The derivation of Swarming models: Mean-Field Limit and Wasserstein distances

TL;DR

This paper reviews the derivation of swarming models from particle systems to continuum equations using mean-field limits and Wasserstein distances, highlighting techniques and bounds for singular potentials.

Contribution

It provides a comprehensive summary of the mean-field derivation for swarming models, including new bounds for singular potentials and the propagation of chaos.

Findings

Established qualitative bounds for initial data approximation.

Extended mean-field limit results to singular potentials.

Demonstrated propagation of chaos for specific potential classes.

Abstract

These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models. We first make a summary of the kinetic models derived as continuum versions of second order models for swarming. We focus on the question of passing from the discrete to the continuum model in the Dobrushin framework. We show how to use related techniques from fluid mechanics equations applied to first order models for swarming, also called the aggregation equation. We give qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit for radial singular (at the origin) potentials up to the Newtonian singularity. We also show the propagation of chaos for more restricted set of singular potentials.