Statistical Self-similarity Research Articles

Exploring anisotropic mechanical properties of lobster claw exoskeleton through fractal models

The outstanding mechanical properties of lobster claw exoskeletons are intricately tied to their internal microstructure. Investigating this relationship can offer vital insights for designing high-performance additive manufacturing structures. Fractal theory, with its fractional dimensional perspective, suits the complexity of real-world phenomena. Our study examines fully hydrated lobster claw exoskeletons using a multifaceted approach: four-point bending tests, scanning electron microscopy observations, and fractal models. Test results reveal superior mechanical properties in longitudinal specimens. Scanning electron microscopy shows non-uniform fiber helical structures and porous elements in the exoskeleton. Fracture mechanisms involve both breaking fiber fragments perpendicular to the cross-section and tearing between these fragments. The observed crack propagation paths exhibit statistical self-similarity. Consequently, we develop fractal models for the crack propagation paths in longitudinal and transverse specimens, calculating crack extension forces. Using the box-counting method and its improved variant, we determine the fractal dimensions of specimen sections. The fractal dimension of longitudinal models exceeds that of transverse models, and calculated crack extension forces are higher in longitudinal models. These findings align well with experimental data, demonstrating fractal theory's efficacy in analyzing the lobster claw exoskeleton's anisotropic mechanical properties.

Growth kinetics and morphology characterization of binary polymeric fluid under random photo-illumination.

We present a comprehensive study using dissipative particle dynamics simulations to investigate phase separation kinetics (PSK) in three-dimensional (3d) polymeric fluids under random photo-illumination. We consider two scenarios: polymer blends with active radicals at one end of each immiscible chain and block copolymer (BCP) melts with photosensitive bonds linking incompatible blocks. The phase separation (PS) is induced by temperature quench of the initial homogeneously mixed system. Simultaneously, the system experiences random photo-illumination, simulated by two concurrent random events: (a) the recombination of active radicals in polymer blends and (b) the breaking of photosensitive bonds in BCP chains. Variations in the bond-breaking probability, Pb, mimic the change in light intensity. The length scale follows power law growth, R(t) ∼ tϕ, where ϕ represents the growth exponent. Increasing Pb results in a gradual transition in growth kinetics from micro-PS to macro-PS, accompanied by corresponding transition probabilities for both systems. Micro-PSK dominates the evolution process at low Pb values. The scaling functions exhibit data overlap for most scaled distances, indicating the statistical self-similarity of evolving patterns. Our study enhances the understanding of PSK in polymeric fluids, revealing the impact of photosensitive bonds and active radicals. Furthermore, it suggests the potential for designing novel polymeric materials with desired properties.

Fractal Analysis of the Cerebrovascular System Pathophysiology.

The cerebrovascular system is characterized by parameters such as arterial blood pressure (ABP), cerebral perfusion pressure (CPP), and cerebral blood flow velocity (CBFV). These are regulated by interconnected feedback loops resulting in a fluctuating and complex time course. They exhibit fractal characteristics such as (statistical) self-similarity and scale invariance which could be quantified by fractal measures. These include the coefficient of variation, the Hurst coefficient H, or the spectral exponent α in the time domain, as well as the spectral index ß in the frequency domain. Prior to quantification, the time series has to be classified as either stationary or nonstationary, which determines the appropriate fractal analysis and measure for a given signal class. CBFV was characterized as a nonstationary (fractal Brownian motion) signal with spectral index ß between 2.0 and 2.3. In the high-frequency range (>0.15Hz), CBFV variability is mainly determined by the periodic ABP variability induced by heartbeat and respiration. However, most of the spectral power of CBFV is contained in the low-frequency range (<0.15Hz), where cerebral autoregulation acts as a low-pass filter and where the fractal properties are found. Cerebral vasospasm, which is a complication of subarachnoid hemorrhage (SAH), is associated with an increase in ß denoting a less complex time course. A reduced fractal dimension of the retinal microvasculature has been observed in neurodegenerative disease and in stroke. According to the decomplexification theory of illness, such a diminished complexity could be explained by a restriction or even dropout of feedback loops caused by disease.

