numerical method for solving stochastic differential equations with poissonian white shot noise we propose a numerical integration scheme to solve stochastic differential equations driven by poissonian white shot noise our formula which is based on an integral equation which is equivalent to the stochastic differential equation utilizes a discrete time approximation with fixed integration time step we show that our integration formula approaches the euler formula if the poissonian noise approaches the gaussian white noise the accuracy and efficiency of the proposed algorithm are examined by studying the dynamics of an overdamped particle driven by poissonian white shot noise in a spatially periodic potential we find that the accuracy of the proposed algorithm only weakly depends on the parameters characterizing the poissonian white shot noise this holds true even if the limit of gaussian white noise is approached

multiscale resolution in the computation of crystalline microstructure this paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy it is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations such behavior of minimizing deformations is understood for an extended model that involves surface energies moreover the closely related question of the role of different growth conditions of the employed bulk energy is discussed by explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders for first order laminates the estimates are optimal

approximation of a laminated microstructure for a rotationally invariant double well energy density we give error estimates for the approximation of a laminated microstructure which minimizes the energy int omega phi nabla v x dx for a rotationally invariant double well energy density phi a we present error estimates for the convergence of the deformation in l 2 the convergence of directional derivatives of the deformation in the twin planes the weak convergence of the deformation gradient the convergence of the microstructure or young measure of the deformation gradients and the convergence of nonlinear integrals of the deformation gradient

numerical approximation of young measures in non convex variational problems summary in non convex optimisation problems in particular in non convex variational problems there usually does not exist any classical solution but only generalised solutions which involve young measures in this paper first a suitable relaxation and approximation theory is developed together with optimality conditions and then an adaptive scheme is proposed for the efficient numerical treatment the young measures solving the approximate problems are usually composed only from a few atoms this is the main argument our effective active set type algorithm is based on the support of those atoms is estimated from the weierstrass maximum principle which involves a hamiltonian whose good guess is obtained by a multilevel technique numerical experiments are performed in a one dimensional variational problem and support efficiency of the algorithm

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials the solution of these variational prob lems must be described in terms of a microstructure rather than in terms of a deformation moreover the numerical approximation of the deformation gra dient often does not converge strongly as the mesh is refined nevertheless the probability distribution of the deformation gradients near each material point does converge recently we introduced a metric to analyze this convergence in this paper we give an optimal order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier

on contact anosov flows exponential decay of correlations for co 4 contact anosov flows is established this implies in particular exponential decay of correlations for all smooth geodesic flows in strictly negative curvature

fluctuations in reversible chemical reactions abstract the paper proposes a new approach to describe fluctuations of reversible chemical reactions in closed systems the deterministic rate laws are cast into the form of nonlinear onsager type closed laws by means of nonlinear transport theory a fokker planck equation describing the stochastic process of concentration fluctuations is obtained it is shown that the stochastic formulation reduces to the correct deterministic rate laws in the thermodynamic limit v with the concentrations kept fixed concrete examples of reactions in ideal mixtures are given and the results of the presented approach are compared with those of the usual approach by means of a birth and death type master equation it is shown that both approaches lead to the same stationary probability and exhibit the same natural boundaries reflecting the fact of a restricted state space the proposed fokker planck equation is different from the fokker planck equation obtained from the master equation by truncating its kramers moyal expansion however the two equations are shown to have identical fokker planck coefficients in the vicinity of the deterministic equilibrium state compared with the usual master equation approach the proposed stochastic modeling of chemical reactions has the advantage of allowing for a straightforward extension to reactions in non ideal mixtures

on the computation of the rank one convex hull of a function we report on several numerical experiments where the rank one convexification of an energy density is computed the explicit examples cover a whole spectrum of typical situations one may encounter one of those is especially relevant for the computation of microstructures in crystalline solids

a study on the action of non gaussian noise on a brownian particle abstract we analytically address the non equilibrium problem of a brownian particle in contact with a thermal reservoir by means of a non gaussian langevin noise term t the presence of noise kurtosis is akin to a second temperature reservoir acting on the system and we exploit its consequences by means of studying a converging exact form for the stationary probability distribution

reliable and efficient approximation of polyconvex envelopes an iterative algorithm that approximates the polyconvex envelope fpc of a given function f mathbb r n times m to mathbb r i e the largest function below f which is convex in all minors is established also presented are a rigorous error analysis with a focus on reliability and optimal orders of convergence an efficient strategy that reduces the large number of unknowns as well as numerical experiments

