{
  "id": "1611.06433",
  "version": "v1",
  "published": "2016-11-19T21:29:24.000Z",
  "updated": "2016-11-19T21:29:24.000Z",
  "title": "The probability density function for the Havriliak-Negami relaxation",
  "authors": [
    "K. Górska",
    "A. Horzela",
    "Ł. Bratek",
    "K. A. Penson",
    "G. Dattoli"
  ],
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.mtrl-sci"
  ],
  "abstract": "We study the functions related to the Havriliak-Negami frequency relaxation $\\sim [1+(i\\omega\\tau_{0})^{\\alpha}]^{-\\beta}$ with $\\tau_{0}$ characteristic time, measured in many experiments. We furnish exact and explicit expression for the response function $f_{\\alpha, \\beta}(t/\\tau_{0})$ in time domain and a probability density $g_{\\alpha, \\beta}(u)$ in space domain for $\\alpha = l/k < 1$ and $\\beta < k/l$, with $k$ and $l$ positive integers. For $0 < \\alpha < 1$ and $\\beta=1$ we reproduce the functions related to the Cole-Cole relaxation. We use the method of integral transforms. We show that $f_{\\alpha, \\beta}(t/\\tau_{0})$ with $\\beta = (2-q)/(q-1)$ and $\\tau_{0} = (q-1)^{1/\\alpha}$, $1 < q < 2$, goes over to the one-sided L\\'{e}vy stable distribution when $q$ tends to one. Moreover, applying the self-similar property of $g_{\\alpha, \\beta}(u)$ we introduce two-variable density which satisfies the integral form of evolution equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-19T21:29:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "probability density function",
      "havriliak-negami relaxation",
      "havriliak-negami frequency relaxation",
      "time domain",
      "integral form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}