Studies in Irreversible Thermodynamics. II. A Simple Class of Lattice Models for Open Systems

Abstract

A stochastic model for an ensemble of open, simple lattice systems immersed in a heat bath is presented. All processes available to the systems are first order in the numbers of molecules. Rate equations are derived for means and variances of stochastic variables using the method of cumulant generating functions. There is a close relation between the rate equations for the means and those for the variances. Although the variances decay at a rate comparable to that of the means, they approach their equilibrium functions of the means at roughly twice that rate. Thus there is a time domain, prior to final equilibrium with the surroundings, in which the systems assume the equilibrium probability distribution appropriate to the existing mean values [provided the initial distribution is ``normal,'' i.e., that its central moments have the same order of magnitude (or smaller) as at equilibrium]. The principle of detailed balance at equilibrium is found to be especially useful when applied at the probability (rather than mean value) level. Statistical mechanics provides the equilibrium properties of the ensemble and the connection between chemical potentials and kinetic parameters. Finally, we discuss and solve the mean and variance equations in special cases for two lattice-gas examples, some of whose properties were presented in the first paper of this series. The two models differ in some fundamental respects.