dynamic scaling of disordered ising systems the critical dynamics of site disordered three dimensional ising systems has been studied by conventional local dynamics simulations investing 1015 single spin flips the author has verified that the correlation functions of disordered ising systems obey dynamic scaling behaviour the dynamic scaling functions have been calculated the critical exponent z appears to be strongly dependent on the degree of disorder the author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour the dynamical exponent has been estimated to be z 2 4 or 0 1 which is surprisingly near to the renormalization group result in o in 1 2

on correlations of cm type maass waveforms under the horocyclic flow maass waveforms of cm type are a special kind of eigenfunction of the hyperbolic laplacian whose defining components namely eigenvalue and fourier coefficients are given by simple formulae involving algebraic integers chosen from a suitable number field k in this paper we report on some computer experiments aimed at ascertaining the extent to which the autocorrelation behaviour of cm forms agrees with that of mock i e random waveforms in the limit of high energy our results suggest that no significant differences are seen

local and cluster critical dynamics of the 3d random site ising model abstract we present the results of monte carlo simulations for the critical dynamics of the three dimensional site diluted quenched ising model three different dynamics are considered these correspond to the local update metropolis scheme as well as to the swendsen wang and wolff cluster algorithms the lattice sizes of l 10 96 are analysed by a finite size scaling technique the site dilution concentration p 0 85 was chosen to minimize the correction to scaling effects we calculate numerical values of the dynamical critical exponents for the integrated and exponential autocorrelation times for energy and magnetization as expected cluster algorithms are characterized by lower values of dynamical critical exponent than the local one also in the case of dilution critical slowing down is more pronounced for the metropolis algorithm however the striking feature of our estimates is that they suggest that dilution leads to decrease of the dynamical critical exponent for the cluster algorithms this phenomenon is quite opposite to the local dynamics where dilution enhances critical slowing down

path integral semiclassical and stochastic propagators for markovian open quantum systems we develop the path integral theory for master equations of general lindblad form positive semigroups describing markovian open quantum systems first the hamiltonian path integral expression for the propagator is derived which exhibits nicely the decoherence of pairs of phase space histories a very appealing picture arises in the semiclassical limit where the degree of decoherence is expressible in terms of a phase space decoherence distance functional for the important class of effective hamiltonians quadratic in the momenta we derive the lagrangian version of the path integral propagator we then evaluate the path integral approximately in a stationary phase approximation leading to a van vleck type propagator valid under semiclassical conditions we also derive the propagator for the soluble damped harmonic oscillator in closed form from path integrals finally connections to the active field of stochastic pure state descriptions of open quantum systems are established here in particular to linear quantum state diffusion

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

poincare husimi representation of eigenstates in quantum billiards for the representation of eigenstates on a poincare section at the boundary of a billiard different variants have been proposed we compare these poincare husimi functions discuss their properties and based on this select one particularly suited definition for the mean behavior of these poincare husimi functions an asymptotic expression is derived including a uniform approximation we establish the relation between the poincare husimi functions and the husimi function in phase space from which a direct physical interpretation follows using this a quantum ergodicity theorem for the poincare husimi functions in the case of ergodic systems is shown

critical behavior of a three dimensional random bond ising model using finite time scaling with extensive monte carlo renormalization group method we have investigated the critical behavior of a three dimensional random bond ising model for a series of the disorder strength by a finite time scaling combining with monte carlo renormalization group method in the presence of a linearly varying temperature the method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures the static exponents obtained agree well with most existing results verify both the hyperscaling and the rushbrooke scaling laws and their combined scaling law which in turn validate their asymptotic nature and corroborate the universality of the relevant random fixed point with respect to the forms of disorder the dynamic critical exponent z is estimated to 2 114 51 which is compatible with those obtained from experiments and renormalization group analyses the exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point they also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover our results demonstrate the effectiveness of the finite time scaling method

the three dimensional dilute ising magnet the dilute ising model with a p 0 8 fraction of the sites occupied by spins is simulated on l3 systems for l up to 300 using a single cluster algorithm of wolff in the range of reduced temperature 0 002 t tc tc 0 03 the susceptibility appears to fit a simple power with an effective exponent eff 1 36

summability of the perturbative expansion for a zero dimensional disordered spin model we show analytically that the perturbative expansion for the free energy of the zero dimensional quenched disordered ising model is borel summable in a certain range of parameters provided that the summation is carried out in two steps first in the strength of the original coupling of the ising model and subsequently in the variance of the quenched disorder this result is illustrated by some high precision calculations of the free energy obtained by a straightforward numerical implementation of our sequential summation method

