{
  "id": "2011.04716",
  "version": "v1",
  "published": "2020-11-09T19:45:17.000Z",
  "updated": "2020-11-09T19:45:17.000Z",
  "title": "Local time for run and tumble particle",
  "authors": [
    "Prashant Singh",
    "Anupam Kundu"
  ],
  "comment": "18 pages, 9 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We investigate the local time $(T_{loc})$ statistics for a run and tumble particle in an one dimensional inhomogeneous medium. The inhomogeneity is introduced by considering the position dependent rate of the form $R(x) = \\gamma \\frac{|x|^{\\alpha}}{l^{\\alpha}}$ with $\\alpha \\geq 0$. For $\\alpha =0$, we derive the probability distribution of $T_{loc}$ exactly which is expressed as a series of $\\delta$-functions in which the coefficients can be interpreted as the probability of multiple revisits of the particle to the origin starting from the origin. For general $\\alpha$, we show that the typical fluctuations of $T_{loc}$ scale with time as $T_{loc} \\sim t^{\\frac{1+\\alpha}{2+\\alpha}}$ for large $t$ and their probability distribution possesses a scaling behaviour described by a scaling function which we have computed analytically. In the second part, we study the statistics of $T_{loc}$ till the RTP makes a first passage to $x=M~(>0)$. In this case also, we show that the probability distribution can be expressed as a series sum of $\\delta$-functions for all values of $\\alpha~(\\geq 0)$ with coefficients appearing from appropriate exit problems. All our analytical findings are supported with the numerical simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-09T19:45:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local time",
      "tumble particle",
      "position dependent rate",
      "probability distribution possesses",
      "appropriate exit problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}