Quantum Chaos Level Statistics

The nearest-neighbor spacing distribution p(s) distinguishes integrable from chaotic quantum systems. Poisson statistics arise for integrable systems (no level repulsion); GOE/GUE Wigner-Dyson distributions arise for time-reversal-symmetric / broken-symmetry chaotic systems. The unfolding procedure maps eigenvalues to uniform mean density.

Sampled

Poisson

GOE Wigner

GUE Wigner

Ensemble

Matrix ensemble Matrix size N: 200 Samples: 20

Statistics

Mean spacing ⟨s⟩—

Variance ⟨s²⟩−⟨s⟩²—

Level repulsion β—

Ratio ⟨r⟩—

Theory

Poisson: p(s)=e⁻ˢ, ⟨r⟩≈0.386
GOE (β=1): p(s)=πs/2·exp(−πs²/4), ⟨r⟩≈0.536
GUE (β=2): p(s)=32s²/π²·exp(−4s²/π), ⟨r⟩≈0.603
GSE (β=4): ⟨r⟩≈0.676
β=repulsion exponent: p(s)∼sᵝ as s→0