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

ergodicity of eigenfunctions for ergodic billiards we give a simple proof of ergodicity of eigenfunctions of the laplacian with dirichlet boundary conditions on compact riemannian manifolds with piecewise smooth boundaries and ergodic billiards examples include the bunimovich stadium the sinai billiard and the generic polygonal billiard tables of kerckhoff masur and smillie

critical behavior of the specific heat for pure and site diluted simple cubic ising systems the exponent of the specific heatc is determined for the pure and the site diluted simple cubic ising model concentrationx 0 0 2 0 4 of nonmagnetic sites by a finite size scaling analysis of the peak value cmax l for systems of linear dimensionsl 8 16 32 and 64 the cmax values are obtained by the ferrenberg swendsen algorithm using monte carlo data from a fully vectorized multi spin coding program we obtain 0 11 for x 0 and a crossover to a negative value upon dilution with 0 029 4 both forx 0 2 andx 0 4

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

an axisymmetric ordinary state based peridynamic model for linear elastic solids abstract in this study a new axisymmetric ordinary state based peridynamic pd model for axisymmetric problems of linear elastic solids is presented a fracture criterion based on the pd bond energy density is proposed adaptive dynamic relaxation adr method is adopted to obtain equilibrium solutions and a viable fictitious density of the model is derived and proven to be valid for implementation in the adr method performance and validity of the proposed axisymmetric pd model are demonstrated by three kinds of numerical problems i e compression tests pull out deformation and indentation fracture in the compression tests with focus on constant strain deformations the displacements predicted by the present model are compared with classical analytical solutions and the results show good agreements both m convergence and convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases the recovery of the poisson s ratios in the model is tested in detail as well the capability of capturing general non uniform axisymmetric deformation by the proposed model is verified in the pull out analysis the equilibrium displacement fields predicted by the present model agree very well with those by the finite element method the peridynamic evaluation of strains and stresses also show good match with the finite element ones the proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data the model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state based peridynamics

negativity witness for the quantum ergodic conjecture for an ergodic system the time average of a classical observable coincides with that obtained via the liouville probability density a delta function on the energy shell reinterpreting this distribution as a wigner function that is the weyl representation of a density operator the quantum ergodic conjecture identifies this classical construction as an approximate representation of the quantum eigenstate of the same energy it is found that this reasonable hypothesis as far as expectations for observables are concerned does not satisfy the requirement of positivity so that the delta function on the shell cannot be a true wigner function this result was first presented by n balazs for the special case of a single degree of freedom such that the system is simultaneously integrable and ergodic here it is shown for a delta function on an arbitrary curved energy shell in a dimensional phase space that there exists a positive operator for which the conjectured wigner function predicts a negative expectation

non ordinary state based peridynamic analysis of stationary crack problems an implicit implementation of the non ordinary state based peridynamics formulation for quasi static linearly elastic solids is presented emphasis is placed on assessing the accuracy of the numerical scheme in the vicinity of the crack front and other sources of stress concentration we also present a comparative study of methods used to control the zero energy modes inherent in the nonlocal definition of the strain tensor and reduce the spurious oscillations present particularly in regions of high strain gradients the accuracy of the peridynamics scheme including the impact of the lattice spacing and configuration is assessed by performing an analysis of the near tip stress and displacement fields k fields for 2d problems the manuscript also summarizes a verification study based on the classical 3d penny shaped crack problem and a validation study of a 3d notched fracture specimen

asymptotically compatible schemes and applications to robust discretization of nonlocal models many problems in nature being characterized by a parameter are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit numerical schemes that are convergent in both regimes offer robust discretizations which can be highly desirable in practice the asymptotically compatible schemes studied in this paper meet such objectives for a class of parametrized problems an abstract mathematical framework is established rigorously here together with applications to the numerical solution of both nonlocal models and their local limits in particular the framework can be applied to nonlocal diffusion models and a general state based peridynamic system parametrized by the horizon radius recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter thus it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical dis

