal electron motion with the lower temperature T⊥and the maximum of the stopping power is around V≃vth⊥, whereas the maximum for an ion motion in longitudinal direct ion is atV≃vth/bardbl≫vth⊥. IV. STOPPING IN PLASMAS WITH WEAK MAGNETIC FIELD For the case when the magnetic field is weak, in the sense that t he dimensionless parame- terη=ωc/ωpis much less than unity, the functions GandF, Eqs. (6) and (7), which define the dielectric function, can be expanded about its field free valuesg0(s/A)/A2,f0(s/A)/A2 Eqs. (14) and (15) 10G(s) +iF(s) =1 A2/bracketleftbigg g0/parenleftbiggs A/parenrightbigg +if0/parenleftbiggs A/parenrightbigg/bracketrightbigg +η2sin2α (kλD/bardbl)2[g1(s) +if1(s)], (29) where g1(s) +if1(s) =2 3(1−τ)∞/integraldisplay 0t3dt/parenleftBiggt2 2τsin2α−1/parenrightBigg exp(ist√ 2−A2t2) (30) +is√ 2 6τ∞/integraldisplay 0t4dtexp(ist√ 2−A2t2), s=ω/kv th/bardbl. Substituting this expression (29) into Eq. (8) leads to S=S0+η2S1, (31) whereS0is the stopping power in plasma without magnetic field Eq. (16 ) andη2S1represents the change due to a weak magnetic field. After some simplificat ions it becomes S1=/radicalbiggπ 2Z2e2 24π2λ2 D/bardblV vth/bardbl1/integraldisplay 0dµπ/integraldisplay 0dϕ(1−µ2) cos2Θ A5(32) ×exp −V2 v2 th/bardblcos2Θ 2A2 τ/parenleftbigg 7−V2 v2 th/bardblcos2Θ A2/parenrightbigg −4A2 f2 0/parenleftbigg V vth/bardblcos Θ A/parenrightbigg +g2 0/parenleftbigg V vth/bardblcos Θ A/parenrightbigg. In the temperature isotropic plasma ( τ= 1) Eq. (32) coincides with the results by May and Cramer [7] after integration over ϕ. Note that the additional term S1does not depend on the cutoff parameter kmax. In the next subsections we evaluate Eq. (32) for small and lar ge projectile velocities. A. Small projectile velocities When the projectile ion moves slowly ( V <vth) in plasma Eq. (32) leads to the simplified expression S1=Z2e2 60πλ2 D/radicalbiggπ 2V vthP(ϑ,τ), (33) with 11P(ϑ,τ) =/parenleftbigg1 + 2τ 3/parenrightbigg3/2/bracketleftBig P1(τ) +P2(τ) sin2ϑ/bracketrightBig , (34) P1(τ) =5 6(1−τ)2  14τ+ 25−3(9τ+ 4)/radicalBig |1−τ|p0(τ)  , (35) P2(τ) =5 12τ(1−τ)2  3τ(23τ+ 16)/radicalBig |1−τ|p0(τ)−28τ2−91τ+ 2  . (36) Here, the function p0(τ) is given by Eq. (12). In temperature isotropic plasma with τ= 1 we haveP1(1) =P2(1) = 1. In Fig. 4 the normalized friction coefficient P(ϑ,τ) for the additional stopping power S1 is plotted as a function of τforϑ= 0 (solid line), ϑ=π/6 (dotted line), ϑ=π/3 (dashed line),ϑ=π/2 (dot-dashed line). The general behaviour of P(ϑ,τ) is similar to the friction coefficient of the plasma without magnetic field (see Fig. 1). H ere, the correction P(ϑ,τ) can be also negative at small τandϑ, which then corresponds to a slight decrease of the stopping power, Eq. (31). B. High projectile velocities When the projectile ion moves with large velocity ( V≫vth), Eq. (32), yields S1=−Z2e2ω2 p 8V2/braceleftBigg 2C1(1 + cos2ϑ)−C2B(ϑ,τ)/bracketleftBigg cos2ϑ+sin2ϑ B(ϑ,τ) + 1/bracketrightBigg/bracerightBigg , (37) where B(ϑ,τ) =/parenleftbiggτ τcos2ϑ+ sin2ϑ/parenrightbigg1/2 , (38) C1=1 3√ 2π∞/integraldisplay 0x2exp(−x2/2)dx f2 0(x) +g2 0(x), C2=1 3√ 2π∞/integraldisplay 0x2(7−x2) exp(−x2/2)dx f2 0(x) +g2 0(x).(39) For numbers C1andC2we get the accurate values C1= 1 andC2= 0 (C2≃10−12) respectively. Therefore from Eq. (37) we have finally S1=−Z2e2ω2 p 4V2(1 + cos2ϑ). (40) 12This result is in accord with the results of Honda et al. [6] an d May and Cramer [7], who, however, kept the terms O(V−4) in their work as well. Although the function S1in Eq. (40) is proportional to the plasma density, the full correction t ermη2S1does not depend on the plasma density. In Figs. (5) and (6) we show the velocity dependence of the fun ctionS1forτ= 10−2 andτ= 102respectively. The different curves are ϑ= 0 (solid line), ϑ=π/6 (dotted line), ϑ=π/3 (dashed line), ϑ=π/2 (dot-dashed line). For small and medium projectile veloci ties the weak magnetic field decreases the total stopping power fo r smallτand increases it in the highτlimit. For high projectile velocities the magnetic field alw ays reduces the stopping power independent of the temperature anisotropy, see Eq. (4 0). V. STOPPING IN PLASMAS WITH STRONG MAGNETIC FIELD We now turn to the case when a projectile ion moves in a tempera ture anisotropic plasma with a strong magnetic field, which is on one hand, sufficiently weak to allow a classical description (¯ hωc<kBT⊥or ¯h/mv th⊥<ac, and, on the other hand, comparatively strong so that the cyclotron frequency of the plasma electrons exceed s the plasma frequency ωc≫ωp. This limits the values of the magnetic field itself and values of perpendicular temperature and plasma density. From these conditions we can obtain 3×10−6n1/2 0<B 0<105T⊥, (41) wheren0is measured in cm−3,T⊥is measured in eV, and B0in kG. Conditions (41) are always true in the range of parameters n0<1015cm−3,B0<100kG,T⊥>10−3eV. Then the perpendicular motion of the electrons is completely quench ed and the stopping power de- pends only on the longitudinal electron temperature T/bardbl. The dependence on the transversal temperature will be only introduced by the cutoff parameter E q. (9). In the limit of sufficiently strong magnetic field, Eq. (8) beco mes Sinf=2Z2e2 π2λ2 D/bardblξ/bardbl/integraldisplay 0k3dk1/integraldisplay 0dµπ/integraldisplay 0dϕcos Θf0(s) [k2+g0(s)]2+f2 0(s), (42) withs= (V/vth/bardbl)(cos Θ/µ) andg0,f0from Eqs. (14), which gives after integration over k 13Sinf=Z2e2 2π2λ2 D/bardbl1/integraldisplay 0dµπ/integraldisplay 0dϕcos ΘQ0/parenleftBiggV vth/bardblcos Θ µ,ξ/bardbl/parenrightBigg . (43) Here the function Q0is given by Eq. (17). For further simplification of Eq. (43) we introduce the new variable of integration x= cos Θ/µ. Afterϕintegration in Eq. (43) we finally find the stopping power in the presence of a strong magnetic field a s Sinf(V,ϑ) =Z2e2 8πλ2 D/bardblQ/parenleftBiggV vth/bardbl,ϑ/parenrightBigg , (44) where Q/parenleftBiggV vth/bardbl,ϑ/parenrightBigg = sin2ϑ∞/integraldisplay −∞Q0/parenleftbigg V vth/bardblx,ξ/bardbl/parenrightbigg xdx (x2+ 1−2xcosϑ)3/2. (45) In the previous works [11, 12, 13] only the case of ϑ= 0 the motion of the projectile along the magnetic field direction has been investigated. In this case the integral in Eq. (45) diverges, while prefactor sin2ϑtends to zero. Introducing the new variable of the integrati on in Eq. (45) y= (x−cosϑ)/sinϑwe obtain for vanishing angle ϑ Q/parenleftBiggV vth/bardbl,ϑ→0/parenrightBigg = 2Q0/parenleftBiggV vth/bardbl,ξ/bardbl/parenrightBigg . (46) Thus expression (44) reproduces the known results for the st opping power on an ion which moves along the direction of the magnetic field [11, 12, 13]. In the following paragraphs we will discuss its low and high v elocity limits. A. Small projectile velocities In the low velocity limit ( V≪vth/bardbl) Eq. (45) becomes Q/parenleftBiggV vth/bardbl,ϑ/parenrightBigg ≃2V vth/bardbl/braceleftbigg√ 2πψ(ξ/bardbl)/bracketleftbigg sin2ϑln/parenleftbigg2vth/bardbl Vsinϑ/parenrightbigg + 1−2 sin2ϑ/bracketrightbigg +C1(ξ/bardbl) sin2ϑ/bracerightbigg ,(47) where C1(ξ/bardbl) =1/integraldisplay 0dx x2/bracketleftBig Q0(x,ξ/bardbl)−√ 2πψ(ξ/bardbl)x/bracketrightBig +∞/integraldisplay 1dx x2Q0(x,ξ/bardbl). (48) 14Here, the function ψis defined by Eq. (22). Since we deal with small ion beam-plasm a coupling Z ≪1 we have, ξ/bardbl≫1 in Eqs. (47) and (48) and the function C1(ξ) simplifies C1(ξ/bardbl)≃√ 2πln2 γlnξ/bardbl+ 0.6, (49) whereγ= 0.5772 is Euler’s constant. We note that the friction coefficient Sinf/Vfrom Eqs. (44) and (47) contains a logarith- mically large term which vanishes for ϑ→0. It will be shown in the next section that this behaviour is a characteristic feature of the stopping power at low velocities and the friction coefficient for arbitrary strength of the magnetic field. B. High projectile velocities In the case of high projectile velocities ( V≫vth/bardbl) the general expression (45) becomes Q/parenleftBiggV vth/bardbl,ϑ/parenrightBigg ≃4πv2 th/bardbl V2/braceleftBigg sin2ϑ/bracketleftBigg ln/parenleftBigg2V vth/bardblsinϑ/parenrightBigg +C2(ξ/bardbl)−2/bracketrightBigg + 1/bracerightBigg , (50) where C2(ξ/bardbl) =1 2π1/integraldisplay 0Q0(x,ξ/bardbl)xdx+∞/integraldisplay 1dx x/bracketleftBiggx2 2πQ0(x,ξ/bardbl)−1/bracketrightBigg (51) which gives for ξ/bardbl≫1C2(ξ/bardbl)≃lnξ/bardbl. The stopping power for strong magnetic fields shows in the low and high velocity limits (Eqs. (47) and (50)) an enh ancement for ions moving transversal to the magnetic field compared to the case of the l ongitudinal motion ( ϑ= 0). This effect is in agreement with PIC simulation results [14]. In contrast to the field-free case, at strong magnetic field and for ϑ= 0,V≫vth/bardbl(Eqs. (44) and (50)) we have Sinf≃Z2e2ω2 p/2V2independent of kmax. The cutoff kmaxnecessary at low ion velocities is, however, less well defined here than for the field-free case, w here the cutoff (9) was deduced from the binary collision picture. Now, the electrons are fo rced to move parallel to B0. Since we assumed the motion of the ion in this direction as well the i on and an electron just pass each other along a straight line. For symmetry reasons the to tal momentum transfer and the stopping power is zero. Purely binary interactions cont ribute nothing and the stopping of the ion is only due to the collective response of the plasma , that is, due to modes with 15long wavelengths k <1/λD/bardbl. This suggests taking kmaxof the order of 1 /λD/bardbl, but further investigations are clearly needed here for a more precise de scription in this particular case. In Figs. (7) and (8), the stopping power Sinfis plotted as a function of projectile velocity (in units of vth/bardbl) forn0= 106cm−3,T/bardbl= 10−4eV ,T⊥= 10−5eV (Fig. (7)), T⊥= 0.1eV (Fig. (8)), and for four different values of angle ϑ:ϑ= 0 (solid line), ϑ=π/6 (dotted line),ϑ=π/3 (dashed line) and ϑ=π/2 (dash-dotted line). The enhancement of Sinf(V,ϑ) with respect to Sinf(V,0) in the low and in high velocity limit by increasing of the an gleϑis documented in Fig. (9), for T/bardbl= 10−4eV,T⊥= 0.1eV,n0= 106cm−3,ϑ=π/6 (solid line), ϑ=π/4 (dotted line), ϑ=π/3 (dashed line) and ϑ=π/2 (dash-dotted line). The physical origin of this angular behaviour in the low and high velocity limits is the enhancement of the effective impact parameter for an individual electron-i on collision with increasing ϑ. For medium projectile velocities V≃vth/bardblthe collective excitations in plasma become important and then stopping power is higher for small ϑ. VI. STOPPING AT ARBITRARY MAGNETIC FIELD AND IN LOW-VELOCIT Y LIMIT. ANOMALOUS FRICTION COEFFICIENT We now proceed with a projectile ion at low velocities and at a rbitrary magnetic field. This regime is of particular importance for the electron coo ling process [1, 2, 3]. In the presence of a magnetic field the friction coefficient here cont ains a term which diverges like ln(vth/bardbl/V) in addition to the usual (see e.g. Sec. III) constant one. For this consideration it is convenient to use the Bessel fun ction representation of the dielectric function which has been given e.g. by Ichimaru [1 7], see Appendix A Eq. (A7), and to write the real and imaginary parts of Eq. (A7) separate ly G= 1−√ 2ω |k/bardbl|vth/bardblΛ0(z)Di/parenleftBiggω |k/bardbl|vth/bardbl√ 2/parenrightBigg (52) −√ 2 |k/bardbl|vth/bardbl∞/summationdisplay n=1Λn(z)/braceleftBigg ω/bracketleftBigg Di/parenleftBiggω+nωc |k/bardbl|vth/bardbl√ 2/parenrightBigg +Di/parenleftBiggω−nωc |k/bardbl|vth/bardbl√ 2/parenrightBigg/bracketrightBigg +nωc/parenleftbigg1 τ−1/parenrightbigg/bracketleftBigg Di/parenleftBiggω−nωc |k/bardbl|vth/bardbl√ 2/parenrightBigg −Di/parenleftBiggω+nωc |k/bardbl|vth/bardbl√ 2/parenrightBigg/bracketrightBigg/bracerightBigg , 16F=/radicalbiggπ 2  ω |k/bardbl|vth/bardblΛ0(z) exp −ω2 2k2 /bardblv2 th/bardbl  (53) +2 |k/bardbl|vth/bardbl∞/summationdisplay n=1Λn(z) exp −ω2+n2ω2 c 2k2 /bardblv2 th/bardbl  × ωch nωcω k2 /bardblv2 th/bardbl +nωc/parenleftbigg1 τ−1/parenrightbigg sh nωcω k2 /bardblv2 th/bardbl    . The notations in Eqs. (52) and (53) are explained in Appendix A. For the friction coefficient we have to consider S, given by Eq. (8) in the low-velocity limit and thus the functions GandFgiven by Eqs. (52) and (53), when ω=kV. Now we have to write the Taylor expansion of Eqs. (52) and (53) for sm allω=kV. However, the first term of Eq. (53) exhibits a singular behaviour in the lim it ofω=kV→0 where the k/bardbl integration diverges logarithmically for small k/bardbl. We must therefore keep ω=kVfinite in that integration to avoid such a divergence. This anomalous contribution which arises from the first term of Eq. (53) in low-velocity limit is San≃/parenleftbigg2 π3/parenrightbigg1/2Z2e2 λ2 D/bardblV vth/bardblξ/bardbl/integraldisplay 0k3dk1/integraldisplay 0dµ µπ/integraldisplay 0dϕcos2ΘΛ0(z) exp/parenleftbigg −V2 2v2 th/bardblcos2Θ µ2/parenrightbigg [k2+E2(k,µ)]2, (54) where Λ 0(z) = exp( −z)I0(z) andE2(k,µ) =G(ω= 0) is E2(k,µ) = 1 +2√ 2η kµ/parenleftbigg1 τ−1/parenrightbigg∞/summationdisplay n=1nΛn(z)Di/parenleftBiggnη kµ√ 2/parenrightBigg . (55) Herez= (k2τ/η2)(1−µ2),µ= cosα=k/bardbl/k, and Θ is the angle between kandV. Afterµ andϕintegration, see Appendix B, Eq. (54) reads San≃/parenleftbigg2 π/parenrightbigg1/2Z2e2 4λ2 D/bardblV vth/bardblsin2ϑln/parenleftbiggvth/bardbl V2.26 sinϑ/parenrightbigg F(τ,η), (56) with F(τ,η) =τξ2 /bardbl/integraldisplay 0Λ0(x/η2)xdx [x+ 1 + (τ−1)Λ0(x/η2)]2. (57) The function Fand thusSan(56) vanishes in the limit B0→0 (orη→0) like 17F(τ,η)≃η (2π)1/2/bracketleftBigg arctan(kmaxλD⊥)−kmaxλD⊥ 1 + (kmaxλD⊥)2/bracketrightBigg . (58) The anomalous term Eqs. (56) and (57) therefore represents a new effect arising from the presence of the magnetic field, which is not restricted to anisotropic plasmas. For temperature isotropic plasma ( τ= 1) and for a sufficiently weak magnetic field η <ξ /bardbl (orωc<kmaxvth/bardbl), Eq. (57) takes the form F(τ,η)≃exp/parenleftBigg1 η2/parenrightBigg/bracketleftBigg/parenleftBigg 1 +1 η2/parenrightBigg K0/parenleftBigg1 η2/parenrightBigg −1 η2K1/parenleftBigg1 η2/parenrightBigg/bracketrightBigg , (59) whereK0andK1are the modified Bessel functions of the second kind. In the ca se of very strong magnetic field η >ξ /bardbl√τ(orωc>kmaxλD⊥) the function F(τ,η) reads F(τ,η)≃Ψ(ξ/bardbl) = ln(1 +ξ2 /bardbl)−ξ2 /bardbl 1 +ξ2 /bardbl. (60) The physical origin of such an anomalous friction coefficient may be traced to the spiral motion of the electrons along the magnetic field lines. These electrons naturally tend to couple strongly with long-wavelength fluctuations (i.e., s mallk/bardbl) along the magnetic field. In addition, when such fluctuations are characterized by slo w variation in time (i.e., small ω=kV), the contact time or the rate of energy exchange between the electrons and the fluctuations will be further enhanced. In a plasma such low-f requency fluctuations are pro- vided by the slow projectile ion. The above coupling can ther efore be an efficient mechanism of energy exchange between the electrons and the projectile ion. In the limit of V→0, the frequencyω=kV→0 tends to zero as well. The contact time thus becomes infinite and the friction coefficient diverges. The anomalous friction coefficient (see Eq. (56)) vanishes, h owever, when the ion moves along the magnetic field ( ϑ= 0). Then the friction coefficient is solely given by the secon d term of Eq. (53). The contribution of this term to the stoppin g power leads to the usual friction law in plasma and reads for arbitrary angles ϑ S≃/parenleftbigg2 π/parenrightbigg1/22Z2e2 λ2 D/bardblV vth/bardblξ/bardbl/integraldisplay 0k3dk1/integraldisplay 0dµ µE1(k,µ) [k2+E2(k,µ)]2(61) ×/bracketleftbigg µ2cos2ϑ+1 2/parenleftBig 1−µ2/parenrightBig sin2ϑ/bracketrightbigg 18with E1(k,µ) =∞/summationdisplay n=1Λn(z) exp/parenleftBigg −n2η2 2k2µ2/parenrightBigg/bracketleftBigg 1 +/parenleftbigg1 τ−1/parenrightbiggn2η2 k2µ2/bracketrightBigg (62) andE2(k,µ) as defined by Eq. (55). In Figs. (10) and (11) we compare the anomalous term Sanwith the low velocity stopping without magnetic field S0see Eq. (19), where San/S0is plotted as a function of ωc/ωpfor ϑ=π/6 (solid line), ϑ=π/3 (dotted line), ϑ=π/2 (dashed line), Z= 0.1,V/vth= 0.2, and for two values of the anisotropy parameter τ:τ= 0.1 (Fig. (10)), τ= 10 (Fig. (11)). We conclude that the anomalous term Sangives espessially for strong magnetic fields ( ωc> ωp) and for strongly temperature anisotropic plasma ( T⊥≫T/bardbl) an important contribution to the stopping. It should be noted that the observed enhance ment of stopping due to San forT⊥≫T/bardblcan be potentially interesting for future electron cooling experiments. VII. SUMMARY The purpose of this work was to analyze the stopping power of a n ion in temperature anisotropic magnetized classical plasma. A general expres sion obtained for stopping power was analyzed in four particular cases: in a plasma without ma gnetic field; in a plasma with weak and very strong magnetic fields; and in a plasma with arbi trary magnetic field and for low-velocity projectile. From the results obtained in Secs. III-V, we found that the st opping power essentially depends on the plasma temperature anisotropy. In field-free case and for small ion velocities the effect of the anisotropy results in an enhancement of the s topping power when the ion moves in the direction with low temperature. For small projectile velocities a weak magnetic field slight ly decreases the field-free stop- ping power for small τ, in the opposite case (large τ) the field-free stopping power slightly increases. In the high-velocity limit correction to the fiel d-free stopping power for weak magnetic fields is always negative and the stopping power is r educed by the magnetic field. In the case of strong magnetic fields we demonstrated an enhan cement of the stopping power with increasing of ϑfor low and high-velocity regions compared to the case of an i on which moves along B0. 19In low-velocity limit but for arbitrary magnetic field, we fin d an enhanced stopping power compared to the field-free value mainly due to the strong coup ling between the spiral motion of the electrons and the long-wavelength, low-frequency flu ctuations excited by the projectile ion. This anomalous stopping power increases with the angle ϑ(the angle between ion velocity Vand magnetic field B0) and depends strongly on the temperature anisotropy τ=T⊥/T/bardbl, as seen in Figs. (10) and (11). Although the nature of the ano malous stopping power is conditioned by the external magnetic field the tempe rature anisotropy of the plasma can intensify this effect when T⊥≫T/bardbl(see Fig. (11)). This emphasizes the importance of the special role of fluctua tions with small k/bardbland small ω(small projectile velocity V) and as another significant contribution to the energy excha nge processes arising from the collective modes of plasma. Pote ntially, the electron plasma waves and the ion acoustic waves in a magnetized plasma might provi de a significant energy- exchange mechanism between projectile ion and plasma parti cles. This fact then makes it necessary to consider the influence of plasma collective m odes to anomalous stopping process. This problem will be treated in a subsequent work. Acknowledgments Finally, it is pleasure to thank Prof. Christian Toepffer for helpful discussions. One of the authors (H.B.N.) is grateful to Prof. Christian Toepffer for hospitality at the Institut f¨ ur Theoretische Physik II, Universit¨ at Erlangen-N¨ urn berg, where this work was concluded and would like to thank the Deutscher Akademischer Austausc hdienst for financial support. We are indebted to Claudia Schlechte for her help in preparin g the manuscript. APPENDIX A Here we describe the evaluation of the dielectric function i n the temperature anisotropic case where the velocity distribution of the unperturbated d istribution function was given by Eq. (4). We next introduce the Fourier transformations of f1(r,v,t) with respect to variables randt,f1(k,ω,v). Because of the cylindrical symmetry (around the magnetic field direction b=B0/B0=ˆz) of the problem, we choose 20v=v⊥cosσˆx+v⊥sinσˆy+v/bardblˆz. (A1) Then the Vlasov Eq. (2) for the distribution function become s ∂ ∂σf1(k,ω,v) +i ωc(kv−ω−i0)f1(k,ω,v) =−ie mωcϕ(k,ω)/parenleftBigg k∂f0 ∂v/parenrightBigg , (A2) whereϕ(k,ω) is the Fourier transformation of ϕ(r,t). The positive infinitesimal + i0 in Eq. (A2) serves to assure the adiabatic turning on of the disturb ance and thereby to guarantee the causality of the response. The solution of the Eq. (A2) ha s the form f1(k,ω,v) =−ie mωcϕ(k,ω)σ/integraldisplay ∞dσ2/parenleftBigg k∂f0 ∂v/parenrightBigg σ=σ2exp i ωcσ2/integraldisplay σdσ1[−ω−i0 + (kv)σ=σ1] .(A3) Combining Eq. (A3) with the Poisson equation (3) we find for th e dielectric function ε(k,ω) = 1−4πie2 mωck2∞/integraldisplay 0v⊥dv⊥2π/integraldisplay 0dσ+∞/integraldisplay −∞dv/bardblσ/integraldisplay ∞dσ2/bracketleftBigg k/bardbl∂f0 ∂v/bardbl+k⊥cos(ϕ−σ2)∂f0 ∂v⊥/bracketrightBigg ×exp i ωcσ2/integraldisplay σdσ1/bracketleftBig k/bardblv/bardbl−ω−i0 +k⊥v⊥cos(ϕ−σ1)/bracketrightBig , (A4) wherekx=k⊥cosϕ,ky=k⊥sinϕ. After integration by the variables σ1,σ2andσ, and using the expression [20] exp(−izsinθ) =+∞/summationdisplay n=−∞Jn(z) exp(−inθ), (A5) whereJnis the Bessel function of the nth order, we obtain the expression [17] ε(k,ω) = 1−8π2e2 mk2+∞/summationdisplay n=−∞∞/integraldisplay 0v⊥dv⊥+∞/integraldisplay −∞dv/bardbl/parenleftBiggnωc v⊥∂f0 ∂v⊥+k/bardbl∂f0 ∂v/bardbl/parenrightBiggJ2 n(k⊥v⊥/ωc) nωc+k/bardblv/bardbl−ω−i0.(A6) Substituting Eq. (4) for the unperturbed distribution func tionf0into Eq. (A6) we finally results in ε(k,ω) = 1 +1 k2λ2 D/bardbl/braceleftBigg 1 ++∞/summationdisplay n=−∞/parenleftBigg 1 +T/bardbl T⊥nωc ω−nωc/parenrightBigg/bracketleftBigg W/parenleftBiggω−nωc |k/bardbl|vth/bardbl/parenrightBigg −1/bracketrightBigg Λn(β)/bracerightBigg ,(A7) 21whereβ=k2 ⊥v2 th⊥/ω2 c=k2 ⊥a2 c, Λn(z) = exp( −z)In(z),In(z) is the modified Bessel function of thenth order, and W(z) is the plasma dispersion function [18]. To show the identity of the two forms (Eqs. (6) and (A7)) of the dielectric function we will use the expansion in modified Bessel functions [20] exp(zcosθ) =∞/summationdisplay n=−∞In(z) exp(inθ). (A8) This allows to rewrite exp[ −X(t)] withX(t) from Eq. (7) as exp[−X(t)] = exp( −t2cos2α)+∞/summationdisplay n=−∞Λn(β) exp/parenleftBigginωct√ 2 kvth/bardbl/parenrightBigg . (A9) Substituting Eq. (A9) into expression (6) and integration o ver the variable tleads to Eq. (A7). APPENDIX B We now give a more detail derivation of the anomalous term San(Eq. (56)). We start with the expression (see Eq. (54)) Q(k,ϕ,λ ) =1/integraldisplay 0dµ µΦ(µ,k,ϕ ) exp/parenleftBigg −λ2φ2(µ,ϕ) 2µ2/parenrightBigg , (B1) whereφ(µ,ϕ) = cos Θ,λ=V/vth/bardbl, Φ(µ,k,ϕ ) =Λ0(z) cos2Θ [k2+E2(k,µ)]2. (B2) Forλ→0 a leading-term approximation of (B1) leads to Q(k,ϕ,λ )≃Φ(0,k,ϕ) ln√ 2 λ|φ(0,ϕ)|√γ+ O(1), (B3) whereγis the Euler’s constant, |φ(0,ϕ)|= sinϑ|cosϕ|, Φ(0,k,ϕ) =Λ0(k2τ/η2) sin2ϑcos2ϕ [k2+E2(k,0)]2 (B4) and E2(k,0) = 1 + 2/parenleftbigg1 τ−1/parenrightbigg∞/summationdisplay n=1Λn(k2τ/η2). (B5) 22Using the relation [17, 20] +∞/summationdisplay n=−∞Λn(z) = 1, (B6) the function E2(k,0) we finally takes the form E2(k,0) =1 τ+/parenleftbigg 1−1 τ/parenrightbigg Λ0(k2τ/η2). (B7) Substituting Eqs. (B3), (B4) and (B7) into Eq. (54) and integ ration over ϕwe finally come to expression (56). [1] A. H. Sørensen and E. Bonderup, Nucl. Instrum. Methods 215, 27 (1983). [2] H. Poth, Phys. Reports 196, 135 (1990). [3] I. N. Meshkov, Phys. Part. Nucl. 25, 631 (1994). [4] Proceedings of the 12th International Symposium on Heav y Ion Inertial Fusion, (Heidelberg, Germany, Sept. 1997), Nucl. Instrum. Methods A 415, (1998). [5] I. A. Akhiezer, Zh. ´Eksp. Teor. Fiz. 40, 954 (1961) [Sov. Phys. JETP 13, 667 (1961)]. [6] N. Honda, O.Aona, and T.Kihara, J. Phys. Soc. Jpn. 18, 256 (1963). [7] R. M. May and N. F. Cramer, Phys. Fluids 13, 1766 (1970). [8] G. G. Pavlov and D. G. Yakovlev, Zh. ´Eksp. Teor. Fiz. 70, 753 (1976) [Sov. Phys. JETP 43, 389 (1976)]. [9] J. G. Kirk and D. I. Galloway, Plasma Phys. 24, 339 (1982). [10] S. V. Bozhokin and ´E. A. Choban, Fiz. Plazmy 10, 779 (1984) [Sov. J. Plasma Phys. 10, 452 (1984)]. [11] H. B. Nersisyan, Phys. Rev. E 58, 3686 (1998). [12] H. B. Nersisyan and C. Deutsch, Phys. Lett. A 246, 325 (1998). [13] C. Seele, G. Zwicknagel, C. Toepffer, and P.-G. Reinhard , Phys. Rev. E 57, 3368 (1998). [14] M. Walter, C. Toepffer, and G. Zwicknagel, Nucl. Instrum . Meth. B, (1999) (to be published). [15] E. Lifshitz and L. P. Pitaevskij, Physical Kinetics (Pergamon Press, Oxford, 1981). [16] G. Zwicknagel, C. Toepffer, and P.-G. Reinhard, Phys. Re ports309, 117 (1999). [17] S. Ichimaru, Basic Principles of Plasma Physics (Benjamin, Reading, MA 1973), Sec. 7.4. 23[18] D. B. Fried and S. D. Conte, The Plasma Dispersion Function (Academic Press, New York, 1961). [19] Th. Peter and J. Meyer-ter-Vehn, Phys. Rev. A 43, 1998 (1991). [20] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980). 24FIG. 1: Normalized friction coefficient I1+I2sin2ϑ(see Eqs. (19)-(21)) in plasma with Z= 0.2 as a function of τ=T⊥/T/bardblfor four values of ϑ;ϑ= 0 (solid line), ϑ=π/6 (dotted line), ϑ=π/3 (dashed line), ϑ=π/2 (dot-dashed line). FIG. 2: Stopping power (in units of 10−3eV/cm) as a function of projectile velocity V(in units of/angbracketleftvth/angbracketright=vth) in a strongly temperature anisotropic plasma without magn etic field ( T= 0.1eV, n0= 108cm−3,τ= 10−2) for four values of angle ϑ,ϑ= 0 (dotted line), ϑ=π/6 (dashed line), ϑ=π/3 (long-dashed line), ϑ=π/2 (dot-dashed line). Solid line isotropic plasma with temperature T=T= 0.1eV. FIG. 4: The function P(ϑ,τ) (see Eqs. (33)-(36)) as a function of τ=T⊥/T/bardblfor four values of ϑ,ϑ= 0 (solid line), ϑ=π/6 (dotted line), ϑ=π/3 (dashed line), ϑ=π/2 (dot-dashed line). FIG. 5: Additional stopping power S1(in 10−5eV/cm) in plasma ( n0= 108cm−3,T= 0.1eV, τ= 10−2) with weak magnetic field (see Eq. (32)) as a function of proje ctile velocity V(in units of/angbracketleftvth/angbracketright=vth) forϑ= 0 (solid line), ϑ=π/6 (dotted line), ϑ=π/3 (dashed line), ϑ=π/2 (dot-dashed line). FIG. 6: As Fig. 5, but here τ= 102. FIG. 7: Stopping power Sinf(in 10−3eV/cm) in plasma ( n0= 106cm−3,T/bardbl= 10−4eV,τ= 0.1) with strong magnetic field as a function of projectile veloci tyV(in units of vth/bardbl) forϑ= 0 (solid line), ϑ=π/6 (dotted line), ϑ=π/3 (dashed line), ϑ=π/2 (dot-dashed line). FIG. 8: As Fig. 7, but here τ= 103. FIG. 9: The ratio Sinf(V,ϑ)/Sinf(V,0) as a function of projectile velocity V(in units of vth/bardbl) for T/bardbl= 10−4eV,τ= 103,ϑ=π/6 (solid line), ϑ=π/4 (dotted line), ϑ=π/3 (dashed line), ϑ=π/2 (dot-dashed line). FIG. 3: As Fig. 2, but here τ= 102. 25FIG. 10: The ratio of the anomalous stopping power to the stop ping power without magnetic field (San/S0) as a function of ωc/ωpforZ= 0.1,V/vth= 0.2,τ= 0.1,ϑ=π/6 (solid line), ϑ=π/3 (dotted line), ϑ=π/2 (dashed line). FIG. 11: As Fig. 10, but here τ= 10. 2610−1100101102 τ012345I1(τ)+I2(τ)sin2ϑϑ = 0 ϑ = π/6 ϑ = π/3 ϑ = π/20 1 2 3 4 5 6 V/<vth>00.81.62.4STOPPING POWER (in 10−3eV/cm)Isotropic plasma ϑ = 0 ϑ = π/6 ϑ = π/3 ϑ = π/20 1 2 3 4 5 6 V/<vth>00.40.81.2STOPPING POWER (in 10−3eV/cm)Isotropic Plasma ϑ = 0 ϑ = π/6 ϑ = π/3 ϑ = π/210−1100101102 τ−5515253545P(ϑ,τ)ϑ = 0 ϑ = π/6 ϑ = π/3 ϑ = π/20 1 2 3 4 5 6 V/<vth>−5−4−3−2−101S1 (in 10−5ev/cm) ϑ = 0 ϑ = π/6 ϑ = π/3 ϑ = π/20 1 2 3 4 5 6 V/<vth>−0.600.61.21.8S1 (in 10−5eV/cm)ϑ = 0 ϑ = π/6 ϑ = π/3 ϑ = π/20 1 2 3 4 5 V/vth||00.61.21.8Sinf (in 10−3 eV/cm)ϑ = 0 ϑ = π/6 ϑ = π/3 ϑ = π/20 1 2 3 4 5 V/vth||0246Sinf (in 10−3eV/cm)ϑ = 0 ϑ = π/6 ϑ = π/3 ϑ = π/210−1100101 V/vth||110Sinf(V,ϑ)/Sinf(V,0)ϑ = π/6 ϑ = π/4 ϑ = π/3 ϑ = π/20 2 4 6 8 10 ωc/ωp00.20.40.6San/S0 ϑ = π/6 ϑ = π/3 ϑ = π/20 2 4 6 8 10 ωc/ωp051015San/S0ϑ = π/6 ϑ = π/3 ϑ = π/2

arXiv:physics/9912019v1 [physics.plasm-ph] 9 Dec 1999Electric microfield distribution in two-component plasmas . Theory and Simulations J. Ortnera)∗, I. Valuevb), and W. Ebelinga) a)Institut f¨ ur Physik, Humboldt Universit¨ at zu Berlin, Invalidenstr. 110, D-10115 Berlin, Germany b)Department of Molecular and Chemical Physics, Moscow Insti tute of Physics and Technology, 141700 Dolgoprudny, Russia (to be published in Contr. Plasma Phys.) Abstract The distribution of the electric microfield at a charged part icle moving in a two-component plasma is calculated. The theoretical appr oximations are obtained via the parameter integration technique and using the screened pair approximation for the generalized radial distributio n function. It is shown that the two-component plasma microfield distributio n shows a larger probability of high microfield values than the correspondin g distribution of the commonly used OCP model. The theory is checked by quasicl assical molecular-dynamics simulations. For the simulations a cor rected Kelbg pseudopotential has been used. PACS: 52.25.Vy, 52.25.Gj, 52.65.-y, 05.30.-d Keywords: Two-component plasma; Electric microfield; Semi classical molecular dynamics ∗Corresponding author, Tel.: (+4930) 2093 7636, email: jens @physik.hu-berlin.de 1I. INTRODUCTION The purpose of this paper is the investigation of the microfie ld distribution in a two- component plasma at the position of a charged particle. The determination of the distribution of the electric micro field component created by one of the subsystems separately - electron or ion - is a well s tudied problem (for a review see [1]). Holtsmark [2] reduced the line shape problem to the determination of the prob- ability distribution of perturbing ionic electric microfie ld. In recent papers it was argued that the electric microfield low frequency part (due to the io n dynamics) also influences the fusion rates [3] and the rates for the three-body electro n-ion recombination [4] in dense plasmas. Holtsmark’s work on the electric microfield d istribution was restricted to ideal plasmas. The opposite limiting case of infinite coupli ng strength was considered by Mayer [5,6] within the ion sphere model. Within this model th e central ion undergoes harmonic oscillations around the center of the negatively c harged ion sphere. This results in a Gaussian approximation for electric microfields at the i on position. The nonideality of plasmas leads to quantitative corrections to Holtsmark’ s result as shown by Baranger and Mozer [7] and Iglesias [8] for the case of weakly coupled p lasmas and by Iglesias et al. [9] for the case of strongly coupled plasmas. In these papers it is shown that with increas- ing coupling strength Γ the long tailed Holtsmark distribut ion is changed into the fast decaying Gaussian approximation. Here the coupling parame ter Γ = e2/kTd is defined via the electron density ne(d= [3/4πne]1/3is the average distance of the electrons). In the cited papers the electric microfield created by one of t he subsystems has been studied by an almost total neglect of the influence of the othe r subsystem. A common assumption is that the distribution of the high-frequency c omponent (due to the electron dynamics) is the same as that of an electron gas with uniform n eutralizing background. This is the so called model of the one component plasma (OCP). For the ion subsystem, in a first approximation, the electrons are assumed to move fr ee through the plasma. Since the electron motion is much more rapid than the ion one, the electrons are treated 2as a smeared negative charged background. For simplicity th is background charge was assumed to be uniform in the density and not to be distorted by the ion motion. This again is the OCP model. A more realistic model should also take into account the vari ation of the background charge density. A background charge distribution which diff ers from a uniform distribution results in a screening of the ion motion, the screening stren gth is generally frequency dependent, e.g. it depends on the ion velocity. In a first appr oximation one might neglect the frequency dependence of the screening. Then one arrives at the model of an OCP on a polarizable background (POCP). In the theory of microfield s this slightly more involved model is used to describe the low frequency part [7,10]. Howe ver, both the OCP and the POCP fail to describe the correlations between the electron and the ion subsystem. To include the electron-ion correlations one has to conside r the model of a two- component plasma (TCP). This paper is adressed to the electr ic microfield studies in an equilibrium two-component plasma. To our knowledge the e lectric microfield in a TCP has been studied only by Yan and Ichimaru [11]. However, d ue to a couple of flaws contained in the paper of Yan and Ichimaru a further inve stigation is required. For simplicity we will restrict ourselves to the case of a two -component plasma which is anti-symmetric with respect to the charges ( e−=−e+) and therefore symmetrical with respect to the densities ( n+=ni=n−=ne). Further, the theoretical investigations are carried out for arbitrary electron ion mass ratios. To simpl ify the numeric investigations we simulated so far only a mass symmetrical (nonrelativisti c) electron-positron plasma withm=m+=me. We study this - so far unrealistic - case of mass - symmetrica l plasmas in order to save computer time in particle simulatio ns. The mass-symmetrical model is well suited to check the quality of various analytic al approximations. In addition, the results of the simulation are also applicable to the case of an electron-hole plasma in semiconductors. As for the case of the OCP the microfield distribution of a TCP i n the weak coupling 3regime is approximated by the Holtsmark distribution. Howe ver, coupled plasmas are important objects in nature, laboratory experiments, and i n technology [12–14]. Therefore we are interested in the modification of the microfi eld distribution caused by the coupling of plasma particles. Both theoretical inves tigations and semiclassical simulations are performed to study the microfield distribut ion in two-component plasmas. In this paper the free charges (electron and ions) are simula ted by a semiclassical dy- namics based on effective potentials. The idea of the semicla ssical method exploited in the numerical part of this paper is to incorporate quantum-mech anical effects (in particular the Heisenberg and the Pauli principle) by appropriate pote ntials. This method was pi- oneered by Kelbg, Dunn and Broyles, Deutsch and others [15–1 7]. Several investigations were devoted to the simulation of equilibrium two-componen t plasmas [18–22]. Being interested in semiclassical methods we mention explicitel y the semiclassical simulations of two-component plasmas performed by Norman and by Hansen [ 18,19]. Certainly, such a semiclassical approach has several limit s. For example, bound states cannot be described classically, therefore our methods are restricted to the subsystem of the free charges. However, this is not a very serious restric tion since most of the plasma properties are determined by the subsystem of the free charg es. The semiclassical approach may be very usefull to calculate a standard macroscopic property such as the microfield distribution since it has a we ll defined classical limit. The advantage of such an approach is the relative simplicity of t he algorithm. II. MICROFIELD DISTRIBUTION Consider a two-component plasma consisting of Niions of one species and Ne=Ni electrons with masses meandmiand charges ee=−ei=−e. The total number of particles is N=Ne+Ni. The plasma system with the total volume Vand temperature in energy units T= 1/βis described by the Hamilton operator 4ˆH=ˆK+ˆV=/summationdisplay a=e,iNa/summationdisplay i=1ˆp2 a,i 2ma+1 2/summationdisplay a,b=e,iNa/summationdisplay i=1Nb/summationdisplay j=1ˆvab(ra,i,rb,j). (1) The interaction potential between two particles is given by the Coulomb potential vab(ri,rj) =eaeb |ri−rj|. (2) The operator of electric field acting on a certain particle (h ereafter called the first particle) is defined by the sum of single particle Coulomb fiel d operators, E=N/summationdisplay j=2Ej(r1j),Ej(r1j) =−ej r3 1jr1j, rij=|ri−rj|. (3) Define now the electric microfield distribution W(ε) as the probability of measuring an electric field εequal to Eat the probe charge position r1, W(ε) =< δ(ε−E)> , (4) where <ˆA >= (1/Z) Sp/parenleftBigˆAexp[−βˆH]/parenrightBig denotes the quantumstatistical average of the operator ˆA, and Z= Sp exp[ −βˆH] is the partition sum of the plasma system. We assume that our system is isotropic. Then we may rewrite Eq .(4) as [6] P(ε) = 2π−1ε/integraldisplay∞ 0dl l T(l) sin(εl), (5) where P(ε) is related to W(ε) by 4 πW(ε)ε2dε=P(ε)dε, and T(k) =< eik·ε> (6) is the Fourier transform of the microfield distribution func tionW(ε). It is convenient to introduce the dimensionless quantity F=E E0, (7) where E0is defined through the total density n=N/VbyE0= 2π(4/15)2/3e n2/3. The probability distribution for the dimensionless field Fthen becomes with L=lE0, P(F) = 2π−1F/integraldisplay∞ 0dL L T (L) sin(FL). (8) 5Consider now some known limiting cases for the microfield dis tribution. In the weak coupling regime for Γ ≪1 the Holtsmark distribution is applicable for the microfiel d distribution and we have T(L,Γ≪1) =TH(L) = exp[ −L3/2]. (9) The other limiting case of strong coupling Γ ≫1 is known so far only for the one- component plasma model. For the OCP the ion sphere model hold s in the strong coupling regime. Within this model the charge will be attracted towar ds the center of its oppositely charged sphere of radius d= [3/4πne]1/3and with average density ne. The harmonic potential for the displacement of the center leads then to a G aussian approximation for the distribution of the normalized electric field F=E/E 0,OCPat the charge , P(F) = (2 /π)1/2(F2/τ3/2) exp( −F2/2τ), (10) where τ= (bΓ )−1xcothx , x = (T¯h2/4mee4)Γ3, b=4 5/parenleftBigg2π2 5/parenrightBigg1/3 . (11) The normalizing field strength for the OCP case should be expr essed in terms of the electron density neonly, E0,OCP= 2π(4/15)2/3e n2/3 e. In the case of a classical one- component plasma ¯ h→0 the parameter τplaying the role of an effective temperature in the Gaussian distribution Eq.(10) simplifies and reads τcl= 1/(bΓ), and Eq.(10) turns into the expression developed by Mayer [5]. However, there is no commonly accepted generalization of th e ion sphere model for the two-component plasma with charges of different signs. Mo reover we will show that in the case of TCP there is not any analogue for the Gaussian di stribution in strongly coupled OCP. First we mention that the Fourier transform of the Gaussian d istribution for electric microfield applicable in the strong coupling OCP regime equa ls T(L,Γ≫1) =TG(L) = exp/bracketleftBigg −L2τ 2/bracketrightBigg . (12) 6Notice that the Taylor expansion of the Gaussian function TG(L) starts with TG(L) = 1 −L2τ 2. (13) On the other hand it is possible to perform exact calculation s for the leading terms in the small Lexpansion of the Fourier transform T(L). From the definition of T(k) Eq. (6) it follows that T(k→0) = 1 −k2< ε2> 6+k4< ε4> 120±. . . . (14) In Refs. [9] it was argued that it is necessary to incorporate the knowledge of the second moment < ε2>into the calculation of microfield distributions in OCP. One might now try to generalize this idea to the case of a TCP. However, as can be easily seen the coefficient in the k2term< ε2>diverges in the case of a TCP: < ε2>=<N/summationdisplay j=2(Ej(r1j) )2+/summationdisplay j/negationslash=kEj(r1j)·Ek(r1k)> . (15) The first sum on the r.h.s of Eq.(15) can be written in terms of t he partial correlation function of particles aandb, < ε2>1=gab(r) =1 V nanb<Na/summationdisplay i=1Nb/summationdisplay j=1δ(r−rjb+ria)> , (16) and reads 4πnee2/integraldisplay∞ 0dr r2[gee(r) +gei(r) ], (17) which diverges at small distances, since for a fluid quantum s ystem both gei(0)/ne}ationslash= 0 and gee(0)/ne}ationslash= 0. In the classical OCP only gee(r) appears with gee(0) = 0, therefore the above integral and < ε2>are finite. In contrast to what we have found Yan and Ichimaru [11] predict a finite second moment. In Ref. [11] no derivatio n of their second moment expression valid “strictly in the classical limit” [11] is g iven. To isolate a possible error in the derivation of Yan and Ichimaru one may perform semicla ssical calculations of the second moment. Details of the semiclassical model are given in the next Section. We 7mention here only that in the semiclassical model the quantu m system is modeled by a system of classical particles interacting via an effective potential uab(r) =eaeb/r+ us,ab(r), where the short-range part of the effective potential us,ab(r) cuts the short-range divergency of the Coulomb potential. Therefore at short dis tances us,ab(r→0) =−eaeb/r. Within the semiclassical model the second moment reads < ε2>=4πne β(gei(0)−gee(0))−β/summationdisplay a,b/angbracketleftbigg ∇eaeb r∇us,ab(r)/angbracketrightbigg . The first term in the above equation has been reported in Ref. [ 11], the second term has been omitted. However, as may be easily seen this second term is divergent. It may be expressed by an integral similar to that of Eq. (17) which div erges at the lower bound. Thus we have established a qualitative difference between the classical OCP and the TCP system. For the first the second moment of the microfield di stribution is finite and corresponds to the variance of the Gaussian distribution. In contrast to the OCP case the second moment of the TCP system diverges. As a result the TCP microfield distribution does never converge t o a Gaussian distribution. We now generalize a coupling parameter technique which was u sed to calculate the microfield distribution of a classical OCP [8,9] to the case o f a quantum TCP. Consider the function T(l) =Z(l) Z, (18) Z(l)≡Speil·Ee−βˆH. (19) Introduce the “coupling strength” parameter λ, 0≤λ≤1, of the function, Z(λ) = Sp eiλl·Ee−βˆH. From the definition of T(l) in Eq. (18) and assuming the first particle to be an electron one obtains lnT(l) =/integraldisplay1 0d λ∂lnZ(λ) ∂λ =/summationdisplay anaea/integraldisplay1 0d λ/integraldisplay drφ(r)gea(r, λ), (20) 8where φ(r) =il·r r3, (21) gab(r, λ) =1 Z(λ)V nanbSpNa/summationdisplay i=1Nb/summationdisplay j=1δ(r−rjb+ria)eiλl·Ee−βˆH. (22) The functions gab(r, λ) may be considered as generalized partial distribution fun ctions. In the case of a TCP Eq.(20) reads lnT(l) =n 2e/integraldisplay1 0d λ/integraldisplay drφ(r) [gei(r, λ)−gee(r, λ)]. (23) The above expression is still exact. The use of the “exponent ial approximation” (EXP) [9] ansatz leads to the expression gea(r, λ)≃gea(r,0) exp [ Ea(r;λ)], a=e, i , (24) with the “renormalized potential” given as [9] Ea(r;λ) =iλl·E∗ a(r), E∗ a(r) =Ea(r1a) +/summationdisplay bnb/integraldisplay drbEb(r1b) [gab(rab)−1 ]. (25) After substitution of Eq.(25) into Eq.(24) and performing t he integration over λand the angles one gets T(l)≃exp/bracketleftBigg 4π/summationdisplay ana/integraldisplay∞ 0dr r2gea(r)Ea(r) E∗ a(r)[j0(lE∗ a(r))−1]/bracketrightBigg , (26) E∗ a(r) =Ea(r)/bracketleftBigg 1 + 4 π/summationdisplay bnb/integraldisplayr 0r′2d r′[gab(r′)−1 ]/bracketrightBigg , (27) Ea(r) =ea r2, (28) withj0being the Bessel function of order zero. We notice that the us e of the screened Coulomb potential Eq.(27) ensures the divergency of the sec ond moment of the TCP microfield distribution. In this point our theory differs ess entially from the results obtained by Yan and Ichimaru who used a potential of mean force instead of the screened Coulomb field [11]. Eqs.(26)-(28) constitute the so called exponent ial approximation (EXP) [9]. It 9is known that in contrast to the so called adjustable paramet er exponential approximation (APEX) the EXP expression poorly agrees with MD OCP data. In t he APEX [9] one substitutes Eq. (27) by an ad hoc ansatz for E∗ a(r). According to this ansatz the potential E∗ a(r) is approximated by a parametrized Debye potential where th e parameter is choosen to satisfy the second moment. In order to get a generalized AP EX expression for the TCP one should know the second moment of the TCP microfield dis tribution. However, in the above consideration we have shown that the second mome nt of the TCP microfield distribution diverges. Therefore there is not any straight forward generalization of APEX to the TCP case. In the weak coupling limit both approximatio ns, EXP and APEX, reduce to the Debye-H¨ uckel (DH) approximation. Consider therefore the DH limit in the case of TCP. In the weak coupling limit the pair correlation function is given by the screened pair appr oximation which in our case of a two-component plasma reads [12]: gab(r) =S(2) ab(r) exp/bracketleftBigg −βeaeb r/parenleftBig e−κr−1/parenrightBig/bracketrightBigg , (29) where κ= (4πβ/summationtext anae2 a)1/2is the inverse Debye screening length. Further S(2) ab(r) = const ./summationdisplay α′exp (−βEα)|Ψα|2(30) is the two-paricle Slater sum written in terms of the wave fun ctions Ψ αand energy levels Eαof the pair ab, respectively. The prime at the summation sign indicates th at the contribution of the bound states (which is not be considered here) has to be omitted. The Slater sum will be considered in the next section. To calculate the effective field E∗ a(r) in Eq.(27) it suffices to use the linear DH approx- imation gab(r)−1 = −βeaeb rexp [−κr], (31) since the nearest neighbour contribution to E∗ a(r) is already singled out in Eq.(27). In addition, the linear DH approximation leads to a perfect scr eening of the impurity charge, 10which is an important requirement for a consistent approxim ation. The substitution of Eq.(31) into Eq.(27) yields the Debye screened field E∗ a(r) =ea r2(1 +κr) exp(−κr). (32) We put now Eqs. (32) and (29) into Eq. (26) and obtain the DH app roximation for the microfield distribution in a two-component plasma. This app roximation may be expressed in terms of the dimensionless quantities introduced in Eqs. (7) and (8) and reads T(L) =Tee(L)Tei(L), lnTea(L) =15L3/2 4√ 2π/integraldisplay∞ 0dx x2B(x)/parenleftBiggsinB(x) B(x)−1/parenrightBigg exp/bracketleftBigg ZaΓc√ Le−√ 6ΓL/cx/bracketrightBigg ·exp/parenleftBigg β us,ea(d√ L cx)/parenrightBigg , B(x) =x2/parenleftBigg 1 +√ 6ΓL cx/parenrightBigg exp/bracketleftBigg −Γc√ L/bracketrightBigg , c=√ 2π21/3 (5π)1/3, Zi=−Ze= 1 ,(33) where the electron Wigner-Seitz radius dand the coupling constant Γ have been defined in the Introduction. Further in Eq. (33) we have introduced a n effective short range potential exp (−β us,ea(r)) =S(2) ea(r) exp/parenleftBigg −βeea r/parenrightBigg . Equation (8) with T(L) from Eq. (33) constitutes the Debye-H¨ uckel approximatio n for the microfield distribution applicable to the weakly coupled TC P. These equations generalize the corresponding DH approximation used to calculate the OC P microfield distribution [8]. We mention that the approximation Eqs. (33) can be direc tly obtained from Eq. (23) using the nonlinear Debye-H¨ uckel approximation for the ge neralized radial distribution function, gea(r;λ) = exp/bracketleftBigg β/bracketleftBigg 1 +iλl∇ eβ/bracketrightBiggeea re−κr/bracketrightBigg exp [−βus,ea(r)]. (34) In the next section we consider the two-particle Slater sum a nd introduce the semi- classical model employed in the numerical simulations. 11III. SLATER SUM, SEMICLASSICAL MODEL AND MD-SIMULATIONS As pointed out in the Introduction the idea of the semiclassi cal methods is to incor- porate quantum-mechanical effects (in particular the Heise nberg and the Pauli principle) by appropriate effective potentials. An easy way to arrive at effective potentials describing quan tum effects is the use of the so-called Slater sums which were studied in detail by sev eral authors [12,23]. The Slater sum caracterizes the distribution of the system in co ordinate space. Choosing the logarithm of the Slater sum U(N)(r1, . . .,rN) =−TlnS(r1, . . .,rN), (35) as a potential for the classical motion of the particles, we m ap our quantum system onto a classical one. The potentials U(N)(r1, . . .,rN) are often called quantum statistical effective potentials and they are used to calculate the corre ct thermodynamic functions of the original quantum system [12,23,18]. The Slater sum may be considered as an analogue of the classic al Boltzmann factor. The only modification in comparison with the classical theor y is the appearance of many- particle interactions. If the system is not to dense (i.e., i n the nondegenerate limit) one may neglect the contributions of higher order many-particl e interactions. In this case one writes approximately, U(N)(r1, . . .,rN)≈/summationdisplay i<juij(ri,rj), (36) where the effective two-particle potential uabis defined by the two-particle Slater sum Eq. (30). The Slater sum for the pair of charged particles can be approx imated in different ways. Following Kelbg [15] one considers the Coulombic interacti on as a perturbation; in the first order one gets the expression uab(r) =eaeb r/parenleftBig F(r/λab)/parenrightBig , (37) 12with F(x) = 1−exp/parenleftBig −x2/parenrightBig +√πx(1−erf (x)), (38) which we will call the Kelbg potential. Here λab= ¯h/√2mabTis De Broglie wave length of relative motion, m−1 ab=m−1 a+m−1 b,a=e, i,meandmibeing the electron and ion masses, respectively. Further in Eq.(37) we have neglected the exchange contributions. An effective potential similar to Eq. (37) was derived by Deut sch and was used in the simulations by Hansen and McDonald [19]. The Kelbg potential is a good approximation for the two-part icle Slater sum in the case of small parameters ξab=−(eaeb)/(Tλab) if the interparticle distance ris sufficiently large. At small interparticle distances it deviates from the exact value of −T·ln(Sab(r= 0)). In order to describe the right behavior at small distances it is better to use a corrected Kelbg potential defined by [24,25] uab(r) = (eaeb/r)·F(r/λab)−kBT˜Aab(ξab) exp/parenleftBig −(r/λab)2/parenrightBig . (39) In Eq. (39) the temperature-dependent magnitude ˜Aab(T) is adapted in such a way that the Slater sum Sab(r= 0) and its first derivative S′ ab(r= 0) have the exact value at zero distance known from previous works [12,26]. The explic it expressions read [25] ˜Aee=√π|ξee|+ ln/bracketleftBigg 2√π|ξee|/integraldisplaydy yexp (−y2) exp (π|ξee|/y)−1/bracketrightBigg (40) ˜Aei=−√πξei+ ln/bracketleftbigg√πξ3 ie/parenleftbigg ζ(3) +1 4ζ(5)ξ2 ie/parenrightbigg + 4√πξei/integraldisplaydy yexp (−y2) 1−exp (−πξei/y)/bracketrightBigg (41) For low temperatures 0 .1< T r<0.3 one shall use the corrected Kelbg-potential Eq.(39) to get an appropriate approximation for the Slater s um at arbitrary distances. In the region of higher temperatures Tr=T/T I=/parenleftBig 2T¯h2/miee4/parenrightBig >0.3 (42) the Kelbg potential ( Aab= 0) and the corrected Kelbg potential almost coincide. At st ill higher temperatures T/T I>1 the Kelbg potential does not differ from the corrected Kelbg 13potential only in the case of electron-ion interaction. For the interaction of the particles of the same type the correction ˜Aabincludes also the exchange effects , which make the potential unsymmetrical (that means ueidiffer from uee). The potential assymetry becomes apparent at high temperatures ( T >100000 K) only. In the present work we are interested in the regime of interme diate temperatures. Therefore the simulations are performed with the potential Eq.(39) which is presented in Fig. 1 and compared with other potentials approximating the two-particle Slater sum. To check the quality of the predictions from the approximati on given in Sec. II we have performed a series of molecular dynamic simulations fo r comparison. The leap-frog variant of Verlet’s algorithm was used to integrate numeric ally the equations of motions obtained from the effective potential Eq.(39). The simulati ons were performed using a 256-particle system of electrons and positrons with period ical boundary conditions. The temperature of the system was choosen as T= 30 000 K, the coupling has varied from weak coupling (Γ = 0 .2) up to intermediate coupling (Γ = 2). In the investigated ra nge of plasma parameters the size of the simulation box was signific antly greater than the Debye radius. Therefore the long-range Coulomb forces are screen ed inside each box and no special procedure like Ewald summation was implemented to c alculate them. Either MD runs with Langevin source or MC procedures were used to estab lish thermal equilibrium in the system, both methods have led to the same results. In Figs. 2-5 we present the results of the approximation Eqs. (8) and (33) as well as the Holtsmark (Eq. (9)) approximation. The short range pote ntial in Eq. (33) is given by the corrected Kelbg potential without the Coulomb term us,ab(r) = (eaeb/r)· {(F(r/λab)−1)}+Aabexp(−(r/λab)2), (43) withF(x) from Eq.(38). The results of the analytical approximation are compared wi th MD data. It can be seen from the figures that the Debye-H¨ uckel approximation i s in good agreement with the MD data for the case of weak coupling, however, with increasi ng coupling strength this 14agreement becomes poorer. This is not surprising, since the DH approximation is valid only in the weak coupling regime. To get a better agreement fo r the case of intermediate coupling one has to improve the calculation of the radial dis tribution function. From the figures we also see that the MD data show a large probab ility of high mi- crofield values. The long tails in the distribution function reflect the attraction between oppositely charged particles. As a result the probability t o find a particle of opposite charge at small distances from the probe charge and thus prod ucing large microfields is even higher than in the ideal Holtsmark case. This situation is in striking contrast to the OCP case where the repulsion of particles with the same charg e leads to a small prob- ability of high microfield values. As for the TCP the long tail s are still present in the case of an intermediate coupling for which the OCP microfield distribution approaches the Gaussian distribution Eq. (10) [9]. In the DH approximat ion the long tails are less pronounced for the case Γ = 2. Here the Debye-H¨ uckel length i s smaller than the average distance between the particles. Thus the particle interact ions become screened even at short distances. A result of this unphysical screening is th e supression of high microfields within the DH approximation and for large coupling paramete rs. At still higher densities (Γ≥3 atT= 30 000 K) the De-Broglie wavelength becomes comparable wit h the inter- particle distance and the semi-classical approach employe d in the numerical part of the paper fails to describe the quantum two-component plasma pr operly. IV. CONCLUSIONS The electric microfield distribution at a charged particle i n a two-component plasma has been studied. Generalizing the corresponding transfor mation for the case of a classical OCP we have expressed the Fourier transform of the electric m icrofield distribution in terms of generalized partial radial distribution function s. Using a simple Debye-H¨ uckel like generalized radial distribution function (including the unscreened short range part stemming from the effective potential) we have obtained theo retical predictions for the 15electric microfield distribution of the TCP. It has been show n that in contrast to the OCP the second moment of the TCP microfield distribution diverge s. Semiclassical molecular-dynamics simulations of the two- component plasma using ef- fective potentials have been performed . The effective poten tial was choosen to describe the nondegenrate limit of the quantum system appropriately . The microfield distribution for different coupling constants (from Γ = 0 .2 to Γ = 2 .0) has been obtained. With increasing coupling strength the most probable value of ele ctric microfields is shifted to lower fields. However, at all coupling strengths for which th e simulations have been per- formed the microfield distribution shows long tails indicat ing a large probability of high microfields. This behavior is in contrast to the correspondi ng behavior in one-component plasmas. It reflects the divergency of the second moment of th e TCP microfield distribu- tion. At weak coupling there is an overall agreement of the microfie ld distribution obtained by the analytical approximation with the MD data. Although o ur simple approximation fails to provide accurate numerical results for larger coup ling constants, the formalism allows to generalize the results to the case of intermediate and strong coupling. V. ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeisch aft (DFG, Germany) and the Deutscher Akademischer Austauschdienst (DAAD, Ger many). 16REFERENCES [1] J. W. Dufty, in Strongly Coupled Plasmas , ed. by F. J. Rogers and H. E. DeWitt (Plenum, New York, 1987). [2] J. Holtsmark, Ann. Physik, 58577 (1919). [3] M. Yu. Romanovsky and W. Ebeling, Physica A 252, 488-504 (1998) [4] M. Yu. Romanovsky, Zh. Eksp. Teor. Fiz. 114, 1230-1241 (1998) [5] H. Mayer, unpublished work, discussed in Refs. [6]. [6] A. A. Broyles, Phys. Rev. 115, 521 (1955). [7] M. Baranger and B. Mozer, Phys.Rev. 115521 (1959); 118626 (1960). [8] C. A. Iglesias, Phys. Rev. A 27, 2705 (1983). [9] C. A. Iglesias, J. L. Lebowitz, and D. MacGowan, Phys. Rev . A28, 1667 (1983). [10] G. Ecker, Z. Physik 148593 (1957); G. Ecker and K. G. M¨ uller, Z. Physik 153317 (1958). [11] X.-Z. Yan and S. Ichimaru, Phys. Rev. A 34, 2167 (1986). [12] Kraeft, W.D., Kremp, D., Ebeling, W. and R¨ opke, G., “Qu antum Statistics of Charged Particle Systems”. (Akademie-Verlag, Berlin; Ple num Press, New York; russ. transl: Mir, Moscow 1986). [13] Ichimaru, S. “Statistical Plasma Physics: I. Basic Pri nciples, II: Condensed Plasmas”. (Addison-Wesley, Reading, 1992, 1994). [14] Kraeft, W.D. and Schlanges, M. (editors), “Physics of S trongly Coupled Plasmas” (World Scientific. Singapore, 1996). [15] G. Kelbg, Ann. Physik 13354,14394 (1964). 17[16] T. Dunn and A. A. Broyles, Phys. Rev. 157, 156 (1967). [17] C. Deutsch, Phys. Lett. 60A, 317 (1977). [18] Zamalin, V.M., Norman, G.E. and Filinov, V.S., “The Mon te Carlo Method in Sta- tistical Mechanics” (in Russ.) (Nauka, Moscow, 1977). [19] Hansen, J.-P. and McDonald, I.R., Phys. Rev. A 23, 2041, (1981). [20] Pierleoni, C., Ceperley, D.M., Bernu, B. and Magro, W.R ., Phys. Rev. Lett., 73, 2145, (1994). [21] Klakow, D., Toepffer, C. and Reinhard, P.-G., Phys. Lett . A,192, 55 (1994); J. Chem. Phys., 101, 10766 (1994). [22] Penman, J.I., Clerouin, J. and Zerah, P.G., Phys. Rev E, 51, R5224, (1995). [23] Ebeling, W., Ann. Physik, 21, 315 (1968); 22(1969) 33,383,392; Physica 38, 378 (1968); 40, 290 (1968); 43, 293 (1969); 73, 573 (1974). [24] W. Ebeling, G. E. Norman, A. A. Valuev, and I. Valuev, Con tr. Plasma Phys. 39, 61 (1999). [25] J. Ortner, I. Valuev and W. Ebeling, Contr. Plasma Phys. 39, 311 (1999). [26] Rohde, G. Kelbg, W. Ebeling, Ann. Physik 22(1968). 18FIGURE CAPTIONS (Figure 1) Effective potentials Eq.(37)(Kelbg potential) and Eq.(39) (corrected Kelbg potential). The Kelbg potential is drawn for three temperat ures, the corrected Kelbg-potential is explicitely shown at T= 10 000 K for both interactions and atT= 100 000 K for the electron-electron interaction only; in th e other cases the corrected Kelbg potential coincides with the Kelbg pote ntial within the figure accuracy. For comparison we have included also the low-temp erature limit of the effective potential of free charges (the “classical” potent ial-dashed line); the repulsive part of the classical potential coincides with the bare Coul omb potential. (Figure 2) Comparison of microfield distribution P(F) curves at T= 30 000K and Γ = 0 .2 from molecular dynamics (MD) and the analytical approxima tion derived in this work (DH) Eqs. (8) and (33). Figure 3 Same as in Fig. 2 at Γ = 0 .8. Figure 4 Same as in Fig. 2 at Γ = 1 .2. Figure 5 Same as in Fig. 2 at Γ = 2 .0. 19FIGURES 0.0 0.2 0.4 0.6 0.8 r, e2/kT−10.0−5.00.05.010.0 uab(r)/kTCoulomb potential Kelbg−potential classical potential corrected Kelbg−pot.T=10.000 K T=30.000 K T=100.000 K T=100.000 K T=30.000 K T=10.000 K Figure 1. (Microfield distribution in two-component plasma s; Ortner, Valuev, Ebeling) 200.0 2.0 4.0 6.0 8.0 F0.000.100.200.300.400.50P(F)MD data DH approximation Holtsmark Figure 2. (Microfield distribution in two-component plasma s; Ortner, Valuev, Ebeling) 210.0 2.0 4.0 6.0 8.0 F0.000.100.200.300.400.50P(F)MD data DH approximation Holtsmark Figure 3. (Microfield distribution in two-component plasma s; Ortner, Valuev, Ebeling) 220.0 2.0 4.0 6.0 8.0 F0.000.100.200.300.400.50P(F)MD data DH approximation Holtsmark Figure 4. (Microfield distribution in two-component plasma s; Ortner, Valuev, Ebeling) 230.0 2.0 4.0 6.0 8.0 F0.000.200.400.600.80P(F)MD data DH approximation Holtsmark Figure 5. (Microfield distribution in two-component plasma s; Ortner, Valuev, Ebeling) 24

arXiv:physics/9912020v1 [physics.gen-ph] 9 Dec 1999Classical approaches to Higgs mechanism Assen Kyuldjiev∗ Institute of Nuclear Research and Nuclear Energy, Tzarigradsko chauss´ ee 72, Sofia 1784, Bulgaria June 8, 2011 Abstract The standard approach to Higgs mechanism is based on the ex- istence of unitary gauge but, unfortunately, it does not com e from a coordinate change in the configuration space of the initial m odel and actually defines a new dynamical system. So, it is a questiona ble ap- proach to the problem but it is shown here that the final result could still make sense as a Marsden-Weinstein reduced system. (Th is reduc- tion can be seen as completely analogous to the procedure of o btaining the “centrifugal” potential in the classical Kepler proble m.) It is shown that in the standard linearization approximatio n of the Coulomb gauged Higgs model geometrical constraint theo ry offers an explanation of the Higgs mechanism because solving of the Gauss law constraint leads to different physical submanifolds whi ch are not preserved by the action of the (broken) global U(1) group. ∗E-mail: KYULJIEV@INRNE.BAS.BG . Supported by the Bulgarian National Foundation for Science under grant Φ-610. 1Despite the phenomenal success of the Standard Model, Higgs mecha- nism remains yet experimentally unverified. The current pre sentation of the spontaneous symmetry breaking (SSB) is still not quite conv incing. It boils down to applying change of variables (which a closer inspect ion reveals not to be a coordinate change), in order to rearrange the quadrat ic terms in the Lagrangian in a form suggesting presence of certain particl es and absence of others. Elimination of dynamics along the action of the gl obal symmetry group to be broken is done by hand and without justification. I t would be interesting to reanalyse the problem from purely classical viewpoint without appealing to the quantum mystique. To be concrete we shall concentrate on the Lagrangian analys ed in [1] in its form highlighting “radial-angular” decomposition: L=−1 4FµνFµν−1 2∂µR ∂µR−1 2e2R2AµAµ+eR2Aµ∂µθ−1 2R2∂µθ ∂µθ−V(R2) and the corresponding Hamiltonian (when we assume the Coulo mb gauge condition ∇. A= 0) is: H=1 2/bracketleftBig π2 R+ (∇R)2+π2 A+ (∇×A)2−2 (∇A0)2+R2(eA−∇θ)2/bracketrightBig + +π2 θ 2R2+eA0πθ+V(R2) (1) where πR=∂0R,πA=E,Ek=F0kandπθ=R2(∂0θ−eA0) and ( −+ ++) metric signature is assumed. Most of the treatments of Higgs model make use of the so called unitary gauge defining eBµ=eAµ−∂µθand rewriting the Lagrangian as: L′=−1 4FµνFµν−1 2∂µR ∂µR−1 2e2R2BµBµ−V(R2). Like any Lagrangian, Lis a function on a tangent bundle TMand in this case the configuration space Mis the space of potentials Aµand the fields Randθ; while L′is function on the tangent bundle of the new field Bµand the field R. The mixing of variables on the configuration and tangent spa ce means that L′presents a new dynamical system (possibly quite sensible on e) which is not equivalent to the initial one and which cannot be obtained by a mere coordinate change in the configuration space. The stand ard explanation is that after a local gauge transformation the Lagrangian co uld be rewritten 2in the new form but, in general, this is not an allowed procedu re. A natural question arises whether there is a more rigorous explanatio n of this recipe. The present paper claims that the final result coincides with the Marsden- Weinstein (MW) reduction [2] of the initial dynamical syste m (and is actually analogous to the treatment of the classical Kepler problem l eading to the “centrifugal” potential). To remind, when we have a group Gwith a Lie algebra Gacting on a symplectic manifold Pin a (strongly) Hamiltonian manner and defining a Lie algebra homomorphism, we have a mome ntum mapping J:P→ G* given by /angb∇acketleftJ(p), a/angb∇acket∇ight=fa(p)∀a∈ G where fais the Hamiltonian function of the fundamental vector field d efined by the action of aand also satisfying [fa, fb] =f[a,b]∀a, b∈ G then the MW quotient manifold J−1(µ)/Gµhas a unique symplectic struc- ture (provided µis weakly regular value of Jand the action of the isotropy group of µ–GµonJ−1(µ) is free and proper). This is a powerful method for obtaining reduced dynamics on a symplectic space starting f rom symplectic dynamical system with a symmetry. (We shall skip here any tec hnicalities like which group actions admit momentum maps, Ad*-equivariance, clean intersections etc.) In our case the group to be broken U(1) acts as θ→θ+φ (2) and the space J−1(0) is the subspace defined by πθ≡ −R2B0= 0. The group action quotiening of this space amounts to eliminatio n of any residual θ-dependence. Reducing the Hamiltonian (1) (and assuming th at the Gauss law constraint ∆ A0=eπθis solved) we obtain the Hamiltonian corresponding toL′in the B0= 0 gauge, in conformity with the standard interpretation. It is noteworthy to analyse the more general case when we redu ce by a nonzero value µof the dual algebra G* . The MW quotiening would be equivalent to fixing πθ=const/negationslash= 0 and again factoring out θfrom the phase space and the Hamiltonian. As a result the initial potential V(x) =−ax+bx2 witha, b > 0 will be modified by a cx−2term (with c >0) and this will lead to higher values of the Higgs mass without possibility for it s vanishing. (This could also add a new free parameter in possible future experi mental testing of the Higgs mechanism.) 3This is actually not an explanation why θ-symmetry is spontaneously broken—this is just a more rigorous procedure for factoring out (θ,πθ) de- gree of freedom and thus eliminating the movement along θwhich would be the dynamics typical for a massless field. It is precisely thi s movement along the flat bottom of the potential surface which could lead to a m assless (Gold- stone) field. One could still ask what prevents movement in th is direction and hence causing SSB. Being aware that SSB could only exist i n systems with infinite degrees of freedom, one may also wonder where th is property is encoded in the above mentioned procedures. A rigorous approach to these problems could be found e.g. in [ 3] where a structure of Hilbert space sectors (HSS) is found in soluti ons of nonlin- ear classical relativistic field equations. Each sector is i nvariant under time evolution, has a linear structure and is isomorphic to a Hilb ert space; and may be labeled by conserved dynamical charges. Different HSS define “dis- joint physical worlds” which could be considered as a set of c onfigurations which are accessible in a given laboratory starting from a gi ven ground state configuration. Then any group which maps a HSS into another HS S is spon- taneously broken and only “stability” groups which map a HSS into itself would be proper symmetry groups. Despite the nontriviality of existence of stable linear str uctures in the set of solutions of certain nonlinear equations and the possibi lity to explain in principle the existence of SSB this approach does not seem ve ry practical. Another possible route is offered by the use of geometrical co nstraint theory. Higgs model is a beautiful example of a constrained system. L agrangian does not depend on θbut only on ∂µθ, thus allowing solutions with θ-rotations but (θ,πθ) degree of freedom remains coupled with the potential Aµ. The assumption that θ-rotations are frozen (and consequent writing them off) obviously seems ungrounded. In what follows we shall return to our model in the Coulomb gau ge. This case was successfully tackled [4] by linearisation of the eq uations leading to massive wave equation for θ. Here we shall be interested not so much in the (linearised) equations but in the symmetry breaking. We have primary constraint π0≡∂L ∂(∂0A0)= 0 and the condition of its preserving gives the Gauss law constraint equation ∆ A0=eπθ. To be precise, this equation is not the proper constraint – the submanifolds determined by i ts solutions will give the surfaces on which the dynamics will be constrained a fter factoring out the ( A0,π0) degree of freedom. Obviously, the solutions of the equatio n 4are A0=G∗eπθ+f where Gis a Green function for the Laplacian, ∗denotes convolution i.e. (g∗h)(x) =/integraltextg(x−y)h(y)dyandfis any function satisfying ∆ f(x, t) = 0. Solutions of this equation would define different physical submanifolds labeled by solutions of this equation. After differentiatio n we obtain ˙A0=eG∗˙πθ+˙f=−eG∗∂k(R2Bk) +˙f and taking into account that ∂k(R2Bk) = ∆( R2θ) in the linearisation ap- proximation [4], we have: ˙A0=eR2θ+˙f This shows that we will have different dynamics on different ph ysical sub- manifolds because the general form of the “massive wave equa tion” for θ would be ✷θ=e2R2θ+e˙f(as long as linearisation approximation could be trustworthy). More interestingly, transformations (2) do es not act by lifting from the “configuration space”: A0→A0˙A0→˙A0+eR2φ and does not preserve the chosen submanifold. Transformati on actions of this kind are not very typical in physics (the standard N¨ oth er theorem, for example, assumes only lifted transformations ). Thus we hav e a geometrical analog of HSS and its origin could be traced to the requiremen ts to the phys- ical constrained submanifolds in infinite dimensions [5].( In this reference one could also see how this phenomenon appears in an exactly solv able model.) Acknowledgements The author is indebted to Prof. F. Strocchi for an illuminati ng and inspiring discussion. References [1] P. Higgs, Phys. Rev. 145 (1966) 1156 [2] J. Marsden and A. Weinstein, Rep. Math. Phys. 5 (1974) 121 5[3] C. Parenti, F. Strocchi and G. Velo, Phys. Lett. B 59 (1975 ) 157; C. Parenti, F. Strocchi and G. Velo, Comm. Math. Phys. 53 (1977) 65; F. Strocchi, SISSA preprint 34/87/FM, Trieste, 1987 [4] J. Bernstein, Rev. Mod. Phys. 46 (1974) 7 [5] A. Kyuldjiev, Phys. Lett. B 293 (1992) 375 6

physics/9912021 9 Dec 1999Mathematical Model of Attraction and Repulsion Forc es Alexei Krouglov Matrox Graphics Inc. 3500 Steeles Ave. East, Suite 1300, Markham, Ontario L3 R 2Z1, Canada Email: Alexei.Krouglov@matrox.com This article represents the author’s personal view and not the view of Matrox Graphics Inc.2ABSTRACT Here I introduce the model in an attempt to describe t he underlying reasons of attraction and repulsion forces between two ph ysical bodies. Both electrical and gravitational forces are considered. Result s are based on the technique developed in the Dual Time-Space Model of W ave Propagation. Keywords : Wave Equation, Field Theory31. Introduction Developed recently [1] the Dual Time-Space Model of W ave Propagation (DTSMWP) proved to be a useful tool in investigating the wave nature of matter. In present paper the model is applied to the phenomen a of attraction and repulsion forces between physical bodies, which constitutes t he subject of the field theory [2]. 2. Model Assumptions According to [1], the DTSMWP has the following assumptio ns. In the time domain, (1) The second derivative of energy’s value with respect to time is inversely proportional to energy’s disturbance. (2) The first derivative of energy’s level with respect to time is directly proportional to energy’s disturbance. In the space domain, (3) The second derivative of energy’s value with respect to direction is inversely proportional to energy’s disturbance. (4) The first derivative of energy’s level with respect to direction is directly proportional to energy’s disturbance.43. Impact of Energy’s Discrepancy in Space I assume we have the jump of energy’s value ()txU, at point 0x, () ( ) . 0, 0 ,, 00 00 0 >≥ >>≥ ≤  =∆+= ττ tandxxfortandxxfor UtxUEUtxU (1) I also assume that energy’s level was initially stable, () . 0 ,0 xandtforUtx ∀< =Φ (2) Then according to [1], we have the propagation of ene rgy’s disturbance in space. Therefore we can write () ( ) ( ) ∆+=Φ= 10 0 10 1 ,, xEUtxUtxU (3) where 01xx>, τ≥t, and ()10xE∆ is the energy’s disturbance propagated from point 0x to point 1x. From [1] we can see that energy’s disturbance caused by th e energy’s level decreases in the space domain and retains the same si gn. Thus the following takes place, () ( ) . 0, 0 00 00 10 010 0 <∆>∆ <∆<∆>∆>∆ EwhenEwhen xEExEE (4)54. Compact Energy Body at Rest Consider we have two jumps of the energy’s values as fol lows, () ( ) ( ) . 0, 0, 0 ,,, 22 11 01 00 >≥ >>≥ ≤≤>≥ <  =∆+== τττ tandxxfortandxxxfortandxxfor UtxUEUtxUUtxU (5) and the energy’s level was initially at rest, () . 0 ,0 xandtforUtx ∀< =Φ (6) We assume that compact energy body lies at rest within 2 1xxx≤≤. We can conclude from [1] that there are forces ()txF,1 and ()txF,2 applied respectively to the points 1x and 2x of compact body with the following magnitudes, ()()1 2 1 , , E txFtxF ∆⋅= = µ, (7) where τ≥t, and 0>µ is a constant. To balance these forces we have to complement them with the opposite forces ()1xF′ and ()2xF′, therefore for 0>t, ()() ( )( )  =′+=′+ . 0 ,, 0 , 2 21 1 xFtxFxFtxF (8) Note that forces ()txF,1 and ()txF,2 act in the direction of disturbance propagation, and forces ()1xF′ and ()2xF′ act in the opposite direction.65. Impact of Energy’s Discrepancy on Compact Energy Body Let me consider three points of interest where we have the jumps of energy’s values, () ( ) ( ) ( ) . 0, 0, 0, 0 ,,,, 22 11 00 01 000 0 >≥ >>≥ ≤≤>≥ <<>≥ ≤    =∆+==∆+= ττττ tandxxfortandxxxfortandxxxfortandxxfor UtxUEUtxUUtxUEUtxU (9) and the energy’s level was initially at rest, () . 0 ,0 xandtforUtx ∀< =Φ (10) We assume that the compact energy body lies at rest until the energy’s disturbance from point 0x reaches the point 1x of compact body at time τ>1t. Let me describe the forces applied to the point 1x of compact body at the time ttt∆+=1. At first I will consider the case when magnitudes of ener gy’s disturbances are close to each other, i.e. 1 0EE∆≈∆ , that causes inequality ()1 10 ExE ∆<∆ . There are four possible situations. (a) 00>∆E, 01>∆E () () ( ) ()()1 11 1 10 1 11 , , xFtxFE xEE ttxF ′= =∆⋅< ∆−∆⋅=∆+ µ µ . (11) Therefore the value of force ()1xF′, that has a direction inside the compact body, exceeds the value of an opposite force ()ttxF ∆+11,, and two bodies are repulsed.7(b) 00<∆E, 01>∆E () () ( ) ()()1 11 1 10 1 11 , , xFtxFE xEE ttxF ′= =∆⋅> ∆−∆⋅=∆+ µ µ . (12) Therefore the bodies are attracted. (c) 00>∆E, 01<∆E () () ()()1 11 1 10 1 11 , , xFtxFE xEE ttxF ′= =∆⋅>∆−∆⋅=∆+ µ µ . (13) Therefore the bodies are attracted. (d) 00<∆E, 01<∆E () () ()()1 11 1 10 1 11 , , xFtxFE xEE ttxF ′= =∆⋅<∆−∆⋅=∆+ µ µ . (14) Therefore the bodies are repulsed. Thus we have described so far the forces of electrical attr action and repulsion. Now I will consider the case when 01 0 >∆>>∆ EE . From here we have an inequality () 0 21 10 >∆⋅>∆ E xE , that as we can see soon, creates the force of gravitational attraction, () () ()()1 11 1 10 1 11 , , xFtxFE xEE ttxF ′= =∆⋅>∆−∆⋅=∆+ µ µ . (15) Hence the bodies are attracted.8References 1. A. Krouglov, “Dual Time-Space Model of Wave Propag ation,” Working Paper physics/9909024, Los Alamos National Laboratory, S eptember 1999 (available at http://xxx.lanl.gov ). 2. L.D. Landau and E.M.Lifshitz, “The Classical Theory o f Fields,” Pergamon Press, Oxford, UK, 1971.

arXiv:physics/9912022v1 [physics.gen-ph] 10 Dec 1999About Perpetuum Mobile without Emotions. A.V.Nikulov, Institute of Microelectronics Technology and High Purity M aterials, Russian Academy of Sciences, 142432 Chernogolovka, Moscow Distric t, RUSSIA One of the oldest science problems - possibility of the perpe tuum mobile is discussed. The interest to this problem was provoke a resu lt, published re- cently, which contradicts to the second law of thermodynami cs. According to this result, the thermal fluctuations can induce a voltage wi th direct component in a inhomogeneous superconducting ring at an unaltered tem perature corre- sponded to the resistive transition of the ring segment with the lowest critical temperature. This result arises from obvious statements: 1 ) the switching of a ring segment lbinto and out of the normal state, while the rest of the ring (segment la) remains superconducting, can induced a voltage with dc com po- nent (It is shown that, in spite of the wide spread opinion, th is statement is correct because the superconductivity is a macroscopic qua ntum phenomena); 2) the thermal fluctuations switch the mesoscopic ring segme ntlbwith lowest critical temperature Tsbinto and out of the normal state at T≃Tsb, while the rest of the ring remains superconducting if Tsa> T≃Tsb. In order to resolve the contradiction between these obvious statements and the second law of ther- modynamics a possibility of the second order perpetuum mobi le is considered theoretically. It is shown that from two type of the perpetuu m mobile, only type ”b” and only in quantum systems is possible. According t o the presented interpretation, the total entropy, as the measure of the cha os, may be system- atically reduced in some quantum system because a ”switchin g” between the classical and quantum mechanics is possible. Instruction f or the making of the perpetuum mobile is enclosed. 1 Introduction Physics is not lyricism. Any sentiment is inappropriate her e. Nevertheless some physical problems provoke emotion. The first of these proble ms is the per- petuum mobile. Any statement on a possibility of perpetuum m obile provokes first of all the sense of distrust. According to the dominant o pinion this prob- lem is once and for all decided. Almost all scientists, durin g more than two centuries, are fully confident in the impossibility of any pe rpetuum mobile. I, as well as other grave scientists, was sure that only madman m ay be in earnest about a possibility of any perpetuum mobile. But a result, wh ich I have ob- tained recently, has compelled me to change my point of view. According to this result the chaotic energy of thermal fluctuation can be t ransformed to the electric energy of a direct current at an unaltered temperat ure by means of a 1mesoscopic inhomogeneous superconducting ring. This result arises from obvious statements: 1) the switchin g of a ring seg- ment lbinto and out of the normal state, while the rest of the ring (se gment la) remains superconducting, can induced a voltage with dc com ponent; 2) the thermal fluctuations switch the mesoscopic ring segment lbwith lowest critical temperature Tsbinto and out of the normal state at T≃Tsb, while the rest of the ring remains superconducting if Tsa> T≃Tsb. Both these statements are agreed with our modern knowledge. But it follows directly fr om this statements that the thermal fluctuation can induce a voltage with dc comp onent at an un- altered temperature corresponded to the resistive transit ion of the ring segment with the lowest critical temperature. It is obvious that the dc voltage may be used for an useful work. This means that the useful work can be obtained from the chaotic energy of thermal fluctuation (i.e. from the heat energy) at unal- tered temperature (i.e. in the equilibrium state). This pos sibility contradicts directly to the second law of thermodynamic. And it is well kn own that a vi- olation of the second law of thermodynamics means a possibil ity of the second order perpetuum mobile. In order to resolve this contradiction between the obvious s tatements and the second law of thermodynamics I have investigated the rea son of the firm belief in the impossibility of perpetuum mobile. As a result I conclude that this belief does not have a theoretical substantiation. Only arg ument against the perpetuum mobile is numerous unsuccessful attempts to inve nt it. But it is not strict argument. What could not be made yesterday can be made today. For example, the mesoscopic superconducting ring can not be mad e twenty years ago but it can be made at present. Now I am sure that the statements 1) and 2) are correct. The second order perpetuum mobile is possible beca use the chaos may be systematically reduced in some quantum system. Therefor e I have published in [1, 2] the result which contradicts to the second law of the rmodynamics [3]. But it is not enough to publish such result. No one will straig ht off believe that such result can be correct. Therefore I think that I ough t expound in detail my arguments. I make this in the present article. In the begin ning I remind briefly the history of the considered problem. After that a br ief theoretical consideration of the perpetuum mobile problem is presented . In the section 4, the quantum force is introduced in order to explain the dc vol tage appearance in the superconducting ring segment. In the section 5, I expl ain when and why the total entropy may be systematically reduced. The last se ction is directions for who want to make the perpetuum mobile. I must write that the theoretical result published in [1] was provoked by an experimental result. But this experimental result is not pu blished. Therefore the result [1] ought be considered as the theoretical predic tion but not as an explanation. 22 A few history The perpetuum mobile is one of the oldest problems of science . This problem is more old than almost all foundations of modern physics. Ma ny persons at- tempted to invent a perpetuum mobile during many centuries. Such attempts were known beginning with 13 century. The principle of the im possibility of the perpetuum mobile was postulated first by Stevin (1548-1620 y ears). The Paris Academy of Sciences has decided in 1775 year to do not conside r any project of a perpetuum mobile. This verdict did not have any scientific b asis. The first and second laws of thermodynamics were formulated only in th e next century. Nevertheless, beginning with that time, almost all (with th e exception of few [4]) scientists are sure that a perpetuum mobile is impossib le. Sometimes one says that the impossibility of a perpetuum mob ile is based on the first and second laws of thermodynamics. But it is not qu ite right. It is more right to say that the second law of thermodynamics is b ased on the statement on the impossibility of a perpetuum mobile. The fir st formula of this law - Carnot’s principle - was proposed in 1824 year. Carnot w rote that the useful work can not be obtained from the heat energy at unalte red temperature (in the equilibrium state) because in the opposite case the p erpetuum mobile is possible. Following the Carnot’s idea Rudolf Clausius in 18 50 year and William Thomson (Lord Kelvin) in 1851 year have proposed the formula s which are more known now. According to the Clausius’s formula the heat energy can not be transferred from a cold body to a hot body without an exp ense of an additional energy. According to the Thomson’s formula it is impossible to obtain a power-driven energy (useful work) by means of a cooling of a body with lowest temperature. The second law of thermodynamics is formulate d also as the law of entropy increase. The entropy S was introduced first in 186 5 year by Rudolf Clausius as a value which changes on ∆ S=Q/Tin a reversible process when a thermodynamic body obtains the heat energy Q(Q >0) or gives the heat energy (Q <0). T is the Kelvin’s temperature of the body. Ostwald was for mulated in 1877 year the second law of thermodynamics as the impossibil ity of the second order perpetuum mobile. This formulas are equivalent and ar e used for the present. But the modern interpretation of the second law of t hermodynamics, as well as the entropy, differ in essence from the old one domin ated in the last century. According to the both interpretation the entropy value, S, can not decrease in a closed thermodynamic system. It does not change in the re versible pro- cess and increases in irreversible process. Any thermodyna mic system tends to the equilibrium state corresponding to a greatest Svalue. But there is the difference between old and modern definition of the entropy wh ich causes some contradiction between old and modern interpretation of the second law of ther- modynamics. According to the Clausius’s definition, the temperatures of parts of any ther- modynamic system can not differ in the equilibrium state. If t he temperatures 3differ, T1> T2, the entropy increases ∆ S=Q/T2−Q/T1when the heat energy Qis transferred from a hot (with T=T1) to a cold (with T=T2) part of the system. Consequently, the state with T1> T2is not equilibrium. According to the old interpretation the heat energy flows only from a hot to a cold part and any dynamic process (any transfer of the heat energy) does no t take place in the equilibrium state. Following to L.Boltzmann, J.W.Gibbs and others we define now the entropy by the relation S=kBlnP. Here kBis the Boltzmann constant; Pis the statistical weight proportional to a number of microscopic states and to the probability of macroscopic state. The maximum entropy corr esponds to the maximum probability: P= exp( S/kB). It is easy to show (see [5]) that the number of microscopic states P1atT1=T2is higher than P2atT1> T2 (at the same total internal energy). We could make the conclu sion from this statement that the modern definition of the entropy is agreed with the Clausius’s definition. But on other hand, P1+P2> P1. The thermodynamic system can not be at the same time in the different states, with T1=T2andT1> T2. But it can goes between these states. This process is well known a s the thermal fluctuation. The heat energy is transferred from a part to oth er part of any thermodynamic system at T >0. The Qtransfer from a cold T2to hot T1 part of a system is hardly probable if −∆S=−(Q/T1−Q/T2)≫kB. But the probability ∝exp(Q/k BT1−Q/k BT2) of this transfer in opposite direction does not differ strongly if Q|T2−T1|/kBT1T2<1 (i.e. the Clausius’s formula of the second law of thermodynamics is correct to an accuracy of ∆S=kB). The fluctuations contradict to the old interpretation. Therefo r they were interpreted as the violation of the second law of thermodynamics in the be ginning of our century [6]. Evidence of the fluctuation existence was one of the reasons why the modern interpretation has won the old interpretation. A ccording to the modern interpretation, the second law of thermodynamics ha s probabilistic but not reliable nature. We know now that dynamic processes take place in the equilibrium state at T >0. But these processes are chaotic. According to the modern interpretation, the second law of thermodynamics is the law of chaos increase. And the entropy is interpreted now as a measure of t he chaos. The second law of thermodynamics was formulated in order to d escribe the transformation of a heat energy Qto an useful work Ain the heat engine. Scientists understood that the mechanical (power-driven) energy is dissipated completely in the heat at any work in a consequence of the fric tion,Emech→ A→Q. Proceeding from the impossibility of the perpetuum mobile Carnot has shown in 1824 year that the heat energy can not be transfor med completely to the mechanical energy. A heater with Theatand a cooler with Tcool< Theat should be in any heat engine, according to Carnot. A work body obtains the heat energy Q1from the heater at T=Tmax≤Theatand gives the heat energy to the cooler at T=Tmin≥Tcool. According to the Carnot’s law, the efficiency Ef=A/Qof any heat engine can not exceed Efmax= 1−Tmin/Tmax. This law is considered now as the consequence of the first law of the rmodynamics 4A=Q1−Q2and of the second law of thermodynamics ∆ S=Q2/Tmin− Q1/Tmax≥0. The Emaxis realized in a reversible regime when ∆ S= 0. But these laws of thermodynamics were formulated later than the Carnot’s law. According to the Carnot’s law, Efmax= 0 in the equilibrium state and the total heat energy is systematically increased at the work be cause the mechanical energy dissipated in the heat energy can not be restored comp letely. There- fore we must use a fuel in order to obtained the useful work. Th e Carnot’s law remains without change to our time although it’s substan tiation and in- terpretation were changed qualitatively. According to the old interpretation the work can not be obtained at Tmax=Tminbecause any dynamic pro- cess is absent in the equilibrium state. According to the mod ern knowledge the heat is the chaotic motion. The chaotic dynamic processe s take place in any thermodynamic system at T > 0. The Carnot’s law is connected now with the law of chaos increase. The total entropy, as the meas ure of the chaos, may be systematically reduced if the heat engine is po ssible in which Ef=A/Q 1> Ef max= 1−Tmin/Tmax. The reduction of the entropy at the transformation of the heat energy to the mechanical energy ( or other form of ordered energy Eord),EfQ 1→A=/integraltext dXF ord→Eord, is not completely com- pensated by the transfer of the heat energy Q2from the heater to the cooler if Ef=A/Q 1> Ef max. (Eordis, for example, the kinetic energy of a flywheel or the magnetic energy LI2/2 in a solenoid.) The chaos increase, taken place at the dissipation of the ordered energy Eord→A=/integraltext dXF dis→Qdis, may be completely compensated if Ef > Ef max. (The dissipative force Fdisis, for example, the friction force retarding the flywheel or the for ce decreasing the cur- rent value Iin the solenoid when it’s resistance is not equal zero.) Cons equently, the heat engine with Ef > Ef maxis the second order perpetuum mobile: the useful work could be obtained anyhow long time without any fu el. According to the opinion dominated now, it is impossible because the tota l chaos can not be reduced. This belief, as well as the second law of thermodyna mics in the old in- terpretation, is founded on the postulate of the perpetuum m obile impossibility. This postulate has long and rich history and, may be, therefo re does not have a theoretical substantiation. This problem ought be consid ered theoretically at last. 3 Theory of Perpetuum Mobile A possibility of a perpetuum mobile means that the useful wor k can be obtained anyhow long time T. At the useful work, as well as at any other work, an energy is transferred from a part to other part of the system. The work Ais the product of a force Finto a distance dX,A=/integraltext dXF. Because dX=vdt, A=/integraltextT 0dtFv =T < Fv > . Here < Fv > =/integraltextT 0Fvdt/T is the average by the time T of the product of the force Finto the velocity v. Consequently, a perpetuum mobile is possible if a process exists in which the average < Fv > by 5anyhow long time T is not equal zero. If Fis the total force of a closed system and< Fv >∝negationslash = 0 then the first order perpetuum mobile is possible. < Fv >∝negationslash = 0 contradicts to the law of energy conservation (the first law o f thermodynamics). I can not doubt the this law. Therefore I, as well all other sci entist, am fully confident in the impossibility of the first order perpetuum mo bile. Let turn to the second order perpetuum mobile. Both the conventional heat engine and the second order perpe tuum mobile do not create a new energy. They put in order the chaotic heat e nergy. Because the ordered energy is dissipated at any real work Eord→T < F disv >→Qdis the heat energy should be transformed in the ordered energy Qdis→T < Fordv >→Eordin order the work can take place any long time. < Fordv >∝negationslash= 0 in two cases: a)if < Fordv >∝negationslash=< Ford>< v > , or b) if both < F ord>∝negationslash= 0 and < v >∝negationslash = 0. Thus, two type of both heat engine and second order perpetuum mobile may be: type ”a”, when < F ordv >∝negationslash=< Ford>< v > and type ”b”, when < F ordv >=< F ord>< v > but both < F ord>∝negationslash= 0 and < v >∝negationslash = 0. The case a) takes place if the force Fand the velocity vare corre- lated. This correlation takes place in a conventional heat e ngine. For example, the pressure in a steam-engine has different value when a pist on is moved in opposite directions. Therefore < F ordv >∝negationslash= 0 although < v > = 0 because < F ordv >∝negationslash=< F ord>< v > . But in order to achieve this correlation an con- trolled heat flow is used in any conventional heat engine. Suc h flow is possible only in the inequilibrium state. Therefore the total entrop y increases both at the ordered process and at the dissipation process. This pro cess can not be any- how long (infinite) time because the state of thermodynamic s ystem is changed: the total entropy increases. The heat energy Qdiscan not be transformed com- pletely in the ordered energy. Therefore any conventional h eat engine is not the perpetuum mobile. The total state of the thermodynamic syst em should not change during the work of the second order perpetuum mobile. Therefore it should work in the equilibrium state because only in this sta te the total entropy value does not increase in time. F∝negationslash= 0 in the equilibrium state only because of the fluctuation. T he fluc- tuation is chaotic. In a chaotic process < Fv > =m < vdv/dt > =m < dv/dt >< v > =< F >< v > . There is a mathematical problem to prove that < vdv/dt > −< v >< dv/dt > = 0 if the function v(t) is chaotic. This proof is evidence of the impossibility of the type ”a” perpetuum mobi le. Our last hope to invent the perpetuum mobile is the case ”b”. I t is obvious that< v > = 0 in any classical (no quantum) system where all states are p er- mitted. (There is used the reference system in which the tota l momentum of the considered thermodynamic system is equal zero). The pro bability of a state Pis proportional to exp −(E/k BT). The energy Eof a state is function of v2in a consequence of the space symmetry. Therefore the probabil ityP(v) =P(−v) and< v > =/summationtextP(v)v+P(−v)(−v) = 0 if all states are permitted. This ar- gument may be considered as a theoretical substantiation of the verdict made by the Paris Academy of Sciences. The quantum mechanics was n ot known in 61775 year. But it can not be considered as the evidence of the impossibil ity of the type ”b” perpetuum mobile in our time because no all states are per mitted according to the quantum mechanics. Therefore < v >∝negationslash = 0 in some quantum systems. One of such systems is the mesoscopic superconducting ring cons idered in [1, 2]. As a consequence of the relation (see [7]) vs=1 2m(¯hdφ dr−2e cA) =2e mc(Φ0 2πdφ dr−A) (1) the velocity of the superconducting electrons vsalong the circumference of a completely superconducting ring must have fixed values /integraldisplay ldlvs=e mc(Φ0n−Φ) (2) dependent on the magnetic flux because n=/integraltext ldl(1/2π)dφ/dr must be an inte- ger number since the wave function Ψ = |Ψ|exp(iφ) must be a simple function. Here Φ 0=π¯hc/eis the flux quantum; A is the vector potential; m is the elec- tron mass and e is the electron charge; l= 2πris the ring circumference; ris the ring radius; Φ =/integraltext ldlAis the magnetic flux contained within the ring. At Φ/Φ0∝negationslash=nand Φ /Φ0∝negationslash=n+ 0.5 the permitted states with the opposite directed velocity have different values of the kinetic energ yEkin=mv2 s/2. For example at Φ /Φ0= 1/4 the lowest permitted velocities in a homogeneous superconducting ring are equal vs=−¯h/mR 4 atn= 0 and vs= 3¯h/mR 4 atn= 1. The kinetic energy of these states differ in 9 times. There fore the thermodynamic average of the velocity < vs>is not equal zero. It is important that the motion of the superconducting condensate is circul ar. It is obvious the type ”b” perpetuum mobile is possible only at a circular moti on. Only in this case the position of the work body (superconducting condens ate in the case of the considered ring) does not change without limit during an yhow long time at < v >∝negationslash = 0. Thus, the result published in [1, 2] may be correct if an order ed force Ford exists the average value of which < Ford>is not equal zero in the equilibrium state. Such force exists. It acts at the closing of the superc onducting state in the ring and is connected with the quantization of the genera lized momentum of superconducting electrons along the ring circumference . Therefore I will call it as quantum force. 4 A quantum force According to the statement 1) (see Introduction) the switch ing of a ring segment lbinto and out of the normal state, while the rest of the ring (se gment la) remains superconducting, can induced a voltage with dc comp onent < Vb>∝negationslash= 0. la+lb=l= 2πr. This statement means that the dc voltage exists both on 7the switched segment < E b>=< Vb> /l band on the superconducting segment < E a>=< V a> /l a. Because < V a>+< V b>=/integraltext ldl < E > =/integraltext ldl < − ▽V−(1/c)dA/dt > =−(1/c)< dΦ/dt > = 0,< V a>=−< V b>. Here < E > = (/integraltextT 0dtE)/Tis the average value over a long time T. The value of the magnetic flux contained within the ring Φ =/integraltext ldlAmay change iteratively at the iterative switching of the segment lb, but< dΦ/dt >= 0 when the magnetic fluxBSinduced by an external magnet is not changed. Bis the magnetic induction induced by an external magnet; S=πr2is the ring area. < E a>∝negationslash= 0 contradicts to the wide spread opinion that the direct volta ge can not exist in any superconducting region. This opinion is correct if only force of the electric field Fe=eEacts on super- conducting electrons. Then, according to the classical mec hanics, the velocity v should increases without limit ( mv=/integraltextT 0dtmdv/dt =/integraltextT 0dteE=e < E > T ) if < E >∝negationslash = 0. But if we proposed that only Feacts on superconducting elec- trons we should conclude that any quantization is not possib le. It should be the case of the state with infinite conductivity where the a verage value of the generalized electron momentum along the ring circumf erence < p > l= l−1/integraltext ldlp=l−1/integraltext ldl(mv+ (e/c)A) =m < v > l+(e/c)Φ/lcan not change because/integraltext ldlmdv/dt =/integraltext ldleE=/integraltext ldle(− ▽V−(1/c)dA/dt ) =−(e/c)dΦ/dt. Here< v > l=l−1/integraltext ldlvis the average velocity along the ring circumference. But it is well known [7] that superconducting state differs fr om the state with infinite conductivity because the superconductivity is a ma croscopic quantum phenomena. The < p > lvalue can change at the transition to the supercon- ducting state. Electrons are accelerated against the force of the electric field in this case. This takes place, for example, at the Meissner effe ct, the Little-Parks effect and other quantization phenomenon. In order to describe these phenomenon using the language of t he classical mechanics a quantum force Fqshould be introduced,/integraltext dtFq=<∆p > l.< p > l= (e/c)Φ/l= (e/c)BS/l when the ring in the normal state, because < v >l= 0. The < p > lvalue does not change when a ring segment lais switched into the superconducting state and other segment lbis remained in the normal state, nsa>0 and nsb= 0. nsaandnsbare the densities of superconducting electrons in the segments laandlb. This corresponds to the laws of the classical mechanics and does not require of the quantum force. Contrad iction with the classical mechanics appears at the closing of the supercond ucting state in the ring (when nsa>0 and nsb>0) because the momentum 2 p= ¯h▽φ= 2mvs+2e cA(see (1)) and the velocity of superconducting pair (see (2)) are quantized if the superconducting state is closed. The avera ge momentum < 2p > l=l−1/integraltext ldl¯h▽φ= ¯h2πn/l = (2e/c)(Φ0/l)nand the average velocity along the ring circumference < vs>l=l−1/integraltext ldlvs=e mcΦ0n−Φ lcan have only permitted values when nsa>0 and nsb>0. Therefore the average momentum should be change on <∆2p > l= (2e/c)(nΦ0−BS)/lat the closing of the superconducting state if the magnetic flux within the ring is not divisible by the 8flux quantum, Φ = BS∝negationslash=nΦ0). Consequently, the quantum force/integraltext tcldtFq= (2e/c)(nΦ0−BS)/lacts at the closing of the superconducting state. Here tclis a time of the superconducting state closing. The superconducting electrons in the segment laare accelerated against the force of the Faraday electric field/integraltext ldlFe=e/integraltext ldlE=−edΦ/dt=−eLdI/dt . The velocity is changed from vsa= 0 to the value determined in the stationary state, when the current in the ring I=Is=Isa=sajsa=saensavsa=Isb= sbjsb=sbensbvsb, by the relation vsa=e mcnsb (lansb+lbnsa)(Φon−Φ) (3 a) (atsa=sb=s).sis the area of the wall section of the ring. The magnetic flux inside the ring changes from Φ = BSto Φ = BS−LI. In the stationary state I=Is=se2 mcnsansb (lansb+lbnsa)(Φ0n−Φ) (3 b) The momentum <2p >lreturns to the initial value during the decay time L/R nbafter the transition of the segment lbin the normal state. Here Rnb= ρbnlb/sis the resistance of the segment lbin the normal state. This process corresponds to the laws of the classical mechanics. The velo city< ve>lin the segment lbis decreased because of the dissipation. Therefore a charge q=/integraltext dt(Ia−Ib) on the boundaries of the segments laandlband, as a consequence, the potential difference VaandVbappear. The electric field Ea=− ▽Va− (1/c)dA/dt retards the superconducting electrons in the segment laandEb= −▽Vb−(1/c)dA/dt counteracts of the dissipated force Fdis=−eEb+mdv e/dtin the segment lb. The voltage Eahas the same direction after the transition of lb both in the superconducting and normal states: the supercon ducting electrons are accelerated against Eaafter the transition to the superconducting state and are retarded by Eaafter the transition to the normal state. Consequently < Va>∝negationslash= 0. Thus, in spite of the wide spread opinion, the direct volt age can exist on the superconducting segments of the ring. The quantum force acts both in inhomogeneous and in homogene ous rings. At the transition to the superconducting state of a homogene ous ring the elec- trons, which become superconducting, accelerate/integraltext dt(dvs/dt) = (¯h/2nr)(n− Φ/Φ0) [7] against the electric field E=−(L/l)dIs/dt. The work done by the quantum force/integraltext dxFq=/integraltext dtvsFqincreases the kinetic energy/integraltext dtlsn sm(dvs/dt)vs= lsnsmv2 s/2 and the energy of magnetic field FL=/integraltext dt(−lsnseEvs) =/integraltext dtL(dIs/dt)Is= LI2 s/2. This means that the energy of the superconducting state in creases if Φ∝negationslash=nΦ0. The Little-Parks effect [8] is experimental evidence of thi s. Ac- cording to the Tinkham’s explanation [9] of this effect, the c ritical temperature of a superconducting tube with narrow wall depends in a perio dic way on the magnetic flux value within the tube Tc(Φ) = Tc[1−(ξ(0)/r)2(n−Φ/Φ0)2] (4) 9because the |n−Φ/Φ0|tends towards a minimum possible value and therefore the kinetic energy of superconducting electrons changes pe riodically with the magnetic field. It ought be noted that the using of the quantum force is not a principal new in comparison with the Tinkham’s explanatio n. We may say: ”theTcdepends on Φ because the quantum force should be overcome at t he superconducting transition if Φ ∝negationslash=nΦ0”. And we may say: ”the Tcdepends on Φ because the energy of the superconducting state increases i f Φ∝negationslash=nΦ0”. These statements are equivalent. Timkham did not consider the mag netic energy because LI2 s/2 is proportional to n2 sand therefore does not influence on Tc. The ring with narrow wall (the wall thickness w≪r, λ,λis the penetra- tion depth of magnetic field), when LIs≪Φ0and Φ ≃BS, is considered in this paper. In the opposite case w≫λ,vs≃0 and Φ =/integraltext ldlA≃nΦ0in the superconductor interior. nis any integer number if a nonsuperconducting singularity is inside landn= 0 if a singularity is absent. The Meissner effect Φ≃nΦ0= 0 takes place in the later case. In order to describe this effe ct in the classical mechanic, the quantum force may be introduced also. But this de- scription is not so obvious as in the case of the inhomogeneou s superconducting ring. The Meissner effect is more intricate phenomenon than t he quantization of the fluxoid: nis not any integer number, but n= 0. It is obvious that the direct voltage can appear only in the in homogeneous case when the dissipating force Fdisacts only in a segment of the ring. The quantum force Fqaccelerates electrons both in the laand in the lbsegments (only/integraltext tcldtl−1/integraltext ldlFqhas a sense), whereas the dissipating force Fdisretards electrons only in the lbsegment. Therefore the potential difference with dc component is induced in the ring segments. The Fqacts only if the Fdisacts. It returns the average momentum to the same value <2p >l= (2e/c)(Φ0/l)n. The<2p >lvalue changes only at the switching of the segment lbinto and out of the normal state. Any other changes of the superconductin g electron density (nsaandnsb) do not influence on this value. 5 When and Why the Total Entropy may be Systematically Reduced The appearance of the direct voltage means that the inhomoge neous supercon- ducting ring can be used as a direct-current generator. The c urrent in a resistor Rloadloaded on the segment lbisIload=RbIa/(Rb+Rload). After the transition of the segment lbin the normal state with Rb=Rbnthe current Iain the seg- ment ladecreases exponentially from Isdetermined by the relation (3a) during the decay time L/R sys, where Rsys=RbnRload/(Rbn+Rload). Consequently the power-driven energy Asw=/integraltextdtRloadI2 load= (Rbn/(Rbn+Rload))(LI2 s/2) can develop across the load Rloadat the switching of the segment lb. The power Wload=Aswf= (Rbn/(Rbn+Rload))FLf. Here f= 1/Nis the frequency of 10the switching; Nis the average number of the switching in a time unity. Because the power Wloadmay be utilized for a useful work and the segment lbmay be switched by the temperature change, the superconduct ing ring may be used as the heat engine. The ordered force in this heat engine is the quantum force Fq. The Fqdirection coincides with the direction of the velocity vsof superconducting electrons. Whereas the dissipating force is directed against the velocity. The work done by the quantum force/integraltext dxFq=/integraltext dtvsFqincreases the ordered energy: the kinetic energy of superconducting e lectrons and the energy of magnetic field LI2 s/2. A part of this ordered energy may be used for a useful work and other part is dissipated in the ring after th elbswitching into the normal state. The energy is dissipated completely if the load is absent i.e. 1/Rload= 0. There is not a contradiction with the second law of thermodyn amics if the segment lbis switched in a consequence of a temperature change above an d below Tcb[2]. The heat energy Qsw=cb∆Tshould be spent for the heating of the segment lbfromTmin=Tcb−∆TtoTmax=Tcb. Here cbis the heat capacity. Because Is∝nsb∝(Tcb−T), the work Asw∝(Tmax−Tmin)2. Con- sequently, the maximum efficiency Ef=Asw/Qswof the ring as a heat engine is proportional to ( Tmax−Tmin). This is agreed with both the old and modern interpretation of the second law of thermodynamics. But the superconducting ring differs qualitatively from the conventional heat engin e because it can work without correlation between Fordandv, i.e. it is not the type ”a” heat engine, as the conventional heat engine, but is the type ”b” heat engi ne. It can work at both a ordered and chaotic switching of the segment lb. The section lbcan be switched chaotically by the thermal fluctuation at an unaltered temperature (at Tmax=Tmin≃Tcb). It is well known [7, 10] that the resistance of a superconductor R < R nin some region above Tcand R >0 in some region below Tcbecause superconducting droplets (with charac- teristic size ≃ξs(T)) appear in the normal state and normal droplets (phase- slip centers) (with characteristic size ≃ξn(T)) appear in the superconducting state in the consequence of the thermal fluctuation. The cohe rence lengths ξs(T) =ξ(0)(T/T c−1)−0.5atT > T candξn(T) =ξ(0)(1−T/T c)−0.5at T < T cin the linear approximation valid at |T−Tc| ≫GiTc.Giis the Gins- burg number. We are interested here the one-dimensional cas e, in which the transverse dimensions of superconductor are small compare d with the coher- ence length ( w < ξ ,s < ξ2). In this case the ξs(T) has a finite value at T < T c, which increases with temperature decreasing. ξs(T)≃ξn(T) atT≃Tc The probability of the switching at T≃Tcbof the segment lbin the normal state is much bigger than the one of the segment laifTca> Tcb≃T. Therefore the inhomogeneous superconducting ring is switched by the fl uctuation from closed ( nsa>0,nsb>0) to open ( nsa>0,nsb= 0) superconducting state at an unaltered temperature corresponded to the resistive t ransition of the ring segment with the lowest critical temperature. As it was show n above, the voltage appears at this switching if Φ /Φ0∝negationslash=n. The probability of the closing is enough 11high if lb≃ξs(T). The voltage is chaotic at Φ /Φ0=n+ 0.5, because the switching induced by the fluctuation is chaotic in time and th e quantum force acts in opposite directions with equal probability. This ca se does not differ qualitatively from other fluctuation phenomena, for exampl e from the Nyquist’s noise [11]. The power of the chaotic voltage is ”spread” on al l frequencies ωas well as at the Nyquist’s noise, the power of which is proporti onal to a frequency band ∆ ω:< V2 Nyq> /R = 4kBT∆ω[5]. The qualitative difference from the completely chaotic fluct uation effects (such as the Nyquist’s noise) takes place at Φ /Φ0∝negationslash=n+ 0.5. n can be any integer number, but the state with minimum |n−Φ/Φ0|value has a maximum probability, because the energy of this state is minimum. Th erefore the average value of the quantum force by a long time Tis not equal zero:/integraltextT 0dtFq/N=/summationtext sw.P(|n−Φ/Φ0|)(2e/c)(nΦ0−Φ)/lN∝negationslash= 0 at Φ /Φ0∝negationslash=nand Φ /Φ0∝negationslash=n+ 0.5. HereNis the number of the switching during the time T;/summationtext sw.is the sum by these switching; P(|n−Φ/Φ0|) is the probability that/integraltext ldl▽φ/2π=nin the closed superconducting state. Because/integraltextT 0dtFq/N∝negationslash= 0 the voltage Vbis not completely chaotic ( Vb(ω= 0) = < Vb>∝negationslash= 0) although it is induced by the chaotic switching. This obvious consideration leads to res ult published first in [1]. This result means that a part p(Φ/Φ0) of the chaotic electric energy induced by thermal fluctuation can be ordered in the inhomogeneous su perconducting ring. p(Φ/Φ0) = 0 at Φ /Φ0=nor Φ/Φ0=n+ 0.5 and 0 < p(Φ/Φ0)<1 at Φ/Φ0∝negationslash=nand Φ /Φ0∝negationslash=n+ 0.5. The direct voltage V(ω= 0)∝negationslash= 0 can appear only in an inhomogeneous ring because in a homogeneous ring t he switching is chaotic not only in time but also in space. Different segments of the ring are switched in different time (if l≫ξ(T)). Therefore < V > = 0. In a mesoscopic ring with l < ξ(T) the switching takes place simultaneously in the whole ring . Therefore the potential difference is equal zero. The power of the energy regulating in the inhomogeneous supe rconducting ring is Word=p(Φ/Φ0)Aswfsw.Asw< F Land the maximum frequency of the switching fswis determined by the characteristic relaxation time of the superconducting fluctuation: fsw≤1/τrel.. Therefore Word< p(Φ/Φ0)FL/τrel.. The fluctuations induce the magnetic energy FL≃µkBT/(1 +µ), where kBT is the characteristic energy of the thermal fluctuation; µ/(1 +µ) is the part of FLin whole change of the superconductor energy. µhas the maximum value µ= (32 π3/κ2)(Ls/l3 b)(n−Φ/Φ0)2atT=Tc. This relation is valid for a ring withs≪λ2 L. Here κ=λL/ξis the superconductor parameter introduced in the Ginsburg-Landau theory; λLis the London penetration depth of the magnetic field. In the linear approximation region at T > T c,τrel.=τGL= ¯h/8kB(T−Tc) (see [10]). The probability of the switching from into and ou t of the normal state is not small only in the critical region, i.e. at |T−Tc|< GiT c. Therefore the maximum power of the energy regulating in the inhomogeneous superconducting ring may be estimated by the relation 12Wmax≃p(Φ/Φ0)µ 1 +µ8πGi(kBTc)2 ¯h(5) This power is very weak. Even for a high-Tc superconductor wi thTc≃100K, (kBTc)2/¯h≃10−8Wt. Thus, the mesoscopic superconducting ring, switched by the fluctuation into and out of the closing superconducting state, may be conside red as the second order perpetuum mobile of type ”b”. But the ring without a loa d are useless perpetuum mobile because the energy both is ordered and is di ssipated inside it. Moreover, it is perpetuum mobile only in the old interpre tation, which was revised in the beginning of our century together with the int erpretation of the second law of thermodynamics. ”Perpetuum mobile” is litera lly permanent motion. According to modern knowledge the permanent motion takes place at any nonzero temperature. The voltage inside the ring, as wel l as the Nyquist’s noise, is one of the examples of this permanent motion. This permanent motion is not perpetuum mobile according to t he modern interpretation. It contradicts to old but not to modern inte rpretation of the second law of thermodynamics. According to the old interpre tation, the en- tropy decreases at the closing of the superconducting state and increase at the transition to normal state of the segment lb. But according to the modern inter- pretation, the switching by the fluctuation does not change t he entropy value, because both the closed and open superconducting state are i ncluded to the statistical weight P. The entropy, as the measure of chaos, does not change both at < Vb>= 0 and at < Vb>∝negationslash= 0 if whole magnetic energy LI2 s/2 induced by the quantum force is dissipated in the segment lbafter it’s transition to the normal state, i.e. < F q>< v s>+< F dis>< v > = 0. In this case, the situation in the inhomogeneous superconducting ring does n ot differ from the one in a homogeneous ring or at the Meissner effect. Nevertheless the permanent motion in the mesoscopic superc onducting ring differs qualitatively from other types of the permanent moti on because it is partly ordered at < Fq>< v s> >0. Therefore the potential chance exists to utilize this permanent motion for the useful work. In order t o realize this chance the direct potential difference induced in the inhomogeneou s superconducting ring by the quantum force should be put under load. The inhomo geneous su- perconducting ring with a load is the perpetuum mobile not on ly according to the old interpretation, but also the modern interpretati on. It is the useful perpetuum mobile. In this case no whole magnetic energy is di ssipated in the segment lb. A part develops across the load. This process contradicts t o the Clausius’s formula if a temperature of the load is higher tha n the one of the ring. It contradicts to the Carnot’s principle and the Thomson’s f ormula if the load is an electricmotor. The total entropy (total chaos) is redu ced when the heat energy is transformed in a ordered energy, ∆ Q→tWord→Eord, because the reduction of the entropy ∆ S=−∆Q/T=−Eord/Tcan not be compensated in the equilibrium state for the transfer a heat energy from a heater to a cooler 13because Theat=Tcoolin this state. The transformation ∆ Q→tWord→Eord can take place anyhow long time. At tWord≫kBT, ∆S≫kB. The useful work can be obtained in the load because the voltag e induced by fluctuation is partly ordered, Vb(ω= 0) = < Vb>∝negationslash= 0, in the inhomogeneous superconducting ring. There is qualitative difference, for example, from the Nyquist’s noise [11], the power of which can not be used for a u seful work because it is chaotic. It is impossible also to transfer any energy of the Nyquist’s noise from a cold resistor to a hot resistor. The second law of therm odynamics can not be broken when the chaotic voltage is put under a load. The power of the Nyquist’s noise is not summed up: the power of one resistor < V2 Nyq> /R = 4kBT∆ωis equal the one of N resistors < V2 Nyq> /NR = 4kBT∆ω. Whereas the power Wordcan be summed up: the direct voltage on a system of N rings, the segments of which with lowest critical temperature are c onnected in series, is equal N < V b>. The power Wmax(see (5)) corresponds to the power of the Nyquist’s noise in the frequency band ∆ ω=p(Φ/Φ0)(µ/(1 +µ))2Gi(kBTc/¯h). This ∆ ωmay be enough wide. At p(Φ/Φ0)(µ/(1+µ))2Gi≃1, ∆ω≃kBTc/¯h≃ 1013sec−1atTc= 100 K. The statement on the possibility of the perpetuum mobile is n ot new in essence. In the beginning of our century scientists have com e to the conclusion that the perpetuum mobile, as the permanent motion of energy , takes place in any thermodynamic system at T >0. The new statement in the my consid- eration is the possibility of regulating of this motion in so me quantum system. This chaos reduction is connected with the quantum force, wh ich has a funda- mental cause:/integraltext ldlp=/integraltext ldl¯h▽φis ”bad” (no gauge-invariant) quantum value iflis not a closed path, and it is ”good” (gauge-invariant) valu e iflis a closed path. We may say that the quantum force acts at a ”transition” from classical to quantum mechanics. It can be introduced because the duali sm of electron. We can use the relation mdv/dt =Fbecause the superconducting electrons are particles. But these particles can accelerate against the c lassical forces because the electron is wave. It is obvious that the entropy, as measure of the chaos, decre ases at the ”transition” from the classical to quantum mechanics. Plan ck has introduced in 1900 year the quantization in order to reduce the number of microscopic states from infinite to finite value. Therefore one may say tha t the perpetuum mobile is possible because the ”switching” between the clas sical and quantum mechanics is possible. It was become obvious in the beginning of our century that the second law of thermodynamics in the old interpretation is correct to kBT. The violation of the second order of thermodynamics in the modern interpreta tion has also the fundamental limit. Because the characteristic energy of flu ctuation kBTand the time of any cycle can not be shorter than ¯ h/kBTin accordance with the uncertainty relation, the power of any perpetuum mobile Wp.m.<(kBT)2/¯h (6) 14We may say that the second law of thermodynamics is correct to (kBT)2/¯h. 6 How to Make the Perpetuum Mobile In order to make the perpetuum mobile, the modern methods of t he nano- technology should be used. Sizes of the ring segment with low est critical tem- perature should not surpass strongly the coherence length o f the superconduc- tor. Other sizes of the ring should be also enough small. Beca use the power of a ring is very weak, Wmax<(kBTc)2/¯h, a system with big number of the rings should be used in order to obtain an acceptable power. T he useful power Wload=N2< Vb>2Rload/(Rload+NRb)2can be obtained by the system of N rings the segments lbof which are connected in series. This power has maximum value Wload=N < V b>2/4RbatRload=NRb. Because the sign and the value of< Vb>depend on n−Φ/Φ0=n−BS/Φ0, the area Sof all rings should be the same. The fluctuations induce < Vb>only in a narrow region near Tcb. Therefore the critical temperature of all rings should be ap proximately identi- cal. All rings should be identically inhomogeneous Tcb< Tca. The requirement Tcb< T cacan be realized in the ring with difference areas of wall secti onla andlb,sb< sa. In accordance with the Little-Parks effect Tcb(Φ)< Tca(Φ) at sb< saif Φ/Φ0∝negationslash=n. Because Wmax∝T2 ca system of high-Tc superconductor (HTSC) rings has maximum power. In order to obtain Wload≈1Wtthe system of no less than 4 108HTSC rings should be used. But it is very difficult to make the HT SC rings with enough small sizes. The coherence length of all kn own HTSC is very small. Therefore the conventional superconductor rin gs (for example Al rings) ought be used for first experimental investigation. T he expected volt- age< V b> < (RbWmax)1/2≃(p(Φ/Φ0)8Giµ/(1 +µ))1/2R1/2 bkBTc/¯h1/2is small but measurable. For example at Tc= 1Kand a real value Rb= 1 Ω, R1/2 bkBTc/¯h1/2≃10−6V= 1µV. (p(Φ/Φ0)8Giµ/(1 +µ))1/2≤1 in any case. This value depends in the periodic manner on the magnetic fiel d (with period ∆B= Φ0/S) and is not equal zero only at the temperature closed to Tcb. The perpetuum mobile can not solve the energy problem. It’s p ower is very weak. But the system of superconducting rings may be use d in some applications. It can used simultaneously as the direct-cur rent generator [12] and as the micro-refrigerator [13]. The perpetuum mobile ca n work anyhow long time without an expense of any fuel. Therefore it may be e specially useful in self-contained systems. 7 ACKNOWLEDGMENTS I am grateful Jorge Berger for the preprint of his paper and fo r stimulant dis- cussion. I thank for financial support the International Ass ociation for the 15Promotion of Co-operation with Scientists from the New Inde pendent States (Project INTAS-96-0452). References [1] A.V. Nikulov and I.N. Zhilyaev, The Little-Parks effect i n an inhomo- geneous superconducting ring, J. Low Temp.Phys. 112, 227-236 (1998); http://xxx.lanl.gov/abs/cond-mat/9811148. [2] A.V.Nikulov, ”Transformation of Thermal Energy in Elec tric Energy in an Inhomogeneous Superconducting Ring” in Symmetry and Pairing in Super- conductors , Eds. M.Ausloos and S.Kruchinin, Kluwer Academic Publishe rs, Dordrecht, p. 373 (1999); http://xxx.lanl.gov/abs /cond- mat/9901103. [3] A.V.Nikulov, Violation of the second law of thermodynam ics in a supercon- ducting ring, Abstracts of XXII International Conference on Low Tempera- ture Physics , Helsinki, Finland, p.498 (1999). [4] J.Berger, The fight against the second law of thermodynam ics,Physics Es- says7, 281 (1994). [5] Charles Kittel, Thermal Physics. John Wiley and Sons, Inc. New York 1973 [6] M.Smoluchowski, ”Gultigkeitsgrenzen des zweiten Haup tsatzes der Warmetheorie”, in Vortrage uber kinetische Theorie der Materie und der Elektrizitat (Mathematische Vorlesungen an der Universit at Gottingen, VI) . Leipzig und Berlin, B.G.Teubner, p.87 (1914). [7] M.Tinkham, Introduction to Superconductivity. McGraw -Hill Book Com- pany (1975). [8] W.A.Little and R.D.Parks, Phys. Rev. Lett. 9, 9 (1962); Phys. Rev. 133, A97 (1964). [9] M.Tinkham, Phys. Rev. 129, 2413 (1963). [10] W.J.Skocpol and M.Tinkham, Fluctuation near supercon ducting phase transitions, Rep.Prog.Phys. 38, 1049 (1975) [11] H.Nyquist, Phys. Rev. 32, 110 (1928). [12] A.V.Nikulov, A superconducting mesoscopic ring as dir ect-current genera- tor,Abstracts of NATO ASI ”Quantum Mesoscopic Phenomena and Mes o- scopic Devices in Microelectronics”, Ankara, Turkey, p.105, (1999). 16[13] A.V.Nikulov, A system of mesoscopic superconducting r ings as a microre- frigerator, Proceedings of the Symposium on Micro- and Nanocryogenics, Jyvaskyla, Finland, p.68, (1999). 17

