  We discuss the dynamics and thermodynamics of the Brownian Mean Field (BMF)
model which is a system of N Brownian particles moving on a circle and
interacting via a cosine potential. It can be viewed as the canonical version
of the Hamiltonian Mean Field (HMF) model. We first complete the description of
this system in the mean field approximation. Then, we take fluctuations into
account and study the stochastic evolution of the magnetization both in the
homogeneous phase and in the inhomogeneous phase. We discuss its behavior close
to the critical point.
