{
  "id": "1504.01378",
  "version": "v1",
  "published": "2015-04-06T10:29:40.000Z",
  "updated": "2015-04-06T10:29:40.000Z",
  "title": "An extension of the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions by the Caputo fractional derivative",
  "authors": [
    "Minoru Biyajima",
    "Takuya Mizoguchi",
    "Naomichi Suzuki"
  ],
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.quant-gas",
    "hep-ph"
  ],
  "abstract": "In order to generalize the Maxwell-Boltzmann (MB), Bose-Einstein (BE), and Fermi-Dirac (FD) distributions to fractional order, we start with the thermodynamical equation, $\\partial U/\\partial \\beta =-aU-bU^2$, with $\\beta =1/k_BT$ and parameters $a$ ($a>0$) and $b$, which is equivalent to the equation proposed by Planck in 1900. Setting $R=1/U$ and $x=a(\\beta-\\beta_0)$, we obtain the linear partial differential equation $\\partial R/\\partial x = R + b/a$ from the thermodynamical equation. Then, the Caputo fractional derivative of order $p$ ($p>0$) is introduced in place of the partial derivative of $x$. We obtain the fractional MB, BE, and FD distributions, where the exponential function, ${\\rm e}^x$, is replaced by the Mittag-Leffler function, $E_p(x^p)$. The behaviors of the fractional FD distribution are examined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-06T10:29:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "caputo fractional derivative",
      "fermi-dirac distributions",
      "bose-einstein",
      "maxwell-boltzmann",
      "linear partial differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150401378B",
      "inspire": 1358223
    }
  }
}