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T Araújo Lima1, F M de Aguiar1

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Summary
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This study numerically investigates particle dynamics in an elliptical stadium billiard. Researchers found a critical line where chaotic behavior emerges, mirroring phase transitions in liquid helium.

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Area of Science:

  • Physics
  • Dynamical Systems
  • Statistical Mechanics

Background:

  • The elliptical stadium billiard is a model system for studying the transition between regular and chaotic dynamics.
  • Understanding the boundary between regular and chaotic motion is crucial in various fields of physics.

Purpose of the Study:

  • To numerically investigate the dynamics of a particle in an elliptical stadium billiard.
  • To identify and characterize the critical line separating regular and chaotic regions in the phase space.
  • To explore the analogy between this system's behavior and phase transitions in liquid helium.

Main Methods:

  • Numerical investigation of particle dynamics in a reduced phase space with discrete time.
  • Calculation of relative measure r(n) and Shannon entropy s.
  • Fitting of numerically calculated functions with renormalization group formulas.

Main Results:

  • A critical line t(c)=√a(2)-1 was identified in the parameter space.
  • The function ψ(t)=√1-r(∞)(t) critically vanishes at t=t(c).
  • The function c(t)=t(ds/dt) displays a pronounced peak at t=t(c), indicating a transition.
  • Numerical results show remarkable resemblance to the lambda transition in liquid helium.

Conclusions:

  • The identified line t(0)(a) is a strong candidate for the bound for chaos in the elliptical stadium billiard.
  • The observed critical phenomena and the analogy with liquid helium suggest universal behavior in dynamical systems.
  • This research provides insights into the fundamental mechanisms governing chaos in classical systems.