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 quasi-linear integro-differential evolution equation
with Dirichlet boundary conditions. The associated dynamical large deviation
principle is proved. Furthermore, for β 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