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

Body-attitude alignment: first order phase transition, link with rodlike polymers through quaternions, and stability