Modeling Multifractal Features of Soil Particle Size Distributions with Kolmogorov Fragmentation Algorithms

Abstract

We have developed a new type of fragmentation algorithm that was inspired by a theoretical question raised by A.N. Kolmogorov—and still unanswered after 60 yr—regarding the characteristics of fragment size distributions when the size of the fragments, rβ, exhibits a power‐law dependence on the size of the original material, r, with 0 ≤ β ≤ 1. Our fragmentation algorithm uses β and N (which denotes the number of particles produced in the fragmentation) as input parameters and was used for various simulations performed with N values of 2, 3, and 4 and β values from 0 to 1. Simulations with β = 0 resulted in lognormal distributions according to the Kolmogorov–Smirnov goodness‐of‐fit test at a confidence level of 95%. On the other hand, simulations with fractional values of β > 0 gave highly heterogeneous distributions exhibiting multifractal characteristics. Rényi dimensions (Dq) and Hölder exponents [α(q)] at q = 0 and 1 were defined with coefficients of determination R2 > 0.95 in 78.3% of samples. The resulting α(0), box dimension D0, and entropy dimension D1 = α(1) values spanned the ranges 0.55 to 1.82, 0.52 to 1, and 0.48 to 0.94, respectively, and were thus suggestive of a multifractal nature in the simulated fragment size distributions. The multifractal characteristics of the simulated distributions were consistent with similar analyses performed on actual soil particle size distributions. These results suggest that the new algorithm can be useful for modeling natural fragmentation processes.

Intermittent Plurisink Model and the Emergence of Complex Heterogeneity Patterns: A Simple Paradigm for Explaining Complexity in Soil Chemical Distributions

“…It is observed that the influence of the random component diminishes for increasing number of points used in the simulation. Also the Table 3 2 0,5-0,5-0,5-0, 5 1,991 0,9999 0,25-0,25-0,25-0,25 0,7-0,7-0,5-0,5 1,951 0,9999 0,8-0,6-0,7-0, 6 1,924 0,9997 2 values become closer to 1. Figure 4 shows the value of the estimated entropy dimension for increasing number of points.…”

“…The aim of this paper is providing a consistent explanation of the mentioned results via an extremely simple model. Recently fragmentation algorithms were proposed to replicate the multifractal nature of soil PSD (Martín et al [6]). In…”

Land-Use Conversion Changes the Multifractal Features of Particle-Size Distribution on the Loess Plateau of China

“…As previously stated, different multifractal parameters represent different information for soil PSDs. High values of these multifractal parameters indicate that PSDs have wide ranges, high heterogeneities, and are more homogeneous among all intervals [21,43,44]. Increases in multifractal parameters may be attributed to increases in fine-soil particles [40,45,46].…”

Fractal and Multifractal Models Applied to Porous Media

“…Bird et al (2009) present a generalization of Turcotte's (1986) fractal fragmentation model by introducing a time component and relaxing the requirement for a single scale-invariant probability of failure. Martín et al (2009) show how an extension of Kolmogorov's (1941) fragmentation theory can generate multifractal mass-size distributions. One of the missing links in this special section is the connection between soil fragmentation and pore space geometry, that is, packing models.…”

“…Next, Bird et al (2009) and Martín et al (2009) introduce some new mathematical approaches for modeling the dynamic fragmentation of earth materials. Bird et al (2009) present a generalization of Turcotte's (1986) fractal fragmentation model by introducing a time component and relaxing the requirement for a single scale-invariant probability of failure.…”