Impact of symmetry on ergodic properties of triangular billiards

Letter

Impact of symmetry on ergodic properties of triangular billiards

Katerina Zahradova, Julia Slipantschuk, Oscar F. Bandtlow, and Wolfram Just

Phys. Rev. E 105, L012201 – Published 18 January 2022

Abstract

Polygonal billiards constitute some of the simplest yet counterintuitive dynamical systems in physics. Even basic features of the dynamics, such as ergodicity of the microcanonical distribution or the decay of correlations have not been settled in general. In this Letter, we will highlight the importance of symmetries of the billiard table for the resulting dynamics. Although typical triangular billiards appear to show correlation decay, symmetric billiards may not even be ergodic with respect to the uniform distribution in phase space.

Received 7 May 2021
Accepted 2 December 2021

DOI:https://doi.org/10.1103/PhysRevE.105.L012201

Physics Subject Headings (PhySH)

Billiards Dynamical systems

Chaos & nonlinear dynamics Ergodic theory

Nonlinear DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

Katerina Zahradova^*, Julia Slipantschuk^†, Oscar F. Bandtlow , and Wolfram Just

School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

^*K.Zahradova@qmul.ac.uk
^†Present address: Department of Mathematics, Warwick University, Coventry, CV4 7AL, United Kingdom.

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Vol. 105, Iss. 1 — January 2022

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Images

Figure 1
Autocorrelation function of the momentum $p$ on a double logarithmic scale for an asymmetric billiard with inner angles $α = π (\sqrt{2} - 1) / 4$ and $β = π (\sqrt{5} - 1) / 4$ . Data have been computed from a time series of length $2^{30}$ with ensembles of initial conditions of different sizes: dark blue (top) $N = 1$ , light blue (second from top) $N = 10$ , red (third from top) $N = 100$ , yellow (bottom) $N = 1000$ . For visibility, values have been shifted by a factor of $10^{- 1}$ . The solid black line indicates a power law decay with exponent $- 1.4$ . The inset displays the dependence of the plateau for a long time as a function of the ensemble size, measured in terms of the maximum (dark blue circles), the absolute mean (light blue triangles), and the variance (red squares). The additional black lines indicate a power law decay with exponents 1/2 (dashed) and 1 (dashed-dot).
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Figure 2
Order parameter $Φ_{N}$ , see Eq. (2), on a double-logarithmic scale for a symmetric triangle (dark blue, solid dark gray) with $α = β = π (\sqrt{5} - 1) / 4$ and asymmetric triangles with $β = α - π ɛ$ with $ɛ = 10^{- 3}$ (dashed light blue), $10^{- 5}$ (dashed-dot red), and $10^{- 7}$ (yellow, solid light gray), along with the right-angled triangle $α = π (\sqrt{5} - 1) / 4, β = π / 2$ (dotted dark blue, gray). The correlation functions have been computed via a Fourier transform of time series of length $2^{27}$ . The ensemble average has been computed using $10^{4}$ randomly generated initial conditions. The inset shows $1 / Φ_{N}$ for the symmetric case as a function of $N$ on a double-logarithmic scale.
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Figure 3
Distribution of the finite time ergodic average of momentum $p$ , see Eq. (3), on a semilogarithmic scale. Top: symmetric triangle with $α = β = π (\sqrt{5} - 1) / 4$ , middle: distorted symmetric triangle with $β = α - π 10^{- 3}$ , bottom: right-angled triangle with $α = π (\sqrt{5} - 1) / 4, β = π / 2$ . Top lines (dark blue) correspond to $N = 10^{4}$ , middle lines (light blue) to $N = 10^{5}$ , and bottom lines (red) to $N = 10^{6}$ . Data have been computed from a uniform random ensemble of initial conditions with ensemble size $10^{6}$ . The distributions have been generated as a histogram with bin size $4 \times 10^{- 4}$ . The insets show the half-width (dark blue circles) and standard deviation (light blue triangles) for the three values of $N$ , along with the trivial scaling $1 / \sqrt{N}$ (dashed).
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Figure 4
Ergodic averages, see Eq. (4), on a semilogarithmic scale, for a symmetric triangle with $α = β = π (\sqrt{5} - 1) / 4$ (dark blue, dark gray), distorted triangle with $β = α - π 10^{- 7}$ (red, midgray), and a right-angled triangle with $α = π (\sqrt{5} - 1) / 4, β = π / 2$ (yellow, light gray) for fixed initial condition $(s_{0}, p_{0}) = (0.5, 0.64)$ .
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