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Annalen der Physik

Diagram Expansions in Nonequilibrium Statistical Mechanics Classical Limit Considerations

Ralf Der, Siegfried Fritzsche A, Reinhold Haberlandt

Published Jan 1, 1982• 0 citations
Publisher

Abstract

Abstract A diagram expansion is proposed for calculating traces of the kind Tr{Ae−itLB} which are of interest for calculating time correlation functions and expectation values in nonequilibrium statistical mechanics. Arbitrary initial conditions are considered. In the classical limit the diagram expansion of FUJITA is obtained. A systematic method for obtaining quantum corrections including exchange and symmetry effects is proposed.

DOI
10.1002/andp.19824940605

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Abstract A diagram expansion is proposed for calculating traces of the kind Tr{Ae−itLB} which are of interest for calculating time correlation functions and expectation values in nonequilibrium statistical mechanics. Arbitrary initial conditions are considered. In the classical limit the diagram expansion of FUJITA is obtained. A systematic method for obtaining quantum corrections including exchange and symmetry effects is proposed.
Starting from the usual formulation of nonequilibrium quantum statistical mechanics, the expectation value of an operator A in a steady state nonequilibrium quantum system is shown to have the form 〈A〉 =Tr{${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}(\mathit{H}\mathrm{\ensuremath{-}}\mathit{Y})}$A} /Tr{${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}(\mathit{H}\mathrm{\ensuremath{-}}\mathit{Y})}$}, where H is the Hamiltonian, \ensuremath{\beta} is the inverse of the temperature, and Y is an operator which depends on how the system is driven out of equilibrium. Because 〈A〉 is not expressed as a sum of correlation functions integrated over real time, one can now consider performing nonperturbative calculations in interacting nonequilibrium quantum problems.
In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".
The complete rule of the Feynman-diagram expansion of a nonequilibrium process is given by using the path-integral technique. At an initial time tI{\mathit{t}}_{\mathit{I}}tI​, the system is in equilibrium and is brought into the nonequilibrium state by a time-dependent Hamiltonian. All the interactions, including the initial correlation, are taken into account by a diagrammatic expansion. Besides the well-known equilibrium propagators and the conventional 2\ifmmode\times\else\texttimes\fi{}2 nonequilibrium propagator matrix, the propagator becomes a 3\ifmmode\times\else\texttimes\fi{}3 matrix that contains extra elements depending explicitly on tI{\mathit{t}}_{\mathit{I}}tI​. Both the Feynman rule of the coordinate representation and the coherent-state representation are presented. It is shown that the propagator matrix is expressed compactly by the contour representation. The classical limit is also discussed.
A theory of non-equilibrium thermodynamics is developed. The general method to derive the closed set of kinetic equations for expectation values of physical quantities is developed, by making use of the functional Taylor expansion with respect to expectation values of physical quantities around the reference state. For this purpose, a definition of the thermodynamic state in the system whose state changes with time is presented in order to describe the general state of the many-body system from the unified point of view. The statistical mechanical probability distribution functions for the quantum-mechanical expectation values at the stationary state are developed, without making use of the concept of the constant of the motion. The relation between the expectation value and the most probable value for the probability distribution function is discussed, and it is found that both values are equivalent to each other, if the system is extended infinitely in space and time. Thus the condition for the thermodynamic limit may be defined in a generalized way. Our method is looked as a perturbational one in the equilibrium statistical mechanics, though the choice of the reference state is arbitrary.
In a preceding paper [J. Math. Phys. 3, 475 (1962)], some model Hamiltonians were proposed for quantum-mechanical many-body systems with pair forces. For infinite systems in thermal equilibrium, they led to temperature-domain propagator expansions which were formally summable and expressible by closed equations. These expansions were identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models corresponded to ring- and ladder-diagram summations from the true propagator expansion, augmented by infinite classes of self-energy corrections. The model Hamiltonians were called stochastic because they contained parameters whose phases were fixed by random choices. In the present paper, more general models are formulated which yield formally summable propagator expansions for finite systems. The analysis is extended to correlation and Green's functions defined for nonequilibrium ensembles. The nonequilibrium treatment is developed in the Heisenberg representation in such a way that unlinked diagrams do not arise. A basic convergence question associated with the formal closed equations for the model propagators and correlation functions is examined by means of finite-difference integration of the Heisenberg equations of motion. This procedure appears to converge independently of whether the perturbation expansions for the propagators and correlation functions converge. It yields substantial support for the validity of the formal closed model equations.
High-temperature virial expansion is a powerful tool in equilibrium statistical mechanics. In this paper we generalize this method to the study of nonequilibrium dynamics based on contour Green's functions and present the diagrammatic rules to calculate the virial expansion coefficients. As an application of our theory, we recover the dynamics of a Bose gas quenched from noninteracting to finite interaction and obtain analytically the long-time limit of the momentum distribution function of the gas, consistent with previous numerical results.