Study of nonlinear dynamic modes of native DNA via mathematical modeling methods

One of the main problems in the nonlinear dynamics of DNA is the identification of physical mechanisms responsible for the mechanical features of its behaviour. The formation and interaction of open states in the DNA structure may be one such mechanism. In this paper we propose a mathematical model of native DNA, which allows us to describe its thermodynamic and kinetic properties taking into account the collective behaviour of the ensemble of open states. The concept of a microscopic open state in DNA and its associated parameter, the displacement vector of nitrogenous bases, are introduced. By averaging the base displacement vectors over the ensemble of states, the thermodynamic variable is determined. According to the statistical model of DNA in the self-consistent field approximation, the structural parameter of the system thermalization is derived. This parameter reflects the statistical self-similarity in the behaviour of the ensemble of open states. Regularities of "criticality" for different ranges of the structural parameter are established and phenomenological representations of the free energy in the framework of the Ginzburg-Landau approach are proposed. Numerical modelling of the dynamics of native DNA is carried out for different ranges of the structural parameter, which has allowed us to establish the types of self-similar solutions and the corresponding open-state collective modes. It is shown that the latter have the nature of finite-amplitude fluctuations in the form of breahters and autosoliton modes, as well as blow-up dissipative structures. The sensitivity of the model to initial conditions is investigated, and the influence of nonlinearity inherent in the model on the modelling results is established. The limitations and prospects of using the approach proposed in this work for mathematical modelling of native DNA are discussed.

Spherical seepage model of Bingham fluid in rough and low-permeability porous media

Based on the microstructure of porous media that exhibits statistical self-similarity fractal features, this paper investigates the radial flow characteristics of non-Newtonian fluids within rough porous media. The analytical equation of permeability and starting pressure gradient of Bingham fluid in low permeability rough porous media are established. It is found that the relative roughness is inversely proportional to the permeability and proportional to the starting pressure gradient. In addition, it is also found that the permeability of low permeability porous media decreases spherically with the increase of radial distance and curvature fractal dimension, and increases with the increase of pore area fractal dimension and porosity. Furthermore, the staring pressure gradient is directly proportional to the radial distance, yield stress and curvature fractal dimension. By comparing the model in this paper with the existing experimental data, the correctness and rationality of the spherical seepage fractal model are effectively verified.

Revealing the fractal and self-similarity of realistic collective human mobility

Revealing the fractal scaling laws of collective human mobility is benefit to understand the complexity of collective human mobility behaviours and has broad application prospects, such as infectious disease prevention, because fractal networks are potentially more robust against targeted attacks than non-fractal networks. Although many statistical indicators of individual human mobility had been observed in power-law distributions, there is still no evidence so far to verify the fractal or self-similarity of collective human mobility behaviours. This study uncovered the fractal and self-similarity properties on realistic weighted human mobility networks at world-, nation-, and city-wide scales. After observing the necessary conditions of network fractal, i.e., the evident power-law probability distributions of weight-related metrics, the fractal dimensions of weighted human mobility networks were measured to be about 0.67∼0.88. Furthermore, experimental results demonstrated their distinct statistical self-similarity and approximately structural self-similarity. These exciting findings can enhance the fire-new understandings of collective human mobility behaviours, and provide novel solutions for many real-world applications.

Simply solvable model capturing the approach to statistical self-similarity for the diffusive coarsening of bubbles, droplets, and grains.

Aqueous foams and a wide range of related systems are believed to coarsen by diffusion between neighboring domains into a statistically self-similar scaling state, after the decay of initial transients, such that dimensionless domain size and shape distributions become time independent and the average grows as a power law. Partial integrodifferential equationsfor the time evolution of the size distribution for such phase separating systems can be formulated for arbitrary initial conditions, but these are cumbersome for analyzing data on nonscaling state preparations. Here we show that essential features of the approach to the scaling state are captured by an exactly-solvable ordinary differential equationfor the evolution of the average bubble size. The key ingredient is to characterize the bubble size distribution approximately, using the average size of all bubbles and the average size of the critical bubbles, which instantaneously neither grow nor shrink. The difference between these two averages serves as a proxy for the width of the size distribution. Solution of our model shows that behavior is controlled by a signed length δ that is proportional to the width of the initial distribution relative to that in the scaling state. In particular, δ is negative if the initial preparation is too monodisperse, and is positive if it is too polydisperse. To test our approach, we compare with data for quasi-two dimensional dry foams created with three different initial amounts of polydispersity. This allows us to readily identify the critical radius from the average area of six-sided bubbles, whose growth rate is zero by the von Neumann law. The growth of the average and critical radii agree quite well with exact solution, though the most monodisperse sample crosses over to the scaling state faster than expected. A simpler approximate solution of our model performs equally well. Our approach is applicable to 3d foams, which we demonstrate by re-analyzing prior data, as well as to froths of dilute droplets and to phase separation kinetics for more general systems such as emulsions, binary mixtures, and alloys.

