= Brownian radius estimate from endpoint displacements
{c}
{title2=$\widehat a=k_BT/(6\pi\eta\widehat D),\quad\widehat D=\sum_i|\Delta\mathbf r_i|^2/(6nt)$}
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.
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