Simple deterministic dynamical systems with fractal diffusion coefficients

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.

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.