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) theory. Given a Bose–Einstein condensate (BEC) with density $$N\gg 1$$ surrounded by thermal fluctuations with density 1, we assume that the system dynamics is generated by a Hamiltonian with mean-field scaling. We derive a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics, with rigorous error control. 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 $$0<\lambda \ll 1$$ is the coupling constant determining the HFB interaction, and a, $$a^+$$ denote annihilation and creation operators. While the BEC and the HFB fluctuations both evolve at a microscopic time scale $$t\sim 1$$ , the Boltzmann dynamics is much slower, by a factor $$\lambda ^2$$ . 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 $$\lambda \sim \Big (\frac{\log \log N}{\log N}\Big )^{\alpha }$$ , for different values of $$\alpha >0$$ .