Large Deviations for Trapped Interacting Brownian Particles and Paths

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 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. Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany, [email protected] 2School of Theoretical Physics, Dublin Institute for Advanced Studies, 10, Burlington Road, Dublin 4, Ireland 3Fachbereich Mathematik und Informatik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, D-55099 Mainz, Germany, [email protected] 4Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany, [email protected] MSC 2000. 60F10; 60J65; 82B10; 82B26.