{
  "id": "1305.3397",
  "version": "v2",
  "published": "2013-05-15T09:12:54.000Z",
  "updated": "2015-03-03T14:25:02.000Z",
  "title": "The Brownian motion as the limit of a deterministic system of hard-spheres",
  "authors": [
    "Thierry Bodineau",
    "Isabelle Gallagher",
    "Laure Saint-Raymond"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\\varepsilon$ simultaneously goes to $0$, in the fast relaxation limit $\\alpha = N\\varepsilon^{d-1}\\to \\infty $ (with a suitable diffusive scaling of the observation time). As suggested by Hilbert in his sixth problem, we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions: we use indeed the linear Boltzmann equation to describe one tagged particle in a gas close to global equilibrium. Our proof is based on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-15T09:12:54.000Z",
      "abstract": "We provide a rigorous derivation of the brownian motion as the hydrodynamic limit of systems of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\\varepsilon$ simultaneously goes to 0, in the fast relaxation limit $N \\varepsilon^{d-1} \\to \\infty$ (with a suitable scaling of the observation time and length). As suggested by Hilbert in his sixth problem, we use the linear Boltzmann equation as an intermediate level of description for one tagged particle in a gas close to global equilibrium. Our proof relies on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-03T14:25:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian motion",
      "deterministic system",
      "hard-spheres",
      "fast relaxation limit",
      "linear boltzmann equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.3397B"
    }
  }
}