On the steady state solutions to a 2 D Smoluchowski equation

In this paper, we re-examine a 2D Smoluchowski equation employed for modeling nematic liquid crystalline polymers. Specifically, we provide a novel proof concerning the investigation of steadystate solutions to the 2D Smoluchowski equation. We establish that when the intensity constant is less than or equal to 4, a unique (trivial) solution exists. Conversely, when the intensity constant exceeds 4, precisely two solutions emerge, corresponding to the isotropic and nematic phases, respectively. The proof relies solely on calculus, which is transparent and accessible.