Instabilities in the mean field limit

Consider a system of $N$ particles interacting through Newton’s second law with Coulomb interaction potential in one spatial dimension or a ${\mathcal{C}}^2$ smooth potential in any dimension. We prove that in the mean field limit $N\to +\mathrm{\infty }$, the $N$ particles system displays instabilities in times of order $\log N$ for some configurations approximately distributed according to unstable homogeneous equilibria.