long time large scale limit of the wigner transform for a system of linear oscillators in one dimension we consider the long time large scale behavior of the wigner transform w t x k of the wave function corresponding to a discrete wave equation on a 1 d integer lattice with a weak multiplicative noise this model has been introduced in basile et al in phys rev lett 96 2006 to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum the kinetic limit for the wigner transform has been shown in basile et al in arch rat mech 195 1 171 203 2009 in the present paper we prove that in the unpinned case there exists 0 0 such that for any 0 0 the weak limit of w t 3 2 x k as 1 satisfies a one dimensional fractional heat equation partial t w t x hat c partial x 2 3 4 w t x with hat c 0 in the pinned case an analogous result can be claimed for w t 2 x k but the limit satisfies then the usual heat equation

homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

a semi classical trace formula at a non degenerate critical level abstract we study the semi classical trace formula at a critical energy level for a h pseudodifferential operator whose principal symbol has a unique non degenerate critical point for that energy this leads to the study of hamiltonian systems near equilibrium and near the non zero periods of the linearized flow the contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals the new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface

asymptotic approximation of degenerate fiber integrals we study asymptotics of fiber integrals depending on a large parameter when the critical fiber is singular full asymptotic expansions are established in two different cases local extremum and isolated real principal type singularities the main coefficients are computed and invariantly expressed in the most singular cases it is shown that the leading term of the expansion is related to invariant measures on the spherical blow up of the singularity the results can be applied to certain degenerate oscillatory integrals which occur in spectral analysis and quantum mechanics

contributions of non extremum critical points to the semi classical trace formula we study the semi classical trace formula at a critical energy level for a h pseudo differential operator on rn whose principal symbol has a totally degenerate critical point for that energy we compute the contribution to the trace formula of isolated non extremum critical points under a condition of real principal type the new contribution to the trace formula is valid for all time in a compact subset of r but the result is modest since we have restrictions on the dimension

macroscopic behavior of microscopic oscillations in harmonic lattices via wigner husimi transforms we consider the dynamics of infinite harmonic lattices in the limit of the lattice distance tending to 0 we allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by fourier transform and linear algebra methods

hidden structure in the randomness of the prime number sequence abstract we report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers we also check numerically our findings to ensure that they hold for finite sequences of primes that would eventually appear in applications finally our theory allows us to link with three different but important topics the hardy littlewood conjecture the statistical mechanics of spin systems and the celebrated sierpinski fractal

self averaging of wigner transforms in random media we establish the self averaging properties of the wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance the main ingredients in the proof are the error estimates for the semiclassical approximation of the wigner transform by the solution of the liouville equations and the limit theorem for two particle motion along the characteristics of the liouville equations the results are applied to a mathematical model of the time reversal experiments for the acoustic waves and self averaging properties of the re transmitted wave are proved

energy transport in stochastically perturbed lattice dynamics we consider lattice dynamics with a small stochastic perturbation of order varepsilon and prove that for a space time scale of order varepsilon 1 the local spectral density wigner function evolves according to a linear transport equation describing inelastic collisions for an energy and momentum conserving chain the transport equation predicts a slow decay as 1 sqrt t for the energy current correlation in equilibrium this is in agreement with previous studies using a different method

the structure factor of primes although the prime numbers are deterministic they can be viewed by some measures as pseudo random numbers in this article we numerically study the pair statistics of the primes using statistical mechanical methods particularly the structure factor in an interval with m large and l m smaller than unity we show that the structure factor of the prime number configurations in such intervals exhibits well defined bragg like peaks along with a small diffuse contribution this indicates that primes are appreciably more correlated and ordered than previously thought our numerical results definitively suggest an explicit formula for the locations and heights of the peaks this formula predicts infinitely many peaks in any non zero interval similar to the behavior of quasicrystals however primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational we also show numerically that the diffuse part decays slowly as m and l increases this suggests that the diffuse part vanishes in an appropriate infinite system size limit

justification of the cauchy born approximation of elastodynamics we present sharp convergence results for the cauchy born approximation of general classical atomistic interactions for static problems with small data and for dynamic problems on a macroscopic time interval

asymptotics of the solutions of the random schrodinger equation we consider solutions of the schrodinger equation with a weak time dependent random potential it is shown that when the two point correlation function of the potential is rapidly decaying then the fourier transform hat zeta epsilon t xi of the appropriately scaled solution converges point wise in to a stochastic complex gaussian limit on the other hand when the two point correlation function decays slowly we show that the limit of hat zeta epsilon t xi has the form hat zeta 0 xi rm exp ib kappa t xi where b t is a fractional brownian motion

spectral fluctuations of schrodinger operators generated by critical points of the potential starting from the spectrum of schrodinger operators on mathbb r n we propose a method to detect critical points of the potential we argue semi classically on the basis of a mathematically rigorous version of gutzwiller s trace formula which expresses spectral statistics in term of classical orbits a critical point of the potential with zero momentum is an equilibrium of the flow and generates certain singularities in the spectrum via sharp spectral estimates this fluctuation indicates the presence of a critical point and allows to reconstruct partially the local shape of the potential some generalizations of this approach are also proposed

