Anticontinuous limit of discrete Landau-de Gennes theory

We study the dynamics of a discrete Landau-de Gennes theory for nematic liquid crystals in the small intersite coupling regime (``anticontinuous limit'') in some finite lattices and graphs with simple geometries. We consider the case of 3 x 3 Q-tensor systems and extend recent results on small coupling intersite equilibria. to the case of geometries without boundaries, in particular we show that equation for Landau-de Gennes equilibriais reduced to an $SO(3)$ equivariant equation on manifolds parametrized by uniaxial tensors. This result is used to justify the approximation of a class of Landau-de Gennes equilibria by equilibria of a generalized Oseen-Frank theory. The theory connects the Landau-de Gennes equilibria to equilibria of a generalized discrete Oseen-Frank energy and also implies that the gradient flow of the Landau-de Gennes energy has a normally hyperbolic invariant attracting submanifold

where the motion is described by uniaxial Q-tensors. Numerical studies of the Landau-De Gennes gradient flow in simple geometries show a rapid approach to a near-uniaxial state at each site, and a slower decay an equilibria. Of special interest is the periodic chain, where we see two equilibria, corresponding to the two homotopy classes of the projective plane.