Brownian radius estimate from endpoint displacements ($\hat{a}=k_{B}T/(6\pi \eta \hat{D}),\hat{D}={\sum }_i|\Delta {\mathbf{r}}_i{|}^2/(6nt)$)

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.