Statistical self-similarity of detrital zircon U-Pb ages reveals sedimentary provenance: A case study from the Chinese Loess Plateau

A major concern of the geoscience community is often where sediments originally come from — this is a material's provenance. Tracing provenance allows us to determine how a material moved from its source to its sink, and thereby to evaluate the tectonic and paleoclimatic history. Detrital zircon UPb geochronology has long been used for this purpose. However, current approaches to measuring (dis)similarity in zircon age distributions may lead to erroneous geologic interpretations. Here we propose an Age-Number fractal model to characterize zircon age densities and perform intersample comparisons using multidimensional scaling. The advantage of the new method is that compared to such traditional methods as Kolmogorov-Smirnov test, it accentuates holistic rather than partial differences between two age distributions and is less-affected by data biases related to grain-size sorting and uneven sample sizes. We then use this model to unravel the debatable provenance of the Chinese Loess Plateau (CLP) by comparing the zircon ages of Quaternary loess on the CLP with that of loose sediments from potential source regions. The model divides the CLP into four distinct provenance zones, receiving materials primarily from the northeastern Tibetan Plateau, upper Yellow River, and/or Mu Us Desert. Our observations corroborate that the aeolian (East Asian monsoon and Westerlies) and fluvial (Yellow River) processes play a joint role in the dust deposition. A new pattern of spatial heterogeneity in the loess provenance is established and can be interpreted as a combined effect of atmospheric circulation patterns, availability of riverine sediments, and such factors as aridification, implying a more dynamically-evolved CLP than previous thought. Our results will aid future development and interpretation of the dust-based paleoclimatic proxies.

Statistical self-similarity in Lozi and Hénon's strange attractors

Non-classical characterizations of strange attractors assign occupancy times and numbers to each of the hypercubes that cover them, local information, and generate statistical distributions over such quantities, global information, at the expense of losing all the local ones. The confrontation between local and global information would allow statistical quantification, important for the characterization of the self-similarity of the involved attractor: correspondence between global and local properties are expected as manifestations of the self-similarity that characterizes that fractal set (statistical self-similarity). Hence, the importance of clarifying what the local distributions would be. The core of this work is the presentation of statistical distributions of the mentioned times and numbers, and quantities derived from them, linked to each hypercube of a set that corresponds to a small fraction of their total, that confirm the correspondence between local and global statistical properties.

Box dimension of the border of Kingdom of Saudi Arabia

Fractal dimension unlike topological dimension is (usually) a non-integer number which measures complexity, roughness, or irregularity of an object with respect to the space in which the set lies. It is used to characterize highly irregular objects in nature containing statistical self-similarity such as mountains, snowflakes, clouds, coastlines, borders etc.In this article, box dimension (a version of fractal dimension) of the border of Kingdom of Saudi Arabia (KSA) is computed using a multicore parallel processing algorithm based on the classical box-counting method. A power law relation is obtained from numerical simulations which relates the length of the border with the scale size and provides a very close estimate of the actual length of the KSA border within the scaling regions and scaling effects on the length of KSA border are considered. The algorithm presented in the article is shown to be highly scalable and efficient and the speedup of the algorithm is computed using Amdahl's and Gustafson's laws. For simulations, a high performance parallel computer is employed using Python codes and QGIS software.

