CHAPTER 2 - Equilibrium theory of dense fluids: the correlation functions

Summary

Introduction

We saw in the previous chapter how in the limit of low densities we were able to obtain a convergent expansion for the configurational partition function, and thereby determine all the thermodynamic functions of the system including the virial equation of state. However, what we did not have was a detailed knowledge, or indeed any knowledge, of the structure of the system.

Clearly, at liquid densities the series expansions will fail to converge⁽¹⁾, and anyway the computational labour in evaluating the multi-dimensional high-order cluster integrals which would develop in the highly connected fluid would be overwhelming, even in the economical Ree-Hoover formalism.

So, we are forced to adopt a new mathematical approach-the formalism of molecular distribution functions, or correlation functions. Instead of trying to evaluate the N-body configurational integral directly, the theory describes the probability of configurational groupings of two, three, and more particles. Further, it may be shown that we can still obtain the same amount of information concerning the system as is obtained from the study of the statistical integral itself. Moreover, in this way we obtain direct information on the molecular structure of the system being studied. Of course the method of correlation functions applies equally well to gases and solids, but we would not generally adopt that approach by choice since other characteristics of these phases suggest a more direct route to the partition function. The formalism is adopted, however, in the case of amorphous solids.

We shall find that the one- and two-body distribution functions, generally denoted by g₍₁₎ and g₍₂₎ respectively, will be of central importance in the equilibrium theory of liquids.