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THE PROBLEM OF MASTER EQUATIONS

Leo Van Hove

Published Jan 1, 1961• 1 citation
OpenAlex

Abstract

The master equation approach to the ergodic problem is discussed. The quantum master equation was derived under the assumption of random phases at the initial time only. The system consisting of one conduction electron interacting with the phonon gas in a harmonic lattice was considered. The use of the resolvent and the case of small perturbation are discussed. (M.C.G.)

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The master equation approach to the ergodic problem is discussed. The quantum master equation was derived under the assumption of random phases at the initial time only. The system consisting of one conduction electron interacting with the phonon gas in a harmonic lattice was considered. The use of the resolvent and the case of small perturbation are discussed. (M.C.G.)
Techniques previously developed for deriving a master equation to general order in the perturbation for the transition probability are now applied to the interference term. A master equation to general order for the partial interference term is obtained, which, in the limit of small perturbation, leads to a Pauli master equation for the interference term itself. The long-time behavior of the interference term is discussed. By means of these results, a complete master equation can be written without any random phase assumption. The ergodic behavior of a quantum many-body system is demonstrated for a large class of physical quantities, and the approach to microcanonical equilibrium is discussed. (auth)
We discuss the derivation of the Pauli master equation, based on a repeated random phase assumption, and of van Hove's result, based on an initial random phase assumption. For the former we indicate a derivation which is closer to the general approach of stochastic theory than Pauli's original method. For the van Hove result, we show that the diagonal and nondiagonal parts of the evolution operator of the Schrödinger or von Neumann equation are readily obtained by Zwanzig's projection operator method.
Starting from the quantum-Boltzmann equation derived in a previous paper, we study the irreversible evolution of an electron gas in the one-particle phase space. The connection with phase space is established by expressing one-electron states in terms of the overcomplete and nonorthogonal generating system of coherent states. By using the generalized closure relation for coherent states, as well as the fact that a one-particle operator is completely determined by the ensemble of expectation values for all coherent states, we obtain the master equations in a form that allows us to follow the evolution in phase space. This form of the master equations provides a direct link between the quantum-statistical approach and the semi-classical Boltzmann equation. The latter is obtained after a coarse-graining procedure in the one-particle phase space and by using the fact that the electron-electron interaction, as well as the interactions between the electron gas and the bath subsystems provided by phonons or photons, are local in real space.
A new general non-perturbative method for the solution of the generalized master equation for a simple two-level model of the electron-phonon system coupled to a thermodynamic reservoir is suggested. The model includes a two-level system interacting linearly with the single-mode phonons coupled to the thermodynamic bath. For the interaction of the bath with the vibrational mode we use a stochastic model generalized to finite temperatures. The results are compared with a direct numerical solution.