New Series Expansion in Statistical Mechanics

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Abstract
A new approximation scheme is applied to thermodynamic variables which have power-series expansions in terms of real parameters. This method is based upon a previously discussed formula designed to overcome the convergence difficulties of the quantum-mechanical Born (perturbation) series. Some properties of the approximation scheme are obtained, and then the method is used to obtain information from power-series expansions such as those met with in thermodynamics and statistical mechanics. In particular, the virial coefficients for several molecular models are used to obtain numerical approximations to the equations of state. The solution for a one-dimensional system of hard rods is obtained exactly in this scheme. For the two-dimensional hard-square lattice gas, the results agree quite well with recent low- and high-density Padé approximants. The qualitative behavior of the solution for a system of hard spheres resembles that of the Mayer theory of condensation. It is noteworthy that the approximation scheme has the property of generating solutions which are monotonically increasing in the density. Thus, no van der Waals loop is obtained.