Two counter-streaming electron beams (pink vs blue) through a background ion sea. Electrostatic modes with wavenumber k grow exponentially — a classic plasma PIC simulation showing collective instability.
The dispersion relation ω²=ωₚ²[1/(ω-kv₀)²+1/(ω+kv₀)²]=1 admits unstable modes when kv₀/ωₚ is not too large. Growth rate γ∝ωₚ(kv₀/ωₚ)^(−½). Watch phase-space vortices (holes) form as the instability saturates via particle trapping.