Body-attitude coordination in arbitrary dimension

Degond, Pierre; Diez, Antoine; Frouvelle, Amic

doi:10.48550/arxiv.2111.05614

Cited by 3 publications

(33 citation statements)

References 59 publications

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“…There have been several extensions of [36]. In relation to the present work, let us mention [28] which includes attraction-repulsion forces, and [24,31,32,33] where alignment of body attitudes (in-stead of mere self-propulsion velocity) is considered and shown to support topological states [26]. Local existence of solutions for the SOH model was proved in [34,90] and numerical simulations can be found in [28,39,71].…”

Section: Introductionmentioning

confidence: 89%

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Topological states and continuum model for swarmalators without force reciprocity

Degond¹,

Diez²,

Walczak³

2022

Preprint

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Swarmalators are systems of agents which are both self-propelled particles and oscillators. Each particle is endowed with a phase which modulates its interaction force with the other particles. In return, relative positions modulate phase synchronization between interacting particles. In the present model, there is no force reciprocity: when a particle attracts another one, the latter repels the former. This results in a pursuit behavior. In this paper, we derive a hydrodynamic model of this swarmalator system and show that it has explicit doubly-periodic travelling-wave solutions in two space dimensions. These special solutions enjoy non-trivial topology quantified by the index of the phase vector along a period in either dimension. Stability of these solutions is studied by investigating the conditions for hyperbolicity of the model. Numerical solutions of both the particle and hydrodynamic models are shown. They confirm the consistency of the hydrodynamic model with the particle one for small times or large phase-noise but also reveal the emergence of intriguing patterns in the case of small phase-noise.

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Section: Introductionmentioning

confidence: 89%

“…After scaling and neglect of higher order terms in ε, the kinetic model reduces to (3.17) with the BGK type collision operator (3.15). BGK-type models of Vicsek-type dynamics have been investigated in [24,25,26,33,39].…”

Section: Particle System Associated With the Bgk Operatormentioning

confidence: 99%

Topological states and continuum model for swarmalators without force reciprocity

Degond¹,

Diez²,

Walczak³

2022

Preprint

Self Cite

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“…In such case, the coefficients c i are linked with those of the kinetic model by explicit formulas (see e.g. [20,21,22] in dimension 3 and [15] in arbitrary dimension). However, here, our goal is to study the SOHB model (2.1)-(2.5) in full generality, without reference to a specific kinetic model (except in Section 5), so the values of the coefficients will remain unspecified.…”

Section: The Self-organized Hydrodynamics Model For Body Orientationmentioning

confidence: 99%

“…Hence, due to the pressure component of F , the fluid goes down the density gradients and due to its second component, it tends to escape the regions of large variations of body-attitude. In [15], the second term of the expression (2.4) of W is shown to be written…”

Section: The Self-organized Hydrodynamics Model For Body Orientationmentioning

confidence: 99%

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Hyperbolicity and nonconservativity of a hydrodynamic model of swarming rigid bodies

et al. 2023

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We study a nonlinear system of first order partial differential equations describing the macroscopic behavior of an ensemble of interacting self-propelled rigid bodies. Such system may be relevant for the modelling of bird flocks, fish schools or fleets of drones. We show that the system is hyperbolic and can be approximated by a conservative system through relaxation. We also derive viscous corrections to the model from the hydrodynamic limit of a kinetic model. This analysis prepares the future development of numerical approximations of this system.

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

Degond,

Frouvelle

2024

Eur. J. Appl. Math

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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 generalised 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 generalised collision invariant.

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Body-attitude coordination in arbitrary dimension

Cited by 3 publications

References 59 publications

Topological states and continuum model for swarmalators without force reciprocity

Topological states and continuum model for swarmalators without force reciprocity

Hyperbolicity and nonconservativity of a hydrodynamic model of swarming rigid bodies

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

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