= Rotational diffusion of a sphere
{title2=$D_r=k_BT/(8\pi\eta a^3)$}
For a <sphere> of radius $a$ in an otherwise stationary, unbounded <Newtonian fluid> with <dynamic viscosity> $\eta$, impose the <no-slip boundary condition> and the slow-flow assumptions of <Stokes flow>. The resisting <torque> on a <rotating sphere in Stokes flow> is $-\zeta_r\boldsymbol\Omega$, where $\boldsymbol\Omega$ is its <angular velocity> and $\zeta_r=8\pi\eta a^3$. Thermal <rotational diffusion> at absolute <temperature> $T$ then gives
$$
D_r=\frac{k_BT}{\zeta_r}=\frac{k_BT}{8\pi\eta a^3},\qquad
D_t=\frac{k_BT}{6\pi\eta a},\qquad
\frac{D_r}{D_t}=\frac3{4a^2}.
$$
Here $k_B$ is the <Boltzmann constant>; the translational expression is the <Stokes–Einstein relation>. The ratio has units of inverse length squared, so comparing the numerical values of $D_r$ and $D_t$ without their units is meaningless. At fixed <temperature> and <dynamic viscosity>, decreasing radius tenfold increases $D_r$ a thousandfold but $D_t$ only tenfold. This strong size dependence explains why small particles can lose directional memory extremely quickly.
Choose $T=298\,\mathrm K$ and $\eta=0.001\,\mathrm{Pa\,s}$ as illustrative model inputs. Using $k_B=1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}$ gives $k_BT=4.11433402\times10^{-21}\,\mathrm J$ and the following predictions:
$$
\begin{array}{c|c|c|c}
a\ (\mu\mathrm m)&D_r\ (\mathrm{s^{-1}})&D_t\ (\mu\mathrm{m^2\,s^{-1}})&\tau_1=1/(2D_r)\ (\mathrm s)\\
\hline
0.1&163.704&2.18272&0.00305429\\
1&0.163704&0.218272&3.05429
\end{array}
$$
These are bulk three-dimensional <sphere> calculations, explicitly not a fit to the confined <ellipsoid> experiment below. Walls, nonspherical shape and departure from the <no-slip boundary condition> can change rotational resistance. A visible surface mark is still necessary to read the <sphere>'s turning in an image, even when its rotational <diffusion coefficient> is well defined physically.
\Image[/rotational-diffusion-sphere-size-scaling.png]
{title=Bulk <sphere> size controls <rotational diffusion> and three-dimensional directional memory}
{description=Original model curves for $T=298\,\mathrm K$ and $\eta=0.001\,\mathrm{Pa\,s}$. Left: $D_r$ and $D_t$ are each divided by their own value at radius $1\,\mu\mathrm m$, making the plotted ratios <dimensionless>. Right: the three-dimensional marked-vector <autocorrelation> $e^{-2D_rt}$ for the two chosen radii; dots mark its $1/e$ times. The horizontal time axis is logarithmic. These are predictions for <spheres> in unbounded <fluid>, not measurements or confined <ellipsoid> fits.}
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