Zeros of the partition function and phase transition

TL;DR

This paper discusses how analyzing the zeros of the partition function in the complex plane can reveal phase transitions in physical systems, providing insights even for small finite systems.

Contribution

It highlights the significance of partition function zeros in understanding phase transitions, extending the analysis to finite systems and illustrating with examples.

Findings

Zeros approach the real axis at phase transitions

Method applies to small finite systems

Examples include van der Waal and Bose gases

Abstract

The equation of state of a system at equilibrium may be derived from the canonical or the grand canonical partition function. The former is a function of temperature T, while the latter also depends on the chemical potential \mu for diffusive equilibrium. In the literature, often the variables \beta=(k_BT)^{-1} and fugacity z=exp(\beta \mu) are used instead. For real \beta and z, the partition functions are always positive, being sums of positive terms. Following Lee, Yang and Fisher, we point out that valuable information about the system may be gleaned by examining the zeros of the grand partition function in the complex z plane (real \beta), or of the canonical partition function in the complex \beta plane. In case there is a phase transition, these zeros close in on the real axis in the thermodynamic limit. Examples are given from the van der Waal gas, and from the ideal Bose gas, where we show that even for a finite system with a small number of particles, the method is useful.