Theory of Diffusion Controlled Growth

TL;DR

This paper introduces a new theoretical framework for diffusion-controlled growth models, linking them with turbulence and predicting a universal tip scaling exponent, with results confirmed by simulations.

Contribution

It develops a Gaussian truncation approach to analytically calculate properties of diffusion-limited aggregation and dielectric breakdown models, revealing superuniversality of the tip scaling exponent.

Findings

Agreement with simulation data is encouraging

Prediction and confirmation of superuniversality of tip scaling exponent

Analytical calculations of model properties using Gaussian truncation

Abstract

We present a new theoretical framework for Diffusion Limited Aggregation and associated Dielectric Breakdown Models in two dimensions. Key steps are understanding how these models interrelate when the ultra-violet cut-off strategy is changed, the analogy with turbulence and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is both predicted and confirmed.