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arXiv:1612.08033 · Journal of Mathematical Physics

Time reversal invariance of quantum kinetic equations: Nonequilibrium Green functions formalism

Miriam Scharnke, Niclas Schlünzen, M. Bönitz

Published Jun 1, 2017• 12 citations
arXivHTMLPDF

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. In this article, it is shown analytically that quantum kinetic generalizations of the Boltzmann equation that are derived using the nonequilibrium Green functions formalism as well as all approximations that stem from Φ-derivable self-energies are time reversal invariant.

DOI
10.1063/1.4989684

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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. In this article, it is shown analytically that quantum kinetic generalizations of the Boltzmann equation that are derived using the nonequilibrium Green functions formalism as well as all approximations that stem from Φ-derivable self-energies are time reversal invariant.
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.
This is the first part of our work on time reversal symmetry applied to non-equi-librium statistical stationary states from a unified microscopic quantum statistical point of view. In this paper, a formalism for time reversal symmetry is constructed in the framework of the Closed Time Path Green's Functions (CTPGF), which can be applied both to equilibrium and non-equilibrium stationary states. By using the generating functional technique of the CTPGF, symmetry relations for the statistical Green's functions and vertex runctions are derived for systems invariant under time reversal.
The renormalization scheme in nonequilibrium thermal quantum field theories is reexamined. Instead of the self-energy diagonalization scheme, we propose to diagonalize Green’s function at equal time. This eliminates the problem of on-shell definition related to time-dependent energies and spatially inhomogeneous situations, and yields a Boltzmann equation that contains memory effect. The new diagonalization scheme and the derivation of the Boltzmann equation from it can be applied to any thermal situation. It allows the treatment of a nonequilibrium problem beyond perturbational calculations in a self-consistent manner. The results are applicable to both thermo field dynamics and the closed time path formalism.
In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".
We review a non-relativistic current algebra symmetry approach to constructing the Bogolubov generating functional of many-particle distribution functions and apply it to description of invariantly reduced Hamiltonian systems of the Boltzmann type kinetic equations, related to naturally imposed constraints on many-particle correlation functions. As an interesting example of deriving Vlasov type kinetic equations, we considered a quantum-mechanical model of spinless particles with delta-type interaction, having applications for describing so called Benney-type hydrodynamical praticle flows. We also review new results on a special class of dynamical systems of Boltzmann–Bogolubov and Boltzmann–Vlasov type on infinite dimensional functional manifolds modeling kinetic processes in many-particle media. Based on algebraic properties of the canonical quantum symmetry current algebra and its functional representations, we succeeded in dual analysis of the infinite Bogolubov hierarchy of many-particle distribution functions and their Hamiltonian structure. Moreover, we proposed a new approach to invariant reduction of the Bogolubov hierarchy on a suitably chosen correlation function constraint and deduction of the related modified Boltzmann–Bogolubov kinetic equations on a finite set of multi-particle distribution functions. There are also presented results of application of devised methods to describing kinetic properties of a many-particle system with an adsorbent surface, in particular, the corresponding kinetic equation for the occupation density distribution function is derived.