A fast point charge interacting with the screened Vlasov-Poisson system

TL;DR

This paper rigorously proves the validity of the stopping power theory for a fast point charge interacting with a plasma modeled by the screened Vlasov-Poisson equations, showing velocity decay over long times.

Contribution

It provides a rigorous mathematical validation of the stopping power theory in a plasma setting with a smooth interaction potential, extending understanding of long-time charge dynamics.

Findings

Velocity of the point charge decreases as predicted by the stopping power formula.

The result holds for initial velocities above a certain threshold and for exponentially long times.

Long-time behavior is influenced by the plasma's electric field and occurs after the charge passes through regions.

Abstract

We consider the long-time behavior of a fast, charged particle interacting with an initially spatially homogeneous background plasma. The background is modeled by the screened Vlasov-Poisson equations, whereas the interaction potential of the point charge is assumed to be smooth. We rigorously prove the validity of the \emph{stopping power theory} in physics, which predicts a decrease of the velocity $V(t)$ of the point charge given by $\dot{V} \sim -|V|^{-3} V$, a formula that goes back to Bohr (1915). Our result holds for all initial velocities larger than a threshold value that is larger than the velocity of all background particles and remains valid until (i) the particle slows down to the threshold velocity, or (ii) the time is exponentially long compared to the velocity of the point charge. The long-time behavior of this coupled system is related to the question of Landau damping which has remained open in this setting so far. Contrary to other results in nonlinear Landau damping, the long-time behavior of the system is driven by the non-trivial electric field of the plasma, and the damping only occurs in regions that the point charge has already passed.