Continuous- and discrete-time Glauber dynamics. First- and second-order phase transitions in mean-field Potts models

TL;DR

This paper compares continuous- and discrete-time Glauber dynamics in mean-field Potts models, revealing different phase transition behaviors and equilibrium states, including novel second-order transitions in the discrete case.

Contribution

It demonstrates that discrete-time Glauber dynamics can lead to period-2 orbits and second-order phase transitions, unlike continuous-time dynamics which only show first-order transitions.

Findings

Discrete-time dynamics reach equilibrium only for ferromagnetic coupling.

Antiferromagnetic coupling leads to period-2 orbits in discrete-time dynamics.

Second-order phase transitions emerge in the discrete case for antiferromagnetic coupling.

Abstract

As is known, at the Gibbs-Boltzmann equilibrium, the mean-field $q$-state Potts model with a ferromagnetic coupling has only a first order phase transition when $q\geq 3$, while there is no phase transition for an antiferromagnetic coupling. The same equilibrium is asymptotically reached when one considers the continuous time evolution according to a Glauber dynamics. In this paper we show that, when we consider instead the Potts model evolving according to a discrete-time dynamics, the Gibbs-Boltzmann equilibrium is reached only when the coupling is ferromagnetic while, when the coupling is anti-ferromagnetic, a period-2 orbit equilibrium is reached and a stable second-order phase transition in the Ising mean-field universality class sets in for each component of the orbit. We discuss the implications of this scenario in real-world problems.