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Diffusion-limited aggregation as a markovian process: bond-sticking conditions

¹Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, 69978 Ramat Aviv, Israel.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

|November 23, 2000

Summary

This study solves cylindrical lattice diffusion limited aggregation (DLA) using a Markovian matrix method. The research determines steady-state configurations and fractal dimensionality, extrapolating to approximately 1.64.

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Area of Science:

Physics
Materials Science
Complex Systems

Background:

Diffusion Limited Aggregation (DLA) is a fundamental model for pattern formation.
Understanding the growth dynamics and emergent properties of DLA is crucial in various scientific fields.
Previous studies often focused on 2D or 3D DLA, with limited analytical solutions for specific geometries.

Purpose of the Study:

To develop and apply a novel Markovian matrix method for solving cylindrical lattice DLA.
To determine the steady-state configurations and growth rates of cylindrical DLA aggregates.
To calculate the fractal dimensionality of these aggregates and extrapolate key properties.

Main Methods:

A Markovian matrix method was employed, representing growth step probabilities.
The Laplace equation was solved to accurately calculate transition probabilities.
Approximations involving finite rows near the aggregate front were utilized.
The method allowed for the calculation of steady-state configuration weights and approach rates.

Main Results:

The steady-state growing configurations and their weights were determined.
The rate of approaching the steady-state growth stage was calculated.
Average upward growth probability and average steady-state density were derived.
The fractal dimensionality was extrapolated to a value near 1.64.

Conclusions:

The Markovian matrix method provides an effective analytical approach for cylindrical lattice DLA.
The study elucidates the statistical properties and fractal nature of cylindrical DLA aggregates.
The findings contribute to a deeper understanding of pattern formation in constrained geometries.