Two-stream instability (1D PIC plasma)

What you are seeing: two counter-streaming beams of electrons in a 1D plasma. The configuration is unstable: any tiny perturbation in density grows exponentially, the beams form rolling vortices in phase space ("electron holes"), then trap each other, eventually settling into a single thermal distribution. This is the canonical kinetic plasma instability.

The top plot is the (x, v) phase space, sampled by 10000 macro-particles, drawn with persistence so the electron-hole vortices leave trails. The middle strip is the density-mode spectrogram ($|\hat\rho_k|$ for $k = 1\ldots 8$ versus time): mode 1 dominates the linear phase, harmonics appear at saturation. The bottom trace is $\log|\hat\rho_{k=1}|$ with the dashed analytic reference of slope $\gamma = \omega_p/(2\sqrt 2)\approx 0.354$ (plasma units): the measured slope tracks it in the linear regime, then the mode saturates. Default $v_0 = 0.6$ puts the fundamental near the peak-growth wavenumber.