{
  "id": "2101.00261",
  "version": "v1",
  "published": "2021-01-01T16:09:44.000Z",
  "updated": "2021-01-01T16:09:44.000Z",
  "title": "Non-normalizable quasi-equilibrium solution of the Fokker-Planck equation for nonconfining fields",
  "authors": [
    "Celia Anteneodo",
    "Lucianno Defaveri",
    "Eli Barkai",
    "David A. Kessler"
  ],
  "comment": "22 pages, 8 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep compared to temperature, physical observables like the mean square displacement are essentially time-independent over a long time interval, the stagnation epoch. However the standard Boltzmann-Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker-Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, valid for times which are long, but still short compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is a an error function, shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker-Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-01T16:09:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fokker-planck equation",
      "non-normalizable quasi-equilibrium solution",
      "nonconfining fields",
      "bg statistics",
      "exponentially large escape time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}