| time ωpet | |
| measured growth rate γ | |
| cold-fluid theory γ for this k | |
| fastest-growing mode (cold theory) | |
| total energy drift |
A real 1-D electrostatic particle-in-cell simulation: CIC deposition, a spectral Poisson solve, and a leapfrog push. Two counter-streaming electron beams on a fixed ion background. Watch the phase space: ripples → exponential growth → the beams roll into cat's-eye vortices → thermalisation. Nothing here is scripted; the vortices form because the equations say so.