{
  "id": "cond-mat/0211453",
  "version": "v1",
  "published": "2002-11-20T20:26:01.000Z",
  "updated": "2002-11-20T20:26:01.000Z",
  "title": "Persistence in q-state Potts model: A Mean-Field approach",
  "authors": [
    "G. Manoj"
  ],
  "comment": "11 pages in RevTeX, 10 figures, submitted to Phys. Rev. E",
  "journal": "Phys. Rev. E 67: 026115 (2003)",
  "doi": "10.1103/PhysRevE.67.026115",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We study the Persistence properties of the T=0 coarsening dynamics of one dimensional $q$-state Potts model using a modified mean-field approximation (MMFA). In this approximation, the spatial correlations between the interfaces separating spins with different Potts states is ignored, but the correct time dependence of the mean density $P(t)$ of persistent spins is imposed. For this model, it is known that $P(t)$ follows a power-law decay with time, $P(t)\\sim t^{-\\theta(q)}$ where $\\theta(q)$ is the $q$-dependent persistence exponent. We study the spatial structure of the persistent region within the MMFA. We show that the persistent site pair correlation function $P_{2}(r,t)$ has the scaling form $P_{2}(r,t)=P(t)^{2}f(r/t^{{1/2}})$ for all values of the persistence exponent $\\theta(q)$. The scaling function has the limiting behaviour $f(x)\\sim x^{-2\\theta}$ ($x\\ll 1$) and $f(x)\\to 1$ ($x\\gg 1$). We then show within the Independent Interval Approximation (IIA) that the distribution $n(k,t)$ of separation $k$ between two consecutive persistent spins at time $t$ has the asymptotic scaling form $n(k,t)=t^{-2\\phi}g(t,\\frac{k}{t^{\\phi}})$ where the dynamical exponent has the form $\\phi$=max(${1/2},\\theta$). The behaviour of the scaling function for large and small values of the arguments is found analytically. We find that for small separations $k\\ll t^{\\phi}, n(k,t)\\sim P(t)k^{-\\tau}$ where $\\tau$=max($2(1-\\theta),2\\theta$), while for large separations $k\\gg t^{\\phi}$, $g(t,x)$ decays exponentially with $x$. The unusual dynamical scaling form and the behaviour of the scaling function is supported by numerical simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-11-20T20:26:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "q-state potts model",
      "mean-field approach",
      "persistence",
      "scaling function",
      "persistent site pair correlation function"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. E"
    },
    "note": {
      "typesetting": "RevTeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}