Zero-flow steady gyrotactic orientation distribution ($f_0(\theta )=\Lambda e^{\Lambda \cos \theta }/(4\pi \sinh \Lambda )$)

For aligning rate $1/B$, positive rotational diffusivity $D_r$ and $\Lambda =(B{D}_r{)}^{-1}$, the zero-current steady Fokker-Planck equation gives $f_{0,\theta }=-\Lambda \sin \theta f_0$. Normalization over the unit sphere gives the displayed density per unit solid angle. Its mean direction is $⟨\mathbf{p}⟩=(\coth \Lambda -1/\Lambda )\mathbf{k}$, the Langevin function. As $B{D}_r\to 0$, it concentrates weakly into a point mass at the upward pole; the limit is not an ordinary bounded probability density.