Maximization of entropy, kinetic equations, and irreversible thermodynamics

Abstract

By an extension to the dense-fluid regime of a method based upon maximization of entropy subject to constraints, first exploited by Lewis to obtain the Boltzmann equation, kinetic equations for one-particle and two-particle classical distribution functions are obtained. For the hard-sphere potential and a one-particle constraint, the kinetic equation (for the one-particle distribution function) of the revised Enskog theory is obtained; for a two-particle constraint a more general kinetic equation (for the two-particle distribution function) than that studied by Livingston and Curtiss is obtained. For a pair potential with hard-sphere core plus smooth attractive tail, a new mean-field kinetic equation is obtained on the one-particle level. In the Kac-tail limit the equation takes the form of an Enskog-Vlasov equation. The method yields an explicit entropy functional in each case. Explicit demonstration of an $H$ theorem is made for the one-particle theories in a novel way that illustrates the roles of the reversible and irreversible parts of the hard-sphere piece of the collision integral. The latter part leads to the classical form of entropy-production density as described by linear irreversible thermodynamics and so possesses many of the features of the Boltzmann collision integral. The former part introduces new elements into the entropy-production term. It is noted that the kinetic coefficients of the revised Enskog theory exhibit Onsager reciprocity in the linear regime. Upon consideration of the standard Enskog theory in the linear regime, we construct an entropy-production density and identify conjugate fluxes and forces and also kinetic coefficients which are shown to exhibit Onsager reciprocity. The standard theory is in disagreement, however, with the results of phenomenological irreversible thermodynamics for the conventional forms of fluxes and forces.