Describing a universal critical behavior in a transition from order to chaos

Abstract

We present a comprehensive discussion of a transition from integrability to non-integrability in an oval billiard with a static boundary. This transition is controlled by a deformation parameter ǫ, which modifies the boundary shape from circular, corresponding to ǫ = 0 and an integrable dynamics, to oval for ǫ 6 = 0, where non-integrability emerges. The deformation of the circular billiard gives rise to a chaotic layer that develops along a well-defined stripe in phase space. By introducing a set of transformations that isolate this chaotic stripe, we characterise the diffusive spreading of ensembles of trajectories and identify an observable, ωrms,sat, which plays the role of an order parameter for the transition. For small deformations, the saturation value of the diffusion obeys the scaling law ωrms,sat ∝ ǫ˜α, with a critical exponent ˜α = 0.507(2), vanishing continuously as ǫ → 0. The associated susceptibility, χ = dωrms,sat/dǫ, diverges in the same limit, signalling the presence of critical behavior analogous to that observed in second-order (continuous) phase transitions in statistical mechanics.