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Global and local diffusion in the standard map.

Mirella Harsoula1, George Contopoulos1

  • 1Research Center for Astronomy, Academy of Athens Soranou Efesiou 4, GR-115 27 Athens, Greece.

Physical Review. E
|March 18, 2018
PubMed
Summary

The standard map exhibits normal diffusion (μ=1) or anomalous diffusion (μ=2) depending on accelerator mode islands. Even small islands cause anomalous diffusion, though long-term behavior typically reverts to normal diffusion.

Area of Science:

  • Dynamical Systems
  • Statistical Physics
  • Chaos Theory

Background:

  • The standard map is a fundamental model for studying chaotic dynamics and transport phenomena.
  • Understanding diffusion exponents is crucial for characterizing particle or information spread in complex systems.

Purpose of the Study:

  • To investigate the global and local transport and diffusion characteristics of the standard map.
  • To determine the factors influencing diffusion exponents (μ) in different regions of the phase space.

Main Methods:

  • Calculation of the diffusion exponent (μ) across the entire phase space and within specific regions.
  • Analysis of the influence of accelerator mode islands and normal islands of stability.
  • Investigation of the role of nonlinearity parameter (K) and stickiness effects.

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Main Results:

  • Global diffusion is normal (μ=1) without accelerator modes, and anomalous (μ=2) when they exist.
  • Local diffusion exponents are μ=0 in normal islands, μ=2 in accelerator modes, and converge to μ=1 in chaotic regions.
  • Maximum stickiness can lead to prolonged ballistic motion before reverting to normal diffusion.

Conclusions:

  • The presence of accelerator mode islands, however small, fundamentally alters the global diffusion behavior of the standard map.
  • The coexistence of different island types and chaotic regions leads to complex local and global transport dynamics.
  • Analytical solutions for accelerator-type periodic orbits suggest their prevalence, implying frequent, albeit small, intervals of anomalous diffusion.