Journal of Statistical Physics

, Volume 174, Issue 2, pp 494–518 | Cite as

Fick Law and Sticky Brownian Motions

  • Thu Dang Thien NguyenEmail author


We consider an interacting particle system in the interval [1, N] with reservoirs at the boundaries. While the dynamics in the channel is the simple symmetric exclusion process, the reservoirs are also particle systems which interact with the given system by exchanging particles. In this paper we study the case where the size of each reservoir is the same as the size of the channel. We will prove that the hydrodynamic limit equation is the heat equation with boundary conditions which relate first and second spatial derivatives at the boundaries for which we will prove the existence and uniqueness of weak solutions.


Hydrodynamic limits Free boundary problem Sticky random walk 



I greatly appreciate Prof. Errico Presutti for suggesting the problem and offering me a large number of useful ideas. I also would like to express my gratitude to Lorenzo Bertini, Paolo Butta, Anna De Masi, Pablo Ferrari and Frank Redig and Maria Eulalia Vares for their valuable comments and suggestions. In addition, I would like to thank the reviewers for their careful reading of my paper and for their insightful comments.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Gran Sasso Science InstituteL’AquilaItaly
  2. 2.Department of MathematicsUniversity of QuynhonQuy NhonVietnam

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