Fractal Heterogeneous Media

TL;DR

This paper introduces a method to generate compact fractal disordered media with a variable Hurst exponent, enabling better modeling of complex, self-similar heterogeneous systems across various scientific fields.

Contribution

It generalizes the random midpoint displacement algorithm to produce stochastic fractals with a tunable Hurst exponent, linking fractality to topology rather than roughness.

Findings

Generated structures are invasive stochastic fractals with variable Hurst exponent.

The Hurst exponent serves as an estimator of compactness, not roughness.

Potential applications include modeling complex biological and environmental systems.

Abstract

A method is proposed for generating compact fractal disordered media, by generalizing the random midpoint displacement algorithm. The obtained structures are invasive stochastic fractals, with the Hurst exponent varying as a continuous parameter, as opposed to lacunar deterministic fractals, such as the Menger sponge. By employing the Detrending Moving Average algorithm [Phys. Rev. E 76, 056703 (2007)], the Hurst exponent of the generated structure can be subsequently checked. The fractality of such a structure is referred to a property defined over a three dimensional topology rather than to the topology itself. Consequently, in this framework, the Hurst exponent should be intended as an estimator of compactness rather than of roughness. Applications can be envisaged for simulating and quantifying complex systems characterized by self-similar heterogeneity across space. For example, exploitation areas range from the design and control of multifunctional self-assembled artificial nano and micro structures, to the analysis and modelling of complex pattern formation in biology, environmental sciences, geomorphological sciences, etc.