Time-reversal invariance of quantum kinetic equations II: Density operator formalism

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.