New Method for Generating Density Expansions

Abstract

The calculus of finite differences is used to develop a new method for expressing the thermodynamic limit of a reasonably arbitrary statistical-mechanical average as a power series in the number density ρ. The method is simple, straightforward, and purely analytic: it involves no intermediate expansion in powers of the activity and it avoids the use of graph theory. Moreover, the method is developed independently of the prescription for computing the statistical average, a fact which lends to the results an especially wide range of applicability. In particular, these results may be used in classical or quantum statistical mechanics, for intermolecular potentials which are not spherically symmetric or pairwise additive, for molecules of arbitrary internal structure and complexity, and for polar molecules. A general formula is obtained for the coefficient of ρk in the series; as usual, the most difficult problem one need solve in order to compute this coefficient is the evaluation of a k-molecule average. It is shown that if all the coefficients exist and if the density is less than a certain well-defined critical density, then the series converges to the thermodynamic limit of the average in question. The practical use of the method is clarified by examples.