Statistical properties of a single-file diffusion front

Abstract

Statistical properties of the front of a semi-infinite system of single-file diffusion (a one-dimensional system where particles cannot pass each other, but in between collisions each one independently exhibits diffusive motion) are investigated. Exact as well as asymptotic results are provided for the probability density function of (a) the front position, (b) the maximum of the front positions, and (c) the first-passage time to a given position. The asymptotic laws for the front position and the maximum front position are found to be governed by Fisher Tippett Gumbel extreme value statistics. The asymptotic properties of the first-passage time is dominated by a stretched-exponential tail in the distribution. The farness of the front with the rest of the system is investigated by considering (i) the gap from the front to the closest particle, and (ii) the density profile with respect to the front position, and analytical results are provided for late-time behaviours.