Apparent Fractality Emerging from Models of Random Distributions

TL;DR

This paper demonstrates that apparent fractal properties can emerge from models of randomly distributed spheres in various dimensions, especially at low volume fractions, suggesting that observed fractality may often originate from randomness.

Contribution

It provides analytical and numerical evidence that apparent fractal behavior arises from randomness in sphere models, challenging the notion of universal fractal dimensions.

Findings

Apparent fractal behavior occurs at low volume fractions.

The range of apparent fractality spans one to two orders of magnitude.

Fractal dimensions depend on density and are not universal.

Abstract

The fractal properties of models of randomly placed $n$-dimensional spheres ($n$=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using analytical and numerical calculations it is shown that in the regime of low volume fraction occupied by the spheres, apparent fractal behavior is observed for a range of scales between physically relevant cut-offs. The width of this range, typically spanning between one and two orders of magnitude, is in very good agreement with the typical range observed in experimental measurements of fractals. The dimensions are not universal and depend on density. These observations are applicable to spatial, temporal and spectral random structures. Polydispersivity in sphere radii and impenetrability of the spheres (resulting in short range correlations) are also introduced and are found to have little effect on the scaling properties. We thus propose that apparent fractal behavior observed experimentally over a limited range may often have its origin in underlying randomness.