Mean-field approach to random Apollonian packing

Area of Science:

• Physics
• Materials Science
• Statistical Mechanics

Background:

• Growing spheres and random packing are fundamental in various scientific fields.
• Understanding the scaling properties of these systems is crucial for theoretical and applied research.
• Previous models often rely on assumed radius distributions, limiting their predictive power.

Purpose of the Study:

• To investigate the scaling properties of growing spheres in 2, 3, and 4 dimensions.
• To develop a mean-field model for insertion probability without assuming a functional form for the radius distribution.
• To infer the scaling behavior and fractal dimensions of random Apollonian packing.

Main Methods:

• Utilizing a mean-field approach to model sphere insertion probability.
• Comparing model predictions with extensive numerical simulations (256 simulations, 20x10^6 spheres per dimension).
• Analyzing the functional form of insertion probability and its relation to packing structures.

Main Results:

• The developed mean-field model demonstrates unprecedented agreement with numerical simulations across 2, 3, and 4 dimensions.
• The model successfully predicts the scaling behavior of random Apollonian packing.
• Inferred fractal dimensions align with theoretical expectations and simulation data.

Conclusions:

• The mean-field approach provides a robust framework for understanding the scaling properties of growing spheres.
• The model's accuracy validates its ability to capture the complex dynamics of random packing.
• This work offers new insights into the geometric and statistical properties of Apollonian packing in multiple dimensions.