= Solution
The <Stokes–Einstein relation> connects thermal <diffusion> with dissipative <mobility>. For a dilute suspension of isolated rigid <spheres> of <hydrodynamic radius> $a$ in an unbounded <Newtonian fluid> of <viscosity> $\eta$,
$$
\boxed{D=\frac{k_BT}{\zeta}=\frac{k_BT}{6\pi\eta a},}
$$
where $\zeta$ is translational <drag coefficient>, $D$ <diffusion coefficient> and $T$ absolute <temperature>. <Stokes drag> supplies $\zeta=6\pi\eta a$ under no-slip, continuum, low-Reynolds-number conditions. The <Einstein diffusion relation> $D=k_BT/\zeta$ is more general than that particular spherical <drag> law.
A short <equilibrium> argument explains the connection. Let a weak external potential $U$ exert <force> $\mathbf F=-\nabla U$, so drift <velocity> is $\mathbf F/\zeta$. The number <flux> is $\mathbf j=n\mathbf F/\zeta-D\nabla n$. A <Boltzmann distribution> $n\propto e^{-U/(k_BT)}$ has $\nabla n=n\mathbf F/(k_BT)$; requiring zero <equilibrium> <flux> fixes $D=k_BT/\zeta$. <thermal fluctuations> and viscous <dissipation> are therefore linked, rather than independent adjustable effects. The original theory also relates <diffusion> to measured random displacements.<On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-Kinetic Theory of Heat by Albert Einstein>[Sections 3–5, printed pages 9–18 in the linked translation: equilibrium flux balance, spherical drag and displacement variance]{ref}
On times long compared with momentum relaxation, an isotropic diffusing <sphere> has
$$
\langle[\Delta x(t)]^2\rangle=2Dt,\qquad
\langle|\Delta\mathbf r(t)|^2\rangle=6Dt.
$$
<single-particle tracking> can measure the slope. Alternatively, dilute single-scattering <dynamic light scattering> measures a field correlation $g_1(q,t)=e^{-Dq^2t}$; the corresponding intensity-correlation excess is proportional to $e^{-2Dq^2t}$. The scattering <wavenumber> and the distinction between field and intensity decay must be retained.
A <Stokes-Einstein experimental consistency test> independently measures <temperature> and <solvent> <viscosity>, uses a narrow distribution of inert <spheres>, checks decay rate proportional to $q^2$ or displacement variance proportional to time, and extrapolates toward infinite dilution. Subtract drift, account for wall-enhanced <drag>, and verify that aggregation is absent. Tracking additionally needs spatial calibration and correction for finite exposure and <localization precision>. At finite <concentration>, collective <diffusion> is not automatically the single-particle <self-diffusion> in the equation. A <hydrodynamic radius> is not generally identical to a <polymer> <radius of gyration>; nonspherical particles, slip or viscoelastic <solvents> require a different <mobility> model.
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