  This paper is devoted to the nonlinear analysis of a kinetic model introduced
by Saintillan and Shelley to describe suspensions of active rodlike particles
in viscous flows. We investigate the stability of the constant state
$\Psi(t,x,p) = \frac{1}{4\pi} $ corresponding to a distribution of particles
that is homogeneous in space (variable $x \in \mathbb{T}^3$) and uniform in
orientation (variable $p \in \mathbb{S}^2$). We prove its nonlinear stability
under the optimal condition of linearized spectral stability, without any
addition of spatial diffusion. The mathematical novelty and difficulty compared
to previous linear studies comes from the presence of a quasilinear term in $x$
due to nonlinear convection. A key feature of our work, which we hope to be of
independent interest, is an analysis of enhanced dissipation and mixing
properties of the advection diffusion operator $$\partial_t + (p + u(t,x))
\cdot \nabla_x - \nu \Delta_p $$ on $\mathbb{T}^3 \times \mathbb{S}^2$ for a
given appropriately small vector field $u$.