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

critical crossover phenomena in disordered ising systems the author has investigated the critical behaviour of site disordered ising systems over a large range of dilution 0 6 or p or 1 0 investing about 1015 single spin flips with conventional local dynamics critical temperatures have been determined with high precision using the cumulant method the finite size simulations lead to concentration dependent exponents which seem to violate universality the author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region the author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder

numerical experiments on the convergence properties of state based peridynamic laws and influence functions in two dimensional problems abstract peridynamics is widely used as the theoretical basis for numerical studies of fracture evolution propagation and behavior while the theory has been shown to converge to continuum mechanics in the theoretical limit its behavior as a discrete numerical approximation with respect to classic problems has not been shown in this study we use standard analytical solutions to thoroughly test the numerical accuracy and rate of convergence of the spatial discretization obtained by peridynamics we analyze the accuracy and rate of convergence of three different peridynamic constitutive responses of these two involve a state based dilation and the third is based on the estimation of the deformation gradient additionally we study the choice of the peridynamic influence function in each of the constitutive responses the peridynamic materials are solved in the linear elastic regime by solving a linear system of equations obtained by symbolically differentiating the force states we test the methods against standard constant strain solutions for uniaxial compression isotropic compression and simple shear we also apply the methods to a finite material with a pressurized thin crack using the westergaard s solution method to obtain an analytical displacement field for comparison the two dilation based peridynamic constitutive responses are found to only converge to one of the constant strain solutions while the deformation gradient based law converges in all cases with an appropriate choice of the influence functions we show that a cubic influence function is the best choice of those considered in all methods only the deformation gradient based model converges for all three linear deformation problems but is less accurate than the dilation based models for the thin crack problem because of instabilities we demonstrate an ad hoc smoothing technique based on the influence function that is able to alleviate these instabilities and improve the accuracy of the deformation gradient based model

crossover effects in dilute magnetic materials by finite time dynamics method abstract crossover phenomena are ubiquitous in disordered system however the crossover effects from the competition between different fixed points make it hard to identify the asymptotic scaling regime controlled by the random fixed point therefore the analyses on the crossover effects are of great importance to understand phase transition in dilute magnetic materials in this paper we derive the scaling function characterizing the crossovers to probe critical behavior according to these we show an alternative method to locate the asymptotic critical region in the quenched random systems in the finite time dynamics frame for the three dimensional bond diluted ising model the asymptotic critical exponents are identified with 0 686 1 0 356 6 0 057 3 1 345 20 z 2 170 11 at middle bond concentration p 0 7 from the effective ones describing the approach to the asymptotic regime for weak dilution p 0 9 and strong dilution p 0 5 the dynamic critical exponent from our non equilibrium simulation supports distinctly z 2 18 as suggested previously as opposed to the existing larger values

precise determination of the bond percolation thresholds and finite size scaling corrections for the sc fcc and bcc lattices extensive monte carlo simulations were performed to study bond percolation on the simple cubic s c face centered cubic f c c and body centered cubic b c c lattices using an epidemic kind of approach these simulations provide very precise values of the critical thresholds for each of the lattices pc s c 0 2488126 0 0000005 pc f c c 0 1201635 0 0000010 and pc b c c 0 1802875 0 0000010 for p close to pc the results follow the expected finite size and scaling behavior with values for the fisher exponent 2 189 0 002 the finite size correction exponent 0 64 0 02 and the scaling function exponent 0 445 0 01 confirmed to be universal

decoherence of semiclassical wigner functions the lindblad master equation governs the general markovian evolution of a density operator for an open quantum system semiclassical wigner functions represent density operators in phase space in terms of chords on a classical manifold so that the amplitude and phase of each chord contribution is classically defined inserting such a wigner function into a phase space version of the master equation its explicit evolution is derived in the absence of dissipation there results a simple extension of the unitary evolution of the semiclassical wigner function which does not affect the phase of each chord contribution while dampening its amplitude exponentially projecting the wigner function on to an orthogonal position or momentum basis the dampening of long chords emerges as the exponential decay of off diagonal elements of the density matrix

perturbative solution for the generalised anharmonic oscillators serious difficulties have been encountered in the past when attempting the straightforward application of the borel method to the summation of perturbation series in quantum mechanics and quantum field theory because of an analytic continuation of the borel transform series which is usually performed by means of conformal mapping techniques the author shows that this continuation may be accomplished by combining the borel method with the first confluent form of the epsilon algorithm of wynn 1960 1968 for the calculation of the ground state energy eigenvalue of the generalised anharmonic oscillators two different approaches are presented both of which lead to accurate results

quantitative study of scars in the boundary section of the stadium billiard we construct a semiclassically invariant function on the boundary of the billiard taken as the poincare section in birkhoff coordinates based on periodic orbit information as an ansatz for the normal derivative of the eigenfunction defining an appropriate scalar product on the section we can compute the scar intensity of a given periodic orbit on an eigensate as the overlap beetween the constructed function and the normal derivative on the section of the eigenstate in this way we are able to investigate how periodic orbits scar the spectrum and how a given eigenstate decompose into scar functions we use this scheme on the bunimovich stadium

