Alignment of Rods and Partition of Integers

TL;DR

This paper investigates the ordering behavior of rods under competing alignment and diffusion, providing an exact steady-state solution and analyzing how the order parameter depends on diffusion strength.

Contribution

It introduces an exact steady-state solution for the nonlinear, nonlocal kinetic theory of rod alignment, using iterated partitions of integers.

Findings

Order emerges at weak diffusion levels.

Fourier modes decay exponentially with wave number.

Order parameter relates to diffusion constant.

Abstract

We study dynamical ordering of rods. In this process, rod alignment via pairwise interactions competes with diffusive wiggling. Under strong diffusion, the system is disordered, but at weak diffusion, the system is ordered. We present an exact steady-state solution for the nonlinear and nonlocal kinetic theory of this process. We find the Fourier transform as a function of the order parameter, and show that Fourier modes decay exponentially with the wave number. We also obtain the order parameter in terms of the diffusion constant. This solution is obtained using iterated partitions of the integer numbers.