Wigner-Dyson vs Poisson nearest-neighbor spacing distribution
P(s) — nearest-neighbor spacing distribution
Eigenvalue spectrum (unfolded)
100
1.00
Statistics:
⟨r⟩ = —
σ(s) = —
Regime: —
Bohigas-Giannoni-Schmit conjecture (1984): quantum systems whose classical limit is chaotic have energy level statistics matching random matrix theory (GOE/GUE/GSE). Integrable systems show Poisson statistics (no level repulsion). Wigner-Dyson: P(s) ∝ s^β exp(−A_β s²) — levels repel. Poisson: P(s) = e^(−s) — levels cluster. The ratio r=min(Δ,Δ')/(max(Δ,Δ')) distinguishes regimes: ⟨r⟩≈0.53 (GOE) vs 0.39 (Poisson).