{
  "id": "1303.6162",
  "version": "v1",
  "published": "2013-03-25T15:19:40.000Z",
  "updated": "2013-03-25T15:19:40.000Z",
  "title": "Thermalization of Levy flights: Path-wise picture in 2D",
  "authors": [
    "Mariusz Zaba",
    "Piotr Garbaczewski"
  ],
  "comment": "11 pages, 6 figures",
  "journal": "International Journal of Statistical Mechanics, vol. 2013, ID 78345",
  "doi": "10.1155/2013/738345",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We analyze two-dimensional (2D) random systems driven by a symmetric L\\'{e}vy stable noise which, under the sole influence of external (force) potentials $\\Phi (x) $, asymptotically set down at Boltzmann-type thermal equilibria. Such behavior is excluded within standard ramifications of the Langevin approach to L\\'{e}vy flights. In the present paper we address the response of L\\'{e}vy noise not to an external conservative force field, but directly to its potential $\\Phi (x)$. We prescribe a priori the target pdf $\\rho_*$ in the Boltzmann form $\\sim \\exp[- \\Phi (x)]$ and next select the L\\'evy noise of interest. Given suitable initial data, this allows to infer a reliable path-wise approximation to a true (albeit analytically beyond the reach) solution of the pertinent master equation, with the property $\\rho (x,t)\\rightarrow \\rho_*(x)$ as time $t$ goes to infinity. We create a suitably modified version of the time honored Gillespie's algorithm, originally invented in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which consistently sets down at a pre-defined target pdf. We pay special attention to the response of the 2D Cauchy noise to an exemplary locally periodic \"potential landscape\" $\\Phi (x), x\\in R^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-25T15:19:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "levy flights",
      "path-wise picture",
      "thermalization",
      "target pdf",
      "external conservative force field"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Hindawi",
      "journal": "Adv. High Energ. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.6162Z"
    }
  }
}