Fractal heterogeneous media.

A method is presented 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 [A. Carbone, 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 nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology, environmental sciences, geomorphological sciences, etc.