title: Symmetries of the Doi kinetic theory for nematic polymers of arbitrary aspect ratio: at rest and in linear flows.
abstract: The Doi theory has successfully modeled the monodomain shear flow problem for rigid, rodlike nematic polymers. Numerical simulations of the Smoluchowski equation for the orientational probability distribution function (PDF) predict monodomain attractors in regions of nematic concentration N and shear rate gamma. Theoretical work has focused on approximate constructions of PDF solutions in linear flow regimes. Here we develop a collection of simple observations, expressed by symmetries of the Smoluchowski equation, which imply global properties that all PDF solutions must obey. The well-known orientational degeneracy of quiescent nematics is a continuous O(3) symmetry. In simple shear, a discrete reflection symmetry survives that is evident in recent numerical simulations and implies bistability of out-of-plane attractors; and rodlike and discotic nematic liquids of reciprocal aspect ratio respond identically up to a fixed rotation of the PDF. Finally, we show the orientational effects due to varying molecular aspect ratio in any linear flow are equivalent to varying the straining component of the flow field.