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 $${\\mathbb {T}}^d_N$$TNd 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 $$\\Lambda $$Λ on the continuous d-dimensional torus $${\\mathbb {T}}^d$$Td. In this setting, bonds crossing the membrane have jump rate $$\\alpha /N^\\beta $$α/Nβ and all other bonds have jump rate one, where $$\\alpha >0$$α>0, $$\\beta \\in [0,\\infty ]$$β∈[0,∞], and $$N\\in {\\mathbb {N}}$$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 $$\\beta $$β. For $$\\beta \\in [0,1)$$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 $$\\beta \\in (1,\\infty ]$$β∈(1,∞], the hydrodynamic equation is the heat equation with Neumann boundary conditions, meaning that the slow membrane $$\\partial \\Lambda $$∂Λ divides $${\\mathbb {T}}^d$$Td into two isolated regions $$\\Lambda $$Λ and $$\\Lambda ^\\complement $$Λ∁. And for the critical value $$\\beta =1$$β=1, the hydrodynamic equation is the heat equation with certain Robin boundary conditions related to the Fick’s Law.