Anomalous diffusion
Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), ⟨ r 2 ( τ ) ⟩ {\displaystyle \langle r^{2}(\tau )\rangle } , and time. This behavior is in stark contrast to Brownian motion, the typical diffusion process described by Albert Einstein and Marian Smoluchowski, where the MSD is linear in time (namely, ⟨ r 2 ( τ ) ⟩ = 2 d D τ {\displaystyle \langle r^{2}(\tau )\rangle =2dD\tau } with d being the number of dimensions and D the diffusion coefficient).