Hydrodynamic limit for a type of exclusion process with slow bonds in dimension d ≥ 2

Abstract

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus T^d. In the discrete torus N^-1T^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^c, the exchange rate is 1; if one site is in Λ and the other site is in Λ^c, 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 ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. 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 complementary regions.