Tip splittings and phase transitions in the dielectric breakdown model: mapping to the diffusion-limited aggregation...

Area of Science:

• Physics
• Complex Systems
• Materials Science

Background:

• The dielectric breakdown model (DBM) is a fundamental model for fractal growth.
• Understanding phase transitions in DBM is crucial for characterizing complex growth phenomena.
• Multifractal analysis provides insights into the heterogeneity of growth measures.

Purpose of the Study:

• To investigate the phase transition in the multifractal spectrum of the DBM growth measure.
• To identify the underlying mechanism responsible for this phase transition.
• To establish a relationship between the branching angle and the fractal dimension.

Main Methods:

• Analysis of the multifractal spectrum of the growth measure in DBM.
• Investigation of the role of branch tip splitting angle (eta) in the growth process.
• Derivation of an analytic rescaling relation for the branching angle.
• Comparison of analytic predictions with numerical simulations of DBM.

Main Results:

• A phase transition was identified in the multifractal spectrum of the DBM growth measure.
• Branch tip splitting at a fixed angle was found to be the cause of the transition.
• An analytic rescaling relation was derived, showing agreement with numerical simulations.
• Cluster dimension decreases linearly with the angle; non-fractal growth occurs near 74 degrees (eta = 4.0+/-0.3).

Conclusions:

• The fixed angle of branch splitting in DBM induces a phase transition in its multifractal properties.
• The observed angle can be rescaled to an effectively universal angle in diffusion-limited aggregation.
• The study provides a quantitative link between branching geometry and the transition to non-fractal growth.