  We consider the natural Langevin dynamics which is reversible with respect to
the mean-field plane rotator (or classical spin XY) measure. It is well known
that this model exhibits a phase transition at a critical value of the
interaction strength parameter K, in the limit of the number N of rotators
going to infinity. A Fokker-Planck PDE captures the evolution of the empirical
measure of the system as N goes to infinity, at least for finite times and when
the empirical measure of the system at time zero satisfies a law of large
numbers. The phase transition is reflected in the fact that the PDE for K above
the critical value has a stable manifold of stationary solutions, that are
equivalent up to rotations. These stationary solutions are actually unimodal
densities parametrized by the position of their maximum (the synchronization
phase or center). We characterize the dynamics on times of order N and we show
substantial deviations from the behavior of the solutions of the PDE. In fact,
if the empirical measure at time zero converges as N goes to infinity to a
probability measure (which is away from a thin set that we characterize) and if
time is speeded up by N, the empirical measure reaches almost instantaneously a
small neighborhood of the stable manifold, to which it then sticks and on which
a non-trivial random dynamics takes place. In fact the synchronization center
performs a Brownian motion with a diffusion coefficient that we compute. Our
approach therefore provides, for one of the basic statistical mechanics systems
with continuum symmetry, a detailed characterization of the macroscopic
deviations from the large scale limit -- or law of large numbers -- due to
finite size effects. But the interest for this model goes beyond statistical
mechanics, since it plays a central role in a variety of scientific domains in
which one aims at understanding synchronization phenomena.
