= Rotational diffusion
{title2=$D_r$}
{wiki}
Rotational diffusion is the random change of a particle's <rigid body orientation> through <Brownian motion>. A suspended <ellipsoid> can turn while its centre hardly moves; <translational diffusion> instead spreads its position. Random molecular collisions supply fluctuating <torques> as well as <forces>. The rotational <diffusion coefficient> $D_r$ measures how quickly angular uncertainty grows, rather than a steady spin speed. Its units are $\mathrm{s^{-1}}$, often written $\mathrm{rad^2\,s^{-1}}$ to distinguish angular spreading from translational spreading, whose <diffusion coefficient> $D_t$ has units $\mathrm{m^2\,s^{-1}}$. The <radian> is <dimensionless>, but writing angular <variance> in $\mathrm{rad^2}$ remains useful.
First identify what the image can reveal. A marked end of an elongated particle defines an arrow: turning it through $\pi$ reverses that arrow. An unmarked <ellipsoid> with identical ends reveals only its long axis, a <director (liquid crystals)>: angles $\phi$ and $\phi+\pi$ describe the same visible state. An unmarked perfect <sphere> has no visible <rigid body orientation> at all. Tracking its centre cannot measure its <rotational diffusion>; a surface mark or another identifiable internal direction is needed.
This distinction makes <rotational diffusion> useful: it measures how long a particle remembers its pointing direction. For an elongated particle, turning also changes which laboratory direction is easiest for <translational diffusion>. Randomization of <rigid body orientation> can therefore erase directional preferences in the observed displacement, even though the particle's shape has not changed.
Back to article page
