The structure of equilibrium solutions of the one-dimensional Doi equation

Abstract

We analyse the structure of steady state solutions of the one-dimensional Doi model for rod-like molecules. We prove that if the interaction strength parameter U is less than 4, then the constant solution is the only possible steady state solution. If U is larger than 4, then there is a new solution that corresponds to the nematic phase. All other non-constant solutions are obtained from this solution by translation. We prove further that the nematic solutions have period π instead of 2π , a property that signifies the fact the rods are symmetric, i.e. they have orientations but no directions.

Stability of equilibria of nematic liquid crystalline polymers

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Inertial Manifolds for a Smoluchowski Equation on the Unit Sphere

“…However, it is widely accepted that the Maier-Saupe potential affords sufficient degrees of freedom to capture the dynamics of the micro-micro interaction. In a recent development, the bifurcation diagram for the Onsager equation (and therefore also Smoluchowski equation) with the Maier-Saupe potential was confirmed rigorously (see [5], [6], [8], [13], [18], [19]). The equation undergoes two bifurcations.…”

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