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Annales Henri Poincaré

, Volume 12, Issue 8, pp 1601–1612

First Online: 21 May 2011Received: 14 September 2010Accepted: 11 April 2011


In this work we give a positive answer to the following question: does Stochastic Mechanics uniquely define a three-dimensional stochastic process which describes the motion of a particle in a Bose–Einstein condensate? To this extent we study a system of N trapped bosons with pair interaction at zero temperature under the Gross–Pitaevskii scaling, which allows to give a theoretical proof of Bose–Einstein condensation for interacting trapped gases in the limit of N going to infinity. We show that under the assumption of strictly positivity and continuous differentiability of the many-body ground state wave function it is possible to rigorously define a one-particle stochastic process, unique in law, which describes the motion of a single particle in the gas and we show that, in the scaling limit, the one-particle process continuously remains outside a time dependent random -interaction-set- with probability one. Moreover, we prove that its stopped version converges, in a relative entropy sense, toward a Markov diffusion whose drift is uniquely determined by the order parameter, that is the wave function of the condensate.

Communicated by Vieri Mastropietro.

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Author: Laura M. Morato - Stefania Ugolini


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