toward the fourier law for a weakly interacting anharmonic crystal for a system of weakly interacting anharmonic oscillators perturbed by an energy preserving stochastic dynamics we prove an autonomous stochastic evolution for the energies at large time scale with respect to the coupling parameter it turn out that this macroscopic evolution is given by the so called conservative non gradient ginzburg landau system of stochastic differential equations the proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics

nonequilibrium statistical mechanics of hamiltonian rotators with alternated spins we consider a finite region of a d dimensional lattice of nonlinear hamiltonian rotators where neighbouring rotators have opposite alternated spins and are coupled by a small potential of size varepsilon a a ge 1 2 we weakly stochastically perturb the system in such a way that each rotator interacts with its own stochastic thermostat with a force of order varepsilon then we introduce action angle variables for the system of uncoupled rotators varepsilon 0 and note that the sum of actions over all nodes is conserved by the purely hamiltonian dynamics of the system with varepsilon 0 we investigate the limiting as varepsilon rightarrow 0 dynamics of actions for solutions of the varepsilon perturbed system on time intervals of order varepsilon 1 it turns out that the limiting dynamics is governed by a certain autonomous stochastic equation for the vector of actions this equation has a completely non hamiltonian nature this is a consequence of the fact that the system of rotators with alternated spins do not have resonances of the first order the varepsilon perturbed system has a unique stationary measure widetilde mu varepsilon and is mixing any limiting point of the family widetilde mu varepsilon of stationary measures as varepsilon rightarrow 0 is an invariant measure of the system of uncoupled integrable rotators there are plenty of such measures however it turns out that only one of them describes the limiting dynamics of the varepsilon perturbed system we prove that a limiting point of widetilde mu varepsilon is unique its projection to the space of actions is the unique stationary measure of the autonomous equation above which turns out to be mixing and its projection to the space of angles is the normalized lebesque measure on the torus mathbb t n the results and convergences which concern the behaviour of actions on long time intervals are uniform in the number n of rotators those concerning the stationary measures are uniform in n in some natural cases

numerical analysis of oscillations in nonconvex problems we study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer

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

on the relation between white shot noise gaussian white noise and the dichotomic markov process it is shown that the dichotomic markov process converges to a white shot noise interpreted according to the stratonovich integration rule in the joint limit in which the average duration of one of the states goes to zero and the value at this state goes to infinity a further limit procedure allows us to obtain gaussian white noise from white shot noise these results are applied to the problem of noise induced transitions it is shown that white shot noise can give rise to transitions which do not occur for gaussian white noise the above results are finally generalized in introducing compound dichotomic markov processes

numerical approximation of non homogeneous non convex vector variational problems in this paper we study a numerical scheme for non convex vector variational problems allowing for microstructure based on the approximation of gradient young measures we present a convergence result and some numerical experiments

green kubo formula for weakly coupled systems with noise we study the green kubo formula kappa varepsilon varsigma for the heat conductivity of an infinite chain of d dimensional finite systems cells coupled by a smooth nearest neighbor potential varepsilon v the uncoupled systems evolve according to hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength varsigma noting that kappa varepsilon varsigma exists and is finite whenever varsigma 0 we are interested in what happens when the strength of the noise varsigma to 0 for this we start in this work by formally expanding kappa varepsilon varsigma in a power series in varepsilon kappa varepsilon varsigma varepsilon 2 sum n geq 2 varepsilon n 2 kappa n varsigma and investigating the formal equations satisfied by kappa n varsigma we show in particular that kappa 2 varsigma is well defined when no pinning potential is present and coincides formally with the heat conductivity obtained in the weak coupling van hove limit where time is rescaled as varepsilon 2 t for the cases where the latter has been established liverani and olla in jams 25 555 583 2012 dolgopyat and liverani in commun math phys 308 201 225 2011 for one dimensional systems we investigate kappa 2 varsigma as varsigma to 0 in three cases the disordered harmonic chain the rotor chain and a chain of strongly an harmonic oscillators moreover we formally identify kappa 2 varsigma with the conductivity obtained by having the chain between two reservoirs at temperature t and t delta t in the limit delta t to 0 n to infty varepsilon to 0

constitutive theory for some constrained elastic crystals abstract some of the observations of a 15 superconductors near cubic tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints here we begin to develop a theory of this kind

a mesh transformation method for computing microstructures a numerical method is established to solve the problem of minimizing a nonquasiconvex potential energy convergence of the method is proved both in the case on its own and in the case when it is combined with a weak boundary condition numerical examples are given to show that the method especially when applied together with a continuation method and some other numerical techniques is not only successful and efficient in solving problems with laminated microstructures but also capable of computing more complicated microstructures

