Hydrodynamics for SSEP with Non-reversible Slow Boundary Dynamics: Part I, the Critical Regime and Beyond
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Hydrodynamics for SSEP with Non-reversible Slow Boundary Dynamics: Part I, the Critical Regime and Beyond C. Erignoux1 · P. Gonçalves2
· G. Nahum2
Received: 20 December 2019 / Accepted: 9 September 2020 / Published online: 18 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 et al. (Commun Math Phys 118:31– 59, 1988). We also derive the hydrostatic limit for this model, whose proof is based on the method developed in Landim and Tsunoda (Ann Henri Poincaré 54(1):51–74, 2018) and Tsunoda (Hydrostatic limit for exclusion process with slow boundary revisited, RIMS Kôkyûroku Bessatsu). We observe that we do not make use of correlation estimates in none of our results.
Communicated by Abhishek Dhar. C.E. gratefully acknowledges funding from the European Research Council under the European Unions Horizon 2020 Program, ERC Consolidator GrantUniCoSM (Grant Agreement No. 724939). P.G. thanks FCT/Portugal for support through the project UID/MAT/04459/2013. G.N thanks FCT/Portugal for the support through the project Lisbon Mathematics PhD (LisMath). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (Grant Agreement No. 715734).
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P. Gonçalves [email protected] C. Erignoux [email protected] G. Nahum [email protected]
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Dip. Matematica e Fisica, Univ. Degli Studi Roma Tre, Largo San Leonardo Murialdo, 1 - 00146 Rome, Italy
2
Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
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Keywords Hydrodynamic limits · Exclusion process · Slow boundary · Non reversible boundary dynamics · Heat equation · Non linear Robin boundary conditions
1 Introduction One of the intriguing questions in Statistical Physics is related to the understanding of how local microscopic perturbations of the dynamics of a particle system, carries through its macroscopic description. In recent years, several articles have been dedicated to the understanding of adding a slow bond, a slow site or a slow boundary to the most classical interacting particle system, namely, the exclusion process. For references on this topic, we refer the reader to [1,13,14] and r
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