Derivation of a macroscopic model for Brownian hard needles

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