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Universal energy diffusion in a quivering billiard.

Jeffery Demers1, Christopher Jarzynski2

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Introducing the quivering billiard model, we demonstrate how small boundary wiggles cause deterministic particle dynamics to become stochastic. This leads to universal energy distributions and Fermi acceleration in time-dependent billiard systems.

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Area of Science:

  • Physics
  • Dynamical Systems
  • Statistical Mechanics

Background:

  • Billiard systems model particle motion within boundaries.
  • Time-dependent billiards introduce complex dynamics.
  • The Fermi-Ulam model explores energy gain from moving walls.

Purpose of the Study:

  • To introduce and analyze a "quivering billiard" model for time-dependent systems.
  • To investigate the transition from deterministic to stochastic dynamics in billiards.
  • To explain universal energy distributions and Fermi acceleration.

Main Methods:

  • Simulating particle dynamics in a billiard with infinitesimally wiggling boundaries.
  • Analyzing the energy diffusion process of particle ensembles.
  • Comparing results with the Fermi-Ulam model and static wall approximations.

Main Results:

  • Quivering billiard dynamics become inherently stochastic with small boundary displacements.
  • Particle ensembles evolve to a universal energy distribution.
  • Universal Fermi acceleration is observed regardless of billiard shape or dimensionality.

Conclusions:

  • The quivering billiard model accurately represents time-dependent billiards with small boundary motions.
  • The model resolves discrepancies in previous Fermi-Ulam models.
  • The quivering limit is proposed as the true fixed wall limit for Fermi-Ulam dynamics.