Hydrodynamics of a particle model in contact with stochastic reservoirs

Abstract

We consider an exclusion process with finite-range interactions in the microscopic interval [0, N]. The process is coupled with the simple symmetric exclusion processes in the intervals [−N, −1] and [N + 1, 2N], which simulate reservoirs. We show that an average of the empirical densities of the processes speeded up by the factor N2 converge to solutions of parabolic partial differential equations inside [−N, −1], [0, N], and [N + 1, 2N], which correspond to the macroscopic intervals (−1, 0), (0, 1), and (1, 2). Since the total number of particles is preserved by the evolution, we obtain the Neumann boundary conditions on the external boundaries u = −1, u = 2 of the reservoirs. Finally, a system of Neumann and Dirichlet boundary conditions is derived at the interior boundaries u = 0, u = 1 of the reservoirs.