Past exam of the soft condensed matter course of the University of Cambridge / 2012 / Paper 1 / 1 / a / Solution

Use the squared three-dimensional endpoint displacements, not their arithmetic mean. The mean squared displacement estimate and corresponding diffusion coefficient areThe factor six is for three-dimensional Brownian motion. For an isolated no-slip sphere, the Stokes–Einstein relation gives the Brownian radius estimate from endpoint displacements:Here room temperature is taken as $298\mathrm{K}$; choosing $293\mathrm{K}$ gives $55\mathrm{nm}$, a much smaller difference than the sampling uncertainty. Under independent increments with a normal distribution, $\mathrm{sd}(\hat{D})/D=\sqrt{2/(3n)}\simeq 0.37$ for $n=5$. The sampling uncertainty of an endpoint diffusivity estimate therefore makes this a rough radius estimate. It assumes negligible drift and hydrodynamic coupling between probes, the supplied bulk viscosity, and actual three-dimensional tracking. Localization noise adds to the squared displacements and would bias this inferred radius downward if uncorrected.