Gibbs and non-Gibbs aspects of continuous spin models

PhD ceremony: Ms. W.M. Ruszel, 16.15 uur, Academiegebouw, Broerstraat 5, Groningen

Thesis: Gibbs and non-Gibbs aspects of continuous spin models

Promotor(s): prof. A.C.D. van Enter, prof.dr. C. Külske

Faculty: Mathematics and Natural Sciences

 

A finite or infinite particle system in equilibrium is described by a Boltzmann-Gibbs distribution, or Gibbs measure. Gibbs measures can be constructed for interactions which are reasonably local. Inversely, a probability measure is non-Gibbsian if there exists a configuration which is a discontinuity point of the conditional probabilities w.r.t. the conditioning. In this case there is influence from infinity. In that case it cannot be a Gibbs measure for a reasonably local interaction. This non-Gibbsian property can appear during stochastic evolution, which is modelling a rapid temperature change of the system. One possible interpretation is that during the heating the system loses its temperature. In this thesis we study Gibbs and non-Gibbs aspects of continuous spin models during time-evolution. For every initial temperature and every temperature of the dynamics the time-evolved measure stays Gibbsian. Furthermore we prove that the Gibbs property is preserved for both initial and dynamical temperature being high or infinite. Then we prove loss of the Gibbs property for planar rotors with low initial measure and evolving under infinite-temperature dynamics. The presence of an initial external field is responsible for recovery of Gibbsianness after some time. Finally we give the first example in the literature of Chaotic Temperature Dependence for compact continuous spins.