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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 $\eta$,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 $D=k_{B}T/\zeta$ 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 $n\propto e^{-U/(k_{B}T)}$ has $\nabla n=n\mathbf{F}/(k_{B}T)$; requiring zero equilibrium flux fixes $D=k_{B}T/\zeta$. thermal fluctuations and viscous dissipation are therefore linked, rather than independent adjustable effects. The original theory also relates diffusion to measured random displacements.^[ref]

On times long compared with momentum relaxation, an isotropic diffusing sphere hassingle-particle tracking can measure the slope. Alternatively, dilute single-scattering dynamic light scattering measures a field correlation $g_1(q,t)=e^{-D{q}^{2}t}$; the corresponding intensity-correlation excess is proportional to $e^{-2D{q}^{2}t}$. 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.

Brownian motion is the irregular motion of a mesoscopic particle caused by thermal molecular impacts. A colloidal sphere in a fluid provides a physical example. The random force competes with dissipative drag; averaging many trajectories produces diffusion even though each trajectory fluctuates.

A free-particle Langevin equation with thermal Langevin force normalization isThe noise has zero mean, and its strength is linked to drag by the fluctuation-dissipation theorem. The velocity relaxes on the time scale $\tau =m/\zeta$. At long times $t\gg \tau$, the Einstein diffusion relation isFor a slowly moving no-slip sphere of radius $a$ in a continuum fluid of dynamic viscosity $\eta$, Stokes drag law gives $\zeta =6\pi \eta a$. The particle probability density obeys ${\partial }_{t}P=D{\nabla }^{2}P$; a localized initial position develops a Gaussian profile. The ballistic-to-diffusive Brownian displacement changes character: at $t\ll \tau$, equilibrium velocities instead give ballistic mean-square displacement $3(k_{B}T/m)t^2$.

The drag-noise model assumes a classical bath with a sufficiently short correlation time; a mathematical white-noise trajectory does not resolve individual molecular collisions. Tracking nonliving colloidal particles measures diffusion and tests the connection between thermal fluctuations and dissipation.^[ref]