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 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. KEYWORDSBBGKY-hierarchy, density operators, quantum dynamics, quantum kinetic theory, time reversibility INTRODUCTIONThe time evolution of quantum many-body systems is of great interest currently in many areas of modern physics and chemistry, for example, in the context of laser-matter interaction, non-stationary transport, or dynamics following an interaction or confinement quench. The theoretical concepts used to study these dynamics are fairly broad and include (but are not limited to) wave function-based approaches, density functional theory, and quantum kinetic theory. The latter treats the time dynamics of the Wigner distribution or, more generally, the density matrix and captures the relaxation towards an equilibrium state (see, e.g., Refs. 1-4). The most famous example of a kinetic equation is the Boltzmann equation, along with quantum generalization, but this equation is known to not be applicable to the short-time dynamics. For this reason, generalized quantum kinetic equations were derived that are non-Markovian in nature (e.g., Refs. 1,3,[5][6][7][8][9] and that have a number of remarkable properties, including the conservation of total energy, in contrast to kinetic energy conservation in the Boltzmann equation. It was recently demonstrated that these generalized quantum kinetic equations are well suited to study the relaxation dynamics of weakly and moderately correlated quantum systems, in very good agreement with experiments with ultra-cold atoms (e.g., Refs. 10, 11) and first-principle density matrix renormalization group methods. [12] Contrib. Plasma Phys.

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