{
  "id": "2004.01898",
  "version": "v1",
  "published": "2020-04-04T09:37:48.000Z",
  "updated": "2020-04-04T09:37:48.000Z",
  "title": "Diffusion with resetting in a logarithmic potential",
  "authors": [
    "Somrita Ray",
    "Shlomi Reuveni"
  ],
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.soft",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study the effect of resetting on diffusion in a logarithmic potential. In this model, a particle diffusing in a potential $U(x) = U_0\\log|x|$ is reset, i.e., taken back to its initial position, with a constant rate $r$. We show that this analytically tractable model system exhibits a series of phase transitions as a function of a single parameter, $\\beta U_0$, the ratio of the strength of the potential to the thermal energy. For $\\beta U_0<-1$ the potential is strongly repulsive, preventing the particle from reaching the origin. Resetting then generates a non-equilibrium steady state which is characterized exactly and thoroughly analyzed. In contrast, for $\\beta U_0>-1$ the potential is either weakly repulsive or attractive and the diffusing particle eventually reaches the origin. In this case, we provide a closed form expression for the subsequent first-passage time distribution and show that a resetting transition occurs at $\\beta U_0=5$. Namely, we find that resetting can expedite arrival to the origin when $-1<\\beta U_0<5$, but not when $\\beta U_0>5$. The results presented herein generalize results for simple diffusion with resetting -- a widely applicable model that is obtained from ours by setting $U_0=0$. Extending to general potential strengths, our work opens the door to theoretical and experimental investigation of a plethora of problems that bring together resetting and diffusion in logarithmic potential.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-04T09:37:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "logarithmic potential",
      "subsequent first-passage time distribution",
      "non-equilibrium steady state",
      "general potential strengths",
      "simple diffusion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}