{
  "id": "1809.07911",
  "version": "v1",
  "published": "2018-09-21T01:47:36.000Z",
  "updated": "2018-09-21T01:47:36.000Z",
  "title": "Hydrodynamic Limit for the SSEP with a Slow Membrane",
  "authors": [
    "Tertuliano Franco",
    "Mariana Tavares"
  ],
  "comment": "36 pages, 6 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we consider a symmetric simple exclusion process (SSEP) 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-21T01:47:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "35K55"
    ],
    "keywords": [
      "slow membrane",
      "hydrodynamic limit",
      "hydrodynamic equation",
      "symmetric simple exclusion process",
      "jump rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}