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We investigated chaotic billiards, finding that their average velocity follows a power law. For shape-preserving billiards, the acceleration exponent depends only on rotational properties, yielding specific values.

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Area of Science:

  • Physics
  • Dynamical Systems
  • Statistical Mechanics

Background:

  • Chaotic billiards exhibit complex dynamics.
  • Velocity and energy fluctuations are key characteristics.
  • Shape-preserving transformations introduce unique behaviors.

Purpose of the Study:

  • To theoretically and numerically analyze velocity dynamics in shape-preserving chaotic billiards.
  • To determine the acceleration exponent (β) for such systems.
  • To investigate the relationship between energy fluctuations and billiard transformations.

Main Methods:

  • Theoretical analysis of velocity dynamics.
  • Numerical simulations of chaotic billiards.
  • Adiabatic limit study of energy fluctuations.

Main Results:

  • Average velocity follows a power law 〈v〉 = n(β) with respect to collision number n.
  • For shape-preserving chaotic billiards, β depends on rotational properties: β=0 for uniform rotation, β=1/6 for conserved angular momentum, and β=1/4 otherwise.
  • Three quantities (two scalars, one tensor) determine energy fluctuations for arbitrary shape-preserving transformations.

Conclusions:

  • The rotational properties of shape-preserving chaotic billiards dictate their acceleration exponent.
  • Energy fluctuations are fully characterized by specific conserved quantities.
  • Theoretical predictions align perfectly with numerical results.