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

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.