The fractal dimensions of Laplacian growth: an analytical approach based on a universal dimensionality function

TL;DR

This paper provides an analytical framework for predicting the fractal dimensions of Laplacian growth structures like DLA and DBM using a universal dimensionality function, aligning well with previous numerical results.

Contribution

It introduces a universal dimensionality equation based on an information-function, linking fractal dimensions to entropy measures and system parameters, advancing understanding of morphological transitions.

Findings

The analytical model accurately predicts fractal dimensions of DLA and DBM.

The model connects fractal dimensions to Rényi entropies and generalized dimensions.

DBM dimensions follow a universal pattern independent of initial conditions.

Abstract

Laplacian growth, associated to the diffusion-limited aggregation (DLA) model or the more general dielectric-breakdown model (DBM), is a fundamental out-of-equilibrium process that generates structures with characteristic fractal/non-fractal morphologies. However, despite of diverse numerical and theoretical attempts, a data-consistent description of the fractal dimensions of the mass-distributions of these structures has been missing. Here, an analytical description to the fractal dimensions of the DBM and DLA is provided by means of a recently introduced general dimensionality equation for the scaling of clusters undergoing a continuous morphological transition. Particularly, this equation relies on an effective information-function dependent on the Euclidean dimension of the embedding-space and the control parameter of the system. Numerical and theoretical approaches are used in order to determine this information-function for both DLA and DBM. In the latter, a connection to the R\'enyi entropies and generalized dimensions of the cluster is made, showing that DLA could be considered as the point of maximum information-entropy production along the DBM transition. These findings are in good agreement with previous theoretical and numerical results (two- and three-dimensional DBM, and high-dimensional DLA). Notably, the DBM dimensions can be conformed to a universal description independently of the initial cluster-configuration and the embedding-space.