Hydrodynamic Limit for a type of Exclusion Processes with slow bonds in dimension $\ge 2$

Abstract

Let $\Lambda$ be a connected closed region with smooth boundary contained in the $d$-dimensional continuous torus $\bb T^d$. In the discrete torus $N^{-1} \bb T^d_N$, we consider a nearest neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on $\Lambda$ in the following way: if both sites are in $\Lambda$ or ${\Lambda }^{\complement }$, the exchange rate is one; If one site is in $\Lambda$ and the other one is in ${\Lambda }^{\complement }$ 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 $\pΛ$. We show that this exclusion type process has a non-trivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by $\partial \Lambda$. 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 complementar regions.

18 pages, 1 figure

References in corpus (2)

• Hydrodynamic limit of gradient exclusion processes with conductances on $\bb Z^d$
• Hydrodynamic limit of symmetric exclusion processes in inhomogeneous media