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Simple deterministic dynamical systems with fractal diffusion coefficients.

R Klages1, J R Dorfman

  • 1Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany. R.Klages@physik.tu-berlin.de

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
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This study reveals a fractal structure in the diffusion coefficient of a deterministic model. This finding explains the complex behavior of particles in chaotic scattering systems.

Area of Science:

  • Physics
  • Complex Systems
  • Statistical Mechanics

Background:

  • Deterministic diffusion models offer insights into particle transport.
  • Chaotic scattering systems exhibit complex dynamics influenced by control parameters.

Purpose of the Study:

  • To analyze a simple model of deterministic diffusion.
  • To investigate the relationship between a control parameter and the diffusion coefficient.
  • To explore the fractal nature of diffusion in chaotic systems.

Main Methods:

  • Utilized a one-dimensional array of scatterers with moving point particles.
  • Employed a piecewise linear, expanding, deterministic map on unit intervals.
  • Calculated the diffusion coefficient and largest eigenmodes using Markov partitions and topological transition matrices.

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Main Results:

  • The diffusion coefficient is well-defined and depends on the control parameter.
  • Largest eigenmodes of the map match those of phenomenological diffusion equations.
  • The diffusion coefficient exhibits a fractal structure as a function of the control parameter.

Conclusions:

  • The study successfully explains features of the fractal structure in the diffusion coefficient.
  • Deterministic chaotic scattering models can display complex, fractal diffusion behavior.
  • Findings contribute to understanding particle dynamics in complex physical systems.