Fractal Regime Research Articles

From Zipf to Price and beyond

Consider a society comprising n members that are ranked in a decreasing order of their personal wealths. For this society: Zipf’s Law manifests the case in which the members’ ranks and wealths display an inverse power-law relation; and Price’s Law manifests the case in which the n richest members possess, collectively, half of all the wealth. This paper goes from Zipf’s Law to Price’s Law and beyond, and it does so by introducing and exploring a novel Generalized Price’s Law (GPL): a general allometric scaling of the quantiles of rank distributions. Akin to multifractals, the GPL is governed by spectrums of powers. The GPL spectra are investigated, and are shown to be fractal objects: from a socioeconomic perspective, the spectra are power-law Lorenz curves that characterize poor fractality and rich fractality; from a probabilistic perspective, the spectra are characterized by Pareto statistics and by Lindy Laws. The intersection of Zipf’s Law and of the GPL is also investigated, and it is shown to be: (i) the phase transition between the two markedly different fractal regimes of the GPL spectra; (ii) the phase transition between two markedly different macroscopic regimes of Zipf’s Law; (iii) Price’s Law. Metaphorically, this paper establishes navigation directions in the space of rank distributions: how to get from the ‘Zipf street’ to the ‘Price junction’, from there to the new ‘GPL avenue’, and back.

Kitaev formula for periodic, quasicrystal, and fractal Floquet topological insulators

Topological insulators are attracting researchers to reveal related fundamental physics, whose fractal versions are expanding topological physics into the fractal regime. The fractal topological systems, however, show a challenge to the Kitaev formula, because it leads to a noninteger Chern number. The Chern number within the band gap oscillates more and more actively as the system size increases. We uncover that such a challenge should be attributed to the coupling, brought about by the lattice fractalization, between the internal and external edge states. To confirm this, examples of periodic, quasicrystal, and fractal Floquet topological insulators are compared with each other by analyzing the topological edge states and local Chern markers. The sum of the local Chern markers on all the edges of the fractal lattice presents a fractional power relation with the system size, reflecting the fractalization of the topological edge states. We reveal that once such a coupling is shut down, the Kitaev formula would once again result in integers.

A Numerical Study of Elastic Wave Arrival Behavior in a Naturally Fractured Rock Based on a Combined Displacement Discontinuity-Discrete Fracture Network Model

The arrival behavior of elastic waves in a naturally fractured rock is studied based on numerical simulations. We use the discrete fracture network method to represent the distribution of a natural fracture system and employ the displacement discontinuity method to compute the propagation of elastic waves across individual fractures. We analyze macroscopic wavefield arrival properties collectively arising from the interaction between elastic waves and numerous fractures in the system. We show that the dimensionless angular frequency ῶ = ωZ/κ exerts a fundamental control on the arrival behavior of a plane wave traveling through the fractured rock, where ω, Z, and κ are the angular frequency, seismic impedance, and fracture stiffness, respectively. An asynchronous arrival phenomenon of the wave energy occurs and becomes more significant with an increased ῶ. Two regimes are identified according to the two-branch dependency of the fractal dimension D of the FFAW on ῶ, where the wave arrival behavior is within a non-fractal regime for ῶ smaller than the critical frequency ῶc ≈ 1.0, and enters the fractal regime for ῶ ≥ ῶc. The self-affine properties of the FFAW, i.e., the roughness exponent α and the correlation length lc, both linearly decrease as a function of the exponent ξ (with ῶ = 10ξ) in the fractal regime. Early breakthrough of wave transport occurs in regions with relatively low fracture density, while late-time arrival happens in regions of high fracture density.

“Holographic Implementations” in the Complex Fluid Dynamics through a Fractal Paradigm

Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply “holographic implementations”, through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed.

