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Phase transitions and macroscopic limits in a BGK model of body-attitude coordination

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Title: Phase transitions and macroscopic limits in a BGK model of body-attitude coordination
Authors: Degond, P
Diez, A
Frouvelle, A
Merino Aceituno, S
Item Type: Journal Article
Abstract: In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body attitude of an agent is modelled by a rotation matrix in R3 as in Degond et al. (Math Models Methods Appl Sci 27(6):1005–1049, 2017). The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher-dimensional space from which we deduce the complete description of the possible equilibria. Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated with the stable equilibria in the spirit of Degond et al. (Arch Ration Mech Anal 216(1):63–115, 2015, Math Models Methods Appl Sci 27(6):1005–1049, 2017).
Issue Date: 30-May-2020
Date of Acceptance: 30-Apr-2020
URI: http://hdl.handle.net/10044/1/79919
DOI: 10.1007/s00332-020-09632-x
ISSN: 0938-8974
Publisher: Springer Verlag
Journal / Book Title: Journal of Nonlinear Science
Copyright Statement: © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Sponsor/Funder: The Royal Society
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: WM130048
EP/M006883/1
EP/P013651/1
Keywords: Fluids & Plasmas
0102 Applied Mathematics
Publication Status: Published online
Online Publication Date: 2020-05-30
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics