Low-temperature dynamics of the Curie-Weiss Model: Periodic orbits, multiple histories, and loss of Gibbsianness

Abstract

We consider the Curie-Weiss model at a given initial temperature in vanishing external field evolving under a Glauber spin-flip dynamics corresponding to a possibly different temperature. 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 bigger than one 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, when the initial temperature is smaller than the temperature of the dynamics, and smaller than 1, we prove that the time-evolved measure is Gibbs initially and becomes non-Gibbs 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 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.

31 pages, 24 figures

References in corpus (4)

• Relative entropy and variational properties of generalized Gibbsian measures
• Continuous Spin Mean-Field models: Limiting kernels and Gibbs Properties of local transforms
• Short-time Gibbsianness for Infinite-dimensional Diffusions with Space-Time Interaction
• Introduction to (generalized) Gibbs measures

Cited by in corpus (18)

• Finite-time dynamical phase transition in non-equilibrium relaxation
• Landau theory for finite-time dynamical phase transitions
• Large deviations for Markov jump processes with mean-field interaction via the comparison principle for an associated Hamilton-Jacobi equation
• Variational description of Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model
• Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions
• Gibbsian and non-Gibbsian properties of the generalized mean-field fuzzy Potts-model
• Gibbs-non-Gibbs transitions via large deviations: computable examples
• Asymmetric Thermal Relaxation in Driven Systems: Rotations go Opposite Ways
• Variational description of Gibbs-non-Gibbs dynamical transitions for spin-flip systems with a Kac-type interaction
• Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction: Closing the Ising gap
• Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction
• Critical Fluctuations at Finite-Time Dynamical Phase Transition
• On the prevalence of non-Gibbsian states in mathematical physics
• Attractor properties of non-reversible dynamics w.r.t invariant Gibbs measures on the lattice
• Hamiltonian and Lagrangian for the trajectory of the empirical distribution and the empirical measure of Markov processes
• Dynamical Gibbs-non-Gibbs transitions in the Curie-Weiss Potts model in the regime beta<3
• Large deviations of continuous regular conditional probabilities
• Gibbs-non Gibbs transitions in different geometries: The Widom-Rowlinson model under stochastic spin-flip dynamics