New Method in Non-Equilibrium Statistical Mechanics of Cooperative Systems

Abstract

It is a common every-day experience that a system of macroscopic size composed of a great number of interacting particles exhibits well-organized dynamical behavior which is qualitatively different from the behavior exhibited by individual constituent particles. The aim of the nonequilibrium statistical mechanics has been to understand such macroscopic behavior on the molecular level. The key ideas in this endeavor are (i) the identification of variables which describe such well-organized motions (also referred to as gross behavior) and are referred to variously as the collective coordinates, the macroscopic variables or the gross variables (in this article we adopt the last-mentioned terminology) and (ii) the dissipation of these well-organized motions into heat by the action of rapid random molecular motions (the so-called random forces). Here the underlying assumption is that the time scale characterizing the motion of gross variables is always well separated from the time scale for random forces, which appears to be justified under normal circumstances by the very existence of macroscopic laws such as classic hydrodynamic equations. Theoretically, however, the picture is not so clear cut. Consider, as an example, a gas described by the Boltzmann equation. If the above-mentioned assumption were correct, this implies that the Boltzmann collision operator has an eigenvalue spectrum which consists of five-fold zeros for five collision invariants corresponding to the five gross variables and the background spectrum for the remaining “random force” variables which is well separated from the zeros. This, however, seems to be true only for the gases with intermolecular forces which are Maxwellian or “harder.” For “softer” intermolecular forces, there exists a continuous background spectrum extending to zero 1].

The work supported by the National Science Foundation.