weakly nonlinear schrodinger equation with random initial data it is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data while for the general case a proof of the kinetic limit remains open we report on first progress as wave equation we consider the nonlinear schrodinger equation discretized on a hypercubic lattice since this is a hamiltonian system a natural choice of random initial data is distributing them according to the corresponding gibbs measure with a chemical potential chosen so that the gibbs field has exponential mixing the solution t x of the nonlinear schrodinger equation yields then a stochastic process stationary in x d and t if denotes the strength of the nonlinearity we prove that the space time covariance of t x has a limit as 0 for t 2 with fixed and sufficiently small the limit agrees with the prediction from kinetic theory

a semi classical trace formula for schrodinger operators in the case of a critical energy level let 2 2 v x be a schrodinger operator on rn with smooth potentialv x as x the spectrum of is discrete and one can study the asymptotic of the smoothed spectral density e k ek e as 0 here ek k nis the spectrum of and c 0 r we shall investigate the case whereeis a critical value of the symbolhof and extending the work of brummelhuis paul and uribe in 3 we will prove the existence of a full asymptotic expansion for in terms of and ln and compute the leading coefficient we will consider new weyl type estimates for the counting function nec k n ek e

theory of analogous force on number sets abstract a general statistical thermodynamic theory that considers given sequences of x integers to play the role of particles of known type in an isolated elastic system is proposed by also considering some explicit discrete probability distributions px for natural numbers we claim that they lead to a better understanding of probabilistic laws associated with number theory sequences of numbers are treated as the size measure of finite sets by considering px to describe complex phenomena the theory leads to derive a distinct analogous force fx on number sets proportional to px x t at an analogous system temperature t in particular this leads to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits the theory also allows to establish recursion relations to predict sequences of fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the benford s law in fibonacci numbers a possible relevance to prime numbers is also analyzed

similarity in the statistics of prime numbers abstract we present numerical evidence for regularities in the distribution of gaps between primes when these are divided into congruence families in dirichlet s classification the histograms for the distribution of gaps of families are scale invariant

semiclassical spectral estimates for schrodinger operators at a critical energy level case of a degenerate minimum of the potential abstract we study the semi classical trace formula at a critical energy level for a schrodinger operator on r n we assume here that the potential has a totally degenerate critical point associated to a local minimum the main result which computes the contribution of this equilibrium is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point

asymptotics of the solutions of the stochastic lattice wave equation we consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing the multiplicative noise conserves energy and in the unpinned case also conserves momentum we obtain a time inhomogeneous ornstein uhlenbeck equation for the limit wave function that holds for both square integrable and statistically homogeneous initial data the limit is understood in the point wise sense in the former case and in the weak sense in the latter on the other hand the weak limit for square integrable initial data is deterministic

kinetic limit for wave propagation in a random medium we study crystal dynamics in the harmonic approximation the atomic masses are weakly disordered in the sense that their deviation from uniformity is of the order e the dispersion relation is assumed to be a morse function and to suppress crossed recollisions we then prove that in the limit e 0 the disorder averaged wigner function on the kinetic scale time and space of order e 1 is governed by a linear boltzmann equation

summability of connected correlation functions of coupled lattice fields we consider two nonindependent random fields psi and phi defined on a countable set z for instance z mathbb z d or z mathbb z d times i where i denotes a finite set of possible internal degrees of freedom such as spin we prove that if the cumulants of psi and phi enjoy a certain decay property then all joint cumulants between psi and phi are ell 2 summable in the precise sense described in the text the decay assumption for the cumulants of psi and phi is a restricted ell 1 summability condition called ell 1 clustering property one immediate application of the results is given by a stochastic process psi t x whose state is ell 1 clustering at any time t then the above estimates can be applied with psi psi t and phi psi 0 and we obtain uniform in t estimates for the summability of time correlations of the field the above clustering assumption is obviously satisfied by any ell 1 clustering stationary state of the process and our original motivation for the control of the summability of time correlations comes from a quest for a rigorous control of the green kubo correlation function in such a system a key role in the proof is played by the properties of non gaussian wick polynomials and their connection to cumulants

anomalous energy transport in the fpu β chain we consider the energy current correlation function for the fpu lattice for small nonlinearity one can rely on kinetic theory the issue reduces then to a spectral analysis of the linearized collision operator we prove thereby that on the basis of kinetic theory the energy current correlations decay in time as t 3 5 it follows that the thermal conductivity is anomalous increasing as n2 5 with the system size n 2008 wiley periodicals inc

applications of statistical mechanics in number theory abstract the links between statistical physics and number theory are discussed first the attempts to prove the riemann hypothesis by means of the suitable spin model and the lee yang theorem about zeros of the partition function are shortly reviewed next the analogies between random walks and prime numbers are mentioned in the last section the partition function of the system whose energies are defined by the distances between consecutive primes is calculated the arguments are given that such a prime numbers gas behaves like a set of noninteracting harmonic oscillators

