The Yang-Lee distribution for a class of lattice gases

Abstract

Classical lattice gases moving on a simple cubic lattice are considered. The lattice is assumed to grow only one-dimensionally. The gas particles have hard cores (of diameter greater than the lattice spacing) and are further subject to interactions of finite range and finite order. The interactions outside the hard cores may be represented as the components of av-dimensional vector, ϕ, which is initially allowed to be complex.

Using a transfer matrix technique, an asymptotic expression is obtained for the grand canonical pressure (at complex values of the inverse absolute temperature β and the fugacityz).

Let λ₁ ... λ _M denote the eigenvalues of the transfer matrix. Define ϕ to be aD*-interaction if and only if the quotients, λ _j /λ _k , 1≦j<k≦M, regarded as functions of β,z (with ϕ fixed) arenonconstant. In this paper it is assumed that there exists at least one allowable D*-interaction. With this assumption, the main result is that ifF denotes the set of interaction vectors for which the distribution, Ω, of limit points of zeros of the grand partition function in the complexz-plane at fixed β (res. complex β-plane at fixedz) contains a domain, thenF contains no product setA ₁×...×A _v ,A _k ⊂ℂ, 1≦k≦v unless one or more of theA _k consists of (at most) isolated points. This implies that the set of vectors for which Ω consists of arcs is dense in the set of all allowable vectors (in the usual topology for ℂv).