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).

在本文中，我们研究通过身体-姿势协调在自组织模型中发生的相变现象。在这里，代理人的身体姿势由\（{\ mathbb {R}} ^ 3 \）中的旋转矩阵建模如Degond等。（Math Models Methods Appl Sci 27（6）：1005-1049，2017）。这项研究的起点是一个BGK方程，该方程在动力学水平上建模系统分布函数的演变。这项工作的主要新颖之处在于表明，在空间上均一的情况下，取决于所涉及代理的局部密度，是否可能出现自组织。我们首先展示了人体定向模型与高维空间中聚合物向列取向模型之间的联系，从中我们得出了可能的平衡的完整描述。然后，由于该BGK模型特有的梯度流结构，我们能够证明在不同状态下的稳定性和趋于平衡的收敛性。然后，我们按照Degond等人的精神推导了与稳定平衡有关的宏观模型。（Arch Ration Mech Anal 216（1）：63–115，2015，Math Models Methods Appl Sci 27（6）：1005-1049，2017）。