Bogolyubov kinetic equation for quantum dynamical systems

D. Ya. Petrina

M. Bönitz, Miriam Scharnke, Niclas Schlünzen

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.

Alexander I. Akhiezer, S. V. Peletminskiǐ

N. Barabanenkov

The stochastic process of the motion of a test particle in a dynamical system with short~ range two-body interaction potential is considered. The method of stochastic perturbation theory in the low density and the procedure of renormalization of the Green's function of the test particle and the formalism of collision operators for the Liouville equation are used to give a quantum generalization of Bogolyubov's kinetic equation [I]+ this leading to a nonexponential law of decrease of the correlation function of the velocity of the test particle in a system of classical hard spheres over long time intervals

G. O. Balabanyan

Elmar G. Petrov

Based on the non-equilibrium density matrix method and using the Bogolyubov’s approach for the decoupling of many-particle distribution functions, a closed set of kinetic equations is derived for the state occupancies of closed and open quantum systems. It is shown that the transformation of a non-Markovian master equation into a Markovian one becomes possible at a high-frequency stochastic field alternating the energy levels of a quantum system. At a weak interaction between quantum subsystems composing the open quantum system, the set of linear balance-like kinetic equations for multistate occupancies of the system is transformed into a set of nonlinear kinetic equations for the occupancies of separate quantum subsystems.