  We consider a lattice gas evolving in a bounded cylinder of length 2N + 1 and
interacting via a Neuman Kac interaction of range N, in contact with particles
reservoirs at different densities. We investigate the associated law of large
numbers and large deviations of the empirical current and of the density. The
hydrodynamic limit for the empirical density, obtained in the diffusive
scaling, is given by a nonlocal, nonlinear evolution equation with Dirichlet
boundary conditions.
