  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.
