  In this paper we consider the Glauber dynamics for the one-dimensional Ising
model with dissipation, in a mesoscopic regime obtained by letting inverse
temperature and volume go to infinity with a suitable scaling. In this limit
the magnetization has a periodic behavior. Self-organized collective
periodicity has been shown for many mean-field models but, to our knowledge,
this is the first example with short-range interaction. This supports the view
that self-organized periodicity is not linked with the mean-field assumption
but it is a thermodynamic phenomenon compatible with short range interactions.
