Anomalous Fluctuations of Extremes in Many-Particle Diffusion

Over one hundred years ago Einstein created a remarkably simple and powerful theory describing the behavior of a single diffusing particle. That theory has since been applied countless times to successfully model widely disparate systems. However, this theory neglects the effects a shared environment has on the particles. As a consequence, the Einstein theory dramatically fails to predict the behavior of extreme diffusion, i.e. outlier particles which have moved the farthest from their starting points. We study particles undergoing a random walk in a beta distributed environment and provide theoretical predictions, which we confirm numerically, of the behavior of the maximally displaced particle. By introducing a shared environment, we find three scaling regimes relating to the KPZ equation for the variance of the maximally displaced particle, contrary to the Einstein diffusion model which predicts a single scaling regime. Understanding the behavior of outliers will have wide ranging applicability to physical, biological, epidemiological, economic, and social systems where outliers often determine behavior.