CHAPTER 5 - Numerical methods in the theory of liquids

Summary

Introduction

In the previous chapters approximate integral equations were developed for the pair distribution. From this function we were able to establish virtually all the thermodynamic functions of interest, although it was clear that certain approximations had, of necessity, to be invoked either on the grounds of mathematical expediency as in the KBGY class of equations, or on more physical grounds in the case of the PY-HNC cluster expansions. The development of approximate integral equations was, of course, enforced by the impossibility of obtaining a direct evaluation of the N-body partition function for dense fluids, and as yet there appears to be no general method of calculating this quantity. It will be recalled that the difficulty was mathematical rather than conceptual, originating in the collective nature of the total potential Φ_N even in the pairwise additive approximation. Powerful numerical techniques have been brought to bear upon the various integral equations, and these are generally evaluated by iterative techniques. The question arises as to whether these complex calculational techniques should not be used directly in rigorous variants of the theory. An ideal case would, of course, be the possibility of calculating directly the configuration integral of a system of a large number of particles.

It appears that with the advent of large electronic computers we have at our disposal a means of calculating an ensemble average in terms of the accessibility of states of the system. The ‘states’ of the system may be purely configurational and determinate as in the molecular dynamics approach, where the representative point evolves in accordance with a classical Liouville equation, subject to constraints of microscopic reversibility and ergodicity.