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

the phonon boltzmann equation properties and link to weakly anharmonic lattice dynamics for low density gases the validity of the boltzmann transport equation is well established the central object is the one particle distribution function f which in the boltzmann grad limit satisfies the boltzmann equation grad and much refined cercignani argue for the existence of this limit on the basis of the bbgky hierarchy for hard spheres at least for a short kinetic time span the argument can be made mathematically precise following the seminal work of lanford in this article a corresponding program is undertaken for weakly nonlinear both discrete and continuum wave equations our working example is the harmonic lattice with a weakly nonquadratic on site potential we argue that the role of the boltzmann f function is taken over by the wigner function which is a very convenient device to filter the slow degrees of freedom the wigner function so to speak labels locally the covariances of dynamically almost stationary measures one route to the phonon boltzmann equation is a gaussian decoupling which is based on the fact that the purely harmonic dynamics has very good mixing properties as a further approach the expansion in terms of feynman diagrams is outlined both methods are extended to the quantized version of the weakly nonlinear wave equation the resulting phonon boltzmann equation has been hardly studied on a rigorous level as one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal wigner functions for three phonon processes such a result requires extra conditions on the dispersion law we also outline the reasoning leading to fourier s law for heat conduction

localization lengths and boltzmann limit for the anderson model at small disorders in dimension 3 we prove lower bounds on the localization length of eigenfunctions in the three dimensional anderson model at weak disorders our results are similar to those obtained by schlag shubin and wolff j anal math 88 2002 for dimensions one and two we prove that with probability one most eigenfunctions have localization lengths bounded from below by o left lambda 2 over log 1 over lambda right where is the disorder strength this is achieved by time dependent methods which generalize those developed by erdos and yau commun pure appl math liii 667 753 2003 to the lattice and non gaussian case in addition we show that the macroscopic limit of the corresponding lattice random schrodinger dynamics is governed by a linear boltzmann equation

a proof of the gutzwiller semiclassical trace formula using coherent states decomposition the gutzwiller trace formula links the eigenvalues of the schrodinger operator as planck s constant goes to zero the semiclassical regime with the closed orbits of the corresponding classical mechanical system gutzwiller gave a heuristic proof of this trace formula using the feynman integral representation for the propagator of later using the theory of fourier integral operators mathematicians gave rigorous proofs of the formula in various settings here we show how the use of coherent states allows us to give a simple and direct proof

limit of fluctuations of solutions of wigner equation we consider fluctuations of the solution w e t x k of the wigner equation which describes energy evolution of a solution of the schrodinger equation with a random white noise in time potential the expectation of w e t x k converges as e 0 to bar w t x k which satisfies the radiative transport equation we prove that when the initial data is singular in the x variable that is w e 0 x k x f k and f in mathcal s mathbb r d then the laws of the rescaled fluctuation z varepsilon t varepsilon 1 2 w varepsilon t x k bar w t x k converge as e 0 to the solution of the same radiative transport equation but with a random initial data this complements the result of 6 where the limit of the covariance function has been considered

a semi classical trace formula at a totally degenerate critical level we study the semi classical trace formula at a critical energy level for an h pseudo differential operator on open image in new window n whose principal symbol has a totally degenerate critical point for that energy this problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point

semi classical spectral estimates for schrodinger operators at a critical level case of a degenerate maximum of the potential we study the semi classical trace formula at a critical energy level for a schrodinger operator on r n we assume here that the potential has a totally degenerate crit ical point associated to a local maximum the main result which establishes the contribution of the associated equilibrium in the trace formula is valid for all time in a compact subset of r and includes the singularity in t 0 for these new contributions the asymptotic expansion involves the logarithm of the parameter h depending on an explicit arithmetic condition on the dimension and the order of the critical point this logarithmic contribution can appear in the leading term

quantum diffusion of the random schrodinger evolution in the scaling limit we consider random schrodinger equations on r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and psi t the solution with initial data psi 0 the space and time variables scale as x sim lambda 2 varkappa 2 text and t sim lambda 2 varkappa text with 0 varkappa varkappa 0 left d right we prove that in the limit 0 the expectation of the wigner distribution of psi t converges weakly to the solution of a heat equation in the space variable x for arbitrary l 2 initial data

sharp semiclassical estimates for the number of eigenvalues below a totally degenerate critical level we consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than e for elliptic operators in l2 rd we describe a method of obtaining remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the intervals e e h where e is close to e if the volume of this region is o h then we obtain remainder estimates o h1 d with no assumptions on the hessian of the principal symbol at the critical level moreover we do not assume that the coefficients are smooth all results hold if second order derivatives of coefficients are holder continuous

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure