Hydrodynamic Limit for the SSEP with a Slow Membrane

In this paper we consider a symmetric simple exclusion process (SSEP) on the $d$-dimensional discrete torus $\mathbb{T}^d_N$ with a spatial non-homogeneity given by a slow membrane. The slow membrane is defined here as the boundary of a smooth simple connected region $\Lambda$ on the continuous $d$-dimensional torus $\mathbb{T}^d$. In this setting, bonds crossing the membrane have jump rate $\alpha/N^\beta$ and all other bonds have jump rate one, where $\alpha>0$, $\beta\in[0,\infty]$, and $N\in \mathbb{N}$ is the scaling parameter. In the diffusive scaling we prove that the hydrodynamic limit presents a dynamical phase transition, that is, it depends on the regime of $\beta$. For $\beta\in[0,1)$, the hydrodynamic equation is given by the usual heat equation on the continuous torus, meaning that the slow membrane has no effect in the limit. For $\beta\in(1,\infty]$, the hydrodynamic equation is the heat equation with Neumann boundary conditions, meaning that the slow membrane $\partial \Lambda$ divides $\mathbb{T}^d$ into two isolated regions $\Lambda$ and $\Lambda^\complement$. And for the critical value $\beta=1$, the hydrodynamic equation is the heat equation with certain Robin boundary conditions related to the Fick's Law.