On the mean-field limit of the Cucker-Smale model with Random Batch Method

TL;DR

This paper analyzes the mean-field limit of the Random Batch Method for the Cucker-Smale model, introducing a new approach to handle chaos at discrete times and combining it with gPC for stochastic equations.

Contribution

It provides a novel analysis of the mean-field limit with discrete-time chaos propagation and introduces the RBM-gPC method for stochastic mean-field equations.

Findings

Wasserstein distance quantifies approximation accuracy.

Discrete-time chaos propagation is rigorously analyzed.

RBM-gPC conserves positivity and momentum.

Abstract

In this work, we focus on the mean-field limit of the Random Batch Method (RBM) for the Cucker-Smale model. Different from the classical mean-field limit analysis, the chaos in this model is imposed at discrete time and is propagated to discrete time flux. We approach separately the limits of the number of particles $N\to\infty$ and the discrete time interval $\tau\to 0$ with respect to the RBM, by using the flocking property of the Cucker-Smale model and the observation in combinatorics. The Wasserstein distance is used to quantify the difference between the approximation limit and the original mean-field limit. Also, we combine the RBM with generalized Polynomial Chaos (gPC) expansion and proposed the RBM-gPC method to approximate stochastic mean-field equations, which conserves positivity and momentum of the mean-field limit with random inputs.