On a class of solvable stationary non equilibrium states for mass exchange models

TL;DR

This paper identifies a class of solvable stationary non-equilibrium states in diffusive mass exchange models, explicitly computing transport coefficients and large deviation functionals using Macroscopic Fluctuation Theory.

Contribution

It provides explicit solutions for the Hamilton-Jacobi equation for a class of models, including generalized zero range and misanthrope processes, advancing understanding of non-equilibrium stationary states.

Findings

Explicit transport coefficients for reversible models.

Solvable Hamilton-Jacobi equations for large deviations.

Complete characterization of reversible misanthrope processes.

Abstract

We consider a family of models having an arbitrary positive amount of mass on each site and randomly exchanging an arbitrary amount of mass with nearest neighbor sites. We restrict to the case of diffusive models. We identify a class of reversible models for which the product invariant measure is known and the gradient condition is satisfied so that we can explicitly compute the transport coefficients associated to the diffusive hydrodynamic rescaling. Based on the Macroscopic Fluctuation Theory \cite{mft} we have that the large deviations rate functional for a stationary non equilibrium state can be computed solving a Hamilton-Jacobi equation depending only on the transport coefficients and the details of the boundary sources. Thus, we are able to identify a class of models having transport coefficients for which the Hamilton-Jacobi equation can indeed be solved. We give a complete characterization in the case of generalized zero range models and discuss several other cases. For the generalized zero range models we identify a class of discrete models that, modulo trivial extensions, coincides with the class discussed in \cite{FG} and a class of continuous dynamics that coincides with the class in \cite{FFG}. Along the discussion we obtain a complete characterization of reversible misanthrope processes solving the discrete equations in \cite{CC}.