Carmo R B do, Lima T Araújo
Laboratório de Física, Instituto Federal de Alagoas, AL 57460-000, Brazil.
Departamento de Física, Universidade Federal Rural de Pernambuco, Recife, PE 52171-900, Brazil.
Phys Rev E. 2024 Jan;109(1-1):014224. doi: 10.1103/PhysRevE.109.014224.
The present work consists of a numerical study of the dynamics of irrational polygonal billiards. Our contribution reinforces the hypothesis that these systems can be strongly mixing, although never demonstrably chaotic, and discusses the role of rotational symmetries on the billiards boundaries. We introduce a biparametric polygonal billiard family with only C_{n} rotational symmetries. Initially, we calculate through the relative measure r(ℓ,θ;t) the phase space filling. This is done for some integer values of n and for a plane of parameters ℓ×θ. From the resulting phase diagram, we can identify the fully ergodic systems. The numerical evidence that symmetrical polygonal billiards can be strongly mixing is obtained by evaluating the position autocorrelation function Cor_{x}(t), which follows a power-law-type decay t^{-σ}. The strongly mixing property is indicated by σ=1. For odd, small values of n, the exponent σ≃1 is found. On the other hand, σ<1 (weakly mixing cases) for small, even values of n. Intermediate n values present σ≃1 independently of parity. For larger values of symmetry parameter n, the biparametric family tends to be a circular billiard (integrable case). For such values of n, we identified even less ergodic behavior at the pace at which n increases and σ decreases.
本研究工作包括对无理多边形台球动力学的数值研究。我们的研究进一步支持了这样一种假设，即这些系统虽绝无明显混沌，但可能具有强混合性，并讨论了旋转对称性在台球边界上所起的作用。我们引入了一个仅具有(C_{n})旋转对称性的双参数多边形台球族。首先，我们通过相对测度(r(ℓ,θ;t))来计算相空间填充情况。针对(n)的一些整数值以及参数平面(ℓ×θ)进行了此项计算。从所得的相图中，我们能够识别出完全遍历的系统。通过评估遵循幂律型衰减(t^{-σ})的位置自相关函数(Cor_{x}(t))，获得了对称多边形台球可具有强混合性的数值证据。强混合性质由(σ = 1)表示。对于奇数且较小的(n)值，发现指数(σ≃1)。另一方面，对于偶数且较小的(n)值，(σ < 1)（弱混合情况）。中间的(n)值呈现出(σ≃1)，与奇偶性无关。对于较大的对称参数(n)值，双参数族趋向于圆形台球（可积情况）。对于这样的(n)值，我们发现在(n)增加且(σ)减小的过程中，遍历行为更少。
