Unifying averaged dynamics of the Fokker-Planck equation for Paul traps

TL;DR

This paper applies averaging theory to the Fokker-Planck equation in Paul traps, revealing a simplified 2D phase portrait that characterizes plasma dynamics and the conditions for bounded or unbounded solutions.

Contribution

It introduces a novel averaging approach to analyze the Fokker-Planck equation in Paul traps, simplifying the understanding of plasma dynamics and solution stability.

Findings

Averaged dynamics are represented by a simple 2D phase portrait.

The phase portrait is independent of the rf field amplitude.

Boundaries between solution behaviors are characterized by a parabola.

Abstract

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.