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- Title
Low-Temperature Dynamics of the Curie-Weiss Model: Periodic Orbits, Multiple Histories, and Loss of Gibbsianness.
- Authors
Ermolaev, Victor; Külske, Christof
- 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 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, 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.
- Subjects
COMBINATORIAL dynamics; VECTOR fields; LAGRANGE equations; HEAT radiation &; absorption; GIBBS phenomenon; GIBBS' equation; GIBBS' free energy; PHASE transitions
- Publication
Journal of Statistical Physics, 2010, Vol 141, Issue 5, p727
- ISSN
0022-4715
- Publication type
Article
- DOI
10.1007/s10955-010-0074-x