{
  "id": "1405.5081",
  "version": "v2",
  "published": "2014-05-20T13:45:02.000Z",
  "updated": "2015-05-06T19:08:02.000Z",
  "title": "Scalings limits for the exclusion process with a slow site",
  "authors": [
    "Tertuliano Franco",
    "Patrícia Gonçalves",
    "Gunter M. Schütz"
  ],
  "comment": "In this version of the paper we corrected a mistake that we found in our previous version in the regime beta less or equal than one. Now, we establish a phase transition both at the level of hydrodynamics and equilibrium fluctuations, but the asymptotic behavior of the system for some regimes of beta remains open",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the symmetric simple exclusion processes with a slow site in the discrete torus with $n$ sites. In this model, particles perform nearest-neighbor symmetric random walks with jump rates everywhere equal to one, except at one particular site, the slow site, where the jump rate of entering that site is equal to one, but the jump rate of leaving that site is given by a parameter $g(n)<1$. Two cases are treated, namely $g(n)=1-b/n$ with $b>0$, and $g(n)=\\alpha n^{-\\beta}$ with $\\beta>1$, $\\alpha>0$. In the former, both the hydrodynamic behavior and equilibrium fluctuations are driven by the heat equation (with periodic boundary conditions when in finite volume). In the latter, they are driven by the heat equation with Neumann's boundary conditions. We therefore establish the existence of a dynamical phase transition. The critical behavior remains open.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-20T13:45:02.000Z",
      "title": "Hydrodynamic limit for the exclusion process with a slow site",
      "abstract": "We consider the symmetric simple exclusion processes with a slow site in the discrete torus with $n$ sites. In this model, particles perform nearest-neighbor symmetric random walks with jump rates everywhere equal to one, except at one particular site, {\\em{the slow site}}, where the jump rate of entering that site is equal to one, but the jump rate of leaving that site is equal to $\\alpha n^{-\\beta}$, with $\\alpha>0$ and $\\beta\\geq 0$. Hence this site works as a trap, since the higher the value of $\\beta$, the more difficult a particle leaves the slow site. We prove that there is a phase transition in the sense that the hydrodynamic limit is given by three different partial differential equations, depending on the range of the parameter $\\beta$. The hydrodynamic behavior is given by the heat equation with periodic, Robin's or Neumann's boundary conditions, if $\\beta\\in[0,1)$, $\\beta=1$ or $\\beta\\in{(1,\\infty)}$, respectively.",
      "comment": "32 pages, 3 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-05-06T19:08:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "26A24",
      "35K55"
    ],
    "keywords": [
      "slow site",
      "hydrodynamic limit",
      "jump rate",
      "perform nearest-neighbor symmetric random walks",
      "particles perform nearest-neighbor symmetric random"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.5081F"
    }
  }
}