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As is known, at the Gibbs-Boltzmann equilibrium, the mean-field $q$-statePotts model with a ferromagnetic coupling has only a first order phasetransition when $q\geq 3$, while there is no phase transition for anantiferromagnetic coupling. The same equilibrium is asymptotically reached whenone considers the continuous time evolution according to a Glauber dynamics. Inthis paper we show that, when we consider instead the Potts model evolvingaccording to a discrete-time dynamics, the Gibbs-Boltzmann equilibrium isreached only when the coupling is ferromagnetic while, when the coupling isanti-ferromagnetic, a period-2 orbit equilibrium is reached and a stablesecond-order phase transition in the Ising mean-field universality class setsin for each component of the orbit. We discuss the implications of thisscenario in real-world problems.

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