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Capturing correlations in chaotic diffusion by approximation methods.

Georgie Knight1, Rainer Klages

  • 1School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK. g.knight@qmul.ac.uk

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Summary
This summary is machine-generated.

We developed three methods to approximate diffusion coefficients in random walks with complex correlations. All methods accurately converge to the exact result, offering practical tools for analyzing chaotic diffusion.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Statistical Mechanics

Background:

  • Deterministic random walks can exhibit complex dynamical correlations.
  • Approximating diffusion coefficients in such systems is crucial for understanding their behavior.
  • Irregular parameter variations complicate the analysis of diffusion.

Purpose of the Study:

  • To systematically approximate the diffusion coefficient for deterministic random walks with dynamical correlations.
  • To evaluate the effectiveness of three distinct approximation methods.
  • To assess the potential for identifying fractal instabilities in parameter-dependent diffusion.

Main Methods:

  • Expansion of the Taylor-Green-Kubo formula for diffusion.
  • Approximation of Markov partitions and transition matrices using escape rate theory.
  • Analytical and numerical verification using a one-dimensional map.

Main Results:

  • All three investigated methods converge to the analytically exact diffusion coefficient.
  • The methods effectively capture complex dynamical correlations.
  • The study confirms the practicability of the analytical and numerical approaches.

Conclusions:

  • The developed methods provide reliable approximations for diffusion coefficients in complex systems.
  • These techniques are valuable for studying parameter-dependent diffusion and potential fractal instabilities.
  • The findings offer insights into chaotic diffusion dynamics where exact solutions are unavailable.