  In this note, we study the phase transitions arising in a modified
Smoluchowski equation on the sphere with dipolar potential. This equation
models the competition between alignment and diffusion, and the modification
consists in taking the strength of alignment and the intensity of the diffusion
as functions of the order parameter. We characterize the stable and unstable
equilibrium states. For stable equilibria, we provide the exponential rate of
convergence. We detail special cases, giving rise to second order and first
order phase transitions, respectively. We study the hysteresis diagram, and
provide numerical illustrations of this phenomena.
