MSD & Diffusion Coefficient

Mean-squared displacement (MSD): the canonical observable for diffusion. For normal (Brownian) diffusion in 2D, MSD(τ) = 4Dτ — a linear rise. The diffusion coefficient D = kBT/(6πηr) (Stokes-Einstein) connects molecular size r, viscosity η, and temperature. Anomalous diffusion occurs in crowded media: sub-diffusion (MSD ~ τ^α, α < 1) arises from obstacles, caging, or viscoelastic cytoplasm; super-diffusion (α > 1) from active transport or Lévy flights. In practice, D is extracted by fitting the linear (or power-law) regime of the ensemble-averaged MSD. The MSD log-log plot (right) shows the fitted slope α and intercepts. This technique is standard in single-particle tracking, NMR relaxation, and neutron scattering.