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Growth of the Wang-Casati-Prosen counter in an integrable billiards
by Z. Hwang, C. A. Marx, J. Seaward, S. Jitomirskaya, M. Olshanii
Submission summary
| As Contributors: | Maxim Olshanii |
| Preprint link: | scipost_202108_00061v2 |
| Date accepted: | 2022-08-01 |
| Date submitted: | 2022-06-24 05:34 |
| Submitted by: | Olshanii, Maxim |
| Submitted to: | SciPost Physics |
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
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, a class of numbers with properties lying in between irrational and rational. 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 $\ang{45} \!\! : \!\ang{45} \!\! : \! \ang{90}$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Author comments upon resubmission
(a) completely changed the order of presentation; (b) expanded the introduction by a factor of two; (c) clearly presented the position of our system of interest on the integrable->rational->Liouvillian->generic axis for right-triangular billiards. We want to express our wholehearted thank you: our manuscript improved visibly. Hoping for an affirmative answer.