A FIRST-ORDER REDUCTION OF THE CUCKER-SMALE MODEL ON THE REAL LINE AND ITS CLUSTERING DYNAMICS

Abstract

We present a first-order reduction for the Cucker-Smale (C-S) model on the real line, and discuss its clustering dynamics in terms of spatial configurations and system parameters. In previous literature, flocking estimates for the C-S model were mainly focused on the relaxation dynamics of the particle's velocities toward the common velocity. In contrast, the relaxation dynamics of spatial configurations was treated as a secondary issue except for the uniform boundedness of the spatial diameter. In this paper, we first derive a first-order system for the spatial coordinate that can be rewritten as a gradient flow, and then use this first-order formulation to derive several sufficient conditions on the clustering dynamics based on the spatial positions depending on the natural velocities characterized by initial position-velocity configurations.