On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

Abstract

In this paper, we study the quantum fluctuation dynamics in a Bose gas on a torus Λ Ă R 3 that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a mean-field Hamiltonian and Bose-Einstein condensate (BEC) with density N , we extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the HFB dynamics and the BEC dynamics. Using a Fock-space approach, we provide explicit error bounds. Given an approximately quasi-free initial state, we determine the time evolution of the centered correlation functions xay, xaay ´xay 2 , xa `ay ´|xay| 2 for mesoscopic time scales. For large but finite N , we consider both the case of fixed system size |Λ| " 1, and the case |Λ| " N 6 53 ´. In the case |Λ| " 1, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in Q; this phenomenon is reminiscent of the Talbot effect. For the case´, we prove that the collision operator is well approximated by the expression predicted in the literature. THOMAS CHEN AND MICHAEL HOTT 7.1. Discrete case 78 7.2. Approximation by continuous limit 81 Appendix A. Introductory observations 83 Appendix B. Proof of convergence to mean field equations 85 B.1. Error bounds due to HFB evolution 89 B.2. Various error bounds 90 References 95

The derivation of the compressible Euler equation from quantum many-body dynamics

“…On the other hand, the derivation of the quantum Boltzmann equation is at a rudimentary stage. (See, for example, [10,15,27] and the references within.) Not to mention the possibility that one might need to pass to another classical Boltzmann equation if one takes such a route.…”