arXiv:physics/9912023v1 [physics.atom-ph] 10 Dec 1999Mechanisms of positron annihilation on molecules G. F. Gribakin∗ School of Physics, University of New South Wales, Sydney 205 2, Australia Abstract The aim of this work is to identify the mechanisms responsibl e for very large rates and other peculiarities observed in low-energy positron annihilation on molecules. The two mechanisms considered are: (i) Direct annihilation of the incoming positron with one of the molecular electrons. T his mechanism dominates for atoms and small molecules. I show that its cont ribution to the annihilation rate can be related to the positron elastic sca ttering cross sec- tion. This mechanism is characterized by strong energy depe ndence of Zeffat small positron energies and high Zeffvalues (up to 103) for room temperature positrons, if a low-lying virtual level or a weakly bound sta te exists for the positron. (ii) Resonant annihilation, which takes place wh en the positron un- dergoes resonant capture into a vibrationally excited quas ibound state of the positron-molecule complex. This mechanism dominates for l arger molecules capable of forming bound states with the positron. For this m echanism Zeff averaged over some energy interval, e.g., due to thermal pos itron energy distri- bution, is proportional to the level density of the positron -molecule complex, which is basically determined by the spectrum of molecular v ibrational states populated in the positron capture. The resonant mechanism c an produce very large annihilation rates corresponding to Zeff∼108. It is highly sensitive to molecular structure and shows a characteristic ε−1/2behaviour of Zeffat small positron energies ε. The theory is used to analyse calculated and measured Zefffor a number of atoms and molecules. 34.50.-s, 78.70.Bj, 71.60.+z, 36.10.-k Typeset using REVT EX 1I. INTRODUCTION The aim of this work is to develop the framework for the descri ption of low-energy positron annihilation on molecules, and to analyse its two m ain mechanisms: direct and res- onant annihilation. There are a number of remarkable phenom ena associated with this pro- cess: very large annihilation rates [1–3], high sensitivit y of the rates to small changes in the molecular structure [4], large ionization-fragmentation cross sections for organic molecules at sub-Ps-threshold positron energies [5], and rapid incre ase of the fragmentation and anni- hilation rates towards small positron energies [6,7]. In sp ite of decades of study, there is no consistent physical picture or even general understanding of these processes, and there have been very few calculations [8], which leaves too much room fo r speculations [9]. My main objective is to consider real mechanisms of positron annihi lation on molecules, describe their characteristic features, make estimates of the correspond ing annihilation rates, and formu- late the terms in which positron-molecule annihilation sho uld be described and analysed. In recent work [7], Iwata et al. describe new experiments to study positron annihilation on molecules. Some of these experiments test specific featur es of the annihilation processes described in the present paper. Though some aspects of the ex perimental work are discussed here, futher details and comparison with theory and various models of positron annihilation can be found in Ref. [7]. The annihilation rate λfor positrons in a molecular or atomic gas is usually express ed in terms of a dimensionless parameter Zeff: λ=πr2 0cZeffn, (1) where r0is the classical radius of the electron, πr2 0cis the non-relativistic spin-averaged rate of electron-positron annihilation into two γquanta, and nis the number density of molecules [10]. Equation (1) implies that Zeffis the effective number of target electrons contributing to the annihilation process. In terms of the annihilation cros s section σathe rate is λ=σanv, so by comparison with Eq. (1), we have σa=πr2 0cZeff/v, (2) where vis the positron velocity. Accordingly, the spin-averaged c ross section of annihilation of a non-relativistic positron on a single electron corresp onds to Zeff= 1, see e.g. [11]. If the annihilation occurs during binary positron-molecule c ollisions, as in the experiments of the San Diego group [3,4,12] who use a positron trap and work a t low gas densities, the parameter Zeffis independent of the density. It characterizes the annihil ation of a positron on a single molecule. One could expect that Zeffis comparable to the number of electrons Zin an atom or molecule. Moreover, low-energy positrons do not penetra te deep into the atom, and annihilate most probably with the valence electrons only. H owever, even for hydrogen Zeff= 8 at low energies [13]. This is a manifestation of correlatio n effects. The most important of them is polarization of the atom by the positron and, as a resu lt, an attractive −αe2/2r4 positron-atom potential, αbeing the atomic dipole polarizability. An additional shor t-range contribution to the positron-atom attraction comes from vi rtual Ps formation, i.e., hopping, or rather, tunneling of an electron between the atomic ion an d the positron. The electron 2density on the positron is also enhanced due to the Coulomb at traction between them. These effects make atomic Zefflarge, e.g., Zeff= 401 for room temperature positrons on Xe [14]. Even compared with this large number, annihilation rates fo r low-energy (room tem- perature) positrons on polyatomic molecules are huge. They increase very rapidly with the molecular size, and depend strongly on the chemical composi tion of the molecules, see Fig. 1. This has been known for quite a while, after early measuremen ts for CCl 4,Zeff= 2.2×104 [1], butane, 1 .5×104[2], and Zeffranging between 104and 2×106for large alkanes C nH2n+2, n=4–16 [15] (see also [4]). The largest Zeffvalues measured so far are 4 .3×106for antracene C14H10[16] and 7 .5×106for sebacic acid dimethyl ester C 12H22O4[15]. Thus, while Zeff up to five orders of magnitude greater than Zhave been observed, the physical processes responsible for these anomalously large annihilation rate s have not been really understood. In other words, if the observed Zeffare parametrically large, compared to the number of available electrons, then what are the parameters that dete rmine large annihilation rates for positrons on molecules? In this work I consider two basic mechanisms of positron-mol ecule annihilation. The first mechanism is direct annihilation of the incoming positron with one of the molecular electrons. The contribution of this mechanism to the annihi lation rate is proportional to the number of valence electrons available for annihilation. It can be enhanced by the positron- molecule interaction which distorts the positron wave. In p articular, the positron density in the vicinity of the molecule increases greatly if a low-ly ing virtual state ( ε0>0) or a weakly bound level ( ε0<0) exists for the s-wave positron. In this case Z(dir) eff∝1/(ε+|ε0|) for small positron energies ε<∼|ε0|[17–19]. This type of enhancement is responsible for largeZeffvalues observed in heavier noble gas atoms, where successiv ely lower virtual levels exist for the positron ( Zeff= 33.8, 90.1 and 401 for Ar, Kr and Xe, respectively [4,14]). This understanding is confirmed by the temperature dependen ces of the annihilation rates measured for the noble gases in [20]. Note that for room-temp erature positrons, ε∼kBT, even for ε0→0 the size of the enhancement due to virtual/weakly bound sta tes is limited. The second mechanism is resonant annihilation . By this I mean a two-stage process. The positron is first captured into a Feshbach-type resonanc e, where positron attachment is accompanied by excitation of some molecular degrees of fr eedom. Such process is well known for electrons [21]. The positron in the quasi bound sta te then annihilates with a molecular electron. Enhancement of annihilation due to a si ngle resonance was considered theoretically in [22,23]. The possibility of forming such r esonances by excitation of the vibrational degrees of freedom of molecules was proposed by Surko et al. [3] to explain high annihilation rates and their strong dependence on the m olecular size observed for alkanes. It was also considered in relation to the problem of fragmentation of molecules by positron annihilation [24]. However, its contribution t o the annihilation have never been properly evaluated. To make this mechanism work for low -energy positrons one must assumed that positrons can form bound states with large neut ral molecules, i.e., the positron affinity of the molecule is positive, εA>0 [3]. The capture is then possible if the energy of the incoming positron is in resonance with the vibrationall y excited state of the positron- molecule complex [25]. The density of the vibrational excit ation spectrum of this complex can be high, even if the excitation energy supplied by positr on binding, Ev=εA+ε, is only few tenths of an eV (it is reasonable to assume that the pr esence of the positron does not change the vibrational spectrum of the molecule by t oo much). For positrons 3with thermal Maxwellian energy distribution the contribut ion of the resonant annihilation mechanism averaged over a number of resonances Z(res) effis observed. The magnitude of Z(res) effis determined by three parameters of the positron-molecule resonant states: their annihilation width Γ a, the autodetachment width Γ c, which also determines the probability of positron capture, and the level density ρ(Ev) of the positron-molecule resonant states populated in positron capture. The magnitude of Γ afor positron-molecule bound states is comparable to the spin-averaged annihilation width of the P s atom (Γ a/¯h∼5×10−10s). Note that Γ adoes not increase with the size of the molecule, because the i ncrease in the number of electrons is accompanied by thinning of the positr on density in the (quasi)bound positron-molecule state. It turns out (see Sec. II) that for Γc≫Γathe magnitude of Z(res) eff is simply proportional to ρ(Ev). This density increases rapidly with the size of the molecu le, ρ(Ev)∝(Nv)n, where Nvis the number of vibrational modes, n∼εA/ωis the effective number of vibrational quanta excited in positron capture, a ndωis a typical molecular vibrational frequency. Thus, the resonant annihilation me chanism can explain the rapid increase of Zeffwith the size of the molecule shown in Fig. 1. Moreover, my est imates show that for thermal positrons Z(res) effup to 108could be observed. A necessary condition for the resonant annihilation to occu r is the existence of positron- molecule bound states. Until recently there was almost no po sitive information about the possibility of positron binding to neutral atomic species. The experimental results and their interpretation by Surko et al. [3] could be viewed as the strongest, albeit indirect, evide nce of positron binding to large molecules. This situation has c hanged now. Many-body theory calculations of Dzuba et al. [26] indicated strongly that positrons can be bound by Mg, Zn, Cd, and Hg and, possibly, many other atoms. Recently the v ariational calculations of Ryzhikh and Mitroy proved rigorously that positrons form bo und states with Li atoms, and demonstrated that bound states also exist for Na, Be, Mg, Zn, Cu and Ag [27]. Molecules are much larger potential wells for the positron, and it seem s natural that many of them should be capable of binding positrons. Ideas about different mechanisms in positron-molecule anni hilation have been discussed earlier in a number of theoretical [17,23] and experimental [3,22] works. However, there is a need to re-examine this question using a unified approach to t he annihilation mechanisms, and define clearly the physical variables which determine th e observed annihilation rates. The latter is especially important for the present work whic h aims to provide understanding of a whole variety of phenomena, including the origins of the high values of Zefffor molecules and their dependence on the chemical composition and positr on energy. II. ANNIHILATION MECHANISMS In this section a derivation of the positron annihilation ra te within a standard scattering theory formalism is presented. I show how to estimate the con tributions of the direct and resonant mechanisms, and examine specific features of these mechanisms. 4A. General expressions The effective number of electrons Zeffrelated to the annihilation rate through Eq. (1) is determined by the positron density on the electrons Zeff=/integraldisplayZ/summationdisplay i=1δ(r−ri)|Ψk(r1, . . .,rZ,r)|2dr1. . .drZdr, (3) where Zis the number of target electrons, riandrare the coordinates of the electrons and positron, respectively, and Ψ k(r1, . . .,rZ,r) is the total wave function of the system. It describes scattering of the positron with initial momentum kfrom the atomic or molecular target in the ground state Φ 0, and is normalized as Ψk(r1, . . .,rZ,r)≃Φ0(r1, . . .,rZ)eikr(r≫Ra), (4) where Rais the radius of the target (atomic units are used throughout ). Note that for molecular targets Ψ kand Φ 0should, strictly speaking, depend on the nuclear coordinat es as well. Let us first assume that the electron-positron degrees of fre edom are completely decoupled from the nuclear motion. The scattering wave function is the n determined by the positron interaction with the charge distribution of the ground-sta te target and electron-positron cor- relation interaction (polarization of the target, virtual Ps formation, etc.). Let us denote the corresponding wave function Ψ(0) k. At positron energies of a few electron Volts the molecule can be excited electronically, and the positron may find itse lf trapped in electronically ex- cited Feshbach resonance states. This may result in rapid re sonant energy dependence of the Ψ(0) kwave function. However, at small sub-eV or room-temperatur e positron energies electron excitations cannot be produced, and Ψ(0) kbehaves smoothly. On the other hand, if the positron affinity of a molecule is positive, the system ‘ molecule+positron’ is capable of forming a stable “positronic ion”, whose lifetime is only limited by positron annihilation. This system will also have a number of excited bound states Φ νcorresponding to vibrational excitations of the positron-molecule complex. Their typic al energies are of the order of 0.1 eV and smaller, as determined by the vibrational spectrum of the molecule. If we now turn on the coupling Vbetween the electron-positron and nuclear degrees of freedom the total scattering wave function will be given by |Ψk∝angbracketright=|Ψ(0) k∝angbracketright+/summationdisplay ν|Φν∝angbracketright∝angbracketleftΦν|V|Ψ(0) k∝angbracketright E−Eν+i 2Γν. (5) The first term on the right-hand side describes direct, or potential [28], scattering of the positron by the ground-state molecule. The second term d escribes positron capture into bound positron-molecule states. Equation (5) has the a ppearance of a standard perturbation-theory formula. The energy of the system is E=E0+k2/2, where E0is the target ground state energy. The energies of the positron -molecule (quasi)bound states Φνin the denominator are complex, Eν−i 2Γν, because these states are, in fact, unstable against positron annihilation with one of the target electr ons, and against positron emis- sion, a process inverse to positron capture. Therefore, the total width of state νis the 5sum of the annihilation and emission (or capture) widths: Γ ν= Γν a+ Γν c[29]. These states manifest as resonances in positron-molecule scattering. T hey may not give a sizeable con- tribution to the scattering cross section, but, as I show bel ow, they can contribute a lot to the positron-molecule annihilation rate. The contribution of a particular resonant state νto the wave function is proportional to the corresponding capture amplitude ∝angbracketleftΦν|V|Ψ(0) k∝angbracketright, which also determines the capture width Γν c= 2π/integraldisplay |∝angbracketleftΦν|V|Ψ(0) k∝angbracketright|2kdΩk (2π)3=k π|∝angbracketleftΦν|V|Ψ(0) k∝angbracketright|2, (6) where the latter formula is valid for the positron swave which dominates at low positron energies (see below). If the positron interaction with vibr ations cannot be described by perturbations Eqs. (5) and (6) remain valid, provided we rep lace the first-order capture amplitudes ∝angbracketleftΦν|V|Ψ(0) k∝angbracketrightwith their non-perturbative values. The annihilation width of the positron-molecule state Φ νis a product of the spin-averaged electron-positron annihilation cross section σ2γ=πr2 0c/v, the positron velocity v, and the density factor, Γν a=σ2γv∝angbracketleftΦν|Z/summationdisplay i=1δ(r−ri)|Φν∝angbracketright =πr2 0c/integraldisplayZ/summationdisplay i=1δ(r−ri)|Φν(r1, . . .,rZ,r)|2dr1. . .drZdr (7) ≡πr2 0cρν ep, where ρν epis the average positron density on the target electrons in th eνth bound state. For the ground state positronium ρPs ep= (8πa3 0)−1. One can use this value to estimate the annihilation width of the positron-molecule complex. The p resence of many electrons in a large molecule does not lead to an increase of the width, beca use the positron is spread over a larger volume due to the normalization condition /integraldisplay |Φν(r1, . . .,rZ,r)|2dr1. . .drZdr= 1. Therefore, using the Ps estimate of the density one obtains Γν a∼0.5×10−7a.u.∼1µeV, which corresponds to the annihilation lifetime τa∼5×10−10s. To calculate Zeffwave function (5) is substituted into Eq. (3), which yields Zeff=∝angbracketleftΨk|Z/summationdisplay i=1δ(r−ri)|Ψk∝angbracketright =∝angbracketleftΨ(0) k|Z/summationdisplay i=1δ(r−ri)|Ψ(0) k∝angbracketright+/braceleftBigginterference terms/bracerightBigg +2π2 k/summationdisplay µνA∗ µ∝angbracketleftΦµ|/summationtextZ i=1δ(r−ri)|Φν∝angbracketrightAν (E−Eµ−i 2Γµ)(E−Eν+i 2Γν), (8) where Aνis the capture amplitude introduced as Γν c= 2π|Aν|2[cf. Eq. (6)]. The terms on the right-hand side correspond to the contributions of di rect annihilation, resonant an- nihilation (i.e., annihilation of the positron captured in to the positron-molecule quasibound state), and the interference between the two. 6B. Direct annihilation The direct annihilation term in Eq. (8) Z(dir) eff=∝angbracketleftΨ(0) k|Z/summationdisplay i=1δ(r−ri)|Ψ(0) k∝angbracketright (9) is a smooth function of the positron energy. Let us estimate i ts magnitude and find its energy dependence at small positron energies. When the posi tron is outside the atomic system, r > R a, the wave function Ψ(0) kcontains contributions of the incoming and scattered positron waves Ψ(0) k(r1, . . .,rZ,r) = Φ 0(r1, . . .,rZ)/bracketleftBigg eikr+f(Ω)eikr r/bracketrightBigg , (10) where f(Ω) is the scattering amplitude. Due to positron repulsion f rom the atomic nuclei the low-energy positron does not penetrate deep inside the a tomic system. Accordingly, the positron annihilates mostly with the outer valence elec trons, where the electron and positron densities overlap. This takes place “on the surfac e” of the atomic system, and Eq. (10) essentially determines the amplitude of finding the positron there. Of course, due to short-range electron-positron correlations the tru e wave function at small distances cannot be factorized similarly to Eq. (10). The Coulomb inte raction between the positron and electron increases the probability of finding both at the same point in space, as required by the δ-function in Eq. (3). This effect enhances the annihilation r ate [19]. However, since small distances and relatively large interactions ar e involved, these correlations do not depend on the momentum of the incoming positron at low energi es. On the other hand, to participate in the annihilation event the positron must firs t approach the target, and this is described by Eq. (10). Unlike the short-range correlation e ffects, the scattering amplitude can be very sensitive to the positron energy. This effect is fu lly accounted for by Eq. (10), and I use it to evaluate the energy dependence and magnitude o fZ(dir) eff. After substitution of expression (10) into Eq. (9) one obtai ns Z(dir) eff=/integraldisplay ρ(r)/bracketleftBigg eikr+f(Ω)eikr r/bracketrightBigg /bracketleftBigg e−ikr+f∗(Ω)e−ikr r/bracketrightBigg r2drdΩ, (11) where ρ(r)≡ ∝angbracketleftΦ0|/summationtextZ i=1δ(r−ri)|Φ0∝angbracketrightis the electron density in the ground state of the system. The electron density drops quickly outside the atom , and the positron density decreases rapidly inside the atom. Therefore the integrati on in Eq. (11) should be taken over a relatively thin shell of thickness δRaenclosing the atomic system. Let us approximate the integration domain by a spherical shell of radius r=Ra, where Rais the typical distance between the positron and the target during the annihilation , comparable to the size of the atom or molecule. For small positron momenta, kRa<1, Eq. (11) then yields Z(dir) eff= 4πρeδRa/parenleftbigg R2 a+σel 4π+ 2RaRef0/parenrightbigg , (12) where ρeis the electron density in the annihilation range (which can be enhanced due to short-range electron-positron correlations), σelis the elastic cross section, σel=/integraltext|f(Ω)|2dΩ, 7andf0is the spherically symmetric part of the scattering amplitu de,f0= (4π)−1/integraltextf(Ω)dΩ. For positron interaction with an atom the latter is simply eq ual to the s-wave scattering amplitude. Its real part is expressed in terms of the swave phase shift δ0as Re f0= sin 2δ0/2k. The swave gives a dominant contribution to the cross section σelat low projectile energies [28]. For k→0 it is determined by the scattering length a,σel= 4πa2, asf(Ω) =−a in this limit. A similar description is also valid for positr on scattering from a molecule at small momenta. Note that the relation between Z(dir) effand elastic scattering given by Eq. (12) could also be derived by matching the true many-body wave function of the p ositron-target system at low energy ( E≈0) with the asymptotic form (10). In this case Rawill be the matching radius, and the factor before the brackets will remain a free atomic- sized parameter. However, even in the form (12) the electron density ρeand the overlap δRaare effective parameters, and the accurate value of the pre-factor can only be found by compari son with numerical calculations (see Sec. IIIA). Nevertheless, Eq. (12) is very useful for th e analysis of direct annihilation. The three terms in brackets are due to the incoming positron p lane wave, the scattered wave, and the interference term, respectively, cf. Eqs. (10) and ( 11). Even if the cross section σelis zero or very small, as in the case of a Ramsauer-Townsend mi nimum, the annihilation rateZ(dir) effis nonzero. Its magnitude is determined by the effective anni hilation radius Ra, electron density ρeandδRa, which gives Z(dir) eff∼1–10, since the quantities involved have “normal”, atomic-size values. Equation (12) shows that the annihilation rate for slow posi trons is greatly enhanced if the scattering cross section is large. This occurs when the s cattering length is large, because the positron-target interaction supports a low-lying virt ualslevel ( a <0) or a weakly bound sstate ( a >0) [28]. Their energies, ε0=±1/2a2, respectively, must be much smaller than typical atomic energies, |ε0| ≪1 Ryd. For |a| ≫Rathe scattering cross section at low energies is much greater than the geometrical size of the tar get. This effect leads to strong enhancement of Z(dir) eff[17–19]. Theoretically, this gives a possibility of infinit ely large cross sections and annihilation rates at zero positron energy, if |a| → ∞ . However, for nonzero momenta the swave cross section does not exceed the unitarity limit σel= 4π/k2(for the swave). This fact puts a bound on the enhancement of Z(dir) eff. For example, for thermal positrons with k2/2∼kBTat room temperature ( k∼0.05 a.u.) we obtain Z(dir) eff∼103 from Eq. (12). Consequently, much higher values of Zeffcannot be produced by the direct annihilation mechanism. A more detailed discussion of this point and illustrations of the validity of Eq. (12) are presented in Sec. IIIA. C. Resonant annihilation Unlike the direct annihilation term, the interference and t he resonant terms on the right- hand side of Eq. (8) are rapidly varying functions of energy. The energy scale of this variation is given by the mean spacing Dbetween the resonances. If the resonances are due to vibrational excitations of a single mode of the positron- molecule complex then D=ω, withω<∼0.1 eV for a typical vibrational frequency. In a complex molecu le the positron attachment energy is sufficient for excitation of several mod es, and Dcan be much smaller. To describe the annihilation rates observed in experiments with non-monochromatic, e.g., 8thermal, positrons, one needs to average the interference a nd resonance terms over an energy interval ∆ Ewhich contains many resonances: 1 ∆E/integraldisplay ∆EdE 2/radicalBigg 2π2 kRe/summationdisplay ν∝angbracketleftΨ(0) k|/summationtextZ i=1δ(r−ri)|Φν∝angbracketrightAν E−Eν+i 2Γν +2π2 k/summationdisplay µνA∗ µ∝angbracketleftΦµ|/summationtextZ i=1δ(r−ri)|Φν∝angbracketrightAν (E−Eµ−i 2Γµ)(E−Eν+i 2Γν)/bracketrightBigg (13) Upon averaging the first, interference term vanishes. In the second, resonance term the diagonal items in the sum ( µ=ν) dominate. Averaging is then reduced to the integral over the Breit-Wigner resonant profiles. The number of reson ances within ∆ Eis ∆E/D. Therefore, the total annihilation rate is the sum of the dire ct and resonant contributions, Zeff=Z(dir) eff+Z(res) eff, (14) with the resonant contribution given by Z(res) eff=2π2 k/angbracketleftBiggρν epΓν c D[Γν a+ Γν c]/angbracketrightBigg , (15) where the angular brackets stand for averaging over the reso nances, and Γ ν= Γν a+ Γν c substituted for the total width. Below I will show that the re sonant term in Eq. (14) can be much greater than the direct one, and very high Zeffvalues can be achieved. It is easy to see that the resonant contribution could also be derived from standard resonant scattering theory developed originally to descri be neutron scattering via compound nucleus resonances ( [28], Ch. 18) . The maximal s-wave capture cross section is given by σ= πλ2≡πk−2. The true capture cross section is smaller than σ, because the capture takes place only when the positron energy matches the energy of the reson ance. For positrons with finite energy spread (e.g., thermal ones), the capture cross secti on is then σc∼(Γc/D)σ, where Dis the mean energy spacing between the resonances. More accu rately, σc= (2πΓc/D)σ [28]. If we are concerned with the annihilation process, the capture cross section must be multiplied by the probability of annihilation, Pa= Γ a/(Γc+ Γa), which gives the energy- averaged resonance annihilation cross section σa=2π2 k2ΓaΓc D(Γc+ Γa), (16) where averaging over resonances similar to that in Eq. (15) i s assumed. By comparison with Eqs. (2) and (7), the resonant contribution to Zeff, Eq. (15), is recovered. The way Eq. (15) has been derived implies that the positrons a re captured in the s wave. Otherwise, an additional factor of (2 l+1), where lis the positron orbital momentum, appears in the formula [28]. At low positron energies the cap ture widths behave as Γc∝(kR)2l+1(17) for resonances formed by positron capture with the orbital m omentum l[28] (Ris the typical radius of the target). So, the swave capture indeed dominates in the resonant annihilation 9of slow positrons. At higher energies contributions of seve ral lowest partial waves should be added in Z(res) eff. Let us estimate the rate of resonant annihilation and compar e it with the maximal direct contribution Z(dir) eff∼103for room-temperature positrons. The typical annihilation widths for positron-molecule (quasi)bound states are very small, Γν a∼1µeV (see Sec. IIA). If one assumes that the positron capture width is much greater, Γν c≫Γν a, (18) the total width Γ ν≈Γν ccancels the capture width in Eq. (15), and the resonant contr ibution is given by Z(res) eff=2π2 k/angbracketleftBiggρν ep D/angbracketrightBigg =2π2 kρepρ(Ev). (19) In the last equality I use the fact that electron-positron de grees of freedom are almost unaffected by the vibrational motion of the nuclei. Hence, fo r a given molecule the positron density on the target electrons ρepis the same for different vibrational resonances. I have also introduced the density of resonances ρ(Ev) =D−1, where Ev=εA+εis the vibrational excitation energy due to positron-molecule binding. Equat ion (19) shows that for Γ c>1 µeV the contribution of the resonant mechanism is independent of the capture width, and is determined by the density of positron-molecule resonant states populated by positron capture. Suppose that only a single mode with D∼0.1 eV is excited. Equation (19) then yields Z(res) eff∼4×103, if I use the estimates ρep=ρPs ep, and k= 0.05 for room-temperature positrons. The resonance spacing Dcannot be smaller than the widths of the resonances, which ar e limited by the annihilation width Γ a. Thus, one can obtain an upper estimate of the resonant annihilation rate from Eq. (15) by putting Γ c≈Γa∼0.5×10−7a.u., and D∼2πΓc, which gives the maximal possible capture cross section σ. These estimates yield Z(res) eff∼5×107 at room temperature (cf. Zeff= 7.5×106for C 12H2204[15]). This theoretical maximum of Z(res) effcorresponds to the unitarity limit of the swave capture cross section. However, this estimate of Zeffis not trivial. The resonance mechanism shows that such larg e cross sections can be achieved for the annihilation process, in spite of the fact that it is suppressed by the relativistic factor πr2 0c=π/c3∼10−6, in atomic units [see Eq. (2)]. Equation (19) predicts unusual low-energy threshold behav iourZ(res) eff∝1/k∝1/√ T (the latter for thermal positrons). In a standard situation the cross section of an inelastic process involving a slow projectile in the initial state beh aves as σ∝1/k. This dependence is characteristic of the swave scattering, which dominates at low projectile energie s, and is valid in the absence of long-range forces between the targ et and the projectile. It is known as the “1 /v” law, and its examples are numerous: from the ( n, γ) nuclear reaction to dissociative electron attachment to molecules, where it is observed below 1 meV [30]. Therefore, one would expect the positron annihilation cros s section to behave as σa∝1/k. Accordingly, Zeff, which is proportional the annihilation rate, is expected t o be constant at low positron energies. The anomalous threshold dependence of Eq. (19) clearly cont radicts this general state- ment. This “puzzle” is easily resolved if we recall conditio n (18) that has lead to Eq. (19). 10For very low positron momenta the s-wave capture width behaves as Γ c∝kR, so that (18) is clearly violated, and the resonant contribution in Eq. (1 4) becomes constant as k→0. However, at higher positron energies the 1 /kbehaviour of Zeffmay be observed. This de- pendence corresponds to the 1 /εdrop of the cross section which is reported in some electron attachment experiments (see, e.g., [31]). The fact that positron-molecule resonances give a large con tribution to the annihilation rate, as compared to the direct annihilation, does not mean t hat they also contribute much to the elastic scattering cross section. In analogy with Eq. (16), the resonant contribution to the elastic scattering is given by σ(res) el=2π2 k2Γ2 c D(Γc+ Γa), (20) and for Γ c≪Dit is much smaller than the direct, or potential, scattering cross section. III. ILLUSTRATIONS AND COMPARISON WITH EXPERIMENT A. Effect of virtual or weakly bound states on direct annihila tion If low-energy positron scattering is dominated by the prese nce of a virtual or weakly bound state at ε0=±κ2/2, the corresponding cross section has the form (for scatter ing by a short-range potential [28]) σel=4π κ2+k2, (21) where κ=a−1. According to Eq. (12) a similar maximum should appear in the momentum dependence of the annihilation rate. Its magnitude at k= 0 can be arbitrarily large if κ→0 (|a| → ∞ ), which corresponds to a level at zero energy. However, for n onzero momenta the maximal cross section is finite, σel∼4π/k2, which corresponds to the unitarity limit for the s-wave cross section. Real atomic and molecular targets have nonzero electric dip ole polarizabilities α, which give rise to the long-range polarization potential −α/2r4for the positron. Its effect is taken into account by the modified effective-range formula for the s-wave phase shift [32], tanδ0=−ak/bracketleftBigg 1−παk 3a−4αk2 3ln/parenleftbiggC 4√αk/parenrightbigg/bracketrightBigg−1 , (22) σel=4πa2 /bracketleftBig 1−(παk/3a)−(4αk2/3) ln/parenleftBig C 4√αk/parenrightBig/bracketrightBig2+a2k2, (23) the latter formula being valid when the scattering length is large and the s-wave scattering dominates at small k. In equations (22) and (23) Cis a dimensionless positive constant. Note that for α= 0, Eq. (21) is immediately recovered. The polarization pot ential modifies the behaviour of the cross section at low energies. For examp le, it leads to a more rapid decrease of the cross section for a <0,σel= 4πa2[1 + 2παk/3a+O(k2lnk)]. However, this 11does not change the estimates of the maximal values of Zeffthat could be produced in direct annihilation. To illustrate the relation between direct annihilation and elastic scattering, and the enhancement of both due to the presence of a low-lying virtua l level, let us compare the behaviour of Zeffandσelfor Ar and Kr. The results shown in Fig. 2 were obtained within the polarized-orbital method [33], which takes into accoun t the polarization of the target by the positron. These calculations yield large negative valu es of the scattering length for Ar, Kr and Xe (see Table I), indicating the presence of positron- atom virtual levels formed due to strong positron-atom attraction. The increase of |a|correlates with the increase of the dipole polarizability in these atoms. Similar values of ahave been obtained in the many- body theory calculations of Dzuba et al. [19]. Figure 2 shows that both σelandZeffare enhanced at low momenta due to the presence of the virtual slevels. This effect is stronger for Kr, which has a greater absolute value of the positron sca ttering length. As illustrated by Fig. 2a for Kr, Eq. (23) provides a good description of the c ross section at small k. The visible difference between Zeffandσelin Fig. 2 is due to the background given by the energy-independent term R2 ain Eq. (12). Figure 3 provides a direct comparison between Zeffand the right-hand side of Eq. (12), and shows that this relation is valid at low positron energie s. The comparison is based on the polarized-orbital method results for the noble-gas ato ms [33], and the values of Zeffand σelobtained for the ethylene molecule (C 2H4) by the Schwinger multichannel method [8]. In this comparison I have considered Raand the pre-factor 4 πρeδRain Eq. (12) as fitting parameters. Their values are listed in Table I together with the values of aobtained in those calculations. Note that the theoretical results used to pro duce this plot are not necessarily “exact” or accurate (although, experimental data confirm th at they are reasonable [7,20]). It follows from the derivation that Eq. (12) holds for any cal culation, as long as the same wave function is used in the scattering and annihilation cal culations [35]. In agreement with the estimates made in Sec. IIB, Fig. 3 shows that direct annihilation is indeed strongly enhanced by the presence of low-lying vir tual levels. Nevertheless, even for targets with very large scattering lengths, such as Xe or C2H4, the annihilation rates do not exceed Zeff∼103for room-temperature positron momenta (0.05 a.u.). Direct annihilation is the only annihilation mechanism for atoms and molecules which do not form bound states with positrons. It will also dominat e for small molecules which do form a weakly bound state with the positron, but whose vibrat ional frequencies are high. In this case the energy εA+εis simply insufficient for the excitation of the resonant quas ibound states at low impact positron energies ε. For large molecules the difference between the resonant and d irect mechanisms is proba- bly most obvious when one compares the experimental values o fZefffor alkanes and perflu- orinated alkanes shown in Fig. 1. The large annihilation rat es of the alkane molecules with more than two carbon atoms cannot be explained by direct anni hilation. They also display a very rapid increase with the size of the molecule, which is t ypical of resonant annihilation. On the other hand, the Zeffvalues of the perfluorinated alkanes remain comparatively s mall, in spite of their softer vibrational spectra. Thus, one is le ad to conclude that the resonant mechanism is switched off for them. The latter is explained by the very weak attraction be- tween the positron and fluorine atoms [7], insufficient to prov ide positron-molecule binding. Let us examine the effect of fluorination on Zefffor the lightest molecule of the series, 12methane. The experimental data at room temperature are: Zeff= 158 .5, 715, 411, 127, and 38, for CH 4, CH 3F, CH 2F2, CHF 3, and CF 4, respectively (data from [7,12] normalized to the given value for methane). These values are small enough t o be accounted for by the direct mechanism. Within its framework the increase and sub sequent drop of Zeffcould be explained by the existence of a loosely bound state for the po sitron on methane, which turns into a virtual level as the number of substitute fluorine atom s increases [36]. In terms of κparameter this would mean that κis small and positive for CH 4, and then goes through zero, and becomes negative upon fluorination. Accordingly, both the cross section and the annihilation rate peak for the molecule with the smallest ab solute value of κ, namely CH 3F. This picture is considered in Ref. [7] in more detail using th e zero-range potential model for positron-molecule interaction. Besides having a larger value of Zeff, the molecule with a smaller |κ|(i.e., larger |a|) should have a more rapid dependence of the annihilation rate on the positron energy, cf. Figure 3. If the experiment is done with thermal positrons th is should manifest in a stronger temperature dependence of the Maxwellian average of Zeff(k) Zeff(T) =/integraldisplay∞ 0e−k2/2kBT (2πkBT)3/2Zeff(k)4πk2dk (24) on the positron temperature T. The overbar is usually omitted, as it is clear from the conte xt whether one is dealing with Zeff(k) at a specific positron momentum, or with a thermal average Zeff(T). The temperature dependences of the annihilation rates fo r methane and fluoromethane measured in Ref. [7] are shown in Fig. 4. Also sh own are low-temperature theoretical fits obtained using Eqs. (12), (23) and (24). The ir parameters are given in the caption. The dipole polarizability of CH 3Fα= 16.1 a.u. is close to that of methane, α= 17.6 a.u., and I use the latter for both molecules. The constant Cappears in Eqs. (22) and (23) under the logarithm, and the result is not very sensitiv e to it, so C= 1 has been chosen. The value of the characteristic radius Ra= 4 a.u. is similar to those for noble gas atoms and ethylene (table I), and the pre-factor 4 πρeδRa= 1 is between those for noble gas atoms and C 2H4. Of course, the number of independent parameters ( a,C,Ra and 4πρeδRa) is too large to enable their unique determination from the e xperimental data. However, the fits clearly demonstrate that very different Zeff(T) curves can be obtained only due to different κvalues ( κ= 0.045 and 0.01, for CH 4and CH 3F, respectively. These values imply that both molecules have bound states with the positro n. The binding energy for CH4isεA=κ2/2 = 1.0×10−3a.u.= 0 .028 eV, and the binding energy corresponding to κ= 0.01 is just 1 meV. There is a large uncertainty in the latter val ue, because measurements performed at and above room temperature, T= 0.0253 eV, are not really sensitive to such small κ. This can be seen, e.g., from Eq. (21), which becomes κ-independent for κ≪k. Zero-range model calculations presented in Ref. [7] show th at the last three members of the fluoromethane sequence have negative κ, corresponding to virtual levels with increasing energies. This causes the decrease of their Zeffvalues. As seen in Fig. 4, equation (12) for the direct annihilation c ombined with the modified effective range formula (23) works well in the low-energy par t of the graph. However, the data for methane clearly show an abrupt departure from this l aw at higher T, and the formation of some kind of a plateau in Zeff(T). In principle, one could think that this is 13due to contributions of higher partial waves, not included i nσel, Eq. (23). However, their contribution has been included via the Raterm of Eq. (12). Also, the contributions of higher partial waves to Zeffemerge as εl, which is a manifestation of the Wigner threshold law [28]. For thermally averaged rates this corresponds to Tl. Thus, it cannot be responsible for this sudden feature. On the other hand, if the methane molecule forms a bound state with the positron the system can also have vibrationally excited positron-molec ule resonant states. The positron bound state on CH 4must belong to the A1symmetry type of the molecule. Since the positron swave dominates at low energies, its capture into the A1state can result in the excitation of A1vibrational modes of the molecule. The frequency of this mod e for methane isω= 2916 cm−1= 0.361 eV. Assuming that the positron binding does not change th is frequency much, the lowest vibrationally excited positron -molecule resonance will occur at ε=ω−εA≈0.33 eV. It is easy to estimate the contribution of a single narrow vib rational resonance located at positron energy ενto the thermally averaged Zeff[38], ∆Zeff(T) =8π3ρν epΓν c Γνa+ Γνce−εν/kBT (2πkBT)3/2≃8π3ρν epe−εν/kBT (2πkBT)3/2, (25) the latter formula valid for Γν c≫Γν a, which implies that the resonance has a capture width greater than 1 µeV. Figure 4 shows the effect of the lowest vibrational A1resonance at εν= 0.33 eV on Zefffor methane (chain curve). Its onset is indeed quite rapid, d ue to the exponent in Eq. (25), which makes ∆ Zeff(T) very small for kBT < ε ν. To fit the experimental data the density ρν epis chosen to be 25% of ρPs ep. One could expect that for a weakly bound state ( εA= 0.028 eV), where the positron spends most of its time outside th e molecule, its density on the electrons is reduced below that of Ps (binding energy 6.8 eV) [39]. B. Resonant annihilation: molecular vibrations and temper ature dependence 1. Vibrations. Equation (19) derived in Sec. IIC shows that the annihilatio n rate due to positron capture into resonances is determined by the level density o f these quasibound vibrationally excited states of the positron-molecule complex. This dens ity depends on the excitation energy available, as defined by the positron kinetic energy a nd positron affinity, Ev=εA+ε, and also on the structure of the molecular vibrational spect rum. Suppose that the molecule possesses a particular symmetry, which is true for most of th e molecules where positron annihilation has been studied so far [4]. The electronic gro und state wave function of the molecule is usually nondegenerate and invariant under all s ymmetry transformations. Let us call this symmetry type A. Depending on the actual symmetry of the molecule this can beA1,Ag, orA1g. If the positron can be bound by such molecule, the electron- positron part of the wave function of the positron-molecule complex will a lso be fully symmetric, i.e., of theAsymmetry type. Consider now the capture of a continuous spectrum positron i nto the bound positron- molecule state. At low positron energies this process is dom inated by the incident positron 14swave, higher partial waves being suppressed as ( kR)2l, compared to the swave [cf. Eq. (17)]. As a result, the electron-positron part of the wave fu nction of the initial (molecule and the s-wave positron) and final (bound positron-molecule complex ) states of the capture process are characterized by the same full molecular symmet ryA. This imposes a selection rule on the nuclear vibrations which can be excited during th e capture process. They must also belong to the Asymmetry type. Therefore, the selection rule limits the spectrum of possib le vibrationally excited reso- nances which could in principle be formed. It allows arbitra ry excitations and combinations of the Amodes. It also allows overtones and combinations of the othe r symmetry types, provided such excitations contain the Asymmetry type, i.e., the (symmetric) product of the symmetry types involved contains Aamong its irreducible representations [28]. This does not mean that all such vibrations will contribute to the dens ity factor ρ(Ev) in Eq. (19) for Zeff. Some of them may have extremely weak coupling to the electro n-positron degrees of freedom, with capture widths much smaller than 1 µeV. In this case they will be effectively decoupled from the positron capture channel, and hence, wil l not contribute to Zeff. Of course, this can only be found out by doing detailed calculat ions for specific molecules. Nevertheless, it is instructive to compare Eq. (19) with exp erimental data. This com- parison enables one to extract the effective mean spacing Dbetween the positron-molecule resonances. For experiments with thermal positrons Eq. (19 ) must be averaged over the Maxwellian positron momenta distribution, Z(res) eff=2π2ρep D/angbracketleftbigg1 k/angbracketrightbigg T=2π2ρep D/parenleftbigg2 πkBT/parenrightbigg1/2 . (26) Let us use the Ps value, ρep= 1/8π, to estimate the electron-positron density, and apply Eq. (26) to simple symmetric molecules with Zeff>∼104, where resonant annihilation must be the dominant mechanism. The effective spacings D= 4.51×106/Zeff(in cm−1) obtained from the experimental Zeffvalues measured with room-temperature positrons [4] are li sted in Table II. They are compared with the low frequency vibrational mod es of the Asymmetry type of these molecules taken from Ref. [37]. As discussed above, vi brations of the Asymmetry type also occur in overtones and combinations of other modes. How ever, their frequencies scale with the size and chemical composition of the molecule in a wa y similar to the Amodes, and the Amode frequencies listed in the table are representative of t he lower vibrational modes on the whole. For molecules with moderate Zeffat the top of the table, such as CCl 4, the effective resonance spacing Dis comparable to the frequencies of single modes. With the in crease of the size of the molecule (alkanes), or masses of the constitu ents (e.g., CBr 4), the vibrational modes are softened, and the number of low-frequency modes in creases. At the same time one can expect that the positron binding energy increases fo r these molecules. These effects, and especially the increase of the number of modes, facilita te multimode excitations, whose density is much greater that the level density of the individ ual modes. Accordingly, we see thatDbecomes much smaller that the frequencies of the individual modes at the bottom of the table. In the simplest model this effect can be estimated as follows. Suppose the vibrational modes in question are characterized by some typical frequen cyω, and the molecule has Nv such modes. Suppose, the positron binding energy is εA=nω, where nis the number of 15vibrational quanta excited due to positron binding. If we ne glect the small kinetic energy of the positron, Ev≈εA, the total number of various vibrational excitations at ene rgyEv is given by ( Nv+n−1)!/[n!(Nv−1)!] (number of ways to distribute nvibrational quanta among Nvmodes). For large molecules εAremains finite, whereas Nvincreases linearly with the size of the molecule, the total number of vibrational mod es being 3 N−6, where Nis the number of atoms. Therefore, the number of vibrational ex citations available, and the density of the resonant vibrational spectrum, increase as ( Nv)n∝Nn. Such rapid increase is indeed observed for alkanes and aromatic hydrocarbons, see Fig. 1. The effective number of vibrational modes excited in the capture process, n= 6.1 and 8.2, respectively, is compatible with the positron binding energy of few tens of an electron Vo lt. For example, if I use the lowest Agmode frequency of hexane (Table II), the positron affinity is εA∼6ω≈0.25 eV. This number looks reasonable, compared with positron bindi ng energies on single atoms, e.g.,εA= 0.08, 0.15, and 0.38, for Be, Cu and Mg, respectively [27,39]. Apart from the rapid growth, Zefffor alkanes shows clear signs of saturation, when the number of carbon atoms becomes greater than 8 or 10. Apparent ly, this takes place well before the unitarity limit derived in Sec. IIC is reached. Th is behaviour can be understood if we recall that Eq. (19) is valid only when the capture width Γcis greater than the annihilation width Γ a. With the increase of the number of vibrational modes their c oupling to the electron-positron degrees of freedom decreases. Thi s coupling is represented by Γ c, and for small capture widths, Γ c<Γa,Z(res) efffrom Eq. (15) is estimated as Z(res) eff≃2π2 k/angbracketleftBiggρν epΓν c DΓν a/angbracketrightBigg =2πc3 k/angbracketleftbiggΓν c D/angbracketrightbigg , (27) where Eq. (7) is used together with r0=c−2, in atomic units. The decrease of Γ cis a simple consequence of sum rules, because the total strength of posi tron coupling is distributed among larger number of possible vibrational excitations. I n this regime Γ cis proportional to D, and the increase of Z(res) effrelated to the increase of the density of vibrational excita tion spectrum stops. The relation Γ c∝Dwhich characterizes this regime is well known in neutron capture into compound resonances [40]. It takes pla ce in complex atomic spectra, e.g., in rare-earths, where the oscillator strengths are di stributed among very large numbers of transitions [41]. It also emerges in the unimolecular rea ction treatment of dissociative electron attachment [21], where it is responsible for very l arge lifetimes (i.e., small state widths) of transient molecular anions. 2. Dependence on the positron energy or temperature. Let us now look at the energy dependence of the resonant annih ilation rate. At very small positron energies Z(res) effmust be constant (see discussion at the end of Sec. IIC). Howe ver, as soon as the s-wave capture width becomes greater that 1 µeV, the corresponding annihilation rate shows a 1 /k∼ε−1/2dependence on positron energy, as predicted by Eq. (19). For a thermally averaged rate this is described by Eq. (26). Figu re 5 presents a comparison between the 1 /√ Tlaw and the experimental temperature dependence of Zefffor C 4H10[7]. This molecule has Zeff∼104. Within the present theoretical framework this large value must be due to the resonant annihilation process. 16The theory and experiment agree well at low temperatures. On e may notice that the measured Zeffshow a slightly steeper rise towards small T. However, the difference is not large, both in relative and absolute terms. It could be expla ined by a direct contribution Z(dir) effin Eq. (14), which peaks sharply at small energies, if the pos itron-molecule scattering length is large (see Sec. IIIA). In spite of the dominance of t he resonant contribution, Z(res) eff∼104for butane, the addition of Z(dir) eff∼103at small positron energies would still be noticeable. A more pronounced feature of the experimental data, which is not accounted for by Eq. (26), is the plateau observed at higher temperatures, T >0.05 eV, where Zeffgoes well above the 1/√ Tcurve. To find its possible origins let us first take a closer lo ok at Eq. (26) and its predecessor, Eq. (19). For small impact positron energies εthe vibrational excitation energy is given by Ev≈εA. Accordingly, the resonance density ρ(Ev) in Eq. (19), and the mean spacing Din Eq. (26) are approximately constant. As the positron ener gy, or temperature, increase, the resonance density factor should also increas e, since ρ(Ev) is a strong function of the excitation energy for multimode vibrational spectra . Therefore, the decrease of Z(res) eff should be slower than 1 /k, or 1/√ T. Moreover, the density factor may even produce a rise in the energy dependence of Z(res) eff. Besides this, contributions of higher positron partial waves which emerge as T,T2, etc., at small T, may also contribute to Z(res) effin the plateau region. It might even seem that these effects could lead to a ra pid increase of Z(res) effwith positron energy. However, there is an effect that suppresses the increase of re sonant annihilation. Throughout the paper I have assumed that the positron-molec ule resonances have only two decay channels, annihilation and detachment, the latte r being the reverse of positron capture. When the positron energy rises above the threshold of molecular vibrational exci- tations, the resonances can also decay into the ‘positron + v ibrationally excited molecule’ channels. In this situation the total width of a resonance wi ll be given by Γ ν= Γν a+Γν c+Γν v, where Γν vis the decay width due to positron detachment accompanied by the vibrational excitation of the molecule. This leads to a modification of Eq . (19), which now reads Z(eff) eff=2π2ρep kρ(Ev)/angbracketleftBiggΓν c Γν c+ Γν v/angbracketrightBigg . (28) This equation shows that as soon as the positron energy excee ds another inelastic vibrational- excitation threshold, the factor in brackets drops, thereb y reducing the resonant annihilation contribution. Such downward step-like structures at vibra tional thresholds are well known in dissociative electron attachment experiments (see, e.g ., Refs. [30,42]). When the positron energy is well above the lowest inelastic vibrational thres hold the “elastic” width Γ cwill become much smaller than the “inelastic” width Γ v, due to a large number of open inelastic vibrational-excitation scattering channels, and due to a k inematic increase of Γ vabove the respective thresholds. This will strongly suppress the res onant annihilation contribution (28) with respect to that of Eq. (19) at larger positron energ ies. One may speculate that it is precisely the increase of Γν vthat counteracts the rise of ρ(Ev), and prevents rapid growth ofZ(eff) effwith positron energies. It may also be true that a similar mec hanisms is behind the dramatic drop of the dissociative attachment cross section s for projectile energies above few lower vibrationally inelastic thresholds [21,30]. 17IV. SUMMARY AND OUTLOOK In this work I have considered two possible mechanisms of low -energy positron annihila- tion in binary collisions with molecules. The first mechanisms is direct annihilation. It describes po sitron annihilation with atoms and small molecules, as well as molecules which do not form bo und states with the positron. The annihilation rate due to this mechanism has been related to the positron elastic scatter- ing properties. In particular, it is enhanced when the posit ron has a low-lying virtual s-type level or a weakly bound state at ε0=±κ2/2. For zero-energy positrons the direct annihi- lation rate is inversely proportional to |ε0|. Small κ, together with the dipole polarizability of the target, also determine the rapid energy dependence of Zeffat small positron energies. Estimates show that for room-temperature positrons Zeffof up to 103can be produced due the virtual/weakly bound state enhancement. The second mechanism is resonant annihilation. It is operat ional when the positron forms temporary bound states with the molecule. As a necessary con dition, the positron affinity of the molecule must be positive. The positron capture is a re sonant process, whereby the energy of the positron is transferred into vibrational exci tations of the positron-molecule complex. The contribution of this mechanisms to the annihil ation rate is proportional to the level density of the positron-molecule resonances ρ. These resonances are characterized by the capture width Γ cand annihilation width Γ a∼1µeV. For Γ c>Γaits contribution is independent of Γ c, and is basically determined by the density ρ. The resonant mechanism can give very large annihilation rates (up to 108). Through its dependence on the vibrational excitation spectrum of the positron-molecule complex, thi s mechanism shows high sensitivity to the chemical composition of the target, and the size of the molecule. Both are essential features of the experimental data [4]. The difference between the two mechanisms is illustrated mos t clearly by compari- son of the annihilation rates of alkanes and perfluoroalkane s. For example, C 6H14has Zeff= 120 000, whereas for C 6F14,Zeffis only 630. The present theory attributes this huge difference to the fact that perfluorocarbons do not form bound states with the positrons, and hence, the resonant annihilation is switched off for them . On the other hand, this mechanism is behind the the high Zeffvalues of alkanes. The experimental group at San Diego has performed a number of measurements on pro- tonated and deuterated molecules to test the sensitivity of Zeffto the molecular vibrational modes [4,7]. For example, their data for benzene show that a r eplacement of a single hy- drogen atom with deuterium changes the annihilation rate fr omZeff= 15 000 for C 6H6to Zeff= 36 900 for C 6H5D. On the other hand, the data on fully protonated vs fully deu ter- ated alkanes shows very little difference between the two cas es. Such behaviour is natural for smaller alkanes, e.g., methane, where direct annihilat ion is the dominant mechanism. However, observed for large alkanes, it cannot be readily in terpreted by means of Eq. (19) or alike. It is possible that the vibrational excitations ar e dominated by low-lying C −C modes which are weakly affected by deuteration. On the other h and, deuteration may also influence positron coupling to the molecular vibrations, wh ich will most likely lead to a reduction of Γ cin Eq. (15). If the system is in the regime where Γ c∼Γa, this effect may offset the decrease of the vibrational spacings. In spite of these difficulties, which could only be resolved by doing calculations for specific 18molecules, the present theory offers a consistent descripti on of positron-molecule annihila- tion in real terms, through some well defined parameters whic h characterize the system. It clearly identifies the two basic mechanisms of positron an nihilation and discusses their specific features. It also shows that studies of positron ann ihilation on molecules may give a unique insight into the physics of molecular reactions whic h go through formation of vibra- tionally excited intermediate states. Such processes are v ery likely to be responsible for large dissociative electron attachment cross sections observed for molecules such as SF 6. They are also of key importance for the whole class of chemical reacti ons, namely, for unimolecular reactions (see, e.g., [43]). ACKNOWLEDGMENTS This work was strongly stimulated by the vast experimental d ata of the San Diego group, and I very much appreciate numerous discussions with its mem bers, especially C. Surko and K. Iwata. I am thankful to my colleagues at the University of New South Wales, V. Flambaum, A. Gribakina, M. Kuchiev, and O. Sushkov for their encouragement and useful discussions. My thanks also go to S. Buckman for the informat ion on vibrational excitations and dissociative attachment. 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P. Lima, Phys. Rev . Lett. 77, 1028 (1996). TheAgsymmetry (“ swave”) dominates in both σelandZeffat low positron energies, and I use σ≈4πsin2δ0/k2to extract the swave phase shift and amplitude f0necessary for implementation of Eq. (12). [9] G. Laricchia and C. Wilkin, Phys. Rev. Lett. 79, 2241 (1997). [10] In this paper the word ‘molecules’ is often used interch angeably with ‘atoms’, or ‘atoms or molecules’. However, there are specific phenomena, e.g., low-energy positron capture (Sec. IIC), which involve vibrational degrees of freedom, a nd hence, apply to molecules only. [11] A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Interscience Publish- ers, New York, 1965). [12] Koji Iwata, Positron Annihilation on Atoms and Molecules , PhD dissertation (Univer- sity of California, San Diego, 1997). [13] J. W. Humberston and J. B. G. Wallace, J. Phys. B 5, 1138 (1972). [14] T. J. Murphy and C. M. Surko, J. Phys. B 23, L727 (1990). [15] M. Leventhal, A. Passner, and C. 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[25] The energy of positrons at room temperature, kBT≈25 meV, is too small to excite any electronic degrees of freedom. The formation of Ps is als o impossible if we consider molecules with ionization potentials greater than the Ps bi nding energy of 6.8 eV. [26] V. A. Dzuba, V. V. Flambaum, G. F. Gribakin, and W. A. King , Phys. Rev. A 52, 4541 (1995). [27] G. G. Ryzhikh and J. Mitroy, Phys. Rev. Lett. 79, 4124 (1997). J. Phys. B 31, L401 (1998); 31, 4459 (1998); 31, 5013 (1998); 32, 1375 (1999); G. G. Ryzhikh, J. Mitroy, and K. Varga, J. Phys. B 31, 3965 (1998). [28] L. D. Landau and E. M. Lifshitz, Quantum mechanics , 3rd. ed. (Pergamon Press, Ox- ford, UK, 1977). [29] At low positron energy annihilation and emission are th e only decay channels of the resonances. At higher energies positron emission accompan ied by the vibrational (and then electronic) excitation of the molecule becomes possib le, see Sec. IIIB. [30] D. Klar, M.-W. Ruf, and H. Hotop, Aust. J. Phys. 45263 (1992). [31] A. Kiendler, S. Matejcik, J. D. Skalny, A. Stamatovic, a nd T. D. M¨ ark, J. Phys. B 29 6217 (1996). [32] T. F. O’Malley, L. Spruch, and L. Rosenberg, J. Math. Phy s.2, 491 (1961). [33] R. P. McEachran, D. L. Morgan, A. G. Ryman, and A. D. Stauff er, J. Phys. B 10, 663 (1977); 11, 951 (1978); R. P. McEachran, A. G. Ryman, and A. D. Stauffer, J . Phys. B11, 551 (1978); 12, 1031 (1979); R. P. McEachran, A. D. Stauffer, and L. E. M. Campbell, ibid13, 1281 (1980). [34] A. A. Radtsig and B. M. Smirnov, Parameters of Atoms and Atomic Ions: Handbook (Energoatomizdat, Moscow, 1986). [35] The most accurate theoretical data on positron scatter ing and annihilation are available for hydrogen. To avoid cluttering the plot the correspondin g results are not presented in Fig. 3. For hydrogen Eq. (12) provides a good fit of Z(dir) eff[J. W. Humberston, Adv. At. Mol. Phys. 15, 101 (1979)] with Ra= 3.2 a.u., 4 πρeδRa= 0.344 a.u., when variational scattering phase shifts are used to obtain σelandf0[A. K. Bhatia et al., Phys. Rev. A 3, 1328 (1971); 9, 219 (1974); D. Register and R. T. Poe, Phys. Lett. A 51, 431 (1975)]. [36] Experimental data for a variety of molecules show that a non-zero dipole moment of the molecule does not have any direct effect on Zeff, Ref. [12]. For example, Zeff= 319, 1090, and 1600, for H 20, NO 2, and NH 3, whereas their dipole moments are 1.85, 0.32, and 1.47 Debye, respectively. There also seems to be no corre lation between Zeffvalues of CH 3F, CH 2F2, CHF 3, quoted in the text, and their respective dipole moments, 1. 85, 1.97, and 1.65 Debye. [37] L. M. Sverdlov, M. A. Kovner, and E. P. Krainov, Vibrational Spectra of Polyatomic Molecules (John Wiley & Sons, New York, 1974). [38] The contribution of a particular resonance to Zeffis given by the Breit-Wigner for- mula, cf. terms with µ=νin Eq. (8). For the purpose of thermal averaging, the contribution of the νth resonance can be approximated by the δ-function, ∆ Zeff= (2π2/k)ρν ep(Γν c/Γν)δ(E−Eν), without any loss of accuracy. [39] Numerical calculations of positron-atom bound states show that for atoms with ioniza- 21tion potentials greater than 6.8 eV and positron affinities in the range (1–5) ×10−3a.u. (Zn, Be, Cu, and Ag) the annihilation rates are 20–30% that of Ps, see G. Ryzhikh and J. Mitroy, J. Phys. B 31, 5013 (1998); J. Mitroy and G. Ryzhikh, J. Phys. B 32, 1375 (1999). [40] A. Bohr and B. Mottelson, Nuclear structure, Vol. 1 (Benjamin, New York, 1969). [41] I. I. Sobelman, Atomic Spectra and Radiative Transitions (Springer, Berlin, 1992). [42] H. Hotop, D. Klar, J. Kreil, M.-W. Ruf, A. Schramm, and J. M. Weber, in The Physics of Electronic and Atomic Collisions, edited by L. J. Dub´ e et al.(AIP, New York, 1995), p. 267. [43] R. G. Gilbert and S. C. Smith, Theory of Unimolecular and Recombination Reactions (Blackwell Scientific, Boston, 1990). 22TABLES TABLE I. Scattering lengths and fitting parameters for the re lation between Z(dir) effandσel, Eq. (12) Atom or a R a 4πρeδRa molecule (a.u.) (a.u.) (a.u.) He −0.52a3.9 0.21 Ne −0.61a5.0 0.23 Ar −5.30a4.3 0.42 Kr −10.4a4.2 0.41 Xe −45.3a4.2 0.41 C2H4 −18.5b4.4 3.0 aCalculated in Ref. [33]. bObtained from the calculations of da Silva et al. [8]. TABLE II. Annihilation rates and vibrational frequencies o f molecules Molecule Formula ZeffaDb(cm−1) Symmetry Frequenciesc(cm−1) Carbon tetrachloride CCl 4 9 530 473 A1 459 Butane C 4H10 11 300 399 Ag 429, 837, 1057, ... Cyclohexane C 6H12 20 000 226 A1g 384, 802, 1158, ... Pentane C 5H12 37 800 119 A1 179, 401, 863, .. . Carbon tetrabromide CBr 4 39 800 113 A1 269 Hexacloroethane C 2Cl6 68 600 65.7 A1g 164, 431, 976 Hexane C 6H14 120 000 37.6 Ag 305, 371, 901, .. . Heptane C 7H16 242 000 18.6 − − aExperimental values obtained for room-temperature positr ons in the trap, Ref. [4]. bEffective spacing for the resonances in Z(res) eff, Eq. (26), corresponding to experimental data. cLowest molecular vibrational frequencies of the given symm etry from Ref. [37]. 23FIGURES 5 10 50101001000 10 100101001000 FIG. 1. Annihilation rates Zefffor alkanes, C nH2n+2(solid circles, n= 1–10, 12, and 16), perfluorinated alkanes, C nF2n+2(solid squares, n= 1–3, 6, and 8) and aromatic hydrocarbons, benzene, naphthalene, and antracene, C nHn/2+3(open hexagons, n= 6, 10, and 14), as functions of the number of electrons in the molecule Z(a), and number of atoms N(b). Data are taken from Ref. [12], Tables B1, 4.3, 4.9 and 4.11 (see also [4]). Al so shown are power-law fits for alkanes, Zeff∝N6.1(solid line), perfluorinated alkanes, Zeff∝N1.75(dashed line), and aromatic hydrocarbons, Zeff∝N8.2(dot-dashed line). 240 0.05 0.1 0.15 0.2 0.2505001000 0 0.05 0.1 0.15 0.2 0.25020406080 FIG. 2. Elastic scattering cross section σel(a) and annihilation rates Zeff(b) for Ar (dashed curves) and Kr (solid curves), as calculated in Ref. [33]. Al so shown in (a) are the analytical approximations of σelfor Kr by the short-range potential formula (21) (dotted lin e with crosses), and the modified effective range formula (23), which accounts for the dipole polarization of the target (open circles). Here I have used the calculated value ofa=−10.4 a.u., experimental dipole polarizability α= 16.74 a.u. [34], and C= 0.4 obtained from the s-wave phase shift of Ref. [33]. Note that the modified effective range formula (open circles) gives an accurate description of the cross section shown by the solid curve. 250 0.05 0.1 0.15 0.2 0.25101001000 FIG. 3. Relation between Zeffdue to direct annihilation and the elastic scattering cross section. Calculated Zeffvalues for He (open triangles), Ne (solid triangles), Ar (op en squares), Kr (solid squares), and Xe (solid circles) [33], and C 2H4(open circles) [8] are compared with the predictions of Eq. (12), shown by solid curves. In the latter I have used th e scattering cross sections and amplitudes calculated in the same theoretical papers, and c onsidered Raand the pre-factor 4 πρeδRa as fitting parameters. 260.02 0.04 0.06 0.08 0.1 0.2 0.4501005001000 FIG. 4. Annihilation rates for methane and fluoromethane. Ex perimental data for CH 4(solid circles) and CH 3F (open circles) [7] have been normalized to Zeff= 158 .5 for methane at room temperature. Thermal-averaged direct annihilation fits ob tained from Eqs. (12) and (23) using 4πρeδRa= 1,Ra= 4,C= 1,α= 17.6 a.u., are shown for CH 4(κ= 0.045, solid curve), and CH3F (κ= 0.01, dashed curve). Also shown for methane is the sum of the dir ect contribution and that of the first vibrational A1resonance at εν= 0.33 eV, obtained using ρν ep= 0.25ρPs ep, Eq. (25) (chain curve). 27FIG. 5. Dependence of Zeffon positron temperature for butane, C 4H10. Solid circles, exper- imental data [7], normalized at room temperature to Zeff= 11300 [4]. Solid curve is the 1 /√ T dependence, Eq. (26), with ρep=ρPs ep, and effective resonance spacing D= 1.90×10−3a.u.= 417 cm−1. 28

arXiv:physics/9912024v1 [physics.ins-det] 10 Dec 1999COMMENT ON A TONOMURA EXPERIMENT : LOCALITY OF THE VECTOR POTENTIAL OLIVIER COSTA DE BEAUREGARD AND GEORGES LOCHAK∗ Abstract Three predictions for additional tests in a Tonomura experi ment: 1,2: The Fresnel frin-ges displayed outside and inside the geome tric shadow of a toroidal magnet should subsist intact, the ones if the othe rs are masked, and vice versa ; 3 : Placing the registering film just before th e magnet and thus uncovering the entire fringe pattern should display th e curved fringes connecting the outer and inner straight ones. Physicality o f the vector po- tential expressed in the source adhering gauge will thus be u nequivocally proved. 1 An unconventional testable claim De Broglie has tersely stated [1] that his [2] universal form ula Pi≡µUi−eAi=/planckover2pi1ki(1) relating the canonical 4-momentum Piof a point charge of charge −e, rest mass µ, 4-velocity Ui, to the 4-frequency kiof the associated wave, selects uniquely the electromagnetic gauge. The point is : In absenc e of external elec- tromagnetic sources, adding to the 4-potential Aian arbitrary 4-gradient would entail indefiniteness of the 4-frequency -a cardiac arythmy of the electron, so to speak. This is not observed -and is denied by crystal diffract ion unequivocally displaying (in standard notation) the formula p≡mv=/planckover2pi1k (2) What then of gauge invariance of the Dirac equation? Adding t o the canonical 4-momentum operator i/planckover2pi1∂i−eAian arbitrary 4-gradient can be compensated by substracting this same 4-gradient from the wave function ’s phase. All right -this is like cashing a cheque. An invariance law of a differer ential equation need not subsist in its solutions, which imply integration c onditions. What de ∗Fondation LOUIS DE BROGLIE 23 rue MARSOULAN 75012 PARIS 1Broglie means is that in the expression of a free electron’s 4 -momentum the 4-potential is identically zero : Ai≡0 in absence of electromagnetic sources. This is unquestionable. Corollary 1 Any sort of electron interference experiment performed in p res- ence of a toroidal magnet displays the curlless vector poten tialA(r)as expressed in the source adhering gauge -this being tantamount to a meas urement of the vector potential. So we claim (a big step forward !) that : A locally observable e ffect underlies the A.B. effect. 2 Proof via a Tonomura experiment Tonomura [3] has combined an ’electron biprism inter-feren ce’ with an Aharonov- Bohm one. A very perfect toroidal magnet of trapped flux Φ quan tized in h/2e units [4] placed downstream of a ’biprism’ has its axis z orth ogonal to the planes displaying ’normally’ the Fresnel fringes. A registering fi lm, placed ’normally’ after the magnet, displays outside and inside its circular s hadow straight Fres- nel fringes which are either identical to each other or black -to-white exchanged, depending on the flux Φ being an even or an odd multiple of h/2e. The fact is that the magnet’s shadow, now termed the black ring, is ’ge ometric’ style showing no circular fringes, and thus no explicit A.B. effect. This precludes any observable interference between the external and inter nal fringes -which can be tested. Obturating the inside of the black ring should not affect in an y way the ex- ternal fringes which, displayed as parallel straight lines , are identical to those existing in absence of the magnet -because the magnet’s influ ence is asymptoti- cally zero. Slipping transversally the magnet out of the pic ture will just uncover the genuine Fresnel fringes. Similarly, for the reasons sta ted, obturating the out- side of the black ring should not affect in any way the internal fringes -a very crucial test of locality ! How the outside and inside fringes are linked together is hid den by the black ring. Between them exists a phase shift amounting to a multip le ofh/e, due to addition of −eAto the two kinetic momenta mvcombined with obliquity of the two interfering kvectors vizz the axis z. The hidden curved fringes can be recovered by placing the registering film just before the mag net, thus wiping off the black ring. The fringe pattern displayed will not be the g enuine Fresnel one, but one where curved fringes now connect the external and int ernal straight Tonomura ones -a very strong proof of local physicality of Aahead of electron impact ! 23 Direct mesurement of the vector potential If the vector potential Ais a locally measurable magnitude a precise measure- ment of a curlless vector potential is possible. Rather than a spatially extended ’electron biprism’ one should then use as interference gene rator a small diffract- ing crystal. Performed inside the curlless vector potential A(r) generated by a toroidal mag- net, crystal diffraction will evidence, instead of formula ( 2), the formula /planckover2pi1k=mv−eA (3) yielding a measurement of Aexpressed in the source adhering gauge. The maximal and neater effect will obtain if the magnet’s cent er coincides with that of the crystal and its axis with that of the gun. Then turn ing the magnet around its center will modify the intensity along the circul ar rings. 4 Resurrection of the potentials ’assassinated’ by Heaviside Electromagnetic gauge invar-iance states : Forces, linear or angular, depend on the fields, not the potentials. All right, this is very true. But the integrals of forces -over space, energies, or over ti me, momenta (linear or angular ; the 6-component angular momentum including the boost) do de- pend on the potentials. As interaction energies and linear o r angular momenta of bound systems are measurable mag-nitudes the attached po tentials also are -with expressions selected as integration conditions [4]. This is well known but underestimated in the case of Einstein ’s energy-mass equivalence : the electrostatic mass defect of a bound syste m is part of its total mass. By relativistic covariance there follows [4] that action-r eaction (linear or an- gular) also selects the source adhering gauge as an integrat ion condition ; an example is afforded by the Wheeler-Feynman electrodynamics . Electromagnetically induced inertia thus emerges as a gene ral concept to be discussed elswhere. De Broglie [1] [5] has stated that both the Einstein W=c2menergy-mass and the Planck W=hνenergy-frequency equivalences select the electromagneti c gauge ; covariant expressions of these statements have been produced [5]. References [1] L. de Broglie, Optique Electronique et Corpusculaire (Hermann, Paris, 1950) p. 45-49. [2] L. de Broglie, Annales de Physique 3 (1925) 22 ; see p. 55-5 6. [3] A. Tonomura, Ann. New York Acad. Sci. 225 (1995) 227. 3[4] O. Costa de Beauregard and J.M. Vigoureux, Phys. Rev. D 9 ( 1974) 1626. [5] O. Costa de Beauregard in Advanced Electrodynamics, T. W . Barrett and D. M. Grimes eds (World Scientific, Singapore, 1995) p. 77-10 4 ; Physics Essays 10 (1997) 492 and 646 ; Ann. Fond. L. de Broglie 23 (1998 ) 135. [6] L. de Broglie, C. R. Ac. Sci. 225 (1947) 163. 4

1A ‘New Formula’ to Provide the Compatibility between the Special Theory of Relativity (STR), Black Holes and Strings Ali Riza AKCAY TUBITAK-UEKAE P.K. 21, 41470 - Gebze Kocaeli-TURKEY E-mail: aakcay@yunus.mam.gov.tr Abstract This paper describes a ‘New Formula’ in place of Einstein’s Famous Formula (EFF) to provide the compatibility between the Special Theory of Relativity (STR), black holes and strings. The ‘New Formula’ can also predict and describes the space-time singularities without the distribution of mass and energy. According to the ‘New Formula’, any particle can reach to and exceed the speed of light ( )cv≥. The EFF ( )2mcE= is only valid and applicable in the vacuum (the mediums which have low current density: outside the string, outside black hole), but is not valid and applicable for inside string and inside black hole including space-time singularities. However, the ‘New Formula’ is valid and applicable in all mediums including inside string and inside black hole. Keywords: ‘New Formula’, Einstein’s Famous Formula (EFF), Black Holes, Superconductivity, Superconducting Strings, Space-time Singularities. 1. Introduction As known, the Einstein’s Famous Formula [ ] Emc= − 02 21b is the general result of the Special Theory of Relativity (STR). According to this formula ; the energy2()E approaches infinity as the velocity v approaches the velocity of light ( c). The velocity (or speed) must therefore always remain less than c. This means that it is not allowed by the STR to travel faster than light [1]. A black hole is a region of space from which it is impossible to escape if one is traveling at less than the speed of light. But the Feynman sum over histories says that particles can take any path through space-time. Thus it is possible for a particle to travel faster than light [2]. What is meant was that matter could curve a region in on itself so much that it would effectively cut itself off from the rest of the Universe . The region would become what is called a black hole. Objects could fall into the black hole, but noting could escape. To get out, they would need to travel faster than the speed of light, which is not allowed by the theory of relativity. Thus the matter inside the black hole would be trapped and would collapse to some unknown state of very high density. Einstein was deeply disturbed by the implications of this collapse, and he refused to believe that it happened. But, Robert Oppenheimer showed in 1939 that an old star of more than twice the mass the sun would inevitably collapse when it had exhausted all its nuclear fuel [2]. The fact that Einstein’s general theory of relativity turned out to predict singularities led to a crisis in physics. The equations of general relativity, which relate to curvature of space-time with the distribution of mass and energy, cannot be defined as a singularity. This means that general relativity cannot predict what comes out of a singularity. In particular, general relativity cannot predict how the universe should begin at the big bang. Thus, general relativity is not a complete theory. It needs an3added ingredient in order to determine how the universe should begin and what should happen when matter collapses under its own gravity [2]. Stars which collapse into black holes generally posses a magnetic field. In addition, black holes swallow electrically charged particles from the interstellar medium such as electrons and protons. It is therefore reasonable to expect black holes to have electromagnetic properties. H. Reissner in 1916, and independently G. Nordstrom in 1918, discovered an exact solution to Einstein’s equations for the gravitational field caused by an electrically charged mass. This solution is generalized version of Schwarzschild’s solution, with one other parameter: the electric charge. It describes space-time outside the event horizon of an electrically charged black hole [3]. The ‘New Formula’ can describe the electrically charged black holes by using the current density as a parameter. The most remarkable discovery including a semi-classical gravitational effect is the Hawking radiation, which is concluded by treating matter fields on space-time as quantum while a black hole metric as classical. According to this theory, a black hole radiates particle flux of a thermal spectrum, whose temperature is pk2 where kis the surface gravity [4]. The existence of Hawking radiation is closely related to the fact that a particle which marginally escapes from collapsing into a black hole is suffered from infinite redshift. In other words, the particle observed in the future infinity had a very high frequency when it was near the event horizon. Gamma-ray bursts (GRBs) appear as the brightest transient phenomena in the Universe. The nature of the central engine in GRBs is a missing link in the theory of fireballs to their stellar mass progenitors. It is shown that rotating black holes produce4electron-positron outflow when brought into contact with a strong magnetic field. The outflow is produced by a coupling of the spin of the black hole to the orbit of the particles. For a nearly extreme Kerr black hole, particle outflow from an initial state of electrostatic equilibrium has a normalized isotropic emission of ()( )q2 2 2 48sin7 105~ O cMMBB× erg/s, where B is the external magnetic field strength, ,104.413G Bc×= and Mis the mass of the black hole. This initial outflow has a half-opening angle BBc3≥q . A connection with fireballs in g-ray bursts is given [12]. Cosmic strings can be turned into superconductors if electromagnetic gauge invariance is broken inside the strings. This can occur, for example, when a charged scalar field develops a non-zero expectation value in the vicinity of the string core. The electromagnetic properties of such strings are very similar to those of thin superconducting wires, but they are different from the properties of bulk superconductors [11]. Strings predicted in a wide class of elementary particle theories behave like superconducting wires. Such strings can carry large electric currents and their interactions with cosmic plasmas can give rise to a variety of astrophysical effects [11]. To provide the compatibility between STR, black holes and strings a ‘New Formula’ has been developed by adding the ratio ( )maxJJ as a new parameter (or dimension) to the EFF. 2. Superconductivity The discovery of superconductivity started from the finding of Kamerlingh Onnas in 1911 that the resistance of mercury has an abrupt drop at a temperature of 4.2 0K and5has practically a zero dc-resistance value at temperatures below 4.2 0K. This new phenomenon of zero-resistance at low temperature was soon found in many other metals and alloys. An important characteristic of the loss of dc-resistance observed is the sharpness of the transition. The temperature at which superconductivity first occurs in a material is thus termed the critical (or transition) temperature of the material and is denoted by T c [5]. A superconductor is simply a material in which electromagnetic gauge invariance is spontaneously broken. Detailed dynamical theories are needed to explain why and at what temperatures this symmetry breaking occurs, but they are not needed to drive the most striking aspects of superconductivity: exlusion of magnetic fields, flux quantization, zero resistivity, and alternating currents at a gap between superconductors held at different voltages [6]. 2.1 The discovery of High-T c Superconductors The first of a new family of superconductors, now usually known as the High-Tc or cuprate superconductors, was discovered in 1986 by Bednorz and Müller. It was a calcium-doped lanthanum cuprate perovskite. When optimally doped to give the highest T c, it had the formula La 1.85Ca0.15CuO4, with a T c of 30 0K. This was already sufficiently high to suggest to the superconductivity community that it might be difficult to explain using the usual forms of BCS theory, and a large number of related discoveries followed quickly. In the following year Wu et al found that the closly related material Yba 2Cu3O7-δ, now known as YBCO, has a T c of about 93 0K when δ≅0.10, well above the boiling point of liquid nitrogen [7].62.2 Basic Superconductivity: The Order Parameter A relativistic version of a superconductor the abelian Higgs model ()()()Φ−Φ Φ+ −= V DDFF L m m mn mn 41(1) mnF is the electromagnetic field strength, mD is the covariant derivative ( )Φ −∂=Φ m m miqA D (2) and ()ΦV the potential of the scalar field ()( )22 41m−ΦΦ=ΦV (3) If 2m is positive the field Φ has a nonzero vacuum expectation value. A convenient parametrization of Φ is qieqr=Φ 0>=rr (4) Under gauge transformations am m m∂−→AA aqq+→ (5) The covariant derivative (2) reads in this notation ( ) [ ]rqm m mq m∂− −∂=Φ Aiq e Dqi(6) The quantity qm m m∂−=AA~ is gauge invariant. Moreover m n nm m n nm mnAA AA F~~∂−∂=∂−∂= (7) The equation of motion reads, neglecting loop corrections (or looking L as an effective lagrangian) 0~ 2~2 = +∂ nmn mAmF ()Φ=q m2~(8) In the gauge 00=A a static configuration has 00=∂Ar , 00=Φ∂ so that 00==i iFE . Eq.(8) implies that70~ 2~2 = +Λ∇ AmHr rr (9) The term Amr~2~2 in Eq.(9) is a consequence of spontaneous symmetry breaking and is an stationary electric current (London current). A persistent current with ,0=Er means 0=r since Ejrr =r and hence superconductivity. The curl of Eq. (9), reads 02~2 2= −∇ HmHr r (10) The magnetic field has a finite penetration depth m~1, and this nothing but the Meissner effect. The key parameter is Φ, which is the order parameter for superconductivity: it signals spontaneous breaking of charge conservation [8]. 2.3 Critical current density The following set of differential equations are called Ginzburg-Landau equations: ( )021 2* *2 =Α−∇− + + y yyb ay eimh inV (11) ( )Α −∇−∇ =2 *2* * * ** 2y yyy yme mieJhinV (12) We have written J for Js (supercurrent density) since in thermodynamic equilibrium there are no normal currents. By noting () y y q =expi, equation for the supercurrent density may also be written as    Α−∇ =hh* *2* e meJ qy(13) which shows that the gradient of the phase of the wave function y determines the observable quantity, the supercurrent density [5].8We shall now apply G-L (Ginzburg-Landau) equations to calculate the critical current in a superconducting thin film at which superconductivity breaks down. First, we consider thin film x<<d and l<<d so that y and sJ may be supported to be a constant and uniform over the sample cross section of the thin films. We can set ()xieqy y= with y being independent of x. Equation (13) for the supercurrent yields ( ) se AemeJ ny y q2* 2 * ** = −∇ =h (14) where sn denotes the mean velocity of the superconducting pair of electrons [5]. The mixed-state critical current density as a function of B : The critical current density Jc is the transport current density at which pinning can no longer hold the flux at rest in the face of the thermodynamic driving force. At this point the frictional pinning force per unit volume flf/3 is equal to JBcΛ, so we have ( )( ) [ ]2 02 2 32/ /11p BBBBlJ c c cm zh − ≅ . (15) It is interesting to write down the ratio of critical current density predicted by (15) to the ideal critical current density Jmax of a thin film at which superconducting state itself collapses [5]. ( )32 2 max/1lp BBBBJJ c cclzh −≅ . (16) 2.4 High-frequency conductivity Unfortunately, in considering the response of superconductors to high-frequency fields, there are many situations, especially with conventional superconductors, where the non-locality of the response is important, and calculations must be based on the full Mattis-Bardeen equation.9In dirty limit (where d<<l and 0x<<l ) the relative shortness of l means that we may replace ( ) IRTw,, by ( ) ITw,,0. Assuming that the dominant scattering is elastic scattering by impurities we know that l is the same in the normal and superconducting states. We then have an effective complex conductivity s usually written as ss1 2−i and given by ( ) wpw ss s hiTIi n−=− ,0, 2 1. (17) The real part of the Mattis-Bardeen conductivity ()s1T corresponds to a current of normal excitations. At low temperatures it is exponentially small at low frequencies, varying as ekT−Δ/, but it rises rapidly as soon as w exceeds the gap frequency h/2Δ=gw (of order 1011 to 1012 Hz for conventional superconductors and 1013 Hz in cuprates), the frequency at which creation of pairs of excitations becomes possible. The imaginary part of the conductivity ()s2T, corresponding to the superelectrons, is proportional to 1w, as one would expect for an inertia-dominated response, almost up to gap frequency. In fact, if we use the Pippard equation as an approximation we find that )()0( 1 0 02 Tn ΛΛΔ=Λ≅wpswxshl. (18) This dirty limit conductivity may be compared with the clean limit London conductivity ()TΛw1 . Near the gap frequency this approximation fails and 2s falls more rapidly with frequency. In the context of high-frequency conductivity the two-fluid model means a system whose conductivity may be written as10sw t w=+++  ne mf if is en s2 1(19) where fn and fs represent the fractions of the electrons which are normal and superfluid respectively (with ffn s+ =1), t is a relaxation time for the normal electrons and s is an infinitesimal. In this simple model the normal electrons have both inertia and damping, with the usual Drude conductivity at high frequencies, and the superelectrons have inertia but no dumping . We notice that the model obeys the conductivity sum rule swwp'()−∞∞ ∫ =dne me2 (20) which applies to all systems of mobile electrons [7]. 2.5 High Current in General Relativity Physical nature of equilibrium of current-carrying filaments is studied on the basis of Einstein equations of General Relativity. Considering a conducting filament as an element of the structure of universe, one has to take into account both electromagnetic and gravitational interactions of charges [9]. Intergalactic currents are playing a very important role in modern plasma astrophysics. Understanding that the universe is largely a Plasma Universe came from the fact that electromagnetic forces exceed gravitational forces by a factor 3610, and even if neutral as a whole system a relatively small electromagnetic fluctuation can lead to non-uniform distribution of matter [9]. Another topic that requires General Relativity is the old Alfven’s problem of a limiting current. If we ignore the effect of General Relativity, then self-consistent theory does not impose any limitations on the current values of equilibrium relativistic beams. In General Relativity the matter curves the space-time, and this results in11gravitational self-attraction of matter. If total energy (or mass) of matter exceeds some limit, the forces of contraction cannot be balanced by the pressure. In this case equilibrium is not possible, and the matter undergoes infinite contraction, which is called gravitational collapse [9]. We show that the current of an equilibrium filament cannot exceed 25 max1094.0⋅=I A. Currents 2010≈I A in the Galactic and Intergalactic Medium are discussed by Peratt. Nevertheless solution of the problem of limiting current in General Relativity is interesting in principle, especially taking into account filamentary structure of the Universe. Our analysis realizes common physical nature of gravitational and electromagnetic collapse, and displays peculiarities of space distribution of matter and gravitational field near the collapse boundary [9]. 3. Background for the Evolution of the ‘New Formula’ This chapter gives some background information that upholds the ‘New Formula’. 3.1 Acceleration of Ultra High Energy Particles by Black Holes and Strings (Currents in High Energy Astrophysics) It is well known that charged black holes can have a magnetic dipole moment (indeed for a rotating charged black hole, the gyromagnetic ratio is 2, the same as for a Dirac particle). Such a black hole can thus also interact with a particle having a magnetic moment. The interaction energy in this case is given by: factorsspace curvedrPBHE × ≅3 intm m (21) Here BHm and Pm are the magnetic dipole moments of the black hole and the particle respectively.12For PmBm≈ (the Bohr magneton) and for maximally charged hole this gives a maximal energy (at srr=) of: 234 max MGBcEm ≅ (22) For a 1710 gm. primordial Hawking black hole this gives (note ME1a ) eV E23 max10≅ . If the black hole is embedded in a magnetic field such high energy particles accelerated by the hole can also emit ultra high frequency gamma radiation (suppressed by 41m). We next consider the acceleration of particles by cosmic strings and fundamental superstrings. Superstrings are produced near the Planck scale (energy plE or 1910≈plM GeV). They are characterized by a tension GcTpl2≈ (mass per unit length). 1 2810−≈gcm Tpl strings produced by a symmetry breaking at any other energy (mass scale) M≈ have a tension given by: 22    ≅ pls MM GcT (23) In addition, one can have conducting cosmic strings which are essentially topological line defects [10]. There are some nice analogies between vortex lines in a Type II superconductor (carrying a quantized flux ec2h ) and conducting cosmic strings. For instance, the field vanishes everywhere in a superconductor (Meissner effect), i.e. 0=abF, everywhere except along Abrikosov vortex lines carrying a confined quantized flux ec2h . Inside a superconductor we have the Landau equations:13JBBvvv=+Δ2 2l (24) The vanishing of the field inside a superconductor is an effect of the Landau- Ginzburg theory where we have the Maxwell field coupled to a scalar field as: ( )2 2 l faffm m− =DD (25) lf≅ near the broken symmetric state. Far from the flux tube: ( )0= += f d f m m mieA D (26) and [ ] 0 , = =f f mn n mieF DD (27) So either f or mnF must vanish. This has the solution: ()q m mlffdie e A = −= , 1 (28) The Higgs field responsible for these defects is described by a relativistic version of the Landau-Ginzburg model and consequently it can be shown that conducting strings also carry flux. ecnh=f (29) The flux can be shown to give rise to an electric field given by: 2 2 pls pl sMTeGceGMcTVhh≅ ≅ (30) Thus charged particles can be accelerated to a maximal energy given by: (corresponding to a critical current): pl sMGecTE2121≅ (31) For a string tension, corresponding to a GUT scale GeV M1510≅ , (the corresponding tension being given by eq. (23)):14eV E2110≅ (32) A higher string tension sT gives rise to a higher value of E. For a GUTs scale .10, 1022 16eV EGeV M ≈ ≈ It must be noted that Ultra High Energy (UHE) particles can be spontaneously generated by Evaporating Black Holes (EBH) [10]. 3.2 Electron-Positron Outflow from Black Holes Gamma-ray bursts (GRBs) appear as the brightest transient phenomena in the Universe. The nature of the central engine in GRBs is a missing link in the theory of fireballs to their stellar mass progenitors. It is shown that rotating black holes produce electron-positron outflow when brought into contact with a strong magnetic field. The outflow is produced by a coupling of the spin of the black hole to the orbit of the particles. For a nearly extreme Kerr black hole, particle outflow from an initial state of electrostatic equilibrium has a normalized isotropic emission of ()( )q2 2 2 48sin7 105~ O cMMBB× erg/s, where B is the external magnetic field strength, ,104.413G Bc×= and Mis the mass of the black hole. This initial outflow has a half-opening angle BBc3≥q . A connection with fireballs in g-ray bursts is given [12]. A theory is decribed for electron-positron pair-creation powered by a rapidly spinning black hole when brought into contact with a strong magnetic field. The magnetic field is supplied by the surrounding matter as in forementioned black hole/torus or disk systems. A rapidly spinning black hole couples to the surrounding matter by Maxwell stresses [12].15Pair-creation can be calculated from the evolution of wave-fronts in curved spacetime, which is well-defined between asymptotically flat in- or out-vacua. By this device, any inequivalence between them becomes apparent, and generally gives rise to particle production. It is perhaps best known from the Schwinger process, and in dynamical spacetimes in cosmological scenarios. Such particle production process is driven primarily by the jump in the zero-energy levels of the asymptotic vacua, and to a lesser degree depends on the nature of the transition between them. The energy spectrum of the particles is ordinarily nonthermal, with the notable exception of the thermal spectrum in Hawking radiation from a horizon surface formed in gravitational collapse to a black hole. There are natural choices of the asymptotic vacua in asymptotically flat Minkowski spacetimes, where a time-like Killing vector can be used to select a preferred set of observers. This leaves the in- and out-vacua determined up to Lorentz transformations on the observers and gauge transformations on the wave- function of interest. These ambiguities can be circumvented by making reference to Hilbert spaces on null trajectories – the past and future null infinities ±J in Hawking’s proposal – and by working with gauge covariant frequencies. The latter received some mention in Hawking’s original treatise, and is briefly as follows [12]. Hawking radiation derives from tracing wave-fronts from +J to −J, past any potential barrier and through the collapsing matter, with subsequent Bogolubov projections on the Hilbert space of radiative states on −J. This procedure assumes gauge covariance, by tracing wave-fronts associated with gauge-covariant frequencies in the presence of a background vector potential aA. The generalization to a rotating black hole obtainsby taking these frequencies relative to real, zero-angular momentum observers, whose world-lines are orthogonalto the azimuthal Killing vector as given by16( )f ff fx ∂ −∂=∂ ggt t aa. Then ta∂~x at infinity and aa∂x assumes corotation upon approaching the horizon, where abg denotes the Kerr metric. This obtains consistent particle-antiparticle conjugation by complex conjugation among all observers, except for the interpretation of a particle or an antiparticle. Consequently, Hawking emission from the horizon of a rotating black hole gives rise to a flux to infinity ()1 21 22 +Γ=−k wpp wFVe dtdnd, for a particle of energy w at infinity [12]. Here, M41=k and HΩ are the surface gravity and angular velocity of the black hole of mass M, Γ is the relevant absorption factor. The Fermi-level FV derives from the (normalized) gauge-covariant frequency as observed by a zero-angular momentum observer close to the horizon, namely, eV eV V H ZAMO F+Ω−=+ =− nw w w for a particle of charge e− and azimuthal quantum number n, where V is the potential of the horizon relative to infinity. The results for antiparticles (as seen at infinity) follow with a change of sign in the charge, which may be seen to be equivalent to the usual transformation rule ww−→ and nn−→. 3.3 Superconducting Strings Superconductivity can be understood as a spontaneously broken electromagnetic gauge invariance. When the gauge invariance is broken, the photon acquires a mass and any magnetic field applied at the boundary of the superconductor decays exponentially towards its interior. The magnetic field is screened by a non-dissipative superconducting current flowing along the boundry – the well-known Mei βner effect [11].17Cosmic strings can be turned into superconductors if electromagnetic gauge invariance is broken inside the strings. This can occur, for example, when a charged scalar field develops a non-zero expectation value in the vicinity of the string core. The electromagnetic properties of such strings are very similar to those of thin superconducting wires, but they are different from the properties of bulk superconductors [11]. Strings predicted in a wide class of elementary particle theories behave like superconducting wires. Such strings can carry large electric currents and their interactions with cosmic plasmas can give rise to a variety of astrophysical effects [11]. The idea that strings could become superconducting was first suggested in a pioneering paper by Witten [1985a]. Later it was realized that the role of the superconducting condensate could be played not only by a scalar field, but also by a vector field whose flux is trapped inside a non-abelian string [Preskill, 1985; Everett, 1988]. If the vector field is charged, the gauge invariance is again spontaneously broken inside the string. Witten also proposed another mechanism for string superconductivity, which operates in models where some fermions acquire their masses from a Yukawa coupling to the Higgs field of the string [11]. 3.3.1 Bosonic string superconductivity The simplest example of scalar string superconductivity occurs in a toy model with two complex scalar fields f and s interacting with separate ()1~U and ()QU1 gauge fields mA~ and mA, respectively [Witten, 1985a ]. The first ()1~U is broken and gives rise to vortices. The second ()QU1 which we identify with electromagnetism, although18unbroken in vacuum, can provide a charged scalar condansate in the string interior. The Lagrangian is merely a replicated version of the abelian-Higgs model, () mnmn mnmn m msf s f FF FF V D DL41 ~ 41,~ 2 2 − − − + = , (33) where f f f m m mAig D~ ~−∂= and s s s m m mieA D −∂= . From the Lagrangian (33) we can derive the usual electromagnetic current density, ( )ssss m m mD Diej − = (34) The total current J can than be found by integrating over the string cross-section [11]. Assuming that the vector potential mA remains approximately constant across the string we have ( ) ∑ +∂ = z zeA e J q 2 , (35) where 2 ∑∫= s dxdy . (36) Using the expression for the current (35), we obtain the total current flowing around the loop [Witten, 1985a ] ()()RN R eeJd pln 12 2 ∑∑ += . (37) 3.3.2 String electrodynamics The defining property of a superconducting string is its response to an applied electric field: the string develops an electric current which grows in time, ( ). ~/2EcedtdJ h (38) Here, E is the field companent along the string and e is the elementary charge ( )2 210~−e .19The charge carries in the string can be bosons or fermions. We consider first the case of fermionic superconductivity. Models of this type have fermions which are massless inside the string and have a finite mass m outside the string. Particles inside the string can be thought of as a one-dimensional Fermi gas. When an electric field is applied, the Fermi momentum grows as ,eEpF=& and the number of fermions per unit length, hp2Fpn= , also grows [11]: h &eEn~. (39) The particles move along the string at the speed of light. The resulting current is encJ=, and dtdJ is given by (38). The current continues to grow until it reaches a critical value h2~emcJc, (40) when mcpF=. At this point, particles at the Fermi level have sufficient energy to leave the string. Consequently, in this simplified picture, the growth of the current terminates at cJ and the string starts producing particles at the rate (39). The fermion mass m is model-dependent, but it does not exceed the symmetry breaking scale of the string, h. Hence, ( )213 max~ hceJJcm ≤ , (41) where we have used the relation hc2hm= . Grand unification strings can carry enormous currents, sesu J31 max10~ , while for electroweak-scale strings sesu J17 max10~ . Note that the actual value of the critical current cJ is highly model- dependent [11]. Superconducting strings can also have bosonic charge carriers. This occurs when a charged scalar or gauge field develops a vacuum expectation value inside the string. As20a result the electromagnetic gauge invariance inside the string is broken, indicating superconductivity. The critical current cJ for this type of string is determined by the energy scale at which the gauge invariance is broken. It is model-dependent, but is still bounded by maxJ from (41) [11]. Superconducting strings can develop currents not only electric, but also in magnetic fields. Consider a segment of string moving at a speed v in magnetic field B. In its rest frame the string ‘sees’ an electric field ()BcvE~ , and so the current grows at the rate ()vBedtdJ h2~ . (42) A closed loop of length L oscillating in a magnetic field acts as an ac generator and develops an ac current of amplitude ()BLe J h21.0~ (43) The factor of 0.1 appears because the area of the loop is typically of the order 21.0~LA . An oscillating current-carrying loop in vacuum emits electromagnetic waves. For a loop without kinks or cusps the radiation power is cJ E em em2~Γ& , (44) where the numerical factor emΓ depends on the loop’s shape, but not on its length; typically, .100~emΓ The ratio of the power in electromagnetic waves to that in gravitational waves is 2 max1 22 ~       − JJ cG ce EE gem m h&& , (45) and we see that for sufficiently large electromagnetic radiation can become the dominant energy loss mechanism for the loop [11].21For a loop with kinks, emΓ in (44) has a weak logarithmic dependence on the current; its characteristic range is 3 210 10 ≤Γ≤em. If the loop has cusps, then for maxJJ<< the radiation power is dominated by the emission of short periodic bursts of highly directed energy from near-cups regions. An estimate of E& in this case is complicated by the fact that the string motion near the cups is strongly affected by radiation back-reaction. Only an upper bound on this radiation power has been obtained, cJJEem max≤& . (46). 4. The Evolution of the ‘New Formula’ 4.1 Theoretical Approach The Einstein’s Famous Formula ( ) Emcmc == −2 02 21b is a general result of the Special Theory of Relativity (STR). Here, m0denotes the rest mass, m denotes the relativistic mass and b=vc. According to the Einstein’s Famous Formula (EFF) the energy ( E) approaches infinity as the velocity ( v) approaches the velocity of light (c). The velocity must therefore always remain less than c, however great may be energies used to produce the acceleration [1]. This means that, according to the EFF it is impossible for a particle to travel faster than light, and it is therefore impossible to escape from black holes. As given in section (3.3.2) in detail; When an electric field is applied, the Fermi momentum grows as ,eEpF=& and the number of fermions per unit length, hp2Fpn= , also grows [11]: h &eEn~.22The particles move along the string at the speed of light. The resulting current is encJ=, and dtdJ is given by ( )EcedtdJ h2= . The current continues to grow until it reaches a critical value h2~emcJc, when mcpF=. At this point, particles at the Fermi level have sufficient energy to leave the string. Consequently, in this simplified picture, the growth of the current terminates at cJ and the string starts producing particles at the rate (39). The fermion mass m is model-dependent, but it does not exceed the symmetry breaking scale of the string, h. Hence, ( )213 max~ hceJJcm ≤ , where we have used the relation hc2hm= . This means that, the speed of particles approaches the speed of light ( )cv→ when the current inside string approaches the critical value ( )h2emcJJc=→ . At this point, particles at the Fermi level can leave the string. Considering this result, it is quite clear that there is a direct relation between the speed of particles and the current value inside string. When we look the Einstein’s Famous Formula (EFF), there is no the relation between the current value of mediums at which the particles move and the speed of particles. Thus, we can say that the Special Theory of Relativity (STR) and string theory are not compatible. Therefore, this paper describes a ‘New Formula’ by enhancing the EFF to provide the compatibility between STR and strings. In addition, as given in sections 3.1 and 3.2, it is shown that rotating black holes produce electron-positron outflow when brought into contact with a strong magnetic field. It is quite clear that the current value inside black holes increases when they23brougth into contact with a strong magnetic field. This means that, the particles can escape from black holes when the current inside black holes approaches a critical value ( maxJJJc≤→ ). Thus, it is possible for particles to escape from black holes and consequently to reach to the speed of light ()cv= which is not allowed by the Special Theory of Relativity. Therefore, It should be allowed by ‘New Formula’ that particles can reach and exceed the speed of light. Considering the above requirements a ‘New Formula’ has been developed by enhancing the EFF to provide the compatibility between the Special Theory of Relativity (STR), black holes and strings. Here, maxJJ has been added to the EFF as a new parameter, and the ‘New Formula’ has been developed as    − −= max222 0 1 1JJ cvcmE . (47) In case h2emcJJc== and ( )213 maxhceJ m= the ‘New Formula’ can also be written as ()    − −= 2132 222 0 1 1 hh ceemc cvcmE m. (48) As the loop radiates away its energy by emitting electromagnetic and gravitational waves, it shrinks and the dc current in the loop grows as 1−∝LJ . (49) If the loop has no cusps or kinks, then its electromagnetic radiation power is 2~J EemΓ& , (50)24with 100~emΓ . (The power is not much different for a kinky loop, but the presence of cusps can change it drastically.) The ratio of electromagnetic and gravitational power output is of the order 22 ~mGJ EE gem && . (51) If the current evolves according (49), then emE& gradually grows, and the net fraction of the loop’s mass radiated electromagnetically during its entire lifetime is (Ostriker, Thompson and Witten, 1986) ( )i if ff 1tan1−= , (52) where if is the initial value of .g emEE&& Eventually the current reaches the critical level maxJJ≈. As the loop shrinks further, the current remains near critical and all the extra charge carriers are expelled from the strings [11]. For the loop with cusps, the electromagnetic radiation power is dominated by bursts of radiation from near-cusp regions. The motion of the string in these regions is strongly affected by the radiation back-reaction, and the resulting power is difficult to estimate. It is expected to be much greater than the power for a cuspless loop (50). An upper bound for emE& is given by mJEem≤& . (53) In the vicinity of a cusp, the current tends to become super-critical. An invariant measure of the current is ()( )2 221 2f f g &−′ −=− qJJa a. (54) Near a cusp 02→′=−xg and ∞→a aJJ (unless 0=′±ff& ). The physical origin of this effect is very simple: a moving string becomes contracted by a factor25( )212 1 1− − −=′ x x & , and the density of charge carriers increases by the same factor. The right-hand side of (54) can be estimated, ()2 ~zJLJJa a. The current becomes super- critical in the region maxJLJ≤z . As a result the loop will lose a fraction max~JJ of its charge carriers. If the motion of the loop were strictly periodic, then during the next period the current near the cusp would be exactly maxJ [11]. Considering the above information, it is quite clear that the current value ()J in a superconducting medium (e.g. inside string and black hole) can reach to a super critical value ()maxJ. This means that J can be equal to maxJ in the ‘New Formula’, and in this state of the ‘New Formula’ ( )maxJJ= will allow that particles can reach to and exceed the speed of light and consequently can leave black holes and strings. State: maxJJ= v Emc mm= = =    0 02 0, vc Emc mm= = =    02 0 and v Emc mm=∞ = =    02 0. There is no the limitation for the speed of a particle in this state. In addition, the energy and mass of a particle do not change when speed changes in this state. This state can predict and describe the space-time singularities without the distribution of mass and energy. 4.2 Practical Approach (Experiments) The EEFF (Enhanced Einstein’s Famous Formula) which is completely same as the ‘New Formula’ has been experimentally proved and justified. The detailed information about the practical approach of the’New Formula’ (or EEFF) including experiments are given in the References [13] (cond-mat/9909373) and [14] (gr-qc/9909077).265. Conclusion A ‘New Formula’ has been theoretically developed and described in this paper. The ‘New Formula’ has been developed in place of Einstein’s Famous Formula (EFF) to provide the compatibility between the Special Theory of Relativity (STR), black holes and strings. The ‘New Formula’ can also predict and describes the space-time singularities without the distribution of mass and energy. It is allowed by the ‘New Formula’ that any particle can reach to and exceed the speed of light ()cv≥. A very important conclusion of this paper is that the EFF ( )2mcE= is only valid and applicable in the vacuum (the mediums which have low current density: outside the string, outside black hole), but is not valid and applicable for inside string and inside black hole including space-time singularities. However, the ‘New Formula’ is valid and applicable in all mediums including inside string and inside black hole.27References [[1]]A. Einstein , Relativity The Special and General Theory (London, 1994), p. 32-33, 45. [[2]]S.W. Hawking , Black Holes and Baby Universes and other Essays (Bantam Books, 1994), p.62, 68-71, 73, 76. [[3]]J-P Luminet , Black Holes (Cambridge 1995), p. 155-157 [[4]]S.W. Hawking , Commun. Math. Phys. 43 199 (1975) [[5]] Shu∼∼Ang Zhou , Electrodynamic theory of superconductors (London, 1991), p.157-158, 161, 180-181, 186-187. [[6]] S. Weinberg , The Quantum Theory of Fields, Volume II (Cambridge University Press 1996), p.332. [[7]] J.R. Waldram , Superconductivity of Metals and Cuprates (Institute of Physics Publishing, Bristol and Philadelphia 1996), p.84-86. [[8]]A. Di Giacomo , the dual superconductor picture for confinement (NATO ASI Series B: Physics Vol.368), p.415-437. [[9]]B.E. Meierovich , High Current in General Relativity (Particles, Fields and Gravitation), AIP Conference Proceedings 453, p. 498-517. [[10]]C. Sivaram , Production and Acceleration of Ultra High Energy Particles by Black Holes and Strings (Currents in High-Energy Astrophysics), NATO ASI Series (Series C-Vol.458), p.177- 182. [[11]]A. Vilenkin and E.P.S. Shellard , Cosmic Strings and Other Topological Defects (Cambridge University Press 1994), p. 122-153, p. 343-367. [[12]]Maurice H.P.M. van Putten , Electron-Positron outflow from black holes, astro-ph/9911396 (1999). [[13]]A.R. Akcay, Transmission of the Infinite Frequencies by using High-Tc Superconductors, Proceedings UHF-99 (International University Conference “Electronics and Radiophysics of Ultra-High Frequencies”, St. Petersburg State Technical University St. Petersburg, Russia May 24-28, 1999), p.352-359 and cond-mat/9909373 (1999).28[[14]]A.R. Akcay , The Enhancement of the Special Theory of Relativity Towards the Prediction of the Space-time Singularities, gr-qc/9909077.

arXiv:physics/9912026v1 [physics.flu-dyn] 13 Dec 1999Detection of a flow induced magnetic field eigenmode in the Rig a dynamo facility Agris Gailitis, Olgerts Lielausis, Sergej Dement’ev, Erne sts Platacis, Arnis Cifersons Institute of Physics, Latvian University LV-2169 Salaspils 1, Riga, Latvia Gunter Gerbeth, Thomas Gundrum, Frank Stefani Forschungszentrum Rossendorf P.O. Box 510119, D-01314 Dresden, Germany Michael Christen, Heiko H¨ anel, Gotthard Will Dresden University of Technology, Dept. Mech. Eng. P.O. Box 01062, Dresden, Germany (Submitted to Phys. Rev. Lett., December 10, 1999) In an experiment at the Riga sodium dynamo facility, a slowly growing magnetic field eigenmode has been detected over a period of about 15 seconds. For a slightly decreased propeller rotation rate, additional measurements showed a slow decay of this mode. The measured results correspond satisfactory with numerical predictions for the growth rat es and frequencies. PACS numbers: 47.65.+a, 52.65.Kj, 91.25.Cw Magnetic fields of cosmic bodies, such as the Earth, most of the planets, stars and even galaxies are believed to be generated by the dynamo effect in moving elec- trically conducting fluids. Whereas technical dynamos consist of a number of well-separated electrically con- ducting parts, a cosmic dynamo operates, without any ferromagnetism, in a nearly homogeneous medium (for an overview see, e.g., [1] and [2]). The governing equation for the magnetic field Bin an electrically conducting fluid with conductivity σand the velocity vis the so-called induction equation ∂B ∂t=curl(v×B) +1 µ0σ∆B (1) which follows from Maxwell equations and Ohms law. The obvious solution B=0of this equation may be- come unstable for some critical value Rmcof the mag- netic Reynolds number Rm=µ0σLv (2) if the velocity field fulfills some additional conditions. HereLis a typical length scale, and va typical velocity scale of the fluid system. Rmcdepends strongly on the flow topology and the helicity of the velocity field. For self-excitation of a magnetic field it has to be at least greater than one. For typical dynamos as the Earth outer core,Rmis supposed to be of the order of 100. The last decades have seen an enormous progress of dynamo theory which deals, in its kinematic version,with the induction equation exclusively or, in its full ver- sion, with the coupled system of induction equation and Navier-Stokes equation for the fluid motion. Numerically, this coupled system of equations has been treated for a number of more or less realistic models of cosmic bodies (for an impressive simulation, see [3]). Quite contrary to the success of dynamo theory, ex- perimental dynamo science is still in its infancy. This is mainly due to the large dimensions of the length scale and/or the velocity scale which are necessary for dynamo action to occur. Considering the conductivity of sodium as one of the best liquid conductors ( σ≈107(Ωm)−1 at 100◦C) one gets µ0σ≈10 s/m2. For a very efficient dynamo with a supposed Rmc= 10 this would amount to a necessary product Lv= 1 m2/s which is very large for a laboratory facility, even more if one takes into ac- count the technical problems with handling sodium. His- torically notable for experimental dynamo science is the experiment of Lowes and Wilkinson where two ferromag- netic metallic rods were rotated in a block at rest [4]. A first liquid metal dynamo experiment quite similar to the present one was undertaken by some of the authors in 1987. Although this experiment had to be stopped (for reasons of mechanical stability) before dynamo action oc- curred the extrapolation of the amplification factor of an applied magnetic field gave indication for the possibility of magnetic field self-excitation at higher pump rates [5]. Today, there are several groups working on liquid metal dynamo experiments. For a summary we refer to the workshop ”Laboratory Experiments on Dynamo Action” held in Riga in summer 1998 [6]. After years of preparation and careful velocity profile optimization on water models, first experiments at the Riga sodium facility were carried out during November 6-11, 1999. The present paper comprises only the most important results of these experiments, one of them being the observation of a dynamo eigenmode slowly growing in time at the maximum rotation rate of the propeller. A more comprehensive analysis of all measured data will be published elsewhere. 1/0/0/0/0/1/1/1/1/0/0/0/1/1/1 /0/0/0/0/0/0 /1/1/1/1/1/1 /0/0/0/0/0 /1/1/1/1/1 /0/0/0/0/0/0/0 /1/1/1/1/1/1/1/0/0/1/1 /0/0/1/1 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The Riga dynamo facility. Main parts comprising: 1 - Two motors (55 kW each), 2 - Propeller, 3 - Helical flow region, 4 - Back-flow region, 5 - Sodium at rest, 6 - Sodium storage tanks, ∗- Position of the flux-gate sensor, ×- Posi- tions of the six Hall sensors. The principal design of the dynamo facility, together with some of the most important dimensions, is shown in Fig. 1. The main part of the facility consists of a spiral flow of liquid sodium in an innermost tube (with a velocity up to the order of 15 m/s) with a coaxial back- flow region and a region with sodium at rest surrounding it. The total amount of sodium is 2 m3. The sodium flow up to 0 .6 m3/s is produced by a specially designed propeller which is driven by two 55 kW motors. All three sodium volumes play an important role in the magnetic field generation process. The spiral flow within the immobile sodium region amplifies the mag- netic field by stretching field lines [7]. The back-flow is responsible for a positive feedback [8]. The result is an axially non-symmetric field (in a symmetric flow geome- try!) slowly rotating around the vertical axis. Hence, a low frequency AC magnetic field is expected for this con- figuration. Concerning the azimuthal dependence of the magnetic field which includes terms of the type exp( imϕ) with in general arbitrary m it is well-known that for those Rm available in this experiment only the mode with m= 1 can play any role [8]. A lot of details concerningthe solution of the induction equation for the chosen ex- perimental geometry and the optimization of the whole facility in general and of the shape of the velocity profiles in particular can be found in [8], [9], and [10]. For the magnetic field measurements we used two dif- ferent types of sensors. Inside the dynamo, close to the innermost wall and at a height of 1/3 of the total length from above, a flux-gate sensor was positioned. Addition- ally, 8 Hall sensors were positioned outside the facility at a distance of 10 cm from the thermal isolation. Of those, 6 were arranged parallel to the dynamo axis with a relative distance of 50 cm, starting with 35 cm from the upper frame. Two sensors were additionally arranged at different angles. After heating up the sodium to 300◦C and pumping it slowly through the facility for 24 hours (to ensure good electrical contact of sodium with the stainless-steel wall s) various experiments at 250◦C and around 205◦C at dif- ferent rotation rates of the propeller were carried out. According to our numerical predictions, self-excitation was hardly to be expected much above a temperature of 200◦C since the electrical conductivity of sodium de- creases significantly with increasing temperature. Nev- ertheless, we started experiments at 250◦C in order to get useful information for the later dynamo behaviour at lower temperature, i.e. at higher Rm. Although the experiment was intended to show self-excitation of a magnetic field without any noticeable starting magnetic field, kick-field coils fed by a 3-phase current of variable low frequency were wound around the module in order to measure the sub-critical amplification of the applied magnetic field by the dynamo. This measurement phi- losophy was quite similar to that of the 1987 experiment [11] and is based on generation theory for prolongated flows as the length of our spiral flow exceeds its diame- ter more then ten times. Generation in such a geometry should start as exponentially high amplification of some kick-field (known as convective generation) and should transform, at some higher flowrate, into self-excitation without any external kick-field. As a typical example of a lot of measured field ampli- fication curves, Fig. 2 shows the inverse relation of the measured magnetic field to the current in the excitation coils for a frequency of 1 Hz and a temperature of 205◦C versus the rotation rate of the propeller. The two curves (squares and crosses) correspond to two different settings of the 3-phase current in the kick-field coils with respect to the propeller rotation. An increasing amplification of the kick-field can be clearly observed in both curves until a rotation rate of about 1500 rpm. These parts of both curves point to about 1700 rpm which might be inter- preted as the onset of convective generation [8], [9]. If the excitation frequency would be exactly the one the system likes to generate as its eigenmode, the curves should fur- ther approach the abscissa axis up to the self-excitation point. As 1 Hz does not meet exactly the eigenmode fre- 2quency the points are repelled from the axis for further increasing rotation rates as it is usual for externally ex- cited linear systems passing the point of resonance (for this interpretation, see also Fig. 5). 020406080100 0 500 1000 1500 2000I/B [A/mT] Rotation rate [1/min] FIG. 2. Dependence of the magnetic field amplification on the propeller rotation rate for T=205◦C and f=1 Hz. The ordinate axis shows the inverse relation of the measured mag - netic field to the current in the kick-field coils. Squares and crosses correspond to two different settings of the 3-phase current in the kick-field coils with respect to the propeller rotation. It should be underlined that all points on Fig. 2 ex- cept the rightmost one are calculated from sinusoidal field records showing the same 1 Hz frequency as the kick-field. However, the rightmost point at 2150 rpm is exceptional. Let us analyse the magnetic field signal at this rotation rate in more detail. Fig 3a shows the magnetic field mea- sured every 10 ms at the inner sensor in an interval of 15 s. Evidently, there is a superposition of two signals. Numerically, this signal (comprising 1500 data points) has been analyzed by means of a non-linear least square fit with 8 free parameters according to B(t) =A1ep1tsin (2πf1t+φ1) +A2ep2tsin (2πf2t+φ2) The curve according to this ansatz (which is also shown in Fig. 3a) fits extremely well into the data giving the following parameters (the errors are with respect to a 68.3 per cent confidence interval): A1= (0.476±0.004) mT , p1= (−0.0012±0.0003) s−1 f1= (0.995±0.00005) s−1, φ1=−0.879±0.012 A2= (0.133±0.001) mT , p2= (0.0315±0.0009) s−1 f2= (1.326±0.00015) s−1, φ2= (0.479±0.009) The positive parameter p2= 0.0315s−1together with the very small error gives clear evidence for the appear- ance of a self-exciting mode at the rotation rate of 2150 rpm. Fig. 3b shows in a decomposed form the two con- tributing modes, the larger one reflecting the amplifiedfield of the coils and the smaller one reflecting the self- excited mode. (a) -0.6-0.4-0.200.20.40.60.8 02468101214B [mT] t [s]Measured Data Fitting curve (b) -0.6-0.4-0.200.20.40.60.8 02468101214B [mT] t [s]0.995 Hz 1.326 Hz FIG. 3. Measured magnetic field and fitting curve (a). De- composition of the fitting curve into two curves with differen t frequencies (b). For reasons of some technical problems, this highest ro- tation rate could be hold only for 20 seconds after which it fell down to 1980 rpm. At that lower rotation rate the coil current was switched of suddenly. Figure 4 shows the magnetic field behaviour at three selected Hall sen- sors positioned outside the dynamo. This mode has a fre- quency of f= 1.1 s−1and a decay rate of p=−0.3 s−1. A similar signal was recorded by the inner fluxgate sen- sor, too. It is interesting to compare the frequencies and growth or decay rates at the two different rotation rates 2150 rpm and 1980 rpm with the numerical predictions. These are based on the outcomes of a two-dimensional time depen- dent code which was described in [9]. As input velocity for the computations an extrapolated velocity field based on measurements in water at two different heights and at three different rotation rates (1000, 1600, and 2000 rpm) was used. Fig. 5 shows the predicted growth rates and frequencies for the three temperatures 150◦C, 200◦C, and 250◦C which are different due to the dependence of the electrical conductivity on temperature. The two pairs of 3points in Fig. 5 represent the respective measured val- ues. Having in mind the limitations and approximations of the numerical prognosis [9] the agreement between pre- calculations and measured values is good, particularly re- garding the frequencies of the magnetic field eigenmode. -0.015-0.01-0.00500.0050.010.015 0 1 2 3 4 5 6B [mT] t [s]Sensor at 0.85 m at 1.85 m at 2.85 m FIG. 4. Magnetic fields measured at 3 selected positions outside the dynamo module after switching off the coil cur- rent. The main part of the experiment was originally planned at T=150◦C where self-excitation with a much higher growth rate was expected. Unfortunately, the safety rules required to stop the experiment at T=205◦C since technical problems with the seal of the propeller axis against the sodium flow-out have been detected. It is worth to be noted that the overall system worked stable and without problems over a period of about five days. The sealing problem needs inspection, but represents no principle problem. -2.5-2-1.5-1-0.500.511.5 1200 1400 1600 1800 2000 220000.511.5Growth rate p [1/s] Frequency f [1/s] Rotation rate [1/min]p pred. at 150 °C 200°C 250°C p meas. at 205 °C f pred. at 150 °C 200°C 250°C f meas. at 205 °C FIG. 5. Numerical predictions for growth rates pand fre- quencies fof the dynamo eigenmode in dependence on the rotation rate for three different temperatures, and measure d values. For the first time, magnetic field self-excitation was ob- served in a liquid metal dynamo experiment. Expectedly,the observed growth rate was still very small. The corre- spondence of the measured growth rates and frequencies with the numerical prognoses is convincing. The general concept of the experiment together with the fine-tuning of the velocity profiles [9] have been proven as feasible and correct. The facility has the potential to exceed the threshold of magnetic field self-excitation by some 20 per cent with respect to the critical magnetic Reynolds num- ber. The experiment will be repeated at lower temper- ature when the technical problems with the seal will be resolved. For lower temperature, a higher growth rate will drive the magnetic field to higher values where the back-reaction of the Lorentz forces on the velocity should lead to saturation effects. We thank the Latvian Science Council for support under grant 96.0276, the Latvian Government and In- ternational Science Foundation for support under joint grant LJD100, the International Science Foundation for support under grant LFD000 and Deutsche Forschungs- gemeinschaft for support under INK 18/A1-1. We are grateful to W. H¨ afele for his interest and support, and to the whole experimental team for preparing and running the experiment. [1] F. Krause and K.-H. R¨ adler, Mean-field magnetohydro- dynamics and dynamo theory , (Akademie-Verlag, Berlin, and Pergamon Press, Oxford, 1980) [2] H. K. Moffat, Magnetic field generation in electrically conducting fluids , (Cambridge University Press, 1978) [3] G. A. Glatzmeier and P. H. Roberts, Nature 377, 203 (1995) [4] F. J. Lowes and I. Wilkinson, Nature 198, 1158 (1963) [5] A. Gailitis, B. G. Karasev, I. R. Kirillov, O. A. Lielausi s, S. M. Luzhanskii, A. P. Ogorodnikov, G. V. Preslitskii, Mag. Gidrodin., No.4, 3, (1987) [6]Proceedings of the International Workshop on Laboratory Experiments on Dynamo Action, Riga, June 14-16, 1998 , edited by O. Lielausis, A. Gailitis, G. Gerbeth, F. Stefani, (FZ Rossendorf, 1998) [7] Yu. B. Ponomarenko, Zh. Prikl. Mekh. Tekhn. Fiz.(USSR) 6, 47 (1973) [8] A. Gailitis, Mag. Gidrodyn. 32, 63 (1996) [9] F. Stefani, G. Gerbeth, A. Gailitis, in Transfer Phe- nomena in Magnetohydrodynamic and Electroconducting Flows, edited by A. Alemany, Ph. Marty, P. Thibault, (Kluwer Academic Publishers, Dordrecht, 1999), p. 31 [10] M. Christen, H. H¨ anel, and G. Will, in Beitr¨ age zu Flu- idenergiemaschinen , Bd. 4, (Verlag und Bildarchiv W. H. Faragallah, Sulzbach/Ts. 1998), p. 111 [11] A. Gailitis, B. G. Karasev, I. R. Kirillov, O. A. Lielaus is, A. P. Ogorodnikov, in Liquid Metal Magnetohydrody- namics , edited by J. Lielpeteris and R. Moreau, (Kluwer Academic Publishers, Dordrecht, 1989), p. 413 4

arXiv:physics/9912028v1 [physics.atom-ph] 14 Dec 1999Calculation of parity and time invariance violation in the r adium atom. V.A.Dzuba, V.V.Flambaum and J.S.M. Ginges School of Physics, University of New South Wales, Sydney 205 2,Australia (December 22, 2013) Abstract Parity ( P) and time ( T) invariance violating effects in the Ra atom are strongly enhanced due to close states of opposite parity, th e large nuclear charge Z and the collective nature of P,T-odd nuclear moments. We have performed calculations of the atomic electric dipole momen ts (EDM) pro- duced by the electron EDM and the nuclear magnetic quadrupol e and Schiff moments. We have also calculated the effects of parity non-co nservation pro- duced by the nuclear anapole moment and the weak charge. Our r esults show that as a rule the values of these effects are much larger than t hose considered so far in other atoms (enhancement is up to 105times). PACS: 11.30.Er,31.15.Ar Typeset using REVT EX 1I. INTRODUCTION The lower energy levels of radium corresponding to configura tions of different parity have very close energies. This leads to a strong enhancement of th e various parity ( P) and time (T) invariance violating effects. In our previous paper [1] we c onsidered the states 7 s6d3D2 withE= 13993.97cm−1and 7s7p3P1withE= 13999.38cm−1, which are separated by a very small interval of ∼5 cm−1(∼10−3eV). Simple estimates showed that the effects of nuclearP- andT-odd moments such as the magnetic quadrupole moment (MQM), t he Schiff moment (SM) and the anapole moment (AM) are many times l arger than in all atomic systems considered before. In the present paper we present m ore accurate calculations of these and other parity and time invariance non-conserving e ffects in those states of the radium atom where the effects are large. We use a relatively si mpleab initio approximation to perform the calculations. The approximation is a reasona ble compromise between the simplicity of the calculations and the accuracy of the resul ts. It is based on relativistic Hartree-Fock (RHF) and configuration interaction (CI) meth ods. A minimum number of basis states are used at the CI stage of the calculations. How ever, important many-body effects, such as polarization of the atomic core by an externa l field and correlations between core and valence electrons, are included in the calculation s of single-electron matrix elements. To control the accuracy of the calculations we also calculat ed hyperfine structure intervals and lifetimes of lower states of radium and its lighter analo g barium. Our calculations confirm the estimates done in the previous w ork [1] and show that the value of most P- andT-odd effects in radium is much higher than in other atoms consi dered before. The parity non-conserving (PNC) electric dipole tr ansition amplitude between the ground and3D1even states is about E1PNC≈0.8×10−9(QW/N)iea0,which is 100 times larger than the measured PNC amplitude in cesium [2] and abou t 5 times larger than the corresponding amplitude in francium [3]. The enhancement o f the electron electric dipole moment (EDM) in the3D1state of Ra is about 5400, which is again many times larger tha n corresponding values for the ground states of Fr (910) and Au (260) [4]. The transition amplitude between the ground and3D2even states induced by the nuclear anapole moment is about 10−9ea0, which is more than 103times larger than a similar amplitude in Cs [2]. Also, the EDM of the Ra atom in the3D2state induced by the nuclear Schiff and magnetic quadrupole moments is strongly enhanced. Both contributio ns (SM and MQM) are about 10−19η e·cm(ηis the dimensionless constant of the P-,T-odd nucleon-nucleon interaction). This is again about 105times larger than the EDM of the Hg atom which currently gives the best limit on η[5]. All this makes radium a very promising candidate for the experimental study ofP- andT-odd forces by means of atomic physics. II. METHOD We use relativistic Hartree-Fock (RHF) and configuration in teraction (CI) methods to construct two-electron wave functions of the ground and low er excited states of barium and radium. The calculations start from the RHF method for a clos ed shell system corresponding to the ground state configuration (6 s2for Ba and 7 s2for Ra). Since nsnp andns(n−1)d configurations, with n= 6 for Ba and n= 7 for Ra, do not correspond to a closed-shell system, we calculate panddbasis states in a model HF potential. For example, to calcula te 27pand 6dstates of Ra, we keep all other states frozen, remove the cont ribution of one 7s-electron from the direct HF potential and use this potentia l to calculate the required states. The same procedure applies for Ba. Thus, we have five s ingle-electron basis states for the CI calculations ( ns1/2,np1/2,np3/2,(n−1)d3/2,(n−1)d5/2). It turns out, however, that this simple CI approximation significantly overestima tes the relative value of spin-orbit intervals for the odd-parity states and underestimates it f or the even-parity states. This affects the accuracy of the calculation of P- andT-odd effects because most of them involve transitions with a change of spin which are sensitive to the v alue of the relativistic effects. We found that the spin-orbit intervals are sensitive to the scr eening of the Coulomb interaction between two external electrons (recall that Coulomb integr als contribute to the spin-orbit splitting due to the difference between the single-particle radial wave functions belonging to different components of the single-particle doublets). T o improve the quality of the wave functions we introduce fitting factors fkto the Coulomb interaction in the CI calculations ( k is the multipolarity of the Coulomb interaction). It was fou nd that multiplying all Coulomb integrals of multipolarities 0, 1 and 2 by factors f0= 0.7,f1= 0.75,f2= 0.9 significantly improves the energies and fine structure intervals of lower o dd and even states of barium and radium. These factors simulate the effect of the screenin g of the Coulomb interaction between valence electrons and core electrons. They also com pensate to some extent the effect of the incompleteness of the basis set. To calculate values other than energy, such as the effect of el ectron interaction with photons and nuclear P- andT-odd fields, we also include core polarization effects (direc t and exchange RPA-type corrections) and core-valence corre lation effects (the Bruckner-type correlation corrections). These two effects are very import ant for the considered states of radium. Indeed, consider mixture of the3DJand3PJ′states by the P- andT-odd interac- tionW. Corresponding dominant configurations (7 s6dand 7s7p) can only be mixed by a /angb∇acketleft7p|W|6d/angb∇acket∇ightmatrix element. However, this matrix element is extremely s mall in the Hartree- Fock approximation. This is because the electron interacti on withP- andT-odd nuclear moments is localized in the vicinity of the nucleus where the d-electron does not penetrate due to the centrifugal barrier. On the other hand, the polari zation of the electron core by these moments produces a long-range correction δVto the HF potential which effectively renormalizes the interaction of an external electron with t he nucleus. The corresponding matrix element /angb∇acketleft7p|W+δV|6d/angb∇acket∇ightis not small even in the case of the p−dtransition due to the long-range of the renormalized interaction W+δV. Note that /angb∇acketleft7s|W|7p/angb∇acket∇ightmatrix elements also contribute to the mixture of the3DJand3PJ′states due to the configuration interaction. Thus, there is an interference of several fact ors: thes−pmatrix elements are large but their contribution is suppressed due to the smalln ess of the configuration mixing. Thep−dmatrix elements are considerably smaller (although not neg ligible) but they ap- pear in the dominating configurations. It cannot be said in ad vance which transitions are more important and as we see from our calculations there are c ases whens−ptransitions dominate over p−dand vice-versa (see below). The Bruckner-type correlation corrections (the correlation corrections to the single-electron wave f unctions) are also important, since they increase the density of an external electron on the nucl eus by ∼30% (see e.g. [6]). The full scale inclusion of the core polarization and correl ation effects into the CI cal- culations (see, e.g. [7]) lies beyond the framework of this r esearch. We adopted a simplified approach in which the corresponding corrections are calcul ated for the single-particle matrix 3elements. The relative values of the renormalization of the matrix elements by the core po- larization and core-valence correlations have been extrap olated from accurate calculations of the core polarization and Bruckner-type correlation cor rections for the radium positive ion. Ra+has a simple electronic structure - one electron above close d shells - and the corresponding procedures are well defined for it [8]. To check our method and the accuracy of our results, we calcul ated the hyperfine struc- ture (hfs) constants of213Ra and137Ba. The results for the energies and hfs constants are presented in Table I. One can see that even for this very simpl e CI approximation the accuracy of the energies and fine structure intervals is very good. The accuracy of the hfs constants is also good for the most important states3D2and3P1. Table II shows the effect of the core polarization (RPA) and Bruckner-type correlati ons (Σ) on the single-electron matrix elements. One can see that these effects play a crucial role in the p−dmatrix elements. However, their contribution to the s−pmatrix elements is also very important. III. PARITY VIOLATION IN 7S2→7S6DTRANSITIONS A. Spin-independent parity non-conservation The Hamiltonian HPNCof the interaction of an electron with the nuclear weak charg eQW (formula (A6) in the appendix) mixes states of the same total momentum Jand opposite parity. Thus, electric dipole transitions between states o f initially equal parity become possible. In particular, the transition between the ground state1S0and the excited3D1 state is enhanced due to the closeness of the opposite parity state1P1. The dominating contribution to this transition is given by E1PNC=/angb∇acketleft7s2 1S0|dz|7s7p3P1/angb∇acket∇ight/angb∇acketleft7s7p3P1|HPNC|7s6d3D1/angb∇acket∇ight E(3D1)−E(3P1). (1) Apart from the enhancement, there are several suppression f actors in (1). First, the electric dipole matrix element is small because of a change of spin. It is 3 to 5 times smaller than most of those amplitudes which do not change atomic spin (see Table III). Second, in the matrix element of the PNC interaction, leading configuratio ns produce only the p3/2−d3/2 single-electron matrix element which is small. It is not zer o mostly due to core polarization. However, it is about 25 times smaller than the s1/2−p1/2matrix element. The latter contribute to the PNC amplitude due to configuration mixing. Our calculations show that the contribution of the s−ptransition to the PNC amplitude is about 7 times larger than the contribution of the p−dtransition. In spite of some suppression, the final answer is quite large: 225Ra :E1PNC= 0.77×10−9(QW/N)iea0, (2) 223Ra :E1PNC= 0.76×10−9(QW/N)iea0. (3) This is one hundred times larger than the measured PNC amplit ude in cesium [2] and about 5 times larger than the corresponding amplitude in francium [3]. Even radium isotopes have close values of the amplitudes (approximately, the effect is proportional to the number of neutronsN). 4B. Anapole moment The Hamiltonian of the electron interaction with the nuclea r anapole moment is presented in the appendix (A9). Similar to the spin-independent PNC in teraction, it mixes states of opposite parity and leads to non-zero E1-transition amplit udes between states of initially equal parity. However, it can also mix states with ∆ J= 1 and it depends on the nuclear spin, so that its contribution to transitions between different hy perfine structure components are different. The corresponding expression is very similar to ( 1). However, dependence on the hyperfine structure must be included (see formula (A12) in th e appendix for details). This amplitude is proportional to the /angb∇acketleft3P1||/vector αρ(r)||3D2/angb∇acket∇ightmatrix element. Contributions of different single-electron transitions into this matrix element are p resented in Table IV. Note the strong cancellation between terms corresponding to s−pandp−dtransitions. This means that an accurate inclusion of the core polarization and core -valence correlation effects is very important indeed, as has been discussed above. We believe th at the fitting of the energies helps to stabilize this matrix element similar to the case of theE1-transition amplitude. The results for1S0−3D1and1S0−3D2transitions are presented in Table V. Note that the contribution of the anapole moment to the PNC amplitude ( 3) can be measured by comparing the amplitudes between different hyperfine struct ure components similar to what was done for cesium [2]. However, it may be much more efficient t o measure the effect of the anapole moment in the1S0−3D2transition because it is about ten times larger due to the small energy denominator and because the nuclear spin indep endent PNC interaction does not contribute to this amplitude at all due to the large chang e of the total electron angular momentum ∆ J= 2. IV. ATOMIC ELECTRIC DIPOLE MOMENTS A. Electron EDM An electron electric dipole moment interacting with an atom ic field mixes states with the same total momentum Jand opposite parity. As a result, an atomic EDM appears. The EDM of radium in the3D1state is strongly enhanced due to the closeness of the opposi te parity state3P1. In an approximation when only the mixture of the closest sta tes is included, the EDM is given by d= 2/angb∇acketleft7s6d3D1| −er|7s7p3P1/angb∇acket∇ight/angb∇acketleft7s7p3P1|HEDM|7s6d3D1/angb∇acket∇ight E(3D1)−E(3P1). (4) Calculations using formulae from the appendix give the foll owing result d= 5370de. (5) Note that a very strong enhancement is caused by the small ene rgy denominator E(3D1)− E(3P1) = 0.001292 a.u. 5B. Schiff moment Electron interaction with the nuclear Schiff moment also pro duces an atomic EDM. The EDM of Ra caused by Schiff moment is strongly enhanced in the3D2state. Its value is approximately given by dz= 2/angb∇acketleft7s6d3D2|dz|7s7p3P1/angb∇acket∇ight/angb∇acketleft7s7p3P1|HSM|7s6d3D2/angb∇acket∇ight E(3D2)−E(3P1). (6) More detailed formula which include the dependence of (6) on the hyperfine structure is presented in the appendix (A19). Table IV shows single-electron contributions to the /angb∇acketleft3P1||HSM||3D2/angb∇acket∇ightmatrix element. Note thats−ptransitions strongly dominate here. However, the contribu tion of the p−d transitions is not negligible and should be included for acc urate results. Calculated values of the radium EDM induced by the Schiff mome nt are presented in Table VI. C. Magnetic quadrupole moment Electron interaction with the nuclear MQM can also produce a n EDM of an atom. However, in contrast with the case of the Schiff moment, the MQ M of isotopes where the nuclear spin I <1 (like225Ra, whereI= 1/2) is zero. The EDM of Ra in the3D2state is given by a formula similar to (6) dz= 2/angb∇acketleft7s6d3D2|dz|7s7p3P1/angb∇acket∇ight/angb∇acketleft7s7p3P1|HMQM|7s6d3D2/angb∇acket∇ight E(3D2)−E(3P1). (7) Again, more detailed formula can be found in the appendix (A2 4). Table IV shows single-electron contributions to the /angb∇acketleft3P1||Amk||3D2/angb∇acket∇ightmatrix element. Note that in contrast to the cases of the Schiff and anapole moments ,p−dtransitions dominate overs−ptransitions in this matrix element. For the anapole moment t hese two types of transitions contribute almost equally, while for the Schiff moments−ptransitions dominate. Note thats−ptransitions appear due to configuration mixing only, while c ontribution of thep−dtransitions is extremely small if core polarization is not i ncluded. This indicates once more that even for a rough estimation of the time or parit y invariance violating effects in Ra an inclusion of the appropriate many-body effects is ess ential. Calculated values of the radium EDM induced by the magnetic q uadrupole moment are presented in Table VII. V. LIFETIMES To plan experimental measurements of space and time invaria nce violation in radium it is important to know the lifetimes of the states of interest. Apart from that, comparison of the calculated and experimental lifetimes can serve as a goo d test of the method used for calculation of P- andT- invariance violation since the same dipole transition amp litudes 6contribute in either case. As far as we know, none of the radiu m lifetimes have been measured so far. On the other hand, some experimental data is availabl e for barium. Therefore, we calculated lifetimes of lower states of both atoms. Results for dipole transition amplitudes are presented in Table III and the corresponding lifetimes a re in Table VIII. For the purpose of the present work, the most important state s of radium are3P1,3D1and 3D2states. The decay rate of the3P1state is strongly dominated by the3P1-1S0transition. Transitions to the3D1and3D2states are suppressed due to small frequencies. The3P1-1S0 dipole transition amplitude involves a change of the atomic spin and therefore is sensitive to the value of the relativistic effects. This makes the ampli tude numerically unstable. This probably explains the poor agreement between different calculations (see Table VIII). However, we believe that the fitting of the fine structure whic h we have done for Ba and Ra (see Section II) brings the amplitude close to the correct value. This is supported by similar calculations for barium. The3P1-1S0amplitude contributes 38% to the decay rate of the3P1state of barium. Good agreement between calculated and expe rimental lifetimes of this state (see Table VIII) means that all transition ampl itudes, including the3P1-1S0 amplitude, are calculated quite accurately. The lifetime of the3D1state of Ra is determined by the3D1-3P0transition which is numerically stable. The lifetime of this state calculated b y us is in good agreement with the estimations done by Budker and DeMille [9]. The3D2state of radium is a metastable state. It decays only via elec tric quadrupole (E2) transition to the ground state. Calculations similar t o the electric dipole transitions show that the lifetime of this state in the absence of externa l fields is about 15 seconds. However, measurements of the atomic EDM involve placing the atoms in a strong electric field. It is important to know how the lifetimes of the3D2and3D1states of Ra are affected by this field. The electric field mixes states of different pari ty and ∆J= 0,±1. If only an admixture of the nearest state is taken into account, the amp litude which determines the decay rate of a3DJstate is given by A=/angb∇acketleft1S0|dzE|3P1/angb∇acket∇ight/angb∇acketleft3P1|dz|3DJ/angb∇acket∇ight E(3P1)−E(3DJ). (8) Where Eis the electric field. This leads to the following decay rates W(3D2) = 0.21E2(9) W(3D1) = 0.25×10−4E2(10) For an electric field of 10 kV/cm, the lifetime of the3D2state is 30 µs, while the lifetime of the3D1state is 240 ms. This latter result is in good agreement with e stimations done by Budker and DeMille [9]. Note that the state3D2, with maximum or minimal possible projection of the total momentum on the direction of the elec tric field (M=±2), cannot be mixed by this field with the3P1state. Therefore its lifetime is much less affected. VI. CONCLUSION The radium atom turns out to be a very promising candidate for the study of time and space invariance violating effects. All such effects cons idered in this paper are strongly 7enhanced due to the high value of the nuclear charge Zand the closeness of the opposite parity states of the atom. Moreover, the contribution of diff erent mechanisms to the time and space invariance violating effects can be studied separatel y if measurements are performed for different states and different isotopes of the radium atom . For example, the atomic EDM induced by the electron EDM is strongly enhanced in the3D1state, while contributions of the nuclear Schiff and magnetic quadrupole moments are stron gly enhanced in the3D2state. On the other hand, the magnetic quadrupole moment is zero for isotopes with nuclear spin I= 1/2, like225Ra, while the Schiff moment for these isotopes is not zero. Calculations of the space and time invariance violating effe cts in radium reveal the im- portance of relativistic and many-body effects. The accurac y achieved in the present work is probably 20-30 %. However, a further improvement in accur acy is possible if such a need arises from the progress in measurements. This work was supported by the Australian Research Council. APPENDIX A: WAVE FUNCTIONS AND MATRIX ELEMENTS 1. Radium wave functions Two-electron wave functions of the ground (1S0) and three excited (3P1,3D1and3D2) states of radium used in this work for the calculation of spac e and time invariance violation have the following form |7s2J= 0,L= 0,M= 0/angb∇acket∇ight= −0.9757|7s1 2,−1 27s1 2,1 2/angb∇acket∇ight −0.1150|7p1 2,−1 27p1 2,1 2/angb∇acket∇ight − −0.0752(|7p3 2,−3 27p3 2,3 2/angb∇acket∇ight − |7p3 2,−1 27p3 2,1 2/angb∇acket∇ight) + (A1) +0.0658(|6d3 2,−3 26d3 2,3 2/angb∇acket∇ight − |6d3 2,−1 26d3 2,1 2/angb∇acket∇ight) + +0.0702(|6d5 2,−5 26d5 2,5 2/angb∇acket∇ight − |6d5 2,−3 26d5 2,3 2/angb∇acket∇ight+|6d5 2,−1 26d5 2,1 2/angb∇acket∇ight), |7s7p J= 1,L= 1,M= 1/angb∇acket∇ight= −0.9010|7s1 2,1 27p1 2,1 2/angb∇acket∇ight −0.3537|7s1 2,−1 27p3 2,3 2/angb∇acket∇ight+ 0.2042|7s1 2,1 27p3 2,1 2/angb∇acket∇ight − −0.0976|7p1 2,−1 26d3 2,3 2/angb∇acket∇ight+ 0.0563|7p1 2,1 26d3 2,1 2/angb∇acket∇ight+ 0.0512|7p3 2,−1 26d3 2,3 2/angb∇acket∇ight − (A2) −0.0591|7p3 2,1 26d3 2,1 2/angb∇acket∇ight+ 0.0512|7p3 2,3 26d3 2,−1 2/angb∇acket∇ight −0.0018|7p3 2,−3 26d5 2,5 2/angb∇acket∇ight+ +0.0014|7p3 2,−1 26d5 2,3 2/angb∇acket∇ight −0.0010|7p3 2,1 26d5 2,1 2/angb∇acket∇ight+ 0.0006|7p3 2,3 26d5 2,−1 2/angb∇acket∇ight, |7s6d J= 1,L= 2,M= 1/angb∇acket∇ight= −0.8660|7s1 2,−1 26d3 2,3 2/angb∇acket∇ight+ 0.5000|7s1 2,1 26d3 2,1 2/angb∇acket∇ight+ 0.0002|6d3 2,−3 26d5 2,5 2/angb∇acket∇ight − (A3) −0.0001|6d3 2,−1 26d5 2,3 2/angb∇acket∇ight+ 0.0001|6d3 2,1 26d5 2,1 2/angb∇acket∇ight −0.0001|6d3 2,3 26d5 2,−1 2/angb∇acket∇ight − −0.0021|7p1 2,−1 27p3 2,3 2/angb∇acket∇ight −0.0012|7p1 2,1 27p3 2,1 2/angb∇acket∇ight, |7s6d J= 2,L= 2,M= 2/angb∇acket∇ight= −0.8087|7s1 2,1 26d3 2,3 2/angb∇acket∇ight −0.5366|7s1 2,−1 26d5 2,5 2/angb∇acket∇ight+ 0.2400|7s1 2,1 26d5 2,3 2/angb∇acket∇ight − −0.0084|6d3 2,1 26d3 2,3 2/angb∇acket∇ight −0.0059|6d3 2,−1 26d5 2,5 2/angb∇acket∇ight+ 0.0053|6d3 2,1 26d5 2,3 2/angb∇acket∇ight − (A4) −0.0032|6d3 2,3 26d5 2,1 2/angb∇acket∇ight −0.0068|6d5 2,−1 26d5 2,5 2/angb∇acket∇ight+ 0.0091|6d5 2,1 26d5 2,3 2/angb∇acket∇ight+ 8+0.0130|7p1 2,1 27p3 2,3 2/angb∇acket∇ight+ 0.0038|7p3 2,1 27p3 2,3 2/angb∇acket∇ight. We use the following form for the single-electron wave funct ion ψ(r)jlm=1 r/parenleftBigg f(r)Ω(r/r)jlm iαg(r)˜Ω(r/r)jlm/parenrightBigg . (A5) Hereα= 1/137.036 is the fine structure constant, ˜Ω(r/r)jlm=−(/vector σ·n)Ω(r/r)jlm. 2. Spin-independent weak interaction The Hamiltonian of the spin-independent weak interaction o f an electron with the nucleus is given by [10] HPNC=−G 2√ 2ρ(r)QWγ5, (A6) whereG= 2.22255 ×10−14a.u. is the Fermi constant, ρis the nuclear density (/integraltextρdV= 1), QW≈ −N+Z(1−4 sin2θW) is the nuclear weak charge, and γ5is a Pauli matrix. The matrix element of (A6) with wave functions (A5) has a form /angb∇acketleftj1l1m1|HPNC|j2l2m2/angb∇acket∇ight=−iG 2√ 2QWRPNCδj1j2δl1˜l2δm1m2, (A7) RPNC=α/integraltextρ(f1g2−g1f2)dr−radial integral , ˜l= 2j−l. However, it is often more convenient to express (A8) in a form /angb∇acketleftj1l1m1|HPNC|j2l2m2/angb∇acket∇ight= (−1)j1−m1/parenleftBigg j10j2 −m10m2/parenrightBigg (−i)αG 2√ 2QWCPNCRPNC,(A8) CPNC=/radicalBig 2j1+ 1δj1j2δl1˜l2−angular coefficient for the reduced matrix element. 3. Anapole moment The Hamiltonian of the interaction of an electron with the nu clear anapole moment has the form [11] HAM=G√ 2(I·/vector α) I(I+ 1)Kκaρ(r), (A9) whereIis the nuclear spin, K= (I+1 2)(−1)I+1 2−l,lis the orbital momentum of the outermost nucleon,κais a dimensionless constant proportional to the strength of the nucleon-nucleon PNC interaction [12]. The matrix elements of the Hamiltonia n (A9) between the many- electron states of the atoms depend on the hyperfine structur e (see, e.g. [19]) 9/angb∇acketleftIJ′F|HAM|IJF/angb∇acket∇ight=G√ 2Kκa I(I+1)(−1)F+I+J′/braceleftBigg I I 1 J J′F/bracerightBigg × ×/radicalBig I(I+ 1)(2I+ 1)/angb∇acketleftJ′||/vector αρ(r)||J/angb∇acket∇ight, (A10) F=I+J,J−atomic total momentum . The electron part of the operator (A9) is /vector αρ(r). Its single-electron matrix elements over states (A5) have a form /angb∇acketleftj1l1m1|/vector αρ(r)|j2l2m2/angb∇acket∇ight= (−1)j1−m1/parenleftBigg j11j2 −m1q m 2/parenrightBigg (C1AMR1AM+C2AMR2AM),(A11) C1AM= (−1)j1+l2+1 2/radicalBig 6(2j1+ 1)(2j2+ 1)/braceleftBigg1 2j1l2 j21 21/bracerightBigg δ˜l1l2, C2AM= (−1)j1+l1+3 2/radicalBig 6(2j1+ 1)(2j2+ 1)/braceleftBigg1 2j1l1 j21 21/bracerightBigg δl1˜l2, R1AM=−4πα/integraltextg1f2dr, R2AM=−4πα/integraltextf1g2dr. The dominating contribution to the z-component of the parity non-conserving electric dipole transition amplitude between the1S0and3D1states of Ra induced by the anapole moment is given by E1PV= (−1)F−f/parenleftBigg F1F′ −f0f/parenrightBigg (−1)4F′+J+J′+2I+1G√ 2Kκa/radicalBig2I+1 I(I+1)× ×/radicalBig (2F+ 1)(2F′+ 1)/braceleftBigg J′I F′ F1J/bracerightBigg/braceleftBigg I I 1 J J′F′/bracerightBigg /angb∇acketleft7s2||E1||7s7p/angb∇acket∇ight/angb∇acketleft7s7p||/vector αρ(r)||7s6d/angb∇acket∇ight E7s6d−E7s7p. (A12) HereF=I+J,f= min(F,F′). 4. Electron EDM The Hamiltonian of the interaction of the electron EDM dewith the atomic electric fieldEhas the form [10] HEDM=−deβ(Σ·E), (A13) where β=/parenleftBigg 1 0 0−1/parenrightBigg ,Σ=/parenleftBigg /vector σ0 0/vector σ/parenrightBigg ,E=−∇V(r). The atomic EDM induced by (A13) can be calculated as an averag e value of the operator of the dipole moment over states mixed by an operator similar to (A13) H′ EDM=−de(β−1)(Σ·E). (A14) Its single-electron matrix elements have a form 10/angb∇acketleftj1l1m1|H′ EDM|j2l2m2/angb∇acket∇ight= (−1)j1−m1/parenleftBigg j10j2 −m10m2/parenrightBigg deCEDMREDM, (A15) CEDM=√2j1+ 1δj1j2δl1˜l2, REDM= 2α2/integraltextg1dV drg2dr. Note that the selection rules and the angular coefficients are the same as for the spin inde- pendent weak interaction (A6), while the radial integrals a re different. 5. Schiff moment The Hamiltonian of the interaction of an electron with the nu clear Schiff moment has the form [13] HSM= 4πS· ∇ρ(r), (A16) S=SI/I,S is Schiff moment. Many-electron matrix elements of (A16) dep end on the hyperfine structure similar to (A9) /angb∇acketleftIJ′F|HSM|IJF/angb∇acket∇ight= (−1)F+I+J′/braceleftBigg I I 1 J J′F/bracerightBigg S/radicalBig I(I+1)(2 I+1) I/angb∇acketleftJ′||4π∇ρ(r)||J/angb∇acket∇ight. (A17) The electron part of the operator (A16) is 4 π∇ρ(r). Its single-electron matrix elements over states (A5) have the form /angb∇acketleftj1l1m1|4π∇ρ(r)|j2l2m2/angb∇acket∇ight= (−1)j1−m1/parenleftBigg j11j2 −m1q m 2/parenrightBigg CSMRSM, (A18) CSM= (−1)j2+3 2/radicalBig (2j1+ 1)(2j2+ 1)/parenleftBigg j1j21 1 21 20/parenrightBigg ξ(l1+l2+ 1), ξ(x) =/braceleftBigg 1,ifxis even 0,ifxis odd, RSM=−4π/integraltext(f1f2+α2g1g2)dρ drdr. The EDM of Ra induced by the nuclear Schiff moment for a particu lar hyperfine structure component of the3D2state is approximately given by dz= 2/parenleftBigg F1F −F0F/parenrightBigg (−1)2F+2I+J+J′/braceleftBigg J′I F F1J/bracerightBigg/braceleftBigg I I 1 J J′F/bracerightBigg/radicalBig (I+1)(2 I+1) I× (2F+ 1)S/angb∇acketleft7s6d3DJ||E1||7s7p3PJ′/angb∇acket∇ight/angb∇acketleft7s7p3PJ′||4π∇ρ(r)||7s6d3DJ/angb∇acket∇ight E7s6d−E7s7p. (A19) 6. Magnetic quadrupole moment The Hamiltonian of the interaction of an electron with the nu clear magnetic quadrupole moment has the form [13] 11HMQM=−M 4I(2I−1)tmkAmk, tmk=ImIk+IkIm−2 3δkmI(I+ 1), (A20) Amk=ǫnimαn∂i∂k1 r. Its many-electron matrix element is /angb∇acketleftIJ′F|HMQM|IJF/angb∇acket∇ight=3M 8I(2I−1)/radicalBig 5(2F+ 1)(2I+ 3)(I+ 1)(2I+ 1)I(2I−1)×   2 2 0 J′I F J I F  /angb∇acketleftJ′||Amk||J/angb∇acket∇ight. (A21) (A22) The single-electron matrix elements of the operator Amkhave the form /angb∇acketleftj1l1m1|Amk|j2l2m2/angb∇acket∇ight= (−1)j1−m1/parenleftBigg j12j2 −m1q m 2/parenrightBigg (C1MQM+C2MQM)RMQM, (A23) C1MQM= (−1)j2−1 24 3/radicalBig (2j1+ 1)(2j2+ 1)/parenleftBigg j1j22 1 21 20/parenrightBigg ξ(l1+l2+ 1), C2MQM= (−1)j1+j2+l2+14/radicalBig 5(2j1+ 1)(2j2+ 1)(2l1+ 1)(2l2+ 1)× /parenleftBigg l11l2 0 0 0/parenrightBigg  1l1l2 2j1j2 11 21 2  , RMQM=α/integraltextF(r)(g1f2+f1g2)dr, whereF(r) =/braceleftBigg r/r4 N,ifr≤rN 1/r3,ifr>r N, rN−nuclear radius . The EDM of Ra induced by the nuclear MQM for a particular hyper fine structure component of the3DJstate is approximately given by dz= 2/parenleftBigg F1F −F0F/parenrightBigg (−1)F+I+J(2F+ 1)3 23√ 5 4M/radicaltp/radicalvertex/radicalvertex/radicalbt(2I+ 3)(I+ 1)(2I+ 1) I(2I−1)× /braceleftBigg J′I F F1J/bracerightBigg  2 2 0 J I F J′I F  /angb∇acketleft7s6d3DJ||E1||7s7p3PJ′/angb∇acket∇ight/angb∇acketleft7s7p3PJ′||Amk||7s6d3DJ/angb∇acket∇ight E7s6d−E7s7p.(A24) For the case of the EDM in the3DJstate,J′= 1,J= 2 in (A24). 12REFERENCES [1] V. V. Flambaum, Phys. Rev. A 60, R2611 (1999). [2] C.S. Wood, S.C. Bennett, D. Cho, B.P. Masterson, J.L. Rob erts, C.E. Tanner, and C.E. Wieman, Science 275, 1759 (1997). [3] V.A. Dzuba, V.V. Flambaum, and O.P. Sushkov, Phys. Rev. A 51, 3454 (1995). [4] T.M.R. Byrnes, V.A. Dzuba, V.V. Flambaum, and D.W. Murra y, Phys. Rev. A 59, 3082 (1999). [5] J.P. Jacobs, W.M. Klipstein, S.K. Lamoreaux, B.R. Hecke l, and E.N. Fortson, Phys. Rev. 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A. Ahmad, W. Klempt, R. Neugart, E. W. Otten, K. Wendt, C. Ekstr¨ om, and the ISOLDE Collaboration, Phys. Lett. 133B , 47 (1983). [16] A. A. Radzig, B. M. Smirnov, Reference Data on Atoms, Mol ecules and Ions (Springer, Berlin, 1985). [17] P. Hafner and W. H. E. Schwarz, J. Phys. B 11, 2975 (1978). [18] J. Bruneau, J. Phys. B 17, 3009 (1984). [19] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonski i, Quantum Theory of Angular Momentum (World Scientific, 1988). [20] V. Spevak, N. Auerbach, and V.V. Flambaum, Phys. Rev. C 56, 1357 (1997). [21] V.V. Flambaum, Phys. Lett. B 320, 211 (1994). 13TABLES TABLE I. Energies and hyperfine structure constants of lower excited states of137Ba (I=3/2 µ=0.937365) and213Ra (I=1/2, µ=0.6133). Atom State Energies (cm−1) hfs constant A (MHz) Calc. Exper. [14] Calc. Exper. [15] Ba 6 s5d3D1 9225 9034 -632 -520.5 3D2 9346 9216 357 415.9 3D3 9554 9596 504 456.6 1D2 12147 11395 -26 -82.18 6s6p3P0 12203 12226 3P1 12577 12637 1233 1150.59 3P2 12464 13514 878 1P1 18042 18060 -48 -109.2 Ra 7 s7p3P0 12971 13078 3P1 13926 13999 8058 3P2 16660 16689 4637 1P1 21033 20716 -1648 -2315 7s6d3D1 13893 13716 -4108 3D2 14042 13994 1749 3D3 14299 14707 2744 1D2 17750 17081 -320 14TABLE II. Single-electron matrix elements of the P- and T-odd interactions for radium (pre- sented reduced matrix elements of an electron part of the Ham iltonian as specified in the table, see Appendix for details). All values are in atomic units. Matrix element Approximation Even or RHFaRHF+RPA+Σboddc Spin-independent PNC interaction, H=ρ(r)γ5 /angb∇acketleft7s1/2||H||7p1/2/angb∇acket∇ight -2769 -3832 Odd /angb∇acketleft7p3/2||H||6d3/2/angb∇acket∇ight 0.004 -146.8 Odd Nuclear Anapole moment, H=/vector αρ(r) /angb∇acketleft7s1/2||H||7p1/2/angb∇acket∇ight -503 -577 Odd /angb∇acketleft7s1/2||H||7p3/2/angb∇acket∇ight -0.508 20.26 Even /angb∇acketleft7p1/2||H||6d3/2/angb∇acket∇ight -0.024 -66.29 Even /angb∇acketleft7p3/2||H||6d3/2/angb∇acket∇ight 0.0006 -29.99 Odd /angb∇acketleft7p3/2||H||6d5/2/angb∇acket∇ight 0 11.71 Even Electron dipole moment, H= (β−1)ΣE /angb∇acketleft7s1/2||H||7p1/2/angb∇acket∇ight 12.06 17.05 Even /angb∇acketleft7p3/2||H||6d3/2/angb∇acket∇ight 0.556 2.082 Even Nuclear Schiff moment, H= 4π∇ρ(r) /angb∇acketleft7s1/2||H||7p1/2/angb∇acket∇ight -44400 -63027 Even /angb∇acketleft7s1/2||H||7p3/2/angb∇acket∇ight -32550 -56730 Odd /angb∇acketleft7p1/2||H||6d3/2/angb∇acket∇ight -1497 1873 Odd /angb∇acketleft7p3/2||H||6d3/2/angb∇acket∇ight -0.03 2767 Even /angb∇acketleft7p3/2||H||6d5/2/angb∇acket∇ight -0.07 8163 Even Nuclear Magnetic quadrupole moment, H=Amk /angb∇acketleft7s1/2||H||7p3/2/angb∇acket∇ight 17.28 25.06 Odd /angb∇acketleft7p1/2||H||6d3/2/angb∇acket∇ight 2.831 2.933 Odd /angb∇acketleft7p1/2||H||6d5/2/angb∇acket∇ight -0.2017 6.631 Even /angb∇acketleft7p3/2||H||6d5/2/angb∇acket∇ight 0.5389 4.011 Odd aRelativistic Hartree-Fock bCore polarization and core-valence correlation interacti on are included cEven means that /angb∇acketlefti||H||j/angb∇acket∇ight=/angb∇acketleftj||H||i/angb∇acket∇ight; odd means that /angb∇acketlefti||H||j/angb∇acket∇ight=−/angb∇acketleftj||H||i/angb∇acket∇ight. 15TABLE III. E1-transition amplitudes for Ba and Ra ( |/angb∇acketlefti||dz||j/angb∇acket∇ight|a0). Transition Ba Ra i j Amplitude Frequency ǫi−ǫj(a.u.) Amplitude Frequency ǫi−ǫj(a.u.) 3P03D1 2.3121 0.01473 3.0449 -0.002904 3P11S0 0.4537 0.05758 1.0337 0.06379 3P13D1 2.0108 0.01641 2.6389 0.001292 3P13D2 3.4425 0.01559 4.4399 0.0000247 3P11D2 0.1610 0.00566 0.0467 -0.01404 3P23D1 0.5275 0.02042 0.7166 0.01354 3P23D2 2.024 0.01959 2.7283 0.01228 3P23D3 4.777 0.01785 6.3728 0.009027 3P21D2 0.1573 0.00966 0.1499 -0.001790 1P11S0 5.236 0.08229 5.4797 0.09439 1P13D1 0.1047 0.04113 0.4441 0.03189 1P13D2 0.4827 0.04030 1.188 0.03063 1P11D2 1.047 0.03037 2.4053 0.01656 TABLE IV. Single-electron contributions to the two-electr on matrix element /angb∇acketleft3P1||H||3D2/angb∇acket∇ight. Dash means no contribution due to selection rules. Zero mean s very small contribution. Same units as in Table II. Transition H=−eraH=αρ(r)bH= 4π∇ρ(r)cH=Amkd 7s1/2−7p1/2 -0.3677 58.78 6421 - 7p1/2−7s1/2 0.0215 3.431 -375 - 7s1/2−7p3/2 -0.1306 -0.515 1441 -1.565 7p3/2−7s1/2 -0.0585 0.230 645 -0.289 7p1/2−6d3/2 0.0020 0.029 -1 -0.003 6d3/2−7p1/2 3.856 -54.08 -1528 -1.848 7p1/2−6d5/2 - - - 0 6d5/2−7p1/2 - - - -2.032 7p3/2−6d3/2 0.0004 0.006 -1 - 6d3/2−7p3/2 -0.2470 3.522 325 - 7p3/2−6d5/2 0 0 0 0 6d5/2−7p3/2 1.364 2.442 -1702 -0.736 Total 4.4399 13.85 5226 -6.473 aForE1 transition amplitude bFor anapole moment contribution cFor Schiff moment contribution dFor Magnetic quadrupole moment contribution 16TABLE V. Parity non-conserving E1-transition amplitude in duced by nuclear anapole moment I F F′/angb∇acketleftdz/angb∇acket∇ightin units 10−10κaiea0 1S0−3D11S0−3D2 0.5 0.5 1.5 2.05 -20.3 1.5 1.5 0.5 -0.58 5.7 1.5 1.5 -1.4 13.8 1.5 2.5 1.3 -12.9 TABLE VI. EDM of Ra atom in the3D2state induced by nuclear Schiff moment I F d z(a.u.) dz(ecm) 0.5 1.5 −0.94×108S −0.19×10−11ηa−0.36×10−19η 1.5 0.5 −0.16×108S −0.42×10−11ηb−0.80×10−19η 1.5 1.5 −0.30×109S −0.81×10−11ηb−0.15×10−18η 1.5 2.5 −0.28×109S −0.76×10−11ηb−0.14×10−18η aNuclear Schiff moment Sis assumed to be S= 400 ×108η efm3[20] bS= 300 ×108η efm3[20] TABLE VII. EDM of223Ra isotope ( I= 3/2) in the3D2state induced by nuclear magnetic quadrupole moment F d zadzb 0.5 1344 Mm e 7.4×10−20η e·cm 1.5 1292 Mm e 7.0×10−20η e·cm 2.5 −806Mm e −4.4×10−20η e·cm aIn terms of nuclear magnetic quadrupole moment M bMis assumed to be M= 10−19(η/m p)e·cm, [21] where mpis the proton mass 17TABLE VIII. Lifetimes of lower short-living states of Ba and Ra Atom State Lower states to decay to Lifetime via E1-transitions This work Other data Ba3P03D1 2.83µs 3P11S0,3D1,3D2,1D2 1.37µs 1.2 µsa 3P23D1,3D2,3D3,1D2 1.41µs 1P11S0,3D1,3D2,1D2 9.1 ns 8.5 nsa Ra3P11S0,3D1,3D2 505 ns 420 nsb, 250 nsc 3P23D1,3D2,3D3 74.6 ns 3D13P0 617µs 800 µsd 1D23P1,3P2 38 ms 1P11S0,3D1,3D2,1D2 5.5 ns aReference [16] bReference [17] cReference [18] dEstimation, Reference [9] 18

