Large deviation principles for nongradient weakly asymmetric stochastic lattice gases

TL;DR

This paper establishes large deviation principles for weakly asymmetric lattice gases, characterizing their hydrodynamic limits and stationary measure behaviors under external fields, using nonlinear diffusion equations and variational analysis.

Contribution

It introduces a rigorous large deviation framework for nongradient weakly asymmetric lattice gases, including the analysis of the quasi-potential and stationary measures.

Findings

Hydrodynamic limit described by nonlinear driven diffusion equation

Large deviation principle for the empirical density established

Stationary large deviation rate function independent of external field E

Abstract

We consider a lattice gas on the discrete d-dimensional torus $(\mathbb{Z}/N\mathbb{Z})^d$ with a generic translation invariant, finite range interaction satisfying a uniform strong mixing condition. The lattice gas performs a Kawasaki dynamics in the presence of a weak external field E/N. We show that, under diffusive rescaling, the hydrodynamic behavior of the lattice gas is described by a nonlinear driven diffusion equation. We then prove the associated dynamical large deviation principle. Under suitable assumptions on the external field (e.g., E constant), we finally analyze the variational problem defining the quasi-potential and characterize the optimal exit trajectory. From these results we deduce the asymptotic behavior of the stationary measures of the stochastic lattice gas, which are not explicitly known. In particular, when the external field E is constant, we prove a stationary large deviation principle for the empirical density and show that the rate function does not depend on E.