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 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 model of the fractal dimensions of the DBM and DLA is provided by means of a recently introduced 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 Rényi 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. The results are in good agreement with previous theoretical and numerical estimates for two- and three-dimensional DBM, and high-dimensional DLA. Notably, the DBM dimensions conform to a universal description independently of the initial cluster-configuration and the embedding-space.
Highlights
The establishment of a unified and comprehensive theory of fractal growth constitutes a great challenge given the great diversity and complexity of the out-of-equilibrium processes that give origin to fractal morphologies in nature[1,2,3,4]
There are diverse works that have rigorously proven the consistency of the two-dimensional diffusion-limited aggregation (DLA) within a self-similar picture, with a fractal dimension very close to D = 1.7111,13,15,20, there are other results based on multifractal analysis where this is not conclusive
We present the results of using equation (2), along with other theoretical and numerical results, in order to find the general solution to the fractal dimensions D(d, η)
Summary
The establishment of a unified and comprehensive theory of fractal growth constitutes a great challenge given the great diversity and complexity of the out-of-equilibrium processes that give origin to fractal morphologies in nature[1,2,3,4]. This is the case of Laplacian growth, with its emblematic diffusion-limitied aggregation (DLA) model and the more general dielectric breakdown model (DBM), which constitute a paradigm of out-of-equilibrium growth[5] These models have received significant attention in diverse scientific and technological fields, from the oil industry, through bacterial growth, to cosmology[5,6,7], even with relevant applications in current neuroscience and cancer research[8,9,10]. There are diverse works that have rigorously proven the consistency of the two-dimensional DLA within a self-similar picture, with a fractal dimension very close to D = 1.7111,13,15,20, there are other results based on multifractal analysis where this is not conclusive In these studies, the DLA cluster is found to be a weak mass-multifractal, that in terms of the generalized dimension, Dq and momenta q, goes from Dq→−∞ ≈ 1.75 to Dq→∞ ≈ 1.6535,36; while in others, it is a monofractal with Dq ≈ 1.7 for all q37–40. Mean-field approaches[55,56,57] have provided the closed expression, DMF(d, η)
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