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
of the point
charge given by
,
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 the particle slows down to the threshold
velocity or 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 nontrivial electric field of the plasma, and the damping only occurs in regions
that the point charge has already passed.
