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Canadian Journal of Physics

Master hierarchy of kinetic equations for binary interactions in the adiabatic approximation

Pagiotis Vasilopoulos, C. M. Van Vliet

Published Jan 1, 1983• 4 citations
Publisher

Abstract

The inhomogeneous master equation obtained in a previous paper is employed to derive a hierarchy of moment equations (quantum mechanical) analogous to the BBGKY hierarchy; the method used is the generating function according to Laplace. In this master equation, and in all these moment equations as well, the effect of the interaction between the particles of the system is represented by relaxation terms; further, there are streaming terms due to the external field. The interactions considered are of a binary nature, fermion–boson or boson–boson. The resulting equations have a simple form (especially for fermion–boson interaction) since the terms representing the interaction can be cast in a form in which only the Boltzmann operator (appropriate to the one-body description) and the second order Fokker–Planck moment appear. The Boltzmann equations obtained by the new method are the same as before (K. M. van Vliet. J/Math. Phys.).

DOI
10.1139/p83-017

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The inhomogeneous master equation obtained in a previous paper is employed to derive a hierarchy of moment equations (quantum mechanical) analogous to the BBGKY hierarchy; the method used is the generating function according to Laplace. In this master equation, and in all these moment equations as well, the effect of the interaction between the particles of the system is represented by relaxation terms; further, there are streaming terms due to the external field. The interactions considered are of a binary nature, fermion–boson or boson–boson. The resulting equations have a simple form (especially for fermion–boson interaction) since the terms representing the interaction can be cast in a form in which only the Boltzmann operator (appropriate to the one-body description) and the second order Fokker–Planck moment appear. The Boltzmann equations obtained by the new method are the same as before (K. M. van Vliet. J/Math. Phys.).
This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.
We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iℒ0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.
Abstract This chapter deals with deriving Fokker–Planck equations (FPEs) that govern the behaviour of phase space distribution functions (normalised and unnormalised) for boson and fermion systems due to dynamical or thermal evolution. Specific forms are obtained for master equation dynamics, relating the drift vector and diffusion matrix elements in the FPE to coefficients specifying terms in the Hamiltonian and relaxation operators. The approach used is to derive correspondence rules, which map changes in the density operator when multiplied from left or right by mode annihilation or creation operators (as in evolution equations) onto corresponding changes in distribution functions, these being differentiations and or multiplications by phase space variables (left or right for fermions). Derivations are carried out via changes to characteristic functions or to canonical forms of the density operator. The correct successive application of correspondence rules for fermions is treated, and alternative forms obtained for bosons to correspondence rules and the FPE.
In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.
By means of the time-smoothing method developed in the previous papers of this series, the quantum kinetic equation is derived from the principles of statistical mechanics. In this paper, we avoid the BBGKY hierarchy; instead we use the Liouville-von Neumann equation for the time-smoothed density matrix. The Boltzmann distribution function is defined in terms of this time-smoothed density matrix. The present scheme may be applied to strong-coupling systems even if they are not in near equilibrium.
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case of an adiabatically slow external perturbation. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of with cubic repulsion. Equilibrium statistical mechanics of this gas allows to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e. approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and inter-level transition amplitudes, they allow to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N, where N is the number of the Pechukas gas particles (i.e. energy levels in the system).