arXiv:physics/9912029v1 [physics.gen-ph] 14 Dec 1999THE BAG MODEL OF NUCLEI Nguyen Tuan Anh1 Department of Theoretical Physics, Faculty of Physics, Hanoi National University, Hanoi, Vietnam. (June, 1993) Abstract The basic assumptions and the general results of our bag mode l for nuclei are presented in detail. Nuclei are considered in a unified integ ration of the mean field theory and the MIT bag model. Bachelor Thesis, 1993. 1Present Address: Institute for Nuclear Science and Technique, Hanoi, Vietna m. 1Table of Contents Chapter 1. The Mean Field Theory of Nuclei and The Bag Model of Baryon 1. Mean Field Theory of Nuclei 2. Relativistic Nuclear Physics 3. Bag Model of Baryon Chapter 2. The Bag Model of Nuclei ( Z=N) 1. Main Assumption 2. Wave Function for Baryon 3. Radius and Binding Energy of Nuclei Chapter 3. The Bag Model of Nuclei ( Z/ne}ationslash=N) 1. Basic Assumption 2.A-dependence of Nuclear Radius 3. Weizssacker Formula Conclusion and Discussion 2Chapter 1 The Mean Field Theory of Nuclei and The Bag Model of Baryon 1.1 Mean Field Theory of Nuclei (MFT) Two model field theories of the nuclear system were studied in detail by Serot and Walecka [1]. The first is based on baryons and neutral scalar and vecto r mesons (model QHD-I). The quanta of the fields are the nucleons ( n,p) and the sigma ( σ) and omega ( ω) mesons. The neutral scalar meson couples to the scalar density of bar yons through gsψψφand the neutral vector meson couples to the conserved baryon curren t throughgvψγµψVµ. The Lagrangian density for this model is L=ψ[γµ(i∂µ−gvVµ)−(M−gsφ)]ψ +1 2/parenleftbig ∂µφ∂µφ−m2 sφ2/parenrightbig −1 4FµνFµν+1 2m2 vVµVµ, (1.1) where Fµν=∂µVν−∂νVµ. Lagrange’s equations yield the field equations: [γµ(i∂µ−gvVµ)−(M−gsφ)]ψ= 0, (1.2) (∂µ∂µ−m2 s)φ=gsψψ, (1.3) ∂µFµν−m2 vVν=gvψγνψ, (1.4) Eq. (1.2) is the Dirac equation with the scalar and vector fiel ds. Eq. (1.3) is simply the Klein-Gordon equation with a scalar source. And Eq. (1.4 ) looks like massive QED with the conserved baryon current, Aν=ψγνψ. (1.5) 3The energy-momentum tensor is: Tµν=1 2/bracketleftbigg −∂λφ∂λφ+m2 sφ2+1 2FλσFλσ−m2 vVλVλ/bracketrightbigg gµν +iψγµ∂νψ+∂µφ∂νφ+∂νVλFλµ. (1.6) Lagrange’s equations ensure that this tensor is conserved a nd satisfies ∂µTµν= 0. It follows that the energy-momentum Pνdefined by Pν=/integraldisplay d3xT0ν(1.7) is a constant of motion. We observe that at high baryon density, the scalar and vector field operators can be replaced by their expectation values, which then serve as cl assical, condensed fields in which the baryons move, φ→ /an}b∇acketle{tφ/an}b∇acket∇i}ht ≡φ0, (1.8) Vµ→ /an}b∇acketle{tVµ/an}b∇acket∇i}ht ≡δµ0V0. (1.9) For a static, uniform system, the quantities φ0andV0are constants independent of xµ. Rotational invariance implies that the expectation value /an}b∇acketle{t− →V/an}b∇acket∇i}htvanishes. The condensed, constant, classical fields φ0andV0are directly related to the baryon sources. The source for V0is simply the baryon density ρA=A/V. Since the baryon current is conserved, the baryon number A=/integraldisplay Vd3xψψ (1.10) is a constant of motion. For a uniform system of Abaryons in a volume V, the baryon density is also a constant of motion. In contrast, the source forφ0involves the expectation value of the Lorentz scalar density /an}b∇acketle{tψψ/an}b∇acket∇i}ht ≡φ0. This quantity is dynamical that must be calculated self-consistently using the thermodynamic arg ument that an isolated system at fixedAandV(and zero temperature) will minimize its energy, ∂E(A,V;φ0) ∂φ0= 0. (1.11) In the MFT, the Lagrangian density is LMFT=ψ/bracketleftbig iγµ∂µ−gvγ0V0−(M−gsφ0)/bracketrightbig ψ −1 2m2 sφ2 0+1 2m2 vV2 0. (1.12) Hence, the Dirac equation is linear, /bracketleftbig iγµ∂µ−gvγ0V0−(M−gsφ0)/bracketrightbig ψ= 0, (1.13) 4and may be solved directly. We seek normal-mode solutions of the formψ(xµ) =ψ(/vector x)e−iEt. This leads to Hψ(/vector x) =Eψ(/vector x), (1.14) H≡[−i/vector α.∇+gvV0+β(M−gsφ0). The effective mass M∗is defined by M∗=M−gsφ0. (1.15) The condensed scalar field φ0thus serves to shift the mass of the baryons. Evidently, the condensed vector field V0shifts the frequency (or energy) of the solutions, E=gvV0+E∗. (1.16) In the MFT, the energy-momentum tensor is Tµν MFT=iψγµ∂ν−1 2/parenleftbig m2 vV2 0−m2 sφ2 0/parenrightbig gµν. (1.17) It then follows that Pν=/integraldisplay Vd3xT0ν. (1.18) The second nuclear model is more realistic. To discuss nucle i withZ/ne}ationslash=N, it is necessary to extend model QHD-I to include ρmesons, which couple to the isovector current, and the coulomb interaction. Since we are also interested in comparing quantitative pred ictions with experiment, we will extend the model to include the ρmeson and photon fields (QHD-II). The Lagrangian density is L=ψ/bracketleftbigg γµ/parenleftbigg i∂µ−gvVµ−1 2gρ/vector τ./vectorbµ−1 2e(1 +τ3)Aµ/parenrightbigg −(M−gsφ)/bracketrightbigg ψ +1 2/parenleftbig ∂µφ∂µφ−m2 sφ2/parenrightbig −1 4FµνFµν+1 2m2 vVµVµ −1 4/vectorGµν./vectorGµν+1 2m2 ρ/vectorbµ./vectorbµ−1 4HµνHµν, (1.19) where Fµν=∂µVν−∂νVµ, /vectorGµν=∂µ/vectorbν−∂ν/vectorbµ, Hµν=∂µAν−∂νAµ. The field equations (1.2 to (1.4) must also be extended to incl ude contributions from the rho and photon fields, (∂µ∂µ−m2 ρ)/vectorbµ=gρψ/vector τγ µψ,(1.20) /squareAµ=eψ1 +τ3 2γµψ,(1.21) /bracketleftbigg γµ/parenleftbigg i∂µ−gvVµ−1 2gρ/vector τ./vectorbµ−1 2e(1 +τ3)Aµ/parenrightbigg −(M−gsφ)/bracketrightbigg ψ= 0 (1.22) 5IfN/ne}ationslash=Z, the neutral field ρ0corresponding to b(3) µcan develop a classical, constant ground-state expectation value in nuclear matter accordin g to /an}b∇acketle{tb(j) µ/an}b∇acket∇i}ht=δµ0δj3b0. (1.23) The baryon field, however, obeys a Dirac equation analogous t o (1.13) namely, /bracketleftbigg iγµ∂µ−gvγ0V0−1 2gρτ3γ0b0−1 2e(1 +τ3)γ0A0−(M−gsφ0)/bracketrightbigg ψ= 0 (1.24) Although the baryon field is still an operator, the meson field s are classical; hence, Eq. (1.24) is linear, and we may seek normal-mode solutions of th e formψ(xµ) =ψ(/vector x)e−iEt. This leads to hψ(/vector x) =Eψ(/vector x), (1.25) h≡/bracketleftbigg −i/vector α.∇+gvV0+1 2gρτ3b0+1 2e(1 +τ3)A0+β(M−gsφ0)/bracketrightbigg , which defines the single-particle Dirac Hamiltonian h. The single-particle wave functions in a central, parity-co nserving field may be written as ψα(/vector x) =ψnkmτ(/vector x) =/parenleftBigg Gnkτ(r) rΦkm iFnkτ(r) rΦ−km/parenrightBigg ζτ. (1.26) The equations for the baryon wave functions follow immediat ely upon substituting (1.26) into (1.25): dGa dr+k rGa−/bracketleftbigg Ea−gvV0−1 2gρτab0−1 2e(1 +τa)A0+M−gsφ0/bracketrightbigg Fa= 0,(1.27) dFa dr−k rGa+/bracketleftbigg Ea−gvV0−1 2gρτab0−1 2e(1 +τa)A0−M+gsφ0/bracketrightbigg Ga= 0.(1.28) For a given set of meson fields, the Dirac equations (1.27) and (1.28) may be solved by integrating outward from the origin and inward from large r, matching the solutions at some intermediate radius to determine the eigenvalues Ea. Analytic solutions in the regions of large and small rallow the proper boundary conditions to be imposed. 1.2 Relativistic Nuclear Physics Relativistic nuclear physics was first pioneered and develo ped by Shakin and Celenza [2]. The systematic application of this theory is able to resolve some long-standing puzzles in the theory of nuclear structure such as the binding energy an d the saturation density of nuclear matter, the effective force in nuclei, and the nucleo n self-energy for bound and 6continuums. Its success is based upon the use of the Dirac equ ation for the description of motion of a nucleon. The potentials appearing in the Dirac equation are assumed to contain large (Lorentz) scalar and vector fields. The scalar fields enter the Dirac equation in the same way as th e nucleon mass. Since these fields are quite large ( −400 MeV), they have the effect of including a major reduction of the nucleon mass when the nucleon is in the nuclear matter. It is the description of this change of mass of the nucleon that is an essential element in t he success of the relativistic approach. It is further necessary to understand that the vec tor field seen by a nucleon is large and repulsive, so that the energy of the nucleon in th e nuclear matter does not differ very much from the energy of a nucleon moving in the weak fields which appear in the standard Schrodinger description. More precisely, t he description relation relating the energy and momentum of a nuclear quasiparticle is simila r to that of the Schrodinger theory. Therefore, the system may be said to exhibit “hidden ” relativity. Indeed, for decades the Schrodinger approach to nuclear str ucture physics provided a reasonably satisfactory model of nuclear dynamics. It is on ly in the last decade that the true relativistic features of the system have become appare nt. We write the Dirac equation for a nucleon in the nuclear matte r as [/vector α./vector p+γ0m+V(/vector p)]φ(/vector p,s) =ǫφ(/vector p,s), whereV(/vector p) is the potential. It will be useful to introduce the self-en ergy Σ(/vector p) =γ0V(/vector p) and rewrite this equation as [/vector γ./vector p+m+ Σ(/vector p)]φ(/vector p,s) =γ0ǫφ(/vector p,s). Now let us assume that the self-energy is of the form Σ(/vector p) =A+γ0B, so that we have [/vector γ./vector p+ (m+A)]φ(/vector p,s) =γ0(ǫ−B)φ(/vector p,s). A positive-energy spinor solution of this equation is φ(/vector p,s) =/parenleftBigg /tildewidem /tildewideE(/vector p)/parenrightBigg1/2 u(/vector p,s,/tildewidem), where u(/vector p,s,/tildewidem) =/tildewideE(/vector p) +/tildewidem 2/tildewidem/parenleftBigg χs /vector σ./vector p /tildewideE(/vector p)+/tildewidemχs/parenrightBigg . Hereu(/vector p,s,/tildewidem) is the positive-energy solution of the Dirac equation with out interac- tion, except for the fact that the nucleon mass mhas been replaced by /tildewidem=m+Aand /tildewideE(/vector p) =/radicalbig /vector p2+/tildewidem2. The normalization chosen here is u†u=/tildewideE//tildewidem, 7so that φ†φ= 1. We further note that the energy eigenvalue is ǫ=B+/radicalbig /vector p2+/tildewidem2 =m+B+A+/vector p2 2/tildewidem+· · ·. Now, as we have mentioned, Ais large and negative ( −400 MeV) and Bis large and positive (300 MeV). Therefore AandBlargely cancel and the dispersion relation is essentially the same as that which one could find in a nonrel ativistic model. The development of this simple relativistic nuclear model lead s to two categories. The first we will call Dirac Phenomenology. This category is distingu ished by having several free parameters which are adjusted to fit nuclear date. The second category will be called Relativistic Brueckner-Hartree-Fock (RBHF) theory and is characterized as having no free parameters other than those introduced in fitting free - space nucleon - nucleon scattering data. Interest in the development of the RBHF app roximation grew out of the successful application of Dirac phenomenology to the de scription of nucleon - nucleus scattering data. We have so far presented only the main idea and the materials n ecessary to our con- sideration. For those who are interested to the detailed res ults of the relativistic nuclear physics, please, see the monograph of Celenza and Shakin quo ted above and the references herein. 1.3 Bag Model of Baryons It is well accepted QCD is the theory of strong interactions. However, in general, QCD is never solvable: at low energies and small momenta transfer t he running coupling constant αs>1. The bag model, outlined for the first time by the group of M.I .T. theorists [3], is a phenomenological approach, in which two basic features of QCD are incorporated: asymptotic freedom and confinement. The main assumption of the M.I.T. bag model states that, bary on is considered to be a bag of spherical shape, in which the constituent quarks mov e freely and are described by the Dirac equation Hψ=i∂ψ ∂t, with the Hamiltonian H=/vector α./vector p+βM. Consider the case k=−1, which is the S1/2level. The solution of this equation has 8the form ψn,−1(/vector r,t) =Nn,−1 /radicalBig E+M Ej0/parenleftbigωr R/parenrightbig χm −1 −i/radicalBig E−M Ej1/parenleftbigωr R/parenrightbig χm 1 e−iEt. If we parametrize the energy levels as /tildewideEnk=ωnk/R,/tildewideEnk=√ E2−M2, the density of quarks is readily calculated as J0=ψγ0ψ∼/bracketleftbigg j2 0/parenleftBigωr R/parenrightBig +E−M E+Mj2 1/parenleftBigωr R/parenrightBig/bracketrightbigg θV, where θV=/braceleftbigg 1r≤R 0r>R. Thus, the density certainly does not vanish at r=R. Clearly, although the lower component is suppressed for small r, it does make a sizeable contribution near the surface of the bag. Of course it is natural to ask whether this is not un usual in comparison with nonrelativistic experience, where ψ(R) would be zero. However, such a solution would not be consistent with the linear Dirac equation. What count s is that there should be no current flow through the surface of the confining region. For e xample, in the MIT bag model it is required that nµψγµψ= 0 at the surface - where nµis a unit four vector normal to the surface of the confining reg ion. In the MIT bag model this condition is imposed through a linea r boundary condition iγ.nψ =ψ at the surface. This implies ψ†=−iψ†γ†.n, and hence ψ=−iψγ.n, because γµ=γ0㵆γ0. Consider now the normal flow of current through the bag surfac e: inµJµ=inµψγµψ = (iψγ.n)ψ=ψ(iγ.nψ) =−ψψ=ψψ= 0. 9Thus, it is not the density, but ψψwhich should vanish at the boundary in the rela- tivistic theory, ψψ/vextendsingle/vextendsingle r=R=E+M Ej2 0(ω)−E−M Ej2 1(ω) = 0. That is, the matching condition is exactly equivalent to the linear boundary condition (l.b.c.) for the static spherical MIT bag, iγ.nψ =−iγ.ˆrψ=ψ, where nµ= (0,ˆr). We consider the energy-momentum tensor for a model, Tµν V=TµνθV, andTµνis the familiar energy-momentum tensor for a free Dirac field Tµν=iψγµ∂νψ. The condition for overall energy and momentum conservation is that the divergence of the energy-momentum tensor should vanish, and this is cer tainly true for Tµν, as is easily proven from the free Dirac equation ∂µTµν= 0. However, the fact that these quarks move freely only inside t he restricted region of spaceVleads to problems. Indeed, ∂µθV=nµ∆s, where ∆ sis a surface delta function ∆s=−n.∂(θV). In the static spherical case we find that ∆ sis simplyδ(r−R). Putting all these together we obtain ∂µTµν V=iψγ.n∂νψ∆s, and using the l.b.c. ∂µTµν V=−1 2∂ν(ψψ)/vextendsingle/vextendsingle s∆s=−Pnν∆s, wherePis the pressure exerted on the bag wall by the contained Dirac gas P=−1 2n.∂ν(ψψ)/vextendsingle/vextendsingle s. Clearly, this model violates energy-momentum conservatio n. Furthermore, this violation is an essential result of the confinement process. 10The resolution of this problem, we add an energy density term BθVto the Lagrangian density. Then (since Tµνinvolves Lgµν) the new energy-momentum tensor Tµν MIThas the form Tµν MIT= (Tµν+Bgµν)θV. Therefore, the divergence of the energy-momentum tensor is ∂µTµν MIT= (−P+B)nν∆s, which will vanish if B=P=− −1 2n.∂ν(ψψ)/vextendsingle/vextendsingle s. Therefrom a relativistic bag model of nuclei will be propose d, theA-dependence of nuclear radius will be calculated and the Weizssacker formu la for nuclear binding energy will be derived exactly if the corresponding parameters of t he model are adequately fitted. 11Chapter 2 Bag Model of Nuclei ( Z=N) 2.1 Main Assumption We consider the simplest possible case of Abaryons moving inside a spherical volume of radiusR, outside of which there is a pressure exerted on the nuclear s urface. Let us therefore begin with the Dirac equation for a baryon of massM: (iγµ∂µ−M−Σ)ψ(xµ) = 0, (2.1) where Σ is the baryon self-energy having the form Σ =φ+γµVµ. (2.2) Inserting (2.2) into (2.1) we obtain the equation [γµ(i∂µ−Vµ)−(M+φ)]ψ(xµ) = 0. (2.3) Eq. (2.3) is nonlinear quantum field equation and its exact so lution is very com- plicated. We have therefore made little progress by writing down this equation with a suitable method for solving it. In the MFT, φ→ /an}b∇acketle{tφ/an}b∇acket∇i}ht ≡φ0, (2.4) Vµ→ /an}b∇acketle{tVµ/an}b∇acket∇i}ht ≡δµ0V0. (2.5) For a static, uniform system the quantities φ0andV0are constants. Hence, the Dirac equation is linear, [iγµ∂µ−γ0V0−(M+φ0)]ψ(xµ) = 0, (2.6) and may be solved directly. Our basic assumption is formulated: the nucleus Ais considered to be a MIT bag, inside of which the motion of nucleon is described by the Dira c equation (2.6). The quantitiesφ0andV0can be determined only after we have fitted to experimental da ta. 12Let us next consider the energy-momentum conservation for n uclear bag. For stable nuclei, there should be no current flow through the surface of the confining region. In the MIT bag model it is required that inµJµ=−ψψ=ψψ= 0 (2.7) at the surface, where nµis a unit four vector normal to the surface of the confining reg ion. Thus,ψψwhich should vanish at the boundary in a relativistic theory . The matching condition of the present model is exactly equivalent to the l inear boundary condition for the static spherical MIT bag. The Lagrangian density for the present model is L=ψ[iγµ∂µ−γ0V0−(M+φ0)]ψθV+BθS, (2.8) whereBθSis a energy density term. Then the energy-momentum tensor has the form Tµν Bag=iψγµ∂νψθV+BgµνθS, (2.9) whereθVandθSdefine the bag volume and the surface θV=/braceleftbigg1r≤R 0r>R, θ V=  0r<R 1r=R 0r>R(2.10) andBis the constant surface tension. Therefore, the divergence of the energy-momentum tensor is ∂µTµν Bag=/bracketleftbigg1 2n.∂(ψψ)/vextendsingle/vextendsingle S+ 2B/bracketrightbigg nν∆S = (−PS+ 2B)nν∆S. (2.11) The condition for energy and momentum conservation is ∂µTµν Bag= 0, (2.12) hence B=−1 4n.∂(ψψ)/vextendsingle/vextendsingle S, (2.13) wherePSis the pressure exerted on the bag wall by the contained Abaryons. E. (2.13) involves the square of the baryon fields, and is refe rred to as the nonlin- ear boundary condition of the MIT bag model of nuclei. Becaus e of this condition the introduction of a constant surface tension Binvolves no new parameters. Eq. (2.6) and the condition (2.13) constitute the basic ingr edients of our model. 132.2 Wave Function for Baryon The Dirac equation for the present model is linear and may be s olved directly. We seek normal-mode solutions of the form ψ(xµ) =ψ(/vector r)e−iEtθV. (2.14) The Dirac equation then becomes Hψ(/vector r) =Eψ(/vector r), (2.15) H≡[−i/vector α.∇+V0+β(M+φ0)]. The effective mass M∗is defined by M∗=M+φ0. (2.16) The scalar field φ0thus serves to shift the mass of the baryons. Evidently, the v ector field V0shifts the frequency (or energy) of the baryon, E∗=E−V0. (2.17) Hence, Eq. (2.15) becomes (−i/vector α.∇+βM∗)ψ(/vector r) =E∗ψ(/vector r). (2.18) The single-particle wave functions in a central, parity-co nserving field may be written as ψα(/vector r) =ψnkmτ(/vector r) =/parenleftBigg Gnkτ(r) rΦkm iFnkτ(r) rΦ−km/parenrightBigg ζτ. (2.19) Their angular momentum and spin parts are simply spin spheri cal harmonics Φkm=/summationdisplay ml,ms/an}b∇acketle{tlml1 2ms|l1 2jm/an}b∇acket∇i}htYlmlχms, (2.20) k=/braceleftbiggl= +(j+ 1)>0 −(l+ 1) = −(j+ 1)<0, (2.21) whereYlmlis a spherical harmonic and χmsis a two-component Pauli spinor. The label α,{α}={a;m}={nkτ;m}, specifies the full set of quantum numbers describing the single-particle solutions. Since the system is assumed sph erically symmetric and parity conserving, αcontains the usual angular-momentum and parity quantum num bers.ζτis a two-component isospinor. The principal quantum number is denoted by n. The phase choice in (2.19) leads to real bound-state wave functions GandFfor real potentials in Hamiltonian (2.15). 14The equations for the baryon wave functions follow immediat ely upon substituting (2.19) into (2.18)1: /parenleftbiggd dr+k r/parenrightbigg G−(E∗+M∗)F= 0, (2.22) /parenleftbiggd dr−k r/parenrightbigg F+ (E∗−M∗)G= 0. (2.23) These equations contain all information about the static gr ound-state nucleus. They are coupled linear differential equations that may be solved exactly for a given set of potentials. Consider the case k=−1 which is the level S1/2. Eq. (2.22) implies F= (E∗+M∗)−1/parenleftbiggd dr−1 r/parenrightbigg G, (2.24) so that defining W2=E∗2−M∗2, (2.25) the equation for the upper component of ψα(/vector r) is /parenleftbiggd2 dr2+W2/parenrightbigg G= 0 (2.26) The solution of this equation has the form G(r) =CsinWr, (2.27) and hence [from Eq. (2.24)] F(r) =C(E∗+M∗)−1(WcosWr−sinWr/r ). (2.28) The solutions of the Dirac equation (2.18) come from (2.19) i s written as ψα(/vector r) =C/parenleftbiggj0(Wr)Φ−1m −iW E∗+M∗j1(Wr)Φ1m/parenrightbigg ζτ. (2.29) The normalization condition that yields the numbers of bary ons contained in the nucleus A, /integraldisplay d3x ψ†ψ=A. (2.30) Now let us assume that the bag has a spherical shape with radiu sR.ψψwhich should vanish at the boundary in a relativistic theory. Eq. (2.29) i mplies that [see Eq. (2.7)], ψψ/vextendsingle/vextendsingle r=R=j2 0(WR)−W2 (E∗+M∗)2j2 1(WR) = 0 (2.31) 1We use /vector σ.∇(GΦkm/r) =−(1/r)(d/dr+k/r)GΦ−km, and a similar relation for F. 15and hence j0(WR) =/radicalbigg E∗−M∗ E∗+M∗j2 1(WR). (2.32) This is appropriate boundary condition for confined baryons . Thus, the boundary condition of the MIT bag model is used, which provided the eig enfrequency of baryon ωa, if we parametrize the energy levels (wavenumber) as Wa=ωa/R; {a}={nkτ}, k =−1, (2.33) wherenis the principal quantum number and ωasatisfies the equation [from Eq. (2.32)] tanωa=ωa 1−M∗R−/radicalbig ω2 a+M∗2R2. (2.34) 2 Hence, the eigenvalues Eamay be determined by matching the solutions at some intermediate radius. Analytic solutions in the restricted region of space Vallow the proper boundary conditions to be imposed. Taking into consi deration (2.33) we get the energy spectra for baryon, Ea=±/radicalbig W2 a+M∗2+V0 =±/radicalbigg ω2 a R2+M∗2+V0. (2.35) For convenience, the sign ( −) drops out in what follows. The single-particle wave functions now has the form ψα(/vector r) =C/parenleftbiggj0/parenleftbigωa Rr/parenrightbig Φ−1m −iWa E∗a+M∗j1/parenleftbigωa Rr/parenrightbig Φ1m/parenrightbigg ζτ. (2.36) Given the general form of the solutions in (2.36), we may now e valuate the local baryon density. Assume that the nuclear ground state consists of fil led shells up to some value of nandk. This is consistent with spherical symmetry and is appropri ate for magic nuclei. With these assumptions, the local density of baryons is read ily calculated as ρA=ψ†ψθV =C2/bracketleftbigg j2 0/parenleftBigωar R/parenrightBig +E∗ a−M∗ E∗ a+M∗j2 1/parenleftBigωar R/parenrightBig/bracketrightbigg θV. (2.37) Substituting Eq. (2.37) into Eq. (2.30), we can calculate th e normalization constant which is defined by Eq. (2.30) for k=−1, C2=A 4πR3j2 0(ωa)/parenleftbiggE∗ a+M∗ E∗a/parenrightbiggE∗ a(E∗ a−M∗)R 2E∗2aR−2E∗a+M∗. (2.38) 2We use (2.25) and (2.33). 163 Finally, the single-particle wave functions may be written as ψα(/vector r) =N /radicalBig E∗a+M∗ E∗aj0/parenleftbigωa Rr/parenrightbig i/radicalBig E∗a−M∗ E∗a/vector σ./hatwider j1/parenleftbigωa Rr/parenrightbig Φm 1/2ζτ, (2.39) where N2=A 4πR3j2 0(ωa)E∗ a(E∗ a−M∗)R 2E∗2aR−2E∗a+M∗. (2.40) By taking the explicit solutions of the Dirac equation ψα(/vector r) =Nα /radicalBig E∗a+M∗ E∗ajk∓1/parenleftbigωa Rr/parenrightbig i/radicalBig E∗a−M∗ E∗a/vector σ./hatwider jk/parenleftbigωa Rr/parenrightbig Φkmζτ, (2.41) where the upper (or lower) sign refers to kpositive (or negative), it is easily verified that onlyk= 1 (ork=−1) leads to an angle-independent result on the right-hand si de of Eq. (2.13). Thus only states with j= 1/2 can satisfy the nonlinear boundary condition as given. 2.3 Radius and Binding Energy of Nuclei We have seen that the only change in the calculation of the ene rgy in the MIT bag model for nuclei is the addition of a surface term, BS. It is assumed that Bis a universal constant, chosen to fit one piece of data. Once Bis chosen, because of the nonlinear boundary condition the radius of the bag is uniquely determi ned for each nuclei. The meaning of this addition to energy-momentum tensor can b e clarified by consid- ering the total energy of the bag state, P0≡E(A) =/integraldisplay d3x T00 Bag=/integraldisplay d3x(T00θV+BθS), (2.42) which we shall label E(A) as a precursor to our discussion of binding energy later. Ba sed on (2.35) and (2.42) the nuclear energy E(A) is derived immediately E(A) =AEa+ 4πR2B =A/radicalbigg ω2a R2+M∗2+AV0+ 4πR2B. (2.43) The first term is the kinetic energy, while the second is a surf ace term. Essentially it implies that it cost an energy BSto make this tension at the bag surface within which 3We use/integraltextR 0dr r2j2 m(ωr/R) =R3 2/bracketleftbig j2 m(ω) +j2 m±1(ω)−2m+1 ωjm(ω)jm±1(ω)/bracketrightbig . 17the baryons move. It should be intuitively clear that energy -momentum conservation is related to pressure balance at the bag surface, so that a smal l change in radius should not significantly increase E(A). Nevertheless, the nonlinear boundary condition implies that ∂E ∂R= 0. (2.44) We wish to stress that it is an assumption of the model that Bshould be constant for all nuclei. As all nuclear bags have radii in the region (1 .0÷1.2)A1/3fm, this assumption will be severely tested. Generalizing Eq. (2.43) to include exited states, the nonli near boundary condition implies ∂E(A) ∂R=−Aω2 a R2/radicalbig ω2a+M∗2R2+ 8πRB= 0, (2.45) and hence A=8πB ω2 aR3/radicalbig ω2a+M∗2R2 (2.46) The real and positive root Rof Eq. (2.46) is found out after an algebraic manipulation, R=r0A1/3, (2.47) where r0=/parenleftBigωa 4πB/parenrightBig1/3 α1/2;n= 0, k=−1, (2.48) α=(β/2)1/4 [1−(β/2)3/2]1/2+ (β/2)3/4, β=/bracketleftBigg/parenleftbigg256a 27/parenrightbigg1/2 + 1/bracketrightBigg1/3 −/bracketleftBigg/parenleftbigg256a 27/parenrightbigg1/2 −1/bracketrightBigg1/3 , a=/parenleftbiggAM∗3 8πBω2 a/parenrightbigg2 , (2.48) shows that r0actually depends weakly on A. The above obtained formula (2.47) is well known in nuclear ph ysics. It is one of the main successes of our model. Using Eq. (2.47) we can then simplify the expression for E(A): E(A) =AV0+/bracketleftBigg/radicalbig ω2a+r2 0M∗2A2/3 r0+ 4πBr2 0/bracketrightBigg A2/3. (2.49) Hence the binding energy per nucleon is obtained ε(A) =−(M−φ0) +/bracketleftBigg/radicalbig ω2 a+r2 0M∗2A2/3 r0+ 4πBr2 0/bracketrightBigg A−1/3. (2.50) 18Clearly, the remarkable result obtained for the binding ene rgy per nucleon of the bag model of nuclei was indeed a coincidence. As was known, the semi-empiric formula of Weizssacker [8, 11 ] for binding energy per nucleon reads ε(A) =−a1+a2A−1/3+a3/parenleftbiggZ−N 2A/parenrightbigg2 +a4Z2 A4/3, (2.51) in whicha1,a2,a3, anda4take the following values, in the energy unit equal to 0 .9311 MeV, a1= 16.9177, a 2= 19.120, a 3= 101.777, a 4= 0.7627. Now let us indicate that (2.51) is possibly derived from our m odel if the parameters φ0,V0andBare fitted adequately. Next confronting (2.50) with (2.51) w e conclude that the above mentioned parameters must fulfil equalities, M−V0=a1, (2.52)/bracketleftBigg/radicalbig ω2a+r2 0M∗2R2 r0+ 4πBr2 0/bracketrightBigg =a2. (2.53) It is worth to notice that parameter V0is explicitly defined by (2.52). The equation (2.53) constrains two unknown parameters of the theory, φ0andV0. The final term in Eq. (2.50) represents the contribution to th e surface energy from the positive-frequency states, where the mass has been shif ted by the constant, condensed scalar fieldφ0.φ0(or the effective mass M∗=M+φ0depends explicitly on the scalar field) is a dynamical quantity that must be calculated self-consis tently using the thermodynamic argument that an isolated system at fixed AandV(and zero temperature) will minimize its energy: ∂E(A,V;φ0) ∂φ0= 0. (2.54) However,φ0is related to ωaby the relation (2.34). As a consequence, (2.54) is replaced by conditions: ∂ ∂ωa[E(A) +λϕ] = 0, (2.55) ∂ ∂φ0[E(A) +λϕ] = 0, (2.56) in which ϕ= 1−M∗R−/radicalbig ω2a+M∗2R2−ωa tanωa. Substituting λfrom (2.55) and (2.56) one gets finally M∗R=[M∗R+ (ω2 a+M∗2R2)]ωatan2ωa ωatan2ωa+ (tanωa−ωa−ωatan2ωa)/radicalbig ω2a+M∗2R2, (2.57) 19withφ0andωaare the roots of the system (2.34) and (2.57). This model problem is exactly solvable. It retains the essen tial features of the bag model for nuclei: A-dependence of nuclear radius and the formula for nuclear bi nding energy. Furthermore, it yields a simple solution to the field equation. This solution and model problem thus provide a meaningful starting point f or describing the nuclear many-body system as well as a consistent basis for consideri ng nuclei with N/ne}ationslash=Zusing relativistic nuclear physics, the bag model and standard ma ny-body techniques. We proceed to investigate nuclei with N/ne}ationslash=Z. 20Chapter 3 Bag Model of Nuclei ( N/ne}ationslash=Z) To realistically discuss nuclei N/ne}ationslash=Z, it is necessary to extend bag model of nuclei N=Z to include neutral field, which couple to the isovector curre nt, and the coulomb interaction. IfZ/ne}ationslash=N, the neutral charged, isovector field corresponding to b(3) µcan develop a classical, constant ground-state expectation value in nuclear matter according to /an}b∇acketle{tb(3) µ/an}b∇acket∇i}ht=δµ0δj3b3. The only change in the results of Chapter 2 is that there are no w separate solutions for protons and neutrons, with the appropriate frequency mo difications: V0→/braceleftbiggV0+1 2b3+eA0 for proton, V0+1 2b3 for neutron. The field equation for uniform nuclear matter must also be ext end to include contri- bution from the classical field b3and the coulomb potential A0:/bracketleftbigg iγµ∂µ−γ0V0−1 2τ3γ0b3−1 2e(1 +τ3)γ0A0−(M+φ0)/bracketrightbigg ψ= 0. The Lagrangian density, which is obtained from Eq. (2.8) by r eplacing the classical fields. Thus in bag model of nuclei with N/ne}ationslash=Z L=ψ/bracketleftbigg iγµ∂µ−γ0V0−1 2τ3γ0b3−1 2e(1 +τ3)γ0A0−(M+φ0)/bracketrightbigg ψθV+BθS, which is a generalization of Eq. (2.8) to allow for classical , constant fields φ0,V0,b3, and A0. Hence, the energy-momentum tensor is still (2.9). 3.1 Basic Assumptions As was know, in Celenza and Shakin theory [2] the motion of nuc leon in nuclear matter is described by the Dirac equation, (iγµ∂µ−M−Σ)ψ(xµ) = 0, (3.1) 21where Σ is the nucleon self-energy having the form Σ =φ0+γ0V0, (3.2) withφ0andV0constants. Taking into account the isotopic degree of freedom and the Co ulomb interaction of protons, in our model, we assume Σ has the generalized from Σ =φ0+γ0V0+1 2τ3γ0b3+1 2e(1 +τ3)γ0A0, (3.3) in whichφ0,V0,b3, andA0are constants. Inserting (3.3) into (3.1) we obtain the equa tion, /bracketleftbigg iγµ∂µ−γ0V0−1 2τ3γ0b3−1 2e(1 +τ3)γ0A0−(M+φ0)/bracketrightbigg ψ= 0, (3.4) where the mass matrix Mis, of course, M=/parenleftbigg Mp0 0Mn/parenrightbigg Now our basic assumption is formulated: the nucleus Ais considered to be a M.I.T. bag, inside which the motion of nucleon is described by the Di rac equation (3.4). The parameters φ0,V0,b3, andA0will be fitted to experimental data. Let us next consider the energy-momentum conservation for n uclear bag. Let Tµνbe the energy-momentum tensor of nucleon, described by Eq. (3. 4), inside the bag. Then the total energy-momentum tensor Tµν Bagof the nucleus is clearly given by Tµν Bag=TµνθV+BθS, (3.5) whereθVandθSare the well-known step functions for volume and surface of t he bag, respectively, and Bis the surface tension. From (3.5) it follows that ∂µTµν Bag=/bracketleftbigg1 2n.∂(ψψ)/vextendsingle/vextendsingle S+ 2B/bracketrightbigg nν∆S. The energy-momentum conservation, ∂µTµν Bag= 0, leads to B=−1 4n.∂(ψψ)/vextendsingle/vextendsingle S, (3.6) which resembles the nonlinear boundary condition in the bag model for baryon [3]. Equation (3.4) and the relation (3.6) constitute the basic i ngredients of out model. 223.2A-dependence of Nuclear Radius The solution of (3.4) can be found in the form ψ(xµ) =/parenleftbiggψp(xµ) ψn(xµ)/parenrightbigg , (3.7) ψp(xµ) =ψp(/vector x)e−iEpt, ψn(xµ) =ψn(/vector x)e−iEnt. (3.8) Substituting (3.7) and (3.8) into (3.4) we arrive at the equa tions for protons and neutron, separately, /bracketleftbigg −i/vector α.▽+V0+1 2b3+eA0+β(Mp+φ0)/bracketrightbigg ψp(/vector x) =Epψp(/vector x), (3.9) /bracketleftbigg −i/vector α.▽+V0−1 2b3+β(Mn+φ0)/bracketrightbigg ψn(/vector x) =Enψn(/vector x). (3.10) For convenience, let us define M∗ p=Mp+φ0, M∗ n=Mn+φ0, E∗ p=Ep−V0−1 2b3−eA0, E∗ n=En−V0+1 2b3. With this in mind we rewrite (3.9) and (3.10) as follows /bracketleftbig −i/vector α.▽+βM∗ p/bracketrightbig ψp(/vector x) =E∗ pψp(/vector x), (3.11) [−i/vector α.▽+βM∗ n]ψn(/vector x) =E∗ nψn(/vector x). (3.12) It is known that the solutions of (3.11) and (3.12) read, resp ectively, ψp(/vector x) =Np /radicalBig E∗p+M∗p E∗pjk−1(Wpr)Φkm i/radicalBig E∗p−M∗p E∗pjk(Wpr)Φ−km , (3.13) ψn(/vector x) =Nn /radicalBig E∗n+M∗n E∗njk−1(Wnr)Φkm i/radicalBig E∗n−M∗n E∗njk(Wnr)Φ−km , (3.14) wherekandk−1 are the indices of the eigenfunctions corresponding to the eigenvalues of operator K, K=β(/vector σ./vectorl+ 1), W2 p=E∗2 p−M∗2 p, W2 n=E∗2 n−M∗2 n, (3.15) 23and the normalization constants Np,Nnare defined by /integraldisplay d3x ψ† pψpθV=Z, (3.16) /integraldisplay d3x ψ† nψnθV=N, (3.17) ZandNare the numbers of protons and neutrons contained in the nucl eusA. Now let us assume that the bag has a spherical shape with radiu sR. Then the boundary condition of the M.I.T. bag model is used, which pro vides the eigenfrequencies of proton and neutron, Ω pand Ω n, correspondingly, Ωp=WpR, Ωn=WnR. (3.18) As was known, Ω pand Ω nsatisfies the equations tanΩ p=Ωp 1−M∗ pR−/radicalbigΩ2 p+M∗2 pR2, (3.19) tanΩ n=Ωn 1−M∗nR−/radicalbig Ω2n+M∗2nR2. (3.20) It is worth to remember that (3.19) and (3.20) are derived for k= 1, to which corre- spond the only states satisfying (3.6). Taking into conside ration (3.15) and (3.18) we get the energy spectra for proton and neutron, respectively, Ep=±/radicalbigg Ω2p R2+M∗2p+V0+1 2b3+eA0, (3.21) En=±/radicalbigg Ω2 n R2+M∗2n+V0−1 2b3. (3.22) For convenience, the sign ( −) drops out in what follows. Based on (3.21), (3.22), and (3.5) the nuclear energy E(A) is derived immediately E(A) =Z/radicalbigg Ω2p R2+M∗2 p+N/radicalbigg Ω2n R2+M∗2 n+AV0+Z−N 2b3+eZA 0+ 4πBR2.(3.23) Next let us introduce the mean mass /an}b∇acketle{tM/an}b∇acket∇i}ht, the effective mass M∗and the mean fre- quency /an}b∇acketle{tΩ/an}b∇acket∇i}htof nucleons contained in nucleus A, /an}b∇acketle{tM/an}b∇acket∇i}ht=ZM p+NM n A, M∗=/an}b∇acketle{tM/an}b∇acket∇i}ht+φ0, A/radicalbigg /an}b∇acketle{tΩ/an}b∇acket∇i}ht2 R2+M∗2=Z/radicalbigg Ω2p R2+M∗2 p+N/radicalbigg Ω2n R2+M∗2 n. (3.24) 24It is easily prove that /an}b∇acketle{tΩ/an}b∇acket∇i}ht, defined by (3.24), really exists. Substituting of (3.24) into (3.23) leads to E(A) =A/radicalbigg /an}b∇acketle{tΩ/an}b∇acket∇i}ht2 R2+M∗2+AV0+Z−N 2b3+eZA 0+ 4πBR2. (3.25) The nonlinear boundary condition (3.6) requires ∂E(A) ∂R= 0, which yields A=8πB /an}b∇acketle{tΩ/an}b∇acket∇i}htR3/radicalBigg 1 +/parenleftbiggM∗R /an}b∇acketle{tΩ/an}b∇acket∇i}ht/parenrightbigg2 . (3.26) The real and positive root Rof (3.26) is found out after an algebraic manipulation, R=r0A1/3, (3.27) where r0=/parenleftbigg/an}b∇acketle{tΩ/an}b∇acket∇i}ht 4πB/parenrightbigg1/3 δ1/2, (3.28) δ1/2=(ξ/2)1/4 [1−(ξ/2)3/2]1/2+ (ξ/2)3/4, ξ=/bracketleftBigg/parenleftbigg256a 27+ 1/parenrightbigg1/2 + 1/bracketrightBigg1/3 −/bracketleftBigg/parenleftbigg256a 27+ 1/parenrightbigg1/2 −1/bracketrightBigg1/3 , a=/parenleftbiggAM∗3 8πB/an}b∇acketle{tΩ/an}b∇acket∇i}ht/parenrightbigg2 , (3.28) shows that r0actually depends weakly on A. The above obtained formula (3.27) is well known in nuclear ph ysics. It is one of the main successes of our model. Finally, the normalization constant NpandNngiven by (3.13) and (3.14), are calcu- lated fork= 1, Np=/parenleftbiggZ 4πR3/parenrightbigg1/2/bracketleftbig E∗ p(E∗ p−M∗ p)R/bracketrightbig1/2 j0(Ωp)/bracketleftbig 2E∗2 pR−2E∗ p+M∗ p/bracketrightbig1/2, Nn=/parenleftbiggN 4πR3/parenrightbigg1/2[E∗ n(E∗ n−M∗ n)R]1/2 j0(Ωn) [2E∗2 nR−2E∗ n+M∗ n]1/2(3.29) 253.3 Weizssacker Formula As was known, the semi-empiric formula of Weizssacker [7, 10 ] for binding energy per nucleon reads f=−a1+a2A−1/3+a3(Z−N)2 4A2+a4Z2 A4/3, (3.30) in whicha1,a2,a3, anda4take the following values, in the energy unit equal to 0 .9311 MeV, a1= 16.9177, a 2= 19.120, a 3= 101.777, and a4=3e2 5r0= 0.7627. The charge distribution radius r0for almost nuclei is fitted to be rc= 1.2162 10−13cm, (3.30) agrees well with experimental data for most nuclei. Now let us indicate that (3.30) is possibly derived from our m odel if the parameters φ0,V0,b3,A0, andBare fitted adequately. For this end, let us substitute (3.27) into (3.25), E(A) =AV0+/bracketleftBigg/radicalbig /an}b∇acketle{tΩ/an}b∇acket∇i}ht2+r2 0M∗2A2/3 r0+ 4πBr2 0/bracketrightBigg A2/3+b3(Z−N) 2+A0eZ. (3.31) Therefrom, the binding energy per nucleon is obtained f=V0− /an}b∇acketle{tM/an}b∇acket∇i}ht+/bracketleftBigg/radicalbig /an}b∇acketle{tΩ/an}b∇acket∇i}ht2+r2 0M∗2A2/3 r0+ 4πBr2 0/bracketrightBigg A−1/3+b3(Z−N) 2A+A0eZ A.(3.32) Next confronting (3.32) with (3.30) we conclude that the abo ve mentioned parameters must fulfil equalities V0− /an}b∇acketle{tM/an}b∇acket∇i}ht=−a1, (3.33)/radicalbig /an}b∇acketle{tΩ/an}b∇acket∇i}ht2+r2 0M∗2A2/3 r0+ 4πBr2 0=a2, (3.34) b3=a3Z−N 2A, (3.35) A0=a4Z eA1/3=3r0 5rceZ R. (3.36) It is clear that (3.33) and (3.36) express directly the physi cal meaning of b3andA0: – The mean field value b3is proportional to the relative ratio of the numbers of proto ns and neutrons, contained in nucleus A. 26– The mean value of Coulomb potential created by Zprotons equals to that created by a sphere of charge Ze, embedded in a nuclear medium, the dielectric coefficient of which is 3 r0/5rc. It is worth to notice that three parameters V0,b3, andA0are explicitly defined by (3.33), (3.35), and (3.36). The equation (3.34) contains tw o unknown parameters of the theory,φ0andB. As was shown in the Walecka theory [1], φ0is a dynamical quality and therefore it is defined self-consistently. Namely, we use th e thermodynamic argument that an isolated system with fixed baryon number Aand volume Vwill minimize its energy, ∂E(A,V;φ0) ∂φ0= 0. (3.37) However,φ0is related to Ω pand Ω nby the relations (3.19) and (3.20). As a conse- quence, (3.37) is replaced by the conditions: ∂ ∂Ωp[E(A) +α1ϕ1+α2ϕ2] = 0, (3.38) ∂ ∂Ωn[E(A) +α1ϕ1+α2ϕ2] = 0, (3.39) ∂ ∂φ0[E(A) +α1ϕ1+α2ϕ2] = 0. (3.40) in which ϕ1= 1−M∗ pR−/radicalBig Ω2 p+M∗2 pR2−Ωp tan Ω p, ϕ1= 1−M∗ nR−/radicalbig Ω2 n+M∗2 nR2−Ωn tanΩ n. Eliminating α1andα2from (3.38) and (3.39) and substituting them into (3.40) one gets finally 2ZΩpsin2Ωp sin2Ωp+ 2Ω p−ZM∗ pR/radicalbigΩ2p+M∗2pR2 M∗pR+/radicalbigΩ2p+M∗2pR2+2NΩnsin2Ωn sin2Ωn+ 2Ω n−NM∗ nR/radicalbig Ω2n+M∗2nR2 M∗ nR+/radicalbig Ω2 n+M∗2 nR2= 0, (3.41) φ0and Ω p, Ωnare the roots of the system (3.19), (3.20) and (3.41). 27Chapter 4 Conclusion and Discussion In the previous sections the basic assumptions and the gener al results of our bag model for nuclei are presented in detail. The model is built on a sim ple hypothesis: the nucleus is considered to be a MIT bag, in which the motion of nucleons i s described by the Dirac equation and the mean field values φ0,V0,b3, andA0are supposed to be constants. In addition to these mean fields, there exists the surface tensi onBof the bag that guarantees the energy-momentum conservation. Two major successes are: the formula (3.27) for the nuclear r adiusRand the Weizs- sacker formula with the parameters verified in (3.33-36). Al l the parameters appearing in the theoryφ0,V0,b3,A0, andBare, in principle, determined by the equations (3.33-36), (3.19, 20), and (3.41). Thus, our theory is a mathematically closed system. The devel- opment of the formalism suggested here will be carried out fo r various concrete nuclei in next papers. 28Acknowledgement I would like to thank Prof. Dr. Tran Huu Phat for his valuable c onducts on the final draft of the text. I am very thankful to Dr. Nguyen Xuan Han and Phan Huy Thien for useful helps and interest in the work. 29Bibliography [1] B. D. Serot and J. D. Walecka, Adv. in Nucl. Phys. ,16, 1, (1986). [2] L. S. Calenza and C. M. Shakin, Relativistic Nuclear Physics , World Scientific, (1986). [3] A. W. Thomas, Adv. in Nucl. Phys .,13, 1, (1983). [4] A. Arima and F. Iachello, Ann. Phys. (N.Y.), 99, 253, (1976). [5] Y. K. Gambhir, P. Ring, and A. Thicnet, Ann. Phys .,198, 132, (1990). [6] I. Tanihata, T. Kobayashi, S. Shimaura, and T. Minaminos o,Proceedings of I Intern. Conf. on PRadioactive Nuclear Beams , Berkerley, (1989), p. 429. [7] D. Hirata, H. Toki, T. Watabe, I. Tanihata, and B. V. Calso n,Phys. Rev. C44, 1467, (1991). [8] C. F. Weizssacker, Zs. f. Phys. ,96, 431, (1935). [9] P. A. Seeger and W. M. Haward, Nucl. Phys. ,A238 , 491, (1975). [10] A. E. S. Green, Phys. Rev. ,95, 1006, (1954). [11] Neumark, Solution of Cubic and Quartic Equation , (1965). 30

arXiv:physics/9912030v1 [physics.ao-ph] 14 Dec 1999Radiation of mixed layer near-inertial oscillations into the ocean interior J. Moehlis1, Stefan G. Llewellyn Smith2 February 2, 2008 1Department of Physics, University of California, Berkeley , CA 94720 2Department of Mechanical and Aerospace Engineering, Unive rsity of California, San Diego 9500 Gilman Drive, La Jolla, CA 92093-0411 Abstract The radiation from the mixed layer into the interior of the oc ean of near-inertial oscillations excited by a passing storm in the presence of th e beta effect is reconsidered as an initial-value problem. Making use of the fact that the m ixed layer depth is much smaller than the total depth of the ocean, the solution is obt ained in the limit of an ocean that is effectively infinitely deep. For a uniform initi al condition, analytical results for the velocity, horizontal kinetic energy densit y and fluxes are obtained. The resulting decay of near-inertial mixed layer energy in the p resence of the beta effect occurs on a timescale similar to that observed. 1 Introduction There is much observational evidence, starting with Webste r (1968) and Pollard and Millard (1970), that storms can excite near-inertial currents in th e mixed layer of the ocean. This phenomenon is evident in observations from the Ocean Storms Experiment (D’Asaro et al. 1995, Levine and Zervakis 1995, Qi et al. 1995). Simple model s which treat the mixed layer as a solid slab have been quite successful at explaining the pro cess by which wind generates such currents (see, e.g., Pollard and Millard (1970), D’Asaro (1 985)). A weakness of the model of Pollard and Millard (1970) is that it explains the decay of th ese currents with an arbitrary decay constant. Much subsequent work has attempted to deter mine the detailed character- istics of this decay, with possible mechanisms including no nlinear interactions which transfer energy to other frequencies (Henyey et al. 1986), turbulent dissipation (Hebert and Moum 1993), and the radiation of downward propagating near-iner tial oscillations (NIOs) excited 1by inertial pumping into the interior of the ocean (Gill 1984 ). The downward radiation of NIOs will be the focus of this paper. Observations give a timescale for the decay of the energy dep osited by the passing storm on the order of ten to twenty days (D’Asaro et al. 1995, Levine and Zervakis 1995, Qi et al. 1995). This timescale stands in contrast with estimates suc h as that by Gill (1984) that near- inertial currents decaying through the downward propagati on of NIOs and with a horizontal length scale typical of the atmospheric forcing mechanism c an remain in the mixed layer for longer than a year. To account for this difference, several me chanisms for the enhancement of vertical propagation of NIOs have been suggested. D’Asar o (1989) demonstrated that the β-effect causes a reduction of horizontal scales because the m eridional wavenumber evolves according to l=l0−βt, wherel0is the initial wavenumber, and l <0 corresponds to southward propagation; this accelerates the rate of inerti al pumping of energy out of the mixed layer, thereby enhancing the decay. The decay is also e nhanced through interaction with background geostrophic or quasigeostrophic flow (e.g. Balmforth et al. 1998, Balmforth and Young 1999, and van Meurs 1998). This paper reconsiders the vertical propagation of near-in ertial energy deposited into the mixed layer by a storm, in the presence of the β-effect, using a different approach from that of D’Asaro (1989). The analysis uses the formalism of Young a nd Ben Jelloul (1997) which is outlined in Section 2. In Section 3, a simplified model with three main assumptions is presented. First, the background flow is assumed to be consta nt in the zonal direction (i.e. independent of longitude with zero vorticity). Second, the buoyancy frequency is taken to be small in the mixed layer, and constant in the ocean interior ( i.e. beneath the mixed layer). Third, it is assumed that the storm has moved very rapidly acr oss the ocean and has created a horizontally uniform near-inertial current to the east co ncentrated within the mixed layer: it is the subsequent evolution of this motion that is examine d. Section 4 uses the fact that the depth of the ocean is very much larger than the mixed layer depth to formulate and solve the model for an ocean which is effectively infinitely deep. Se ction 5 discusses the results and suggests directions for further investigation. 2 The NIO equation We consider an ocean of infinite horizontal extent and depth D, with the mixed layer compris- ing the region −Hmix<z < 0, and the rest of the water column occupying −D<z< −Hmix. Thexandyaxes are taken to point to the east and north, respectively. T he buoyancy fre- quencyN=N(z) is an arbitrary piecewise continuous function of depth z. Young and Ben Jelloul (1997) derive an evolution equation fo r a complex field A(x,y,z,t ) which governs leading-order NIO motion in the presence of a s teady barotropic background 2flow and the β-effect: LAt+∂(ψ,LA ) ∂(x,y)+i 2f0∇2A+i/parenleftbigg βy+1 2ζ/parenrightbigg LA= 0, (1) where LA=∂ ∂z/parenleftBiggf2 0 N2∂A ∂z/parenrightBigg , (2) ψis the streamfunction for the background flow, ζ≡ ∇2ψis the associated vorticity, and the Coriolis parameter is f=f0+βy. Here ∇is the horizontal gradient, and ∇2=∂2 x+∂2 y. Subscripts denote partial differentiation. The NIO velocit y field (u,v,w ), buoyancy b, and pressurepare given by u+iv=e−if0tLA, w=−1 2f2 0N−2(Axz−iAyz)e−if0t+c.c., b=i 2f0(Axz−iAyz)e−if0t+c.c., p=i 2(Ax−iAy)e−if0t+c.c. The buoyancy bis related to the density ρby ρ=ρ0/bracketleftBigg 1−1 g/integraldisplayz 0N2(z′)dz′−b g/bracketrightBigg , whereρ0is the reference density at the top of the ocean. The pressure phas been normalized byρ0. The boundary conditions are that Az= 0 atz= 0 andz=−D. This ensures that w vanishes at the top and bottom of the ocean. Using these bound ary conditions, /integraldisplay0 −D(u+iv) = 0. (3) Thus barotropic motion is not included in the analysis. Howe ver Gill (1984) has shown that the barotropic response to a storm is instantaneous and the a ssociated currents are weak. 3 A Simplified Model To simplify the analysis, we assume that Aandψdo not vary in the x-direction, and that ζ= 0. The analysis thus neglects the effect of background barot ropic vorticity but crucially keeps theβ-effect. The buoyancy frequency profile is taken to be N2=ǫ2N2 0,−Hmix<z < 0, N2=N2 0,−D<z < −Hmix, 3whereǫ≪1. Finally, the storm is assumed to have produced an initial c ondition of a horizontally uniform near-inertial current to the east con centrated within the mixed layer. Instead of approaching this problem by use of an integral ope rator as in D’Asaro (1989) or by projecting onto normal modes (e.g., Gill 1984, Balmforth et al. 1998), the problem will be formulated as an initial value problem on a semi-infinite dom ain corresponding to an ocean that is effectively infinitely deep. In order to formulate the problem properly for this limit, this section considers an ocean of finite depth. In Section 4 t he solution in the limit that the depth of the interior is much greater than the mixed layer dep th will be found. This formulation as a radiation problem which ignores the pr esence of the ocean bottom requires the projection of the initial condition to be sprea d across all the normal modes. This is certainly true for small mixed layer depths in the mod el of Gill (1984), as shown in Table 1 of that paper; also see Table 1 of Zervakis and Levin (1995). For deeper mixed layers, this is no longer true since half the initial energy b ecomes concentrated in the first two or three modes. However, as pointed in Section 7 of Gill (1 984), the depth of the ocean “influences the rate of loss of energy by imposing modulation s on the rate, but the average rate of loss is not affected very much by depth changes”. Hence the results presented here should be qualitatively relevant even when the continuum as sumption is not valid. 3.1 Nondimensionalization Quantities are nondimensionalized according to ˆy=y/Y, ˆz= 1 +z/H mix,ˆt= Ωt, ˆN=N/N 0, where Y≡/parenleftBiggH2 mixN2 0 βf0/parenrightBigg1/3 ,Ω≡/parenleftBiggβ2H2 mixN2 0 f0/parenrightBigg1/3 . Typical values β= 10−11m−1s−1,Hmix= 100 m,f0= 10−4s−1,N0= 10−2s−1giveY= 105 m and Ω = 10−6s−1. The relevant timescale is thus Ω−1= 11.5 days. Also, the velocity and the fieldAare nondimensionalized by (ˆu,ˆv) =(u,v) U, ˆA=f2 0 UN2 0H2 mixA, whereUis a characteristic value of the initial velocity. The hats are now dropped for ease of notation. With this nondi mensionalization, the buoyancy frequency profile is N2=ǫ2,0<z < 1, N2= 1,−H≡1−D/H mix<z < 0, 4and the NIO equation (1), the boundary conditions, and initi al condition become Azzt+i 2N2Ayy+iyAzz= 0, (4) Az= 0, z =−H, z = 1, (5) Azz=N2(u+iv), t = 0. (6) The requirement that uandvremain finite imply the jump conditions Az|z=0+=ǫ2Az|z=0−, A yy|z=0+=Ayy|z=0−, (7) wherez= 0+andz= 0−are the limits as z→0 from positive and negative zvalues, respectively. This nondimensionalization allows some immediate conclus ions to be drawn about the propagation of NIO energy downwards. Most importantly, if Hmixincreases, then the timescale Ω−1decreases. Thus, assuming that the storm causes a uniform ne ar-inertial current throughout the whole mixed layer, energy transfer w ill be faster for a deeper mixed layer. This confirms the results of Gill (1984), which associ ated the more efficient transfer with a larger projection of the initial velocity profile on th e first vertical mode. 3.2 Boundary Condition at the Base of the Mixed Layer Expanding A(y,z,t) =A0(y,z,t) +ǫ2A2(y,z,t) +O(ǫ4) for 0<z < 1, (4) becomes at O(ǫ0) A0zzt+iyA0zz= 0. Integrating this subject to the boundary condition that Azand thusA0zvanishes at z= 1 implies that A0is independent of z. AtO(ǫ2), A2zzt+iyA2zz+i 2A0yy= 0, (8) which may be integrated subject to the boundary condition th atA2zvanishes at z= 1 to give A2zt+iyA2z+i 2A0yy(z−1) = 0. Evaluating at z= 0+and usingAyy=A0yy+O(ǫ2) andAz=ǫ2A2z+O(ǫ4), Azt+iyAz−iǫ2 2Ayy=O(ǫ4), z = 0+. Finally, applying (7) gives the upper boundary condition fo r the NIO field in the ocean interior to leading order in ǫ: Azt+iyAz−i 2Ayy= 0z= 0−. (9) 5Results obtained in the ocean interior using (9) are in fact l eading-order solutions. We shall continue to use the notation A, even though it is really the leading-order term in the expansion. 3.3 Initial Condition Suppose that in a short time compared with the NIO wave propag ation time, the passing storm induces near-inertial currents in the mixed layer wit h a horizontal scale that is much larger than the one under consideration, and which can hence be taken to be uniform. For simplicity, the initial velocity (consistent with equatio n (3)) is assumed to be piecewise constant with depth: (u,v) = (1,0) 0<z< 1, = (−H−1,0),−H <z< 0. The weak flow in the ocean interior is necessary to ensure that the flow has no barotropic component. Integrating equation (6) with respect to zand using the boundary conditions (5) gives at t= 0 Az=ǫ2(z−1),0<z< 1, (10) Az=−(z+H)/H, −H <z< 0. (11) 4 Solution for an Infinitely Deep Ocean The total depth of the ocean is typically on the order of a hund red times the depth of the mixed layer. Thus, the limit of infinite depth is considered. The initial condition is taken to be equation (11) with H→ ∞. The boundary condition for z→ −∞ is taken to be Azz→0, corresponding to the near-inertial velocities vanishin g at infinite depth. Of course, this limit excludes the possibility of reflections off the bot tom of the ocean which may be important. Finally, the boundary condition for z= 0−given by equation (9) is used. Hence the problem to be solved for the semi-infinite domain z<0 becomes Azzt+i 2Ayy+iyAzz= 0, z < 0, Azt+iyAz−i 2Ayy= 0, z = 0−, Azz→0, z → −∞, Az=−1, t = 0. 64.1 NIO velocity field These equations may be solved using Laplace transforms. Her e we present only the major results; further details are given in Moehlis (1999). We mak e the transformations A(y,z,t) = e−iyt˜B(z,T),T≡t3/3, andα≡(1 +i)/2 and define the Laplace transform of ˜Bby b(z,p)≡ L[˜B]≡/integraldisplay∞ 0˜B(z,T)e−pTdT. (12) Then b(z,p) =−1 α1√p+αexp/parenleftBiggαz√p/parenrightBigg . (13) This Laplace transform and its derivatives with respect to zmust be inverted numerically for the ocean interior ( z<0). For the top of the ocean interior ( z= 0−) however, they may be obtained in closed form. For example, Azz(y,0−,t) =e−iyt/bracketleftBigg eit3/6erfc/parenleftBigg1 +i 2√ 3t3/2/parenrightBigg −1/bracketrightBigg . (14) We now consider the back-rotated velocity Azz=eif0t(u+iv), which filters out purely inertial motion at frequency f0. Back-rotated velocities may be represented by hodographs which show the vector (Re( Azz),Im(Azz)) as curves parametrized by time. For f0>0, if these curves are traced out in a clockwise (counterclockw ise) fashion, the corresponding motion has frequency larger (smaller) than f0. Figure 1 shows the back-rotated velocity at different locations. A common characteristic is that the m agnitude of the back-rotated velocity starts at zero, reaches a peak value shortly after t he storm, then decays away. The depth dependence of the back-rotated velocity is seen by com paring Figure 1 (a) and (b), where both have y= 0 and thus the same value of the Coriolis parameter f. Qualitatively the results are the same, but closer to the mixed layer the direct ion change of the back-rotated velocity becomes slower, meaning that the frequency is clos er tof0. An idea of the latitudinal dependence is seen by comparing Figure 1 (a,c,d): at y= 1 the hodograph is traced out in a clockwise fashion as for y= 0, but at y=−2 it is traced out in a counterclockwise fashion. 4.2 Kinetic energy density and fluxes The horizontal kinetic energy (HKE) per unit area contained within the mixed layer is /integraldisplay1 0dz/vextendsingle/vextendsingle/vextendsingle/vextendsingleAzz N2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≡/integraldisplay1 0dz/vextendsingle/vextendsingle/vextendsingle/vextendsingleAzz ǫ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =/integraldisplay1 0dz|A2zz|2. Expanding ˜B(z,T) =˜B0(z,T) +ǫ2˜B2(z,T) +O(ǫ4) in the mixed layer, (8) may be used to show that pb2zz−˜B2zz(z,0)−i 2b0= 0, (15) 7-0.4-0.200.20.4 -0.4-0.2 00.20.4-0.4-0.200.20.4 -0.4-0.200.20.4 -0.4-0.200.20.4 -0.4-0.2 00.20.4-0.4-0.200.20.4 -0.4-0.2 00.20.4(a) (c) (d)(b)Im/( Az z /) Im/( Az z /)Im/( Az z /) Re/( Az z /)Re/( Az z /) Im/( Az z /)Re/( Az z /)Re/( Az z /) Figure 1: Back-rotated velocity for (a) z=−1,y= 0, (b)z=−0.5,y= 0, (c)z=−1, y= 1, and (d) z=−1,y=−2. The diamonds are drawn at t= 0,5,10,15,20. 800.20.40.60.81 012345 0.010.11 0.1 1 10 tt eMLeML Figure 2: Horizontal kinetic energy per unit volume (HKE) in the mixed layer, eML, for linear and logarithmitc axes. The solid line shows the exact result and the dashed line the asymptotic result. whereb2=L[˜B2] andb0=L[˜B0]. The initial condition within the mixed layer is ˜B2zz(z,0) = 1. NowAis continuous across z= 0, and ˜B0is independent of z(see Section 3.2). Hence b2zz=1 p−i 2αp1√p+α, which may be inverted to give A2zz(y,t) =e−iyteα2t3/3erfc/parenleftBiggα√ 3t3/2/parenrightBigg . (16) Therefore the HKE within the mixed layer is eML≡/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleerfc/parenleftBigg1 +i 2√ 3t3/2/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . The time dependence of eMLis shown in Figure 2. Asymptotic results from Abramowitz and Stegun (1972) for the complementary error function impl y that eML∼1−2√ 3πt3/2, t ≪1, eML∼6 πt3, t → ∞. 9Since the energy which leaves the mixed layer enters the inte rior of the ocean, this implies that for short times the energy in the interior increases lik et3/2. This does not contradict the result from D’Asaro (1989) that for short times the thermocl ine energy grows like t6. That result assumes that the wind persists to generate a constant inertially oscillating velocity, and that there is no propagating inertial motion. Here, the w ind has an instantaneous effect, causing an initial horizontally uniform inertial current, and propagating inertial motion is included fully. Another quantity of interest is the flux of HKE. Using (4) and i ts complex conjugate gives ∂ ∂tHKE =∂ ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingleAzz N2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =i 2N2∂ ∂y(AzzA∗ y−A∗ zzAy) +i 2N2∂ ∂z(A∗ yzAy−AyzA∗ y). (17) AssumingAzzA∗ y−A∗ zzAyvanishes for |y| → ∞ and using equation (5), d dt/integraldisplay−d −Hdz/integraldisplay∞ −∞dx/integraldisplay∞ −∞dy|Azz|2=/integraldisplay∞ −∞/integraldisplay∞ −∞FE(y,t;d)dxdy, (18) where FE(y,t;d)≡i 2(A∗ yzAy−AyzA∗ y)|z=−d (19) gives the flux of HKE from the region z >−dto the region z <−d. For this model, we consider the flux per unit area. Integrating (18) with respec t to time shows that the quantity E(t;d)≡/integraldisplayt 0FEdt gives the total amount of HKE which has penetrated into the re gionz <−d. Note that E(t;d)→1 corresponds to all the energy originally in the mixed layer having reached depths below z=−d. Results for FE(t;d) andE(t;d) obtained by numerically inverting the appropriate Laplace transforms are shown in Figure 3. FEpeaks at the nondimensionalized timet≈0.62; for the typical values quoted in Section 3.1, this corres ponds to about a week after the storm. From Figure 3(b) and using the fact that what ever energy flows through z= 0−must have initally been in the mixed layer, we see that by t= 1 (about 11.5 days after the storm) nearly half of the energy associated with ho rizontal NIO currents caused by the storm has left the mixed layer; however, only about 38% of the total energy has penetrated below z=−1. Byt= 2 (about 23 days after the storm), 82% of the total energy has left the mixed layer, but only 58% has penetrated below z=−1. Thus, at t= 2 nearly a quarter of the total energy is contained in the distance Hmiximmediately beneath the mixed layer. This is reminiscent of the accumulation of NIO energy below the mixed layer seen in Balmforth, Llewellyn Smith and Young (1998). This model thu s gives reasonable estimates for the timescale for which the decay of NIO energy occurs: fo r example, D’Asaro et al. 1000.10.20.30.40.50.6 012345d=0 d=0.5 d=1 d=2 d=5 d=10 00.20.40.60.81 012345d=0 d=0.5 d=1 d=2 d=5 d=10 t /(a/) FEEt /(b/) Figure 3: (a) FE(t;d) and (b)E(t;d) for different depths dbelow the base of the mixed layer. These show instantaneous and time-integrated fluxes of HKE. (1995) found that the mixed layer inertial energy was reduce d to background levels by 21 days after the storm. Figure 4 shows the vertical dependence of the HKE and FEat different times. As time increases the instantaneous distribution of HKE becomes mo re sharply peaked near the base of the mixed layer, but remains bounded (asymptotically app roaching unity) because of energy conservation. 4.3 Large-time behavior The asymptotic behavior of near-inertial properties may be derived using the method of steepest descents (see Moehlis 1999 for details). This show s that in the limit of large ξ≡ z2/3t, and along the “rays” z=−η3 0t3/3, u2+v2∼2 (1 +η2 0)πη2 0t3, F E∼2η0 π(1 +η2 0)t. A useful way to represent the asymptotic results is to write η0in terms of zandtand then draw contour plots of quantities of physical interest in the (z,t) plane: this is shown in Figure 5. In the asymptotic limit for large ξ, withzconstant,u2+v2andFEdecrease as time increases. Note that ξis large for sufficiently large zand/ort. Finally, Moehlis (1999) also obtained results for the verti cal shearu2 z+v2 z. To leading order inǫ, the vertical shear within the mixed layer is zero. The resul ts for vertical shear for 11-1.5-1-0.500.5 00.20.40.60.8 1t=1 t=2 t=5 t=10 -5-4-3-2-10 00.10.20.30.40.5t=1 t=1.5 t=2 t=2.5 t=3 /(a/) zu /2/+ v /2z/(b/)FE Figure 4: Vertical profiles of (a) u2+v2and (b)FE(t,|z|) aty= 0 for different times showing the decay of energy from the mixed layer (0 <z < 1) and resultant behavior in the interior (z<0). Note the different vertical scales. the interior of the ocean lack physical realism because the m odel allows the shear to grow forever as a consequence of the initial infinite shear due to t he discontinuity in the initial velocity profile. 5 Conclusion A simplified model has been developed to examine the decay due to theβ-effect of near- inertial currents excited in the mixed layer by a passing sto rm. This decay occurs due to the radiation of downward propagating NIOs into the inter ior of the ocean. The main assumptions of the model are that the background flow does not vary in the longitudinal direction and has no associated vorticity, that the ocean ha s a simple (piecewise constant) buoyancy frequency profile, and that the storm has moved very quickly over the ocean causing a horizontally uniform near-inertial current concentrate d in the mixed layer. The β-effect is included in the analysis and is responsible for the radiatio n of NIOs. Because the depth of the mixed layer is much smaller than the total depth of the oce an, the problem is formulated in the limit of an effectively infinitely deep ocean; the resul tant initial value problem is solved by Laplace transforms. Analytical results are given for the horizontal kinetic energy density 1222.533.544.55-5-4-3-2-122.533.544.55-5-4-3-2-1t zzt /(a/)/(b/) Figure 5: Contour plots of the asymptotic results for (a) u2+v2and (b)FE. Darker shading corresponds to smaller values. in the mixed layer, and results from the numerical inversion of the appropriate Laplace transforms are given for horizontal kinetic energy, energy flux, and back-rotated velocity. The asymptotic behavior is also investigated. Although this simplified model cannot be expected to capture the full complexity of the aftermath of a storm passing the ocean, it does capture much o f the observed behavior. Most importantly, in the presence of the β-effect the decay of near-inertial mixed layer energy is found to occur on the appropriate timescale (appro ximately twenty days), which confirms the analysis of D’Asaro (1989) and observations by D ’Asaro et al. (1995), Levine and Zervakis (1995), and Qi et al. (1995). The main advantage of the approach described in this paper is that many aspects of the decay in the mixed layer are analytically obtained for all times, unlike D’Asaro (1989) which predicts the timesca le for the decay in a short time limit or estimates it in terms of the time it takes normal mode s to become out of phase (cf. Gill 1984). Extensions to a more realistic ocean and storm wo uld involve including a more realistic buoyancy frequency profile (for example, the profi le used by Gill 1984), considering the effect of different initial velocities (including both ho rizontal and vertical structure), and considering the effect of background flow. The study of all of t hese could use the same formalism of Young and Ben Jelloul (1997) and an approach sim ilar to that presented here. 13Acknowledgments The majority of this work was carried out at the 1999 Geophysi cal Fluid Dynamics program at the Woods Hole Oceanographic Institution. The authors wo uld particularly like to thank W. R. Young for many useful discussions regarding this work. References [1] Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathe matical Functions, Wiley Interscience Publications, 1046 pp. [2] Balmforth, N. J., Llewellyn Smith, S. G. and Young, W. R. ( 1998) Enhanced dispersion of near-inertial waves in an idealized geostrophic flow. J. Mar. Res. , 56:1–40. [3] Balmforth, N. J. and Young, W. R. (1999) Radiative dampin g of near-inertial oscillations in the mixed layer. J. Mar. Res. , 57:561–584. [4] D’Asaro, E. A. (1985) The energy flux from the wind to near- inertial motions in the surface mixed layer. J. Phys. Oceanogr. , 15:1043–1059. [5] D’Asaro, E. A. (1989) The decay of wind-forced mixed laye r inertial oscillations due to theβeffect. J. Geophys. Res. , 94:2045–2056. [6] D’Asaro, E. A., Eriksen, C. C., Levine, M. D., Niiler, P., Paulson, C. A., and van Meurs, P. (1995) Upper-ocean inertial currents forced by a s trong storm. Part I: Data and comparisons with linear theory. J. Phys. Oceanogr. , 25:2909–2936. [7] Garrett, C. (1999) What is the “near-inertial” band and w hy is it different? Unpublished manuscript. [8] Gill, A. E. (1984) On the behavior of internal waves in the wakes of storms. J. Phys. Oceanogr. , 14:1129–1151. [9] Hebert, D. and Moum, J. N. (1993) Decay of a near-inertial wave. J. Phys. Oceanogr. , 24:2334–2351. [10] Henyey, F. S., Wright, J. A., and Flatt´ e, S. M. (1986) En ergy and action flow through the internal wave field: an eikonal approach. J. Geophys. Res. , 91:8487–8495. [11] Levine, M. D. and Zervakis, V. (1995) Near-inertial wav e propagation into the pycn- ocline during ocean storms: observations and model compari son.J. Phys. Oceanogr. , 25:2890–2908. 14[12] Moehlis, J. (1999) Effect of a simple storm on a simple oce an, in Stirring and Mixing, 1999 Summer Study Program in Geophysical Fluid Dynamics , Woods Hole Oceanogr. Inst. Unpublished manuscript. [13] Pollard, R. T. and Millard, R. C. Jr. (1970) Comparison b etween observed and simulated wind-generated inertial oscillations. Deep-Sea Res. , 17:813–821. [14] Qi, H., De Szoeke, R. A., Paulson, C. A., and Eriksen, C. C . (1995) The structure of near-inertial waves during ocean storms. J. Phys. Oceanogr. , 25:2853–2871. [15] van Meurs, P. (1998) Interactions between near-inerti al mixed layer currents and the mesoscale: the importance of spatial variabilities in t he vorticity field. J. Phys. Oceanogr. , 28:1363–1388. [16] Webster, F. (1968) Observation of inertial-period mot ions in the deep sea. Rev. Geo- phys., 6:473–490. [17] Young, W. R. and Ben Jelloul, M. (1997) Propagation of ne ar-inertial oscillations through a geostrophic flow. J. Mar. Res. , 55:735–766. [18] Zervakis, V. and Levine, M. D. (1995) Near-inertial ene rgy propagation from the mixed layer: theoretical considerations. J. Phys. Oceanogr. , 25:2872–2889. 15

arXiv:physics/9912031v1 [physics.ed-ph] 14 Dec 1999Hydrogen atom in a spherical well David Djajaputra and Bernard R. Cooper Department of Physics, West Virginia University, PO BOX 631 5, Morgantown, WV 26506, USA (February 21, 2014) We discuss the boundary effects on a quantum system by examini ng the problem of a hydrogen atom in a spherical well. By using an approximation method wh ich is linear in energy we calculate the boundary corrections to the ground-state energy and wav e function. We obtain the asymptotic dependence of the ground-state energy on the radius of the we ll. The hydrogen atom occupies a unique place in atomic physics because it is the only atom for which the Schr¨ odinger equation can be solved analytically. The cal- culation of the energy spectrum of the hydrogen atom is a standard exercise in a physicist’s education and is dis- cussed in detail in many textbooks on quantum physics. [1–3] Textbook discussions normally consider a hydrogen atom in free space, with vanishing eigenfunction at infin- ity as one of the boundary conditions. Experiments in atomic physics, of course, are normally done by position- ing the atoms in a well-controlled cavity. One could then ask what effects does the presence of the finite boundary have on the wave functions and the energy levels of the atoms. A typical answer that one may get is that for com- mon cavities used in actual experiments which are much larger than the characteristic atomic distance, the Bohr radiusa0, the boundary gives rise only to an “exponen- tially small” correction because the eigenfunctions of the atom decay exponentially with distance. One, however, rarely gets a more quantitative answer than this and it is therefore an interesting challenge to obtain such an an- swer. Moreover, modern technology has opened the pos- sibility of constructing interesting structures in atomic and molecular scales for which the question of boundary effects has become more than mere academic. In this paper we will examine the boundary corrections for a hydrogen atom situated at the center of a spherical cavity of radius Sas shown in Fig.1. We will assume the wall of the cavity to be impenetrable and consider the following spherically-symmetric potential: V(r) =/braceleftbigg −e2/r, r<S, ∞, r>S.(1) The radius of the cavity will be assumed to be much larger than the Bohr radius: S≫a0.In the remainder of the paper we shall use the atomic units: ¯h=e2 2= 2m= 1. (2) The unit of length is the Bohr radius a0= ¯h2/me2and the unit of energy is the Rydberg: Ry = e2/2a0= 13.6 eV. The Schr¨ odinger equation takes the following form: HΨ(r) =/parenleftBig − ∇2−2 r/parenrightBig Ψ(r) =EΨ(r). (3)/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/BnZr/j45+/j83 FIG. 1. Hydrogen atom in a spherical well of radius S. The wave function Ψ( r) satisfies the Schr¨ odinger equa- tion for the hydrogen atom for r < S , in particular it should still be regular at the origin. The only difference from the free-space case is that now we have to impose a different boundary condition: the wave function should vanish atr=Sinstead of at r=∞. ForS≫a0, the changes in the ground-state wave func- tion and energy due to the presence of the wall are ex- pected to be “small” because the wave function is concen- trated at the center of the cavity, far away from the con- fining wall. Standard perturbation technique, however, is not useful in this case because the infinite potential at the wall prevents the calculation of the required ma- trix elements. The Rayleigh-Ritz variational method is one viable alternative, but it is not clear how one should choose the best set of variational functions to be used. Furthermore, it cannot conveniently be used to calculate the corresponding corrections for the excited states. In the following we shall use an approximation method which is linear in energy to calculate these corrections. This is a well-known method in solid-state physics and has been widely used in electronic structure calculations since its initial introduction by O. K. Andersen in 1975. [4,5] The method is best applied to the calculations of the wave functions of a hamiltonian with energies which are in close vicinity of the energy of a known wave function. The present problem of a hydrogen atom in a spheri- 1cal well can be used to illustrate the application of this method. In the absence of the confining cavity, the hy- drogen atom has a well-known spectrum: εn=−1 n2, n= 1,2,... . (4) In the presence of the cavity, we write En=εn+ ∆εn. (5) We use small letters ( ε,ψ, etc.) to denote quantities for the free-space problem and capital letters ( E,Ψ, etc.) for the corresponding quantities in the cavity problem. The dimensionless parameter (∆ εn/εn) is expected to be small forn2a0≪S.In the linear method, the (unnor- malized) wave function at energy Enis approximated by Ψ(En,r) =ψ(εn,r) + ∆εn˙ψ(εn,r). (6) Here ˙ψ(εn,r) is the derivative with respect to energy of ψ(ε,r) evaluated at ε=εn: ˙ψ(εn,r) = [∂ψ(ε,r)/∂ε] (ε=εn). (7) The eigenfunctions in the cavity problem are then ob- tained by imposing the boundary condition at r=S: Ψ(En,S,ˆr) = 0, (8) which gives an expression for the energy correction: ∆εn=−ψ(εn,S,ˆr) ˙ψ(εn,S,ˆr). (9) Hereˆr= (θ,φ) is a unit vector in the direction of r. To apply this simple approximate method we need the general solution to the Schr¨ odinger equation at an arbi- trary energy E. Since we are dealing with a spherically- symmetric system, we can separate the variables: Ψ(r) =R(r)Ylm(ˆr). (10) The resulting radial differential equation is d2R dr2+2 rdR dr+/bracketleftBig E+2 r−l(l+ 1) r2/bracketrightBig R= 0. (11) Transforming the variables by defining ω=√ −E, ρ = 2ωr, (12) and using the ansatz R(ρ) =ρle−ρ/2u(ρ), (13) then gives us the following differential equation [6] ρu′′+/bracketleftBig 2(l+ 1)−ρ/bracketrightBig u′−/bracketleftBig l+ 1−1 ω/bracketrightBig u= 0,(14)5 10 15 20r a0 -1-0.8-0.6-0.4-0.20.20.4rR /j64r /j68 /j119/j32/j32= 1 /j119/j32/j32= 1/2 /j119/j32/j32= 0.98 FIG. 2. The function rRl(ω, r) as a function of r/a0for l= 0 and ω= 1, 0.98, and 0.50. The ω= 1 curve is nodeless. Asωis decreased from 1 to 0.50, the node of the wave function moves from r=∞tor= 2a0. which is the equation for the confluent hypergeometric function. The general solution of this equation, which is regular at the origin, is [6] u(ρ) =A1F1/parenleftBig l+ 1−1 ω; 2l+ 2;ρ/parenrightBig , (15) whereAis a normalization constant. The radial part of the general solution to the Schr¨ odinger equation Eq.(3) with energy E=−ω2therefore is Rl(ω,r) =A(2ωr)le−ωr 1F1/parenleftBig l+ 1−1 ω; 2l+ 2; 2ωr/parenrightBig . (16) The free-space solution is obtained by requiring that R(r)→0 asr→ ∞.From the properties of the hyperge- ometric functions, [6] this can only happen if ( l+1−1/ω) is a nonpositive integer which implies that 1 ω=n, l = 0,1,...,n, (17) withna positive integer. This directly leads to the Ryd- berg spectrum in Eq.(4). The function Rl(ω,r) is plotted in Fig.2 for l= 0 andω = 1, 0.98, and 0.50. The ω= 1 curve is the ground-state wave function of the hydrogen atom in free space and is nodeless. As ωis reduced below 1, the wave function ac- quires a single node which moves from r=∞tor= 2a0 atω= 0.50.where it becomes the ( n,l) = (2,0) eigen- state of the hydrogen atom in free space. One therefore can obtain the ground-state wave function and energy of the hydrogen atom in a cavity of radius Sby numerically searching for the energy which gives a wave function with a single node at r=S. This provides a useful comparison for our approximation. Since the spherical harmonics are independent of the energy we can recast Eq.(9) into ∆εnl= 2ωnRl(ωn,S) ˙Rl(ωn,S). (18) 2whereωn=√−εnand ˙Rl(ωn,S) = [∂Rl(ω,S)/∂ω] (ω=ωn). (19) Substituting the radial function Rl(ω,r) in Eq.(16) into Eq.(18) then gives us an explicit formal expression for ∆εnwhich should be valid for R≫n2a0.Note that the presence of the finite boundary lifts the azimuthal degeneracy of the states with different orbital quantum numberl(and the same radial quantum number n). As in the case of the screened Coulomb potential, this oc- curs because one no longer deal with the pure Coulomb potential. In group theoretical language, modifications to the pure Coulomb potential break the SO(4) symme- try of the hydrogen atom: the Runge-Lenz operator no longer commute with the hamiltonian. [7] This should be contrasted with the classical case where the Runge-Lenz vector is still a good constant of motion and the presence of the boundary does not have any effect on the orbit of the particle if it is greater than the orbit’s aphelion. To gain an insight into Eqs.(18)-(19), we shall consider the ground state ( n= 1), which is a special case of the zero angular momentum ( l= 0) states. We have R0(ω,r) =A e−ωr 1F1/parenleftBig 1−1 ω; 2; 2ωr/parenrightBig . (20) For the ground state ( n= 1), this is R0(1,r) =Ae−r1F1/parenleftBig 0; 2; 2r/parenrightBig =A e−r. (21) We are interested in obtaining a simple analytical ex- pression of the correction to the ground-state energy for S≫a0, therefore we need to calculate the limiting form of˙R0(ω,r) forr≫a0.The asymptotic expansion of the hypergeometric function 1F1(a,b,z) for largezis [8] 1F1(a,b,z) Γ(b)=eiπa zaI1(a,b,z) Γ(b−a)+ezza−bI2(a,b,z) Γ(a),(22) with I1(a,b,z) =R−1/summationdisplay n=0(a)n(1 +a−b)n n!eiπn zn+O(|z|−R),(23) I2(a,b,z) =R−1/summationdisplay n=0(b−a)n(1−a)n n!1 zn+O(|z|−R).(24) The Pochhammer symbol ( a)nis defined by [6] (a)n=a(a+ 1)···(a+n−1) =Γ(a+n) Γ(a).(25) We need to calculate the derivative of this function at a= (1−1/ω) withω= 1.In this case the dominant term comes from the derivative of Γ( a) in the second term in Eq.(22). The first term can be neglected because it does246810-1-0.8-0.6-0.4-0.2 S a0E Exact Linear Limit FIG. 3. Dependence of the ground-state energy of a hy- drogen atom confined in a spherical cavity on the radius of the cavity S. The topmost curve is the exact result which is obtained by numerically searching for the node of the wave function.The middle curve is obtained from the linear appro x- imation, Eq.(18), using the exact wave function Eq.(21). Th e lowest curve is obtained using the limiting formula Eq.(31) . not have the exponential term ezwhich dominates the derivative at large distances. Keeping only the largest term, we get ∂ ∂a1F1(a,b,z)≈ −ezza−bΓ(b)I2(a,b,z)ψ(a) Γ(a).(26) Hereψ(a) is the digamma function: ψ(a) = Γ′(a)/Γ(a). [8] Its ratio with Γ( a) asa→0 is lim a→0ψ(a) Γ(a)= lim a→0−γ−1/a −γ+ 1/a=−1, (27) whereγis the Euler constant. This then gives /bracketleftBig∂ ∂a1F1(a,b,z)/bracketrightBig (a→0)≈ezza−bΓ(b)I2(a,b,z).(28) Using this expression, and keeping only the first two terms inI2(a,b,z), we can obtain the limiting form of ˙R0(ω,r) at largerandω→1: ˙R0(ω,r)≈Ae−ωr ω2/braceleftBige2ωr (2ωr)1+1/ω/bracketleftBig 1 +Γ(2 + 1/ω) 2ωrΓ(1/ω)/bracketrightBig/bracerightBig . (29) Exactly at ω= 1, this expression becomes ˙R0(1,r)≈Aer 4r2/bracketleftBig 1 +1 r/bracketrightBig . (30) 3Finally, using this equation and Eq.(21) in Eq.(18), we get the boundary correction to the ground-state energy: ∆ε0(S)≈8S(S−1)e−2S, S≫a0. (31) Fig.3 displays this asymptotic dependence of the energy correction on the radius of cavity, together with the exact curve and the one obtained from Eq.(18) using the exact wave function Eq.(21). It is seen that the asymptotic formula, Eq.(31), is fairly accurate for radii greater than about four Bohr radius. Note that the exact energy at S= 2a0is equal to1 4Ry, which is the energy of the first excited state ( n,l) = (2,0) of the hydrogen atom in free space. This is because the corresponding wave function has a node at r= 2a0as can be seen in Fig.2. Knowing the dependence of the ground-state energy on the cavity radius, Eq.(31), allows us to calculate the pressure needed to “compress” a hydrogen atom in its ground state to a certain size. This is given by p(S) =−∂∆ε0 ∂V≈4e−2S π/parenleftBig 1−2 S/parenrightBig . (32) AtS= 4a0this has a value of 2 .13×10−4eV/a3 0= 1.47×104GPa. At this radius, the change of the ground- state energy is 0.032 Ry which is only three percent of the binding energy of a free hydrogen atom. In conclusion, we have used a linear approximation method to calculate the asymptotic dependence of the ground-state energy of a hydrogen atom confined to a spherical cavity on the radius of the cavity. The bound- ary correction to the energies of the excited states can be obtained using the same method. Acknowledgements —D. D. is grateful to Prof. David L. Price (U. Memphis) for introducing him to Andersen’s linear approximation method and for many useful discus- sions. This work has been supported by AF-OSR Grant F49620-99-1-0274. [1] R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles , 2nd Edition, John Wiley and Sons, New York, 1985. [2] M. A. Morrison, T. L. Estle, and N. F. Lane, Quantum States of Atoms, Molecules, and Solids , Prentice-Hall, En- glewood Cliffs, NJ, 1976. [3] M. Weissbluth, Atoms and Molecules , Academic Press, New York, 1978. [4] O. K. Andersen, “Linear methods in band theory,” Phys. Rev. B 12, 3060 (1975). [5] V. Kumar, O. K. Andersen, and A. Mookerjee, Lectures on Methods of Electronic Structure Calculations , World Scientific, Singapore, 1994.[6] J. B. Seaborn, Hypergeometric Functions and Their Ap- plications , Springer-Verlag, New York, 1991, Chapter 6. [7] W. Greiner and B. M¨ uller, Quantum Mechanics: Symme- tries, Springer-Verlag, Berlin, 1994, Chapter 14. [8] M. Abramowitz and I. A. Stegun, Handbook of Mathemat- ical Functions , Dover, New York, 1965, Formula 13.5.1. 4

arXiv:physics/9912032v1 [physics.atom-ph] 16 Dec 1999Stabilization not for certain and the usefulness of bounds C. Figueira de Morisson Faria,∗A. Fring†and R. Schrader† ∗Max-Planck-Institut f¨ ur Physik komplexer Systeme, N¨ othnitzer Str. 38, D-01187 Dresden, Germany †Institut f¨ ur Theoretische Physik, Freie Universit¨ at Berlin, Arnimallee 14, D-14195 Berlin, Germany Abstract. Stabilization is still a somewhat controversial issue conc erning its very existence and also the precise conditions for its occurrenc e. The key quantity to set- tle these questions is the ionization probability, for whic h hitherto no computational method exists which is entirely agreed upon. It is therefore very useful to provide var- ious consistency criteria which have to be satisfied by this q uantity, whose discussion is the main objective of this contribution. We show how the sc aling behaviour of the space leads to a symmetry in the ionization probability, whi ch can be exploited in the mentioned sense. Furthermore, we discuss how upper and lowe r bounds may be used for the same purpose. Rather than concentrating on particular a nalytical expressions we obtained elsewhere for these bounds, we focus in our discuss ion on the general princi- ples of this method. We illustrate the precise working of thi s procedure, its advantages, shortcomings and range of applicability. We show that besid es constraining possible values for the ionization probability these bounds, like th e scaling behaviour, also lead to definite statements concerning the physical outcome. The pulse shape properties which have to be satitisfied for the existence of asymptotica l stabilization is the van- ishing of the total classical momentum transfer and the tota l classical displacement and not smoothly switched on and off pulses. Alternatively we support our results by general considerations in the Gordon-Volkov perturbation theory and explicit studies of various pulse shapes and potentials including in particu lar the Coulomb- and the delta potential. INTRODUCTION There is considerable interest in the high intensity regime (intensities larger than 3.5 ×Wcm−2for typical frequencies), because since the early nineties it may be realized experimentally. The perturbative description, w hich was a very successful approach in the low intensity regime, breaks down for such hi gh intensities. Thus, this regime constitutes a new challenge to theorists. Compa ring the status of the understanding and clarity of the description of the two r egimes one certainly 0)To appear in the Proceedings of the ICOMP8 (Monterey (USA), O ctober 1999)observes a clear mismatch and should probably conclude that the challenge has not been entirely met so far. One also observes a clear imbalance between numerical calculations and analytical descriptions. In particular, the issue of stabilization has led to several controversies and there are still several recent computations which are in clear con tradiction to each other. Since it is not very constructive simply to count the numbers of numerical results which agree and those which do not1, our investigations aim at analytical descrip- tions which unravel the physical assumptions and might serv e to pinpoint possible errors. In view of the panel discussion at this meeting the main purpo se of this con- tribution is to summarize our findings [1-6] and in particula r explain the working and limitations of our method in the hope to dispel a few misun derstandings and misconceptions which have occurred. FRAMEWORK AND PHYSICAL PROPERTIES We start by stating our physical assumptions. We consider an atom with poten- tialV(/vector x) in the presence of a sufficiently intense laser field, such tha t it may be described in the non-relativistic regime by the time-depen dent Schr¨ odinger equa- tion in the dipole approximation i∂ψ(/vector x,t) ∂t=/parenleftbigg −∆ 2+V(/vector x) +/vector x·/vectorE(t)/parenrightbigg ψ(/vector x,t) =H(/vector x,t)ψ(/vector x,t). (1) We will use atomic units throughout this article. We take the pulse to be of the general form /vectorE(t) =/vectorE0f(t) (2) wheref(t) is assumed to be a function whose integral over tis well behaved with f(t) = 0 unless 0 ≤t≤τ. This means τconstitutes the pulse duration, f(t) the pulse shape function and E0the amplitude of the pulse, which we take to be positive without loss of generality. Important quantities for our discussion are the total class ical momentum transfer /vectorb(τ), the classical displacement /vector c(τ) and the classical energy transfer a(τ) defined through the relations /vectorb(t) =/integraldisplayt 0ds/vectorE(s), /vector c (t) =/integraldisplayt 0ds/vectorb(s), a (t) =1 2/integraldisplayt 0dsb2(s). (3) The quantity of interest, which one aims to compute, is the io nization probability P(ϕ) defined as 1)Panel discussion at this meeting.P(ϕ) =/bardbl(1−P)U(τ,0)ϕ/bardbl2= 1− /bardblPU(τ,0)ϕ/bardbl2. (4) HerePdenotes the orthogonal projection in the space L2(R3) of square integrable wave functions onto the subspace spanned by the bound states ϕofH(/vector x,t= 0), /bardbl·/bardblis the usual Hilbert space norm and the time evolution operat or is defined by U/parenleftBig t,t′/parenrightBig ≡T[Exp(−i/integraldisplayt t′H(/vector x,s)ds)], (5) withTdenoting the time ordering. The question one is interested i n is: How does P(ϕ) behave as a function of E0? In particular is it possible that P(ϕ) decreases when the field amplitude E0increases, in other words does stabilization exist? Quantitatively this means we should find a behaviour of the fo rm dP(ϕ)(E0) dE0≤0 for P(ϕ)/negationslash= 1 (6) with 0 ≤E0≤ ∞ on a finite interval for E0. We refer to a behaviour in (6) for the equal sign as weak stabilization and for strict inequali ty we call this strong stabilization. We stress once more that this description is entirely non-re lativistic. The rel- ativistic regime surely poses a new challenge and a full quan tum field theoretical treatment is desirable, but it should be possible to settle t he question just raised within the framework outlined above, since stabilization i s not claimed to be a relativistic effect. In particular it is not clear which cons equences on the physics in this regime one expects from a description in the form of th e Klein-Gordon equation2. Furthermore, appealing to a more formal description3in terms of scat- tering matrices4instead of the time evolution operator U/parenleftbig t,t′/parenrightbig will not shed any new light on the question raised, unless one deals with non-t rivial asymptotics. The time-ordering in (5) poses the main obstacle for the expl icit computations of P(ϕ). To get a handle on the issue, one can first resort to general a rguments which provide analytical expressions constraining the outcome. The least such arguments are good for is to serve as consistency checks for results obt ained by other means. This is especially useful when one has a controversy as in the case at hand. In addition we will demonstrate that they also allow some defini te statements and explain several types of physical behaviour without knowin g the exact expression of the quantities which describe them. CONSTRAINTS FROM SCALING PROPERTIES More details concerning the arguments of this section may be found in [5]. Denot- ing byλ>0 the dilatation factor and by ηthe scaling dimension of the eigenfunc- 2)See contribution to the panel discussion at this meeting by F .H.M. Faisal. 3)See contributions to the panel discussion at this meeting by F.H.M. Faisal and H. Reiss. 4)For pulses of the form (2) the scattering matrix S= lim t±→±∞exp(it+H+)·U(t+, t−)· exp(−it−H−) and U(τ,0) coincide in the weak sense. (see e.g. [1] for a more detaile d discussion)tionϕ(/vector x) :=ψ(/vector x,t= 0) of the Hamiltonian H(/vector x,t= 0), we consider the following scale transformations5 /vector x→/vector x′=λ/vector x andϕ(/vector x)→ϕ′(/vector x′) =λ−ηϕ(/vector x). (7) As the only two physical assumptions we now demand that the Hi lbert space norm, i.e./bardblϕ(/vector x)/bardbl=/bardblϕ′(/vector x′)/bardbl, remains invariant and that the scaling of the wavefunction is preserved for all times. From the first assumption we deduc e immediately that the scaling dimension has to be η=d/2 withdbeing the dimension of the space. The scaling behaviour (7) may usually be realized by scaling the coupling con- stant. Considering for instance the wavefunction ϕ(x) =√αexp(−α|x|) of the only bound state when the potential in (1) is taken to be the on e-dimensional delta-potential V(x) =αδ(x), equation (7) imposes that the coupling constant has to scale as α→α′=λ−1α. Choosing instead the Coulomb potential in the form V(/vector x) =α/rrequires the same scaling behaviour of the coupling constan t for (7) to be valid. This is exhibited directly by the explicit express ions of the corresponding wavefunctions ϕnlm(/vector x)∼α3/2(αr)lexp(−αr/n)L2l+1 n+l(2αr/n). From the second assumption we conclude ψ(/vector x,t)→ψ′(/vector x′,t′) =U′(t′,0)ϕ′(/vector x′) =λ−d/2ψ(/vector x,t) =λ−d/2U(t,0)ϕ(/vector x).(8) Consequently this means that the time evolution operator sh ould be an invariant quantity under these transformations U(t1,t0) =T/parenleftBig e−i/integraltextt1 t0H(/vector x,s)ds/parenrightBig →U′(t′ 1,t′ 0) =T/parenleftbigg e−i/integraltextλ2t1 λ2t0H′(/vector x,s)ds/parenrightbigg =U(t1,t0).(9) Equation (9) then suggests that the scaling of the time has to be compensated by the scaling of the Hamiltonian in order to achieve invarianc e. Scaling therefore the time as t→t′=ληtt , (10) equation (9) only holds if the Stark Hamiltonian of equation (1) scales as H(/vector x,t)→H′(/vector x′,t′) =ληHH(/vector x,t) with ηH=−ηt. (11) The properties (10) and (11) could also be obtained by demand ing the invariance of the Schr¨ odinger equation (1). The overall scaling behavio ur ofH(/vector x,t) is governed by the scaling of the Laplacian, such that we obtain the furth er constraint ηH=−2. (12) 5)More formally we could also carry out all our computations by using unitary dilatation opera- torsU(λ), such that the transformation of the eigenfunction is desc ribed by U(λ)ϕ(/vector x) =ληϕ′(λ/vector x) and operators Oacting on ϕ(/vector x) transform as U(λ)OU(λ)−1=O′.As a consequence we can read off the scaling properties of the p otential as V(/vector x)→V′(/vector x′) =λ−2V(/vector x). (13) Considering for instance the one-dimensional delta-poten tial and the Coulomb po- tential in the forms specified above, equation (13) imposes t hat the coupling con- stant has to scale as α→α′=λ−1αin both cases. This behaviour of the coupling constant is in agreement with our earlier observations for t he corresponding wave- functions. We will now discuss the constraint resulting from equation ( 11) on the scaling behaviour of the pulse. We directly observe that /vectorE(t)→/vectorE′(t′) =λ−3/vectorE(t). (14) This equation is not quite as restrictive as the one for the po tential, since in the latter case we could determine the behaviour of the coupling whereas now a certain ambiguity remains in the sense that we can only deduce /vectorE0→/vectorE′ 0=ληEo/vectorE0, f(t)→f′(t′) =ληff(t),withηE0+ηf=−3.(15) Thus, under the assumptions we have made, it is not possible t o disentangle the contribution coming from the scaling of the amplitude or the pulse shape function. However, there might be pulse shape functions for which ηfhas to be 0, since no suitable parameter, analogously to the coupling constan t for the potential, is available in its explicit form to achieve the expected scali ng. Finally, we come to the scaling behaviour of the ionization p robability. Noting that the projection operator has to be a scale invariant quan tity, i.e.P→P′=P, we obtain together with (7) and (9) that the ionization proba bility remains an invariant quantity under the scaling transformation P(ϕ) =/bardbl(1−P)U(τ,0)ϕ/bardbl2→ P′(ϕ′) =P(ϕ). (16) We have therefore established that transforming the length scale corresponds to a symmetry in the ionization probability P(ϕ). This symmetry can be exploited as a consistency check in various approximation methods in n umerical or analytical form as outlined in [5]. In this sense the arguments of this se ction are similar in spirit to those of the next section. Nonetheless, scaling pr operties may also be used to explain directly certain types of physical behaviour, as for instance the behaviour ofP(ϕ) as a function of the coupling constant (see [5]). CONSTRAINTS FROM BOUNDS In this section we wish to comment on the method of computing b ounds which is alternative to computing P(ϕ) exactly. This means we estimate expressions of the form/bardbl(1−P)U(τ,0)ϕ/bardbl2≤ P u(ϕ) and /bardblPU(τ,0)ϕ/bardbl2≤1− P l(ϕ) (17) such that Pl(ϕ)≤ P(ϕ)≤ P u(ϕ). (18) How does this work? We can not go into all the technical details, but we would like to illustrate the general principle of the computation al steps involved. First one should note that from a mathematical point of view there a re seldom general principles for deriving such inequalities, except for a few elementary theorems (see e.g. [7]). Therefore the steps in the derivations very often do not always appear entirely compelling. In mathematics, absolute inequaliti es, i.e. those which hold for all real numbers, are important in analysis especially in co nnection with techniques to prove convergence or error estimates, and in physics they have turned out to be extremely powerful for instance in proving the stability of matter [8] or to establish properties of the entropy [9]. The basic ingredients which are always exploited are the Min kowski and H¨ older inequalities /bardblψ+ψ′/bardbl ≤ /bardblψ/bardbl+/bardblψ′/bardbl,/bardblψψ′/bardbl ≤ /bardblψ/bardbl · /bardblψ′/bardbl, (19) used in the form /bardblψ−ψ′+X−X/bardbl ≤ /bardblψ−X/bardbl+/bardblX−ψ′/bardbl, (20) /bardblψXX−1ψ′/bardbl ≤ /bardblψX/bardbl · /bardblX−1ψ′/bardbl, (21) whereψandψ′are meant to be formal objects. The aim and sometimes the art o f all considerations is now to choose Xsuch that the loss in accuracy is minimized. One should resort here to as much physical inspiration as pos sible, for instance if there is a conjecture or a result from other sources which sug gests a dynamics one can compare with. There exist also more sophisticated possi bilities to estimate the norm, as for instance to relate the Hilbert space norm to diffe rent types of norms, e.g. the operator norm6or the Hilbert-Schmidt norm7 /bardblAψ/bardbl ≤ /bardblA/bardblop/bardblψ/bardbl ≤ /bardblA/bardblH.S./bardblψ/bardbl. (22) Where do we start? In fact, the starting point is identical to the one of perturb a- tion theory, that is the Du Hamel formula involving the time e volution operator associated to two different Hamiltonians H1(t) andH2(t) U1(t,t′) =U2(t,t′)−i/integraldisplayt t′ds U 1(t,s) (H1(s)−H2(s))U2(s,t′). (23) 6)The operator norm is defined as /bardblA/bardblop=sup ϕ:/bardblϕ/bardbl=1/bardblAϕ/bardbl. 7)Denoting by α1≥α2≥. . .the positive eigenvalues of the operator T= (A∗A)1/2the Hilbert- Schmidt norm of the operator Ais defined as /bardblA/bardblH.S.= (/summationtext∞ n=1α2 n)1/2.For instance, identifying the Stark Hamiltonian in (1) with H1(s), one chooses H2(s) =−∆/2 +/vector x·/vectorE(t) orH2(s) =−∆/2 +V(/vector x) in the high- or low intensity regime, respectively. Instead of iterating (23) and ending up with a power series in Vin the former or a power series in E0in the second case one inserts (23) into (17) and commences with the estimation of the norm in the way just o utlined. Most conveniently these considerations are carried out in a diffe rent gauge, for the high intensity regime in the Kramers-Henneberger gauge. Where do we stop? The whole procedure may be terminated when one arrives at expressions which may be computed explicitly. When can we apply bounds? In general in all circumstances. In particular problems occurring in the context of perturbative considerations, l ike the convergence, are avoided completely. Especially when the strength of the pot ential and the field are comparable, e.g. in the turn-on and off region, this method is not limited in its applicability, as is for instance the case for the Gordon-Vo lkov series. What can we deduce? IdeallyPl(ϕ) andPu(ϕ) are very close to each other, in which case we are in the position of someone solving the probl em numerically withPl(ϕ) andPu(ϕ) related to the numerical errors. If the lower bound tends to 1 for an increasing finite realistic value of E0there will be little room left for P(ϕ) to decrease and one may deduce that stabilization is absent (see figure 9 in [2]). Furthermore, we can always make statements about the e xtreme limits. For instance for the extreme frequency limit we obtain d dE0/parenleftBig lim ω→∞P(ϕ)/parenrightBig = 0. (24) This relates our discussion to the seminal paper on the stabi lization issue by Gavrila and Kaminski [10]. For the extreme field amplitude limit we fo und lim E0→∞P(ϕ) = 1 − |/angbracketleftϕ,ψ GV(τ)/angbracketright|2forb(τ) =c(τ) = 0 (25) lim E0→∞P(ϕ) = 1 otherwise , (26) whereψGV(τ) =UGV(τ,0)ϕis the Gordon-Volkov wave function. For the definition ofUGVsee (39). We would like to stress that this limit is not merely of mathematical interest8. The result (25) is a clear indication of weak stabilization , though it is still desirable to find the precise onset of this behaviour. I t is also clear that as a consequence of (25) a value of P(ϕ) which is equal or larger than the r.h.s. of (25) forany finite and experimentally realisable value of E0immediately implies the existence of strong stabilization. What are the shortcomings? For realistic values of the parameters involved the expressions sometimes yield Pl(ϕ) = 0 or Pu(ϕ) = 1 (27) 8)See contribution to the panel discussion at this meeting by F .H.M. Faisal.in which case the constraint is of course not very powerful. I n that situation it simply means that we have lost too much accuracy in the deri vation for that particular parameter setting. One should note, however, th ere is no need to give up in that situation since as is evident the expressions for the bounds are by no means unique . It should then be quite clear that one can not deduce9thatthe bound is useless if one encounters the situation (27). Even m ore such a conclusion seems very much astray in the light of [1,2,4,6], where we pre sented numerous examples for which the bounds are well beyond the values in (2 7). Sometimes this could, however, only be achieved for extremely short pulses . As we pointed out in [2] this can be overcome at the cost of having to deal with high er Rydberg states10, which is a direct consequence of the scaling behaviour outli ned in the previous section. How do typical expressions look like? In [1] we derived for instance the expression Pl(ϕ) = 1−/braceleftbigg/integraldisplayτ 0/bardbl(V(/vector x−c(t)ez)−V(/vector x))ϕ/bardbldt+ 2 2E+b(τ)2/bardbl(V(/vector x−c(τ)ez)−V(/vector x))ϕ/bardbl+2|b(τ)| 2E+b(τ)2/bardblpzϕ/bardbl/bracerightbigg2 (28) for a lower bound. For given potentials and pulse shapes term s involved in (28) may be computed at ease. As stated in [1], it is important to pa y attention to the fact that (28) is derived for the condition b(τ)2/2>−E≡binding energy11. Such restrictions which at first emerge as technical require ments in the derivations usually indicate at some physical implications. In this cas e it points at the different physical situation we encounter when the total momentum tra nsfer is vanishing (see also (25)). What still needs to be done? Probably it is unrealistic to expect to find a bound which is universally applicable and restrictive at the same time, rather one should optimize the bounds for particular situations. For instanc e it would be highly desirable to find more powerful bounds for the situations b(τ) = 0,c(τ)/negationslash= 0 and b(τ) =c(τ) = 0. For the latter case we expect in hindsight from (25) that the loss in the estimations may be minimized if in (23) we chose to compare the Stark Hamiltonian with the free Hamiltonian −∆/2 instead of H=−∆/2 +/vector x·/vectorE(t) as was done in [1]. 9)As was done by J.H. Eberly at the panel discussion at this meet ing. 10)This should not lead to the conclusion that bounds in general are exclusively applicable to higher Rydberg states, see contribution to the panel discus sion at this meeting by M. Gavrila. 11)During the panel discussion at this meeting J.H. Eberly exhi bited a plot of our result for Pl(ϕ) involving a pulse which did not satisfy this condition. As h e confirmed to a question from the audience his pulse satisfied b(τ) = 0. The conclusions drawn by J.H. Eberly concerning the usefulness of bounds based on this plot are therefore meanin gless. (See also footnote 9.)IMPORTANCE OF PULSE SHAPES From our previous discussion it is evident that the physical outcome differs for different pulse shapes. However, the fact that a pulse is adia batically switched on or off is not very important, rather the precise values of b(τ) andc(τ) are the determining quantities. In particular the case b(τ) =c(τ) = 0 (29) is very special, since then asymptotically weak stabilizat ion is certain to exist. An adiabatically switched on or off pulse sometimes satisfies (2 9), but this condition is by no means identical to it. We found no evidence for stabil ization for an adi- abatically switched on field when b(τ)/negationslash= 0. To our knowledge the importance of (29) was first pointed out by Grobe and Fedorov [11], using int uitive arguments, who employed a trapezoidal enveloping function with symmet rical turn-on and turn-off time T, which has the nice feature that for Tandτbeing integer cycles b(τ) =c(τ) = 0 and for Thalfτbeing integer cycles b(τ) = 0,c(τ)/negationslash= 0. There- after, this observation seems to have been widely ignored in the literature since many authors still employ pulses which do not have this prope rty, trading (29) for the condition of an adiabatic smooth turn-on or/and turn -off12. For instance a sine-squared switch on and off with Tandτbeing integer cycles has b(τ) = 0, c(τ)/negationslash= 0, an entire sine-squared envelope for τbeing integer cycles satisfies b(τ) = 0, c(τ)/negationslash= 0. Using gaußian envelopes or gaußian switch on and no switc h off usually yieldsb(τ)/negationslash= 0,c(τ)/negationslash= 0. A pulse which has the nice features that it allows a theoretical investigation of all possible cases for the val ues ofb(τ) andc(τ) is the tripleδ-kick in the form f(t) =δ(t) +β1δ(t−τ/2) +β2δ(t−τ), (30) which we employed in [6]. This pulse obviously satisfies b(τ) =E0(1 +β1+β2/2) and c(τ) =E0(1 +β1/2) (31) such that by tuning the constants β1,β2we may realise any desired value of b(τ) andc(τ). How do real pulses look like13? The quantity which is experimentally accessible is the Fourier transform of the pulse (2) /tildewideE(ω) =/integraldisplay∞ −∞E(t)eiωtdt=∞/summationdisplay n=0αnωn. (32) 12)As may be supported by numerous publications, this observat ion appears not to have become common knowledge as claimed by M. Gavrila in the introductio n to the panel discussion at this meeting. 13)We acknowledge that the following argument was initiated, t hough not agreed upon in this form, by an e-mail communication with H.G. Muller.withαnbeing constants. For finite pulses this quantity coincides w ith the total momentum transfer for vanishing frequency ω /tildewideE(ω= 0) =/integraldisplay∞ −∞E(t)dt=/integraldisplayτ 0E(t)dt=b(τ). (33) Provided that α0=b(τ) = 0, the Fourier transform of the momentum transfer /tildewideb(ω) =/integraldisplay∞ −∞b(t)eiωtdt (34) is on the other hand related to the total displacement for van ishing frequency /tildewideb(ω= 0) =/integraldisplay∞ −∞b(t)dt=/integraldisplayτ 0b(t)dt=c(τ) (35) such that /tildewideE(ω) =b(t)eiωt|∞ −∞−iω/tildewideb(ω)∼ −iωc(τ) +O(ω2). (36) This means that when the experimental outcome is /tildewideE(ω) =α2ω2+α3ω3+α4ω4+... (37) the total momentum transfer and the total displacement are z ero. Experimentally, the observed fall off is expected to be even stronger [12]. COMPARISON WITH GV-PERTURBATION THEORY It is instructive to compare our findings with other standard methods as for instance the Gordon-Volkov (GV) perturbation theory. Usin g now in (23) for H2 the Hamiltonian just involving the field and the free particl e Hamiltonian in the Kramers-Henneberger frame subsequent iteration yields U1(t,t′) =UGV(t,t′)−i/integraldisplayt t′ds U GV(t,s)VUGV(s,t′) −/integraldisplayt t′ds/integraldisplayt sds′UGV(t,s′)VUGV(s′,s)VUGV(s,t′) +... (38) whereUGVcorresponds to the free-particle evolution operator in the KH frame UGV(t,t′) =e−ia(t)e−ib(t)zeic(t)pze−i(t−t′)p2 2e−ic(t′)pzeib(t′)zeia(t′). (39) As was explained in [4] we may use these expressions together with the Riemann- Lebesgue theorem in order to obtain the extreme frequency an d intensity limit, finding (24), (25) and (26). For these arguments to be valid we have to assumethat the Gordon-Volkov series makes sense, so in particular we have to assume its convergence. The latter assumption may be made more rigorous when conside ring the one- dimensional delta potential V(x) =−αδ(x) which is well known to possess only one bound state. In that case the problem of computing ioniza tion probabilities is reduced to the evaluation of P(ϕ) = 1− |/angbracketleftϕ,ψ GV(τ)/angbracketright+/angbracketleftϕ,Ψ(τ)/angbracketright|2(40) with /angbracketleftϕ,ψ GV(τ)/angbracketright=2 πe−ia(τ)/integraldisplay∞ −∞dpexp/parenleftBig −iτα2p2 2−ic(τ)αp/parenrightBig /parenleftbig 1 + (p+b(τ)/α)2/parenrightbig (1 +p2)(41) /angbracketleftϕ,Ψ(τ)/angbracketright=ie−ia(τ)/radicalbigg α5 2π3/integraldisplayτ 0/integraldisplay∞ −∞ψI(s)ei(c(τ)−c(s))pe−i 2p2(τ−s)dsdp (α2+ (p+b(τ))2). (42) Here the only unknown is the function ψI(t) which can be obtained as a solution of the Volterra equation ψI(t) =/integraldisplay∞ −∞dpψ GV(p,t) +α/radicalbigg i 2π/integraldisplayt 0dsψI(s)ei(c(t)−c(s))2 2(t−s) √t−s. (43) Iteration of this equation is a well controllable procedure and in [6] we found that the series converges for all values of α. The results obtained from the analysis of this equation match the results obtained from bounds. CONCLUSIONS The main outcome of our investigations is that the classical momentum trans- feranddisplacement caused by a laser pulse on an electron are the essential parameters determining the existence of weak asymptotic st abilization. In fact, we obtained evidence for stabilization only for pulses for whi ch these two quantities vanish at the end of the pulse, i.e., with b(τ) = 0 andc(τ) = 0. Using purely analytical methods, we have shown that, for a wi de range of po- tentials, namely Kato and one- and three-dimensional delta potentials, we always have lim E0→∞P(ψ) = 1 unless b(τ) = 0 andc(τ) = 0, in which case the ionization probability tends to the lowest order in GV-perturbation th eory, which corresponds simply to the free particle Green’s function (39). Furtherm ore, for infinite frequen- cies, the high-frequency condition of [10] is a way to obtain b(t) = 0 andc(t) = 0 foralltimes.Clearly, smooth pulses in general do not necessarily fullfil the above conditions, and therefore will not provide a mechanism for stabilizatio n, but just prolong the onset of ionization. In fact, we have observed no stabilizat ion for adiabatically switched on and off pulses of several shapes, for which analyt ic expressions for lower bounds of ionization probabilities lead to conclusive statements concerning the existence or absence of stabilization. Therefore, as an overall conclusion: Bounds are useful indeed, also in the context of high intensity laser physics! REFERENCES 1. Fring A., Kostrykin V. and Schrader R., J. Phys. B: At. Mol. Opt. Phys. 29(1996) 5651. 2. Figueira de Morisson Faria C. , Fring A. and Schrader R., J. Phys. B: At. Mol. Opt. Phys.31(1998) 449. 3. Fring A., Kostrykin V. and Schrader R., J. Phys. A: Math. Gen. 30(1997) 8599. 4. Figueira de Morisson Faria C., Fring A. and Schrader R., Laser Physics 9(1999) 379. 5. Figueira de Morisson Faria C., Fring A. and Schrader R. ” Existence Criteria for Sta- bilization from the Scaling Behaviour of Ionization Probab ilities”, physics/9911046. 6. Figueira de Morisson Faria C., Fring A. and Schrader R. ” Momentum Transfer, Displacement and Stabilization ”, in preparation. 7. Hardy G.H., Littlewood J.E. and Polya G., Inequalities , Cambridge, CUP, 1934; Levin V.I. and Stechkin S.B. Amer. Math. Soc. Transl. 14 (1960) 1. 8. Lieb E., Rev. Mod. Phys. 48(1976) 553. 9. Wigner E.P. and Yanase M., Proc. Nat. Acad. Sci. US 49(1963); Wehrl A., Rev. Mod. Phys. 50(1978) 221. 10. Gavrila M. and Kaminski J.Z., Phys. Rev. Lett. 53(1984) 613. 11. Grobe R. and Fedorov M.V., Phys. Rev. Lett. 68(1993) 2592. 12. H.G. Muller, private communication.

