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

TL;DR

This paper extends the circle theorem to continuum systems, calculates exact zeros of the partition function for finite systems with asymmetric Gaussian peaks, and characterizes the zeros' locus and density at first order transitions.

Contribution

It generalizes the circle theorem to continuum systems and provides explicit formulas for the zeros' distribution at first order phase transitions.

Findings

Derived the locus of zeros as $r = e^{(rac{ riangle c}{2l}) heta^2}$

Calculated the angular density of zeros as $2\pi g( heta)=l(1+rac{3}{2}(rac{ riangle c}{l})^2 heta^2)$

Extended the understanding of zeros distribution in complex plane for phase transition analysis.

Abstract

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 $r = e^{(\Delta c/2l)\theta^2}$ and $2\pi g(\theta)=l(1+\frac{3}{2}(\Delta c/l)^2\theta^2)$ respectively in the complex $z(\equiv re^{i\theta})$-plane where $l$ is the reduced latent heat, $\Delta c$ is the discontinuity in the reduced specific heat and $z= \exp(1-T_c/T)$.