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

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.