Random walks on fractals and stretched exponential relaxation

Area of Science:

• Statistical Physics
• Complex Systems Dynamics
• Fractal Geometry

Background:

• Stretched exponential relaxation is a widespread phenomenon observed across diverse scientific fields.
• Despite its prevalence, a fundamental explanation for stretched exponential behavior has remained elusive.
• Understanding this relaxation mechanism is crucial for characterizing complex systems.

Purpose of the Study:

• To elucidate the underlying mechanism of stretched exponential relaxation.
• To investigate the role of fractal geometry in relaxation dynamics.
• To connect relaxation exponents to established theories like percolation theory.

Main Methods:

• Simulating random walks on percolation clusters within curved spaces.
• Analyzing relaxation dynamics across dimensions ranging from 2 to 7.
• Correlating observed decay exponents with fractal properties and percolation theory exponents.

Main Results:

• Demonstrated that random walks on curved percolation clusters exhibit accurate stretched exponential relaxation.
• Established a direct relationship between the relaxation decay exponent and the fractal nature of these clusters.
• Found that the decay exponent in each dimension relates to known exponents from flat-space percolation theory.

Conclusions:

• The fractal character of configuration spaces is proposed as the origin of stretched exponential behavior in complex systems.
• This finding offers a unifying explanation for relaxation phenomena in systems like polymers, colloids, and glasses.
• The study bridges concepts from fractal geometry, percolation theory, and relaxation dynamics.