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

Abstract

Let $Λ$ 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 $Λ$ in the following way: if both sites are in $Λ$ or $Λ^\complement$, the exchange rate is one; If one site is in $Λ$ and the other one is in $Λ^\complement$ 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 $\pΛ$. 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 $\partialΛ$. 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.