arXiv:physics/9912033v1 [physics.hist-ph] 16 Dec 1999On the Gravitational Field of a Sphere of Incompressible Fluid according to Einstein’s Theory † by K. Schwarzschild (Communicated February 24th, 1916 [see above p. 313].) (Translation ‡by S. Antoci∗) §1. As a further example of Einstein’s theory of gravitation I have calculated the gravitational field of a homogeneous sphere of finite radius, which consists of incompressible fluid. The addition “of incompressible fluid” is necessary, since in the theory o f relativity gravitation depends not only on the quantity of matter, but also on its energy, and e. g.a solid body in a given state of tension would yield a gravitation different from a fluid. The computation is an immediate extension of my communicati on on the gravitational field of a mass point (these Sitzungsberichte 1916, p. 189), that I sh all quote as “Mass point” for short. §2. Einstein’s field equations of gravitation (these Sitzung sber. 1915, p. 845) read in general: /summationdisplay α∂Γα µν ∂xα+/summationdisplay αβΓα µβΓβ να=Gµν. (1) The quantities Gµνvanish where no matter is present. In the interior of an incom pressible fluid they are determined in the following way: the “mixed energy t ensor” of an incompressible fluid at rest is, according to Mr. Einstein (these Sitzungsber. 1914 , p. 1062, the Ppresent there vanishes due to the incompressibility): T1 1=T2 2=T3 3=−p, T4 4=ρ0,(the remaining Tν µ= 0). (2) Herepmeans the pressure, ρ0the constant density of the fluid. The “covariant energy tensor” will be: Tµν=/summationdisplay σTσ µgνσ. (3) Furthermore: T=/summationdisplay σTσ σ=ρ0−3p (4) and κ= 8πk2, †Sitzungsberichte der K¨ oniglich Preussischen Akademie de r Wissenschaften zu Berlin, Phys.-Math. Klasse 1916, 424-434. ‡The valuable advice of A. Loinger is gratefully acknowledge d. ∗Dipartimento di Fisica “A. Volta”, Universit` a di Pavia, Vi a Bassi 6 - 27100 Pavia (Italy). 1where k2is Gauss’ gravitational constant. Then according to Mr. Ein stein (these Berichte 1915, p. 845, Eq. 2a) the right-hand sides of the field equations rea d: Gµν=−κ(Tµν−1 2gµνT). (5) Since the fluid is in equilibrium, the conditions /summationdisplay α∂Tα σ ∂xα+/summationdisplay µνΓµ σνTν µ= 0 (6) must be satisfied (ibidem Eq. 7a). §3. Just as in “Mass point”, also for the sphere the general equ ations must be specialised to the case of rotation symmetry around the origin. Like there, it is convenient to introduce the polar coordinates of determinant 1: x1=r3 3, x2=−cosϑ, x 3=φ, x 4=t. (7) Then the line element, like there, must have the form: ds2=f4dx2 4−f1dx2 1−f2dx2 2 1−x2 2−f2dx2 3(1−x2 2), (8) hence one has: g11=−f1, g22=−f2 1−x2 2, g33=−f2(1−x2 2), g44=f4 (the remaining g µν= 0). Moreover the fare functions only of x1. The solutions (10), (11), (12) reported in that paper hold al so for the space outside the sphere: f4= 1−α(3x1+ρ)−1/3, f2= (3x1+ρ)2/3, f1f2 2f4= 1, (9) where αandρare for now two arbitrary constants, that must be determined afterwards by the mass and by the radius of our sphere. It remains the task to establish the field equations for the in terior of the sphere by means of the expression (8) of the line element, and to solve them. For the right-hand sides one obtains in sequence: T11=T1 1g11=−pf1, T22=T2 2g22=−pf2 1−x2 2, T33=T3 3g33=−pf2(1−x2 2), T44=T4 4g44=ρ0f4. G11=κf1 2(p−ρ0), G22=κf2 21 1−x2 2(p−ρ0), G33=κf2 2(1−x2 2)(p−ρ0), G44=−κf4 2(ρ0+ 3p). The expressions of the components Γα µνof the gravitational field in terms of the functions fand the left-hand sides of the field equations can be taken without ch ange from “Mass point” ( §4). If one again restricts himself to the equator ( x2= 0), one gets the following overall system of equations: 2First the three field equations: −1 2∂ ∂x1/parenleftbigg1 f1∂f1 ∂x1/parenrightbigg +1 41 f2 1/parenleftbigg∂f1 ∂x1/parenrightbigg2 +1 21 f2 2/parenleftbigg∂f2 ∂x1/parenrightbigg2 +1 41 f2 4/parenleftbigg∂f4 ∂x1/parenrightbigg2 =−κ 2f1(ρ0−p),(a) +1 2∂ ∂x1/parenleftbigg1 f1∂f2 ∂x1/parenrightbigg −1−1 21 f1f2/parenleftbigg∂f2 ∂x1/parenrightbigg2 =−κ 2f2(ρ0−p), (b) −1 2∂ ∂x1/parenleftbigg1 f1∂f4 ∂x1/parenrightbigg +1 21 f1f4/parenleftbigg∂f4 ∂x1/parenrightbigg2 =−κ 2f4(ρ0+ 3p). (c) In addition comes the equation for the determinant: f1f2 2f4= 1. (d) The equilibrium conditions (6) yield the single equation: −∂p ∂x1=−p 2/bracketleftbigg1 f1∂f1 ∂x1+2 f2∂f2 ∂x1/bracketrightbigg +ρ0 21 f4∂f4 x1. (e) From the general considerations of Mr. Einstein it turns out that the present 5 equations with the 4 unknown functions f1,f2,f4,pare mutually compatible. We have to determine a solution of these 5 equations that is fr ee from singularities in the interior of the sphere. At the surface of the sphere it must be p= 0, and there the functions f together with their first derivatives must reach with contin uity the values (9) that hold outside the sphere. For simplicity the index 1 of x1will be henceforth omitted. §4. By means of the equation for the determinant the equilibri um condition (e) becomes: −∂p ∂x=ρ0+p 21 f4∂f4 ∂x. This can be immediately integrated and gives: (ρ0+p)/radicalbig f4=const. =γ. (10) Through multiplication by the factors −2, +2f1/f2,−2f1/f4the field equations (a), (b), (c) trans- form into: ∂ ∂x/parenleftbigg1 f1∂f1 ∂x/parenrightbigg =1 2f2 1/parenleftbigg∂f1 ∂x/parenrightbigg2 +1 f2 2/parenleftbigg∂f2 ∂x/parenrightbigg2 +1 2f2 4/parenleftbigg∂f4 ∂x/parenrightbigg2 +κf1(ρ0−p), (a′) ∂ ∂x/parenleftbigg1 f2∂f2 ∂x/parenrightbigg = 2f1 f2+1 f1f2∂f1 ∂x∂f2 ∂x−κf1(ρ0−p), (b′) ∂ ∂x/parenleftbigg1 f4∂f4 ∂x/parenrightbigg =1 f1f4∂f1 ∂x∂f4 ∂x+κf1(ρ0+ 3p). (c′) If one builds the combinations a′+2b′+c′anda′+c′, by availing of the equation for the determinant one gets: 0 = 4f1 f2−1 f2 2/parenleftbigg∂f2 ∂x/parenrightbigg2 −2 f2f4∂f2 ∂x∂f4 ∂x+ 4κf1p (11) 30 = 2∂ ∂x/parenleftbigg1 f2∂f2 ∂x/parenrightbigg +3 f2 2/parenleftbigg∂f2 ∂x/parenrightbigg2 + 2κf1(ρ0+p). (12) We will introduce here new variables, which recommend thems elves since, according to the results of “Mass point”, they behave in a very simple way outside the s phere. Therefore they must bring also the parts of the present equations free from ρ0andpto a simple form. One sets: f2=η2/3, f4=ζη−1/3, f1=1 ζη. (13) Then according to (9) one has outside the sphere: η= 3x+ρ, ζ=η1/3−α, (14) ∂η ∂x= 3,∂ζ ∂x=η−2/3. (15) If one introduces these new variables and substitutes γf−1/2 4forρ0+paccording to (10), the equations (11) and (12) become: ∂η ∂x∂ζ ∂x= 3η−2/3+ 3κγζ−1/2η1/6−3κρ0, (16) 2ζ∂2η ∂x2=−3κγζ−1/2η1/6. (17) The addition of these two equations gives: 2ζ∂2η ∂x2+∂η ∂x∂ζ ∂x= 3η−2/3−3κρ0. The integrating factor of this equation is ∂η/∂x . The integration gives: ζ/parenleftbigg∂η ∂x/parenrightbigg2 = 9η1/3−3κρ0η+ 9λ(λ integration constant ). (18) When raised to the power 3 /2, this gives: ζ3/2/parenleftbigg∂η ∂x/parenrightbigg3 = (9η1/3−3κρ0η+ 9λ)3/2 If one divides (17) by this equation, ζdisappears, and it remains the following differential equat ion forη: 2∂2η ∂x2/parenleftbig∂η ∂x/parenrightbig3=−3κγη1/6 (9η1/3−3κρ0η+λ)3/2. Here∂η/∂x is again the integrating factor. The integration gives: 2/parenleftbig∂η ∂x/parenrightbig= 3κγ/integraldisplayη1/6dη (9η1/3−3κρ0η+λ)3/2(19) and since:2 δη δx=2δx δη 4through a further integration it follows: x=κγ 18/integraldisplay dη/integraldisplayη1/6dη (η1/3−κρ0 3η+λ)3/2. (20) From here xturns out as function of η, and through inversion ηas function of x. Then ζfollows from (18) and (19), and the functions fthrough (13). Hence our problem is reduced to quadratures. §5. The integration constants must now be determined in such a way that the interior of the sphere remains free from singularities and the continuous j unction to the external values of the functions fand of their derivatives at the surface of the sphere is reali sed. Let us put r=ra,x=xa,η=ηa, etc. at the surface of the sphere. The continuity of ηand ζcan always be secured through a subsequent appropriate dete rmination of the constants αandρ in (14). In order that also the derivatives stay continuous a nd, in keeping with (15), ( dη/dx )a= 3 and (dζ/dx )a=η−2/3 a, according to (16) and (18) it must be: γ=ρ0ζ1/2 aη−1/6 a, ζa=η1/3 a−κρ0 3ηa+λ. (21) From here follows: ζaη−1/3 a= (f4)a= 1−κρ0 3η2/3 a+λη−1/3 a. Therefore γ=ρ0/radicalbig (f4)a. (22) One sees from the comparison with (10) that in this way also th e condition p= 0 at the surface is satisfied. The condition ( dη/dx )a= 3 yields the following determination for the limits of inte gration in (19): 3dx dη= 1−κγ 6/integraldisplayηa ηη1/6dη (η1/3−κρ0 3η+λ)3/2(23) and therefore (20) undergoes the following determination o f the limits of integration: 3(x−xa) =η−ηa+κγ 6/integraldisplayηa ηdη/integraldisplayηa ηη1/6dη (η1/3−κρ0 3η+λ)3/2. (24) The surface conditions are therefore completely satisfied. Still undetermined are the two constants ηaandλ, which will be fixed through the conditions of continuity at t he origin. We must first of all require that for x= 0 it should be also η= 0. If this were not the case, f2in the origin would be a finite quantity, and an angular variat iondφ=dx3in the origin, which in reality means no motion at all, would give a contribution t o the line element. Hence from (24) follows the condition for fixing ηa: 3xa=ηa−κγ 6/integraldisplayηa 0dη/integraldisplayηa ηη1/6dη (η1/3−κρ0 3η+λ)3/2. (25) λwill be fixed at last through the condition that the pressure a t the center of the sphere shall remain finite and positive, from which according to (10) it fo llows that there f4must remain finite and different from zero. According to (13), (18) and (23) one h as: f4=ζη−1/3=/parenleftbigg 1−κρ0 3η2/3+λη−1/3/parenrightbigg/bracketleftbigg 1−κγ 6/integraldisplayηa ηη1/6dη (η1/3−κρ0 3η+λ)3/2/bracketrightbigg2 . (26) 5One provisorily supposes either λ >0 orλ <0. Then, for very small η: f4=λ η1/3/bracketleftbigg K+κγ 7η7/6 λ3/2/bracketrightbigg2 , where one has set: K= 1−κγ 6/integraldisplayηa 0η1/6dη (η1/3−κρ0 3η+λ)3/2. (27) In the center ( η= 0)f4will then be infinite, unless K= 0. But, if K= 0,f4vanishes for η= 0. In no case, for η= 0,f4results finite and different from zero. Hence one sees that the hypothesis: either λ >0 orλ <0, does not bring to physically practicable solutions, and i t turns out that it must be λ= 0. §6. With the condition λ= 0 all the integration constants are now fixed. At the same tim e the integrations to be executed become very easy. If one intr oduces a new variable χinstead of η through the definition: sinχ=/radicalbiggκρ0 3·η1/3/parenleftbigg sinχ a=/radicalbiggκρ0 3·η1/3 a/parenrightbigg , (28) through an elementary calculation the equations (13), (26) , (10), (24), (25) transform themselves into the following: f2=3 κρ0sin2χ, f 4=/parenleftbigg3cosχ a−cosχ 2/parenrightbigg2 , f1f2 2f4= 1. (29) ρ0+p=ρ02cosχ a 3cosχ a−cosχ(30) 3x=r3=/parenleftbiggκρ0 3/parenrightbigg−3/2/bracketleftbigg9 4cosχ a/parenleftbig χ−1 2sin2χ/parenrightbig −1 2sin3χ/bracketrightbigg . (31) The constant χais determined by the density ρ0and by the radius raof the sphere according to the relation: /parenleftbiggκρ0 3/parenrightbigg3/2 r3 a=9 4cosχ a/parenleftbig χa−1 2sin2χa/parenrightbig −1 2sin3χa. (32) The constants αandρof the solution for the external region come from (14): ρ=ηa−3xaα=η1/3 a−ζa and obtain the values: ρ=/parenleftbiggκρ0 3/parenrightbigg−3/2/bracketleftbigg3 2sin3χa−9 4cosχ a/parenleftbig χa−1 2sin2χa/parenrightbig/bracketrightbigg (33) α=/parenleftbiggκρ0 3/parenrightbigg−1/2 ·sin3χa. (34) 6When one avails of the variables χ,ϑ,φinstead of x1,x2,x3(ix), the line element in the interior of the sphere takes the simple form: ds2=/parenleftbigg3cosχ a−cosχ 2/parenrightbigg2 dt2−3 κρ0[dχ2+sin2χdϑ2+sin2χsin2ϑdφ2]. (35) Outside the sphere the form of the line element remains the sa me as in “Mass point”: ds2= (1 −α/R)dt2−dR2 1−α/R−R2(dϑ2+ sin2ϑdφ2) where R3=r3+ρ.(36) Nowρwill be determined by (33), while for the mass point it was ρ=α3. §7. The following remarks apply to the complete solution of ou r problem contained in the previous paragraphs . 1. The spatial line element ( dt= 0) in the interior of the sphere reads: −ds2=3 κρ0[dχ2+sin2χdϑ2+sin2χsin2ϑdφ2]. This is the known line element of the so called non Euclidean g eometry of the spherical space. Therefore the geometry of the spherical space holds in the in terior of our sphere . The curvature radius of the spherical space will be/radicalbig 3/κρ0. Our sphere does not constitute the whole spherical space, but only a part, since χcan not grow up to π/2, but only up to the limit χa. For the Sun the curvature radius of the spherical space, that rules the g eometry in its interior, is about 500 times the radius of the Sun (see formulae (39) and (42)). That the geometry of the spherical space, that up to now had to be considered as a mere possibility, requires to be real in the interior of gravitat ing spheres, is an interesting result of Einstein’s theory. Inside the sphere the quantities: /radicalbigg3 κρ0dχ,/radicalbigg3 κρ0sinχdϑ,/radicalbigg3 κρ0sinχsinϑdφ, (37) are “naturally measured” lengths. The radius “measured ins ide” from the center of the sphere up to its surface is: Pi=/radicalbigg 3 κρ0χa. (38) The circumference of the sphere, measured along a meridian ( or another great circle) and divided by 2π, is called the radius “measured outside” Po. It turns out to be: Po=/radicalbigg3 κρ0sinχ a. (39) According to the expression (36) of the line element outside the sphere this Pois clearly identical with the value Ra= (r3 a+ρ)1/3that the variable Rassumes at the surface of the sphere. With the radius Poone gets for αfrom (34) the simple relations: α Po=sin2χa, α=κρ0 3P3 o. (40) 7The volume of our sphere is: V=/parenleftbigg/radicalbigg 3 κρ0/parenrightbigg3/integraldisplayχa 0dχsin2χ/integraldisplayπ 0dϑsinϑ/integraldisplay2π 0dφ = 2π/parenleftbigg/radicalbigg3 κρ0/parenrightbigg3/parenleftbigg χa−1 2sin2χa/parenrightbigg . Hence the mass of our sphere will be ( κ= 8πk2) M=ρ0V=3 4k2/radicalbigg3 κρ0/parenleftbigg χa−1 2sin2χa/parenrightbigg . (41) 2. About the equations of motion of a point of infinitely small mass outside our sphere, which maintain tha same form as in “Mass point” (there equations (1 5)-(17)), one makes the following remarks: For large distances the motion of the point occurs according to Newton’s law, with α/2k2 playing the rˆ ole of the attracting mass. Therefore α/2k2can be designated as “gravitational mass” of our sphere. If one lets a point fall from the rest at infinity down to the sur face of the sphere, the “naturally measured” fall velocity takes the value: va=1/radicalbig 1−α/RdR ds=/radicalbiggα Ra. Hence, due to (40): va=sinχ a. (42) For the Sun the fall velocity is about 1/500 the velocity of li ght. One easily satisfies himself that, with the small value thus resulting for χaandχ(< χa), all our equations coincide with the equations of Newton’s theory apart from the known second ord er Einstein’s effects. 3. For the ratio between the gravitational mass α/2k2and the substantial mass Mone finds α 2k2M=2 3sin3χa χa−1 2sin2χa. (43) With the growth of the fall velocity va(=sinχ a), the growth of the mass concentration lowers the ratio between the gravitational mass and the substantia l mass. This becomes clear for the fact thate. g. with constant mass and increasing density one has the transi tion to a smaller radius with emission of energy (lowering of the temperature throug h radiation). 4. The velocity of light in our sphere is v=2 3cosχ a−cosχ, (44) hence it grows from the value 1 /cosχ aat the surface to the value 2 /(3cosχ a−1) at the center. The value of the pressure quantity ρ0+paccording to (10) and (30) grows in direct proportion to the velocity of light. At the center of the sphere ( χ= 0) velocity of light and pressure become infinite when cosχ a= 1/3, and the fall velocity becomes/radicalbig 8/9 of the (naturally measured) velocity of light. Hence there is a limit to the concentration, above which a sphere of incom pressible fluid can not exist. If one would apply our equations to values cosχ a<1/3, one would get discontinuities already outside 8the center of the sphere. One can however find solutions of the problem for larger χa, which are continuous at least outside the center of the sphere, if one g oes over to the case of either λ >0 orλ <0, and satisfies the condition K= 0 (Eq. 27). On the road of these solutions, that are clearly not physically meaningful, since they give infinite pressure at the center, one can go over to the limit case of a mass concentrated to one point, and retrie ves then the relation ρ=α3, which, according to the previous study, holds for the mass point. It is further noticed here that one can speak of a mass point only as far as one avails of the variable r, that otherwise in a surprising way plays no rˆ ole for the geometry and for the motion inside our g ravitational field. For an observer measuring from outside it follows from (40) that a sphere of g iven gravitational mass α/2k2can not have a radius measured from outside smaller than: Po=α. For a sphere of incompressible fluid the limit will be 9/8α.(For the Sun αis equal to 3 km, for a mass of 1 gram is equal to 1 .5·10−28cm.) 9

arXiv:physics/9912034v1 [physics.data-an] 16 Dec 1999Kalman Filter Track Fits and Track Breakpoint Analysis Pierre AstieraAlessandro CardinibRobert D. Cousinsb,1 Antoine Letessier-SelvonaBoris A. Popova,2 Tatiana Vinogradovab aLPNHE, Laboratoire de Physique Nucl´ eaire et des Hautes Ene rgies, Universit´ es de Paris 6 et 7, 75252 Paris Cedex 05, France. bDepartment of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, U.S.A. Abstract We give an overview of track fitting using the Kalman filter met hod in the NOMAD detector at CERN, and emphasize how the wealth of by-product information can be used to analyze track breakpoints (discontinuities in tr ack parameters caused by scattering, decay, etc.). After reviewing how this informa tion has been previously exploited by others, we describe extensions which add power to breakpoint detection and characterization. We show how complete fits to the entire track, with breakpoint parameters added, can be easily obtained from the informati on from unbroken fits. Tests inspired by the Fisher F-test can then be used to judge breakpoints. Signed quantities (such as change in momentum at the breakpoint) ca n supplement un- signed quantities such as the various chisquares. We illust rate the method with electrons from real data, and with Monte Carlo simulations o f pion decays. Key words: breakpoints, Kalman filter, track fitting PACS code: 07.05.Kf 1 Introduction The Kalman filter is an efficient algorithm for fitting tracks in particle spec- trometers with many position-sensing detectors [1–6]. It c ures many of the problems of traditional χ2track fitting using Newton steps, which becomes 1Email address: cousins@physics.ucla.edu 2on leave from the Laboratory of Nuclear Problems, JINR, 1419 80 Dubna, Russia. Accepted for publication in Nuclear Instruments and Method s 14 December 1999more and more unwieldy as the number of position measurement s increases. In such situations where Kalman filtering is naturally applied , it can be possible to detect and characterize track breakpoints , defined as locations where one or more of the track parameters is discontinuous. Obvious brea kpoints, such as a large kink due to a particle decay, are often found before a f ull track fit is performed. More subtle breakpoints may only manifest thems elves when the track is fit to obtain the track parameters. Fr¨ uhwirth[3] has investigated the detection of breakpoin ts using information which is a natural by-product of a Kalman filter track fit. When fitting a drift- chamber track with Nposition measurements (“hits”), the idea has been the following. At the location of hit k, one has the best-fit track parameters (a) using hits 1 through kand (b) using hits k+1 through N. One constructs a χ2 for the consistency of these two sets of track parameters. Th isχ2, along with other χ2’s at hand from the two fits, can be combined to form test statis tics for breakpoints. Such methods suffer a loss of power because of two defects. Fir st,χ2’s by con- struction throw away information about the arithmetic sign s of differences; such information is relevant since a track’s momentum shoul d normally de- crease when the particle decays. Second, appropriate constraints of a breakpoint hypothesi s are not incorpo- rated. For example, if a particle decays at hit k, a desired quantity is the mismatch in track parameters describing the momentum vector, under the constraint that the track position vector from fits (a) and (b) is identical. In this paper, we describe a procedure which uses informatio n from the Kalman filter fit to construct the result of a fulltrack fit which has additional track parameters to account for the discontinuities at a breakpoi nt. It is natural to allow for one, two, or three discontinuous parameters in o rder to describe different physical processes: Type I: An electron emitting a bremsstrahlung photon generally cha nges only its momentum magnitude , since the photon is essentially collinear with the electron direction. Type II: A particle with a hard elastic scatter may have momentum mag- nitude essentially unchanged, while changing the two angle s specifying di- rection. Type III: A charged pion or kaon decaying to µνin general changes mo- mentum magnitude as well as the two angles. With the method we describe, trial breakpoint fits at every hi t (away from the ends) of the track can be quickly obtained, and used to sea rch for and characterize breakpoints. 2Billoir [1] has investigated Type II breakpoints, performi ng fits which do not assume an existing breakpointless fit. We show in this paper h ow the by- products of a Kalman filter fit allow one to avoid refitting the h its while incorporating the constraints. We discuss these tools in the context of the NOMAD [7] neutrin o experiment at CERN, within which this development took place. However, the results are generally applicable to any experiment in which the numb er of position measurements is large enough to fit the track on both sides of t he potential breakpoint. In Sec. 2 we introduce some notation and describe our track pa rameters and track models, including energy loss. In Sec. 3 a review of the traditional (non- Kalman) track fit is given. Sec. 4 describes its replacement b y the Kalman filter. In Sec. 5, we briefly review previous work on breakpoin t variables. In Sec. 6, we introduce the new breakpoint variables, and in Sec . 7, we present some indicative results of their use. We conclude in Sec. 8. 2 Track parameters and track models 2.1 Parameters When fitting a track, one typically describes its location in 6D phase space by choosing a fixed reference surface (a vacuum window, chamb er plane, etc.) and then fitting for the 5 independent parameters of the track at the position where the track intersects this surface . We let the vector xcontain these track parameters. In a fixed-target experiment with beam directio n along the zaxis, the parameterization of xis often taken to be x= (x, y,dx/dz,dy/dz, q/p) (1) at a reference plane at fixed z, where q/pis the charge/momentum. NOMAD is a fixed-target experiment with drift chamber planes perpendicular to the zdirection (nearly aligned with the neutrino beam). However , the chambers are immersed in a uniform magnetic field3, so that soft tracks often loop back, and a helical parameterization similar to collid er experiments is 3Sense wires of one chamber make angles of +5, 0 and –5 degrees w ith respect to the magnetic field direction providing a space measuremen t along coordinates designated u,y, and v. 3YParticle's trajectoryB tan λϕ R ZX(x,y,z,t0) VU Fig. 1. Definition of the helix parameters used to describe ch arged particle trajectory in the NOMAD setup. more appropriate. We maintain the reference surface as a pla ne with fixed z (thezof the first hit of the track), and specify a track there by x= (x, y,1/R,tanλ, φ, t), (2) where we have introduced the three parameters of a helical cu rve in the uni- form magnetic field: the signed4inverse radius of curvature 1 /R, the dip angle tanλ, and the angle of rotation φ(see Fig. 1). In addition to these 5 traditional parameters, NOMAD has a sixth parameter, the zero-time-offs et for the drift chamber measurements, called t. It has been introduced because of trigger time jitters. In the following, unless specified otherwise, track parameters will refer to this second parameterization. The parameter 1 /Ris related to the momentum pby 1 R∝B p/radicalBig 1 + (tan λ)2, (3) while the sign of (1 /R) only reflects the particle charge if the time runs the right way along the track. In NOMAD the sign convention we use d implied that the product R·φincreases with time along the assumed time direction given by the ordering of the measurements along the track. 4In NOMAD, 1 /Rhas a sign opposite to the particle charge. 4Throughout our work, we use a change in 1 /Ras an indicator of a change inp; this is strictly true only when the change in tan λis negligible, but is an adequate approximation. (Inhomogeneity in the magnetic field B does not matter, since we compare 1 /Restimates at the same point.) With this parameterization, the three physical processes e numerated in the Introduction have the following breakpoint signatures: Type I: An electron emitting a bremsstrahlung photon has a disconti nuity in 1/R. Type II: A pion with a hard elastic scatter has discontinuities in tan λand φ. Type III: A charged pion or kaon decaying to µνhas discontinuities in 1 /R, tanλ, and φ. From the estimation of the track parameters at the reference plane one needs a transformation, called the track model, with which one com putes the ex- pected measurements at any position in the detector. This mo del describes the dependence of the measurements on the initial values in t he ideal case of no measurement errors and of deterministic interactions of the particle with matter. Use of a correct description is of the utmost importa nce for the per- formance of the fitting procedure, be it traditional or not. 2.2 Equation of motion in a magnetic field The trajectory of a charged particle in a (static) magnetic fi eld is determined by the following equation of motion: d2r/d2s= (kq/p)·( dr/ds)×B(r(s)), (4) where ris the position vector, sis the path length, kis a constant of propor- tionality, qis the (signed) charge of the particle, pis the absolute value of its momentum, and B(r) is the static magnetic field. With our parameterization it proved convenient to use φas the running param- eter rather than zsince particles may loop back in the detector and cross the same measurement plane several times. The equation of motio n can be readily integrated along a trajectory step (from position 0 to posit ion 1) where Rand tanλare assumed constant. x1=x0+R0·tanλ·(φ1−φ0) (5) y1=y0+R0·(cosφ1−cosφ0) (6) z1=z0+R0·(sinφ1−sinφ0) (7) 5R1=R0 (8) tanλ1= tan λ0 (9) t1=t0+R0·(φ1−φ0)/(βcosλ0). (10) In the last equation, βis the particle velocity and R·∆φhas the right sign following our convention. In NOMAD, detector measurement planes are located at fixed zand Eqn. 7 can be solved, sinφ1= sinφ0+ (z1−z0)·(1/R0), (11) to obtain φ1at the desired z. In practice, among all the possible solutions, our track model returns the one corresponding to the next cro ssing in the requested time direction. The magnetic field strength varies by a few percent in the trac king volume. This was accommodated by ignoring the minor components of th e field and updating 1 /Rat every tracking step so that the product R·Bremains constant, up to energy losses which are now discussed. 2.3 Energy losses The ionization losses are accounted for by updating 1 /Rat every tracking step5: ∆(1/R) =d(1/R) dEdE dx∆x=|1/R| 0.3BβdE dx∆φ (12) where ∆ φ=φend−φstart, and where d E/dxis given by the Bethe-Bloch equation, evaluated (by default) with the pion mass, and a lo cal matter density extracted from the detector model used in the GEANT [8] simul ation of the experiment. In the central tracker part of the detector, the matter density is about 0.1 g/cm3: the ionization loss model does not need to include detailed relativistic corrections. In NOMAD, bremsstrahlung losses should be accounted for (on average) in the track model for electron tracks, because the central tracke r amounts to about 1 radiation length ( X0). In the electron (and positron) track model, one adds 5The absolute value derives from our sign conventions, so tha t a track gains energy if tracked backwards in time. 6to the ionization losses (evaluated with the electron mass) bremsstrahlung losses: ∆(1/R) =∆φ β2X0cosλ, (13) where we can readily approximate β= 1. As expressed, these losses include the whole radiated photon spectrum, although beyond a certa in threshold, the radiated photons can be detected in the downstream electrom agnetic calorime- ter, or sometimes as a conversion pair in the tracker. But no c onvincing way was found to define a threshold that separates continuous sma ll losses from accidental big ones. 3 Traditional track fits Given the track model and an estimate ( x0) of the parameters at the refer- ence position, one can compute at each measurement location in the detector (labeled k= 1, . . .N ) the theoretical “ ideal measurements ” which would be made in the absence of fluctuations due to the two categories o f “noise”: pro- cess noise (multiple scattering, bremsstrahlung, etc.) an d measurement noise (detector resolution). This computation is represented by the system equation in the absence of noise, xk=fk(x0), (14) where fis a deterministic function (the track model) giving the val ues of the parameters at each location k. In general the set of parameters xkis not measured directly by the apparatus; only a function of it, hk(xk), is observed. (In our case hkis a drift chamber position measurement.) Let mk=hk(xk) +εk (15) be the measurement equation where εkrepresents the measurement errors. By convention, ∝an}bracketle{tmk∝an}bracketri}ht=hk(xk) and ∝an}bracketle{tεi∝an}bracketri}ht= 0. In practice, one has to check that the track model (involved in the calculation of xk) and the measurement function hkfulfill this convention. 7In a traditional (non-Kalman) track fit, one calculates the N×Ncovariance matrix of the measurements6: V(m) ij=∝an}bracketle{t(mi−hi(xi))(mj−hj(xj))∝an}bracketri}ht, (16) where the angle-brackets represent an average over an ensem ble of tracks with the same track parameters x. According to the Gauss-Markov theorem, the minimization of χ2 trk(x) = (m−h(x))T[V(m)]−1(m−h(x)) (17) yields parameter estimates with minimum variance among all linear unbiased estimates. Note that when the parameter evolution is not affected by any s tochastic noise (as assumed by Eq. 14), V(m) ij=∝an}bracketle{tεiεj∝an}bracketri}htis diagonal (at least block diagonal, if the apparatus provides multidimensional correlated measu rements), and may depend weakly on xvia the detector response (for example, the spatial reso- lution of drift chambers depends on the track angle w.r.t the anode plane). The system equation becomes non deterministic when the trac k experiences stochastic processes such as multiple scattering, bremsst rahlung or ionization losses. The system equation (Eq. 14) becomes : xk=fk(x0) +wk (18) where wkis a random vector representing the fluctuation of the parame ters along the path from location 0 to location k. This process noise translates, via the track model, into off-diagonal elements in V(m)(the noise from location 0 tokdepends on the noise from location 0 to k−1 and k−1 tok). More importantly, V(m)now depends on the reference location z0at which the parameters will be estimated. Of course the average value of the perturbing effects have to be included in fkbut the fluctuations (e.g. track scattering or energy loss straggling when relevant) have to be described b y the p.d.f of wk. In fact only the covariance matrix of the wkis needed in practice. (See Sec. 4.) Given particular data m, the track fit consists in finding the value of xwhich minimizes Eq. 17. Minimization is typically an iterative pr ocess with some convergence criteria to decide when to stop iterating. The m atrixV(m)can be calculated by Monte Carlo techniques or sometimes analyt ically. If one is 6The superscript in parentheses in V(m)is to make clear which vector it corre- sponds to; this will be the convention in this paper. 8fortunate, the calculation depends only weakly on x, so it can be done once per fit using the initial guess for x, and not changed at each iteration. Note that the N×Nmatrix V(m)must be inverted, where Nis the number of measurement positions. Letχ2 trk(written without explicit argument x) be the minimum value after convergence of the minimization procedure and /hatwidexthe value of the estimated parameters (the value of xat the minimum). The covariance matrix of the estimates is then approximated by the inverse of the curvatu re matrix at the minimum : V(/hatwidex) ij=/bracketleftBigg1 2∂2χ2 trk ∂xi∂xj/bracketrightBigg−1 (19) A traditional fit gives the track parameters /hatwidexonlyat the fixedreference z0, say at the beginning of the track. In order to find the best-fit trac k parameters at theendof the track, one has to recompute V(m)using the new reference and perform a completely independent fit. Due to multiple scatte ring, the results of two fits using different reference z0’s are notrelated by the track model, and cannot be obtained from one another. This effect is alread y present in a perfectly linear ideal case, where the detector resolution does not depend on x. It just reflects the fact that the weights of measurements to estimate the track parameters depend (eventually strongly) on z0. This is unfortunate, since in practice, track extrapolatio n is often desired from both ends of the track. Furthermore, optimal track estimate s at every possible sensor position can be quite useful, either to detect outlie rs efficiently, or to optimally collect hits left over during a first pass. Finally, if one attempts such a traditional track fit in an exp eriment with large N(such as NOMAD, where Nranges up to 150) Ncan be too big to make inversion of V(m)practical. Given these difficulties, in NOMAD the Kalman filter was implem ented in- stead [9]. 4 The Kalman Filter The Kalman filter is a least-squares stepwise parameter esti mation technique. Originally developed in the early 60’s to predict rocket tra jectories from a set of their past positions, it can be used to handle multiple scattering while estimating track parameters. We try here to briefly shed ligh t on the features of 9the Kalman filter for track fitting and refer to the literature for more details [1– 4]. The Kalman filter technique gives, mathematically speaking , exactly the same result as a standard least squares minimization. In the fram ework of track fit- ting, it essentially avoids big matrix inversion and provid es almost for free an optimal estimate of track parameters at any location, allow ing the detection of outlying measurements, extrapolation and interpolatio n into other subde- tectors. The set of parameters xis called the state vector in Kalman filtering. Starting from Eq. 18 we rewrite the system equation in a stepw ise form, where the state vector at location kis obtained from its value at previous location7 k−1. xk=fk(xk−1) +wk (20) We shall assume in the following that both wkandεk(the measurement errors from Eq. 15) are independent random variables with me an 0 and a finite covariance matrix. Linearizing the system in the vicinity of xk−1, one obtains: fk(xk−1) =Fk·xk−1 (21) hk(xk) =Hk·xk (22) for the track model and the measurement equation. We can now recall keywords used in the Kalman filter estimatio n technique: •Prediction is the estimation of the state vector at a “future” time, that is the estimation of the state vector at time or position ( k+ 1) using all the measurements up to and including mk. •Filtering is estimating the “present” state vector based upon all pres ent and “past” measurements. For Forward filtering, this means estimating track parameters at kusing measurements up to and including mk. ForBackward filtering, this means estimating track parameters at kusing the measure- ments mNdown to mk. 7One has to assume that the measurements are ordered with resp ect to time to handle multiple scattering because the covariance matrix o f measurement residuals depends on this order. Without multiple scattering, the ord ering does not affect the filter result as in the case of a traditional fit. 10•Filter. The algorithm which performs filtering is called a filter and i s built incrementally: filtering m1tomkconsists in filtering m1tomk−1, propa- gating the track from mk−1tomkand including mk. A filter can proceed forward ( kincreases) or backward ( kdecreases). •Smoothing means using all the measurements to provide a track param- eter estimate at any position. The smoothed estimate can be o btained as a weighted mean of two filtered estimates: the first one using m1tomk (forward), the other using mNtomk+1(backward).8 One can understand the basic idea of the Kalman filter in the fo llowing way. If there is an estimate of the state vector at time (location) tk−1, it is extrap- olated to time tkby means of the system equation. The estimate at time tk is then computed as the weighted mean of the predicted state v ector and of the actual measurement at time tk, according to the measurement equation. The information contained in this estimate can be passed bac k to all previous estimates by means of a second filter running backwards or by t he smoother. The main formulas for our linear dynamic system are the follo wing: System equation: xk=Fk·xk−1+wk (23) E{wk}= 0,cov{wk}=Qk(1≤k≤N) (24) Measurement equation: mk=Hk·xk+εk (25) E{εk}= 0,cov{εk}=Vk=G−1 k(1≤k≤N) (26) where the matrices QkandVkrepresent the process noise (multiple scatter- ing, bremsstrahlung, etc.) and measurement noise (detecto r resolution) re- spectively. The details of Qkcalculation for the parameterization adopted in NOMAD can be found in Ref. [10]. As an example we include here the formulas for making a predic tion: 8This leads to a subtlety in practice, when we actually have in hand the forward and backward filter estimates at k; averaging these would lead to double-counting the information from mk. Hence, to be proper, one must unfilter mkfrom one of the estimates. This small correction is implemented in our smoo ther and in the quantity χ2 (FB) kdiscussed below, but was deemed negligible and never implem ented in the other breakpoint quantities. 11•Extrapolation of the state vector: xk−1 k=Fkxk−1 •Extrapolation of the covariance matrix: Ck−1 k=FkCk−1FT k+Qk •Predicted residuals: rk−1 k=mk−Hkxk−1 k •Covariance matrix of the predicted residuals: Rk−1 k=Vk+HkCk−1 kHT k Using the Kalman filter, the computer time consumed for a trac k fit is propor- tional to the number of hits on the track, while with the tradi tional technique it is proportional to the cube of the same number in case of mul tiple scattering. After the Kalman fitting procedure one has the following avai lable informa- tion : •/hatwidexF kand/hatwidexB k: the Forward and Backward estimates of the state vector at position k, i.e., the estimate of the track parameters at location kusing measurements 1 up to k(forward) or Ndown to k(backward). •χ2 (F) kandχ2 (B) k: the minimum χ2value of the forward and backward fits up to measurement k. •V(/hatwidexk,F)andV(/hatwidexk,B): the covariance matrices of /hatwidexF kand/hatwidexB krespectively. •/hatwidexk,χ2 trkandV(/hatwidexk): the same quantities, determined from the smoothed estimates, the equivalent of a full fit done at location k. Note that the χ2 trk minimum does not depends on the location at which the paramet er are estimated ( χ2 trkfor/hatwidexkis independent of k). Thus, much information exists as the by-product of a track fit : at every hit on the track away from the ends, we have the results of three fit s for the track parameters at that hit: a fit to the part of the track upstream, a fit to the part of the track downstream, and a fit to the whole track. This info rmation is the input to breakpoint analyses. 5 Earlier Applications to Breakpoint Searches A natural way to compare /hatwidexF kand/hatwidexB kis discussed by R. Fr¨ uhwirth [3] and was implemented in NOMAD tracking [9,11] before developing our extensions. One simply constructs the χ2of the mismatch of all the forward-backward parameters at each hit k: χ2 (FB) k = (/hatwidexB k−/hatwidexF k)T[V(/hatwidexk,B)+V(/hatwidexk,F)]−1(/hatwidexB k−/hatwidexF k). (27) 12The value of χ2 (FB) k is easily computed from the following relationship which holds for any k: χ2 trk=χ2 (F) k+χ2 (B) k+χ2 (FB) k. (28) After the track fit, we find the hit kfor which χ2 (FB) k is a maximum, i.e., for which the forward-backward mismatch in track parameter s has greatest significance; we call this maximum value /tildewideχ2 (FB). One can assign a breakpoint at that kthere if /tildewideχ2 (FB)is above some threshold. Fr¨ uhwirth also investigated various combinations of /tildewideχ2 (FB)withχ2 (F) k,χ2 (B) k, and degrees of freedom in the track fits, but concluded that /tildewideχ2 (FB)was his best breakpoint test statistic [3,12]. 6 Some Additional Breakpoint Variables Based on Constraine d Fits to the Specific Breakpoint Types One may suspect that previously defined breakpoint tests do n ot make optimal use of the available information. Any χ2quantity is by definition insensitive to the arithmetic sign of differences, while in the processes of interest, a decrease in the momentum is expected. Furthermore, χ2 (FB) k mixes all the parameter mismatch information together. The signed forward-backwa rd mismatch in single quantities such as 1 /Rcan be examined, but it has the problem that the other 5 parameters are not constrained to be the same. (Ph ysical changes in 1/Rcan result, for example, in fitted mismatches in φas well as 1 /R.) Finally, an optimal test should use a more fully developed breakpoint hypothesis, so that a more meaningful comparison of χ2’s, with and without breakpoints, can take place. 6.1 Constrained Fits to Breakpoints Here we show how to obtain and examine the result that one woul d get by doing a traditional fit which uses allhits,but which allows a subset of the track parameters to be discontinuous at a particular hit k. We parameterize the full track with 1 to 3 added parameters in order to incorporate bre akpoints of Types I, II, and III at hit k. E.g., for Type I, we replace the parameter 1 /R by two parameters, a forward value 1 /RFjust before hit kand a back value 1/RBjust after hit k. Thus, our fits to the 3 breakpoint types have 7, 8, and 9 parameters, respectively. We denote these sets with breakp oints by αrather thanx, and they are, respectively for the three types: 13αI={x, y,1/RF,1/RB,tanλ, φ, t }, (29) αII={x, y,1/R,tanλF,tanλB, φF, φB, t}, (30) αIII={x, y,1/RF,1/RB,tanλF,tanλB, φF, φB, t}. (31) For definiteness, we discuss here the concept in terms of a Typ e I breakpoint. One can imagine a cumbersome procedure whereby one puts a Typ e I break- point at a particular hitk, and performs a traditional χ2track fit (with 7 parameters in our case) to allthe hits of the track, minimizing the full Type I track’s χ2, which we call χ2 full,I,k(αI). (32) One would obtain the best estimate of αI, its covariance matrix, and the min- imum value of χ2 full,I,k, all for a breakpoint at that particular hit k. One could then repeat this for each possible value of k, obtaining numerous potential track-with-breakpoint fits. Essentially the same set of results can be obtained far more economically by starting from the results /hatwidexF kand /hatwidexB kat each kwhich already exist as by- products of the Kalman filter fit. These results carry all the i nformation that we need, since their error matrices contain the information (up to linear ap- proximation) on how χ2 (F) kandχ2 (B) kchange when the track parameters change. We need only perform a linear χ2minimization in which {/hatwidexF k,/hatwidexB k} (now playing the role of the “measured data”) is compared to t he Type I breakpoint model prediction HIαI.HIis the model matrix with 12 rows and 7 columns containing only zeros and ones : HIαI={(x, y,1/RF,tanλ, φ, t),(x, y,1/RB,tanλ, φ, t)}. (33) Or, introducing the two 6 ×7 submatrices HF IandHB IofHI: HIαI= (HF IαI, HB IαI). (34) Since our 12 pieces of “measured data” {/hatwidexF k,/hatwidexB k}are two independent sets of 6 parameters with their corresponding covariance matrices , the appropriate chisquare can be written as : χ2 (FB) k(α) = ( /hatwidexF k−HFα)T[V(/hatwidexk,F)]−1(/hatwidexF k−HFα) + (/hatwidexB k−HBα)T[V(/hatwidexk,B)]−1(/hatwidexB k−HBα). (35) (Here and below, we suppress the subscript I since the equati ons are true for all breakpoint types, with Eqns. 33 and 34 suitably changed. ) 14The full χ2of Eq. 32 can be written as : χ2 full,I,k(α) = (m−h(Hα))T/bracketleftBig V(m) k/bracketrightBig−1(m−h(Hα)) (36) where V(m) kis the block diagonal matrix containing the covariance matr ix V(m,F) k of the measurements ( m1. . .m k) and the covariance matrix V(m,B) k of the measurement ( mk+1. . .m N). As shown in Appendix A, χ2 (FB) k(α) of Eqn. 35 is related to χ2 full,I,k(α) of Eqn. 36 in a revealing way. At each hit k, χ2 full,I,k(α) =χ2 (F) k+χ2 (B) k+χ2 (FB) k(α), (37) subject to sufficient linearity in the fits. Thus by finding the m inimum of χ2 (FB) k(α), we find the minimum of χ2 full,I,k(α), since χ2 (F) kandχ2 (B) kare known. The minimum of χ2 (FB) k(α) is obtained without iteration since the model relating αto{/hatwidexF k,/hatwidexB k}is linear. The set of estimated parameters is given by : /hatwideαk=V(/hatwideαk)HT(V(/hatwidex) k)−1X (38) whereX={/hatwidexF k,/hatwidexB k};V(/hatwidex) kis the block diagonal matrix containing V(/hatwidexk,F)and V(/hatwidexk,B); and where the covariance matrix for the new estimate, V(/hatwideαk), is given by : V(/hatwideαk)=/bracketleftbigg HT(V(/hatwidex) k)−1H/bracketrightbigg−1 . (39) The value of χ2 (FB) k at its minimum is χ2 (FB) k(/hatwideα) =−/parenleftbigg HT(V(/hatwidex) k)−1X/parenrightbiggT ·/hatwideα. (40) Since the elements of Hare mostly 0, and the rest equal to 1, the multiplica- tions by HandHTwere done by hand before coding the software; elements of (V(/hatwidex) k)−1have only one or two simple terms. The computer time to perform the Type I, II, and III breakpoin t fits at all hits k(away from the track ends) was a negligible addition (few per cent) to the NOMAD track finding and fitting software. This added time was m ore than paid back by the speedup in matrix inversion which was obtain ed by explicitly unrolling the loops in the DSINV routine from CERNLIB [13]. 156.2 Breakpoint Variables Based on the Constrained Fits From the wealth of information thus available at each hit, we discuss two of the most useful categories: 1) signed differences, in sigm a, of breakpoint parameters, and 2) χ2comparisons based on the Fisher F-test. As an example, for Type III breakpoint fits, we let DIII,k(1/R) be the forward- backward difference in 1 /R, divided by its standard deviation (sigma, com- puted from the covariance matrix taking account of errors in both quantities and their correlations). This signed quantity effectively gives the significance, in sigma, of the jump in momentum at that hit. Similar quantit ies, with analo- gous notation, are calculated for all components of α, for all breakpoint Types. Thus, for bremsstrahlung studies, DI,k(1/R) gives the momentum change un- der the constraint that all other track parameters are conti nuous at hit k. TheFisher Fstatistic [14] is appropriate for testing if adding parameters yields a statistically significant reduction in the χ2of a fit. It is simply the ratio of the respective χ2/dof for the two versions of the fits. Thus we naturally apply it to our track fits with and without breakpoint parameters. F or example, for NOMAD’s Type I fits, we have at each hit, FI,k= (χ2 full,I,k/(N−7))/slashBig (χ2 trk/(N−6)); (41) we similarly define FII,kandFIII,k. For a true breakpoint at a given hitk, each Fstatistic in principle obeys the standard significance-table distributions [14]. However, we normally search through all hits in a track for the lowest value of F, denoted/tildewideF. This lowest value does not of course follow the usual significance-table values. Hence, as a practical matter, we use data or Monte Carlo events to measur e the distribu- tion of/tildewideFand the effect of tests based on it. 7 Effectiveness of the additional breakpoint variables The effectiveness of any breakpoint search is of course highl y dependent on details of hardware (e.g., number and quality of the positio n measurements) and software (e.g., how many tracks with breakpoints are rec onstructed as a single track). We present for illustration some experience with the NOMAD detector, using both simulated and real data events. 1610-410-310-210-1 0 10 20 30 40 50 60 70 80 90 100 Breakpoint χ2 (FB) kArbitrary units 00.010.020.030.040.050.060.070.08 -10 -8 -6 -4 -2 0 2 4 6 8 10 DI,k(1/R)Arbitrary units Fig. 2. Test of breakpoint search criteria using real data (m uons producing δ-electrons in the NOMAD setup). Comparison of breakpoint ch isquare ( χ2 (FB) k) and normalized difference between curvatures in backward an d forward directions ((1/RB−1/RF)/(σ1/R)) for muons (solid line) and electrons (dashed line). In bot h electron distributions, the excess on the right is evidence of potential breakpoints. 7.1 Electron identification and reconstruction Algorithms developed for electron identification and recon struction have been checked under running conditions using δ-electrons produced by straight- through muons (5 GeV < pµ<50 GeV) crossing the NOMAD detector during slow-extracted beam between neutrino spills. This sample o f selected electrons from real data can be used to check the subdetector responses compared to simulations and to tune breakpoint search criteria taking i nto account the effect of the drift chambers alignment quality (see Fig. 2). A special approach to deal with electrons emitting bremsstr ahlung photons (Type I breakpoints) in the NOMAD detector has been develope d [10]. If one has identified a reconstructed track in drift chambers as bei ng an electron9, 9Transition Radiation Detector (TRD) is used for electron id entification in the 17Fig. 3. A reconstructed event from real data (the projection onto the yzplane, in which tracks bend) before an attempt to apply the breakpoint search algorithm. The track at the bottom was identified as an electron by TRD. Th e triangles are track extrapolations used to search for more hits and to matc h information from different subdetectors. Fig. 4. The same event as before after applying a recursive br eakpoint search al- gorithm. Two breakpoints are found along the electron traje ctory and they are associated with two photons (dashed lines): one built out of a conversion inside the drift chambers fiducial volume and the other from a stand-alo ne cluster in electro- magnetic calorimeter. The bars on the right are proportiona l to energy deposition in the electromagnetic calorimeter. then hard bremsstrahlung photons can be looked for and neutr al tracks can be created requiring further matching with preshower and el ectromagnetic calorimeter. A successful application of this approach to a n event from real data is shown in Fig. 3 and Fig. 4. A recursive breakpoint sear ch algorithm has been applied to a track identified as an electron. As a result t wo breakpoints NOMAD experiment. 18have been found, each associated with an observed hard brems strahlung pho- ton (one of which converts to a reconstructed e+e−pair). When applying the breakpoint search algorithm to an electron track one must ke ep in mind a potential problem related to a possible presence of several kinks along the tra- jectory (since it can bias the calculation of variables used for the breakpoint search). Details are in Ref. [10]. 7.2 Studies using simulated pion decays As another example of using the breakpoint variables descri bed in Sec. 6.2, we compare several tests for detecting the Type III breakpoi nt (discontinuity of 1/R, tanλ, and φ) of the decay π→µν, and for locating the position of the decay. The results shown here are for Monte Carlo (MC) simulation [7 ] of muon neu- trino interactions in the NOMAD detector. From the collabor ation’s standard MC samples, we selected two samples of reconstructed tracks : •25000 pions which did not decay and which were reconstructed . •2000 pions which did decay via π→µν, and for which a single track was reconstructed consisting of hits left by boththe pion and the muon. Neither of these two samples contains the pion decays which w ere broken into separate tracks by the track-finding algorithm, with a v ertex assigned at the decay point. The selected tracks were chosen to have more than 20 hits (N > 20) and no backward looping, and with the MC pion decays withi n these hits. Track-finding mistakes (for example use of hits f rom other tracks in the event) were included, but to obtain the pion track, we r equired that at least 90% of the hits be correctly assigned. Figure 5 contains histograms of momentum ×charge for the two samples. In order to have samples with similar momenta, we consider here only tracks with momentum less than 6 GeV. Figure 6 shows, for one of the decaying negative pion tracks, the values of χ2 (FB) k,FIII,k,DIII,k(1/R) and DIII,k(φ) at every hit where they are computed. They are plotted at the z-positions of the hits, which in the NOMAD detector are every few cm. The decay point in the MC is indicated by the l ine at 360 cm. The dotted lines are at the z-positions of the extreme values (maximum or minimum, as relevant for a breakpoint) of the respective v ariables. Near the MC decay point, both χ2 (FB) k andFIII,kreach their extreme values, indicating respectively: a large forward-backward mismatch in the Kal man fits, and a marked improvement in the track χ2by adding a three-parameter breakpoint. The sign of the change in 1 /Rcorresponds to a decrease in momentum, as 19050010001500200025003000 -20 -10 0 10 20Pion tracks Momentum × q, GeVEntries 02004006008001000 -20 -10 0 10 20Pion decays Momentum × q, GeVEntries Fig. 5. Histogram of momentum ×charge for simulated pions without decay (left) and with decay (right). expected in a decay. DIII,k(φ) also shows an extremum near the breakpoint; however, we have generally not used it as the primary breakpo int indicator. 7.3 Tests for Existence of a Breakpoint For testing the existence of a breakpoint, we compare the fol lowing test statis- tics: •χ2/dof for the breakpointless fit. (This test gives no additiona l information on the location of the breakpoint.) •/tildewideχ2 (FB). The breakpoint is located at the maximum value of the forwar d- backward mismatch chisquare ( χ2 (FB) k) for breakpointless fit among hits in the track; •/tildewideF. The breakpoint is located at the minimum value Fisher Famong hits in the track (applicable separately to the different breakpoin t types if desired). •/tildewideFIIIcombined with the forward-backward mismatch in radius of th e curva- tureDIII,k(1/R) or angle DIII,k(φ)./tildewideFIIIgives the breakpoint location and DIII,k(1/R),DIII,k(φ) are computed at that location. Fig. 7 contains histograms of χ2 trk/dof, /tildewideχ2 full,I/dof, /tildewideχ2 full,II/dof, and /tildewideχ2 full,III/dof for decaying and non-decaying pions, with the means shown. T heχ2/dof is in general reduced by adding breakpoint parameters. A measu re of the signif- icance of the improvement is the Fisher F(Eq. 41), which is shown in Fig. 8 for the same fits. Also shown in Fig. 8 is a histogram of /tildewideχ2 (FB), the maximum forward-backward mismatch from the breakpointless fit. 200102030405060708090100 200 250 300 350 400 Zhit , cmχ∼2 (FB) 0.30.40.50.60.70.80.91 200 250 300 350 400 Zhit , cmFIII,k -10-8-6-4-20 200 250 300 350 400 Zhit , cmDIII,k(1/R) -10-7.5-5-2.502.557.510 200 250 300 350 400 Zhit , cmDIII,k(φ) Fig. 6. /tildewideχ2 (FB),FIII,k,DIII,k(1/R) and DIII,k(φ) as a function of the zof the DC hit. The MC decay point is shown in solid line. From the histograms in Fig. 8, one can calculate the efficiency of labeling tracks as pion decays using “cuts” on these variables. For tabulati ng a comparison, we choose a cut value for/tildewideFsuch that 10% of non-decaying pions are falsely called pion decays. We then find the efficiency for identifying real pion decays, i.e., the percentage of pion decay tracks which are called pi on decays when using the same cut value. This cut value in principle depends on the dof, but for illustration we use the same cut value for all track lengt hs. The results are given in the first five columns of Table 1. We include for com parison the result for testing for pion decay simply by using χ2 trk/dof, i.e., the test one would naturally use if one had only a traditional non-Kalman track fit. As observed by Fr¨ uhwirth [3], this test is quite competitive w ith the /tildewideχ2 (FB)-test for detecting the existence of a breakpoint, even though it g ives no information about the location. The highest efficiency is obtained using/tildewideFIII. We find this to be true for various other comparisons, although one is cau tioned that the 21050100150200250300350400 0 1 2 3 χ∼2 trk /dofEntriesMeandecays 1.28 Meanno decays 1.00 050100150200250300350400 0 1 2 3 χ∼2 full,I /dofEntriesMeandecays 1.03 Meanno decays 0.93 050100150200250300350400 0 1 2 3 χ∼2 full,II /dofEntriesMeandecays 1.07 Meanno decays 0.91 050100150200250300350400 0 1 2 3 χ∼2 full,III /dofEntriesMeandecays 0.99 Meanno decays 0.90 Fig. 7. Histograms of χ2 trk/dof,/tildewideχ2 full,I/dof,/tildewideχ2 full,II/dof, and /tildewideχ2 full,III/dof for the sample of pion decays (shaded) and pions with no decay (white). The s amples are normal- ized to the same number of events. particular efficiencies listed are for the sample of decaying pions which were not detected by the standard track finding/fitting, and are he nce expected to be highly experiment-dependent. Next we add the signed information available in the new fits: t he difference in radius of curvature DIII,k(1/R), and/or the angular differences DIII,k(φ) andDIII,k(tanλ). We recommend that DIII,k(1/R) not be used to locate the breakpoint, but rather we evaluate this difference at the loc ation dictated by/tildewideFIII. Fig. 9 contains scatter plots of/tildewideFIIIvsDIII,k(1/R) for decays and non-decays. Because true decays have a decrease in momentum , a judicious cut on this scatter plot is more effective than a cut solely on/tildewideFIII. An even more powerful technique is to construct likelihood functio ns based on these 2D densities, and use the constructed likelihood ratio as a tes t of pion decay. The 220100200300400500 0 20 40 60 80 100 χ∼2 (FB)Entries-decays -no decays 02004006008001000 0 0.5 1 1.5 F∼ IEntries 0100200300400500600700800 0 0.5 1 1.5 F∼ IIEntries 0100200300400500600700 0 0.5 1 1.5 F∼ IIIEntries Fig. 8. Histograms of the /tildewideχ2 (FB)and/tildewideFI,/tildewideFII,/tildewideFIIIfor pion decays (shaded) and pions with no decay (white). Table 1 Efficiency of the detection of pion decays, using the various t est statistics. The first five columns are for simple cuts on the respective variables. The last column is for a cut on a 2D likelihood-ratio test for/tildewideFIIIvsDIII,k(1/R) described in the text. The cut value for each test is chosen so that 10% of non-decaying p ions are wrongly called decays. The efficiency is computed with respect to the s ample which contains only tracks which were not detected as decaying by the origin al track-finding/fitting algorithms. χ2 trk/dof /tildewideχ2 (FB)/tildewideFI/tildewideFII/tildewideFIII/tildewideFIIIvsDIII,k(1/R) 45% 41% 38% 40% 49% 56% 2300.20.40.60.811.21.4 -10 -5 0 5 10 DIII,k (1/R)F∼ IIIdecays 00.20.40.60.811.21.4 -10 -5 0 5 10 DIII,k (1/R)F∼ IIIno decays 00.20.40.60.811.21.4 -10 -5 0 5 10 DIII,k (φ)F∼ IIIdecays 00.20.40.60.811.21.4 -10 -5 0 5 10 DIII,k (φ)F∼ IIIno decays Fig. 9./tildewideFIIIvsDIII,k(1/R) and/tildewideFIIIvsDIII,k(φ) histograms for the pion decays and pion with no decays simulated tracks. In order to superimpos e negative and positive tracks on the same plots, we have changed the sign of DIII,k(1/R) for one charge. (Recall that 1/R is a signed quantity reflecting the charge of the track.) result of such a procedure (applied with simple smoothing of the 2D densities, and tested on an independent sample) is given in the last colu mn of Table 1. The improvement over previous breakpoint tests [3] is most s ignificant. Nearly as good efficiencies are obtained from scatter plots of/tildewideFIIIvsDIII,k(φ) (also shown in Fig. 9), and from/tildewideFIIIvsDIII,k(tanλ). 7.4 Finding the Location of the Breakpoint One may also ask which of the variables gives the best determi nation of the location of the breakpoint. We studied in particular the diff erence between the 24050100150200250300350400450 -200 0 200 Z(χ∼2 (FB)) - Z(MC), cmEntries 050100150200250300350 -200 0 200 Z(F∼ III) - Z(MC), cmEntries Fig. 10. /tildewideχ2 (FB)and/tildewideFIIIresolutions for MC pion decay sample in z. The white histograms are for the initial sample; the shaded histogram s are for events in which detected breakpoints passed the selection criteria. MC decay point and the zposition of the hit corresponding to the extrema /tildewideχ2 (FB)or/tildewideFIII. These are histogrammed in Fig. 10 for pions which decay. The white histograms show the resolution for the initial sample s, with no selection made using /tildewideχ2 (FB)or (/tildewideFIIIvsDIII,k(1/R)). The shaded histograms show the (more relevant) resolutions for tracks remaining after dec ay selection using the respective variables. There was not a significant difference , within the limited scope of this study. 8 Conclusions Replacement of mismatch chisquare for all the forward-back ward parameters by the breakpoint variables introduced in Sec. 6.2 can give a dded power to breakpoint detection in the framework of Kalman filtering te chnique. We show in particular above that this is the case in a realistic simul ation of pion decays in the NOMAD detector. In addition, these breakpoint variab les have been successfully used to reconstruct electron hard bremsstrah lung in real data. As expected on theoretical grounds, our most powerful breakpo int detection is based on a scatter plot of a Fisher F-test vs. an appropriate signed difference of a track parameter across the breakpoint. 25Acknowledgements This work was performed within the NOMAD collaboration and h ence bene- fited from numerous aspects of NOMAD’s simulation and event r econstruction codes. The authors are grateful to Emmanuel Gangler, Kyan Sc hahmaneche, and Jean Gosset for their contributions to the NOMAD Kalman fi lter. We gratefully acknowledge some early investigations by Mai Vo [11] on track breakpoints in NOMAD. References [1] P. Billoir, Nucl. Instr. Meth. 225, 352 (1984); [2] P. Billoir, R. Fr¨ uhwirth and M. Regler, Nucl. Instr. Met h. A241 (1985) 115-131. [3] F. Fr¨ uhwirth, “Application of Filter Methods to the Rec onstruction of Tracks and Vertices in Events of Experimental High Energy Physics” , HEPHY-PUB 516/88 (Vienna, Dec. 1988) [4] P. Billoir and S. Qian, Nucl. Instr. Meth. A294 (1990) 219 -228; P. Billoir and S. Qian, Nucl. Instrum. Methods A295 (1990) 492-500. [5] D. Stampfer, M. Regler and R. Fr¨ uhwirth, “Track fitting w ith energy loss,” Comput. Phys. Commun. 79 (1994) 157-164. [6] M. Regler, R. Fr¨ uhwirth and W. Mitaroff, “Filter methods in track and vertex reconstruction,” J. Phys. G G22, 521 (1996). [7] NOMAD Collaboration, J. Altegoer et al., Nucl. Instr. an d Meth. A 404 (1998) 96; NOMAD Collaboration, J. Altegoer, et al., Phys. Lett. B 4 31 (1998) 219; NOMAD Collaboration, P. Astier, et al., Phys. Lett. B 453 (19 99) 169. [8] GEANT : Detector Description and Simulation Tool, CERN P rogramming Library Long Writeup W5013, GEANT version 3.21 [9] P. Astier, A. Letessier-Selvon, B. Popov, M. Serrano, “N OMAD Reconstruction Software: Drift Chamber Package”, version 5 release 2, unpu blished (1994), and later releases with additional authors. [10] Boris Popov, Ph.D. thesis, U. Paris VII, (unpublished, 1998). http://www-lpnhep.in2p3.fr/Thesards/lestheses.html http://nuweb.jinr.ru/ ∼popov [11] Mai Vo, NOMAD collaboration communication (unpublish ed, 1995) and Ph.D. thesis, Saclay, (unpublished, 1996). [12] Fr¨ uhwirth refers to /tildewideχ2 (FB)asCk. He defines a related quantity, Fk= (Ck/n1)/((χ2 (F) k+χ2 (B) k)/n2), where n1is the d.o.f. for /tildewideχ2 (FB)andn2is 26the sum of the d.o.f. for χ2 (F) kandχ2 (B) k. Fr¨ uhwirth concludes that the Fk-test is less powerful than the Ck-test, and that the χ2is almost as good an indicator of the existence of a kink as the Ck-test. [13] CERN Program Library routine number F012. A modified ver sion of the routine was used to write the source code of the unrolled loops. [14] P.R. Bevington and D.K. Robinson, Data Reduction and Error Analysis for the Physical Sciences , (New York: McGraw-Hill, 1992), pp. 205-209, 261-267. A Derivation of Eq. 37: χ2 full,I,k=χ2 (F) k+χ2 (B) k+χ2 (FB) k In Eqn. 36, V(m) kis the block diagonal matrix containing the covariance matr ix V(m,F) k of the first kmeasurements mF= (m1. . .m k) and the covariance matrix V(m,B) k of the last N−kmeasurements mB= (mk+1. . .m N). The right-hand side of Eqn. 36 can thus be split into two terms: χ2 full,I,k(α) =/bracketleftBig mF−h(HFα)/bracketrightBigT[V(m,F)]−1/bracketleftBig mF−h(HFα)/bracketrightBig +/bracketleftBig mB−h(HBα)/bracketrightBigT[V(m,B)]−1/bracketleftBig mB−h(HBα)/bracketrightBig , (A.1) where one recognizes, in analogy with Eqn. 17, the forward an d backward χ2 terms. We can expand each around their respective minima /hatwidexF kand/hatwidexB k, and recall that covariance matrices are the inverse of curvatur e matrices, giving : χ2 full,I,k(α) =χ2 (F) k+ (∆xF)T[V(/hatwidexk,F)]−1∆xF +χ2 (B) k+ (∆xB)T[V(/hatwidexk,B)]−1∆xB, (A.2) where ∆ xF=/hatwidexF k−HFαand ∆xB=/hatwidexB k−HBα. Combining this with Eqn. 35 yields Eq. 37. 27

arXiv:physics/9912036v1 [physics.atom-ph] 17 Dec 1999Relativistic semiclassical approach in strong-field nonli near photoionization J. Ortnera)and V. M. Rylyukb) a)Institut f¨ ur Physik,Humboldt Universit¨ at zu Berlin, Inv alidenstr. 110, 10115 Berlin, Germany b)Department of Theoretical Physics, University of Odessa, D vorjanskaja 2, 270100, Odessa, Ukraine (to be published in Phys. Rev. A) Nonlinear relativistic ionization phenomena induced by a s trong laser radiation with elliptically polarization are considered. The starting point is the clas sical relativistic action for a free electron moving in the electromagnetic field created by a strong laser beam. The application of the relativis- tic action to the classical barrier-suppression ionizatio n is briefly discussed. Further the relativistic version of the Landau-Dykhne formula is employed to conside r the semiclassical sub-barrier ioniza- tion. Simple analytical expressions have been found for: (i ) the rates of the strong-field nonlinear ionization including relativistic initial and final state e ffects; (ii) the most probable value of the components of the photoelectron final state momentum; (iii) the most probable direction of pho- toelectron emission and (iv) the distribution of the photoe lectron momentum near its maximum value. PACS numbers:32.80.Rm, 32.90.+a, 42.50.Hz, 03.30.+p I. INTRODUCTION Relativistic ionization phenomena induced by strong laser light have become a topic of current interest [1–8]. In the nonrelativistic theory it is assumed that the electron v elocity in the initial bound state as well as in the final state is small compared with the speed of light. However, the elect rons may be accelerated up to relativistic velocities in an intense electromagnetic field produced by modern laser de vices. If the ponderomotive energy of the electron is of the order of the rest energy a relativistic consideration is required. Relativistic effects in the final states become important for an infrared laser at intensities of some 1016W cm−2. The minimal intensity required for relativistic effects increases by two orders of magnitude for wavelength c orresponding to visible light. Ionization phenomena connected with relativistic final state effects have been stu died for the cases of linearly and circularly polarized lase r radiation both in the tunnel [6,8] and above-barrier regime s [1,2,7]. The main relativistic effects in the final state are [1,2,5–8]: (i) the relativistic energy distribution and (i i) the shift of the angular distribution of the emitted elect rons towards the direction of propagation of incident laser beam . It has been shown that a circularly polarized laser light produces a large amount of relativistic electrons [1,2,8]. On the contrary, it has been found that the ionization rate for relativistic electrons is very small in the case of linea r polarization [6]. Relativistic effects have also to be taken into account if the binding energy Ebin the initial state is comparable with the electron rest energy [3,4]. A relativistic formula tion is necessary for the ionization of heavy atoms or singly or multiply charged ions from the inner K-shell. In Refs. [3, 4] the relativistic version of the method of imaginary time has been employed to calculate the ionization rate for a boun d system in the presence of intense static electric and magnetic fields of various configurations. Analytical expre ssions have been found which apply to nonrelativistic bound systems as well as to initial states with an energy correspon ding to the upper boundary of the lower continuum. The present paper is aimed to consider the nonlinear photoio nization connected with relativistic final states velociti es and/or low lying initial states from a unique point of view. F urther the current work extends the investigation of relativistic ionization phenomena to the case of arbitrary elliptical polarization. Certainly this may be done in the framework of the so-called s trong field approximation [9]. In the papers of Reiss and of Crawford and Reiss [1,2,7] a relativistic version of t his approximation has been given for the ionization of an hydrogen atom with linearly and circularly polarized light . Within this approximation one calculates the transition amplitude between the initial Dirac state for the hydrogen a tom and the final state described by the relativistic Volkov wave function. Coulomb corrections are neglected in the Vol kov state. Therefore the final results are obtained only within exponential accuracy. Analytical results for the io nization rate applying to above barrier cases as well as to tunneling cases have been given in Refs. [1,2,7]. However, t he corresponding expressions are complicated and contains infinite sums over all multiphoton processes. Numerical cal culations are needed to present the final results. In contrast to the more sophisticated investigations, such as the solution of the Dirac equation or the strong field approximation, we are aimed to obtain simple analytical exp ressions. From our final formulas the explicit dependence of the ionization rate and of the photoelectron spectrum on t he parameters, such as binding energy of the atom, field strength, frequency and ellipticity of the laser radiation may be understood without the need of numerical calculation s. In this sense our approach resembles that of Popov et al. [3,4] and of Krainov [6,8]. 1II. RELATIVISTIC ACTION AND CLASSICAL BARRIER-SUPRESSION IONIZATION Let us start with the classical relativistic action for an el ectron of charge emoving in the field of an electromagnetic plane wave with the vector potential A(t−x/c). Here and below Adenotes a two-dimensional vector in the y-z plane. The action may be found as a solution of the Hamilton-Jacobi e quation and reads [10] Sf(ξ;ξ0) =mc2/braceleftBigg f·r c−αx c−1 +α2+f2 2α(ξ−ξ0) +e mc2αf/integraldisplayξ ξ0Adξ−e2 2m2c4α/integraldisplayξ ξ0A2dξ/bracerightBigg , (1) where αandf= (a1, a2) are constants, r= (y, z); further is ξ=t−x/c,ξ0is the initial value. Assuming a harmonic plane wave of elliptic polarization with the electric field E=F[eycosωξ+gezsinωξ] we find the following expression for the relativistic action Sf(ξ;ξ0) =mc2/braceleftBigg f·r c−α 2/parenleftBig t+x c/parenrightBig −β2 2α(ξ−ξ0) +ǫ αω[a1(cosωξ−cosωξ0) +a2g(sinωξ−sinωξ0)]−ǫ2 8αω/parenleftbig g2−1/parenrightbig (sin 2ωξ−sin 2ωξ0)/bracerightBigg , (2) where the notation β2= 1 + a2 1+a2 2+ ((1 + g2)/2)ǫ2has been introduced, the parameter ǫ=eF/ωmc characterizes the strength of relativistic effects. Further the vector pot ential of the laser radiation has been choosen in the form Ax= 0, A y=−cF ωsinωξ , A z=gcF ωcosωξ . (3) By applying the usual Hamilton-Jacobi method we take the der ivative of the action Sfwith respect to the constants a1, a2andαand set the result equal to new constants β1, β2andβ3in order to obtain the electron trajectory under the influence of the wave field. We obtain that the electron mot ion in the field and in the laboratory coordinate system is given by the equations ( ξ0= 0), α2(t+x/c)−β2ξ+2ǫ ω(a1cosωξ+ga2sinωξ) +1−g2 4ωǫ2sin2ωξ=β3, v x=cf(ξ)−1 f(ξ) + 1, y=β1+ca1 αξ−cǫ αωcosωξ , v y=2c α(1 +f(ξ)){a1+ǫsinωξ}, z=β2+ca2 αξ−gcǫ αωsinωξ , v z=2c α(1 +f(ξ)){a2−ǫgcosωξ}, f(ξ) =δ2 α2+2ǫ α2(a1sinωξ−ga2cosωξ) +1−g2 α2ǫ2sin2ωξ, (4) where β1, β2andβ3together with a1, a2andαhave to be determined from the initial conditions for positi on and velocity. Further we have introduced the notation δ2= 1 + a2 1+a2 2+g2ǫ2. Quantum effects may be neglected, for strong enough fields, i. e.,F≫FB=E2 0/4Z(in a.u., where E0is the electron energy in the initial state and Zis the effective charge of the atomic core). In this case the io nization process may be described by an electron trajectory given in Eqs. (4). In a pure classical task the constants of motion may be determined from the initial velocity and position of the e lectron at the beginning of the laser action. However, the initial state is given by quantum mechanics. According t o a simple classical picture of ionization, the barrier supression ionization (BSI) [12], the transition occurs fr om the bound state to that continuum state which has zero velocity at the time twith the phase ξof the vector potential A(ξ). From this condition we have to choose the constants as α=/radicalBig δ2+ 2ǫ(a1sinωξ−ga2cosωξ) + (1−g2)ǫ2c2sin2ωξ , a1=−ǫsinωξ , a 2=gǫcosωξ . (5) The maximal ionization rate occurs at the maximum of the elec tric field of the laser radiation. For our choice of the gauge (see Eqs. (3)) the electric field has its maximum at the p haseξ= 0. (For the sake of simplicity we neglect the second maximum at ξ=π). From Eqs. (5) we conclude that the most probable final state is described by the constants 2α= 1, a1= 0, a2=ǫg , (6) Two important results follow from this derivation. First, consider the components of the final electron drift mo mentum along the beam propagation, px=c(1−α2+ a2 1+a2 2)/2α, along the major axis, py=c a1, and along the small axis of the polarization ellipse, pz=c a2, respectively. From Eq. (6) we see that the photoelectrons are preferably pr oduced with the drift momentum px=ǫ2g2 2c , p y= 0, p z=ǫgc . (7) For a laser wavelength of λ= 780 nm and for laser intensities of about I = (1018−1019) W/cm2, the parameter ǫ is equal to: ǫ1= 0.65−2.1 for the linearly polarized wave ( g= 0) and ǫ2= 0.46−1.46 for the circularly polarized wave ( g2= 1). According to the classical barrier-supression ioniza tion model the photoelectrons are emitted with a relativistic drift momentum [12] at these laser intensitie s and for sufficiently large ellipticity g. On the contrary, in the case of linear polarization the photoelectrons have a ze ro drift momentum. Second, the angle between the electron drift momentum compo nents along and perpendicular to the direction of the laser beam propagation is shifted toward the forward dir ection and reads tanθ=p⊥ |pz|=|px| |pz|=ǫ|g| 2, (8) For linearly polarized laser light we obtain tan θ= 0. For the case of circularly polarized light (where tan θ=ǫ/2) our result coincide with that of previous works [2,8]. III. RELATIVISTIC SEMICLASSICAL APPROACH Consider now the process of nonlinear ionization of a strong ly bound electron with a binding energy Ebcomparable with the rest energy. Recently the ionization process in sta tic crossed electric and magnetic fields has been considered [3,4]. The results of this paper may be applied to the ionizat ion in laser fields only for the case of very strong fields ǫ≫1. With an increasing frequency of the laser light (especial ly for a tentative x-ray laser) very high laser intensities are required to satisfy this condition. Therefore it is nece ssary to generalize the result of [3,4] to the case of nonzero frequencies. We consider the sub-barrier ionization. The c ondition to be satisfied is the opposite to the case of pure classical ionization, F≪FB, in addition we have the quasiclassical condition ¯ hω≪Eb. No restrictions are applied to the parameter ǫ. Thus we will cover both the regime of relativistic tunnel an d multiphoton ionization. We employ the relativistic version of the Landau-Dykhne for mula [3,5]. The ionization probability in quasiclassical approximation and with exponential accuracy reads W∝exp/braceleftbigg −2 ¯hIm (Sf(0;t0) +Si(t0))/bracerightbigg , (9) where Si=E0t0is the initial part of the action, Sfis given by Eq. (1) (or Eq. (2)). The complex initial time t0has to be determined from the classical turning point in the comp lex half-plane [3,5]: Ef(t0) =mc2/braceleftBigg 1 +α2+f2 2α−e mc2αfA(t0)+e2 2m2c4αA2(t0)/bracerightBigg =E0=mc2−Eb. (10) Explicitely we obtain for λ0=−iωt0the following relation in the case of an elliptically polari zed planar wave, sinh2λ0−g2/parenleftbigg coshλ0−sinhλ0 λ0/parenrightbigg2 =γ2(α), (11) where γ(α) =η√1 +α2−2αε0, orγ2(α) = (1 −α)2η2+αγ2, with the dimensionless initial energy εo=E0/mc2 and the relativistic adiabatic parameter η=ǫ−1=ωmc/eF . Eq. (9) together with Eqs. (2) and (11) expresses the transition rate between the initial state and the final Vo lkov state with abitrary momentum within exponential accuracy. It applies for the case of sub-barrier ionization with elliptically polarized laser light. We are now interested in the totalionization rate. Within exponential accuracy it suffices to fi nd the maximum of the transition rates between initial state and all possible final states. Aquivalently, one has to find the minimum of the imaginary part of the action as a function of the final stat e momentum. The minimization of the imaginary part 3of the action with respect to the components of the final state momentum leads to the following boundary conditions [13] (x,r)(t0) = 0,Im (x,r)(t= 0) = 0 . (12) From these conditions we obtain that the most probable final s tate is characterized by the parameters α2= 1 +1 2η2/parenleftbigg 1 +g2−2g2sinh2λ0 λ2 0−1−g2 2λ0sinh2λ0/parenrightbigg , (13) a1= 0 (14) a2= (g/η)(sinh λ0/λ0). (15) Substituting the values λ0=−iωt0andαinto the final state action we obtain the probability of relat ivistic quasiclassical ionization in the field of elliptically pola rized laser light. Within exponential accuracy we get W∝exp/parenleftbigg −2Eb ¯hωf(γ, g, E b)/parenrightbigg , (16) where f(γ, g, E b) =/parenleftbigg 1 +1 +g2 2γ2α+mc2 Eb(1−α)2 2α/parenrightbigg λ0−/parenleftbigg 1−g2+ 2g2tanhλ0 λ0/parenrightbiggsinh2λ0 4γ2α. (17) The magnitudes αandλ0has to be taken as the solution of Eqs. (11) and (13). Further γ=√2mEbω/eF is the common adiabatic Keldysh parameter from nonrelativistic t heory [5]. Equation (16) is the most general expression for the relativistic ionization rate in the quasiclassical regime and for field strength smaller than the above-barrier threshold. It describes both the tunnel as well as the multip hoton ionization. It is the relativistic generalization of previous nonrelativistic results [13]. A. Relativistic tunnel ionization Consider now some limiting cases. In the limit of tunnel ioni zation η≪1 we reproduce the static result of Refs. [3,4] and obtain the first frequency correction Wtunnel∝exp/braceleftbigg −FS FΦ/bracerightbigg Φ =2√ 3(1−α2 0)3/2 α0−3√ 3(1−α2 0)5/2 5α0η2(1−g2/3) +O(η4), (18) where Fs=m2c3/e¯h= 1.32·1016V/cm is the Schwinger field of quantum electrodynamics [15] and α0= (ε0+/radicalbig ε2 0+ 8)/4. In the nonrelativistic regime, εb=Eb/mc2≪1, the parameter α0= 1−εb/3+ε2 b/27 and the probability of nonrelativistic tunnel ionization including the first re lativistic and frequency corrections reads Wtunnel∝exp/braceleftBigg −4 3√ 2mE3/2 b e¯hF/bracketleftbigg 1−γ2 10(1−g2/3)−Eb 12mc2/parenleftbigg 1−13 30γ2(1−g2/3)/parenrightbigg/bracketrightbigg/bracerightBigg . (19) Here the first two terms in the brackets describe the familiar nonrelativistic ionization rate including the first freque ncy correction, the next two terms are the first relativistic cor rections. It follows from Eq. (18) that the account of relativistic effects increases the ionization rate in compa rison with the nonrelativistic rate. However, even for bind ing energies of the order of the electron rest energy the relativ istic correction in the exponent is quite small. In the “vacuum” limit Eq. (19) results into W∝exp{−9FS/2F/parenleftbig 1−9/40η2(1−g2/3)/parenrightbig }. We find a maximal deviation of about 18% in the argument of the exponential from the nonrela tivistic formula. Here the “vacuum” limit shall not be confused with the pair creation from the vacuum. It is know n that there are no nonlinear vacuum phenomena for a plane wave [15]. In contrast to that we deal here with the ion ization of an atom being in rest in the laboratory system of coordinates. Nevertheless, the “vacuum” limit sh ould be considered only as the limiting result of the present semiclassical approach where the effects of pair pro duction have been neglected. The polarization of the vacuum becomes important if the binding energy of the atom ex ceeds the electron rest energy. At the binding energy 4Eb= 2mc2the single particle picture employed in this paper breaks do wn ultimately. The electron energy is decreased up to the upper limit for the energy of a free positron, and the threshold energy for the production of an electron- positron pair becomes zero. On the contrary, for a weak relat ivistic initial state εb≪1 we expect only a small influence of pair production effects on the ionization proces s. An appropriate consideration of vacuum polariztion effects can be given only in the framework of quantum electrod ynamics. However, this is beyond the scope of the present paper. B. Relativistic multiphoton ionization Consider now the multiphoton limit η≫1. In this case the parameters λ0= ln (2 γ//radicalbig 1−g2) (orλ0= lnγ√2 lnγ forg=±1) and α= 1−εb/2λ0and the ionization probability in the relativistic multiph oton limit reads Wmulti−ph∝exp/parenleftbigg −2Eb ¯hωf(γ≫1, g, ε b)/parenrightbigg , (20) f(γ≫1, g, ε b) = ln2γ/radicalbig 1−g2−1 2−Eb 8mc2ln 2γ//radicalbig 1−g2, g∝ne}ationslash=±1, (21) f(γ≫1, g, ε b) = ln 2 γ/radicalbig 2 lnγ−1 2−Eb 8mc2ln 2γ√2 lnγ, g =±1. (22) Again the first two terms in the function f(γ≫1, g, ε b) reflect the nonrelativistic result [13], the relativistic effects which lead to an enhancement of the ionization probability a re condensed in the third term. It has been shown that there is an enhancement of ionization r ate in the relativistic theory for both large and small η. This should be compared with the results found by Crawford a nd Reiss. In their numerical calculations they also found an enhancement of relativistic ionization rate for a c ircularly polarized field and for η≫1, but for η≪1 their results suggest a strong reduction of the ionization p robability [2]. For the case of linearly polarized light the ionization rate is found to be reduced by relativistic effect s [7]. However, Crawford and Reiss studied the above-barrie r ionization of hydrogen atom within the strong-field approxi mation. In contrast to that we have investigated the sub- barrier ionization from a strongly bound electron level, wh ich yields an enhancement of the ionization rate. This enhancement is connected with a smaller initial time t0. As a result the under barrier complex trajectory becomes shorter and the ionization rate increases in comparison wit h the nonrelativistic theory. Figure 1 shows the relativist ic ionization rate Eq. (16) and the nonrelativistic Keldysh fo rmula as a function of the binding energy eband for the case of linear polarization. The figure should be considered only as an illustration of the enhancement effect. The frequency and intensity parameters used for the calculatio ns are still not available for the experimentalists. C. The case of weak relativistic initial state The switch from the multiphoton to the tunnel regime with inc reasing field strength may be studied in the nonrel- ativistic limit εb≪1. Within first order of εbthe ionization probability is found to be Wweak−rel∝exp/braceleftbigg −2Eb ¯hωf(γ, g, ε b≪1)/bracerightbigg , (23) where f(γ, g, ε b≪1) =f(0)(γ, g) +εbf(1)(γ, g). Here f(0)(γ, g) =/parenleftbigg 1 +1 +g2 2γ2/parenrightbigg λ(0)−/parenleftbigg 1−g2+ 2g2tanhλ(0) λ(0)/parenrightbiggsinh2λ(0) 4γ2(24) represents the nonrelativistic result [13], and λ(0)satisfies the equation: sinh2λ(0)−g2/parenleftbigg coshλ(0)−sinhλ(0) λ(0)/parenrightbigg2 =γ2. 5Besides, f(1)(γ, g) =B 8γ4/braceleftbiggB+ 4γ2 A/bracketleftbigg γ2+1 +g2 2−cosh2λ(0) 2/parenleftbigg 1−g2+ 2g2tanhλ(0) λ(0)/parenrightbigg −g2tanhλ(0) λ(0)/parenleftbigg 1−sinh2λ(0) 2λ(0)/parenrightbigg/bracketrightbigg −/parenleftbigg 1 +g2+ 2g2sinh2λ(0) λ(0)2+1−g2 2λ(0)sinh2λ(0)/parenrightbigg λ(0)+/parenleftbigg 1−g2+ 2g2tanhλ(0) λ(0)/parenrightbigg sinh2λ(0)/bracerightbigg ,(25) is the first relativistic correction, with A= sinh2 λ(0)−2g2/parenleftbigg coshλ(0)−sinhλ(0) λ(0)/parenrightbigg /bracketleftbigg sinhλ(0)−1 λ(0)/parenleftbigg coshλ(0)−sinhλ(0) λ(0)/parenrightbigg/bracketrightbigg , (26) B= 1 + g2−2g2sinh2λ(0) λ(0)2−1−g2 2λ(0)sinh2λ(0). (27) Equation (23) is valid in the whole γ-domain, i.e., in the multiphoton regime γ <1 as well as in the tunnel limit γ >1. For small adiabatic parameters, i.e., γ→0, it coincides with Eq. (19); in the case of large γ→ ∞ it transforms to Eq.(20). We mention that Eq. (23) reproduces the full rela tivistic formula Eq. (16) with very high accuracy for Eb< mc2. The expression for the rate of ionization of a weak relativis tic initial state essentially simplifies in the case of linear polarization. Then we have Wweak−rel∝exp/braceleftBigg −2Eb ¯hωf(γ, g= 0, εb≪1)/bracerightBigg , f(γ, g= 0, εb≪1) = arsinh γ+1 2γ2/bracketleftBig arsinh γ−γ/radicalbig 1 +γ2/bracketrightBig −εbγ4+γ2−2γ/radicalbig 1 +γ2arsinh γ+ arsinh2γ 8γ4arsinh γ.(28) The terms in f(γ, g= 0, εb≪1) which do not vanish as εb→0 represent the nonrelativistic quasiclassical ionizatio n rate found by Keldysh [14]; the terms proportional to εbare the first relativistic correction to the Keldysh formula . IV. RELATIVISTIC PHOTOELECTRON SPECTRUM Consider now the modifications of the energy spectrum induce d by relativistic effects. First we will characterize the most probable final state of the ejected electron. The classi cal nonrelativistic barrier-supression ionization predi cts a nonzero leaving velocity of the photoelectron. However, r elativistic effects as well as frequency corrections modify this result of the classical BSI picture. In the relativisti c semiclassical theory employed in this paper we may set the constants a1= 0 and a2= (g/η)(sinh λ0/λ0) according to Eqs. (14) and (15). From Eqs. (4) we obtain then for the most probable emission velocity in the laboratory system of coordinates vx,leaving =c1−α2+g2 η2/parenleftBig 1−sinhλ0 λ0/parenrightBig2 1 +α2+g2 η2/parenleftBig 1−sinhλ0 λ0/parenrightBig2(29) vy,leaving = 0, (30) vz,leaving =2αc 1 +α2+g2 η2/parenleftBig 1−sinhλ0 λ0/parenrightBig2g η/parenleftbiggsinhλ0 λ0−1/parenrightbigg . (31) where αhas to be taken from the Eq. (13). In the tunnel limit ( η << 1) we obtain: vx,leaving =c1−α2 0 1 +α2 0+O(η2), (32) vz,leaving =gcηα 01−α2 0 1 +α2 0+O(η3). (33) Here and below α0= (ε0+/radicalbig ε2 0+ 8)/4. The first term in the x-component of the leaving velocity is independent from both the frequency and the intensity of the laser light. It co incides with the static result of Mur et al.[4]. The leading 6term in the z-component is proportional to the frequency and inverse pro portional to the electric field strength of the laser radiation. From Eqs. (32) and (33) it also follows that thex-component of the leaving velocity vanishes in the nonrelativistic limit, whereas the z-component has a nonzero nonrelativistic limit. For a nonre lativistic atom, we get: vx,leaving =v2 6c/braceleftbig 1 +O(v2/c2, γ2)/bracerightbig , (34) vz,leaving =v 6gγ/braceleftbig 1 +O(v2/c2, γ2)/bracerightbig , (35) where v=/radicalbig 2Eb/mis the initial ”atomic” velocity of the electron. In the ”vac uum” limit ( α0= 1/2), we have: vx,leaving =3 5c+O(η2), (36) vz,leaving =3 10gcη+O(η3). (37) It follows from these equations that a strongly bound electr on has a relativistic emission velocity along the direction of the laser beam propagation. For a nonrelativistic initia l state, εb≪1, the emission velocity along the beam propa- gation is small. Nevertheless, the mean emission velocity s eems to be the most sensitive measure of the appearance of relativistic effects in the initial states. In Fig. 2 the x-component of the leaving velocity is plotted versus the bin ding energy of the initial state. Though we have choosen the same p arameters of the laser beam as in Fig. 1 it should be mentioned that the dependence of the emission velocity x-component on the laser parameters is rather weak. The main parameter determining the leaving velocity along the p ropagation of the laser beam is the binding energy of the atom. From Eqs. (13)-(15) we also obtain the most probable value fo r each component of the final state drift momentum (which is the full kinetic momentum minus the field momentum) . We put a1=py,m/mc,a2=pz,m/mcand α= (−px,m+/radicalBig m2c2+p2x,m+p2y,m+p2z,m)/mcand get px,m=mc 2α/braceleftbigg 1−α2+g2 η2sinh2λ0 λ2 0/bracerightbigg , (38) py,m= 0 (39) pz,m=mcg ηshλ0 λ0(40) The leading terms in the tunnel limit ( η << 1) read px,m=mc 2α0g2 η2, (41) pz,m=mcg η. (42) For a nonrelativistic initial state and within the tunnel re gime ( γ≪1) we obtain px,m=e2F2g2 2ω2mc/parenleftbigg 1 +γ2 3g2+ 1 g2/parenrightbigg , (43) pz,m=eF ωmg/parenleftbigg 1 +γ2 6/parenrightbigg , (44) where we have given the leading terms and the first frequency c orrections. From Eqs. (38)-(40) we easily obtain the most probable angle of electron emission. Denote by θthe angle between the polarization plane and the direction of the photoelectr on drift motion; and by ϕthe angle between the projection of the electron drift momentum onto the polarization plane a nd the smaller axis of the polarization ellipse. In the case of a nonrelativistic atom the most probable angles read tanθm=px,m |pz,m|=eF|g| 2mcω/parenleftbigg 1 +g2+ 2 g2γ2 6/parenrightbigg , ϕ m= 0. (45) We conclude that relativistic effects produce a nonzero comp onent of the mean electron drift momentum along the axis of beam propagation. As a result the mean angle of electr on emission is shifted to the forward direction. 7However, in the case of linear polarization the appearance o f a nonzero x-component of the photoelectron drift momentum is connected with relativistic effects in the initi al state. The latter are typically small except the case of ionization from inner shells of heavy atoms. Notice that f or the nonrelativistic atom the most probable value for the drift momentum components as well as the expression for t he peak of the angular distribution coincide with the corresponding expressions within the BSI model (see Sec. II ) if one neglects the frequency corrections. Consider now the relativistic final state spectrum, i.e., th e momentum distribution near the most probable final state drift momentum. The calculations will be restricted t o the tunnel regime γ≪1. Assuming weak relativistic effects in the initial state, εb≪1, and putting δpx= (px−px,m)≪mc,δpz= (pz−pz,m)≪mcandpy≪mc, one obtains Wp∝Wtunnelexp/bracketleftBigg −γ ¯hω/bracketleftbig δp2 x−2δpxδpzǫg+δp2 z/parenleftbig 1 + 2ǫ2g2+ǫ4g4/4/parenrightbig/bracketrightbig m(1 +ǫ2g2/2)2/bracketrightBigg ·exp/bracketleftBigg −p2 y,m 3mγ3(1−g2) ¯hω−p4 y,m 4m3c2(1 +ǫ2g2/2)2γ ¯hω/bracketrightBigg , (46) where Wtunnelis the total ionization rate Eq. (19) in the weak relativisti c tunnel regime. In Eq. (46) only the leading contributions in δpxandδpzhave been given; in the pydistribution an additional relativistic term proportiona l top4 y has been maintained which becomes the leading term in the cas e of static fields with γ= 0. In the non-relativistic limitǫ≪1 and p≪cwe reproduce the results of Ref. [13]. For the cases of linear (g= 0) and circular ( g=±1) polarization our results are in agreement with recent deriv ations of Krainov [6,8]. The first exponent in Eq.(46) describes the momentum distrib ution in the plane perpendicular to the major axis of the polarization plane. In the nonrelativistic theory ( ǫ= 0) the width of the momentum distribution in pxcoincide with the width of the pzdistribution. The relativistic effects (which are measured byǫg) destroy this symmetry in the (x,z)-plane. The distribution of pxbecomes broader, the pzdistribution becomes narrower. We also mention the appearance of a cross term proportional to the product δpxδpzwhich is absent in the nonrelativistic theory. In Fig. 3 the distribution of the projection of the photoelectr on drift momentum on the axis of the beam propagation is shown. We consider electrons which are produced in the cre ation of Ne8+(Eb= 239 eV) ions by an elliptically polarized laser radiation with wave length λ= 1.054µm, field strength 2 .5×1010V/cm and ellipticity g= 0.707. The relativistic momentum distribution is compared with the di stribution of nonrelativistic theory. From the figure we see that the main effect is the shift of the maximum of the momen tum distribution, the broadening remains small for the parameters we have considered. The first term in the second exponent of Eq.(46) determines th e nonrelativistic energy spectrum for the low energetic electrons moving along the major polarization axis, wherea s the second, relativistic term becomes important for the high energy tail. A detailed analysis of the photoelectron s pectrum will be given elsewhere [16]. In conclusion, in this paper relativistic phenomena for the ionization of an atom in the presence of intense ellipticall y polarized laser light have been considered. The cases of rel ativistic classical above-barrier and semiclassical sub- barrier ionization have been investigated. Simple analytic expres sions for the ionization rate and the relativistic photoele ctron spectrum have been obtained. These expressions apply for re lativistic effects in the initial state as well as in the final state. We have shown that relativistic initial state effects lead to a weak enhancement of the ionization rate in the sub-barrier regime. The mean emission velocity has been sho wn to be a more sensitive measure for the appearance of relativistic effects in the initial state. The more importan t relativistic final state effects may cause a sharp increase of the electron momentum projection along the propagation o f elliptically polarized laser light. This results in a shif t of the most probable angle of electron emission to the forwar d direction. Finally, the expressions obtained in this paper within expo nential accuracy may be improved by taking into account the Coulomb interaction through the perturbation theory. T he results of this paper may be also used in nuclear physics and quantum chromodynamics. V. ACKNOWLEDGEMENTS This research was partially supported by the Deutsche Forsc hungsgemeinschaft (Germany). 8[1] H. R. Reiss, J. Opt. Soc. Am. B 7, 574 (1990). [2] D. P. Crawford and H. R. Reiss, Phys. Rev. A 50, 1844 (1994). [3] V. S. Popov, V. D. Mur and B. M. Karnakov, Pis’ma Zh. Eksp. T eor. Fiz. 66, 213 (1997) [JETP Lett. (USA), 66229 (1997)]. [4] V. D. Mur, B. M. Karnakov and V. S. Popov, Zh. Eksp. Teor. Fi z.114, 798 (1998) [J. Exp. Theor. Phys. 87, 433 (1998)]. [5] N. B. Delone and V. P. Krainov, Uzp. Fiz. Nauk 168, 531 (1998) [Phys. Usp. 41, 469 (1998)]. [6] V. P. Krainov, Opt. Express 2, 268 (1998). [7] D. P. Crawford and H. R. Reiss, Opt. Express 2, 289 (1998). [8] V. P. Krainov, J. Phys. B 32, 1607 (1999). [9] H. R. Reiss, Phys. Rev. A 22, 1786 (1980). [10] L. D. Landau and E. M. Lifshitz, The classical theory of fields (Pergamon, Oxford, 1977). [11] V. B. Beresteskii, E. M. Lifshitz and L. P. Pitaevskii, Relativistic quantum theory (Pergamon, Oxford, 1958). [12] P. B. Corkum, N. H. Burnett, and F. Brunel, in Atoms in Intense Laser Fields , edited by M. Gavrila (Academic Press, New York, 1992), p. 109. [13] V. S. Popov, V. P. Kuznezov and A. M. Perelomov, Zh. Eksp. Teor. Fiz. 53, 331 (1967). [14] L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. JETP 20, 1307 (1965)]. [15] J. Schwinger, Phys. Rev. 82, 664 (1951). [16] J. Ortner, J. Phys. B, submitted. 9FIGURE CAPTIONS (Figure 1) Absolute value of the logarithm of the ionization rate −lnWversus the binding energy of initial level eb=Eb/mc2. The solid line shows the relativistic rate Eq.(16), the das hed line is the nonrelativistic Keldysh formula (Eq. (23) without the relativistic correction term ). The curves are shown for a frequency ω= 100 and an intensity I= 8.5·107(in a.u.). (Figure 2) Thex-component of the emission velocity vx/cversus the binding energy of initial level eb=Eb/mc2. The emission velocity in the nonrelativistic theory is zero . The curve is shown for a frequency ω= 100 and an intensity I= 8.5·107(in a.u.). (Figure 3) Spectrum of the electron momentum projection along the beam propagation for electrons produced in the creation of Ne8+by an elliptically polarized laser radiation with wave leng thλ= 1.054µm, field strength 2.5×1010V/cm and ellipticity g= 0.707; the relativistic spectrum is taken from Eq. (46), the no nrelativistic one is Eq. (46) with ǫ= 0. 100.0 0.5 1.0 1.5 2.0 eb0.0200.0400.0600.0800.01000.0− ln W FIG. 1. 110.00 0.50 1.00 1.50 2.00 E b /mc 20.000.200.400.600.80v x /c FIG. 2. 12−10.0 0.0 10.0 20.0 30.0 40.0 drift momentum px (in a.u.)0.00.20.40.60.81.0Electron yield (in arbitrary units)relativistic nonrelativistic FIG. 3. 13

