Dynamical properties of single-file diffusion

TL;DR

This paper analyzes the statistical behavior of a tagged particle in single-file diffusion, deriving correlation functions, laws analogous to Brownian motion, and a formalism for full trajectory statistics.

Contribution

It introduces a comprehensive formalism using macroscopic fluctuation theory to obtain the full statistics of a tagged particle in single-file diffusion.

Findings

Derived two-time correlation functions for tagged particles.

Discussed single-file analogs of classical Brownian motion laws.

Developed a cumulant generating functional for full trajectory statistics.

Abstract

We study the statistics of a tagged particle in single-file diffusion, a one-dimensional interacting infinite-particle system in which the order of particles never changes. We compute the two-time correlation function for the displacement of the tagged particle for an arbitrary single-file system. We also discuss single-file analogs of the arcsine law and the law of the iterated logarithm characterizing the behavior of Brownian motion. Using a macroscopic fluctuation theory we devise a formalism giving the cumulant generating functional. In principle, this functional contains the full statistics of the tagged particle trajectory---the full single-time statistics, all multiple-time correlation functions, etc. are merely special cases.