Body-Attitude Alignment: First Order Phase Transition, Link with Rodlike Polymers Through Quaternions, and Stability

Frouvelle, Amic

doi:10.1007/978-3-030-82946-9_7

Cited by 3 publications

(2 citation statements)

References 25 publications

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“…[44]. Phase transitions in kinetic models have also received a great deal of attention [20,22,23,34,35].…”

Section: Introductionmentioning

confidence: 99%

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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“…[44]. Phase transitions in kinetic models have also received a great deal of attention [20,22,23,34,35].…”

Section: Introductionmentioning

confidence: 99%

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

Degond,

Frouvelle

2024

Eur. J. Appl. Math

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“…The rigorous derivation of the kinetic model has been investigated in [28]. With a slight modification this kinetic model produces phase transitions which have been studied in [16,34]. The SOHB model exhibits topologically non-trivial solutions [17].…”

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confidence: 99%

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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From kinetic to fluid models of liquid crystals by the moment method

Degond

Frouvelle

Liu

2022

KRM

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<p style='text-indent:20px;'>This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.</p>

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Body-Attitude Alignment: First Order Phase Transition, Link with Rodlike Polymers Through Quaternions, and Stability

Cited by 3 publications

References 25 publications

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

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

Hyperbolicity and nonconservativity of a hydrodynamic model of swarming rigid bodies

From kinetic to fluid models of liquid crystals by the moment method

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