Classical and quantum mechanics of a strongly chaotic billiard system
type: Item
instance of: scholarly article
title: Classical and quantum mechanics of a strongly chaotic billiard system
zbMATH Open document ID: 0706.58038
DOI: 10.1016/0167-2789(90)90058-W
author: Martin Sieber
author: Frank Steiner
published in: Physica D
publication date: 1990-01-01T00:00:00Z
review text: Periodic orbits for the hyperbola billiard (a prominent chaotic conservative dynamical system) are studied numerically using an extremum principle and Newton's method. More than 50000 primitive periods are obtained. The length spectrum behaves as if it would result from a one- dimensional random-walk process. Most strikingly, the length spacings seem to follow a Poisson distribution and the stochastic model agrees well with the asymptotic law \(e^{\ell \tau}/\ell \tau\) for the number of (primitive) periodic orbits of length less than \(\ell\). The topological entropy \(\tau\) (close to 0.6) may be expressed in terms of the parameters of the random-walk model. Finally, the classical system is compared with the corresponding quantum system: the periodic-orbit theory due to Gutzwiller evaluated with 36 terms agrees at low energies with the Gaussian level density obtained from a numerical solution of the Schrödinger equation.
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 70G99
Mathematics Subject Classification ID: 81Q50
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4159693
zbMATH Keywords: chaotic behavior
zbMATH Keywords: billiard systems
zbMATH Keywords: quantum chaos
zbMATH Keywords: Periodic orbits
zbMATH Keywords: chaotic conservative dynamical system
reviewed by: Gert Roepstorff
MaRDI profile type: Publication
cites work: Q3396669
cites work: Unstable periodic orbits and semiclassical quantisation
cites work: On the periodic orbits of a strongly chaotic system
cites work: Periodic-orbit sum rules for the Hadamard-Gutzwiller model
full work available at URL: 0167-2789(90)90058-w
OpenAlex ID: W2025244865
description: scientific article; zbMATH DE number 4159693