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With $x=\cos \theta$ and $D_r=0$, the Fokker-Planck equation becomes the probability-transport equationIts characteristic obeys $\dot{x}=(1-x^2)/B$. Writing $s=t/B$ and $q=\tanh s$ givesConservation of orientation probability pushes the initially uniform density forward by this Jacobian. ThusSince $q=(1-\alpha )/(1+\alpha )$ with $\alpha =e^{-2t/B}$, the deterministic alignment of an initially isotropic orientation distribution isAt $t=0$, $\alpha =1$ and $f=1/(4\pi )$; the characteristic Jacobian preserves $2\pi {\int }_{-1}^{1}fdx=1$ at every finite time. Its reciprocal is quadratic in $x$, as in the suggested substitution.

To compute the mean, integrate explicitly:ConsequentlyThe apparent singularity at $t=0$ is removable: $⟨x⟩=2t/(3B)+O((t/B{)}^3)$, so the initial mean is zero. At late times,Except for the exactly downward initial orientation, which has zero probability in the initial smooth distribution, every characteristic aligns upward. The infinite-time orientation distribution is a point mass at the upward pole.

Close to the upward pole, the two Cartesian tilt components of bottom-heavy spherical-cell orientation dynamics obey the displayed Ornstein-Uhlenbeck process. Thus $d⟨|\mathbf{q}{|}^2⟩/dt=-2⟨|\mathbf{q}{|}^2⟩/B+4D_r$. The stationary spread is $2B{D}_r$, and late axisymmetric variance perturbations relax on $B/2$ in this weak-noise approximation. The zero-noise mean from deterministic alignment of an initially isotropic orientation distribution has the same exponential scale but a different transient prefactor. Reaching a tiny diffusion-supported mean-deficit floor $B{D}_r$ from an initially isotropic population takes order $B\log [(B{D}_r{)}^{-1}]$, with logarithmic corrections. Equality of limiting mean directions does not make the full transients identical, and the alignment time is not automatically the pure rotational-diffusion time $D_r^{-1}$.