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arXiv · 2304.07860

Generic alignment conjecture for systems of Cucker-Smale type

Roman Shvydkoy
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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 $\R^n$ 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.

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Roman Shvydkoy. 2023-04-16. Generic alignment conjecture for systems of Cucker-Smale type. https://arxiv.org/abs/2304.07860

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