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Journal of the Physical Society of Japan

A Contribution to the Theory of Nonequilibrium Statistical Mechanics. I. The Initial Density Matrix

Masakazu Ichiyanagi

Published Jul 15, 1989• 9 citations
Publisher

Abstract

What kind of ensemble has to be used for describing general nonequilibrium behavior is one of the unsolved problems of nonequilibrium statistical mechanics. We would like to propose a formalism for nonequilibrium many-body systems which follows the spirit of Gibbs' ensemble theory. We shall set forth a new perspective of Kirkwood's time-smoothing operation to define the initial density matrix. We will show that such an initial density matrix may be employed to prove the positivity of entropy production.

DOI
10.1143/jpsj.58.2297

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What kind of ensemble has to be used for describing general nonequilibrium behavior is one of the unsolved problems of nonequilibrium statistical mechanics. We would like to propose a formalism for nonequilibrium many-body systems which follows the spirit of Gibbs' ensemble theory. We shall set forth a new perspective of Kirkwood's time-smoothing operation to define the initial density matrix. We will show that such an initial density matrix may be employed to prove the positivity of entropy production.
The concept proposed by Kirkwood that a macroscopic quantity is the time average of the ensemble average of the corresponding dynamical operator is examined in detail. This concept provides us with a new possibility of studying nonequilibrium macroscopic systems. The method to be used here is based on the fact that a time-smoothed density matrix also satisfies the original Liouville-von Neumann equation for the unsmoothed one. Forms of nonequilibritum density matrix are given as exact solutions to it. In characterizing thermodynamically the density operators the use of the Gibbs-Duhem relation is made. A generalized version of Kubo formula is obtained for nonequilibrium systems in the hydrodynamical stage. The present method is also applicable to nonlinear dynamical systems.
In this paper, we formally demonstrate that the nonequilibrium density matrix developed by Hershfield for the steady state has the form of a McLennan-Zubarev nonequilibrium ensemble. The correction term in this pseudoequilibrium Gibbs-like ensemble is directly related to the entropy production in the quantum open system. The fact that both methods state that a nonequilibrium steady state can be mapped onto a pseudoequilibrium, permits us to develop nonequilibrium quantities from formal expressions equivalent to the equilibrium case. We provide an example: the derivation of a nonequilibrium distribution function for the electron population in a scattering region in the context of quantum transport.
We develop a theoretical framework for describing steady-state quantum transport phenomena, based on the general maximum-entropy principle of nonequilibrium statistical mechanics. The general form of the many-body density matrix is derived, which contains the invariant part of the current operator that guarantees the nonequilibrium and steady-state character of the ensemble. Several examples of the theory are given, demonstrating the relationship of the present treatment to the widely used scattering-state occupation schemes at the level of the self-consistent single-particle approximation. The latter schemes are shown not to maximize the entropy, except in certain limits.
Entropy in nonequilibrium statistical mechanics is investigated theoretically so as to extend the well-established equilibrium framework to open nonequilibrium systems. We first derive a microscopic expression of nonequilibrium entropy for an assembly of identical bosons/fermions interacting via a two-body potential. This is performed by starting from the Dyson equation on the Keldysh contour and following closely the procedure of Ivanov, Knoll and Voskresensky [Nucl. Phys. A {\bf 672} (2000) 313]. The obtained expression is identical in form with an exact expression of equilibrium entropy and obeys an equation of motion which satisfies the HHH-theorem in a limiting case. Thus, entropy can be defined unambiguously in nonequilibrium systems so as to embrace equilibrium statistical mechanics. This expression, however, differs from the one obtained by Ivanov {\em et al}., and we show explicitly that their ``memory corrections'' are not necessary. Based on our expression of nonequilibrium entropy, we then propose the following principle of maximum entropy for nonequilibrium steady states: ``The state which is realized most probably among possible steady states without time evolution is the one that makes entropy maximum as a function of mechanical variables, such as the total particle number, energy, momentum, energy flux, etc.'' During the course of the study, we also develop a compact real-time perturbation expansion in terms of the matrix Keldysh Green's function.
The book is devoted to the fundamental aspects of the non-equilibrium statistical mechanics of many-particle systems. The concept of Zubarev's approach, which generalizes the notion of Gibbs' ensembles, and introduces a nonequilibrium statistical operator, providing an adequate basis for dealing with strongly correlated systems that are governed by nonperturbative phenomena, such as the formation of bound states, quantum condensates and the instability of the vacuum. Besides a general introduction to the formalism, this book contains contributions devoted to the applications of Zubarev's method to the solution of modern problems in different fields of physics: transport theory, hydrodynamics, high-energy physics, quark-gluon plasma and hadron production in heavy-ion collisions. The book provides valuable information for researchers and students in these fields, requiring powerful concepts to solve fundamental problems of non-equilibrium phenomena in strongly
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.