 Solution

For $D_r>0$, the steady polar flux satisfies ${\partial }_{\theta }(\sin \theta j_{\theta })=0$. Regularity at both poles makes its integration constant zero. ThusNormalize with $x=\cos \theta$:The zero-flow steady gyrotactic orientation distribution is thereforeDifferentiating the logarithm of its normalization integral gives the Langevin function meanAs $\Lambda \to 0$, $f_0\to 1/(4\pi )$ and the mean tends to zero; as $\Lambda \to \infty$, the distribution concentrates at the upward pole and the mean tends to $V_s\mathbf{k}$. At zero diffusivity the pole-concentrated steady state is a probability measure, not a regular density of this formula.