Effective Fractal Dimension Research Articles

Quantitative analysis of pore-size influence on granite residual soil permeability using CT scanning

In the field of soil hydrology, pore structure is a key factor affecting soil permeability. This paper uses micro-CT scanning technology to visualize and quantitatively analyze the impact of different pore sizes on the permeability of granite residual soil through scanning, image processing, pore removal, and permeability simulation. The results show that granite residual soil samples have a well-developed connected pore structure, with connected pore diameters ranging from 25-400 μm, while isolated pores are mostly smaller than 50 μm. Medium-sized pores bridge small and large pores, forming an important component of pore connectivity. Pore size significantly affects permeability; the removal of medium-sized pores greatly reduces the connectivity, fractal dimension, and permeability of the remaining pores, indicating that pores in the 100–300 μm range contribute the most to permeability. In contrast, pores smaller than 50 μm and larger than 300 μm have a minimal impact on overall permeability. Effective porosity and fractal dimension are crucial parameters affecting permeability, with a significant linear correlation between the two. This study supply crucial data that delineates the relationship between the structure of soil pores and their permeability, which is invaluable for developing improvement strategies aimed at reducing soil permeability.

Coastlines violate the Schramm–Loewner Evolution

Mandelbrot’s empirical observation that the coast of Britain is fractal has been confirmed by many authors, but it can be described by the Schramm–Loewner Evolution? Since the self-affine surface of our planet has a positive Hurst exponent, one would not expect a priori any critical behavior. Here, we investigate numerically the roughness and fractal dimension of the isoheight lines of real and artificial landscapes. Using a novel algorithm to take into account overhangs, we find that the roughness exponent of isoheight lines is consistent with unity regardless of the Hurst exponent of the rough surface. Moreover, the effective fractal dimension of the iso-height lines decays linearly with the Hurst exponent of the surface. We perform several tests to verify if the complete and accessible perimeters would follow the Schramm–Loewner Evolution and find that the left passage probability test is clearly violated, implying that coastlines violate SLE.

Fractional cosmic strings

Topological defects are investigated in the framework of quantum gravity models based on the hypothesis of an effective fractal dimension of the Universe. From a minimal coupling procedure, the tools of fractional calculus are used to determine the geometry associated to a fractional cosmic string. Several results for the propagation of light are discussed, notably the light-deviation angle due to the defect and the geodesics of light.

Lattice Boltzmann modeling of the effective thermal conductivity in plant fiber porous media generated by Quartet Structure Generation Set

This study proposes a novel approach that combines a Quartet Structure Generation Set (QSGS) algorithm with the Lattice Boltzmann Method (LBM) to investigate effective thermal conductivity in plant fiber materials. The QSGS algorithm is utilized to construct a two-dimensional digital structure of the porous media, enabling the determination of effective thermal conductivity through LBM simulations. The simulation results demonstrate good agreement with experimental data, exhibiting a maximum relative error of only 1.7%. The study examines the influence of key factors on effective thermal conductivity, including temperature, porosity, fiber distribution, and fractal dimension. The effective thermal conductivity increased by approximately 6.1% for each 10 °C increase in temperature between 10 °C and 50 °C. Moreover, the effective thermal conductivity showed an average decrease of 16.3% with each 10% increment in porosity. The increase in thermal conductivity becomes more significant with increasing temperature and decreasing porosity. Conversely, an increased fractal dimension correlates with a decline in effective thermal conductivity. The effective thermal conductivity is influenced synergistically by the directional growth probability of the solid phase and the direction of heat flow. The findings offer valuable insights into thermal conductivity behavior and heat transfer mechanisms in plant fiber porous media.

Epidemic propagation risk study with effective fractal dimension.

In this article, the risk of epidemic transmission on complex networks is studied from the perspective of effective fractal dimension. First, we introduce the method of calculating the effective fractal dimension of the network by taking a scale-free network as an example. Second, we propose the construction method of administrative fractal network and calculate the . using the classical susceptible exposed infectious removed (SEIR) infectious disease model, we simulate the virus propagation process on the administrative fractal network. The results show that the larger the is, the higher the risk of virus transmission is. Later, we proposed five parameters P, M, B, F, and D, where P denotes population mobility, M denotes geographical distance, B denotes GDP, F denotes , and D denotes population density. The new epidemic growth index formula was obtained by combining these five parameters, and the validity of I in epidemic transmission risk assessment was demonstrated by parameter sensitivity analysis and reliability analysis. Finally, we also confirmed the reliability of the SEIR dynamic transmission model in simulating early COVID-19 transmission trends and the ability of timely quarantine measures to effectively control the spread of the epidemic.

