Quantum Chaos — Level Spacing & Brody Distribution

Crossover from Poisson (integrable) to Wigner-Dyson (chaotic) statistics

Level Spacing Distribution P(s)

Energy levels (unfolded)

Spectral rigidity Δ₃(L)

Number variance Σ²(L)

Brody parameter fit

Random Matrix Theory: Integrable systems → Poisson spacing P(s)=e⁻ˢ. Chaotic systems (GOE) → Wigner surmise P(s)=(πs/2)e^(-πs²/4). Brody distribution P(s)=A·sᵝ·exp(-αs^(β+1)) interpolates with β∈[0,1]. Spectral rigidity Δ₃(L) measures long-range correlations. KAM crossover visible in β.
Level repulsion exponent β: — | Mean spacing ⟨s⟩: — | Regime: —