IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v178y2024ics096007792301233x.html
   My bibliography  Save this article

Statistical mechanical characterization of billiard systems

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S096007792301233X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2023.114331?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Oliveira, Diego F.M. & Leonel, Edson D., 2014. "Statistical and dynamical properties of a dissipative kicked rotator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 498-514.
    2. de Souza Filho, Edson E. & Mathias, Amanda C. & Viana, Ricardo L., 2021. "Fractal structures in the deflection of light by a pair of charged black holes," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    3. Oliveira, Diego F.M. & Roberto Silva, Mario & Leonel, Edson D., 2015. "A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 909-915.
    4. da Costa, Diogo Ricardo & Fujita, André & Batista, Antonio Marcos & Sales, Matheus Rolim & Szezech Jr, José Danilo, 2022. "Conservative generalized bifurcation diagrams and phase space properties for oval-like billiards," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    5. Anze Mraz & Michele Diego & Andrej Kranjec & Jaka Vodeb & Peter Karpov & Yaroslav Gerasimenko & Jan Ravnik & Yevhenii Vaskivskyi & Rok Venturini & Viktor Kabanov & Benjamin Lipovšek & Marko Topič & Ig, 2023. "Manipulation of fractionalized charge in the metastable topologically entangled state of a doped Wigner crystal," Nature Communications, Nature, vol. 14(1), pages 1-8, December.
    6. Oliveira, Diego F.M. & Robnik, Marko & Leonel, Edson D., 2011. "Dynamical properties of a particle in a wave packet: Scaling invariance and boundary crisis," Chaos, Solitons & Fractals, Elsevier, vol. 44(10), pages 883-890.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:178:y:2024:i:c:s096007792301233x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.