Mathematical Physics © Springer-Verlag 2002 Possible Loss and Recovery of Gibbsianness During the Stochastic Evolution of Gibbs Measures

Abstract: We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure µ = ν. Both ν and µ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following: (1) For all ν and µ, νS(t) is Gibbs for small t. (2) If both ν and µ have a high or infinite temperature, then νS(t) is Gibbs for all t> 0. (3) If ν has a low non-zero temperature and a zero magnetic field and µ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t. (4) If ν has a low non-zero temperature and a non-zero magnetic field and µ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where µ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios. 1