= Planar rotational diffusion
{title2=$d\phi=\sqrt{2D_r}\,dW$}
For free planar turning with constant $D_r$, no imposed <torque> and negligible <inertia>, model an unwrapped <angle> by the <stochastic differential equation>
$$
d\phi=\sqrt{2D_r}\,dW,\qquad
\Delta\phi=\phi(t)-\phi(0),\qquad
\langle\Delta\phi\rangle=0,\qquad
\langle(\Delta\phi)^2\rangle=2D_rt.
$$
Here $W$ is standard one-dimensional <Brownian motion>, with increment <variance> $dt$, and brackets denote <expected values> over repeated realizations. The angular <variance> is conventionally reported in $\mathrm{rad^2}$. “Unwrapped” means recording accumulated turns continuously, rather than resetting the <angle> at each revolution. Directly subtracting wrapped camera angles creates artificial jumps; for an unmarked <ellipsoid> the wrapping period is $\pi$, and resolving successive turns additionally requires sufficiently frequent images.
For a planar arrow $\mathbf u=(\cos\phi,\sin\phi)$, the orientational <autocorrelation> is
$$
C_{\rm arrow}(t)=\langle\mathbf u(t)\cdot\mathbf u(0)\rangle
=\langle\cos\Delta\phi\rangle=e^{-D_rt}.
$$
For an unoriented planar axis, use an <autocorrelation> unchanged by replacing either endpoint <angle> by that <angle> plus $\pi$:
$$
C_{\rm axis}(t)=\langle\cos(2\Delta\phi)\rangle=e^{-4D_rt}.
$$
The same $D_r$ thus gives arrow and axis <correlation times> $1/D_r$ and $1/(4D_r)$, respectively, where a <correlation time> here is the lag at which the relevant curve reaches $1/e$. A short explanation is enough: $\Delta\phi$ has a <Gaussian distribution> of <variance> $2D_rt$, so its <characteristic function> gives $\langle\cos(n\Delta\phi)\rangle=e^{-n^2D_rt}$. The integer $n$ selects what the observable can distinguish; it does not change the underlying angular spreading.
For an original numerical example, choose $D_r=0.25\,\mathrm{s^{-1}}$. After $2\,\mathrm s$, the angular <variance> is $1\,\mathrm{rad^2}$ and its <root mean square> displacement is $1\,\mathrm{rad}$, although the mean displacement is zero. The planar arrow <autocorrelation> is $e^{-0.5}\simeq0.607$, while the planar axis <autocorrelation> is $e^{-2}\simeq0.135$. Calling both “the orientation correlation” without specifying the observable would hide a factor of four in decay rate.
\Image[/rotational-diffusion-planar-orientation.png]
{title=Planar <rotational diffusion>: arrows, unoriented axes and loss of angular memory}
{description=Original schematics, three simulated <Brownian motion> angle paths and exact ensemble <autocorrelation> curves for the chosen $D_r=0.25\,\mathrm{s^{-1}}$. The paths use independent <Gaussian distribution> increments of <variance> $2D_r\Delta t$ with $\Delta t=0.01\,\mathrm s$. The curves are model predictions, not fits to these three paths or experimental measurements. Both curves concern planar turning; neither is the three-dimensional unit-vector law.}
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