second order elliptic operators and heat kernels on lie groups arendt batty and robinson proved that each second order strongly elliptic operator c associated with left translations on the lp spaces of a lie group g generates an interpolating family of semigroups t whenever the coefficients of c are sufficiently smooth we establish that t has an integral kernel k satisfying the bounds a it ee kt g h at e g te where d is the dimension of g igh 1 l is the right invariant distance from h to g and a b co etc are positive constants both bounds are derived by generalization of nash s arguments for pure second order operators on rd

nonequilibrium fluctuations for a tagged particle in mean zero one dimensional zero range processes we prove a non equilibrium functional central limit theorem for the position of a tagged particle in mean zero one dimensional zero range process the asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation

analysis of dynamics in magnetotelluric data by using the fisher shannon method abstract the fisher shannon information fs plane defined by the fisher information measure and the shannon entropy power is used to investigate the complex dynamics of magnetotelluric data of three stations in taiwan in the fs plane the electric and magnetic components are significantly separated characterized by different degrees of order further investigation shows that signals measured in areas with very high level of seismic activity are well discriminated

derivation of the hydrodynamical equation for one dimensional ginzburg landau model the hydrodynamical behavior of one dimensional scalar ginzburg landau model with conservation law is investigated the dynamics of the system is given by solving a stochastic partial differential equation under appropriate space time scaling a deterministic limit is obtained and the limit is described by a certain nonlinear diffusion equation

derivation of a hydrodynamic equation for ginzburg landau models in an external field the lattice approximation to a time dependent ginzburg landau equation is investigated in the presence of a small external field the evolution law conserves the spin but is not reversible a nonlinear diffusion equation of divergence type is obtained in the hydrodynamic limit the proof extends to certain stochastically perturbed hamiltonian systems

fluctuations of one dimensional ginzburg landau models in nonequilibrium we study the fluctuation of one dimensional ginzburg landau models in nonequilibrium along the hydrodynamic diffusion limit the hydrodynamic limit has been proved to be a nonlinear diffusion equation by fritz guo papanicolaou varadhan etc we proved that if the potential is uniformly convex then the fluctuation process is governed by an ornstein uhlenbeck process whose drift term is obtained by formally linearizing the hydrodynamic equation

hydrodynamical limit for a hamiltonian system with weak noise starting from a general hamiltonian system with superstable pairwise potential we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles we prolve that in the scaling limit the time conserved quantities energy momenta and density satisfy the euler equation of conservation laws up to a fixed timet provided that the euler equation has a smooth solution with a given initial data up to timet the strength of the noise term is chosen to be very small but nonvanishing so that it disappears in the scaling limit

multiscale temporal integrators for fluctuating hydrodynamics following on our previous work s delong b e griffith e vanden eijnden and a donev phys rev e 87 033302 2013 we develop temporal integrators for solving langevin stochastic differential equations that arise in fluctuating hydrodynamics our simple predictor corrector schemes add fluctuations to standard second order deterministic solvers in a way that maintains second order weak accuracy for linearized fluctuating hydrodynamics we construct a general class of schemes and recommend two specific schemes an explicit midpoint method and an implicit trapezoidal method we also construct predictor corrector methods for integrating the overdamped limit of systems of equations with a fast and slow variable in the limit of infinite separation of the fast and slow time scales we propose using random finite differences to approximate some of the stochastic drift terms that arise because of the kinetic multiplicative noise in the limiting dynamics we illustrate our integrators on two applications involving the development of giant nonequilibrium concentration fluctuations in diffusively mixing fluids we first study the development of giant fluctuations in recent experiments performed in microgravity using an overdamped integrator we then include the effects of gravity and find that we also need to include the effects of fluid inertia which affects the dynamics of the concentration fluctuations greatly at small wave numbers

functional inequalities on arbitrary riemannian manifolds abstract for any connected not necessarily complete riemannian manifold we construct a probability measure of type e v x d x where d x is the riemannian volume measure and v is a function c smooth outside a closed set of zero volume satisfying poincare sobolev type functional inequalities in particular v is c smooth on the whole manifold when the poincare and the super poincare inequalities are considered the sobolev inequality for infinite measures are also studied

