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

TL;DR

This paper proves the hydrodynamic and hydrostatic limits of the symmetric simple exclusion process with non-reversible boundary dynamics, using entropy methods, and characterizes the boundary conditions in different regimes.

Contribution

It provides a simplified proof of hydrodynamic and hydrostatic behavior for SSEP with slowed reservoirs, including non-linear Robin and Neumann boundary conditions.

Findings

Hydrodynamic limit given by heat equation with non-linear Robin boundary conditions.

Hydrostatic limit derived without using correlation estimates.

Boundary conditions depend on reservoir speed, with Neumann in the slow regime.

Abstract

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with slowed reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size K placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The hydrodynamic limit is given by the heat equation with non-linear Robin boundary conditions or Neumann boundary conditions, the latter being in the case when the 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 observe that we do not make use of correlation estimates in none of our results.