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Statistical mechanical characterization of billiard systems

Author

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  • Cetin, Kivanc
  • Tirnakli, Ugur
  • Oliveira, Diego F.M.
  • Leonel, Edson D.

Abstract

Area-preserving maps play an important role in diverse fields as they are widely used for modeling complex systems. In addition, these maps provide rich observations by presenting stable orbits and chaotic behavior separately or together in the phase space depending on the control parameter. In recent years, several studies on these maps, drawing inspiration from the phase space dynamics, have shown that nonextensive statistical mechanics provides appropriate instruments to characterize these systems. In this study, we perform a rigorous numerical analysis to delve into the statistical mechanical properties of a billiard system. Our primary goal is to confirm the presence of a q-Gaussian distribution, with an estimated q value of approximately 1.935. We accomplish this by examining the probability distribution of the cumulative sum of system iterates, focusing specifically on initial conditions within the stability islands. Our findings align seamlessly with the latest research in this field. Furthermore, we show that a multi-component probability distribution containing both Gaussian and q-Gaussians describes the entire system for some parameter regions where the phase space consists of stability islands together with the chaotic sea.

Suggested Citation

  • Cetin, Kivanc & Tirnakli, Ugur & Oliveira, Diego F.M. & Leonel, Edson D., 2024. "Statistical mechanical characterization of billiard systems," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:chsofr:v:178:y:2024:i:c:s096007792301233x
    DOI: 10.1016/j.chaos.2023.114331
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    References listed on IDEAS

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    1. Harle, M. & Feudel, U., 2007. "Hierarchy of islands in conservative systems yields multimodal distributions of FTLEs," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 130-137.
    2. Hansen, Matheus & da Costa, Diogo Ricardo & Caldas, Iberê L. & Leonel, Edson D., 2018. "Statistical properties for an open oval billiard: An investigation of the escaping basins," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 355-362.
    3. Oliveira, Diego F.M. & Leonel, Edson D., 2010. "Suppressing Fermi acceleration in a two-dimensional non-integrable time-dependent oval-shaped billiard with inelastic collisions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(5), pages 1009-1020.
    4. Diego F. M. Oliveira & Rafael A. Bizão & Edson D. Leonel, 2009. "Scaling Properties of a Hybrid Fermi-Ulam-Bouncer Model," Mathematical Problems in Engineering, Hindawi, vol. 2009, pages 1-13, March.
    5. Cetin, Kivanc & Afsar, Ozgur & Tirnakli, Ugur, 2015. "Limit behaviour and scaling relations of two kinds of noisy logistic map in the vicinity of chaos threshold and their robustness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 424(C), pages 269-282.
    6. Jan Ravnik & Yevhenii Vaskivskyi & Jaka Vodeb & Polona Aupič & Igor Vaskivskyi & Denis Golež & Yaroslav Gerasimenko & Viktor Kabanov & Dragan Mihailovic, 2021. "Quantum billiards with correlated electrons confined in triangular transition metal dichalcogenide monolayer nanostructures," Nature Communications, Nature, vol. 12(1), pages 1-8, December.
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    Cited by:

    1. Tsallis, Constantino & Borges, Ernesto P., 2024. "Nonlinear dynamical systems: Time reversibility versus sensitivity to the initial conditions," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).

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