on entropy minimization and convergence we examine the minimization of information entropy for measures on the phase space of bounded domains subject to constraints that are averages of grand canonical distributions we describe the set of all such constraints and show that it equals the set of averages of all probability measures absolutely continuous with respect to the standard measure on the phase space with the exception of the measure concentrated on the empty configuration we also investigate how the set of constrains relates to the domain of the microcanonical thermodynamic limit entropy we then show that for fixed constraints the parameters of the corresponding grand canonical distribution converge as volume increases to the corresponding parameters derivatives when they exist of the thermodynamic limit entropy the results hold when the energy is the sum of any stable tempered interaction potential that satisfies the gibbs variational principle e g lennard jones and the kinetic energy the same tools and the strict convexity of the thermodynamic limit pressure for continuous systems valid whenever the gibbs variational principle holds give solid foundation to the folklore local homeomorphism between thermodynamic and macroscopic quantities

hydrodynamic scaling limits with deterministic initial configurations we discuss the hydrodynamic scaling limits starting with deterministic configurations for different models certain estimates on the entropy of the system are derived

hydrodynamics and large deviation for simple exclusion processes we prove the hydrodynamical limit for weakly asymmetric simple exclusion processes a large deviation property with respect to this limit is established for the symmetric case we treat also the situation where a slow reaction creation and annihilation of particles is present

equilibrium fluctuations for interacting brownian particles we consider an infinitely extended system of brownian particles interacting by a pair force gradv their initial distribution is stationary and given by the gibbs measure associated with the potentialv with fugacityz we assume thatv is symmetric finite range three times continuously differentiable superstable and positive and that the fugacity is small in the sense that 0 z 0 28 eedq 1 e v q in addition a certain essential self adjointness property is assumed we prove then that the time dependent fluctuations in the density on a spatial scale of order 1 and on a time scale of order 2 converge as 0 to a gaussian field with covariance edqg q e 2 t f q withp the density and the compressibility

scaling limit for interacting ornstein uhlenbeck processes the problem of describing the bulk behavior of an interacting system consisting of a large number of particles comes up in different contexts see for example 1 for a recent exposition in 4 one of the authors considered the case of interacting diffusions on a circle and proved that the density of particles evolves according to a nonlinear diffusion equation the interacting particles evolved according to a generator that was symmetric in equilibrium in this article we consider interacting ornstein uhlenbeck processes here the diffusion generator is not symmetric relative to the equilibrium and the earlier methods have to be modified considerably we use some ideas that were employed in 3 to extend the central limit theorem from the symmetric to nonsymmetric cases

equilibrium fluctuations for interacting ornstein uhlenbeck particles we study the hydrodynamic density fluctuations of an infinite system of interacting particles on d the particles interact between them through a two body superstable potential and with a surrounding fluid in equilibrium through a random viscous force of ornstein uhlenbeck type the stationary initial distribution is the gibbs measure associated with the potential and with a given temperature and fugacity we prove that the time dependent density fluctuation field converges in law under diffusive scaling of space and time to the solution of a linear stochastic partial differential equation driven by white noise

discrete approximation of the radial contribution to the schrodinger time evolution operator in three dimensional euclidean space the laplace operator 2 on n dimensional euclidean space in spherical coordinates is the free particle schr dinger time evolution operator may be constructed by exponentiation denoting a central finite difference approximation of r by the matrix with is investigated and explicitly evaluated for n 3 provides an approximation of the leading term of the radial component of the kinetic energy contribution to the evolution operator an unconditionally stable numerical algorithm for quantum mechanical scattering is proposed based on this approximation when n 3

on the diffusive nature of entropy flow in infinite systems remarks to a paper by guo papanicolau varadhan the hydrodynamic behaviour of interacting diffusion processes is investigated by means of entropy free energy arguments the methods of 13 are simplified and extended to infinite systems including a case of anharmonic oscillators in a degenerate thermal noise following 14 15 and 3 5 we derive a priori bounds for the rate of entropy production in finite volumes as the size of the whole system is infinitely extended the flow of entropy through the boundary is controlled in much the same way as energy flow in diffusive systems 4

large deviations for a stochastic model of heat flow we investigate a one dimensional chain of 2n harmonic oscillators in which neighboring sites have their energies redistributed randomly the sites n and n are in contact with thermal reservoirs at different temperature and kipnis et al j statist phys 27 65 74 1982 proved that this model satisfies fourier s law and that in the hydrodynamical scaling limit when n the stationary state has a linear energy density profile bar theta u u 1 1 we derive the large deviation function s u for the probability of finding in the stationary state a profile u different from bar theta u the function s has striking similarities to but also large differences from the corresponding one of the symmetric exclusion process like the latter it is nonlocal and satisfies a variational equation unlike the latter it is not convex and the gaussian normal fluctuations are enhanced rather than suppressed compared to the local equilibrium state we also briefly discuss more general models and find the features common in these two and other models whose s is known

