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Journal of physics. A, Proceedings of the Physical Society. General

A general approach to nonequilibrium quantum statistics

Edward R. Pike, S. Swain

Published Jul 1, 1971• 20 citations
Publisher

Abstract

Direct use of a generalized Laplace transform is proposed for applications in nonequilibrium quantum statistics. The method includes the equilibrium 'double-time' Green's function method as a special case. An example is given in optical theory in which non-Markoffian behaviour and Poincare recurrences have been explicitly found.

DOI
10.1088/0305-4470/4/4/019

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Direct use of a generalized Laplace transform is proposed for applications in nonequilibrium quantum statistics. The method includes the equilibrium 'double-time' Green's function method as a special case. An example is given in optical theory in which non-Markoffian behaviour and Poincare recurrences have been explicitly found.
A Laplace transform relation between the distribution function $\mathrm{exp}{\ensuremath{-}\ensuremath{\beta}H}$ in quantum statistical mechanics and the Green's function of the wave equation has been noted by Koppe. This relationship may be extended to the stationary-state scattering equation. General properties of the transformation are discussed. Applications to the evaluation of the virial coefficients are considered. A further application of the perturbation methods of Brueckner and his collaborators is described.
This paper is intended to provide the axiomatic study of nonequilibrium quantum statistical mechanics with some simple and rigorously solvable models. The models considered here are obtained as generalizations of the Ising model. They illustrate and allow a rational discussion of the following concepts relevant to the theory of irreversible phenomena: coarse-graining and time-smoothing, ergodicity, recurrences, impossibility of a Markovian description of the approach to equilibrium for some physical systems, justification of the various random phase assumptions, properties of the interaction responsible for the approach to equilibrium, master equations, etc.
The Green's function method is one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has proved invaluable in many research fields. This book provides a unique, self-contained introduction to nonequilibrium many-body theory. Starting with basic quantum mechanics, the authors introduce the equilibrium and nonequilibrium Green's function formalisms within a unified framework called the contour formalism. The physical content of the contour Green's functions and the diagrammatic expansions are explained with a focus on the time-dependent aspect. Every result is derived step-by-step, critically discussed and then applied to different physical systems, ranging from molecules and nanostructures to metals and insulators. With an abundance of illustrative examples, this accessible book is ideal for graduate students and researchers who are interested in excited state properties of matter and nonequilibrium physics.
We describe the large applicability of the Nonequilibrium Statistical Operator Method (NSOM) for the study of dissipative dynamic systems far from equilibrium. It is shown that the NSOM can be encompassed by a unifying variational principle, which produces a large family of NSO that contains existing examples as particular cases. Further, we review the application of the NSOM for the construction of a nonlinear quantum theory of large scope, and for the generation of a response function theory, for far-from-equilibrium Hamiltonian systems. An accompanying non-equilibrium thermodynamic Green's function theory is briefly described. Also it is shown that the NSOM provides mechano-statistical foundations for phenomenological irreversible thermodynamics, and for the important question of stability of far-from-equilibrium steady states and the emergence of self-organized dissipative structures in condensed matter.
A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman.The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later.In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform.The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.