Zeros of the partition function for a continuum system at first-order transitions.

We extend the circle theorem on the zeros of the partition function to a continuum system. We also calculate the exact zeros of the partition function for a finite system where the probability distribution for the order parameter is given by two asymmetric Gaussian peaks. For the temperature driven first order transition in the thermodynamic limit, the locus and the angular density of zeros are given by r = e(∆c/2l)θ 2 and 2πg(θ) = l(1+ 3 2(∆c/l) 2θ2) respectively in the complex z(≡ reiθ)-plane where l is the reduced latent heat, ∆c is the discontinuity in the reduced specific heat and z = exp(1− Tc/T ). 02.30.Dk, 02.50.Cw, 05.70.Fh, 64.60.-i Typeset using REVTEX 1 One of the fascinating subjects of equilibrium statistical mechanics is to understand how an analytic partition function acquires a singularity when the system undergoes a phase transition. For the last three decades main focus has been on the second order transition. Only recently a renewed interest in the first order phase transition began to emerge [1]. Since Yang and Lee [2] first published their celebrated papers on the theory of phase transitions and the circle theorem on the zeros of the partition function, there have been many attempts to generalize the theorem [3]. Fisher [4] initiated the study of zeros of the partition function in the complex temperature plane, and Jones [5] proposed a scenario for the first order transition for a continuum system. However very little is known about the distribution of zeros for the continuum case. This is because the partition function for the continuum system is not a polynomial in general and the original proof of the circle theorem relied heavily on particular properties of the coefficients of a polynomial. Recently we have been able to prove the theorem in a quite different approach [6] and this approach allows us to extend the theorem to the continuum case. We found that the circle theorem follows from a certain mathematical relation which exists between a probability density function and the zeros of its characteristic function. In this paper we first prove three theorems and a corollary without referring to the partition function. When these theorems and the corollary are translated in physical terms we find that, (1) the zeros of the partition function can be expressed in terms of the discontinuities in the derivatives of the free energy across the phase boundary if there is a non-vanishing discontinuity in the first order derivative; (2) there is no zeros in the single phase region where the probability distribution is given by a single Gaussian peak; (3) the zeros of the partition function are calculable exactly at the two phase coexistence point where the probability distribution is given by two asymmetric Gaussian peaks; and (4) the zeros lie on the unit circle if the transition is symmetric. Furthermore we find the finite size scaling very much similar to that of the discrete system [6]. Therefore this result can again be used, for a continuum system, (1) to resolve the recent controversy over equal weight vs. equal height of the probability distribution 2 functions [7–9]; and (2) distinguish the first order transition from the second [10,6], just as we have done for the discrete system in ref. [6]. Consider a probability density function f(x) of a random variable X of continuous type, which satisfy i) f(x) ≥ 0 and ii) ∫∞ −∞ f(x)dx = 1. The characteristic function [11] of a random variable X is defined by