{
  "id": "1808.02737",
  "version": "v1",
  "published": "2018-08-08T12:18:01.000Z",
  "updated": "2018-08-08T12:18:01.000Z",
  "title": "Infinite Ergodic Theory for Heterogeneous Diffusion Processes",
  "authors": [
    "N. Leibovich",
    "E. Barkai"
  ],
  "comment": "16 pages, 12 figures, 2 tables",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. We concentrate on a spatially dependent diffusion coefficient that behaves as ${D(x)}\\sim |x-\\tilde{x}|^{2-2/\\alpha}$ in the vicinity of a point $\\tilde{x}$, where $\\alpha$ can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables, are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation; It\\^o, Stratonovich, or H\\\"anggi-Klimontovich, so the existence of an infinite density, and the density's shape, are both related to the considered interpretation and the structure of $D(x)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-08T12:18:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite ergodic theory",
      "heterogeneous diffusion processes",
      "infinite density",
      "langevin equation",
      "multiplicative noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}