{
  "id": "1005.0120",
  "version": "v2",
  "published": "2010-05-02T03:53:22.000Z",
  "updated": "2021-08-06T12:29:08.000Z",
  "title": "Persistence Exponent for the Simple Diffusion Equation: The Exact Solution for any Integer Dimension",
  "authors": [
    "Devashish Sanyal"
  ],
  "comment": "5 pages",
  "journal": "Journal of Modern Physics, Vol-12, No-10, 2021",
  "doi": "10.4236/jmp.2021.1210083",
  "categories": [
    "cond-mat.stat-mech",
    "physics.data-an"
  ],
  "abstract": "The persistence exponent $\\theta_o$ for the simple diffusion equation ${\\phi}_t({\\it x},t) = \\triangle \\phi (x,t)$ , with random Gaussian initial condition {\\color{red},} has been calculated exactly using a method known as selective averaging. The probability that the value of the field $\\phi$ at a specified spatial coordinate remains positive throughout for a certain time $t$ behaves as $t^{-\\theta_o}$ for asymptotically large time $t$. The value of $\\theta_o$, calculated here for any integer dimension $d$, is $\\theta_o = \\frac{d}{4}$ for $d\\leq 4$ and $1$ otherwise. This exact theoretical result is being reported possibly for the first time and is not in agreement with the accepted values $ \\theta_o = 0.12, 0.18,0.23$ for $d=1,2,3$ respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-02T03:53:22.000Z",
      "title": "Persistence Exponent for Diffusion: The Exact Solution",
      "abstract": "The persistence exponent for diffusion equation with random gaussian initial conditions has been calculated for any dimension $d$. The value of the exponent in the asymptotic limit for large time comes out to be $d/4$ . The result is at variance with the generally accepted values of $0.12$,$0.18$ and $0.23$ for $d=1,2,3$ respectively.",
      "comment": "4 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2021-08-06T12:29:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "persistence exponent",
      "exact solution",
      "random gaussian initial conditions",
      "large time comes",
      "diffusion equation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.0120S"
    }
  }
}