Universal second-order phase transition from integrability to chaos

Abstract

We report a dynamical phase transition from integrability to non-integrability in a simple oval- like billiard with boundary R(θ) = 1 + ǫ cos(pθ). For ǫ = 0, the phase space is foliated by invariant curves corresponding to periodic or quasiperiodic motion, whereas for small ǫ a thin chaotic layer separates rotational and librational trajectories. As ǫ increases, this layer grows according to a well-defined scaling law whose chaotic dispersion follows ωrms,sat ∼ ǫ˜α, where the exponent ˜α coincides with those of the Fermi-Ulam model, periodically corrugated waveguides, and a family of discrete mappings, revealing a universal mechanism for the onset of chaos in weakly perturbed integrable systems. The deviation of the reflection angle in the billiard, ωrms,sat, acts as an order parameter: it vanishes continuously as ǫ → 0, signalling an ordered (integrable) phase, while its susceptibility χ = dωrms,sat/dǫ diverges, indicating a second-order phase transition. A symmetry breaking and an analytically solvable diffusion process complete the near-critical phenomenology. These results establish a unified framework for the emergence of chaos from integrability.