Low-Temperature Dynamics of the Curie-Weiss Model: Periodic Orbits, Multiple Histories, and Loss of Gibbsianness

Abstract

We consider the Curie-Weiss model at initial temperature 0 < β −1 ≤ ∞ in vanishing external field evolving under a Glauber spin-flip dynamics with temperature 0 < β −1 ≤ ∞. We study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity (bad points). We provide a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending earlier work for the case of independent spin-flip dynamics.For initial temperature β −1 > 1 we prove that the time-evolved measure stays Gibbs forever, for any (possibly low) temperature of the dynamics.In the regime of heating to low-temperatures from even lower temperatures, 0 < β −1 < min{β −1 , 1} we prove that the time-evolved measure is Gibbs initially and becomes nonGibbs after a sharp transition time. We find this regime is further divided into a region where only symmetric bad configurations exist, and a region where this symmetry is broken. In the regime of further cooling from low-temperatures, β −1 < β −1 < 1 there is always symmetry-breaking in the set of bad configurations. These bad configurations are created by a new mechanism which is related to the occurrence of periodic orbits for the vector field which describes the dynamics of Euler-Lagrange equations for the path large deviation functional for the order parameter.To our knowledge this is the first example of the rigorous study of non-Gibbsian phenomena related to cooling, albeit in a mean-field setup.

A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit

“…The non-reversible time-evolutions we consider here suggest another set of questions, namely whether there are any non-Gibbsian pathologies along the trajectories depending on starting measures as found for reversible dynamics in [16,18,39,36,20,17,14,33,22]. Acknowledgement: This work is supported by the Sonderforschungsbereich SFB | TR12-Symmetries and Universality in Mesoscopic Systems.…”

Variational Description of Gibbs-Non-Gibbs Dynamical Transitions for Spin-Flip Systems with a Kac-Type Interaction

“…In Fernández, den Hollander and Martínez [13], building on earlier work by Külske and Le Ny [15] and Ermolaev and Külske [12], we showed that this program can be fully carried out for the Curie-Weiss model subject to an infinite-temperature dynamics. The goal of the present paper is to extend this work away from the mean-field setting by considering a model with a Kac-type interaction, i.e., Ising spins with a long-range interaction.…”

The Widom–Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality

“…Proof of Proposition 4.9. Considering the proof of Proposition 4.5, note that the estimates (10), (11) and (12) also hold at the critical time. In particular we still have supσ ω B ρ(ω C∩B ) ≤ 1.…”