Stochastic Description of a Bose–Einstein Condensate

Abstract

Abstract. 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.

A Doob h-Transform of the Gross–Pitaevskii Hamiltonian

“…This process also coincides with the one rigorously arising from the application of the Nelson map to the N-body Hamiltonian [26]. For the Nelson map see [6,7,28,29].…”

“…The Nelson map in fact does not work in the case of non linear Hamiltonians and so it cannot apply directly to the GP Hamiltonian. Anyway in [26] we proved that the (unique in law) one-particle interacting process (4.3), properly stopped before it enters in a time-dependent random interaction-subset of R 3 converges, in a relative entropy sense, to the stopped version of this ground state process (5.3). Since in the GP scaling limit the Lebesgue measure of this random interaction subset converges to zero for all t [26], the process (5.3) is a very good (non interacting) approximation of the process associated to the GP Hamiltonian (3.3).…”

“…Remark 1 As remarked in [26], the above assumptions on 0 N are satisfied under regularity assumptions on V and v ( [30] Sect. XIII 11).…”

“…First we recall some results from [18,26] and, in part, from [35], regarding the notable behavior in the GP limit of the N interacting diffusion processes (4.2) associated to the ground state of the N-body Hamiltonian, as a consequence of the complete Bose-Einstein Condensation (BEC). Besides Theorem 1 and 2, a third important theorem [18] says that asymptotically the interaction energy localizes into small balls surrounding each particle.…”

“…This ground state process (5.3) was firstly introduced in [26] as related in some sense to the GP Hamiltonian or to the GP equation (3.3). The Nelson map in fact does not work in the case of non linear Hamiltonians and so it cannot apply directly to the GP Hamiltonian.…”