A new angle on complex dynamics

Random walks arise in many areas of physics and other sciences. They don’t all look the same. A particle moving freely behaves differently from a particle in a static potential, which behaves differently from a particle that’s being actively transported, and so on. In biological systems, from atoms in a protein to birds in a flock, the complex interactions between components give rise to an especially rich variety of possible dynamics. The usual way to characterize random-walk trajectories, such as the one shown in the figure, is by looking at the displacement vectors V(t; Δ) describing the particle’s motion from time t to time t + Δ. One particular favorite measure is the mean-square displacement, the average of V²(t; Δ) over all t for a particular Δ. In simple Brownian motion, in which each time step is completely uncorrelated with all the...