Spatial Distribution and Scaling Properties of Lidar‐Derived Snow Depth in the Extratropical Andes

AbstractWe characterize elevational gradients, probability distributions, and scaling patterns of lidar‐derived snow depth at the hillslope scale along the extratropical Andes. Specifically, we analyze snow depth maps acquired near the date of maximum accumulation in 2018 at three experimental sites: (i) the Tascadero catchment (31.26°S, 3,270–3,790 m), (ii) the Las Bayas catchment (33.31°S, 3,218–4,022 m); and (iii) the Valle Hermoso (VH) catchment (36.91°S, 1,449–2,563 m). We examine two subdomains in the latter site: one with (VH West) and one without (VH East) shrub cover. The comparison across sites reveals that elevational gradients are site‐dependent, and that the gamma and normal distributions are more robust than the lognormal function to characterize the spatial variability of snow depth. Multiscale behavior in snow depth is obtained in all sites, with up to three fractal regimes, and the magnitude of primary scale breaks is found to be related to the mean separation distance between local snow depth peaks. The differences in snow depth fractal parameters between VH West—the only vegetated subdomain—and the remaining sites suggest that local topographic and land cover properties are dominant controls on the spatial structure of snow, rather than average hydroclimatic conditions. Overall, the results presented here provide, for the first time, insights into the spatial structure of snow depth along the extratropical Andes Cordillera, showing notable similarities with other mountain regions in the Northern Hemisphere and providing guidance for future snow studies.

Desiccation and cracking behaviour of clay loam subjected to different irrigation methods during wetting–drying cycles

AbstractDesiccation cracking is a common phenomenon of soil, which potentially affects irrigation and crop growth in farmland. Understanding cracking behaviour aids in developing an effective means of crack prevention or utilization. In the present paper, we report on a phenomenological investigation aimed at exploring the effects of irrigation methods on the cracking behaviour of soil during wetting–drying cycles. Nine identical specimens were prepared and subjected to three irrigation methods (surface irrigation, sprinkling irrigation and drip irrigation) and three wetting–drying cycles. Specimen cracking was monitored by a digital camera and crack patterns were delineated by geometric parameters calculated using image processing techniques. The results showed that the evaporation process of specimens could be roughly divided into three stages according to the variation of evaporation rate. Most cracks were generated during the constant rate stage, although the soil matrix zone was nearly saturated. Surface irrigation increased the cracking water content and cracking degree, in contrast to sprinkling irrigation and drip irrigation. The contribution of crack length propagation and crack width broadening to crack area growth was nearly the same when the soil water content decreased to low values. Stabilized crack networks varied from fractal regimes to uniform distribution regimes with the increasing‐length scale, and the crossover length was independent of irrigation methods and wetting–drying cycles. Under sprinkling irrigation and drip irrigation, cracks formed in the first drying remained as failure zones and most of them reappeared during subsequent drying. However, after surface irrigation most cracks did not propagate in previously cracked zones during subsequent drying, giving rise to shifting crack networks between wetting–drying cycles. Our results provide new insights regarding crack development and evolution in agricultural soil, which will be useful in crack management and irrigation arrangement during crop production.Highlights Effects of irrigation methods and wetting–drying cycles on cracking were investigated. Most cracks initiate and propagate when the soil matrix zone is nearly saturated. Different irrigation methods result in reappearing or shifting crack networks between cycles. This study provides new insights regarding crack development and evolution in agricultural soil.

A Universal Model of Unsaturated Hydraulic Conductivity With Complementary Adsorptive and Diffusive Process Components

AbstractAccurate estimation of unsaturated hydraulic conductivity (HC) is one of the most challenging problems in soil science. Here, we propose a novel approach to model HC using percolation theory. Transient behavior of water transport phenomena at low moisture contents requires additional physical process representation, beside capillary conductivity, to ensure accurate prediction of unsaturated HC. We augment the capillary model from percolation theory with two additional components, namely, (1) film flow, which is the product of volumetric flow rate per perimeter by specific perimeter of solid particles, and (2) isothermal vapor HC, derived from the Fick's law of vapor diffusion and relative humidity. The fractal characteristics of last fractal regime are used to model tortuosity and ultimately HC of vapor flow. Since the typical pressure head range of universal scaling from percolation theory is analogous to the range of vapor flow, we demonstrate that the universal scaling presented in previous studies is not sufficient to model HC for water contents below a crossover point. We also, by analyzing the scaled water retention properties, demonstrate that most studied soils exhibit three fractal regimes. Therefore, a piecewise HC function of capillary flow is developed to account for three fractal regimes, providing more flexibility for soils with multimodal characteristics. The proposed joint HC function is more accurate compared to the model of Peters‐Durner‐Iden and predecessor percolation theory models.

