On the van der Waals Theory of the Vapor-Liquid Equilibrium. II. Discussion of the Distribution Functions

For the same one-dimensional fluid model discussed in Part I, we have derived general expressions for the two- and three-particle distribution functions. It is seen that these distribution functions depend on all the eigenvalues and eigenfunctions of the basic Kac integral equation, and the dependence is so transparent that the generalization to s particles is obvious. The fluctuation and virial theorems are discussed and shown to be consequences of our general formula. In the van der Waals limit, the behavior of the two-point distribution function is discussed, both for distances of the order of the hard core and for distances of the order of the range of the attractive force. The long-range behavior is, in first approximation, equivalent to the one-dimensional version of the Ornstein-Zernike theory, but only in the one-phase region and not too near the critical point. In the two-phase region, all distribution functions are linear combinations of the two corresponding distribution functions of the saturated vapor and liquid, with coefficients proportional to the mole fractions of vapor and liquid. This is shown for our model; we also give arguments for our belief that these relations are general, and express the geometrical separation of the two phases. The relation to the Ornstein-Zernike theory is discussed in more detail, especially in connection with a recent formulation of this theory by Lebowitz and Percus. We conclude with some comments on the relevance of our results for the three-dimensional problem.