  We present a simple model of alignment of a large number of rigid bodies
(modeled by rotation matrices) subject to internal rotational noise. The
numerical simulations exhibit a phenomenon of first order phase transition with
respect the alignment intensity, with abrupt transition at two thresholds.
Below the first threshold, the system is disordered in large time: the rotation
matrices are uniformly distributed. Above the second threshold, the long time
behaviour of the system is to concentrate around a given rotation matrix. When
the intensity is between the two thresholds, both situations may occur. We then
study the mean-field limit of this model, as the number of particles tends to
infinity, which takes the form of a nonlinear Fokker--Planck equation. We
describe the complete classification of the steady states of this equation,
which fits with numerical experiments. This classification was obtained in a
previous work by Degond, Diez, Merino-Aceituno and the author, thanks to the
link between this model and a four-dimensional generalization of the
Doi--Onsager equation for suspensions of rodlike polymers interacting through
Maier--Saupe potential. This previous study concerned a similar equation of BGK
type for which the steady-states were the same. We take advantage of the
stability results obtained in this framework, and are able to prove the
exponential stability of two families of steady-states: the disordered uniform
distribution when the intensity of alignment is less than the second threshold,
and a family of non-isotropic steady states (one for each possible rotation
matrix, concentrated around it), when the intensity is greater than the first
threshold. We also show that the other families of steady-states are unstable,
in agreement with the numerical observations.
