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Transport Theory and Statistical Physics

Variational Principle for Kadanoff-Baym Kinetic Equations

Anatoly Yu. Zakharov

Published Dec 18, 2008• 1 citation
Publisher

Abstract

A system of quasi-particles with scattering on arbitrary static potential is considered by means of non-equilibrium Green function method in Kadanoff-Baym formulation. The problem of kinetic coefficients evaluation can be solved with Kadanoff-Baym quantum kinetic equations for the system in the presence of an external field. A variational method of solution for linearized over-external field quantum kinetic equations is proposed.

DOI
10.1080/00411450802515726

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A system of quasi-particles with scattering on arbitrary static potential is considered by means of non-equilibrium Green function method in Kadanoff-Baym formulation. The problem of kinetic coefficients evaluation can be solved with Kadanoff-Baym quantum kinetic equations for the system in the presence of an external field. A variational method of solution for linearized over-external field quantum kinetic equations is proposed.
We have implemented time-propagation of the non-equilibrium Green function for atoms and molecules, by solving the Kadanoff-Baym equations within a conserving self-energy approximation. We here demonstrate the usefulnes of time-propagation for calculating spectral functions and for describing the correlated electron dynamics in a non-perturbative electric field. We also demonstrate the use of time-propagation as a method for calculating charge-neutral excitation energies, equivalent to highly advanced solutions of the Bethe-Salpeter equation.
Utilizing a non-equilibrium Green function like the generalized Kadanoff-Baym ansatz, a systematic perturbative method is presented to calculate the expectation value of an arbitrary physical quantity under the restriction that the Wigner distribution function is fixed. It is shown that, in the diagrammatic expression of the quantity, a certain part of contributions can be eliminated due to the restriction. Together with the quantum kinetic equation, this method provides a basis for the kinetic-theoretical description.
The following issues are discussed inspired by the recent paper of Kadanoff (arXiv: 1403:6162): (a) Construction of a generalized one-particle Wigner distribution (GWD) function (analog of the classical distribution function) from which the quantum kinetic equation due to Kadanoff and Baym (KB) is derived, often called the Quantum Boltzmann Equation (QBE); (b) The equation obeyed by this has a collision contribution in the form of a two-particle Green function. This term is manipulated to have Kinetic Entropy in parallel to its counterpart in the classical Boltzmann kinetic equation for the classical distribution function. This proved to be problematic in that unlike in the classical Boltzmann kinetic equation, the contribution from the kinetic entropy term was non-positive; (3) Kadanoff surmised that this situation could perhaps be related to quantum entanglement that may not have been included in his theory. It is shown that GWD is not positive everywhere (indicating dynamical quantumness) just like the commonly recognized property of the Wigner function (negative property indicating quantumness of the state). The issue of non-positive feature appearing in approximate evaluation of patently positive entities in many particle systems is here pointed to an early discussion of this issue (Phys. Rev. A10, 1852 (1974)) in terms of a theorem on truncation of cumulant expansion of a probability distribution function due to Marcinkeiwicz. The last issue of presence or absence of entanglement in an approximate evaluation of a many particle correlation poses a new problem; it is considered here in terms of fermionic entanglement theory in the light of density matrix and Green function theory of many-fermion systems. The clue comes from the fact that the Hartree-Fock approximation exhbits no entantanglement in two-particle fermion density matrix and hence also in two-particle Green function.
In this paper, we extend the method of Kadanoff-Baym equations for open quantum systems to arbitrary kinds of systems and heat baths, either fermionic or bosonic. This includes three spacial dimensions and different potentials for the system-bath interaction or external traps. We study the quantum-mechanical formation of bound states in one and also in three dimensions with the full Kadanoff-Baym equations and compare them to more simplified approaches with and without memory effects. An in-depth examination of the thermodynamics of open systems is performed, showing perfect equilibration of the system's degrees of freedom along with a comprehensive investigation of the influence of the heat bath on the system's wave functions. The formation time, decay time, and regeneration of bound states and their dependence on the temperature and coupling strength is explored. We evaluate the nonequilibrium Kadanoff-Baym equations for the system particles, assuming that interactions are elastic two-particle collisions with the heat-bath particles. Finally, we describe in detail the method used to numerically solve the corresponding spatially heterogeneous integrodifferential equations for the set of one-particle Green's functions.