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Journal of the Physical Society of Japan

A Contribution to the Theory of Nonequilibrium Statistical Mechanics. II. Nonequilibrium Density Matrix

Masakazu Ichiyanagi

Published Jul 15, 1989• 3 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.58.2305

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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.
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.
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This is the second in a series of papers which intends to develop a statistical mechanical theory capable of independently treating responses to thermal as well as mechanical disturbances. The essential idea is to develop a method of solving the Liouvill-von Neumann equation for the density matrix of a many-body system in a time-dependent Gibbsian form. The theory is made determinate by defining the relative entropy, which will reflect the character of the entropy production in our model. In this paper, rigorous expressions are obtained for nonlinear responses, which contain all the most general consequences of statistical mechanics regarding the macroscopic variables. In particular, the Onsager reciprocity relations follow from the formula in nonlinear regions. An attempt to treat a stationary state where heat is produced is made in terms of a nonequilibrium density matrix satisfying the Liouville-von Neumann equation.
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.
Exact equations of motion for the space- and time-dependent thermodynamic coordinates of a many-body system are derived directly from the Liouville equation. This is done by definining a generalized canonical density operator depending only upon present values of the thermodynamic coordinates. This operator is used no matter how far the system is from equilibrium. An explicit expression for the entropy of a system possibly not in equilibrium is given in terms of this operator. The equation of motion for the operator is derived, and the coupled, nonlinear, integrodifferential equations of motion for the thermodynamic coordinates follow immediately.