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This study compares smooth scattering systems to billiard approximations. For regular Sinai scatterers, the billiard model is accurate, but singular Sinai scatterers show key differences, impacting fractal dimensions.

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Area of Science:

  • Classical mechanics
  • Nonlinear dynamics
  • Mathematical physics

Background:

  • Scattering problems involve particles interacting with potentials.
  • Billiard systems approximate smooth potentials with hard-wall obstacles.
  • The validity of billiard approximations for nonlinear, far-from-integrable systems is investigated.

Purpose of the Study:

  • To determine when billiard approximations accurately represent smooth scattering systems.
  • To analyze differences between smooth scattering and billiard problems for singular Sinai scatterers.
  • To understand the impact of potential steepness and shape on scattering properties.

Main Methods:

  • Analysis of a classical two-dimensional scattering problem with a multi-mountain potential.
  • Comparison of smooth scattering systems with corresponding billiard systems.
  • Investigation of regular and singular Sinai scatterers.
  • Study of the influence of steepness parameters on fractal dimensions.

Main Results:

  • For regular Sinai scatterers, smooth scattering properties converge to billiard properties.
  • Singular Sinai scatterers exhibit significant differences from billiard models, including potential stable periodic orbits in the smooth flow.
  • The fractal dimension of the scattering function is sensitive to the ratio of steepness and billiard deviation parameters.
  • Corners in scatterer shapes significantly impact scattering functions.

Conclusions:

  • Billiard approximations are valid for regular Sinai scatterers but not universally applicable.
  • Singular Sinai scatterers present distinct behaviors requiring careful analysis beyond simple billiard models.
  • The study highlights the complex interplay between potential shape, steepness, and scattering dynamics.