Gyrotactic orientation Fokker-Planck equation

For orientation probability density $f$ on the unit sphere, deterministic drift $\dot{\mathbf{p}}$ and isotropic rotational diffusion coefficient $D_R$ give ${\partial }_{t}f+{\nabla }_p\cdot (f\dot{\mathbf{p}}-D_R{\nabla }_{p}f)=0$. The derivatives are tangential to the sphere, and normalization is $\int fd\Omega =1$. The spherical divergence of $\mathbf{k}-p_3\mathbf{p}$ is $-2p_3$.

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.

For rigid fluid rotation with vorticity $-2\Omega {\mathbf{e}}_2$, $B\Omega \gg 1$, and $B{D}_R$ of order one, the normalized steady density is $f=(4\pi {)}^{-1}[1-2p_1/(B\Omega )]+O((B\Omega {)}^{-2})$. The result follows by acting with the rotation generator on $p_1$ and using $\int p_i{p}_{j}d\Omega =4\pi {\delta }_{ij}/3$. Reversing the imposed rotation reverses the mean horizontal swimming direction.

With ${\beta }^2=\Omega R^2/D$ and $\epsilon =2V_{s}R/(3B\Omega D)$, the first angular harmonic of the steady cell conservation equation satisfies $a^{\prime \prime }+a^{\prime }/r-a/r^2-i{\beta }^{2}a=0$ and $a^{\prime }(1)=-1$. The regular solution is $a=-I_1(qr)/[q{I}_1^{\prime }(q)]$, $q=e^{i\pi /4}\beta$. At large $\beta$ it gives the displayed layer with width $1/\beta$. The exact Modified Bessel function of the first kind expression remains regular at the axis, unlike extrapolation of the boundary-layer formula there.