{
  "id": "1502.06143",
  "version": "v1",
  "published": "2015-02-21T21:13:19.000Z",
  "updated": "2015-02-21T21:13:19.000Z",
  "title": "On the Mean-Field and Classical Limits of Quantum Mechanics",
  "authors": [
    "François Golse",
    "Clément Mouhot",
    "Thierry Paul"
  ],
  "comment": "36 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The main result in this paper is a new inequality bearing on solutions of the $N$-body linear Schr\\\"{o}dinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of $N$ identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of $C^{1,1}$ interaction potentials. The quantity measuring the approximation of the $N$-body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent $2$. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the $N$-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-21T21:13:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C10",
      "35Q55",
      "82C05",
      "35Q83"
    ],
    "keywords": [
      "quantum mechanics",
      "mean field limit",
      "classical limit",
      "mean-field",
      "mean field hartree equation"
    ],
    "publication": {
      "doi": "10.1007/s00220-015-2485-7",
      "journal": "Communications in Mathematical Physics",
      "year": 2016,
      "month": "Apr",
      "volume": 343,
      "number": 1,
      "pages": 165
    },
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016CMaPh.343..165G"
    }
  }
}