Ensemble and effective dispersion in three-dimensional isotropic fractal media

We determine the time-dependent behavior of the dispersion coefficient for transport in formations with isotropic log-conductivity fields showing fractal behavior.We consider two different dispersion coefficients for point-like injection: (1) the ensemble dispersion coefficients, defined as half the rate of change of the second central moments of the ensemble-averaged concentration distribution and (2) the effective dispersion, which is half the rate of change of the expected second central moments. Our results show, that the two longitudinal macrodispersion coefficients steadily grow with time and remain different at all times in a fully fractal regime, indicating that no Fickian transport regime is ever reached. The resulting effective longitudinal transport model is consequently a fractional advection–dispersion equation. In the semi-fractal regime, a Gaussian transport regime is reached eventually. However, compared to the case of a classic non-fractal regime, the transient non-Gaussian regime lasts much longer. In the transverse direction, the two dispersion coefficients approach the same large-time limit also in fractal media highlighting the fundamental difference between longitudinal and transverse dispersion.

Applicability of fractal capillary pressure models to sandstones

The microscopic fractal theory plays an important role in the prediction of macro petrophysical parameters. However, the significant fractal dimension changes with the fractal regime or the capillary pressure model. We adopt three traditional capillary pressure models and a modified capillary pressure model to characterize the fractal behavior of six sandstones from the Songliao Basin. The various models indicated an overall nonfractal behavior for the sandstones in a pore-throat size regime; however, they showed a fractal behavior in the small pore-throat regime and a nonfractal behavior in the large pore-throat regime. The modified model was identified as the most appropriate capillary-pressure model to characterize the fractal behavior of sandstones, since it avoids the oversimplifications of pore morphologies and the unreasonable omissions typical of traditional models. This scope of this study was to promote a correct understanding of the fractal behavior of sandstones by using capillary pressure models.

A fractal analysis of the water retention curve

AbstractThe dependence of the soil water content ϑ upon the matric potential ψ is studied within a fractal approach that regards the water retention curve as a sequence of well defined fractal regimes. Each of such regimes accounts for a given functional dependence ϑ≡ϑ(ψ), which in turn is characterized by a fractal dimension. The difference between the double fractal (observed into sandy soils) and multifractal (observed into clay soils) regime is explained by recalling that, for a sandy soil, the transition from saturated to dry conditions is driven by a steep reduction of ψ. To the contrary, for a clay (where the change from the highest water contents to the smallest ones is characterized by a large range of the matric potential), the multifractal behaviour is observed. These results are also confirmed by the analysis of experimental data. In particular, we show that the intermediate regime, generally accounting for the fractal multimodality, is due to the sandy nature of the soil at stake, practically immaterial. Finally, we demonstrate that our model can be also regarded as the straightforward generalization of that of Millán and González‐Posada ().

Does a Fractal Microstructure Require a Fractional Viscoelastic Model?

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity.

Magistral and fractal regimes of the hydro-fracturing

The paper describes the specific difference between two fracture growth regimes, when it forms the radial fracture network (fractal regime) and when it forms one big fracture in certain direction (magistral regime). A continuum model is used, which represents the fracture network in the media with the permeability tensor, rapidly increasing with the fracture appearance. The fracture front instability is shown, linked with the Saffman-Taylor instability, even in isotropic case. By analogy with Saffman-Taylor instability in porous media, the fractal nature of the fracture network is derived. The dimensionless parameters of fracture front speed and anisotropy are found, which would help to numerical and experimental modeling of the reservoir.

