Hydrodynamics of a particle model in contact with stochastic reservoirs

TL;DR

This paper analyzes a particle exclusion process with finite-range interactions coupled with reservoirs, showing convergence of empirical densities to PDE solutions with specific boundary conditions, revealing how microscopic dynamics lead to macroscopic boundary behaviors.

Contribution

It introduces a microscopic model with reservoirs and proves the convergence of empirical densities to PDEs with Neumann and Dirichlet boundary conditions, linking particle interactions to boundary phenomena.

Findings

Empirical densities converge to PDE solutions as system size grows.

Neumann boundary conditions emerge at external reservoir boundaries.

Mixed Neumann and Dirichlet conditions arise at internal reservoir boundaries.

Abstract

We consider an exclusion process with finite-range interactions in the microscopic interval $[0,N]$. The process is coupled with the simple symmetric exclusion processes in the intervals $[-N,-1]$ and $[N+1,2N]$, which simulate reservoirs. We show that the empirical densities of the processes speeded up by the factor $N^2$ converge to solutions of parabolic partial differential equations inside the intervals $[-N,-1]$, $[0,N]$, $[N+1,2N]$. Since the total number of particles is preserved by the evolution, we obtain the Neumann boundary conditions on the external boundaries $x=-N$, $x=2N$ of the reservoirs. Finally, a system of Neumann and Dirichlet boundary conditions is derived at the interior boundaries $x=0$, $x=N$ of the reservoirs.