{
  "id": "1912.01753",
  "version": "v1",
  "published": "2019-12-04T00:45:36.000Z",
  "updated": "2019-12-04T00:45:36.000Z",
  "title": "A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions",
  "authors": [
    "Hayate Suda"
  ],
  "comment": "43 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider one-dimensional infinite chains of harmonic oscillators with stochastic perturbations and long-range interactions which have polynomial decay rate $|x|^{-\\theta}, x \\to \\infty, \\theta > 1$, where $x \\in \\mathbb{Z}$ is the interaction range. We prove that if $2< \\theta \\le 3$, then the time evolution of the macroscopic thermal energy distribution is superdiffusive and governed by a fractional diffusion equation with exponent $\\frac{3}{7-\\theta}$, while if $\\theta > 3$, then the exponent is $\\frac{3}{4}$. The threshold is $ \\theta = 3$ because the derivative of the dispersion relation diverges as $k \\to 0$ when $\\theta \\le 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-04T00:45:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional diffusion equation",
      "stochastic harmonic chains",
      "long-range interactions",
      "macroscopic thermal energy distribution",
      "polynomial decay rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}