correlations of chaotic eigenfunctions a semiclassical analysis we derive a semiclassical expression for an energy smoothed autocorrelation function defined on a group of eigenstates of the schr odinger equation the system we consider is an energy conserved hamiltonian system possessing time invariant symmetry the energy smoothed autocorrelation function is expressed as a sum of three terms the first one is analogous to berry s conjecture which is a bessel function of the zeroth order the second and the third terms are trace formulae made from special trajectories the second term is found to be direction dependent in the case of spacing averaging which agrees qualitatively with previous numerical observations in high lying eigenstates of a chaotic billiard

three dimensional randomly dilute ising model monte carlo results we perform a high statistics simulation of the three dimensional randomly dilute ising model on cubic lattices l 3 with l le 256 we choose a particular value of the density x 0 8 for which the leading scaling corrections are suppressed we determine the critical exponents obtaining nu 0 683 3 eta 0 035 2 beta 0 3535 17 and alpha 0 049 9 in agreement with previous numerical simulations we also estimate numerically the fixed point values of the four point zero momentum couplings that are used in field theoretical fixed dimension studies although these results somewhat differ from those obtained using perturbative field theory the field theoretical estimates of the critical exponents do not change significantly if the monte carlo result for the fixed point is used finally we determine the six point zero momentum couplings relevant for the small magnetization expansion of the equation of state and the invariant amplitude ratio r xi that expresses the universality of the free energy density per correlation volume we find r xi 0 2885 15

on the crossover to universal criticality in dilute ising systems monte carlo simulations of the critical behaviour of disordered ising systems have led to concentration dependent critical exponents which seem to violate universality we apply the renormalization group to investigate the crossover effects which cause the observed behaviour we improve the three loop expansion of the callan symanzik and wilson functions by a pade borel approximation and solve the flow equations for various initial points the exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

universality in the off equilibrium critical dynamics of the three dimensional diluted ising model we study the off equilibrium critical dynamics of the three dimensional diluted ising model we compute the dynamical critical exponent z and we show that it is independent of the dilution only when we take into account the scaling corrections to the dynamics finally we will compare our results with the experimental data

amplitude distribution of eigenfunctions in mixed systems we study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space for an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and a good agreement with numerical computations for the family of limacon billiards is found the natural extension of our result to more general systems e g with a potential is also discussed

critical behaviour of 3d systems with long range correlated quenched defects a field theoretic description of the critical behaviour of systems with quenched defects obeying power law correlations x a for large separations x is given directly for three dimensional systems and for different values of the correlation parameter 2 a3 a renormalization analysis of the scaling function in the two loop approximation is carried out and the fixed points corresponding to the stability of various types of critical behaviour are identified the obtained results essentially differ from results evaluated by a double expansion the critical exponents in the two loop approximation are calculated with the use of the pade borel summation technique

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

finite time scaling via linear driving application to the two dimensional potts model we apply finite time scaling to the q state potts model with q 3 and 4 on two dimensional lattices to determine its critical properties this consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields from the finite time scaling of the order parameter the coercivity and the hysteresis area and its derivative we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature the static critical exponents obtained in general and the magnetic exponent ensuremath delta in particular agree reasonably with the conjectured ones the dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short time dynamic results for q 4 our results also suggest an alternative way to characterize the weak universality

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

determination of the dynamic and static critical exponents of the two dimensional three state potts model using linearly varying temperature we employ a successive monte carlo renormalization group procedure in the presence of a linearly varying temperature to study the three state q 3 potts model on square lattices by matching correlation functions of different lattice sizes at different renormalization levels a rate exponent r associated with the temperature sweep rate r and the correlation length exponent ensuremath nu are obtained the dynamic critical exponent z is then obtained by the scaling law z r ensuremath 1 ensuremath nu derived in our method the results of z 2 171 62 for q 3 in this work and the previously obtained z 2 15 13 for q 2 seem to support the extension of the weak universality hypothesis to dynamic critical behavior with these calculated exponents the dynamic scaling forms of both specific heat and order parameter at various r are presented and all the other static critical exponents are determined which verify our method discussions are made on how to improve the accuracy of the estimation of the dynamic critical exponent

