Universal fractality of morphological transitions in stochastic growth processes

Abstract

Stochastic growth processes give rise to diverse and intricate structures everywhere in nature, often referred to as fractals. In general, these complex structures reflect the non-trivial competition among the interactions that generate them. In particular, the paradigmatic Laplacian-growth model exhibits a characteristic fractal to non-fractal morphological transition as the non-linear effects of its growth dynamics increase. So far, a complete scaling theory for this type of transitions, as well as a general analytical description for their fractal dimensions have been lacking. In this work, we show that despite the enormous variety of shapes, these morphological transitions have clear universal scaling characteristics. Using a statistical approach to fundamental particle-cluster aggregation, we introduce two non-trivial fractal to non-fractal transitions that capture all the main features of fractal growth. By analyzing the respective clusters, in addition to constructing a dynamical model for their fractal dimension, we show that they are well described by a general dimensionality function regardless of their space symmetry-breaking mechanism, including the Laplacian case itself. Moreover, under the appropriate variable transformation this description is universal, i.e., independent of the transition dynamics, the initial cluster configuration, and the embedding Euclidean space.

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

“…For further details see Methods. For comparison purposes, previously computed fractal dimensions provided by the radial density correlation [18], D r (p), will be used. In the case of the DBM, all numerical data for D(η) is obtained from the literature and listed in detail below.…”

“…As described in previous work [18], in all simulations, each particle has a diameter equal to one (the basic unit of distance of the system). For BA or MF (see Fig 1b), we follow a standard procedure in which particles are launched at random from the circumference of a circle of radius r L = 2r max + δ, with equal probability in position and direction of motion.…”

“…While these DBM clusters define one of the most fundamental morphological transitions, they are not unique. In fact, the identification of the most basic growth mechanisms that give origin to fractal clusters has lead to the generation of diverse fractal/non-fractal morphological transitions [18,19]. For instance, the combination of the stochastic growth dynamics of the fundamental diffusionlimited aggregation (DLA) [20] or ballistic aggregation (BA) [21] models, with the highly energetic growth dynamics of the mean-field (MF) aggregation model (see Fig.…”

“…For instance, the combination of the stochastic growth dynamics of the fundamental diffusionlimited aggregation (DLA) [20] or ballistic aggregation (BA) [21] models, with the highly energetic growth dynamics of the mean-field (MF) aggregation model (see Fig. 1b), have led to the creation of the non-trivial BA-MF and DLA-MF morphological transitions [18]. In these systems, the transition is controlled through the probability of aggregation under MF dynamics or mixing parameter, p, that continuously goes from p = 0 (BA/DLA) to p = 1 (MF), see Fig.…”

“…Special attention is paid to the analytical treatment that D(D 0 , Φ) provides to the angular fractal dimensions as a function of the corresponding control parameters. These results are compared to the radial scaling description (obtained in a previous analysis [18]) in order to determine their degree of equivalence and complementarity. In particular, we also look into their levels of predictability, this is, the extent to which the knowledge of just one of the radial or angular measurements predicts the other one.…”