{
  "id": "2012.03142",
  "version": "v1",
  "published": "2020-12-05T23:25:52.000Z",
  "updated": "2020-12-05T23:25:52.000Z",
  "title": "Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls",
  "authors": [
    "Thomas Vojta",
    "Alex Warhover"
  ],
  "comment": "26 pages, 11 figures included",
  "categories": [
    "cond-mat.stat-mech",
    "physics.bio-ph"
  ],
  "abstract": "Fractional Brownian motion and the fractional Langevin equation are models of anomalous diffusion processes characterized by long-range power-law correlations in time. We employ large-scale computer simulations to study these models in two geometries, (i) the spreading of particles on a semi-infinite domain with an absorbing wall at one end and (ii) the stationary state on a finite interval with absorbing boundaries at both ends and a source in the center. For both stochastic processes and both geometries, we find that the probability density close to an absorbing wall behaves as $P(x) \\sim x^\\kappa$ with the distance $x$ from the wall in the long-time limit. In the case of fractional Brownian motion, $\\kappa$ varies with the anomalous diffusion exponent $\\alpha$ as $\\kappa=2/\\alpha -1$, as was conjectured previously. The probability density of the fractional Langevin equation can be mapped onto that of fractional Brownian motion driven by the same noise by replacing $\\alpha$ with $2-\\alpha$. We also contrast the behavior near absorbing walls with the strong accumulation and depletion effects recently observed for fractional Brownian motion and the fractional Langevin equation near reflecting boundaries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-05T23:25:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional langevin equation",
      "absorbing wall",
      "employ large-scale computer simulations",
      "fractional brownian motion driven",
      "long-range power-law correlations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}