Accessibility of the surface fractal dimension during film growth.

Fractal properties on self-affine surfaces of films growing under nonequilibrium conditions are important in understanding the corresponding universality class. However, measurement of the surface fractal dimension has been intensively investigated and is still very problematic. In this work, we report the behavior of the effective fractal dimension in the context of film growth involving lattice models believed to belong to the Kardar-Parisi-Zhang (KPZ) universality class. Our results, which are presented for growth in a d-dimensional substrate (d=1,2) and use the three-point sinuosity (TPS) method, show universal scaling of the measure M, which is defined in terms of discretization of the Laplacian operator applied to the height of the film surface, M=t^{δ}g[Θ], where t is the time, g[Θ] is a scale function, δ=2β,Θ≡τt^{-1/z},β, and z are the KPZ growth and dynamical exponents, respectively, and τ is a spatial scale length used to compute M. Importantly, we show that the effective fractal dimensions are consistent with the expected KPZ dimensions for d=1,2, if Θ≲0.3, which include a thin film regime for the extraction of the fractal dimension. This establishes the scale limits in which the TPS method can be used to accurately extract effective fractal dimensions that are consistent with those expected for the corresponding universality class. As a consequence, for the steady state, which is inaccessible to experimentalists studying film growth, the TPS method provided effective fractal dimension consistent with the KPZ ones for almost all possible τ, i.e., 1≲τ<L/2, where L is the lateral size of the substrate on which the deposit is grown. In the growth of thin films, the true fractal dimension can be observed in a narrow range of τ, the upper limit of which is of the same order of magnitude as the correlation length of the surface, indicating the limits of self-affinity of a surface in an experimentally accessible regime. This upper limit was comparatively lower for the Higuchi method or the height-difference correlation function. Scaling corrections for the measure M and the height-difference correlation function are studied analytically and compared for the Edwards-Wilkinson class at d=1, yielding similar accuracy for both methods. Importantly, we extend our discussion to a model representing diffusion-dominated growth of films and find that the TPS method achieves the corresponding fractal dimension only at steady state and in a narrow range of the scale length, compared to that found for the KPZ class.

Fractal Fluctuations at Mixed-Order Transitions in Interdependent Networks.

We study the critical features of the order parameter's fluctuations near the threshold of mixed-order phase transitions in randomly interdependent spatial networks. Remarkably, we find that although the structure of the order parameter is not scale invariant, its fluctuations are fractal up to a well-defined correlation length ξ^{'} that diverges when approaching the mixed-order transition threshold. We characterize the self-similar nature of these critical fluctuations through their effective fractal dimension d_{f}^{'}=3d/4, and correlation length exponent ν^{'}=2/d, where d is the dimension of the system. By analyzing percolation and magnetization, we demonstrate that d_{f}^{'} and ν^{'} are the same for both, i.e., independent of the symmetry of the process for any d of the underlying networks.

Effective Fractal Dimension at 2d-3d Crossover

This article is aimed at reviewing and studying the effects of the 2d-3d crossover on the effective fractal and spatial dimensions, as well as on the critical exponents of the physical properties of bulk and bounded systems at criticality. Here we consider the following problems: (1) the two types of dimensional crossovers and the concept of the universality classes; (2) a smooth 2d-3d crossover and the calculation of the effective fractal and spatial dimensions, as well as the effective critical indices; (3) the fractal dimension, its connection with the random mean square order-parameter fluctuations and a new phase formation; (4) the fractal nuclei of a new phase and the medical consequences of carcinogenesis and nucleation isomorphism.