hydrodynamic limit for a system with finite range interactions we study a system of interacting diffusions the variables present the amount of charge at various sites of a periodic multidimensional lattice the equilibrium states of the diffusion are canonical gibbs measures of a given finite range interaction under an appropriate scaling of lattice spacing and time we derive the hydrodynamic limit for the evolution of the macroscopic charge density

a comparison of different propagation schemes for the time dependent schro dinger equation abstract a comparison of three widely used time propagation algorithms for the time dependent schrodinger equation is described a typical evolution problem is chosen to demonstrate the efficiency and accuracy of the various methods on a numerical grid using a pseudo spectral fft spatial representation for scattering and bound state evolution the methods used second order differencing split operator propagation chebyshev polynomial expansion are discussed in terms of their applicability to various classes of dynamic problems a new method is introduced which is based upon a low order lanczos technique this method appears to offer an accurate and flexible alternative to the existing techniques overall the chebyshev method is recommended for time independent potentials and the lanczos method for time dependent potentials

relative entropy and mixing properties of infinite dimensional diffusions let be a diffusion process taking values on the infinite dimensional space t z where t is the circle and with components satisfying the equations d i i dw i b i dt for some coefficients i and b i i z suppose we have an initial distribution and a sequence of times t n such that lim n s tn exists where s t is the semi group of the process we prove that if i and b i are bounded of finite range have uniformly bounded second order partial derivatives and inf i i 0 then is invariant

local stability and hydrodynamical limit of spitzer s one dimensional lattice model an infinite system of ordinary differential equations is considered the right hand side is just the negative gradient of potential energy of a one dimensional system of unbounded spins interacting by a symmetric and convex pair potential constant configurations are stationary points and the mean spin is conserved it is shown that each of these stationary points has its own domain of attraction the initial distribution need not be translation invariant as a consequence we obtain that the mean spin satisfies the heat equation in the hydrodynamical limit

scaling limit for a mechanical system of interacting particles a system of a large number of classical particles moving on a onedimensional segment with virtually reflecting boundaries is studied the particles interact with one another through repulsive pair potential forces and are subjected to resistance proportional to their velocities because of the latter it is only the number of particles that is conserved under the evolution of the system it is proved that in the hydrodynamic limit of diffusion type scaling the normalized counting measure of particle locations converges and its limiting density is governed by a non linear diffusion equation which in typical cases is of porous media equation type

large deviations of interacting particle systems in infinite volume we prove a large deviation theorem from the hydrodynamical limit for the empirical measure of ginzburg landau and zero range processes in infinite volume starting from deterministic initial configurations in the ginzburg landau case the main tool is the study of the evolution of the h 1 norm and in the zero range case the attractiveness which allows couplings

systems of interacting diffusions with partial annihilation through membranes we introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface ii this system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition a related interacting random walk model with discrete state spaces has recently been introduced and studied in chen and fan 2014 in this paper we establish the functional law of large numbers for this new system thereby extending the hydrodynamic limit in chen and fan 2014 to reflected diffusions in domains with mixed type boundary conditions which include absorption harvest of electric charges we employ a new and direct approach that avoids going through the delicate bbgky hierarchy

on the hydrodynamic limit of a ginzburg landau lattice model the law of large numbers in arbitrary dimensions a lattice system of interacting diffusion processes is investigated the evolution is attractive and time reversible the spin satisfies a conservation law it is shown that the rescaled spin field converges in probability to the corresponding solution to a nonlinear diffusion equation

fisher information and nonlinear dynamics abstract the past decade has witnessed an impressive growth in the scope and variety of physical applications of fisher s information measure fim as a consequence fim is widely regarded today as providing one with a unifying principle for physics in the present paper we investigate its usefulness in connection with the description of important aspects of the behavior of dynamical systems and find that the unification referred to above can be extended to the field of these systems

