= Deterministic alignment of an initially isotropic orientation distribution
{title2=$\langle p_z\rangle=\coth(t/B)-(t/B)\operatorname{csch}^2(t/B)$}
Without rotational <diffusion>, $x=p_z$ follows $\dot x=(1-x^2)/B$, hence $x=(x_0+\tanh(t/B))/(1+x_0\tanh(t/B))$. Pushing forward the initial uniform orientation distribution gives $f=\alpha/[\pi(1+\alpha-(1-\alpha)x)^2]$, $\alpha=e^{-2t/B}$, per unit solid angle. Integrating $2\pi xf$ yields the displayed mean, interpreted continuously as zero at $t=0$. Its late deficit is $(4t/B-2)e^{-2t/B}+O((t/B)e^{-4t/B})$. At each finite time the density is smooth and normalized; the infinitely late distribution is a point mass at the upward pole.
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