On The Mean Field limit for Cucker-Smale models

TL;DR

This paper rigorously derives the mean-field limit for generalized Cucker-Smale models using an Eulerian approach in Wasserstein topology, providing explicit constants and insights into flocking behavior.

Contribution

It introduces a novel Eulerian method for mean-field derivation of Cucker-Smale models, avoiding empirical measures and analyzing flocking properties.

Findings

Derived mean-field limit in Wasserstein topology.

Provided explicit constants in the limit process.

Gave insights into flocking behavior for non-strict Cucker-Smale dynamics.

Abstract

In this note, we consider generalizations of the Cucker-Smale dynamical system and we derive rigorously in Wasserstein's type topologies the mean-field limit (and propagation of chaos) to the Vlasov-type equation introduced in [12].Unlike previous results on the Cucker-Smale model, our approach is not based on the empirical measures, but, using an Eulerian point of view introduced in [8] in the Hamiltonian setting, we show the limit providing explicit constants.%Using an Eulerian point of view introduced in \cite{gmp} in the Hamiltonian setting, we don't use empirical measures and provide explicit constants. Moreover, for non strictly Cucker-Smale particles dynamics, we also give an insight on what induces a flocking behavior of the solution to the Vlasov equation to the - unknown a priori - flocking properties of the original particle system.