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Area of Science:

  • Mathematical Physics
  • Dynamical Systems
  • Quantum Chaos

Background:

  • Billiards in polygonal domains are models for studying classical and quantum dynamics.
  • Understanding the relationship between classical chaos and quantum spectral statistics is a key challenge.
  • Hexagonal billiards with C3 symmetry offer a unique system for exploring these dynamics.

Purpose of the Study:

  • To introduce and numerically investigate a biparametric family of hexagonal billiards.
  • To map the phase diagram and identify fully ergodic systems.
  • To characterize the spectral properties of quantum counterparts and their relation to classical dynamics.

Main Methods:

  • Numerical investigation of relative measure in reduced phase space for discrete time.
  • Analysis of position autocorrelation function decay rates (σ).
  • Calculation of energy eigenvalues, nearest neighbor spacing distribution, and cumulative spacing function for quantum billiards.

Main Results:

  • Identification of fully ergodic systems within the hexagonal billiard family.
  • Observation of position autocorrelation function decay |Cq(t)|∼t−σ, with 0 < σ ≤ 1.
  • Quantum spectra exhibit Gaussian Unitary Ensemble (GUE) and Gaussian Orthogonal Ensemble (GOE) behaviors for σ∼1, akin to chaotic systems.
  • Formulas for intermediate quantum statistics derived for 0 < σ < 1.
  • Ergodic parameter α identifies regimes of quantum dynamical localization.

Conclusions:

  • Hexagonal billiards, despite being non-chaotic (zero Lyapunov exponents), can exhibit near strongly mixing dynamics.
  • Quantum spectral statistics in these billiards mimic those of chaotic systems, providing evidence for universality.
  • The study establishes a quantitative link between classical phase space properties and quantum spectral behavior.