Definition and Time Evolution of Correlations in Classical Statistical Mechanics

Abstract

The study of dense gases and liquids requires consideration of the interactions between the particles and the correlations created by these interactions. In this article, the N-variable distribution function which maximizes the Uncertainty (Shannon’s information entropy) and admits as marginals a set of (N−1)-variable distribution functions, is, by definition, free of N-order correlations. This way to define correlations is valid for stochastic systems described by discrete variables or continuous variables, for equilibrium or non-equilibrium states and correlations of the different orders can be defined and measured. This allows building the grand-canonical expressions of the uncertainty valid for either a dilute gas system or a dense gas system. At equilibrium, for both kinds of systems, the uncertainty becomes identical to the expression of the thermodynamic entropy. Two interesting by-products are also provided by the method: (i) The Kirkwood superposition approximation (ii) A series of generalized superposition approximations. A theorem on the temporal evolution of the relevant uncertainty for molecular systems governed by two-body forces is proved and a conjecture closely related to this theorem sheds new light on the origin of the irreversibility of molecular systems. In this respect, the irreplaceable role played by the three-body interactions is highlighted.

Keywords

• information theory
• high order correlations
• entropy
• many-body interactions
• irreversibility