Superdiffusion and hydrodynamic phase separation in a uniaxial active suspension

We examine, analytically and to some extent numerically, the dynamics of the concentration
field in an active suspension with permanent uniaxial anisotropy. We show that in the
homogeneous phase the advection of fluctuations in the concentration, by the active flow
they themselves generate, leads to a superdiffusive dynamic exponent $z = d/2$ for
dimension $d<4$. We uncover a novel flow-induced contribution to active-particle currents
that can lead to a negative effective diffusivity on a cone of directions, and hence to a
purely hydrodynamic mechanism for phase separation. We suggest experimental tests of our
predictions.