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The purpose of this article is to provide a simple proof of the hydrodynamicand hydrostatic behavior of the SSEP in contact with slowed reservoirs whichinject and remove particles in a finite size windows at the extremities of thebulk. More precisely, the reservoirs inject/remove particles at/from any pointof a window of size K placed at each extremity of the bulk and particles areinjected/removed to the first open/occupied position in that window. Thehydrodynamic limit is given by the heat equation with non-linear Robin boundaryconditions or Neumann boundary conditions, the latter being in the case whenthe reservoirs are too slow. The proof goes through the entropy method of Guo,Papanicolaou and Varadhan. We also derive the hydrostatic limit for this model,whose proof is based on the method developed by Landim and Tsunoda. We observethat we do not make use of correlation estimates in none of our results.

Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part I, the critical regime and beyond