Macroscopic limit of a Fokker-Planck model of swarming rigid bodies

TL;DR

This paper derives a macroscopic hydrodynamic model for a system of self-propelled rigid bodies with local alignment, extending previous work to higher dimensions and overcoming significant mathematical challenges using advanced geometric and algebraic tools.

Contribution

It introduces a novel derivation of the macroscopic limit for a high-dimensional swarming model, utilizing the geometry of the rotation group and representation theory.

Findings

Derived a hyperbolic hydrodynamic system in any dimension n ≥ 3.

Identified the generalized collision invariant using geometric and algebraic methods.

Extended previous models to higher dimensions with new mathematical techniques.

Abstract

We consider self-propelled rigid-bodies interacting through local body-attitude alignment modelled by stochastic differential equations. We derive a hydrodynamic model of this system at large spatio-temporal scales and particle numbers in any dimension $n \geq 3$. This goal was already achieved in dimension $n=3$, or in any dimension $n \geq 3$ for a different system involving jump processes. However, the present work corresponds to huge conceptual and technical gaps compared with earlier ones. The key difficulty is to determine an auxiliary but essential object, the generalized collision invariant. We achieve this aim by using the geometrical structure of the rotation group, namely, its maximal torus, Cartan subalgebra and Weyl group as well as other concepts of representation theory and Weyl's integration formula. The resulting hydrodynamic model appears as a hyperbolic system whose coefficients depend on the generalized collision invariant.