Large deviations of the empirical current for the boundary driven Kawasaki process with long range interaction

TL;DR

This paper studies the large deviations of the empirical current and density in a boundary-driven Kawasaki process with long-range interactions, deriving hydrodynamic limits and nonlocal evolution equations.

Contribution

It introduces a detailed analysis of large deviations for a Kawasaki process with long-range interactions and boundary reservoirs, extending existing models.

Findings

Hydrodynamic limit described by a nonlocal nonlinear PDE.

Large deviation principles established for empirical current and density.

Characterization of the empirical current fluctuations in the model.

Abstract

We consider a lattice gas evolving in a bounded cylinder of length 2N + 1 and interacting via a Neuman Kac interaction of range N, in contact with particles reservoirs at different densities. We investigate the associated law of large numbers and large deviations of the empirical current and of the density. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by a nonlocal, nonlinear evolution equation with Dirichlet boundary conditions.