The Annals of Probability

Large deviations for trapped interacting Brownian particles and paths

Stefan Adams, Jean-Bernard Bru, and Wolfgang König

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Abstract

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

Article information

Source
Ann. Probab., Volume 34, Number 4 (2006), 1370-1422.

Dates
First available in Project Euclid: 19 September 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1158673322

Digital Object Identifier
doi:10.1214/009117906000000214

Mathematical Reviews number (MathSciNet)
MR2257650

Zentralblatt MATH identifier
1105.60021

Subjects
Primary: 60F10: Large deviations 60J65: Brownian motion [See also 58J65] 82B10: Quantum equilibrium statistical mechanics (general) 82B26: Phase transitions (general)

Keywords
Large deviations interacting Brownian motions occupation measure energy functionals Gross–Pitaevskii functional

Citation

Adams, Stefan; Bru, Jean-Bernard; König, Wolfgang. Large deviations for trapped interacting Brownian particles and paths. Ann. Probab. 34 (2006), no. 4, 1370--1422. doi:10.1214/009117906000000214. https://projecteuclid.org/euclid.aop/1158673322


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