arXiv:physics/9912037v1 [physics.ed-ph] 17 Dec 1999A Procura das Leis Fundamentais (In search of fundamental laws) V. Pleitez Instituto de F´ ısica Te´ orica Universidade Estadual Paulista Rua Pamplona, 145 011405-900–S˜ ao Paulo, SP Brazil RESUMO Uma das atividades importantes do ensino de ciˆ encias em ger al, e de f´ ısica em particular, ´ e a discuss˜ ao de problemas n˜ ao apenas atua is mas aqueles cuja solu¸ c˜ ao ´ e urgente. Quer dizer que deveria-se transm itir aos estudantes, principalmente aos da terceira s´ erie, a imagem de uma ciˆ en cia ativa, viva; deixando claro os seus sucessos e seus fracassos, suas dificu ldades para seguir adiante. Um ponto central dessa problem´ atica ´ e a carateri za¸ c˜ ao do que deve entender-se por leis fundamentais da natureza . Em particular fazemos ˆ enfase neste trabalho no fato que esse tipo de leis podem existir em ´ areas diferentes das tradicionalmente reconhecidas. Numa discuss˜ ao desse tipo ´ e imposs´ ıvel (e nem mesmo desej´ avel) evitar a perspectiva hist´ orica do desenvolvimento cient´ ıfico.ABSTRACT One of the main activities in science teaching, and in partic ular in Physics teaching, is not only the discussion of both modern problems and problems which solution is an urgent matter. It means that the picture of an active and alive science should be transmitted to the students, mainly to the College students. A central point in this matter is the issue which ch aracterizes the Fundamental Laws of Nature. In this work we emphasize that th is sort of laws may exist in areas which are different from those usually considered. In this type of discussion it is neither possible nor desirab le to avoid the historical perspective of the scientific development. 21 Introdu¸ c˜ ao ´E frequente ouvir dizer ou ler que a ciˆ encia em geral, e a f´ ıs ica em par- ticular, est´ a passando por momentos dif´ ıceis. Por exempl o o n´ umero de estudantes de f´ ısica est´ a diminuindo nos Estados Unidos [ GO99, AI99] e provavelmente, isso ocorra no mundo todo [PF99]. No entanto quando anal- izada cuidadosamente ´ e f´ acil se convencer de que a situa¸ c ˜ ao n˜ ao deveria ser essa. Paradoxalmente, isso acontece justamente no momento em que a pre- sen¸ ca da ciˆ encia ´ e mais contundente na sociedade moderna [MA99]. N˜ ao d´ a para entender que n˜ ao seja amplamente reconhecido que a s contribui¸ c˜ oes da ciˆ encia, e da f´ ısica em particular, em todos os aspectos da vida nas so- ciedades modernas tˆ em sido, s˜ ao ´ e ser˜ ao essenciais para o desenvolvimento. Alguns dos problemas que a f´ ısica enfrenta, s˜ ao comuns ` a c iˆ encia em geral. Alguns cientistas acreditam que existem dificuldades nos pr ´ oprios projetos da f´ ısica; outros que esses problemas n˜ ao est˜ ao nos temas de estudo da f´ ısica mas nas suas rela¸ c˜ oes com a sociedade [MA99]. De qualquer f orma, a vis˜ ao pesimista sobre as ´ areas de pesquisa na f´ ısica enfraquece as rela¸ c˜ oes dela com a sociedade. Assim, discutir em n´ ıvel estrictamente ci ent´ ıfico quais os rumos e dificulades da f´ ısica, contribui a melhorar o di´ alo go com a sociedade. Assim podemos perguntar-nos o que seria necess´ ario fazer p ara manter em bom estado a pesquisa, o ensino e a influˆ encia cultural da ciˆ encia em geral, e da f´ ısica em particular? [MA99, SC90]. ´E necess´ ario que, entre outras coisas, se fa¸ ca ˆ enfase na i mportˆ ancia da ciˆ encia f´ ısica b´ asica1de maneira que se promova a f´ ısica orientada, mo- tivada pela curiosidade; temos que reconhecer a importˆ anc ia de educar e informar ao p´ ublico, isto ´ e, a divulga¸ c˜ ao cient´ ıfica; t amb´ em em melhorar o ensino da f´ ısica e o jornalismo cient´ ıfico. Apenas as moti va¸ c˜ oes de curto prazo e econˆ omicas n˜ ao s˜ ao suficientes. Devemos sempre re ssaltar que os conceitos f´ ısicos s˜ ao a base dos microprocessadores, do l aser e da fibra ´ otica; somente para mencionar algumas das contribui¸ c˜ oes import antes baseadas em princ´ ıpios b´ asicos. Por´ em poderiamos retroceder at´ e o s´ eculo passado e mencionar muitas outras contribui¸ c˜ oes da ciˆ encia ou en t˜ ao tentar prever quais ser˜ ao os futuros impactos quando as revolu¸ c˜ oes do m inilaser [GO98], da computa¸ c˜ ao quˆ antica sejam realidade [CO99] ou, mesmo os avan¸ cos im- previs´ ıveis em outras ´ areas como as Ciˆ encias da Terra: ou ser´ a que descobrir qual ´ e o mecanismo respons´ avel pelo movimento das placas t ectˆ onicas n˜ ao ´ e 1Por “ciˆ encias f´ ısicas” entendemos a totalidade das ciˆ en cias f´ ısicas: astronomia, as- trof´ ısica, cosmologia, materia condensada, f´ ısica do me io ambiente, f´ ısica de part´ ıculas elementares, etc. 3fundamental? acaso n˜ ao ter´ a impacto no desenvolvimento f uturo conhecer melhor a evolu¸ c˜ ao interna da Terra? [MA98]. Acreditamos que uma discus˜ ao sobre o que ´ e a procura de leis funda- mentais da natureza possa contribuir um pouco para o esclarescimento dessa problem´ atica complicada. Afinal, a curiosidade continuar ´ a a ser uma mo- tiva¸ c˜ ao para alguns estudantes seguirem uma carreira cie nt´ ıfica, a f´ ısica por exemplo. 1.1 Un pouco de hist´ oria Pode-se dizer que, em certo sentido, a f´ ısica contemporˆ an ea come¸ cou com Cop´ ernico, Galileo e outros [CO98]. Por outro lado, a prime ira s´ ıntese con- ceitual na descri¸ c˜ ao dos fenˆ omenos observados na ´ epoca , foi a de Newton no s´ eculo XVII. As leis de Newton e outros princ´ ıpios gerai s, como as leis de conserva¸ c˜ ao, permitiram a descri¸ c˜ ao de todos os fenˆ omenos conhecidos at´ e a ´ epoca de Newton e nos anos posteriores.2Quando afirmamos que a teoria de Newton foi uma s´ ıntese queremos dizer que ela permitiu que processos aparentemente n˜ ao relacionados, como o movimen to dos planetas e os observados aqui na Terra, fossem descritos por um ´ unico conjunto de princ´ ıpios. No s´ eculo seguinte (Sec. XVIII) foi realizado o desenvolvi mento ma- tem´ atico da mecˆ anica cl´ assica newtoniana. Hamilton, La grange e outros nomes bem conhecidos. Quase que concomitantemente, no come ¸ co do s´ eculo XIX, foram feitos uma s´ erie de experimentos sobre fenˆ omen os el´ etricos e magn´ eticos que culminaram com a descoberta, por Faraday, A mp` ere e ou- tros pesquisadores, das leis da eletricidade e do magnetism o as quais logo seriam unificadas , junto com a ´ otica, na teoria do campo eletromagn´ etico de Maxwell. (Esta foi a segunda grande s´ ıntese nas leis dos fenˆ omenos naturais.) Temos ent˜ ao que no come¸ co do s´ eculo XX, uma dinˆ amica de pa rt´ ıculas relativ´ ıstica (Einstein) e a eletrodinˆ amica de Maxwell ( tamb´ em relativ´ ıstica) formabam os pilares do nosso conhecimento cient´ ıfico das le is b´ asicas da natureza. Essas teorias constituem o que se conhe¸ ce hoje pe lo nome de F´ ısica Cl´ assica . No final do s´ eculo passado ainda a existˆ encia dos ˆ atomos n˜ ao era ampla- mente aceita, ou seja que n˜ ao se acreditava que a microf´ ısica fosse constitu´ ı- 2Havia algumas discrepˆ ancias mas, para a exposi¸ c˜ ao sucin ta que estamos fazendo isto n˜ ao tem importˆ ancia. Isto nos levaria a considerar a quest ˜ ao delicada de quando um experimento ´ e crucial [PL99]. 4da de fenˆ omenos diversos dos observados em escalas macrosc ´ opicas. Apenas em 1913 as experiˆ encias de J. Perrin mostraram que os ´ atomo s, os quais os qu´ ımicos usavam apenas como uma maneira de descri¸ c˜ ao d as rea¸ c˜ oes qu´ ımicas, tinham existˆ encia real [NY72]. Tampouco havia nessa ´ epoca uma vis˜ ao do universo como um todo, isto ´ e, o conceito de que o un iverso evolue a partir de um estado inicial.3 Por outro lado, o chamado problema do corpo negro , isto ´ e, a lei que descreve a intera¸ c˜ ao da radia¸ c˜ ao em equilibrio t´ ermic o com a mat´ eria, estava em aberto, e os experimentos n˜ ao confirmavam os modelos te´ o ricos para explicar esse fenˆ omeno. A resolu¸ c˜ ao do problema levaria , no percurso das d´ ecadas seguintes, ` a descoberta da mecˆ anica quˆ antica , a teoria que substitui a mecˆ anica de Newton no caso de fenˆ omenos na escala atˆ omic a (da ordem de 10−8cm). Entre 1895 e 1897, foram feitas 3 descobertas experimentais que teriam grandes implica¸ c˜ oes ao longo de todo o s´ eculo XX: •a descoberta dos Raios-X por R¨ ontgen, •a descoberta da radioactividade natural por Becquerel, •a descoberta do el´ etron por J. J. Thompson. As duas primeiras foram feitas por acaso. Nos anos seguintes ficaria claro que os raios-X s˜ ao ondas eletromagn´ eticas de grande energ ia e que a radio- atividade era um fenˆ omeno atˆ omico ou, melhor, nuclear. Is to n˜ ao era evi- dente mas foi demostrado por Rutherford nas primeiras d´ eca das deste s´ eculo. A descoberta de Thompson e outras experiˆ encias posteriore s, mostraram que os portadores da eletricidade negativa s˜ ao constituentes universais da mat´ eria. Estava assim, descoberta a primeira part´ ıcula e lementar [PL97]. Podemos dizer, de maneira resumida, que os f´ ısicos no come¸ co do nosso s´ eculo estudavam experimentalmente a radioactividade, o s te´ oricos propu- nham modelos do ´ atomo, outros pesquisadores experimentai s estudavam os raios c´ osmicos ou tentavam obter baixas temperaturas. T e´ oricos como Einstein estudavam a generaliza¸ c˜ ao da relatividade rest rita e que o levaria ` a proposta da relatividade geral. At´ e o come¸ co dos anos 30 pensava-se que todos os fenˆ omenos naturais tinham origem em apenas duas for¸ cas fundamentais: a gravit a¸ c˜ ao e a eletro- magn´ etica. Estas teorias eram descritas como campos cl´ as sicos preenchendo 3´E interessante observar que, ainda que muitas das id´ eias na f´ ısica moderna tenham, de alguma forma, um conceito an´ alogo na Grecia antiga, este n˜ ao ´ e o caso de um universo em expans˜ ao. Este ´ ultimo ´ e um conceito que nasce no nosso s ´ eculo. 5o espa¸ co todo. As suas fontes eram a massa e a carga el´ etrica , respectiva- mente. No caso gravitacional as equa¸ c˜ oes de Einstein desc revem a gravita¸ c˜ ao em condi¸ c˜ oes especiais, mas a teoria de Newton ´ e usada na m aioria das aplica¸ c˜ oes do dia-a-dia. Pouco tempo depois, ainda nos anos 30, foi reconhecido que pa ra explicar fenˆ omenos atˆ omicos e sub-atˆ omicos (nucleares) era nece ss´ ario admitir a exis- tˆ encia de duas outras for¸ cas: a fraca e a forte. A primeira, a for¸ ca fraca, para explicar o decaimento radioativo βe a segunda, para garantir a estrutura nuclear. Nenhuma das duas for¸ cas ´ e observada macrosc´ opi camente e, con- trariamente ` as for¸ cas gravitacionais e eletromagn´ etic as, devem ter alcance muito curto. At´ e hoje, as 4 for¸ cas podem ser tratadas separadamente. Em termos ob- servacionais, isso significa quatro escalas diferentes par a as se¸ c˜ oes de choque e vidas m´ edias dos diferentes processos entre as part´ ıcul as elementares at´ e agora observadas. A descri¸ c˜ ao atual das for¸ cas fracas e f ortes est´ a baseada em teorias de calibre (ou de gauge) locais que tˆ em como exemplo a eletro- dinˆ amica quˆ antica (QED). Todo este esquema n˜ ao foi obtid o sem reservas. Afinal a ciˆ encia tem de ser c´ etica e o preconceito, sejam pos itivos (adiantam o reconhecimento de um fato ou teoria) ou negativos (dificult am o mesmo) mora ao lado. 1.2 Motiva¸ c˜ oes Acreditamos que uma discuss˜ ao sobre o que poderia ser a proc ura de leis fundamentais ajudar´ a a estudantes de pos-gradua¸ c˜ ao na escolha ou na va lo- riza¸ c˜ ao das suas respectivas ´ areas de pesquisa e aos da gr adua¸ c˜ ao a escolher sua futura ´ area de trabalho. Se o n´ umero de estudantes de F´ ısica esta diminuindo, como poderia se reverter essa tendˆ encia? Qual quer que seja a resposta a este desafio uma das suas componentes ser´ a a moti va¸ c˜ ao dos estudantes sobre o que ´ e importante pesquisar. Ent˜ ao, se f az necess´ ario uma discuss˜ ao sobre onde e como podemos procurar esse tipo d e leis fun- damentais. Este ´ e um ponto importante e esperamos que este a rtigo possa contribuir, ainda que modestamente, a repensar o assunto. S im, repens´ a-lo porque j´ a existe uma resposta tradicional ` a pergunta de on de podemos iden- tificar as leis fundamentais. No momento que novos fatos ou pr opriedades da mat´ eria s˜ ao descobertos, essa resposta n˜ ao ´ e mais aprop riada. Precissamos ent˜ ao redescobrir qual o sentido das leis fundamentais. Quer dizer que no ensino de ciˆ encias os aspectos pedag´ ogic os n˜ ao s˜ ao mais suficientes. Se ensinar o que sabemos ´ e dif´ ıcil, n˜ ao o ´ e menos ensinar 6o que n˜ ao sabemos. N˜ ao saber no sentido amplo do termo: cois as que a ciˆ encia est´ a ainda pesquisando ou mesmo n˜ ao tem condi¸ c˜ oes, no momento, de responder. Existem outros aspectos do problema como a educa¸ c˜ ao do p´ u blico em geral. Convencer ` as pessoas que a f´ ısica continuar´ a a ser a base da ciˆ encia e a tecnologia no futuro y que tambi´ en jogar´ a un papel import ante na an´ alise e resolu¸ c˜ ao de problemas energ´ eticos e do meio ambiente. Mas antes de chegar ao p´ ublico, precissamos convencer os estudantes so bre quais s˜ ao os problemas fundamentais que devem ser atacados por eles. Que existem problemas fundamentais em ´ areas n˜ ao reconhecidas por uma mentalidade infantil que infelizmente ainda permeia os nossos meios aca dˆ emicos. Um aspecto que n˜ ao ser´ a tratados aqui ´ e o fato que as diretriz es metodol´ ogicas n˜ ao s˜ ao suficientes para caraterizar a atividade cient´ ıfi ca [PL96, PL99]. 2 Rompendo barreiras O m´ etodo cient´ ıfico, qualquer coisa que entendamos por iss o, n˜ ao tem um ant´ ıdoto contra os preconceitos. Por exemplo, mesmo no com e¸ co do pre- sente s´ eculo f´ ısicos como Lord Kelvin (e Mach como veremos mais adiante) nao acreditavam na existˆ encia dos ´ atomos. Segundo eles os ´ atomos seriam apenas abstra¸ c˜ oes ´ uteis para os qu´ ımicos. No entanto, o mesmo Lord Kelvin escreveu no pref´ acio do livro de Hertz [HE62] The explanation of the motion of the planets by a mechanism of crystal cycles and epicyles seemed natural and intelligi ble, and the improvement of this mechanism invented by Descartes in h is vortices was no doubt quite satisfactory to some of the greatest of Newton’s scientific contemporaries. Descartes’s doctri ne died hard among the mathematicians and philosophers of continen tal Europe; and for the first quarter of last century belief in uni versal gravitation was insularity of our countrymen. Segundo Weinberg [WE93a] The heroic past of mechanism gave it such prestige that the followers of Descartes had trouble accepting Newton’s theo ry of the solar system. How could a good Cartesian, believing that all natural phenomena could be reduced to the impact of material bodies or fluids on one another, accept Newton’s view that the sun exerts a force on the earth across 93 million miles of empt y 7space? It was not until well the eighteen century that Contin ental philosophers began to feel comfortable with the idea of acti on at a distance. In the end Newton’s ideas did prevail on the Contin ent as well as in Britain, in Holland, Italy, France, and Germany (in that order) from 1720 on. Apenas em 1728 ap´ os uma viagem de Voltaire a Londres a escola Newtoniana come¸ cou a ter disc´ ıpulos em Paris [WE93b]. N˜ ao ´ e surpree ndente que o conceito de a¸ c˜ ao a distˆ ancia n˜ ao era aceito pela comunidade. ´E interessante que o pr´ oprio Newton disse [WH51] ...that one body may act upon another at a distance through vacuum, without the mediation of anything else ... is to me so great absurdity, that I believed no man, who has in philosoph ical matters a competent faculty for thinking, can ever fall into . Por alguns anos, depois de 1687 (ano da publica¸ c˜ ao dos Principia ), mesmo em Cambridge, continuo-se a ensinar o Cartesianismo. Apenas ocor- reu que no Continente as ideais de Newton demoraram um pouco m ais para serem aceitas [WH51]. Voltaire escrevia em 1730 [WH51] A Frenchman who arrives in London will find philosophy, like everything else, very much changed there. He has left th e wordls a plenum, and now he find a vacuum. It is the language used, and not the thing in itself, that irritates the humand m ind. If Newton had not used the world attraction in his admirable philosophy, every one in our Academy would have open his eyes to the light; but unfortunately he used in London a word to whi ch an idea of ridicule was attached in Paris... Segundo Whittaker [WH51] In Germany, Leibnitz described the Newton formula as a re- turn to the disacredited scholastic concept of occult quali ties and a late as the middle eighteenth century Euler and two of the Bernoullis based the explanation of magnetism on the hypoth esis of vortices. Deve-se lembrar tamb´ em que essa oposi¸ c˜ ao entre disc´ ıpu los de Newton e Descartes fez que os primeiros rejeitassem, posteriorment e, a id´ eia de ´ eter nos fenˆ omenos el´ etricos e magn´ eticos. Vemos que como dis semos antes, o preconceito mora ao lado, a verdade aparece sempre com dific uldades! Neste caso o curioso ´ e que posteriormente a vis˜ ao newtonia na passou a ser o preconceito contra novas formas de descrever o mundo f´ ısi co. 83 Desafios Usualmente, descreve-se o desenvolvimento da f´ ısica como a evolu¸ c˜ ao da ex- plica¸ c˜ ao de fenˆ omenos relativos a uma determinada escal a das dimens˜ oes es- paciais e do tempo, em termos de processos mais elementares c aracter´ ısticos de uma escala espa¸ co-temporal menor. Foi o que aconteceu co m a descoberta da estrutura atˆ omica da mat´ eria a qual sabemos agora que ´ e composta de ´ atomos e mol´ eculas. Logo se constatou que os ´ atomos por su a vez s˜ ao cons- tituidos por el´ etrons e pelo n´ ucleo atˆ omico. Este ´ ultim o ´ e formado pelos n´ ucleons que por sua vez s˜ ao formados pelos quarks. Foi est a hierarquia de fenˆ omenos que levou os cientistas a acreditar que as leis fu ndamentais eram apenas aquelas que permitiam descer na escala das dimens˜ oe s espaciais e do tempo. Isto ´ e, o desenvolvimento da ciˆ encia, e em particul ar o da f´ ısica, foi at´ e pouco tempo totalmente reducionista . At´ e onde vai esta cadeia? Sem d´ uvida a resposta a esta pergu nta faz parte da chamada pesquisa b´ asica . Por´ em, este tipo pesquisa est´ a restrita ` a procura de novas leis carater´ ısticas de escalas menores? N ˜ ao h´ a novas leis fundamentais , por exemplo, na escala humana ou a n´ ıvel atˆ omico? Se a resposta for positiva, como podemos reconhecer leis fundam entais? Se for negativa, por quˆ e ? N˜ ao ´ e f´ acil uma defini¸ c˜ ao de lei fundamental . De fato, nenhum defini¸ c˜ ao ´ e f´ acil. Mas podemos reconhec´ e-la. Qu ando um conceito ou lei n˜ ao depende de outro de maneira direta que o explica po demos dizer que o primeiro ´ e um conceito ou lei fundamental. Assim, a qu´ ımica tem conceitos e leis que n˜ ao podem ser reduzidos ` a f´ ısica. Ist o ´ e, a qu´ ımica tem seu estatus particular como ciˆ encia da natureza mesmo que s eus fundamen- tos estejam baseados nas leis da f´ ısica. Mas existem ainda m esmo ´ areas da f´ ısica onde as leis cl´ assicas ou quˆ anticas ajudam pouco p ara se estabelecer suas leis e conceitos. Um exemplo, a ser discutido mais adian te, ´ e o caos determin´ ıstico . Al´ em das dificuldades intr´ ınsecas, a resposta ` a pergunta acima ´ e parti- cularmente delicada, porque a situa¸ c˜ ao atual da f´ ısica t e´ orica ´ e, em certo sentido, de crise. As palavras recentes de Schweber resumem a problem´ atica atual [SC93] A deep sense of unease permeates the physical sciences...Tr a- ditionally, physics have been highly reductionist, analyz ing na- ture in terms of smaller and smaller building blocks and reve al- ing underlying, unifying fundamental laws...Now, however , the reductionist approach that has been the hallmark of theoret ical 9physics in the 20th century is being superseded by the invest iga- tion of emergent phenomena, the study of the properties of co m- plexes whose ‘elementary’ constituents and their interact ions are known. Physics, it coul be said, is becoming like chemistry. As pesquisas cient´ ıficas s˜ ao divididas segundo Weisskopf emintensivas eextensivas . As do primeiro tipo teriam a ver com a procura de leis funda- mentais, as do segundo tentam descrever os fenˆ omenos em ter mos das leis fundamentais conhecidas [WE67a]. Neste sentido a f´ ısica d a mat´ eria con- densada, f´ ısica de plasma e outras ´ areas seriam do tipo ext ensivo, entanto que a f´ ısica de altas energias e parte da f´ ısica nuclear ser iam intensivas. Tomada literalmente ´ e uma maneira de desenvolvimento “bar roca”, isto ´ e, uma disciplina ´ e separada numa multid˜ ao de ´ areas, uma quantidade de detalhes e complexidades desorganizados. Isto pode ocorre r em ciˆ encias matematizadas ou mesmo nas ciˆ encias emp´ ıricas. Na verdade estamos numa ´ epoca de grandes mudan¸ cas em que as assun- ¸ c˜ oes b´ asicas da pesquisa nas diversas ´ areas da f´ ısica p arecem deslocadas com rela¸ c˜ ao as anteriores: a complexidade e aemergˆ encia (o da turbulˆ encia por exemplo) parecem ser os objetivos principais a serem tratad os [SC93]. Outra ´ area de grande futuro s˜ ao as t´ ecnicas de ´ optica quˆ antic a.´E poss´ ıvel prever at´ e onde nos levara os novos testes dos princ´ ıpios da mecˆ a nica quˆ antica? Desde os experimentos de Aspect e colaboradores [AS82] que t estaram as desigualdades de Bell e mostraram que a interpreta¸ c˜ ao ort odoxa era confir- mada, pasando pelos efeitos “superluminares” de Chiao et al.[ST93, CH93], at´ e testes mais recentes [GH99], indicam que podemos estar assistindo ` a des- coberta de novos fenˆ omenos quˆ anticos e isso ter´ a importa ntes conseq¨ uˆ encias em computa¸ c˜ ao (que cada vez est´ a atingindo distˆ ancias m enores) e, por isso, em todas as outras ´ areas da ciˆ encia e da tecnologia. Tudo is so n˜ ao parece t˜ ao fundamental e b´ asico como outras leis da natureza? Paradoxalmente, a situa¸ c˜ ao, no caso da f´ ısica de part´ ıc ulas elementares, ´ e uma consequˆ encia do sucesso da teoria quˆ antica de campo s e do uso das simetrias, locais e globais. fica dif´ ıcil de prever qual ser ´ a o formalismo que substituir´ a ao atual. No entanto, quando apropriadamente considerada, a situa¸ c˜ ao atual ´ e empolgante. Acreditamos apenas que a f ´ ısica entrou numa nova fase de maturidade nas diversas ´ areas. Por exempl o, a f´ ısica de neutrinos est´ a numa fase de muita coleta de dados experimen tais dos quais poder´ a sair dados definitivos das propiedades dos neutrino s [NU98, GE99]. O sentimento de dificuldade acima mencionado, n˜ ao ocorre ap enas na f´ ısica de part´ ıculas elementares. O mesmo ocorre em ´ area s como a mat´ eria 10condensada e a cosmologia, mesmo (ou justamente por causa de les) com os dados recentes do COBE [SM92]), parecem estar numa situa¸ c˜ ao de aparente falta de perspectivas. No caso da mat´ eria condensada n˜ ao t em havido avan¸ cos na compreens˜ ao dos fenˆ omenos cr´ ıticos e a super conditividade a altas temperaturas ainda n˜ ao tem uma teoria bem estabeleci da [SC93]. Mas podemos assinalar para a descobertas experimentais da c ondensa¸ c˜ ao de Bose-Einstein com diversos tipos de ˆ atomos, inclusive o hi drogˆ enio [CO98b, KL99]. Mesmo em ´ areas de grande desenvolvimento recente co mo o caos determin´ ıstico e fenˆ omenos relacionados, parece ter-se alcan¸ cado uma es ta- bilidade nas descobertas te´ oricas e experimentais [RU93] . Osfractais tam- pouco produziram uma renova¸ c˜ ao da nossa vis˜ ao da naturez a (pelo menos por enquanto) e servem (quase) apenas para produzir figuras e x´ oticas com ajuda de computadores [KA86]. A ´ area da programa¸ c˜ ao, a de speito dos grandes avan¸ cos, continua na sua crise peremne [GI94]. Tudo isso est´ a relacionado com o que esperamos da f´ ısica co mo um todo e, em particular, da f´ ısica te´ orica. Considero que transm itir esse tipo de ansiedade ´ e fundamental no ensino de f´ ısica . Precissamos ensinar n˜ ao ape- nas o conhecido mas tamb´ em o desconhecido, o que est´ a sendo pesquisado no momento pelos especialistas das diversas ´ areas. Deve-s e fazer ˆ enfase na ignorˆ ancia da ciˆ encia em certos assuntos. Isso coloca a pr ioridade da atual- iza¸ c˜ ao dos professores com rela¸ c˜ ao ` as necessidades pu ramente pedag´ ogicas. Mais n˜ ao apenas isso. Na atualidade a vida das pessoas ´ e cad a vez mais afetada pela ciˆ encia e a t´ ecnica. Elas precissam entender melhor em que consiste o m´ etodo cient´ ıfico ou melhor, em que consiste a maneira cient´ ıfica de pensar e agir (e tamb´ em quais s˜ ao as suas limita¸ c˜ oes). Essa necessidade ´ e fazer ‘compreender’ a ciˆ encia pelos estudantes (e o p´ ubli co geral) ´ e mais im- portante que apenas a mera atualiza¸ c˜ ao dos resultados obt idos pela ciˆ encia e a tecnologia. Do ponto de vista dos pr´ opios pesquisadores e dos estudante s de p´ os- gradua¸ c˜ ao as medita¸ c˜ oes s˜ ao mais delicadas, mas nem po r isso menos ur- gentes ou necess´ arias. ´E urgente e/ou necess´ ario obter uma fun¸ c˜ ao de onda para o universo (mesmo que o universo primordial)? pode-se o bter uma teoria de tudo (“theory of everything”) com os conhecimento s emp´ ıricos at- uais? A prioriza¸ c˜ ao dos objetivos da pesquisa ´ e essencia lmente uma escolha pessoal, ainda que outros fatores influenciem nela (como o fin anciamento, mercado de trabalho, a influˆ encia do orientador na pos-grad uac˜ ao). Uma interrogante importante sempre ser´ a sobre o quˆ e estamos e m capacidade de verificar experimentalmente. A especula¸ c˜ ao ´ e valida m as temos de ter cuidado em n˜ ao cair numa situa¸ c˜ ao grega, isto ´ e, uma situa¸ c˜ ao onde apenas 11o conceito de teoria matematicamente “bela” ´ e o que importa . Esse conceito ´ e certamente relativo. Nesse quadro geral, o problema ´ e colocado aos pesquisadore s e, em par- ticular aos estudantes que come¸ cam sua p´ os-gradua¸ c˜ ao, de escolher rumos nas suas pesquisas. A escolha ´ e certamente um assunto pesso al. Todos est˜ ao sozinhos ao fazˆ e-la. Vale a pena, no entanto, fazer an´ alis es que possam, pelo menos, colocar o assunto em discuss˜ ao de maneira que v´ ario s crit´ erios pos- sam ser levados em conta na hora de escolher. Um aspecto que atrai os pesquisadores para determinados cam pos da pesquisa ´ e o fato de ela ser considerada ampla e “fundamenta l”. Isto ´ e, base de tudo o resto, que seria constitu´ ıdo apenas de detalh es. Os conceitos de “importˆ ancia”, “beleza” e “consistˆ encia” s˜ ao tamb´ e m, frequentemente trazidos ` a tona. Se uma ´ area ´ e “fundamental” ou, aceitando que essa palavra seja sinˆ onima de “importante”, ent˜ ao ela deve ser relevante para ´ areas v izinhas. Por ex- emplo, parece indiscut´ ıvel que h´ a varios anos a biologia m olecular ´ e a ´ area mais fundamental das ciˆ encias biol´ ogicas. Assim, um estu dante pode ser motivado a escolher essa ´ area de pesquisa. Os objetivos des sa ciˆ encia (com- preender melhor a transmiss˜ ao da informa¸ c˜ ao gen´ etica) s˜ ao, aparentemente, mais f´ aceis de identificar. Sua importˆ ancia com rela¸ c˜ ao a doen¸ cas como c´ ancer, aids e outras, assim como a sua utiliza¸ c˜ ao em t´ ec nica recentes de produtos transgˆ enicos e clonagens ´ e indiscut´ ıvel. Se a f ´ ısica de altas ener- gias ´ e vista como uma maneira de entender melhor as for¸ cas n ucleares ent˜ ao poderia ser comparada com a biologia molecular. No entanto, esse objetivo foi deixado de lado, no que se refere aos fatos principais a se rem explica- dos, e se procura uma unifica¸ c˜ ao das for¸ cas observadas (at ´ e o momento) na natureza. Ainda que isto possa ser uma motiva¸ c˜ ao para at rair jovens talentosos, poderia ser uma maneira, a curto prazo, de frust r´ a-los e perder quadros valiosos. Em 1964 Alan Weinberg [WE64] observara que o afastamento da f ´ ısica de altas energias do resto das outras ´ areas da f´ ısica dimin ue a sua importan- cia como ciˆ encia fundamental. Claro que como ciˆ encia tem o bjetivos bem definidos e ambiciosos. O problema, ´ e que ´ e cara. Por isso su as verbas s˜ ao cada vez mais dif´ ıceis de serem obtidas nos paises do primei ro mundo. Em parte porque tem de competir com ´ areas e/ou temas de pesquis a novos, isto ´ e, que n˜ ao existiam 10 ou 15 anos atr´ as (pelo menos n˜ ao de m aneira estru- turada). Por outro lado, devemos lembrar que a ciˆ encia ´ e um a s´ o. Assim, se um projeto ´ e cancelado no primeiro mundo vai nos afetar tamb ´ em. Nos n˜ ao podemos ficar, pelo menos na ´ area de f´ ısica te´ orica, resol vendo problemas 12diferentes dos da comunidade internacional. N˜ ao devemos a ceitar a divis˜ ao do trabalho internacional. O desafio, levando em considera¸ c˜ ao as diferen¸ cas de recursos, ´ e o mesmo. A “beleza” e a “consistˆ encia” s˜ ao fatores muitas vezes mai s determi- nantes que a observa¸ c˜ ao experimental na aceita¸ c˜ ao de um a determinada teoria. Isso serve para decidir entre duas teorias com difer entes graus de beleza. Mas, este adjetivo tem unicidade? i.e., podemos formular apenas “uma” ´ unica teoria bela? Pelo menos por enquanto este parec e ser o caso da Relativiadade Geral e da Mecˆ anica Quˆ antica. O caso dest a ´ ultima ´ e mais impressionante. Podemos fazer corre¸ c˜ oes ` a Relativ idade Geral acres- centando termos ` a lagrangeana mas n˜ ao sabemos como modific ar apenas “um pouco” a mecˆ anica quˆ antica!4Por outro lado, esta n˜ ao determina o tipo de part´ ıculas e suas intera¸ c˜ oes. A criterio de beleza sempre foi utilizado pelos cientistas. Segundo Chan- drasehkar [CH79] Science, like arts, admits aesthetic criteria; we seek theo ries that display a proper conformity of the parts to one another a nd to the whole while still showing some strange in their propor tion. O problema ´ e que mesmo na arte o criterio de beleza ´ e cultura l e depende tamb´ em do tempo. Pior, na ciˆ encia como na arte os preconcei tos tˆ em um papel, para bem o para mal, importante. A beleza manifesta, p ara n´ os, da teoria atˆ omica n˜ ao era evidente para grandes f´ ısicos de s ´ eculo pasado, como Lord Kelvin e outros. Mach por exemplo dizia que [WE93a]: If believed in the reality of atoms is so crucial, then I renou nce the physicsl way of thinking. I will not be a professional phy sicist, and I hand back my scientific reputation. Qualquer que seja a defini¸ c˜ ao de beleza para teorias cient´ ıficas, a “simpli- cidade” deve fazer parte dela. Mas, como se mede a simplicida de? Segundo Weinberg [WE93a], ´ e a simplicidade de id´ eias o que importa . Rubbia ´ e mais enf´ atico: o “script” ´ e mais importante que os “atores”. A t eoria de Newton ´ e constituida por 3 equa¸ c˜ oes entanto que a do Einstein tem 10! Mas sem d´ uvida nenhuma a ´ ultima ´ e considerada, pela maioria da co munidade de f´ ısicos, como sendo mais bela (e fundamental) que a primeir a! Assim, n˜ ao devemos identificar a simplicidade con o n´ umero m´ ınimo de q ualquer coisa. ´E interessante que o chamado “modelo padr˜ ao” das f´ ısica de part´ ıculas el- ementares a despeito de contar com um n´ umero grande de parˆ a metros ´ e 4Isto ´ e, podemos sim modificar as rela¸ c˜ oes de comuta¸ c˜ ao. 13de uma grande simplicidade na descric˜ ao das intera¸ c˜ oes e ntre part´ ıculas ele- mentares. E, o que ´ e mais importante, o modelo n˜ ao depende f ortemente dos valores que esses parˆ ametros venham a ter na realidade. Na e letrodinˆ amica cl´ assica, alguns dos parˆ ametros como o´ ındice de refra¸ c ˜ ao tem que ser obtidos experimentalmente. Isso n˜ ao tira beleza ` a teoria de Maxwe ll. Por outro lado, ´ e bom frisar que a explica¸ c˜ ao de porquˆ e ce rtos parˆ ametros tˆ em os valores observados ´ e um problema fundamental apena sseeles es- tiverem relacionados com objetos fundamentais. Talvez os q uarks n˜ ao sejam os objetos fundamentais da natureza. Por exemplo, os chamad os ˆ angulos de Cabibbo-Kobayashi-Maskawa s˜ ao equivalentes ` a orienta¸ c˜ ao de certas ´ orbitas planet´ arias. Esta orienta¸ c˜ ao ´ e de fundamental importˆ ancia para n´ os: ela determina as esta¸ c˜ oes na Terra. No entanto, n˜ ao consider amos como funda- mental explicar por primeiros princ´ ıpios as orienta¸ c˜ oe s das ´ orbitas porque eles (os planetas) h´ a muito tempo deixaram de ser considera dos objetos de estudo das leis fundamentais. N˜ ao era este o caso na ´ epoca d e, digamos, Kepler (ver mais adiante). Por enquanto consideramos os qua rks como sendo fundamentais. Ser´ a isso mantido com o desenvolvimen to da f´ ısica nos pr´ oximos d´ ecadas ou s´ eculos? N˜ ao sabemos. A “inevitabilidade” ´ e outra carater´ ıstica que Weinberg a tribue ` a beleza de uma teoria [WE93a]. A teoria da relatividade geral parece inevit´ avel uma vez adotados os princ´ ıpios (simples) de Einstein. No en tanto Wein- berg subestima a inevitabilidade dos dados experimentais. Os dados as- tronˆ omicos tornaram inevit´ avel a lei do inverso do quadra do da distˆ ancia. Nas outras intera¸ c˜ oes a inevitabilidade ´ e obtida dando p rioridade ` as sime- trias em vez de a mat´ eria. Um terceiro aspecto para Weinberg que deve ser incorporado ` a beleza ´ e a sua “rigidez” [WE93a]. Pode-se descrever uma grande var iedade de fenˆ omenos construindo-se teorias o mais flex´ ıveis poss´ ı veis. N˜ ao ´ e isto o que esperamos de uma teoria dita de fundamental. A rigidez da s teorias em f´ ısica de part´ ıculas elementares ´ e dada pela simetria e pela consistˆ encia matem´ atica como por exemplo renormalizabilidade e o cance lamento das anomalias. 4 Ca¸ ca ao universo A procura da “f´ ormula do mundo” implica uma defini¸ c˜ ao do mu ndo. Isto ´ e, precissamos a priori definir o sujeito a ser explicado. H´ a apenas alguns s´ eculos, o “mundo” era restrito aos planetas. Ainda que hoj e em dia nosso 14“mundo” ´ e mais complexo e amplo, n˜ ao vemos nenhuma raz˜ ao p ela qual j´ a tenham sido observados todas as suas carater´ ısticas a s erem explicadas. Surpressas podem aparecer mesmo naquelas escalas espa¸ co- temporais nas quais atualmente pensamos ja ter estudado em detalhe. A postura adotada freq¨ uentemente pelos f´ ısicos, ´ e reflet ida na vis˜ ao de Dirac. Segundo ele, a mecˆ anica quˆ antica estava completa e m 1929 e as imperfei¸ c˜ oes relativas ` a sua s´ ıntese com a relatividad e restrita eram [SC93] ...of no importance in the consideration of atomic and mole- cular structure and ordinary chemical reactions...the und erlaying physical laws necessary for the mathematical theory of a lar ge part of physics and the whole of chemistry are thus completel y known, and the difficulty is only that the exact application of these laws lead to equations much too complicated to be solub le. Estas palavras de Dirac foram motivadas pelo sucesso da mecˆ anica quˆ antica n˜ ao relativ´ ıstica na explica¸ c˜ ao da estrutura do ´ atomo e mol´ eculas. A vis˜ ao de Dirac ´ e atualmente compartilhada pela maioria d os f´ ısicos. De fato, como deixa claro acima Schweber, o reducionismos ´ e a m arca da f´ ısica te´ orica deste s´ eculo. Mais ainda, ´ e uma carater´ ıstica, at´ e recentemente do- minante, de toda a ciˆ encia moderna. N˜ ao ´ e poss´ ıvel negar os bons resultados obtidos. Ainda segundo Schweber [SC93] These conceptual developments in fundamental physics have revealed a hierarchical structures of the physical words. E ach layer of the hierarchy successfully represented while rema ining largely decoupled from other layers. These advanced have su p- ported the notion of the existence of objective emergent pro perties and have challenged the reductionist approach. They have al so given credence to the notion that to a high degree of accuracy our theoretical understanding of some domains have stabilized , since the foundational aspects are considered known. Quantum mechanics reasserted that the physical world prese nt itself hierarchically. The world was not carved up into terr estial, planetary and celestial spheres but layered by virtue of cer tain constants of nature... Planck’s constant allow us to parse t he world into microscopic and macroscopic realms, or more pre- cisely into the atomic and molecular domains and the macro- scopic domains composed of atoms and molecules. The story repeated itself with the carring out of the nucleon domain: q ua- sistable entities–neutrons and protons–could be regarded as the 15building blocks of nuclei, and phenomenological theories c ould account for many properties and interactions of nuclei. Os f´ ısicos te´ oricos s˜ ao as vezes otimistas demais com rel a¸ c˜ ao aos objetivos a serem alcan¸ cados a curto pra¸ co. Por´ em os f´ ısicos tamb´ em s˜ ao c´ eticos, Weisskopf por exemplo se pergunta [WE91] Is it really an aim of theoretical physics to get the world for - mula? The greatest physicists have always thought that ther e was one, and that everything else could be derived from it. Ei n- stein believed it, Heisenberg believed it, I am not such a gre at physicist, I do not beleive it... This I think, is beacuse nat ure is inexhaustible. Devemos perguntar-nos se o desenvolvimento futuro implica uma continua¸ c˜ ao nessa dire¸ c˜ ao ou uma pausa para reorganizar todos os conhe cimentos adquiri- dos at´ e hoje, antes de ser poss´ ıvel a proposta de uma nova or dem. Por outro lado, acreditamos que o problema n˜ ao ´ e se devemos ou n˜ ao reconhecer se a f´ ısica de part´ ıculas elementares ´ e a ´ uni ca ´ area fundamental da f´ ısica. O que estamos tratando ´ e mais profundo. ´E se existem leis ver- daderamente fundamentais a serem descobertas (ou que j´ a o t enham sido) em estruturas diferentes daquelas das pequenas escalas sub-n ucleares ou no uni- verso primordial. ´E curioso observar que esse tipo de estruturas hier´ arquica s na dimens˜ ao espa¸ co-temporal foram obtidas sempre que os i nstrumento de observa¸ c˜ ao eram refinados para poder atingir distˆ ancias cada vez menores. Por exemplo, o processo de dete¸ c˜ ao e estudo de partes cada v ez menores ocorre tamb´ em na biologia. Depois de estudar doen¸ cas bact erianas, com o advento do microsc´ opio eletrˆ onico, foram detetadas doen ¸ cas virais. Podem existir agentes produtores de doen¸ cas menores ( prions ) ainda n˜ ao deteta- dos? [GA94]. ´E poss´ ıvel que, al´ em de refinamentos na sensitividade dos aparelhos, que sem d´ uvida foi o eixo do desenvolvimento das ciˆ encias, o re- finamento das capacidades de c´ alculo possa introduzir novo s conceitos. O caos pode ter sido um dos primeiros exemplos. A f´ ısica poder ia entrar numa fase n˜ ao reducionista (poderiamos dizer holista mas este termo j´ a ´ e usado com outros prop´ ositos; ou global ou usar tamb´ em n˜ ao-reducionista ). Em todo caso pode ser que n˜ ao seja uma reviravolta completa. Os aspectos globais tem suas dificuldades tamb´ em e seu progresso n˜ ao de ver´ a ser t˜ ao r´ apido como alguns podem pensar. (Mencionamos antes que me smo ´ areas como o caos passam pelas mesmas dificuldades.) Por outro lado , a tradi¸ c˜ ao reducionista ainda n˜ ao foi esgotada e dever´ a dar resultad os importantes nas pr´ oximas d´ ecadas. Segundo Weinberg [WE93a] 16At this moment in the history of science it appears that the best way to approach these laws is through the physics of elem en- tary particle, but is an incidental aspect of reductionism a nd may change. Argumento te´ oricos falharam as vezes redondamente. Vejam os por ex- emplo os seguintes argumentos de Maxwell [MA54] ...to explain electromagnetic phenomena by means of mechan - ical action transmitted from one body to another by means of a medium occupying the space between them. The ondulatory the - ory of light also assume the existence of a medium. To fill all space with a new medium whenever any new phe- nomena is to be explained is by no means philosophical, but if the study of two different branches of science has independen tly suggested the idea of a medium, and if the properties which mu st be attibutted to the medium...are the same...the evidence f or the physical existence of the medium will be considerably stren gth- ened. O que Maxwell n˜ ao sabia era que a estrutura matem´ atica da te oria dis- pensava a existˆ encia de qualquer meio para a transmiss˜ ao d e ondas eletro- magn´ eticas. Por outro lado, historicamente a existˆ encia do medio para os fenˆ omenos eletromagn´ eticos foi importante. N˜ ao apen as para Maxwell. Faraday, consideraba o v´ acuo como uma substˆ ancia. Isso aj udava-o a ver o campo eletromagn´ etico como sendo transmitido pelo meio. Isto foi um avan¸ co com rela¸ c˜ ao ` a a¸ c˜ ao a distˆ ancia de Newton. A f´ ısica de part´ ıculas elementares tem sido mesmo reducio nista. E ´ e a isso que deve seu sucesso. O ponto ´ e se devecontinuar sendo, ou se chegou o momento de dar ˆ enfase aos aspectos globais ou n˜ ao- reducionistas. Assim colocada, esta discuss˜ ao deixa de ser algo vazio. Ela pode determinar o sucesso ou o fracasso de novas gera¸ c˜ oes de pesquisadores . Como foi dito acima, ´ e pos´ ıvel que nas pr´ oximas d´ ecadas a tendˆ encia n a f´ ısica de part´ ıculas elementares seja a mesma que a dos ´ ultimos 50 anos. Tem muito s dados a serem obtidos antes de acharmos que devemos voltar a proble mas mais fundamentais deixados para tr´ as (se ´ e que isso acontecer´ a algun dia). Por outro lado, devemos ter sempre em mente, a historicidade dos pro- blemas e de suas solu¸ c˜ oes. Na metade do Sec. XIX discutia-s e se a cria¸ c˜ ao espontˆ anea da vida era poss´ ıvel. Poderia a vida ter surgid o da n˜ ao-vida? As experiˆ encias de Pasteur mostraram que o fenˆ omeno de putre fa¸ c˜ ao era provo- cado pelos micro-organismos presentes no ar. A biologia era assim, separada 17da qu´ ımica. Esta separa¸ c˜ ao foi positiva nas d´ ecadas pos teriores com am- bas disciplinas se desenvolvendo separadamente. Mas, depo is da mecˆ anica quˆ antica passou-se a acreditar que todos os processos biol ´ ogicos s˜ ao reduzi- dos a processos qu´ ımicos que pela sua vez s˜ ao manifesta¸ c˜ oes das leis da f´ ısica elementar. Por´ em, em alg´ un momento da evolu¸ c˜ ao do universo (ou da Terra) a vida surgiu da n˜ ao-vida num processo ainda n˜ ao con hecido. Apenas n˜ ao s˜ ao os processos simples do dia-a-dia nos quais acredi tavam os defen- sores da gera¸ c˜ ao espontˆ anea pre-Pasteur, por exemplo pe la fermenta¸ c˜ ao e putrefa¸ c˜ ao como acreditava F. A. Pouchet. Um outro exempl o, ´ e a lei da gravita¸ c˜ ao de Newton. Como vimos na Sec. 2, os f´ ısicos eur opeos (Descartes principalmente), e o pr´ oprio Newton, n˜ ao aceitavam do con ceito de “a¸ c˜ ao a distˆ ancia” e do de “espa¸ co absoluto”. Mas, a lei de Newton d a gravita¸ c˜ ao foi superior que a dos v´ ortices de Descartes para preparar o cam inho da teoria da gravita¸ c˜ ao geral de Einstein. Poderiamos colocar vari os exemplos onde fica claro que uma solu¸ c˜ ao a um determinado problema permit e o desen- volvimento de uma ´ area mesmo que posteriormente se verifiqu e que aquela solu¸ c˜ ao n˜ ao era correta ou apenas o era de maneira aproxim ada. O objetivo da ciˆ encia continua a ser a ca¸ ca ao universo . A discuss˜ ao ´ e qual o passo mais imediato a ser dado na dire¸ c˜ ao certa. 5 Part´ ıculas elementares: al´ em do modelo padr˜ ao Na d´ ecada dos anos 70 na ´ area de f´ ısica de part´ ıculas elem entares ficou com- pleto (do ponto de vista te´ orico) o chamado modelo padr˜ ao no qual o mundo subnuclear ´ e composto em termos de gluons, b´ osons vetoria is intermedia- rios, o f´ oton, quarks e leptons e o escorregadi¸ co b´ oson de Higgs [WE67b, WI72, HI64]. Depois disso podemos perguntar-mos se haver´ a uma outra camada de estrutura. Como mencionado acima, n˜ ao sabemos. ´E por isso que a procura continua. As id´ eias te´ oricas que permitiram ` a f´ ısica chegar ao est abelecimento de uma s´ erie de dom´ ınios hier´ arquicos quase aut´ onomos s˜ a o: o grupo de renor- maliza¸ c˜ ao (que nos indica como podemos fazer extrapola¸ c ˜ oes), o teorema de desacoplamento (que nos permite esquecer ao fazer as extr apola¸ c˜ oes, part´ ıculas de massa maior que a escala de energia relevante para as ex- periˆ encias), a liberdade assintˆ otica (que nos permite us ar teoria de per- turba¸ c˜ oes) e a quebra espˆ ontanea de simetria (que nos per mite gerar massa para as diferrentes part´ ıculas sem estragar a consistˆ enc ia matem´ atica da 18teoria). O sucesso deste modelo na descri¸ c˜ ao das intera¸ c˜ oes entr e part´ ıculas e- lementares coloca o problema de se determinar quais as leis d a f´ ısica al´ em deste modelo. A despeito da impressionante concordˆ ancia c om os dados experimentais, existe um concenso entre os f´ ısicos de que e ste modelo n˜ ao ´ e a teoria final. O modelo deixa muitas coisas sem resposta e t em muitos parˆ ametros a serem determinados pela experiˆ encia. Como m encionamos antes, isso poderia n˜ ao ser um problema j´ a que qualquer teo ria f´ ısica vai precissar sempre de um n´ umero (finito) de parˆ ametros de ent rada a serem determinados pela experiˆ encia. O ponto de vista reducionista implica que tudo na natureza ´ e controlado por um mesmo conjunto de leis fundamentais. O modelo padr˜ ao estaria na base de tudo o resto mas e depois, o que ´ e que explica esse mode lo? quais os princ´ ıpios gerais que explicariam porque esse modelo e n˜ a o outro ´ e o que ´ e v´ alido at´ e as energias dos aceleradores atuais. Constitu e este um problema fundamental a servir de guia para as futuras gera¸ c˜ oes? A re sposta usual a esta pergunta est´ a no esp´ ıritu das palavras de Einstein qu em, em 1918, dizia The supreme test of the physicist is to arrive at those univer - sal elementary laws from which the cosmos can be buildt by pur e deduction . Tarefa dif´ ıcil, nem mesmo sabemos como construir por primeiros princ´ ıpios hadrons partindo de quarks e gluons! (Para n˜ ao falar de n´ uc leos em ter- mos de nucleons, mol´ eculas em termos de ´ atomos.) Esse tipo de afirma¸ c˜ ao emotiva, mesmo vindo de f´ ısicos como Einstein devem ser ana lisadas cuida- dosamente. Principalmente pelos estudantes que est˜ ao com e¸ cando a sua p´ os-gradua¸ c˜ ao. Atualmente a ´ area de neutrino ´ e uma das mais ativas da f´ ısi ca de part´ ıculas elementares fornecendo muitos dados experimentais que per mitem testar hip´ oteses do que seria a f´ ısica al´ emdo modelo padr˜ ao. De maneira geral as observa¸ c˜ oes astrof´ ısicas [GE99] se unem aos dados de a celeradores e de experimentos com energias baixas que est˜ ao medindo com m aior pre- ciss˜ ao efeitos bem conhecidos para a procura da nova f´ ısic a. Novos dados de efeitos h´ a muito tempo procurados como por exemplo a viola¸ c˜ ao da sime- tria CP [FE99, CE99], ou novas possibilidades permitidas po r novas t´ ecnicas experimentais como o estudo da anti-mat´ eria com a produ¸ c˜ ao e armazena- mento de anti-pr´ otons e mesmo de anti-hidrogˆ enio [CE99]. Onde est´ a a crise? 196 Dire¸ c˜ ao ´ unica? Por ser reducionista a ciˆ encia moderna ´ e tamb´ em unificado ra. Unificadora no sentido que pretende uma descri¸ c˜ ao unificada dos fenˆ om enos f´ ısicos e reducionista no sentido que pretende reducir o n´ umero de co nceitos inde- pendentes com os quais seriam formuladas as leis da natureza . Esse ponto de vista foi criticado por Anderson alguns anos atr´ as. Segu ndo ele [AN72] The main fallacy in this kind of thinking is that the reductio n- ist hypothesis does not by any means imply the “construction ist” one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and recons truct the universe. In fact, the more the elementary particle phys ics tell us about the nature of fundamental laws, the less releva nce they seem to have to the very real problems of the rest of scien ce, much less to those of society. The constructionist hypothesis breaks down when confronte d with the twin difficulties of scale and complexity. The behavi or of large and complex aggregates of elementary particles, it tu rns out, is not to be understood in terms of a simple extrapolation of t he properties of a few particles. Instead, at each level of comp lexity enterily new properties appear, and the understanding of th e new behaviors requieres research which I think is as fundamenta l in its nature as any other. O quadro do percurso desde o “menos fundamental” at´ e o “mais funda- mental” pode ser resumido na Tabela 1 na qual os elementos de u ma ciˆ encia Xobedecem as leis de uma ciˆ encia Y[AN72]. A hierarquia mostrada na Tabela 1, por´ em, n˜ ao implica que a ciˆ encia X seja apenas aplica¸ c˜ ao da ciˆ encia Y. Em cada n´ ıvel novas leis, concei tos, generaliza¸ c˜ oes e mesmo novos m´ etodos de pesquisa s˜ ao ne cess´ arios. Mesmo que saibamos que ap´ os o aquescimento as mol´ eculas se afast am at´ e que a forma s´ olida se dissocie, as mol´ eculas agora obedecem as l eis dos fluidos que n˜ ao podem ser deduzidas a partir das leis dos s´ olidos. Por e xemplo, a vida ( a biologia em geral) ´ e em seu n´ ıvel mais fundamental, qu´ ı mica. Isso n˜ ao implica que seja apenas qu´ ımica. O mesmo pode ser dito da qu´ ımica, ela ´ e basicamente f´ ısica mas as leis da f´ ısica ajudam pouco no es tabelecimento de 20X Y Estado s´ olido ou Muitos Corpos Part´ ıculas Elementares Qu´ ımica Muitos corpos Biologia Molecular Qu´ ımica Biologia Celular Biologia Molecular ...... Psicologia Fisiologia Ciˆ encias Sociais Psicologia Table 1: “Hierarquia” das ciˆ encias de Anderson. novas leis qu´ ımicas. Claro que essas novas leis da qu´ ımica n˜ ao devem violar as leis da f´ ısica. Mas, fazer qu´ ımica n˜ ao ´ e fazer f´ ısica . E nunca ser´ a. Na pr´ atica temos “disconnected clumps” nos diferentes dom ´ ınios das ciˆ encias. Isso acontece nos dois sentido referidos acima: um dom´ ınio de sub- estrutura n˜ ao ajuda na explica¸ c˜ ao da maioria dos process os da estrutura acima dela. Para refor¸ car o assunto enfatissemos que at´ e p ouco tempo atr´ as a f´ ısica atˆ omica entra como um fator de corre¸ c˜ ao da f´ ısi ca nuclear. Esta pela sua vez n˜ ao ´ e “construida” (no sentido de Anderson) pe la f´ ısica de quarks. Mas acreditava-se que por sua vez a f´ ısica nuclear n ˜ ao teria nada a ver com a f´ ısica atˆ omica. No entanto, recentemente foi des coberto um efeito que contradiz esta ´ ultima afirma¸ c˜ ao: foi encontrado que a orienta¸ c˜ ao do spin do n´ ucleo de uma mol´ ecula de H2afeta o espalhamento dessa mol´ ecula biatˆ omica na superficie de um cristal [BE98]. Isso vai ajuda r a estudar estrutura do campo el´ etrico em superf´ ıcies. At´ e onde iss o pode ir? isto ´ e, ser´ a que um dia estaremos observando efeitos do conte´ u do de quarks em f´ ısica do estado s´ olido? n˜ ao sabemos, isso depende de m elhoramentos na t´ ecnica que est˜ ao fora da nossa capacidade de previs˜ ao . Mas, se isso acontecer ent˜ ao Anderson estaria errado! ´E usual acreditar que quando encontradas, verdades univers ais devem ser explicadas em termos de outras mais profundas,..., at´ e ati ngirmos a chamada teoria final . Este ´ e de fato umdos projetos para a ciˆ encia. Mas n˜ ao ´ e o ´ unico. E nem mesmo talvez seja o mais interessante. Um princ ´ ıpio cient´ ıfico explica outro se este ´ ultimo n˜ ao viola as leis do primeiro. Por´ em, temos de entender que as leis do princ´ ıpio mais b´ asico n˜ ao ajudam a determinar as leis do segundo. Apenas servem como referencial subjacente .´E por isso que continuar´ a havendo qu´ ımica independentemente de que de s eus fundamentos sejam f´ ısicos. Mais ainda, as leis da qu´ ımica ou da mat´ eri a condensada, 21para pˆ or dois exemplos, podem ter uma generalidade vertica l (no sentido de Weisskopf acima). Podemos colocar a evolu¸ c˜ ao do progre sso cient´ ıfico da forma mostrada na Tabela 2 [DR98] 1) observa¸ c˜ oes, fenˆ omenos complexos, infinidade ↓ de objetos, ↑ 2) Organiza¸ c˜ ao em termos Introdu¸ c˜ ao de conceitos emp´ ıricos de detalhes Redu¸ c˜ oes conceitos “´ uteis” sucessivas 3) Leis emp´ ıricas–no¸ c˜ ao de objetos compostos ↓ 4) As leis emp´ ıricas ↑ podem ser expressas como rela¸ c˜ oes formais especializa¸ c˜ ao 5) Poucos objetos simples, leis mais gerais 6) Abstra¸ c˜ oes, matematiza¸ c˜ ao, idealiza¸ c˜ ao, generaliza¸ c˜ ao 7) Objetos simples irredut´ ıveis, ↑ conceitos e rela¸ c˜ oes O problema O fim universais, inverso: a redu¸ c˜ ao leis da redu¸ c˜ ao completa ` a composi¸ c˜ ao Table 2: Seq¨ uˆ encias do reducionismo vs composi¸ c˜ ao. A vis˜ ao de Dirac continua na tradi¸ c˜ ao da f´ ısica de part´ ı culas e campos. Na d´ ecada dos anos 80 as teorias de supercordas que tinham si do elaboradas desde 1974 por Veneziano, pasando por Nambu e outros, aparec eram como fortes candidatas para a teoria que unificasse as quatro inte ra¸ c˜ oes conheci- das. Essa seria ent˜ ao a culmina¸ c˜ ao da vis˜ ao reducionist a da f´ ısica. Segundo Witten [OV91] a teoria das supercordas s˜ ao ...a piece of twenty-first-century physics that has fallen i nto the twentiesth century, and would probably require twenty- second- century mathematics to understand . 22Em 1980 Hawking disse [HA81] que existia ...the possibility that the goal of the theoretical physics might be achieved in the not too distant future, say, by the end of th e century. By this I mean that we might have a complete, consis- tent and unified theory of the physical interactions which wo uld describe all possible observations. Isso tem mais de pessimista que de otimista. Significa que tod o o conheci- mento te´ orico e em particular novos dados experimentais n˜ ao ser˜ ao capazes, nos pr´ oximos s´ eculos, de indicar uma outra dire¸ c˜ ao para as leis da natureza. Esse ´ e o ponto fraco de todo o paradigma de “unifica¸ c˜ ao”. Essa posi¸ c˜ ao come¸ ca a mudar. Um exemplo radical ´ e o de Geo rgi quem afirma que It is true that in chemistry and biology one does not encounte r any new physical principles. But the systems on which the old principles act differ in such a way drastic and qualitative wa y in the different fields that it is simply not useful to regard one a s a branch of another. Indeed the system are so different that ‘pr in- ciples’ of new kinds must be developed, and it is the principl es which are inherently chemical or biological which are impor tant. In the same way, to study phenomena at velocities much less thancand angular momentum much greater than ¯h, it is simply not useful to regard them as special cases of phenomena for ar bi- trary velocity and angular momentum. We do not need relativi ty and quantum mechanics for small velocity and large angular m o- menta...if we had to discover the laws of relativistic quant um me- chanics from the beginning, we probably would never have gon e anywhere. Nenhuma forma de estudar a natureza ´ e compar´ avel ` a pesqui sa cient´ ıfica a partir (principalmente) de Galileo. Entendemos quantita tivamente os fe- nˆ omenos. Isso faz, entre outras coisas, a diferen¸ ca entre nossos ´ atomos e os de Dem´ ocrito. Entender quantitativamente os fenˆ omenos d iz respeito a que podemos fazer predi¸ c˜ oes quantitativas que podem ser confi rmadas ou n˜ ao pela experiˆ encia.5Sem estas ´ ultimas n˜ ao podemos dizer se uma teoria ´ e 5N˜ ao pretendemos que todos os aspectos de uma teoria tenham q ue ser testados pela experiˆ encia. Esta era a posi¸ c˜ ao dos positivistas. As teo rias segundo eles tˆ em de estar baseadas apenas em observ´ aveis. 23correta ou n˜ ao. Claro, as coisas n˜ ao s˜ ao t˜ ao simples como parecem dado que podem existir segundo os dados experimentais v´ arias te orias poss´ ıveis. Aqui a simplicidade ´ e ´ util. Mas apenas isso, ´ util, n˜ ao de finitiva. Assim, a divis˜ ao aristot´ elica de movimentos naturais en˜ ao naturais n˜ ao passa de uma descri¸ c˜ ao cuja plausabilidade n˜ ao pode ser testada. Mes mo que a f´ ısica mo- derna fizesse uso de tais conceitos (como o faz do ´ atomo) deve mos distinguir uma opini˜ ao de uma pesquisa metodol´ ogica (mais ou menos) b em definida. Neste sentido, a referˆ encia aos ´ atomos de Dem´ ocrito ´ e ap enas aned´ otica. ´E bom lembrar que ainda que f´ ısicos como Newton e Faraday tin ham em mente uma “teoria final”, agora sabemos que o contexto te´ ori co e experi- mental da ´ epoca era bem restrito para tal efeito. Isso ´ e mai s um exemplo de que o m´ etodo cient´ ıfico (qualquer coisa que isso signifique ) n˜ ao ´ e suficiente para explicar as motiva¸ c˜ oes, as escolhas e os preconceito s dos cientistas. As- sim, acreditamos que a quantidade de especula¸ c˜ ao ´ e restr ita por fatos al´ em das opini˜ oes da comunidade cient´ ıfica e a priori n˜ ao est´ a bem definida. Os exemplos da teoria geral da relatividade e da predi¸ c˜ ao d a radia¸ c˜ ao de fundo s˜ ao exemplos de extrapola¸ c˜ oes que deram certo e i sso motivou a extrapola¸ c˜ ao dos resultados te´ oricos al´ em das possibi lidades de verifica¸ c˜ ao experimental. Mas, quantas extrapola¸ c˜ oes falharam? No m ´ ınimo para ser- mos consistentes com a estat´ ıstica devemos considerar iss o quando fizermos escolhas pessoais sobre o tema de pesquisa. O caso contr´ ari o tamb´ em acon- tece. Achar que tudo j´ a ´ e conhecido e que n˜ ao h´ a mais espa¸ co para espec- ula¸ c˜ oes. ´E bem conhecida a opini˜ ao no final do s´ eculo pasado (atribu´ ıda a Lord Kelvin) e mesmo no come¸ co deste s´ eculo (como Michels on) sobre o fato que tudo que tinha de ser descoberto j´ a o tinha sido fei to. Assim, podemos nos perguntar se a luta de Einstein ´ e ainda a nossa. N ˜ ao no sentido escatol´ ogico no qual n˜ ao temos a menor d´ uvida que ´ e. Mas n o sentido de uma escolha pessoal da linha de pesquisa de um(uma) jovem cie ntista. Que existe um sentido nas explica¸ c˜ oes n˜ ao h´ a d´ uvida: as leis de New- tonexplicam as de Kepler, as de Einstein as de Newton, etc. O p onto ´ e, se esse sentido ´ e ´ unico ou, existem ramifica¸ c˜ oes? Quando um a teoria final no sentido · · · → mol´ eculas →´ atomos →n´ ucleos →n´ ucleons →quarks → · · · for obtida, ainda fenˆ omenos como a turbulˆ encia e supercon dutividade a altas temperaturas precissar˜ ao ser explicados e o que esteja par a al´ em dos quarks poder´ a n˜ ao ser importante para isso. Se podemos dizer que as verdades mais fundamentais s˜ ao aque las mais abrangentes devemos, no entanto, aceitar que existem verda des fundamen- 24tais “horizontais” (“extensivas” no sentido de Weisskopf) , isto ´ e, n˜ ao fazem parte de uma mesma cadeia de explica¸ c˜ oes em ordem crescent e da escala de determinadas grandezas (massa, velocidade ou energia). As sim, as leis de Newton podem ser mais fundamentais que as de Kepler e as de Ein stein, pela sua vez, mais fundamentais que as de Newton. Mas, ser´ a q ue isso ajudaria na compreens˜ ao das propriedades do ADN? ser´ a que apenas seria necess´ ario um grande computador para explicar essas propr iedades resol- vendo equa¸ c˜ oes da mecˆ anica quˆ antica para os el´ etrons e os n´ ucleos? Talvez n˜ ao. Do contr´ ario teriamos voltado ao mecanicismo pre-Ma xwell, apenas substituindo a mecˆ anica cl´ assica pela quˆ antica. Podem e xistir quest˜ oes que n˜ ao possam ser resolvidas com as nossas ferramentas atuais , te´ oricas ou experimentais. Alguns fatos, como a origem da vida, parecem ser devidos a aci dentes hist´ oricos. Se, contudo, alg´ um dia as condi¸ c˜ oes inicia is passassem a ser parte das leis da f´ ısica isso pode ser feito n˜ ao necessaria mente no sentido ´ atomo →n´ ucleon · · ·mas, mesmo com fenˆ omenos macrosc´ opicos. Novos princ´ ıpios que n˜ ao contradigam as leis microsc´ opicas po der˜ ao encontrar no- vas generalidades n˜ ao deduz´ ıveis daqueles. Por exemplo, a universalidade do caos ´ e suficientemente abr angente e n˜ ao depende (por enquanto) de leis mais gerais em escalas menore s.´E este tipo de universalidade que acreditamos existir em diferentes n´ ıveis de organiza¸ c˜ ao independentes uns dos outros. Por outro lado, atualmente ex istem teorias t˜ ao especulativas (a teoria dos “baby universes” e outras) n˜ ao completa- mente formuladas matematicamente e sem suporte experiment al (mesmo a longo prazo) que podemos at´ e compar´ a-las com a formula¸ c˜ ao aristot´ elica (a situa¸ c˜ ao grega mencionada antes). N˜ ao ´ e obvio que vai acontecer com o caos o que aconteceu com a ter- modinˆ amica. Esta come¸ cou como ciˆ encia autˆ onoma mas foi depois funda- mentada na mecˆ anica estat´ ıstica. Muito menos obvio ´ e oca so da biologia ou do problema da conciˆ encia [HO94, PE94]. Ainda que a mec´ ani ca estat´ ıstica “explica” a termodinˆ amica apenas no sentido que a incorpor a.6 Mas devemos ser cr´ ıticos tamb´ em com rela¸ c˜ ao a essas posi ¸ c˜ oes. ´E ver- dade que n˜ ao adianta muito para os qu´ ımicos saber que a mate ria ´ e formada por quarks. Mas de alguma maneira esse conhecimento ´ e subja cente a toda a qu´ ımica. Gostemos ou n˜ ao. Na pr´ atica nossos m´ etodos te ´ oricos s˜ ao muito limitados. ´E sempre dif´ ıcil considerar as situa¸ c˜ oes limites como aq uele entre 6As cr´ ıticas a Boltzmann estavam corretas porque apenas a me cˆ anica quˆ antica permi- tiria uma formula¸ c˜ ao coerente das leis estat´ ısticas mas ela n˜ ao era conhecida nos primeiros anos do s´ eculo [KU87]. 25a mecˆ anica quˆ antica e a cl´ assica, ou como diz Georgi acima , entre a mecˆ anica relativ´ ısta e a n˜ ao-relativ´ ısta. Mas essas dificuldades devem ser vistas como limita¸ c˜ oes nossas e n˜ ao s˜ ao ´ unicas nessas ´ areas. Acontecem mesmo na mecˆ anica cl´ assica n˜ ao-relativ´ ısta . Por exemplo, sabemos que as diferentes maneiras de formular a mecˆ anica c l´ assica como 1) leis de Newton, 2) princ´ ıpio de D’Alembert, 3) princ´ ıpi o dos deslocamen- tos virtuais, 4) princ´ ıpio de Gauss, 5) princ´ ıpio de Hamil ton, 6) princ´ ıpio de a¸ c˜ ao m´ ınima, 7) coordenadas generalizadas e equa¸ c˜ o es de Lagrange, 8) equa¸ c˜ oes canˆ onicas de Hamilton, 9) equa¸ c˜ oes de Hamilt on-Jacobi e teoria das trasforma¸ c˜ oes. Todos estes formalismos s˜ ao complet amente equivalentes no sentido que, qualquer problema de mecˆ anica cl´ assica po de, em princ´ ıpio, ser resolvido por qualquer um desses m´ etodos. (Na pr´ atica porque todos levam ` as equa¸ c˜ oes de Newton.) As vezes, alguns deles s˜ ao mais ou menos apropriados para um problema particular. Outro, tˆ em a vant agem de per- mitir uma aprecia¸ c˜ ao mais profunda dos sistemas dinˆ amic os. Finalmente, alguns deles s˜ ao mais apropriados na respectiva extens˜ ao quˆ antica [LO87]. Contudo, nem toda informa¸ c˜ ao ´ e a mesma em cada um destes fo rmalismos. Por exemplo, com rela¸ c˜ ao as simetrias e leis de conserva¸ c ˜ ao. A conserva¸ c˜ ao da energia, momento linear e momento angular aparecem em qua lquer dos formalismos acima mencionados. Mas, em geral as leis de cons erva¸ c˜ ao po- dem ser diferentes. As simetrias do sistema s˜ ao diferentes quando se usam as equa¸ c˜ oes do movimento ou a Lagrangeana. Qu´ al deles ser ia mais fun- damental? Lembremos que algumas equa¸ c˜ oes do movimento na o tˆ em uma Lagrangeana ou Hamiltoniana correspendente. N˜ ao existem respostas defini- tivas dentro dos nossos conceitos te´ oricos atuais para ess e tipo de pergunta. Nem por isso achamos que eles n˜ ao descrevem a mesma mecˆ anic a cl´ assica. Por outro lado, devemos lembrar que algumas ciˆ encias s˜ ao p or natureza pr´ opria globais por exemplo as chamadas Ciˆ encias da Terra [BR92]. Nesta nova s´ ıntese a Terra ´ e considerada como sendo um sistema cuja dinˆ amica regˆ e-se por causas m´ ultiplas que se ligam e regulam entre s i [AL88]. A moral da hist´ oria ´ e que Terra n˜ ao pode ser tratada de jeito nenhu m de maneira reducionista. Constitue um problema suficiente geral.7 7As suas leis poder˜ ao ser verificadas em planetas diferentes da Terra quando forem estudados. Recentemente foram encontradas evidˆ encias de que no planeta Marte houve invers˜ oes do campo magn´ etico o que implicaria uma tectˆ on ica de placas semelhante ` a da Terra [CO99b] 267 Caos: leis fundamentais? A ferramenta em f´ ısica de part´ ıculas elementares para a ex trapola¸ c˜ ao das leis de uma determinada escala para escalas menores ´ e o grupo de renor- maliza¸ c˜ ao [WI83]. Sabemos, ent˜ ao, como extrapolar leis conhecidas a uma determinada escala de distˆ ancias para escalas menores. Po r´ em, se novas leis ser˜ ao descobertas no futuro, e n˜ ao vemos nenhum princ ´ ıpio geral que o proiba, ent˜ ao deveremos ir atualizando nossas extrapola¸ c˜ oes. Assim, qual- quer afirma¸ c˜ ao relativa ao futuro do universo como um todo d eve ser enten- dida apenas como uma predi¸ c˜ ao dos nossos conhecimentos atuais das leis fundamentais. Essas afirma¸ c˜ oes mudar˜ ao quando novas lei s fundamentais sejam descobertas nas escalas intermedi´ arias ou mesmo na d ire¸ c˜ ao hori- zontal. Quem poderia ter previsto a descoberta da radioativ idade? ou a mecˆ anica quˆ antica poderia ter sido postulada apenas por m ´ etodos formais? Al´ em disso, tudo est´ a baseado numa hip´ otese que mesmo raz o´ avel pode- ria n˜ ao ser verdadeira: a que as leis da natureza foram sempr e as mesmas. Claro, n˜ ao existe uma proposta razo´ avel para uma poss´ ıve l varia¸ c˜ ao tempo- ral dessas leis. A proposta de Dirac, que as constantes da nat ureza podem variar com o tempo n˜ ao foi confirmada at´ e agora e pode n˜ ao se r a mais interessante [DI37]. Devemos perceber que, se nem a arg´ ucia nem a estupidez s˜ ao p revis´ ıveis muito menos o s˜ ao as futuras descobertas te´ oricas e/ou exp erimentais. De qualquer forma, a Natureza ´ e mais imaginativa do que n´ os. A elegˆ ancia matem´ atica n˜ ao ´ e suficiente. Podemos imaginar quais seri am as estru- turas matem´ aticas se os f´ ısicos do s´ eculo passado tivess em tentado unificar, mais ou menos no sentido que conhecemos hoje, a eletrodinˆ am ica de Neu- mann e Weber com a gravita¸ c˜ ao de Newton? Nessa eletrodinˆ a mica as for¸ cas eletromagn´ eticas se propagam de um corpo a outro com veloci dade infinita. Teriam resistido essas estruturas matem´ aticas ` as descob ertas experimentais do final do se´ culo XIX? ´E bem prov´ avel que n˜ ao. De fato, ´ e interessante observar que Faraday queria mostrar que o eletromagnetismo estava rela- cionado com a gravita¸ c˜ ao [HO94]. Isso mostra, repetimos, que as motiva¸ c˜ oes pessoais dos cientistas n˜ ao tem nada a ver com os resultados reais obtidos. Faraday ficou longe de atingir seu desejo. Mas, visto restros pectivamente, ser´ a que precissava dele? Em 1950 John Von Neumann construia se computador Johnniac ( sic). Acreditava Von Neuman que a metereologia seria a ´ area princ ipal do uso dos computadores [DY88]. Segundo Von Neumann os fenˆ omenos met ereol´ ogicos eram de dois tipos: os est´ aveis e osinst´ aveis . Os primeiros s˜ ao aque- 27les que suportam pequenas perturba¸ c˜ oes, os segundos n˜ ao . Por´ em, assim que os computadores estivessem funcionando todos os proble mas relativos ` a predi¸ c˜ ao do tempo seriam resolvidos. Todos os processo s est´ aveis seriam previstos e os inst´ aveis controlados. Von Neumann n˜ ao imaginou que n˜ ao ´ e poss´ ıvel classificar a desloca¸ c˜ ao de fluidos em previs´ ıveis e control´ aveis. N˜ ao previu a des coberta do caos de- termin´ ıstico [RU93]. Este fenˆ omeno ´ e caraterizado por uma depˆ endenci a hipersens´ ıvel das condi¸ c˜ oes inicias, quaisquer que sej am estas condi¸ c˜ oes. Isso quer dizer que neste tipo de sistemas, pequenas perturb a¸ c˜ oes implicam grandes efeitos a longo prazo. O movimento regido pelas leis da mecˆ anica newtoniana ´ e det erminado sem ambig¨ uidade pela condi¸ c˜ ao inicial, no entanto, exis te, em geral, uma limita¸ c˜ ao na predi¸ c˜ ao de sua trajet´ oria. Temos ent˜ ao , ao mesmo tempo determinismo e impreditibilidade a longo prazo. O que define um sistema dinˆ amico ´ e uma evolu¸ c˜ ao temporal determinista bem defin ida. Talvez seja interessante observar que toda a f´ ısica desde os gregos at´ e poucos anos atr´ as foi baseada na geometria cl´ assica (euclideana ou n˜ ao) na q ual os elementos b´ asicos das formas s˜ ao as linhas, planos, c´ ırculos, esfe ras, cones, etc. No entanto a geometria fractal [MA77] parte de um universo mais parecido ao real: irregular e ´ aspero. Podemos nos perguntar quais seri am as leis b´ asicas se este tipo de geometria fosse o paradigma desde o come¸ co. S er´ a que o caos, seria um fato incorporado nas pr´ oprias leis do movimento, e m vez de sˆ e-lo nas condi¸ c˜ oes iniciais, como ocorre quando consideramos as leis de Newton? De qualquer forma o caos e a geometria fractal da natureza ´ e a vingan¸ ca de Simplicio sobre Sartori [GA85]. O movimento real n˜ ao ´ e t˜ a o simpels como acreditava Galileu. (Este ´ e mais um exemplo de que a escolha de teorias ou resultados tem um car´ ater hist´ orico. A teoria de Galileo s e mostrou frut´ ıfera entanto que a vis˜ ao global n˜ ao o foi. Mas acabariam se encon trando!). Do ponto de vista conceitual a descoberta do caos ´ e uma revol u¸ c˜ ao como o foram as teorias da relatividade e a mecˆ anica quˆ antica. N o entanto trata-se de fenˆ omenos a grandes escalas, inclusive com rela¸ c˜ ao a e scala humana. As- sim, vemos que este poderia ser um exemplo de que as “leis fund amentais” aparecem n˜ ao necessariamente quando estudamos processos carater´ ısticos de dimens˜ oes cada vez menores. Um aspecto a ser levado em con ta ´ e a “uni- versalidade” de qualquer coisa que possamos chamar de “lei f undamental”. A dependˆ encia hipersens´ ıvel das condi¸ c˜ oes iniciais fo i descoberta no fi- nal do s´ eculo XIX por Jacques Hadamard. Contribui¸ c˜ oes im portantes foram feitas por Duhem e Poincar´ e. No entanto apenas com o advento dos com- putadores r´ apidos foi poss´ ıvel fazer um estudo quantitat ivo riguroso. Assim, 28podemos dizer que a coloca¸ c˜ ao do caos como um novo paradigm a ´ e um feito que come¸ cou na d´ ecada dos anos 60. Isso significa que foram p recissos mais de trˆ es s´ eculos para que novas fenˆ omenos com suas res pectivas leis fossem descobertos “dentro das leis fundamentais” de Newto n. Neste sen- tido, poderiamos comparar as experiˆ encias realizadas a al tas energias como equivalentes ` a experiˆ encia de Cavendish: apenas est˜ ao t entando descobrir generalidades sobre leis fundamentais. O estudo detalhado fica como tarefa para as pr´ oximas d´ ecadas (s´ eculos ?). Assim, voltando a von Neumann, ele n˜ ao imaginou que em algun s anos seria descoberto que o movimento ca´ otico que geralmente ´ e imprevis´ ıvel e incontrol´ avel ´ e que ´ e a regra n˜ ao a exce¸ c˜ ao. Vemos ent˜ ao, e poderiamos dar muitos mais exemplos, que a preditividade ´ e pequena mesmo p ara mentes como as de Von Neumann. 8 Que biologia ´ e essa? Nos dias de hoje ´ e frequente escutar que “assim como a f´ ısic a foi a ciˆ encia do s´ eculo XX a biologia ser´ a a ciˆ encia do s´ eculo XXI”. De f ato, da d´ ecada de 50 para c´ a os avan¸ cos na biologia molecular s˜ ao impress ionantes. Atu- almente os projetos de sequˆ enciamento dos genomas de v´ ari os organismos, em particular o projeto Genoma Humano [TE99, JA99] permite v isualizar um sim fim de aplica¸ c˜ oes da genˆ omica nas ´ areas da sa´ ude e a gropecuaria. At´ e tem sido dito que os f´ ısicos deveriam fazer biologia. A fortunadamente, quando a f´ ısica estava realizando as suas hoje famosas desc obertas nas trˆ es primeiras d´ ecadas deste s´ eculo os biologos continuaram . .. a fazer biologia! Um fato, no entanto, deve ser enfatizado. A biologia realiza ndo essa espe- tacular revolu¸ c˜ ao fica menos biologia no sentido tradicio nal. A biologia est´ a se convertindo cada vez mais em uma ciˆ encia quantitativa co mo a qu´ ımica e a f´ ısica. A matem´ atica e a inform´ atica s˜ ao cada vez mais impressind´ ıveis para continuar o seu desenvolmimento. Sem os programas sequ enciadores n˜ ao teria sido poss´ ıvel realizar os projetos Genoma. Crai g Venter da Celera Genomics est´ a instalando o segundo maior conglomerado de c omputadores do mundo (somente inferior ao do Departamento de Energia dos Estados Unidos) [TE99]. Segundo Leroy Hood “a biologia se tornou inf orma¸ c˜ ao”, metade dos cientistas que trabalhar˜ ao no instituto que ele est´ a montando na Universidade de Seattle ser˜ ao matem´ aticos, f´ ısicos, ci entistas da computa¸ c˜ ao e qu´ ımicos [TE99]. O genoma humano n˜ ao diz como os 100 mil ge nes tra- balham juntos para formar o organismo humano. A compreens˜ a o disso ´ e 29uma tarefa que n˜ ao pode ser levada adiante somente pelos bio logos. Esse ´ e um empreendimento multidisciplinario no qual os f´ ısicos p oder˜ ao fazer con- tribui¸ c˜ oes importantes. N˜ ao apenas eles, para entender o modo como as diferentes partes de qualquer genoma interagem entre si ser ˜ ao necess´ arios n˜ ao apenas computadores cada vez mais r´ apidos e programas cada vez mais sofisticados, o que da origem a uma nova ´ area a bioinform´ atica , mas tamb´ em ser´ a necess´ ario construir modelos matem´ aticos e estat´ ısticos, compreender melhor a intera¸ c˜ ao entre as mol´ eculas, tarefa para qu´ ım icos. De fato a influˆ encia dos f´ ısicos em outras ´ areas das ciˆ enc ias fica evidente quando vemos que: M. F. Perutz, ganhou o prˆ emio Nobel de Qu´ ı mica em 1962 pelos seus estudos da estrutura das proteinas globular es. No mesmo ano F. H. C. Crick ganhava o prˆ emio Nobel de Fisiologia e Medi cina pela descoberta da estrutura da dupla h´ elice do DNA. Em 1962 foi a vez de M. Delbr¨ uck pela descoberta do mecanismo de replica¸ c˜ ao e a e strutura gen´ etica dos virus (em f´ ısica temos o espalhamento Delbr¨ uck). W. Gi lbert ganhou em 1980 o prˆ emio Nobel de Qu´ ımica pelos estudos na bioqu´ ım ica dos ´ acidos nucleicos em particular do DNA recombinante (em f´ ısica ´ e c onhecido por sua demostra¸ c˜ ao do teorema de Goldstone) e apenas para citar o mais recente, em 1998 o prˆ emio Nobel de qu´ ımica teve um f´ ısico entre os ga nhadore, W. Kohn pelas suas contribui¸ c˜ oes ` a qu´ ımica computacion al. Sim, alguns f´ ısicos continuar˜ ao a fazer biologia mas a forma¸ c˜ ao tra dicional de biol´ ogos (e m´ edicos e outras carreiras afins) ter´ a de ser reformulad a. 9 Computa¸ c˜ ao quˆ antica As contribui¸ c˜ oes dos f´ ısicos ` a ´ area da computa¸ c˜ ao tˆ em sido tamb´ em im- pressionantes. E n˜ ao devemos esquecer que isso foi obtido s em ter como motiva¸ c˜ ao a aplica¸ c˜ ao que posteriormente apareceu. A W orld Wide Weg (WWW) foi desenvolvida no CERN (usando a j´ a 25 anos de velha I nternet) com outras finalidades [BI99]. A revolu¸ c˜ ao somente aconte ceu, no entanto, quando foi desenvolvido o Mosaic no NCSA (National Center fo r Supercom- puter Applications) Todas as ´ areas sem exe¸ c˜ ao tˆ em sido influenciadas pela rev olu¸ c˜ ao da in- form´ atica. Isso continuar´ a ocorrendo sempre que a capaci dade de tratar informa¸ c˜ ao aumente. No entanto, o crecimento da rapidez d as computado- ras est´ a associada a uma maior capacidade de miniaturiza¸ c ˜ ao. Aqui vale lembrar que o ponto de partida de tudo foi a descoberta do efei to transis- tor [AD76]. Mais ainda, o primeiro transistor tinha dimens˜ oes macrosc´ opicas 30e seu pre¸ co era da ordem de USA$ 1. Dai para c´ a por esse pre¸ co podem-se comprar milh˜ oes deles! Foi isso que permitiu a revolu¸ c˜ ao da inform´ atica n˜ ao prevista mesmo por von Neumann (veja discus˜ ao na Sec. 7). De fato a densidade de transistores em cada chip aumentou exp onencial- mente nos ´ ultimos 24 anos. Manter esse ritmo nos pr´ oximos a nos implicar´ a em confrontar, em algun momento, as barreiras da mecˆ anica q uˆ antica. Toda a ciˆ encia e a tecnologia nanom´ etrica ´ e dominada pelos efe itos quˆ anticos. A escrita de dimens˜ oes nanom´ etricas est´ a cada vez mais des envolvida [HO99]; as suas aplica¸ c˜ oes v˜ ao desde a qu´ ımica (onde as t´ ecnica s poderiam ser us- adas para controlar a distˆ ancia entre os reagentes numa rea ¸ c˜ ao qu´ ımica) at´ e dispositivos eletrˆ onicos com dimens˜ oes moleculare s. A ciˆ encia aplicada chega cada vez mais perto da ciˆ encia b´ asica. Neste dom´ ıni os os fenˆ omenos quˆ anticos ser˜ ao cada vez mais importantes. Assim entender melhor essa estranha e bela teoria ser´ a um do s mais importantes temas que a f´ ısica vai brindar ao resto das cieˆ encias e, em geral, a todas as outras formas das atividades humanas. Por outro la do e n˜ ao menos espetacular ser´ a o controle da computa¸ c˜ ao quˆ anti ca [PR99]. A maioria das ´ areas do conhecimento puderam ter grande dese nvolvi- mento nas ultimos anos apenas pelos avan¸ cos na inform´ atic a e esta contou e continuar´ a a contar, direta ou indiretamente, com a partic ipa¸ c˜ ao dos f´ ısicos. ´E pos isso que muitos f´ ısicos continuar˜ ao a fazer f´ ısica e muitos biol´ ogos passar˜ ao a pensar cada vez mais ... como f´ ısicos! A f´ ısica est´ a longe de estar esgotada [DA99]. 10 S´ ıntese versus diversidade Algumas vezes a f´ ısica se encontra em situa¸ c˜ oes de s´ ınte se, enquanto na maior parte das vezes ´ e a diversidade a que prevalece. De fat o, a diversidade ´ e uma carater´ ıstica das ciˆ encias desenvolvidas. Segundo Dyson [DY88] por per´ ıodos longos as diversas ciˆ en cias per- manecem dominadas pela concretitude . Por exemplo, na maior parte do s´ eculo XIX e nas d´ ecadas posteriores aos anos 30 deste s´ ec ulo. Em out- ras ocasi˜ oes, ´ e a abstra¸ c˜ ao que domina. Os pesquisadores de uma ´ epoca determinada n˜ ao podem escolher entre qual a tendˆ encia que domine. Isso est´ a definido por fatores externos e, muitas vezes pelo acas o. Depois das revolu¸ c˜ oes da mecˆ anica quˆ antica e relatividade restri ta e geral, como pode- riamos esperar o desenvolvimento de esquemas te´ oricos mai s gerais ainda num breve per´ ıodo de tempo? 31No entanto, progressos importantes foram conseguidos no pe r´ ıodo de 1960-1980. O chamado modelo padr˜ ao das intera¸ c˜ oes eletrofracas e fortes foi uma conquista do ponto de vista da teoria quˆ antica de cam pos, acrescen- tadas de descobertas te´ oricas como a liberdade assintˆ otica e omecanismo de Higgs mencionados antes. A partir da´ ı, uma s´ erie de extrapola¸ c ˜ oes dessas estruturas levaram ` a maioria dos f´ ısicos a pensar que a s´ ı ntese final estaria chegando ao fim. Isso fica evidente nas palavras de Hawking cit adas anteri- ormente. A vis˜ ao de Dirac dominou a f´ ısica nas d´ ecadas passadas. Em 1970 L´ eon van Hove dizia [SC93] ... physics now look more like chemistry in the sense that... a much larger fraction of the total research deals with comple x sys- tems, structure and processes, as against a smaller fractio n con- cerned with the fundamental laws of motion and interactions ...we all believe that the fundamentals of classical mechanics, o f the electromagnetic interaction, and of statistical mechanic s dom- inate the multifarious transitions and phenomena you discu ss this week; and I assume that none of you expects his work on such problem to lead to modifications of these laws. You known the equations more than the phenomena... Agora sabemos que n˜ ao foi bem assim. Novas leis fundamentai s estavam sendo descobertas pelos te´ oricos e em pouco tempo testadas pelos f´ ısicos experimentais. (Lembremos do caos na mecˆ anica cl´ assica. ) No fundo, temos a esperanza que isso aconte¸ ca de novo. ´E, no entanto, pouco prov´ avel a curto prazo. No paradigma das teorias de grande unifica¸ c˜ ao , supersimetria e supercordas o problema ´ e que as predi¸ c˜ oes n˜ ao ambiguas destas teorias s´ o ocorrem a escalas que dificilmente ser˜ ao atingidas pela f´ ı sica experimental em curto prazo. A possibilidade seria uma mudan¸ ca de paradi gma. Por´ em, n˜ ao h´ a nenhuma proposta te´ orica que traga uma luz nesse se ntido. Mas, como dissemos antes, a natureza ´ e mais esperta do que n´ os. A ciˆ encia progride lentamente sem se importar com nossas pr essas e angustias. Em 1896, ou seja antes da descoberta do n´ ucleo at ˆ omico e da mecˆ anica quˆ antica, Emil Wiechert disse [DY88] A mat´ eria que supomos ser o principal componente do uni- verso ´ e formada por tijolos independentes, os ´ atomos qu´ ı micos. Nunca ser´ a demais repetir que a palavra “´ atomo” est´ a hoje em dia separada de qualquer especula¸ c˜ ao filos´ ofica antiga: s abemos 32precisamente que os ´ atomos com os quais estamos lidando n˜ a o s˜ ao em nenhum sentido os mais simples componentes conceb´ ı veis do universo. Ao contr´ ario, diversos fenˆ omenos, especial mente na ´ area da espectroscopia, levam ` a conclus˜ ao de que os ´ atom os s˜ ao estruturas bastante complexas. At´ e onde vai a ciˆ encia mod erna, devemos abandonar por completo a id´ eia de que penetrando no limiar do pequeno conseguiremos alcan¸ car as funda¸ c˜ oes fi nais do universo. Acredito que podemos abandonar essa id´ eia sem nenhum remorso. O universo ´ e infinito em todas as dire¸ c˜ oes , n˜ ao apenas acima de n´ os, na grandeza, mas tamb´ em abaixo de n´ os, na pequenhez. Se partirmos da nossa escala humana de existˆ encia e explorarmos o conte´ udo do universo al´ em e al ´ em, chegaremos finalmente, tanto no reino do pequeno quanto no reino do grande, a distˆ ancias obscuras onde primeiro nos no ssos sentidos e depois nossos conceitos nos falhar˜ ao. Depois disso fica dif´ ıcil entender as palavras do Mach acima ! Definitiva- mente os f´ ısicos hoje em dia n˜ ao pensam mais como Mach. Por e xemplo, ´ e interessante a posi¸ c˜ ao de Dyson [DY88]: ...a Natureza ´ e complexa. J´ a n˜ ao ´ e mais nossa a vis˜ ao que Einstein conservaria at´ e sua morte, de um mundo objetivo de espa¸ co, tempo e mat´ eria, independente do pensamento e da o b- serva¸ c˜ ao humanos. Einstein esperava encontrar um univer so dotado do que chamava ‘realidade objetiva’, de um universo d e picos montanhosos que ele poderia compreender por meio de um conjunto finito de equa¸ c˜ oes. A natureza, como em fim se desco - briu, vive n˜ ao nos cumes elevados, mas nos vales l´ a embaixo . Este tipo de posicionamento ainda que mais frequentes na atu alidade, n˜ ao s˜ ao majorit´ arias. Sabe-se agora que h´ a milhares de teorias de supercordas que s˜ ao mate- maticamente consistentes da mesma maneira que as duas teori as de Green e Schwarz. Esta consistˆ encia matem´ atica ´ e garantida pel a invariˆ ancia con- forme. A menos que seja mostrado que essa diversidade de teor ias s˜ ao equi- valentes, a ´ unica e importante consequˆ encia das teorias d e supercordas ´ e que as simetrias do espa¸ co-tempo e internas n˜ ao s˜ ao colocada s a m˜ ao. De fato, em 1985 j´ a se tinham reduzido a 5 as teorias de supercordas di ferentes. Logo depois, a introdu¸ c˜ ao de um novo tipo de simetria chamada de dualidade-S (o exemplo cl´ assico ´ e a dualidade dos campos el´ etrico e ma gn´ etico) reduz 33a apenas o n´ umero a 3. Mais recentemente, com a descoberta de novas dualidades as 5 de supercordas em 10 dmens˜ oes e uma teoria de campos em 11 dimens˜ oes s˜ ao consideradas apenas a manifesta¸ c˜ ao de apenas uma teoria– M, ainda que n˜ ao exista uma formula¸ c˜ ao completa desse tipo de teo- ria [BE99, DU98, HI95]. O problema ´ e que “ no one knows how to w rite down the equation of this theory” [WE99]. De qualquer forma, o limite de baixas energias tem de ser escolhido antes. Por exemplo, se f or confirmado um modelo que inclua o modelo padr˜ ao (de maneira unificada ou n˜ ao) ent˜ ao deve haver uma teoria de supercordas cujo limite se baixas en ergias seja esse modelo e n˜ ao outro. (Um das teorias de Green e Schwarz tem um l imite de baixas energias perto do modelo padr˜ ao. Mas at´ e o moment o n˜ ao foi encontrada uma teoria que reprodu¸ ca a baixas energias os qu arks e leptons conhecidos.) Mesmo que algu´ em descobrisse qual ´ e essa teo ria de supercor- das, n˜ ao saberiamos explicar porque esta teoria ´ e a que des creve o mundo real. O objetivo da f´ ısica n˜ ao ´ e apenas descrever o mundo m as explicar porque ele ´ e como ´ e [WE93a]. Devemos por tanto dar maior importˆ ancia aos detalhes. Por e xemplo, n˜ ao ´ e qualquer conjunto de equa¸ c˜ oes diferenciais parci ais que descreve o campo eletromagn´ etico. S˜ ao apenas as equa¸ c˜ oes de Maxwe ll que o fazem. Da mesma maneira n˜ ao ´ e qualquer teoria n˜ ao-Abeliana que d escreve as intera¸ c˜ oes de quarks e leptons. Em ambos casos, as equa¸ c˜ oes de Maxwell e o modelo padr˜ ao, sempre podemos estudar maneiras de genera liz´ a-los. Mas, n˜ ao ser´ a qualquer generaliza¸ c˜ ao que ser´ a seguida pela natureza. Mesmo que dispuss´ essemos de uma teoria de supercordas real ´ ıstica, que explicasse tudo o que modelo padr˜ ao deixa em aberto (as mass as dos fermions por exemplo) ainda teriamos que explicar porque essa teoria e suas asun¸ c˜ oes importantes s˜ ao escolhidas pela natureza. Em outras palav ras, ´ e mais prov´ avel que essa teoria de supercordas precisse de princ´ ıpios ainda mais profundos para ser explicada. Usualmente Einstein ´ e considerado como um dos defensores d a procura de leis unificadas da natureza. No entanto, ele via isso, pelo menos num per´ ıodo da sua vida, como um processo sem fim. Em 1917 ele escr eveu para Felix Klein [PA82] However we select from nature a complex [of phenomena] us- ing the criterion of simplicity, in no case will its theoreti cal treat- ment turn out to be forever appropriate (sufficient). Newtons ’s theory, for example, represent the gravitational field in a s eem- ingly complete way by means of the potential φ. This description 34proves to be wanting; the functions gµνtake its place. But I do not doubt that the day will come when that description, too, w ill have to yield to another one, for reason which at present we do not yet surmise. I belive that this process of deeping the the ory has no limits. Klein escreveu para Einstein nesse mesmo ano falando sobre a invariˆ ancia conforme na eletrodinˆ amica. Einstein respondeu [PA82] It does seem to me that you highly overrate the value of formal point of view. These may be valuable when an already found truth needs to be formulated in a final form, but fail almost al ways as heuristics aids. Segundo Pais [PA82] Nothing is more striking about the later Einstein than his change of position in regard to this advice, give when he was i n his late thirties. Quer dizer que segundo esta vis˜ ao inicial de Einstein, a cad eia de ex- plica¸ c˜ oes em termos de princ´ ıpios cada vez mais gerais e p rofundos, n˜ ao teria fim. Assim, uma teoria de tudo n˜ ao seria poss´ ıvel. Por outro lado, segundo Weinberg [WE93a], o fato que nossos princ´ ıpios tˆ e m-se tornado mais simples e econˆ omicos, poderia indicar que devahaber uma tal teoria. No entanto, o que queremos enfatizar aqui n˜ ao ´ e se concorda mos ou n˜ ao com as posi¸ c˜ oes do tipo das de Weinberg. Queremos ´ e coloca r se a nossa compreens˜ ao da estrutura´ ıntima da mat´ eria ser´ a, em ´ ul tima instˆ ancia, mel- horada se dedicarmos mais esfor¸ cos ` a f´ ısica de 1 TeV ou ` a d a escala de Planck (1019GeV) ou, mesmo se novas leis “macrosc´ opicas” poder˜ ao ser d e utilidade na compreens˜ ao ´ ultima do universo. Uma “teoria de tudo” n˜ ao seria eficaz com rela¸ c˜ ao ao proble ma da com- plexidade organizada existente na natureza. De fato, ´ e pos s´ ıvel que as leis que regem a complexidade e que seriam v´ alidas para qualquer sistema com- plexo, incluindo o universo, n˜ ao sejam do tipo das leis da na tureza conheci- das at´ e o momento [BA94]. Pesquisar os detalhes de “vales e montanhas”, para usar a ana logia de Deyson, pode ser mais interessante que pesquisar os picos. N o m´ ınimo vai se encontrar coisas diferentes e n˜ ao menos importantes. Pi or, n˜ ao temos escolha. Se precissamos pesquisar detalhes ou n˜ ao, depend e do desenvolvi- mento de uma determinada ciˆ encia. Os vales n˜ ao s˜ ao apenas um ponto 35de referˆ encia para medir a altitude dos picos. Eles tˆ em a su a pr´ opria di- versidade, suas pr´ oprias leis e metodologia. Suas pr´ opri as surpressas. N˜ ao devemos ter medo de que a f´ ısica, pelo momento, se torne uma b otˆ anica ou uma qu´ ımica. Depois, uma nova ordem geral vir´ a. De novo p odemos chamar a aten¸ c˜ ao aqui para o tectˆ onica de placas. Por muit os s´ eculos os estudos da Terra eram “chatos”. Procurava-se classificar as rochas! mas sem essa fase aborrecida n˜ ao teriamos a s´ ıntese atual. Apenas agindo podemos ver o que realmente acontecer´ a. Deve mospar- ticipar do processo cient´ ıfico. ´E por isso que a decis˜ ao pessoal mencionada acima ´ e importante. N˜ ao ´ e apenas uma quest˜ ao de opini˜ ao . Segundo a de- cis˜ ao tomada seguiremos um ou outro caminho na nossa pesqui sa e segundo esse caminho poderemos ser melhor ou pior sucedidos. Melhor , se acredi- tamos que o processo cient´ ıfico, longe de acabar, est´ a apen as come¸ cando devemos tornar-nos participes dele o quanto antes. Se ainda ´ e o sonho de alguns f´ ısicos a formula¸ c˜ ao de uma te oria que unifique todas as intera¸ c˜ oes conhecidas [GE89a], devemos ter sempre pre- sente que uma teoria de tudo ´ e em princ´ ıpio imposs´ ıvel. To da teoria tem sua componente fenomenologico, aspectos que n˜ ao podem ser calcul´ aveis usando os conceitos da mesma teoria. Todas as teorias s˜ ao e s er˜ ao aproxi- madas, estaremos sempre numa “unended quest”. E isso ´ e empo lgante. Nos ´ ultimos anos a nossa compreens˜ ao da teoria quˆ antica d e campos mudou consideravelmente. A descri¸ c˜ ao das part´ ıculas em teoria quˆ antica de campos depende da energia na qual estudamos essas intera¸ c˜ oes. Assim todas as teorias podem ser consideradas como teorias efetivas , levando em conta apenas as part´ ıculas relevantes na escala de energia considerada. Este ´ e uma realiza¸ c˜ ao do fato que podemos estudar fenˆ ome nos ou pro- cessos f´ ısicos apenas num intervalo limitado de energia. A parecem infinitos pela exigˆ encia de localidade que significa que a cria¸ c˜ ao e/ou aniquila¸ c˜ ao de part´ ıculas ocorre num ponto do espa¸ co-tempo. O processo d erenormal- iza¸ c˜ ao foi interpretado at´ e recentemente como uma maneira de abso rver os infinitos nos parˆ ametros f´ ısicos. Isto ´ e, introduzind o um cut-off Λ e modificando a teoria para distˆ ancias menores que Λ−1, aparecem apenas quantidades pass´ ıveis de serem medidas experimentalment e como a massa e a carga el´ etrica. Finalmente faz-se Λ → ∞. A teoria fica independente do cut-off . Segundo Dirac, quem nunca aceitou o processo de renormaliz a¸ c˜ ao, a f´ ısica te´ orica tomou um pista errada com esse desenvolvim ento. Hoje em dia essa vis˜ ao (de Dirac) ´ e considerada muito restrita. Afinal a eletrodinˆ amica quˆ antica ´ e apenas uma parte do modelo eletrofraco que cert amente ´ e parte de uma teoria mais abrangente. O problemas dos infinitos ser´ a resolvido 36quando tivermos uma teoria final (se ´ e que ela existe). Este c aso ´ e interes- sante para refletir. As vezes os problemas apenas existem por que estamos supervalorizando nossos objetos de estudo. Por exemplo, Ke pler propˆ os um modelo das ´ orbitas dos planetas baseado em simetrias. Agor a sabemos que as simetrias fundamentais n˜ ao aparecem nesse tipo de siste mas. Assim a proposta de Kepler era interessante mais aplicada no proble ma errado. Os planetas n˜ ao s˜ ao mais objetos fundamentais para a formula ¸ c˜ ao de teorias f´ ısicas b´ asicas. Esta discus˜ ao parece vanal mas n˜ ao ´ e. Muitas escolhas de estudantes ou mesmo de pesquisadores ser˜ ao feitas segundo o aspecto que valorizem na pesquisa. Isto ´ e, o que seja considerado um pro blema impor- tante. Se algu´ em concorda com Dirac que “a eletrodinˆ amica quˆ antica atual n˜ ao corresponde ao elevado padr˜ ao de beleza matem´ atica q ue seria de es- perar de uma teoria f´ ısica fundamental” e tentar modificar apenas a QED poder´ a encontrar problemas insoluv´ eis mesmo para mentes bem preparadas e privilegiadas. As teorias de grande unifica¸ c˜ ao tentavam esclarecer melho r o modelo padr˜ ao. Por exemplo dar resposta ao problema da quantiza¸ c ˜ ao da carga e tentar calcular o ˆ angulo de mistura eletrofraco (sin θW). Certamente n˜ ao foram propostas como continua¸ c˜ ao das ideias de Einstein [ GE89a]. Segundo Georgi Einstein’s attempts at unification were rearguard action wh ich ignored the real physics of quantum mechanical interaction s be- tween particles in the name of philosophical and mathematic al elegance. Unfortunately, it seems to me that many of my col- leages are repeating the Eintein’s mistake. The progress of the fields is determined, in the long run, by the progress of experimental physics. Theorists are, after all, pa- rasites. Without our experimental friends to do the real wor k, we might as well be mathematicians or philosophers. When the science is healthy, theoretical and experimental particle physics track along together, each reforcing the other. But there ar e of- ten short period during which one or other aspect of the field gets away ahead. Then theorists tend to lose contact with re- ality ...During such periods without experiments to exited them, theorists tend to relax back into their grounds states, each doing whatever come most naturally. As a result, since different th e- orists have different skills, the field tends to fragment into little subfields. Finally, when the crucial ideas or the crucial exp eri- 37ments come along and the field regains its vitality, most theo rists find that they have been doing irrelevants things...But the w on- derful thing about physics is that good theorists don’t keep doing irrelevant things after experiment has spoken. The useless sub- fields are pruned away and everyone does more or less the same thing for a while, until the next boring period. Segundo Heisenberg [HE93], em f´ ısica te´ orica podem-se, i ) formular teo- rias fenomenol´ ogicas, ii) esquemas matem´ aticos rigoros os, ou iii) usar a filo- sofia como guia. No primeiro tipo ficam as pesquisas de Heisenb erg e o grupo de Sommerfeld (Pauli, Land´ e, H¨ oln e outros). Eles inventa vam f´ ormulas que reproduzissem os experimentos. No entanto, mesmo quando be m sucedidas essa teorias fenomenol´ ogicas n˜ ao fornecem nenhuma infor ma¸ c˜ ao real sobre o conte´ udo f´ ısico do fenˆ omeno [HE93]. As vezes os c´ alculo s feitos com ambos esquemas coincidem. Isto n˜ ao ´ e de se estranhar. A equivalˆ encia pode ser matem´ atica mas n˜ ao f´ ısica. No entanto, as vezes resultados rigorosos tamb´ em levam a re sultados em discordˆ ancia do observado. Isto ´ e, eles tˆ em tamb´ em li mita¸ c˜ oes. A tese de doutorado de Heisenberg versou sobre o c´ alculo da estabi lidade de um fluxo entre duas paredes fixas. O resultado foi que para um cert o n´ umero de Reynolds o fluxo torna-se inst´ avel e turbulento. Um ano depo is E. Noether mostrou rigorosamente que o problema de Heisenberg n˜ ao tin ha solu¸ c˜ ao: o fluxo devia ser est´ avel em toda parte. Outro exemplo mais rec ente sobre as limita¸ c˜ oes das demostra¸ c˜ oes gerais ´ e o do mecanismo de Higgs, que nada mais ´ e do que uma evas˜ ao do teorema de Goldstone. Este fora r ecebido com descren¸ ca por muitos te´ oricos pois o teorema de Goldst one tinha sido mostrado rigorosamente pelos axiom´ aticos. Antes do semin ario que daria em Princeton, Higgs conta que segundo Klaus Hepp ... what I going to say must be nonsense because axiomatic field theorist had proved the Goldstone theorem rigorously u sing the methos of C∗-algebras. However, I survive questions from Arthur Wightman and others, so I conclude that perhaps the C∗ algebraists should look again [HI91]. Sem coment´ arios. Heisenberg disse que nunca se soube onde estava o erro na demo stra¸ c˜ ao de Noether mas em 1944 (vinte anos depois) Dryden e colaborad ores fizeram experiˆ encias precissas do fluxo laminar entre duas paredes e da transi¸ c˜ ao para a turbulˆ encia e descobriram que os c´ alculos de Heisen beg estavam em 38concordˆ ancia com a experiˆ encia. Lin do MIT simulou a exper iˆ encia (von Neumann sugeriu usar computadores) e confirmou de novo os res ultados de Heisenberg. Poderiamos colocar outros exemplos, mas esses s˜ ao suficientes para mostrar as limita¸ c˜ oes dos m´ etodos matem´ aticos rig urosos. Devemos lembrar, para fazer justi¸ ca que com os m´ etodos fenomenol´ ogicos somos obri- gados a usar sempre os velhos conceitos mesmo para uma situa¸ c˜ ao nova. Bom, isto quer dizer que qualquer que seja o m´ etodo usado na p esquisa te´ orica apenas quando verificado experimentalmente8podemos dizer que a teoria funcionou. Tamb´ em devemos enfatizar que o passo dec isivo ´ e sempre discont´ ınuo. Isso aconteceu, por exemplo, com a mecˆ anica quˆ antica [HE93]. A filosofia como guia da pesquisa te´ orica teve seu apogeu com o posi- tivismo. Mach, e depois o circulo de Viena (recentemente te´ oricos como Chew), insistiram que uma teoria devia ser formulada em term os de quan- tidades observ´ aveis. Mach, e muitos dos seus contempor´ an eos, acreditava que os ´ atomos eram apenas uma quest˜ ao de conveniˆ encia, n˜ ao creditavam na existˆ encia real deles. Einstein fora influenciado por es ta vis˜ ao no come¸ co da sua carreira mas logo mudou de id´ eia. Ele diria que [HE93] ... a posibilidade que se tem de observar ou n˜ ao uma coisa depende da teoria que se usa. ´E a teoria que decide o que pode ou n˜ ao ser observado . 11 Conclus˜ oes Pode a comunidade cient´ ıfica errar o rumo? ´E uma quest˜ ao delicada. Uma teoria ´ e aceita pela comunidade dependendo de v´ arios fato res como: sua exatid˜ ao nas predi¸ c˜ oes, seu contexto e o grau em que est´ a determinada pela experiˆ encia. Por´ em, a curto prazo, existem outros fatore s que n˜ ao ´ e f´ acil de reconhecer como sendo esp´ urios: modismo, ideologia e estu pidez generali- zada. As vezes o prazo n˜ ao ´ e t˜ ao curto assim: o modelo solar de Aristarco passou despercevido pelos astronˆ omos ao longo de 17 s´ ecul os e por quase 200 anos a teoria da via L´ actea de Kant tampouco foi popular n os meios acadˆ emicos. Recentemente alguns aspecto da mecˆ anica quˆ antica est˜ ao sendo esclarecidos, as conclus˜ oes de N. Bohr seriam corretas mas pelos argumentos errados? [DU99]. 8Aqui essa “verifica¸ c˜ ao ” ´ e entendida de maneira ampla. Pod e ser apenas indireta, por exemplo, uma consistˆ encia global da teoria com os dados experimentais. Isto ´ e o que ocorre com o modelo padr˜ ao da f´ ısica das part´ ıculas eleme ntares. 39Deve-se insistir com os estudantes que um aspecto que import a (certa- mente n˜ ao o ´ unico) no que-fazer cient´ ıfico ´ e a “emo¸ c˜ ao” , qualquer coisa que isso signifique. Segundo Kadanoff [RU93] ´E uma experiˆ encia como nenhuma outra que eu possa des- crever; a melhor coisa que pode acontecer a um cientista, com - preender que alguma coisa que ocorreu em sua mente correspon de exatamente a alguma coisa que acontece na natureza. ´E sur- preendente, todas as vezes que ocorre. Ficamos espantados c om o fato de que um construto de nossa pr´ opria mente possa real- mente materializar-se no mundo real que existe l´ a fora. Um grande choque e uma alegria muito grande. ou, de maneira mais dramatica nas palavras de Einstein The years of anxious searching in the dark, with their intens e longing, their alternations of confidence and exhaustion an d the final emegence into the light—only those who have experience d it can understand it. Para sentir essa emo¸ c˜ oes n˜ ao precissamos obter resultad os t˜ ao importantes quanto os de Kadanoff e Einstein! Apenas devem ser resultados nossos . Nos Estados Unidos os estudantes est˜ ao deixando a academia para tra- balhar na empressa privada. Isso n˜ ao seria problema se entr e eles, segundo Anderson [AN99], n˜ ao estivessem os melhores. Os menos cria tivos ficam nas posi¸ c˜ oes permanentes em f´ ısica. Segundo Anderson a N ational Science Foundation (NSF) e outras agˆ encias de fomento est˜ ao incen tivando a falta de criatividade, talvez, influenciados pelo “Horganism”.9As causas disso reside pelo menos em parte tamb´ em no sistema de “peer-revie w” mas este ´ e um aspecto que n˜ ao vai ser discutido neste artigo, serve ape nas como uma confirma¸ c˜ ao de que a vis˜ ao pessoal que temos sobre o que-fa zer cient´ ıfico e o futuro de ciˆ encia tˆ em implica¸ c˜ oes no desenvolvimento da pr´ opria ciˆ encia. Uma carater´ ıstica do nosso tempo ´ e a pressa. N˜ ao apenas na ciˆ encia. Umberto Eco trata do problema da rapidez [EC94] Quando enalteceu a rapidez, Calvino preveniu: ‘N˜ ao quero dizer que a rapidez ´ e um valor em si. O tempo narrativo pode se r lento, c´ ıclico ou im´ ovel...Esta apologia da rapidez n˜ ao pretende negar os prazeres da demora. Se algo importante ou absorvent e est´ a ocorrendo, temos de cultivar a arte da demora. 9A cren¸ ca que o fim da ciˆ encia est´ a pr´ oximo e, o que fica ´ e ape nas per´ ıodos de “ciˆ encia normal” segundo a vis˜ ao de Kunh [KU62]. 40Isto ´ e v´ alido n˜ ao s´ o na fic¸ c˜ ao mas tamb´ em na ciˆ encia. Como podemos ser pesimistas se nos ´ ultimos anos a f´ ısica fo i capaz de i) encontrar algumas leis da natureza novas, ii) obter novos co mportamentos da natureza, iii) desenvolver instrumentos que permitiram ob servar fenˆ omenos em condi¸ c˜ oes completamente diferentes das estudadas no p asado? [WE91]. As experiˆ encias em Stanford no fim dos anos 60, e que desvenda ram a es- trutura do n´ ucleob, n˜ ao foram meras repeti¸ c˜ oes da exper iˆ encia de Geiger- Marsden. O conceito de visualiza¸ c˜ ao dos fenˆ omenos tinha mudado. De fato novas maneiras de estudar (“ver”) a natureza s˜ ao t˜ ao impor tantes quanto as leis fundamentais que surgir˜ ao desses estudos. Chegamos ao fim?...que direito temos de supor que os nucle- ons, el´ etrons e neutrinos s˜ ao realmente elementares e n˜ a o po- dem ser subdivididos em pares constituintes ainda menores? H´ a apenas meio s´ eculo, n˜ ao se supunha que os ´ atomos eram indi - vis´ ıveis?... embora seja imposs´ ıvel prever o desenvolvi mento fu- turo da ciˆ encia da mat´ eria, temos atualmente raz˜ oes para acredi- tar que nossas part´ ıculas elementares s˜ ao na verdade as un idades b´ asicas e n˜ ao podem ser novamente subdivididas...parece , assim, que chegamos ao fim de nossa pergunta dos elementos b´ asicos que formam a mat´ eria. Estas palavras foram escritas por George Gamow em 1960 [GA62 ].´E in- teressante que elas continuem sendo, em parte verdadeiras. Os nucleons, el´ etrons e neutrinos continuam a ser indivis´ ıveis no sent ido direto. Como podia imaginar Gamow que os nucleons seriam divis´ ıveis “em certo sentido”? Somos capazes de estudar a estrutura dos nucleons mas seus co nstituentes est˜ ao, aparentemente, confinados! Segundo Weinberg, ... it is foolhardy to assume that one knows even the terms in which a future final theory will be formulated. ´E dif´ ıcil usar argumentos gerais sobre a utopia que serviri a de guia para o avan¸ co da f´ ısica de altas energias. Segundo Bohr [HE93] Quando se tem uma formula¸ c˜ ao correta, o oposto dela ´ e, e- videntemente uma formula¸ c˜ ao errada. Mas quando se tem uma verdade profunda, ent˜ ao seu oposto pode ser igualmente uma ver- dade profunda . 41Assim, assumir que devem existir leis gerais com as quais pos sam ser descri- tas, pelo menos em princ´ ıpio, todas as coisas parece uma ver dade profunda. O seu oposto, que no fundo n˜ ao existem leis fundamentais tam b´ em o ´ e (como arg¨ uem Wheeler e Nielsen [WE93a]). Desde 1859 sabia-se que havia um problema com a ´ orbita de mer curio se interpretada dentro da teoria da gravita¸ c˜ ao de Newton. Este problema, como ´ e bem conhecido, foi resolvido pela teoria da relativi dade geral de Einstein. Mas, em 1916 al´ em dessa discrepˆ ancia haviam tam b´ em mais duas. Uma referia-se a anomalias relativas aos movimentos dos com etas Halley e Encke. A outra era a respeito do movimento da lua [WE93a]. Em t odos estes casos, como no da ´ orbita de mercurio, os movimentos n˜ ao concordavam com as previs˜ oes da teoria de Newton. Agora, no entanto, sab e-se que as anomalias nos movimentos dos cometas s˜ ao devidas ` a press˜ ao de escape dos gases ja que o cometa ´ e esquentado quando passa perto do sol. O movimento da lua foi melhor comprendido quando se levou em conta o seu ta manho que implica em complicadas for¸ cas tidais. Assim, segundo Wein berg [WE93a] ...there is nothing in any single disagreement between theo ry and experiment that stands up and waves a flag and says “ I am an important anomaly” . Assim, n˜ ao sabemos em geral quando estamos lidando com um ve rdadeiro sinal de f´ ısica nova. Usualmente a maneira de fazer f´ ısica de Dirac ´ e considerad a com a maneira matem´ atica. Mas ele tinha uma posi¸ c˜ ao mais ampla e um con- hecimento das limita¸ c˜ oes dessa maneira de trabalhar [HE9 3] Em qualquer parte da f´ ısica em que se saiba muito pouco, so- mos obrigados a nos prender ` a base experimental, sob pena de mergulharmos em especula¸ c˜ oes estravagantes, que quase c erta- mente estar˜ ao erradas. N˜ ao desejo condenar completament e a especula¸ c˜ ao. Ela pode ser divertida e indiretamente ´ uti l, mesmo que acabe por se mostrar errada...mas ´ e precisso tomar cuid ado para n˜ ao se deixar envolver demais por ela. Onde fica a intui¸ c˜ ao em tudo isto? quais os limites do m´ etod o cient´ ıfico? Vale a pena se preocupar com isto? as respostas s˜ ao pessoais . Um exemplo da importˆ ancia da emo¸ c˜ ao ´ e expressa por Thomas Mann quan do escreveu Astronomy—a great science—teaches us to consider the earth as a comparison of an insignificant star in the giant cosmic 42turnoil, roving about at the the periphery of our galaxy. Thi s is, no doubt, correct. But I doubt that such correctness reve als the whole truth. In the depth of my soul I belive—that this ear th has a central significance in the universe. In the depth of my soul I entertain the presumption that the act of creation whi ch called forth the inorganic world, from nothingness, and the pro- creation of life from the inorganic world, was aimed at human ity. A greart experiment was initiated, whose failure by human ir re- sponsability would mean the failure of the act of creation it self, its very refutation. May be it is so, mat be it is not. It would b e good if humanity behaved as if it were. As vezes os artistas enxergam mais longe que os cientistas. U m deles ja disse, s´ eculos atr´ as There are more things in Heaven and Earth. Horatio. Than are dreamt of in your philosophy. Precissamos convencer os estudantes que existem (e que semp re exis- tir˜ ao) muitas coisas a serem descobertas, talvez virando ` a esquina. Con- vencˆ e-los que o progresso cient´ ıfico e tecnol´ ogico foi ob tido lentamente, e por vezes de maneira ca´ otica, e que n˜ ao existe uma raz˜ ao pa ra que surpres- sas n˜ ao ocorram de novo. Que a pressa n˜ ao serve para queimar etapas. Que previs˜ oes s˜ ao dif´ ıceis de se fazer. N˜ ao apenas para n´ os mas que tamb´ em era dif´ ıcil para von Neumann. Precisamos colocar em discus˜ ao a maneira como se processa o desenvolvimento das ideias cient´ ıfica, seus a casos, atrasos e acelera¸ c˜ oes devido a preconceitos que n˜ ao fazem parte do m´ etodo cient´ ıfico, mas est˜ ao sempre presente para bem ou para mal. Isso implica a valoriza¸ c˜ ao da perpectiva hist´ orica no ensino de ciˆ encias. 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arXiv:physics/9912038v1 [physics.bio-ph] 17 Dec 1999A Model of Convergent Extension in Animal Morphogenesis Mark Zajac∗and Gerald L. Jones University of Notre Dame, Department of Physics Notre Dame, IN 46556 February 20, 2014 Abstract In this paper we argue that the pattern of cell movements in th e mor- phogenetic process known as convergent extension can be und erstood as a energy minimization process, provided the cell-cell adhes ive energy has a certain kind of anisotropy. This single simple property is s ufficient cause for the type of cell elongation, alignment, and intercalati on of a cellular array that is the characteristic of convergent extension. W e describe the type of anisotropy required. We show that the final aspect rat io of the array of cells is independent of the initial configuration an d of the degree of cell elongation. We find how it depends on the anisotropy. In the development of the animal embryo great changes of form (morpho- genesis) take place [1]. This is certainly true during gastr ulation - a period of embryonic development during which axial structures are fo rmed by extensive cell rearrangement. During these rearrangements groups of cells move coher- ently over distances very large compared to cell dimensions . This process has been extensively investigated by experiments, particular ly on embryos of the frog,Xenopus laevis , and particularly by R.E. Keller and his collaborators (see [2] for a brief review and extensive references). One characteristic and widespread type of rearrangement ha s been termed “convergent extension” and occurs, for example, in the deve lopment of axial structure such as precursors to the vertebrate spinal colum n. Here an active group of cells undergoes a threefold process. The individua l cells, originally roughly isodiametric (Fig. 1a), elongate and their axes of elongation become aligned . If these were the only motions the final configuration would b e as in Fig. 1b. But at the same time, though on a somewhat slower ti me scale, the cells intercalate between each other. The intercalation is in the direction of alignment so that the number of cells in that direction dec reases while the number of cells in directions perpendicular to the alignmen t increases producing ∗mzajac@krypton.helios.nd.edu 1ab c Figure 1: Intercalation. Isodiametric cells (a) become elo ngated and aligned (b)while simultaneously intercalating (c) so that an array of cells extends at right angles to the direction of cell motion. a final configuration as in Fig. 1c. The elongation process ten ds to increase the overall length of the group of cells in the direction of align ment and tends to decrease the length in orthogonal directions (since the v olume stays roughly constant). The process of intercalation does just the rever se but is the dominant effect so that the axis of net extension of the group of cells is at right angles to the axis of individual cell elongation . In this paper we argue that certain important aspects of convergent extension can be understoo d as a tendency of the active cells to minimize their total energy, provided th at one assumes that they interact with a non-uniform surface (adhesive) energy satisfying certain conditions which we explicate. We also develop a mean field th eory of this process. Minimum energy principles have been used to explain cell rea rrangement since Steinberg’s [3] suggestion that differential cell adh esion plus cell motility can account for cell sorting patterns in mixtures of two or mo re cell types (see [4] for a review and extensive references to the literature) . Goel and Lieth [5] have considered cell sorting for a simple geometrical model in the presence of anisotropic surface adhesion between cells of fixed shape. C ell sorting, driven by energy minimization, has also been the subject of many compu ter simulations [6, 7]. Drasdo, Kree, and McCaskill [8] have done simulation s with anisotropic surface adhesion. Anisotropic surface adhesion has not, so far, been used to explain the convergent extension of a homogeneous group of c ells. We do not model here the dynamics of convergent extension. We assume, as in [3] and [5], that cell motility will allow the system to explore its possi ble configurations and that, as a strongly dissipative system, it will evolve towar ds the configuration of minimum energy. In the embryo convergent extension usually takes place in an asymmetric environment where the inactive cells bounding the active re gion are not the same on all sides of that region. Thus one can argue that the ex tension, and its orientation, may be determined by the interactions at th e boundaries which “channel” the active cells, rather than being an intrinsic c ollective property of the group of active cells. Under these experimental circums tances there is little doubt that the boundaries strongly influence active cell mov ements. Indeed, in the physical model of Weliky, et.al. [9] the extension is driven by the observation 2that active cells behave differently at the boundaries paral lel to the elongation from those at the boundaries perpendicular to the elongatio n. A subsequent and elegant experiment by Shih and Keller [10] however stron gly suggests that, in addition, the active cells have a strong intrinsic collec tive mechanism driving their convergent extension. In these experiments a layer (essentially a monolayer) of ac tive cells was excised from a frog embryo, at a stage before convergent exte nsion had begun, and cultured on a uniform surface in a medium which allowed th eir normal development. Subsequently the layer showed strong converg ent extension in the plane of the substrate - and this in the absence any plausible lateral anisotropy either in the substrate or in the culture medium. This behavi or thus appears to be an example of “broken symmetry” so well known in condens ed matter physics, and asks for an explanation based on collective beh avior induced by cell-cell interactions. To explain this behavior as an energy minimization process w e assume that cell-cell interactions take place through surface adhesio n, which can be charac- terized by an energy per unit contact area. We assume that the cell rearrange- ments take place with negligible cell division and little ch ange in cell volume, as is observed in the later stages of the above experiment. Ther e seems to be no clear understanding in the literature of the trigger for the cell elongation which initiates the convergent extension, and our model does not p rovide this. Our main assumption is that the adhesive energy of the contact su rface between two cells will depend on how that surface is oriented relative to the axes of elongation of the two cells. This would be the case, for example, were the surface density of adhesive binding sites to be different on the long side of a c ell (parallel to the axis of elongation) from that on the short sides (perpendicu lar to the axis of elongation). We can find in the literature no compelling evid ence either for or against this assumption. We argue here that a specific type of this assumption is a sufficient cause of the elongation, alignment, and interc alation resulting in convergent extension. We give here a two dimensional version of our proposal since t he convergent extension takes place in the plane of the substrate and the he ight of the cells does not seem to play an important role. Hence we consider a co llection of two dimensional cells of (nearly) the same fixed area. We first want to find the conditions which favor alignment. We assume a compact ar ray of cells, so large that array surface effects can (for this argument) be ne glected. Fig. 2a is a cartoon of a few elongated cells in such a large ordered ar ray of cells and Fig. 2b is for a disordered array. Suppose that we can roughly distinguish, for each cell, two long sides (parallel to the axis of elongation ) and two short sides (perpendicular). It is clear in Fig. 2a that in the ordered ar ray the cell-cell contact surfaces are, for the most part, either roughly para llel to the common axes of alignment or roughly perpendicular to that axis. We s hall term these as long-long ( ll) or short-short ( ss) contacts since they occur, primarily, at contacts between a pair of long sides or a pair of short sides. In the dis ordered array of Fig. 2b there are many contact surfaces that make intermedia te angles with the now different axes of adjacent cells. We term these long-shor t (ls) contacts since 3a b Figure 2: Cell alignment. For an ordered array (a) most cell a ttachments are either end to end or side to side while a disordered array (b) e xhibits significant binding between poles and lateral surfaces. they tend to occur when the contact surfaces are between a lon g side of one cell and a short side of a neighbor. If the energy density (per unit length) of the ls contacts is enough larger than those of llandsscontacts then the ordered array will have the lower energy per cell (we assume that the array i s large enough that we can neglect the effect of the array boundaries on the bulk or dering). More quantitatively, let landsbe the average long and short side lengths of each cell, which, for the moment, we take as fixed. Suppose that all cell-cell contacts can be characterized as ll,ss, orlsand that the total length of each type in the array is Lll,Lss, and Lls. In a large array of Ncells we have 2 Nl= 2Lll+Lls and 2 Ns= 2Lss+Lls(again neglecting array boundaries where cells do not contact other cells). Since N,landsare fixed these equations provide two constraints between the three contact lengths. We assume th at three energy densities ( Jll,Jss, and Jls) are adequate to characterize the interactions at the various surfaces. Then the bulk energy of an array due to cell -cell interactions is E=LllJll+LssJss+LlsJls (1) = (2 Nl−Lls)Jll/2 + (2 Ns−Lls)Jss/2 +LlsJls = (Jls−Jll/2−Jss/2)Lls+N(lJll+sJss). This energy is an increasing function of Llsif the ordering condition γls=Jls−(Jll+Jss)/2>0 (2) is satisfied. In this event ordered arrays ( Lls= 0) will have lower bulk energies than disordered ( Lls>0) arrays. Note that condition (2) is just that the ls surface tension γlsbe positive. The above argument is exact if the cells are assumed (unreali stically) to be identical rectangles arranged in arbitrary tesselations o f the plane and is similar to that used in [5] in the cell sorting problem. For realistic cells it is a crude but plausible representation of the assumed anisotropy of t he surface adhesion. It is interesting to consider the case where the adhesive ene rgy density of a two cell contact is the product of a factor from each cell. Thi s might be so, for 4Figure 3: Anisotropic binding. Adhesive energy at the point of contact between cells is assumed to depend on ( ˆn·ˆa)2where ˆn is the local unit normal while ˆa gives alignment, assumed com- mon to all cells. Figure 4: Forfeited bonds. At an inter- face with uniformly inert surroundings, missing adhesive energy will vary with the orientation of the surface cells, rel- ative to the boundary. example, if the variation in adhesive energy were caused by a variation in the density of binding sites on the cell surface. If we make the na tural assumption that the density of adhesive bonds is proportional to the pro duct of the density of binding sites on the cell surfaces in contact, then we woul d have in the above model Jll=−jljl,Jss=−jsjs, and Jls=−jljs, where the sign is chosen make allJ <0 when all j >0. It is easy to show that this choice satisfies the ordering condition Eq. (1) whenever jlandjsare positive and are not equal. In addition to Eq. (1) let us suppose that the llenergy density is lower than thessenergy density. Jll< Jss(orjl> js). (3) Now the energy Eq. (1) of the array can be reduced by increasin g the cell long side lengths land decreasing the short side length scausing, or at least favoring, elongation of the cells. At equilibrium these surface effect s will presumably be balanced by internal cellular forces opposing further elon gation. We can also argue that Eq. (3) will produce intercalation in t he direction of elongation. To see this we consider the effect of the boundary on a finite array of N cells. Suppose that there is no adhesive energy between t he boundary cells and the culture medium. Then the expression Eq. (1) und erestimates the array energy because it assumes all cell surfaces are in c ontact with other cell surfaces and so overestimates the contact lengths LllandLss. From Eq. (1) we should subtract the (negative) adhesive energy that i s not present at the contacts between the boundary cells and the surrounding medium. Fig. 1 shows arrays of twelve elongated cells. In Fig. 1c the array e xtension is at right angles to the cell elongation and in Fig. 1b it is along the cel l elongation. It 5is clear that in 1c the boundary contacts are primarily throu gh short cell sides whereas in Fig. 1b they are primarily though long cell sides. Since the long sides have lower (more negative) energy than the short, the e nergy (corrected for boundaries) of the configuration shown in Fig. 1b is highe r than that of Fig. 1c. Thus if we start with any compact initial array of unelong ated cells we expect cell motility and energy minimization to produce con figuration of type 1c by cell elongation, alignment, and intercalation parall el to the alignment. In order for these processes to produce net extension in the dir ection perpendicular to alignment the effects of intercalation must dominate thos e of elongation. In the case of a rectangular array of a large number of rectangul ar cells one can show that this will be the case independent of the degree of el ongation. One can also show that the ratio of the array dimensions in the direct ions perpendicular and parallel to the elongation is just Jll/Jss. We shall derive these results more generally below. The above arguments concerning surface effects can be made so mewhat more realistic and quantitative by the following mean field type o f modeling. We assume that we have a large array of Nelongated and aligned cells. The total energy of the array is the bulk energy due to cell-cell intera ctions plus the surface correction for the absence of cells outside the boundary. Th e bulk energy is proportional to N, or equivalently, the array area A, so we write it as λA, where λis the (negative) bulk energy per unit area in the aligned arr ay. To model the anisotropic cell-cell interaction we assume that Jdepends on the angle between the direction of alignment, specified by the unit vector ˆa, and the unit vector ˆn normal to the contact segment between the cells (see Fig. 3). More explicitly, we assume that J(ˆn·ˆa) is negative, an even function (since ˆa,−ˆaandˆn,−ˆn specify the same physical situations), and is minimum at ˆn·ˆa= 0 (so that ll interactions have the lowest energy). Figure 4 shows part of a finite array of vertically aligned cells and their boundary with an externa l medium with which we assume they have no adhesive energy. To get the energy of th e finite array we must subtract from the bulk energy half the energy the boun dary cells would have had with cells external to the array had the boundary bee n absent. Half, since adhesive energy is shared between two cells. So E=λA−1 2/contintegraldisplay J(ˆn·ˆa)dl (4) where the integral is taken around a closed boundary. We want to minimize this over all closed boundaries enclosing the same area A. Alternatively we can interpret λas a Lagrange multiplier and find the extrema of 4 over all clos ed curves at fixed λ. To do this we assume the curves are parameterized as r(u) with 0 ≤u≤1, and r(0) = r(1). Then, since d l= ( ˙x2+ ˙y2)1/2du(where ˙r=dr/du), while ( ˆn·ˆa) = (ay˙x−ax˙y)/( ˙x2+ ˙y2)1/2andA=/integraltext1 0y˙xduwe can write the energy as/integraltext1 0L(r,˙r)duwith L(r,˙r) =λy˙x−J(ˆn·ˆa)( ˙x2+ ˙y2)1/2/2. The extremal curves are solutions of the usual Euler-Lagran ge equations for L and are degenerate with respect to translations in the x-yplane. This gives rise to two first integrals and two constants of integration (whic h we choose to be 6zero), which fix the position of the extremal curve. The integ rated equations have the form 2λr=ˆaJ′(ˆn·ˆa) +ˆn[J(ˆn·ˆa)−(ˆn·ˆa)J′(ˆn·ˆa)], (5) where J′is the derivative of J. These are two coupled first order differential equations whose solutions depend on the particular choice o f the function J. We have not been able to find complete analytic solutions for any interesting choice ofJbut some properties of the solutions can be found. First we no te that for ˆa= 0, or equivalently J= constant, the solution is a circle of radius J/(2λ). Secondly, the turning points of any solution curve are where d(r·r)/du= 0. Nowd du(r·r) = 2λ(˙r·r) = (˙r·ˆa)J′(ˆn·ˆa) (6) where we have used (5) and that ˙r·ˆn= 0 for any curve. From (6) we see that there are two types of turning points. 1) At ˙r·ˆa= 0, that is, where the boundary is perpendicular to the alignment so that ˆn=±ˆa. For any simple closed curve this condition will be satisfied at two points on the curve. Be cause Jis even andJ′is odd we have from (5) that these two points lie at ±J(1)/(2λ) on the line through the origin and parallel to ˆa. 2) At ˆn·ˆa= 0 where J′= 0 and the boundary is parallel to the alignment. For these (5) shows th at 2λr=ˆnJ(0), thus there are turning points at ±J(0)/(2λ) along a line through the origin and perpendicular to ˆ a. If we let D⊥andD/bardblbe the distances between the turning points aligned respectively perpendicular and parallel to then the aspect ratio of the boundary is D⊥/D/bardbl=J(0)/J(1). (7) If|J(0)|>|J(1)|, then the elongation is in the direction perpendicular to th e alignment as is observed in convergent extension. From Fig. 3 we see that J(0) corresponds to our previous Jllwhile J(1) corresponds to Jss. We have also studied the minimization of the energy function al Eq. (4) numerically for the case where Jis chosen to be a gaussian function. We ap- proximate the boundary curve by a polygon of at least 100 side s and use an iterative process that moves down the energy gradient at con stant area. We have started from many initial configurations, all of which a re simple closed polygons. The final boundary curve is always the same and with the correct aspect ratio Eq. (7). This could also be viewed as a model for t he dynamics of convergent extension. Indeed, with the addition of additiv e random forces, the method would be essentially a Langevin dynamics for the evol ution of a highly dissipative system. In conclusion, we have argued that convergent extension can be understood as a energy minimization process, provided the cell-cell ad hesive energy has a certain kind of anisotropy. This single simple property is s ufficient cause for the cell extension, alignment, and intercalation in the dir ection of alignment, that are the characteristics of convergent extension. We ha ve characterized the anisotropy required [Eq. (2) and Eq. (3)]. We have shown that the final aspect 7ratio is independent of the initial configuration and have sh own how it depends on the anisotropy Eq. (7). We believe our arguments are plausible but realize that they are not conclu- sive. Our modeling neglects many degrees of freedom associa ted with cell shape and arrangement, which we think, but cannot prove, are not cr ucial. Our pro- cedure of separately minimizing the bulk and surface energi es is accurate only for a large array of cells. We do not see much possibility of do ing a lot better by purely analytic methods. We have initiated simulations o f convergent exten- sion, using the Potts model and Metropolis dynamics methods of references [6] and [7], with anisotropic adhesive energies of the type desc ribed in this paper. The use of anisotropic adhesive energies introduces techni cal difficulties in that the energy becomes non-local on the scale of the size of a cell , which consider- ably increases the simulation time. Nevertheless we believ e the simulations will eventually substantiate our conclusions. Even so, the more difficult question of whether this is the correct explanation of convergent exten sion remains. Experi- ments that probe the possible anisotropy of cell adhesive en ergy would be useful, as would experiments that show the final configuration is larg ely independent of the initial configuration. References [1] L. Wolpert et al.,Principles of Development (Oxford University Press, New York, 1998). [2] R. Keller and J. Shih, in Interplay of Genetic and Physical Processes in the Development of Biological Form at the Frontier of Physics an d Biology ,Les Houches , edited by D. Beysens, G. Forgacs, and F. Gail (World Scienti fic, Singapore, 1995), pp. 143–153. [3] M. S. Steinberg, Science 141, 401 (1963). [4] F. Graner, J. theor. Biol. 164, 455 (1993). [5] N. S. Goel and A. G. Leith, J. theor. Biol. 28, 469 (1970). [6] F. Graner and J. A. Glazier, Phys. Rev. Lett. 69, 2013 (1992). [7] J. A. Glazier and F. Graner, Phys. Rev. E 47, 2128 (1993). [8] D. Drasdo, R. Kree, and J. S. McCaskill, Phys. Rev. E 52, 6635 (1995). [9] M. Weliky, S. Minsuk, R. Keller, and G. Oster, Developmen t113, 1231 (1991). [10] J. Shih and R. Keller, Development 116, 887 (1992). 8a b

arXiv:physics/9912039v1 [physics.plasm-ph] 20 Dec 1999Semiclassical dynamics and time correlations in two-compo nent plasmas J. Ortnera), I. Valuevb), and W. Ebelinga) a)Institut f¨ ur Physik, Humboldt Universit¨ at zu Berlin, Invalidenstr. 110, D-10115 Berlin, Germany b)Department of Molecular and Chemical Physics, Moscow Insti tute of Physics and Technology, 141700 Dolgoprudny, Russia (August 9, 2013) The semiclassical dynamics of a charged particle moving in a two-component plasma is consid- ered using a corrected Kelbg pseudopotential. We employ the classical Nevanlinna-type theory of frequency moments to determine the velocity and force autoc orrelation functions. The constructed expressions preserve the exact short and long-time behavio r of the autocorrelators. The short-time behavior is characterized by two parameters which are expre ssable through the plasma static correla- tion functions. The long-time behavior is determined by the self-diffusion coefficient. The theoretical predictions are compared with the results of semiclassical molecular dynamics simulation. PACS numbers:52.25.Vy, 52.25.Gj, 52.65.-y, 05.30.-d I. INTRODUCTION The purpose of this paper is the investigation of the dynamic s of force on a charged particle in a two component plasma. Boercker et al. have shown the effect of ion motion on the spectral line broade ning by the surrounding plasma [1,2]. In recent papers it was argued that the microfie ld dynamics influence the fusion rates [3] and rates for three-body electron-ion recombination [4] in dense plasma s. Generally speaking, to calculate the plasma effect on rates and spectral line broadening one needs a theory of aver age forces and microfields, including the resolution in space and time. Basic results in this field were obtained by Si lin and Rukhadze [5], Klimontovitch [6], Alastuey et. al. [7], and Berkovsky et. al. [8]. The determination of the static distribution of the ion or el ectron component of the electric microfield is a well studied problem (for a review see [9]). The corresponding in vestigations are performed on the basis of the one- component plasma (OCP) model. A straightforward generaliz ation of the OCP model is the model of a two-component plasma (TCP), consisting of electrons and ions. In a recent p aper [10] the probability distribution for the electric microfield at a charged point has been studied. It was shown th at the two-component plasma microfield distribution shows a larger probability of high microfield values than the corresponding distribution of the OCP model. The dynamics of the electric microfield is a less understood p roblem than that of the static microfield distribution even for the case of an OCP. Recently some progress has been ma de for both the case of electric field dynamics at a neutral point [7,11,12] and the dynamics of force on a charge d impurity ion in an OCP [8]. This paper is aimed to extend the studies of electric microfie ld dynamics in OCP to the case of an equilibrium two component plasma. For simplicity we consider a two-compone nt plasma which is anti-symmetrical with respect to the charges ( e−=−e+) and therefore symmetrical with respect to the densities ( ni=ne). To simplify the numeric investigations we simulated a mass symmetric (nonrelativi stic) electron-positron plasma with m=mi=me. The theoretical investigations are carried out for arbitrary e lectron-ion mass ratios. 1Dedicated to the 75th birthday of Youri L. Klimontovich 1In this paper we will study the dependence of the force dynami cs on the coupling constant Γ = e2/kBTaof the plasma, where Tis the temperature, and a= (3/4πne)1/3is the average distance between the electrons. Coupled plasmas with a plasma parameter of the order or greater than u nity are important objects in nature, laboratory experiments, and in technology [13–16]. Recent lasers allo w to create a coupled plasma within femtoseconds [17]. Laser generated plasmas are nonequlibrium plasmas with an i nitial electron temperature much higher than the ion temperature. However, in this paper we restrict our conside rations to the model object of an equilibrium two- component plasma (TCP). Several investigations were devoted to the simulation of eq uilibrium two-component plasmas Being interested in quasi-classical methods we mention explicitely the quasi- classical simulations of two-component plasmas performed by Norman and by Hansen [18,19]. In this paper the free charges (electron and ions) are simula ted by a semi-classical dynamics based on effective potentials. The idea of the semi-classical method is to inco rporate quantum-mechanical effects (in particular the Heisenberg and the Pauli principle) by appropriate potenti als. This method was pioneered by Kelbg, Deutsch and others [20,21]. Certainly, such a quasi-classical approac h has several limits. For the calculation of a standard macro - scopic property as the microfield dynamics which has a well de fined classical limit the semi-classical approach may be very useful. The advantage of such an approach is the relativ e simplicity of the algorithm. II. THE SLATER SUM AND THE SEMICLASSICAL MODEL A familiar derivation of effective potentials describing qu antum effects is based on the Slater sums which are defined by the N - particle wave functions, S(r1, . . . ,rN) = const/summationdisplay exp(−β En)|Ψn(r1, . . .,rN)|2, (1) where Enand Ψ nare the energy levels and corresponding wave functions of th e ensemble of Nparticles with coor- dinates r1, . . . , r N. Here we consider a two-component plasma consisting of Neelectrons with mass meandNi=Ne ions with mass mi. The properties of the Slater sums for Coulombic systems wer e studied in detail by several authors [13,22]. Choosing the effective potential U(N)(r1, . . .,rN) =−kBTlnS(r1, . . . ,rN). (2) we may calculate the correct thermodynamic functions of the original quantum system [13,22,18] from the thermody- namic functions of a classical reference system. The Slater sum may be considered as an analogue of the classic al Boltzmann factor. Therefore it is straightforward to use the Slater sum for the definition of an effective potential . The only modification in comparison with classical theory is the appearance of many-particle interactions. If the sys tem is not to dense (i.e., neΛ3 e≪1, Λ e= ¯h/√2mekBT) one may neglect the contributions of higher order many-particl e interactions. In this case one writes approximately, U(N)(r1, . . . ,rN)≈/summationdisplay i<juij(ri,rj), (3) where the effective two-particle potential uabis defined by the two-particle Slater sum, S(2) ab(r) = exp ( −βuab(r)) = const ./summationdisplay α′ exp (−βEα)|Ψα|2. (4) Here Ψ αandEαdenote the wave functions and energy levels of the pair ab, respectively. The prime at the summation sign indicates that the contribution of the bound states (which is not be considered here) has to be omitted. Principal it is possible to calculate the Slater sum for a pai r of particles directly from the known two-particle Coulomb wavefunctions. To simplify the simulations it is be tter to have an analytic expression for the potential. A possible candidate is the so called Kelbg potential obtaine d by a perturbational expansion It reads [20] uab(r) =eaeb rF(r/λab), (5) where λab= ¯h/√2mabkBTis De Broglie wave length of relative motion, m−1 ab=m−1 a+m−1 b,a=e, i. In Eq.(5) F(x) = 1−exp/parenleftbig −x2/parenrightbig +√πx(1−erf(x)). (6) 2Another analytic approximation for the exact two-particle effective potential is the expression derived by Deutsch which was used in the simulations by Hansen and McDonald [19] . The Kelbg potential is a good approximation for the two-part icle Slater sum in the case of small parameters ξab=−(eaeb)/(kBTλab) if the interparticle distance ris sufficiently large. However, at small interparticle dista nces it shows a deviation from the exact value of −kBT·ln(Sab(r= 0)). In order to describe the right behavior also at small distances it is better to use a corrected Kelbg potenti al defined by [24] uab(r) = (eaeb/r)·/braceleftbigg F(r/λab)−rkBT eaeb˜Aab(ξab)exp/parenleftbig −(r/λab)2/parenrightbig/bracerightbigg . (7) In Eq. (7) the coefficient Aab(T) is adapted in such a way that Sab(r= 0) and his first derivative S′ ab(r= 0) have the exact value corresponding to the two-particle wave func tions of the free states [13,24,23]. The corresponding coefficients for the elctron-electron and for the electron-i on interaction read ˜Aee=√π|ξee|+ ln/bracketleftBigg 2√π|ξee|/integraldisplaydy yexp/parenleftbig −y2/parenrightbig exp (π|ξee|/y)−1/bracketrightBigg (8) ˜Aei=−√πξei+ ln/bracketleftbigg√πξ3 ie/parenleftbigg ζ(3) +1 4ζ(5)ξ2 ie/parenrightbigg + 4√πξei/integraldisplaydy yexp/parenleftbig −y2/parenrightbig 1−exp (−πξei/y)/bracketrightBigg (9) We mention that in the region of high temperatures Tr=T/T I=/parenleftbig 2kBT¯h2/miee4/parenrightbig >0.3. (10) the Kelbg potential ( Aab= 0) almost coincide with the corrected Kelbg potential Eq. ( 7). In the region of intermediate temperatures 0 .1< Tr<0.3 the Kelbg potential does not give a correct description of t he two-particle Slater sum at short distances. Instead we may use the corrected Kelbg-p otential Eq.(7) to get an appropriate approximation for the Slater sum at arbitrary distances. The effective potentials derived from perturbation theory d o not include bound state effects. The other limiting case of large ξabor small temperature Tr<0.1, where bound states are of importance, can be treated by ano ther approach [22]. Here a transition to the chemical picture is m ade, i.e. bound and free states have to be separated. In the present work we are interested in the regime of interme diate temperatures. In this regime the simulations of the dynamics may be performed with the potential Eq.(7). III. FORCE-FORCE AUTOCORRELATION FUNCTION The system under consideration is a two-component plasma co nsisting of electrons and ions which is described by the semiclassical model introduced in Sec II. Let us choose t he position of one of the charged particles (for example an electron) as a reference point. Hereafter we call this par ticle the first one. The semiclassical force acting on the first particle equals F=−∆1N/summationdisplay j=2u1j(r1−rj) (11) uijbeing the effective pair potential between the ith and jth particles, defined in Eq. (7). Define now two functions characterizing the dynamics of the fi rst particle. The first one C(t) =<v(t)·v(0)> < v2>(12) is the velocity-velocity autocorrelation function (veloc ity acf), the second function C(t) =<F(t)·F(0)> < F2>(13) is the force-force autocorrelation function (force acf). I n the above equations the brackets < . . . > denote averaging over the equilibrium ensemble of the semiclassical system. The velocity acf is formally a function expressing the singl e particle properties. However, it is connected with the forc e acf which involves the collective properties by the relati on 3∂2C(t) ∂t2+ω2 1D(t) = 0, (14) where ω2 1=< F2>/3mkBT. Define the one-side Fourier transform of the velocity and for ce acf, ˆC(ω) =/integraldisplay∞ 0dteiωtC(t),ˆD(ω) =/integraldisplay∞ 0dteiωtD(t). (15) The Fourier transform of Eq.(14) reads ˆD(ω) =ω2ˆC(ω)−iω ω2 1. (16) In order to construct the both autocorrelation functions it is useful to consider the frequency moments of the real part of the velocity acf Fourier transform Mn=1 2π/integraldisplay∞ −∞ωnˆCr(ω)e−iωtdω , n = 0,1,2, . . . . (17) The zeroth moment is the initial value of the velocity acf, M0=C(0) = 1 . (18) Due to the parity of the function ˆCr(ω), all moments with odd numbers are equal to zero. The second moment is expressable through the initial value o f the force acf, M2=1 2π/integraldisplay∞ −∞ω2ˆCr(ω)e−iωtdω=ω2 1D(0) = ω2 1. (19) The fourth moment includes the correlation function of the t ime derivative of the force, M4=1 2π/integraldisplay∞ −∞ω4ˆCr(ω)e−iωtdω=ω2 1ω2 2, (20) where we have introduced the magnitude ω2 2=<˙F2>/< F2>. The Nevanlinna formula of the classical theory of moments [2 5,26] expresses the velocity acf Fourier transform 1 π/integraldisplay∞ ∞ˆCr(ω) z−ωdω=−iˆC(z) =En+1(z) +qn(z)En(z) Dn+1(z) +qn(z)Dn(z)(21) in terms of a function qn=qn(z) analytic in the upper half-plane Im z >0 and having a positive imaginary part there Imqn(ω+iη)>0, η > 0, it also should satisfy the limiting condition: ( qn(z)/z)→0 asz→ ∞ within the sector θ <arg(z)< π−θ. In Eq.(21) we have employed the Kramers-Kronig relation co nnecting the real and imaginary part of ˆC(ω). The polynomials Dn(andEn) can be found in terms of the first 2 nmoments as a result of the Schmidt orthogonalization procedure. The first orthogonal polynom ials read D1=z , D 2=z2−ω2 1, D3=z(z2−ω2 2), (22) E1= 1, E2=z , E 3=z2+ω2 1−ω2 2). (23) Consider first the approximation n= 1 leading to the correct frequency moments M0andM2. Using the Nevanlinna formula and Eq. (16) we obtain ˆC(z) =iz+q1(z) z2−ω2 1+q1z, ˆD(z) =iz z2−ω2 1+q1z. (24) The physical meaning of the function q1(z) is that of a memory function [8] since the inverse Fourier tr ansform of Eq. (24) is ∂2C(t) ∂t2+ω2 1C(t) +/integraldisplayt 0ds q1(t−s)∂C(s) ∂s= 0. (25) 4We have no phenomenological basis for the choice of that func tionq1(z) which would provide the exact expression for ˆC(z) and ˆD(z). A simple approximation is to put the function q1(z) equal to its static value q1(z) =q1(0) = iν (26) and Eq. (25) simplifies to the equation of a damped oscillator with frequency ω1and damping constant ν. ∂2C(t) ∂t2+ω2 1C(t) +ν∂C(t) ∂t= 0. (27) The static value q1(z= 0) is connected with the self-diffusion coefficient D. The latter is defined by the time integral of the velocity acf D=1 βm1/integraldisplay∞ 0dtC(t) =1 βm1ˆC(0), (28) where β= 1/(kBT) and m1is the mass of the first particle. With the use of Eqs. (28) and ( 26) we obtain from Eq. (21) that ν=ω2 1βm1D. The inverse Fourier transform of Eq. (21) with the static app roximation Eq. (26) expresses the velocity and force acf’s as a linear combination of two exponential functions e xp(z1t) and exp( z2t), where z1/2=−ν/2±/radicalbig ν2−4ω2 1/2. Within this approximation we may distinguish between two re gimes. In the first regime - the “diffusion-regime” - one deals with a large diffusion constant. As a result ν=βm1Dω2 1>2ω1and Eq. (27) is the equation of an overdamped oscillator. In this regime the velocity autocorrelation fu nction goes monotoneously to zero. With decreasing diffusio n constant the damping constant νbecomes smaller. At certain thermodynamical conditions ju st the opposite inequality ν <2ω1holds. This corresponds to an “oscillatory-regime” and at l east one of the autocorrelation functions should show an oscillatory behavior. The existence of the two regim es have been established for the case of an OCP [8] and has been confirmed by our molecular-dynamics simulation for the case of a TCP. To obtain not only a qualitative but also a quantitative correspondence with the results of MD si mulations one has to go beyond the simple approximation n= 1 in the Nevanlinna formula Eq. (21). Consider therefore the case n= 2 in Eq. (21). Then the autocorrelation functions are expre ssed via the function q2(z) as ˆC(z) =iz2+ω2 1−ω2 2+q2(z)z z(z2−ω2 2) +q2(z2−ω2 1), ˆD(z) =iz(z+q2) z(z2−ω2 2) +q2(z2−ω2 1). (29) Eq. (29) reproduces the exact freqency moments from M0up to M4. For the function q2(z) we choose again a static approximation q2(z)≡q2(0)≡ih , (30) where hhas to be taken from the relation h=/parenleftbiggω2 2 ω2 1−1/parenrightbigg /βm1D (31) in order to obtain the exact low frequency value ˆC(0) given by Eq. (28). From Eq. (29) we find that the autocorrelation functions are n ow given by the linear combination of three expo- nentials, C(t) =3/summationdisplay i=1CieiΩit, D(t) =3/summationdisplay i=1dieiΩit. (32) The complex frequencies Ω iare the poles of the expressions Eq. (29). They are defined as t he solutions of the cubic equation, Ω(Ω2−ω2 2) +ih(Ω2−ω2 1) = 0. (33) The coefficients Ci(di) characterizes the strength of the ith mode, 5Ci=ω2 1 Ω2 idi, i= 1,2,3, (34) d1=i(h+iΩ1)Ω1(Ω2−Ω3)/N (35) d2=i(h+iΩ2)Ω2(Ω3−Ω1)/N (36) d3=i(h+iΩ3)Ω3(Ω1−Ω2)/N (37) N= (Ω 1−Ω2)(Ω3−Ω1)(Ω2−Ω3). (38) Equations (32) constitute the basic approximation of our pa per. The frequencies Ω iand the coefficients Ci(or di, respectively) are expressed by three parameters - the diffu sion constant D, and the frequencies ω1andω2. The constructed autocorrelation functions satisfy the follow ing conditions: (i) the exact short time behavior for the vel ocity acf is reproduced to the orders t2andt4, (ii) the short time behavior of the force acf is reproduced t o the order t2, (iii) the long time behavior of the velocity acf generates th e exact diffusion constant, and (iv) the connection between the velocity and force acf’s Eq. (14) is satisfied. The parameters D,ω1andω2may be calculated by another approximations. The both frequ encies ω1andω2 are expressable via the partial correlation functions of ou r semiclassical system. The parameter ω1is given by the electron-ion and electron-electron partial pair correlat ion functions. To calculate the frequency ω2one needs the knowledge of the partial ternary distribution functions. T he diffusion constant may be obtained from kinetic theory. In contrast to the case of an OCP [8] the parameters to be calcu lated are very sensitive to the approximations used to calculate the static distribution functions. Therefore in this paper we take the “input” parameters directly from the computer simulations. To check the quality of the predictions from our approximati on we have performed molecular dynamics simulations for comparison. The equations of motions obtained with the e ffective potential Eq.(7) were integrated numerically for the case of equal masses me=miusing the leap-frog variant of Verlet’s algorithm. The simu lations were performed for 128 electrons and 128 positrons moving in a cubic box with periodic boundary conditions. In the investigated range of plasma parameters ( T= 30 000 K, the coupling parameter has varied from Γ = 0 .2 up to Γ = 3) the size of the simulation box was significantly greater than the Deby e radius. Therefore the long-range Coulomb forces are screened inside each box and there was no need to use the Ewald summation instead the simple periodic boundary conditions. The thermal equilibrium in the system was estab lished (and maintained) by a Langevin source. Such simulations has been recently used to obtain the static dist ribution of the electric microfield at a charged particle [10 ]. In this paper we extract the velocity and force autocorrelat ion functions as the main characteristics of the microfield dynamics. TABLE I. The Γ dependence of the parameters ω1,ω2andD.ω1andω2are given in units of electron plasma frequency ωpe=/radicalbig 4πnee2/me,Dis given in units of 1 /(meωpeβ) Γ ω1 ω2 D 0.2 0.84 13.6 10.3 1.5 0.88 3.3 4.41 3.0 0.61 2.1 5.75 In Figs. 1-3 we present the results of the MD data. The simulat ion results are compared with our analytical approximation Eqs. (29). The three input parameters for the analytical approximation are taken from the MD simulations. The diffusion constant is obtained from the tim e integral of the velocity acf (Eq. (28)). Since the velocity acf is a slowly decaying function it requires a long simulation time to extract the diffusion constant. For our model system with equal electron and ion masses it is possibl e to perform the necessary simulations. The frequency ω2has been taken from the exact short time behavior of the force acfD(t) = 1−ω2 2t2/2. Finally the frequency ω1was choosen to fit the model to the data. In Table I we show the param etersω1,ω2andDfor three coupling parameters Γ considered in this paper. Except the case of the force acf at Γ = 0 .2 there is a good overall agreement between the theoretical a pproximations and the MD data. We believe that the strong deviation of the MD data from the theoretical predictions for Γ = 0 .2 is a numerical artefact due to the poor statistics in the weak co upling case. From the figures we see that with increasing plasma parameter Γ the dynamics of the charged particles swi tches from the diffusion-like regime at Γ = 0 .2 to the oscillator-like motion at Γ = 3 .0. The value Γ = 1 .5 may be considered as a critical value separating the both re gimes. We may also see from the figures that the oscillator-like moti on is more pronounced for the force acf. At still higher densities (Γ ≥3 atT= 30 000 K) the semi-classical approach employed in this pape r fails to describe the quantum two-component plasma properly. 6IV. CONCLUSIONS The electric microfield dynamics at a charged particle in a tw o-component plasma has been studied. The quantum plasma has been modeled by a semiclassical system with effect ive potentials. The effective potential was choosen to describe the nondegenrate limit of the quantum system appro priately. We have investigated the velocity and force acf’s of the semiclassical system. The starting point for th e theoretical analysis was the exact expression of the autocorrelation functions through the Nevanlinna formula Eq. (29), satisfying three sum rules for the velocity acf. The approximation Eq. (30) together with Eq. (31) expresses the velocity acf in terms of three parameters. Two of them - ω1andω2- describe the exact short time behavior of the velocity acf u p to the order t4, the third parameter, the self-diffusion constant Dis related to the time integral of the velocity acf. Since the force acf can be obtained from the velocity acf by a second time derivative the force ac f is expressed through the same three parameters. The general picture is as follows. At weak coupling the diffusion of the charged particle dominates the collective plasma oscillations and the particle motion is diffusion-like. In t his regime the velocity acf decays exponentially with a deca y rate 1 /D(time in units of the inverse electron plasma frequency ωpe). The force acf has a positive decay at short times (decay rate ω2 1D) and a negative decay at long times (with the rate 1 /D). At strong coupling the diffusion is supressed and a weakly damped oscillatory behavior for the f orce acf developes. The theoretical predictions has been compared with molecular dynamics simulations data. There i s an overall agreement of the force dynamics obtained by the analytical approximation with the MD data. Finally, we mention that there is no one to one correspondenc e of the semiclassical autocorrelation functions with the corresponding characteristics of the quantum system. Neve rtheless, we suspect that the semiclassical force dynamics considered in this paper at least qualitatively reproduces the electric microfield dynamics of the quantum system. Acknowledgments. This work was supported by the Deutsche Forschungsgemeinsc haft (DFG) and the Deutscher Akademischer Austauschdienst (DAAD) of Germany. [1] D. Boercker, C. Iglesias, and J. Dufty, Phys. Rev. A 36, 2254 (1987). [2] D. Boercker, in Spectral Line Shapes 7 , edited by R. Stamm and B. Talin (Nova Science, New York, 1993 ); inSpectral Line Shapes 5 , edited by J. Szudy (Ossolineum, Wroclaw, Poland, 1989). [3] M. Yu. Romanovsky and W. Ebeling, Physica A 252, 488-504 (1998). [4] M. Yu. Romanovsky, Zh. Eksp. Teor. Fiz. 114, 1230-1241 (1998). [5] V.P. Silin, A.A. Rukhadse, Electromagnetic Properties of Plasmas and Plasma-like Media (in Russ.) ( Gosatomizdat, Moscow, 1964). [6] Yu.L. Klimontovich, Kinetic Theory of Electromagnetic Processes (Springer, Berlin-Heidelberg-New York, 1982). [7] A.Alastuey, J.L.Lebowitz, D.Levesque, Phys.Rev. A43(1991) 2673. [8] M. Berkovsky, J. W. Dufty, A. Calisti, R. Stamm, and B. Tal in, Phys. Rev. E 54, 4087 (1996). [9] J. W. Dufty, in Strongly Coupled Plasmas , ed. by F. J. Rogers and H. E. DeWitt (Plenum, New York, 1987). [10] J. Ortner, I. Valuev, and W. Ebeling, Phys. Lett. A (subm itted). [11] J. W. Dufty and L. Zogaib, Phys. Rev. A 44, 2612 (1991). [12] M. Berkovsky, J. W. Dufty, A. Calisti, R. Stamm, and B. Ta lin, Phys. Rev. E 51, 4917 (1995). [13] Kraeft, W.D., Kremp, D., Ebeling, W. and R¨ opke, G., “Qu antum Statistics of Charged Particle Systems”. (Akademie- Verlag, Berlin; Plenum Press, New York; russ. transl: Mir, M oscow 1986). [14] Ebeling,W., F¨ orster,A., Fortov,V.E., Gryaznov,V.K . and Polishchuk,A.Ya., “Thermophysical Properties of Hot Dense Plasmas” (Teubner, Stuttgart-Leipzig 1991). [15] Ichimaru, S. “Statistical Plasma Physics: I. Basic Pri nciples, II: Condensed Plasmas”. (Addison-Wesley, Readin g, 1992, 1994). [16] Kraeft, W.D. and Schlanges, M. (editors), “Physics of S trongly Coupled Plasmas” (World Scientific. Singapore, 199 6). [17] W. Theobald, R. H¨ assner, C.W¨ ulker, and R. Sauerbrey, Phys. Rev. Lett. 77, 298 (1996). [18] Zamalin, V.M., Norman, G.E. and Filinov, V.S., “The Mon te Carlo Method in Statistical Mechanics” (in Russ.) (Nauka , Moscow, 1977). [19] Hansen, J.-P. and McDonald, I.R., Phys. Rev. A 23, 2041, (1981). [20] G. Kelbg, Ann. Physik 13354,14394 (1964). [21] C. Deutsch, Phys. Lett. 60A, 317 (1977). [22] Ebeling, W., Ann. Physik, 21, 315 (1968); 22(1969) 33,383,392; Physica 38, 378 (1968); 40, 290 (1968); 43, 293 (1969); 73, 573 (1974). [23] Rohde, G. Kelbg, W. Ebeling, Ann. Physik 22(1968). 7[24] W. Ebeling, G. E. Norman, A. A. Valuev, and I. Valuev, Con tr. Plasma Phys. 39, 61 (1999). [25] V. M. Adamyan, T. Meyer, and I. M. Tkachenko, Fiz. Plazmy 11, 826 (1985) [Sov. J. Plasma Phys. 11, 481 (1985)]. [26] J. Ortner and I. M. Tkachenko, Phys. Rev. A 46, 7882 (1992). 8FIGURE CAPTIONS (Figure 1) Time dependence of velocity acf C(t) and force acf D(t) at Γ = 0 .2. Time is in units of inverse electron plasmafrequency ω−1 pe. Solid lines: present theoretical approximation; Points: results of molecular-dynamics simulations. (Figure 2) Same as in Fig. 1 at Γ = 1 .5. (Figure 3) Same as in Fig. 1 at Γ = 3 .0. 90.0 2.0 4.0 6.0 t−0.50.00.51.0C(t), D(t)C(t) MD data D(t) MD data Γ= 0.2 C(t) theory D(t) theory 100.0 2.0 4.0 6.0 8.0 10.0 t−0.50.00.51.0C(t), D(t)C(t) MD data D(t) MD data Γ = 1.5 C(t) theory D(t) theory 110.0 2.0 4.0 6.0 8.0 10.0 t−0.50.00.51.0C(t), D(t)C(t) MD data D(t) MD data Γ=3.0 C(t) theory D(t) theory 12

arXiv:physics/9912040 v2 21 Dec 1999 Exploring the physics of the relativistic energy-momentum relationship A.C.V. Ceapa PO Box 1-1035, R-70700 Bucharest, Rumania E-mail: alex_ceapa@yahoo.com Considerations on the complementary time-dependent coordinate transformations emboding Lorentz transformation (LT) show that the relativistic energy-momentum relationship, implicitly the relativistic mass and energy, do not depend on the β appearing in LT, being associated to the absolute motion of a particle and related to its inner structure. Results concerning the concept of operational theory and its application to the electromagnetic and gravitational field theories, as well as to the quantum mechanics are given in appendixes. 1. Complementary Time Dependent Coordinate Transformations* 2. Transformation Laws for Energy and Linear Momentum* 3. Contravariant and Covariant Four-Vectors* 4. Four-Momentum, Proper Frame* 5. The Relativistic Energy-Momentum Relationship* 6. The True Derivation of Standard Energy-Momentum Relationship* Appendix 1: Four-Dimensional Consequences of CTs* Appendix 2: Operational Theories* Appendix 3: Absolute Coordinate Systems* References* It is interesting to examine in more detail the quantity pµpµ=E2/c2-p2 ,(1) where E=mc2, p=mv(2) are, respectively, the relativistic energy and linear momentum of a free particle of relativistic mass m=βmo,(3) rest mass mo and velocity v, in relation to the laws of transformation of the energyand linear momentum of a free particle under the coordinate transformation equations x'=x-vt, y'=y, z'=z, t'=t-vx/c2(4) and x'=β(x-vt), y'=y, z'=z, t'=β(t-vx/c2),(5) called complementary time dependent coordinate transformations. 1. Complementary Time Dependent Coordinate Transformations We distinguish between ordinary time dependent coordinate transformations (OTs) and complementary time dependent coordinate transformations (CTs). The OTs are simply obtained by changing angles and lengths in time independent coordinate transformations into time dependent quantities. They are represented by spatial rotations and translations. CTs are related to the tracing of radii vectors by physical signals traveling through space with constant velocity υ. This tracing is required by our need of knowing the length and the direction of the radius vector of any geometrical point belonging to a "stationary" subspace before drawing and projecting it onto the coordinate axes of a stationary coordinate system (K) in space which is at absolute rest (see also Sect.1.1 in ref.1) as long as such points are aimed by an uniform translatory motion. The CTs are established for points of a subspace coinciding at an instant of time with space points. Unlike OTs which can be written whenever after the radii vectors of a geometrical point were traced by a pencil, the CT can be written only after the radii vectors we trace by a pencil have previously been traced by physical signals of identical nature. Depending on the nature of the physical signals tracing radii vectors, we have a CT or another. For light signals we have LT as a particular CT in the three-dimensional space. The preference for LT is related to the large value of c in comparison with the speeds of all the known physical signals, to the propagation of the electromagnetic and gravitational fields at speed c, but especially to the fact, pointed out elsewhere2, that c is also a subquantum velocity. The equations of any CT are those of LT with c changed to the speed υ of the used physical signal. Specific to all CTs is their time equation obtained in their preliminary form as the time equivalent of their spacial equation written along the direction of motion of k relative to K. The manner in which we use the physical signals to establish a CT is just that used to obtain LT in Sect.4 of ref.1. Like LT, any CT reduces to GT in the "low-velocity" approximation. This only means that in such a situation OO' becomes negligible in comparison with OP1 (OP') and O'P1 (O'P') in the diagram in Fig.5 (10) in ref.1, ct* reduces to ct and, implicitly, t* reduces to the time t on the time axis. As concerns the homogeneity of the CTs, it originates in the initial superposition of the coordinatesystems k and K required to obtain the geometry in Figs.5 and 10. The most simple CT is that given by Eqs.(38) in ref.1. It follows from the first of Eqs.(5) and (21) related to the upper diagrams in Figs.1 and 2 in ref.1. The raising of Eq.(21) was largely discussed in Sect.4.1. Like LT, Eqs.(38) form a group. For v=c, Eqs.(38) reduce to x'=x-ct, t'=t-x/c.(6) Fig.1 Eqs.(6) are related to the diagram in Fig1. Since k is carried by the tip of a light signal, only geometrical points P(x',x)∈ (O',O), where O' and O are, respectively, the origins of k and K, can be joined by light signals. Naturally, Eqs.(6) do not form a group; this because, carried by light signals leaving simultaneously O, the coordinate systems kA and kB are always superposed to each other. Moreover, the time component of Eqs.(6) should not be identified with the time relation t'=t-r/c which, connecting two synchronous clocks, does not belong to a coordinate transformation (for consequences of CT see Appendixes 1 to 3 below). 2. Transformation Laws for Energy and Linear Momentum Assume for the beginning that we do not know that the energy and the linear momentum form a four-vector. Also assume that we do not know the transformation laws satisfied by the covariant and contravariant components of a four-vector. So that we propose to establish the transformation laws of the two from the invariance of the action E't'-p'x'=Et-px(7) under Eqs.(4) and (5), connecting the coordinate system K at absolute rest to the parallel coordinate system k in uniform rectilinear motion along the common x', x axis of coordinates. Denote by E, p and E', p' the energies and linear moments of a free particle in relation to K and k, respectively. Substituting Eqs.(4), (5) and their inverses in Eq.(7), we get, respectively, the equations E=E'+p'v, p=p'+E'v/c2,(8')E=β(E'+p'v), p=β(p'+E'v/c2) (8") and E'=E-pv, p'=p-Ev/c2,(9') E'=β(E-pv), p'=β(p-Ev/c2).(9") Eqs.(8) and (9) constitute the searched laws of transformation of the energy and linear momentum under the CT Eqs.(4) and (5). Each of these laws is analogous to the inverse of the CTs taken into account as a consequence of the last. 3. Contravariant and Covariant Four-Vectors It is well-known that the contravariant and covariant components of a four-vector, respectively Aµ and Aµ, are mathematically given by the transformation laws3 Aµ =(∂ xµ /∂x'ν )A'ν , Aµ =(∂ x'ν /∂xµ )A'ν ,(10) where Greek indices run from 0 to 3, with the coordinates x'µ and xµ connected by LT. The derivation of the transformation laws of the contravariant and covariant components pµ and pµ of the four-momentum from the invariant called action in Sect.2 makes explicit the way in which the mixture of times and coordinates in the LT equations raises Eqs.(10). Continuing this line of thought, we further consider a physical quantity which is a differential function of x', x'o(=ct') that in their turn, by the LT equations x'=β (x-vxo/c), x'o=β (xo-vx/c), are continuous functions of x, xo(=ct) with partial derivatives. The differential of this function is df=(∂ f/∂ x)dx+(∂ f/∂ xo)dxo= =[(∂ f/∂ x')(∂ x'/∂ x)+(∂ f/∂ x'o)(∂ x'o/∂ x)]dx+ +[(∂ f/∂ x')(∂ x'/∂ xo)+(∂ f/∂ x'o)(∂ x'o/∂ xo)]dxo= =β [(∂ f/∂ x'-v/c)(∂ f/∂ x'o)]dx+β [-(v/c)(∂ f/∂ x')+∂ f/∂ x'o]dxo. With the notations ∂ f/∂ x=A, ∂ f/∂ xo=Ao, ∂ f/∂ x'=A', ∂ f/∂ x'o=A'o, we regain the first of Eqs.(10). This result is worthwhile because it infers that the components of any four-vector are always derivatives of a function whichmust be identified for its physical meaning and consequences to be well- determined. Unfortunately, there is the common tendency of endowing the four- vectors with a mysterious physical existence which, by their transformation law analogous with LT, extends onto the last. 4. Four-Momentum, Proper Frame The four-momentum was defined by3 pµ=mocuµ , where uµ=dxµ/ds is the four-velocity, ds=(ηµνdxµdxν)1/2 is the metric of the Minkowskian space and ηµν=(-1,-1,-1,+1) is the suitable metric tensor. When written with respect to a coordinate system K at absolute rest (see also Sect.3.2 in ref.1), for which ds=β-1cdt, the four-momentum is given by pµ=moβ dxµ/dt=(moβv, moβc), in agreement with the classical definition of the linear momentum and the dependence on velocity of the mass. When written with respect to the "stationary" coordinate system k' in which a particle is at rest (v=0)-called proper frame, the four-momentum takes the preliminary form pµ=modxµ/dτ by virtue of ds=cdτ, where τ is the proper time, and a final form pµ=(moβv, moβc), identical to that relative to K, by the equation dx=vdt=vβdτ , following from the standard LT equations under the condition dx'=0 required to measure dτ . Thus, against the appearances, we obtain the natural result that whenever a free particle moves with respect to K with constant velocity v or is at rest with respect to a coordinate system moving with the same velocity relative to K (its proper frame), it possesses the same mass moββ, the same energy moββc2 and (although we cannot define a non-zero velocity in this case) the same quantity of motion. Stating that the mass and the energy of a particle are, respectively, mo and moc2 in its proper frame is false and misleading as long as that particle is carried by its proper frame. The values mo and moc2 are true only for a particle at rest in a stationary coordinate system. If Einstein connected these values to the proper frame, he did it only because missing the meaning of Ξ in his original paper on relativity (see also Sect.3.7 in ref.1), and believing that he eliminated the coordinate system at absolute rest from his theory of "relativity", he was compelled to introduce the concept of proper frame just as he was compelled to extend the L- principle to "stationary" coordinate systems. Thus, whenever we use the proper frame we must keep in mind that the true quantities defining a particle at rest with respect to it are a non-zero quantity of motion, a mass m=moβ and an energy E'=moβc2 (here β having nothing in common, as concerns its origin, with βoccurring in the Lorentz transformation!). In fact, the quantities moβ and moβc2 are always associated to the absolute motion of a particle. This can be explained by that any state of motion of a particle alters its subquantum basic state. 5. The Relativistic Energy-Momentum Relationship Let us write Eqs.(8) and (9) in relation to the proper frame of a free particle. Assuming p'=0, they are E=E',(11') E=βE',(11") and E'=β-2E,(12') E'=β-1E.(12") The last of Eqs.(8) and (9) are |p|=Ev/c2.(13) Since Eqs.(4) and (5) [the inverses of Eqs.(4) and (5)] connect a coordinate system k (K) in uniform rectilinear motion with respect to a coordinate system K (k) at absolute rest, whenever k represents a moving (rest) proper frame, the energy E' appearing in Eqs.(11)[(12)] (see also Sect.4) is E'=βmoc2 [E'=moc2]. Thus Eqs.(11) and (12) become E=βmoc2,(14') E=β2moc2(14") and E=β2moc2,(15') E=βmoc2.(15") The quantity (1) reduces by Eq.(13) to E2/c2-p2=β-2E2/c2. Further, by Eqs.(14) and (15) it takes the forms E2/c2-p2=mo2c2(16') E2/c2-p2=m2c2.(16")We recognize in Eq.(16') the standard relativistic energy-momentum relationship. We also see that Eq.(16"), which is β2 times Eq.(16') and embodies a change of origin on the energy scale, has previously been missed by assuming that E'=moc2 for particles at rest in their proper frames, irrespective of the state of rest or uniform translatory motion of the last. Therefore, obtained by Eqs.(14) and (15) as well, Eqs.(16) do not depend on the presence of β in the CT taken into account. Implicitly, the dependence on ββ of Eqs.(2) and (3) is, in accord with the experiment, not determined by LT. The coincidence of Eq.(16') [(16")] with that resulting from Eqs.(2) and (3) for a free particle moving relative to a K at absolute rest [in uniform rectilinear motion] assures the invariance of pµpµ in relation to LT. 6. The True Derivation of Standard Energy-Momentum Relationship The true derivation of the standard energy-momentum relationship is related to a particle at absolute rest with respect to a stationary coordinate system K. Its suitable energy is E=moc2. Its linear momentum is p=0. Inserting these values in Eq.(8") we obtain p'=-E'v/c2, E=β-1E'. Thus the energy and the linear momentum of this particle relative to a coordinate system k in uniform translatory motion with respect to K, as well as those of a particle moving with the same velocity relative to K, are E'=βmoc2, p'=βmov. The relationship (16') is immediate. Observe that, by replacing the stationary coordinate system K by a "stationary" coordinate system K, and denoting the energy and the linear momentum of a free particle respectively by mc2 [with m given by Eq.(3)] and p=0, we also deduce by Eq.(8") the energy-momentum relationship (16"). Unlike their derivation by means of Eqs.(11) and (12), Eqs.(16) have now a precise physical significance. Appendix 1: Four-Dimensional Consequences of CTs The LT equations, as well as the equations of any other CT defined in Sect.1, predict a four-dimensional metric and, implicitly, a four-dimensional space-time physically determined but with no physical significance. When the physical signals are light signals, the space-time is just that of Minkowski. There results that the Minkowski space-time is a consequence of the tracing by light signals of radii vectors of geometrical points belonging to moving subspaces. By the four-quantities and invariants also having their origin in the tracing of radii vectors by light signals, the Minkowski space-time appears to be a rigorous framework to describe the physical reality filling space. The events are determined in relation to four-dimensional coordinate systems. Beside the Minkowski space-time there is the four-space, also formal, associated to the four-momentum pµ=(p,E/c) just as the former was attached to the four-vector xµ=(x,ct). This is the energy-momentum space which Caianiello joined4 with the Minkowski space-time into an eight- dimensional space, and which metric enabled him to deduce the maximum acceleration aM. The endowing of phase space with a metric just expresses the physical determination of the quantum theory. Other aspects to be noted concern the metric of the Minkowski space-time itself. Thus, since t* denotes the time in which a physical signal traces the radius vector of a moving geometrical point P, ct does not define the x of P as a function of time but just as a length. Therefore, we can not write dx=cdt, but either dx=vdt or (dx2+dy2+dz2)1/2=vdt. It is for this reason that the metric s, reduced to (y2+z2)1/2 or 0 by x=ct, was used by Einstein only in connection with the light cone, while the infinitesimal metric ds (which always differs from zero) to get the main predictions of his theory. Appendix 2: Operational Theories Whenever physicists searched for solutions of an equation or a set of equations mapping to a less understood physical reality, they resorted to additional mathematical constraints on the quantities related by those equations to reach their goal. Often the solutions of these equations were in accord with experiments. Sometimes they were not, and the confidence in them diminished and delayed clear experimental results. The last is the case with the weak gravitational waves predicted by Einstein's theory of general relativity on the analogy with the plane electromagnetic waves. Like transverse waves, they were mathematically defined by imposing the transverse-traceless conditions5 Ψµ ν uν =0, Ψµ µ=0 (17) to their potentials Ψµν=hµν-(1/2)ηµνhσ σ, where hµν are the infinitely small components of the metric tensor gµν=ηµν +hµν and uν is the four-velocity of the wave, that satisfy Einstein's linearized equations of the gravitational field. Nevertheless, we see that actually the transformation laws of Ψµν are implied bythe transformation laws h'µν=(∂x'µ/∂xα)(∂x'ν/∂xβ)hαβ (18) of hµν under the inverse of the CT Eqs.(6) that connect a coordinate system moving with velocity c with respect to one at absolute rest. By a little algebra, Eqs.(18), (6) and (17) show that the only non-zero potentials of a weak gravitational wave traveling along the x axis are Ψ '2 2≡Ψ2 2=-Ψ3 3≡-Ψ'3 3, Ψ '2 3≡Ψ2 3=-Ψ3 2≡Ψ'3 2, (19) i.e., just those deduced by the traceless condition mathematically imposed. That is to say, Eqs.(18) connect wave potentials defined in relation to both the coordinate system carried by the wave and the stationary coordinate system of an observer recording the wave. Going on, since the quantities characterizing electromagnetic and gravitational fields are attached to geometrical points of subspaces traveling at speed c through empty space, and the CT Eqs.(6) refer points of such subspaces to coordinate systems at rest in this space, we focus our attention on the theories of electromagnetic and gravitational fields which exhibit retarded quantities for revealing the mathematical constraints historically imposed to obtain them6. Consider for the beginning the mathematical quantities f and ξµ appearing, respectively, in the electromagnetic and gravitational theories7 by the gauge transformation of their four-potentials. Observe that the dependence on t-x/c of f and ξµ has been obtained by imposing the Lorentz condition ∂Aµ/∂xµ=0 and its gravitational counterpart ∂Ψµν/∂xν=0 on the four-potentials of the plane electromagnetic and gravitational waves, respectively, Aµ and Ψµν. Since Aµ and Ψµν are defined in a coordinate system k moving with velocity c with respect to a coordinate system at absolute rest-the working coordinate system of the observer-their time dependence, as well as that of f and ξµ, follows from the second of Eqs.(6) just as a result of the velocity of k relative to K. Moreover, because f and ξµ belong to the mathematical basis of the two theories, the time dependence previously imposed on them is equivalent to the more general requirement, namely that these theories are developed in the working coordinate systems of the observers performing measurements of physical quantities.Therefore, the last of Eqs.(6) accounts for the retarded potentials, whose omnipresence has been until now only in agreement with experiment8, in that they are defined in coordinate systems k traveling at speed c with respect to a K in empty space and measured by observers with respect to such systems by the diagram in Fig.4 in ref.1. Extending these conclusions about the role played by Eqs.(6) in founding the two theories to the full class of complementary time dependent coordinate transformations (i.e., to those obtained by other physical signals than light), we get the concept of operational theory stated as6,9: A physical theory is an operational theory if and only if the quantities entering into its equations are expressed in the working coordinate systems of the observers performing measurements. The main consequence of this concept is that the modern physics becomes a system of operational theories valid in the working systems of the observers performing measurements. This means implementing the new theories as operational ones and removing from the older theories those mathematical constraints historically imposed by the process of knowledge exclusively for obtaining the time dependence of the physical quantities required by experiment. Both the definition of pµ investigated in Sect.4, and the subsequent rewriting of the equations of the relativistic quantum mechanics in a moving Ki (pointed out in the next Appendix), as well as the new derivation and meaning of the Hubble law in relation to Eqs.(4)6,10, which applied to the furthest quasars recently discovered does not support the receding of the galaxies from the Earth, represent first steps done in this aim. In addition, by obtaining the potentials (19) of a weak (plane) gravitational wave6,11, and converting the equation t'=t(1+2φ/c2)1/2(1-v2/c2)1/2, which predicts the effect that a Newtonian gravitational field of potential φ has on an elapsing of time, to12 t'K=t(1+2φ/c2)1/2≅ t(1+φ/c2), by the last of Eqs.(6) (t' being defined in the proper frame of the field and t'K in the coordinate system of a moving observer), and not by imposing prior mathematical constraints (e.g., dxi=0 in the metric12), the operational approach also extends to the weak field approximation of general relativity. Appendix 3: Absolute Coordinate Systems A main result obtained in Sect.4 was that the energy and the quantity of motion ofa particle at rest with respect to its proper frame moving with constant velocity v relative to K are given by Eqs.(2) without an observer isolated in that proper frame to be able to identify them by calculation or measurements. We concluded that (see also ref.6) the energy moc2 is specific to a particle at rest with respect to a coordinate system at absolute rest. It appears that by their defining relationships the "relativistic" energy and momentum were from the beginning estimated in relation to a coordinate system at absolute rest (we here denote it by Kabs). Since the invariant pµpµ predicts the standard energy-momentum relationship (6'), the equations of the relativistic quantum mechanics and, implicitly, the coupling constant moc2 between the systems of subquantum particles they involve, were defined in relation to Kabs. But as the invariant pµpµ was estimated in relation to the proper frame of the particle, and the energy and momentum of the last are given by Eqs.(2) and (3), the equations of the relativistic quantum mechanics, involved by Eq.(16") with mo changed to mi=βmo and the suitable coupling constant mic2 are defined in relation to the coordinate systems Ki moving with velocities vi relative to Kabs, as well. And, since the observers are associated to Ki not to Kabs, the writing of these equations in Ki corresponds to the foundation of the relativistic quantum mechanics as an operational theory, in full agreement with the definition of such a theory given in Appendix 2. Concerning the new coupling constant mic2 between the systems of subquantum particles2, it reveals an extra-coupling between these systems and the state of motion of a quantum particle defined by vi. It is this extra-factor that which predicted by the energy-momentum relationship is related to the existence of an absolute coordinate system in physics despite our impossibility of identifying it in nature as a system of reference bodies. The usual work with Eq.(16') and, implicitly, with the equations of the relativistic quantum mechanics defined in relation to Kabs mathematically originates in our possibility to drop β2 in Eq.(16"), while physically in that the coordinate systems Ki of the observers can actually be related to the stationary coordinate systems Ξi as shown in Sect.3.6 in ref.1. References 1A.C.V. Ceapa, General Physics/9911067. 2A.C.V. Ceapa, Physical Grounds of Einstein's Theory of Relativity. Roots of the Falsification of 20th Century Physics. (3rd Edition, Bucharest, 1998), Part III. 3L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (Pergamon, N.Y.,1980) Ch.6. 4E.R. Caianiello, in The 5th Marcel Grossman Meeting (Eds. D.G. Blair and M.J. Buckingham, World Scientific, Singapore, N. Jersey, London, Hong Kong, 1989) vol.1, p.94. 5C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973) p.946. 6A.C.V. Ceapa, Phys. Essays 4 (1991) 60. 7L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Ch.18, 46, 102. 8P. Fortini, F. Fuligni and C. Gualdi, Lett. Nuovo Cim. 23(1978) 345. 9A.C.V. Ceapa, in Abstr. Contrib. Papers. 12th Internatl. Conf. on Gen. Relativity and Gravitation (Boulder,Colorado,1989), vol.1, p.158. 10A.C.V. Ceapa, Ann. N.Y. Acad. Sci. 470 (1986) 366. 11A.C.V. Ceapa, in Abstr. Contrib. Papers. 10th Internatl. Conf. on Gen. Relativity and Gravitation (Eds. B. Bertotti, F. de Felice and A. Pascolini, Rome, 1983), vol.2, p.904. 12L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Chs.84, 88.

