Large deviations for interacting Brownian Particles and Paths in trap potentials

We introduce two probabilistic models for $N$ interacting Brownian motions moving in a trap in $ \R^d $ under the presence of 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 repellence, while the second one imposes a path repellency.

We analyse both models in the limit of diverging time with fixed number $ N $ of Brownian motions. In particular, we prove large deviations principles for the normalised occupation measures. The minimisers of the rate functions in the two cases are given in terms of the ground state, respectively the ground product-states, of a certain associated operator, the Hamilton operator for a system of $ N $ interacting trapped particles (bosons). 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 toy 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-$N$ behaviour of the unrestricted ground states in terms of the well-known Gross-Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-$N$ behaviour of the product-state ground states is also described by the Gross-Pitaevskii formula, however with the scattering length of the pair potential replaced by its integral.