a new fourth order fourier bessel split step method for the extended nonlinear schrodinger equation fourier split step techniques are often used to compute soliton like numerical solutions of the nonlinear schrodinger equation here a new fourth order implementation of the fourier split step algorithm is described for problems possessing azimuthal symmetry in 3 1 dimensions this implementation is based in part on a finite difference approximation fda of 1 r r r r that possesses an associated exact unitary representation of e i 2 fda the matrix elements of this unitary matrix are given by special functions known as the associated bessel functions hence the attribute fourier bessel for the method the fourier bessel algorithm is shown to be unitary and unconditionally stable the fourier bessel algorithm is employed to simulate the propagation of a periodic series of short laser pulses through a nonlinear medium this numerical simulation calculates waveform intensity profiles in a sequence of planes that are transverse to the general propagation direction and labeled by the cylindrical coordinate z these profiles exhibit a series of isolated pulses that are offset from the time origin by characteristic times and provide evidence for a physical effect that may be loosely termed normal mode condensation normal mode condensation is consistent with experimentally observed pulse filamentation into a packet of short bursts which may occur as a result of short intense irradiation of a medium

convergence to equilibrium of conservative particle systems on mathbb z bm d we consider the ginzburg landau process on the lattice zd whose potential is a bounded perturbation of the gaussian potential we prove that the decay rate to equilibrium in the variance sense is t d 2 up to a logarithmic correction for any function u with finite triple norm that is u x zd xu

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

large deviation principle for interacting brownian motions we prove the large deviation principle for the empirical process in a system of locally interacting brownian motions in the nonequilibrium such a phenomenon has been proven only for two lattice systems the symmetric simple exclusion process and the zero range process therefore we have achieved the third result in this context and moreover the first result for the diffusion type interacting particle system 2016 wiley periodicals inc

efficient finite difference solutions to the time dependent schrodinger equation the matrix elements of the exponential of a finite difference realization of the one dimensional laplacian are found exactly this matrix is used to formulate an efficient algorithm for the numerical solution to the time dependent quantum mechanical scattering of a single particle from a time independent potential in one space and one time dimension the method generalizes to higher spatial dimensions as well as to multiparticle problems

scaling limits for interacting diffusions we consider a large number of particles diffusing on a circle interacting through a drift resulting from the gradient of a pair potential whose support is of the order of the interparticle distance we derive a nonlinear bulk diffusion equation for the density of the particle distribution on the circle the diffusion coefficient is determined as a function of density in terms of standard thermodynamical objects

associated bessel functions and the discrete approximation of the free particle time evolution operator in cylindrical coordinates a central finite difference approximation for the radial contribution r to the laplacian 2 r r r is considered in a three dimensional cylindrical coordinate system r z a free particle schrodinger time evolution operator is constructed by exponentiation e i 2 2 e 1 2 2 r r r e i 2 r r e i 2 r e i 2 r denoting the central finite difference approximation of r by 1 r2 t the matrix s e i 2 t with r2 is shown to be similar to a particular unitary representation uvk of the group of motions on euclidean three space that has been described by vilenkin and klimyk the matrix elements of uvk generalize the bessel function and provide an approximation of the leading term in the radial contribution to the evolution operator

explicit constants for gaussian upper bounds on heat kernels on utilise des inegalites logarithmiques de sobolev pour obtenir des bornes superieures gaussiennes sur les noyaux de la chaleur associes a des operateurs d ordre 2 elliptiques et hypoelliptiques

analysis of electromagnetic pre seismic emissions using fisher information and tsallis entropy abstract in this work we investigate the dynamics of electromagnetic precursors recorded prior to significant earthquakes in greece the analysis of these signals is performed using fisher information which is a powerful tool for investigating complex and non stationary signals our results point to a decrease of the precursor complexity as the main tectonic event is approaching the results are compatible to those derived using tsallis entropy the sensitivity of fisher information and tsallis entropy are compared on the basis of the features of the underlying fracture process which they reveal

the fisher information measure and shannon entropy for particulate matter measurements abstract we investigated the dynamics of particulate matter data recorded in tito a small industrial area of southern italy the analysis of these signals was performed using the fisher information measure fim which is a powerful tool for investigating complex and nonstationary signals and the shannon entropy which is a well known tool for investigating the degree of disorder in dynamical systems our results point to an increase of disorder and complexity from fine to coarse particulates

derivation of cahn hilliard equations from ginzburg landau models the generalized cahn hilliard equation is obtained as the hydrodynamic limit from a stochastic ginzburg landau model the associated large deviation principle is also proved in the one dimensional case we prove a related result about the scaling limit of conservative langevin dynamics of an sos surface