arXiv:physics/9912041v1 [physics.plasm-ph] 21 Dec 1999Dynamic structure factor and collective excitations of neutral and Coulomb fluids J.Ortner to be published in Phys. Scripta Institut f¨ ur Physik, Humboldt Universit¨ at zu Berlin, Inv alidenstr. 110, D-10115 Berlin, Germany Abstract The dynamic sructure factor as the basic quantity describin g the collective excitations in a fluid is considered. We consider the cases of neutral and Coulombic fluids. The classical method of mome nts is applied to construct the dynamic structure factor satisfyi ng all known sum rules. An interpolational formula is found which expres ses the dynamic characteristics of a classical or quantum fluid syst em in terms of its static correlation parameters. The analytical resul ts based on 1the theory of moments are compared with Molecular dynamics d ata for various model systems. 1 Introduction In the past there has been considerable interest in the time d ependence of correlation functions or equivalently of the frequency dep endence of structure factors. These functions has been studied in neutral and Cou lomb fluids both theoretically and by molecular-dynamic simulations [1]. U nder a neutral fluid we understand here a fluid of particles interacting via a short-ranged potential. That means the termin neutral fluids includes suc h nonneutral systems as dusty plasmas [2] and charged colloidal suspensi ons [3] where the interaction between the charged particles of one subsystem is screened by the motion of particles from another subsystem. Several approaches are devoted to the study of dynamic prope rties of strongly interacting fluid systems. In a rather incomplete l ist we mention the approaches in Refs. [4, 5] based on the memory function fo rmalism, the approaches based on the theory of moments [6, 7], and the appr oach based on the quasilocalized charge approximation [8]. It is interes ting to note that all 2these approaches succeeded by exploiting the method of coll ective variables [9] in various modifications. This paper gives a short overview of the application of the me thod of mo- ments to the determination of dynamic properties of coupled fluid systems. As the main quantity describing the dynamics of a systems we c onsider the dynamic structure factor. The dynamic structure factor may be measured in scattering experiments. The peaks in the dynamic structure factor determine the collective excitations of the system. There may propaga te different col- lective excitations depending on the type of the system (neu tral or Coulomb). Generally speaking in neutral fluids we deal with sound modes whereas the plasma mode is a finite frequency mode. The different behavior of the modes is connected with the different behavior of the interaction p otential Fourier transform at small wavenumbers k. In a neutral fluid the Fourier transform is finite, in the Coulomb case it diverges for k→0. 2 Dynamic properties of neutral fluids We consider a system of Nparticles of one species with masses mand in- teracting via a pair potential V(r). The Fourier transform of the interaction 3potential satisfies the inequality V(k= 0)<∞. The Hamiltonian of the neutral fluid reads: H=N/summationdisplay i=1p2 i 2m+1 2/summationdisplay i/negationslash=jV(xi−xj), (1) where piis the ith particle momentum. In what follows we will use a classical notation, though all the calculations are easily generaliz ed to the quantum case. Define the particle density and its Fourier transform n(r, t) =/summationdisplay iδ(r−x(t)), nk(t) =/summationdisplay ieik·xi(t)(2) ,the density-density correlation function g(r, t) =/an}bracketle{tn(r, t)n(0,0)/an}bracketri}ht. (3) and the dynamic structure factor S(k, ω) =1 2πn/integraldisplay∞ −∞ei(ωt−k·r)g(r, t)dt dr (4) In order to construct the dynamic structure factor as a centr al function for the determination of the dynamic properties of the system it is useful to consider the frequency moments of the dynamic structure fac tor: Mn(k) =/integraldisplay∞ −∞ωnS(ω,k) =in N/angbracketleftiggdn dtnnk(t)n−k(0)/angbracketrightigg t=0(5) 4Due to the parity of the structure factor all moments with odd numbers are equal to zero. The zeroth and second moments read M0(k) =S(k), (6) M2(k) =k2 mkBT . (7) where S(k) = (1 /N)/angbracketleftig nkn−k/angbracketrightig is the static structure factor of the fluid. The fourth moment includes particle correlations and reads, M4(k) = 3 k4(kBT)2/m2+Mpot 4(k) Mpot 4(k) =N V m2k4V(k)kBT+1 V m2(8) /summationdisplay q/negationslash=−k[S(k+q)−S(q)] (k·q)2kBT V(q). (9) The Nevanlinna formula of the classical theory of moments [6 ] expresses the dynamic structure factor S(k, z) =1 πImEn+1(k, z) +qn(k, z)En(k, z) Dn+1(k, z) +qn(k, z)Dn(k, z)(10) in terms of a function qn=qn(k, z) analytic in the upper half-plane Im z >0 and having a positive imaginary part there Im qn(k, ω+iη)>0, η > 0, it also should satisfy the limiting condition: ( qn(k, z)/z)→0 asz→ ∞ within the sector θ <arg(z)< π−θ. The polynomials Dn(andEn) can be found in terms of the first 2 nmoments as a result of the Schmidt orthogonalization 5procedure. The first orthogonal polynomials read [6] D1=z , D 2=z2−ω2 1, D 3=z(z2−ω2 2), (11) E1=M0, E 2=M0z , E 3=M0(z2+ω2 1−ω2 2), (12) where ω2 1(k) =M2(k)/M0(k) and ω2 2(k) =M4(k)/M2(k). Consider first the approximation n= 1 leading to the correct frequency moments M0andM2. Using the Nevanlinna formula Eq. (10) we obtain ( q1=q1,r+iq1,i), S(k, ω) =S(k) πq1,i(k, ω)ω2 1 [ω2−ω2 1(k) +q1,r(k, ω)ω]2+q2 1,i(k, ω)ω2.(13) We have no phenomenological basis for the choice of that func tionq1(z) which would provide the exact expression for S(k, z). We mention that the physical meaning of the function h1(z) =−iq1(z) is that of a memory function since from Eq. (13) it follows that the inverse Fourier transform o f the function C(k, z) = (1 /iπ)/integraltext∞ −∞S(k, ω)/(z−ω) obeys the equation ∂2C(k, t) ∂t2+ω2 1C(k, t) +/integraldisplayt 0ds h1(k, t−s)∂C(k, s) ∂s= 0. (14) A simple approximation is to put the function q1(z) equal to its static value q1(z) =q1(0) = iν(k,) and Eq. (14) simplifies to the equation of a damped oscillator with frequency ω1and damping constant ν. ∂2C(k, t) ∂t2+ω2 1C(k, t) +ν(k,)∂C(k, t) ∂t= 0. (15) 6From Eq. (15) follows the dispersion relation of collective excitations in a classical neutral fluid, ω2 c(k) =ω2 1(k) =M2(k) M0(k)=k2kBT mS(k). The corresponding generalization to the quantum case ( T= 0) reads ω0(k) =¯hk2 2mS(k)[10]. Consider now the long-wavelength behavior k→0. In this case the static structure factor S(k→0) =nkBTκTis determined by the compressibility κT=−(1/V) (∂V /∂P )T. Then the dispersion relation reads ω2 c(k) =u2k2, u2=/parenleftigg∂P ∂ρ/parenrightigg T=v2 s γ, γ=cp cv, (16) which differs from the familiar dispersion equation for the s ound wave by the factor γ. For a model of independent oscillators: cp=cvandγ= 1. Therefore the above approximation for the static structure factor based on the Nevanlinna equation with n= 1 represents the model of independent damped quasiparticles. To go beyond this approximation one has to choose the 3-momen t ap- proximation n= 2 in the Nevanlinna hierarchy reproducing the moments M0,M2andM4. Within this approximation and choosing q2(k, ω) =h(k) we obtain the following expression for the dynamic structur e factor: S(k, ω) =S(k) πh(k)ω2 1(k) (ω2 2(k)−ω2 1(k)) ω2(ω2−ω2 2)2+h2(k)(ω2−ω2 1)2, (17) 7where h(k) has to be taken from the relation h(k) = (ω2 2−ω2 1)/ν(k) = (S(k)/π)((ω2 2/ω2 1−1)/S(k,0)) (18) in order to satisfy the exact low freqency behavior S(k,0). The value S(k,0) may be taken from elastic scattering experiments, from anot her theory or it may be used as a fit parameter. Consider again the long wave-length limit k→0. Then the frequencies ω2 1(k) =u2k2, u2=/parenleftigg∂P ∂ρ/parenrightigg T(19) ω2 2(k) =v2k2, v2=n mV(0) + 3kBT m. (20) At small temperatures kBT≪nV(0) we have u2=v2and we obtain the dynamic structure factor for a “classical” fluid at low tempe rature S(k, ω) =πkBT mu2{δ(ω−ku) +δ(ω+ku)}, (21) representing undamped sound waves. The corresponding gene ralization to a quantum fluid reads S(k, ω) =π¯hk mu(1−exp(−¯hku/k BT))/braceleftig δ(ω−ku) +e−¯hku/k BTδ(ω+ku)/bracerightig ,(22) At zero temperature the system may only absorb energy and we o btain the simple equation for the dynamic structure factor. S(k, ω) =π¯hk muδ(ω−ku). (23) 83 Dynamic properties of Coulomb fluids Consider a one component plasma (OCP) consisting of Nparticles with charges Zeand masses minteracting via the Coulomb potential Vc(r) = Z2e2/rand embedded in a neutralizing homogeneous background. The clas- sical OCP may be characterized by the coupling parameter Γ =e2 akBT,a= (3V/4πN)1/3being the Wigner-Seitz radius. The quantum plasma has an ad- ditional parameter - the degeneration parameter θ=√2mkBT/¯h2(3π2n)2/3. In what follows for the case of simplicity we concentrate on a classical plasma. For Γ≪1 we deal with an ideal (or Vlasov) plasma, for Γ ≫1 the plasma is called a strongly coupled one. The Vlasov approximation tak es into account only the mean field part of the interaction and the dispersion relation for the longitudinal plasmons is predicted as ω2 c(k) =ω2 p/parenleftbigg 1 + 3k2 k2 D/parenrightbigg with the plasma frequency ω2 p=4πZ2e2n mand the squared inverse Debye-length k2 D=4πZ2e2n kBT. The Vlasov theory predicts a strong positive dispersion of t he plasmons, i.e., dω/dk > 0. However, in a coupled plasma the potential energy plays an im- portant role and the Vlasov approximation is not longer vali d. To construct the dynamic structure factor for a coupled plasma consider t he frequency moments of the dynamic structure factor S(k, ω). The frequency moments formally coincides with that of a neutral plasma (Eqs. (6)-( 8)). The only dif- 9ference is that the interaction potential of the neutral flui d has to be replaced by that of the Coulomb system. The application of the Nevanli nna formula leads then to a corresponding hierarchy of approximations f or the dynamic structure factor. If one is interested in the structure fact or of a quantum system again Eqs. (10) hold, if one replaces S(k, ω) on the left hand side of the Eqs. (10) by the loss function R(k, ω= [(1−exp(−β¯hω))/β¯hω]S(k, ω). However, in the quantum case additional contributions to th e zeroth and fourth frequency moment occur [6, 7]. Consider the 3-moment approximation Eq. (17). If one is inte rested in the investigation of the high-frequency collective excita tion spectrum only it is sufficient to neglect the function h(k) since the damping (described by the function h) is small in strongly coupled plasmas. If one puts h(k) = 0 Eq. (17) provides the expression of the dynamic structure f actor for a strongly coupled plasma obtained within the QLC approach [8 ], if the thermal contributions may be neglected with respect to the correlat ion contributions. Within the simple approximation h(k) = 0 the dynamic structure factor has δpeaks at the frequencies ωcwhich in the classical case are determined by 10the equation ω2 c(k) =M4 M2=ω2 p 1 + 3k2 k2 D+1 N/summationdisplay q/negationslash=−k[S(k+q)−S(q)](k·q)2 k2q2 .(24) Fork→0 the dispersion relation simplifies and we get ω2 c(k) =ω2 p/parenleftigg 1 + 3k2 k2 D+4 45Ec kBTnΓk2a2/parenrightigg withEcbeing the correlation energy density. Using the simple esti mation Ec kBTn=−0.9Γ valid in the strong coupling regime one obtains the disper - sion relation ω2 c(k) =ω2 p[1 +k2a2(−0.08 + Γ−1)] and one predicts a negative dispersion for Γ >13. To study the dynamic structure factor one has to go beyond the simple approximation h= 0. To satisfy the low frequency behavior one may choose the approximation Eq. (18). To check the quality of the predi ctions from our approximation molecular dynamic simulations have been performed for comparison [7]. The semiclassical simulations were perfor med to model a quantum gas of 250 electrons moving in a cubic box with period ic bound- ary conditions.The thermal equilibrium was established by a Monte Carlo procedure. A detailed description of the semiclassical mod el used in the simulations may be found elsewhere [7]. In Figs. 1 and 2 we hav e plotted the loss function R(q, ω) (q=ka) for various values of wavenumbers qfor 11the cases of strong (Γ = 10 ) and very strong coupling (Γ = 100 ) [ 7]. In both cases we obtain a sharp plasmon peak at small q values, wi th increas- ing wavenumber the plasmon peak widens. Almost no dispersio n has been observed at Γ = 10. This is in good agreement with the above est imation for the critical value Γ = 13 separating regimes with positiv e dispersion from that with negative dispersion. For the case of very strong co upling Γ = 100 we have found a strong negative dispersion. In Figs. 3 and 4 we present the results of the MD data and compare them with our analytical ap proxima- tion Eqs. (17) and (18). To calculate the parameters ω1(k) and ω2(k) we have used the static structure factor obtained from the HNC e quations. The value S(k,0) determining the parameter h(k) might be taken from the MD simulations. However, the dynamic structure factor at the z ero frequency can be obtained with the necessary accurazy only from long ti me simula- tions. Alternatively we have choosen the value S(k,0) to fit the model to the MD data. It should be mentioned that the value S(k,0) mainly determines the width of the plasmon peak, the peak position is quite inse nsitive to the choice of the value S(k,0). From the figures it can be seen that there is a reasonable agreement between the MD data and the present app roxiamtion based on the sum rules. The peak position is reproduced with h igh accuracy, 12the agreement in the width of the peaks is less satisfactory. One concludes that the static approximation q2(k, ω) =ih(k) undersetimates the damping of the quasiparticles. 4 Conclusions In this paper we have shown that the application of the classi cal theory of moments gives a satisfactory description of many propertie s of neutral and Coulomb fluids. The Nevanlinna formula generates approxima te expressions for the dynamic structure factor in terms of their static cor relations. The quality of the Nevanlinna expression mainly depends on the q uality of the model used to calculate the static properties of the fluid. Th e presented results may be improved by a specification of the interpolati on function q2(k, ω). In conclusion, the present approach has been also used to cal culate the dynamic structure factor of two-dimensional electron gas [ 11], of binary ionic mixtures [12] and of two-component plasmas [13]. It had been extended to magnetized plasmas [14] and can be generalized to calculate partial dynamic structure factors. Here, the matrix form of the Nevanlinna f ormula becomes 13helpful. References [1] For a review of earlier papers, see M. Baus and J. P. Hansen , Phys. Rep. 59, 1 (1980).. [2] H. Thomas, G. E.Morfill, V. Demmel, J. Goree, B. Feuerbach er, and D. Mohlmann, Phys. Rev. Lett. 73, 652 (1994). [3] R. T. Farouki and S. Hamaguchi, Appl. Phys. Lett. 61, 2973 (1992). [4] J.P. Hansen and I.K. McDonald, Phys. Rev. A 23, 2041 (1981). [5] P. John and L.G. Suttorp, Physica A 197, 613 (1993). [6] V.M. Adamyan and I.M. Tkachenko, Teplofiz. Vys. Temp. 21, 417 (1983) [High Temp. (USA) 21, 307 (1983)]. [7] J. Ortner, F. Schautz, and W. Ebeling, ibid.56, 4665 (1997); W. Ebeling and J. Ortner, Phys. Scr. 75, 93 (1998). [8] G. Kalman and K.I. Golden, Phys. Rev. A 41, 5516 (1990);an appli- cation of QLC approximation to dusty plasmas my be found in: M . Rosenberg and G. Kalman, Phys. Rev. E 56, 7166 (1997). 14[9] J. Ortner, Phys. Rev. E, 59, 6312 (1999). [10] Feynman, R.P., Phys. Rev 91, 1291 (1953). [11] J. Ortner and I.M. Tkachenko, Phys. Rev. A 46, 7882 (1992). [12] S. V. Adamjan and I. M. Tkachenko, Ukr. Fiz. Zh. 361336 (1991). [13] S. V. Adamjan, I. M. Tkachenko, J. L. Munoz-Cobo Gonzale s and G. Verdu Martin, Phys. Rev. E 48N3 (1993). [14] J. Ortner, V.M. Rylyuk, and I.M. Tkachenko, Phys. Rev. E 50, 4937 (1994). 15FIGURE CAPTIONS (Figure 1) The simulation data for the loss function R(q, ω) versus fre- quency ω/ω pfor different wavevectors q=kaat Γ = 10 and θ= 1. (Figure 2) Same as in Fig. 1 at Γ = 100 and θ= 50. Figure 3 Comparison of the loss function R(q, ω) within the present sum rules approach (Eqs. (17 and (18) with S(k, ω) replaced by R(k, ω)) versus frequency ω/ω pwith the corresponding MD data at Γ = 100 andθ= 50 for wavevector q= 0.619. Figure 4 Same as Fig.3; at Γ = 100 and θ= 50 for wavevector q= 1.856, [7] . 160.0 1.0 2.0 3.0 4.0 w (in units of the plasmasfrequency)0.010.020.030.040.050.0R(q,w) / R(q,0)q=0.619 q=1.238 q=1.856 q=2.475 Figure 1: The simulation data for the loss function R(q, ω) versus frequency ω/ω pfor different wavevectors q=kaat Γ = 10 and θ= 1. 170.0 0.5 1.0 1.5 ω/ωp0.010.020.030.040.050.0R(q,ω)/R(q,0)q=0.619 q=1.237 q=1.856 q=3.094 Figure 2: Same as in Fig. 1 at Γ = 100 and θ= 50 180.0 0.5 1.0 1.5 2.0 ω/ωp0.010.020.030.040.050.0R(q,ω)/R(q,0)sum rules approach MD results Figure 3: Comparison of the loss function R(q, ω) within the present sum rules approach (Eqs. (17 and (18) with S(k, ω) replaced by R(k, ω)) versus frequency ω/ω pwith the corresponding MD data at Γ = 100 and θ= 50 for wavevector q= 0.619. . 190.0 0.5 1.0 1.5 2.0 w ( in units of the plasmafrequency )0.05.010.015.0R(q,w) / R(q,0)"quantum" MD results sum rules approach Figure 4: same as Fig.3; at Γ = 100 and θ= 50 for wavevector q= 1.856, [7] . 20

arXiv:physics/9912042v1 [physics.comp-ph] 21 Dec 1999A Monte Carlo code for full simulation of a transition radiation detector M.N. Mazziotta1 Dipartimento di Fisica dell’Universit´ a and INFN Sezione d i Bari, via Amendola, 173, I-70126 Bari (Italy) Abstract A full simulation of a transition radiation detector (TRD) b ased on the GEANT, GARFIELD, MAGBOLTZ and HEED codes has been developed. This s imulation can be used to study and develop TRD for high energy particle i dentification using either the cluster counting or the total charge measurement method. In this article it will be also shown an application of this simulation to the di scrimination of electrons from hadrons in beams of momentum of few GeV/c or less, assuming typical TRD configuration, namely radiator–detector modules. Key words: Monte Carlo; Full Simulation; Transition Radiation; TRD; C harge Measurement; Cluster Counting. (To be submitted to Computer Physics Communication) 1 Introduction Transition radiation (TR) is an electromagnetic radiation produced by ultrarelativistic charged particles crossing the interfa ce between two materials with different dielectric properties [1,2]. The TR spectrum is peaked in the X-ray region and the probability of a X-ray photon being emit ted at each interface is of the order of α≃1/137. The transition radiation yield is proportional to the Lorentz factor γof the incident charged particle and is independent on the kind of particle. That offers an attractiv e alternative to identify particles of given momentum with a non destructive method. 1fax: +39 080 5442470; e-mail: mazziotta@ba.infn.it Preprint submitted to Elsevier Preprint 2 February 2008In order to enhance the TR X-ray production, radiators consi sting of several hundred foils regularly spaced or irregular radiators of fe wcmof thickness consisting of carbon compound foam layers or fiber mats are us ually adopted. The “multilayer” radiator introduces significant physical constraints on the radiation yield, because of the so-called “interference eff ects”. It has been established that the radiation emission threshold occurs a t a Lorentz factor γth= 2.5ωpd1, whereωpis the plasma frequency (in eV units) of the foil material, and d1is its thickness in µm. Forγ≥γththe radiation yield increases up to a saturation value given by γsat∼γth(d2/d1)1/2, whered2is the width of the gap between the foils [3]. The conventional method of TR detection is the measurement o f the sum of the energy released by ionization and from photoelectrons p roduced by TR X-rays. The radiating particle, if not deflected by magnetic fields, releases its ionization energy in the same region as the X-ray photons , introducing a background signal that can be reduced if a gaseous detector i s used. Since the gas must provide efficient conversion of the TR photons, th e use of high- Z gases is preferred. The detector usually consists of propo rtional chambers filled with argon or xenon with a small addition of quenching g ases for gain stabilization ( CO2,CH4). The measurement of TR using proportional chambers is genera lly based on one or both of the following methods: •the “charge measurement” method, where the signal collecte d from a chamber wire is charge analyzed by ADCs [4]; •the “cluster counting” method, where the wire signal is shar ply differentiated in order to discriminate the X-ray photoelec tron clusters producing pulses (hits) exceeding a threshold amplitude fr om theδ-ray ionization background [5]. In both cases a cut on the analyzed charge or on the number of cl usters is needed in order to discriminate radiating particles from sl ower nonradiating ones. Multiple module TRDs, with optimized gas layer thickn ess, are normally employed to improve background rejection. A reduced chambe r gap limits the particle ionizing energy losses, while the X-rays escaping detection may be converted in the downstream chambers. Transition radiation detectors are presently of interest i n fast particle identification, both in accelerator experiments [6,7] and i n cosmic ray physics [8]-[16]. A TRD is used to evaluate the underground cosmic ra y muon energy spectrum in the Gran Sasso National Laboratory [17]. In spit e of their use in several high energy experiments, a simulation code is not ye t available in the standard simulation tools. Several codes based on parameterizations of test beam measu rements have 2been developed to simulate the TRDs [18,19]. Lately a TRD has been proposed in a Long Base Neutrino Oscillation Experiment [20], in whic h a simulation has been developed using a GEANT interface [21]. The results achieved in the last experience have been rather satisfactory, in spite of s ome difficulties to track low energy photons in GEANT. In this paper a full simulation of a TRD is described. The prog ram is based on GEANT [22], GARFIELD [23], MAGBOLTZ [24] and HEED [25] cod es in order to exploit the best performances in each one. In this wa y a full simulation has been developed tracking the particles into the detector and producing the pulse shape from each proportional tubes. 2 Transition radiation emission Extensive theoretical studies have been made about TR. The b asic properties of the TR production as well as the interference phenomena in multifoil radiator stacks are rather well understood and well describ ed with classical electromagnetism (for instance see [26]). There was also an attempt to give a quantum description of TR [27]. The quantum corrections to t he TR intensity become interesting for the emission of very high energy phot ons, namely when the TR photon energy is comparable with the energy of the radi ating particle. Therefore they are no longer significant in the X-ray region f or incident charged particle of momenta of few GeV/c and the expressions derived are similar to the classical theory. Therefore, the TR emission is descr ibed for practical purposes by classical formulation, and the TR energy is cons idered carried out by photons (quanta). As shown by Artru et al. [3] the TR energy Wemitted from a stack of Nfoils of thickness d1at regular distances d2, without taking into account absorption effects, can be written as: d2W dω dθ2=η4 sin2φ1 2 sinNφ 2 sinφ 2 2 (1) Where η=α π/parenleftigg1 γ−2+θ2+ξ2 1−1 γ−2+θ2+ξ2 2/parenrightigg2 θ2(2) is the energy emitted at each interface. In eq. (1) and (2) θis the angle between the incident particle and the TR X-ray, and ξi=ωi/ωwhereωis the 3TR quantum energy (in eV units) and ωiare the plasma energies of the two media “1” (foil) and “2” (gap). The factor 4 sin2φ1 2in eq. (1) is due to the coherent superposition of TR fields generated at the two interfaces of one foil, with the phase an gleφ1=d1/z1 being the ratio of the foil thickness d1(inµm units) to the “formation zone” z1of the foil material: z1=/parenleftig 2.5ω(γ−2+θ2+ξ2 1)/parenrightig−1(3) The last factor of eq. (1) describes the coherent interferen ce of TR in a stack composed of Nfoils and gaps at regular distances d2.φ=φ1+φ2is the total phase angle of one basic foil plus gap, with φ2being defined in analogy to φ1. The TR X-ray energy distribution can be obtained by taking t he ratio of equation (1) to ω. Since the TR yield from multifoil stack is described as an int erference phenomenon due to whole radiator, in order to calculate the t otal TR quanta emitted by the particle crossing the radiator, one needs to k nown the total number of foils crossed. Therefore it is not possible to foll ow the particle into radiator in order to calculate the probability to emit a quan tum in a given step, i.e. we do not have a cross section for the TR effect. That may introduce some difficulties to simulate the TR process. Moreover, the TR intensity is a complex function of the thicknesses d1andd2, of the plasma energies ω1and ω2for a given γLorentz factor. This behaviour may introduce an additional difficulty to calculate the TR spectra for any kind of radiator s. The energy of the TR photons depends on the radiator material and its structure. In ref. [3] it is shown that the average TR energy c arried out by quanta is given by: <ω> ≃0.3γthω1 (4) Assumingd1= 10µmandω1= 20eVone obtains γth∼500 and < ω > ∼3keV. This may introduce some difficulties to track soft X-ray photons in a medium. The ability to identify particles by a TRD is determined by th e relative amounts of TR and ionization energy loss in the proportional chambers. Large fluctuations of ionization loss in thin gas layers limit this methods. Therefore, in order to better understand the performance of a TRD, one ne eds careful calculations of ionization energy loss and its fluctuations , producing knock-on orδ-electrons. On the other hand, if one would like to use the clu ster counting method to separate the TR X-ray from the track ionization bac kground, then the range and the size of δ-electron and of photoelectron, the number of 4electron–ion pairs produced in the gas and their arrival tim e on the wire need to be taken into account. Finally the current produced on the anode wire of the gas chambers and its pulse shape fed to discriminator by t he front end electronics also play an important role in this method. 3 TRD full simulation On the basis of the above discussion, the approach followed t o simulate a TRD is based on the codes GEANT, GARFIELD, MAGBOLTZ and HEED (the last two codes are used by GARFIELD). The geometric descript ion of the detector has been given by GEANT, including the simulation o f all physical processes that occur in the materials crossed by the particl es. The ionization energy loss and the photoelectric process in the gas have bee n not considerated in the GEANT code, because they are simulated by HEED. When charged particles cross the gas of proportional chambe rs, or photons are entering into these volumes, the HEED package is called. In this way the ionization energy loss and the electron–ion pairs distr ibution along the track are calculated. The photoelectric absorption of phot ons in the gas is also simulated, including the evaluation of the photoelect rons produced and the total number of electron–ion pairs. Finally the current pulse produced on the anode wire is evaluated by the GARFIELD code using the gas properties as its drift velocity and gain calculated by the MAGBOLTZ pro gram as a function of the electric field. 3.1 TR process The GEANT code does not simulate transition radiation. In or der to produce the TR photons in GEANT, a physical process has been introduc ed whenever a relativistic charged particle crosses the radiator. The TR photon energy spectrum and the mean number of X-ray are calculated for the input radiator and for the energy of primary particle which one simulates. When the charged particle crosses the radiator a nd comes out the TR process is activated. The total number of TR photons is generated according to a Poisson distribution if their average number is less than 10, otherwise a Gaussian distribution may be used. The energy of each TR X-ray is randomly generated according to a calculated spectrum an d its position is generated along the radiating particle path at the end of a radiator. The produced TR photons are then treated as secondary particles in GEANT and they are stored in the common block GCKING. In order to be tran sported 5by GEANT, these photons are stored in the data structure JSTA K by the GSKING routine. 3.1.1 TR formulas used in the code The TR production relations used in this simulation take int o account the photon absorption in the radiator. This effect has been simul ated using the GEANT absorption lengths of the photons calculated for this material. Regular radiator The energy distribution of TR photons for a stack of plates ta king into account the absorption in the foils and gaps is given by [3]: d2N dω dθ2=1 ωη4 sin2φ1 2 sin2Nφ 2+ sinh2Nσ 2 sin2φ 2+ sinh2σ 2 e−N−1 2σ(5) whereσ=d1/λ1+d2/λ2is the absorption in one foil + one gap and λ1andλ2 are the absorption lengths for the emitted radiation in two m edia as calculated by GEANT (see paragraph 3.2). For large values of the number of foils N, theδfunction can be assumed to approximate the last two factors of the above expression. Making this approximation and integrating over θ2, equation (5) becomes: dN dω=1 ω4α Nequ 1 +τ/summationdisplay nθn/parenleftigg1 ρ1+θn−1 ρ2+θn/parenrightigg2 (1−cos (ρ1+θn)) (6) where ρi= 2.5d1ω(γ−2+ξ2 i); τ=d2/d1; θn=2π n−(ρ1+τρ2) 1 +τ>0; Nequ=1−e−Nσ 1−e−σ. Nequis the number of equivalent foils when the absorption is take into account. To evaluate the total number of TR photons the numerical calc ulation of equation (6) has been carried out at selected X-ray energies (ω), from 1keV to 100keV, with a precision better than 10−3. In Fig. 1 the TR spectra for a 6regular radiator, evaluated taking into account the absorp tion in the radiator, are shown. They are calculated from the eq. (6). This figure sh ows a broad peak around 3 −5keVenergy, corresponding to TR mean energy produced by the regular radiator adopted. Irregular radiator The transition radiation has been observed in irregular mat erials consisting for instance of plastic foams. A general formulation of the s pectral distribution of the number of TR X-ray quanta produced in a irregular mediu m, consisting of randomly parralel plates of arbitrary thickness, is give n by Garibian et al. [28]. This formulation has been given with the plates arrang ed in vacuum. It has been modified to take into account the presence of a materi al in the gap. The average number of radiation quanta taking into account t he absorption of the radiation is given by: <d2N dω dθ>=2α π ω/parenleftigg1 1−β2ǫ1+θ2−1 1−β2ǫ2+θ2/parenrightigg2 θ3I (7) Here I= 21−pN 1−pRe(1 +p 2−h1)−(p−h11 +p 2)h2 1−h1h2+ (8) 2Re(1−h1) (p−h1)h2(pN−hN 1hN 2) (1−h1h2) (p−h1h2) is the factor due to the superpositions of the radiation field s in the plates and in the gap. The other parameters are: ǫk= 1−(ωk/ω)2+i/(5λkω); hk=<e−i φkdk>; φk= 5ω/parenleftbigg β−1−/radicalig ǫk−sin2θ/parenrightbigg =φ′ k+iφ′′ k; p=<e−d1/λ1> < e−d2/λ2>. The angle brackets denote the averaging of random quantitie s with a distribution determined by the distributions of d1andd2. For most of foam radiators the random foil and the gap thickne ss can be described by a gamma distribution [29]. In this way one finds t hat [28]: 7hk=|hk|ei ψk; |hk|= /parenleftigg 1 +<dk> 2λkαk/parenrightigg2 +/parenleftiggφ′ k<dk> αk/parenrightigg2 −αk/2 ; ψk=−αkarctgφ′ k<dk> αk+<dk>/(2λk); p=/parenleftigg 1 +<d1> λ1α1/parenrightigg−α1/parenleftigg 1 +<d2> λ2α2/parenrightigg−α2 . The parameters αkrepresent the degree of irregularity: αk= (<dk>/σk)2 where< dk>andσkare the mean values and the mean squared deviations respectively of foil ( k= 1) and gap ( k= 2) thickness distributions. 3.2 Use of the GEANT package The GEANT 3.21 code is used to describe the geometrical volum es inside the detector and to define the materials. It has been done by the st andard GEANT routine taking care of tracking parameters in order to define the active physical processes and the cuts (GSTPAR). In this way, the photons are tracked using the GEANT absorption coefficients and the gamma cuts have been lowered to 1keVin all the materials. The materials used, which are not defined in the default GEANT program, have been implemented using the standard routine (GSMATE or GSMIXT). The radiators have been defined as a mixture composed by the fo il material and the gap material (air) containing the proportion by weig hts of each material. The foil materials and the gas chamber walls have b een defined as compounds containing the proportion by number of atoms of each kind [22]. In Fig. 2 the photon attenuation lengths calculated by GEANT for polyethylene ( CH2,ρ= 0.93g/cm3), kapton (C22H10N2O5,ρ= 1.42g/cm3) and mylar ( C5H4O2,ρ= 1.05g/cm3) are shown. In this figure one can see that the kapton photon attenuation length is always less tha n polyethylene and the photon attenuation length for kapton is the same as fo r mylar. The gas chambers are the sensitive volume of the TRD and for ea ch charged particle crossing the gas or for each photons absorbed insid e, a GEANT HITS structure is defined to describe the interaction between par ticle and detector. In the HITS structure the following information are stored: •HITS(1) = number of volume level (by GEANT); •HITS(2) = energy loss in the gas (by HEED); 8•HITS(3) = input time in the volumes (by GEANT); •HITS(4:6) = x, y and z of entry point in the volume (by GEANT); •HITS(7:9) = x, y and z of exit point in the volume (by GEANT); •HITS(10) = number of cluster produced in the gas (by HEED); •HITS(11) = number of electron–ion pairs produced in the gas ( by HEED); •HITS(12:111) = current pulse on the wire for 100 time slices ( by GARFIELD). The DIGIT structure is similar to the HITS one, where the info rmation are stored as a sum of all particles crossing that volume, while t he input and the output coordinate are relative to the primary particle whic h has crossed the chamber. The event processing is a highly CPU consuming job. To optimi ze CPU usage DST files are produced to be analyzed at a later time. For each e vent the GEANT ZEBRA data structures containing the geometrical defi nition, the input kinematics, the tracking banks (JXYZ) and the simulat ed detector response (HITS and DIGIT banks) are stored in DST files which p rovide the input data set for the analyses to be performed. In this wa y, the electronic response of the chamber front end can be implemented startin g by the anode current impulse. In order to save some run informations the H EADER bank is also used by the GSRUNG routine. 3.3 Use of the GARFIELD package The GARFIELD program has been developed to simulate gaseous wire chambers operating in proportional mode. It can be used for i nstance to calculate the field maps and the signals induced by charged pa rticles, taking both electron pulse and ion tail into account. An interface t o the MAGBOLTZ program is provided for the computation of electron transpo rt properties in nearly arbitrary gas mixtures. Starting from version 6, GAR FIELD has also an interface with the HEED program to simulate ionization of gas molecules by particles traversing the gas chamber. A few examples of GA RFIELD results can find via WWW [23,25]. The HEED program computes in detail the energy loss of fast ch arged particles in gases, taking δ-electrons and optionally multiple scattering of the incom ing particle into account. The program can also simulate the abs orption of photons through photo-ionization in gaseous detectors. Fr om this program, the distribution of electron–ion pairs along the particle trac k length in the gas has been computed by GARFIELD. Some modifications have been incl uded in the GARFIELD default version in order to calculate the cluster s ize distribution of photons absorbed in the gas by HEED. Starting from these cl uster size 9distributions the current anode wire signal is calculated b y GARFIELD. In Fig. 3 the pair distribution produced by 5 .9keVphotons (55Fe) in 1cm of xenon at NTP is shown. From this figure one can see the presen ce of a mean peak of about 270 electron–ion pairs due to the photoele ctron and the Auger electron. There is also a secondary peak due to occasio nal detection of a photoelectron whitout Auger emission. In this TRD simulation the GARFIELD 6.27 version has been use d. From the source files of GARFIELD program, written in Fortran 77 an d Patchy as pre-processor, the main routines have been included in th e code together with the GEANT routines. Some modifications have been introd uced in order to skip interactive input information used by GARFIELD. All information to run the program are given via FFREAD data card. The cell defini tion and the gas composition of the chambers to be simulated have been processed in initialization of the program. 4 Program description The main items of this simulation have already been describe d in the above discussion. In this section an example of how the program wor ks is given. It has been written in Fortran by patchy as pre-processor on a PC 166 MHz, 80MB of RAM, in the LINUX system (RedHat 5.2 version). It is transp ortable on any system changing some patchy control flags in cradle files. There are two codes: the first is dedicated to event simulatio n for DST production; the second one is used to analyze the DST files inc luding a graphical interface too. The input of these program is given via data cards by FFREAD facility. The user inputs for the first program are s tored in the run header bank after the initialization to be used by the sec ond one. 4.1 Geometry The geometry used to simulate a TRD consists of 10 radiator-p roportional chamber modules. The radiator consists of 250 polyethylene foils of 5µmof thickness at regular distances of 200 µmin air. The chamber consists of two planes of 16 cylindrical proportional tubes each of 2 mmof radius (straw tubes) to form a double layer close pack configuration. These tubes a re widely used in recent high energy physics experiments [19,30]. Since th e typical materials used for the tube wall are made by carbon compounds (kapton, m ylar and polycarbonate) and their thickness are typically 30 −50µm, the straw tubes 10are good candidate to be used as X-ray detector due to the redu ced attenuation length of the wall. In this simulation the straw tube walls are made of kapton of 3 0µmthickness internally coated with copper of 0 .3µmthickness. The anode wire used is of 25µmthickness. The gas mixture used is based on Xe(80%) −CO2(20%) at atmospheric pressure. The anode voltage used is 1450 Voltwhich corresponds to a gas gain of about 2 ·104. 4.2 Front end electronic The front end electronic used in this simulation consists of a simply amplifier which is described by a low band-pass transfer function with a bandwidth of 50MHz and an overall gain of 10: ˜A(ω) =A01 1 +iω ω0(9) whereA0= 10mV/µA andω0= 50MHz. The anode current produced in the proportional tubes as a fun ction of the timeI(t) is converted in the output voltage amplitude V(t) by: V(t) =/integraldisplay∞ tI(t′)A(t−t′)dt′(10) whereA(t) is the Fourier transform of ˜A(ω): A(t) =  ω0e−ω0t, if t ≥0 0, otherwise(11) In this example no noise is assumed. Of course a real electron ics is described by a more complex transfer function with an electronic noise . In Fig. 4 a typical anode signal from a tube produced by a X-ray of 5.9keV (55Fe) is shown. When this signal is processed by the low band-pa ss it assumes the shape reported in Fig. 5. From this figure one can see that t he electronics performed a formation of the input signal with a FWHM of about 25nsec. In Fig. 6 is shown a typically anode signal produced by a charg ed particle crossing a tube. In this figure one can see two peaks are produc ed by two clusters. The low band-pass cannot allow to distinguish the two clusters since the second one is superimposed to first one (signal pile-up) a s shown in Fig. 117, because their time distance is lower than the FWHM of the el ectronic resolution. 4.3 Results In this paragraph the results achieved by the TRD geometry de fined above are shown. In Fig. 8 the average energy loss (summed over 10 plane s) as function of the Lorentz factor is shown. This result has been obtained by simulating pions and electrons of different energies with or without rad iators. For each energy 1000 events have been simulated. In this figure one can see that the yield increases with γwhen the radiators are arranged before the proportional tubes. The TR saturati on is achieved at γ≃8000. Forγless than 100–500 only the ionization is released in the gas, as is shown in the same figure. In Fig. 9 the average energy loss distributions (summed over 10 planes) for electrons of 4 GeV/c and pions of 255 MeV/c are shown. From this figure it is possible to see that the average value of the electron dist ribution is greater than the average for pions. This is due to presence of the X-ra y TR produced in the radiator by the electrons. In order to perform the cluster size analysis, one needs to kn ow the relationship between the output signal amplitude and the energy loss in th e tube. Therefore an analysis of voltage amplitude has been done using X-rays o f 5.9keV(55Fe). In Fig. 10 is shown the output voltage amplitude distributio n produced by a 55Fe X-rays absorbed in a proportional tube. From this figure on e can see that the energy loss of 5 .9keVcorresponds to 170 mVof output voltage amplitude. In order to count the number of hits produced for instance by T R photons and byδ-ray with energy greater than 5 keV, a cut of 145 mVis imposed to the voltage amplitude signal produced in each tube. In Fig. 11 th e average total number of hits (summed over all fired tubes) when the output si gnal is greater than 145mVas function of γis shown. The behaviour of the TRD when is analyzed by the cluster counting method is similar to the cha rge measurement one. In Fig. 12 the distributions of the total number of hits for el ectrons of 4GeV/c and pions of 255 MeV/c are shown. Again we can observe that the average value of the electron distribution is greater th an the one of the pion distribution, due to presence of the X-ray TR produced i n the radiator by the electrons. In order to discriminate electrons from pions at given momen tum by charge 12measurement or by cluster counting, we can use this simulati on to optimize the gas thickness, the radiator, the threshold and the numbe r of modules. In this way, we can optimize one of these methods or we can use m ore sophisticated ones, for example analyzing the pulse shape a s function of the drift time or using the likelihood and/or neural network ana lysis by the pattern information, namely the fired tube configuration in the TRD. 5 Conclusions A full simulation of a transition radiation detector (TRD) b ased on the GEANT, GARFIELD, MAGBOLTZ and HEED codes has been developed . The simulation can be used to study and develop TRD for high en ergy particle identification using either the cluster counting or the tota l charge measurement method. The program works very well according to the design e xpectations. It is quite flexible and it can be used to simulate any detector which is based on proportional counters, providing a very useful simulati on tool. Acknowledgements I am grateful to Prof. P. Spinelli for useful discussions, su ggestions and continuous support. I would like to thank my colleagues of Ba ri University and INFN for their contributions. References [1] V. L. Ginzburg and I. M. Frank, JETP 16(1946) 15 [2] G. M. Garibian, Sov. Phys. JETP 6(1958) 1079 [3] X. Artru et al., Phys. Rev. D 12 (1975) 1289 [4] J. Fischer et al., Nucl. Instr. and Meth. 127(1975) 525 [5] T. Ludlam et al., Nucl. Instr. and Meth. 181(1981) 413 [6] C. Camps et al., Nucl. Instr. and Meth. 131(1975) 411 [7] B. Dolgoshein, Nucl. Instr. and Meth. A 326 (1993) 434 [8] T. A. Prince et al., Nucl. Instr. and Meth. 123(1975) 231 [9] G. Hartman et al., Phys. Rev. Lett. 38(1977) 368 13[10] S. P. Swordy et al., Nucl. Instr. and Meth. 193(1982) 591 [11] K. K. Tang, The Astroph. Journ. 278(1984) 881 [12] J. L’Heureux, Nucl. Instr. and Meth. A 295 (1990) 245 [13] S. W. Barwick et al., Nucl. Instr. and Meth. A 400 (1997) 34 [14] R. L. Golden et al., The Astr. Journ. 457(1996) L103 [15] E. Barbarito et al. Nucl. Instr. and Meth. A 313 (1992) 295 [16] E. Barbarito et al. Nucl. Instr. and Meth. A 357 (1995) 588 [17] E. Barbarito et al. Nucl. Instr. and Meth. A 365 (1995) 214; The MACRO Collaboration (M. Ambrosio et al.), Proc. XXIV ICRC, Rome, 1(1995) 1031; The MACRO Collaboration (M. Ambrosio et al.), Proc. XXV ICRC, Du rban, (1997); The MACRO Collaboration (M. Ambrosio et al.), Nuclear Physi cs61B(1998) 289; The MACRO Collaboration (M. Ambrosio et al.), Astropar ticle Physics 10 (1999) 10; The MACRO Collaboration (M. Ambrosio et al.), Pro c. XXVI ICRC, Salt Lake City, (1999), hep-ex 9905018 [18] M. Castellano et al., Comput. Phys. Commun. 61(1990) 395 [19] T. Akesson et al., Nucl. Instr. and Meth. A 361 (1995) 440 [20] G. Barbarino et al., The NOE detector for a long baseline neutrino oscillation experiment , INFN/AE-98/09 (1998) [21] P. Bernardini et al, GNOE: GEANT NOE simulation , Internal note 2/98 (1998) (unpublished) [22] R. Brun et al., CERN Publication DD/EE/84-1 (1992) [23] R. Veenhof, GARFIELD, a drift-chamber simulation program ,W5050 (1999); http://consult.cern.ch/writeup/garfield/ [24] S. Biagi MAGBOLTZ, a program to compute gas transport parameters W5050 (1997) [25] I. Smirnov, HEED, an ionization loss simulation program W5060 (1995); http://consult.cern.ch/writeup/heed/ [26] C.W. Fabjan and W. Struczinski, Phys. Rev. Lett. 57B(1975) 483 [27] G.M. Garibian, Sov. Phys. JETP 12(1961) 1138 [28] G.M. Garibian et al., Sov. Phys. JETP 39(1974) 265 [29] C.W. Fabjan, Nucl. Instr. and Meth. 146(1977) 343 [30] E. Barbarito et al. Nucl. Instr. and Meth. A 361 (1996) 39 14List of Figures 1 The TR spectra generated by 250 foils of polyethylene (d1= 5µmandω1= 20eV) at regular distances d2= 200µm in air (ω2= 0.7eV). Solid line: γ= 5000; dashed line: γ= 1000 and dotted line: γ= 500. 16 2 Photon attenuation length for different materials as calcu lated by GEANT routines in the range from 1 keVto 100keV. Solid line: polyethylene; dashed line: kapton and dotted li ne: mylar. 17 3 Electron–ion pairs distribution for 1 cmof xenon at NTP produced by photons of 5 .9keV(55Fe). 18 4 Anode current signal produced by a X-ray of 5 .9keVabsorbed in a tube. 19 5 Output amplitude voltage produced by a X-ray of 5 .9keV absorbed in a tube as processed by the low band-pass electronic. 20 6 Anode current signal produced by a charged particle crossi ng a tube. 21 7 Output amplitude voltage produced by a charged particle crossing a tube as processed by the low band-pass electronic . 22 8 Average energy loss (summed over 10 planes) as a function of the Lorentz factor. The error bars have been evaluated as rat io of the RMS over the square root of the number of events. 23 9 Average energy loss (summed over 10 planes) distribution f or twoγvalues. Solid line: pions of 255 MeV/c ; dashed line: electrons of 4 GeV/c 24 10 Output voltage amplitude distribution (histogram) prod uced by X-rays of 5 .9keV. The line is the result of a Gaussian fit 25 11 Total number of hits with a signal greater than 145 mVas a function of the Lorentz factor. The error bars have been evaluated as ratio of the RMS over the square root of the number of events. 26 12 Hits distribution for two γvalues. Solid line: pions of 255MeV/c ; dashed line: electrons of 4 GeV/c 27 1510-410-310-210-1 1 10 102 TR X-ray energy (keV)dN/dω (keV)-1 Fig. 1. The TR spectra generated by 250 foils of polyethylene (d1= 5µmand ω1= 20 eV) at regular distances d2= 200 µmin air ( ω2= 0.7eV). Solid line: γ= 5000; dashed line: γ= 1000 and dotted line: γ= 500. 1610-410-310-210-1110 1 10 102 Photon energy (keV)Attenuation length (cm) Fig. 2. Photon attenuation length for different materials as calculated by GEANT routines in the range from 1 keVto 100 keV. Solid line: polyethylene; dashed line: kapton and dotted line: mylar. 170255075100125150175200 0 50 100 150 200 250 300 350 400Entries Mean RMS 1000 249.6 63.61 Total number of electron-ion pairs in XeNumber of events Fig. 3. Electron–ion pairs distribution for 1 cmof xenon at NTP produced by photons of 5 .9keV(55Fe). 18-160-140-120-100-80-60-40-200 0 20 40 60 80 100 120 140 Time (nsec)Wire Current ( µA) Fig. 4. Anode current signal produced by a X-ray of 5 .9keVabsorbed in a tube. 19-160-140-120-100-80-60-40-200 0 20 40 60 80 100 120 140 Time (nsec)Output Voltage (mV) Fig. 5. Output amplitude voltage produced by a X-ray of 5 .9keVabsorbed in a tube as processed by the low band-pass electronic. 20-30-25-20-15-10-50 0 20 40 60 80 100 120 140 Time (nsec)Wire Current ( µA) Fig. 6. Anode current signal produced by a charged particle c rossing a tube. 21-80-70-60-50-40-30-20-100 0 20 40 60 80 100 120 140 Time (nsec)Output Voltage (mV) Fig. 7. Output amplitude voltage produced by a charged parti cle crossing a tube as processed by the low band-pass electronic. 2222.533.544.555.566.57 1 10 102103104Electrons with radiator Electrons without radiator Pions with radiator Lorentz factor γAverage energy (keV) Fig. 8. Average energy loss (summed over 10 planes) as a funct ion of the Lorentz factor. The error bars have been evaluated as ratio of the RMS over the square root of the number of events. 230100200300400500 0 2.5 5 7.5 10 12.5 15 17.5 20 Average energy (keV)Number of events Fig. 9. Average energy loss (summed over 10 planes) distribu tion for two γvalues. Solid line: pions of 255 MeV/c ; dashed line: electrons of 4 GeV/c 240102030405060708090 0 50 100 150 200 250 300 119.3 / 45 Constant 76.43 Mean 170.1 Sigma 15.79 Output voltage amplitude (mV)Number of events Fig. 10. Output voltage amplitude distribution (histogram ) produced by X-rays of 5.9keV. The line is the result of a Gaussian fit 2500.511.522.533.54 1 10 102103104 Lorentz factor γAverage total number of hitsElectrons with radiator Electrons without radiator Pions with radiator Fig. 11. Total number of hits with a signal greater than 145 mVas a function of the Lorentz factor. The error bars have been evaluated as rat io of the RMS over the square root of the number of events. 26110102103 0 2 4 6 8 10 12 14 16 Total number of hitsNumber of events Fig. 12. Hits distribution for two γvalues. Solid line: pions of 255 MeV/c ; dashed line: electrons of 4 GeV/c 27

arXiv:physics/9912043v1 [physics.bio-ph] 21 Dec 1999DNA Transport by a Micromachined Brownian Ratchet Device Joel S. Bader∗,†, Richard W. Hammond,†Steven A. Henck,†Michael W. Deem,†,‡Gregory A. McDermott,† James M. Bustillo,§John W. Simpson,†Gregory T. Mulhern,†Jonathan M. Rothberg† (February 21, 2014) ∗Author to whom correspondence should be addressed. †CuraGen Corporation, 555 Long Wharf Drive, New Haven, CT 06511. ‡Present address: Department of Chemical Engineering, University of California, Los Angeles, CA 90095. §Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720. Classification Biological Sciences: Biophysics Physical Sciences: Chemistry Corresponding Author Joel S. Bader, CuraGen, 555 Long Wharf Drive, New Haven, CT, 06511. Tel. (203)401-3330x236; Fax (203)401-3351; Email js- bader@curagen.com Manuscript information: 15 text pages, 4 figures, no ta- bles. Counts: 92 words in abstract; 27,800 characters in paper (counting spaces); 35,720 characters counting figure and equation requirements. Abbreviations: all standard. 1We have micromachined a silicon-chip device that trans- ports DNA with a Brownian ratchet that rectifies the Brown- ian motion of microscopic particles. Transport properties for a DNA 50mer agree with theoretical predictions, and the DNA diffusion constant agrees with previous experiments. This type of micromachine could provide a generic pump or sepa- ration component for DNA or other charged species as part of a microscale lab-on-a-chip. A device with reduced featur e size could produce a size-based separation of DNA molecules , with applications including the detection of single nucleo tide polymorphisms. 2I. INTRODUCTION The Human Genome Project aims to provide the com- plete sequence of the 3 billion base-pairs of the human genome. While the dominant method for analyzing DNA fragments remains gel electrophoresis, new technologies that have the potential to increase the rate and decrease the cost of DNA sequencing and analysis, such as mass spectrometry and hybridization arrays, are critical to the project’s success [1]. Here we describe a novel method of DNA transport and separation based on a Brownian ratchet. As described originally by Smoluchowski and noted by Feynman, a Brownian particle can undergo net transport on a poten- tial energy surface that is externally driven to fluctuate between several distinct states [2,3]. Brownian ratchets have attracted theoretical attention [4–12] due to their description of molecular motors [13–15] and to their sim- ilarity with phenomena termed stochastic resonance and resonance activation [16,17]. Brownian ratchets have been demonstrated to trans- portµm- to mm-sized particles using dielectrophoresis [18], optical tweezers [19], and electrocapillary forces [ 20] to generate ratchets. Other devices based on entropic ratchets [21] or physical barriers [22,23] have been pro- posed as well. More recently, a geometrical sieve device has been used to separate phospholipids [24]. Despite these successes, the Brownian ratchet mecha- nism has not before proved capable of transporting DNA fragments in the size ranges applicable to DNA sequenc- ing (<1000 nt) because the interactions used to estab- lish the ratchet potential were too weak. In contrast to previous devices using polarization interactions to gen- erate ratchets [18,19], we have fabricated a device that uses charge-charge interactions to generate the ratchet potential. As seen below, the charge-charge interactions have sufficient strength to establish ratchets that can trap small DNA fragments. The ratchet-like wells that trap DNA are generated by charging a series of patterned electrodes. When the elec- trodes are discharged, the traps vanish and the molecules undergo Brownian motion. Next the potential is re- applied, and the particles again collect in the traps. A spatial asymmetry in the shape of each ratchet-shaped well rectifies the Brownian motion and produces net transport as the on-off cycle is repeated. Each molecule’s transport rate depends on its diffusion constant, allowing the possibility of size-based separations. We have imple- mented the device by microfabrication on a silicon chip. This report describes the Brownian ratchet theory, provides a derivation of the transport rate, and presents experimental results for a single sized DNA oligomer. Greater details regarding the fabrication methods and amore extensive presentation of the experimental results for a variety devices and oligomer sizes are available else- where [25]. II. THEORY AND METHODS A silicon wafer with six micromachined devices is shown in Fig. 1, with a schematic design below. The electric potential that creates the ratcheting traps is gen - erated by two arrays of interdigitated electrodes that are perpendicular to the transport axis. The two sets of ar- rays each extend from bond pads on opposite sides of the device. The spacing between electrodes extending from the same bond pad is l. The asymmetric pattern creates two different spacings, randl−r, between electrodes extending from opposite bond pads. The smaller spacing rdefines the feature size of the device. As shown in also shown in Fig. 1, a simplified one- dimensional description of the potential approximates the electrodes as infinitely thin wires. To simplify the theoretical model, we have neglected the finite width of the electrode, the dependence of the electric potential on the distance normal to the surface, and the corresponding detailed calculation of the potential along the transport axis. When a voltage difference Vis applied across the two electrode arrays during the on-state, with duration ton, sawtooth-shaped ratcheting traps are created for charged particles. The electrodes are discharged to V= 0 during the off-state, with duration toff, and particles undergo isotropic Brownian diffusion. When the potential is re- applied, the particles are again trapped in potential wells . The times tonandtoffare within the low-frequency, quasi- static approximation; nonadiabatic effects and current reversals, reported elsewhere [26,27], are not applicable in this regime. Due to the asymmetry of the sawtooth shape and the choice of toff, a particle starting from well 0 has a measurable probability to be trapped in well 1 and virtually zero probability to be trapped in well −1. Transport can be generated by repetitive cycling between the on-state and the off-state. The transport properties of a particle are determined by its diffusion constant D and its charge Q, along with the thermal energy kBTand the device parameters defined previously. Each cycle begins by applying the potential Vfor a timetonthat is sufficient to localize particles at the bot- toms of the trapping wells. A particle at the barrier-top drifts down the side of the sawtooth of length l−rto find the bottom of the well, which defines the time required for complete trapping, ton= (l−r)2kBT/QV D , (1) 3according to overdamped Brownian motion. At the end of the trapping, the particle distribution at the bottom of a well is assumed to be much narrower than the feature sizer. In the next phase of the cycle, the potential is turned off for time toff. When the potential is re-applied, parti- cles that have diffused further than the barrier to the left (roughly distance laway) will hop to the previous well, and particles that have diffused further than the barrier to the right (roughly distance raway) will be transported to the next well. Since r≪l, we can choose an intermedi- ate time toffsuch that toffr2/2D≪l2/2Dand particles always move right, never left. To compute the probability αthat a particle moves one well to the right in a single cycle, we note that the effective distance reffit must travel is between r(the inner edge-to-edge distance between electrodes) and 3 r (the outer edge-to-edge distance). The probability dis- tribution (or equivalently the Greens function) for a par- ticle starting at the origin undergoing one-dimensional Brownian motion for time toffis P(x;toff) =exp[−x2/4Dtoff]√4πDtoff. (2) An expression for αis then obtained by integrating P(x;toff) from reffto infinity: α=1 2erfc(/radicalBig r2 eff/4Dtoff) =1 2erfc(/radicalbig tr/2toff).(3) The distance r2 effhas been written in terms of the char- acteristic diffusion time 2 Dtr, where tr=r2 eff/2Dis the time required to diffuse a distance equal to the short sidereffof the trapping well. In this derivation, we assumed that toffis short enough that particles diffuse less than a single ratchet, i.e. P(x;toff)≈0 for x > l. An expression for αvalid in the limit of large toffis α=/summationtext∞ k=−∞k/integraltextkl+reff (k−1)l+reffdxP(x;toff). After each cycle, the particle distribution shifts to the right the distance αl. After ncycles, the envelope of the distribution of particles in each well evolves as a Gaussian with center x(n) and square width σ2(n): x(n) =nlα, (4a) σ2(n) =nl2α(1−α). (4b) Here we have assumed that all the particles are in well 0 at the start of the first cycle. Both αand the steady-state flux of particles through the device, flux =α ton+toff, (5)are plotted in Fig. 2. The transport fraction α(black line) increases with toffand approaches a maximum value of 1/2 when back-diffusion is neglected. The flux in units of the characteristic time trisαtr/(ton+toff) and is shown forton=toff/3. The flux is non-monotonic and exhibits a maximum when toff≈tr. Other voltage modulations more complicated than the on-off pattern described here are also possible and can change the direction of particle flow according to particle size [4]. Other similar types of non-monotonic behavior have been termed stochastic resonance and resonance activation [16,17] although they may also be described by dispersion and linear response [28]. Devices were fabricated from Pt, a non-reactive, corrosion-resistant metal chosen to avoid electrolysis of water, using relatively standard micromachining tech- nologies [25]. The fabrication began with thermally oxi- dized 100 mm diameter silicon wafers. A 200 ˚A thick Ti layer was used as an adhesion layer between the subse- quent Pt layer and the silicon dioxide. Next, a 200 nm thick layer of Pt was deposited on top of the Ti. The electrodes were defined in the metal layers using pho- tolithography and ion milling. For the devices used in this work, the electrodes and the gaps between nearest electrodes were 2 µm and the spatial period was 20 µm. Oligomers labeled with fluorescent rhodamine dye (Amitof Biotech Inc., Boston, MA) were placed on the surface of the chip at 4 pmole/ µl in deionized water. A microscope slide cover glass (Macalaster Bicknell, New Haven, CT) cut to size was used to confine the solution to a uniform thickness of approximately 10 µm. A seal- ing compound was used to obtain a liquid tight seal along the edges, leaving the ends open. Square wave modula- tion (Synthesized Function Generator DS345, Stanford Research Systems, Sunnyvale, CA) with an amplitude of 1.6 V and offset of 0.8 V (one set of electrodes at 0 V and the second set at 1.6 V) was applied to the elec- trodes to generate the flashing ratchet potential. Fre- quencies ranging from 0.7 Hz to 8 Hz were used, with a ratioton/toff= 1/3. Video images of the analyte fluorescence were used to record the DNA transport. Images were captured us- ing a low light imaging CCD camera (MTI VE1000SIT) mounted on an epi fluorescence microscope (Zeiss Ax- ioskop, Germany) using a 10 ×Fluor objective. The chip was illuminated using the output of a 50 W mer- cury lamp filtered with a green band pass filter. The brightfield fluorescence was imaged through a red low pass filter. The video images were recorded on video tape and transferred to a PC using a composite color PCI bus frame grabber (DT3153 Data Translation, Inc., Marlboro, MA). The fluorescence intensity resulting from the fluorescently labeled DNA fragments was analyzed in 4a line across the video image synchronized to the tonpe- riod (HL Image++97, Western Vision Software, Utah). According to Eq. 1, the expected time required to focus a DNA 50mer is 0.006 sec, based on a thermal energy of 26 meV, a charge of 1 |e|−per nucleotide yielding QV= 80 eV, and a diffusion constant of 1 .8×10−7cm2/sec (es- timated from a rhodamine-labeled 30mer on a quartz sur- face [29]). We visually ascertained that the trapping time tonwas sufficient to permit complete focusing of the DNA on the positive electrodes, even for the fastest switching rate of 8 Hz ( ton= 0.03 sec). III. RESULTS AND DISCUSSION In Fig. 3 we show three images from a typical ex- periment using a device with 2 µm electrodes and a 0.7 Hz switching frequency to transport a rhodamine- labeled DNA 50mer. These images were saved during the trapping phase of the cycle, and fluorescence maxima are clearly seen from DNA molecules captured on the posi- tive electrodes. At the start of the experiment, the DNA oligomers are focused on left-most three electrodes. As the potential cycles between on and off states, the packet moves to the right and broadens. The transport rate α can be estimated by noting that the intensity maximum moves from electrode 1 to electrode 3 after 10 cycles, then to electrode 5 after 10 more cycles, yielding α≈0.2. For a more precise value, we extracted the intensity profile across the image, set a baseline at the 85thpercentile of intensity, calculated the average position x(n) from the intensity above baseline, normalized x(n) by the 10-pixel spacing between electrodes, then measured the slope of x(n) to obtain α. For this experiment α= 0.18. The square width also increases linearly with the num- ber of cycles (data not shown); because we used a base- line threshold that narrows the width of the distribution, however, the formula of Eq. 4b no longer provides an accurate relationship between σ2(n) and α. Fig. 4 shows the experimental results for α, along with ±1σerror bars from repeated runs. As predicted, αde- creases with increasing frequency. Also shown in Fig. 4 is a theoretical curve from Eq. 3. Least-squares fitting to the data from all but the highest frequency yielded D(r/reff)2= 3.5×10−8cm2/sec. The good agreement be- tween the experimental results and the single-parameter theoretical fit supports our conclusion that transport is due to a Brownian ratchet. Furthermore, since reff≈(2–3)r, we find that D= 1.4– 3×10−7cm2/sec. This indirect measure of the diffusion constant brackets the estimated diffusion constant for a DNA 50mer close to a glass surface, 1 .8×10−7cm2/sec [29].The experimental results demonstrating transport per- mit an examination of the feasibility fabricating a device to provide useful size-based separations of DNA. Separat- ing two chemical species requires that they have different diffusion constants DandD′and different hopping prob- abilities αandα′. The resolution between the species, defined as resolution =|x(n)−x′(n)| 0.5·[σ(n) +σ′(n)], (6) improves as n1/2. The number of cycles required to reach the resolution of 1 typical for DNA separation applica- tions is n=α(1−α) (α−α′)2, (7) where we have assumed that the two packets acquire a similar width. The separation parameter toff(and, through toff, the quantities αandα′) can be selected to optimize various quantities associated with a resolved separation, for example the device length, approximately l×[α/(α−α′)]2, or the total separation time, n×(ton+ toff). A typical application requiring the separation of DNA fragments is the analysis of single nucleotide polymor- phisms (SNPs). An SNP is a position in the genome where multiple nucleotides are likely. Characterizing ge- netic diversity through SNPs has applications including the identification of genes for disease inheritance and sus- ceptibility, the development of personalized medicines, and the documentation of human evolutionary history and migrations through a genetic record [30–33]. Prelim- inary sets of thousands of SNPs, identified by the 12 nt on either side of a polymorphism, have been reported [34]. Validating and detecting these SNPs can be ac- complished by resequencing specific 25 nt regions of the genome. Here we investigate the use of a Brownian ratchet de- vice for the resequencing and detection application. Cal- culations are based on a device with feature size r= 0.1µm, periodicity 1 µm, and a potential difference of 0.1Vbetween electrodes in the on-state. We estimate the effective DNA diffusion constant D(r/reff)2from the the- oretical scaling for a self-avoiding walk, D∼(length)−0.6 [35,36], and our experimental results for the 50mer. The persistence length of single-stranded DNA is 4 nm, or 13.6 nt [37], indicating that the self-avoiding walk should be adequate (although not quantitative) for fragment sizes we consider. The detection of an SNP requires, at most, the abil- ity to sequence the 25 nt region surrounding the poly- morphism, which can be accomplished by separating a 5DNA 24mer from a 25mer respectively. Using the the- oretical scaling of diffusion constant with DNA length, we extrapolate D(r/reff)2of 5.44×10−8cm2/sec and 5.31×10−8cm2/sec for the 24mer and 25mer. The time required to focus the DNA at the start of each cycle is ton≈1.8×10−4sec. The optimized separation parame- ters, calculated using Eq. 7, require toff= 2.5×10−4sec and 12,000 cycles for a total separation time of 5.4 sec on a 1.25 cm chip. In conclusion, we have fabricated a Brownian ratchet device that is capable of transporting small DNA molecules in aqueous solution, rather than the inconve- nient gel and polymer solutions required for electrophore- sis. This type of device could be used as a pump compo- nent for transport or separation of charged species in a microfabricated analysis chip. Multiple miniaturized de- vices can also be arrayed side-by-side for high-throughput operation. Based on experimental measurements, we suggest the feasibility of using this type of device for bio- logical applications, for example the validation of SNPs. ACKNOWLEDGMENTS We wish to acknowledge the support of SBIR grant 1 R43 HG01535-01 from the National Human Genome Research Institute and Advanced Technology Program award 1996-01-0141 from the National Institute of Stan- dards and Technology. We acknowledge the assistance of Rajen Raheja for image analysis. ∗Author to whom correspondence should be addressed. †CuraGen Corporation, 555 Long Wharf Drive, New Haven, CT 06511. ‡Present address: Department of Chemical Engineering, University of California, Los Angeles, CA 90095. §Department of Electrical Engineering and Computer Sci- ence, University of California, Berkeley, CA 94720 [1] Rowen, L., Mahairas, G., & Hood, L. (1997) Science 278, 605. [2] von Smoluchowksi, M. (1912) Physik. Z. XIII, 1069. [3] Feynman, R. P., Leighton, R. B., & Sands, M. (1966) in Feynman Lectures in Physics (Addison-Wesley, Reading, MA), p. 46-1. [4] Bier, M. & Astumian, R. D. (1996) Phys. Rev. Lett. 76, 4277. [5] Astumian, R. D. & Bier, M. (1993) Phys. Rev. Lett. 72, 1766. [6] Astumian, R. D. & Bier, M. (1996) Biophys. J. 70,637.[7] Astumian, R. D. (1997) Science 276,917. [8] Magnasco, M. O. (1993) Phys. Rev. Lett. 71,1477. [9] Doering, C. R., Horsthemke, W., & Riordan, J. (1994) Phys. Rev. Lett. 72,2984. [10] H¨ anggi, P. & Bartussek, R. (1996) in Lecture Notes in Physics 476 , eds. Parisi, J., M¨ uller, S. C., & Zimmerman, W. (Springer, New York, Berlin), p. 294. [11] Prost, J., Chauwin, J.-F., Peliti, L., & Ajdari, A. (199 4) Phys. Rev. Lett. 72,2652. [12] J¨ ulicher, F., Ajdari, A., & Prost, J. (1997) Rev. Mod. Phys. 69,1269. [13] Howard, J., Hudspeth, A. J., & Vale, R. D. (1989) Nature 342,154. [14] Kuo, S. C. & Sheetz, M. P. (1993) Science 260,232. [15] Svoboda, K., Schmidt, C. H., Schnapp, B. J., & Block, S. M. (1993) Nature 365,721. [16] Gammaitoni, L., H¨ anggi, P., Jung, P., & Marchesoni, F. (1998) Rev. Mod. Phys. 70, 223. [17] Doering, C. J. & Gadoua, J. C. (1992) Phys. Rev. Lett. 69,2318. [18] Rousselet, J., Salome, L., Ajdari, A., & Prost, J. (1994 ) Nature 370,446. [19] Faucheux, L. P., Bourdieu, L. S., Kaplan, P. D., & Libch- aber, A. J. (1995) Phys. Rev. Lett. 74,1504. [20] Gorre, L., Ioannidis, E., & Silberzan, P. (1996) Europhys. Lett.33, 267; Gorre-Tallini, L., Spatz, J. P., & Silberzan, P. (1998) Chaos 8,650. [21] Slater, G. W., Guo, H. L. & Nixon, G. I., (1997) Phys. Rev. Lett. 78,1170. [22] Ertas, D. (1998) Phys. Rev. Lett. 80,1548. [23] Duke, T. A. J. & Austin, R. H. (1998) Phys. Rev. Lett. 80,1552. [24] van Oudenaarden, A., & Boxer, S. G. (1999) Science 285, 1046. [25] Hammond, R. W., Bader, J. S., Henck, S. A., Deem, M. W., McDermott, G. A., Bustillo, J. M., & Rothberg, J. M. (1999) Electrophoresis in press. [26] Bartussek, R., H¨ anggi, P., & Kissner, J. G. (1994) Euro - phys. Lett. 28, 459. [27] Jung, P., Kissner, J. G., and H¨ anggi, P. (1996) Phys. Rev. Lett. 76, 1166. [28] Robertson, B., & Astumian, R. D. (1991) J. Chem. Phys. 94,7414. [29] Xu, X.-H. & Yeung, E. S. (1997) Science 275, 1106. The diffusion constant of dye-labeled 30mer on quartz was measured to be approximately 2.6 ×smaller than the free-solution value of 6.2 ×10−7cm2/sec. Similarly, the diffusion constant of the bare dye was 7 ×smaller than the bulk value of 2.8 ×10−6cm2/sec. We estimate the dif- fusion constant of a 50mer based on the self-avoiding walk scaling, D50mer/D30mer = (50 /30)−0.6, which gives D= 1.8×10−7cm2/sec. [30] Risch, N. & Merikangas, K. (1996) Science 273,1516; [31] Lander, E. S. (1996) Science 274,536; [32] Collins, F. S., Guyer, M. S., & Chakravarti, A. (1997) 6Science 278,1580; [33] Kruglyak, L. (1997) Nature Genet. 17,21. [34] Wang, D. et al. (1998) Science 280,1077. [35] Flory, P. J. (1953) Principles of Polymer Chemistry (Cor- nell University Press, New York); [36] Doi, M. (1986) The Theory of Polymer Dynamics (Clarendon Press, New York). [37] Grosberg, A. Y. (1994) Statistical Physics of Macro- molecules (AIP Press, New York). FIG. 1. The Brownian ratchet device is shown in schematic. Modulating the electric potential at the electr odes generates a ratchet-like potential energy surface for char ged molecules like DNA. Cycling the ratchet between an on-state and an off-state generates transport. FIG. 2. The probability αthat a particle hops one well to the right during a single cycle of device operation is shown as a function of the ratio tr/toff(black line). For large toff,α approaches 0.5 as we do not consider back-diffusion. The flux in units of the characteristic time trisαtr/(ton+toff) and is shown for ton=toff/3 (grey line). The flux is reminiscent of a stochastic resonance with a maximum when toff≈tr. FIG. 3. Three images are shown from a typical experiment using a device with 2 µm electrodes and a 0.7 Hz switch- ing frequency to transport a rhodamine-labeled DNA 50mer. These images were saved during the trapping phase of the cycle, and fluorescence maxima are clearly seen from DNA molecules captured on the positive electrodes. At the start of the experiment, the DNA oligomers are focused on left-most three electrodes. As the potential cycles between on and off states, the packet moves to the right and broadens. FIG. 4. Experimental results for the transport of a DNA 50mer by a device with 2 µm electrodes and 20 µm periodic- ity (points with 1 σerror bars) are compared with theoretical predictions (line). The theory requires a single adjustabl e parameter related to the diffusion constant of DNA. 7Bader et al. / DNA Transport Fig. 1 +− Transport Axisr lSchematic of interdigitated electrodes, On-state of first cycle Theoretical model, On-state of first cycle Off-state of first cycle On-state of second cycleQV Free diffusion toff Retrapping tonReturn to original wellRatchet forward Silicon wafer with 6 devices Bader et al. / DNA Transport Fig. 2 00.10.20.30.40.5 0.01 0.1 1 10 100 tr/toff α flux Bader et al. / DNA Transport Fig. 3 0 cycles 10 cycles 20 cycles distance → time ↓ Bader et al. / DNA Transport Fig. 4 00.10.20.30.4 0.1 1 10 100 Frequency / Hz α

arXiv:physics/9912044v1 [physics.atom-ph] 22 Dec 1999Relativistic photoelectron spectra in the ionization of at oms by elliptically polarized light J. Ortner Institut f¨ ur Physik,Humboldt Universit¨ at zu Berlin, Inv alidenstr. 110, 10115 Berlin, Germany (February 2, 2008) Relativistic tunnel ionization of atoms by intense, ellipt ically polarized light is considered. The relativistic version of the Landau-Dykhne formula is emplo yed. The general analytical expression is obtained for the relativistic photoelectron spectra. The m ost probable angle of electron emission, the angular distribution near this angle, the position of th e maximum and the width of the energy spectrum are calculated. In the weak field limit we obtain the familiar non-relativistic results. For the case of circular polarization our analytical results ar e in agreement with recent derivations of Krainov [V.P. Krainov, J. Phys. B, 32, 1607 (1999)]. PACS numbers:32.80.Rm, 32.90.+a, 42.50.Hz, 03.30.+p I. INTRODUCTION Recently an increasing interest in the investigation of rel ativistic ionization phenomena has been observed [1–10]. Relativistic effects will appear if the electron velocity in the initial bound state or in the final state is comparable wit h the speed of light. The initial state should be considered re lativistic in the case of inner shells of heavy atoms [3,4]. In a recent paper [10] the photoionization of an atom from a sh ell with relativistic velocities has been considered for the case of elliptically polarized laser light. In the prese nt paper we will study the effect of a relativistic final-state of an electron on the ionization of an atom by elliptically po larized light. The initial state will be considered as nonrelativistic. The final-state electron will have an ener gy in the laser field measured by the ponderomotive energy. If the ponderomotive energy approaches the electron rest en ergy, then a relativistic treatment of the ionization proce ss is required. For an infrared laser the necessary intensitie s are of the order of 1016W cm−2. Ionization phenomena influenced by relativistic final state effects have been studied for the cases of linearly and circularly polarized laser radiation both in the tunnel [6, 9] and above threshold regimes [2,7]. The ionization rate for relativistic electrons has been found to be very small fo r the case of linear polarization [6,7]. On the contrary a circularly polarized intense laser field produces mainly re lativistic electrons [2,9]. In the papers of Reiss and of Crawford and Reiss [1,2,7] a cova riant version of the so-called strong field approximation [11] has been given for the cases of linear and circular polar ization. Within this approximation one calculates the transition amplitude between the initial state taken as the solution for the Dirac equation for the hydrogen atom and the final state described by the relativistic Volkov solutio n. Coulomb corrections are neglected in the final Volkov state. Analytical results for the ionization rate have been given in Refs. [1,2,7] These results apply to above barrier cases as well as to tunneling cases. However, the correspond ing expressions are complicated and numerical calculation s are needed to present the final results. The present paper is aimed to investigate the relativistic e lectron energy spectra in the ionization of atoms by intense elliptically polarized laser light. In contrast to the more sophisticate d strong field approximation we would like to obtain simple analytical expressions from which the dependence of the ionization process on the parameters, such as binding energy of the atom, field strength, frequency and ellipticity of the laser radiation may be understood without the need of numerical calculations. Therefore we re strict the considerations to the case of tunnel ionization. Our results will be applicable only for laser field strengths smaller than the inner atomic field F≪Fa. In order to observe relativistic effects, the inequality ǫ=F/ωc > 0.1 should be fulfilled. (The atomic system of units is used throughout the paper, m=e= ¯h= 1.) Both inequalities yield a limitation for the laser freq uency ωfrom above. For the ionization of multi-charged ions an infrared laser sati sfies this condition. The non-relativistic sub-barrier ionization with ellipti cally polarized light was studied in [12]. In the tunnel limi t the simple expression Wnonrel∝exp/braceleftbigg −4 3γ ωEb/bracketleftbigg 1−1 10/parenleftbigg 1−g2 3/parenrightbigg/bracketrightbigg/bracerightbigg exp/braceleftBigg −γ ω/bracketleftBigg/parenleftbigg pz−gF ω/parenrightbigg2 +p2 x/bracketrightBigg/bracerightBigg (1) has been derived for the electron momentum spectrum within e xponential accuracy. Here pxandpzare the projections of the drift momentum on the direction of the wave propagatio n and along the smaller axis of the polarization ellipse, 1respectively; Ebis the ionization energy of the atomic state, F,ωandgare the field amplitude, frequency and ellipticity of the laser radiation, respectively; and γ=ω√2Eb/F≪1 is the Keldysh adiabatic parameter. From Eq. (1) one concludes that the electrons are mainly ejec ted in the polarization plane along the smaller axis of polarization; the most probable momentum at the time of ejec tion has the components: px=py= 0 and pz=gF/ω. (For the sake of simplicity of the notations we neglect throu gout the paper the second symmetric maximum for the component pz,pz=−gF/ω.) II. RELATIVISTIC SEMICLASSICAL APPROACH We shall now generalize the non-relativistic result Eq. (1) to the case of relativistic final state effects, when gF/ω becomes comparable with the velocity of light. Our derivati on starts with the relativistic version of the Landau- Dykhne formula [3,5,10]. The ionization probability in qua siclassical approximation and with exponential accuracy reads W∝exp{−2 Im ( Sf(0;t0) +Si(t0))}, (2) where Si=−E0t0is the initial part of the action, Sfis the final-state action. In the latter we will neglect the influence of the atomic core. Then the final-state action may b e found as a solution of the Hamilton-Jacobi equation and reads [13] Sf(0;ξ0) =c/braceleftBigg rξ0+ǫc qω[−pycosωξ0−pzgsinωξ0] +ǫ2c2 4q/bracketleftbigg/parenleftbigg 1 +g2)ξ0+g2−1 2ωsin2ωξ0/bracketrightbigg/bracerightbigg . (3) Here the vector potential of the laser radiation has been cho sen in the form Ax= 0, A y=−cF ωsinωξ , A z=gcF ωcosωξ, , (4) where ξ=t−x/c,ξ0is the initial value. Further the notations r=/radicalbig c2+p2, q =r−px (5) have been introduced; px,pyandpzare the components of the final electron momentum along the be am propagation, along the major and along the small axis of the polarization e llipse, respectively; p2=p2 x+p2 y+p2 z. The complex initial time t0has to be determined from the classical turning point in the c omplex half-plane [3,5,9,10]: Ef(t0) =c/braceleftBigg r+ǫc q[pysinωt0−gpzcosωt0] +ǫ2c2 2q/bracketleftbiggg2+ 1 2+g2−1 2cos2ωt0/bracketrightbigg/bracerightBigg =E0=c2−Eb. (6) Eq. (2) together with Eqs. (3) and (6) is the most general expr ession for the relativistic rate of sub-barrier ionization by elliptically polarized laser light. We consider now the lim it of a nonrelativistic initial state, i.e. Eb≪c2. Furthermore the considerations will be restricted to the tunnel regime λ=−iωt0≪1, or, equivalently, the Keldysh adiabatic parameter should satisfy the inequality γ≪1. Under these conditions we may expand the sine and cosine fu nctions in Eqs. (3) and (6) in Taylor series. Then we obtain the rate of tunnel ionization for arbitrary final-state momenta. Expanding this expression near its maximum value in terms of the parameters q,pyandpzone arrives at the following general expression Wrel∝exp/braceleftbigg −4 3γ ωEb/bracketleftbigg 1−γ2 10/parenleftbigg 1−g2 3/parenrightbigg −Eb 12c2/bracketrightbigg/bracerightbigg exp/braceleftBig −γ ω/bracketleftBig (pz−pz,m)2+ (q−qm)2/bracketrightBig/bracerightBig (7) for the tunnel ionization rate (first exponent) and the momen tum distribution of the photoelectron (second exponent) within exponential accuracy. In Eq. (7) the ionization rate and the most probable value for each component of the electron momentum, py,m= 0 pz,m=F ωg/parenleftbigg 1 +γ2 6/parenrightbigg , q m=c−Eb 3c(8) are given including the first frequency and relativistic cor rections in the initial state. In the distribution near the maximum momenta only those terms have been maintained which do not vanish at zero frequency. Equation (7) agrees 2with the relativistic angular-energy distribution of Krai nov [9] in the case of circular polarization g=±1, vanishing frequency corrections γ2≪1 and negligible relativistic effects in the initial state Eb≪c2. In the nonrelativistic limit,i.e., p≪c,F/wc ≪1 and Eb≪c2, we have q−qm=pxand Eq. (7) reduces to Eq. (1) as it should. From Eq. (8) we easily obtain the most probable value for the c omponent of the electron momentum along the beam propagation px,m=F2g2 2ω2c+Eb 3c/parenleftbig g2+ 1/parenrightbig , (9) the peak value of the angular distribution tanθm=px,m |pz,m|=F|g| 2cω/parenleftbigg 1 +g2+ 2 g2γ2 6/parenrightbigg , ϕ m= 0, (10) and the value of the most probable electron energy Em=p2 m, with pm=/radicalBig p2x,m+p2z,m=F|g| ω/radicalBigg 1 +/parenleftbiggFg 2ωc/parenrightbigg2 +γ2 3+Eb 3c2(g2+ 1). (11) Hereθis the angle between the polarization plane and the directio n of the photoelectron motion; ϕis the angle between the projection of the electron momentum onto the pol arization plane and the smaller axis of the polarization ellipse. For the ellipticity 0 <|g|<1 the most probable momentum pmof the ejected electron is situated in the plane perpendicular to the maximum value of the electric fiel d strength; for |g|= 1 the electron output in the ( y, z) plane is isotropic. Notice that the most probable total elec tron momentum pmcontains relativistic final state effects, frequency corrections and weak relativistic initial state effects. Relativistic effects do not contribute to the projec tion of the momentum along the smaller axis of the polarization el lipse. On the contrary both relativistic final and initial state effects increase the electron momentum projection alo ng the propagation of elliptically polarized laser radiati on. The increase due to relativistic initial state effects is pro portional to ( Eb/c2)(1 +g2). It is typically small (except for the ionization from K shells of heavy atoms [3,10]) and does n ot vanish in the case of linear polarization of the laser light. In contrast to that the relativistic increase due to fi nal state effects which is measured by Fg/2ωcis absent in the case of linear polarized laser radiation. In what follows we will neglect the frequency corrections an d the relativistic initial state effects in order to compare with previous works. In this case and for the case of circular polarization the expressions for the angle θmand the most probable electron momentum pmcoincide with the corresponding expressions of Krainov [9] . Moreover, though our calculations are valid only in the tunnel regime, our value for the most probable angle of electron ejection coincides with an approximation given by Reiss for the case o f circular polarization [1,2] and valid in the above-barrie r ionization regime. In Ref. [2] it has been shown that the simp le estimate tan θm=F/2cωis in good agreement with the numerical calculations based on the strong-field approx imation and performed for above threshold conditions with circularly polarized light. Therefore we expect that our fo rmula Eq. (10) predicts, at least qualitatively, the locati on of the peak in the relativistic angular distribution for the case of above barrier ionization with elliptically polariz ed light. This statement is supported by a semiclassical considerati on of the above barrier ionization. According to the semiclassical model [14] the transition occurs from the bou nd state to that continuum state which has zero velocity at the time twith the phase ξof the vector potential A(ξ). From this condition we have q=/radicalBig c2+p2y+p2z+ǫ2c2g2+ 2ǫc(pysinωξ−gpzcosωξ) + (1 −g2)ǫ2c2sin2ωξ , (12) py=−F ωsinωξ , p z=gF ωcosωξ . (13) The ionization rate becomes maximal at the maximum of the ele ctric field of the laser beam. Due to our choice of the gauge (see Eqs. (4)) this maximum occurs at the phase ξ= 0. From Eqs. (12) and the relation px= (c2−q2+p2 y+p2 z)/2qwe conclude that the most probable final state has the momentu m with the components px=F2g2 2cω2, p y= 0, p z=F ω, (14) which agrees with the above estimations Eqs. (8) and (9) deri ved for the case of tunnel ionization if one neglects the frequency corrections and the relativistic initial state e ffects. 3III. RESULTS AND CONCLUSIONS It is now straightforward to obtain the probability distrib ution for the components of the final state momentum. Neglecting again the frequency corrections and the relativ istic initial state effects we get from Eq. (7) Wrel∝exp/braceleftbigg −4 3γ ωEb/bracerightbigg exp/braceleftBigg −γ ω/bracketleftbig δp2 x−2δpxδpzǫg+p4 y/4c2+δp2 z/parenleftbig 1 + 2ǫ2g2+ǫ4g4/4/parenrightbig/bracketrightbig (1 +ǫ2g2/2)2/bracerightBigg , (15) where only the leading contributions in δpx= (px−px,m),pyandδpz= (pz−pz,m) have been given. In the non- relativistic limit ǫ≪1 and p≪cwe obtain Eq. (1). For the case of linear polarization g= 0 we reproduce the momentum distribution of Krainov [6] including the rela tivistic high energy tail for electrons emitted along the polarization axis. The latter is described by the term exp/braceleftbig −(γ/ω)(p4 y/4c2)/bracerightbig . However, the high energy tail contains only a very small part of the ejected electrons. From Eq. (15) we see that in the case of linear polarization most of the electrons have nonrelativistic velocities. This is i n agreement with recent numerical calculations based on the strong-field approximation [8]. In contrast to the case o f linear polarized laser radiation, the intense elliptical ly polarized laser light with ǫ|g|of the order of unity produces mainly relativistic electron s. For the sake of comparison we shall give the angular distribu tion at the maximum of the electron energy spectrum and the energy spectrum at the peak of the angular distributi on. We obtain both distributions from Eq. (7) by putting px=psinθandpz=pcosθ, where we have taken into account that the ionization rate is maximal for the emission in the ( x, z) plane. Choosing the peak value of the angular distribution θ=θm= arctan ǫ|g|/2 we obtain Wrel∝exp/braceleftBigg −2 3(2Eb)3/2 F/bracerightBigg exp/braceleftBigg −/parenleftbiggp−pm ∆p/parenrightbigg2/bracerightBigg (16) for the energy distribution along the most probable directi on of electron ejection. Here ∆p=/radicalBigg F√2Eb1 + (g2/2)(F/ωc)2 /radicalBig 1 +g2(F/ωc)2, (17) is the width of the relativistic energy distribution. From E q. (17) we conclude that the relativistic width is broader than the nonrelativistic one, it increases with increasing field strength. The relativistic broadening has its maximum for circular polarization, there is no relativistic broade ning of the energy width for the case of linear polarization. In Fig. 1 the electron momentum spectrum from Eq. (16) is show n for electrons born in the creation of Ne8+(Eb= 239 eV) ions by an elliptically polarized laser radiation wi th wave length λ= 1.054µm, field strength 2 .5×1010V/cm and ellipticity g= 0.707. The relativistic spectrum is compared with the spectru m of nonrelativistic theory. From the figure one sees the shift of the energy spectrum to higher e nergies, the relativistic broadening of the spectrum is too small to be observed from the figure. Putting in equation (7) p=pm= (F|g|/ω)/radicalBig 1 + (Fg/2ωc)2we obtain the angular distribution for the most probable photoelectron energy, Wrel∝exp/braceleftBigg −2 3(2Eb)3/2 F/bracerightBigg exp/braceleftBigg −/parenleftbiggθ−θm ∆θ/parenrightbigg2/bracerightBigg , (18) where the width of the angular distribution equals ∆θ=ω |g|/radicalBigg 1 F√2Eb1/radicalbig 1 +g2(F/2ωc)2. (19) We see that the relativistic theory predicts a narrower angu lar distribution as the nonrelativistic theory. The infinit e width in the case of linear polarization is an artefact of the calculations using the angle between the polarization plane and the direction of electron movement. For the linear polarization the electrons are ejected preferably along the polarization axis if one neglects relativistic initial state effects. For the case of circular polarization the ener gy- angular distributions Eqs. (16) and (18) coincide with the c orresponding expressions of Krainov [9]. Notice that our notations slightly differ from those of Krainov. In Fig. 2 we have plotted the relativistic and non-relativis tic angular distributions for the electrons produced by the same process as in Fig. 1. The relativistic distribution has its maximum at the angle θm= arctan ǫ|g|/2 = 16 .71◦and 4the nonrelativistic theory predicts a peak at the zero angle . Again the relativistic reduction of the angular distribut ion width is not observable for the parameters we have chosen. Fr om Figs. 1 and 2 one concludes that the appearance of a nonzero mean of the drift momentum component along the beam p ropagation and the shift of the mean emission angle into the forward direction are the most important indicatio ns for a relativistic ionization process. On the contrary the width of the energy-angle distributions as well as the to tal ionization rate are less sensitive to the relativistic fi nal state effects. In conclusions, in this paper the relativistic semiclassic al ionization of an atom in the presence of intense elliptica lly polarized laser light has been considered. Simple analytic expressions for the relativistic photoelectron spectrum h ave been obtained. For the cases of linear and circular polariza tion our results agree with previous studies. We have shown that the location of the peak in the relativistic angul ar distribution is shifted toward the direction of beam propagation. The theoretical approach employed in the pape r predicts that the maximum of the electron energy spectrum is increased due to relativistic effects. The valid ity of the simple expressions is formally limited to the tunnel regime. Nevertheless, a part of the results, such as t he most probable angle for electron emission, is shown to be valid in the above barrier ionization regime as well. The r esults obtained in this paper within exponential accuracy may be improved by the account of Coulomb corrections. Howev er, whereas the Coulomb corrections may strongly influence the total ionization rate [15,16] we expect only a s mall influence of the atomic core on the electron spectrum. IV. ACKNOWLEDGEMENTS I gratefully acknowledge usefull discussions with V.M. Ryl yuk. This work is financially supported by the Deutsche Forschungsgemeinschaft (Germany) under Grant No. Eb/126- 1. [1] H. R. Reiss, J. Opt. Soc. Am. B 7, 574 (1990). [2] D. P. Crawford and H. R. Reiss, Phys. Rev. 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Gavrila (Academic Press, New York, 1992), p. 109. [15] M.V.Ammosov, N.B.Delone, and V.P.Krainov, Zh.Eksp.T eor.Fiz 91, 2008 (1986) [Sov.Phys.JETP 64, 1191 (1986)]. [16] D. Bauer and P. Mulser, Phys. Rev. A 59, 569 (1999). 5FIGURE CAPTIONS (Figure 1) Electron momentum spectra for electrons produced in the cre ation of Ne8+by an elliptically polarized laser radiation with wave length λ= 1.054µm, field strength 2 .5×1010V/cm and ellipticity g= 0.707 and ejected at the most probable angle θ=θm; the relativistic spectrum is taken from Eq. (16) with θm= 16.71◦, the non-relativistic one from Eq. (1) with θm= 0. (Figure 2) Electron angular distribution at the most probable electro n momentum p=pm. The other parameters are the same as in Fig. 1; the relativistic angular distribut ion is taken from Eq. (18) with pm= 85.91, the non-relativistic one from Eq. (1) with pm= 82.28. 670.0 80.0 90.0 100.0 Electron momentum (in a.u.)0.00.20.40.60.81.0Electron yield (in arbitrary units)relativistic nonrelativistic FIG. 1. 7−10.0 0.0 10.0 20.0 30.0 Angle (in deg)0.00.20.40.60.81.0Electron yield (in arbitrary units)relativistic nonrelativistic FIG. 2. 8

arXiv:physics/9912045v1 [physics.atm-clus] 22 Dec 1999EPJ manuscript No. (will be inserted by the editor) Static Electric Dipole Polarizabilities of Na Clusters S. K¨ ummel1, T. Berkus2, P.-G. Reinhard2, and M. Brack1 1Institute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany 2Institute for Theoretical Physics, University of Erlangen , D-91077 Erlangen, Germany Received: date / Revised version: date Abstract. The static electric dipole polarizability of Na Nclusters with even N has been calculated in a collective, axially averaged and a three-dimensional, fini te-field approach for 2 ≤N≤20, including the ionic structure of the clusters. The validity of a collectiv e model for the static response of small systems is demonstrated. Our density functional calculations veri fy the trends and fine structure seen in a recent experiment. A pseudopotential that reproduces the experim ental bulk bond length and atomic energy levels leads to a substantial increase in the calculated polarizab ilities, in better agreement with experiment. We relate remaining differences in the magnitude of the theoret ical and experimental polarizabilities to the finite temperature present in the experiments. PACS. 36.40.-c Atomic and molecular clusters – 31.15.E Density fu nctional theory in atomic and molecular physics – 33.15.Kr Properties of molecules, electric polar izability 1 Introduction The measurement of the static electric polarizability of sodium clusters [1] and its interpretation in terms of the jellium model [2] was one of the triggers for the research activities that today form the field of modern metal clus- ter physics. The first theoretical studies were followed by several others with different methods and aims: density functional calculations using pseudopotentials [3,4] or t ak- ing all electrons into account [5] aimed at a quantitative description of the experimentally observed effects, semi- classical approaches [6] focused on size-dependent trends , and the static electric polarizability served to test and compare theoretical concepts [7,8,9]. Recently, the field received new inspiration from a second experimental de- termination of the static polarizability of small, uncharg ed Na clusters [10]. Whereas a qualitative understanding of the experi- ments can be obtained with relatively simple models, a quantitative theoretical determination of the polarizabi l- ity requires knowledge of the ionic and electronic config- urations of the clusters. Great effort has been devoted in the past to determine these [4,11,12,13]. However, taking all ionic and electronic degrees of freedom into account in a three-dimensional calculation is a task of consider- able complexity. Therefore, most of these studies were re- stricted to clusters with not more than nine atoms. To reduce the computational expense, approximations for in- cluding ionic effects were developed [14,15,16,17,18,19]. A second problem, however, is the great number of close- lying isomers that are found in sodium clusters. This effect is especially pronounced when stabilization of an overallshape through electronic shell effects is weak, i.e. for the “soft” clusters that are found between filled shells. In the present work we present calculations for the static electri c polarizability that include the ionic structure in a realis tic way. We take into account a great number of isomers for clusters with up to 20 atoms, especially for the soft cluster s that fill the second electronic shell. The theoretical con- cepts that we used in this study are introduced in section 2, where we also discuss the relevant cluster structures. In section 3 we present our results and compare with other calculations and experimental work. Our conclusions are summarized in section 4. 2 Theoretical concepts The starting point for the theoretical determination of the polarizability of a cluster is the calculation of the ionic a nd electronic configuration of the ground-state and close ly- ing isomers. In the present work, this was done in two steps. First, we calculated low-energy ionic geometries fo r a wide range of cluster sizes with an improved version [19] of the “Cylindrically Averaged Pseudopotential Scheme” (CAPS) [18]. In CAPS the ions are treated fully three- dimensionally, but the valence electrons are restricted to axial symmetry. The cluster ground state is found by si- multaneously minimizing the energy functional with re- spect to the set of ionic positions (simulated annealing) and the valence-electron density. For the exchange and correlation energy we used the local-density approxima- tion (LDA) functional of Perdew and Wang [20], and for the pseudopotential we employed the recently developed2 S. K¨ ummel et al.: Static Electric Dipole Polarizabilitie s of Na Clusters phenomenological smooth-core potential that reproduces low temperature bulk and atomic properties [19]. Detailed comparisons with ab initio calculations have shown [19] that CAPS predicts ionic geometries of sodium clusters rather accurately since truly triaxial deformations are ra re. Furthermore, in a second step we performed fully three- dimensional (3D) Kohn-Sham (KS) calculations to check the ordering of isomers and to calculate polarizabilities without axial restriction on the electrons, and also in- cluded configurations from 3D geometry optimizations into our analysis as discussed below. Fig. 1 schematically depicts the most important ionic geometries for neutral clusters with even electron num- bers between 2 and 20. (We have calculated the polariz- abilities also for many further and higher isomers which, however, are not shown in Fig. 1 for the sake of clarity. They were omitted from the discussion since they do not lead to qualitatively different results.) For the small clus - Na4 Na12 a Na12 bNa12 c Na14 aNa14 b Na14 c Na16 a Na 18 a Na 18 b Na18 c Na 20 a Na20 b Na20 cNa8Na 6 Na10 Na16 cNa16 d Na16 b Fig. 1. Cluster structures Na 4to Na 20. See text for discussion.ters Na 2, Na4and Na 6, many other theoretical predictions have been made [3,4,10,12,13], and our geometries are in perfect agreement with them. In addition, due to the con- struction of the pseudopotential [19], the bond lengths are close to the experimental ones, as e.g. seen in the dimer, where our calculated bond length is 5 .78a0and the exper- imental one [21] is 5 .82a0. For Na 8and Na 10, our results are in agreement with 3D density functional calculations [3,4,13]. For Na 12, we do not now of any ab initio calcu- lations. Therefore, besides two low-energy configurations from CAPS [(a) and (b)], we also included a locally re- optimized low-energy geometry from a 3D, H¨ uckel model calculation [15] in our analysis (c). Our 3D calculations confirm our CAPS results and find structures (a) and (b) quasi degenerate with a difference in total energy of 0.05 eV, whereas structure (c) is higher by 0.4 eV. Two of the three geometries considered for Na 14[(b) and (c)] were also found very similar in 3D H¨ uckel model calculations [15,17], and both CAPS and the 3D KS calculations find all of them very close in energy. For Na 16, we find as the CAPS-ground state structure (a), and in our 3D calcula- tions structure (b) is quasi degenerate with (a), whereas structures (c) and (d) are higher by 0.08 eV and 0.5 eV. Due to their very different overall shapes, these isomers span a range of what can be expected for the polarizabil- ity. For Na 18and Na 20, our structures are again in close agreement with the 3D density functional calculation of [13], and all three structures are quasi degenerate. The static electric polarizability was calculated in two different ways. The first is based on a collective description of electronic excitations. It uses the well known equality α= 2m−1, (1) which relates the negative first moment m−1(Q) =/integraldisplay∞ 0E−1SQ(E)dE=/summationdisplay ν(¯hων)−1|/angbracketleftν|Q|0/angbracketright|2 (2) of the strength function SQ(E) =/summationdisplay ν|/angbracketleftν|Q|0/angbracketright|2δ(Eν−E0−E), (3) to the static electric polarizability αin the direction spec- ified by the external (dipole) excitation operator Q. In the evaluation of the strength function, the excited states |ν/angbracketrightare identified with collective excitations. A dis- cussion of this approach can be found in [22]. The collec- tive calculations were carried out using the cylindrically averaged densities and the “clamped nuclei approxima- tion” [4], i.e.the ionic positions were taken to be the same with and without the dipole field. We have also checked this widely used approximation in the context of our stud- ies and find it well justified, as discussed below. The static polarizability can also be calculated directly from the derivative of the induced dipole moment µin the presence of an external electric dipole field F(“finite field method”): αij=µj(+Fi)−µj(−Fi) 2Fi, i, j =x, y, z, (4)S. K¨ ummel et al.: Static Electric Dipole Polarizabilities of Na Clusters 3 where µj(F) = −e/integraldisplay rjn(r,F)d3r+eZ/summationdisplay RRj (5) for ions with valence Z. Here one has to make sure that the numerically applied finite dipole field Fis small enough to be in the regime of linear response, but that it is on the other hand large enough to give a numerically stable signal. We have carefully checked this and found that the used field strengths between 0 .00001 e/a02and 0.0005e/a02 meet both requirements. Applied to the axial calculations, this approach allows to obtain the polarizability in the z- direction. By employing this method with the 3D KS cal- culations we have checked the influences of the axial av- eraging and the collective model on the polarizability and found that the z-polarizabilities from the axial and the 3D finite-field calculations agree within 1% on the aver- age for the low-energy isomers. This shows that the axial averaging is a good approximation for the clusters dis- cussed here. The performance of the collective model will be discussed below in Section 3.1. The orientation of our coordinate system was chosen such that the z-axis is in that principal direction of the tensor of inertia in which it deviates most from its average value. The average static electric polarizability ¯α:=1 3tr(α) (6) of course is independent of the choice of coordinate system. 3 Results 3.1 Comparison of different theoretical results Since all density functional calculations that we know of agree on the geometry of the smallest sodium clusters, these clusters can serve as test cases to compare differ- ent theoretical approaches. In Table 1 we have listed the averaged static dipole polarizability as obtained in dif- ferent calculations, together with the value obtained in the recent experiment of Rayane et al. [10]. All calcula- tions reproduce the experimental trend and give the cor- rect overall magnitude. But also, all calculations under- estimate the polarizability. The magnitude of this under- estimation, however, varies considerably for the different approaches. Whereas our results are closest to the exper- iment and close to the theoretical ones of Ref. [10], with the largest difference to the experiment being 8 % for Na 8, a difference of 27 % is found for this cluster in the calcu- lation based on the ab initio Bachelet, Hamann, Schl¨ uter (BHS) Pseudopotential [4]. A good part of this difference can be explained by comparing the bond lenghts of the clusters. The BHS pseudopotential considerably underes- timates the bond lengths [4], leading to a higher electron density and a lower polarizability. Our empirical smooth- core pseudopotential, on the other hand, was constructed to reproduce the experimental low-temperature bulk bondlength (together with the compressibility and the atomic 3s-level) when used with the LDA, and correspondingly results in a higher polarizability, in better agreement wit h experiment. It is further interesting to note that also the polarizabilities calculated with the empirical Bardsley p o- tential [3], which was constructed to reproduce atomic energy levels, are noticeably higher than the BHS-based values. This shows that the cluster polarizability is also sensitive to atomic energy levels, and the fact that our values are closest to the experiment thus is a natural con- sequence of the combination of correct atomic energy lev- els and bond lengths. Table 1. Averaged static electric polarizability of small sodium clusters in ˚A3. Brd: density functional (DF) calculation with empirical non-local Bardsley pseudopotential [3]. BH S: DF calculation with ab initio non-local Bachelet, Hamann, Schl¨ uter pseudopotential [4]. All el.: all-electron DF ca lcula- tion including gradient corrections to the exchange-corre lation functional [5]. TM: DF calculation with Troullier-Martins non- local pseudopotential [9]. GAUSS.: DF calculation based on the GAUSSIAN94 program with SU basis set [10]. Present: present work, values from three-dimensional approach. Exp R: recent experiment [10]. Brd. BHS All el. TM GAUSS. Present ExpR Na237.7 33.1 35.9 36.2 38.2 37.0 39.3 Na476.3 67.1 71.4 77.2 78.4 78.7 83.8 Na6100.3 89.4 94.8 not 104.4 107.3 111.8 Na8111.7 97.0 not 117.6 119.2 123.0 133.6 From comparison with the calculations that went be- yond the LDA [5,10], it however becomes clear that the empirical pseudopotentials by construction ”compensate” some of the errors that are a consequence of the use of the LDA. Therefore, it would be dangerous to argue that the inclusion of gradient corrections, which have been shown to increase the polarizability, could bring our calculated values in agreement with experiment: going beyond the LDA but keeping the empirical LDA pseudopotentials could lead to a double counting of effects. We therefore conclude that, on the one hand, a considerable part of the earlier observed differences between theoretical and experimen- tal polarizabilities can be attributed to effects associate d with errors in the bond lengths or atomic energy levels, but on the other hand, further effects must contribute to the underestimation with respect to experiment. We will come back to this second point below. In Table 2 we have listed the polarizabilities of Na 2to Na20for the geometries shown in Fig. 1. The left half gives the polarizability as computed from the 3D electron den- sity with the finite-field method, and the right half lists the values obtained in the axially averaged collective ap- proach. For the clusters up to Na 8, the two methods agree well and the differences for the averaged polarizabilities are less than 1% for Na 2and Na 8, and 3% for Na 4and Na6. This shows that the collective description is rather accurate, which is remarkable if one recalls that we are4 S. K¨ ummel et al.: Static Electric Dipole Polarizabilitie s of Na Clusters dealing with only very few electrons. Beyond Na 8, the dif- ferences are 6 % on the average, which is still fair, but ob- viously higher. This looks counter-intuitive at first sight , because the collective description should become better for larger systems. However, for N > 8 there comes an increasing number of particle-hole states close to the Mie plasmon resonance [22], leading to increasing fragmenta- tion of the collective strength. m−1and thus αis sensitive to energetically low-lying excitations since their energi es enter in the denominator in Eq. 2, and this can lead to an underestimation of the polarizability. Table 2. Static electric polarizability in ˚A3for the cluster geometries of Fig. 1. Left half: three-dimensional, finite fi eld calculation. Right half: cylindrically averaged, collect ive calcu- lation. 3D, finite field cyl., coll. mod. αx αy αz ¯α αρ αz ¯α Na2 29.9 29.9 51.3 37.0 30.2 51.5 37.3 Na4 47.0 59.0 130.1 78.7 53.7 122.6 76.7 Na6 129.8 129.8 62.2 107.3 124.9 62.1 103.9 Na8 118.1 118.5 132.3 123.0 117.5 131.4 122.1 Na10 126.3 126.3 219.3 157.3 125.3 194.5 148.4 Na12a213.4 213.4 155.7 194.2 199.5 143.6 180.9 Na12b156.5 158.6 261.1 192.1 151.3 234.7 179.1 Na12c158.6 145.1 285.3 196.3 148.7 252.0 183.1 Na14a183.5 183.5 278.9 215.3 177.5 251.7 202.2 Na14b271.3 273.9 137.8 227.7 264.2 132.7 220.4 Na14c175.2 179.3 291.3 215.3 171.1 262.5 201.6 Na16a193.5 193.5 394.5 260.5 190.3 318.4 233.0 Na16b235.2 235.2 231.5 234.0 227.3 220.3 225.0 Na16c212.5 213.4 272.2 232.7 206.0 255.1 222.4 Na16d272.2 260.8 239.3 239.3 260.6 185.1 235.4 Na18a260.3 262.5 291.3 271.4 236.4 266.7 246.5 Na18b251.8 250.7 283.8 262.1 232.6 253.4 239.5 Na18c281.6 280.5 250.7 270.9 255.9 227.7 246.5 Na20a285.7 284.5 309.4 293.2 269.7 279.7 273.0 Na20b275.0 275.0 311.8 287.3 261.8 282.9 268.8 Na20c267.9 271.5 295.2 278.2 283.2 280.3 282.2 Comparing the polarizabilities of clusters with the same number of electrons but different geometries shows the in- fluence of the overall shape of the cluster. For Na 14, e.g., isomers (a) and (c) have a valence electron density which is close to prolate, whereas (b) has a more oblate one. The averaged polarizability for the two prolate isomers is equa l, although their ionic geometries differ. The oblate isomer, however, has a noticeably higher averaged polarizability. This is what one expects, because for oblate clusters there are two principal directions with a low and one with a high polarizability, whereas for prolate clusters the reverse i s true. The fact that different ionic geometries can lead to very similar averaged polarizabilities is also seen for Na 12. It thus becomes clear that contrary to what was believed earlier [3] one cannot necessarily distinguish between de- tails in the ionic configuration by comparing theoretical values to experimental data that measure the averaged polarizability.3.2 Comparison with experiments Fig. 2 shows ¯ αfor our ground state structures as obtained in the axial, collective approach and the 3D finite-field calculations, in comparison to the two available sets of ex- perimental data. The absolute values for the experiments were calculated from the measured relative values with an atomic polarizability of 23 .6˚A3[10]. To guide the eye, the polarizabilities from each set of data are connected by lines. Both experiments and the theoretical data show that, overall, the polarizability increases with increasi ng cluster size. The polarizability from the axial collective model qualitatively shows the same behavior as the one from the 3D finite field calculation. Comparison of the 50100150200250300 2468101214161820Nα Fig. 2. Static electric dipole polarizability in ˚A3versus number of electrons. Crosses with thin, long dashed line: experime nt of Rayane et al. [10]; stars with strong, long dashed line: exper- iment of Knight et al. [1]; open squares with full line: present work, three-dimensional finite-field calculation for lowes t iso- mer; filled squares with short dashed line: present work, axi ally averaged collective calculation for lowest isomer. See tex t for discussion of error bars. 3D values with the experimental data shows that for the smallest clusters, the theoretical and experimental value s agree as discussed before, and the values obtained in the two experiments are comparable up to Na 10. Beyond Na 10, the discrepancies between the two experiments become larger, and also the differences between theoretical and ex- perimental polarizabilities increase. For Na 12, Na14, Na16 and Na 18the experiment of Knight et al.gives lower values than the experiment of Rayane et al., and the calculated averaged polarizability is lower than both experiments for Na12, Na14, and Na 16. For Na 18the finite-field value ob- tained for our ground-state structure matches the value measured by Knight el al., and for Na 20, our ground-state polarizability is very close to the measurement of Rayane et al. In this discussion one must keep in mind, however, that the experimental uncertainty is about +/-2 ˚A3per atom [10], i.e. the uncertainty in the absolute value in- creases with the cluster size, as indicated by the errors bars in Fig. 2. Comparisons are made easier if the linearS. K¨ ummel et al.: Static Electric Dipole Polarizabilities of Na Clusters 5 0.50.550.60.650.70.750.80.850.90.95 2468101214161820Nαn Fig. 3. Normalized static electric dipole polarizability. Crosse s with thin dashed line: experiment of Rayane et al. [10]; stars with strong dashed line: experiment of Knight et al.[1]; squares with full line: present work, three-dimensional finite-fiel d cal- culation for lowest isomer; open triangle, filled triangle a nd upside-down triangle: second, third and fourth isomer, res pec- tively; filled circles: Jellium results from [2,8]. See text for dis- cussion. growth in ¯ αis scaled away. Therefore, one should rather look at the normalized polarizability ¯αn:=¯α Nαatom, (7) which is shown in Fig. 3, because it allows to identify trends and details more clearly. From Fig. 3 it becomes clear that for Na 2to Na 8, the trend seen in the two experiments is similar up to one exception: For Na 6, the experiment of Rayane et al. pre- dicts a noticeably smaller value than the one by Knight et al. Comparison with our theoretical data shows that, although the values of the older experiment are closer to the theory with respect to magnitude for Na 2, Na4and Na8, the trend that our data show corresponds clearly to the one seen in the new experiment since the two curves are parallel. Going from Na 8to Na 10, both experiments predict a steep rise in the polarizability. This rise due to the shell closing at Na 8is also seen in the theoretical data, but it is less pronounced than in the experiments (as we will discuss below). For Na 12, a higher ¯ αnthan for Na 10 is predicted by the data of Rayane, whereas the reverse ordering is seen in the data of Knight et al. Again, our calculations support the finding of the new experiment, and all isomers lead to similar ¯ αn. For Na 14, both experi- ments show a decrease. Our prolate ground state and iso- mer reproduce this trend. That it is the prolate structures that fit to the experiment is consistent with the ab initio molecular dynamics calculations of H¨ akkinen et al. [23]. The next step to Na 16again reveals a slight difference between the two experiments: both predict an increase compared to Na 14, but whereas the older experiment sees ¯αnsmaller for Na 16than for Na 12, the new experiment shows the opposite ordering. Once more, our ground statestructure leads to a polarizability that follows the trend o f the new experiment. (The other isomers, however, lead to smaller polarizabilities, and an explanation for the differ - ence between the experiments thus might be that different ensembles of isomers were populated due to slightly dif- ferent experimental conditions.) Going to Na 18leads to a decrease in the polarizability in both experiments. Our calculation shows this decrease, which is a manifestation o f the nearby shell closing. But whereas the old experiment actually sees the shell closing at Na 18and an increase in the polarizability for Na 20, the new experiment and our data find an absolute minimum at Na 20. A comparison with the polarizability obtained in the spherical jellium model [2,8] for Na 8and Na 20, also indi- cated in Fig. 3, shows the improvement that is brought about by the inclusion of the ionic structure. 3.3 Discussion As just discussed, our calculations reproduce the fine struc - ture seen in the new experiment. However, there is no obvious explanation for why the results of the two exper- iments differ [24]. Also, there is a characteristic change in the magnitude of the difference between our theoreti- cal results and the experimental values of Rayane et al.: whereas the calculated values for Na 2to Na 8and Na 20on the average differ only by 5 % from the new experiment, the open-shell clusters from Na 10to Na 18show 18 % dif- ference for the ground state. The increase in polarizabilit y when going from Na 8to Na 10is considerably underesti- mated, whereas the following steps in the normalized po- larizability are nearly reproduced correctly, i.e. it look s as if the theoretical curve for Na 10to Na 18should be shifted upwards by a constant. A first suspicion might be that this “offset” could be due to the use of CAPS in the geometry optimization. But it should be noted that the step occurs at Na 10, and that also Na 14and Na 18are off by the same amount. Since the low-energy structures of these clusters are well established, as discussed in Section 2, and since also the geometry for Na 12from the 3D calculation does not lead to qualitatively different results, we can conclude that the differences are not due to limits in the geometry optimization with CAPS. Also, the neglection of the re- laxation of the nuclei in the presence of the electric dipole field has been investigated earlier [4] for the small clus- ters and was shown to be a good approximation. We have counter checked this result for the test case Na 10and find corrections of less than 1%. An obvious limitation of our approach is the neglect of the core polarization. However, the all-electron calcu- lations of Guan et al. [5] treat the core electrons explic- itly and do not lead to better agreement with experiment, as discussed in Section 3.1. From this, one already can conclude that core polarization cannot account for all of the observed differences. Its effect can be estimated from the polarizability of the sodium cation. Different measure- ments [25] find values between 0 .179˚A3and 0.41˚A3, lead- ing to corrections of - roughly - 1-2% in ¯ αn. Since the core polarizability leads to a shift in ¯ αnthat is the same for all6 S. K¨ ummel et al.: Static Electric Dipole Polarizabilitie s of Na Clusters cluster sizes, it contributes to the difference that is also seen for the smallest clusters, but it cannot explain the jump in the difference seen at Na 10. Another principal limitation of our approach is the use of the LDA. As, e.g., discussed in [5], the LDA can af- fect the polarizability in different and opposing ways. On the one hand it may lead to an overscreening and thus an underestimation of the polarizability, and early calcu- lations within the spherical jellium model reported that indeed the static polarizability was increased if one went beyond LDA using self-interaction corrections [7]. On the other hand, self-interaction corrections can lead to more negative single particle energies and thus to smaller polar - izabilities [5], and Refs. [8] and [26] give examples where the overall effect of self-interaction corrections on the op - tic response is very small. One cannot directly conclude from the jellium results to our ionic structure calculation s, because the sharp edge of the steep-wall jellium model can qualitatively lead to differences. But in any case it is highl y implausible that the LDA affects the clusters from Na 10 to Na 18much stronger than the other ones, and it should also be kept in mind that the worst indirect effects of the LDA as, e.g., underestimation of bond lengths, are com- pensated by using our phenomenological pseudopotential. One might also ponder about possible uncertainties in the experimental determination of the polarizabilities. A considerable underestimation could be explained if one as- sumes that while passing through the deflecting field, the clusters are oriented such that one always measures the highest component of the polarizability. In that case, we would not have to compare the averaged value to the ex- periment, but the highest one. One could imagine that the cluster’s rotation be damped, since angular momen- tum conservation is broken by the external field and the energy thus could be transferred from the rotation to in- ternal degrees of freedom (vibrations). The time scale of this energy transfer is not known, but since the clusters are spending about 10−4s in the deflecting field region, it seems unlikely that there should be no coupling over such a long time. One could further argue that for statistical reasons it is less likely that a larger cluster will lose its orientation again through random-like thermal motion of its constituent ions than a smaller cluster. However, the maximal energy difference between different orientations is very small, for Na 10, e.g., it is 0 .3K kBfor the typical field strength applied in the experiment [1]. Thus, thermal fluctuations can be expected to wipe out any orientation. From another point of view, however, the finite tem- perature explains a good part of the differences that our calculations (and other calculations for the small cluster s) show in comparison to the experimental data. Whereas our calculations were done for T=0, the supersonic nozzle expansion used in the experiment produces clusters with an internal energy distribution corresponding to about 400 - 600 K [27,28]. An estimate based on the thermal ex- pansion coefficient of bulk sodium leads to an increase in the bond lengths of about 3 %, and a detailed finite- temperature CAPS calculation [29] for Na+ 11at 400 K also shows a bond length increase of 3 %. This will only be alower limit, since in neutral clusters one can expect a large r expansion than in the bulk due to the large surface, and also a larger expansion than for charged clusters. Thus, to get an estimate for the lower limit of what can be expected from thermal expansion, we have scaled the cluster coor- dinates by 3 % and again calculated the polarizabilities, finding an increase of about 3 % for the planar and 5 % for three-dimensional structures. This finding is consiste nt with the results of Guan et al. Together with the correc- tions that are to be expected from the core polarizability, this brings our results for the small and the closed shell clusters in quantitative agreement with the experimental data. 4 Summary and Conclusion We have presented calculations for the static electric dipo le polarizability for sodium clusters with atom numbers be- tween 2 and 20, covering several low-energy structures for each cluster size beyond Na 10. By comparing our re- sults to previous calculations for the smallest clusters, w e have shown that a pseudopotential which correctly repro- duces atomic and bulk properties also improves the static response considerably. We have shown that a collective model for the excited states of sodium clusters, whose va- lidity for the dynamical response was established previ- ously, works reasonably also for the static response in real - istic systems. Over the whole range of cluster sizes studied in the present work, we confirm the fine structure seen in a recent experiment. By comparing the calculated averaged polarizability of different isomers for the same cluster siz e to the measured polarizability, we showed that completely different ionic geometries can lead to very similar averaged polarizabilities. By considering higher isomers we furthe r- more took a first step to take into account the finite tem- perature present in the experiment. Our results show that for the open shell clusters from Na 10to Na 18, also higher lying isomers do not close the remaining gap between the- ory and experiment. This shows that it is a worthwhile task for future studies to investigate the influence of finite temperatures on these “soft” clusters explicitly. For Na 2 to Na 8and Na 20, we showed that quantitative agreement is already obtained when the effects of thermal expansion and the core polarizability are taken into account. One of us (S. K¨ ummel) thanks K. Hansen for several clar- ifying discussions concerning the experimental temperatu res and time-scales, especially with respect to the “orientati on question”, and the Deutsche Forschungsgemeinschaft for fin an- cial support. References 1. W. D. Knight, K. Clemenger, W. A. de Heer, and W. A. Saunders, Phys. Rev. B 31, (1985) 2539. 2. W. Ekardt, Phys. Rev. Lett. 52, (1984) 1925.S. K¨ ummel et al.: Static Electric Dipole Polarizabilities of Na Clusters 7 3. I. Moullet, J. L. Martins, F. Reuse, and J. Buttet, Phys. Rev. Lett. 65, (1990) 476. 4. I. Moullet, J. L. Martins, F. Reuse, and J. Buttet, Phys. Rev. B 42, (1990) 11589. 5. J. 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Ekardt, and J. M. Pacheco, Phys. Rev. B50, (1994) 11079. 17. A. Yoshida, T. Dossing, and M. Manninen, J. Chem. Phys. 101, (1994) 3041. 18. B. Montag and P.-G. Reinhard, Z. Phys. D 33, (1995) 265. 19. S. K¨ ummel, M. Brack, and P.-G. Reinhard, Phys. Rev. B 58, (1998) 1774; S. K¨ ummel, P.-G. Reinhard, and M. Brack, to appear in Eur. Phys. J. D 9. 20. J. P. Perdew and Y. Wang, Phys. Rev. B 45, (1992) 13244. 21. K. K. Verma, J. T. Bahns, A. R. Rajaei-Rizi, W. C. Stwal- ley, and W. T. Zemke, J. Chem. Phys. 78, (1983) 3599. 22. P.-G. Reinhard, O. Genzken, and M. Brack, Ann. Phys. (Leipzig) 51, (1996) 576. 23. H. H¨ akkinen and M. Manninen, Phys. Rev. B 52, (1995) 1540. 24. Private communication by F. Chandezon and P. Durgourd. 25. J. R. Tessmann, A. H. Kahn, and W. Shockley, Phys. Rev. 92, (1953) 890. 26. C. A. Ullrich, P.-G. Reinhard, and E. Suraud, J. Phys. B 31, (1998) 1871. 27. S. Bjørnholm, J. Borggreen, O. Echt, K. Hansen, J. Ped- ersen, and H. D. Rasmussen, Z. Phys. 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1Compendium of vector analysis with applications to continuum mechanics compiled by Valery P. Dmitriyev Lomonosov University P.O.Box 160, Moscow 117574, Russia e-mail: dmitr@cc.nifhi.ac.ru 1. Connection between integration and differentiation Gauss-Ostrogradsky theorem We transform the volume integral into a surface one: ∫∂ ViPdV = ∫∂ VkjiidxdxPdx = ()()() Pxxx xxxdxdxkji kjiVSkj, ,|+ −∫ = = ()()( )()( )  − +− ∫ kjkji kjkji VSkj xxxxxPxxxxxPdxdx ,,, ,,, = =∫ +Scos dSPext+θ ∫− −ScosdSP− intθ = ∫ ScosdSPextθ = ∫⋅PdSien Here the following denotations and relations were used: P is a multivariate function ()kjixxxP,,, i ix∂∂=∂/, V volume, S surface, ie a basis vector, ijji/=⋅ee , n the external normal to the element dS of closed surface with dS dxdxi kjen⋅= , θcos=⋅ien . Thus ∫∂ ViPdV = ()∫⋅ VSdSiPen (1.1) Using formula (1.1), the definitions below can be transformed into coordinate representation.2Gradient ()∫ VSdSPn = () ()∫⋅ VSienPdSie = dVP Vii∫∂e where summation over recurrent index is implied throughout. By definition Pgrad = P∇ = iiPe∂ Divergence ()∫⋅ VSdSnA=() ()∫⋅ VSiendSAi = dVA Vii∫∂ (1.2) By definition Adi# = A⋅∇ = iiA∂ Curl ()dS VSAn×∫ = () ()∫⋅ VSien dSAjjiee× = dVA Vjiji∫×∂ee (1.3) By definition Acurl = A×∇ = jijiAee×∂ Stokes theorem follows from (1.3) if we take for the volume a right cylinder with the height 0→h. Then the surface integrals over the top and bottom areas mutually compensate each other. Next we consider the triad of orthogonal unitvectors m, n, 2 where m is the normal to the top base and n the normal to the lateral face nm2×= Multiplying the left-hand side of (1.3) by m gives dS lateral×⋅∫Anm =()dS lateralAnm⋅×∫ =dS lateralA2⋅∫ = dlh lA2⋅∫ where 2 is the tangent to the line. Multiplying the right-hand side of (1.3) by m gives ∫⋅ ShmdSAcurl where m is the normal to the surface. Now, equating both sides, we come to the formula sought for dl lA2⋅∫ = ∫⋅ SmdSAcurl The Stokes theorem is easily generalized to a nonplanar surface (applying to it Ampere's theorem). In this event, the surface is approximated by a polytope.Then mutual compensation of the line integrals on common borders is used.32. Elements of continuum mechanics A medium is characterized by the volume density ()t,xρ and the flow velocity ()t,xu. Continuity equation The mass balance in a closed volume is given by ()0=⋅ ∂∫+∫dS dV VS Vtnuρρ where tt∂∂=∂/. We get from (1.2) ()∫∫∂=⋅iiu dSρ ρnu dV Thereof the continuity equations follows ()0=∂+∂iituρρ Stress tensor We consider the force fdon the element dS of surface in the medium and are interested in its dependence on normal n to the surface ()nfd where ()()nfnfd d−=− With this purpose the total force on a closed surface is calculated. We have for the force equilibrium at the coordinate tetrahedron ()()()()03 2 1 =+++ nfnfnfnf dddd where the normals are taken to be external to the surface ()1 1en n ⋅−=sign1e , ()2 2en n ⋅−=sign2e , ()3 3en n ⋅−=sign3e Thence ()()jsignd ennf ⋅= ()jdef (2.1)4The force densit y ()n1 is defined by dSd 1f= Insofar as dS dSjjen⋅= we have for (2.1) () ( )jsign d ennf ⋅= ()je1 ( )j jsigndS en⋅=jen⋅()je1 ()jjdS e1en⋅= dS i.e. () ()jje1enn1 ⋅= ()jiij1eeen⋅= The latter means that ()n1 possesses the tensor propert y. The elements of the stress tensor are defined by ()jiijeσ σ= Now, using (1.2), the force on a closed surface can be co mputed as a volu me integral ()j Vj jjdS dS e1 nee1n1 ∫ ∫∫∂=⋅ = dV (2.2) Euler equation The momentum balance is given b y the relation () ()∫+∫∂ VS VtdV u u ρ ρ ()dS dS VS∫=⋅ 1nu (2.3) We have for the second ter m by (1.2) ()∫uρ ()∫=⋅ unu ρdS ( )∫∂=⋅ u nejj jju dSu ρdV Using also (2.2) gives for (2 .3) () ( ) ()jj jj tu e1u u ∂=∂+∂ ρ ρ or ()jjjj tu e1uu ∂=∂+∂ρ ρ (2.4)5Hydrodynamics The stress tensor in a fluid is defined from the pressure as ij ijpδσ−= That gives for (2.4) 0=∂+∂+∂ puuujijjitρρ Elasticity The solid-like medium is characterized by the displacement ()t,xs. For small displacements sut∂= and the quadratic terms in the left-hand part of (2.4) can be dropped. For an isotropic homogeneous medium the stress tensor is determined from the Hooke's law as () ()ijji kkij jisss ∂+∂+∂= µ λδσe where λand µ are the elastic constants. That gives () ()()µλ µ λσ +=∂+∂∂+∂∂=∂ijjji kkijijss s2eijjjiss2∂+∂∂µ and ()()µλ+=∂jje1 graddi #ss2∇+µ ()( ) µλµλ ++∇+= s22 curlcurls λ=graddi #µ−scurlcurls where curlcurl graddi #2+∇= was used. Substituting it to (2.4) we get finally Lame equation ()µλρ+=∂s2 tgraddi #ss2∇+µ where ρ is constant.

