Stochastic dynamics for quantum billiards: Bridging integrability, chaos, and freezing transitions

Area of Science:

• Quantum chaos
• Statistical physics
• Complex systems

Background:

• Understanding the transition from regular to chaotic dynamics is crucial in quantum systems.
• Quantum billiards provide a valuable platform for studying this transition.
• Existing models often struggle to capture the full complexity of the crossover regime.

Purpose of the Study:

• To present a novel stochastic framework for describing the transition from regular to chaotic dynamics in quantum billiards.
• To develop analytic expressions for key statistical measures across different dynamical regimes.
• To investigate the emergence of non-Gaussian statistics and multiscale phenomena during this transition.

Main Methods:

• Incorporation of a background scale of fluctuations in level spacing evolution.
• Derivation of analytic expressions for nearest-neighbor spacing distribution and power spectral density.
• Comparison with numerical simulations for quantum limaçon and mushroom billiards.

Main Results:

• The framework accurately captures dynamics in integrable, chaotic, and mixed regimes.
• A plateau formation in level repulsion strength is predicted as chaos is approached.
• The model aligns with phenomena observed in many-body localization and Gaussian free fields.

Conclusions:

• The stochastic framework provides a reliable description of the order-to-chaos transition in quantum billiards.
• It offers a complementary perspective to microscopic methods.
• The approach is valuable for understanding emergent non-Gaussian statistics and multiscale phenomena.