We consider self-propelled rigid bodies interacting through local body-attitude alignment modelled by stochastic differential equations. We derive a hydrodynamic model of this system at large spatio-temporal scales and particle numbers in any dimension $n \geq 3$ . This goal was already achieved in dimension $n=3$ or in any dimension $n \geq 3$ for a different system involving jump processes. However, the present work corresponds to huge conceptual and technical gaps compared with earlier ones. The key difficulty is to determine an auxiliary but essential object, the generalised collision invariant. We achieve this aim by using the geometrical structure of the rotation group, namely its maximal torus, Cartan subalgebra and Weyl group as well as other concepts of representation theory and Weyl’s integration formula. The resulting hydrodynamic model appears as a hyperbolic system whose coefficients depend on the generalised collision invariant.

中文翻译：

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集群刚体 Fokker-Planck 模型的宏观极限

我们考虑通过随机微分方程建模的局部身体姿态对齐来相互作用的自驱动刚体。我们推导出该系统在大时空尺度和任意维度的粒子数下的流体动力学模型 $n \geq 3$ 。这个目标已经在维度上实现了 $n=3$ 或在任何维度 $n \geq 3$ 对于涉及跳转过程的不同系统。然而，与早期的工作相比，目前的工作在概念和技术上存在巨大差距。关键的困难在于确定一个辅助但重要的对象，即广义碰撞不变量。我们利用旋转群的几何结构，即其最大环面、嘉当子代数和Weyl群以及表示论和Weyl积分公式的其他概念来实现这一目标。所得的流体动力学模型显示为双曲系统，其系数取决于广义碰撞不变量。