{
  "id": "2208.08792",
  "version": "v1",
  "published": "2022-08-18T12:12:19.000Z",
  "updated": "2022-08-18T12:12:19.000Z",
  "title": "Extremal statistics of a one dimensional run and tumble particle with an absorbing wall",
  "authors": [
    "Prashant Singh",
    "Saikat Santra",
    "Anupam Kundu"
  ],
  "comment": "25 pages, 6 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We study the extreme value statistics of a run and tumble particle (RTP) in one dimension till its first passage to the origin starting from the position $x_0~(>0)$. This model has recently drawn a lot of interest due to its biological application in modelling the motion of certain species of bacteria. Herein, we analytically study the exact time-dependent propagators for a single RTP in a finite interval with absorbing conditions at its two ends. By exploiting a path decomposition technique, we use these propagators appropriately to compute the joint distribution $\\mathscr{P}(M,t_m)$ of the maximum displacement $M$ till first-passage and the time $t_m$ at which this maximum is achieved exactly. The corresponding marginal distributions $\\mathbb{P}_M(M)$ and $P_M(t_m)$ are studied separately and verified numerically. In particular, we find that the marginal distribution $P_M(t_m)$ has interesting asymptotic forms for large and small $t_m$. While for small $t_m$, the distribution $P_M(t_m)$ depends sensitively on the initial velocity direction $\\sigma _i$ and is completely different from the Brownian motion, the large $t_m$ decay of $P_M(t_m)$ is same as that of the Brownian motion although the amplitude crucially depends on the initial conditions $x_0$ and $\\sigma _i$. We verify all our analytical results to high precision by numerical simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-18T12:12:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tumble particle",
      "dimensional run",
      "extremal statistics",
      "absorbing wall",
      "marginal distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}