arXiv:physics/9912047v1 [physics.space-ph] 23 Dec 1999MSUPHY99.07 Determination of meteor showers on other planets using come t ephemerides Shane L. Larson† Department of Physics, Montana State University, Bozeman, Montana 59717 (23 December 1999) Abstract Meteor showers on the Earth occur at well known times, and are associated with the decay of comets or other minor bodies whose orbital p aths have crossed the Earth’s trajectory. On the surface, determinin g whether or not two orbital paths intersect appears to be a computationally int ensive procedure. This paper describes a simple geometric method for determin ing if the orbital paths of two bodies ( i.e., a comet and a planet) in the solar system cross from the known ephemerides of the objects. The method is used to determine whether or not meteor showers on other planets in the solar sy stem could be associated with any of 250 known comets. The dates and radian ts of these meteor showers are calculated. Typeset using REVT EX 1I. INTRODUCTION As they traverse their orbits about the Sun, comets slowly ev aporate and fragment, leav- ing small bits of cometary debris along their orbital tracks . Some comet orbits intersect the Earth’s path, and the planet sweeps up a portion of these part iculates each year. Generally, these particles are drawn into the atmosphere, where they bu rn up at high altitudes, produc- ing the yearly meteor showers. A sample of the meteor showers expected on a regular basis for Earth-bound observers is given in Table I. A very detaile d list of meteor streams en- countered by the Earth has been composed based on ground-bas ed observations of amateur astronomers around the world [1]. Given the large number of meteor showers seen on the Earth, it seems natural to ask about the possibility of meteor showers on other planets. It may be impractical for a sky- observer of the future to view meteor showers from some world s: Mercury has no atmosphere, the clouds of Venus are so thick most meteors will likely burn up before a planetbound observer could see them, Jupiter has no solid surface to sit o n while viewing the shower, and so forth. Never-the-less, predicting regular meteor sh owers on other worlds may be important for protecting explorers and spacecraft from inc oming particles, and could be useful for planning expeditions and experiments to collect cometary material. A great deal of modern research has been devoted to analysis o f the evolution of meteor streams in the solar system, particularly those that inters ect the Earth’s orbit (for example, detailed analyses of the evolution of the Quadrantid stream can be found in [2], [3] and [4]; the Geminid stream is analyzed in [5] and [6]). These analyses ta ke into account perturbations to the orbits of the parent bodies, as well as the subsequent e volution of the debris trail after the comet or minor body has continued on in its orbit. Over tim e, streams may wander into a planet’s path causing new meteor storms, or may wander out the planet’s path quenching a shower which has been periodic for decades or cen turies ( e.g., [4] estimates that the Quadrantid shower will vanish by the year 2100). To a first approximation, however, meteor showers will occur if the orbit of a planet and 2the orbit of a minor body intersect (or pass close to one anoth er). One way to determine if this occurs is to evolve the two orbits on a computer and wat ch for an intersection. Alternatively, the methods described in this paper approac h the problem of determining orbit intersections in a completely analytical fashion, re quiring only geometrical methods and matrix algebra. Section II describes the basic parameters and coordinate sy stems used to characterize or- bits in this paper. Section III describes the rotations used to correctly orient two orbits with respect to each other, and applies the rotations to essentia l vectors needed for the analysis. Section IV uses the rotated vectors to determine the interse ction between two orbital planes, and computes the distance between the orbital paths when the planes intersect. Section V proposes a criteria for the existence of a meteor shower base d on the distance between the orbits at intersection. The radiant and the “date” of shower s meeting the criterion is deter- mined. Section VI applies the condition of Section V to 250 kn own comets, summarizes the results, and discusses the limitations of determining mete or showers using this method. Throughout this paper, SI ( Syst` eme Internationale ) units are employed, except where the size of the units makes it convenient to work in standard u nits employed in astronomy (e.g., on large scales, astronomical units (AU) will be used, rath er than meters). II. DESCRIBING ORBITS As is well known, one of the great discoveries of Johannes Kep ler was that the planets travel on elliptical paths, with the Sun at one focus of the el lipse (Kepler’s First Law of Planetary Motion, published in 1609). Since then, an enormo us body of knowledge has been developed regarding the analysis of orbital motion (se e, for example [7]), allowing the determination of the position of virtually any object in the solar system at any moment in time. For the work presented here, a time dependent analysis of the orbital motion is not 3necessary1. The only information which is required is a knowledge of the trajectory of the orbit through space. The distance of the orbital path from th e Sun may be written for elliptical orbits as r=a(1−e2) 1 +ecosθ, (1) whereais the semi-major axis of the orbit, eis the eccentricity, and θis the angle (called the anomaly ) between the body and the axis defined by perihelion, as measu red in the orbital plane. The perihelion distance for the object can be found fr om Eq. (1) by taking θ= 0, yielding rp=a(1−e). (2) The distance expressed in Eq. (1) describes the correct size and shape of an elliptical orbit for any object around the Sun, but more information is n eeded to correctly orient the orbit in three-dimensional space. This information is typi cally collected in a set of numbers known as the orbital ephemeris . The reference for orienting orbits is the plane which is coin cident with the orbital plane of the Earth, known as the ecliptic . This paper will use a reference coordinate system defined in the ecliptic plane as shown in Figure 1. The + zaxis is defined perpendicular to the ecliptic and in the right handed sense with respect to the Ear th’s orbital motion ( i.e., when viewed looking down the + zaxis, the Earth’s motion is counter-clockwise in the xy-plane). The +xaxis is defined along the direction of the Earth’s perihelion . The orbital ephemeris of any body describes its orbit relati ve to the ecliptic plane, and locates the object along its orbital path as a function of tim e. For the problem of determining the possible intersection of two orbital paths, only three e lements of the full ephemeris for a 1We are interested in knowing only whether two orbits cross. A n interesting (but ultimately more difficult) question to address is whether two bodies might act uallycollide because their orbits intersect. 4body will be needed: Ω o(amodified longitude of the ascending node), ι(inclination), and ω (argument of perihelion). Each of these parameters is descr ibed below, and shown in Figure 2. The longitude of the ascending node, Ω, is the angle in the ecl iptic plane between the vernal equinox (called the first point of Aries ) and the point at which the orbit crosses the ecliptic towards the + zdirection (“northward” across the ecliptic). The paramete r, Ωo, used in this paper, is an offset longitude measured from the perihe lion of Earth, rather than the first point of Aries (see Figure 3). The inclination, ι, is the angle between the normal vector of the orbit and the no rmal vector of the ecliptic. Lastly, the argument of perihelion, ω, is the angle between the position of the body as it crosses the ascending node and the position at perihelion, a s measured in the orbital plane of the body . In addition to these three angles, it will be useful to define t wo vectors for each orbit of interest: ˆn, the unit normal vector to the plane of the orbit, and /vector rp, the vector pointing to perihelion in the plane of the orbit. III. ROTATIONS FOR ORBITAL ORIENTATION In order to correctly orient an orbit with respect to the ecli ptic, assume (initially) that the orbit of interest is in the plane of the ecliptic, with the perihelion of the orbit aligned along the + xaxis (i.e., the orbit is co-aligned with the Earth’s orbit). A series of three rotations, based on the angles {ω,ι,Ω}from the orbital ephemeris will produce the correct orientation. The first rotation will set the value of the asce nding node with respect to perihelion, the second rotation will set the inclination to the ecliptic, and the third rotation will move the ascending node to the correct location in the ec liptic plane. A useful method for describing rotations is in terms of matri ces. While it is possible to construct a rotation matrix for rotations about a general axis, it is more convenient to 5conduct rotations about the coordinate axes shown in Figure 1. The matrices describing rotations about the x-,y-, andz-axes will be denoted ˜Mx(φ),˜My(ξ), and ˜Mz(ψ), respectively. To demonstrate the rotations needed to orient the orbit, con sider a general vector, /vectorA, which is rigidly attached to the orbital plane, maintaining its orientation as the plane is rotated. The first rotation locates the ascending node with respect to perihelion; the rotation depends on the value of the argument of perihelion, ω. This is done by rotating around the z-axis byψ=ω. In terms of rotating a general vector /vectorA, this can be written /vectorA1=˜Mz(ω)/vectorA . (3) When this operation is applied to the orbit, the ascending no de will be located on the + x axis. The orbit is inclined around an axis which passes through the ascending node and through the Sun (at one focus of the orbit). Since the first rotation pl aced the ascending node on the +xaxis, and the Sun lies at the origin of coordinates, a rotatio n around the x-axis by the inclination angle, φ=ι, will correctly incline the orbit. In terms of the vector /vectorA1(resulting from Eq. (3)), this yields /vectorA2=˜Mx(ι)/vectorA1. (4) Before the final rotation, it will be convenient to offset the l ongitude of the ascending node such that it is measured from the perihelion of the Earth, rat her than the vernal equinox (this makes thex-axis the origin for measuring the longitude of the ascendin g node). The angle between the vernal equinox and perihelion of Earth is simply the argument of perihelion for Earth,ω⊕, giving (see Figure 3) Ωo= 2π−ω⊕+ Ω. (5) After the second rotation, the ascending node is still locat ed on the + xaxis. Rotation about thez-axis by the offset longitude, ψ= Ω o, will rotate the longitude of the ascending 6node to its correct location in the ecliptic plane. In terms o f the vector /vectorA2(resulting from Eq. (4)), this yields /vectorA3=˜Mz(Ωo)/vectorA2. (6) The vector /vectorA3(which is rigidly attached to the orbit) is correctly orient ed with respect to the ecliptic. The two vectors which will be of use later are the unit normal v ector to the orbit, ˆ n, and the perihelion vector, /vector rp. When the orbital plane is co-aligned with the Earth’s (befo re any rotations have been performed), these vectors have the form ˆn= 0 0 1 , /vector r p= rp 0 0 . (7) The rotation operations described by Eqs. (3), (4), and (6) m ust be applied to these vectors so they correctly describe the orbit with respect to the ecli ptic. Conducting the rotation procedure yields ˆn′= sinιsin Ω o −sinιcos Ω o cosι , (8) and /vector rp′=rp cosωcos Ω o−sinωcosιsin Ω o cosωsin Ω o+ sinωcosιcos Ω o sinωsinι . (9) IV. INTERSECTION OF ORBITS The procedure described in Section III will correctly orien t any orbit with respect to the ecliptic. One could take any planet’s ephemeris ( e.g., from the ephemerides given in Table 7II) and construct the normal vector ˆ nand perihelion vector /vector rpin accordance with Eqs. (8) and (9)2. Similar vectors could be generated for cometary ephemerid es. The real question of interest is not how the orbital planes of planets and comets are related to the ecliptic, but rather how they are oriented wit h respect to each other, and in particular where they intersect. The line defining the inter section of the orbital planes can be used to determine whether or not the orbits actually inter sect. Hereafter, assume that vectors related to a comet’s orbit wi ll bear the subscript ‘ c’ and vectors related to a planet’s orbit will bear the subscri pt ‘+’. Further, suppose the components of the normal vector for a comet’s orbit are ˆ nc= (a,b,c), and the components of the normal vector of a planet’s orbit are ˆ n+= (e,f,g). Both orbital planes automatically share one point in common: the origin, which lies at the focus of each orbital ellipse. Given this point and the two vectors ˆ ncand ˆn+, the equations describing the two orbital planes are CometPlane ax+by+cz= 0 PlanetPlane ex+fy+gz= 0. (10) The intersection of the two planes is a line which is the commo n solution of the two expressions in Eq. (10). Using determinants, the common sol ution to these equations is found to be x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleb c f g/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=−y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea c e g/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=z/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea b e f/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=k , (11) wherekis an arbitrary constant. The solutions {x,y,z }of Eq. (11) will be points along the line of intersection. It is useful to use these values to d efine a new vector, /vectorλ, called the 2To ease the notation, we will drop the primed notation for rot ated vectors from here on. It will be understood that the normal vectors and perihelion vector s have been correctly oriented with respect to the ecliptic. 8‘node vector.’ It points along the line of nodes (the interse ction of the two planes), and has components /vectorλ=k bg−cf ce−ag af−be . (12) To determine if the orbital paths intersect, one must know th e radii of the orbits along the line of nodes. An orbital radius may be determined from Eq . (1) if the value of the anomaly,θ, is known. In terms of two orbits inclined with respect to eac h other, the angles of interest will be the angle between the perihelion vector f or each orbit, /vector rp, and the node vector,/vectorλ. For each orbit, the angle is defined in terms of the dot produc t of the two vectors, yielding cosθ=/vector rp·/vectorλ |/vector rp| ·/vextendsingle/vextendsingle/vextendsingle/vectorλ/vextendsingle/vextendsingle/vextendsingle. (13) The orbits have two opportunities to intersect: at the ascen ding node, and at the descending node. Eq. (13) gives the angle at a single node. To obtain the v alue of the anomaly at the other node, dot the perihelion vector, /vector rp, into the negative of the node vector, −/vectorλ. Once the anomaly is known, the distance between the orbital p aths when the planes intersect is simply ∆ =|r+−rc|, (14) wherer+andrcare computed using Eq. (1) with the anomaly defined by Eq. (13) and the appropriate orbital parameters derived from tabulated eph emerides. V. IS THERE A METEOR SHOWER? The occurrence of a meteor shower associated with a particul ar comet will depend on the value of the separation between the orbital paths, ∆. In t his paper, the criteria for an orbit intersection causing a meteor shower will be 9∆≤κRl, (15) whereRlis the “Roche-lobe radius”, defined as the radius of a sphere w hich has the same volume as the planet’s Roche lobe, and κis a scaling factor. The Roche-lobe radius can be approximated by Rl∼0.52·a/bracketleftBiggm+ M⊙+m+/bracketrightBigg0.44 , (16) wherem+andM⊙are the mass of the planet and the Sun, and ais the semi-major axis of the planet’s orbit [8]. Once an intersection (in the sense of Eq. (15)) has been found , one would like to identify the associated meteor shower in some way. Meteor streams whi ch produce showers on Earth are named for the constellation the shower radiates from. A s imilar naming practice could be implemented for the predicted showers on other planets, i f a radiant could be identified. One method of determining the location of the radiant would b e to locate it in the apparent direction of the relative velocity of the particle s which comprise the shower. If the particles in the stream follow trajectories which are appro ximately the same as the parent body, and have a velocity /vector vc, then an observer on the surface of a planet moving through the stream with velocity /vector v+will measure the velocity vector of the meteors to be /vector vo=/vector vc−/vector v+. (17) The radiant of the shower is at the point on the sky where the ve ctor/vector vooriginates from. Determining an analytic description for the instantaneous speed of a body along an elliptical orbit is a notoriously difficult problem in orbital mechanics . For simplicity, here it will be assumed that the tangent vector to the planet’s orbit, /vector τpoints toward the radiant of the shower (the ‘radiant vector’). The tangent vector of an elliptical orbit in the xy-plane,/vector τ, is given by /vector τ= −asin Υ bcos Υ 0 , (18) 10whereb=a√ 1−e2is the semi-minor axis of the orbit, and the argument Υ is defin ed by3 tanΥ =rsinθ ea+rcosθ. (19) To represent the tangent vector for an orbit which has been pr operly oriented with respect to the ecliptic, /vector τmust be rotated using the procedure described in Section III . Once the radiant vector has been found, it can be used to deter mine which constella- tion the shower originates from by converting its direction al information into conventional astronomical coordinates. /vector τwill point toward some direction in the three dimensional sp ace described by the cartesian coordinates in Figure 1. The cart esian coordinates shown are based on the location of the Earth’s perihelion vector, and n ot on the origin of a particular astronomical coordinate system. The coordinates of intere st for locating the radiant of the shower on a star chart are ecliptic (and ultimately equatorial ) coordinates, with the origin at the first point of Aries, located in the xy-plane at an angle ψ=ω⊕preceding the + xaxis. The components of the radiant vector may be described in cart esian coordinates which have the +xaxis coincident with the first point of Aries by applying a rot ation: /vector τ′=˜Mz(ω⊕)/vector τ . (20) After this rotation, the components of the radiant vector ar e the projections of /vector τonto a cartesian coordinate system coincident with the ecliptic c oordinates. The components may be reduced to two angles which describe the vector’s orienta tion to the plane, Λ (ecliptic longitude) and β(ecliptic latitude), defined by tan Λ =τ′ y τ′x,sinβ=τ′ z |/vector τ|. (21) where (τ′ x,τ′ y,τ′ z) are the cartesian components of the radiant vector, and |/vector τ|is the magnitude. 3Υ has a simple geometric interpretation: it is the angle betw een the x-axis and the radius vector of the ellipse for a coordinate system with its origin at the center of the ellipse, rather than at one focus. 11To locate the radiant in a particular constellation, it is us eful to convert the ecliptic coordinates {Λ,β}to conventional equatorial coordinates, right ascension αand declination δ. The transformation between these two coordinate systems i s described by [9] sinδ= sin(β) cos(ǫ) + cos(β) sin(ǫ) sin(Λ) sinβ= sin(δ) cos(ǫ)−cos(δ) sin(ǫ) sin(α) cos(Λ) cos(β) = cos(α) cos(δ), (22) whereǫis the angular distance between the north ecliptic pole and t he north celestial pole (equivalent to the tilt of the Earth’s axis with respect to th e ecliptic,ǫ= 23.45◦). Since there are no established calendars on other planets, t here is no well defined way of dating a meteor shower. Here, a scheme will be adopted rela ting to the calendar on Earth, illustrated in Figure 4. The ‘months’ for each planet will be defined in terms of the right ascension of Earth at the start of each month on the terr estrial calendar, αdate, and are shown in Table III. The node vector, /vectorλ(which points to the planet’s encounter with a meteor stream), will point toward a particular value of the right ascension. The right ascension defined by each node vector is compared to the value s ofαdate, and the meteor shower is dated. The critical parameters related to determining the occurre nce, radiant, and date of a possible meteor shower are illustrated in Figure 5. VI. RESULTS & DISCUSSION A search for comet-planet orbital intersections in the sola r system was carried out using the planetary ephemerides shown in Table II [10]4, and the comet ephemerides provided 4There is a printed error in the table listing the ephemerides of the planets in this reference: the column ωis actually ˜ ω, which is the ‘longitude of perihelion’, defined as the sum of the longitude of the ascending node (Ω) and the argument of perihelion( ω). Here, Table II lists ω. 12in the Jet Propulsion Laboratory’s DASTCOM (Database of AST eroids and COMets) [11]. The DASTCOM is a collection of orbital parameters and physic al characteristics for the numbered asteroids, unnumbered asteroids, periodic comet s, and other selected comets, used for analyses of solar system dynamics. Before analyzing the data, it is useful to have an idea of what kind of results one might expect to see. Table IV shows a breakdown of the DASTCOM comet database, where comparisons of the orbital perihelia and aphelia of the come ts and planets were used to produce a simple estimate of the number of comets from the dat abase which have the possibility of intersecting the orbit of each planet. Ais the number of comets which have perihelion at radii less than a given planet’s aphelion ( i.e.the closest approach of a comet to the Sun is at least as small as the planet’s greatest distan ce from the Sun), and Bis the number of comets which have aphelion radii which are grea ter than a given planet’s perihelion ( i.e.the greatest distance from the Sun reached by a comet is at lea st the as large as the planet’s closest approach to the Sun), and ηis an estimate of the number of possible comets a planet’s orbit could intersect5. The results of the search for comet-planet orbital intersec tions are listed in Table V, which specified an encounter distance of ∆≤5Rl. (23) In all, 128 possible showers were detected: 3 at Earth, 1 at Ma rs, 106 at Jupiter6, 17 at Saturn, and 1 at Uranus. If one reduces the encounter distanc e to ∆ ≤1Rl, only 32 possible 5These estimates makes no account for the relative orientati on of orbits; it assumes only that the semi-major axes of the planets and comets are aligned. Possi ble encounters enumerated by ηonly reflect a comparison of the radial scales of the orbits. 6In fact, the results of Table V show that Jupiter’s orbit inte rsects the path of comet P/Spahr (1998 U4) twice: one intersection at a separation of ∆ ≃1.6Rl, and a second intersection (at the other node) with a separation of ∆ ≃3.3Rl. 13showers are detected (shown at the top of Table V): 1 at Earth, 28 at Jupiter, 2 at Saturn, and 1 at Uranus. If one allows the encounter distance to expan d to ∆ ≤10Rl, 188 possible showers are detected (data not shown in Table V): 4 at Earth, 5 at Mars, 148 at Jupiter, 24 at Saturn, 6 at Uranus, and 1 at Neptune. As was shown in Table IV, Jupiter has the opportunity to inter sect the orbits of more comets in the database than any other planet in the solar syst em. It is thought that comets with orbital scales smaller than the solar system (‘short pe riod comets’) have evolved largely under the influence of perturbations due to Jupiter (the mass of Jupiter is greater than the mass of the other planets combined), giving a large popul ation of comets which cross Jupiter’s orbit. The search for the origin of these “Jovian f amily comets” has been a matter of much numerical simulation and debate (see, for example, [ 12]). The disproportionately large number of showers detected for Jupiter can be attribut ed to this feature of the comet population. An examination of Table V shows that the showers occur very cl ose to the ecliptic, mostly in the constellations of the zodiac. This should not be surpr ising, since the inclination of the planetary orbital planes is relatively small. The tange nt vectors of the planetary orbits (which were used to define the radiants) will always point clo se to the ecliptic. A good check of the procedure described in this paper is to con sider the predicted showers at Earth. In particular, the method outlined in this work pre dicts two meteor streams which can be identified with known showers. The first is the stream fr om Comet Tempel-Tuttle, originating in Leo in November. This stream can be identified with the Leonid meteor shower (known to be a stream from Tempel-Tuttle), which occurs in mi d-November each year. The second is a stream from Comet Swift-Tuttle, originating in A ries in August. This stream can be identified with the Perseid meteor shower (known to be a stream from Swift-Tuttle), which radiates from Perseus in August, just to the north and w est of Aries. The error in the Perseid radiant demonstrates the limitations of using the t angent vectors of the planetary orbital planes to accurately locate the radiant of a shower. The possible showers computed here have all assumed that the orbits of the comets are 14static and do not precess. Further, it is assumed that the met eor streams remain attached to those static orbits without wandering under the influence of gravitational perturbations in the solar system. In addition, the influences of ‘local’ bo dies around each planet ( e.g., Earth’s moon, or the Galilean satellites around Jupiter) ha ve been ignored. Never-the-less, the method provides a useful way for determining the possibi lity that a given planet will encounter a meteor stream from minor bodies in the solar syst em. ACKNOWLEDGMENTS I would like to thank M. B. Larson for comments and suggestion s, and E. M. Standish who provided helpful discussions regarding planetary ephe merides. I would also like to acknowledge the hospitality of the Solar System Dynamics Gr oup at the Jet Propulsion Laboratory during the time this work was completed. This wor k was supported in part by NASA Cooperative Agreement No. NCC5-410. 15REFERENCES †electronic mail address: shane@physics.montana.edu. [1] P. Jenniskens, Astron. Astrophys. 287, 990 (1994). [2] D. W. Hughes, I. P. Williams and C. D. Murray, Mon. Not. R. A str. Soc. 189, 493 (1979). [3] I. P. Williams, C. D. Murray and D. W. Hughes, Mon. Not. R. A str. Soc. 189, 483 (1979). [4] C. D. Murray, D. W. Hughes and I. P. Williams, Mon. Not. R. A str. Soc. 190, 733 (1980). [5] K. A. Fox, D. W. Hughes and I. P. Williams, Mon. Not. R. Astr . Soc.200, 313 (1982). [6] J. Jones, Mon. Not. R. Astr. Soc. 217, 523 (1985). [7] J. B. Marion and S. T. Thornton, Classical dynamics of particles and systems (Harcourt Brace Jovanovich, New York, 1988). [8] I. Iben, Jr. and A. V. Tutukov, Astrophys. J. Suppl. 54, 335 (1984). [9] W. Schlosser, T. Schmidt-Kaler and E. F. Milone, Challenges of astronomy: hands-on experiments for the sky and laboratory (Springer-Verlag, New York, 1991). [10] E. M. Standish, XX Newhall, J. G. Williams and D. K. Yeoma ns, inExplanatory supple- ment to the Astronomical Almanac , P. K. Seidelmann, Ed., University Science Books, Mill Valley, CA. [11] http://ssd.jpl.nasa.gov/dastcom.html [12] T. Quinn, S. Tremaine and M. Duncan, Astrophys. J. 355, 667 (1990). 16TABLES TABLE I. Some of the yearly meteor showers seen from Earth. Shower Name Date Quadrantids early January Lyrids mid April ηAquarids early May δAquarids late July Perseids mid August Orionids mid October Leonids mid November Geminids mid December 17TABLE II. The mean ephemerides for the planets of the solar sy stem (epoch J2000). The data are semi-major axis a, eccentricity e, inclination ι, longitude of ascending node Ω, and the argument of perihelion ω. Planet a e ι Ω ω (AU) (◦) (◦) (◦) Mercury 0.38709893 0.20563069 7.00487 48.33167 29.12478 Venus 0.72333199 0.00677323 3.39471 76.68069 54.85229 Earth 1.00000011 0.01671022 0.00005 -11.26064 114.20783 Mars 1.52366231 0.09341233 1.85061 49.57854 286.4623 Jupiter 5.20336301 0.04839266 1.3053 100.55615 -85.8023 Saturn 9.53707032 0.0541506 2.48446 113.71504 -21.2831 Uranus 19.19126393 0.04716771 0.76986 74.22988 96.73436 Neptune 30.06896348 0.00858587 1.76917 131.72169 -86.750 34 Pluto 39.348168677 0.24880766 17.14175 110.30347 113.763 29 18TABLE III. The right ascension, αdate, for the months of the year. Month Start αdate Endαdate (decimal h) (decimal h) January 6.7397 8.7780 February 8.7780 10.6191 March 10.6191 12.6575 April 12.6575 14.6301 May 14.6301 16.6685 June 16.6685 18.6411 July 18.6411 20.6794 August 20.6794 22.7178 September 22.7178 0.6904 October 0.6904 2.7287 November 2.7287 4.7013 December 4.7013 6.7397 19TABLE IV. The estimated number of comets from the DASTCOM dat abase which have the possibility of intersecting the orbit of each planet. Ais the number of comets which have perihelia less than each planet’s aphelion, Bis the number of comets which have aphelia which are greater than a planet’s perihelion, and ηis the estimate of the number of possible comets a planet’s or bit could intersect. Planet A B η Mercury 5 250 5 Venus 11 250 11 Earth 30 250 30 Mars 99 250 99 Jupiter 241 206 211 Saturn 250 97 97 Uranus 250 78 78 Neptune 250 72 72 Pluto 250 65 65 20TABLE V. The results of a meteor shower search using the comet s in the JPL DASTCOM database. The first 32 entries (above the line) are for inters ections having ∆ < R l. All other encounters are for ∆ <5Rl. Planet Comet ∆ /Rlδ α Constellation αdate Month (◦) (decimal h) (decimal h) Earth 109P/Swift-Tuttle 0.50 17.75 3.17 Aries 21.46 Aug Jupiter P/LONEOS-Tucker (1998 QP54) 0.03 -20.74 16.34 Scor pius 10.70 Mar Jupiter 117P/Helin-Roman-Alu 1 0.06 -16.07 15.02 Libra 4.5 3 Nov Jupiter 43P/Wolf-Harrington 0.11 15.46 2.89 Aries 17.02 Ju n Jupiter P/Hergenrother (1998 W2) 0.13 -22.82 17.45 Ophiuch us 11.63 Mar Jupiter C/Hale-Bopp (1995 O1) 0.16 6.55 1.25 Pisces 18.90 Ju l Jupiter 78P/Gehrels 2 0.21 22.58 5.25 Taurus 14.59 Apr Jupiter 75P/Kohoutek 0.22 10.50 1.92 Pisces 18.12 Jun Jupiter P/Spahr (1998 W1) 0.24 4.65 0.94 Pisces 18.87 Jul Jupiter 124P/Mrkos 0.33 -20.44 20.21 Ophiuchus 0.01 Sep Jupiter 14P/Wolf 0.33 23.24 6.25 Gemini 13.67 Apr Jupiter 53P/Van Biesbroeck 0.37 15.79 9.47 Leo 10.71 Mar Jupiter 59P/Kearns-Kwee 0.40 -3.95 23.60 Aquarius 20.79 Au g Jupiter 91P/Russell 3 0.42 9.26 10.73 Leo 4.65 Nov Jupiter 76P/West-Kohoutek-Ikemura 0.44 -2.72 23.79 Pisce s 17.53 Jun Jupiter 26P/Grigg-Skjellerup 0.45 19.47 8.52 Cancer 2.30 O ct Jupiter 132P/Helin-Roman-Alu 2 0.51 22.42 7.29 Gemini 12.7 4 Apr Jupiter P/Kushida (1994 A1) 0.54 16.19 3.05 Aries 16.83 Jun Jupiter 16P/Brooks 2 0.58 22.32 7.36 Gemini 12.68 Apr Jupiter 83P/Russell 1 0.59 16.19 9.38 Leo 3.16 Nov 21Jupiter D/Kowal-Mrkos (1984 H1) 0.62 8.94 10.79 Leo 4.71 Dec Jupiter 135P/Shoemaker-Levy 8 0.68 17.76 9.00 Cancer 2.77 N ov Jupiter 139P/Vaisala-Oterma 0.73 15.58 2.91 Aries 16.99 Ju n Jupiter 86P/Wild 3 0.74 -15.87 14.98 Libra 4.58 Nov Jupiter 104P/Kowal 2 0.82 17.59 3.39 Aries 16.47 May Jupiter P/LINEAR-Mueller (1998 S1) 0.87 -22.75 17.38 Ophiu chus 11.57 Mar Jupiter P/Shoemaker-Levy 6 (1991 V1) 0.88 -19.59 20.49 Capr icornus 14.14 Apr Jupiter 18P/Perrine-Mrkos 0.95 18.94 3.75 Taurus 16.08 May Jupiter 85P/Boethin 0.97 -23.17 17.89 Sagittarius 11.99 Ma r Saturn P/Jager (1998 U3) 0.07 11.62 10.42 Leo 20.32 Jul Saturn 126P/IRAS 0.53 -21.96 17.46 Ophiuchus 11.79 Mar Uranus C/Li (1999 E1) 0.84 13.27 2.29 Aries 20.73 Aug Earth 55P/Tempel-Tuttle 4.15 12.87 9.88 Leo 3.53 Nov Earth 26P/Grigg-Skjellerup 4.52 -23.19 17.40 Ophiuchus 14 .07 Apr Mars C/LINEAR (1998 U5) 1.99 12.98 10.22 Leo 4.28 Nov Jupiter 47P/Ashbrook-Jackson 1.00 -23.06 17.71 Ophiuchus 11.84 Mar Jupiter 15P/Finlay 1.01 -21.91 19.59 Sagittarius 13.37 Apr Jupiter 97P/Metcalf-Brewington 1.10 22.31 7.36 Gemini 12. 68 Apr Jupiter 81P/Wild 2 1.10 20.79 4.36 Taurus 22.50 Aug Jupiter 121P/Shoemaker-Holt 2 1.10 3.71 0.80 Pisces 18.70 J ul Jupiter 54P/de Vico-Swift 1.11 -22.22 17.02 Ophiuchus 11.2 8 Mar Jupiter 56P/Slaughter-Burnham 1.12 -20.78 16.36 Scorpius 10.71 Mar Jupiter P/Korlevic-Juric (1999 DN3) 1.15 -20.95 20.02 Sagi ttarius 0.17 Sep Jupiter P/Mueller 4 (1992 G3) 1.16 19.11 3.80 Taurus 21.97 Au g Jupiter 52P/Harrington-Abell 1.20 -12.43 22.16 Aquarius 2 2.27 Aug Jupiter 69P/Taylor 1.24 7.45 1.40 Pisces 19.40 Jul Jupiter P/Larsen (1997 V1) 1.27 19.86 4.03 Taurus 15.79 May 22Jupiter 46P/Wirtanen 1.32 -4.11 23.58 Aquarius 17.28 Jun Jupiter 100P/Hartley 1 1.32 -22.79 17.41 Ophiuchus 2.30 Oct Jupiter 87P/Bus 1.34 20.61 8.14 Cancer 1.93 Oct Jupiter C/Ferris (1999 K2) 1.38 -0.39 0.16 Pisces 20.17 Jul Jupiter 77P/Longmore 1.44 -22.53 19.22 Sagittarius 0.82 Oc t Jupiter C/Mueller (1997 J1) 1.45 8.58 1.59 Pisces 18.51 Jun Jupiter P/Hartley-IRAS (1983 V1) 1.51 -23.23 18.06 Sagitta rius 12.13 Mar Jupiter 119P/Parker-Hartley 1.53 16.89 3.22 Aries 16.65 Ma y Jupiter 114P/Wiseman-Skiff 1.54 10.49 1.92 Pisces 18.12 Jun Jupiter 102P/Shoemaker 1 1.59 -20.60 16.29 Scorpius 10.66 M ar Jupiter P/Spahr (1998 U4) 1.61 21.43 7.81 Gemini 12.27 Mar Jupiter 33P/Daniel 1.65 -10.00 22.60 Aquarius 16.19 May Jupiter 62P/Tsuchinshan 1 1.70 2.38 0.59 Cetus 18.46 Jun Jupiter 70P/Kojima 1.79 12.54 2.30 Aries 20.41 Jul Jupiter 60P/Tsuchinshan 2 1.84 4.84 0.97 Pisces 19.23 Jul Jupiter 4P/Faye 1.92 23.24 6.21 Gemini 13.71 Apr Jupiter 67P/Churyumov-Gerasimenko 1.95 -17.50 21.07 Capr icornus 14.66 May Jupiter 6P/d’Arrest 2.01 10.31 10.55 Leo 9.60 Feb Jupiter 36P/Whipple 2.03 22.25 7.40 Gemini 12.65 Mar Jupiter C/LINEAR (1998 U1) 2.09 21.13 7.94 Gemini 1.74 Oct Jupiter C/Spacewatch (1997 BA6) 2.16 -7.16 23.08 Aquarius 2 1.34 Aug Jupiter P/Levy (1991 L3) 2.22 -18.03 15.50 Libra 9.95 Feb Jupiter 116P/Wild 4 2.28 -20.88 20.04 Sagittarius 0.14 Sep Jupiter P/Shoemaker-Levy 1 (1990 V1) 2.40 -15.18 21.61 Capr icornus 15.18 May Jupiter 9P/Tempel 1 2.41 -17.42 15.35 Libra 4.20 Nov Jupiter C/LINEAR (1999 H3) 2.60 -19.64 15.96 Libra 10.37 Feb Jupiter 31P/Schwassmann-Wachmann 2 2.63 17.03 3.25 Aries 2 1.42 Aug 23Jupiter P/LINEAR (1999 J5) 2.70 -0.25 12.26 Virgo 7.67 Jan Jupiter C/LINEAR (1998 W3) 2.71 3.74 11.63 Virgo 8.40 Jan Jupiter 21P/Giacobini-Zinner 2.73 22.93 6.88 Gemini 13.11 Apr Jupiter 40P/Vaisala 1 2.84 17.14 3.28 Aries 21.45 Aug Jupiter 108P/Ciffreo 2.88 -15.33 21.57 Capricornus 15.15 Ma y Jupiter 136P/Mueller 3 2.93 10.97 10.43 Leo 9.73 Feb Jupiter 42P/Neujmin 3 2.93 18.27 8.87 Cancer 11.30 Mar Jupiter 7P/Pons-Winnecke 3.14 -8.03 13.50 Virgo 6.22 Dec Jupiter 103P/Hartley 2 3.19 22.21 5.01 Taurus 14.82 May Jupiter 128P/Shoemaker-Holt 1-B 3.22 21.72 4.76 Taurus 15. 07 May Jupiter D/van Houten (1960 S1) 3.26 -18.52 20.80 Capricornu s 23.50 Sep Jupiter P/Kushida-Muramatsu (1993 X1) 3.27 -1.95 23.91 Pis ces 17.67 Jun Jupiter C/Zhu-Balam (1997 L1) 3.27 20.87 4.39 Taurus 15.43 M ay Jupiter P/LINEAR (1998 VS24) 3.27 19.53 8.50 Cancer 11.64 Ma r Jupiter P/Spahr (1998 U4) 3.30 23.24 6.32 Gemini 0.27 Sep Jupiter 61P/Shajn-Schaldach 3.38 20.61 8.14 Cancer 11.97 M ar Jupiter 65P/Gunn 3.44 -17.58 15.39 Libra 4.16 Nov Jupiter 120P/Mueller 1 3.60 -23.03 17.67 Ophiuchus 11.81 Ma r Jupiter 129P/Shoemaker-Levy 3 3.66 0.12 0.23 Pisces 20.08 J ul Jupiter 30P/Reinmuth 1 3.67 12.38 2.27 Aries 20.38 Jul Jupiter 22P/Kopff 3.67 5.55 11.34 Leo 8.72 Jan Jupiter 112P/Urata-Niijima 3.70 -20.61 20.14 Sagittarius 13.84 Apr Jupiter 110P/Hartley 3 3.70 4.81 0.97 Pisces 19.23 Jul Jupiter 49P/Arend-Rigaux 3.78 12.37 2.26 Aries 20.38 Jul Jupiter P/Lagerkvist (1996 R2) 3.98 -23.06 18.72 Sagittari us 12.66 Apr Jupiter 19P/Borrelly 4.00 -6.29 23.22 Aquarius 16.88 Jun Jupiter 17P/Holmes 4.00 -17.66 15.41 Libra 9.86 Feb 24Jupiter 137P/Shoemaker-Levy 2 4.02 18.67 3.67 Taurus 16.16 May Jupiter 131P/Mueller 2 4.05 22.39 5.12 Taurus 14.72 May Jupiter P/Jedicke (1995 A1) 4.08 1.25 12.02 Virgo 7.94 Jan Jupiter D/Tritton (1978 C2) 4.23 0.66 0.32 Pisces 19.98 Jul Jupiter 48P/Johnson 4.32 2.03 11.90 Virgo 8.09 Jan Jupiter 106P/Schuster 4.41 -15.82 21.46 Capricornus 15.04 May Jupiter C/LINEAR (1998 M5) 4.67 -12.99 22.05 Aquarius 22.37 Aug Jupiter P/LONEOS (1999 RO28) 4.84 14.99 9.65 Leo 10.54 Feb Jupiter 98P/Takamizawa 4.89 5.64 11.33 Leo 8.74 Jan Jupiter C/Spacewatch (1997 P2) 4.89 -0.74 0.10 Pisces 20.23 Jul Jupiter 73P/Schwassmann-Wachmann 3 4.92 -17.02 15.25 Libr a 4.30 Nov Jupiter P/Helin-Lawrence (1993 K2) 4.99 -8.87 13.64 Virgo 6 .06 Dec Saturn C/Catalina (1999 F1) 1.01 -22.07 19.24 Sagittarius 1 3.30 Apr Saturn P/Gehrels (1997 C1) 1.27 -4.44 13.11 Virgo 17.07 Jun Saturn P/Hermann (1999 D1) 1.36 21.43 7.67 Gemini 23.10 Sep Saturn C/LINEAR (1998 Q1) 1.37 -21.90 19.37 Sagittarius 10. 99 Mar Saturn P/Shoemaker 4 (1994 J3) 2.46 -0.51 0.31 Pisces 6.06 De c Saturn P/Montani (1997 G1) 2.90 3.09 11.90 Virgo 18.56 Jun Saturn P/Helin (1987 Q3) 3.17 -22.38 18.90 Sagittarius 11.3 9 Mar Saturn 63P/Wild 1 3.27 22.22 7.10 Gemini 23.61 Sep Saturn 140P/Bowell-Skiff 3.31 17.75 9.03 Cancer 21.80 Aug Saturn D/Bradfield 1 (1984 A1) 3.58 22.37 6.92 Gemini 23.76 Se p Saturn 134P/Kowal-Vavrova 3.65 14.63 9.80 Leo 3.37 Nov Saturn C/Spacewatch (1997 BA6) 3.95 -14.20 14.90 Libra 9.34 Feb Saturn C/LINEAR (1999 N4) 3.96 -21.44 17.13 Ophiuchus 11.49 Mar Saturn C/LINEAR (1999 H3) 4.20 19.72 8.42 Cancer 22.40 Aug Saturn P/Lagerkvist-Carsenty (1997 T3) 4.90 -20.42 20.15 S agittarius 14.09 Apr 25FIGURES FIG. 1. The reference coordinate system in the ecliptic plan e. The z-axis is defined in the right-handed sense with respect to the Earth’s motion, and t hex-axis points towards the perihelion of the Earth. FIG. 2. The three essential angles for correctly orienting o rbits in three dimensional space are (a) Ω o, the ( modified ) longitude of the ascending node; (b) ι, the inclination; and (c) ω, the argument of perihelion. FIG. 3. The modified longitude of the ascending node, Ω o, defined in terms of the Earth’s argument of perihelion, ω⊕, and the conventional value of Ω for the orbit. υindicates the first point of Aries. FIG. 4. The determination of a date for a meteor shower is base d on the right ascension of the planet, as measured from the Sun, at the time it encounters th e meteor stream (defined by the direction of the node vector, /vectorλ). The ‘months’ of the ‘year’ are defined by the right ascensio n of the Earth at the start of each month. The grey area shown would be March for any planet within the March values of αdate(shown). FIG. 5. The intersection of two orbits, showing the essentia l quantities for defining the occur- rence of a meteor shower: the separation of the orbits at cros sing, ∆; the tangent vector to the planet’s orbit, /vector τ, which defines the ‘radiant’ of the shower; and the node vecto r,/vectorλ, which defines the intersection of the two orbits and is used to ‘date’ the me teor shower. 26+x +y +z FIGURE 1r p +x +y +z ecliptic ω ι n Ω o FIGURE 2 +x /K55ascending node ω Ω Ω o FIGURE 312.6575 h 10.6191 h FIGURE 4 λ τ } ∆ FIGURE 5

arXiv:physics/9912048v1 [physics.plasm-ph] 23 Dec 1999Coulomb crystals in the harmonic lattice approximation D. A. Baiko and D. G. Yakovlev Ioffe Physical–Technical Institute, 194021 St.–Petersbur g, Russia H. E. De Witt Lawrence Livermore National Laboratory, CA 94550 Livermor e W. L. Slattery Los Alamos National Laboratory, NM 87545 Los Alamos (December 18, 2013) The dynamic structure factor ˜S(k, ω) and the two-particle distribution function g(r, t) of ions in a Coulomb crystal are obtained in a closed analytic form using the harmonic lattic e (HL) approximation which takes into account all processes of multi-phonon excitation and absorption. The static ra- dial two-particle distribution function g(r) is calculated for classical ( T>∼¯hωp, where ωpis the ion plasma frequency) and quantum ( T≪¯hωp) body-centered cubic (bcc) crys- tals. The results for the classical crystal are in a very good agreement with extensive Monte Carlo (MC) calculations at 1.5<∼r/a<∼7, where ais the ion-sphere radius. The HL Coulomb energy is calculated for classical and quantum bcc and face-centered cubic crystals, and anharmonic correcti ons are discussed. The inelastic part of the HL static structure factor S′′(k), averaged over orientations of wave-vector k, is shown to contain pronounced singularities at Bragg diffrac- tion positions. The type of the singularities is different in classical and quantum cases. The HL method can serve as a useful tool complementary to MC and other numerical meth- ods. PACS numbers: 52.25.Zb I. INTRODUCTION A model of a Coulomb crystal of point charges in a uniform neutralizing background of charges of oppo- site sign is widely used in various branches of physics. The model was originally proposed by Wigner [1] who showed that zero-temperature electron gas immersed into uniform background of positive charges crystallizes into body-centered cubic (bcc) Coulomb crystal at sufficiently low density. Since then the model has been used in solid state physics for describing electron-hole plasma (e.g., Ref. [2]) and in plasma physics for describing dusty plas- mas and ion plasmas in Penning traps (e.g., Ref. [3]). Finally, Coulomb crystals of ions on almost uniform back- ground of degenerate electron gas are known to be formed in the cores of white dwarfs and the envelopes of neutron stars. Consequently, properties of Coulomb crystals are important for studying structure and evolution of these astrophysical objects (e.g., Ref. [4]). As classical examples of strongly coupled systems, theCoulomb crystals have been the subject of extensive stud- ies by various numerical methods, mostly by Monte Carlo (MC; e.g., [5], and references therein), and also by molec- ular dynamics (MD; e.g., Ref. [6]), and path-integral Monte Carlo (PIMC; e.g, Ref. [7]). Although the results of these studies are very impressive, the numerical meth- ods are time consuming and require the most powerful computers. The aim of the present article is to draw attention to a simple analytic model of Coulomb crystals. It has been employed recently in Ref. [8] in connection with trans- port properties of degenerate electrons in strongly cou- pled plasmas of ions. We will show that this model is a useful tool for studying static and dynamic properties of Coulomb crystals themselves. II. STRUCTURE FACTORS IN HARMONIC LATTICE APPROXIMATION For certainty, consider a Coulomb crystal of ions im- mersed in a uniform electron background. Let ˆ ρ(r, t) =/summationtext iδ(r−ˆri(t)) be the Heisenberg representation opera- tor of the ion number density, where ˆri(t) is the operator of the ith ion position. The spatial Fourier harmonics of the number density operator is ˆ ρk(t) =/summationtext ie−ık·ˆri(t). The dynamic structure factor ˜S(k, ω) of the charge density is defined as ˜S(k, ω) =1 2π/integraldisplay+∞ −∞dt e−ıωtS(k, t), (1) S(k, t) =1 N/angbracketleftBig ˆρ† k(t)ˆρk(0)/angbracketrightBig T−Nδk,0 =1 N/summationdisplay ij/angbracketleftBig eık·ri(t)e−ık·rj(0)/angbracketrightBig T −(2π)3nδ(k), (2) where Nis the number of ions in the system, nis the ion number density, ∝angbracketleft. . .∝angbracketrightTmeans canonical averaging at temperature T, and the last term takes into account con- tribution from the neutralizing background. The above definition is equally valid for liquid and solid states of the ion system. In the solid regime, it is natural 1to set ˆri(t) =Ri+ˆui(t), where Riis a lattice vector, andˆui(t) is an operator of ion displacement from Ri. Accordingly, S(k, t) =1 N/summationdisplay ijeık·(Ri−Rj)/angbracketleftBig eık·ˆui(t)e−ık·ˆuj(0)/angbracketrightBig T −(2π)3nδ(k). (3) The main subject of the present paper is to discuss theharmonic lattice (HL) model which consists in re- placing the canonical averaging, ∝angbracketleft. . .∝angbracketrightT, based on the ex- act Hamiltonian, by the averaging based on the corre- sponding oscillatory Hamiltonian which will be denoted as∝angbracketleft. . .∝angbracketrightT0. In order to perform the latter averaging we expand ˆui(t) in terms of phonon normal coordinates: ˆui(t) =/summationdisplay ν/radicalbigg ¯h 2mNω νeν× /parenleftBig eıq·Ri−ıωνtˆbν+e−ıq·Ri+ıωνtˆb† ν/parenrightBig , (4) where mis the ion mass, ν≡(q, s),s= 1,2,3 enumer- ates phonon branches; q,eν,ωνare, respectively, phonon wavevector (in the first Brillouin zone), polarization vec- tor, and frequency; ˆbνandˆb† νrefer to phonon annihilation and creation operators. The averaging over the oscilla- tory Hamiltonian, H0=/summationtext ν1 2¯hων(ˆbνˆb† ν+ˆb† νˆbν), reads ∝angbracketleftˆF∝angbracketrightT0=/summationdisplay ν∞/summationdisplay nνf(nν)Fnνnν, (5) where nνis the number of phonons in a mode ν,f(nν) = e−nνzν(1−e−zν) is the phonon density matrix in thermo- dynamic equilibrium, zν= ¯hων/T,Fnνnνis a diagonal matrix element of the operator ˆF. Inserting Eq. (4) into (3) we can perform the averaging (5) using the technique described, for instance, in Kittel [9]. The resulting structure factor S(k, t) takes into ac- count absorption and emission of anynumber of phonons; it can be decomposed into the time-independent elastic (Bragg) part and the inelastic part, S(k, t) =S′(k) + S′′(k, t). The elastic part is [9]: S′(k) =e−2W(k)(2π)3n/summationdisplay G′ δ(k−G), (6) whereGis a reciprocal lattice vector; prime over the sum means that the G= 0 term is excluded (that is done due to the presence of uniform electron background). In Eq. (6) we have introduced the Debye-Waller factor, e−W(k)=∝angbracketleftexp(ık·ˆu)∝angbracketrightT0, W(k) =3¯h 2m/angbracketleftbigg(k·eν)2 ων/parenleftbigg ¯nν+1 2/parenrightbigg/angbracketrightbigg ph =¯hk2 2m/angbracketleftbigg1 ων/parenleftbigg ¯nν+1 2/parenrightbigg/angbracketrightbigg ph, (7)where ¯ nν= (ezν−1)−1is the mean number of phonons in a mode ν. The brackets ∝angbracketleftfν∝angbracketrightph=1 3N/summationdisplay νfν=1 24π3n3/summationdisplay s=1/integraldisplay dqfν (8) denote averaging over the phonon spectrum, which can be performed numerically, e.g., Ref. [10]. The integral on the rhs is meant to be taken over the first Brillouin zone. The latter equality in Eq. (7) is exact at least for cubic crystals discussed below. For these crystals, W(k) =r2 Tk2/6, where r2 T=∝angbracketleftˆu2∝angbracketrightT0is the mean-squared ion displacement (e.g., [9,10]). The inelastic part of S(k, t) (e.g., [9]) can be rewritten as S′′(k, t) =/summationdisplay Reik·R−2W(k)/bracketleftBig evαβ(R,t)kαkβ−1/bracketrightBig ,(9) vαβ(R, t) =3¯h 2m/angbracketleftbiggeναeνβ ωνcos(ωνt+izν/2) sinh (zν/2)eiq·R/angbracketrightbigg ph. (10) Eqs. (6) and (9) result in the HL dynamical structure factor ˜S(k, ω) =−(2π)3n δ(ω)δ(k) +1 2π/integraldisplay+∞ −∞dt e−iωt−¯hω/2T ×/summationdisplay Reik·R−2W(k)+vαβ(R,τ)kαkβ, (11) where tis real and τ=t−i¯h/(2T). Along with the HL model we will also use the simpli- fied model introduced in Ref. [8]. It will be called HL1 and its results will be labelled by the subscript ‘1’. It consists in replacing S′′(k, t) given by Eq. (9) by a sim- plified expression S′′ 1(k, t) equal to the first term of the sum,R= 0: S1(k, t) =S′(k) +S′′ 1(k, t), S′′ 1(k, t) =e−2W(k)/parenleftBig ev(t)k2−1/parenrightBig , (12) where vis defined by the equation vαβ(0, t) =v(t)δαβ, which is the exact tensor structure for cubic crystals (see above). The accuracy of this approximation, as discussed in Ref. [8], is good for evaluating the quantities obtained by integration over k(e.g., transport properties of degen- erate electrons in Coulomb crystals of ions). III. STATIC CASE. HL VERSUS MC In this section we compare our analytic models with MC simulations of Coulomb crystals. For this purpose we introduce the function 2g(r) = 1 +1 n/integraldisplaydΩr 4π/integraldisplaydk (2π)3[S(k,0)−1]e−ik·r,(13) which may be called the static two particle radial distri- bution function. This function is the result of an angular and a translation average of the static two particle dis- tribution function. In this expression dΩ ris the solid angle element in the direction of r. One can see that 4πr2ng(r)dris the ensemble averaged number of ions in a spherical shell of radius rand width d rcentered at a given ion. Thus g(r) is just the quantity determined from MC simulations [5]. First let us use the HL1 model. From Eqs. (6) and (12) we easily obtain g1(r) =g′(r) +g′′ 1(r), where g′(r) = 1 +/summationdisplay G′e−2W(G)sinGr Gr, g′′ 1(r) =−3√ 3π 8π2nr3 Texp/parenleftbigg −3r2 4r2 T/parenrightbigg . (14) Calculation of g′′(r) in the HL model is more cumber- some. After integration over k=|k|and Ω rthe result can be written as g(r) =g1(r) +/summationdisplay R′/summationdisplay σ=±1/bracketleftbigg√π (2π)3rn ×/integraldisplaydΩk x2γ e−γ2+√ 3πσ 8π2nrRr Te−η/bracketrightBigg , (15) where γ= (r+σRµ)/x,η= 3(r+σR)2/(4r2 T),µ= cosϑ,ϑis an angle between kandR,x2= 4[r2 T/3− (kαkβvαβ(R,0)/k2)], and dΩ kis the solid angle element in the direction of k. Therefore, we need to evaluate a rapidly converging lattice sum (15) of 2D integrals in which xis known once the matrix elements vαβ(R,0) are calculated from Eq. (10). We have performed the integration over the first Brillouin zone required in Eq. (10) using the 3D Gauss integration scheme described in Ref. [11]. The function g(r) depends on the lattice type and on two parameters: the classical ion coupling parameter Γ = Z2e2/(aT) and the quantum parameter θ= ¯hωp/Tthat measures the importance of zero-point lattice vibrations. In this case Zeis the ion charge, a= (4πn/3)−1/3is the ion sphere radius, and ωp=Ze/radicalbig 4πn/m the ion plasma frequency. First consider a classical Coulomb crystal, θ→0, for which ¯ nν≈T/(¯hων). The functions g(r) calculated using the HL and HL1 models for body-centered cubic (bcc) crystals at Γ = 180 and 800 are presented in Figs. 1 and 2. The pronounced peak structure corresponds to the bcc lattice vectors. These results are compared with extensive MC simulations. The MC method is described, e.g., in Ref. [5]. The simulations have been done with 686 particles over nearly 108MC configurations.FIG. 1. g(r) for a bcc Coulomb crystal at Γ = 180. One can observe a very good agreement of HL and MC results for both values of Γ at 1 .5<∼r/a<∼7. The MC results for g(r) are limited to half the size of the basic cell containing the Ncharges due to the bias from particles in the image cells adjacent to the basic cell. ForN= 686 the basic cell length is 14.2 a. Hence the MCg(r) results for this simulation are valid only out tor≈7awhile g(r), given by the HL model, remains accurate as r→ ∞. At small particle separations, r<∼ 1.5a, where g(r) becomes small, the HL g(r) deviates from the MC g(r). It is clear that the HL model cannot be reliable at these r, where strong Coulomb repulsion of two particles dominates, and the MC data (available down to r>∼1.1a) are more accurate. The HL1 model is quite satisfactory at r>∼2.5a, beyond the closest lattice peak. The HL model improves significantly HL1 at lower r. It is interesting that for Γ = 180 the HL1 model agrees slightly better with MC for the range 2 .5<∼r/a<∼6 than the HL model does. With increasing Γ, however, the HL model comes into better agreement with MC at these r, although the difference between the HL and HL1 models becomes very small. This good agreement of the HL models with the MC simulations after the first peak of g(r) indicates that we have a very good description of Coulomb crystals for which the HL model may be used in place of MC simulations. The HL model enables one to analyse quantum effects. Figs. 1 and 2 exhibit also g(r) in the quantum regime atθ= 10. Zero-point lattice vibrations tend to reduce lattice peaks. The simplicity of the implementation of the HL model in the quantum regime is remarkable given the complexity of direct numerical studies of the quantum effects by MC, PIMC or MD simulations (see, e.g., Ref. [7]). 3FIG. 2. Same as in Fig. 1 but at Γ = 800. IV. COULOMB ENERGY To get a deeper insight into the HL and HL1 models let us use them to calculate the electrostatic energy Uof the crystal. Writing this energy as the sum of Coulomb energies of different pairs of ions complemented by the interaction energy of ions with the electron background and the Coulomb energy of the background itself, we ar- rive at the standard expression U N= 2πn/integraldisplay∞ 0r2drZ2e2 r[g(r)−1], (16) where g(r) is given by Eq. (13). Therefore, we can use the function g(r) calculated in Sect. 3 to analyse U. For the HL1 model from Eqs. (14) we get U1 NT=/summationdisplay G′ e−2W(G)2πnZ2e2 TG2−/radicalbigg 3 4πZ2e2 TrT= Γ/bracketleftBigg ζ+r2 T 2a2−/summationdisplay R′a 2Rerfc/parenleftBigg√ 3R 2rT/parenrightBigg/bracketrightBigg , (17) where ζis the electrostatic Madelung constant [= −0.895929 for bcc, and −0.895873 for face-centered cu- bic (fcc) lattice], and erfc( x) is the complementary error function. The second line of this equation is obtained us- ing the formula for the Madelung constant derived with the Ewald method (see, e.g., Ref. [12]) ζ=/summationdisplay R′a 2Rerfc/parenleftbiggAR a/parenrightbigg +3 2/summationdisplay G′e−G2a2/(4A2) G2a2 −3 8A2−A√π, (18)where Ais an arbitrary number. In the particular case of Eq. (17) A=√ 3a/(2rT). For the HL model, using Eq. (15), we have U NT= Γ/braceleftbigg ζ+r2 T 2a2 −/summationdisplay R′/bracketleftbigga 2R−/integraldisplaydΩk 4π2√πa xexp/parenleftbigg −R2µ2 x2/parenrightbigg/bracketrightbigg/bracerightBigg .(19) First, consider the classical crystal at zero tempera- ture,T→0. Then rT→0,x→0, and we reproduce the Madelung energy, U/N→U1/N→ζZ2e2/a. In the limit of small TbothU1/NandU/Ncontain the main term that can be expanded in powers of Tplus an ex- ponentially small term (non-analytic at T= 0). For the classical crystal at any Twe have r2 T/a2=u−2/Γ, where us=∝angbracketleft(ων/ωp)s∝angbracketrightphdenotes a phonon spectrum moment (u−2=12.973 for bcc and 12.143 for fcc). The sum over R∝negationslash= 0 in the last expression for U1 in Eq. (17) is exponentially small. Thus the analytic part of U1in the HL1 model is given only by two terms, U1/(NT) =ζΓ+u−2/2. We see that the HL1 model fails to reproduce correctly the harmonic part of the potential energy: u−2/2 appears instead of conventional 3 /2. On the contrary, the expansion of U/(NT) in the HL model, Eq. (19), contains all powers of T. To analyse this expansion, let us take any term of the sum over R, and introduce a local coordinate frame with z-axis along R. Then /integraldisplay dΩk. . .=/integraldisplay+1 −1dµ/integraldisplay2π 0dφ . . ., (20) where φis an azimuthal angle of kin the adopted frame. Since x→0 asT→0 in the denominator of the exponent under the integral in Eq. (19), only a narrow interval of µin the vicinity of µ= 0 contributes, and we can extend the integration over µto the interval from −∞to +∞. Furthermore, using the definition of x, Eq. (15), we can rewrite xas x2=x2 0(1 +ǫ), ǫ=x2 µ x2 0, (21) x2 0=4 3r2 T−4/parenleftbig vxxcos2φ+vyysin2φ+vxysin2φ/parenrightbig , x2 µ= 4µ2/parenleftbig vxxcos2φ+vyysin2φ+vxysin 2φ−vzz/parenrightbig −8µ/radicalbig 1−µ2(vxzcosφ+vyzsinφ), where vαβ=vαβ(R,0). Accordingly, we can treat ǫas small parameter and expand any integrand in Eq. (19) in powers of ǫand further in powers of µ. This generates the expansion in powers of T. We have been able to evaluate three first terms of this expansion. In particular, the term linear in Tcontains the expression 43T 2/angbracketleftBigg ω2 p ω2ν1 4πn/summationdisplay R′R2−3(R·eν)2 R5eiq·R/angbracketrightBigg ph =3T 2/angbracketleftBigg ω2 p ω2ν/bracketleftbigg Dαβ(q)eναeνβ−1 3/bracketrightbigg/angbracketrightBigg ph, (22) where Dαβis the dynamical matrix. Combining this expression with r2 T/(2a2) and taking into account that Dαβeναeνβ=ω2 ν/ω2 p(according to the basic equa- tion for the phonon spectrum) we see that the HL ex- pansion of the analytic part of Uin powers of Tis U/(NT) =ζΓ + 3 /2 +δUT/(NT); it reproduces not only the Madelung term, but also the correct oscilla- tory term 3 /2, and contains a higher-order contribution δUT/(NT) =AHL 1/Γ +AHL 2/Γ2+. . .that can be called “anharmonic” contribution in the HL model. After some transformations the coefficients AHL 1andAHL 2are re- duced to the sums over Rcontaining, respectively, bi- linear and triple products of vαβ(with integration over µandφdone analytically). Numerically the sums yield AHL 1= 10.64 and AHL 2=−62.4. The anharmonic terms occur since U, as given by Eq. (16), includes exact Coulomb energy (without expanding the Coulomb potential in powers of ion displacements u). However, we use g(r) in the HL approximation and thus neglect the anharmonic contribution in ion-ion cor- relations. Therefore, the HL model does not include all anharmonic effects. Let us compare the HL calculation of δUTwith the exact calculation of the first anharmonic term in the Coulomb energy of classical Coulomb crystals by Dubin [13]. The author studied the expansion δUexact T/(NT) = Aexact 1/Γ +Aexact 2/Γ2+. . .and expressed the first term as Aexact 1= Γ/bracketleftbigg∝angbracketleftU2 3∝angbracketright 72NT2−∝angbracketleftU4∝angbracketright 24NT/bracketrightbigg , (23) where Un/n! is the nth term of the Taylor expansion of the Coulomb energy over ion displacements, while angu- lar brackets denote averaging with the harmonic Hamil- tonian H0. According to Dubin Aexact 1=10.84 and 12.34 for bcc and fcc crystals, respectively. (The same quantity was computed earlier by Nagara et al. [14] who reported Aexact 1=10.9 for bcc.) It turns out that our δUTsums up a part of the infinite series of anharmonic corrections to the energy, denoted by Dubin as/summationtext∞ n=3∝angbracketleftUn∝angbracketright/(n!), so that AHL 1= Γ∝angbracketleftU4∝angbracketright/(24NT),AHL 2= Γ2∝angbracketleftU6∝angbracketright/(6!NT), etc. (The fact that this summation can be performed in a closed ana- lytic form was known from works on the so called self- consistent phonon approximation, e.g., [15] and refer- ences therein.) Our numerical value for the bcc lattice AHL 1= 10.64 is very close to the value of Γ ∝angbracketleftU4∝angbracketright/(24NT) reported by Dubin as ≈10.69 (his Table 3) which confirms accuracy of both calculations. The fact that AHL 1= 10.64 is close to Aexact 1= 10.84 for bcc is acci- dental (Dubin found Γ ∝angbracketleftU2 3∝angbracketright/(72NT2)≈21.53 for bcc).For instance, from the results of Ref. [13] for fcc one in- fers,AHL 1≈5.63 which differs strongly from the exact anharmonic coefficient Aexact 1= 12.34. Now let us set T= 0 and analyse the quantum effects. We can expand Eqs. (17) and (19) in powers of rT/a. For T= 0 the quantity rTtends to the rms amplitude of zero- point vibrations, rT=/radicalbig 3¯hu−1/(2mωp), where u−1is another phonon spectrum moment (=2.7986 and 2.7198 for bcc and fcc, respectively). The expansion of U1/N givesζZ2e2/a+u−1¯hωp/4 plus small non-analytic terms. In the same manner as in Eq. (22) we find that U/N= ζZ2e2/a+ 3u1¯hωp/4 +δU0/N. The second term gives half of the total (kinetic + potential) zero-point harmonic energy of a crystal, as required by the virial theorem for harmonic oscillator ( u1=0.51139 and 0.51319 for bcc and fcc, respectively), while the third term, δU0, represents zero-point anharmonic energy in the HL approximation. To make the above algebra less abstract let us esti- mate the accuracy of the HL model and the relative im- portance of the anharmonicity and quantum effects. In the classical case, taking Γ = 170 (close to the melt- ing value Γ m= 172 for bcc), we estimate the anhar- monic contribution to the total electrostatic energy as |δUT/U| ≈Aexact 1/(|ζ|Γ2)≈4.2×10−4and 4 .8×10−4 for bcc and fcc, respectively. The relative error into Uintroduced by using the HL model is Aexact 2/(|ζ|Γ3)≈5.7×10−5for bcc (if we adopt an estimate of Aexact 2≈247 from the MD data on the full electrostatic energy presented in Table 5 of Ref. [6]) and [Aexact 1−AHL 1]/(|ζ|Γ2)≈2.6×10−4for fcc. We see that Coulomb crystals can be regarded as highly harmonic, and the accuracy of the HL model is sufficient for many practical applications. Obviously, the accuracy becomes even better with decreasing T. The quantum effects can be more important (than the anharmonicity) in real sit- uations. Let us take12C matter at density ρ= 106g cm−3typical for the white dwarf cores or neutron star crusts. The quantum contribution into energy is mea- sured by the ratio 3 u1¯hωp/(4|ζ|Z2e2/a) which is equal to 4.7×10−3at given ρ(and grows with density as ρ1/6). For completeness we mention that the compressibility of the electron background also contributes to the electro- static energy. The relative contribution in the degenerate electron case for12C atρ= 106g cm−3is∼10−2(e.g., Ref. [16]). Another point is that the HL model takes into account zero-point lattice vibrations but neglects ion ex- change which becomes important at very high densities (e.g., Ref. [4]). V. STRUCTURE FACTORS Finally, it is tempting to use the HL model for an- alyzing the ion structure factors themselves. Con- sider the angle-averaged static structure factor S(k) =/integraltext dΩkS(k, t= 0)/(4π). For the Bragg part, from Eq. (6) we obtain the expression 5S′(k) =e−2W(k)2π2n/summationdisplay G′ δ(k−G)/G2, (24) containing delta-function singularities at k=G, lengths of reciprocal lattice vectors G. Direct HL calculation of S′′(k) from Eq. (9) is complicated by the slow conver- gence of the sum and complex dependence of vαβonR. However, the main features of S′′(k) can be understood from two approximations. First, in the HL1 model we havevαβ(0,0)kαkβ= 2W(k), and S′′ 1(k) = 1−e−2W(k) as shown by the dashed line in Fig. 3. The second, more realistic approximation will be called HL2 (and labelled by the subscript ‘2’). It consists in adopting a simplified tensor decomposition ofvαβ(R,0) of the form vαβ(R,0) = F(R)δαβ+ RαRβJ(R)/R2. If so, we can immediately take the fol- lowing integrals/integraltext dΩRvαα(R,0)/(4π) = 3F(R) +J(R) and/integraltext dΩRvαβ(R,0)RαRβ/(4πR2) =F(R) +J(R) (as- suming summation over repeating tensor indices αand β). On the other hand, we can calculate the same inte- grals taking vαβ(R,0) from Eq. (10) at t= 0. In this way we come to two linear equations for F(R) and J(R). Solving them, we obtain F(R) =3¯h 2m/angbracketleftbigg1 ων/parenleftbigg ¯nν+1 2/parenrightbigg/braceleftbigg j0(y)−j1(y) y −(q·eν)2 q2/bracketleftbigg j0(y)−3j1(y) y/bracketrightbigg/bracerightbigg/angbracketrightbigg ph, J(R) =3¯h 2m/angbracketleftbigg1 ων/parenleftbigg ¯nν+1 2/parenrightbigg/bracketleftbigg j0(y)−3j1(y) y/bracketrightbigg ×/bracketleftbigg3(q·eν)2 q2−1/bracketrightbigg/angbracketrightbigg ph, (25) where y=qR, and j0(y) and j1(y) are the spherical Bessel functions. Note that F(0)k2= 2W(k),J(0) = 0. In the limit of large Rthe functions j0(qR) and j1(qR) in Eqs. (25) strongly oscillate which means that the main contribution into the phonon averaging (integration over q) comes from a small vicinity near the center of the Bril- louin zone. Among three branches of phonon vibrations in simple Coulomb crystals, two ( s=1, 2) behave as trans- verse acoustic modes, while the third ( s=3) behaves as a longitudinal optical mode ( ω≈ωp) near the center of the Brillouin zone. Owing to the presence of ω−1 νin the denominator of Eqs. (25), the main contribution at large Rcomes evidently from the acoustic modes. Thus we can neglect optical phonons and set ω=csqfor acoustic modes, where csis the mean ion sound velocity. In the high-temperature classical limit, (¯ nν+1 2)→T/(¯hcsq). Then from Eqs. (25) at R→ ∞ we approximately obtain F(R)≈T 4π2nmR/integraldisplay∞ 0dy/bracketleftbigg j0(y)−j1(y) y/bracketrightbigg2/summationdisplay s=11 c2s =T 16πnmR2/summationdisplay s=11 c2s,J(R)≈ −T 4π2nmR/integraldisplay∞ 0dy/bracketleftbigg j0(y)−3j1(y) y/bracketrightbigg2/summationdisplay s=11 c2s =T 16πnmR2/summationdisplay s=11 c2s. (26) Our analysis shows that an appropriate value of c−2 1+c−2 2 for bcc lattice would be 67 .85/(aωp)2. From Eq. (26) we see that F(R) and J(R) decrease as R−1with increasing R. In the quantum limit θ≫1 we have (¯ nν+1 2)→1 2; applying the same arguments we deduce that F, J∝R−2 asR→ ∞. Using Eq. (9) we have S′′ 2(k) =/integraldisplaydΩk 4π/summationdisplay Reık·R−2W(k) ×/bracketleftBig ek2F(R)+(k·R/R)2J(R)−1/bracketrightBig = 1−e−2W(k) +1 2/summationdisplay R′/integraldisplay+1 −1dµ e−2W(k)+ıkRµ ×/bracketleftBig ek2F(R)+k2J(R)µ2−1/bracketrightBig . (27) A number of the first terms of the sum, say for |R|< R0, where R0/ais sufficiently large, can be calculated exactly. To analyse the convergence of the sum over R at large Rlet us expand the exponential in the square brackets on the rhs. All the terms of the expansion which behave as R−nwithn≥2 lead to nicely convergent contributions to S′′ 2(k). The only problem is posed by the linear expansion term in the classical case. The tail of the sum,/summationtext |R|>R0, for this term can be regularized and calculated by the Ewald method (e.g., Ref. [12]) with the following result /integraldisplaydΩk 4π/summationdisplay |R|>R0eık·R−2W(k)/bracketleftBig ek2F+(k·R/R)2J−1/bracketrightBig ≈2Tk2e−2W(k) 16πnm2/summationdisplay s=11 c2s /summationdisplay |R|>R0sinkR kR2erfc/parenleftbiggAR a/parenrightbigg +4πn k2e−k2a2/(4A2)+/summationdisplay′ |R|<R0sinkR kR2erf/parenleftbiggAR a/parenrightbigg +/summationdisplay G′/summationdisplay τ=±1πnτ kGEi/parenleftbigg −[k+τG]2a2 4A2/parenrightbigg +2A a√π/bracketrightBigg ,(28) where Ei( −x) is the exponential integral, and Ais a num- ber to be chosen in such a way the convergence of both infinite sums (over direct and reciprocal lattice vectors) be equally rapid. Letting A→ ∞ we obtain a much more transparent, although slower convergent formula /bracketleftBigg . . ./bracketrightBigg =4πn k2+ 2πn/summationdisplay G′/bracketleftbigg1 kGln/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+G k−G/vextendsingle/vextendsingle/vextendsingle/vextendsingle−2 G2/bracketrightbigg −/summationdisplay′ |R|<R0sinkR kR2+2ζ a. (29) 6This expression explicitly reveals logarithmic singular- ities at k=G. They come from inelastic processes of one-phonon emission or absorption in the cases in which given wave vector kis close to a reciprocal lattice vec- torG. To prove this statement let us perform Taylor expansions of both exponentials in angular brackets in Eq. (3). The one-phonon processes correspond to those expansion terms which contain products of one creation and one annihilation operator. Thus, in the one-phonon approximation S′′(k, t= 0) reads S′′ 1ph(k, t= 0) =e−2W(k) N/summationdisplay ijeık·(Ri−Rj) ×∝angbracketleft(ik·ˆui)(−ik·ˆuj)∝angbracketrightT0 =e−2W(k) N/summationdisplay ij/summationdisplay ν¯h(k·eν)2 2mNω νeı(k−q)·(Ri−Rj)(2¯nν+ 1) =e−2W(k)/summationdisplay s¯h(k·eqs)2 mωqs/parenleftbigg ¯nqs+1 2/parenrightbigg , (30) where the last summation is over phonon polarizations, q=k−Gis the phonon wave vector which is the given wave vector kreduced into the first Brillouin zone by subtracting an appropriate reciprocal lattice vector G. In addition, in Eq. (30) we have introduced an over- all factor e−2W(k)which comes from renormalization of the one-phonon probability associated with emission and absorption of any number of virtual phonons (e.g., Ref. [9]). Now let us assume that |k−G|a≪1 and average Eq. (30) over orientations of k[integrate over dΩ k/(4π)]. One can easily see that the important contribution into the integral comes from a narrow cone Ω 0aligned along G. Let θ0≪1 be the cone angle chosen is such a way thatGθ0a≪1, but Gθ0≫ |G−k|. Integrating within this cone, we can again adopt approximation of acoustic and longitudinal phonons and neglect the contribution of the latters. For simplicity, we also assume that the sound velocities of both acoustic branches are the same: ων=cs|k−G|. Then, in the classical limit we come to the integral of the type /integraldisplay Ω0dΩk 4π2/summationdisplay s=1(k·eqs)2 ω2qs≈1 4c2s/braceleftbigg ln/bracketleftbiggkGθ2 0 (k−G)2/bracketrightbigg −1/bracerightbigg , (31) which contains exactly the same logarithmic divergency we got in Eq. (29). Note that in the quantum limit we would have similar integral but with ωinstead of ω2in the denominator of the integrand. The integration would yield the expression proportional to |k−G|, i.e., the log- arithmic singularity would be replaced by a weaker kink- like feature. Therefore, the k=Gfeatures of the inelastic structure factor S′′(k) in the quantum limit are expected to be less pronounced than in the classical limit but could be, nevertheless, quite visible. Actually, at any finite temperature, even deep in the quantum regime T≪¯hωpthere are still phonons excited thermally near the very center of the Brillouin zone, where the energy of acous- tic phonons is smaller than temperature. Due to these phonons the logarithmic singularity always exists on top of the kink-like feature at T∝negationslash= 0. After this simplified consideration let us return to qual- itative analysis. We have calculated S′′ 2(k) in the classical limit using the HL2 approximation as prescribed above and verified that the result is indeed independent of R0 (in the range from ∼30ato 100 a) and A. The resulting S′′ 2(k) is plotted in Fig. 3 by the solid line. FIG. 3. Inelastic part of the structure factor at Γ = 180 for classical bcc crystal. Thus, in a crystal, the inelastic part of the structure factor, S′′(k), appears to be singular in addition to the Bragg (elastic) part S′(k). The singularities of S′′(k) are weaker than the Bragg diffraction delta functions in S′(k); the positions of singularities of both types coin- cide. The pronounced shapes of the S′′(k) peaks may, in principle, enable one to observe them experimentally. The structure factor S(k) in the Coulomb liquid (see, e.g., Ref. [17] and references therein) also contains significant but finite and regular humps associated with short-range order. This structure has been studied in detail by MC and other numerical methods. In contrast, the studies of singular structure factors in a crystal by MC or MD methods would be very complicated. Luckily, they can be explored by the HL model. Finally, it is instructive to compare the behavior of S′′(k) at small kin the HL1 and HL2 models. It is easy to see that the main contribution to inelastic scatter- ing at these kcomes from one-phonon normal processes [withq=kin Eq. (30)]. At these kthe HL2 S′′ 2(k) co- incides with the one-phonon S′′ 1ph(k) and with the static 7structure factor of Coulomb liquid (at the same Γ) and reproduces correct hydrodynamic limit [18], S(k)∝k2. The HL1 model, on the contrary, overestimates the im- portance of the normal processes. Let us mention that we have also used the HL2 model to calculate g(r). HL2 appears less accurate than HL but better than HL1. We do not plot g2(r) to avoid obscuring the figures. VI. CONCLUSIONS Thus, the harmonic lattice model allows one to study static and dynamic properties of quantum and classical Coulomb crystals. The model is relatively simple, espe- cially in comparison with numerical methods like MC, PIMC and MD. The model can be considered as com- plementary to the traditional numerical methods. More- over, it can be used to explore dynamic properties of the Coulomb crystals and quantum effects in the cases where the use of numerical methods is especially complicated. For instance, the harmonic lattice model predicts singu- larities of the static inelastic structure factor at the pos i- tions of Bragg diffraction peaks. We expect also that the HL model can describe accurately non-Coulomb crystals whose lattice vibration properties are well determined. Acknowledgements. We are grateful to N. Ashcroft for discussions. The work of DAB and DGY was sup- ported in part by RFBR (grant 99–02–18099), INTAS (96–0542), and KBN (2 P03D 014 13). The work of HEDW and WLS was performed under the auspices of the US Dept. of Energy under contract number W-7405- ENG-48 for the Lawrence Livermore National Labora- tory and W-7405-ENG-36 for the Los Alamos National Laboratory. [1] E.P. Wigner, Phys. Rev. 46, 1002 (1934). [2] S.Ya. Rakhmanov, Zh. Eksper. Teor. Fiz. 75, 160 (1978). [3] W.M. Itano, J.J. Bollinger, J.N. Tan, B. Jelenkovi´ c, X. - P. Huang, and D.J. Wineland, Science 279, 686 (1998); D.H.E. Dubin and T.M. O’Neil, Rev. Mod. Phys. 71, 87 (1999). [4] G. Chabrier, Astrophys. J. 414, 695 (1993); G. Chabrier, N.W. Ashcroft, and H.E. DeWitt, Nature 360, 48 (1992). [5] G.S. Stringfellow, H.E. DeWitt, and W.L. Slattery, Phys . Rev.A41, 1105 (1990); W.L. Slattery, G.D. Doolen, and H.E. DeWitt, Phys. Rev. A21, 2087 (1980). [6] R.T. Farouki and S. Hamaguchi, Phys. Rev. E47, 4330 (1993). [7] S. Ogata, Astrophys. J. 481, 883 (1997). [8] D.A. Baiko, A.D. Kaminker, A.Y. Potekhin, and D.G. Yakovlev, Phys. Rev. Lett. 81, 5556 (1998).[9] C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963). [10] D.A. Baiko and D.G. Yakovlev, Astron. Lett. 21, 702 (1995). [11] R.C. Albers and J.E. Gubernatis, preprint of the LASL LA-8674-MS (1981). [12] M. Born and K. Huang, Dynamical theory of crystal lattices (Claredon Press, Oxford, 1954). [13] D.H.E. Dubin, Phys. Rev. A42, 4972 (1990). [14] H. Nagara, Y. Nagata, and T. Nakamura, Phys. Rev. A36, 1859 (1987). [15] R.C. Albers and J.E. Gubernatis, Phys. Rev. B23, 2782 (1981). [16] G. Chabrier and A.Y. Potekhin, Phys. Rev. E58, 4941 (1998). [17] D.A. Young, E.M. Corey, and H.E. DeWitt, Phys. Rev. A44, 6508 (1991) [18] P. Vieillefosse and J.P. Hansen, Phys. Rev. A12, 1106 (1975) 8

arXiv:physics/9912049v1 [physics.class-ph] 23 Dec 1999Exploring a rheonomic system Antonio S de Castro UNESP - Campus de Guaratinguet´ a - Caixa Postal 205 - 1250000 0 Guaratinguet´ a - SP - Brasil Abstract. A simple and illustrative rheonomic system is explored in th e Lagrangian formalism. The difference between Jacobi’s integral and ene rgy is highlighted. A sharp contrast with remarks found in the literature is pointed out . The non-conservative system possess a Lagrangian not explicitly dependent on tim e and consequently there is a Jacobi’s integral. The Lagrange undetermined multipli er method is used as a complement to obtain a few interesting conclusions. PACS number: 03.20.+i Submitted to: Europ. J. Phys.Exploring a rheonomic system 2 Constraints are restrictions that limit the motion of the pa rticles of a system. The forces necessary to constrain the motion are said to be fo rces of constraint. The constraints expressible as algebraic equations relating t he coordinates of the particles and the time variable are called holonomic, if not they are ca lled nonholonomic. Furthermore, in each type of constraints, holonomic or nonh olonomic, the time variable could appear explicitly. If the time variable does not appea r explicitly in the relations of constraint they are further classified as scleronomic, ot herwise they are said to be rheonomic. Holonomic constraints, and in fact a very restrict class of n onholonomic constraints (those expressible as first-order differential forms relati ng the coordinates and the time variable), are amenable to straightforward general treatm ent in analytical mechanics. These sorts of constraints allow us to describe the motion wi thout paying any explicit reference to the forces of constraint. In addition, holonom ic constraints can be used to reduce the number of coordinates required to the complete de scription of the motion, although this is not always desirable. Simple systems subject to rheonomic constraints are not wid espread in the textbooks on analytical mechanics. Nevertheless, there is a traditional system which is very simple, indeed. It consists of a bead of mass msliding along a frictionless straight horizontal wire constrained to rotate with constant angula r velocity ωabout a vertical axis [1][2][4][3][5]. This simple system presents a wealth of physics not fully explored in the literature. The main purpose of this paper is to make an eff ort for filling this gap, motivated by the strong pedagogical appeal of this illustra tive system. Furthermore, this paper takes the opportunity of doing criticisms on the remar ks in Griffths’s textbook [6] concerning general systems containing rheonomic holonomi c systems: “ ...the rheonomic constraints must be used to reduce the number of generalised coordinates and so the configuration of the system must necessarily depend explici tly on time as well as the n generalized coordinates. In this case a time dependence thu s enters explicitly into the Lagrangian. It may therefore also be concluded that systems which contain a rheonomic constraint possess neither an energy integral nor a Jacobi i ntegral. ” First, one can note that the motion of the bead is caused by a fo rce of constraint perpendicular to the wire, whereas the actual displacement of the bead is in an oblique direction and its virtual displacement satisfying the cons traint is in a parallel direction. Therefore, the force of constraint does actual work but not v irtual work. The vanishing of the virtual work characterizes the constraint as ideal. Since the motion of the bead takes place on the horizontal pla ne one can eliminate the dependence on the vertical coordinate and consider only the coordinates on the plane of the motion. The coordinates are suitably chosen wit hrbeing the distance to the rotation axis and θthe angular position relative to an arbitrary axis on the pla ne of the motion. The Lagrangian of the system is nothing but the kinetic energy of the bead:Exploring a rheonomic system 3 L=1 2m/parenleftBig ˙r2+r2˙θ2/parenrightBig (1) The constraint on the motion of the bead is expressed by φ/parenleftBig˙θ/parenrightBig =˙θ−ω= 0 (2) This relation can be immediately integrated, yielding Φ (θ, t) =θ−ωt+θ0= 0 (3) where θ0is a constant. This form of the condition of constraint allow us to classify it as a holonomic and rheonomic constraint. Now one can use this condition of constraint to eliminate the coordinate θin the Lagrangian, so that one is left with ras generalized coordinate: L=1 2m/parenleftBig ˙r2+ω2r2/parenrightBig (4) At this point the author dares to utter the first criticism on G riffths’s conclusions. The configuration of the system is just given by the coordinate r, which clearly depends on the time variable, but neither the kinetic energy nor the Lag rangian are explicitly time- dependent. In general rheonomic constraints give rise to ex plicitly time dependent terms in the Lagrangian. There are two of these terms, one of them is linear in the generalized velocities and the other one is velocity-independent. Due t o these terms it may tempting to conclude that the Lagrangian for a rheonomic system is alw ays explicitly time dependent, but it is very dangerous because it may be certain cancellations. For the particular system approached in this paper and with the part icular choice of generalized coordinates, the variables combine in such a way that the lin ear term vanishes whereas the independent term does not involve the time explicitly. T hat is the reason why the Lagrangian has no explicit time dependence. Using the Lagrangian given by (4) Lagrange’s equation gover ning the motion of the bead d dt/parenleftBigg∂L ∂˙r/parenrightBigg −∂L ∂r= 0 (5) takes on the form m¨r−mω2r= 0 (6) The energy function h, generally given byExploring a rheonomic system 4 h=/summationdisplay i˙qi∂L ∂˙qi−L (7) obeys the relation dh dt=−∂L ∂t(8) and for the present system it is given by h= ˙r∂L ∂˙r−L (9) Since the Lagrangian is not an explicit function of time h=1 2m/parenleftBig ˙r2−ω2r2/parenrightBig (10) turns out to be Jacobi’s integral, a constant of the motion. N ow arises the second criticism on Griffiths’s comments: although this system cont ains a rheonomic constraint it in fact possess a Jacobi’s integral. The necessary and suffi cient condition for the existence of Jacobi’s integral is that the Lagrangian does n ot depend explicitly on time. It is seen that here Jacobi’s integral is not the energy of the system. The only difference between them is due to the velocity-independent term in the L agrangian. The energy of the system is only kinetic energy and has a time derivative given by dE dt=m˙r/parenleftBig ¨r+ω2r/parenrightBig (11) The insertion of the equation of motion (6) into the last rela tion leads to dE dt=d dt/parenleftBig mω2r2/parenrightBig = 2mω2r˙r (12) which implies that the energy is not a constant of the motion. As we have already seen, the energy can not be a constant of the motion due to the nonvan ishing of the actual work of the force of constraint. It should be obvious that the energy function hand the energy Eare distinctly different functions, subject to distinct conservation laws , but there are special circumstances for which they are identical. This happens if the constraints are scleronomic and the potential energy is velocity-independ ent. If, further, the potential energy does not depend explicitly on time Ebecomes the energy integral and hcomes to be Jacobi’s integral. In addition to these comments is appro priated to keep in mind that the energy function hmust not be confused with the Hamiltonian H, even though they are expressed by similar mathematical structures and their conservation laws rest uponExploring a rheonomic system 5 the very same condition (not depend explicitly on time). The difference between hand His subtler than that one between handE, they are functions of different independent variables. As a matter of fact, in some cases it may not be poss ible to obtain one of them from the knowledge from the other. Usually one must use the Lagrange undetermined multiplier m ethod to obtain the force of constraint. In this method the coordinates randθare not treated as independent coordinates, therefore one has to use the Lagrangian given b y (1) instead of that one given by (4). Now Lagrange’s equations incorporate the cond ition of constraint d dt/parenleftBigg∂L ∂˙r/parenrightBigg −∂L ∂r=λ∂Φ ∂r(13) d dt/parenleftBigg∂L ∂˙θ/parenrightBigg −∂L ∂θ=λ∂Φ ∂θ(14) where λis the Lagrange undetermined multiplier. The generalized f orces of constraint are to be identified as λ∂Φ/∂randλ∂Φ/∂θ. The condition of constraint (3) implies that only the torque of constraint τis nonvanishing. These Lagrange’s equations yield m¨r−mω2r= 0 (15) τ=dpθ dt=mr/parenleftBig 2˙r˙θ+r¨θ/parenrightBig (16) where pθ=∂L ∂˙θ=mr2˙θ (17) happens to be the angular momentum. Combining (16) and (17) w ith (3) one gets τ= 2mωr˙r (18) pθ=mωr2 That the angular momentum is not a constant of the motion come s from the fact that the force of constraint is not central. The constraint force can now be obtained from (18) reckonizing that it acts on the bead directed normal to t he wire. It is an easy matter to check that Fθ= 2mω˙r (19)Exploring a rheonomic system 6 In conclusion, this paper shows that the system considered i s of great value for beginning students of analytical mechanics. In addition, i t is very useful to remove some misunderstandings found in the literature. It should b e emphasized that is the rheonomic nature of the constraint and the particular choic e of generalized coordinates that make the energy to be different from Jacobi’s integral. J acobi’s integral is here the first integral of the motion instead of the energy. The Lagran ge undetermined multiplier method has been used for obtaining the force of constraint in a natural way. Nonethe- less, the force of constraint can also be obtained from (12) b y invoking the principle of work and energy. Only the Lagrangian formalism has been cons idered in this paper but this simple system can also be easily approached by other for malisms of the analytical mechanics. This task is left to the readers. [1] Osgood W F 1937 Mechanics (New York: Dover) [2] Konopinski E J 1969 Classical Descriptions of Motion (San Francisco: Freeman) [3] Goldstein H 1980 Classical Mechanics 2nd ed. (Reading: Addison-Wesley) [4] Lindenbaum S D 1994 Analytical Dynamics (Singapore: World Scientific) [5] Chow T L 1995 Classical Mechanics (New York: Wiley) [6] Griffths J B 1985 The Theory of Classical Dynamics (Cambridge: Cambridge) p 254

Differential criterion of a bubble collapse in viscous liquids Vladislav A. Bogoyavlenskiy * Low Temperature Physics Department, Moscow State University, 119899 Moscow, Russia ~Received 11 January 1999 ! The present work is devoted to a model of bubble collapse in a Newtonian viscous liquid caused by an initial bubble wall motion. The obtained bubble dynamics described by an analytic solution significantly depends onthe liquid and bubble parameters. The theory gives two types of bubble behavior: collapse and viscousdamping. This results in a general collapse condition proposed as the sufficient differential criterion. Thesuggested criterion is discussed and successfully applied to the analysis of the void and gas bubble collapse.@S1063-651X ~99!01207-6 # PACS number ~s!: 47.55.Bx, 78.60.Mq I. INTRODUCTION Formation and collapse of bubbles in liquids are used in many technical applications such as sonochemistry, litho- tripsy, ultrasonic cleaning, bubble chambers, and laser sur-gery @1–4#. Bubble dynamics has been the subject of inten- sive theoretical and experimental studies since Lord Ray-leigh found the well-known analytic solution of this problemfor inviscid liquids @5#. The advanced theory of cavitation developed by Plesset gives the differential Rayleigh-Plesset ~RP!equation for the bubble radius R(t)@6#. The RP equa- tion describes the dynamics of a spherical void or gas bubblein viscous liquids @7–10 #and is also used as a first approxi- mation in more complex problems such as cavitation nearsolid boundaries @11–14 #, collapse of asymmetric bubbles @15,16 #, and sonoluminescence @17–23 #. The main difficulties involved in theoretical investigations of the RP equation are that ~i!the solutions can be obtained only numerically and ~ii!the bubble wall velocity increases to infinity as the bubble collapses. Thus, computer simula-tions of the bubble motion take a great deal of time and maylead to significant errors in the numerically calculated solu-tions, especially when the bubble achieves supersonicspeeds. Unfortunately, the analytically described bubble dy-namics was obtained only for the collapse in inviscid liquids@5#. In this paper we present a way to avoid the above diffi- culties for viscous liquids. The concept is based on the factthat the RP equation is analytically integrable in the case ofthe following restrictions: ~i!the bubble is void and ~ii!the ambient hydrostatic pressure is absent. The imposed restric-tions are valid as the bubble collapses because the gas andthe ambient pressures are negligible in comparison to thevelocity pressure at the bubble wall. The model gives an analytic solution for the bubble radius R(t) and a collapse criterion in the differential form. This differential criterion isconsidered to be a sufficient condition of the bubble collapsein viscous liquids. The present paper is organized as follows. In Sec. II the general model of the void bubble collapse in a viscous liquidcaused by an initial bubble wall motion is formulated andsolved. The subject of Sec. III is the application of the pro- posed differential collapse criterion to the Rayleigh problemin a viscous liquid and to the collapse of an air bubble inwater caused by periodic acoustic pressure. II. GENERAL MODEL A. Problem formulation Let us consider a void bubble immersed in an infinite Newtonian viscous liquid. We assume that the bubble is al-ways spherical. Taking into account the symmetry of thisproblem, we write all the equations in the spherical coordi- nate system ( r, w,u) whose origin is at the center of the bubble. Then the liquid motion is governed by the followingequations @1#: ]srr ]r12srr2suu2sww r5rS]vr ]t11 2]vr2 ]rD, ~1! ]vr ]r12vr r50, ~2! srr52p12m]vr ]r,suu5sww52p12mvr r,~3! whereris the radial coordinate, vris the radial liquid veloc- ity,r5const is the liquid density, sis the stress tensor, pis the hydrostatic pressure, and m5const is the shear viscosity. For this set of equations to be complete, we add the initialand the boundary conditions. Let us assume that the ambientpressure and the surface tension are negligible. These restric-tions result in the following conditions on the bubble surfaceand at infinity: srr$r5R~t!%50,srr$r5`%50. ~4! The initial conditions are chosen to be nonstandard. Usually the initial bubble wall motion is ignored, but in this model the bubble is considered to have a radial velocity V0: R$t50%5R0,dR dt$t50%52V0. ~5!*Electronic address: bogoyavlenskiy@usa.netPHYSICAL REVIEW E JULY 1999 VOLUME 60, NUMBER 1 PRE 60 1063-651X/99/60 ~1!/504~5!/$15.00 504 ©1999 The American Physical SocietyThe system of Eqs. ~1!–~5!completely describes the model. To find the solution, let us use the method proposedby Rayleigh @5#. According to the incompressibility condi- tion given by Eq. ~2!, the radial liquid velocity can be written as vr5dR dtSR rD2 . ~6! After the substitution of Eqs. ~3!and~6!into Eq. ~1!and its subsequent integration in the range ( R,`) we write the ex- pression E R`S]srr ]r212mR2 r4dR dtDdr5rE R`FR2 r2d2R dt212R r2SdR dtD2 22R4 r5SdR dtD2Gdr. ~7! Taking into account Eq. ~4!, we obtain the modified RP equation where the initial conditions are given by Eq. ~5!: Rd2R dt213 2SdR dtD2 14m rRdR dt50. ~8! B. Solution and analysis Let us define the following dimensionless variables and constants: R˜[R R0,t˜[tV0 R0,m˜[m rR0V0,a[1 8m˜21.~9! HereR˜,t˜, and m˜are the dimensionless bubble radius, time, and viscosity, respectively. Then Eqs. ~8!and~5!can be represented as R˜d2R˜ dt˜213 2SdR˜ dt˜D2 14m˜ R˜dR˜ dt˜50, R˜$t˜50%51,dR˜ dt˜$t˜50%521. ~10! The differential equation ~10!is integrable and the obtained analytic solution is the following: t˜52~a11!S1 4~12R˜2!2a 3~12R˜3/2! 1a2 2~12R˜!2a3~12R˜1/2!2a4lna1R˜1/2 a11D.~11! The most illustrative way to discuss the bubble dynamics is by analysis of the kinetic energy accumulated by the liquidnear the bubble wall. The dimensionless expression of thisenergy is given by the relationE ˜[R˜2SdR˜ dt˜D2 5R˜2 ~a11!2S2R˜1aR˜1/22a21a3R˜21/22a4R˜21/2 a1R˜1/2D2. ~12! The bubble collapse corresponds to the condition E˜!`. The overall picture of the bubble behavior given by Eqs. ~11!and~12!is summarized by Fig. 1, which shows the time dependence of the bubble radius R˜(t˜)@Fig. 1 ~a!#and the accumulated energy E˜(t˜)@Fig. 1 ~b!#. The behavior of the curves significantly depends on the value of the dimension- less viscosity m˜. Analysis of Eq. ~11!shows that the bubble radiusR˜(t˜) decreases to zero only if m˜,1 8. In this case, the accumulated energy E˜(t˜) increases to infinity. That is, the bubble collapse takes place: FIG. 1. Bubble radius R˜~a!and accumulated energy E˜~b!as functions of time t˜. Values of parameter m˜are shown at curves.PRE 60 505 DIFFERENTIAL CRITERION OF A BUBBLE COLLAPS E...lim R˜!0E˜5a2 ~a11!2lim R˜!01 R˜5`. ~13! Analysis of the bubble motion at R˜(t˜)!0 gives the fol- lowing approximation formula: R˜5S5a 2~a11!~t˜2t˜C!D2/5 , ~14! t˜C[t˜$R˜50%52~a11!S1 42a 31a2 22a32a4lna a11D. ~15! Heret˜Cis the collapse time, which varies from 0.4 ( m˜ !0) to 0.5 ( m˜51 8). The bubble dynamics described by Eqs. ~14!and~15!is basically similar to one obtained by Rayleigh @5#. The other case m˜.1 8corresponds to the viscous damping. The bubble radius R˜(t˜) smoothly decreases to the equilib- rium value R˜eqand the accumulated energy E˜(t˜) descends to zero: R˜eq5a25S1 8m˜21D2 , ~16! lim R˜!R˜eqE˜51 a2~a11!2lim R˜!R˜eq~R˜1/21a!250. ~17! III. APPLICATIONS OF THEORY The condition of the bubble collapse m˜,1 8is the main result obtained in the preceding section. Let us rewrite thisinequality as S2dR dt$R5R0%DrR0 8m.1. ~18! We should emphasize the special features of the model pre- sented that result from the boundary conditions. The above inequality contains only one variable R(t) and two liquid constants randm. Moreover, Eq. ~18!is a local, differential condition, which means there is no information about thepreceding bubble motion. The condition ~18!is insensitive to the substitution R 0$R(t). Thus, the above collapse condi- tion can be represented as c˜~t!.1,c˜~t![S2dR~t! dtDrR~t! 8m, ~19! wherec˜(t) is the dimensionless collapse variable . The physics of the differential condition ~19!is quite simple. Let us focus on a bubble motion governed by thesystem of equations ~1!–~3!~see Sec. II !in the presence of an ambient hydrostatic pressure and a gas pressure inside the bubble. Instead of the curve R(t), the behavior of the curve c˜(t) is analyzed. If the value of c˜(t) achieves the number one, the collapse takes place. This is the sufficient conditionfor a void bubble, since the ambient hydrostatic pressureadditionally forces the bubble to the collapse. The case of the gas bubble is more complicated because the gas pressureslows down the bubble wall motion. However, in most casesthe gas pressure is negligible in comparison to the velocitypressure as the criterion ~19!is realized. Two applications of the proposed criterion are presented below. A. Rayleigh’s problem in a viscous liquid The Rayleigh problem is the study of void bubble motion in a liquid caused by a constant ambient pressure @5#. In this case the boundary and initial conditions are transformedfrom Eqs. ~4!and~5!to the following: srr$r5R%50,srr$r5`%5p0, ~20! R$t50%5R0,dR dt$t50%50, ~21! wherep05const is the ambient hydrostatic pressure. After repeating the sequence of procedures described in Sec. II, weobtain the RP equation @6# Rd 2R dt213 2SdR dtD2 14m rRdR dt52p0 r. ~22! The problem has the well-known analytic solution for in- viscid liquids found by Rayleigh in 1917 @5#. In the case of a Newtonian viscous liquid, the numerical solution was ob-tained by Zababakhin @10#. The computer simulations of the bubble motion show two types of bubble behavior: a collapseand a smooth decrease, where the collapse condition can bewritten as m˜p,0.119, m˜p[m R0Arp0. ~23! To illustrate the advantages of the differential criterion ~19!obtained, the time dependence of the bubble radius R(t) and the collapse variable c˜(t) at various values of m˜pare presented in Fig. 2. Calculation of the bubble radius R(t) shows that the critical value of m˜pcorresponding to the col- lapse criterion lies within the interval 0.1–0.12 @see Fig. 2~a!#. More precise estimates are hampered by instabilities and errors in the numerical procedure as R(t)!0. We propose finding the collapse condition by analyzing the collapse variable c˜(t)@see Fig. 2 ~b!#. When the maxi- mum of the curve c˜(t) is less than 1, the curve corresponds to the viscous damping of the bubble wall motion. The bubble collapse is realized when the curve c˜(t) exceeds 1. Thec˜(t) analysis significantly reduces the numerical error in comparison to the R(t) analysis. This results in the most precise collapse criterion: m˜p,~0.11463 60.00001 !. ~24! B. Collapse of an air bubble in water caused by sound The condition we consider is an air bubble in water sub- jected to a periodic spherical sound wave of ultrasonic fre-506 PRE 60 VLADISLAV A. BOGOYAVLENSKIYquency @7,8#. Assuming the symmetry is spherical, the bubble radius R(t) obeys the following equations: Rd2R dt213 2SdR dtD2 14m rRdR dt 5pg2pa2p0 r1R rcd dt~pg2pa!, ~25! R$t50%5R0,dR dt$t50%50. ~26! HereR0is the equilibrium bubble radius, ris the water density, mis the shear viscosity of the water, cis the speed of sound in water, and p05const is the ambient hydrostatic pressure. The acoustic pressure pais considered to be peri- odic:pa52pa0sin2pvt, ~27! wherepa0andvare the amplitude and the frequency of the sound wave, respectively. Assuming adiabatic conditions in- side the bubble, the gas pressure pgfollows from the van der Waals equation, pg5p0SR032a3 R32a3Dg , ~28! whereais the van der Waals hard core and gis the ratio of specific heats. The set of equations ~25!–~28!describes the nonlinear bubble oscillations that can concentrate the average sound FIG. 2. Bubble radius R/R0~a!and collapse variable c˜~b!as functions of time t/t0wheret0[R0Ar/p0. Values of parameter m˜p are shown at curves. The dotted line corresponds to the collapse criterionc˜(t)51. FIG. 3. Bubble radius R~a!and collapse variable c˜~b!as func- tions of time tduring one acoustic period. These are results for the following parameters: r51.0 g/cm, c51481 m/s, m50.07 g/(cm •s),R054.5mm,R0/a58.5,g51.4,v526.5 kHz, p051.0 atm. Curves 1 and 2 correspond to pa050.98 atm and pa051.06 atm, re- spectively. The dotted line corresponds to the collapse criterion c˜(t)51.PRE 60 507 DIFFERENTIAL CRITERION OF A BUBBLE COLLAPS E...energy by over 12 orders of magnitude @19#. During the acoustic cycle the bubble absorbs the energy from the sound field and its radius expands from the equilibrium value R0to a maximum value. The subsequent compressional part of thesound field causes the bubble to collapse. Heating of thebubble surface caused by the compression may lead to theemission of a pulse of light as the bubble approaches a mini-mum radius. This phenomenon is known as sonolumines-cence @3,4#. Let us illustrate the application of the collapse criterion ~19!to this problem. The calculations of the bubble motion are performed for an air bubble in water where all the valuesof the parameters in Eqs. ~25!–~28!are taken from Refs. @20–22 #. For this problem the bubble behavior is basically governed by the amplitude of the acoustic pressure p a0and by the equilibrium bubble radius R0. Let us consider that R0 5const. Therefore, the only variable of the problem is the amplitude of the sound wave pa0. The calculated bubble radius R(t) and the criterion vari- ablec˜(t) at various values of pa0are presented in Fig. 3. The figure shows that the features of the bubble oscillations are determined by the behavior of c˜(t). The viscous damping@curve 1 in Fig. 3 ~a!#corresponds to the inequality c˜(t),1 during the acoustic cycle @curve 1 in Fig. 3 ~b!#. As a result, the sound wave energy dissipates only by the shear viscosityof the water. Thus, the increase of the air temperature insidethe bubble is negligible. The bubble behavior significantly changes when c ˜(t) exceeds 1 @curve 2 in Fig. 3 ~b!#. In this case the acoustic energy is focused on the bubble and com-presses the air within it to high pressures and temperatures@curve 2 in Fig. 3 ~a!#. It is significant that the bubble collapse is not the suffi- cient condition for sonoluminescense. The emission of lightoccurs only when the energy of the sound field achieves acritical value. For this set of parameters the focused energy drastically increases with the increase of p a0. The edge of sonoluminescence corresponds to the value of pa0;1.2 atm @20#. ACKNOWLEDGMENTS I would like to thank Dr. N. A. Chernova and Dr. D. V. Georgievskii for useful discussions. I would also like to ac-knowledge Ms. Tana Mierau for helpful comments. @1#F. R. Young, Cavitation ~McGraw-Hill, London, 1989 !. @2#M. A. Margulis, Sonochemistry and Cavitation ~Gordon and Breach Publishers, Langhorne, 1995 !. @3#A. J. Walton and G. T. Reynolds, Adv. Phys. 33, 595 ~1984!. @4#B. P. Barber and S. J. Putterman, Nature ~London !352, 318 ~1991!. @5#Lord Rayleigh, Philos. Mag. 34,9 4~1917!. @6#M. S. Plesset, J. Appl. Mech. 16, 277 ~1949!. @7#B. E. Noltingk and E. A. Neppiras, Proc. Phys. Soc. London, Sect. B63, 674 ~1950!. @8#E. A. Neppiras and B. E. Noltingk, Proc. Phys. Soc. London, Sect. B64, 1032 ~1951!. @9#L. J. Trilling, J. Appl. Phys. 23,1 4~1952!. @10#E. I. Zababakhin, Prikl. Mat. Mekh. 24, 1129 ~1960!. @11#C. F. Naude and A. T. Ellis, J. Basic Eng. 83, 648 ~1961!. @12#T. B. Benjamin and A. T. Ellis, Philos. Trans. R. Soc. London, Ser. A260, 221 ~1966!. @13#A. Shima, J. Basic Eng. 90,7 5~1968!.@14#M. S. Plesset and R. B. Chapman, J. Fluid Mech. 47, 283 ~1971!. @15#T. M. Mitchell and F. G. Hammit, J. Fluid Mech. 95,2 9 ~1973!. @16#K. Nakajima and A. Shima, Arch. Appl. Mech. ~Ingenieur Ar- chiv.!46,2 1~1977!. @17#J. B. Keller and M. Miksis, J. Acoust. Soc. Am. 68, 628 ~1980!. @18#A. Prosperetti and A. Lezzi, J. Fluid Mech. 168, 457 ~1986!. @19#R. Hiller, S. J. Putterman, and B. P. Barber, Phys. Rev. Lett. 69, 1182 ~1992!. @20#B. P. Barber and S. J. Putterman, Phys. Rev. Lett. 69, 3839 ~1992!. @21#C. C. Wu and P. H. Roberts, Phys. Rev. Lett. 70, 3424 ~1993!. @22#L. Kondic, J. I. Gersten, and C. Yuan, Phys. Rev. E 52, 4976 ~1995!. @23#K. R. Weninger, B. P. Barber, and S. J. Putterman, Phys. Rev. Lett.78, 1799 ~1997!.508 PRE 60 VLADISLAV A. BOGOYAVLENSKIY

arXiv:physics/9912051v1 [physics.class-ph] 27 Dec 1999RUTHERFORD SCATTERING WITH RETARDATION Alexander A. Vlasov High Energy and Quantum Theory Department of Physics Moscow State University Moscow, 119899 Russia Numerical solutions for Sommerfeld model in nonrelativist ic case are pre- sented for the scattering of a spinless extended charged bod y in the static Coulomb field of a fixed point charge. It is shown that differential cros s section for ex- tended body preserves the form of the Rutherford result with multiplier, not equal to one (as in classical case), but inversely proportional to the value of the size of the body, i.e. the greater is the value of body’s size, the s maller is the value of cross section. 03.50.De Here we continue [1] our numerical investigation of Sommerf eld model in classical electrodynamics. It is convenient to remind that Sommerfeld model of charged rigid sphere [2] is the simplest model to take into consideration the influence of self-electromagnetic field of a radiating exten ded charged body on its equation of motion (in the limit of zero body’s size we get the known Lorentz- Dirac equation with all connected with it problems: renorma lization of mass, preacceleration, run-away solutions, etc.). In the previous article the effect of classical tunneling was considered - due to retardation moving body recognize the existence of poten tial barrier too late, when this barrier is overcomed ([1], see also [3]). Physical considerations bring us to another conclusion. Du e to retardation Rutherford scattering of a charged extended body in the stat ic Coulomb field of a fixed point charge must differ from classical scattering of p oint-like particle. That is the scattering angle for the same value of impact para meter for extended particle must be smaller then that for the point-like partic le without radiation (for Lorentz-Dirac equation Rutherford scattering was num erically investigated in [4]). For the case of simplicity here we consider the nonrelativis tic, linear in ve- locity, version of Sommerfeld model. Let the total charge of a uniformly charged sphere be Q, mechanical mass - m, radius - a. Then its equation of motion reads: m˙/vector v=/vectorFext+η[/vector v(t−2a/c)−/vector v(t)] (1) hereη=Q2 3ca2, /vector v=d/vectorR/dt, /vectorR- coordinate of the center of the shell. 1External force /vectorFext, produced by fixed point charge e(placed at /vector r= 0), is /vectorFext=/integraldisplay d/vector rρ ·e/vector r r3 and for ρ=Qδ(|/vector r−/vectorR| −a)/4πa2 reads /vectorFext=e/vectorR R3, R > a (2) In dimensionless variables /vectorR=/vectorY·2L, ct =x·2Lequation (1-2) takes the form¨/vectorY=K/bracketleftBig˙/vectorY(x−δ)−˙/vectorY(x)/bracketrightBig +λ·/vectorY· |/vectorY|−3(3) with K=2Q2 3mc2L, λ=eQ 2mc2L, δ=a/L or K= (4/3)·(rcl/2L), λ= (e/Q)·K, r cl=Q2 mc2 Taking the X−Yplane to be the plane of scattering ( /vectorY= (X, Y) ), we split equation (3) into two: ¨Y=K/bracketleftBig ˙Y(x−δ)−˙Y(x)/bracketrightBig +λ·Y·(X2+Y2)−3/2 ¨X=K/bracketleftBig ˙X(x−δ)−˙X(x)/bracketrightBig +λ·X·(X2+Y2)−3/2(4) The starting conditions at x= 0 are: Xi= 1000 , Yi=b(−impact parameter ) ˙Xi=vi,˙Yi= 0. We take vito be 0 .1,K= 0.4/3.0 and λ= 0.1 (i.e. e=QandL= 5rcl). A. Numerical results are expressed on figs. 1,2. On fig. 1. one can see how the scattering angle varies from poin t-like particle (classical scattering, curve 1) to extended body (curve 2). Hereb= 60.0, δ= 4.0, vertical axis is Y, horizontal - X. So due to retardation the scattering angle θfor extended body is smaller than that for point-like partic le. Differential cross section dσis given by the formula dσ= 2πρ(θ)|dρ(θ) dθ|dθ 2where ρ=b·2L,or 1 2π(2L)2·dσ dξ=db2 dξ(4) where ξ=1 + cos θ 1−cosθ Classical Rutherford result is that R.H.S. of eq. (4) is cons tant: b2·(vi)4·(λ)−2=ξ (5) or (λ)2 2π(2L)2(vi)4·dσ dξ= 1 (6) This classical result is derived from eq.(4) in standard man ner for K= 0. In the case of extended body ( K= 0.4/3.0,λ= 0.1 and δ/negationslash= 0 in eq.(4) ) numerical calculations for various values of b,1.0< b < 60.0 show that Rutherford formula (5,6) changes in the following way: b2·(vi)4·(λ)−2=ξ·[1 +const ·δ]−1(7) or (λ)2 2π(2L)2(vi)4·dσ dξ= [1 + const ·δ]−1(8) where the multiplier const is equal approximately to 0 .30. Thus differential cross section for extended body preserves the form of the Rutherford result with multiplier, not equal to one (as in cl assical case), but inversely proportional to the value of the size of the body, i .e. the greater is the value of body’s size, the smaller is the value of cross sectio ndσ. On fig. 2 we see how the direct proportionality between b2·(vi)4·(λ)−2 andξchanges in accordance to formula (7). Vertical axis is b2·(vi)4·(λ)−2 and horizontal - ξ. Values of retardation δ(or dimensionless body’s size) are taken to be δ= 0 (Rutherford scattering ),1,2,3,4, and curves are marked accordingly as 0 ,1,2,3,4 (for taken starting conditions the classical result is reproduced by numerical calculations with accuracy <3%). REFERENCES 1. Alexander A.Vlasov, physics/9905050. 2. A.Sommerfeld, Gottingen Nachrichten, 29 (1904), 363 (19 04), 201 (1905). L.Page, Phys.Rev., 11, 377 (1918) T.Erber, Fortschr. Phys., 9, 343 (1961) P.Pearle in ”Electromagnetism”,ed. D.Tepliz, (Plenum, N. Y., 1982), p.211. 3A.Yaghjian, ”Relativistic Dynamics of a Charged Sphere”. L ecture Notes in Physics, 11 (Springer-Verlag, Berlin, 1992). F.Rohrlich, Am.J.Phys., 65(11), 1051(1997). 3. Alexander A.Vlasov, physics/9905050. F.Denef et al, Phys.Rev. E56, 3624 (1997); hep-th/9602066. Alexander A.Vlasov, Theoretical and Mathematical Physics , 109, n.3, 1608(1996). 4. J.Huschielt and W.E.Baylis, Phys.Rev. D17, N 4, 985 (1978 ). 41 2 0.00e03.00e16.00e19.00e11.20e21.50e21.80e22.10e22.40e22.70e23.00e2 -6.08e2 -4.47e2 -2.86e2 -1.26e2 3.52e1 1.96e2 3.57e2 5.18e 2 6.78e2 8.39e2 1.00e3 Fig. 1 50 1 2 3 4 0.00e04.00e08.00e01.20e11.60e12.00e12.40e12.80e13.20e13.60e14.00e1 0.00e0 4.00e0 8.00e0 1.20e1 1.60e1 2.00e1 2.40e1 2.80e1 3.2 0e1 3.60e1 4.00e1 Fig. 2 6

arXiv:physics/9912052v1 [physics.atom-ph] 28 Dec 1999Variational methods, multiprecision and nonrelativistic energies V.I. Korobov Joint Institute for Nuclear Research, 141980, Dubna, Russia It is known that the variational methods are the most powerfu l tool for studying the Coulomb three–body bound state problem. However, they often suffer f rom loss of stability when the number of basis functions increases. This problem can be cured by ap plying the multiprecision package designed by D.H. Bailey. We consider the variational basis f unctions of the type exp( −αnr1−βnr2− γnr12) with complex exponents. The method yields the best availab le energies for the ground states of the helium atom and the positive hydrogen ion as well as man y other known atomic and molecular systems. 1.The development of the variational method for the Coulomb bound state problem can be traced using as an example the ground state of the helium atom. In early days when computers were big and very expensive the search proceeded mainly in the direction of making ex- pansion of the variational wave function as compact as possible (in a sense of number of variational parameters and/or basis sets). At first, the explicitly correlated basi s were introduced [1,2] now called as the Hylleraas basis ψ(r1,r2) =e−1 2s/summationdisplay clmnslumtm, s=r1+r2, u=r12, t=−r1+r2, then it became clear that at least for the ground state of the helium atom it is essential to incorporate into the wave function such peculiarity as the logarithmic be- haviour of the type RlnRatR= (r2 1+r2 2)1 2→0, first analytically derived independently by Bartlett and Fock [3]. In 1966, Frankowski and Pekeris (see Table II) introduced the compact representation [4] of the form ψ(r1,r2) =e−κs/summationdisplay clmnijslumt2m(s2+t2)i/2(lns)j, and later, in 1984, Freund and co-workers [5] reported even more compact expansion of the same form. Inclu- sion of the logarithmic term into the variational wave function brought substantial improvement of nonrela- tivistic energies for the two electron atoms. In 1994, Thakkar and Koga [6] have found a compact expansion without logarithms which uses powers that are not inte- gers nor even half integers. As far as we know none of these compact expansions has been used for analytical evaluation of matrix elements of the Breit interaction. With advance of computer power basis sets became simplified that allowed for calculation of numerous ma- trix elements required for relativistic and QED correc- tions. The efforts were concentrated on a choice of a strategy that defines a sequence of basis functions genereated. In [7] the double basis set method with gen- eralyzed Hylleraas basis functionsψ(r1,r2) =/summationdisplay cijkri 1rj 2rk 12e−αr1−βr2 +/summationdisplay cijkri 1rj 2rk 12e−αr1−βr2 were used. This double basis set technique along with full optimization of nonlinear parameters at each basis set yield substantial progress in accuracy. However, the main factor that hinder further advance become the nu- merical instability due to almost linear dependence of the basis set at large N. The work of Goldman [8] is a bit apart of the main path. It recovers the idea of Pekeris [2] to use uncoupled coordinates and orthogonal Laguerre and Jacoby polyno- mials as basis functions. The method expounded in our work is a continuation of efforts by Drake and Yan to utilize as much simple basis functions (geminals) as possible. 2.Expansion we want to consider here is very similar to the generalized Hylleraas basis set, but instead of us- ing the polynomials over Hylleraas variables we generate nonlinear parameters in the exponents in a quasi-random manner, rli 1rmi 2rni 12e−αr1−βr2−γr12=⇒e−αir1−βir2−γir12.(1) This method has been successfully used in calculations [9,10] previously. Obviously, the matrix elements can be evaluated in the same way as for the generalized Hyller- aas basis set (1). Moreover, if one replaces real exponents by complex exponents the integrals will remain exactly the same as for the real case. In its strategy the method is very close to the SVM method by Varga, Suzuki [11], where gaussians are exploited instead. In a formal way, a variational wave function is ex- panded in a form ψ0=∞/summationdisplay i=1/braceleftBig UiRe/bracketleftbig exp (−αir1−βir2−γir12)/bracketrightbig +WiIm/bracketleftbig exp (−αir1−βir2−γir12)/bracketrightbig/bracerightBig YLM l1l2(ˆr1,ˆr2). Hereαi,βiandγiare complex parameters generated in a quasi-random manner [13,14]: 1αi=/floorleftbigg1 2i(i+ 1)√pα/floorrightbigg [(A2−A1) +A1]+ +i/braceleftbigg/floorleftbigg1 2i(i+ 1)√qα/floorrightbigg [(A′ 2−A′ 1) +A′ 1]/bracerightbigg , ⌊x⌋designates the fractional part of x,pαandqαare some prime numbers, [ A1,A2] and [A′ 1,A′ 2] are real vari- ational intervals which need to be optimized. Parameters βiandγiare obtained in a similar way. An important feature of the method is that it demon- strates a very fast convergence. The general rule which can be inferred experimentally from the use of the method is that increasing of the basis by about 200 func- tions yields about one additional digit in the variational energy. The minor deficiency is that the basis quickly degenerates when Nincreases. Already for moderate N∼250−400 a quadruple precision is required. Multiprecision package of Fortran routines MPFUN has been designed by David H. Bailey [12] for computa- tions with floating point numbers of an arbitrary length. Usually it is necessary to make significant changes into Fortran source code in case if Fortran-77 language is used. Fortunately, the author of MPFUN package has devel- oped a translator program that facilitate converting the programs to multiprecision drastically. In general, two directives incorporated as comments in a source code are required per one routine. For example a source code for the considered variational method has been transformed to multiprecision version within two hours of manual work. Eventually a code we’ve gotten has been tested on a personal computer with the Celeron 500 MHz pro- cessor. For one run with the basis of N= 1400 functions and 40 decimal digits it requires about 3 hours. For users of Fortran–90 no preprocessor is needed due to new advanced features of Fortran–90, such as derived data types and operator extensions. N E (a.u.) 1400 −2.90372437703411959629 1600 −2.903724377034119597843 1800 −2.9037243770341195981964 2000 −2.9037243770341195982713 2200 −2.9037243770341195982955 extrapolation −2.903724377034119598306(10) TABLE I. Variational energy (in a.u.) of the helium ground state as a function of N, the number of basis func- tions. In our calculations for the helium ground state four ba- sis sets with independently optimized nonlinear param- eters were used. These sets were built up like a pine tree. The first layer was tuned to approximate the gen- eral behaviour of the solution at intermediate and large r1andr2. The second layer was chosen to be flexible in a smaller region of r1andr2and so forth. A detailed opti- mization was performed for the sets with total N= 1400andN= 1600. Quadruple precision was not sufficient at theseNand we used the multiprecision version of the program with 40 significant decimal digits. Further cal- culations with N= 1800 −2200 were performed with 48 significant digits and only partial optimization of the parameters of the last layer (corresponding to the region where the logarithmic behaviour is the most essential) was done. Some optimization of a distribution of nibe- tween the layers ( N=n1+n2+n3+n4) was carried out as well. As can be seen from the Table II the present result extends the accuracy of the nonrelativistic ground state energy for the helium atom by as much as 3 decimal dig- its. N E (a.u.) Frankowski and 246 −2.9037243770326 Pekeris [4] Freund, Huxtable, 230 −2.9037243770340 and Morgan III [5] Thakkar and Koga [6] 308 −2.9037243770341144 Drake and Yan [7] 1262 −2.90372437703411948 Goldman [8] 8066 −2.903724377034119594 This work 2200 −2.903724377034119598296 TABLE II. Comparison of the ground state energy of the helium atom obtained in this work with other theoretical cal - culations. Second case is the hydrogen molecular ion ground state that represent an other limit of mass distribution of con- stituents with one light and two heavy particles. For this case it is especially essential that we introduce com- plex exponents, because it is the most natural way to suit the oscillatory behaviour of the vibrational motion in the wave function. In this case (see Table III) again 40 decimal digits have been used for N= 1400 −1800 and 48 decimal digits for large Nto provide the numer- ical stability of the calculations. Table IV demonstrates progress in obtaining variational nonrelativistic energy for this state. The accuracy is extended by as much as 4 additional digits. N E (a.u.) 1400 −0.597139063123404975 1600 −0.597139063123405047 1800 −0.5971390631234050655 2000 −0.5971390631234050710 2200 −0.5971390631234050740 extrapolation −0.597139063123405076(2) TABLE III. Variational energy (in a.u.) of the positive hydrogen ion ground state as a function of N, the number of basis functions. 2N E (a.u.) Gr´ emaud, Delande 31746 −0.597139063123 and Billy [15] Rebane and Filinsky [16] −0.59713906312340 Moss [17] −0.5971390631234 This work 2200 −0.597139063123405074 TABLE IV. Comparison of the ground state energy of the positive hydrogen molecular ion obtained in this work with other theoretical calculations. mp= 1836 .152701 me. In Table V the other examples are summarized. A negative positronium ion demonstrates a limit of three particles of equal masses. The second and third cases are applications of the method to the states with nonzero angular momentum. The last example in this Table is of special proud. That is the last vibrational state in a series ofS-states of the hydrogen molecular cation, and that is the first variational confirmation of the existence of this state (the binding energy corresponding to the cited value is 0.74421(2) cm−1). The accuracy of the artificial channels scattering method [21] is presumably better, however, wave functions are not forthcoming with this method that makes difficult calculation of physical properties of the state other than energy. system E e−e−e+This work −0.2620050702329801077(3) [18]−0.262005070232976 He(23P) This work −2.13316419077928310(2) [19]−2.13316419077927(1) 4He+¯p(L=35, v=0) This work −2.98402095449725(1) [20]−2.98402094 H+ 2(L=0,v=19) This work −0.4997312306 [21]−0.49973123063 TABLE V. Other examples of three–body calculations. ( L is the total angular momentum, vis the vibrational quantum number.) 3. One may say that this high accuracy is redundant and has no physical meaning. But obviously, it shows the power of modern computers and theirs ability to solve the quantum three–body problem to any required accuracy. On the other hand, uncertainty in the variational wave function approximately as much as the square root of the uncertainty in the variational energy and is about 10−9−10−10. This accuracy does not look redundant. These results prove that the nonrelativistic bound state three–body problem is now satisfactorily solved and the main efforts should be addressed to relativistic and QED effects. The other advantage of the method is the simplicity of the basis functions that allows for evaluate analyticallyrelativistic matrix elements of the Breit Hamiltonian. It is possible as well to evaluate analytically the vacuum polarization term (Uehling potential) [22] and to build up an effective numerical scheme for the one–loop self– energy corrections [23]. These features make the consid- ered variational method to be highly powerful universal tool for studying the three–body problem. This work has been partially supported by INTAS Grant No. 97-11032, which is gratefully acknowledged. [1] E.A. Hylleraas, Z. Physik 54, 347 (1929); S. Chan- drasekhar and G. Herzberg, Phys. Rev. 98, 1050 (1955); T. Kinoshita, Phys. Rev. 105, 1490 (1956). [2] C.L. Pekeris, Phys. Rev. 112, 1649 (1958); 115, 1216 (1959). [3] J.H. Bartlett, Phys. Rev. 51, 661 (1937); V.A. Fock, Izvest. Akad. Nauk S.S.S.R. Ser. Fiz. 18, 161 (1954). [4] K. Frankowski and C.L. Pekeris, Phys. Rev. 146, 46 (1966); 150, 366(E) (1966). [5] D.E. Freund, B.D. Huxtable, and J.D. Morgan III, Phys. Rev. A 4, 516 (1971) [6] A.J. Thakkar and T. Koga, Phys. Rev. A 50, 854 (1994) [7] G.W.F. Drake and Zong-Chao Yan, Chem. Phys. Lett. 229, 486 (1994). [8] S.P. Goldman, Phys. Rev. A 57, R677 (1998). [9] S.A Alexander, H.J. Monkhorst, Phys. Rev. A 38, 26 (1988). [10] A.M. Frolov and V.D. Efros Pis. Zh. Eksp. Teor. Fiz. 39, 544 (1984) [Sov. Phys.–JETP Lett. 39, 449 (1984)]; A.M. Frolov, and V.H. Smith, Jr., J. Phys. B 28, L449 (1995). [11] K. Varga and Y. Suzuki, Phys. Rev. C 52, 2885 (1995); Phys. Rev. A 53, 1907 (1996). [12] D.H. Bailey, ACM Trans. Math. Softw. 19, 288 (1993); 21, 379 (1995); see also web-site: www.netlib.org . [13] A.M. Frolov, and V.H. Smith, Jr., J. Phys. B 28, L449 (1995). [14] V.I. Korobov, D. Bakalov, and H.J. Monkhorst, Phys. Rev A 59, R919 (1999). [15] B. Gr´ emaud, D. Delande, and N. Billy, J. Phys. B: At. Mol. Opt. Phys. 31, 383 (1998). [16] T.K. Rebane and A.V. Filinsky, Phys. At. Nuclei 60, 1816 (1997). [17] R.E. Moss, J. Phys. B: At. Mol. Opt. Phys. 32, L89 (1999) [18] A.M. Frolov, Phys. Rev. A 60, 2834 (1999). [19] G.W.F. Drake and Zong-Chao Yan, Phys. Rev. A 46 2378 (1992). [20] Y. Kino, M. Kamimura and H. Kudo, Nucl. Phys. A, 631, 649 (1998). [21] R.E. Moss, Mol. Phys. 801541 (1993). [22] P. Petelenz and V.H. Smith, Jr., Phys. Rev. A 35, 4055 (1987); 36, E4529 (1987). [23] V.I. Korobov and S.V. Korobov, Phys. Rev A 59, 3394 (1999). 3