critical behavior of the pure and random bond two dimensional triangular ising ferromagnet we investigate the effects of quenched bond randomness on the critical properties of the two dimensional ferromagnetic ising model embedded in a triangular lattice the system is studied in both the pure and disordered versions by the same efficient two stage wang landau method in the first part of our study we present the finite size scaling behavior of the pure model for which we calculate the critical amplitude of the specific heat s logarithmic expansion for the disordered system the numerical data and the relevant detailed finite size scaling analysis along the lines of the two well known scenarios logarithmic corrections versus weak universality strongly support the field theoretically predicted scenario of logarithmic corrections a particular interest is paid to the sample to sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality

density and correlation functions of vortex and saddle points in open billiard systems we present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random wave model rwm in particular due to an improvement of the experimental setup we determine experimentally the spatial correlation of saddle points of the current field an asymptotic expression for the vortex saddle and saddle saddle correlation functions based on the rwm is derived with experiment and theory agreeing well we also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the rwm and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying dirichlet conditions

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

semiclassical spatial correlations in chaotic wave functions we study the spatial autocorrelation of energy eigenfunctions psi n q corresponding to classically chaotic systems in the semiclassical regime our analysis is based on the weyl wigner formalism for the spectral average c epsilon q q e of psi n q psi n q defined as the average over eigenstates within an energy window epsilon centered at e in this framework c epsilon is the fourier transform in the momentum space of the spectral wigner function w x e epsilon our study reveals the chord structure that c epsilon inherits from the spectral wigner function showing the interplay between the size of the spectral average window and the spatial separation scale we discuss under which conditions is it possible to define a local system independent regime for c epsilon in doing so we derive an expression that bridges the existing formulas in the literature and find expressions for c epsilon q q e valid for any separation size q q

on the k property of some planar hyperbolic billiards thek property is demonstrated for a class of planar billiards satisfying wojtkowski s principles their boundary may consist of convex scattering concave and linear pieces earlier wojtkowski showed that these billards had non zero lyapunov exponents

pure state correlations chords in phase space the intensity of the overlap of a quantum state with all its phase space translations defines its quantum correlations in the case of pure states these are invariant with respect to fourier transformation the overlaps themselves are here studied in terms of the wigner function and its fourier transform i e the characteristic function or chord function unlike the wigner function the chord function need not be real but eventual symmetry with respect to reflections about a phase space point may relate these representations semiclassical approximations for the classical like region of small chords and for large chords are derived these lead to an interpretation of the fourier invariance in terms of conjugate chords the interrelation of large and small sub planck phase space structures previously noted in the literature is thus reinterpreted

on the number of bouncing ball modes in billiards we study the number of bouncing ball modes in a class of two dimensional quantized billiards with two parallel walls using an adiabatic approximation we show that asymptotically for where depends on the shape of the billiard boundary in particular for the class of two dimensional sinai billiards which are chaotic one can get arbitrarily close from below to which corresponds to the leading term in weyl s law for the mean behaviour of the counting function of eigenstates this result shows that one can come arbitrarily close to violating quantum ergodicity we compare the theoretical results with the numerically determined counting function for the stadium billiard and the cosine billiard and find good agreement

critical exponents of the dilute ising model from four loop expansions four loop expansions of the coefficients of the callan symanzik equation for 3d and 2d dilute ising models are calculated fixed point coordinates and critical exponents are estimated by two methods of summation of double series a generalisation of the pade borel approximation and the first confluent form of the e algorithm of wynn summation of the double series for the 2d dilute ising model gives exponents very close to exponents of the pure ising model in accordance with the exact solution

convergence adaptive refinement and scaling in 1d peridynamics we introduce here adaptive refinement algorithms for the non local method peridynamics which was proposed in j mech phys solids 2000 48 175 209 as a reformulation of classical elasticity for discontinuities and long range forces we use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected we discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1d in the limit of the horizon going to zero continuous micromoduli lead to optimal rates of convergence independent of the grid used while discontinuous micromoduli produce optimal rates of convergence only for uniform grids examples for static and dynamic elasticity problems in 1d are shown the relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement for the same number of nodes copyright 2008 john wiley sons ltd

a generalized ordinary finite deformation constitutive correspondence model for peridynamics abstract a finite deformation constitutive framework for peridynamics that uses generalized kirchhoff stress measures as intermediate quantities in computing the peridynamic material point interactions is proposed it is shown to be a generalization of a previous finite deformation constitutive correspondence theory and has additional useful properties the new model is shown to be an ordinary material model and therefore conserves angular momentum without additional restrictions additionally it does not suffer from well known surface effects present in some peridynamic material models has improved material stability properties and allows for inclusion of elasto plastic behavior in a straightforward manner