EVERAL NOTES ON THE LATTICE THERMALCONDUCTIVITY OF FRACTAL-SHAPED NANOPARTICLES

Using the additive technologies in the production of nanoparticle-fabricated three-dimensional materials has become one of the most promising ways of obtaining effective and low-cost thermoelectric energy converters. Nanostructuring provides a route to modifying selectively the transport properties which determine the materials thermoelectric efficiency. In this paper, we have shown one more effect which consists in a significant dependence of the contribution of lattice vibrations to the thermal conductivity coefficient of a nanoparticle (its reducing is required in practice) on its morphology for nanoparticles of a pure substance. The particle morphology has been specified by the values of its effective diameter, fractal dimension and surface roughness.Using nanoparticles of pure bismuth at low temperatures as an example, we have demonstrated a notable decrease in the lattice thermal conductivity in “complicating” the particle morphology. In the final section, we have presented methods of calculating characteristics of nanoparticle ensembles, the methodology of measuring the fractal dimension experimentally also being discussed.

Unveiling the connection between the global roughness exponent and interface fractal dimension in EW and KPZ lattice models

A connection between the global roughness exponent and the fractal dimension of a rough interface, whose dynamics are expected to be described by stochastic continuum models, still needs more support from simulations in lattice models, which are key to provide completeness for the characterization of a given universality class. In this work, we investigate the asymptotic fractal dimension of interfaces that evolve according to some specific lattice models in d + 1 dimensions (d = 1, 2), which are expected to belong to the Edwards–Wilkinson or Kardar–Parisi–Zhang universality classes. Our results, based on the Higuchi method and on the extrapolation of the effective fractal dimension, allow one to achieve dependence between the asymptotic fractal dimension and global roughness exponent, in which the latter is expected to be hardly accessible for experimentalists. Conversely, we also use a two-points correlation function, which gives the time evolution of the local roughness exponent. As a byproduct, our results suggest that, for d = 1, the fractal dimension converges faster than the global roughness exponents to the asymptotic ones. Therefore, the analysis of the fractal dimension, for d = 1, is suggested to be more accessible than the global roughness exponents to determine the universality class. Corrections for the fractal dimensions in d = 2 were found to be stronger than for d = 1.

Power-law spectrum of energetic particles in classical thermal equilibrium by pitch-angle scattering process

The Boltzmann–Gibbs thermodynamic equilibrium state of charged particles pitch-angle scattered by weak plasma waves is discussed. Degrees of freedom of these waves play a fundamental role in constructing the grand canonical ensemble. Via the gyro-resonance condition, fast particles have an inverse break power-law spectrum for ɛ−μ≪T, where ɛ is the particle energy, μ is the chemical potential, T is the temperature. The break energies are the rest energy and −μ. For ɛ≪−μ≪T, the energy spectral index α is δ/2+1 and δ+1 for non- and ultra-relativistic particles, respectively, with δ an effective fractal dimension of background magnetic field lines. The spectral index for −μ≪ɛ≪T is α+1. This thermal equilibrium scenario, combined with the leaky-box model and cosmic-ray observations, seems to suggest that the Galactic magnetic field is super-diffusive with δ≈1.4.

The fractal characteristics of atmospheric coated soot: Implication for morphological analysis

In this work, we have numerically investigated the variation of fractal characteristics of soot particles as soot is coated. We found that the ideal fractal law can fit the morphologies of coated soot only when the radius of gyration (Rg) of coated soot is identical to that of the soot core. In that case, the fractal dimension (Df) of coated soot is close to that of the soot core, which can explain why the unchanged fractal dimension was observed by the previous study as soot was evenly coated. As the voids among monomers is commonly filled firstly, the Rg of coated soot obtained from the two-dimensional (2D) method would be close to that of the soot core, so the measured fractal dimension would be close to that of the soot core. Our results also show that the fractal parameters of coated soot at different size scales can be varied, so the ideal fractal law is difficult to fit the structures of unevenly coated soot universally. Besides, we also found that the effective fractal dimension can increase as soot is unevenly coated. The results of the Q-space analysis show that as soot particles are unevenly coated, the power-law exponent of the structure factor (S(q)) versus the scattering wave vector (q) tends to -4 for heavily coated soot. This means that heavily, unevenly coated soot exhibits the characteristics of nearly spherical particles.

Patterns formed by chains of magnetic beads

Magnetic beads attract each other forming rather stable chains. We consider such chains formed by magnetic beads and push them into a Hele-Shaw cell either from the boundary or from the center. When such a chain is pushed into a cavity, it bends and folds spontaneously forming interesting unreported patterns. These patterns are self-similar and an effective fractal dimension can be defined. As found experimentally and with numerical simulations, the numbers of beads, loops and contacts follow power laws as a function of packing fraction and, depending on the injection procedure, even energetically less favorable triangular configurations can be stabilized.

