Stability of Trajectories of Classical, Weakly Interacting Particles

Abstract

The Vlasov-Poisson equation describes the macroscopic time evolution of a system with a large number of particles interacting by a force, typically Coulomb or gravitational force.

Although the equation has established itself as a helpful tool yielding strong results, it still lacks a rigorous mathematical justification. So far, the mathematical results all either worked with a cut-off on the force, used a slightly weaker interaction all together, or assumed monokineticity of the initial data. In these settings, however, one was able to prove that the mean-field trajectories arising from the Vlasov-Poisson equation are typically a very good approximation for the real trajectories.

In this paper we consider a similar set-up where we can build on these previous results to help us answer a new but related question. We consider a system of N initially i.i.d. particles with Coulomb interaction. If we move the initial position of one particle by a small δ in ℝ^6 in phase-space, what is the expected impact on the whole system in the time frame [0,T]? Due to the pair interaction, the effect is highly non-trivial. We prove that, under suitable conditions on the starting density and with a cut-off at approximately N^(-1/3), the system as a whole typically stays stable. More precisely, for all the particles that we did not disturb, the distance between the original trajectory and the trajectory in the disturbed system stays smaller than N^(-α)|δ|, for suitable α>0. This α>0 depends on whether the particle has a collision with another particle and on its distance to the disturbed particle. Depending on these characteristics, α takes on values between N^(-1/18+2σ) and N^(-1/3+2σ). The best estimates are obtained for particles that are far away from the disturbed particle and do not have a collision.