Statistical Mechanical Theory of Condensation

The cluster expansion for the free energy of the Ising model is reinterpreted as a cluster expansion for the pressure of the lattice gas. It is observed that the free energy of the Ising model is a function of $1-R^2$, where $R$ is the long-range order, so that the pressure of the lattice gas is a function of $\rho (1-\rho )$, where $\rho$ is density. The factor $1-\rho$ comes from the prevention of more than one particle occupying a lattice site. This idea has motivated the development of a real hard-core gas with a weak attractive tail. A cluster expansion is developed in terms of the tail alone and the hard core is treated as a metric (i.e., the hard-core part of the potential is treated exactly in all integrals). Pressure-volume isotherms are calculated explicitly in the zeroth order or molecular field approximation using the Lennard-Jones (6-12) potential and good quantitative results are found for the critical parameters as well as a qualitative understanding of the condensation phenomena. The theory of the first-order correction (spherical model) is then outlined and fluctuations of the Ornstein-Zernicke type in the local density are found. The theory of condensation is qualitatively understood in the sense that the Weiss field gives an understanding of ferromagnetism. The theory of the detailed fluctuations in the critical region is equally difficult for both phenomena since the problems are put on the same footing in the present paper.