# The analytic structure of thermodynamic systems with repulsive interactions

## TL;DR

This paper analyzes the complex analytic structure of thermodynamic systems with repulsive interactions, revealing singularities and limitations of series expansions, with applications to hadron gases and lattice QCD.

## Contribution

It provides a detailed analysis of the analytic structure of the excluded-volume model using Lambert W-function and compares it with other models, highlighting universal features.

## Key findings

- The pressure function can be expressed via Lambert W-function.
- The model exhibits a branch cut singularity in the complex plane.
- Universal behavior of net baryonic density in terms of polylogarithms.

## Abstract

Thermodynamic properties of systems with repulsive interactions, are considered in the grand canonical ensemble. The analytic structure of the excluded-volume model in the complex plane of the system chemical potential (fugacity) is elaborated, based on the fact that the pressure function can be given in terms of the Lambert W-function. Even though the excluded volume model has no phase transitions at real values of the chemical potential, it does exhibit a branch cut singularity in the complex plane, thus limiting the convergence range of the Taylor expansion in the chemical potential. Close similarities to analytic properties of the other models with repulsive interactions, such as a cluster expansion model, the mean-field model, and the ideal Fermi gas model, are pointed out. As an example, repulsive baryonic interactions in a hadron gas, with a focus on the fugacity/virial and Taylor expansion methods used in lattice QCD, are presented. The asymptotic behavior of the Fourier expansion coefficients in these various models suggests that the singular part of net baryonic density can to leading order be universally expressed in terms of polylogarithms.