On Mean Field Limits for Dynamical Systems

We present a purely probabilistic proof of propagation of molecular chaos for $N$-particle systems in dimension $3$ with interaction forces scaling like $1/\vert q\vert^{\lambda}$ with $\lambda<2$ and cut-off at $q = N^{-1/3}$. The proof yields a Gronwall estimate for the maximal distance between exact microscopic and approximate mean-field dynamics. This can be used to show propagation of molecular chaos, i.e. weak convergence of the marginals to the corresponding products of solutions of the respective mean-field equation without cut-off in a quantitative way. Our results thus lead to a derivation of the Vlasov equation from the microscopic $N$-particle dynamics with force term arbitrarily close to the physically relevant Coulomb- and gravitational forces.