In this paper, we study the macroscopic behavior of the inertial spin (IS) model. This model has been recently proposed to describe the collective dynamics of flocks of birds, and its main feature is the presence of an auxiliary dynamical variable, a sort of internal spin, which conveys the interaction among the birds with the effect of better describing the turning of flocks. After discussing the geometrical and mechanical properties of the IS model, we show that, in the case of constant interaction among the birds, its mean-field limit is described by a nonlinear Fokker–Planck equation, whose equilibria are fully characterized. Finally, in the case of non-constant interactions, we derive the kinetic equation for the mean-field limit of the model in the absence of thermal noise, and explore its macroscopic behavior by analyzing the mono-kinetic solutions.

中文翻译：

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群体惯性旋转模型和相关动力学方程的某些方面

在本文中，我们研究了惯性自旋（IS）模型的宏观行为。这个模型最近被提出来描述鸟群的集体动力学，其主要特征是存在一个辅助动力学变量，一种内部自旋，它传达了鸟类之间的相互作用，更好地描述了转弯。的羊群。在讨论了 IS 模型的几何和机械特性之后，我们表明，在鸟类之间不断相互作用的情况下，其平均场极限由非线性 Fokker-Planck 方程描述，其平衡已完全表征。最后，在非恒定相互作用的情况下，我们推导出了在没有热噪声的情况下模型的平均场极限的动力学方程，