Current reservoirs in the simple exclusion process

We consider the symmetric simple exclusion process in the interval [−N,N] with additional birth and death processes respectively on (N − K,N], K > 0, and [−N,−N +K). The exclusion is speeded up by a factor N2, births and deaths by a factor N. Assuming propagation of chaos (a property proved in a companion paper, De Masi et al., http://arxiv.org/abs/1104.3447) we prove convergence in the limit N→∞to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non-linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.