random matrix elements and eigenfunctions in chaotic systems the expected root mean square value of a matrix element a ensuremath alpha ensuremath beta in a classically chaotic system where a is a smooth ensuremath elzxh independent function of the coordinates and momenta and ensuremath alpha and ensuremath beta label different energy eigenstates has been evaluated in the literature in two different ways by treating the energy eigenfunctions as gaussian random variables and averaging a ensuremath alpha ensuremath beta 2 over them and by relating a ensuremath alpha ensuremath beta 2 to the classical time correlation function of a we show that these two methods give the same answer only if berry s formula for the spatial correlations in the energy eigenfunctions which is based on a microcanonical density in phase space is modified at large separations in a manner that we previously proposed

star graph expansions for bond diluted potts models we derive high temperature series expansions for the free energy and the susceptibility of random bond q state potts models on hypercubic lattices using a star graph expansion technique this method enables the exact calculation of quenched disorder averages for arbitrary uncorrelated coupling distributions moreover we can keep the disorder strength p as well as the dimension d as symbolic parameters by applying several series analysis techniques to the new series expansions one can scan large regions of the p d parameter space for any value of q for the bond diluted four state potts model in three dimensions which exhibits a rather strong first order phase transition in the undiluted case we present results for the transition temperature and the effective critical exponent gamma as a function of p as obtained from the analysis of susceptibility series up to order 18 a comparison with recent monte carlo data chatelain et al phys rev e 64 036120 2001 shows signals for the softening to a second order transition at finite disorder strength

evidences of the instability fixed points of first order phase transitions we study the dynamics of the first order phase transitions in the two dimensional potts model driven by a linearly varying temperature using a finite time scaling with extended dynamic monte carlo renormalization group method it is found that for sufficiently large lattice sizes the flows of apparent exponents of different temperature sweep rates upon renormalization show characteristics that are markedly distinct from those of continuous transitions and are argued to result from the instability fixed points involved and provide a method for estimating the associated instability exponents

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

self averaging distribution of pseudocritical temperatures and finite size scaling in critical disordered systems we evaluate by monte carlo simulations various singular thermodynamic quantities x for ensembles of quenched random ising and ashkin teller models the measurements are taken at t c and we study how the distributions p x and in particular their relative squared width r x over the ensemble depend on the system size l the ashkin teller model was studied in the regime where bond randomness is irrelevant and we found weak self averaging r x ensuremath sim l ensuremath alpha ensuremath nu ensuremath rightarrow 0 where ensuremath alpha l0 and ensuremath nu are the exponents of the pure model fixed point governing the transition for the site dilute ising model on a cubic lattice known to be governed by a random fixed point we find that r x tends to a constant as predicted by aharony and harris we tested whether this constant is universal however this constant is different for canonical and grand canonical disorder we identify the pseudocritical temperature of each sample i t c i l as the temperature at which the susceptibility reaches its maximal value ensuremath chi mathrm max the distribution of these sample dependent t c i l was investigated we found that its variance scales as ensuremath delta t c l 2 ensuremath sim l ensuremath 2 ensuremath nu our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold we did observe deviations from a single function which imply that sample dependent scaling functions are needed these deviations are however relatively small and hence to obtain a fixed statistical error it may be more computationally efficient to measure ensuremath chi mathrm max than the commonly used ensuremath chi t c ensuremath infty

behaviour of boundary functions for quantum billiards we study the behaviour of the normal derivative of eigenfunctions of the helmholtz equation inside billiards with dirichlet boundary condition these boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest therefore they form a reduced representation of the quantum system analogous to the poincare section of the classical system for the normal derivatives we introduce an equivalent to the standard green function and derive an integral equation on the boundary based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies the first term is universal and independent of the shape of the billiard the second one is proportional to the curvature of the boundary the asymptotic behaviour is compared with numerical results for the stadium billiard different limacon billiards and the circle billiard and good agreement is found furthermore we derive an asymptotic completeness relation for the boundary functions

