The Yang-Lee distribution of zeros for a classical one-dimensional fluid

Abstract

The interaction potential is assumed to satisfy (γ) = + infinity if γ 2a, where a is a constant greater than zero, so that only nearest neighbours can interact. At any fixed temperature T let kTπ(z) be the thermodynamic pressure at fugacity z gt-or-equal, slanted 0, as calculated from the equation of state. Let Π (z) be the complete analytic function obtained by analytic continuation of π(z) into the complex z plane, and G be the set of values of z for which one branch of Π (z), say Π max(z), is regular and has a larger real part than all the others. It is proved that, in the limit where the length L of the system tends to infinity, the zeros of the grand partition function Ξ (z, L) approach a point set Z which consists of analytic arcs and is the complement of G. It is also proved that G is simply connected, that for all z in G, and that the limiting line density of zeros of Ξ along any arc of Z (each zero being given the weight L−1) is (2π)−1 times the discontinuity in Imπmax(z) across the arc. As an illustration, (a result of Hemmer et al.), that Z is - infinity < z less-than-or-eq, slant -1/ea for the hard-rod system is confirmed rigorously.