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

Abstract

In this work, we study the quantum fluctuation dynamics in a Bose gas on a torus $\Lambda =(L\mathbb{T}{)}^3$ that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a Bose-Einstein condensate (BEC) with density $N$ surrounded by thermal fluctuations with density $1$, we assume that the system is described by a mean-field Hamiltonian. We extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics. Using a Fock-space approach, we provide explicit error bounds. It is known that the BEC and the HFB fluctuations both evolve at microscopic time scales $t\sim 1$. Given a quasifree initial state, we determine the time evolution of the centered correlation functions $⟨a⟩$, $⟨aa⟩-⟨a{⟩}^2$, $⟨a^{+}a⟩-|⟨a⟩{|}^2$ at mesoscopic time scales $t\sim {\lambda }^{-2}$, where $0<\lambda \ll 1$ denotes the size of the HFB interaction. For large but finite $N$, we consider both the case of fixed system size $L\sim 1$, and the case $L\sim {\lambda }^{-2-}$. In the case $L\sim 1$, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in $\mathbb{Q}$; this phenomenon is reminiscent of the Talbot effect. For the case $L\sim {\lambda }^{-2-}$, we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have $\lambda \sim (\frac{\log \log N}{\log N}{)}^{\alpha }$, for different values of $\alpha >0$.

120 pages. Added comments after referee report. Accepted for publication in Journal of Statistical Physics

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