Hydrodynamic Limit for a type of Exclusion Processes with slow bonds in dimension $\ge 2$

TL;DR

This paper studies a symmetric exclusion process with slow bonds at the boundary of a region in a high-dimensional torus, showing non-trivial hydrodynamic behavior where particles pass through a permeable membrane without reflection.

Contribution

It introduces a new exclusion process model with slow bonds at the boundary of a region and analyzes its hydrodynamic limit in dimensions greater than or equal to two.

Findings

Particles exhibit non-trivial hydrodynamics under diffusive scaling.

Particles are not blocked or reflected by the boundary in the continuum limit.

Abstract

Let $\Lambda$ be a connected closed region with smooth boundary contained in the $d$-dimensional continuous torus $\bb T^d$. In the discrete torus $N^{-1} \bb T^d_N$, we consider a nearest neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on $\Lambda$ in the following way: if both sites are in $\Lambda$ or $\Lambda^\complement$, the exchange rate is one; If one site is in $\Lambda$ and the other one is in $\Lambda^\complement$ and the direction of the bond connecting the sites is $e_j$, then the exchange rate is defined as $N^{-1}$ times the absolute value of the inner product between $e_j$ and the normal exterior vector to $\p\Lambda$. We show that this exclusion type process has a non-trivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by $\partial\Lambda$. Thus the model represents a system of particles under hard core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementar regions.