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

modeling tumor cell migration from microscopic to macroscopic models it has been shown experimentally that contact interactions may influence the migration of cancer cells previous works have modelized this thanks to stochastic discrete models cellular automata at the cell level however for the study of the growth of real size tumors with several million cells it is best to use a macroscopic model having the form of a partial differential equation pde for the density of cells the difficulty is to predict the effect at the macroscopic scale of contact interactions that take place at the microscopic scale to address this we use a multiscale approach starting from a very simple yet experimentally validated microscopic model of migration with contact interactions we derive a macroscopic model we show that a diffusion equation arises as is often postulated in the field of glioma modeling but it is nonlinear because of the interactions we give the explicit dependence of diffusivity on the cell density and on a parameter governing cell cell interactions we discuss in detail the conditions of validity of the approximations used in the derivation and we compare analytic results from our pde to numerical simulations and to some in vitro experiments we notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses supercooled liquids and jamming systems

maximum norm error bound of a linearized difference scheme for a coupled nonlinear schrodinger equations in this article a linearized conservative difference scheme for a coupled nonlinear schrodinger equations is studied the discrete energy method and an useful technique are used to analyze the difference scheme it is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm numerical experiments are presented to support the theoretical results

microscopic theory of isothermal elastodynamics this paper examines random perturbations of the anharmonic chain of coupled oscillators the microscopic system has two conservation laws and its hyperbolic scaling limit results in the quasi linear wave equation p system of isothermal isentropic elasticity in the shock regime the compensated compactness method is used lastly results from j w shearer and d serre are applied

scaling limits of interacting diffusions with arbitrary initial distributions it is remarked that for brownian particles interacting with a smooth repulsive pair potential the nonlinear diffusion equation which s varadhan has derived under an entropy bound for initial densities is valied whatever initial distribution they start with

entropy large deviations and scaling limits we often model the evolution of a complex system at the lowest or micro level because that is what makes the most physical sense but if the system is large we invariably want answers to questions posed on the larger macro or global scale this then involves the study of equations with a large number of variables and extracting useful information from their solutions the macroscopic quantities of interest may not contain complete information about the microscopic variables that drive their evolution something has to be done to obtain a closed system of equations for quantities of interest different contexts require different approaches we are concerned here with the evolution of large systems of interacting particles or field variables that have some built in noise and outline some of the work done at the institute during the last 25 years 2013 wiley periodicals inc

diffusion of color in the simple exclusion process we prove a diffusion scaling limit for the macroscopic densities of colored particles performing the simply excluded random walk and relate this to the limiting behavior of a test particle in equilibrium

uniform convergence of proliferating particles to the fkpp equation abstract in this paper we consider a system of brownian particles with proliferation whose rate depends on the empirical measure the dependence is more local than a mean field one and has been called moderate interaction by oelschlager 16 17 we prove that the empirical process converges uniformly in the space variable to the solution of the fisher kolmogorov petrowskii piskunov equation we use a semigroup approach which is new in the framework of these systems and is inspired by some literature on stochastic partial differential equations

stationary states of random hamiltonian systems we investigate the ergodic properties of hamiltonian systems subjected to local random energy conserving perturbations we prove for some cases e g anharmonic crystals with random nearest neighbor exchanges or independent random reflections of velocities that all translation invariant stationary states with finite entropy per unit volume are microcanonical gibbs states the results can be utilized in proving hydrodynamic behavior of such systems

the hydrodynamic limit for a system with interactions prescribed by ginzburg landau type random hamiltonian as a microscopic model we consider a system of interacting continuum like spin field overr d its evolution law is determined by the ginzburg landau type random hamiltonian and the total spin of the system is preserved by this evolution we show that the spin field converges under the hydrodynamic space time scalling to a deterministic limit which is a solution of a certain nonlinear diffusion equation this equation describes the time evolution of the macroscopic field the hydrodynamic scaling has an effect of the homogenization on the system at the same time

hydrodynamic limit for a spin system on a multidimensional lattice the hydrodynamic limit for a markov process of 0 valued spin fields on a periodic multidimensional lattice is studied in the process a positive real number called energy is attached to each site of the lattice and each couple of adjacent sites exchange thier energy by random amounts at random times the law of the exchange is such that the sum of the total energy is conserved and that the process is reversible and of gradient type for the energy distribution we show that under diffusion type scaling of space and time the macroscopic energy distribution converges to a deterministic limit which is characterized by a non linear diffusion equation t 2 1 p wherep is an increasing function which in a typical case equals const 2