Application of Smart Grid Communication Service Flow Modeling Based on Poisson Model in Grid Operation

The early Poisson distribution and Markov autoregressive models can no longer reflect the characteristics of smart grid business flow. The long correlation characteristics of burst flow are simulated by the heavy tail ON/OFF model with multi-source convergence, while the Poisson model with short correlation and no memory is used for random flow. The simulation results show that the synthesized flow is still has a certain statistical self-similarity, and the coefficient H decreases after synthesis.That the long correlation and burst of self-similar business flows will not be smoothed by the convergence and synthesis of the self-similar business flows with short correlation services. In the actual operation process, the station-level power grid will result in load aggravation and significantly affect the network performance when it acts in response to faults. Therefore, it is necessary to consider the changes of network performance under different burst degrees.

Self-similar geometries within the inertial subrange of scales in boundary layer turbulence

The inertial subrange of turbulent scales is commonly reflected by a power law signature in ensemble statistics such as the energy spectrum and structure functions – both in theory and from observations. Despite promising findings on the topic of fractal geometries in turbulence, there is no accepted image for the physical flow features corresponding to this statistical signature in the inertial subrange. The present study uses boundary layer turbulence measurements to evaluate the self-similar geometric properties of velocity isosurfaces and investigate their influence on statistics for the velocity signal. The fractal dimension of streamwise velocity isosurfaces, indicating statistical self-similarity in the size of ‘wrinkles’ along each isosurface, is shown to be constant only within the inertial subrange of scales. For the transition between the inertial subrange and production range, it is inferred that the largest wrinkles become increasingly confined by the overall size of large-scale coherent velocity regions such as uniform momentum zones. The self-similarity of isosurfaces yields power-law trends in subsequent one-dimensional statistics. For instance, the theoretical 2/3 power-law exponent for the structure function can be recovered by considering the collective behaviour of numerous isosurface level sets. The results suggest that the physical presence of inertial subrange eddies is manifested in the self-similar wrinkles of isosurfaces.

Image Recognition–Based Identification of Multifractal Features of Faults

Geologists have made several advances in applying multifractal theory in geology; however, some questions such as a large statistical workload and low efficiency remain unanswered. Thus, this study proposes an image recognition–based method for calculating fault multifractality. First, grayscale processing and binarization of the fault distribution map were performed. The image was then gridded, and the grids were numbered. Subsequently, computer image recognition technology was used to count the number of faults in each grid as a replacement for manual counting. Finally, the fractal dimensions of the faults were calculated using a multifractal box-counting algorithm. This method was successfully applied to fracture studies of the Maokou Formation in southeast Sichuan. Compared to the conventional approach, the proposed method demonstrated considerably improved work efficiency and accuracy. The results showed that the faults in the study area exhibited good statistical self-similarity in the scale range, indicating fractal characteristics. The fractal dimensions of faults with different orientations and the planar distribution of the fractal dimension contours indicate tectonic stages and stress magnitude in the study area. The results indicate that the tectonic setting of southeast Sichuan was formed primarily during the Indosinian, Yanshanian, and Himalayan periods. From the Indosinian to the early Yanshanian periods, NE-trending faults with relatively large fractal dimensions developed under NW–SE compressional tectonic stress. From the Late Yanshanian to Early Himalayan, EW-trending faults were formed by relatively weak N–S compressional stress and had the lowest fractal dimensions. The NW-trending faults formed by intense NE–SW compressional tectonic stress in the Late Himalayan region had the highest fractal dimensions. To promote oil and gas migration and ensure that faults do not destroy the caprock, oil and gas reservoirs must be in a relatively mild tectonic environment. Thus, the fractal dimensions of faults in favorable areas should be neither too high nor too low. The relationship between the fractal dimensions of faults and well test results in southeast Sichuan indicates that the region along the wells “ls1–xia14–guan3” (with fractal dimensions of 1.49–1.57) in the study area is a relatively favorable region for oil and gas preservation.

