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Anomalous diffusion in infinite horizon billiards.

Douglas N Armstead1, Brian R Hunt, Edward Ott

  • 1Department of Physics and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20904, USA. dna2@physics.umd.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 15, 2003
PubMed
Summary

This study reveals how particle displacement moments in infinite horizon billiards scale over long times. The displacement exponent gamma(q) is independent of initial particle distribution, showing a piecewise linear relationship with q.

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Area of Science:

  • Statistical Mechanics
  • Dynamical Systems
  • Mathematical Physics

Background:

  • Understanding particle dynamics in complex systems like billiards is crucial.
  • Previous studies have explored time dependence of particle displacement, but long-time behavior and initial condition independence remain areas of interest.

Purpose of the Study:

  • To investigate the long-time dependence of displacement moments <|r|(q)> for infinite horizon billiards.
  • To determine the time exponent gamma(q) and its relation to initial particle distributions.

Main Methods:

  • Analysis of moments of displacement <|r|(q)> for various billiard models.
  • Derivation of scaling results for the time evolution of the velocity angle distribution function.

Main Results:

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  • Displacement moments <|r|(q)> exhibit approximate power-law time dependence: <|r|(q)> ~ t^(gamma(q)).
  • The time exponent gamma(q) is piecewise linear, with gamma(q) = q/2 for q<2 and gamma(q) = q-1 for q>2.
  • The derived scaling result demonstrates independence from the initial particle distribution.

Conclusions:

  • The long-time displacement exponent in infinite horizon billiards is robust and independent of initial conditions.
  • This finding resolves prior discrepancies and highlights a universal scaling behavior in these systems.