Single-file Diffusion with Random Diffusion Constants

TL;DR

This paper studies how randomly distributed diffusion constants affect the behavior of particles in a single-file system, revealing dependence on the distribution law and identifying different asymptotic behaviors for edge and central particles.

Contribution

It introduces a detailed analysis of single-file diffusion with random diffusion constants, showing how the distribution law influences particle front dynamics and central particle localization.

Findings

Edge particle fronts can remain non-shrinking for broad or exponential D-distributions.

Central particle distribution is Gaussian if certain moments of D are finite.

Effective diffusion constants scale with N depending on the D-distribution law.

Abstract

The single-file problem of N particles in one spatial dimension is analyzed, when each particle has a randomly distributed diffusion constant D sampled in a density $\rho(D)$. The averaged one-particle distributions of the edge particles and the asymptotic ($N\gg 1$) behaviours of their transport coefficients (anomalous velocity and diffusion constant) are strongly dependent on the D-distribution law, broad or narrow. When $\rho$ is exponential, it is shown that the average one-particle front for the edge particles does not shrink when N becomes very large, as contrasted to the pure (non-disordered) case. In addition, when $\rho$ is a broad law, the same occurs for the averaged front, which can even have infinite mean and variance. On the other hand, it is shown that the central particle, dynamically trapped by all others as it is, follows a narrow distribution, which is a Gaussian (with a diffusion constant scaling as $N^{-1}$) when the fractional moment $<D^{-1/2}>$ exists and is finite; otherwise ($\rho(D)\propto D^{\alpha-1}, \alpha\le\case12$), this density is, far from the origin, a stretched exponential with an exponent in the range $]0, 2]$; then the effective diffusion constant scales as $N^{-\beta}$, with $\beta = 1/(2\alpha)$.