89.
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 ν
S(
t) at time
t and show the following:
(1) For all ν and μ, ν
S(
t) is Gibbs for small
t.
(2) If both ν and μ have a high or infinite temperature, then ν
S(
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 ν
S(
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 ν
S(
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.
Received: 26 April 2001 / Accepted: 10 October 2001
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