Charged particles sit in a rotationally-symmetric electric field that oscillates in time: a(t) = −(qE₀/m)·cos(Ωt)·r. Because the field points inward for half the cycle and outward for the other half, no static arrangement of this force could ever trap a charge (Earnshaw's theorem forbids it) — but a fast enough oscillation produces a net time-averaged restoring force, the same "pseudopotential" trick used by real RF ion traps (Paul traps) and quadrupole mass filters.
The motion obeys a Mathieu-type equation. Writing it in the standard form gives a dimensionless stability parameter:
q = 2·E0 / Ω²
stable (bounded) motion roughly requires q < 0.908
Push q above that threshold — by raising the field amplitude or lowering the frequency — and trajectories grow without bound until particles cross the chamber wall and are lost, exactly as unstable ions are ejected from a real Paul trap. This simulation integrates the oscillating force directly (not just the pseudopotential approximation), so the eject event you see is the genuine dynamics, not a scripted animation.
- Confinement trace — fraction of particles still trapped over time.
- Speed histogram — the live kinetic-energy spread of the confined cloud.
- Coulomb repulsion — particles also push each other apart, so a dense cloud self-heats and loses particles faster than a single test charge would.