On the Emergence of Quantum Boltzmann Fluctuation Dynamics near a Bose–Einstein Condensate

In this work, we study the quantum fluctuation dynamics in a Bose gas on a torus Λ =(L𝕋)^3 that exhibits Bose–Einstein condensation, beyond the leading order Hartree–Fock–Bogoliubov (HFB) theory. Given a Bose–Einstein condensate (BEC) with density N≫ 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 ⟨ a⟩ , ⟨ aa⟩ -⟨ a⟩ ^2 , ⟨ a^+a⟩ -|⟨ a⟩ |^2 at mesoscopic time scales t∼λ ^-2 , where 0<λ≪ 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∼ 1 , the Boltzmann dynamics is much slower, by a factor λ ^2 . For large but finite N , we consider both the case of fixed system size L∼ 1 , and the case L∼λ ^-2- . In the case L∼ 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 L∼λ ^-2- , we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have λ∼ (loglog N/log N )^α , for different values of α >0 .