Uniform-in-time propagation of chaos for the Cucker--Smale model

TL;DR

This paper provides a straightforward proof demonstrating that, under strong interaction conditions, the Cucker--Smale model's particle system maintains a consistent approximation of the mean-field limit over time, with explicit convergence rates.

Contribution

It introduces an elementary method combining existing estimates to achieve uniform-in-time propagation of chaos with explicit convergence rates for the Cucker--Smale model.

Findings

Uniform-in-time propagation of chaos established

Explicit convergence rate in the number of particles

Method combines stability and consistency principles

Abstract

This paper presents an elementary proof of quantitative uniform-in-time propagation of chaos for the Cucker--Smale model under sufficiently strong interaction. The idea is to combine existing finite-time propagation of chaos estimates with existing uniform-in-time stability estimates for the interacting particle system, in order to obtain a uniform-in-time propagation of chaos estimate with an explicit rate of convergence in the number of particles. This is achieved via a method that is similar in spirit to the classical 'stability + consistency implies convergence' approach in numerical analysis.