{
  "id": "1303.5193",
  "version": "v1",
  "published": "2013-03-21T08:57:14.000Z",
  "updated": "2013-03-21T08:57:14.000Z",
  "title": "From the Boltzmann $H$-theorem to Perelman's $W$-entropy formula for the Ricci flow",
  "authors": [
    "Xiang-Dong Li"
  ],
  "categories": [
    "math.DG",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In 1870s, L. Boltzmann proved the famous $H$-theorem for the Boltzmann equation in the kinetic theory of gas and gave the statistical interpretation of the thermodynamic entropy. In 2002, G. Perelman introduced the notion of $W$-entropy and proved the $W$-entropy formula for the Ricci flow. This plays a crucial role in the proof of the no local collapsing theorem and in the final resolution of the Poincar\\'e conjecture and Thurston's geometrization conjecture. In our previous paper \\cite{Li11a}, the author gave a probabilistic interpretation of the $W$-entropy using the Boltzmann-Shannon-Nash entropy. In this paper, we make some further efforts for a better understanding of the mysterious $W$-entropy by comparing the $H$-theorem for the Boltzmann equation and the Perelman $W$-entropy formula for the Ricci flow. We also suggest a way to construct the \"density of states\" measure for which the Boltzmann $H$-entropy is exactly the $W$-entropy for the Ricci flow.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-21T08:57:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ricci flow",
      "entropy formula",
      "boltzmann equation",
      "thurstons geometrization conjecture",
      "poincare conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.5193L"
    }
  }
}