Boundary driven Kawasaki process with long range interaction: dynamical large deviations and steady states

TL;DR

This paper studies a boundary-driven particle system with long-range interactions, deriving large deviation principles and analyzing steady states through integro-differential equations and hydrodynamic limits.

Contribution

It introduces a novel analysis of a Kawasaki process with long-range interactions, establishing dynamical large deviations and characterizing steady states.

Findings

Empirical density solves a quasi-linear integro-differential equation.

Large deviation principle for the empirical density is proved.

For small interaction strength, a law of large numbers for stationary measures is established.

Abstract

A particle system with a single locally-conserved field (density) in a bounded interval with different densities maintained at the two endpoints of the interval is under study here. The particles interact in the bulk through a long range potential parametrized by $\beta\ge 0$ and evolve according to an exclusion rule. It is shown that the empirical particle density under the diffusive scaling solves a quasi-linear integro-differential evolution equation with Dirichlet boundary conditions. The associated dynamical large deviation principle is proved. Furthermore, for $\beta$ small enough, it is also demonstrated that the empirical particle density obeys a law of large numbers with respect to the stationary measures (hydrostatic). The macroscopic particle density solves a non local, stationary, transport equation.