Body-attitude alignment: first order phase transition, link with rodlike polymers through quaternions, and stability

TL;DR

This paper models the alignment of rigid bodies using rotation matrices, revealing a first order phase transition with thresholds, and connects the behavior to rodlike polymers through a mean-field nonlinear Fokker-Planck equation.

Contribution

It introduces a simplified model exhibiting phase transition phenomena and links it to polymer physics via a quaternion-based generalization of the Doi--Onsager equation.

Findings

Numerical simulations show a first order phase transition with two thresholds.

The mean-field limit leads to a nonlinear Fokker--Planck equation with classified steady states.

Exponential stability is proven for disordered and aligned steady states.

Abstract

We present a simple model of alignment of a large number of rigid bodies (modeled by rotation matrices) subject to internal rotational noise. The numerical simulations exhibit a phenomenon of first order phase transition with respect the alignment intensity, with abrupt transition at two thresholds. Below the first threshold, the system is disordered in large time: the rotation matrices are uniformly distributed. Above the second threshold, the long time behaviour of the system is to concentrate around a given rotation matrix. When the intensity is between the two thresholds, both situations may occur. We then study the mean-field limit of this model, as the number of particles tends to infinity, which takes the form of a nonlinear Fokker--Planck equation. We describe the complete classification of the steady states of this equation, which fits with numerical experiments. This classification was obtained in a previous work by Degond, Diez, Merino-Aceituno and the author, thanks to the link between this model and a four-dimensional generalization of the Doi--Onsager equation for suspensions of rodlike polymers interacting through Maier--Saupe potential. This previous study concerned a similar equation of BGK type for which the steady-states were the same. We take advantage of the stability results obtained in this framework, and are able to prove the exponential stability of two families of steady-states: the disordered uniform distribution when the intensity of alignment is less than the second threshold, and a family of non-isotropic steady states (one for each possible rotation matrix, concentrated around it), when the intensity is greater than the first threshold. We also show that the other families of steady-states are unstable, in agreement with the numerical observations.