Extreme Diffusion Measures Statistical Fluctuations of the Environment

We consider many-particle diffusion in one spatial dimension modeled as "random walks in a random environment." A shared short-range space-time random environment determines the jump distributions that drive the motion of the particles. We determine universal power laws for the environment's contribution to the variance of the extreme first passage time and extreme location. We show that the prefactors rely upon a single extreme diffusion coefficient that is equal to the ensemble variance of the local drift imposed on particles by the random environment. This coefficient should be contrasted with the Einstein diffusion coefficient, which determines the prefactor in the power law describing the variance of a single diffusing particle and is equal to the jump variance in the ensemble averaged random environment. Thus a measurement of the behavior of extremes in many-particle diffusion yields an otherwise difficult to measure statistical property of the fluctuations of the generally hidden environment in which that diffusion occurs. We verify our theory and the universal behavior numerically over many random walk in a random environment models and system sizes.