# An exact study of phase transitions in mean field Potts models

## TL;DR

This paper derives an exact partition function for the mean field Potts model with external fields, revealing nematic phase transitions and interpreting them via shock dynamics, applicable to any number of states.

## Contribution

It introduces a method to exactly solve the Potts model's partition function using a diffusion equation approach, generalizable to any number of states.

## Key findings

- Identifies nematic phase transitions at non-zero external fields.
- Links phase transitions to shock dynamics in thermodynamic space.
- Provides an exact solution method for the Potts model.

## Abstract

We construct the exact partition function of the Potts model on a complete graph subject to external fields with linear and nematic type couplings. The partition function is obtained as a solution to a linear diffusion equation and the free energy, in the thermodynamic limit, follows from its semiclassical limit. Analysis of singularities of the equations of state reveals the occurrence of phase transitions of nematic type at not zero external fields and allows for an interpretation of the phase transitions in terms of shock dynamics in the space of thermodynamics variables. The approach is shown at work in the case of a q-state model for q=3 but the method generalises to arbitrary q.