arXiv:physics/9912053v1 [physics.bio-ph] 29 Dec 1999Effective interaction between helical bio-molecules E.Allahyarov1,2, H.L¨ owen1 1 Institut f¨ ur Theoretische Physik II,Heinrich-Heine-Un iversit¨ at D¨ usseldorf,D-40225 D¨ usseldorf, Germany 2 Institute for High Temperatures, Russian Academy of Scien ces, 127412 Moscow, Russia (February 9, 2008) The effective interaction between two parallel strands of he lical bio-molecules, such as deoxyri- bose nucleic acids (DNA), is calculated using computer simu lations of the “primitive” model of electrolytes. In particular we study a simple model for B-DN A incorporating explicitly its charge pattern as a double-helix structure. The effective force and the effective torque exerted onto the molecules depend on the central distance and on the relative orientation. The contributions of non- linear screening by monovalent counterions to these forces and torques are analyzed and calculated for different salt concentrations. As a result, we find that th e sign of the force depends sensitively on the relative orientation. For intermolecular distances smaller than 6 ˚Ait can be both attractive and repulsive. Furthermore we report a nonmonotonic behavi our of the effective force for increasing salt concentration. Both features cannot be described with in linear screening theories. For large distances, on the other hand, the results agree with linear s creening theories provided the charge of the bio-molecules is suitably renormalized. PACS: 87.15.Kg, 61.20Ja, 82.70.Dd, 87.10+e I. INTRODUCTION Aqueous solutions of helical bio-molecules like deoxyri- bose nucleic acids (DNA) are typically highly charged such that electrostatic interactions play an important role in many aspects of their structure and function [1–6]. Understanding the total effective interaction be- tween two helical molecules is important since this gov- erns the self-assembly of bio-molecules, like bundle for- mation and DNA condensation or compaction which in turn is fundamental for gene delivery and gene therapy. In aqueous solution, such rod-like polyelectrolytes relea se counterions in the solution which ensure global charge neutrality of the system. Together with these counteri- ons, there are, in general, added salt ions dissolved in the solution. The thermal ions screen the bare electrostatic interactions between the bio-molecules, such that the ef- fective interaction between them is expected to become weaker than the direct Coulomb repulsion. For very high concentrations of bio-molecules or short distances even a mutual attraction due to counterion ”overscreening” is conceivable [7–24]. In this paper, we study the effective interaction be- tween two parallel helical bio-molecules. In particular, we investigate how the electrostatic interactions are in- fluenced by details of the charge pattern on the biological macromolecules. In fact, in many cases, as e.g. for DNA molecules, the charge pattern on the molecules is not uniform but exhibits an intrinsic helix structure. If two parallel helical molecules are nearby, this helix structur e will induce an interaction that depends on the relative orientation of the two helices. Our studies are based on computer simulation of the “primitive” model of elec- trolytes [25]. In particular we study a simple model for B-DNA. This model explicitly takes into account thedouble-helical charge pattern along the DNA-strand, it also accounts for the molecular shape by modeling the major and minor grooves along the strand. The charged counter- and salt ions in the solutions are explicitly in- corporated into our model. On the other hand, the wa- ter molecules only constitute a continuous background with a dielectric constant ǫscreening the Coulomb in- teractions. Hence the discrete nature of the solvent is neglected as well as more subtle effects as image charges induced by dielectric discontinuities at the DNA-water boundary [26–29], hydration effects due to the affection of the hydrophilic surface to the interfacial layers of wa- ter [30–35], and spatial dependent dielectric constants resulting from the decreasing water mobility in confining geometries and from saturation effects induced by water polarization near the highly charged molecular surfaces [36–41]. Our motivation to consider such a simple ”primitive” model is threefold: First, though solvent effects seem to be relevant they should average out on a length scale which is larger than the range of the microscopic sizes. Hence the electrostatic effects are expected to dominate the total effective interactions. Second, it is justified to study a simple model completely and then adapt it by introducing more degrees of freedom in order to better match the experimental situation. Our philosophy is in- deed to understand the principles of a simple model first and then turn step by step to more complicated models. Third, even within the “primitive” approach, there are many unsolved problems and unexpected effects such as mutual attraction of equally charged particles. Our com- puter simulation method has the advantage that “exact” results are obtained that reflect directly the nature of the model. Hence we get rid of any approximation in- herent in a theoretical description. Consequently, the 1 Typeset using REVT EXdependence of the effective interactions on a model pa- rameter can systematically be studied and the trends can be compared to experiments. In this respect our model is superior to previous studies that describe the counterion screening by linear Debye-H¨ uckel [39,42,4,43] or nonlin- ear Poisson-Boltzmann theory [26,4,44–51] and even to recent approaches that include approximatively counte- rion correlations [52,53]. We also emphasize that one main goal of the paper is to incorporate the molecular shape and charge pattern explicitly which is modelled in many studies simply as a homogeneously charged cylin- der [39,4,54,55]. In fact we find that the double-helix structure has an important influence on the effective in- teraction for surface-to-surface separations smaller tha n 6˚A. In detail, the interaction can be both repulsive and attractive depending on the relative orientation and the mutual distance between two parallel DNA strands. This effect which is typically ignored in the charged-cylinder model for DNA will significantly affect the self-assembly of parallel smectic layers of DNA fragments and may re- sult in unusual crystalline structures at high concentra- tions. Let us also mention that many theoretical studies in- volve only a single DNA molecule [56–59,3]. To extract the effective interaction, however, one has at least to in- clude two molecules in the model which is the purpose of the present paper. In this study we only consider mono- valent counterions. Multivalent counterions and a more detailed survey on the influence of model parameters on the effective interactions will be considered in a subse- quent publication. The remainder of paper is organized as follows. In chapter II, we present the details of the model used in this paper. Chapter III describes the target quantities of the applied model. Simulation details are presented in chap- ter IV. Theories based on linear screening approaches such as the homogeneously charged cylinder model, the Yukawa segment model and the Kornyshev-Leikin theory [60] are shortly discussed in chapter V. Results of the sim- ulation and their comparison to linear screening theories are contained in sections VI-VIII for the point-charge model, the grooved model and added salt respectively. We conclude in section IX. II. THE MODEL The charge pattern and the shape of a single B-DNA molecule is basically governed by the phosphate groups which exhibit a double helix structure with right-hand helicity. We model this by an infinitely long neutral hard cylinder oriented in zdirection with additional charged hard spheres whose centers are located on top of the cylindrical surface. Each charged sphere describes a phosphate group and hence the spheres form a double helix structure. In detail, the effective cylindrical diam- eterDis commonly chosen to be D= 20˚A[61,62,49].The spheres are monovalent, i.e. their charge qp<0 corresponds to one elementary charge e >0,qp=−e, and they have an effective diameter dp. We do not fix dpbut keep it as an additional (formal) parameter in the range between dp= 0.2˚A(practically the point-like charge limit) to dp= 6˚A(to incorporate a groove ge- ometry for the molecule). Furthermore, the helical pitch length is P= 34˚A; the number of charged spheres per pitch length (or per helical turn) is 10. Consequently, successive charges on the same strand are displaced by an azimuthal angle of 36◦corresponding to a charge spac- ing of 3 .4˚Ainzdirection. In a plane perpendicular to the zdirection, phosphate groups of the two different helices are separated by an azimuthal angle of φs= 144◦, see Figure 1, fixing the minor and the major helical groove along the DNA molecule. R2 R1minor groove major grooveφs ) )φ0 φ0) )φ 1 2 yR x FIG. 1. A schematic picture explaining the positions of DNA molecules and the definition of the different azimuthal angles φ0, φ, φ s. For further information see text. We place the discrete charges on the two different he- lices such that two of them fall in a common plane per- pendicular to the zaxis, see again Figure 1. The to- tal line charge density along the DNA molecule is then λ=−0.59e/˚A. The second DNA molecule is considered to be paral- lel to the first one in our studies. The separation be- tween the two cylinder origins is R, we also introduce the surface-to-surface separation h=R−D. The position of the two double helices can be described by a relative angle difference φbetween the two azimuthal angles de- scribing the position of the bottom helix with respect to a fixed axis in the xyplane. This is illustrated in Figure 1. The relative orientation φis the key quantity in describ- 2 Typeset using REVT EXing the angle dependence of the forces induced by the helical structure. We remark that we only study a situ- ation where the discrete phosphates from different DNA strands possess the same zcoordinates for φ= 0. Small shifts in the zcoordinate are not expected to change the results significantly. A further parameter characterizing the discrete location of the phosphate charges along the strands is the azimuthal angle φ0of a phosphate charge with respect to the cylinder separation vector, see again Figure 1. All results are periodic in φ0with a periodicity of 36◦. In addition to the DNA molecules we describe the counterions by charged hard spheres of diameter dcand charge qc. The counterions are held at room temper- ature T= 298 K. Their concentration is fixed by the charge of the DNA molecules due to the constraint of global charge neutrality. Also, additional salt ions with charges q+andq−, modelled as charged hard spheres of diameters d+andd−, are incorporated into our model. The salt concentration is denoted by Cs. The discrete nature of the solvent, however, is neglected completely. The interactions between the mobile ions and phos- phate charges are described within the framework of the primitive model as a combination of excluded volume and Coulomb interactions screened by the dielectric constant ǫof the solvent. The corresponding pair interaction po- tential between the different charged hard spheres is Vij(r) =/braceleftbigg∞ forr≤(di+dj)/2 qiqje2 ǫrforr >(di+dj)/2. (1) where ris the interparticle separation and i, jare indices denoting the different particles species. Possible values foriandjarec(for counterions), + ,−(for positively and negatively charged salt ions), and p(for phosphate groups). In addition, there is an interaction potential V0 ibetween the DNA hard cylinder and the free ions i=c,+,−which is of simple excluded volume form such that these ions cannot penetrate into the cylinder. Due to the length of this paper and the large number of quantities, we summarize most of our notation in Table I. III. TARGET QUANTITIES Our target quantities are equilibrium statistical av- erages for the local counter- and salt ion densities and the effective forces and torques exerted onto the bio- molecules. For that purpose we consider a slightly more general situation with Nparallel DNA molecules con- tained in a system of volume V. The cylinder centers are fixed at positions /vectorRi(i= 1, ..., N ) in the xy-plane. We further assume that there are Nccounterions and N+, N− salt ions in the same system. By this we obtain partial concentrations nc=Nc/V, n +=N+/V, n −=N−/Vof counter and salt ions.First we define the equilibrium number density profiles ρj(/vector r) (j=c,+,−) of the mobile ions in the presence of the fixed phosphate groups via ρj(/vector r) =/an}b∇acketle{tNj/summationdisplay i=1δ(/vector r−/vector rj i)/an}b∇acket∇i}ht, (2) Here {/vector rj i}denote the positions of the ith particle of species j. The canonical average < ... > over an {/vector rj i}- dependent quantity Ais defined via the classical trace /an}b∇acketle{tA/an}b∇acket∇i}ht=1 Z/braceleftBigNc/productdisplay k=1/integraldisplay d3rc k/bracerightBig/braceleftBigN+/productdisplay m=1/integraldisplay d3r+ m/bracerightBig/braceleftBigN−/productdisplay n=1/integraldisplay d3r− n/bracerightBig exp(−β/summationdisplay i=c,+,−[V0 i+/summationdisplay j=c,p,+,−Uij])× A (3) Hereβ= 1/kBTis the inverse thermal energy ( kBde- noting Boltzmann’s constant) and Uij= (1−1 2δij)Ni/summationdisplay l=1Nj/summationdisplay k=1Vij(|/vector ri l−/vector rj k|), (4) is the total potential energy of the counter- and salt ions provided the phosphate groups are at positions {/vector rp n}(n= 1, ..., N p). Finally the prefactor 1 /Zin eq.(3) ensures correct normalization, <1>= 1. Note that the density profiles ρj(/vector r) also depend parametrically on the positions {/vector rp n}of all the fixed phosphate groups ( n= 1, ..., N p). Now we define the total effective force /vectorFiper pitch length acting onto the ith DNA molecule ( i= 1, ..., N ). As known from earlier work [63,64,11,65] it contains three different parts /vectorFi=/vectorF(1) i+/vectorF(2) i+/vectorF(3) i. (5) The first term, /vectorF(1) i, is the direct Coulomb force acting onto all phosphate groups belonging to one helical turn of the ith DNA molecule as exerted from the phosphate groups of all the other DNA molecules: /vectorF(1) i=−/summationdisplay k′ /vector∇/vector rp kNp/summationdisplay n=1;n/negationslash=kVpp(|/vector rp k−/vector rp n|)  (6) where the sum/summationtext′ konly runs over 10 phosphates belong- ing to one helical turn of the ith DNA molecule. This term is a trivial sum of direct interactions. The second term /vectorF(2) iinvolves the electric part of the interaction between the phosphate groups and the counter- and salt ions. Its statistical definition is /vectorF(2) i=−/summationdisplay k′ /an}b∇acketle{t/summationdisplay i=c,+,−Ni/summationdisplay l=1/vector∇/vector rp kVpi(|/vector rp k−/vector ri l|)/an}b∇acket∇i}ht (7) and describes screening of the bare Coulomb interaction (6) by the counter and salt ions. 3 Typeset using REVT EXTABLE I. List of key variables D DNA diameter dc counterion diameter dp phosphate diameter d+, d− salt ion diameters P helical pitch length L length of simulation box ǫ dielectric constant of DNA and water T temperature Np number of phosphates in the simulation box Nc number of counterions in the simulation box Ns number of salt ion pairs in the simulation box Cs salt concentration qc counterion valency qp phosphate valency q+, q− salt ion valencies λ linear charge density of the DNA molecule λB Bjerrum length Γpc coupling parameter between phosphates and counterions F interaction force per pitch length F0 used unit for force , F0= (e 4D)2 M torque acting onto the DNA molecules R interaxial separation between DNA molecules h surface-to-surface separation between DNA molecules φ relative orientational angle between two DNA molecules φ0 reference orientational angle for one DNA molecule F(HC)interaction force per pitch length within the homogeneousl y charged cylinder model λD Debye screening length F(Y S)interaction force per pitch length within the Yukawa segmen t model r∗ p effective phosphate radius in the Yukawa segment model q∗ p effective phosphate charge in the Yukawa segment model ζ size correction factor in the Yukawa segment model F(KL)interaction force per pitch length within Kornyshev-Leiki n theory θ condensation parameter of counterions Finally, the third term /vectorF(3) idescribes a contact (or de- pletion) force arising from the hard-sphere part in Vpi(r) andV0 i(i=c,+,−). It can be expressed as an inte- gral over the molecular surface Siassociated with the excluded volume per one helical turn of the ith DNA molecule: /vectorF(3) i=−kBT/integraldisplay Sid/vectorf /summationdisplay j=c,+,−ρj(/vector r) , (8) where /vectorfis a surface normal vector pointing outwards the DNA molecule. This depletion term is usually ne- glected in any linear electrostatic treatment but becomes actually important for strong Coulomb coupling Γ pcas conveniently defined by [11,66,65] Γpc=|qp qc|2λB dp+dc, (9) with the Bjerrum length λB=q2 ce2/ǫkBT. When Γ pc is much larger than one, the Coulomb interaction dom- inates thermal interactions and counterion condensationmay occur. For DNA molecules this is relevant as dp+dc= 4−6˚AandλB= 7.14˚Afor a monovalent counterion in water at room temperature, resulting in a coupling parameter Γ pclarger than one. Our final target quantity is the total torque per pitch length acting onto the ith DNA molecule. Its component Mialong the z-direction (with unit vector /vector ez) can also be decomposed into three parts Mi=M(1) i+M(2) i+M(3) i (10) with M(1) i=−/vector ez·/summationdisplay k′ /vector rp k× /vector∇/vector rp kNp/summationdisplay n=1;n/negationslash=kVpp(|/vector rp k−/vector rp n|)  (11) M(2) i=−/vector ez·/summationdisplay k′ /vector rp k× /an}b∇acketle{t/summationdisplay i=c,+,−Ni/summationdisplay l=1/vector∇/vector rp kVpi(|/vector rp k−/vector ri l|)/an}b∇acket∇i}ht  (12) 4 Typeset using REVT EXand M(3) i=kBT/vector ez·/integraldisplay Sid/vectorf×/vector r /summationdisplay j=c,+,−ρj(/vector r)  (13) IV. COMPUTER SIMULATION Our computer simulation was performed within a sim- ple set-up which is schematically shown in Figure 2. We consider two parallel DNA molecules in a cubic box of length Lwith periodic boundary conditions in all three directions. Lis chosen to be three times the pitch length Psuch that there are Np= 120 phosphate charges in the box. The number of counterions Nc= 120 in the box is fixed by charged neutrality while the number of salt ions,Ns, is governed by its concentration Cs. The sep- aration vector between the centers of the two molecules points along the x-direction of the simulation box. The relative orientation is described according to our notatio n presented in chapter II, see again Figure 1. We performed a standard Molecular Dynamic (MD) code with velocity Verlet algorithm [67]. System param- eters used in our simulations are listed in Table II. The time step △tof the simulation was typically chosen to be 10−2/radicalbig m d3m/e2, with mdenoting the (fictitious) mass of the mobile ions, such that the reflection of counteri- ons following the collision with the surface of DNA core cylinder and phosphates is calculated with high precision. For every run the state of the system was checked during the simulation time. This was done by monitoring the temperature, average velocity, the distribution function of velocities and total potential energy of the system. On average it took about 104MD steps to get into equilib- rium. Then during 5 ·104−5·106time steps, we gathered statistics to perform the canonical averages for calculate d quantities. The long-ranged nature of the Coulomb interaction was numerically treated via the efficient method pro- posed by Lekner [68]. A summary of this method is given in Appendix A. In order to save CPU time, the Lekner forces between pair particles were tabulated in a separate code before entering into the main MD cycle. The tabulation on a 510 ×510×510 grid with spatial step =0.1˚Awas done in the following manner. The first parti- cle was fixed at the origin (0,0,0) while the second charge was successively embedded on sites of the generated grid. Then the force components acting onto the first charge were calculated via the Lekner method. A force data file was created which was used as a common input for all subsequent MD runs. To decrease error coming from a finite grid length, the forces in the simulations were calculated using the four-step focusing technique [69].zdp rpD dc R2R1rc RPL FIG. 2. Schematic view of the set-up: Two cylindrically shaped DNA molecules with a distance Rat positions /vectorR1and /vectorR2are placed parallel to the z-axis inside a cube of length L. The large gray spheres are counterions of diameter dc. The black spheres of diameter dp, connected by the solid line, are phosphate charges on the cylindrical surface of diameter D. Pis the pitch of DNA. Arrays /vector rpand/vector rcpoint to positions of phosphates and counterions. For sake of clarity, the positi ons of added salt ions are not shown. There are periodic boundary conditions in all three directions. V. LINEAR SCREENING THEORY Linear screening theory can be used to get explicit an- alytical expressions for the effective interactions betwee n helical bio-molecules. These kind of theories, however, should only work for weak Coulomb coupling and thus represent a further approximation to the primitive model. Depending on the form of the fixed charge pattern char- acterizing the biomolecules, one obtains different approx- imations. A. Homogeneously charged cylinder The simplest approach is to crudely describe the biomolecule as a homogeneously charged cylinder. In this case, the effective interaction force per pitch length be- tween two parallel rods reads [25] /vectorF≡/vectorF(HC)=2λ2PλDK1(r/λD) ǫ(D/2)2K2 1(D/(2λD))/vector r r(14) 5 Typeset using REVT EXTABLE II. Parameters used for the different simulation runs. The Debye screening length λD, as defined by Eqn.(15), and the Coulomb coupling Γ pcare also given. Run dc(˚A) dp(˚A) Ns Cs(M) λD(˚A) Γ pc A 1 0.2 - - 9.6 12 B 2 2 - - 9.6 3.6 C 2 6 - - 9.6 1.8 D 1 0.2 15 0.025 8.6 12 E 1 0.2 60 0.1 6.8 12 F 1 0.2 120 0.2 5.6 12 G 1 0.2 440 0.73 3.3 12 H 1 0.2 1940 3.23 1.7 12 I 2 2 120 0.2 5.6 3.6 Hereris the axis-to-axis separation distance between cylinders, λDis the Debye-H¨ uckel screening length fixed by λD=/radicalBigg ǫkBT 4πγ(nc(qce)2+n+(q+e)2+n−(q−e)2)(15) where the factor γ= 1−Vcyl/Vis a correction due to the fact that the mobile ions cannot penetrate into the cylin- dric cores which excludes a total volume Vcyl. Further- more, K1(x) is a Bessel function of imaginary argument. Obviously, the torque is zero for this charge pattern. B. Yukawa segment model It is straightforward to generalize the traditional Debye-H¨ uckel approach to a general charge pattern re- sulting in a Yukawa-segment (YS) model [27,70–74]. One phosphate charge interacts with another phosphate charge via an effective Yukawa potential [75] U(r) =(qpζ)2e2 ǫrexp(−r/λD) (16) Here, ζdescribes a size correction due to the excluded volume of the phosphate groups. This term is assumed to be of the traditional Derjaguin-Landau-Verwey-Overbeek (DLVO) form ζ= exp( r∗ pλD)/(1 +r∗ pλD) (17) where r∗ p= (dp+dc)/2 is an effective phosphate radius for the phosphate counterion interaction. We remark that nonlinear screening effects and the excluded volume of the cylinder can also be incorporated by replacing the bare phosphate charge qpwith an effective phosphate charge q∗ p[27,71,76]. Using the same notation as in chapter III, the total effective force per pitch length acting onto the ith bio- molecule is /vectorFi≡/vectorF(Y S) i=−/summationdisplay k′ /vector∇/vector rp kNp/summationdisplay n=1;n/negationslash=kU(|/vector rp k−/vector rp n|) (18)within in the Yukawa segment model where the sum/summationtext′ has the same meaning as in Eqn.(6). Note that the con- tact term (8) is typically neglected in linear screening the - ory. Furthermore, the effective torque per pitch length is Mi≡M(Y S) i=−/vector ez·/summationdisplay k′ /vector rp k× /vector∇/vector rp kNp/summationdisplay n=1;n/negationslash=kU(|/vector rp k−/vector rp n|)  (19) There are also analytical expressions for the equilibrium density profiles of the mobile ions involving a linear su- perposition of Yukawa orbitals around the phosphate charges [77] which, however, we will not discuss further in the sequel. C. Kornyshev-Leikin theory The linear Debye-H¨ uckel screening theory was recently developed further and modified to account for dielectric discontinuities and counterion adsorption in the grooves of the DNA molecule by Kornyshev and Leikin (KL) [60,78–81]. An analytical expression for the effective pair potential VKL(R, φ) per pitch length between two paral- lel rods of separation Rwith relative orientation φwas given for separations larger than R > D +λD. Here we only discuss the leading contribution in the special case of no dielectric discontinuity which reads VKL(R, φ) =8Pλ2 ǫD2∞/summationdisplay n=−∞(−1)nP2 ncos(nφ)K0(knR) k2n(1−βn)2(K′ n(knD/2))2 (20) and corresponds to the interaction of helices whose strands form continuously charged helical lines. In Eqn.(20), βn=ng knKn(knD/2)I′ n(ngD/2) K′ n(knD/2)In(ngD/2), (21) 6 Typeset using REVT EXkn=/radicalBig 1/λ2 D+ (ng)2, g=2π P, (22) KnandInare modified Bessel functions of nth order, andK′ n(x) =dKn(x)/dx,I′ n(x) =dIn(x)/dx. We emphasize that the KL-theory does not assume a priori the double helical phosphate charge pattern as de- fined in chapter II. There are rather more possible charge patterns considered including a condensation of counteri- ons in the minor and major groove along the phosphate strands, and on the cylinder as a whole. This involves four phenomenological parameters as a further input for the KL theory which makes a direct comparison to the simulation data difficult. In fact, for the charge pattern given in chapter II, the KL-theory reduces to the Yukawa- segment model. In detail, the charge pattern is characterized by the form factor Pn Pn= (1−f1−f2−f3)θδn,0+ f1θ+f2(−1)nθ−(1−f3θ)cos(nφs). Hereδn,mis the Kronecker’s delta function; θis the first phenomenological input parameter which describes the fraction of counterions that are condensed on the whole cylinder. The three numbers fidenote the fractions of counterions in the middle of the minor groove ( f1), in the middle of the major groove ( f2), and on the phosphate strands ( f3) with respect to all condensed counterions. We note that the sum in (20) rapidly converges, such that it can safely be truncated for |n|>2. It is straight- forward to obtain the effective force and torque per pitch length between two molecules from (20) by taking gradi- ents with respect to Randφ. VI. RESULTS FOR POINT-LIKE CHARGES AND NO ADDED SALT In what follows, we consider the set-up of two parallel bio-molecules with periodic boundary conditions shown in Figure 2. We projected /vectorF1onto the vector /vectorR, defining F=/vectorF1·(/vectorR1−/vectorR2)/|/vectorR1−/vectorR2|. Hence a negative sign ofFimplies attraction, and a positive sign repulsion. The torque is given for the first DNA molecule, hence M≡M1. We start with the case of no added salt. First, we assume the counterion and phosphate diameters to be small, in order to formally investigate the system with a high coupling parameter Γ pc>10. A. Distribution of the counterions around the DNA molecules We calculated the equilibrium density field (2) of the counterions in the vicinity of the DNA molecules by com- puter simulation. In detail, we considered three different paths to show the counterion density profile around thefirst DNA molecule: along a phosphate strand and along the minor and major groove. In order to reduce the sta- tistical error we course-grained this density field further in a finite volume which is illustrated in Figure 3. ξδ 0o180oP FIG. 3. A schematic picture to explain the procedure of counterion density calculations along one pitch length of a DNA molecule. The filled circles connected with solid line are phosphate groups. The shaded areas correspond to a path along the major groove and along one phosphate strand. The considered volume has a height ξand width δ. The neigh- bouring DNA molecule is assumed to be on the right hand side. This volume is winding around the molecules with a height ξand width δ. We choose ξ= 3.4˚Aandδ= 2˚A+dc/2. In Figure 4 we plot this coarse-grained density fieldρc(ϕ) versus the azimuthal angle angle ϕfrom 0◦to 360◦where ϕis 0◦resp. 360◦in the inner region between the DNA molecules. Obviously, the counterion density profile has maxima in the neighbourhood of the fixed phosphate charges. Furthermore the concentration of counterions is higher in the minor than in the major grooves with the ϕ- dependence reflecting again the position of the phosphate charges. Also in the inner region between the two DNA molecules, there are on average more counterions than in the outside region. 7 Typeset using REVT EX0 60 120 180 240 300 360 ϕ [degrees]00.511.522.53ρc(ϕ) hDδ FIG. 4. Equilibrium counterion density profile ρc(ϕ) in units of 1 /hDδ versus azimuthal angle ϕfor the parameters of run A, φ= 0◦and a rod separation of R= 30˚A. Solid line: counterion density profile along a phosphate strand (due to symmetry, the counterion density profiles on the two phos- phate strands are the same). Dashed line: counterion densit y profile along the major groove. Dot-dashed line: counterion density profile along the minor groove. B. Nearly touching configurations Let us now consider very small surface-to-surface sep- arations between the DNA molecules. In this case one expects that the dependence of the forces and torques on the relative orientation φis most pronounced. For such nearly touching configurations, however, the dis- creteness of the phosphate charges, as embodied in the parameter φ0, strongly influences the results as well. The qualitative behaviour of the φdependence can be under- stood from Figure 5. Here two touching DNA molecules are shown for different relative orientations φwhere the phosphate strands are schematically drawn as continuous lines. For certain angles φwhich we call touching angles, two neighbouring phosphate charges hit each other. Pos- sible touching angles are φ= 36◦,180◦,324◦. Ifφ0is chosen to be zero, then two point charges are opposing eachother directly. Hence a strong dependence on φand onφ0is expected near touching angles. Results from computer simulation and YS-theory are presented in Figure 6. The parameters are from run A (see Table II) but with dc= 0.8˚A. The surface-to-surface separation is h= 2˚A.36 108 180 252 3240 0 0 0 0 0 φ [degrees]first DNA second DNA FIG. 5. Schematic picture of a DNA-DNA configuration for close separation distances. The abscissa corresponds t o the rotation angle of the first DNA molecule. The second DNA molecule is fixed. For touching angles, the interaction force becomes strongly repulsive. The strongest repulsion is achieved forφ= 180◦since two phosphate strands are meeting simultaneously. For relative orientations different from a touching angle, the force becomes smaller and can be both, attractive and repulsive. YS-theory always predicts a repulsive force. Again there are strong peaks for touch- ing angles in qualitative agreement with the simulation. The actual numbers predicted by YS-theory, however, are much too large and off by a factor of 6-7 around touching angles. The torque shows an even richer structure as a func- tion of φ. Near a touching angle it exhibits three zeroes corresponding to an unstable minimum exactly at the touching angle and two stable minima near the touching angles. The YS-theory shows 2 times larger values for the torque as compared to the simulation data. A qualitatively different force-angle behavior is ob- served for a larger counterion diameter. Results for dc= 1˚Aare shown in Figure 7. Here at touching angles, the interaction force is attrac- tive. The physical reason for that are the contact forces as given by Eqn.(8). Caused by the larger counterion diameter, counterions are stronger depleted in the zone between the DNA molecules. The torque has qualita- tively the same behaviour as before. We emphasize that the results do also depend strongly onφ0. Forφ0= 18◦, for instance, the force Fpractically 8 Typeset using REVT EXvanishes for any relative orientation φas compared to the same data for φ0= 0◦. C. Distance-resolved forces We now discuss in more detail the distance-resolved effective forces. For the parameters of run A, simulation results for Fare presented in Figure 8. Forφ0= 0, the force depends on the relative orien- tation φup to a surface-to-surface separation h≈6˚A in accordance with Figure 7. On the other hand, for φ0= 18◦, there is no φdependence at all for any sepa- ration. This supports the conclusion of previous works [57,55], that the effect of discreteness of the DNA phos- phate charges on the counterion concentration profile is small in general and dwindles a few Angstroms from the DNA surface. In fact, for h >6˚A, there is neither a φ nor a φ0dependence of the force, and the total force is repulsive. Furthermore we compare our simulation results with the prediction of linear screening theories in Figure 9. First of all, our simulation data for the total force (solid circles) are decomposed into the electrostatic part 0 60 120 180 240 300 360 φ [degrees] M / (F0 D )F / F0 −5 −10051030405060 −10−5051030405060 FIG. 6. Interaction force F( left y-axis) and torque M (right y-axis) for fixed surface distance h= 2˚Aversus relative orientation φin degrees. The unit of the force is F0= (e 4D)2. The solid (dashed) line is the simulation result for F(M) while the dot-dashed (dotted) line are data from YS-theory forF(M).φ0is chosen to be zero. The counterion diameter isdc= 0.8˚A.0 60 120 180 240 300 360 φ [degrees]F / F0 −20−10 M / (F0 D) −20−100 05 530405060 30405060 FIG. 7. Same as Figure 6 but now for dc= 1˚A. 20 25 30 35 40 DNA−DNA distance R [A o ]−15−10−50510F / F0 FIG. 8. Effective interaction force Facting onto a DNA pair versus the center-to-center distance R. The solid line is forφ0= 18◦. In this case there is no significant φ-dependence. The meaning of the symbols, that correspond to φ0= 0, is : circles- φ= 180◦, squares- φ= 36◦, triangles- φ= 45◦. F(1)+F(2)(diamonds) and the contact (or depletion) partF(3)(open circles). While the latter is strongly re- 9 Typeset using REVT EXpulsive, the electrostatic part is attractive such that the net force is repulsive. Linear screening theories aim to describe the pure electrostatic force only. Results for linear screening theories on different lev- els are also collected in Figure 9. If one compares with thetotal force, the prediction obtained by a homoge- neously charged cylinder is repulsive and off by a factor of roughly 1.5. A simulation with a homogeneously charged rod yields perfect agreement with linear screening theory since the Coulomb coupling is strongly reduced as the rod charges are now in the inner part of the cylinder. The Yukawa-segment theory is repulsive and off by a factor of 3. It is understandable that the YS model leads to a stronger repulsion than the charged cylinder model as the separation of the phosphate charges in the inner re- gion between the DNA molecules is shorter than the rod center separation. 20 25 30 35 40 DNA−DNA distance R[A o ]−10−5051015202530F / F0 FIG. 9. Theoretical and simulation results for interaction forceFversus separation distance R. The unit of the force is F0= (e 4D)2. The parameters are from run A and φ0= 18◦. Symbols: •- simulation data for all DNA rotation angles, ◦- the entropic part /vectorF(3),⋄- the pure electrostatic part/parenleftbig/vectorF(1)+/vectorF(2)/parenrightbig . Solid line: YS theory. Dot-dashed line: homo- geneously charged cylinder model. Dashed line: the predic- tions of KL theory with f1= 0.1, f2= 0.1, f3= 0.7, θ= 0.71. The Kornyshev-Leikin theory requires four counterion condensation fractions θ,f1,f2,f3as an input. We have tried to determine these parameters from our sim- ulation in order to get a direct comparison without any fitting procedure. In order to do so, we introduce a small shell around the cylinder of width δand determine θasthe fraction of counterions which are condensed onto the DNA within this shell. The actual value for δis some- what arbitrary, we first took a microscopic shell of width δ= 2.5˚Aas well as δ=λB= 7.1˚A. Data for θversus the rod separation are included in Figure 10 for three differ- ent combinations of counterion and phosphate diameters. It becomes evident that the fraction θof condensed coun- terions decreases with the rod distance but saturates at large separations. θalso depends on the size of the coun- terions and phosphate charges. If the width of the shell δis enhanced towards δ=λB= 7.1˚A,θincreases again. On the other hand, θis independent of the relative orien- tation φ. The actual data are consistent with Manning’s condensation parameter [82,83] θ0=λ/|qc|λB= 0.71 particularly if the width δis taken as one Bjerrum length. Our data are also in semiquantitative accordance with other computer simulations [38] and nuclear magnetic resonance (NMR) experiments which show that the con- densed counterion fractions are in the range of 0.65 to 0.85 [84] or 0.53 to 0.57 [85,45]. 20 25 30 35 40 DNA−DNA distance R[A 0 ]00.250.50.751θθ=0.71 FIG. 10. The condensation parameter θversus separation distance R. From top to bottom: solid line- run A ( dc= 1˚A, dp= 0.2˚A), dot-dashed line-run B ( dc= 2˚A,dp= 2˚A), dashed line- run C ( dc= 2˚A,dp= 6˚A). The horizontal line atθ= 0.71 indicates the saturation value at large distances for a larger δ=lB= 7.1˚A. This saturation value is the same for run A,B, and C. According to our results for the counterion density distribution (see Figure 4) we fix the minor and major groove fractions to f1= 0.1, f2= 0.1, and the strand fraction to f3= 0.7. Thus, (1 −f1−f2−f3) = 0.1 is the 10 Typeset using REVT EXfraction of the condensed counterions which is distributed neither on the phosphates strands nor on the minor and major grooves. The force in KL theory depends sensi- tively on θbut is rather insensitive with respect to f1, f2,f3, and φ. If the Bjerrum length is taken as a width for the condensed counterions, θ= 0.71, then the KL the- ory underestimates the total force. If, on the other hand, a reduced value of θ= 0.545 is heuristically assumed, then the KL theory reproduces the total force quite well. A serious problem of the comparison with linear screen- ing theories is that the contact term is not incorporated in any theory apart from recent modifications [86,64]. In fact, one should better compare the pure electrostatic part which is attractive in the simulation. Consequently, none of the linear screening theories is capable to describe the force well. This is due to the neglection of correla- tions and fluctuations in linear screening theories. From a more pragmatic point of view, however, one may state that a suitable charge renormalization leads to quanti- tative agreement with the totalforce. In fact, all three theories yield perfect agreement if the phosphate charges resp. the condensation parameter θis taken as a fit pa- rameter. For instance, the YS-model yields perfect agree- ment with the simulation for distances larger than 26 ˚Aif in Eqn.(16) a renormalized phosphate charge q∗ p=−0.6e is taken replacing the bare charge qp. But this is still unsatisfactory from a more principal point a view. VII. RESULTS FOR THE GROOVED MODEL The groove structure of DNA is expected to be of in- creasing significance as one approaches its surface [87]. We incorporate this in our model by increasing the phos- phate diameter towards dp= 2˚A(run B) and dp= 6˚A (run C). Results for the condensation parameter θare shown in Figure 10. θis decreasing with increasing dp since the coupling parameter Γ pcis decreasing which weakens counterion binding to the phosphate groups. Also the qualitative shape of the counterion density pro- files depends sensitively on the groove nature as can be deduced from Figure 11 as compared to Figure 4. The counterion density along the phosphate strands now ex- hibits minima at the phosphate charge positions while it was maximal there in Figure 4. Furthermore, the coun- terion density in the minor grooves is now higher than along the strands due to the geometrical constraints for the counterion positions which is similar to results of Ref. [55]. In fact, recent X-ray diffraction [88–90] and NMR spectroscopy [91,92] experiments, as well as molecular mechanics [93,94] and Monte Carlo simulations [5] sug- gest that monovalent cations selectively partition into th e minor groove. This effect is present also in our simple model and can thus already be understood from electro- statics and thermostatics.0 60 120 180 240 300 360 ϕ [degrees]00.10.20.30.40.50.60.7ρc(ϕ) hDδ FIG. 11. Same as Figure 4 but now for run C and φ= 45◦, δ= 3˚A. An increasing phosphate and counterion size increases the effective forces which is shown in Figure 12. Here, as φ0was chosen to be 18◦, there is no notable dependence on the relative orientation φ. A similar behavior was ob- served in a hexagonally ordered DNA system via Monte Carlo calculations [24]. This is understandable as coun- terion screening is becoming less effective. We have tried to fit the simulation data using a renormalized charge in the YS theory. A good fit was obtained for large sepa- rations while there are increasing deviations at shorter distances. This is different from our results for small ion sizes also shown in Figure 12 where the fit was valid over the whole range of separations. The adjustable parame- terq∗ pis shown versus the effective phosphate radius r∗ p of the YS model in the inset of Figure 12. It is increasing with increasing r∗ pin qualitative agreement with charge renormalization models [95]. We also note that the physical nature of the electro- static part of the interaction force undergoes a trans- formation upon decreasing the coupling parameter Γ pc. For strong coupling, Γ pc= 12 (run A), the electrostatic partF(1)+F(2)is attractive (see Figure 9). For mod- erate coupling, Γ pc= 3.6 (run B), it is nearly zero for all distances. Finally, for weak coupling,Γ pc= 1.8 (run C) the electrostatic part is elsewhere repulsive. The en- tropic part F(3)for these three runs is always repulsive and does not undergo a significant change. 11 Typeset using REVT EX20 25 30 35 40 DNA−DNA distance R[A o ]0204060F / F0 0.5 1.5 2.5 3.5 4.5 rp* [A o ]0.250.50.751 | qp*| FIG. 12. Interaction force Fversus separation distance R. The open circles are simulation data for all relative orien- tations φwithφ0= 18◦. From bottom to top: dc= 1˚A, dp= 0.2˚A(run A); dc= 2˚A,dp= 2˚A(run B); dc= 2˚A, dp= 6˚A(run C). The dashed lines are fits by the YS model. From bottom to top: fit for the parameters of run A with q∗ p=−0.6e; fit for the parameters of run B with q∗ p=−0.75e; fit for the parameters of run C with q∗ p=−0.85e. The inset is the vari- ation of the renormalized phosphate charge q∗ pversus effective phosphate radius rp∗. VIII. RESULTS FOR ADDED SALT Interactions involving nucleic acids are strongly depen- dent on salt concentration. Indeed, the strength of bind- ing constants can change by orders of magnitude with only small changes in ionic strength [96,97]. Our simula- tions show a similar strong salt impact on the interaction force. When salt ions are added, there is a competition be- tween two effects. The first one is the increasing of the direct repulsion between molecules as a consequence of delocalizing the adsorbed counterions. The second stems from the osmotic pressure of added salt that pushes the salt ions to occupy the inner molecular region and to screen the DNA-DNA repulsion. As we shall show be- low, these two effects result in a novel non-monotonic behaviour of the force as a function of salt concentra- tion.20 25 30 35 40 DNA−DNA distance R[A o ]02040F / F0 −3 −2 −1 0 log10(Cs [M])051015F / F0123 4 5 FIG. 13. Interaction force Facting onto a DNA pair ver- sus distance for φ= 0◦andφ0= 18◦. The unit of the force isF0= (e 4D)2. The solid lines are for increasing salt con- centration: 1- run D, 2 - run E, 3 - run F, 4 - run G, 5 - run H. Dashed line: reference data without salt from run A. The inset shows the force versus salt concentration at fixed separation R= 26˚A. Simulation results for Fversus distance for increas- ing salt concentration are presented in Figure 13. In our simulations, counter and equally charged salt ions are in- distinguishable. We take d+=d−=dc,|q+|=|q−|=e. It can be concluded from Figure 13 that even a small amount of salt ions (line 1, run D, Cs= 0.025M) signifi- cantly enhances the DNA-DNA repulsion (compare with the dashed line corresponding to run A, Cs= 0M). Upon increasing the salt concentration, at large separations, h >10˚A, the screening is increased in accordance with the linear theory. However, at intermediate and nearly touching separations, a non-monotonic behaviour as a function of salt concentration is observed as illustrated i n the inset of Figure 13. In the inset, the maximum of Foc- curs for Cs= 0.2M. The physical reason for that is that added salt ions first delocalize bound counterions which leads to a stronger repulsion. Upon further increasing the salt concentration, the electrostatic screening is en- hanced again and the force gets less repulsive. In order to support this picture we show typical microion config- urations and investigate also the fraction θof condensed counterions as a function of salt concentration. Simulation snapshots are given in Figure 14, where the positions of the mobile ions are projected onto the xy- plane. 12 Typeset using REVT EXa bc FIG. 14. Two-dimensional microion snapshots projected to a plane perpendicular to the helices for φ= 0◦,φo= 18◦, R= 30˚A. The filled circles are the positions of the counterions and positive salt ions, the open circles are the positions fo r the negative salt ions (coions). a- run A, b- run F, c-run G. A comparison of the salt-free case (Figure 14a) with that of moderate salt concentration ( Cs= 0.2M, Fig- ure 14b) reveals that the total number of adsorbed coun- terions decreases with increasing Cs. Furthermore, for Cs= 0.2M(Figure 14b), there are no coions in the inner DNA-DNA region. Thus salt ions do not participate in screening. Consequently, the DNA-DNA interaction, due to delocalization of counterions, will be enhanced. Con- trary to that, for Cs= 0.73M, (Figure 14c) the salt co- and counterions enter into the inner DNA-DNA region and effectively screen the interaction force. Further information is gained from the fraction θof condensed counterions which is plotted as a function of R for different salt concentrations Csin Figure 15. We de- fineθas the ratio of condensed counterions coming from the molecules with respect to the total number of counte- rions stemming from the molecules. As Csincreases, the saturation of θoccurs at smaller distances. In the inset of Figure 15 a non-monotonic behaviour of θas a function of the added salt concentration is visible which again is a clear signature of the scenario discussed above. The increase of θabove a certain threshold of salt concentra- tion is mainly due to a counterion accumulation outside the grooves. A similar trend was predicted by Poisson- Boltzmann [98] and Monte Carlo [61,47] calculations in different models. 13 Typeset using REVT EX20 25 30 35 40 DNA−DNA distance R[A 0 ]0.250.50.751θ−3 −2 −1 0 1 log10(Cs[M])0.20.40.6θ FIG. 15. Same as Figure 10, but now with added salt. Sym- bols: △- run A, •- run D, ◦- run E, ⋄- run F, ∗- run G, × - run H. The inset shows θfor fixed distance as a function of salt concentration: solid line- for R= 26˚A; dashed line- for R= 30˚A. More details of the forces and the comparison to linear screening theories are shown in Figures 16, 17 and 18. For run F, the different parts of the total force are pre- sented in Figure 16. As compared to the salt-free case (Figure 9) the pure electrostatic part is again attractive but much smaller, while the depletion part is repulsive and dominates the total force. All three linear models, homogeneously charged cylinder model, YS, and KL the- ory, underestimate the force. Note that the KL-theory with a θparameter corresponding to a width δof one Bjerrum length and the homogeneously charged cylinder model give the same results. Again with a suitable scal- ing of the prefactor by introducing a renormalized phos- phate charge q∗ presp. by fitting the condensed fraction θ, one can achieve good agreement with the simulation data for distances larger than 24 ˚A. The fitting parameter q∗ p used for the YS-model is −1.1e, while the optimal con- densed fraction θfor the KL-theory is 0 .2. The optimal renormalized phosphate charge q∗ pis shown versus salt concentration in Figure 17. Note that the usual DLVO size correction factor ζis already incorporated in the in- teraction, so what one sees are actual deviations from DLVO theory. The renormalized charge q∗ pincreases with increasing Cswhich is consistent with the works of Del- rowet al[73] and Stigter [27]. If one simulates the force within the homogeneously charged rod model, one finds good agreement with our simulation data for large sepa-rations. Consequently, the details of the charge pattern do not matter for large salt concentrations. 20 25 30 35 40 DNA−DNA distance R[A o ]−100102030405060F / F0 FIG. 16. Same as Figure 9 but now for run F and φ= 0◦,φ0= 18◦. The KL theory was adjusted to f1= 0.1, f2= 0.1, f3= 0.7, θ= 0.71. The results for KL theory and homogeneously charged cylinder models coincide exactly. We also note that our simulations give no notable de- pendence of the force on the relative orientation φfor h >6˚A. Only for small separations, h <6˚Athere is a slight dependence in agreement with Ref. [57]. Finally we show the influence of the ion and phosphate size on the effective force (for the parameters of run I) in Figure 18. The electrostatic part of the force is now repulsive but the total force is still dominated by the de- pletion part. As far as the comparison to linear screening theories is concerned, one may draw similar conclusions as for Figure 16. The fitting parameter q∗ pneeded to de- scribe the long-distance behaviour within the YS model does not depend sensitively on the phosphate and ion sizes. With a suitable scaling of the prefactor one can achieve good agreement with the simulation data for dis- tance larger than 26 ˚A. The fitting parameter q∗ pused for the YS-model is −1.1e, while the optimal condensed fraction θfor the KL-theory is 0 .19. Here again, simu- lations of the homogeneously charged cylinder model are in good agreement with our results obtained for a double stranded DNA molecule. 14 Typeset using REVT EXIX. COMMENTS AND CONCLUSIONS In conclusion, we have calculated the interaction be- tween two parallel B-DNA molecules within a “primi- tive” model. In particular, we focussed on the distance- and orientation-resolved effective forces and torques as a function of salt concentration. Our main conclusions are as follows: 0 2 4 6 8 10 λD [A o ]00.20.40.60.811.21.4|qp*| FIG. 17. Fitted renormalized phosphate charge q∗ pin the YS model, versus Debye screening length λDfor runs D-H. 20 25 30 35 40 DNA−DNA distance R[A o ]0102030405060F / F0 FIG. 18. Same as Figure 9 but now for run I and φ= 0◦,φ0= 18◦. The KL theory was adjusted to f1= 0.1, f2= 0.1, f3= 0.7, θ= 0.71. Note that the KL and homogeneously charged cylinder models produce the same curves.First, the interaction force for larger separations is re- pulsive and dominated by microion depletion. The ori- entational dependence induced by the internal helical charge pattern is short ranged decaying within a typi- cal surface-to-surface separation of 6 ˚A. For shorter sep- arations there is a significant dependence on the rela- tive orientation φand on the discreteness of the charge distribution along the strands. As a function of φ, the force can be both attractive and repulsive. This may lead to unusual phase behaviour in smectic layers of par- allel DNA molecules. Details of the molecular shape and counterion size are important for small separations as well. The torque is relatively small except for small sep- arations where it exhibits a complicated φ-dependence. Second, as a function of added salt concentration we predict a non-monotonic behaviour of the force induced by a competition between delocalization of condensed counterions and enhanced electrostatic screening. This effect can in principle be verified in experiments. Third, linear screening theories describe the simula- tion data qualitatively but not quantitatively. Having in mind that the total force is dominated by the deple- tion term which is typically neglected in linear screen- ing theory, such theories need improvement. On the other hand, the different theories predict the correct long- distance behaviour, if a phenomenological fit parameter - as the renormalized phosphate charge q∗ pfor the Yukawa- segment model or the condensation fraction θfor the Kornyshev-Leikin model - is introduced. The Yukawa- segment model can even predict the orientational depen- dence of the force and the torque at smaller distances in the case of small counterion and phosphate sizes. Hence, a phenomenological Yukawa segment model can be used in a statistical description of the phase behaviour of many parallel DNA strands in a smectic layer. Future work should focus on an analysis for divalent counterions which are expected to lead to a qualitatively different behaviour since the Coulomb coupling is en- hanced strongly in this case. Also, one should step by step increase the complexity of the model in order to take effects such as dielectric discontinuities [38,41,27, 99], chemical bindings of counterions in the grooves and dis- crete polarizable solvents into account. ACKNOWLEDGMENTS We thank A. A. Kornyshev, S. Leikin, G. Sutmann, H. M. Harreis, and C. N. Likos for stimulating discus- sions and helpful comments. Financial support from the Deutsche Forschungsgemeinschaft within the project Lo 418/6-1 (“Theory of Interaction, recognition and assem- bling of biological helices”) is gratefully acknowledged. 15 Typeset using REVT EXAPPENDIX A: LEKNER SUMMATION METHOD FOR FORCES In our simulations we account for the long-range nature of th e Coulomb interactions via the efficient method proposed by Lekner [68]. This method has been successfully applied to partially periodic systems [14,100]. For an assembly of Nions in a central cubic cell of dimension L, the Coulomb force /vectorF(c) iexerted onto particle iby particle j, and by all repetitions of particle jin the periodic system, is /vectorF(c) i=qiqj ǫ/summationdisplay all cells/vector ri−/vector rj |/vector ri−/vector rj|3. (A1) Because of x, y, z symmetry it is sufficient to consider only one component of the force. For the x-component of the force we have /vectorF(c) ix=qiqj ǫL28π∞/summationdisplay l=1lsin(2πl∆x L) ∞/summationdisplay m=−∞∞/summationdisplay n=−∞K0/parenleftBigg 2πl/parenleftbigg (∆y L+m)2+ (∆z L+n)2/parenrightbigg1/2/parenrightBigg (A2) Here, ∆ x=xi−xj,∆y=yi−yj,∆z=zi−zj, and K0(z) is the modified Bessel function of zero order. For a pair of particles not aligned parallel to the x-axis, the convergence of the sum in (A2) is fast. Thus an evaluation of just 20 terms in the sum is enough to get a part-p er-million accuracy. 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arXiv:physics/9912054v1 [physics.atom-ph] 30 Dec 1999Atom optics hologram in time domain A. V. Soroko∗ National Centre of Particle and High Energy Physics, Belaru sian State University, Bogdanovich Street 153, Minsk 220040, Belarus A temporal evolution of atomic wave packet interacting with object and reference electromagnetic waves is investi - gated beyond the linear respond. Under this condition the diffraction of ultracold atomic beam on inhomogeneous laser radiation is interpreted as beam’s passing through a three- dimensional hologram, which thickness is proportional to t he interaction time. It is shown that diffraction efficiency of su ch a hologram may reach 100% and is determined by the time domain. 03.75.Be, 42.50.Vk, 32.80.Lg, 81.15.Fg I. INTRODUCTION The achievements of the last decade in the field of laser light cooling below the recoil limit [1,2] have opened a new chapter of atom optics which objective is to ma- nipulate atomic beams in a way similar to conventional optics by exploiting the wave properties of the particles. Indeed, if momenta of cooled atoms verge to the photon ones, diffraction effects may manifest themselves espe- cially strongly during atomic interaction with space in- homogeneous radiation. For the corresponding part of the de Broglie wave spectrum, this provides a possibil- ity of supplementing the traditional atom optics set of elements such as mirrors [3], diffraction gratings [4,5] or lenses [6] with holograms of different objects, the conven- tional optics analogues of which have been well known for several decades [7]. The destination of such atomic holo- grams is to create matter waves with intended amplitude and phase characteristics. Since these characteristics ar e the same as for the object wave one obtains a strong and convenient implement for holographic imaging with atoms. The latter may have useful practical applications from atom lithography [8] to the manufacturing of mi- crostructures, or quantum microfabrication. One of possibilities to make atomic hologram consists in creation a mechanical mask with appropriate trans- parency for the incident atomic beam (analogue of two- dimensional optical hologram). Such a hologram has the advantage of being permanent, however, up to now only the masks with binary transparency have been prepared. So in the experiment [9] the mask was written onto a thin silicon nitride membrane and allowed for complete or vanishing transmission of the beam at a given point. Evidently this reduces resolution in the reconstructed im- age, because correct holographic storage of information requires the gradually varying transmission of the beam.Very interesting proposal has recently been reported in the work [10], where the authors suggest to use a Bose- Einstein condensate (BEC) as a registration media for atomic hologram. It illustrates the wide potential ap- plicability of condensates which after having been real- ized experimentally [11] are available almost routinely in several laboratories. In this method desired information is encoded into the condensate in the form of density modulations by using object and reference laser beams that form writing optical potential. The reconstruction of matter wave arises due to s-wave scattering of the reading-beam atoms on condensate inhomogeneities. In the previous paper [12] we have shown that atomic hologram may also be constructed (at certain conditions) as a superposition of reference and object electromag- netic waves to be common for optical holography. The creation of intended matter wave arises when ultracold atomic beam is diffracted from this hologram which in a fact represents itself the inhomogeneous light field. Main advantages of the proposed scheme are it’s simplicity be- cause of skipping the recording process and, as a con- sequence, absence of aberrations in the stored informa- tion. In some sense our approach is close to the non- holographic scheme of wave front engineering [13], which implies to arbitrarily shape the center-of-mass wave func- tion of an atom by means of a sequence of suitably shaped laser pulses, because both methods are based only on atom-laser interactions. Main assumption being employed in our holographic scheme is the linear respond of atomic system on the laser-field inhomogeneity. It requires, in particular, the weak perturbation of the incident atomic beam and sets an upper limit on the object wave amplitude (see Eq. (50) in the Ref. [12]). As a result only a little part of atoms in the beam can be transferred into the reconstructed mat- ter wave. Linear respond operation decreases the diffrac- tion efficiency of an atomic hologram, i.e. the ratio of the intensity of diffracted atomic waves to the intensity of reading beam, what may be crucial for practical ap- plications. In conventional optical holography such a sit- uation corresponds to the kinematical regime of writing information [7]. On the other hand, the coupled wave theory of Kogelnik [14] and the theories based on dy- namical approximation [15–17], give one a recipe how to create a hologram with high (up to 100%) diffraction effi- ciency. In this purpose it is necessary to control, besides others, such a parameter as the thickness of the holo- gram. Unfortunately the thickness control is difficult to perform in the scheme of atom holography without regis- tration medium like our one (see Figure 1 for details). So 1the purpose of present paper is to suggest a new approach for creation of atom optics holograms which will inherit the advantages of our previous scheme and also allow high diffraction efficiencies. We will show that desired approach can be realized if one restricts the existence of atomic hologram rather in a time than space domain, so that the hologram will work in a pulsed regime pump- ing atoms from the beam or initial wave packet into the reconstructed wave. Note that suggested regime is well compatible with the Raman cooling methods [2] (includ- ing laser cooling below the gravitational limit [18]) and the recent realization of an atom laser [23], which in a fact repeatedly reproduce coherent or almost coherent atomic wave packets necessary for actual implementation of a reading beam. FIG. 1. Typical for atomic holography layout design of laser beams and matter wave packets. Another important prerequisite for successful wave- front reconstruction with massive particles concerns the compensation for potentially detrimental influence of gravitational effects. Fortunately, the bulk of atoms has the magnetic moment, and all one has to do is use the Stern-Gerlach effect. Superimposing the weakly inhomo- geneous magnetic field onto the path of prepolarized par- ticles and appropriately adjusting the field gradient, it is possible to suspend the ground state atoms everywhereexcept the region of interaction with radiation. But if the laser frequency is far from all atom transitions, the contribution to the total force induced by spatially de- pendent shifts of the Zeeman levels is negligible. Under this condition, atoms move like free particles being af- fected only by the electromagnetic waves. In Sec. II we derive a system of equations which de- scribe interaction of atomic wave packet with object and reference electromagnetic waves when the gravity is com- pensated. Approximate solution of this system is found not assuming weak perturbation of initial state, and do- main of its validity is determined. For reasonable exper- imental conditions the solution admits an atom-optics interpretation that is done in Sec. III. Namely, the in- homogeneous laser radiation is shown to behave like a three-dimensional hologram in respect to the impinging wave packets. A numerical simulation of such a holo- gram created with 31-mode object beam is presented, and high diffraction efficiency is demonstrated. Section IV concludes with a summary of the obtained results. Most cumbersome expressions are placed into Appendix. II. BASIC FORMULAS A. Compensation for the gravity Consider for definiteness an atom with a J=1 2to J=3 2transition, e.g., sodium or cesium. The magnetic fieldB(r) applied to compensate for the gravity is sup- posed to contain a homogeneous component B0directed along the gravity acceleration B0↑↑g. The remain- ing inhomogeneous part of the field B1(r) =B(r)−B0 should be small compared to this component |B1(r)| ≪B0=|B0|. (2.1) As we will see below, to fulfil this condition it is neces- sary to take B0in the range 103÷104G. In practice such a field is strong enough to induce Zeeman shifts which considerably exceed the hyperfine splitting inter- vals∼¯hωHFS(but not the multiplet ones). Therefore an internal atomic eigenstate |J,I,M J,mI∝an}b∇acket∇i}htmay be well described using the set of quantum numbers consisting of the angular momenta of electronic shell Jand nucleus I, and their local projections MJ,mI, on the direction of the magnetic field. The corresponding energy eigenvalue is determined not only by the multiplet level EJbut also by the magnetic field B(r) =|B(r)|and therefore is spa- tially dependent E|J,I,M J,mI/an}bracketri}ht(r) =EJ+aMJmI +(µBgLMJ−µnucmI)B(r),(2.2) whereais the hyperfine coupling constant ( a∝¯hωHFS, e.g., for Na a/¯h= 885.8 MHz),gLdenotes the Lande factor, and µnucis the nuclear magnetic moment. Be- cause of condition (2.1) such a spatial dependence, how- ever, mainly arises from the longitudinal ( B/bardbl 1(r) =B0· 2B1(r)/B0), rather than the transverse ( B⊥ 1(r)) compo- nent of the vector B1(r), provided that the components are defined relative to B0. This is evident from the ex- pression B(r) =/radicalbigg/bracketleftBig B0+B/bardbl 1(r)/bracketrightBig2 +/bracketleftbig B⊥ 1(r)/bracketrightbig2 ≃B0+B/bardbl 1(r) +/bracketleftbig B⊥ 1(r)/bracketrightbig2/(2B0), (2.3) where the term containing B⊥ 1(r) is small and can be neglected. Consequently, by adjusting the gradient of the fieldB/bardbl 1(r) one can achieve translation invariance of the ground state |g∝an}b∇acket∇i}ht=|1/2,I,−1/2,I∝an}b∇acket∇i}ht(or another state withJ= 1/2) in three dimensions: E|g/an}bracketri}ht(r)−Mg·r=const. (2.4) For example, to balance the gravitational force in this way for sodium it is necessary to create a gradient ∇B/bardbl 1(r) =b1g/|g|, whereb1=−4.033 G/cm. This condition does not contradict the Maxwell equation ∇ · B1(r) = 0, because variation of B⊥ 1(r) is not restricted. Note also that the choice B0= 103÷104G maintains condition (2.1) very well within a spatial region of the size∼10 cm. All the other levels are affected by the residual external potential. In particular, the force feacting on the atoms in the excited state, e.g., |e∝an}b∇acket∇i}ht=|3/2,I,−3/2,I∝an}b∇acket∇i}ht, may be estimated from Eqs. (2.2) and (2.4) as |fe| ∼Mg. B. Interaction with laser beams In our scheme, we use pulses of laser light at frequency ωwhich is roughly tuned to the |g∝an}b∇acket∇i}ht → |e∝an}b∇acket∇i}httransition. If the typical size 2 Lof atomic sample is restricted by the condition L≪a/(Mg), one may regard E|e/an}bracketri}ht(r) as the closest to resonance excited level within the whole interaction domain. Indeed, the maximal spatial shift of the level ∼MgL induced by the force feappears to be much less than the hyperfine splitting intervals (MgL ≪a∼¯hωHFS), and the hierarchy of detunings retains. Therefore an atom initially in |g∝an}b∇acket∇i}htstate behaves as a two-level system with respect to the processes with stimulated emission of photons. Each laser beam is considered as a discrete superposi- tion of plane monochromatic electromagnetic waves. In particular, we use the following decomposition of the elec- tric field in the object beam Es(r,t) =/summationdisplay m≥1Emexp(ikm·r−iωt) +c.c., (2.5) where Emandkmstand for the complex amplitude of the modemand its wave vector respectively. Such an approach, does not somehow restrict the generality of consideration, because in our experimental setup atommoves inside a superposition of reference and object laser beams during all the interaction time, and the expression (2.5) must well describe the real laser field only in the atom-laser interaction region. Evidently, the latter re- quirement can always be satisfied by decreasing the min- imal angle between the mode wave vectors. In this case we can also regard the reference beam as a single mode (with the index m= 0) Er(r,t) =E0exp(ik0·r−iωt) +c.c., (2.6) which is a typical arrangement for optical holography. Since the atomic dipole momentum operator ˆdis di- agonal in quantum numbers IandmI, the transitions which change mIare allowed only due to hyperfine inter- action. As a consequence, the excited state |e∝an}b∇acket∇i}htdecays to the lower ones preferentially in the channel |e∝an}b∇acket∇i}ht → |g∝an}b∇acket∇i}ht (with the rate γ). This circumstance makes it possi- ble to deal with an atom as a two-level system even if spontaneous photon emission takes place. However, to simplify the consideration the coherent scattering pro- cesses are assumed to dominate the spontaneous emis- sion, i.e., the regime |∆| ≫γis kept [20,21], where ∆ =ω+ [E|g/an}bracketri}ht(0)−E|e/an}bracketri}ht(0)]/¯his the detuning from resonance in the center of atom-laser interaction region (r= 0). Under such a condition the one-particle density matrix in momentum representation [22] has an obvious time evolution ρab(p1,p2,t) =/integraldisplay dp′ 1/integraldisplay dp′ 2/summationdisplay a′b′Gaa′(p1,p′ 1,t) ×G∗ bb′(p2,p′ 2,t)ρa′b′(p′ 1,p′ 2,t= 0),(2.7) where indices a,b... span the internal atomic states (e,g) andGaa′(p1,p′ 1,t) is the Green function of two- component Shr¨ odinger equation describing atomic dy- namics during the |g∝an}b∇acket∇i}ht ↔ |e∝an}b∇acket∇i}httransitions. In rotating wave approximation this equation rewritten for slowly time dependent ground- and excited-level wave functionsψg(p,t) andψe(p,t) takes the form i∂ ∂tψg(p,t) = [t(p) + ∆]ψg(p,t) −/summationdisplay m≥0Ω∗ mψe(p+ ¯hkm,t), (2.8a) i∂ ∂tψe(p,t) = [t(p)−ife· ∇]ψe(p,t) −/summationdisplay m≥0Ωmψg(p−¯hkm,t), (2.8b) where Ω m=∝an}b∇acketle{te|ˆd·Em|g∝an}b∇acket∇i}ht/¯his the Rabi frequency of modem, and the terms t(p) =p2/(2M¯h) and −ife· ∇ arise in momentum space from the kinetic and potential energy ( −fe·r) correspondingly. For the situation at hand, the upper electronic state can be adiabatically eliminated from Eqs. (2.8) provided that the detuning ∆ is large enough [5,20] 3|∆| ≫ |Ωm|,|fe|L/¯h. (2.9) The route by which one can do it implies a self-consistent assumption |ψe| ≪ |ψg|leading to the zero-order solution of the Eq. (2.8a): ψg(p,t)≃exp{−i[t(p)+∆]t}ψg(p,t= 0). After substitution of this expression into Eq. (2.8b) the latter may be solved in the framework of perturbation theory developed in respect to the potential energy term. In this case, the excited-level wave function acquires a representation ψe(p,t)≃ −/summationdisplay m≥0Ωmψg(p−¯hkm,t) t(p)−t(p−¯hkm)−∆+..., (2.10) where the dots denote omitted terms which include a small ( ∝ |fe|L/|¯h∆|) first-order correction to ψe(p,t) and also summands which oscillate with the non-resonant fre- quencyt(p) and therefore give a negligible contribution when one uses above expression within the context of Eq. (2.8a). For ultracold atomic sample one can further discard the kinetic energy terms in denominators of the expres- sion (2.10). As a result the motion of the ground-state atom is described with the equation i∂ ∂tψg(p,t) = [t(p) + ∆ +f0]ψg(p,t) +/summationdisplay m≥1/braceleftbigg/summationdisplay n≥1 n/ne}ationslash=mfmnψg[p−¯h(km−kn),t] +gmψg[p−¯h(km−k0),t] +g∗ mψg[p+ ¯h(km−k0),t]/bracerightbigg , (2.11) where f0=1 ∆/summationdisplay m≥0|Ωm|2, (2.12) fmn=ΩmΩ∗ n ∆, (2.13) and gm=ΩmΩ∗ 0 ∆(2.14) stand for the effective Rabi frequencies. C. Evolution of wave packets It is known from the theory of thick optical holograms that reconstruction of the original (conjugate) object wave arises only if the reading beam is directed along (opposite) the reference wave and has the same wave- length. Relying on analogy with conventional optics let us consider for definiteness the evolution of atomic wavepacket whose spectrum is initially concentrated around the mean momentum of photons in the reference beam. In such a case one can anticipate creation of the matter wave being similar to the forward object wave. Therefore it is convenient to look for the solution of Eq. (2.11) as a sum of wave packets approaching the plane modes of hologram [16] ψg(p,t) =/summationdisplay m≥0ψm(p−¯hkm,t). (2.15) Initially there are no wave packets corresponding to the object beam, so that ψm(p,t= 0) = 0, m≥1, (2.16) and as a consequence ψ0(p−¯hk0,t= 0) =ψg(p,t= 0)≡ψg(p).(2.17) Population of these atomic motional states ( m≥1) arises due to coupling with ψ0(p,t), the wave packed corre- sponding to the reference beam: i∂ ∂tψm(p,t) =tm(p)ψm(p,t) +gmψ0(p,t),(2.18) where tm(p) =t(p+ ¯hkm) + ∆ +f0. (2.19) Depletion of the state with m= 0 is governed by the equation i∂ ∂tψ0(p,t) =t0(p)ψ0(p,t) +/summationdisplay m≥1g∗ mψm(p,t) +χ(p,t), (2.20) which one can obtain after substituting Eqs. (2.15), (2.18) into Eq. (2.11). So we bring the Eq. (2.11) to the system of equations (2.18), (2.20). The advantage of such a step becomes obvious after making a self-consistent assumption about momentum spectrum of ψm(p,t),m≥0, the validity of which was verified for two-mode case in Ref. [18]. Namely, we will suppose below that all non-vanishing functions have narrow distributions around p= 0 and, as a result, do not overlap in the expression for χ(p,t) χ(p,t) =/summationdisplay m≥1/braceleftbigg/summationdisplay n≥1gnψm[p+ ¯h(2k0−kn−km),t] +/summationdisplay n≥0 n/ne}ationslash=mg∗ mψn[p+ ¯h(km−kn),t] +/summationdisplay n≥1 n/ne}ationslash=m/summationdisplay l≥0fmn ×ψl[p+ ¯h(k0−kl−km+kn),t]/bracerightbigg .(2.21) 4In such circumstances different parts of this term give incoherent contributions, which appear to be small at lowgmand can be taken into account within the theory of perturbation. In zero-order approximation one omits χ(p,t) so that the system (2.18), (2.20) becomes homo- morphic with the rate equations describing a ( m+1)-level atom. Note, the stationary solutions of this truncated system exactly coincide with eigenmodes of correspond- ing optical hologram [16]. To go further it is convenient to perform the Laplace transformation ( m≥0) ψm(p,λ) =/integraldisplay∞ 0dte−λtψm(p,t) (2.22) with the initial conditions (2.16), (2.17). Then the equa- tions for the Laplace transforms will allow an easy zero- order solution ψ(0) 0(p,λ) =−i T(p,λ)ψg(p+ ¯hk0), (2.23a) ψ(0) m(p,λ) =−gm tm(p)−iλψ(0) 0(p,λ), m≥1,(2.23b) where T(p,λ) =t0(p)−iλ+/summationdisplay m≥1−|gm|2 tm(p)−iλ. (2.24) Similarly, next iteration reproduces the first-order so- lution ψ(1) 0(p,λ) =ψ(0) 0(p,λ) +−χ(0)(p,λ) T(p,λ), (2.25a) ψ(1) m(p,λ) =−gm tm(p)−iλψ(1) 0(p,λ), m≥1,(2.25b) whereχ(0)(p,λ) is obtained from the expression (2.21) after making the substitutions ψm(p,t)→ψ(0) m(p,λ), m≥0. In principle, we can get the solution with any preassigned accuracy by repeating the iterations but it will be sufficient to restrict ourselves to the first-order formulas for the following consideration. Desired time-dependent wave functions arise then as the inverse Laplace transforms of ψm(p,λ) in agreement with the Mellin formula ψm(p,t) = 2πi/integraldisplayǫ+i∞ ǫ−i∞dλeλtψm(p,λ), ǫ> 0.(2.26) Finally, using the Eqs. (2.25) and (2.26) one can easily write down an expression for the ground-state component Ggg(p,p′,t) of the Green function appearing in the for- mula (2.7). We placed this expression into the Appendix.D. Validity of the solution At first let us check that the zero-order solution (2.23) indeed has a narrow momentum spectrum around p= 0, provided the initial conditions are chosen properly, and the effective Rabi frequency gmis small enough. Doing it we may restrict ourselves to examination of only a re- gionD={p: [tm(p)−t0(p)]2<∼|gm|2,∀m}, where all the functions in truncated system of equations (2.18), (2.20) have a possibility to influence each other reso- nantly. In this region one can identify the kinetic-energy termstm(p) related to different modes of the object beam (m≥1) without any damage for the result of estimation: tm(p)≈tn(p)≈˜t(p), where ˜t(p) =t(p+ ¯h˜k) + ∆ +f0, and˜kis some typical wave vector in the object beam. Under this condition the integral in the Eq. (2.26) can be calculated explicitly, and the wave functions ψ(0) m(p,t) get a simple analytical representation ψ(0) 0(p,t)≃Ar(p,t)e−ib(p)tψg(p+ ¯hk0),(2.27a) ψ(0) m(p,t)≃As(p,t)gm gΣe−ib(p)tψg(p+ ¯hk0), m≥1. (2.27b) In these formulas Ar(p,t) =ia(p) d(p)sin[d(p)t] + cos[d(p)t], (2.28a) As(p,t) =−igΣ d(p)sin[d(p)t], (2.28b) where a(p) = [˜t(p)−t0(p)]/2, (2.29) b(p) =a(p) +t0(p), (2.30) d(p) =/radicalBig a(p)2+g2 Σ, (2.31) and gΣ= /summationdisplay m≥1|gm|2 1/2 (2.32) stands for the overall effective Rabi frequency. It is seen from Eqs. (2.27) that initial atomic wave packet transforms into motional states with m≥1 at a timeτn(time of the nπpulse [21]) τn=π 2gΣ(2n−1), n∈ N. (2.33) 5This transition is velocity-selective with the most ef- ficiency determined by the Bragg resonance condition p·∆k= 0, where ∆k= (˜k−k0) denotes a typical dif- ference between wave vectors in the object and reference beams (cf. Ref. [20]). The width of a peak in momen- tum distribution along the direction of vector ∆k(the interval from maximum to the first minimum) depends on interaction time, and for t≤2τ1is δp(t) =2MgΣ ∆k/radicalbigg 4/parenleftBigτ1 t/parenrightBig −1. (2.34) For given value ∆ k=|∆k|it decreases with gΣ. There- fore the smaller the effective Rabi frequencies gmthe nar- rower the momentum spectrum of ψ(0) m(p,t). Evidently, to prevent all non-vanishing functions com- posing the term χ(p,t) from being overlapped in momen- tum space their spectra must be concentrated within the domain |p|<¯hδkatt∼τ1, where δk= min m,n≥0|km−kn| (2.35) is the minimal distance between different wave vectors of the laser beams (see Figure 2). FIG. 2. Definitions of the values ∆k,δk, andδpin the simplest case of two-mode object wave. Since the spectral extent along the direction of vector ∆kis characterized by δp(t), we immediately get the first of sufficient conditions δp(τ1)≪¯hδk. (2.36)In agreement with the Eq. (2.34) it sets an upper limit on the overall effective Rabi frequency gΣ≪¯hδk∆k/(2√ 3M). (2.37) In the transverse direction the spectra are the same as that of initial wave packet ψg(p+ ¯hk0). Therefore one must impose another condition |(p′−¯hk0)×∆k|<¯hδk∆k, (2.38) which restricts allowed values of p′in the domain of the Green function Ggg(p,p′,t). When the inequalities (2.36), (2.38) are met, the main correction to the zero-order solution ψ(0) m(p,t) caused by the termχ(p,t) arises outside the near-resonance region Dand depends on geometry of laser beams. So for t∼τ1 and 2D holographic setup like that in the Fig. 2 (i.e., all kmare coplanar vectors) the relative correction has the order of magnitude εr=δp(τ1)/(¯hδk)≪1, what can be seen from Eqs. (2.21), (2.27b). Note, while making this estimation we discarded the third summand in the expression (2.21), because it is proportional to fmnand, consequently, is much less than the first and second ones (∝ |gm|), provided the standard holographic restriction on the intensities of laser beams |E0|2≫ |Em|2,m≥1, leading to the inequality |gm| ≫ |fmn|, is applied here. To illustrate the said in the case of two-mode object wave let us regard absolute values of zero-order solution |ψ(0) g|and first-order correction to it δ(0)=|ψ(1) g−ψ(0) g| as functions of the momentum component pxand the angleθbetween k1andk2, assuming a Cartesian coor- dinate system is introduced in momentum space, p= (px,py,pz), withx(y) axis chosen along (opposite) the vector k0(k1). Figure 3 shows corresponding depen- dences after π-pulse time calculated for sodium atoms, provided the initial wave packet has the Gaussian profile ψg(p)∝exp/bracketleftbig −L2(p−p0)2/(2¯h2)/bracketrightbig with mean momen- tump0= ¯hk0,|k0|=k0= 1.07×105cm−1, spatial extension 2 L= 0.4 cm, and 2D norm equal to 1. The peaks in central region of each plot correspond to for- bidden values of θ∝δk< δp(τ1). Outside these peaks (|θ|>10−4) the relative correction goes down approach- ing 0.15 at large θ, what is below its estimation value εr≈0.5. 6-0.4 -0.2 0 0.2 0.4-0.2-0.100.10.2 00.10.2 0.4 -0.2 0 0.2 -0.4 -0.2 0 0.2 0.4-0.2-0.100.10.2 00.050.1 0.4 -0.2 0 0.2a) b) ) 0 (/c100) 0 ( g/c121 /c113[mrad]/c113[mrad] FIG. 3. Absolute values of zero-order solution |ψ(0) g|(a) and first-order correction to it δ(0)(b) as functions of the momentum component pxand the angle θbetween k1and k2. The rest components of p,py= ¯hk0andpz= 0. The geometry of laser beams is as in the Fig. 2. The effective Rabi frequencies g1=g2= 10 Hz,f12= 0.1 Hz. In the worst case, e.g., km⊥k0,∀m≥1, one obtains more danger estimation for εr εr<∼gΣ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0sin2(gΣτ)exp(iδωτ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (2.39) whereδω= ¯hδ2 k/(2M) stands for the minimal kinetic energy an atom can get due to transition between the laser modes. Nevertheless, the term χ(p,t) may still be treated as a perturbation if the overall effective Rabi fre- quency satisfies more rigorous than (2.37) condition gΣ≪δω. (2.40) Otherwise, gΣ>∼δω, the interaction time should be lim- ited so that t≪τ1. Figure 4 shows time dependencesofδ(0),|ψ(0) g|, and|ψ(1) g|for two-mode object beam with δp(τ1)/(¯hδk) = 0.1 andgΣ= 74δω. We see that in con- sidered unfavorable configuration εrdoes not exceed 0 .1 even ift= 0.5τ1. 0 0.5 1 1.5 200.020.040.060.080.10.120.14 FIG. 4. Time dependences of δ(0)(solid line), |ψ(0) g|(long dashed line), and |ψ(1) g|(short dashed line) for two-mode ob- ject beam with k1,2⊥k0. The components of p,py= ¯hk0 andpx=pz= 0. Cartesian coordinate axes and other pa- rameters are the same as in the Fig. 3. In most of practical cases, however, only a small part of laser modes has a geometry leading to the condition (2.40), and the requirement (2.37) appears to be suffi- cient. III. ATOM OPTICS INTERPRETATION A. General consideration In an idealized situation, one may imagine that all atoms are initially in a pure state determined by the Gaussian profile g(p;p0) =L3/2 ¯h3/2π3/4exp/bracketleftbigg−L2(p−p0)2 2¯h2−i ¯hp·r0/bracketrightbigg (3.1) with mean momentum p0being close to ¯ hk0, space po- sitionr0, and very small dispersion [ L≫¯h/δp(τ1)]. Ac- cording to the Eq. (2.7), after interaction with the laser beams within time domain τ<∼τ1and subsequent free propagation during time tthe atoms remain in pure state, and their wave function can be represented as a superpo- sition of some useful signals ψ(s,r)(p,τ,t;p0) and a back- groundψ(b)(p,τ,t;p0), where ψ(σ)(p,τ,t;p0) =e−it(p)t/integraldisplay dp′G(σ)(p,p′,τ)g(p′;p0), (3.2) 7σ∈ {s,r,b}. The functions G(σ)(p,p′,τ) are defined by Eqs. (A2). In considered case they admit of explicit an- alytical expressions relying on analogy with the formulas (2.27) and being exact at L→ ∞, and κ→0, where κ=p0/¯h−k0. So, omitting inessential common phase factor exp[ −ik0·r0−if0τ−(i/¯h)E|g/an}bracketri}ht(0)τ] we can readily ascertain that Fourier transform of ψ(r)(p,τ,t;p0), ψr(r,τ,t;p0) =Ar(¯hκ,τ)γ0(r,κ)eik0·r−it(k0)t,(3.3) propagates like the reference beam, whereas the trans- form ofψ(s)(p,τ,t;p0), ψs(r,τ,t;p0) =As(¯hκ,τ) ×/summationdisplay m≥1γm(r,κ)gm gΣeikm·r−it(km)t,(3.4) gives birth to matter wave, which inherits amplitude and phase characteristics of the object beam because gm∝ Emas it follows from the Eq. (2.14) and definition of the Rabi frequencies Ω m. The last assertion also takes into account that all functions γm(r,κ) =L3/2 π3/4σ3exp/bracketleftbigg −L2(r−˜rm t)2 2|σ|4+iδφ(r−rm t)/bracketrightbigg , (3.5) used in Eqs. (3.3) and (3.4), slowly depend on rwithin space domains ∼2|σ|2/L, each centered around the point ˜rm t=rm t+ ¯h(t+τ)κ/M, where σ=/radicalbig L2+i¯h(t+τ)/M, (3.6) rm t=r0+¯h(k0+˜k) 2Mτ+¯hkm Mt, (3.7) and introduce a little phase shifts δφ(r−rm t), δφ(r) =1 2|σ|4/bracketleftbigg L4r·κ+¯h(t+τ)(r2−L4κ2) M/bracketrightbigg ,(3.8) disappearing at small κand largeL. If the overall effective Rabi frequency is chosen in agreement with the results of Sec. II D, the background, which represents itself a first-order correction to the wave functionψ(0) g(p,τ), appears to be small at any time τ<∼τ1. So, the states (3.3) and (3.4) being spatially sep- arated after free propagation period tmin= 2LM/(¯h∆k), one may observe a matter wave ψs(r,τ,t;p0) cloning the object beam in a space-time region S={(r,t) : |r−˜rm t|< L, ∀m≥1;t > t min}, where all atomic wave packets related to different modes of this beam still overlap each other. It should be noted, that S ∝ne}ationslash=∅ only when the observation time is limited by the value tmax=LM/[¯h˜ksin(θmax/2)], where θmaxcharacterizes the maximal divergence angle of the object beam, and ˜k=|˜k|. In a given context, physical sense of conditions(2.37), (2.40) consists in the requirement to use more del- icate mechanism (lower laser intensity) in order to restore more detailed information. They have a counterpart in the theory of optical holograms [see e.g., Eq. (4) in the Ref. [16]] which, in turn, is responsible for low intensity of noise in reconstructed wave. In a more realistic case we may expect the initial atomic state to be a statistical mixture well described with the density matrix ρgg(p1,p2,0) =/integraldisplay dp′f(p′)g(p1;p′)g∗(p2;p′),(3.9) wheref(p) denotes a momentum distribution function. If this function is compatible with the condition (2.38), one can readily obtain an expression for ρgg(p1,p2,t) at any time. In the region S, after rewriting into coordinate representation it takes the form ρgg(r1,r2,τ,t) =/integraldisplay dp′f(p′)ψs(r1,τ,t;p′)ψ∗ s(r2,τ,t;p′). (3.10) SinceAs(p,τ) is a sharp-shape function having a nar- row widthδp(τ) along the vector ∆k[see Eq. (2.34)], the integral in Eq. (3.10) is limited in this direction. Let us assume that integration in transversal directions is also restricted within a little domain ∼δp(τ) but due to finite spectral width of f(p). Then on analyzing Eq. (3.10) in the region |r1−r2| ≪¯h/δp(τ) under condition t≪tcoh=M/[˜ksin(θmax/2)δp(τ)] one finds the density matrix to factorize as a product of coherent states φ(r,τ,t) =C1/2(τ)/summationdisplay m≥1γm(r,0)gm gΣeikm·r−it(km)t, (3.11) where C(τ) =/integraldisplay dp|As(p,τ)|2f(p+ ¯hk0). (3.12) While getting the formula (3.11) we allow for small val- ues of phase differences |δφ(r1−rm t)−δφ(r2−rn t)| ≪ π,∀m,n≥1, appearing in the integration region at t≪tcoh, whenceγm(r,κ)≃γm(r,0). Note, compatibil- ity of time conditions tmin< t≪tcohrestrains possible structure of the object beam sin(θmax/2)≪¯h∆k 2L˜kδp(τ). (3.13) So, we see that the superposition of laser beams selec- tively acts only on those wave packets in initial represen- tation of the atomic density matrix Eq. (3.9), whose spec- tra are concentrated near the vector ¯ hk0, and restores a pure state (3.11) in such a way. Therefore the inho- mogeneous laser radiation proves to behave like a three- dimensional hologram in respect to the incident atomic beam (impinging wave packets). 8One can further establish a close relation between an atomic hologram, created in a time domain τand a permanent optical hologram with the thickness dτ= ¯hk0τ/M along direction of the reading beam. Indeed, as is known from optics, the passage of reading beam through a three-dimensional hologram can be interpreted as multiple diffraction in which small waves, diffracted from different registration-media layers with equivalent transmission of light, interfere constructively to form a high intensity of reconstructed wave. The same approach can be used to describe an atom optics hologram being a light structure, inducing an optical potential through the atom-laser dipole interaction [12]. Here the role of equivalent-transmission layers in the media is performed by the equipotential surfaces. Since dτis just the dis- tance the impinging wave packet covers during time τ, the numbers of crossed interfaces (layers or surfaces) are equal for atomic and conventional hologram. Therefore if there were no evident difference in initial and bound- ary conditions, the processes of wave front reconstruction would be identical in both cases. It makes possible to classify atomic holograms as thin or thick diffractive optical elements, and use the Talbot lengthLTalbot, i.e. the typical interval between consecu- tive interfaces, as a characteristic scale to distinguish b e- tween the two classes [17]. Namely the hologram can be considered as thick (three-dimensional) if dτ> LTalbot, or in terms of time τ >L TalbotM/(¯hk0). (3.14) For most of holographic setups (for instance, like that in the Fig. 2)LTalbot∼2π/k0, therefore the criterion (3.14) persists in the time domain τlarge than the period of atomic oscillations. Obviously the latter requirement is well satisfied for τ∼τ1, the time of πpulse, provided gΣ is chosen in agreement with the condition (2.37). B. Diffraction efficiency In a regime, where the background is small, we can define diffraction efficiency ηof a hologram as overall intensity of the modes composing the reconstructed wave, provided the initial wave packet is normalized to 1 η(τ,p0) =/integraldisplay d3p/vextendsingle/vextendsingle/vextendsingleψ(s)(p,τ,t;p0)/vextendsingle/vextendsingle/vextendsingle2 . (3.15) It is clear, however, that ηdepends on the shape of initial distribution as well. Therefore to be more specific let us assume the Gaussian profile (3.1) of impinging wave packet with infinitely small dispersion L→ ∞. Then, integration over p′in the Eq. (3.2) becomes trivial, so that η(τ,p0) =/integraldisplay d3p/vextendsingle/vextendsingle/vextendsingleG(s)(p,p0,τ)/vextendsingle/vextendsingle/vextendsingle2 . (3.16)Using approximate expressions (2.27b) and omitting neg- ligible interference terms one readily gets from above equation η(τ,p0) =η(τ,ξ)≃1 ξ2+ 1sin2/parenleftBig τgΣ/radicalbig ξ2+ 1/parenrightBig ,(3.17) where the dimensionless parameter ξ=(p0−¯hk0)·∆k MgΣ(3.18) characterizes deviation of initial atomic momentum from mean momentum of photons in the reference beam. According to this simple formula the diffraction effi- ciency achieves maximum at τ=τn//radicalbig ξ2+ 1 and can reach 100% if ξ= 0 (see Figure 5). 0 1 2 3 00.511.521 0 1 2 3 FIG. 5. Diffraction efficiency ηof atomic hologram as a function of time domain τ(in units of πpulse) and dimen- sionless parameter ξ. C. Numerical example In the following we show two-dimensional results ob- tained for Na assuming experimental setup like that in the Fig. 2 (i.e., all kmare coplanar vectors and ˜k⊥k0). The image to be reconstructed is a thin line of the width λ= 2π/k0being perpendicular to laser beams plane. To decrease the bulk of computational work we reduced the number of object wave modes to 31 and set up θmax=π/4. Such a field well approaches desired sin- gle line wthin region of size ∼60λ, centered around the pointr=0, if all laser modes going into the expres- sion (2.5) have identical amplitudes Em, and their wave vectors kmare equidistant km=k0/braceleftbigg sin/bracketleftbiggπ(m−16) 120/bracketrightbigg ,−cos/bracketleftbiggπ(m−16) 120/bracketrightbigg ,0/bracerightbigg . (3.19) 9Corresponding profile of the object beam intensity distri- butionI(x) is depicted in Fig. 6, where we direct Carte- sianxaxis along the vector k0= (k0,0,0). The off-axis interference fringes, which are a corollary of moderate number of modes, can easily be separated from the cen- tral line and therefore do not contaminate our consider- ation. -5005010015020025000.20.40.60.81 -4-202400.20.40.60.81 FIG. 6. Intensity I(x) of the 31-mode object wave as a function of observation point r= (x,0,0). The inset shows optical image of a single line ∼λcreated in the central region ∼60λ. In numerical simulation the optical pulse duration was taken to be τ1= 2.82×10−2c, to demonstrate the highest diffraction efficiency. The rest laser light pa- rameters were held as follows: Rabi frequencies Ω 0= 1 MHz and Ω m= 0.01Ω0for all 1 ≤m≤31, detun- ing ∆ = −1 GHz (γ/∆≈0.06), the effective Rabi fre- quenciesgm= 10 Hz,fmn= 0.1 Hz, and gΣ= 55.7 Hz. Note, that for considered laser-beams geometry ∆k=√ 2k0= 1.51×105cm−1andδk= 3.28×103 cm−1, so that the background introduces relative correc- tion of the order εr=δp(τ1)/(¯hδk) = 1.4×10−2and can be neglected. The reconstruction of a real image of the object was achieved by impinging Gaussian wave packets (3.1) hav- ing spatial extension 2 L= 0.4 cm upon the superposition of laser beams near the point r0=/bracketleftbigg −¯hk0τ1 M,L tan(θmax/2)−¯hk0τ1 M,0/bracketrightbigg .(3.20) After finishing the interaction with laser radiation these wave packets appear at a distance L/tan(θmax/2) = 0.48 cm from the image. As a result the most intensive mat- ter field in the imaging region may be observed after free propagation time t=tmaxcos(θmax/2) = 0.16 c, which obviously lies within the limits tmin= 9.6×10−2c and tmax= 0.17 c. Figure 7 shows corresponding atomic den- sity profile ρgg(r,r,τ1,t) when the mean momentum of initial wave packet is exactly equal to ¯ hk0. As it is seen from the bottom part of the plot, the atomic profile dis- plays a good coincidence with distribution of the objectbeam intensity. The diffraction efficiency calculated ac- cording to Eq. (3.15) proves to attain 98% in this case. -2 0 2-10010 00.51 -2 0 2 -20-10010200.20.40.60.81 -4-202400.20.40.60.81 x//c108 y//c108x//c108/c114 /c114( , ) ( , )r□r 0□0/gg gg y//c108 x□=0 y□=0 ) 0 () ( Iy I ) 0 () ( Ix I FIG. 7. Atomic density ρgg(r,r) =ρgg(r,r,τ1,t) as a function of observation point r= (x,y,0). The bottom part of the plot compares atomic profile (solid lines) with the objec t beam intensity distributions I(x) andI(y) (dashed lines) in the planesy= 0 andx= 0 correspondingly. When initial state is a statistical mixture (3.9) with momentum distribution function f(p) being uniform alongxaxis, the atomic density profile acquires a shape represented in the Fig. 8. Since condition (3.13) does not hold at chosen laser light parameters the size of re- constructed line appears to be ∼4 times wider than one might expect from coherent reading beam. Nevertheless, such image broadening is not too substantial, so that the atomic hologram can be used even in this unfavorable design. 10-4 -2 0 2 4-40-2002040 00.51 -4 -2 0 2 4 -40-200204000.20.40.60.81 -4-202400.20.40.60.81 x//c108 y//c108x//c108/c114 /c114( , ) ( , )r□r 0□0/gg gg y//c108 x□=0 y□=0) 0 () ( Iy I ) 0 () ( Ix I FIG. 8. Atomic density profile ρgg(r,r) being obtained when initial state is a statistical mixture with uniform mo- mentum distribution along xaxis. Other notations are the same as in the Fig. 7. IV. CONCLUSIONS In this paper we have studied a method of driving the ultracold atom propagation using effective holograms made of laser radiation in specified time domain. We have shown that scattered atomic wave packet may in- herit the features of object electromagnetic wave pro- vided the atomic internal ground state possesses a trans- lation invariance due to compensation of gravity with the Stern-Gerlach effect. We have established a close relation between atomic hologram created in time domain and thick optical hologram prepared in corresponding spatial region and have found a recipe how to control diffraction efficiency of such atomic hologram by means of varying the time domain. Beside adjustment of atom-laser inter- action time a way to enhance diffraction efficiency has proved to consist of cooling the atomic beam so that all particles would get the same momentum as the momen- tum of photons in the reference wave. An extraordinary role here may be played by BEC and coherent atomic- beam generators, which are under development now [23]. The consideration has been performed for dilute atomic sample, i.e. we have not included any many-atominteractions [24], which may lead to nonlinear atom op- tics effects [25] along with raising of the background. The criteria to neglect these interactions were elaborated in our previous paper [12] using mean-field approximation applied to the Maxwell-Bloch equations [26] and are well satisfied when the mean-field interaction energy per par- ticle is much less than the typical kinetic energy of an atom. We have also neglected such possible sources of the background as spontaneous emission of photons and fluc- tuations of the laser frequency. While the first of these sources may be eliminated by keeping the laser detun- ing much bigger than the spontaneous emission rate, the second one is determined by the spectral width of two- time electromagnetic-field correlation functions [12,27] and substantially decreases if all field modes originate from one initial laser mode. Although our scheme of atomic hologram has been de- veloped for co-directed reading and reference beams it can readily be modified for the experimental setup with opposite propagation of the beams. On full analogy with the conventional optics such a hologram will reconstruct the conjugate object wave. So we see that atom optics holograms appear to be a useful implement for solving some of the basic tech- nological problems in the field of atom lithography. For instance, it will be possible to grow 3D circuitry compo- nents depositing an arbitrary multilayer picture of impu- rity atoms on a silicon substrate. APPENDIX: GROUND-STATE GREEN FUNCTION Here we present the first-order approximation to the ground-state component of the Green function determin- ing time evolution of the atomic density matrix according to the formula (2.7): Ggg(p,p′,t) =e(t)/summationdisplay σ∈{r,s,b}G(σ)(p,p′,t), (A1) where common phase multiplier e(t) = exp[iωt− (i/¯h)E|e/an}bracketri}ht(0)t] recovers the solution (2.26) from its slow time dependence, and G(r)(p,p′,t) =M/bracketleftBig φ(0) 0(p,p′,λ)/bracketrightBig , (A2a) G(s)(p,p′,t) =/summationdisplay m≥1M/bracketleftBig φ(0) m(p,p′,λ)/bracketrightBig , (A2b) G(b)(p,p′,t) =/summationdisplay m≥0M/bracketleftBig φ(b) m(p,p′,λ)/bracketrightBig . (A2c) In these formulas the operator Mstands for inverse Laplace transformation and shift of momentum argu- ments 11M/bracketleftBig φ(σ) m(p,p′,λ)/bracketrightBig ≡2πi/integraldisplayǫ+i∞ ǫ−i∞dλeλt ×φ(σ) m(p−¯hkm,p′,λ),(A3) ǫ>0,σ∈ {0,b},m≥0, whereas φ(0) 0(p,p′,λ) =−i T(p,λ)δ3(p+ ¯hk0−p′),(A4a) φ(b) 0(p,p′,λ) =i T(p,λ)2χ(p,p′,λ), (A4b) φ(σ) m(p,p′,λ) =−gm tm(p)−iλφ(σ) 0(p,p′,λ), (A4c) m≥1, and expression for χ(p,p′,λ) is obtained from the formula (2.21) χ(p,p′,λ) =/summationdisplay m≥1/braceleftbigg/summationdisplay n≥0 n/ne}ationslash=mg∗ mφ(b) n[p+ ¯h(km−kn),p′,λ] +/summationdisplay n≥1gnφ(b) m[p+ ¯h(2k0−kn−km),p′,λ] +/summationdisplay n≥1 n/ne}ationslash=m/summationdisplay l≥0fmn ×φ(b) l[p+ ¯h(k0−kl−km+kn),p′,λ]/bracerightbigg . (A5) [1] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988); J. Lawall, S. Kulin, B. Saubamea, N. Bigelow, M. Leduc, and C. Cohen-Tannoudji, ibid.75, 4194 (1995). [2] M. Kasevich and S. Chu, Phys. 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arXiv:physics/9912055v1 [physics.comp-ph] 30 Dec 1999A Second-Order Stochastic Leap-Frog Algorithm for Multipl icative Noise Brownian Motion Ji Qiang1,⋆and Salman Habib2,† 1LANSCE-1, MS H817, Los Alamos National Laboratory, Los Alam os, NM 87545 2T-8, Theoretical Division, MS B285, Los Alamos National Lab oratory, Los Alamos, NM 87545 (February 21, 2014) A stochastic leap-frog algorithm for the numerical integra tion of Brownian motion stochastic differ- ential equations with multiplicative noise is proposed and tested. The algorithm has a second-order convergence of moments in a finite time interval and requires the sampling of only one uniformly dis- tributed random variable per time step. The noise may be whit e or colored. We apply the algorithm to a study of the approach towards equilibrium of an oscillat or coupled nonlinearly to a heat bath and investigate the effect of the multiplicative noise (aris ing from the nonlinear coupling) on the relaxation time. This allows us to test the regime of validit y of the energy-envelope approximation method. PACS Numbers : 11.15.Pg, 11.30.Qc, 05.70.Ln, 98.80.Cq 02.5 0-r LAUR 99-5263 I. INTRODUCTION Stochastic differential equations with multiplicative noise have not only found many applications in physics but also have interesting mathematical properties. Con- sequently they have attracted substantial attention over the years [1–11]. The key point lies in the fundamen- tal difference between additive and multiplicative noises: Additive noise does not couple directly to the system variables and disappears from the noise-averaged form of the dynamical equations. However, in the case of multi- plicative noise, the system variables do couple directly to the noise (alternatively, we may say that the noise am- plitude depends on the system variables). This fact can lead to dramatic changes of system behavior that cannot occur in the presence of additive noise alone. Two classic illustrations are the Kubo oscillator [12] and the existenc e of long-time tails in transport theory [13]. In this paper we will investigate another example, that of an oscillator nonlinearly coupled to a heat bath, in which the effects of multiplicative noise can significantly alter the qualita - tive nature, as well as the rate [2], of the equilibration process (relative to that of an oscillator subjected only to additive noise). The dynamical behavior of systems subjected to noise can be studied in two different ways: we may either solve stochastic differential equations and average over realiza - tions to obtain statistical information, or we may directly solve the Fokker-Planck equation which describes the evolution of the corresponding probability distribution function. Both approaches have their share of advantages and disadvantages. Fokker-Planck equations are partial differential equations and their mathematical properties are still not fully understood. Moreover, they are very expensive to solve numerically even for dynamical sys- tems possessing only a very modest number of degrees of freedom. Truncation schemes or closures (such as cu- mulant truncations) have had some success in extractingthe behavior of low-order moments, but the systematics of these approximations remains to be elucidated. Com- pared to the Fokker-Planck equation, stochastic differ- ential equations are not difficult to solve, and with the advent of modern supercomputers, it is possible to run very large numbers of realizations in order to compute low-order moments accurately. (We may mention that in applications to field theories it is essentially impossible to solve the corresponding Fokker-Planck equation since the probability distribution is now a functional.) However, the extraction of the probability distribution function it - self is very difficult due to the sampling noise inherent in a particle representation of a smooth distribution. Numerical algorithms to solve stochastic differential equations have been discussed extensively in the litera- ture [14–19]. The simplest, fastest, and still widely-used , is Euler’s method which yields first-order convergence of moments for a finite time interval. Depending on the control over statistical errors arising from the necessari ly finite number of realizations, in the extraction of statis- tical information it may or may not pay to use a higher order algorithm especially if it is computationally expen- sive. Because of this fact, it is rare to find high-order schemes being put to practical use for the solution of stochastic differential equations, and second-order con- vergence is usually considered a good compromise be- tween efficiency and accuracy. A popular algorithm with second-order convergence of moments for additive noise but with only first-order convergence of moments for mul- tiplicative noise is Heun’s algorithm (also called stochas - tic RK2 by some authors) [14,17,20]. A stochastic leap- frog algorithm which has the same order convergence of moments as Heun’s method was suggested in Ref. [21] to study particle motion in a stochastic potential without damping. Several other algorithms for particle motion in a quasi-conservative stochastic system were proposed in Ref. [16] and in the book by Allen and Tildesley [22]. At every time step, these methods all require sampling two 1Gaussian random variables which adds to the computa- tional cost. A modified algorithm suggested in Ref. [19] requires only one Gaussian random variable but applies only to white Gaussian noise. In the following sections, we present a new stochastic leap-frog algorithm for mul- tiplicative Gaussian white noise and Ornstein-Uhlenbeck colored noise which not only has second-order conver- gence of moments but also requires the sampling of only one random uniform variable per time step. The organization of this paper is as follows: General numerical integration of a system of stochastic differ- ential equations with Gaussian white noise is discussed in Section II. The stochastic leap-frog algorithms for Brownian motion with Gaussian white noise and colored Ornstein-Uhlenbeck noise are given in Section III. Nu- merical tests of these algorithms using a one-dimensional harmonic oscillator are presented in Section IV. A phys- ical application of the algorithm to the multiplicative- noise Brownian oscillator is given in Section V. Section VI contains the final conclusions and and a short discussion. II. NUMERICAL INTEGRATION OF STOCHASTIC DIFFERENTIAL EQUATIONS A general system of continuous-time stochastic differ- ential equations (Langevin equations) can be written as ˙xi=Fi(x1,· · ·, xn) +σij(x1,· · ·, xn)ξj(t) (1) where i= 1,· · ·, nandξj(t) is a Gaussian white noise with /an}bracketle{tξj(t)/an}bracketri}ht= 0 (2) /an}bracketle{tξj(t)ξj(t′)/an}bracketri}ht=δ(t−t′) (3) and the symbol /an}bracketle{t· · ·/an}bracketri}htrepresents an average over realiza- tions of the inscribed variable (ensemble average). The noise is said to be additive when σijis not a function of the xi, otherwise it is said to be multiplicative. In the case of multiplicative noises, a mathematical subtlety arises in interpreting stochastic integrals, the so-calle d Ito-Stratonovich ambiguity [23]. It should be stressed that this is a point of mathematics and not of physics. Once it is clear how a particular Langevin equation has been derived and what it is supposed to represent, it should either be free of this ambiguity (as in the case of the example we study later) or it should be clear that there must exist two different stochastic equations, one written in the Ito form, the other in Stratonovich, both representing the same physical process and hence yielding identical answers for the variables of interest. (Another way to state this is that there should be only one unique Fokker-Planck equation.) It is important to note that the vast majority of numerical update schemes for Langevin equations use the Ito form of the equation. The integral representation of the set of equations (1) isxi(t) =xi(0) +/integraldisplayt 0dsFi(x1(s),· · ·, xn(s)) +/integraldisplayt 0dsσij(x1(s),· · ·, xn(s))ξj(s) (4) where xi(0) is a given sharp initial condition at t= 0. The infinitesimal update form of this equation may be derived by replacing twith an infinitesimal time step h: xi(h) =xi(0) +/integraldisplayh 0dt′Fi/bracketleftBigg xk(0) +/integraldisplayt′ 0dsFk(x(s)) +/integraldisplayt′ 0dsσkl(x(s))ξl(s)/bracketrightBigg +/integraldisplayh 0dt′σij/bracketleftBigg xk(0) +/integraldisplayt′ 0dsFk(x(s)) +/integraldisplayt′ 0dsσkl(x(s))ξl(s)/bracketrightBigg ξj(t′) (5) Since Fiandσijare smooth functions of the xi, they may be expanded about their values at t= 0, in which case we can write the exact solution for xi(h) as xi(h) =Di(h) +Si(h) (6) where Di(h) and Si(h) denote the deterministic and stochastic contributions respectively. The deterministi c contribution Di(h) is Di(h) =xi(0) +hFi+1 2h2Fi,kFk+O(h3) (7) where Fi,k=∂Fi/∂xk, the summation convention for the repeated indices having being employed. The stochastic contribution Si(h) is Si(h) =σijWj(h) +σij,kσklClj(h) +Fi,kσklZl(h) +σij,kFk(hWj(h)−Zj(h)) +1 2σij,klσkmσlnHmnj(h) +1 2Fi,klσksσltGst(h) +1 2Fkσij,klσlmKmj(h) +1 2Flσij,klσkmKmj(h) +1 6σij,klmσknσloσmpInopj +O(h5/2) (8) The quantities Wi,Cij,Hijk,Zi,Gij,Kij, and Iijkl are random variables which can be written as stochastic integrals over the Gaussian white noise ξ(t): Wi(h) =/integraldisplayh 0dtξi(t)∼O(h1/2) (9) Cij(h) =/integraldisplayh 0dtW i(t)ξj(t)∼O(h) (10) Hijk(h) =/integraldisplayh 0dtW i(t)Wj(t)ξk(t)∼O(h3/2) (11) 2Zi(h) =/integraldisplayh 0dtW i(t)∼O(h3/2) (12) Gij(h) =/integraldisplayh 0dtW i(t)Wj(t)∼O(h2) (13) Kij(h) =/integraldisplayh 0tdtW i(t)ξj(t)∼O(h2) (14) Iijkl(h) =/integraldisplayh 0dtW i(t)Wj(t)Wk(t)ξl(t)∼O(h2) (15) Ito integration has been employed in the derivation of the above equations. Thenth moment of the xiis /an}bracketle{txi(h)n/an}bracketri}ht=/an}bracketle{t(Di(h) +Si(h))n/an}bracketri}ht =Di(h)n+nDi(h)n−1/an}bracketle{tSi(h)/an}bracketri}ht +C2 nDi(h)n−2/an}bracketle{t(Si(h))2/an}bracketri}ht+· · · (16) where Ci n=/parenleftbigg i n/parenrightbigg =n! i!(n−i)!(17) and /an}bracketle{tSi(h)/an}bracketri}ht=1 4Fi ,klσksσlsh2+O(h3) (18) /an}bracketle{tSi(h)Sj(h)/an}bracketri}ht=σilσjlh+1 2σim ,kσklσjm ,pσplh2 +1 2σilFj ,kσklh2+1 2σjlFi ,kσklh2 +1 2σilσjl ,kFkh2+1 2σjlσil ,kFkh2 +1 4σipσjp ,klσkmσlmh2 +1 4σjpσip ,klσkmσlmh2+O(h3) (19) /an}bracketle{tSi(h)Sj(h)Sk(h)/an}bracketri}ht=O(h3) (20) /an}bracketle{tSi(h)4/an}bracketri}ht= 3(σii)4+O(h3) (21) /an}bracketle{t(Si(h))5/an}bracketri}ht=O(h3) (22) Suppose that the results from a numerical algorithm were written as ¯xi(h) =¯Di(h) +¯Si(h) (23) where the ¯ xiare approximations to the exact solutions xi. The nth moment of ¯ xiis /an}bracketle{t¯xi(h)n/an}bracketri}ht=/an}bracketle{t(¯Di(h) +¯Si(h))n/an}bracketri}ht =¯Di(h)n+n¯Di(h)n−1/an}bracketle{t¯Si(h)/an}bracketri}ht +C2 n¯Di(h)n−2/an}bracketle{t(¯Si(h))2/an}bracketri}ht+· · · (24) Comparing Eqns. (16) and (24), we see that if Di(h) and ¯Di(h), and Si(h) and ¯Si(h) coincide up to h2, we will have xi(h)−¯xi(h) =O(h3) (25) and for a finite time interval /an}bracketle{txi(t)n/an}bracketri}ht − /an}bracketle{t¯xi(t))n/an}bracketri}ht=O(h2) (26)III. STOCHASTIC LEAP-FROG ALGORITHM FOR BROWNIAN MOTION The approach to modeling Brownian motion that we consider here is that of a particle coupled to the environ- ment through its position variable [1]. When this is the case, noise terms enter only in the dynamical equations for the particle momenta. In the case of three dimensions, the dynamical equations take the general form: ˙x1=F1(x1, x2, x3, x4, x5, x6) +σ11(x2, x4, x6)ξ1(t) ˙x2=F2(x1) ˙x3=F3(x1, x2, x3, x4, x5, x6) +σ33(x2, x4, x6)ξ3(t) ˙x4=F4(x3) ˙x5=F5(x1, x2, x3, x4, x5, x6) +σ55(x2, x4, x6)ξ5(t) ˙x6=F6(x5) (27) The convention used here is that the odd indices corre- spond to momenta, and the even indices to the spatial coordinate. In the dynamical equations for the momenta, the first term on the right hand side is a systematic drift term which includes the effects due to external forces and damping. The second term is stochastic in nature and describes a noise force which, in general, is a function of position. The noise ξ(t) is first assumed to be Gaussian and white as defined by Eqns. (2)-(3). The stochastic leap-frog algorithm for the Eqns. (27) is written as ¯xi(h) =¯Di(h) +¯Si(h) (28) The deterministic contribution ¯Di(h) can be obtained us- ing the deterministic leap-frog algorithm. The stochastic contribution ¯Si(h) can be obtained by applying Eq. (8) on Eq. (27). The stochastic integration defined by Eqs. (9) to (15) can be approximated so that the moment rela- tionships defined by Eqs. (18) to (22) are satisfied. After some calculation, the deterministic contribution ¯Di(h) and the stochastic contribution ¯Si(h) of the above recur- sion formula for one-step integration are found to be ¯Di(h) = ¯xi(0) +hFi(¯x∗ 1,¯x∗ 2,¯x∗ 3,¯x∗ 4,¯x∗ 5,¯x∗ 6); {i= 1,3,5} ¯Di(h) = ¯x∗ i +1 2hFi[xi−1+hFi−1(¯x∗ 1,¯x∗ 2,¯x∗ 3,¯x∗ 4,¯x∗ 5,¯x∗ 6)] ; {i= 2,4,6} ¯Si(h) =σii√ hWi(h) +1 2Fi,kσkkh3/2˜Wi(h) +1 2σii,jFjh3/2˜Wi(h) +1 4Fi,klσkkσllh2˜Wi(h)˜Wi(h); {i= 1,3,5;j= 2,4,6;k, l= 1,3,5} ¯Si(h) =1√ 3Fi,jσjjh3/2˜Wj(h) 3+1 4Fi,jjσ2 jjh2˜Wj(h)˜Wj(h) {i= 2,4,6;j= 1,3,5} ¯x∗ i= ¯xi(0) +1 2hFi(¯x1,¯x2,¯x3,¯x4,¯x5,¯x6) {i= 1,2,3,4,5,6} (29) where ˜Wi(h) is a series of random numbers with the mo- ments /an}bracketle{t˜Wi(h)/an}bracketri}ht=/an}bracketle{t(˜Wi(h))3/an}bracketri}ht=/an}bracketle{t(˜Wi(h))5/an}bracketri}ht= 0 (30) /an}bracketle{t(˜Wi(h))2/an}bracketri}ht= 1,/an}bracketle{t(˜Wi(h))4/an}bracketri}ht= 3 (31) This can not only be achieved by choosing true Gaus- sian random numbers, but also by using the sequence of random numbers following: ˜Wi(h) =  −√ 3, R < 1/6 0,1/6≤R <5/6√ 3, 5/6≤R(32) where Ris a uniformly distributed random number on the interval (0,1). This trick significantly reduces the computational cost in generating random numbers. Next we consider the case that the noise in Eqs. (27) is a colored Ornstein-Uhlenbeck process which obeys /an}bracketle{tξi(t)/an}bracketri}ht= 0 (33) /an}bracketle{tξi(t)ξi(t′)/an}bracketri}ht=ki 2exp(−ki|t−t′|) (34) where the correlation factor kiis the reciprocal of the correlation time. In the limit of ki→ ∞, the Ornstein- Uhlenbeck process reduces to Gaussian white noise. The above process can be generated by using a white Gaussian noise from a stochastic differential equation ˙ξi(t) =−kiξi(t) +kiζi(t) (35) where ζi(t) is a standard Gaussian white noise. The ini- tial value ξi(0) is chosen to be a Gaussian random number with/an}bracketle{tξi(0)/an}bracketri}ht= 0 and /an}bracketle{tξi(0)2/an}bracketri}ht=ki/2. For the stochastic process with colored noise, the leap- frog algorithm for Eqns. (27) is of the same form as that for white noise (Cf. Eqn. (29)), but with ¯Di(h) = ¯xi(0) +hFi(¯x∗ 1,¯x∗ 2,¯x∗ 3,¯x∗ 4,¯x∗ 5,¯x∗ 6) +hσii(¯x∗ 2,¯x∗ 4,¯x∗ 6)ξ∗ i; {i= 1,3,5} ¯Di(h) = ¯x∗ i +1 2hFi[¯xi−1+hFi−1(¯x∗ 1,¯x2∗,¯x∗ 3,¯x4∗,¯x∗ 5,¯x6∗) +hσi−1i−1(¯x∗ 2,¯x∗ 4,¯x∗ 6)ξ∗ i−1/bracketrightbig ; {i= 2,4,6} ¯Dξi(h) =ξi(0)exp( −kih); {i= 1,3,5}¯Si(h) =1√ 3σii(¯x2,¯x4,¯x6)kih3/2˜Wi(h); {i= 1,3,5} ¯Si(h) = 0; {i= 2,4,6} ¯Sξi=ki√ h˜Wi(h)−1 2k2 ih3/2˜Wi(h); {i= 1,3,5} (36) where ¯x∗ i= ¯xi(0) +1 2h(Fi(¯x1,¯x2,¯x3,¯x4,¯x5,¯x6) +σii(¯x2,¯x4,¯x6)ξi; {i= 1,3,5} ¯x∗ i= ¯xi(0) +1 2hFi(¯x1,¯x2,¯x3,¯x4,¯x5,¯x6); {i= 2,4,6} ξ∗ i=ξi(0)exp( −1 2kih); {i= 1,3,5} (37) 2.082.12.122.142.162.182.22.222.24 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 h<x^2> 2.062.0652.072.0752.082.0852.092.095 0 0.05 0.1 0.15 0.2 0.25 0.3 h<x^2> FIG. 1. Zero damping convergence test. Top: /angbracketleftx2(t)/angbracketrightat t= 6 as a function of step size with white Gaussian noise. Bottom: /angbracketleftx2(t)/angbracketrightatt= 6 as a function of step size with colored Ornstein-Uhlenbeck noise. Solid lines represent quadrati c fits to the data points (diamonds). IV. NUMERICAL TESTS The above algorithms were tested on a one-dimensional stochastic harmonic oscillator with a simple form of the multiplicative noise. The equations of motion were 4˙p=F1(p, x) +σ(x)ξ(t) ˙x=p (38) where F1(p, x) =−γp−η2xandσ(x) =−αx. As a first test, we computed /an}bracketle{tx2/an}bracketri}htas a function of time step size. To begin, we took the case of zero damping constant ( γ= 0), where /an}bracketle{tx2/an}bracketri}htcan be determined analyt- ically. The top curve in Fig. 1 shows /an}bracketle{tx2/an}bracketri}htatt= 6.0 as a function of time step size with white Gaussian noise. Here, the parameters ηandαare set to 1 .0 and 0 .1. The ensemble averages were taken over 106independent simulations. The analytically determined value of /an}bracketle{tx2/an}bracketri}ht att= 6.0 is 2.095222 (The derivation of the analytical results is given in the Appendix). The quadratic con- vergence of the stochastic leap-frog algorithm is clearly seen in the numerical results. We then considered the case of colored Ornstein-Uhlenbeck noise as a function of time step size using the same parameters as in the white Gaussian noise case and with the correlation parameter k= 0.16. The result is shown as the bottom curve in Fig. 1 and the quadratic convergence is again apparent. 0.460.470.480.490.50.510.520.530.54 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 h<x^2> 0.4380.440.4420.4440.4460.4480.450.4520.454 0 0.05 0.1 0.15 0.2 0.25 0.3 h<x^2> FIG. 2. Finite damping ( γ= 0.1) convergence test. Top: /angbracketleftx2(t)/angbracketrightatt= 12 as a function of step size with white Gaussian noise. Bottom: /angbracketleftx2(t)/angbracketrightatt= 12 as a function of step size with colored Ornstein-Uhlenbeck noise. Solid lines repres ent quadratic fits to the data points (diamonds). We verified that the quadratic convergence is present for nonzero damping ( γ= 0.1). At t= 12.0, and with all other parameters as above, the convergence of /an}bracketle{tx2/an}bracketri}ht as a function of time step is shown by the top and bot- tom curves in Fig. 2 (white Gaussian noise and colored Ornstein-Uhlenbeck noise, respectively). As a comparison against the conventional Heun’s al-gorithm, we computed /an}bracketle{tx2/an}bracketri}htas a function of tusing 100,000 numerical realizations for a particle starting from (0 .0,1.5) in the ( x, p) phase space. The results along with the analytical solution and a numerical solution us- ing Heun’s algorithm are given in Fig. 3. Parameters used were h= 0.1,η= 1.0, and α= 0.1. The ad- vantage in accuracy of the stochastic leap-frog algorithm over Heun’s algorithm is clearly displayed, both in terms of error amplitude and lack of a systematic drift. We note that while in general Heun’s algorithm is only linear for multiplicative noise applications, for the part ic- ular problem at hand it turns out to be quadratic. This is due to a coincidence: the stochastic term of xdoes not contain W(h) but does posses a higher order term hW(h). However, this higher order term has a larger coefficient compared with our stochastic leap-frog algo- rithm, and this accounts for the larger errors observed in Fig. 3. -204812 0 100 200 300 400 500 t<x^2>(t)Exact Error: Heun Error: Leapfrog FIG. 3. Comparing stochastic leap-frog and the Heun al- gorithm: /angbracketleftx2(t)/angbracketrightas a function of t. Errors are given relative to the exact solution. V. A PHYSICAL APPLICATION: THE MECHANICAL OSCILLATOR In this section, we apply our algorithm to studying the approach to thermal equilibrium of an oscillator coupled nonlinearly to a heat bath modeled by a set of noninter- acting harmonic oscillators [1]. The nonlinear coupling leads to the introduction of multiplicative noise into the system dynamics. Lindenberg and Seshadri have pointed out that, at weak coupling, multiplicative noise may sig- nificantly enhance the equilibration rate relative to the rate for weak linear coupling (additive noise) [2]. We will choose the same form of the coordinate couplings as in Ref. [2], in which case the additive noise equations are ˙p=−ω2 0x−λ0p+/radicalbig 2D0ξ0(t) ˙x=p (39) 5and for the system with multiplicative noise only: ˙p=−ω2 0x−λ2x2p−/radicalbig 2D0xξ2(t) ˙x=p (40) where the diffusion coefficients Di=λikT, i = 0,2,λiis the coupling constant, kis Boltzmann’s constant, Tis the heat bath temperature, and ω0is the oscillator angular frequency without damping. The approach to thermal equilibrium is guaranteed for both sorts of noises by the fluctuation-dissipation relation /an}bracketle{tξi(t)ξj(s)/an}bracketri}ht=δijδ(t−s) (41) written here for the general case when both noises are simultaneously present. While in all cases, it is clear that the final distribution is identical and has to be the thermal distribution, the precise nature of the approach to equilibrium can certainly be different. We wish to explore this issue in more detail. An important point to keep in mind is that in this particular system of equations there is no noise-induced drift in the Fokker-Planck equa- tion obtained from the Stratonovich form of the Langevin equation, i.e., there is no Ito-Stratonovich ambiguity. It is a simple matter to solve the Langevin equations given above applying the algorithm from Eqs. (29). As our primary diagnostic, we computed the noise-averaged energy /an}bracketle{tE(t)/an}bracketri}htof the oscillator as a function of time t, where E(t) =1 2p2+1 2ω2 0x2. (42) In the weak coupling limit and employing orbit-averaging (valid presumably when the dynamical time scale is much smaller than the relaxation time scale), one finds [2] /an}bracketle{tE(t)/an}bracketri}ht=kT−(kT−E0)e−λ0t(43) in the case of additive noise (a result which can also be directly obtained as a limiting case from the known form of the exact solution given, e.g., in Ref. [24]). The corre- sponding form of the approximate solution in the case of multiplicative noise is /an}bracketle{tE(t)/an}bracketri}ht=E0kT E0+ (kT−E0)exp(−λ2kTt/ω2 0).(44) While in the case of additive noise, the exponential na- ture of the relaxation is already clear from the form of the exact solution (cf. Ref. [24]), the situation in the case of multiplicative noise is not obviously apparent as no exact solution is known to exist. The prediction of a relaxation process controlled by a single exponential as found in (44) is a consequence of the assumption /an}bracketle{tx2(t)/an}bracketri}ht ≃kT/ω2 0at “late” times, this implying a constant damping coefficient in the Langevin equation (40). The timescale separations necessary for the energy- envelope method to be applicable are encoded in the fol- lowing inequalities [2]:λ0 ω0≪1; additive noise (45) kTλ2 ω3 0≪1; multiplicative noise (46) As a first check, we performed simulations with ω0= 1.0, λ0=λ2= 0.01, and kT= 4.5, in which case both the above conditions are satisfied. Moreover, with these choices of parameter values, and within the energy en- velope approximation, the relaxation time predicted for multiplicative noise is substantially smaller than for the case of additive noise. At the same time we also ran a simulation at kT= 200 to see how the energy envelope approximation for multiplicative noise breaks down at high temperatures. 00.20.40.60.811.2 0 50 100 150 200 250 300 350 400 t<E(t)>/kTI II III FIG. 4. Temporal evolution of the scaled average energy /angbracketleftE(t)/angbracketright/kTwith additive noise and multiplicative noise. The dashed lines I and II are the predictions from Eqn. (44) for kT= 200 and kT= 4.5 respectively. The dashed line III is the theoretical prediction for additive noise with kT= 4.5. As predicted, the relaxation proceeds much faster with mult i- plicative noise: The solid lines are numerical results for m ul- tiplicative noise at kT= 200 and kT= 4.5. It is clear that at higher temperatures, the theory grossly underestimates th e relaxation time. In Fig. 4, we display the time evolution of the aver- age energy (scaled by kTfor convenience) with additive and multiplicative noise both from the simulations and the approximate analytical calculations. In the case of weak coupling to the environment (small λ0, λ2), the rate at which the average energy approaches equilibrium is significantly greater for the case of multiplicative nois e relative to the case of additive noise more or less as ex- pected. In addition, the analytic approximation result- ing from the application of the energy-envelope method (44) is seen to be in reasonable agreement with the nu- merical simulations for kT= 4.5. The slightly higher equlibration rate from the analytical calculation is due to the truncation in the energy envelope equation using the /an}bracketle{tE2(t)/an}bracketri}ht ≈2/an}bracketle{tE(t)/an}bracketri}ht2relation which yields an upper bound 6on the rate of equilibration of the average energy [2]. Note that in the case of high temperature ( kT= 200) the relaxaton time computed from the energy envelope method is much smaller than the numerical result, con- sistent with the violation of the condition (46). While the results shown in Fig. 4 do show that the energy envelope approximation is qualitatively correct within its putative domain of validity, it is clear that the actual relaxation process is not of the precise form (44). In Fig. 5 we illustrate this point by plotting E0(kT− /an}bracketle{tE(t)/an}bracketri}ht) /an}bracketle{tE(t)/an}bracketri}ht(kT−E0)= exp( −λ2kTt/ω2 0) (47) [equivalent to (44)] against time on a log scale: the re- laxation is clearly nonexponential. The reason for the failure of the approximation is that despite the fact that equipartition of energy does take place on a relatively short time scale, it is not true that /an}bracketle{tx2(t)/an}bracketri}htcan be treated as a constant even at relatively late times. 0.0010.010.11 0 50 100 150 200 250 300 tAdditive Noise Multiplicative Noise FIG. 5. The LHS of (47) as a function of time (straight line) compared with numerical results for kT= 4.5. Also shown is a numerical result for the case of additive noise which is i n excellent agreement with the predicted exponential relaxa tion with the relaxation timescale = 1 /λ0. VI. CONCLUSIONS We have presented a stochastic leap-frog algorithm for single particle Brownian motion with multiplicative noise. This method has the advantages of retaining the symplectic property in the deterministic limit, ease of im- plementation, and second-order convergence of moments for multiplicative noise. Sampling a uniform distribution instead of a Gaussian distribution helps to significantly reduce the computational cost. A comparison with the conventional Heun’s algorithm highlights the gain in acu- racy due to the new method. Finally, we have applied the stochastic leap-frog algorithm to a nonlinearly coupledoscillator-heat-bath system in order to investigate the ef - fect of multiplicative noise on the nature of the relaxation process. VII. ACKNOWLEDGMENTS We acknowledge helpful discussions with Grant Lythe and Robert Ryne. Partial support for this work came from the DOE Grand Challenge in Computational Ac- celerator Physics. Numerical simulations were performed on the SGI Origin2000 systems at the Advanced Com- puting Laboratory (ACL) at Los Alamos National Lab- oratory, and on the Cray T3E at the National En- ergy Research Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory. ⋆Electronic address: jiqiang@lanl.gov †Electronic address: habib@lanl.gov [1] R. 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Risken, The Fokker-Planck Equation: Methods of So- lution and Applications (Springer, New York, 1989). APPENDIX A: The analytic solution of Eqns. (38) for /an}bracketle{tx2(t)/an}bracketri}ht(with white Gaussian noise) as a function of time in the special case of zero damping, i.e. γ= 0, can be obtained by solving the equivalent Fokker-Planck equation [24] for the probability density f(x, p, t): ∂ ∂tf(x, p, t) = /bracketleftbigg −p∂ ∂x−∂F1(p, x) ∂p+1 2σ2(x)∂2 ∂p2/bracketrightbigg f(x, p, t) (A1) The expectation value of any function M(x, p;t) can be written as /an}bracketle{tM(x, p)/an}bracketri}ht=/integraldisplay+∞ −∞dxdpM (x, p)f(x, p, t) (A2) Equations (A1) and (A2) can be used to yield a BBGKY- like heirarchy for the evolution of phase space moments. Since the system we are considering is linear, this heirar- chy truncates exactly and yields a group of coupled lin- ear ordinary differential equations for the moments /an}bracketle{tx2/an}bracketri}ht, /an}bracketle{txp/an}bracketri}ht, and/an}bracketle{tp2/an}bracketri}ht. These equations can be written as a single third-order time evolution equation for /an}bracketle{tx2/an}bracketri}ht: d3/an}bracketle{tx2/an}bracketri}ht dt3=−4η2d/an}bracketle{tx2/an}bracketri}ht dt+ 2α2/an}bracketle{tx2/an}bracketri}ht (A3) subject to the initial conditions /an}bracketle{tx2(0)/an}bracketri}ht=x2(0) /an}bracketle{t˙x2(0)/an}bracketri}ht= 2x(0)p(0) /an}bracketle{t¨x2(0)/an}bracketri}ht= 2p2(0)−2η2x2(0) (A4) This equation has an analytical solution written as /an}bracketle{tx2(t)/an}bracketri}ht=c1exp(r1t) +c2exp(r2t) +c3exp(r3t) (A5) where c1,c2, and c3are constants depending on initial conditions, and r1,r2andr3are the roots of a third order alegbraic equation2α2−4η2x−x3= 0 (A6) which gives r1=/parenleftBig/radicalbig 64/27η6+α4+α2/parenrightBig1/3 −/parenleftBig/radicalbig 64/27η6+α4−α2/parenrightBig1/3 r2=1 2(1 +√ 3i)/parenleftBig/radicalbig 64/27η6+α4−α2/parenrightBig1/3 −1 2(1−√ 3i)/parenleftBig/radicalbig 64/27η6+α4+α2/parenrightBig1/3 r3=r∗ 2 (A7) where the superscript ∗represents complex conjugation. The positive real root r1implies that /an}bracketle{tx2(t)/an}bracketri}htwill have an exponential growth in time. 8

arXiv:physics/9912056v1 [physics.atom-ph] 31 Dec 1999Analytical Structure Matching and Very Precise Approach to the Coulombic Quantum Three-Body Problem TAN, Shi-Na∗ Institute of Theoretical Physics, CAS, P.O.Box 2735, Beiji ng 100080, P.R.China Abstract A powerful approach to solve the Coulombic quantum three-bo dy problem is proposed. The approach is exponentially convergent and more efficient than the Hyperspherical Coordinate(HC) method and the Correlation Function Hyperspherical Harmon ic(CFHH) method. This approach is numerically competitive with the variational methods, suc h as that using the Hylleraas-type basis functions. Numerical comparisons are made to demonstrate t hem, by calculating the non-relativistic & infinite-nuclear-mass limit of the ground state energy of t he helium atom. The exponentially convergency of this approach is due to the full matching betw een the analytical structure of the basis functions that I use and the true wave function. This fu ll matching was not reached by almost any other methods. For example, the variational method usin g the Hylleraas-type basis does not reflects the logarithmic singularity of the true wave functi on at the origin as predicted by Bartlett and Fock. Two important approaches are proposed in this work to reach this full matching: the coordinate transformation method and the asymptotic serie s method. Besides these, this work makes use of the least square method to substitute complicat ed numerical integrations in solving the Schr¨ odinger equation, without much loss of accuracy; t his method is routinely used by people to fit a theoretical curve with discrete experimental data, but I use it here to simplify the computation. PACS number(s): 1 INTRODUCTION Most approximate methods to solve a linear partial different ial equation, such as the stationary state Schr¨ odinger equation, are actually to choose an N-dimensional subspace of the infinite-dimensional Hilbert space and then to reduce the partial differential equ ation toNlinear algebraic equations defined in this subspace. The efficiency of this kind of methods is mainly determined by whether one ∗E-mail: tansn@itp.ac.cncan use sufficient small Nto reach sufficient high accuracy, i.e., make the vector most c lose to the true solution in this subspace sufficiently close to the true solut ion while keeping the dimension Nnot too large to handle. Most methods to solve the Coulombic quantum three-body prob lem belong to this class, except for some variational methods that make use of some non-linear va riational parameters. The differences between different methods of this kind mainly lie in different choices of the subspaces of the Hilbert space, i.e., different choices of the basis functions to expa nd the wave function. Theoretically, any discrete and complete set of basis funct ions may be used to expand the wave function, and the convergency is easy to fulfilled. But actua lly, the convergency is often slow and makes sufficient accuracy difficult to achieve. The naive hyper spherical harmonic function method[1- 3] in solving the Coulombic quantum three-body problem is su ch an example–this slow convergency can be illustrated by an analogous and simple example: to exp and the function f(x) =√ 1−x2 (−1≤x≤+1) as a series of the Legendre polynomials of x. This series is convergent like N−s, where sis a positive constant not large and Nis the number of Legendre polynomials involved. The reason for this slow convergency is that f(x) is singular at x=±1 but the Legendre polynomials of xare not. I call this the mismatching between the analytical structur es of the basis functions (the polynomials ofx) andf(x). The correlation function hyperspherical harmonic(CFHH) m ethod[4] were proposed to overcome this difficulty. The spirit of this method can be simply illust rated, still using the above example: to dividef(x) by an appropriately selected function(called the correla tion function) to cancel the low order singularities of f(x) atx=±1, then to expand the remaining function by the Legendre polynomials of x. This time, the series is still convergent as N−s, butsis increased by an amount depending on how many orders’ singularities have been cance led. From this simple discussion one can see that the singulariti es of the function f(x) are not completely canceled by the correlation function, although more sophis ticated correlation function can cancel more orders’ singularities. A very simple approach to totally eliminate the singularity is to make an appropriate coordin ate transformation, and in the same time thoroughly give up the o riginal hyperspherical harmonic function method, not just repair it. For example, for f(x) =√ 1−x2, one may write x= sinθ, where −π/2≤θ≤π/2, thenf(x) = cosθand one can expand f(x) as the series about the Legendre polynomials of (2 /π)θ. This time the series is factorially convergent. The reason is that the analytical structures of f(x) andPl((2/π)θ) match–they are both analytical functions on the whole comp lex plane ofθ. Another useful approach to solve this problem is to use the as ymptotic series. Still considering theexamplef(x) =√ 1−x2, one may write the Taylor series f(x) =f0+f1x+f2x2+f3x3+···. Of course, this series is slowly convergent near x=±1. But one can use the following asymptotic series to calculate fnwhennis large: fn= ((−1)n+ 1)(c3/2n−3/2+c5/2n−5/2+c7/2n−7/2+···), or, equivalently, fn= ((−1)n+ 1)1/2+L/summationdisplay s=1/2˜fss! n!(s−n)!, wheres=1 2,3 2,5 2,···,1 2+L, ands!≡Γ(s+1). For a given n≫1, the error of this formula is minimized whenL/n≃2/3, and the minimum error is about/radicalBig 27 2π2n−23−n, which exponentially decreases with nincreasing. Using such kind of asymptotic formulae to calcu late the high order coefficients of the Taylor series, one can expand the singular function f(x) at high precision, with only finite linear parameters, f0,···,fnand˜f1/2,···,˜f1/2+L. Now I introduce an alternative approach to reduce a different ial equation to a given finite di- mensional subspace LNof tbe Hilbert space. Here Nis the dimension of the subspace. The central problem is how to reduce an operator Oin the Hilbert space, e.g., the kinetic energy operator or the potential energy operator, to an N×Nmatrix in the given subspace. For a state Ψ ∈LN, the state Ψ O≡OΨ usually/∈LN. To reduce Ointo anN×Nmatrix means to find a state Ξ ∈LNto approximate Ψ O. The usual approach to select Ξ is to minimize (Ξ−ΨO,Ξ−ΨO), where (,) is the innerproduct of the Hilbert space. This approach wil l reduceOto a matrix with elements Oij= (φi,Oφ j), whereφi∈LNis a set of orthonormal basis in LN, satisfying ( φi,φj) =δij, 1≤i,j≤N. In numerical calculation, the innerproduct is usually computed by numer ical integration, which needs sufficient accuracy and might be complicated. An alternative approach that does not need these integrations is to write the states as wavefunctions under a particular re presentation(e.g., the space-coordinate representation), and then select Ξ to minimize /summationdisplay a[Ξ(xa)−ΨO(xa)]2, wherexais some sample points in the defining area of the wavefunction s. In order to ensure Ξ to be a good approximation of Ψ O, the sample points should be appropriately chosen. Usually the number of the sample points is greater than and approximately proport ional toN, and the separation betweentwo neighboring sample points should be less than the least q uasi-semiwavelength of a wavefunction inLN. This alternative approach (I call it the least square method ) leads to a reduction of the operator O: ˜Oij= (˜φi,O˜φj)′, where (,)′is a pseudo-innerproduct defined as ( φ,ψ)′≡/summationtext a[φ(xa)−ψ(xa)]2for arbitrary φandψ, and ˜φiis a set of pseudo-orthonormal basis in LNsatisfying ( φi,φj)′=δij. We find that this approach is very similar to the usual one, except that a discrete sum ov er sample points takes the place of the usual innerproduct integration. And there is a great degree of freedom in the selection of the sample points. In fact, as soon as the sample points are selected acc ording to the spirit mentioned above, the accuracy of the solution of the differential equation usuall y will not decrease significantly. The major factor that determines the accuracy of the solution is the ch oice of the subspace LN, which has been discussed to some extent in previous pages. In this work, solving the simpliest quantum three-body prob lem, the three methods discussed above are all used: the coordinate transformation method, t he asymptotic series method, and the least square method. A high precision is reached for the grou nd state energy of the ideal helium atom, and the solution has also some merit in comparison with the Hy leraas-type variational solution[5,6]. In section 2 the Bartlett-Fock expansion[7,8,9] is studied , in order to reflect the analytical structure of the wavefunction near the origin. In this study, the asympto tic series are used to represent the hyper- angular dependence of the wavefunction. In section 3 the ( u,w) coordinate system is used to study the hyper-angular dependence of the wavefunction. This coordi nate system cancels the singularity of the hyper-angular functions totally. The relationship betwee n this coordinate system and the Hyleraas- type variational method is also discussed. The least square method is used to reduce the hyper-angular parts of the kinetic energy operator and the potential energ y operator to finite-dimensional matrices. In section 4 the connection of the outer region solution and t he inner region Bartlett-Fock expansion is studied, using the least square method. In section 5 the nu merical result is presented and compared with those of other methods. Some explanations are made. In s ection 6 some discussions are presented and some future developments are pointed out. 2 BARTLETT-FOCK EXPANSION Considering an S state of an ideal helium atom, that is, assum ing an infinite massive nucleus and infinite light speed, one may write the Schr¨ odinger equatio n −2t(∂2 x+∂2 y+∂2 z+1 z∂z)ψ+Vψ=Eψ , (1)wherex=r2 1−r2 2,y= 2r1r2cosθ12,z= 2r1r2sinθ12, andt=r2 1+r2 2=/radicalbig x2+y2+z2.r1 andr2are the distances of the electrons from the nucleus, and θ12is the angle formed by the two electronic position vectors measured from the nucleus. In t his equation, an S state is assumed, so the wavefunction ψis only dependent on r1,r2andθ12, or, equivalently, x,y, andz. The atomic unit, i.e., ¯h=me=e2/(4πε0) = 1, is assumed throughout this paper. The potential energy is V=−2 r1−2 r2+1 r12, (2) wherer12is the distance between the two electrons. r1=/radicalbiggt+x 2, r2=/radicalbiggt−x 2, r12=√t−y . (3) The Bartlett-Fock expansion is ψ=/summationdisplay n,kψn,ktn(lnt)k k!, (4) wheren= 0,1/2,1,3/2,2,···, andk= 0,1,2,···.ψn,konly depends on the two hyper-angles, say, α≡x/tandβ≡y/t, and does not depend on the hyper-radius, ρ≡√ t. Whenk>n,ψn,k≡0. Using the coordinates t,α, andβ, one may rewrite the Schr¨ odinger equation (1) as (∂2 t+3 t∂t+1 t2L0)ψ= (vt−3/2+pt−1)ψ , (5) wherep≡−E/2, and L0= (1−α2)∂2 α−2αβ∂α∂β+ (1−β2)∂2 β−3α∂α−3β∂β; (6) v=−√ 2√1 +α−√ 2√1−α+1/2√1−β. (7) Substituting eq.(4) into eq.(5), and comparing the corresp onding coefficients before tn(lnt)k, one will obtain Lnψn,k+ (2n+ 2)ψn,k+1+ψn,k+2=vψn−1 2,k+pψn−1,k, (8) whereLn≡n(n+ 2) +L0. The functions ψn,kare solved out in the order with n increasing; and for each n, w ith k decreasing. The physical area of ( α,β) is the unit circle: α2+β2≤1. And the function ψn,k(α,β) may has singularities at α=±1 and atβ= 1. The singularities are of these kinds: (1 −α)s, (1 +α)s, and (1−β)s, withs=1 2,3 2,5 2,···. So one may write the Taylor series in the ( α,β) unit circle: ψn,k(α,β) =∞/summationdisplay a,b=0ψn,k,a,bαaβb. (9) The singularities make the usual cutoff, a+b≤Lf+Ls, inappropriate, because the error decreases slowly when Lf+Lsincreases. But since we have known the forms of the singulari ties, we canwrite the asymptotic formulae to calculate those high order Taylor coefficients that have important contributions: ψn,k,a,b =Li−1 2/summationdisplay s=1 2˜ψn,k,b,s/parenleftBigg s a/parenrightBigg [1 + (−1)a] ; (10 −1) ψn,k,a,b =Li−1 2/summationdisplay s=1 2˜˜ψn,k,a,s/parenleftBigg s b/parenrightBigg (−1)b. (10−2) Eq.(10-1) is appropriate when a≫banda≫1, while eq.(10-2) is appropriate when b≫aandb≫1. /parenleftbigs a/parenrightbig≡(s!)/[a!(s−a)!], ands!≡Γ(s+ 1). Here I have assumed the state is a spin-singlet, and thus ψn,k(−α,β) =ψn,k(α,β). For a spin-triplet, the factor [1 + ( −1)a] in eq.(10-1) should be substituted by [1−(−1)a]. In my actual calculation, the ( a,b) plane is divided into four areas: the finite area: 0 ≤a,b≤Lfanda+b≤Lf+Ls(Lf≫Ls≫1), thea-asymptotic area: a>L fandb≤Ls, theb-asymptotic area: b>L fanda≤Ls, and the cutoff area: the remain area. Eq.(10-1) is used in the a-asymptotic area, and eq.(10-2) is used in the b-asymptotic area, while the contribution from the cutoff area is neglected for it is ex tremely tiny when Lf≫Ls≫1. In a word, a relevant hyper-angular function is described by a finite set of parameters up to a high precision. These parameters are some Taylor coefficient s and some asymptotic coefficients. To operate with some functions of this kind means to operate wit h the corresponding sets of parameters. The relevant operations are: addition of two functions–add ing the corresponding parameters of the two sets; multiplying a function by a constant–multiplying each parameter in the set by the constant; multiplying a function by v(α,β)(eq.(7))– an appropriate linear transformation of the set of parameters of the multiplied function; solving an equation Lnf=gwith g known and f unknown–solving a set of linear equations about the parameters corresponding to f. Here, I write the relevant linear equations corresponding to the equation Lnf=g: [n(n+ 2)−(a+b)(a+b+ 2)]fa,b+ (a+ 1)(a+ 2)fa+2,b+ (b+ 1)(b+ 2)fa,b+2=ga,b; (11−0) [n(n+ 2)−(b+s)(b+s+ 2)]˜fb,s+ (s+ 1)(2s+ 2b+ 3)˜fb,s+1+ (b+ 1)(b+ 2)˜fb+2,s= ˜gb,s; (11−1) [n(n+ 2)−(a+s)(a+s+ 2)]˜˜fa,s+ (s+ 1)(2a+ 2s+ 3)˜˜fa,s+1+ (a+ 1)(a+ 2)˜˜fa+2,s=˜˜ga,s.(11−2) The detailed order to solve ψn,kis: Case 1:n= [n] +1 2, where [n] is an integer. In this case, solve ψn,[n]from eq.(8 n,[n]); and then solveψn,[n]−1from eq.(8 n,[n]−1);···; at last solve ψn,0from eq.(8 n,0). For each ψn,k, the order is: first solve the asymptotic coefficients, from s=1 2tos=Li−1 2; then solve the Taylor coefficients, from a+b=Lf+Lstoa+b= 0(i.e.,a=b= 0).Case 2:nis an integer. In this case, the order is more complicated, be cause the operator Lnhas zero eigenvalue(s) in this case. The order is as following: Step 1: set the asymptotic coefficients and the a+b>n Taylor coefficients of ψn,nto zero; step 2:k←n−1; step 3: ifk<0, goto step 8; step 4: solve the asymptotic coefficients and a+b>n Taylor coefficients of ψn,k, from eq.(8 n,k), in the order analogous to that of case 1. step 5: solve the a+b=nTaylor coefficients of ψn,k+1, from eq.(8 n,k). step 6: solve the a+b < n Taylor coefficients of ψn,k+1, from eq.(8 n,k+1), witha+bdecreas- ing(analogous to case 1) to 0. step 7:k←k−1, and goto step 3; step 8: set the a+b=nTaylor coefficients of ψn,0with some free parameters; step 9: solve the a+b < n Taylor coefficients of ψn,0, from eq.(8 n,0), witha+bdecreas- ing(analogous to case 1) to 0. The free parameters in solving eq.(8)(see step 8 of case 2) ar e finally determined by the boundary condition:ψ→0, whent→+∞. In principle, we can use the Bartlett-Fock expansion (eq.( 4)) for arbitraryt, because it is always convergent. But actually, when tis large, the convergency is slow and there is canceling of large numbers before this converge ncy is reached, both of which make the Bartlett-Fock expansion impractical. So I only use this exp ansion when tis relatively small(see ref.[15] for similarity):√ t≤ρ0. In atual calculation, I chose Lf= 100,Ls= 20,Li= 6,nmax= 7.5 (the largest n value of the terms in eq.(4) that are not neglected), and ρ0= 0.4, and found that the numerical error for the calculation of the inner region (√ t≤ρ0) wavefunction is no more than a few parts in 1010. I use this method to test the accuracy of the calculation: set Ein eq.(8) (note that p≡−E/2) equal to an initial value (for example, set Einitial =−2.9037, or set Einitial =−2.903724377), and use the approximate wavefunction ψappthus obtained to calculate the value ( Hψapp)/ψapp, whereHis the exact Hamiltonian operator, and I find it to be almost equal to the initial value Einitial, with a relative error no more than a few parts in 1010. Whentis larger, another approach is used:3 THE HYPER-ANGULAR DEPENDENCE OF THE WAVEFUNC- TION We have seen that the hyper-angular dependence of the wavefu nction, described as a function of (α,β) for each fixed ρ≡√ t≡/radicalBig r2 1+r2 2, has singularities at α=±1 and atβ= 1. Physically, this corresponds to the case that the distance between two of the three particles equals zero. It can be proved that, for a spin-singlet, the following coordi nate transformation will eliminate these singularities thoroughly : u=/radicalbigg1 +α 2+/radicalbigg1−α 2−1, w=/radicalbig 1−β . (12) Equivalently, u=r1+r2 ρ−1, w=r12 ρ. (13) If the energy-eigenstate ψis symmetric under the exchange of r1andr2(spin-singlet), I believe that, for each fixed ρ,ψis aentire function of ( u,w). If the energy-eigenstate ψis antisymmetric under the interchange of r1andr2(spin-triplet), I believe that, for each fixed ρ,ψ=r1−r2 ρφ, whereφis aentire function of ( u,w). This beautiful characteristic makes it especially appropr iate to approximate ψ, for each fixed ρ, by ann-order polynomial of ( u,w), not by an n-order polynomial of ( α,β). The former expansion, a polynomial of ( u,w), matches the analytical structure of ψ; while the latter one, a polynomial of (α,β), does not. The hyper-spherical harmonic function method b elongs to the latter expansion, a polynomial of ( α,β). So the hyper-spherical harmonic function expansion does not correctly reflect the analytical structure of ψ. The slow convergency of the hyper-spherical harmonic func tion expansion is only a consequence of this analytical structure mismatch ing. We expect that the ( u,w) polynomial expansion converges factorially to the true wavefunction. It is worthful to demonstrate a similar example to illustrate t his. Consider a function f(x) = exp(−x),− 1≤x≤+1; expand f(x) by Legendre polynomials: f(x).=/summationtextn l=0flPl(x); it can be proved that the error of this formula is of the order 1 /(2nn!), which factorially approach zero as nincreases. Using the ( ρ,u,w ) coordinates, one can write the Schr¨ odinger equation as: −1 2(∂2 ρ+5 ρ∂ρ+4L0 ρ2)ψ+C ρψ=Eψ , (14) whereL0andCare the hyper-angular parts of the kinetic energy and the pot ential energy, respectively. 4L0= (1−2u−u2)∂2 u+(2−w2)∂2 w−2(1 +u)(1−2u−u2) u(2 +u)(1−w2) w∂u∂w+(1 +u)(4−10u−5u2) u(2 +u)∂u+4−5w2 w∂w; (15) C=−4(1 +u) u(2 +u)+1 w. (16) The physical area Dof (u,w) is:uw O−1√ 2−11√ 2 D AC B In this figure, point Acorresponds to the coincidence of the two electrons, and poi ntBcorresponds to the coincidence of the nucleus and one electron. For a spin-singlet, we can use an n-order polynomial of ( u,w) to approximate ψ. The coefficients of this polynomial are functions of ρ. Denote byLNthe set of all the polynomials of ( u,w) with order no more than n. Here,N= (n+ 1)(n+ 2)/2 is the dimension. In the physical area D, I choose a set of points as sample points: wa=√ 2(a2+ 0.5) n2, (17) ua= (√ 2−1)−[(√ 2−1)−m(wa)](a1+ 0.5) n1, (18) wherem(w) is the minimum physical uvalue for a wvalue.m(w) =√ 2−w2−1, ifw <1; and m(w) =w−1, ifw≥1.a≡(a1,a2), and 0≤a1<n1, 0≤a2<n2. I chosen1=n2= 2n, so there are altogether 4 n2sample points. These sample points define a pseudo-innerpro duct. I constructed a set of pseudo-orthonormal basis in LN, by using the Schmidt orthogonalization method, and then reduce the operators L0andCtoN×Nmatrices under this basis, using the method introduced in section 1. 4 CONNECTION OF THE INNER SIDE AND THE OUTER SIDE In the area ρ < ρ 0(inner region), the Bartlett-Fock expansion is used. In the areaρ > ρ 0/2(outer region),ψis approximated by a vector in LNfor each given ρ, and the partial derivatives with respect toρare substituted by optimized variable-order and variable- step differences, which requires the selection of a discrete set of ρvalues. The overlap region of the inner region and the outer r egion ensures the natural connection of the derivative of ψ, as well as the connection of ψitself. The connection is performed by using the least square method: fo r a polynomial of ( u,w) atρ=ρ0, appropriately choose the values of the free parameters of th e solution of eq.(8) (see section 2) so that the sum of the squares of the differences of the the inner r egion solution and the outer region polynomial at the sample points is minimized. This defines a l inear transformation to calculate thevalues of those free parameters from the given polynomial. W hen the values of these free parameters are determined, one can calculate the values of ψin the region ρ0/2<ρ<ρ 0, using the Bartlett-Fock solution, and further use these ψvalues to construct polynomials of( u,w) atρ0/2<ρ<ρ 0(according to the law of least square), and then use these polynomials in the difference calculation of the partial derivative of ψwith respect to ρatρ≥ρ0. At a sufficient large value ρ=ρ1, the first-class boundary condition is exerted; of course, future development may sub stitute this by a connection with the long range asymptotic solution of ψ. At last, the whole Schr¨ odinger equation is reduced to an eig en-problem of a finite-dimensional matrix. The dimension of the matrix is Nρ×N, whereNρis the number of free ρnodes used in discretizing the partial derivatives with respect to ρ, andNis the number of independent hyper- angular polynomials used. Note that the energy value should be used in solving eq.(8), but it is unknown. The actual calculation is thus an iteration proces s: choose an initial value of E0to solve eq.(8) and form the Nρ×Ndimensional matrix, and calculate the eigenvalue of this ma trix to get a new valueE1, etc.. The final result is the fixed point of this iteration pro cess. In actual calculation, I found that the convergency of this iteration process is ver y rapid ifρ0is relatively small. Choosing ρ0= 0.4, I found that each step of iteration cause the difference bet ween the eigenvalue of the matrix and the fixed point decrease by about ( −160) times, when calculating the ground state. 5 NUMERICAL RESULT AND COMPARISONS Using 20 independent Bartlett-Fock series(up to the t7.5term in eq.(4), neglecting higher order terms), choosingn= 10 (so that N= 66), choosing Nρ= 40, with ρ0= 0.4 andρ1.= 11.32, and with the discrete values of ρequal to 0.4/1.23,0.4/1.22,0.4/1.2,0.4,0.4×1.2,0.4×1.22,0.4×1.23,···,0.4×1.28.= 1.7199,0.4×1.28+0.3,0.4×1.28+0.6,0.4×1.28+0.9,···,0.4×1.28+9.3, and 0.4×1.28+9.6.= 11.32 (the first three points are for the natural connection of the d erivative of ψ, the last point is for the first-class boundary condition, and the remained 40 points a re free nodes), and discretizing the partial derivatives with respect to ρaccording to the complex-plane-division rule(that is: whe n calculating the partial derivatives with respect to ρatρ=l, use and only use those node points satisfying ρ>l/ 2 in the difference format, because the point ρ= 0 is the singular point), I obtained the result for the ground state energy of the ideal helium atom: E=−2.9037243738 , (19) compared with the accurate approximate value: E=−2.9037243770 . (20) So the relative error of the result (19) is about 1 .1×10−9. Since my method is not a variational method, the error of the approximate wavefunction that I obtained sh ould be of a similar order of magnitude,so if one calculate the expectation value of the Hamiltonian under this approximate wavefunction, the accuracy of the energy will be further raised by several orde rs of magnitude. The result (19) is much more accurate than the result of ref.[ 10]:−2.90359, which used the hyper- spherical coordinate method. In ref.[10], the quantum numb ers (l1,l2) (angular momenta of the two electrons) are used and a cutoff for them is made; this cutoff do es not correctly reflect the analytical structure of ψatr12= 0 (equivalently β= 1). This is the major reason causing the inaccuracy of the result of ref.[10]. It is also worthful to compare my result with that of ref.[4], in which the correlation function hyper-spherical harmonic method is used. Note that the resu lt (19) is obtained by using a set of N= 66 hyper-radius-dependent coefficients to expand the wavef unction. For a similar size in ref.[4], N=64, the result is −2.903724300, with relative error about 26 .5×10−9. When N=169, the result of ref.[4] is−2.903724368, with relative error about 3 .1×10−9. Apparently my method converges more rapidly than that of ref.[4]. The major reason is that th e correlation function hyper-spherical harmonic method does not cancel the singularities totally— there is still some discontinuity for the higher order derivatives, although the low order singulari ties, which trouble the naive hyperspherical harmonic method, are canceled by the correlation function. 6 CONCLUSIONS, DISCUSSIONS AND FUTURE DEVELOPMENTS In conclusion, there are several important ideas in my work t hat should be emphasized: first, I use the asymptotic series to compute the Bartlett-Fock series u p to a high precision, with error no more than, for example, a few parts in 1010. Second, I propose an alternative coordinate system, the ( u,w) system, in which the hyper-angular singularities are thoro ughly eliminated, which renders a factorial convergency for the expansion of the hyper-angular functio n. Third, I make use of the least square method to reduce an operator(infinite dimensional matrix) t o a finite dimensional matrix in a finite dimensional subspace of the Hilbert space and to connect the solutions in different regions, avoiding complicated numerical integrations, without much loss of t he accuracy for the solution. Fourth, the optimized difference format —the complex plane division rule—is used to discretize the partial derivatives of the wavefunction with respect to ρ. I calculated the ground state energy of an ideal helium atom concretely and obtained a very high precision, d emonstrating that my method is superior to many other methods and competitive with any sophisticate d methods. About the analytical structure of the stationary wavefunct ion: 1. there are logarithmic singularities atρ= 0, in the forms of ρm(lnρ)k; 2. for a given ρ,ψ(for a spin-singlet) or ψ/[(r1−r2)/ρ](for a spin-triplet) has no singularity, as a function of ( u,w). Here, I must mention the well known variational method based on the Hyleraas-type functions, because it also satisfies the second characteristic of the wa vefunction mentioned in the above paragraph.One can see this by a simple derivation. The Hyleraas-type fu nction is a entire function of r1,r2and r12, or equivalently, a entire function of r1+r2,r1−r2, andr12. For a fixed ρ, one can substitute (r1−r2)2in this function by 2 ρ2−(r1+r2)2, so that, for fixed ρ, the function is a entire function ofr1+r2andr12for spin-singlet, or such kind of entire function times a com mon factor r1−r2for spin-triplet. Equivalently, for fixed ρ, the Hyleraas-type function is a entire function of ( u,w)(spin- singlet) or such kind of entire function timesr1−r2 ρ(spin-triplet). This characteristic is one of the most important reasons that account for the high accuracy of the H yleraas-type variational method. But this variational method also has its shortcoming: the Hy leraas-type function does not reflect the logarithmic singularities with respect to ρ. So, although this method has high precision for the energy levels, the approximate wavefunctions that it rende rs may deviate significantly from the true wavefunctions near the origin. See ref.[4,13,14] for detai led discussions. A central idea of this paper is: devising the calculation met hod according to the analytical structure of the true solution. The ( u,w) coordinates, the Bartlett-Fock expansion and the asympto tic series approach to compute this expansion, and the complex-plane- division rule in calculating the partial derivatives with respect to ρ, all reflect this central idea. The basic principle that ensu res high numerical precision is just this idea. This preliminary work is incomplete in the following aspect s: First, how to prove thatψ(for spin-singlet, or ψ/[(r1−r2)/ρ] for spin-triplet) has no singularity for fixedρ, as a function of ( u,w)? Note that if this function still has singularities outsid e of the physical areaD(see previous figure), the convergency of the expansion of th e hyper-angular function will be only exponential, not factorial. Of course, even if such kin d of singularities do exist, my method will still converge more rapidly than the correlation function h yperspherical harmonic method, because the latter method only converges like N−p, slower than exp( −γ√ N). The rapid convergency of my method make me guess that such kind of singularities do not ex ist. Second, the asymptotic behavior of the wavefunction, when o ne electron is far away from the nucleus, is not studied in this work. This problem will be imp ortant when the highly excited states and the scattering states are studied, a topic that will beco me my next object. Third, how to use the ideas proposed in this work to study a hel ium atom with finite nuclear mass? Besides this, the relativistic and QED corrections must be c alculated, if one want to obtain a result comparable with high-precision experiments. Fourth, I have focused on the S states till now. When the total angular momentum is not zero, there might be more than one distance-dependent functions ( see, for example, ref.[11]). I believe that some important analytical structures of the S states st udied in this work are also valid for those functions. Surely, some important aspects of this work will also play an important role in the highly excited states and the scattering states: the logarithmic singular ities about ρand the method to compute theBartlett-Fock expansion, the non-singularity with respec t to the coordinates ( u,w), and the technique to connect solutions of different regions, etc.. They can be a pplied to the study of the highly excited states and the scattering states. ACKNOWLEDGEMENTS The encouraging discussions with Prof. SUN Chang-Pu and wit h Prof. Zhong-Qi MA are gratefully acknowledged. I thank Prof. HOU Boyuan for providing me some useful references. I am grateful to Prof. C.M. Lee (Jia-Ming Li) and Dr. Jun Yan for their attenti on to this work and their advices. References [1] V.B.Mandelzweig, Phys.Lett. A.78, 25 (1980) [2] M.I.Haftel, V.B.Mandelzweig, Ann.Phys. 150, 48 (1983) [3] R. 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