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A new angle on complex dynamics

Random-walk trajectories 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 undergoing active transport, 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. Researchers often characterize trajectories by looking at the displacement vectors V(t; Δ) describing the particle’s motion from time t to time t + Δ. (Solid and dashed blue arrows in the figure show some displacement vectors for several values of t and two values of Δ; the gray line shows a random-walk trajectory.) The mean-square displacement is a favorite measure; others are also possible. Now Stanislav Burov, Aaron Dinner, and colleagues at the University of Chicago have shown that by analyzing the angles θ(t; Δ) between successive displacement vectors, they can uncover complex dynamics that are otherwise obscured. Because an angle is a function of three points rather than two, the distribution of θ(t; Δ) for a given Δ can be a sensitive measure of whether a system’s direction is influenced by a retained “memory” of its past configurations. Although the method is still in its infancy, the Chicago researchers have found experimental systems in which the distribution peaks at θ = 0, θ = π, or both, and in which the peaks grow, shrink, or remain unchanged with increasing Δ. They’ve also found that systems with similar-looking mean-square displacement data can yield very different patterns in θ. (S. Burov et al., Proc. Natl. Acad. Sci. USA 110, 19689, 2013.)—Johanna L. Miller


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