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 λ². 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∼(loglogNlogN)α, for different values of α> 0.