{
  "id": "1305.6816",
  "version": "v1",
  "published": "2013-05-29T14:23:29.000Z",
  "updated": "2013-05-29T14:23:29.000Z",
  "title": "About an H-theorem for systems with non-conservative interactions",
  "authors": [
    "Umberto Marini Bettolo Marconi",
    "Andrea Puglisi",
    "Angelo Vulpiani"
  ],
  "comment": "15 pages, 4 figures, submitted",
  "journal": "J. Stat. Mech. (2013) P08003",
  "doi": "10.1088/1742-5468/2013/08/P08003",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We exhibit some arguments in favour of an H-theorem for a generalization of the Boltzmann equation including non-conservative interactions and a linear Fokker-Planck-like thermostatting term. Such a non-linear equation describing the evolution of the single particle probability $P_i(t)$ of being in state $i$ at time $t$, is a suitable model for granular gases and is indicated here as Boltzmann-Fokker-Planck (BFP) equation. The conjectured H-functional, which appears to be non-increasing, is $H_C(t)=\\sum_i P_i(t) \\ln P_i(t)/\\Pi_i$ with $\\Pi_i = \\lim_{t \\to \\infty} P_i(t)$, in analogy with the H-functional of Markov processes. The extension to continuous states is straightforward. A simple proof can be given for the elastic BFP equation. A semi-analytical proof is also offered for the BFP equation for so-called inelastic Maxwell molecules. Other evidence is obtained by solving particular BFP cases through numerical integration or through \"particle schemes\" such as the Direct Simulation Monte Carlo.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-29T14:23:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-conservative interactions",
      "direct simulation monte carlo",
      "elastic bfp equation",
      "single particle probability",
      "inelastic maxwell molecules"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2013,
      "month": "Aug",
      "volume": 2013,
      "number": 8,
      "pages": "08003"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013JSMTE..08..003M"
    }
  }
}