Hydrodynamic Limit for the SSEP with a Slow Membrane

In this paper we consider a symmetric simple exclusion process on the d -dimensional discrete torus T N d 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 Λ on the continuous d -dimensional torus T d . In this setting, bonds crossing the membrane have jump rate α / N β and all other bonds have jump rate one, where α > 0 , β ∈ [ 0 , ∞ ] , and N ∈ 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 β . For β ∈ [ 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 β ∈ ( 1 , ∞ ] , the hydrodynamic equation is the heat equation with Neumann boundary conditions, meaning that the slow membrane ∂ Λ divides T d into two isolated regions Λ and Λ ∁ . And for the critical value β = 1 , the hydrodynamic equation is the heat equation with certain Robin boundary conditions related to the Fick’s Law.