Generic alignment conjecture for systems of Cucker–Smale type

Abstract

The generic alignment conjecture states that for almost every initial data on the torus solutions to the Cucker–Smale system with a strictly local communication align to the common mean velocity. In this note, we present a partial resolution of this conjecture using a statistical mechanics approach. First, the conjecture holds in full for the sticky particle model representing, formally, infinitely strong local communication. In the classical case, the conjecture is proved when N, the number of agents, is equal to 2. It follows from a more general result, stating that for a system of any size for almost every data at least two agents align. The analysis is extended to the open space $\mathbb {R}^n$ in the presence of confinement and potential interaction forces. In particular, it is shown that almost every non-oscillatory pair of solutions aligns and aggregates in the potential well.