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

TL;DR

This paper investigates quantum fluctuation dynamics near a Bose-Einstein condensate, deriving a quantum Boltzmann equation beyond standard approximations, and analyzing the effects of system size and interaction strength on these dynamics.

Contribution

It introduces a second-order Duhamel expansion to derive quantum Boltzmann dynamics beyond HFB fluctuations, with explicit error bounds and analysis of size-dependent phenomena.

Findings

Derivation of quantum Boltzmann dynamics from microscopic models.

Identification of subleading terms affecting collision operators depending on system parameters.

Validation of the Boltzmann approximation for large system sizes and specific interaction regimes.

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 $t\sim1$. Given a quasifree 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 $t\sim\lambda^{-2}$, where $0<\lambda\ll1$ denotes the size of the HFB interaction. For large but finite $N$, we consider both the case of fixed system size $L\sim1$, and the case $L\sim \lambda^{-2-}$. In the case $L\sim1$, 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 $L\sim \lambda^{-2-}$, we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have $\lambda\sim \Big(\frac{\log \log N}{\log N}\Big)^{\alpha}$, for different values of $\alpha>0$.