  The random batch method (RBM) proposed in [Jin et al., J. Comput. Phys.,
400(2020), 108877] for large interacting particle systems is an efficient with
linear complexity in particle numbers and highly scalable algorithm for
$N$-particle interacting systems and their mean-field limits when $N$ is large.
We consider in this work the quantitative error estimate of RBM toward its
mean-field limit, the Fokker-Planck equation. Under mild assumptions, we obtain
a uniform-in-time $O(\tau^2 + 1/N)$ bound on the scaled relative entropy
between the joint law of the random batch particles and the tensorized law at
the mean-field limit, where $\tau$ is the time step size and $N$ is the number
of particles. Therefore, we improve the existing rate in discretization step
size from $O(\sqrt{\tau})$ to $O(\tau)$ in terms of the Wasserstein distance.
