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

Pierre Degond, Amic Frouvelle

TL;DR

This work derives a macroscopic hyperbolic hydrodynamic system (SOHB) for self-propelled rigid bodies in ${\mathbb R}^n$ with local body-attitude alignment, valid for all dimensions $n \ge 3$. A central challenge is the generalized collision invariant (GCI), whose explicit construction relies on the geometry of $SO_n$—notably the maximal torus, Cartan subalgebra, and Weyl group—and Weyl's integration formula. The authors establish the existence of GCIs, express the macroscopic coefficients $c_1$–$c_4$ in terms torus-integrals involving a function $\alpha(\Theta)$ solving a torus-bound system, and derive the final hydrodynamic equations with a precise hyperbolic structure. The resulting SOHB system couples a continuity equation for density with an evolution equation for the mean body orientation, incorporating additional rotation terms that reflect self-rotation and orientation gradients. These results extend prior 3D analyses to arbitrary dimensions and highlight the crucial role of representation theory in analyzing collective dynamics on Lie groups, opening doors to generalized models on other compact groups or homogeneous spaces.

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.

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

TL;DR

This work derives a macroscopic hyperbolic hydrodynamic system (SOHB) for self-propelled rigid bodies in with local body-attitude alignment, valid for all dimensions . A central challenge is the generalized collision invariant (GCI), whose explicit construction relies on the geometry of —notably the maximal torus, Cartan subalgebra, and Weyl group—and Weyl's integration formula. The authors establish the existence of GCIs, express the macroscopic coefficients – in terms torus-integrals involving a function solving a torus-bound system, and derive the final hydrodynamic equations with a precise hyperbolic structure. The resulting SOHB system couples a continuity equation for density with an evolution equation for the mean body orientation, incorporating additional rotation terms that reflect self-rotation and orientation gradients. These results extend prior 3D analyses to arbitrary dimensions and highlight the crucial role of representation theory in analyzing collective dynamics on Lie groups, opening doors to generalized models on other compact groups or homogeneous spaces.

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 . This goal was already achieved in dimension , or in any dimension 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.
Paper Structure (36 sections, 27 theorems, 281 equations, 1 table)

This paper contains 36 sections, 27 theorems, 281 equations, 1 table.

Table of Contents

  1. Introduction
  2. Microscopic model and scaling
  3. Preliminaries: rotations and the rotation group
  4. The particle model
  5. The mean-field model
  6. Scaling and statement of the problem
  7. Hydrodynamic limit (I): main results and first steps of the proof
  8. Statement of the results
  9. Equilibria
  10. The continuity equation
  11. Generalized collision invariants: definition and existence
  12. Definition and first characterization
  13. Existence and invariance properties
  14. Maximal torus and Weyl group
  15. Generalized collision invariants associated with the identity
  16. ...and 21 more sections

Key Result

Theorem 3.2

We suppose that there is a smooth solution $f^\varepsilon$ to System eq:sca_kin_2, eq:sca_J_2. We also suppose that $f^\varepsilon \to f^0$ as $\varepsilon \to 0$ as smoothly as needed. Then, there exist two functions $\rho$: ${\mathbb R}^n \times [0,\infty) \to [0,\infty)$ and $\Gamma$: ${\mathbb R Furthermore, for appropriate real constants $c_1, \ldots, c_4$, $\rho$ and $\Gamma$ satisfy the fol

Theorems & Definitions (37)

  • Definition 3.1: von Mises distribution
  • Theorem 3.2
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Corollary 3.7
  • ...and 27 more