a discontinuous finite element method for solving a multiwell problem many physical materials of practical relevance can attain several variants of crystalline microstructure the appropriate energy functional is necessarily nonconvex and the minimization of the functional becomes a challenging problem a new numerical method based on discontinuous finite elements and a scaled energy functional is proposed it exhibits excellent convergence behavior for the energy second order as well as other crucial quantities of interest for general spatial meshes contrary to standard non conforming methods both theoretical analyses and numerical test calculations are presented and contrasted to other current finite element methods for this problem

on the numerical analysis of non convex variational problems we discuss a numerical method for finding young measure valued minimizers of non convex variational problems to have any hope of a convergence theorem one must work in a setting where the minimizer is unique and minimizing sequences converge strongly this paper has two main goals i we specify a method for producing strongly convergent minimizing sequences despite the failure of strict convexity and ii we show how uniqueness of the young measure can be parlayed into a numerical convergence theorem the treatment of ii is done in the setting of two model problems one involving scalar valued functions and a multiwell energy the other from micromagnetics

numerical analysis of oscillations in multiple well problems variational problems which fail to be convex occur often in the study of ordered materials such as crystals in these problems the energy density for the material has multiple potential wells in this paper we study multiple well problems by first determining the analytic properties of energy minimizing sequences and then by estimating the continuous problem by an approximation using piecewise linear finite elements we show that even when there is no minimizer of the energy the approximations still take on a predictable structure

effective relaxation for microstructure simulations algorithms and applications for a wide class of problems in continuum mechanics like those involving phase transitions or finite elastoplasticity the governing potentials tend to be not quasiconvex this leads to the occurrence of microstructures of in principle arbitrarily small scale which cannot be resolved by standard discretization schemes their effective macroscopic properties however can efficiently be recovered with relaxation theory the paper introduces the variational framework necessary for the implementation of relaxation algorithms with emphasis on problems with internal variables in a time incremental setting the methods developed are based on numerical approximations to notions of generalized convexification the focus is on the thorough analysis of numerical algorithms and their efficiency in applications to benchmark problems an outlook to time evolution of microstructures within the framework of relaxation theory concludes the paper 2004 published by elsevier b v

numerical approximations in variational problems with potential wells in this paper some numerical aspects of variational problems which fail to be convex are studied it is well known that for such a problem in general the infimum of the energy the functional that has to be minimized fails to be attained instead minimizing sequences develop oscillations which allow them to decrease the energy it is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed it is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant an estimate of the approximate deformation as it approximates a measure and some numerical results are also given

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

some remarks on nonequilibrium dynamics of infinite particle systems classical mechanics of infinitely many particles in dimensions one and two is considered particles interacting by a superstable pair potential of finite range the group of motion generated by newton s equations is constructed in the space of locally finite configurations with a logarithmic order of energy fluctuations at infinity a core of the liouville operator is also described results of dobrushin and the author and of marchioro pellegrinotti pulvirenti are improved

strong convergence of numerical solutions to degenerate variational problems numerical approximations of strongly degenerate variational problems of the form j u 0 1 f u u f 2 are considered where f is assumed convex but may have intervals where f 0 it is shown that in spite of the degeneracy natural numerical approximations still converge in w 1 p rates in weaker norms and the connection with nonconvex variational problems are also considered

a numerical iterative scheme for computing finite order rank one convex envelopes abstract it is known that the ith order laminated microstructures can be resolved by the kth order rank one convex envelopes with k i so the requirement of establishing an efficient numerical scheme for the computation of the finite order rank one convex envelopes arises in this paper we develop an iterative scheme for such a purpose the first order rank one convex envelope r1f is approximated by evaluating its value on matrixes at each grid point in rmn and then extend to non grid points by interpolation the approximate kth order rank one convex envelope rkf is obtained iteratively by computing the approximate first order rank one convex envelope of the numerical approximation of rk 1f compared with o h1 3 obtained so far for other methods the optimal convergence rate o h is established for our scheme and numerical examples illustrate the computational efficiency of the scheme

stochastic stability of pollicott ruelle resonances kinetic brownian motion on the cosphere bundle of a riemannian manifold mathbb m is a stochastic process that models the geodesic equation perturbed by a random white force of size varepsilon when mathbb m is compact and negatively curved we show that the l 2 spectrum of the infinitesimal generator of this process converges to the pollicott ruelle resonances of mathbb m as varepsilon goes to 0

approximated convex envelope of a function the goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope such a problem arises in various physical situations where the function considered is some energy that has to be minimized this study is a first step toward understanding how to approximate the quasi convex envelope of a function the importance of this issue is due to the various applications that are encountered in particular in the field of material science 1994 copyright society for industrial and applied mathematics

