Brownian motion of a rod threading through a ring with fixed ring-center

TL;DR

This paper analyzes the coupled translational and rotational Brownian motion of a rod threading through a fixed ring, deriving distribution functions, relaxation times, and the effects of inertia.

Contribution

It introduces a coupled diffusion model for a rod-ring system and derives new analytical expressions for distribution functions and relaxation times.

Findings

MSD shows a metastable plateau before reaching equilibrium

Longest sliding relaxation time scales as α^{-1/2}

Rotational relaxation is longer than fixed-center but shorter than fixed-end rods

Abstract

We study the Brownian motion of a rigid rod threading through a small fixed ring while the ring can freely rotate. We derive the distribution function for the sliding displacement and the unit vector along the rod both at equilibrium and non-equilibrium. The equilibrium distribution is quadratic in the sliding displacement and is controlled by the moment of inertia (mass distribution). Applying the Onsager variational principle, we derive a Smoluchowski equation in which sliding and rotational diffusion are coupled. The mean square displacement (MSD) of sliding shows a metastable plateau in a certain time range before it approaches the final equilibrium value. The longest sliding relaxation time scales as $\alpha^{-1/2}$, where $\alpha$ is the dimensionless moment of inertia of the rod. The rotational relaxation time obtained from the orientational correlation function is longer than that of a rod with its center fixed but faster than a rod with one end fixed. These results may be useful in understanding the dynamics of polymers connected by sliding rings.