Maier-Saupe model of liquid crystals: isotropic-nematic phase transitions and second-order statistics studied by Shiino's perturbation theory and strongly nonlinear Smoluchowski equations.

We study the first- and second-order statistical properties of a dynamical Maier-Saupe model for liquid crystals that is given in terms of a nonlinear Smoluchowski equation. Using Shiino's perturbation theory, we analyze the first-order statistics and give a rigorous proof of the emergence of a phase transition from a uniform distribution to a nonuniform distribution, reflecting phase transitions from isotropic to nematic phases, as observed in nematic liquid crystals. Using the concept of strongly nonlinear Fokker-Planck equations, the second-order statistics of the dynamical Maier-Saupe model is studied and an analytical expression for the short-time autocorrelation function of the orientation of the crystal molecules is derived.