Phase transitions and macroscopic limits in a BGK model of body-attitude coordination (Q2022664)

This paper is a new elaboration of [\textit{P. Degond} et al., Math. Models Methods Appl. Sci. 27, No. 6, 1005--1049 (2017; Zbl 1372.35308)]. It is devoted to an analytical study of a model of self-organisation of a large number of agents through body-attitude coordination. The starting point is the BGK equation \[ \partial_t f+(Ae_1\cdot \nabla_x)f=\rho_f M_{J_f}-f\tag{1} \] for a density distribution function $f=f(t,x,A)$ which gives the distribution of agents at time $t\in{\mathbb{R}}^+$ at position $x\in{\mathbb{R}}^3$ with body orientation $A\in SO_3({\mathbb{R}})$. The right-hand side of (1) describes the interactions between the agents. The underlying assumption is that $f$ relaxes towards a `moving equilibrium' $\rho_f M_{J_f}$ which takes the form of a von Mises distribution \[ M_{\kappa \Omega}(\omega )=\frac{e^{\kappa \Omega \cdot \omega}}{\int_{{\mathbb{S}}^{n-1}}e^{\kappa \Omega \cdot \omega '}\,\mathrm{d}\omega '},\qquad \kappa\in{\mathbb{R}}^+,\ \Omega\in SO_3({\mathbb{R}}), \] whereas \[ \rho_f(t,x)=\int_{SO_3({\mathbb{R}})} f(t,x,A)\,\mathrm{d}A\qquad \text{and}\qquad J_f=\int_{SO_3({\mathbb{R}})} f(t,x,A)A\,\mathrm{d}A, \] are the zero and first order moments of $f$, respectively. For the spatially homogeneous version of (1), i.e. the BGK equation \[ \partial_t f=\rho M_{J_f}-f,\tag{2} \] all its equilibria are determined, and their stability is analysed for any given local density $\rho\in \mathbb{R}^+$. The main outcome of the detailed study of the dynamics of (2) is that in the spatially homogeneous case, it is the local density of the agents that determines whether the self-organisation occurs or not. There is a transition region where both the self-organised and disordered states are asymptotically stable. The former state is attributed to the agents tending to achieve a certain mean body attitude $\Lambda\in SO_3({\mathbb{R}})$ and the latter corresponds to the case of equally distributed attitudes. For the spatially inhomogeneous equation (1) its rescaling \[ \partial_t f^\varepsilon +(Ae_1\cdot \nabla_x)f^\varepsilon = \frac{1}{\varepsilon}\left( \rho_{f^\varepsilon}M_{J_{f^\varepsilon}}-f^\varepsilon \right) \tag{3} \] is investigated formally. The two classes of spatially homogeneous stable equilibria are shown to give rise to two types of regions in the macroscopic space-time domain. Different kinds of hydrodynamic models are obtained in each of these cases. In the ordered region, the approach based on the so-called Generalised Collisional Invariants [\textit{P. Degond} and \textit{S. Motsch}, Math. Models Methods Appl. Sci. 18, 1193--1215 (2008; Zbl 1157.35492)] is adopted in order to obtain a closed system.