Theory of the condensation point

This paper is a report of some studies leading to a new mathematical description of the condensation point for a simple class of models of first-order phase transitions. The paper consists of three main parts. In the first part it is pointed out that, although the conventional droplet model of condensation predicts that the free energy has an essential singularity at the condensation point, this singularity is so weak as to be experimentally unobservable. Furthermore, the analytic continuation of the free energy beyond the singularity describes a metastable phase according to the assumptions of the model. The second part of the paper is devoted to the study of a soluble functional integral that exhibits an essential singularity similar to that found in the droplet model. A method is developed for computing the singular properties of such integrals in cases where it is not possible to evaluate the integrals exactly. In the third part of the paper this method is applied to a simple model of a ferromagnet at temperatures well below the Curie point. Most of the really characteristic features of the droplet model are recovered in this calculation. The detailed results have a bearing on problems of phase coexistence, surface energies, and possibly even condensation rates.