  Collective dynamics of a collisional plasma in a Paul trap is governed by the
Fokker-Planck equation, which is usually assumed to lead to a unique asymptotic
time-periodic solution irrespective of the initial plasma distribution. This
uniqueness is, however, hard to prove in general due to analytical
difficulties. For the case of small damping and diffusion coefficients, we
apply averaging theory to a special solution to this problem, and show that the
averaged dynamics can be represented by a remarkably simple 2D phase portrait,
which is independent of the applied rf field amplitude. In particular, in the
2D phase portrait, we have two regions of initial conditions. From one region,
all solutions are unbounded. From the other region, all solutions go to a
stable fixed point, which represents a unique time-periodic solution of the
plasma distribution function, and the boundary between these two is a parabola.

Plasma Physics