Seung-Yeal Ha, Xinyu Wang, Xiaoping Xue
We study clustering dynamics for the infinite Cucker-Smale (ICS) model and its connection to the spectrum of the graph Laplacian. For the ICS model, we overcome the challenge of estimating the velocities of particles and derive a system of dissipative differential inequalities (SDDI) in terms of infinite norms. As in the finite ensemble, we show that mono-cluster flocking emerges exponentially fast, and additionally establish a sufficient framework for algebraic multi-cluster flocking. Moreover, for the CS model with a finite system size, we offer a complete spectral characterization of multi-cluster flocking. Specifically, the emergence of $n$-cluster behavior corresponds to the limit of the $n$-th eigenvalue of the time-varying Laplacian approaching zero. In contrast, the lower bound of the ($n$+1)-th eigenvalue remains strictly positive. Furthermore, we extend this framework to the ICS model by characterizing weak $n$-cluster flocking via spectral asymptotics, where the $n$-th eigenvalue tends to zero, while both the ($n$+1)-th eigenvalue and the infimum of the essential spectrum remain strictly positive. Our results bridge spectral analysis and clustering dynamics, providing indirect evidence for the fast emergence of mono-cluster flocking and the slow relaxation of multi-cluster patterns in both finite and infinite particle systems.
