{
  "id": "2307.02481",
  "version": "v1",
  "published": "2023-07-05T17:55:39.000Z",
  "updated": "2023-07-05T17:55:39.000Z",
  "title": "Non-equilibrium steady state of the symmetric exclusion process with reservoirs",
  "authors": [
    "Simone Floreani",
    "Adrián González Casanova"
  ],
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider the open symmetric exclusion process on a connected graph with vertexes in $[N-1]:=\\{1,\\ldots, N-1\\}$ where points $1$ and $N-1$ are connected, respectively, to a left reservoir and a right reservoir with densities $\\rho_L,\\rho_R\\in(0,1)$. We prove that the non-equilibrium steady state of such system is $$\\mu_{\\text{stat}} = \\sum_{I\\subset \\mathcal P([N-1]) }F(I)\\bigg(\\otimes_{x\\in I}\\rm{Bernoulli}(\\rho_R)\\otimes_{y\\in [N-1]\\setminus I}\\rm{Bernoulli}(\\rho_L) \\bigg).$$ In the formula above $ \\mathcal P([N-1])$ denotes the power set of $[N-1]$ while the numbers $F(I)> 0$ are such that $\\sum_{I\\subset \\mathcal P([N-1]) }F(I)=1$ and given in terms of absorption probabilities of the absorbing stochastic dual process. Via probabilistic arguments we compute explicitly the factors $F(I)$ when the graph is a homogeneous segment.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-05T17:55:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82C22",
      "60K37",
      "82C23"
    ],
    "keywords": [
      "non-equilibrium steady state",
      "open symmetric exclusion process",
      "absorbing stochastic dual process",
      "absorption probabilities",
      "power set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}