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

## TL;DR

This paper investigates the time reversal symmetry of quantum kinetic equations using the density operator formalism, demonstrating invariance in common approximations and explaining how irreversibility emerges in the Markov limit.

## Contribution

It provides a general criterion for time reversibility of quantum BBGKY-hierarchy approximations and connects irreversibility to the Markov limit in quantum kinetic theory.

## Key findings

- Common quantum kinetic approximations are time-reversal invariant.
- Irreversibility arises from the Markov limit, exemplified by the Born approximation.
- The work extends understanding of symmetry properties in quantum kinetic equations.

## 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.