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arXiv · 2601.08130

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

Zhongqiang Xiong·Shigeyuki Komura·Masao Doi
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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.

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Zhongqiang Xiong, Shigeyuki Komura, Masao Doi. 2026-01-13. Brownian motion of a rod threading through a ring with fixed ring-center. https://doi.org/10.1063/5.0322429

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