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Triangle map: A model of quantum chaos

Casati1, Prosen

  • 1International Center for the Study of Dynamical Systems, Universita'degli Studi dell'Insubria, via Lucini, 3, I-22100 Como, Italy and and Istituto Nazionale di Fisica della Materia and INFN, Unita di Milano, Milano, Italy.

Physical Review Letters
|November 4, 2000
PubMed
Summary

This study numerically demonstrates that an area-preserving parabolic map, derived from billiard dynamics in an elongated triangle, exhibits ergodic and mixing properties. The particle motion on a cylinder follows a Gaussian diffusive process.

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Area of Science:

  • * Dynamical Systems
  • * Mathematical Physics
  • * Ergodic Theory

Background:

  • * The study of billiard systems provides insights into classical mechanics and chaos.
  • * Poincaré maps are essential tools for analyzing the long-term behavior of dynamical systems.
  • * Area-preserving maps are fundamental in Hamiltonian mechanics and celestial dynamics.

Purpose of the Study:

  • * To investigate the dynamical properties of an area-preserving parabolic map derived from a specific billiard problem.
  • * To provide numerical evidence for ergodicity and mixing in this map.
  • * To analyze the diffusive behavior of the system when extended to a cylindrical manifold.

Main Methods:

  • * Numerical simulations were employed to track the trajectories of a billiard particle within an elongated triangular boundary.

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  • * The Poincaré map was constructed to represent the discrete dynamics of the system.
  • * Statistical analysis of particle trajectories was performed to assess ergodicity, mixing, and diffusive properties.

    • Main Results:

    • * Numerical evidence strongly suggests that the parabolic map is ergodic and mixing.
    • * The system exhibits chaotic behavior consistent with these properties.
    • * When analyzed on a cylinder, the particle's motion was found to approximate a Gaussian diffusive process.

    Conclusions:

    • * The area-preserving parabolic map derived from the elongated triangle billiard demonstrates complex dynamical behaviors.
    • * The findings support the theoretical understanding of ergodicity and mixing in such systems.
    • * The observed Gaussian diffusion on the cylinder suggests potential connections to statistical mechanics and transport phenomena.