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

¹ Cooperative Association for Internet Data Analysis, San Diego Supercomputer Center, UCSD San Diego, CA, USA
² Statistical Mechanics and Complexity Center (SMC), INFM-CNR SMC - Rome, Italy, EU
³ Department of Computational and Theoretical Sciences, Faculty of Science, IIUM - Kuantan, Malaysia

^(a) ostilli@roma1.infn.it

Received: 10 January 2013
Accepted: 6 March 2013

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 ⩾ 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.