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

TL;DR

This paper investigates the differences in scaling exponents of radial and angular density correlations during fractal to non-fractal transitions, revealing anisotropy effects and unifying results under a fractal dimensionality framework.

Contribution

It clarifies the limits of radial and angular correlation scaling equivalence in fractal morphodynamics, highlighting anisotropy's role and unifying diverse results with a fractal dimension equation.

Findings

Angular scaling follows a critical power-law.

Radial scaling exhibits exponential behavior.

Anisotropies cause the breakdown of radial/angular equivalence.

Abstract

Fractal/non-fractal morphological transitions allow for the systematic study of the physics behind fractal morphogenesis in nature. In these systems, the fractal dimension is considered a non-thermal order parameter, commonly and equivalently computed from the scaling of the two-point radial- or angular-density correlations. However, these two quantities lead to discrepancies during the analysis of basic systems, such as in the diffusion-limited aggregation fractal. Hence, the corresponding clarification regarding the limits of the radial/angular scaling equivalence is needed. In this work, considering three fundamental fractal/non-fractal transitions in two dimensions, we show that the unavoidable emergence of growth anisotropies is responsible for the breaking-down of the radial/angular equivalence. Specifically, we show that the angular scaling behaves as a critical power-law, whereas the radial scaling as an exponential that, under the fractal dimension interpretation, resemble first- and second-order transitions, respectively. Remarkably, these and previous results can be unified under a single fractal dimensionality equation.