Anisotropic diffusion limited aggregation in three dimensions - universality and non-universality

TL;DR

This paper investigates how lattice anisotropy affects three-dimensional diffusion limited aggregation, revealing universal and non-universal asymptotic growth states, with implications for understanding natural snowflake morphologies.

Contribution

It introduces a new dimension-independent DLA simulation method and analyzes anisotropic growth behaviors across different lattice types, identifying stable and tunable anisotropic states.

Findings

Universal asymptotic states for simple cubic and BCC lattices.

Multiple stable fixed points of anisotropy, including matching BCC case.

Hexagonal growths exhibit tunable polar anisotropy resembling snowflakes.

Abstract

We explore the macroscopic consequences of lattice anisotropy for Diffusion Limited Aggregation (DLA) in three dimensions. Simple cubic and BCC lattice growths are shown to approach universal asymptotic states in a coherent fashion, and the approach is accelerated by the use of noise reduction. These states are strikingly anisotropic dendrites with a rich hierarchy of structure. For growth on an FCC lattice, our data suggest at least two stable fixed points of anisotropy, one matching the BCC case. Hexagonal growths, favouring six planar and two polar directions, appear to approach a line of asymptotic states with continuously tunable polar anisotropy. The more planar of these growths visually resemble real snowflake morphologies. Our simulations use a new and dimension-independent implementation of the Diffusion Limited Aggregation (DLA) model. The algorithm maintains a hierarchy of sphere-coverings of the growth, supporting efficient random walks onto the growth by spherical moves. Anisotropy was introduced by restricting growth to certain preferred directions.