We study emergent collective behaviors of a thermodynamic Cucker-Smale (TCS) ensemble on complete smooth Riemannian manifolds. For this, we extend the TCS model on the Euclidean space to a complete smooth Riemannian manifold by adopting the work [30] for a CS ensemble, and provide a sufficient framework to achieve velocity alignment and thermal equilibrium. Compared to the model proposed in [30], our model has an extra thermodynamic observable denoted by temperature, which is assumed to be nonidentical for each particle. However, for isothermal case, our model reduces to the previous CS model in [30] on a manifold in a small velocity regime. As a concrete example, we study emergent dynamics of the TCS model on the unit $ d $-sphere $ \mathbb{S}^d $. We show that the asymptotic emergent dynamics of the proposed TCS model on the unit $ d $-sphere exhibits a dichotomy, either convergence to zero velocity or asymptotic approach toward a common great circle. We also provide several numerical examples illustrating the aforementioned dichotomy on the asymptotic dynamics of the TCS particles on $ \mathbb{S}^2 $.

中文翻译：

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完备的黎曼流形上热力学的Cucker-Smale集合的新兴动力学

我们研究了在完全光滑的黎曼流形上的热力学的Cucker-Smale（TCS）集合的涌现的集体行为。为此，我们通过采用以下方法将欧氏空间上的TCS模型扩展为一个完整的光滑黎曼流形[30CS合奏，并提供足够的框架来实现速度对齐和热平衡。与[30]，我们的模型有一个由温度表示的额外热力学可观测值，对于每个粒子，假设它是不相同的。但是，对于等温情况，我们的模型简化为[30在小速度状态下的流形上。作为一个具体的例子，我们研究在单位$ d $球体$ \ mathbb {S} ^ d $上TCS模型的涌现动力学。我们表明，在$ d $球面上，所提出的TCS模型的渐近动力学表现出二分法，即趋于零速度收敛或朝着一个公共大圆弧渐近。我们还提供了几个数值示例，用于说明上述关于\ mathbb {S} ^ 2 $上TCS粒子渐近动力学的二分法。