Derivation of the Fick's Law for the Lorentz Model in a Low Density Regime

We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics, which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime, the macroscopic current in the stationary state is given by the Fick's law, with the diffusion coefficient determined by the Green-Kubo formula.