  In this paper, we consider particle systems with interaction and Brownian
motion. We prove that when the initial data is from the sampling of Chorin's
method, i.e., the initial vertices are on lattice points $hi\in \mathbb{R}^d$
with mass $\rho_0(hi) h^d$, where $\rho_0$ is some initial density function,
then the regularized empirical measure of the interacting particle system
converges in probability to the corresponding mean-field partial differential
equation with initial density $\rho_0$, under the Sobolev norm of
$L^\infty(L^2)\cap L^2(H^1)$. Our result is true for all those systems when the
interacting function is bounded, Lipschitz continuous and satisfies certain
regular condition. And if we further regularize the interacting particle
system, it also holds for some of the most important systems of which the
interacting functions are not. For systems with repulsive Coulomb interaction,
this convergence holds globally on any interval $[0,t]$. And for systems with
attractive Newton force as interacting function, we have convergence within the
largest existence time of the regular solution of the corresponding
Keller-Segel equation.
