Stability of trajectories for N -particles dynamics with singular potential

TL;DR

This paper proves that for less singular potentials than electrostatic, particle trajectories starting close in phase space remain close over time, uniformly across particle numbers, when initial states follow Gibbs equilibrium.

Contribution

It establishes stability of particle trajectories over finite times for systems with less singular potentials, extending understanding of many-particle dynamics.

Findings

Trajectories remain close over time for less singular potentials.

Results hold uniformly in the number of particles.

Initial states distributed according to Gibbs equilibrium.

Abstract

We study the stability in finite times of the trajectories of interacting particles. Our aim is to show that in 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 potential 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.