Derivation of a macroscopic model for Brownian hard needles

We study the role of anisotropic steric interactions in a system of hard Brownian needles in two dimensions. 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 non-local 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-needle system with that of a hard-sphere system. We further consider spatially homogeneous solutions and find an isotropic to nematic transition as density increases, consistent with Onsager's theory.

Automated summary

We investigate the impact of anisotropic steric interactions on hard Brownian needles in a two-dimensional system. Despite lacking volume, non-overlapping needles exclude volume in configuration space, influencing the macroscopic behavior of the system. By utilizing the method of matched asymptotic expansions and conformal mapping, we derive a nonlinear non-local partial differential equation describing the evolution of population density in position and orientation. We focus on the regime of high rotational diffusion, which yields an equation for spatial density enabling comparison between the effective excluded volumes of hard-needle and hard-sphere systems. Furthermore, we examine spatially homogeneous solutions and observe an isotropic to nematic transition as density increases, consistent with Onsager's theory.