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Hydrodynamics of a particle model in contact with stochastic reservoirs

We consider an exclusion process with finite-range interactions in the microscopic interval $[0,N]$. The process is coupled with the simple symmetric exclusion processes in the intervals $[-N,-1]$ and $[N+1,2N]$, which simulate reservoirs. We show that the empirical densities of the processes speeded up by the factor $N^2$ converge to solutions of parabolic partial differential equations inside the intervals $[-N,-1]$, $[0,N]$, $[N+1,2N]$. Since the total number of particles is preserved by the evolution, we obtain the Neumann boundary conditions on the external boundaries $x=-N$, $x=2N$ of the reservoirs. Finally, a system of Neumann and Dirichlet boundary conditions is derived at the interior boundaries $x=0$, $x=N$ of the reservoirs.