The Derivation of the Boltzmann Equation from Quantum Many-body Dynamics

Abstract

We consider the quantum many-body dynamics at the weak-coupling scaling. We derive rigorously the quantum Boltzmann equation, which contains the classical hard sphere model and, effectively, the inverse power law model, from the many-body dynamics assuming a physical and optimal regularity bound. The regularity bound we find, on the one hand, is satisfied by quasi-free solutions and comes from calculations regarding the local Maxwellian solution, in which we also prove that 2-body molecular chaos never happens unless $N=+\infty$; on the other hand, it arises from the well-posedness threshold of the limiting Boltzmann equation below which we prove ill-posedness. That is, the regularity cannot be higher at the $N$-body level, cannot be lower in the limit, and is hence a double criticality. To work with this borderline case, we analyze all four sides, with respect to the Fourier transform, of the BBGKY hierarchy sequence with new tools and techniques. We prove well-definedness, compactness, convergence, and uniqueness of hierarchies right at the criticality to complete an optimal derivation. In particular, we have proved that, for physical $N$-particle solutions, the Boltzmann equation emerges as the mean-field limit and time is hence irreversible, from first principles of quantum mechanics.