Microscopic dynamics of thin hard rods

TL;DR

This paper derives a hydrodynamic model for the coupled translational and rotational motion of thin hard rods, revealing density-dependent anisotropic diffusion and validating the model with simulations.

Contribution

It introduces a new hydrodynamic equation based on collision rules for hard needles, connecting microscopic interactions to macroscopic diffusion behavior.

Findings

Anisotropic diffusion occurs only at high densities.

The derived model agrees well with simulations at low densities.

The Perrin equation is insufficient for describing binary collision interactions.

Abstract

Based on the collision rules for hard needles we derive a hydrodynamic equation that determines the coupled translational and rotational dynamics of a tagged thin rod in an ensemble of identical rods. Specifically, based on a Pseudo-Liouville operator for binary collisions between rods, the Mori-Zwanzig projection formalism is used to derive a continued fraction representation for the correlation function of the tagged particle's density, specifying its position and orientation. Truncation of the continued fraction gives rise to a generalised Enskog equation, which can be compared to the phenomenological Perrin equation for anisotropic diffusion. Only for sufficiently large density do we observe anisotropic diffusion, as indicated by an anisotropic mean square displacement, growing linearly with time. For lower densities, the Perrin equation is shown to be an insufficient hydrodynamic description for hard needles interacting via binary collisions. We compare our results to simulations and find excellent quantitative agreement for low densities and qualtitative agreement for higher densities.