| time ωpet | |
| measured damping rate γ (fit over the linear window) | |
| linear theory γ (exact root of ε(k,ω)=0) | |
| measured oscillation ωr | |
| theory ωr | |
| predicted recurrence TR = 2π/(k Δv) | |
| bounce time 2π/ωB, ωB=√(kE) | |
| total particles / total energy drift |
Top: the distribution f(x,v) — watch the perturbation shear into ever-finer diagonal filaments. That is the damping: nothing is dissipated, the free energy simply moves into velocity-space structure too fine to see. Bottom: field energy on a log axis against the linear-theory line. The solver is semi-Lagrangian with Strang splitting and cubic interpolation — collisionless, reversible, and exactly the scheme in Chapter 2.