  We study a popular kinetic model introduced by Saintillan and Shelley for the
dynamics of suspensions of active elongated particles where the particles are
described by a distribution in space and orientation. The uniform distribution
of particles is the stationary state of incoherence which is known to exhibit a
phase transition. We perform an extensive study of the linearised evolution
around the incoherent state. We show (i) in the non-diffusive regime
corresponding to spectral (neutral) stability that the suspensions experiences
a mixing phenomenon similar to Landau damping and we provide optimal pointwise
in time decay rates in weak topology. Further, we show (ii) in the case of
small rotational diffusion \(\nu\) that the mixing estimates persist up to time
scale \(\nu^{-1/2}\) until the exponential decay at enhanced dissipation rate
\(\nu^{1/2}\) takes over.The interesting feature is that the usual velocity
variable of kinetic models is replaced by an orientation variable on the
sphere. The associated \emph{orientation mixing} leads to limited algebraic
decay for macroscopic quantities. For the proof, we start with a general
pointwise decay results for Volterra equations that may be of independent
interest. While, in the non-diffusive case, explicit formulas on the sphere
allow to conclude the desired decay, much more work is required in the
diffusive case: here we prove mixing estimates for the advection-diffusion
equation on the sphere by combining an optimized hypocoercive approach with the
vector field method. One main point in this context is to identify good
commuting vector fields for the advection-diffusion operator on the sphere. Our
results in this direction may be useful to other models in collective dynamics,
where an orientation variable is involved.
