Universality of fractal to non-fractal morphological transitions in stochastic growth processes

TL;DR

This paper offers a unified framework for understanding the transition from fractal to non-fractal structures in stochastic growth processes, revealing universal features and the role of anisotropy and screening effects.

Contribution

It introduces a phenomenological dimensionality function and demonstrates the universality of these transitions across different models, including dielectric breakdown.

Findings

Identifies universal characteristics of morphological transitions.

Introduces a dimensionality function to describe fractality.

Shows dielectric breakdown models fit into the unified framework.

Abstract

Stochastic growth processes give rise to diverse intricate structures everywhere and across all scales in nature. Despite the seemingly unrelated complex phenomena at their origin, the Laplacian growth theory has succeeded in unifying their treatment under one framework, nonetheless, important aspects regarding fractal to non-fractal morphological transitions, coming from the competition between screening and anisotropy-driven forces, still lacks a comprehensive description. Here we provide such unified description, encompassing all the known characteristics for these transitions, as well as new universal ones, through the statistical mix of basic models of particle-aggregation and the introduction of a phenomenological physically meaningful dimensionality function, that characterizes the fractality of a symmetry-breaking process induced by a generalized anisotropy-driven force. We also show that the generalized Laplacian growth (dielectric breakdown) model belongs to this class. Moreover, our results provide important insights on the dynamical origins of mono/multi-fractality in pattern formation, that generally occur in far-from-equilibrium processes.