Interacting Brownian motions and the Gross-Pitaevskii formula

TL;DR

This paper reviews probabilistic methods for the Gross-Pitaevskii theory, focusing on large deviations principles for interacting Brownian motions, which yield variational characterizations of large particle systems.

Contribution

It introduces large deviations principles for the mean occupation measure of interacting Brownian motions, providing new variational formulas for infinite particle systems.

Findings

Large deviations principles established for interacting Brownian motions.

Variational rate functions describe the infinite system behavior.

Effective descriptions of dilute particle systems derived.

Abstract

We review probabilistic approaches to the Gross-Pitaevskii theory describing interacting dilute systems of particles. The main achievement are large deviations principles for the mean occupation measure of a large system of interacting Brownian motions in a trapping potential. The corresponding rate functions are given as variational problems whose solution provide effective descriptions of the infinite system.