A Boltzmann model for rod alignment and schooling fish

TL;DR

This paper analyzes a Boltzmann model for fish schooling and rod alignment, demonstrating a bifurcation phenomenon related to noise intensity and providing rigorous mathematical results for finite modes and formal insights for infinite modes.

Contribution

It rigorously proves the existence of a pitchfork bifurcation in the Boltzmann model of alignment, with detailed analysis for finite Fourier modes and formal extension to infinite modes.

Findings

Existence of a pitchfork bifurcation when noise crosses a critical threshold

Critical exponent of the bifurcation is exactly 1/2 for finite modes

Formal analysis suggests similar behavior for infinite Fourier modes

Abstract

We consider a Boltzmann model introduced by Bertin, Droz and Greegoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the equilibria of this Boltzmann model and we rigorously show the existence of a pitchfork bifurcation when a parameter measuring the inverse of the noise intensity crosses a critical threshold. The analysis is carried over rigorously when there are only finitely many non-zero Fourier modes of the noise distribution. In this case, we can show that the critical exponent of the bifurcation is exactly 1/2. In the case of an infinite number of non-zero Fourier modes, a similar behavior can be formally obtained thanks to a method relying on integer partitions first proposed by Ben-Naim and Krapivsky.