Derivation of a macroscopic model for Brownian hard needles

TL;DR

This paper derives a macroscopic nonlinear nonlocal PDE for the density evolution of Brownian needles, highlighting anisotropic interactions and phase transitions, extending Onsager's theory.

Contribution

It introduces a systematic derivation of a PDE for needle systems considering anisotropic steric effects, bridging microscopic interactions with macroscopic behavior.

Findings

Derived a nonlinear nonlocal PDE for needle density evolution

Identified an isotropic to nematic transition at higher densities

Compared excluded volume effects of needles and spheres

Abstract

We study the role of anisotropic steric interactions in a system of hard Brownian needles. Despite having no volume, non-overlapping needles exclude a volume in configuration space that influences the macroscopic evolution of the system. Starting from the stochastic particle system, we use the method of matched asymptotic expansions and conformal mapping to systematically derive a nonlinear nonlocal partial differential equation for the evolution of the population density in position and orientation. We consider the regime of high rotational diffusion, resulting in an equation for the spatial density that allows us to compare the effective excluded volume of a hard-needles system with that of a hard-spheres system. We further consider spatially homogeneous solutions and find an isotropic to nematic transition as density increases, consistent with Onsager's theory.