Strong Kac’s chaos in the mean-field Bose–Einstein Condensation

Abstract

A stochastic approach to the (generic) mean-field limit in Bose-Einstein Condensation is described and the convergence of the ground state energy as well as of its components are established. For the one-particle process on the path space a total variation convergence result is proved. A strong form of Kac's chaos on path-space for the k-particles probability measures are derived from the previous energy convergence by purely probabilistic techniques notably using a simple chain-rule of the relative entropy. The Fisher's information chaos of the fixed-time marginal probability density under the generic mean-field scaling limit and the related entropy chaos result are also deduced.2000 Mathematics Subject Classification. Primary 60J60, 60K35, 81S20, 94A17; Secondary 26D15,81S20, 60G10,60G40.

Mean-field limit for a class of stochastic ergodic control problems

“…Proof. The proof of the theorem can be found in [36] for 0 ≤ β < 1 3 (for any n and a more general class of potentials v 0 than the one considered here) and in [1] for 0 ≤ β < 1 (for n = 3 and positive-definite interaction potential v 0 ). Theorem 38 is the analogue of Theorem 20 and it proves that E δ,N and E δ satisfy the thesis of Theorem 20.…”