on the relations between markovian master equations and stochastic differential equations abstract we have investigated the relationship between markovian master equations m e and the corresponding stochastic differential equations s d e for closed systems i e systems not subjected to external pumping we show that the form of the fluctuations in the s d e i e additive or multiplicative depends upon the properties of the kernel of the m e and the range of the state space of the stochastic variable s i e bounded or unbounded the knowledge of these two properties of the m e permits the determination of the way in which the fluctuations enter into the s d e i e additive or multiplicative and the calculation of their statistics several examples are presented to illustrate the general theory

numerical analysis of compatible phase transitions in elastic solids the variational model of phase transitions for elastic materials based on linearized elasticity leads to a nonconvex minimization problem p in which a minimum need not be attained in the design of advanced materials the main interest is in reliable numerical predictions of certain macroscopic quantities such as the global deformation and the stress field determined in a relaxed problem qp an explicit formula of the quasi convexified energy density in qp due to r v kohn provides us with a well posed numerical problem first a mathematical a priori and a posteriori error analysis is established for the finite element approximation of the stress variable then the residual based error indicator is implemented within an adaptive mesh refinement algorithm numerical examples illustrate that the macroscopic properties of the materials are computed efficiently with appropriate error control

the appearance of microstructures in problems with incompatible wells and their numerical approach the goal of the paper is to analyze the creation of microstructure in problems of calculus of variations with wells more precisely we consider a case with strong incompatibility between the wells this forces the minimizing sequences to use other gradients than the wells in a puzzling way using a p 1 method we are then able to single out discrete minimizing sequences and to give energy estimates in terms of the mesh size

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

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

nonconforming finite element approximation of crystalline microstructure we consider a class of nonconforming nite element approxima tions of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can un dergo either an orthorhombic to monoclinic double well or a cubic to tetrag onal triple well transformation we rst establish a series of error bounds in terms of elastic energies for the l 2 approximation of derivatives of the de formation in the direction tangential to parallel layers of the laminate for the l 2 approximation of the deformation for the weak approximation of the de formation gradient for the approximation of volume fractions of deformation gradients and for the approximation of nonlinear integrals of the deformation gradient we then use these bounds to give corresponding convergence rates for quasi optimal nite element approximations

numerical analysis of a nonconvex variational problem related to solid solid phase transitions the description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems in this paper a two dimensional model of such materials is studied due to the fact that the corresponding functional has two symmetry related martensitic energy wells the numerical approximation of the deformation gradient does not converge but tends to oscillate between the two wells as the size of the mesh is refined these oscillations may be interpreted in terms of microstructures using a nonconforming mathcal p 1 finite element an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient near one of the two martensitic wells

on the exact and phenomenological langevin equations for a harmonic oscillator in a fluid we have derived starting from the liouville equation and using projection operators the three dimensional langevin and fokker planck equations for the time dependence of the momenta and position of two brownian particles of mass m interacting with a harmonic potential i e a harmonic oscillator in a fluid of particles with mass m with m m the resulting langevin equation has a very complicated structure in that the friction coefficients x due to the hydrodynamic interaction between the oscillator particles are functions of r t the time dependent seperation of the oscillator particles and the noise terms though gaussian are non stationary and of a generalized form i e neither additive nor multiplicative in addition there is a term involving the mean force exerted by the fluid on the oscillating brownian particle we then investigated the various approximations which must be made to reduce this exact langevin equation to the frequently used one dimensional phenomenological langevin equations and corresponding fokker planck equations with purely additive and multiplicative noise these approximations are of three types a one must neglect the term arising from the rotational motion of the oscillator in the fluid b one must neglect the r t dependence of x r t leading to purely additive noise or approximate x r t by a taylor series expansion to quadratic order in r t r0 where r0 is the equilibrium separation of the oscillator particles leading to additive and multiplicative noise and c one must either neglect the mean force term or approximate it by a term linear in r t r0 we conclude from these results that one dimensional phenomenological langevin equations with simple noise structure are of doubtful validity for the dynamical description of the relative momenta or vibrational energy of oscillating molecules in a fluid

