The Annals of Applied Probability

Large deviation principles for nongradient weakly asymmetric stochastic lattice gases

Lorenzo Bertini, Alessandra Faggionato, and Davide Gabrielli

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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 $E$.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 1 (2013), 1-65.

Dates
First available in Project Euclid: 25 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1359124381

Digital Object Identifier
doi:10.1214/11-AAP805

Mathematical Reviews number (MathSciNet)
MR3059203

Zentralblatt MATH identifier
1315.60106

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C05: Classical dynamic and nonequilibrium statistical mechanics (general)
Secondary: 60F10: Large deviations 82C22: Interacting particle systems [See also 60K35]

Keywords
Stochastic lattice gases stationary nonequilibrium states large deviations

Citation

Bertini, Lorenzo; Faggionato, Alessandra; Gabrielli, Davide. Large deviation principles for nongradient weakly asymmetric stochastic lattice gases. Ann. Appl. Probab. 23 (2013), no. 1, 1--65. doi:10.1214/11-AAP805. https://projecteuclid.org/euclid.aoap/1359124381


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