Gibbs and non-Gibbs aspects of continuous spin models

The finite-volume microscopic behaviour of a system in equilibrium is de¬termined by the Boltzmann-Gibbs prescription for probability measures in which probabilities are proportional to the exponential of some Hamiltonian, which describes the energy fluctuations while the total number of particles is fixed. For seeing a phase transition mathematically one has to investigate infinite-volume measures. The natural objects for describing dependences (due to e.g. a spin interaction) are conditional probabilities. These are com¬patible with a possibly non-unique measure on an infinite-volume space. The conditioned measures describe the behaviour of a finite set of spins while all others are fixed. This approach was proposed by Dobrushin, Lanford and Ruelle and is since referred to as the DLR approach. Kozlov and Sullivan gave an equivalent characterization of a Gibbs measure. A measure is Gibbs is it is uniformly nonnull and quasilocal. This property is very important in describing physical reality, since one is interested in modeling local systems which are not influenced by events at infinity. A fortiori a measure is not Gibbs, if one finds at least one discontinuity point (in the product topology) of the conditional probabilities w.r.t. the conditioning. Non-Gibbsianness can appear after applying different kind of transfor¬mations on Gibbs measures, such as renormalization group transformations, projections or time-evolution. In this thesis we study Gibbs and non-Gibbs aspects of continuous spin models during time-evolution. We obtain the ma¬jor part of the results for planar rotors which are spins moving on circles, these are stated in Chapters 3-5 and 7. Additionally we prove short-time Gibbsianness for unbounded spins in Chapter 6. In particular we will prove in Chapter 3, first that for every initial and dynamical inverse temperature the measure stays Gibbsian for short times. Secondly, for small inverse initial and dynamical temperatures the measure is Gibbs for all times. The third result states loss of Gibbsianness in a time interval for a planar rotor model without a field. Furthermore in Chapter 4 we prove loss and recovery of Gibbsianness for models in a field. For both cases where Gibbsianness was lost we assumed a low-temperature initial Gibbs measure and infinite-temperature dynamics (the initial measure converges towards an infinite temperature Gibbs mea¬sure). Analogously as in the Ising case, the presence of an initial external field is responsible for the measure to become Gibbs in finite time again. The following Chapter 5 deals with a planar rotor model in an alternat¬ing field in equilibrium, exhibiting a phase transition at low temperatures. Moreover it contains an argument how a similar proof can be used to prove phase coexistence for a planar rotor model in a field which consist of certain periodic or random configurations on crosses. In Chapter 6 we generalize the short-time conservation result for possibly non-Markovian dynamics for unbounded spins. Finally the last Chapter 7, which stands a bit apart, contains the first ex¬ample of Chaotic Temperature Dependence for compact spins. We construct a potential for a planar rotor system, thus a system of compact spins such that no ground state measure can be obtained by lowering the temperature. Thus the model depends chaotically on the temperature