Fractal features of dual temperature/pH-sensitive poly(N-isopropylacrylamide-co-acrylic acid) hydrogels and resultant effects on the controlled drug delivery performances

Using acrylic acid (AA) as pH-responsive monomer, and N-isopropyl acrylamide (NIPAM) as temperature-responsive monomer, the stimuli-responsive P(NIPAM-co-AA) copolymers were prepared via soap-free emulsion polymerization. Their constructed porous structures, swelling-shrinking behaviors, and controlled drug delivery performances were systematically demonstrated via various characterizations. Particularly, the small-angle X-ray scattering (SAXS) method was employed to elucidate their fractal features and the structural evolutions with increasing AA content, showing the occurrences from dense aggregates (mass fractal (Dm) value of 2.92) to loose networks (surface fractal (Ds) value of 2.18) with statistical self-similarity. Meanwhile, the ibuprofen (IBU) was used as a model drug, their temperature/pH-dual sensitive delivery and related mechanism were discussed on the basis of the fractal demonstrations. The results showed that the cross-linking networks with an interconnected open-cell skeleton and abundant large pores were beneficial to improve the adsorption-diffusion performances of the drug delivery, while the drug-releasing kinetics of P(NIPAM-co-AA)-30 followed a non-Fickian diffusion mechanism. Specially, the fractal evolutions of the obtained copolymers along with the drug loading and releasing behaviors were evaluated via SAXS patterns. In which, the drug-loaded copolymers presented the Ds characteristics with the increased amounts of monomer AA added. While, the fractal features varied from Ds value of 2.40 to Dm value of 2.76 along with sustained drug releasing in pH 2.0 at 25 °C, indicating that the structural transformation of the used copolymers occurred from dense (Dm value of 2.86) to loose (Dm value of 2.76) with the extension of release time. These demonstrations suggested that the resultant P(NIPAM-co-AA) hydrogel is a potential intelligent-responsive drug carrier for targeted delivery.

Shell model intermittency is the hidden self-similarity

We show that the intermittent dynamics observed in the inertial interval of Sabra shell model of turbulence can be rigorously related to the property of scaling self-similarity. In this connection, the space-time scaling symmetries (like in the K41 theory) are replaced by the new hidden scaling symmetry, which is an exact symmetry of inviscid dynamics represented in special rescaled coordinates and times. We derive the formulas expressing anomalous scaling exponents in terms of Perron-Frobenius eigenvalues of linear operators based on the self-similar statistics. Theoretical conclusions are verified by extensive numerical simulations.

Experimental Study on SHPB Dynamic Mechanical Response in Different Damage Zones of Rock under Blast Load

In this study, the damage characteristics of the slit charge pack along the rock slitting and the vertical slitting directions were investigated using green sandstone as the subject. Split-Hopkinson pressure bar (SHPB) experiments were performed on green sandstone, and the dynamic mechanical properties of rock specimen were analyzed in the slitting and vertical slitting directions. Based on the SHPB experiments of the rock specimens in different damage zones along the rock slitting and the vertical slitting directions combined with the fractal theory, the distribution pattern of rock fracture fragmentation was investigated at different positions in the kerf direction and vertical kerf direction. The results of this study show that the dynamic strength of the rock corresponds to the damage degree of the rock, which is greater in the direction of the slit than that in the perpendicular direction of the slit. In addition, the core at 150 mm distance from the blast hole perpendicular to the slit direction could easily bear the blast loads. The lumpiness distribution of rock specimens under impact load shows good statistical self-similarity. The evolution law of fractal dimension shows that with increasing distance from the core to the blast hole, the fractal dimension of the specimen after failure increases. The fractal dimension of the specimen perpendicular to the slit direction is greater than that of the specimen along the slit direction at the same distance.

Fractal dimension and area of seismicity in the Baikal Rift System: Implications for modern geodynamics

