Current reservoirs in the simple exclusion process

TL;DR

This paper studies a symmetric simple exclusion process with boundary reservoirs, proving convergence to the heat equation with boundary conditions derived from a nonlinear equation, confirming Fourier's law in the limit.

Contribution

It establishes the hydrodynamic limit of the exclusion process with boundary reservoirs, deriving boundary conditions from a nonlinear equation and confirming Fourier's law.

Findings

Convergence to the heat equation with boundary conditions

Boundary conditions obtained from a nonlinear equation

Fourier's law holds in the hydrodynamic limit

Abstract

We consider the symmetric simple exclusion process in the interval $[-N,N]$ with additional birth and death processes respectively on $(N-K,N]$, $K>0$, and $[-N,-N+K)$. The exclusion is speeded up by a factor $N^2$, births and deaths by a factor $N$. Assuming propagation of chaos (a property proved in a companion paper "Truncated correlations in the stirring process with births and deaths") we prove convergence in the limit $N\to \infty$ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.