Time‐reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle, and not all statistical theories conserve this particular symmetry—most notably, hydrodynamic equations and kinetic equations such as the Boltzmann equation. Here, we consider quantum kinetic generalizations of the Boltzmann equation using the method of reduced density operators, leading to the quantum generalization of the Bogolyubov‐Born‐Green‐Kirkwood‐Yvon (BBGKY) hierarchy. We demonstrate that all commonly used approximations, including Vlasov; Hartree‐Fock; and the non‐Markovian generalizations of the Landau, T‐matrix, and Lenard‐Balescu equations, are originally time‐reversal invariant, and we formulate a general criterion for time reversibility of approximations to the quantum BBGKY hierarchy. Finally, we illustrate, through the example of the Born approximation, how irreversibility is introduced into quantum kinetic theory via the Markov limit, making the connection with the standard Boltzmann equation. This paper is a complement to paper I (Scharnke et al., J. Math. Phys., 2017, 58, 061903), where the time‐reversal invariance of quantum kinetic equations was analysed in the frame of the independent non‐equilibrium Green functions formalism.
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=+∞; 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.