Ensembles of Maximum Entropy

Abstract

A theorem is stated and proved. The theorem considers the various possible probability density functions, W_N, in the phase space of all the N molecules of a system, defined for an ensemble for which known mechanical properties require a given set of reduced probability densities, w_n, for small numbers of molecules n₀. The theorem states that the function, W_N, of maximum entropy is that with the simplest possible correlations between the molecules consistent with the restraint imposed by the fixed w_n, n₀. It is then shown that the course of the negative entropy flow in a real ``isolated'' system undergoing spontaneous change is a flow of negative entropy into the correlation functions of many molecules followed by the destruction of these negative entropy terms due to the random time dependent fluctuations at the container walls.