  The Random Batch Method (RBM) proposed in [Jin et al. J Comput Phys, 2020] is
an efficient algorithm for simulating interacting particle systems. In this
paper, we investigate the Random Batch Method with replacement (RBM-r), which
is the same as the kinetic Monte Carlo (KMC) method for the pairwise
interacting particle system of size $N$. In the RBM-r algorithm, one randomly
picks a small batch of size $p \ll N$, and only the particles in the picked
batch interact among each other within the batch for a short time, where the
weak interaction (of strength $\frac{1}{N-1}$) in the original system is
replaced by a strong interaction (of strength $\frac{1}{p-1}$). Then one
repeats this pick-interact process. This KMC algorithm dramatically reduces the
computational cost from $O(N^2)$ to $O(pN)$ per time step, and provides an
unbiased approximation of the original force/velocity field of the interacting
particle system. In this paper, We give a rigorous proof of this approximation
with an explicit convergence rate. An improved rate is also obtained when there
is no diffusion in the system. Notably, the techniques in our analysis can
potentially be applied to study KMC for other systems, including the stochastic
Ising spin system.
