We develop the method of Maximum Entropy (ME) as a technique to generateapproximations to probability distributions. The central results consist in (a)justifying the use of relative entropy as the uniquely natural criterion toselect a "best" approximation from within a family of trial distributions, and(b) to quantify the extent to which non-optimal trial distributions are ruledout. The Bogoliuvob variational method is shown to be included as a specialcase. As an illustration we apply our method to simple fluids. In a first useof the ME method the "exact" canonical distribution is approximated by that ofa fluid of hard spheres and ME is used to select the optimal value of thehard-sphere diameter. A second, more refined application of the ME methodapproximates the "exact" distribution by a suitably weighed average overdifferent hard-sphere diameters and leads to a considerable improvement inaccounting for the soft-core nature of the interatomic potential. As a specificexample, the radial distribution function and the equation of state for aLennard-Jones fluid (Argon) are compared with results from molecular dynamicssimulations.
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