Long time and large crowd dynamics of discrete Cucker-Smale alignment models

TL;DR

This paper reviews and extends the analysis of long-term flocking and swarming behaviors in discrete Cucker-Smale models, considering time-dependent agent masses and heavy-tailed communication, and explores hydrodynamic limits.

Contribution

It introduces a comprehensive analysis of long-time dynamics for discrete Cucker-Smale models with time-dependent masses and heavy-tailed kernels, including new insights into non-symmetric communication and hydrodynamic limits.

Findings

Decay of energy fluctuations drives flocking behavior.

Heavy-tailed communication kernels influence large crowd dynamics.

Hydrodynamic models provide a macroscopic perspective on alignment phenomena.

Abstract

We provide a bird's eye view on developments in analyzing the long time, large crowd behavior of Cucker-Smale alignment dynamics. We consider a class of (fully-)discrete models, paying particular attention to general alignment protocols in which agents, with possibly time-dependent masses, are driven by a large class of heavy-tailed communication kernels. The presence of time-dependent masses allows, in particular, non-symmetric communication. While revisiting known results in the literature, we also shed new light of various aspects on the long time flocking/swarming behavior, driven by the decay of energy fluctuations and heavy-tailed connectivity. We also discuss the large crowd dynamics in terms of the hydrodynamic description of Euler alignment models.