AI 中文总结
研究两类具抛物线边界的类体育场台球的可积性破缺相变，通过分析粗糙度标度行为识别两种转变并确定临界指数与标度律，为哈密顿台球可积性破缺转变建立统计力学框架，拓展临界现象概念到非线性动力系统。
AI 中文摘要
我们研究了两类具有抛物线边界的类体育场台球中的可积性破缺转变。聚焦边界产生一个混合相空间，其中规则岛与混沌海共存，而分散边界在任何有限边界变形下都会产生一个完全混沌的相空间。通过分析粗糙度ω的标度行为，我们识别出两种定性不同的转变：聚焦几何的连续转变和分散几何的一阶转变。我们确定了相应的临界指数并建立了相关的标度律。对于连续转变，我们在临界现象框架内进一步提供了完整的特征描述。这些结果为描述哈密顿台球中的可积性破缺转变建立了一个统计力学框架，并表明临界现象的概念自然地扩展到确定性非线性动力系统。
英文摘要
We investigate integrability-breaking transitions in two classes of stadium-like billiards with parabolic boundaries. While focusing boundaries generate a mixed phase space in which regular islands coexist with a chaotic sea, dispersing boundaries produce a fully chaotic phase space for any finite boundary deformation. By analyzing the scaling behavior of the roughness $ω$, we identify two qualitatively distinct transitions: a continuous transition for the focusing geometry and a first-order transition for the dispersing one. We determine the corresponding critical exponents and establish the associated scaling laws. For the continuous transition, we further provide a complete characterization within the framework of critical phenomena by identifying the broken symmetry, the order parameter and its diverging susceptibility, the elementary excitations responsible for chaotic diffusion, and the topological defects governing transport. These results establish a statistical-mechanics framework for describing integrability-breaking transitions in Hamiltonian billiards and suggest that the concepts of critical phenomena naturally extend to deterministic nonlinear dynamical systems.
