Time Reversal Invariance of quantum kinetic equations II: Density operator formalism

Abstract

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 by using the method of reduced density operators leading to the quantum generalization of the BBGKY-(Bogolyubov, Born, Green, Kirkwood, Yvon) hierachy. 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, on 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 {\it et al.}, submitted to J. Math. Phys., arXiv:1612.08033] where time-reversal invariance of quantum-kinetic equations was analyzed in the frame of the independent nonequilibrium Green functions formalism.

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