Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds

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 \begin{document}$ d $\end{document} -sphere \begin{document}$ \mathbb{S}^d $\end{document} . We show that the asymptotic emergent dynamics of the proposed TCS model on the unit \begin{document}$ d $\end{document} -sphere exhibits a dichotomy, either convergence to zero velocity or asymptotic approach toward a common great circle. We also provide several numerical examples illustrating the aforementioned dichotomy on the asymptotic dynamics of the TCS particles on \begin{document}$ \mathbb{S}^2 $\end{document} .