An Approximate Method for Evaluating a Class of Partition Functions

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As a continuation to a previous paper, in §1 a grand partition function is rewritten so as to deal with a system of particles with hard core, in §2 is presented an integral formula expressing the inverse of the volume of the intersection of a sphere and several planes, which formula is employed in §3 to represent the mean value of the weighting factor. In §3 a partition function with a fixed number of particles is evaluated by applying the method of the previous paper, taking a point representing a probable distribution of particle for the origin of the integration space, and replacing the weighting factor by its average over the intersection of a sphere and two planes, N the number of division of the volume occupied by the system of particles being fixed. In §4 the limit N→∞ is taken to reach the final, fairly simple, expression (58). In §5 are described three distributions, a uniform distribution, a crystalline distribution and an interpolation distribution. In §6, the pressure of a system of hard spheres is computed based on three distributions and is shown in Fig.1. While the uniform distribution leads to no phase change, the crystalline distribution as well as the interpolation distribution lead to a phase change. The interpolation distribution gives a fairly good result agreeing with the virial expansion and Wainwright-Alders computations, up to the phase change, diverging from them beyond there. To improve the present method, setting up of a better distribution is proposed in §7.
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岩田義一
Department Of Physics Faculty Of Science Ochanomizu University

An Approximate Method for Evaluating a Class of Partition Functions

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