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

In this paper, we study the quantum fluctuation dynamics in a Bose gas on a torus $\Lambda=([-L/2,L/2]^3/\sim)$ 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 $\langle a\rangle$, $\langle aa\rangle-\langle a\rangle^2$, $\langle a^+a\rangle-|\langle a\rangle|^2$ at mesoscopic time scales. For large but finite $N$, we consider both the case of fixed system size $|\Lambda|\sim1$, and the case $|\Lambda|\sim (\log(N)/\log\log(N))^{\frac78}$. In the case $|\Lambda|\sim1$, 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 $|\Lambda|\sim (\log(N)/\log\log(N))^{\frac78}$, we prove that the collision operator is well approximated by the expression predicted in the literature.