Beijing Institute of Mathematical Sciences and Applications

Jie Du , Computational & Applied Mathematics Group , Hui Yu

Woojoo Shim

Friday, April 22, 2022 10:00 AM - 11:00 AM

Tencent 408 2490 3761 ()

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 for a CS ensemble and providing a sufficient framework to achieve velocity alignment and thermal equilibrium. Compared to the work for a CS ensemble, our model has an extra thermodynamic observable denoted by temperature, which is assumed to be nonidentical for each particle. However, for the isothermal case, our model reduces to the previous CS model 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. We show that the asymptotic emergent dynamics of the proposed TCS model on the unit d-sphere exhibit a dichotomy, either convergence to zero velocity or an asymptotic approach toward a common great circle. We also provide several numerical examples illustrating the aforementioned dichotomy in the asymptotic dynamics of the TCS particles on the 2-sphere.