Statistical properties of single-file diffusion front

TL;DR

This paper analyzes the statistical behavior of the front in single-file diffusion, providing exact and asymptotic results for various properties, revealing extreme value statistics and tail behaviors in the process.

Contribution

It offers new exact and asymptotic analytical results for the front's statistical properties in single-file diffusion, including extreme value laws and first-passage time distributions.

Findings

Front-position follows Gumbel extreme value statistics.

First-passage time exhibits stretched-exponential tail.

Farness of the front characterized by gap and density profile analysis.

Abstract

Statistical properties of the front of a semi-infinite system of single-file diffusion (one dimensional system where particles cannot pass each other, but in-between collisions each one independently follow 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 the 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 behaviors.