Stokes–Einstein relation ($D=k_{B}T/\zeta$)

The Stokes–Einstein relation connects translational diffusion and linear drag through $D=k_{B}T/\zeta$. For an unbounded spherical particle, $\zeta =6\pi \mu a$ and $D=k_{B}T/(6\pi \mu a)$.

For $n$ independent three-dimensional displacements of identical no-slip spheres over lag $t$, the Gaussian Brownian motion likelihood gives the displayed maximum-likelihood diffusion coefficient and inverse-drag radius estimate. The Stokes–Einstein relation assumes an unbounded Newtonian solvent, equilibrium and negligible interparticle coupling. Use squared distances, rather than squaring the mean distance. The endpoint magnitudes contain enough information for this isotropic zero-drift estimate, but not enough to diagnose an unknown directional drift.

Each Cartesian Brownian increment has variance $2Dt$. Summing their normalized squares gives a chi-squared distribution with $3n$ degrees of freedom. Hence $\hat{D}$ is unbiased and has relative standard deviation $\sqrt{2/(3n)}$. Its inverse is not unbiased: for $3n>2$, $\mathbb{E}[1/\hat{D}]=[3n/(3n-2)]/D$. Thus the direct inverse-diffusivity radius estimate has appreciable finite-sample bias when only a few trajectories are observed. Measurement noise or drift adds separate systematic error.

For dilute isolated spheres in a Newtonian solvent, $D\eta a/T=k_B/(6\pi )$. An experimental check independently measures temperature, viscosity and particle size, then obtains diffusion from a displacement variance or a scattering decay. Drift, walls, concentration effects, aggregation and a hydrodynamic radius different from a structural radius can defeat an unqualified comparison.