Growth of the Wang-Casati-Prosen counter in an integrable billiard
Growth of the Wang-Casati-Prosen counter in an integrable billiard
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DOI：
10.21468/scipostphys.14.2.017
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发表时间：
2020-11
期刊：
影响因子：
5.5
通讯作者：
Zaijong Hwang;C. Marx;J. Seaward;S. Jitomirskaya;M. Olshanii
中科院分区：
文献类型：
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作者：
Zaijong Hwang;C. Marx;J. Seaward;S. Jitomirskaya;M. Olshanii
这项工作的动机是由王，卡萨蒂，和Prosen的一篇文章[物理评论E卷89，042918（2014）]致力于研究二维无理直角三角形台球的遍历性。数值结果表明，这些台球一般不遍历。然而，当台球角等于\pi/2π/2乘以一个刘维无理数时，它们就变成遍历的，刘维无理数在道德上是一类可以用有理数很好地近似的无理数。特别是，Wang等人研究了一个特殊的整数计数器，它反映了无理数对速度方向的贡献;他们推测这个计数器在一般情况下是局部的，但在Liouvillian情况下会增长。我们提出了一个推广的王-卡萨蒂-Prosen计数器：这种推广允许包括合理的台球考虑。我们证明，在45°的情况下，\！45°\！\！：\！90°45°：45°：90°台球，计数器无限增长，与Wang等人提出的Liouvillian场景一致。
This work is motivated by an article by Wang, Casati, and Prosen [Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicity in two-dimensional irrational right-triangular billiards. Numerical results presented there suggest that these billiards are generally not ergodic. However, they become ergodic when the billiard angle is equal to \pi/2π/2 times a Liouvillian irrational, morally a class of irrational numbers which are well approximated by rationals. In particular, Wang et al. study a special integer counter that reflects the irrational contribution to the velocity orientation; they conjecture that this counter is localized in the generic case, but grows in the Liouvillian case. We propose a generalization of the Wang-Casati-Prosen counter: this generalization allows to include rational billiards into consideration. We show that in the case of a 45°\!\!:\!45°\!\!:\!90°45°:45°:90° billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.