Boundary driven Kawasaki process with long range interaction: dynamical large deviations and steady states

A particle system with a single locally-conserved field (density) in a bounded interval with different densities maintained at the two endpoints of the interval is under study here. The particles interact in the bulk through a long-range potential parametrized by β 0 and evolve according to an exclusion rule. It is shown that the empirical particle density under the diffusive scaling solves a quasilinear integro-differential evolution equation with Dirichlet boundary conditions. The associated dynamical large deviation principle is proved. Furthermore, when β is small enough, it is also demonstrated that the empirical particle density obeys a law of large numbers with respect to the stationary measures (hydrostatic). The macroscopic particle density solves a non-local, stationary, transport equation.