The fractal geometry and extent of seismicity in the Baikal Rift System (BRS) are estimated from data on 52,700 instrumental events of MLH ≥ 2.5 magnitudes for fifty years (1964–2013). The seismic pattern is characterized by the box-counting Hausdorff dimension D0, multifractal spectra f(α), and surface area S of seismicity at three scales: the rift system as a whole, its three zones, and six subzones. The multifractal spectra record a self-similar hierarchical structure of the BRS seismicity pattern. The space and time variations in the fractal dimension (D0) and area of seismicity (S), which are mapped and plotted as a function of time, show good correlation. The two parameters depend on three related factors: progressive increase in the amount of instrumental data (dataset size), structure of seismogenic fault network, and geodynamic activity. They increase as ever more data appear with time and acquire high local values at increasing extent and density of quakes. Moreover, the obtained D0 estimates reflect statistical self-similarity of earthquake patterns being in the range ≈ 1.45–1.55 over most of BRS, except one zone and one subzone in the rift flanks. They are the highest in the southwest and the lowest in the northeast of the rift system (D0 ≈ 1.60 ± 0.02 and D0 ≈ 1.37 ± 0.02 respectively). This dissimilarity indicates that seismogenic faulting occurs by different mechanisms: distributed failure as a result of superposed global-scale collisional compression and regional rifting in the SW flank and quasi-linear rift propagation in the NE flank. In general, D0 decreases toward the northeastern part of the BRS, where the pattern of earthquakes becomes localized along lineaments instead of being distributed over an area. The space and time variations of D0 and S revealed in the earthquake data are consistent with the location and activity pulses of rifting attractors and provide a realistic explanation of BRS geodynamics and tectonophysics. The global lithospheric compression and the regional pulse-like activity of rifting attractors control the network of seismogenic faults which, in turn, govern the fractal geometry and 2D structure of seismicity in the region. The obtained results confirm the oscillatory dynamics of the regional seismicity at a decadal period correlated with activity pulses of rifting attractors. The oscillations stand out against the background of decreasing global low-frequency secular cycle of the BRS seismicity. The BRS lithospheric geodynamics fits the model of a nonlinear oscillator with dissipation. The suggested analysis of the fractal geometry and extent of seismicity as proxies of the faulting evolution provides insights into modern geodynamics of the Baikal Rift System and its constituents.

Quantifying patterns in art and nature

Many different types of artworks mimic the properties of natural fractal patterns – in particular, statistical self-similarity at different scales. Here, we describe examples of abstract art created by us and well-known artists such as Ruth Asawa and Sam Francis that evoke the repetition and variability of biological forms. We review the ‘drip’ paintings of Jackson Pollock that display statistical self-similarity at varying scales, and discuss studies that measured the fractal dimension of Pollock’s drip paintings. The contemporary environmental artist Edward Burtynsky who captures aerial photographs of man-created and man-altered landscapes that resemble natural patterns is also discussed. We measure fractal dimension and a second shape parameter – fractional concavity – for borders in three of Burtynsky’s photographs of man-made landscapes and of biological tissues that resemble his compositions. This specifies the complexity of patterns in Burtynsky’s photographs of diverse man-impacted landscapes and underscores their similarity to fractal patterns found in nature. Graphical : Log Booms # 1. Photograph © Edward Burtynsky, courtesy Robert Koch Gallery, San Francisco / Nicholas Metivier Gallery, Toronto.

On small-scale and large-scale intermittency of Lagrangian statistics in canopy flow

Abstract

Experimental and Fractal Characterization of the Microstructure of Shales from Sichuan Basin, China

Comprehensive characterization and analysis of the microstructure characteristics of shale are significant for understanding its transport mechanisms and determining highly efficient production programs. In this study, composition analysis, scanning electron microscopy imaging, and gas adsorption–desorption measurements are first conducted on 29 marine shale samples from the Longmaxi Formation, Sichuan Basin, China, to obtain their petrophysical properties and microstructure characteristics. Then, surface fractal dimension, pore fractal dimension, lacunarity, and succolarity are introduced to analyze the microstructure of these shale samples. Finally, the correlations between total organic carbon (TOC) content and microstructures and the relationships among the microstructures are analyzed systematically. The results show that the complex pore network in these shales is constituted by microfractures, intergranular pores, intragranular pores, dissolved pores, and organic pores. The typical type IV adsorption isotherm and type H3 hysteresis loop are identified from nitrogen adsorption–desorption data. TOC content has positive influences on the adsorption capacity and nanopore distribution. The surface fractal dimensions under low and high relative pressure are similar in these marine shale samples, which is different from the two-section characteristic of continental and marine-continental shale. The statistical self-similarity characteristic of the pore-size distribution in these shales is further confirmed based on the analytical fractal dimension equation. The normalized lacunarity can quantitatively characterize the heterogeneity of the pore-size distribution of these shales in two-dimensional space. The succolarity analysis in two-dimensional space explains the low fluid transport capacity of these shales. Universal and logically explained correlations are found between TOC content and microstructures. Moreover, logically explained correlations also exist among the microstructures of the shale samples evaluated.