Brownian Motion and the Diffusion Law

An ensemble of independent two-dimensional random walkers spreads out from the origin under the incessant buffeting of solvent molecules. Because each displacement is a sum of many tiny independent kicks, the cloud is Gaussian and its mean-squared displacement grows linearly in time, $\langle r^2 \rangle = 4Dt$ in 2D, the signature of diffusion; a highlighted tracer drags its jagged random-walk trail. The diffusion coefficient is the Einstein-Stokes value $D = \frac{k_B T}{6\pi \eta r}$, so heating the fluid, thinning it, or shrinking the particle all visibly quicken the spread, the link Einstein used to infer molecular reality from Brownian motion. Side panels track the measured mean-squared displacement against the $4Dt$ line and the step histogram against the predicted Gaussian.