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Progress of Theoretical Physics

Inhomogeneous Particle Distributions in Non-Equilibrium Thermofield Dynamics

J. P. Whitehead

Published Nov 1, 1985• 2 citations
Publisher

Abstract

The recent generalization of thermofield dynamics to open systems is extended to include spatially inhomogeneous particle distributions. The relation to the Boltzman equation is established and a reduction formula for the evaluation of the statistical average of time ordered operator products is given.
DOI
10.1143/ptp.74.1168

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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.
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Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (hb), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution Weq generates orthogonal (Hermite) polynomials Hn in momenta. Suitable moments Wn of W (using the Hn’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the Wn’s yield irreversibly approach to equilibrium. The approach is extended (without hb) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical ϕ4 field theory (without hb). The extension to one non-relativistic quantum particle (with hb) employs the non-equilibrium Wigner function (WQ): difficulties related to non-positivity of WQ are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum ϕ4 field theory (a meson gas off-equilibrium, without hb), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum ϕ4 theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical ϕ4 one, yielding an approach to equilibrium.