## Abstract

We study the hydrodynamic and the hydrostatic behavior of the simple symmetric exclusion process with slow boundary. The term slow boundary means that particles can be born or die at the boundary sites, at a rate proportional to N^{-} ^{θ}, where θ> 0 and N is the scaling parameter. In the bulk, the particles exchange rate is equal to 1. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter θ; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on θ. If θ∈ (0 , 1) , we get Dirichlet boundary conditions, (which is the same behavior if θ= 0 , see Farfán in Hydrostatics, statical and dynamical large deviations of boundary driven gradient symmetric exclusion processes, 2008); if θ= 1 , we get Robin boundary conditions; and, if θ∈ (1 , ∞) , we get Neumann boundary conditions.

Original language | English |
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Pages (from-to) | 1112-1142 |

Number of pages | 31 |

Journal | Journal of Statistical Physics |

Volume | 167 |

Issue number | 5 |

DOIs | |

State | Published - 1 Jun 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2017, Springer Science+Business Media New York.

### Funding

The authors A. Neumann and R.R. Souza was partially supported by FAPERGS (Proc. 002063-2551/13-0). A. N. was partially supported through a grant “L’ORÉAL - ABC - UNESCO Para Mulheres na Ciência”.

Funders | Funder number |
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United Nations Educational, Scientific and Cultural Organization | |

Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul | 002063-2551/13-0 |

Academia Brasileira de Ciências |

## Keywords

- Exclusion process
- Hydrodynamic limit
- Hydrostatic limit
- Slow boundary