= Weak-noise relaxation of gyrotactic alignment
{title2=$d\mathbf q=-\mathbf q\,dt/B+\sqrt{2D_r}\,d\mathbf W$}
Close to the upward pole, the two Cartesian tilt components of <bottom-heavy spherical-cell orientation dynamics> obey the displayed <Ornstein-Uhlenbeck process>. Thus $d\langle|\mathbf q|^2\rangle/dt=-2\langle|\mathbf q|^2\rangle/B+4D_r$. The stationary spread is $2BD_r$, 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 $BD_r$ from an initially isotropic population takes order $B\log[(BD_r)^{-1}]$, 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 $D_r^{-1}$.
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