A universal dimensionality function for the fractal dimensions of Laplacian growth

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.

The dynamics of the angular and radial density correlation scaling exponents in fractal to non-fractal morphodynamics

“…This transformation causes all the data to collapse into single curves according to their corresponding description (radial, angular, or coarse-grained). In this way, equations (7) show that the mathematical description of these systems not only is quite simple, but more general than previously found [17,18]. In addition, they demonstrate the existence of a universal description that, in particular, is independent of the fractal dimension of the initial configuration, the geometrical symmetry-breaking process that drives the transition, and the Euclidean dimension of the embedding space.…”

“…Unified fractality framework. The D(D 0 , Φ) model has been successfully applied to the analytical description of other quantification methods [18] and quite recently, to the DBM in any Euclidean dimension [17].…”

“…To understand this problem, let us first recall that in the Laplacian framework -as originally introduced in the dielectric breakdown model (DBM) [14][15][16][17] -the * jnicolas@ifuap.buap.mx; author to whom correspondence should be addressed.…”

“…To understand this problem, let us first recall that in the Laplacian framework -as originally introduced in the dielectric breakdown model (DBM) [14][15][16][17] -the growth is controlled via a growth-probability distribution, σ ∝ |∇φ| η . Here φ is a scalar field associated to the potential energy landscape of the growing surface, and η is a positive real parameter associated to non-linear effects.…”

“…As η changes from zero to infinity, different structures emerge for a given set of initial boundary conditions. For example, considering an initial seed-point in two dimensions, different values of η generate either homogeneous growth of compact clusters (η = 0), unstable growth of dendritic-shaped fractals (η ∼ 1), or highly anisotropic growth of linear clusters (η ≈ 4) [10,17] (see Fig. 1a).…”