  Pattern formation in uniaxial polymeric liquid crystals is studied for
different dynamic closure approximations. Using the principles of mesoscopic
non-equilibrium thermodynamics in a mean-field approach, we derive a
Fokker-Planck equation for the single-particle non-homogeneous distribution
function of particle orientations and the evolution equations for the second
and fourth order orientational tensor parameters. Afterwards, two dynamic
closure approximations are discussed, one of them considering the relaxation of
the fourth order orientational parameter and leading to a novel expression for
the free-energy like function in terms of the scalar order parameter.
Considering the evolution equation of the density of the system and values of
the interaction parameter for which isotropic and nematic phases coexist, our
analysis predicts that patterns and traveling waves can be produced in
lyotropic uniaxial nematics even in the absence of external driving.
