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We consider a lattice gas interacting by the exclusion rule in the presenceof a random field given by i.i.d. bounded random variables in a bounded domainin contact with particles reservoir at different densities. We show, indimensions $d \ge 3$, that the rescaled empirical density field almost surely,with respect to the random field, converges to the unique weak solution of anon linear parabolic equation having the diffusion matrix determined by thestatistical properties of the external random field and boundary conditionsdetermined by the density of the reservoir. Further we show that the rescaledempirical density field, in the stationary regime, almost surely with respectto the random field, converges to the solution of the associated stationarytransport equation.

Lattice gas model in random medium and open boundaries: hydrodynamic and relaxation to the steady state