Body-attitude alignment : first order phase transition, link with rodlike polymers through quaternions, and stability
Frouvelle, Amic (2021), Body-attitude alignment : first order phase transition, link with rodlike polymers through quaternions, and stability, in Salvarani, F. (eds), Recent Advances in Kinetic Equations and Applications, Springer : Berlin Heidelberg, p. 147-181. 10.1007/978-3-030-82946-9_7
Book titleRecent Advances in Kinetic Equations and Applications
Book authorSalvarani, F. (eds)
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We 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.
Subjects / Keywordsphase transition; rotation matrix; Fokker–Planck equation; quaternions; stability
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Degond, Pierre; Diez, Antoine; Frouvelle, Amic; Merino-Aceituno, Sara (2020) Article accepté pour publication ou publié