{
  "id": "1904.08137",
  "version": "v1",
  "published": "2019-04-17T08:44:14.000Z",
  "updated": "2019-04-17T08:44:14.000Z",
  "title": "A microscopic derivation of Gibbs measures for nonlinear Schrödinger equations with unbounded interaction potentials",
  "authors": [
    "Vedran Sohinger"
  ],
  "comment": "77 pages, 5 figures",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study the derivation of the Gibbs measure for the nonlinear Schr\\\"{o}dinger equation (NLS) from many-body quantum thermal states in the high-temperature limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded $L^p$ interaction potentials on $\\mathbb{T}^d$ for $d=1,2,3$. This extends the author's earlier joint work with Fr\\\"{o}hlich, Knowles, and Schlein, where the regime of defocusing and bounded interaction potentials was considered. When $d=1$, we give an alternative proof of a result previously obtained by Lewin, Nam, and Rougerie. Our proof is based on a perturbative expansion in the interaction. When $d=1$, the thermal state is the grand canonical ensemble. As in the author's earlier joint work with Fr\\\"{o}hlich, Knowles, and Schlein, when $d=2,3$, the thermal state is a modified grand canonical ensemble, which allows us to estimate the remainder term in the expansion. The terms in the expansion are analysed using a graphical representation and are resummed by using Borel summation. By this method, we are able to prove the result for the optimal range of $p$ and obtain the full range of defocusing interaction potentials which were studied in the classical setting when $d=2,3$ in the work of Bourgain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-17T08:44:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "60G60",
      "81V70",
      "82B10",
      "35Q40"
    ],
    "keywords": [
      "nonlinear schrödinger equations",
      "unbounded interaction potentials",
      "gibbs measure",
      "authors earlier joint work",
      "microscopic derivation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 77,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}