Multiscale Structure Evolution in Electrically Conductive Nanocomposites Studied by SAXS

The successful introduction of nanostructured materials is hampered by the lack of a quantitative and qualitative understanding of the structure–property relationship within the nanocomposites. Variation in the electrical conductivity of nanocomposite materials depends on the structure evolution of the nanoscale fillers within the polymer phase. This article pertains to the small angle x-ray scattering study of electrically conductive nanocomposites to understand the structure-property relationship. The nanocomposites in this study are comprised of thermoset polyurethane as the matrix material with carbon nanotubes as the filler material. A percolation threshold of 0.5% wt/wt was observed along with substantial changes in the electrical conductivity of the nanocomposites. The small angle x-ray scattering data exhibits mass and surface fractal regimes indicating multiple structure evolution in the nanocomposites. Moreover, the data is interpreted using the crossovers of the scaling laws and sizes are measured to characterize the microstructure with a possible explanation for structural development.

Robustness of the fractal regime for the multiple-scattering structure factor

In the single-scattering theory of electromagnetic radiation, the fractal regime is a definite range in the photon momentum-transfer q, which is characterized by the scaling-law behavior of the structure factor: S(q)∝1/qdf. This allows a straightforward estimation of the fractal dimension df of aggregates in Small-Angle X-ray Scattering (SAXS) experiments. However, this behavior is not commonly studied in optical scattering experiments because of the lack of information on its domain of validity. In the present work, we propose a definition of the multiple-scattering structure factor, which naturally generalizes the single-scattering function S(q). We show that the mean-field theory of electromagnetic scattering provides an explicit condition to interpret the significance of multiple scattering. In this paper, we investigate and discuss electromagnetic scattering by three classes of fractal aggregates. The results obtained from the TMatrix method show that the fractal scaling range is divided into two domains: (1) a genuine fractal regime, which is robust; (2) a possible anomalous scaling regime, S(q)∝1/qδ, with exponent δ independent of df, and related to the way the scattering mechanism uses the local morphology of the scatterer. The recognition, and an analysis, of the latter domain is of importance because it may result in significant reduction of the fractal regime, and brings into question the proper mechanism in the build-up of multiple-scattering.

An extended fractal growth regime in the diffusion limited aggregation including edge diffusion

We have investigated on-lattice diffusion limited aggregation (DLA) involving edge diffusion and compared the results with the standard DLA model. For both cases, we observe the existence of a crossover from the fractal to the compact regime as a function of sticking coefficient. However, our modified DLA model including edge diffusion shows an extended fractal growth regime like an earlier theoretical result using realistic growth models and physical parameters [Zhang et al., Phys. Rev. Lett. 73 (1994) 1829]. While the results of Zhang et al. showed the existence of the extended fractal growth regime only on triangular but not on square lattices, we find its existence on the square lattice. There is experimental evidence of this growth regime on a square lattice. The standard DLA model cannot characterize fractal morphology as the fractal dimension (Hausdorff dimension, DH) is insensitive to morphology. It also predicts DH = DP (the perimeter dimension). For the usual fractal structures, observed in growth experiments on surfaces, the perimeter dimension can differ significantly (DH ≠ DP) depending on the morphology. Our modified DLA model shows minor sensitivity to this difference.

Brightness versus roughness: a multiscale approach

A link between roughness and brightness is sought for brass specimens that were superfinished, sandblasted and brushed. Only the blasting conditions are varied in order to get different roughness and brightness. First, a relation between roughness and brightness is sought for specimens that were superfinished and sandblasted. The best relation is obtained using the mean height of the motifs, calculated using a low-pass filter and cut-off length equal to 30 μm, with a logarithmic–logarithmic model. Then, the same type of relation is determined after superfinishing sandblasting and brushing. The core material volume Vmc, computed using a high-pass filter with a cut-off length of 60 μm and a linear–logarithmic relationship, gives the best results. A relation between roughness and brightness that is common to both the pre-brushing state and post-brushing state is identified: the best roughness parameter is the arithmetic mean deviation Sa using a high-pass filter with a cut-off of 15 μm, with a logarithmic–logarithmic relationship. Finally, it is shown that the use of these filtering conditions enables us to verify the model of Beckmann and Spizzichino for the examined specimens. This scale corresponds to the end of the fractal regime and is close to the end of the signal correlation.

