Maximum Entropy and the Variational Method in Statistical Mechanics: an Application to Simple Fluids

Details Maximum Entropy and the Variational Method in Statistical Mechanics: an Application to Simple Fluids Maximum Entropy and the Variational Method in Statistical Mechanics: an Application to Simple Fluids

2004-11-25

arxiv.org/abs/cond-mat/0411625v3

Physics

Condensed Matter

Statistical Mechanics

5 figures. Additional demonstrations on radial distribution functions and equation of states are added

Scientific paper

We develop the method of Maximum Entropy (ME) as a technique to generate approximations to probability distributions. The central results consist in (a) justifying the use of relative entropy as the uniquely natural criterion to select a "best" approximation from within a family of trial distributions, and (b) to quantify the extent to which non-optimal trial distributions are ruled out. The Bogoliuvob variational method is shown to be included as a special case. As an illustration we apply our method to simple fluids. In a first use of the ME method the "exact" canonical distribution is approximated by that of a fluid of hard spheres and ME is used to select the optimal value of the hard-sphere diameter. A second, more refined application of the ME method approximates the "exact" distribution by a suitably weighed average over different hard-sphere diameters and leads to a considerable improvement in accounting for the soft-core nature of the interatomic potential. As a specific example, the radial distribution function and the equation of state for a Lennard-Jones fluid (Argon) are compared with results from molecular dynamics simulations.

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