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 experience 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 ν that the mixing estimates persist up to time
scale ν−1/2 until the exponential decay at enhanced dissipation rate ν1/2 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 orientation mixing leads to
limited algebraic decay for macroscopic quantities. For the proof, we start with a general pointwise decay result 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.