  Motivated by a phenomenon of phase transition in a model of alignment of
self-propelled particles, we obtain a kinetic mean-field equation which is
nothing else than the Doi equation (also called Smoluchowski equation) with
dipolar potential. In a self-contained article, using only basic tools, we
analyze the dynamics of this equation in any dimension. We first prove global
well-posedness of this equation, starting with an initial condition in any
Sobolev space. We then compute all possible steady-states. There is a threshold
for the noise parameter: over this threshold, the only equilibrium is the
uniform distribution, and under this threshold, there is also a family of
non-isotropic equilibria. We give a rigorous prove of convergence of the
solution to a steady-state as time goes to infinity. In particular we show that
in the supercritical case, the only initial conditions leading to the uniform
distribution in large time are those with vanishing momentum. For any positive
value of the noise parameter, and any initial condition, we give rates of
convergence towards equilibrium, exponentially for both supercritical and
subcritical cases and algebraically for the critical case.