linearized state based peridynamics for 2 d problems summary in this work the authors formulate a 2 d linearized ordinary state based peridynamic model of elastic deformations and compute the stiffness matrix for 2 d plane stress strain conditions this model is then verified by testing the recovery of elastic properties for given poisson s ratios in the range 0 1 0 45 the convergence behavior of peridynamic solutions in terms of the size of the nonlocal region by comparison with the classical local mechanics model is also discussed the degree to which the peridynamic surface effect influences the recovery of elastic properties is examined and stress strain recovery values are found to have a definite influence on the results the technique used here can provide the basis for applying 2 d peridynamic models to the study of fatigue failure and quasi static fracture problems copyright 2016 john wiley sons ltd

ray chaotic footprints in deterministic wave dynamics a test model with coupled floquet type and ducted type mode characteristics ray chaos manifested by the exponential divergence of trajectories in an originally thin ray bundle can occur even in linear electromagnetic propagation environments due to the inherent nonlinearity of ray tracing maps in this paper we present a novel two dimensional test example of such an environment which embodies intimately coupled refractive wave trapping and periodicity induced multiple scattering phenomenologies and which is amenable to explicit full wave analysis though strictly nonchaotic it is demonstrated that under appropriate conditions which are inferred from a comprehensive parametric database generated via the above noted rigorous reference solution the high frequency wave dynamics exhibits trends toward irregularity and other peculiar characteristics these features can be interpreted as ray chaotic footprints and they are usually not observed in geometries characterized by regular ray behavior in this connection known analogies from other disciplines particularly quantum physics are briefly reviewed and related to the proposed test configuration moreover theoretical implications and open issues are discussed and potential applications are conjectured

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

critical dynamics of cluster algorithms in the dilute ising model autocorrelation times for thermodynamic quantities att c are calculated from monte carlo simulations of the site diluted simple cubic ising model using the swendsen wang and wolff cluster algorithms our results show that for these algorithms the autocorrelation timesdecrease when reducing the concentration of magnetic sites from 100 down to 40 this is of crucial importance when estimating static properties of the model since the variances of these estimators increase with autocorrelation time the dynamical critical exponents are calculated for both algorithms observing pronounced finite size effects in the energy autocorrelation data for the algorithm of wolff we conclude that when applied to the dilute ising model cluster algorithms become even more effective than local algorithms for whichincreasing autocorrelation times are expected

static and dynamic critical properties of 3d systems with long range correlated quenched defects a field theoretic description of static and purely relaxational dynamic critical behaviour of systems with quenched defects obeying power law correlations x a for large separations x is given directly for three dimensional systems and for different values of the correlation parameter 2 a3 a renormalization analysis of the scaling functions in the two loop approximation is carried out and the fixed points corresponding to the stability of various types of critical behaviour are identified the obtained results essentially differ from results evaluated by a double expansion the static and dynamic critical exponents in the two loop approximation are calculated with the use of the pade borel summation technique

the dynamic critical exponent of dilute and pure ising systems a massive dynamic field theory is employed to calculate the critical exponent z of the three dimensional dilute ising model to two loop order without expansion in square root 4 d with result z approximately 2 191 applying the same method to the pure ising model at three loop order we find z 2 022 for d 3 and z 2 124 for d 2

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

statistical properties of high lying chaotic eigenstates we study the statistical properties of the high lying chaotic eigenstates 200000 and above which are deep within the semiclassical regime the system we are analysing is the billiard system inside the region defined by the quadratic complex conformal map of the unit disk as introduced by robnik 1983 we are using heller s method of plane wave decomposition of the numerical eigenfunctions and perform extensive statistical analysis with the following conclusions i the local average probability density is in excellent agreement with the microcanonical assumption and all statistical properties are also in excellent agreement with the gaussian random model ii the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with berry s 1977 prediction iii although the scars of unstable classical periodic orbits in such an ergodic regime are expected to exist so far we have not found any around the 200000th state other than a scar like feature resembling the whispering gallery modes

dual horizon peridynamics a stable solution to varying horizons abstract in this paper we present a dual horizon peridynamics formulation which allows for simulations with dual horizon with minimal spurious wave reflection we prove the general dual property for dual horizon peridynamics based on which the balance of momentum and angular momentum in pd are naturally satisfied we also analyze the crack pattern of random point distribution and the multiple materials issue in peridynamics for selected benchmark problems we show that dh pd is less sensitive to the spatial than the original pd formulation

gaussian random eigenfunctions and spatial correlations in quantum dots we show that spatial correlations in wave functions of quantum dots obtained earlier by averaging over a random potential via supermatrix techniques can be computed with much less effort by making use of berry s conjecture that the energy eigenfunctions in a quantized chaotic system are gaussian random variables furthermore in the case of a time reversal invariant system we find a greatly simplified though equivalent formula for these correlations textcopyright 1996 the american physical society

