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

TL;DR

This work analyzes the mean-field limit of the Random Batch Method (RBM) applied to the Cucker-Smale model, treating the discrete-time chaos introduced by batching and the continuous mean-field dynamics separately. By leveraging flocking estimates and combinatorial arguments, the authors derive quantitative Wasserstein-distance bounds that connect finite-$N$ RBM dynamics to the infinite-particle limit and, ultimately, to the Fokker-Planck-type mean-field equation. The paper also introduces RBM-gPC, a positivity- and momentum-preserving coupling of RBM with generalized Polynomial Chaos (gPC) to efficiently approximate stochastic mean-field CS dynamics, and validates the approach through numerical experiments in 1D and 2D settings. The results establish rigorous error scales with batch size $p$, time step $\tau$, and system size $N$, providing a practical framework for scalable, uncertainty-quantified simulations of collective behavior with rigorous convergence guarantees.

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 $τ\to 0$ 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.

Figures (5)

• Figure 1: Evolution of the expected density $\delta(v-u),\,u=0$ for the RBM-gPC scheme.
• Figure 2: (a): The $\text{MSE}_\text{T}$ of expected temperature between reference solution and the RBM-gPC solution. (b): The $\text{Err}_\text{TV}$ between the expected reference distribution and the expected distribution by the RBM-gPC. Both are obtained at $T=0.5,$ with $dt=dv=10^{-2},$$p=2.$
• Figure 3: (a):The $\text{MSE}_\text{T}$ of expected temperature of the RBM-gPC solution, with $dt=10^{-2}$ and batch size $p$ varying. (b): The $\text{MSE}_\text{T}$ of expected temperature of the RBM-gPC solution, with $p=2$ and time step $dt$ varying. Both are obtained at $T=0.5$ with $N=2^8.$
• Figure 4: The expected distribution of the stochastic 1D Cucker-Smale dynamics with $N=10^4$ agents.
• Figure 5: The expected distribution of the stochastic 2D Cucker-Smale dynamics with $N=10^4$ agents.

Theorems & Definitions (43)

• Lemma 1
• Lemma 2: piccoli2015control, Theorem 3.1
• Remark 1
• Lemma 3: The Wasserstein distance is controlled by the weighted total variation
• Proposition 1: Conservation of momentum
• proof
• Proposition 2: Dissipation of kinetic energy
• proof
• Theorem 1
• Theorem 2