The maximum entropy method in statistical physics: An alternative approach to the theory of simple fluids

Theories of physics are always developed based on the reasoning method of “deduction”. The studies of entropy from thermodynamics to information theory, however, show a natural reasoning method, “inference”, to be more appropriate in the field of statistical physics.; Making statistical predictions in statistical physics requires solving three problems: first, what is the information relevant to the problems of interest; secondly, how to codify the relevant information into a probability distribution; and third, how to obtain estimates for various quantities of interest? The identification of the relevant information is at present a matter of trial and error; beyond experience and intuition, there exists no systematic method. The assignment of a probability distribution is commonly carried out using Jaynes' method of maximum entropy—MaxEnt. The main goal of this thesis is to use a form of the method of maximum entropy—the ME method, which includes MaxEnt as a special case—to address the third problem.; The basic idea is to select a family of trial probability distributions and use the ME method to select the member of the family which best approximates the “exact” but intractable distribution. Thus we have developed a method for generating approximations which includes, but goes beyond, the well-known variational method of Bogoliubov.; Our first goal is to systematize the method of maximum entropy into what one might call an ME algorithm. The algorithm follows a sequence of steps: identifying relevant information; codifying this information into a probability distribution; selecting an optimal approximate but calculable distribution from within a family of trial distributions; and finally, improving the approximation by also taking into account the full family of trial distributions.; To test the applicability of the method we apply it to a system that has been extensively studied in the past: the theory of simple fluids. We develop approximations based on two families of trial distributions. The first brief treatment leads to a mean-field approximation. The second, more extended treatment, is an approximation in terms of hard spheres. Numerical calculations of the radial distribution function and the thermodynamic properties of Argon are compared to experimental results, to the results of molecular dynamics simulations, and to the results of various perturbation theories.; Our work indicates that the ME-improved variational approach developed here offers predictions that are already competitive with the best variational theories. The potential of the ME approach for further improvements and further applications is, at this early stage, far from being exhausted.