Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics. After diffusive rescaling of the kinetic equation, we formally show that the distribution function converges to an equilibrium distribution in particle direction, whose local density and mean direction satisfies a cross-diffusion system. We show that the system is consistent with symmetries typical of a nematic material. The derivation is carried over by means of a Hilbert expansion. It requires the inversion of the linearized collision operator for which we show that the generalized collision invariants, a concept introduced to overcome the lack of momentum conservation of the system, plays a central role. This cross-diffusion system poses many new challenging questions.

中文翻译：

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自推进粒子的向列排列：从粒子到宏观动力学

从描述通过向列排列相互作用的自推进粒子的粒子模型开始，我们推导出粒子密度和平均运动方向的宏观模型。我们首先提出了粒子动力学的平均场动力学模型。在对动力学方程进行扩散重新标度后，我们正式证明分布函数在粒子方向上收敛到平衡分布，其局部密度和平均方向满足交叉扩散系统。我们表明该系统与向列材料的典型对称性一致。推导通过希尔伯特展开进行。它需要线性化碰撞算子的反演，我们证明了广义碰撞不变量，这是为了克服系统缺乏动量守恒而引入的概念，起着核心作用。这种交叉扩散系统提出了许多新的具有挑战性的问题。