Cluster Theory of Condensing Systems

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Abstract
In this paper we develop a detailed and rigorous theory of vapour-liquid coexistence in the condensation of a system consisting of a very large number (N) of interacting molecules (in a volume Nv). We start from the Ursell-Mayer expansion of the partition function and make certain ordinary assumptions about the cluster integrals (bl). First we take a function l*(N) which satisfies certain conditions, and we define “small” clusters [for which l ≦ l*(N)] and “large” clusters [for which l > l*(N)]; thus all clusters in the system (for each N) are classified into “small” and “large” clusters, and the partition function (ΩN) of the system (for each N) is expressed in terms of the factors (ϕN) which contain “small” clusters only and the factors (ΦN) which contain “large” clusters only. Then in Theorem I it is proved that an assembly of molecules divided into any number of “large” clusters may be replaced by only one “larger” cluster composed of all the molecules of the assembly with errors as small as we please if N is sufficiently large. Then in Theorem II it is proved that as soon as the density of the system exceeds that of the saturated vapour, the isotherm becomes horizontal and there appears one “huge” cluster, which is defined as a cluster for which l/N is as near to some positive constant as we please if N is sufficiently large. Thus the “huge” cluster is of a macroscopic size and should be considered to represent the liquid phase coexisting with the saturated vapour. In Appendix A, we show that Mayer's theory of condensation has some defects, and we propose an improved approximate theory in which these defects are avoided and which is substantially similar to the rigorous theory given in the text of this paper. In Appendix B, we compare the present theory with the theory of Bose-Einstein condensation, and remark that the present theory is quite free from the faults contained in the saddle-point method.

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