CHAPTER 3 - Numerical solution of the integral equations

Summary

Introduction

Numerical evaluation and comparison of the various integrodifferential equations developed in chapter 2 may be conveniently divided into three sections. First the low density solutions will be discussed. In this case comparison is not made in terms of the form of the pair distribution function but rather by evaluating the equation of state, based on the pressure and compressibility relations, and comparing the virial coefficients so determined with the exact results of chapter I. Of course, if the theory were self-consistent the equation of state and virial coefficients would be independent of the method of approach-the pressure and compressibility equations would yield the same results. Some effort has gone into forcing self-consistency by choosing a form for c(r) which ensures identical results regardless of the approach (the self-consistent approximation: SCA). This device of enforcing thermodynamic consistency cannot be regarded as an advance of physical understanding; nevertheless, the results are excellent at least to the sixth virial coefficient for hard spheres.

At liquid densities direct comparison of the radial distribution function may be made. It will be seen that all the theories discussed in the previous chapter agree in the qualitative form of the pair distribution, but the quantitative discrepancy is large. Again, appeal to the equation of state is made. The extreme sensitivity of the pressure equation to the precise form of g₍₂)(r) (and, indeed, to the assumed form of the pair potential) provides a severe test of the theory. It is computationally convenient to work in terms of an idealized potential such as the hard sphere or square-well interactions- these models have the important advantage that the resulting equation of state may be compared directly with machine simulations.