on the critical fluctuations in superconductors the field theoretical gauge model for a superconductor generalized to an n 2 component complex order parameter is considered the question of the order of the phase transition occurring in this model is discussed previous renormalization group calculations suggest that in such a model a fluctuation induced first order phase transition occurs we re examine previously obtained expressions for the renormalization group functions in the two loop approximation in three dimensions special attention is being paid to the fact that the corresponding series might be the asymptotic ones and therefore have zero radius of convergence we discuss the possible ways of analytical continuation of the series obtained comparing results obtained by direct calculations with those obtained by pade analysis and the pade borel resummation technique the conjecture is made that in the model under consideration there still exists a possibility for a second order phase transition

improved one point quadrature algorithms for two dimensional peridynamic models based on analytical calculations the peridynamics theory is a reformulation of the classical theory of continuum mechanics based on long range interactions suitable for the description of material failure and damage integration plays a central role in this theory we study one point quadrature algorithms for the discretization of two dimensional integrals in peridynamics these algorithms are closely related to meshfree methods they assume an underlying reference lattice where each lattice point within a body is assigned a cell a main challenge in such algorithms is the accurate estimation of the area of the intersecting region between the neighborhood of a point and a neighbor cell we provide a classification of the different types of intersecting regions in square lattices and present analytical derivations for the exact calculation of their areas leading to improved integration accuracy to address convergence issues geometric centers of intersecting regions are taken as quadrature points replacing commonly used cell centers we present analytical derivations for the exact calculation of those geometric centers based on a decomposition of intersecting regions into subdomains of simple geometry by using exact values for the areas and geometric centers of intersecting regions we achieve an asymptotically monotonic convergence in numerical integration in contrast other algorithms from the literature exhibit a highly oscillatory behavior numerical results compare the proposed algorithms with others from the literature for different quantities of interest and demonstrate their improved accuracy and convergence error estimates for the quadrature algorithms are derived and extended and hybrid algorithms are discussed additional numerical studies using influence functions with a finite support and a controlled regularity suggest an alternative means to improve the convergence of discretization algorithms

how chaotic is the stadium billiard a semiclassical analysis the impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described semiclassically by the gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits i show that this belief is erroneous the gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the green s function semiclassical contributions to the trace show an dependent transition from hard chaos to integrable behaviour for trajectories approaching the bouncing ball orbits a whole region in phase space surrounding the marginal stable family acts semiclassically like a stable island with boundaries being explicitly dependent the localized bouncing ball states found in the billiard derive from this semiclassically stable island the bouncing ball orbits themselves however do not contribute to individual eigenvalues in the spectrum an ebk like quantization of the regular bouncing ball eigenstates in the stadium can be derived the stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics the behaviour is generically found at the border of classically stable islands in systems with a mixed phase space structure

nonequilibrium relaxation study on spin glass model ferromagnetic fm transition point and universality class of the j ising model on cubic lattice are studied by computer simulation using the nonequilibrium relaxation ner method at fm bond density p between multicritical one pmc and 1 static exponents take universal values which are different from those of p 1 or p pmc and they are consistent with the values of the random fixed point and of the simulations and experiments for diluted ising systems the dynamical exponent however seems to depend on p and not to be universal although it may be due to finite time correction the estimation of pmc on square lattice is also refined to be 0 8894 9 which is consistent with the nishimori and nemoto s conjecture 0 889972

universal aging properties at a disordered critical point we investigate analytically near the dimension d uc 4 and numerically in d 3 the nonequilibrium relaxational dynamics of the randomly diluted ising model at criticality using the exact renormalization group method to one loop we compute the two times t phantom rule 0 2em 0ex t w correlation function and fluctuation dissipation ratio fdr for any fourier mode of the order parameter of finite wave vector q in the large time separation limit the fdr is found to reach a nontrivial value x ensuremath infty independently of small q and coincide with the fdr associated to the total magnetization obtained previously explicit calculations in real space show that the fdr associated to the local magnetization converges in the asymptotic limit to this same value x ensuremath infty through a monte carlo simulation we compute the autocorrelation function in three dimensions for different values of the dilution fraction p at t c p taking properly into account the corrections to scaling we find according to the renormalization group predictions that the autocorrelation exponent ensuremath lambda c is independent of p the analysis is complemented by a study of the nonequilibrium critical dynamics following a quench from a completely ordered state