Maximum Entropy Approach to the Theory of Simple Fluids

Abstract

We explore the use of the method of Maximum Entropy (ME) as a technique to generate approximations. In a first use of the ME method the "exact" canonical probability distribution of a fluid is approximated by that of a fluid of hard spheres; ME is used to select an optimal value of the hard-sphere diameter. These results coincide with the results obtained using the Bogoliuvob variational method. A second more complete use of the ME method leads to a better descritption of the soft-core nature of the interatomic potential in terms of a statistical mixture of distributions corresponding to hard spheres of different diameters. As an example, the radial distribution function for a Lennard-Jones fluid (Argon) is compared with results from molecular dynamics simulations. There is a considerable improvement over the results obtained from the Bogoliuvob principle.

Entropic criterion for model selection

“…The crux of this model is that we consider whether the correct choice should have been some other value r d =r d + δr rather than the optimal r d =r d in original hard-sphere approximation. As discussed in [17] this is a question about the probability of r d , P d (r d ). Thus, we are uncertain not just about q N but also about r d and what we actually seek is the joint probability of q N and r d , P J (q N , r d ).…”

“…We consider another improved approximation model obtained from method of ME that is a probability model and has been proved to be competitive to those theories [17]. The crux of this model is that we consider whether the correct choice should have been some other value r d =r d + δr rather than the optimal r d =r d in original hard-sphere approximation.…”

“…Several approximation models are, therefore, developed according to researchers's knowledge and experiences in the studies of simple fluids in last three decades. Yet we shall consider only three approximation models, mean field from [15], hard-sphere from [16] or [17] and improved hard-sphere approximation from [17] that are briefly presented in the following to demonstrate the use of entropic criterion for model selection.…”

“…In addition, one have to consider proper local fluctuation effect in model to generate correct liquid structure by requesting N to be effective particle number N ef f (please refer to [17] for more discussions). By recognizing that diameters other thanr d are not ruled out and that a more honest representation is an average over all hard-sphere diameters we are effectively replacing the hard-spheres by a soft-core potential.…”

“…Hard-sphere approximation: One replaces short-range repulsion by hard-sphere potential with an optimal hard-sphere diameterr d , which is determined by ME method as well [17]. Probability distribution given by hard-sphere approximation is…”