  Starting from a particle model describing self-propelled particles
interacting through nematic alignment, we derive a macroscopic model for the
particle density and mean direction of motion. We first propose a mean-field
kinetic model of the particle dynamics. After diffusive rescaling of the
kinetic equation, we formally show that the distribution function converges to
an equilibrium distribution in particle direction, whose local density and mean
direction satisfies a cross-diffusion system. We show that the system is
consistent with symmetries typical of a nematic material. The derivation is
carried over by means of a Hilbert expansion. It requires the inversion of the
linearized collision operator for which we show that the generalized collision
invariants, a concept introduced to overcome the lack of momentum conservation
of the system, plays a central role. This cross diffusion system poses many new
challenging questions.
