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Updated: Jul 6, 2026
11:51
Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions
Published on: February 22, 2018
Chaotic scattering by steep repelling potentials.
1Faculty of Mathematics and Computer Science, Weizmann Institute of Science, P.O. Box 26, [corrected] Rehovot, Israel 76100. anna.rapoport@weizmann.ac.il
Summary
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.
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.
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