= Solution
For a no-slip spherical particle in a <Newtonian fluid> at low <Reynolds number>, <Stokes law> gives $\zeta=6\pi\eta a$. The <fluctuation-dissipation theorem> then gives the <Stokes–Einstein relation> $D=k_BT/\zeta$. Using the conventional <radial Brownian displacement> in three dimensions,
$$
\boxed{d_{\rm rms}=\sqrt{\langle|\Delta\mathbf r|^2\rangle}
=\sqrt{6Dt}=\sqrt{\frac{k_BTt}{\pi\eta a}}.}
$$
Here $k_B$ is the <Boltzmann constant>, $T$ absolute <temperature>, $\eta$ <dynamic viscosity>, $a$ particle radius and $t$ elapsed time. Larger thermal forcing raises <diffusion>; larger viscous drag reduces it. The formula requires times long relative to momentum relaxation and neglects walls, interactions and external drift.
The word “average” needs a statistic: the vector mean is zero, while the mean radial displacement for Gaussian three-dimensional <diffusion> is $\langle|\Delta\mathbf r|\rangle=4\sqrt{Dt/\pi}$, not $\sqrt{6Dt}$. Both scale as $t^{1/2}$. Neither is the total length of a microscopic random path.
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