Mean-field limit for collective behavior models with sharp sensitivity regions

TL;DR

This paper establishes the rigorous mean-field limit for swarming models with sharp local sensitivity regions, including vision cones and discontinuous communication weights, using optimal transport and stability estimates.

Contribution

It introduces a novel framework for analyzing the mean-field limit in models with discontinuous and localized interactions, extending previous results to more realistic sensitivity regions.

Findings

Quantitative bounds on Wasserstein distance between particle and kinetic solutions

Global-in-time solutions constructed via differential inclusions

Applicability to models with sharp vision cones and discontinuous weights

Abstract

We rigorously show the mean-field limit for a large class of swarming individual based models with local sharp sensitivity regions. For instance, these models include nonlocal repulsive-attractive forces locally averaged over sharp vision cones and Cucker-Smale interactions with discontinuous communication weights. We construct global-in-time defined notion of solutions through a differential inclusion system corresponding to the particle descriptions. We estimate the error between the solutions to the differential inclusion system and weak solutions to the expected limiting kinetic equation by employing tools from optimal transport theory. Quantitative bounds on the expansion of the 1-Wasserstein distance along flows based on a weak-strong stability estimate are obtained. We also provide different examples of realistic sensitivity sets satisfying the assumptions of our main results.