  We study in this paper the longtime behavior of some large but finite
populations of interacting stochastic differential equations whose (infinite
population) limit Fokker-Planck PDE admits a stable periodic solution. We show
that the empirical measure for the population of size $N$ stays close to the
periodic solution, but with a random dephasing at the timescale $Nt$ that
converges weakly to a Brownian motion with constant drift.
