Some aspects of the inertial spin model for flocks and related kinetic equations

TL;DR

This paper investigates the macroscopic behavior of the inertial spin model for bird flocks, deriving mean-field and kinetic equations, and analyzing equilibrium and mono-kinetic solutions to better understand collective flock dynamics.

Contribution

It introduces a detailed analysis of the inertial spin model, deriving its mean-field limit as a non-linear Fokker-Planck equation and exploring kinetic and macroscopic behaviors.

Findings

Mean-field limit described by a non-linear Fokker-Planck equation

Equilibria of the model are fully characterized

Analysis of mono-kinetic solutions in non-constant interactions

Abstract

In this paper we study the macroscopic behavior of the inertial spin (IS) model. This model has been recently proposed to describe the collective dynamics of flocks of birds, and its main feature is the presence of an auxiliary dynamical variable, a sort of internal spin, which conveys the interaction among the birds with the effect of better describing the turning of flocks. After discussing the geometrical and mechanical properties of the IS model, we show that, in the case of constant interaction among the birds, its mean-field limit is described by a non-linear Fokker-Planck equation, whose equilibria are fully characterized. Finally, in the case of non-constant interactions, we derive the kinetic equation for the mean-field limit of the model in the absence of thermal noise, and explore its macroscopic behavior by analyzing the mono-kinetic solutions.