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Journal of Statistical Physics

, Volume 175, Issue 2, pp 233–268 | Cite as

Hydrodynamic Limit for the SSEP with a Slow Membrane

  • Tertuliano FrancoEmail author
  • Mariana Tavares
Article
  • 59 Downloads

Abstract

In this paper we consider a symmetric simple exclusion process on the d-dimensional discrete torus \({\mathbb {T}}^d_N\) with a spatial non-homogeneity given by a slow membrane. The slow membrane is defined here as the boundary of a smooth simple connected region \(\Lambda \) on the continuous d-dimensional torus \({\mathbb {T}}^d\). In this setting, bonds crossing the membrane have jump rate \(\alpha /N^\beta \) and all other bonds have jump rate one, where \(\alpha >0\), \(\beta \in [0,\infty ]\), and \(N\in {\mathbb {N}}\) is the scaling parameter. In the diffusive scaling we prove that the hydrodynamic limit presents a dynamical phase transition, that is, it depends on the regime of \(\beta \). For \(\beta \in [0,1)\), the hydrodynamic equation is given by the usual heat equation on the continuous torus, meaning that the slow membrane has no effect in the limit. For \(\beta \in (1,\infty ]\), the hydrodynamic equation is the heat equation with Neumann boundary conditions, meaning that the slow membrane \(\partial \Lambda \) divides \({\mathbb {T}}^d\) into two isolated regions \(\Lambda \) and \(\Lambda ^\complement \). And for the critical value \(\beta =1\), the hydrodynamic equation is the heat equation with certain Robin boundary conditions related to the Fick’s Law.

Keywords

Hydrodynamic limit Exclusion process Non-homogeneous environment Slow bonds 

Mathematics Subject Classification

60K35 35K55 

Notes

Acknowledgements

T.F. was supported through a project Jovem Cientista-9922/2015, FAPESB-Brazil. M.T. would like to thank CAPES for a PhD scholarship, which supported her research.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.UFBA, Instituto de Matemática, Campus de OndinaSalvadorBrazil

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