Insights from Single-File Diffusion into Cooperativity in Higher Dimensions

TL;DR

This paper develops a formalism to calculate displacement correlations in colloidal systems, extending single-file diffusion insights to higher dimensions and revealing the nature of cooperative motions.

Contribution

It introduces an improved formula for displacement correlation in single-file diffusion and extends it to higher dimensions, enhancing understanding of cooperative dynamics.

Findings

Displacement correlation in SFD can be derived from Lagrangian particle interval correlation.

The formula is extendable to higher dimensions and becomes exact for large systems.

A correction to the MSD asymptotic law in SFD is obtained using nonlinear correlation theory.

Abstract

Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the two-dimensional case, the linear theory gives description of vortical cooperative motion.