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Large deviations for trapped interacting Brownian particles and paths
Stefan Adams, Jean-Bernard Bru and Wolfgang König
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 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 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 minimiser is its ground state, and in the path-repellency model, the minimisers 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-$N$ behaviour 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-$N$ behaviour 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.