Classical Nonequilibrium Ensemble Method

Abstract

In the Gibbs ensemble theory [1–3] of equilibrium statistical mechanics, aided by the Liouville equation for a uniform ensemble, we assume an equilibrium ensemble distribution function such as the canonical ensemble distribution function. Mean values are then computed for macroscopic observables therewith. By corresponding the mean values so computed of the mechanical variables such as energy, pressure, and the number of particles to the thermodynamically determined variables obeying the equilibrium Gibbs relation for the Clausius entropy, we not only identify the temperature with the parameter β appearing in the canonical distribution function [i.e.,exp(−βH)/〈exp(−βH)〉] but also obtain the statistical mechanical expression for the equilibrium entropy (Clausius entropy) of the system in terms of the distribution function assumed. When such correspondences have been made between the thermodynamic variables and the statistical mechanical mean values for the variables, all the statistical mechanical mean values are expressible in terms of the canonical partition function, and it is possible to implement equilibrium statistical thermodynamics by means of the distribution function alone, which provides a molecular picture of the system of interest. In the Gibbs ensemble theory the burden of the required computation is shifted to the partition function, calculation of which constitutes the main task in statistical thermodynamics.