Large deviations for trapped interacting Brownian particles and paths

Details Large deviations for trapped interacting Brownian particles and paths Large deviations for trapped interacting Brownian particles and paths

2004-11-30

arxiv.org/abs/math/0411660v2

Annals of Probability 2006, Vol. 34, No. 4, 1370-1422

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000214 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000000214

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.

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