Hydrodynamic limit of mean zero asymmetric zero range processes in infinite volume - Article 33 - Abstract and results

Aug 29, 2026

Uploaded by Theodora

This paper proves the hydrodynamic behaviour of mean zero, asymmetric zero range processes on the infinite lattice Zd. The argument relies on a volume-uniform bound for the entropy production of the dynamics in large finite volumes, then extends the estimates to infinite volume by lower semicontinuity. The method builds on earlier uniform entropy-production bounds by Fritz and uses an approach introduced by Yau, ultimately establishing convergence to the hydrodynamic equation.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages19

Reading Time48 min

AccessFree to download

Introduction

Local equilibrium and macroscopic evolution

Entropy methods for hydrodynamic limits

Hydrodynamic limit of mean zero asymmetric zero range processes

Structure of the infinite-volume process

Main theorem via finite-volume entropy production

Notation and results

ANNALES DE L'I.H.P.,SECTION B C.LANDIM M.MOURRAGUI Hydrodynamic limit of mean zero asymmetric zerorange processesin infinite volume Annales de l'I.H.P,section B,tome 33,n1(1997),p.65-82 ◎ Gauthier-Villars,1997,tous droits réservés. L'accès aux archives de la revue《 Annales de I'I.H.P.,section B》(http://www.elsevier.com/locate/anihpb)implique I'accord avec les condi-tions générales d'utilisation(http://www.numdam.org/conditions).Toute uti-lisation commerciale ou impression systématique est constitutive d'uneinfraction pénale.Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright. NUMDAM Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/ Hydrodynamic limit of mean zero asymmetriczero range processes in infinite volume by C.LANDIM IMPA,Estrada Dona Castorina 110,CEP 22.460,Jardim Botanico,Rio de Janeiro,Brasiland CNRS URA 1378,Université de Rouen,76821 Mont Saint-Aignan Cedex,France. and M.MOURRAGUI CNRS URA 1378,Université de Rouen,76821 Mont Saint-Aignan Cedex,France. ABSTRACT.-We prove the hydrodynamic behaviour of mean zero,asymmetric zero range processes evolving on the infinite lattice Zd.Theproof relies on a bound,uniform in the volume,for the entropy productionof processes in large finite volume.Such an entropy production bound,uniform in the volume,was first proved by Fritz in [6]to extend to infinitevolume the proof of Guo Papanicolaou and Varadhan of hydrodynamicbehaviour of interacting particle systems.Our approach follows a methodintroduced by Yau in [12]. Key words:Particle systems,hydrodynamic limit. RÉSUMÉ.-Nous prouvons le comportement hydrodynamique desprocessus de zero range asymétriques de moyenne nulle en volume infini.La démonstration repose sur une borne,uniforme par rapport au volume,de la production d'entropie du processus en volume fini.Une telle bornesur I'entropie a déjà été démontré par Fritz dans [6]pour étendre auvolume infini la démonstration de Guo,Papanicolaou et Varadhan sur lecomportement hydrodynamique des systemes de particules en interaction.Notre approche suit une méthode introduite par Yau dans [12]. Mots clés:Systemes de particules,limite hydrodynamique # INTRODUCTION The major problem in the theory of hydrodynamic limit of interactingparticle systems consists in describing the macroscopic time evolution ofa gas from the microscopic interaction between molecules.Consider,forinstance,a gas evolving on a d-dimensional volume V and assume thatall equilibrium states of the system are characterized by a macroscopicvariable p(the density,the temperature,etc.). If the system is not in equilibrium,due to the interaction betweenmolecules,we expect the process to be near equilibrium in smallneighborhoods of each macroscopic point u of the volume V.This localequilibrium will be characterized by a parameter p(u),possibly differentat each point u. We expect this local equilibrium state to change smoothly in time,that is,we expect the system,at any time t and around any point u,to be close to anew equilibrium state characterized by a parameter p(t,u).This parameterp(t,u)should evolve smoothly in time according to a differential equation,the so-called hydrodynamic equation. Although physically well understood,this passage from microscopicdynamics to macroscopic behaviour still presents in the general caseimportant mathematical problems.The interacting particle systemsintroduced by Spitzer constitute a class of stochastic models,complexenough,on the one hand,to present interesting macroscopic behaviour andrelatively simple,on the other hand,to allow rigorous mathematical proofs. Until the break through of Guo,Papanicolaou and Varadhan [7],wherethe intensive use of large deviation techniques led to a robust proof of thehydrodynamic behaviour of a large class of finite volume gradient systemswith one conserved qua

FAQs

Which prior methods or results does the paper build upon?

The approach follows a method introduced by Yau, and it relies on uniform entropy-production ideas originally proved by Fritz to extend arguments of Guo, Papanicolaou, and Varadhan.

Aug 29, 2026

Uploaded by Theodora