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

on the convergence of finite element approximations of a relaxed variational problem the accuracy of finite element approximations to a convex but not strictly convex variational problem is considered convergence is proved for a finite element approximation of a particular vector field related to the solution in a special one dimensional case 0 h convergence is shown for a piecewise linear approximation of the derivative h denotes the size of each element domain numerical results are also presented for this one dimensional case

linear programming approach to nonconvex variational problems in nonconvex optimization problems in particular in nonconvex variational problems there usually does not exist any classical solution but only generalized solutions which involve young measures in this paper after reviewing briefly the relaxation theory for such problems an iterative scheme leading to a sequential linear programming slp scheme is introduced and its convergence is proved by a banach fixed point technique then an approximation scheme is proposed and analyzed and calculations of an illustrative 2d broken extremal example are presented

non equilibrium statistical mechanics of strongly anharmonic chains of oscillators we study the model of a strongly non linear chain of particles coupled to two heat baths at different temperatures our main result is the existence and uniqueness of a stationary state at all temperatures this result extends those of eckmann pillet rey bellet epr99a epr99b to potentials with essentially arbitrary growth at infinity this extension is possible by introducing a stronger version of hormander s theorem for kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains

asymptotic localization of energy in nondisordered oscillator chains we study two popular one dimensional chains of classical anharmonic oscillators the rotor chain and a version of the discrete nonlinear schrodinger chain we assume that the interaction between neighboring oscillators controlled by the parameter 0 is small we rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant and we provide strong evidence that it decays faster than any power law in as 0 the weak coupling regime also translates into a high temperature regime suggesting that the conductivity vanishes faster than any power of the inverse temperature to our knowledge it is the first time that a clear connection has been established between kam like phenomena and thermal conductivity 2015 wiley periodicals inc

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

finite element analysis of microstructure for the cubic to tetragonal transformation martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetry related rotationally invariant energy wells we give estimates for the numerical approximation of a first order laminate for such martensitic crystals we give bounds for the l2 convergence of directional derivatives in the twin plane for the l2 convergence of the deformation for the weak convergence of the deformation gradient for the convergence of the microstructure and for the convergence of nonlinear integrals of the deformation gradient

convergence of a reduced integration method for computing microstructures recent interest in materials which form microstructures has led to efforts to determine the material configurations computationally an approximation method using bilinear finite elements with a reduced integration scheme and a local postprocessing projection has been found in numerical experiments to have the flexibility to represent microstructures on various scales and for various orientations with respect to the grid and has produced results which agree with the theory in this paper we give a rigorous proof of the convergence of the microstructure for this method

ten remarks on nonconvex minimisation for phase transition simulations nonconvex minimisation problems are encountered in many applications such as phase transitions in solids or liquids but also in optimal design tasks or micromagnetism in contrast to rubber type elastic materials and many other variational problems in continuum mechanics the minimal energy may be not attained in the sense of sobolev functions the nonrank 1 convex minimisation problem m is ill posed the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures their limit is a measure called gradient young measure gym and describes the effective energy density w qc the quasiconvex envelope of the original energy density w this gives rise to a relaxed minimisation problem r which is well posed in the sense that the minimum is attained the paper compares computational aspects of the two problems m and r neither adaptive finite element methods nor effective solvers may work for discrete versions of m but theoretical and numerical evidence supports that they work for r the drawback of r is that the relaxed energy density w qc is not always given by a known closed form formula instead a numerical relaxation has to be involved which is of the form m and hence yields a computational challenge other semiconvexity notions are requested and give rise to a new category of algorithms the demand for quasiconvexification algorithms is highly motivated by recent models in finite plasticity as the time discretisation in the latter models typically leads to nonrank 1 convex minimisation problems 2004 published by elsevier b v

diffusion in the mean for an ergodic schrodinger equation perturbed by a fluctuating potential diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time if the static random potential is strong enough to induce complete localization in the absence of time dependent noise then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part

gradient dynamics of infinite point systems nonequilibrium gradient dynamics of d dimensional particle systems is investigated the interaction is given by a superstable pair potential of finite range solutions are constructed in the well defined set of locally finite configurations with a logarithmic order of energy fluctuations if the system is deterministic and d 2 then singular potentials are also allowed for stochastic models with a smooth interaction we need d 4 in order to develop some prerequisites for the theory of hydrodynamical fluctuations in equilibrium we investigate smoothness of the markov semigroup and describe some properties of its generator 0 introduction the purpose of this paper is to study existence and regularity properties of solutions to the following infinite system of stochastic differential equations consider a countable set s of d dimensional particles suspended in a fluid where the interaction is given by a pair potential u rgd 00 oo in a quasi microscopic description of such systems the effect of collisions with the particles of the fluid can be represented by uncorrelated stochastic forces and the soft resistance of the liquid medium reduces the order of the equations of motion from two to one see 13 23 and 24 for some further references configurations of these systems are countable subsets of rd such that any bounded domain contains a finite number of points only particles of a configuration will be identified by labelling points by elements of s thus a