A fast point charge interacting with thescreened Vlasov–Poisson system

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 stopping power theory in physics, which predicts a decrease of the velocity V (t) of the point charge given by V ∼ -|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.