Possible Loss and Recovery of GibbsiannessDuring the Stochastic Evolution of Gibbs Measures
Abstract
We consider Ising-spin systems starting from an initial Gibbs measure $\nu$ and evolving under a spin-flip dynamics towards a reversible Gibbs measure $\mu\not=\nu$. Both $\nu$ and $\mu$ are assumed to have a finite-range interaction. We study the Gibbsian character of the measure $\nu S(t)$ at time $t$ and show the following: (1) For all $\nu$ and $\mu$, $\nu S(t)$ is Gibbs for small $t$. (2) If both $\nu$ and $\mu$ have a high or infinite temperature, then $\nu S(t)$ is Gibbs for all $t>0$. (3) If $\nu$ has a low non-zero temperature and a zero magnetic field and $\mu$ has a high or infinite temperature, then $\nu S(t)$ is Gibbs for small $t$ and non-Gibbs for large $t$. (4) If $\nu$ has a low non-zero temperature and a non-zero magnetic field and $\mu$ has a high or infinite temperature, then $\nu S(t)$ is Gibbs for small $t$, non-Gibbs for intermediate $t$, and Gibbs for large $t$. The regime where $\mu$ has a low or zero temperature and $t$ is not small remains open. This regime presumably allows for many different scenarios.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- 2002
- DOI:
- 10.1007/s002200200605
- arXiv:
- arXiv:math-ph/0105001
- Bibcode:
- 2002CMaPh.226..101V
- Keywords:
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- Mathematical Physics;
- Mathematics - Probability;
- 82C22;
- 60K35
- E-Print:
- doi:10.1007/s002200200605