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 {mathbb {R}}^3 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).
Highlights
The model studied in the present work is a new elaboration of the work initiated in Degond et al (2017) to model collective behaviour of agents described by their position and body attitude
In order to use the properties of the Haar measure, we will need the matrices P and Q to belong to S O3(R) ( O3(R)) and we define another decomposition, called the special singular value decomposition (SSVD) in the following
The following proposition and corollary ensure that the SSVD is preserved by the dynamical system which will allow us to restrict the domain on which the ordinary differential equations (ODEs) (28a) is posed
Summary
The model studied in the present work is a new elaboration of the work initiated in Degond et al (2017) to model collective behaviour of agents described by their position and body attitude. This model (Degond and Motsch 2008) has been the starting point of many other models of self-organised dynamics, including Degond et al (2017) for the body-attitude coordination In this context, we define the von Mises distribution of parameter J ∈ M3(R) (a 3 × 3 real matrix) as the following PDF on S O3(R): MJ ( A) :=. 3 and Appendix B) will be the key to reduce the problem to a form that shares structural properties with the models of nematic alignment of polymers, studied in a completely different context to model liquid crystals (Han et al 2015; Wang and Hoffman 2008; Zhou and Wang 2011; Ball and Majumdar 2010; Ball 2017) These two worlds will be formally linked through the isomorphism between S O3(R) and the group of unit quaternions detailed in Sect. For a matrix M ∈ M3(R), the orbit Orb(M) ⊂ M3(R) is defined by: Orb(M) := {P M Q, P, Q ∈ S O3(R)}
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