Stability of trajectories for N-particle dynamics with a singular potential

We study the stability in finite times of the trajectories of interacting particles. Our aim is to show that on average and uniformly in the number of particles, two trajectories whose initial positions in phase space are close remain close enough at later times. For potentials less singular than the classical electrostatic kernel, we are able to prove such a result for initial positions/velocities distributed according to the Gibbs equilibrium of the system.