Growth of the Wang-Casati-Prosen counter in an integrable billiard
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 be...
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| Format: | Article |
| Language: | English |
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SciPost
2023-02-01
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| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.14.2.017 |
| _version_ | 1811167076694360064 |
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| author | Zaijong Hwang, Christoph A. Marx, Joseph J. Seaward, Svetlana Jitomirskaya, Maxim Olshanii |
| author_facet | Zaijong Hwang, Christoph A. Marx, Joseph J. Seaward, Svetlana Jitomirskaya, Maxim Olshanii |
| author_sort | Zaijong Hwang, Christoph A. Marx, Joseph J. Seaward, Svetlana Jitomirskaya, Maxim Olshanii |
| collection | DOAJ |
| description | 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 $\pi/2$ 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 $45°\!\!:\!45°\!\!:\!90°$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al. |
| first_indexed | 2024-04-10T16:03:08Z |
| format | Article |
| id | doaj.art-f1834ffec9064054a4ddf3e35f8a592e |
| institution | Directory Open Access Journal |
| issn | 2542-4653 |
| language | English |
| last_indexed | 2024-04-10T16:03:08Z |
| publishDate | 2023-02-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics |
| spelling | doaj.art-f1834ffec9064054a4ddf3e35f8a592e2023-02-10T11:22:26ZengSciPostSciPost Physics2542-46532023-02-0114201710.21468/SciPostPhys.14.2.017Growth of the Wang-Casati-Prosen counter in an integrable billiardZaijong Hwang, Christoph A. Marx, Joseph J. Seaward, Svetlana Jitomirskaya, Maxim OlshaniiThis 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 $\pi/2$ 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 $45°\!\!:\!45°\!\!:\!90°$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.https://scipost.org/SciPostPhys.14.2.017 |
| spellingShingle | Zaijong Hwang, Christoph A. Marx, Joseph J. Seaward, Svetlana Jitomirskaya, Maxim Olshanii Growth of the Wang-Casati-Prosen counter in an integrable billiard SciPost Physics |
| title | Growth of the Wang-Casati-Prosen counter in an integrable billiard |
| title_full | Growth of the Wang-Casati-Prosen counter in an integrable billiard |
| title_fullStr | Growth of the Wang-Casati-Prosen counter in an integrable billiard |
| title_full_unstemmed | Growth of the Wang-Casati-Prosen counter in an integrable billiard |
| title_short | Growth of the Wang-Casati-Prosen counter in an integrable billiard |
| title_sort | growth of the wang casati prosen counter in an integrable billiard |
| url | https://scipost.org/SciPostPhys.14.2.017 |
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