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.

翻译：本文聚焦于Cucker-Smale模型中随机批处理方法（RBM）的平均场极限问题。与经典平均场极限分析不同，该模型的混沌特性被施加于离散时间点，并沿离散时间流传播。我们分别处理粒子数$N\to\infty$与离散时间间隔$\tau\to 0$的极限过程，运用Cucker-Smale模型的集群特性及组合数学中的观测方法。采用Wasserstein距离量化近似极限与原始平均场极限之间的差异。此外，我们将RBM与广义多项式混沌（gPC）展开相结合，提出RBM-gPC方法以逼近随机平均场方程，该方法在随机输入条件下保持了平均场极限的正性与动量守恒特性。