Hydrodynamic limit of an exclusion process with vorticity

TL;DR

This paper constructs a non-reversible exclusion process with explicit hydrodynamic limits, revealing how antisymmetric components influence current without affecting density evolution, and relates mobility to the diffusion matrix via Einstein relation.

Contribution

It introduces a non-reversible exclusion process with explicit hydrodynamic behavior and analyzes the impact of antisymmetric diffusion components on current and mobility.

Findings

Explicit non-symmetric diffusion matrix computed

Antisymmetric part affects current but not density evolution

Mobility matrix related to symmetric diffusion via Einstein relation

Abstract

We construct a non reversible exclusion process with Bernoulli product invariant measure and having, in the diffusive hydrodynamic scaling, a non symmetric diffusion matrix, that can be explicitly computed. The antisymmetric part does not affect the evolution of the density but it is relevant for the evolution of the current. Switching on a weak external field we obtain a symmetric mobility matrix that is related just to the symmetric part of the diffusion matrix by an Einstein relation. We argue that this fact is typical within a class of generalized gradient models. We consider for simplicity the model in dimension $d=2$, but a similar behavior can be also obtained in higher dimensions.