{
  "id": "2103.04637",
  "version": "v1",
  "published": "2021-03-08T09:50:33.000Z",
  "updated": "2021-03-08T09:50:33.000Z",
  "title": "Condensation transition in the late-time position of a Run-and-Tumble particle",
  "authors": [
    "Francesco Mori",
    "Pierre Le Doussal",
    "Satya N. Majumdar",
    "Gregory Schehr"
  ],
  "comment": "44 pages, 14 figures",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.soft",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study the position distribution $P(\\vec{R},N)$ of a run-and-tumble particle (RTP) in arbitrary dimension $d$, after $N$ runs. We assume that the constant speed $v>0$ of the particle during each running phase is independently drawn from a probability distribution $W(v)$ and that the direction of the particle is chosen isotropically after each tumbling. The position distribution is clearly isotropic, $P(\\vec{R},N)\\to P(R,N)$ where $R=|\\vec{R}|$. We show that, under certain conditions on $d$ and $W(v)$ and for large $N$, a condensation transition occurs at some critical value of $R=R_c\\sim O(N)$ located in the large deviation regime of $P(R,N)$. For $R<R_c$ (subcritical fluid phase), all runs are roughly of the same size in a typical trajectory. In contrast, an RTP trajectory with $R>R_c$ is typically dominated by a `condensate', i.e., a large single run that subsumes a finite fraction of the total displacement (supercritical condensed phase). Focusing on the family of speed distributions $W(v)=\\alpha(1-v/v_0)^{\\alpha-1}/v_0$, parametrized by $\\alpha>0$, we show that, for large $N$, $P(R,N)\\sim \\exp\\left[-N\\psi_{d,\\alpha}(R/N)\\right]$ and we compute exactly the rate function $\\psi_{d,\\alpha}(z)$ for any $d$ and $\\alpha$. We show that the transition manifests itself as a singularity of this rate function at $R=R_c$ and that its order depends continuously on $d$ and $\\alpha$. We also compute the distribution of the condensate size for $R>R_c$. Finally, we study the model when the total duration $T$ of the RTP, instead of the total number of runs, is fixed. Our analytical predictions are confirmed by numerical simulations, performed using a constrained Markov chain Monte Carlo technique, with precision $\\sim 10^{-100}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-08T09:50:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "condensation transition",
      "run-and-tumble particle",
      "late-time position",
      "markov chain monte carlo technique",
      "position distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}