Correlation function: biasing and fractal properties of the cosmic web

Aims. Our goal is to determine how the spatial correlation function of galaxies describes biasing and fractal properties of the cosmic web. Methods. We calculated spatial correlation functions of galaxies, ξ(r), structure functions, g(r) = 1 + ξ(r), gradient functions, γ(r) = d log g(r)/d log r, and fractal dimension functions, D(r) = 3 + γ(r), using dark matter particles of the biased Λ cold dark matter (CDM) simulation, observed galaxies of the Sloan Digital Sky Survey (SDSS), and simulated galaxies of the Millennium and EAGLE simulations. We analysed how these functions describe fractal and biasing properties of the cosmic web. Results. The correlation functions of the biased ΛCDM model samples at small distances (particle and galaxy separations), r ≤ 2.25 h−1 Mpc, describe the distribution of matter inside dark matter halos. In real and simulated galaxy samples, only the brightest galaxies in clusters are visible, and the transition from clusters to filaments occurs at a distance r ≈ 0.8−1.5 h−1 Mpc. At larger separations, the correlation functions describe the distribution of matter and galaxies in the whole cosmic web. The effective fractal dimension of the cosmic web is a continuous function of the distance (separation). Real and simulated galaxies of low luminosity, Mr ≥ −19, have almost identical correlation lengths and amplitudes, indicating that dwarf galaxies are satellites of brighter galaxies, and do not form a smooth population in voids. Conclusions. The combination of several physical processes (e.g. the formation of halos along the caustics of particle trajectories and the phase synchronisation of density perturbations on various scales) transforms the initial random density field to the current highly non-random density field. Galaxy formation is suppressed in voids, which increases the amplitudes of correlation functions and power spectra of galaxies, and increases the large-scale bias parameter. The combined evidence leads to the large-scale bias parameter of L⋆ galaxies the value b⋆ = 1.85 ± 0.15. We find r0(L⋆) = 7.20 ± 0.19 for the correlation length of L⋆ galaxies.

Vug and fracture characterization and gas production prediction by fractals: Carbonate reservoir of the Longwangmiao Formation in the Moxi-Gaoshiti area, Sichuan Basin

A comprehensive knowledge of the development and connectivity of fractures and vugs in carbonate reservoirs plays a key role in reservoir evaluation, ultimately affecting the gas prediction of this kind of heterogeneous reservoir. The carbonate reservoirs with fractures and vugs that are well developed in the Longwangmiao Formation, Sichuan Basin are selected as a research target, with the fractal dimension calculated from the full-bore formation microimager (FMI) image proposed to characterize the fractures and vugs. For this purpose, the multipoint statistics algorithm is first used to reconstruct a high-resolution FMI image of the full borehole wall. And then, the maximum class-variance method (the Otsu method) realizes the automatic threshold segmentation of the FMI image and acquisition of the binary image, which accurately characterizes the fractures and vugs. Finally, the fractal dimension is calculated by the box dimension algorithm, with its small value difference enlarged to obtain a new fractal parameter ([Formula: see text]). The fractal dimensions for four different kinds of reservoirs, including eight subdivided models of vugs and fractures, show that the fractal dimension can characterize the development and the connectivity of fractures and vugs comprehensively. That is, the more developed that the fractures and vugs are, the better the connectivity will be, and simultaneously the smaller that the values of the fractal dimensions are. The fractal dimension is first applied to the gas production prediction by means of constructing a new parameter ([Formula: see text]) defined as a multiple of the effective thickness ([Formula: see text]), porosity (Por), and fractal dimension ([Formula: see text]). The field examples illustrate that the fractal dimensions can effectively characterize the fractures and vugs in the heterogeneous carbonate reservoir and predict its gas production. In summary, the fractals expand the characterization method for the vugs and fractures in carbonate reservoirs and extend its new application in gas production prediction.

