Dimensions, maximal growth sites, and optimization in the dielectric breakdown model
Joachim Mathiesen1, Mogens H Jensen, Jan Oystein Haavig Bakke
1Physics of Geological Processes, University of Oslo, P.O. 1048 Oslo, Norway. joachim.mathiesen@fys.uio.no
Summary
This study explores fractal cluster growth in the dielectric breakdown model (DBM) using conformal mappings. Researchers found no evidence of a phase transition and developed a method to estimate growth exponents from aggregate data.
Area of Science:
- Physics
- Complex Systems
- Applied Mathematics
Background:
- The dielectric breakdown model (DBM) describes fractal pattern formation.
- Understanding fractal dimensions and growth dynamics is crucial in complex systems.
Purpose of the Study:
- Investigate fractal dimension and maximal growth sites in DBM.
- Analyze the relationship between growth exponent (eta) and fractal properties.
- Determine if a phase transition occurs in fractal growth.
Main Methods:
- Utilized iterated conformal mappings to study DBM.
- Examined the Hoelder exponent (alpha_min) as a measure of maximal growth.
- Applied an optimization principle to estimate effective eta values.
Main Results:
- No evidence found for a phase transition from fractal to nonfractal growth at finite eta.
- Observed that nonfractal growth (D-->1) aligns with alpha_min-->1/2.
- Developed a method to estimate effective eta from temporal growth data.
Conclusions:
- Fractal growth in DBM does not exhibit a sharp phase transition with respect to eta.
- The Hoelder exponent provides insights into growth dynamics and fractal limits.
- The proposed method offers a practical approach for analyzing experimental fractal aggregate data.
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