Hydrodynamic Limit for the SSEP with a Slow Membrane

Abstract

In this paper we consider a symmetric simple exclusion process 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.