{
  "id": "math/0411660",
  "version": "v2",
  "published": "2004-11-30T16:07:37.000Z",
  "updated": "2006-09-22T14:36:50.000Z",
  "title": "Large deviations for trapped interacting Brownian particles and paths",
  "authors": [
    "Stefan Adams",
    "Jean-Bernard Bru",
    "Wolfgang König"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117906000000214 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 4, 1370-1422",
  "doi": "10.1214/009117906000000214",
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce two probabilistic models for $N$ interacting Brownian motions moving in a trap in $\\mathbb {R}^d$ under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyze both models in the limit of diverging time with fixed number $N$ of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measures. The minimizers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of $N$ interacting trapped particles. More precisely, in the particle-repellency model, the minimizer is its ground state, and in the path-repellency model, the minimizers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model. This study is a contribution to the search for a mathematical formulation of the quantum system of $N$ trapped interacting bosons as a model for Bose--Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behavior of the ground state in terms of the well-known Gross--Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behavior of the ground product-states is also described by the Gross--Pitaevskii formula, however, with the scattering length of the pair potential replaced by its integral.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-09-22T14:36:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60J65",
      "82B10",
      "82B26"
    ],
    "keywords": [
      "trapped interacting brownian particles",
      "large deviations",
      "pair potential",
      "brownian motions",
      "ground state"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11660A"
    }
  }
}