On the mean-field limit for the Vlasov-Poisson system in two dimensions

We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in two dimensions with Coulomb interaction force of the form $f^N(q)=\pm \frac{q}{\mathrm{\mid }q{\mathrm{\mid }}^2}$ and $N$-dependent cut-off at $\mathrm{\mid }q\mathrm{\mid }>N^{-2}$. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force.