Melting Behaviour of Fractal-Shaped Nanoparticles (the Example of Si–Ge System)

The influence of shape on phase equilibria in a two-phase region between liquidus and solidus temperatures has been simulated using Si–Ge nanoparticles as an example. The shape and volume of a particle have been set by its effective radius and fractal dimension. The temperature dependence of the fractal dimension of the phases has been taken into consideration in terms of a simple geometrical model. It has been shown that decreasing the volume of a nanoparticle and its fractal dimension (which corresponds to nanoparticles of a more complicated shape) leads to narrowing down the temperature range of the heterogeneous region, changes in phase transition temperatures and equilibrium compositions of co-existing phases. It has been found that at different temperatures the dependences of the composition of the liquid phase on a particle’s morphology differ which is explained by implementing different mechanisms of reducing the surface energy.

On Mutual Solubility in Submicron-Sized Particles of the Pt–Au Catalytic System

The equilibrium phase composition of particles in stratifying solid solutions is a parameter responsible for their catalytic activity. In this work, we used the methods of equilibrium chemical thermodynamics to consider the influence of geometric factors on phase equilibriums in submicron-sized particles with a core–shell configuration in the stratifying Pt–Au system. The particle shape and volume were specified by the effective radius and fractal dimension and used as calculation parameters. The dependence of the mutual solubility of the components on the volume and shape of submicron-sized particles was obtained, and a significant change in the region of compositions of stable solid solutions in particles of different shapes was found.

ON A METRIC GENERALIZATION OF THE tt-DEGREES AND EFFECTIVE DIMENSION THEORY

AbstractIn this article, we study an analogue of tt-reducibility for points in computable metric spaces. We characterize the notion of the metric tt-degree in the context of first-level Borel isomorphism. Then, we study this concept from the perspectives of effective topological dimension theory and of effective fractal dimension theory.

Fractal analysis of the pore structure for clay bound water and potential gas storage in shales based on NMR and N2 gas adsorption

Abstract Fractal dimension (D) is a critical parameter to estimate the heterogeneity of complex pore structure in shale gas reservoirs. To quantify the fractal dimension of various pore types and evaluate their implications on shale effective porosity and gas storage capacity in potential, we performed fractal analysis based on experimental results of low-field nuclear magnetic resonance (LF-NMR) and low-pressure N2 gas adsorption (LP-N2-GA) in Permian Carynginia shales. By comparing the calculated fractal dimensions based on the two approaches, we analyzed the 'surface fractal dimension' for ineffective pores occupied by clay bound water (CBW) and the 'volume fractal dimension' for effective pores (Deff) holding removable fluids for the first time in shales. The NMR-based CBW pore fractal dimension (Dcbw) is linear positively correlated with the fractal dimension of micropore surface (D1) (R2 = 0.91) and the volume of CBW (R2 = 0.58), while negatively correlated with effective porosity (R2 = 0.58). The NMR-based effective pore fractal dimension (Deff) is linear positively correlated with the fractal dimension of meso/macropore volume (D2) (R2 = 0.82) and presents a good positive correlation with gas storage capacity (R2 = 0.80). The results indicate that CBW largely complicates the fractal geometry of nanoscaled pore network and potentially resist effective fluid flows in shales. The pore surface of higher heterogeneity (higher D1) associates with larger surficial CBW retention and would further block the effective pore space for fluid transport. The meso/macropore volumes of higher complexity (higher D2) is intimate with the larger heterogeneity in effective pores for the higher potential of hydrocarbon storage capacity in gas shales.

On some peculiarities of stratification of liquid solutions within pores of fractal shape

In analysis of phase transformations in small-volume systems, attention should be paid to some specific effects. Several peculiarities of stratification of liquid solutions within nano-sized pores in cases of the water-butanol-1 and water-phenol solutions have been investigated by methods of equilibrium chemical thermodynamics. Those peculiarities are due to the influence of a pore's size and shape on mutual solubilities of the components inside. The pore size and shape have been modeled by its effective radius and fractal dimension which have been used as calculation parameters. Equilibrium compositions of co-existing phases have been obtained by finding the minima of the Gibbs function. The dependences of mutual solubilities on pore size and shape have turned out to be different in considered mixtures. In case of the water-butanol-1 solution, the solubility of water in butanol-1 and the solubility of butanol-1 in water both decrease with the decrease in the effective radius and fractal dimension. In the water-phenol solution, decreasing the effective radius can lead either to an increase or to a decrease in the solubility of water in phenol depending on the fractal dimension of a pore. In a certain range of fractal dimensions, the dependence of the solubility of water in phenol on the pore size is non-monotonous with an extremum.