Mean-Field Limits of Deterministic and Stochastic Flocking Models with Nonlinear Velocity Alignment

TL;DR

This paper investigates the mean-field limit for flocking models with nonlinear velocity alignment, providing rigorous proofs and quantitative estimates for both deterministic and stochastic cases, extending classical Cucker-Smale theory.

Contribution

It extends the mean-field analysis of flocking models to nonlinear velocity alignments and offers quantitative convergence rates for the propagation of chaos.

Findings

Proved mean-field limit in deterministic and stochastic settings.

Provided convergence rates for k-particle marginals with fat-tailed kernels.

Extended classical Cucker-Smale theory to nonlinear velocity alignment models.

Abstract

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol $\phi$ and a non-linear coupling of velocities given by the power law $A(\bv) = |\bv|^{p-2}\bv$, $p > 2$. The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the $k$-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.