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

We present a stochastic framework for describing the transition from regular to chaotic dynamics in quantum billiards by incorporating a background scale of fluctuations in the level spacing evolution. Analytic expressions for the nearest-neighbor spacing distribution and power spectral density accurately capture the behavior across integrable, chaotic, and mixed-dynamics regimes. Remarkably, our model predicts a plateau formation in the level repulsion strength as the system approaches chaos-a behavior related to the plasma model of many-body localization and freezing transitions in Gaussian free fields. Comparisons to numerical results for quantum limaçon and mushroom billiards confirm the reliable description of the complex crossover between order and chaos. Our approach provides a complementary perspective to microscopic methods, particularly valuable for understanding the emergence of non-Gaussian statistics and multiscale phenomena in the crossover regime.