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Body-attitude alignment : first order phase transition, link with rodlike polymers through quaternions, and stability

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorFrouvelle, Amic
HAL ID: 874
ORCID: 0000-0001-6828-8176
dc.date.accessioned2021-11-03T15:44:41Z
dc.date.available2021-11-03T15:44:41Z
dc.date.issued2021
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/22163
dc.language.isoenen
dc.subjectphase transition
dc.subjectrotation matrix
dc.subjectFokker–Planck equation
dc.subjectquaternions
dc.subjectstability
dc.subject.ddc515en
dc.titleBody-attitude alignment : first order phase transition, link with rodlike polymers through quaternions, and stability
dc.typeChapitre d'ouvrage
dc.description.abstractenWe present a simple model of alignment of a large number of rigid bodies (modeled by rotation matrices) subject to internal rotational noise. The numerical simulations exhibit a phenomenon of first order phase transition with respect the alignment intensity, with abrupt transition at two thresholds. Below the first threshold, the system is disordered in large time: the rotation matrices are uniformly distributed. Above the second threshold, the long time behaviour of the system is to concentrate around a given rotation matrix. When the intensity is between the two thresholds, both situations may occur. We then study the mean-field limit of this model, as the number of particles tends to infinity, which takes the form of a nonlinear Fokker--Planck equation. We describe the complete classification of the steady states of this equation, which fits with numerical experiments. This classification was obtained in a previous work by Degond, Diez, Merino-Aceituno and the author, thanks to the link between this model and a four-dimensional generalization of the Doi--Onsager equation for suspensions of rodlike polymers interacting through Maier--Saupe potential. This previous study concerned a similar equation of BGK type for which the steady-states were the same. We take advantage of the stability results obtained in this framework, and are able to prove the exponential stability of two families of steady-states: the disordered uniform distribution when the intensity of alignment is less than the second threshold, and a family of non-isotropic steady states (one for each possible rotation matrix, concentrated around it), when the intensity is greater than the first threshold. We also show that the other families of steady-states are unstable, in agreement with the numerical observations.
dc.publisher.cityParisen
dc.identifier.citationpages147-181
dc.relation.ispartoftitleRecent Advances in Kinetic Equations and Applications
dc.relation.ispartofeditorSalvarani, F. (eds)
dc.relation.ispartofpublnameSpringer
dc.relation.ispartofpublcityBerlin Heidelberg
dc.relation.ispartofdate2021
dc.relation.ispartofurl10.1007/978-3-030-82946-9
dc.subject.ddclabelAnalyseen
dc.relation.ispartofisbn978-3-030-82945-2
dc.identifier.doi10.1007/978-3-030-82946-9_7
dc.description.ssrncandidatenon
dc.description.halcandidatenon
dc.description.readershiprecherche
dc.description.audienceInternational
dc.date.updated2023-01-03T13:34:23Z
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