Large deviations for trapped interacting brownian particles and paths

• 成果类型: Article
• 署名作者: Adams, Stefan; Bru, Jean-Bernard; Koenig, Wolfgang
• 署名单位: Max Planck Society; Dublin Institute for Advanced Studies; Johannes Gutenberg University of Mainz; Leipzig University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000214
• 发表日期: 2006
• 页码: 1370-1422
• 关键词: bose-einstein condensation; gas; systems; polaron; vortex; energy

摘要

We introduce two probabilistic models for N interacting Brownian motions moving in a trap in 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.