On the Yang-Lee Distribution for a van der Waals Fluid Near the Critical Point

Abstract

The 'largest real part' (L.R.P.) criterion used to determine the Yang-Lee distribution for a class of one-dimensional continuum models and lattice gases is here applied to the system obeying the van der Waals equation of state near the critical temperature T 0 • It is found that L.R.P. is consistent with the existence of a phase transition for T-:o;.T 0 , but that if T>Tc then L.R.P. predicts unphysical behaviour in the system. A similar calculation for the Husimi­ Temperley lattice model (not given here) indicates that the same anomalous behaviour is pre­ dicted by use of the L.R.P. criterion. A possible explanation of these anQJilalies is suggested. § I. Introduction On the basis of the Yang-Lee electrostatic analogue/) Hemmer and Hauge2l have computed the distribution (say Q) of limit points of zeros of the grand par­ tition function in the complex fugacity (z) plane. Their results show clearly how a phase transition can occur for temperatures at or below the critical tern-. perature T 0 • Their prescription for determining Q is, however, known3l to pro­ duce no unique result. Although subsequent work 4l on lattice-gas approximations to the V.D.W. fluid supports their results, the general question as to the correct procedure for determining Q is still open. To avoid the problem of non-uniqueness, Penrose (unpublished) conjectured that the 'physical branch' of the complete analytic function obtained by analytic continuation from the equation of state, is given by the branch of largest real part (L.R.P.) whenever this branch is unique and regular. The set Q is then taken to coincide with the set, say S, of points z where no such unique, regular branch exists. This conjecture has been verified6l for classical one-dimensional systems of particles with hard cores and nearest-neighbour, two-body forces only. It has also been verified6l for a class of lattice gases, and extended6l to cover the Yang-Lee distribution in the complex temperature-plane for these models. The L.R.P. conjecture has not been verified for models obeying the van der Waals equation of state (van der Waals fluid), but, in view of the uniqueness problem,' it seems of interest to see whether 'L.R.P.' is consistent with 'V.D.W.', at least near the critical point. The present work consists of an approximate but rigor*> The main part of this work was done as part of the author's Ph. D. thesis (London, 1968).