Variational Description of Gibbs-non-Gibbs Dynamical Transitions for the Curie-Weiss Model

Abstract

We perform a detailed study of Gibbs-non-Gibbs transitions for the Curie-Weiss model subject to independent spin-flip dynamics (“infinite-temperature” dynamics). We show that, in this setup, the program outlined in van Enter et al. (Moscow Math J 10:687–711, 2010) can be fully completed, namely, Gibbs-non-Gibbs transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the magnetization conditioned on their endpoint. As a consequence, we show that the time-evolved model is non-Gibbs if and only if this set is not a singleton for some value of the final magnetization. A detailed description of the possible scenarios of bifurcation is given, leading to a full characterization of passages from Gibbs to non-Gibbs and vice versa with sharp transition times (under the dynamics, Gibbsianness can be lost and can be recovered).

Our analysis expands the work of Ermolaev and Külske (J Stat Phys 141:727–756, 2010), who considered zero magnetic field and finite-temperature spin-flip dynamics. We consider both zero and non-zero magnetic field, but restricted to infinite-temperature spin-flip dynamics. Our results reveal an interesting dependence on the interaction parameters, including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in time of the initial magnetization of the optimal trajectories. The numerical plots provided are obtained with the help of MATHEMATICA.