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Journal of Mathematical Physics

On the General Theory of the Approach to Equilibrium. I. Interacting Normal Modes

I. Prigogine, Françoise Hénin

Published Sep 1, 1960• 35 citations
Publisher

Abstract

A general method which permits the derivation of the equations which describe the approach to equilibrium correct to an arbitrary finite order in the coupling constant is presented. This method is applied in the present paper to normal modes interacting through three-phonon processes. In a subsequent paper the method will be applied to interacting particles. The distribution function is first Fourier-analyzed with respect to the angle variables. All Fourier components, except the distribution function of action variables, describe correlations among the normal modes. The formal solution for the Fourier components is studied in the limiting case of a very large number of degrees of freedom N → ∞, and for large times by means of a diagram technique. Each component ρ3n can be split into 2 parts: ρ3n′ and ρ3n″; one (ρ′) due to ``scattering'' of the normal modes satisfies diagonal differential equations. The other (ρ3n″) contains the direct interaction between the normal modes involved in the corresponding correlation. It is completely determined by the functions ρ3n′. The study of this set of equations enables us to study the approach to equilibrium.

DOI
10.1063/1.1703666

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A general method which permits the derivation of the equations which describe the approach to equilibrium correct to an arbitrary finite order in the coupling constant is presented. This method is applied in the present paper to normal modes interacting through three-phonon processes. In a subsequent paper the method will be applied to interacting particles. The distribution function is first Fourier-analyzed with respect to the angle variables. All Fourier components, except the distribution function of action variables, describe correlations among the normal modes. The formal solution for the Fourier components is studied in the limiting case of a very large number of degrees of freedom N → ∞, and for large times by means of a diagram technique. Each component ρ3n can be split into 2 parts: ρ3n′ and ρ3n″; one (ρ′) due to ``scattering'' of the normal modes satisfies diagonal differential equations. The other (ρ3n″) contains the direct interaction between the normal modes involved in the corresponding correlation. It is completely determined by the functions ρ3n′. The study of this set of equations enables us to study the approach to equilibrium.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function h(k)h(k)h(k) and direct correlation function c(k)c(k)c(k). The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ gives the symmetrized scattering function $S(k\ensuremath{\omega})$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.
Using a generalized version of the Volterra-Wiener functional technique, we study a model for a many-mode laser action in which the coupling among modes takes place through the intensities. It is shown that the coupled Langevin equations for the field amplitudes can be solved avoiding the infinities at the mode thresholds. In particular, mean photon numbers, variances, and correlation and cross-correlation functions of the modes can be computed. In order actually to carry out the computations, it is necessary to evaluate the solution of a set of coupled deterministic first-order differential equations and to diagonalize a matrix whose dimension is given by the number of modes envisaged. As an application, the two-mode case is treated in detail and, besides a derivation of the stationary properties, in part already known, the transient photon statistics, during the buildup of the two modes, is thoroughly studied. The advantage of this approach is due to its simplicity and to the possibility of dealing with a large number of interacting modes.
The time-dependent behavior of an isolated system consisting of many spins in a solid lattice is studied. An interaction of the form Bjkσjxσkx is assumed between any pair (j, k) of spins. The system is treated without making any approximations; it is one of the few many-body systems whose time-evolution can be treated exactly. The behavior is fully determined by the initial density matrix and the distribution of values Bjk. By considering a variety of distributions for the Bjk, and a variety of initial density matrices, one finds cases (initial random phasing and particular continuous spectra) where the system relaxes monotonically to a metastable equilibrium; in other cases (coherent initial state or nearest-neighbor coupling only) the system oscillates indefinitely without relaxation; in still other cases (initial random phasing and N → ∞) the system relaxes in an oscillatory way to the metastable equilibrium.