Effects of substrate on cracking patterns and dynamics in desiccating clay layers

AbstractDesiccation cracking is ubiquitous in many materials of practical importance, such as in mud, clay soil, pavement, and concrete, and understanding its dynamics and formation is essential for developing an effective means of its prevention or utilization. We report on a phenomenological investigation aimed at delineating the effect of a coarse‐textured substrate on the cracking dynamics and morphology of an overlying kaolinite clay layer. Drying experiments were carried out using a glass container mounted on a digital balance, packed with a thin layer of kaolinite clay overlying seven types of sand that differed in their particle size distributions. An automatic imaging system recorded the dynamics of cracking at the evaporating surface. Dynamic analysis suggests a decreased duration of cracking with increased substrate particle size and substrate‐independent initial crack propagation. Analysis of the cracking morphology indicates a decreased crack density and larger crack length, and an increased crack width mean and standard deviation with increased substrate particle size. Scanning electron microscopy results, used to study the extent of cracking over various length scales, indicate that only macroscopic cracks formed on the kaolinite clay surface. Fractal and density correlation function analysis of the final crack networks indicate the dependency of the fractal dimension on the substrate particle size, and a crossover length scale ξ that separates the fractal regime from the uniform crack density regime. For length scales greater than ξ, the density correlation function asymptotically approaches the crack density at the surface of the clay.

Monte Carlo simulations of enzymatic reactions in crowded media. Effect of the enzyme-obstacle relative size

We perform Monte Carlo simulations in three-dimensional (3D) lattice in order to study diffusion-controlled and mixed activation-diffusion reactions following an irreversible Michaelis–Menten scheme in crowded media. The simulation data reveal the rate coefficient dependence on time for diffusion-controlled bimolecular reactions developing in three-dimensional media with obstacles, as predicted by fractal kinetics approach. For the cases of mixed activation-diffusion reactions, the fractality of the reaction decreases as the activation control increases. We propose a modified form of the Zipf–Mandelbrot equation to describe the time dependence of the rate coefficient, k(t)=k0(1+t/τ)−h. This equation provides a good description of the fractal regime and it may be split into two terms: one that corresponds to the initial rate constant (k0) and the other one correlated with the kinetics fractality. Additionally, the proposed equation contains and links two limit expressions corresponding to short and large periods of time: k1=k0 (for t≪τ) that relates to classical kinetics and the well-known Kopelman’s equation k∼t−h (for t≫τ) associated to fractal kinetics. The τ parameter has the meaning of a crossover time between these two limiting behaviours. The value of k0 is mainly dependent on the excluded volume and the enzyme-obstacle relative size. This dependence can be explained in terms of the radius of an average confined volume that every enzyme molecule feels, and correlates very well with the crossover length obtained in previous studies of enzyme diffusion in crowding media.

Small-angle scattering from three-phase systems: Investigation of the crossover between mass fractal regimes

In this paper, we construct a three-phase model (that is, a system consisting of three homogeneous regions with various scattering length densities), which illustrate the behavior of small-angle scattering (SAS) scattering curves. Here two phases are a deterministic fractal embedded in another deterministic mass fractal, and they altogether are further embedded in a third phase, which can be a solution or solid matrix. We calculate SAS intensities, derive expressions for the crossover position (that is, the point where the power-law scattering exponent changes) as a function of control parameters, including size, concentration, and volumes of each phase. The corresponding SAS intensities from these models describe a succession of power-law regimes in momentum space where both regimes correspond to mass fractals. The models can be applied to SAS data where the absolute value of the scattering exponent of the first power-law regime is higher than that of the subsequent second power-law regime, that is, the scattering curve of "convex" kind near the crossover position.

Two Fractal Regimes of the Soil Hydraulic Properties

A fractal analysis of the soil retention and hydraulic conductivity curves is presented. The retention process is modeled by a two fractal regimes: one pertaining to high water content values, and another accounting for the low water content data. This significantly improves the physical insight of the retention process as compared with the case of one-fractal models. The fractal dimensions characterizing the two regimes are estimated by fitting the retention curve model upon real data, and subsequently they are used to determine the hydraulic conductivity which for the retention curve models of Mualem and Burdine, is obtained in closed form. The reliability of the model is tested against independent conductivity data collected in a field-scale campaign.