Fractal Distribution Research Articles

MAX PRINCIPLE AND UNIVERSAL SPECTRUM OF PERIODS: COMPLEMENTARY FRACTAL DISTRIBUTIONS AS CONSEQUENCE OF RATIONAL AND IRRATIONAL RELATIONS BETWEEN PARTS OF THE WHOLE SYSTEM

The paper discusses the assumption that Mach principle should result in existence of a universal spectrum of periods. It is shown that fragments of such a spectrum were found in time series of fluctuations of various processes. A general approach is considered that demonstrates the emergence of discrete states in the spectra of periods, which is based on two basic concepts: resonance and roughness of a physical system. This approach leads to the existence of two complementary fractal distributions associated with sets of rational and irrational relations between the elements of the whole system. A brief review of works that also consider universal spectra of periods is given.

Dynamic streaming potential coupling coefficient in porous media with different pore size distributions

SUMMARY Seismoelectric signals are generated by electrokinetic coupling from seismic wave propagation in fluid-filled porous media. This process is directly related to the existence of an electrical double layer at the interface between the pore fluid and minerals composing the pore walls. The seismoelectric method attracts the interest of researchers in different areas, from oil and gas reservoir characterization to hydrogeophysics, due to the sensitivity of the seismoelectric signals to medium and fluid properties. In this work, we propose a physically based model for the dynamic streaming potential coupling coefficient (SPCC) by conceptualizing a porous medium as a bundle of tortuous capillaries characterized by presenting different pore size distributions (PSD). The results show that the dynamic streaming potential coupling coefficient is a complex function depending on the properties of pore fluid, mineral–pore fluid interfaces, microstructural parameters of porous media and frequency. Parameters influencing the dynamic SPCC are investigated and explained. In particular, we show that the PSD affects the transition frequency as well as the shape of the SPCC response as a function of frequency. The proposed model is then compared with published data and previous models. It is found that the approach using the lognormal distribution is in very good agreement with experimental data as well as with previous models. Conversely, the approach that uses the fractal distribution provides a good match with published data for sandstone samples but not for sand samples. This result implies that the fractal PSD may not be pertinent for the considered sand samples, which exhibit a relatively narrow distribution of pore sizes. Our proposed approach can work for any PSD, for example, including complex ones such as double porosity or inferred from direct measurements. This makes the proposed models more versatile than models available in literature.

Chaos in the Bose–Hubbard model and random two-body Hamiltonians

We investigate the chaotic phase of the Bose–Hubbard model (Pausch et al 2021 Phys. Rev. Lett. 126 150601) in relation to the bosonic embedded random matrix ensemble, which mirrors the dominant few-body nature of many-particle interactions, and hence the Fock space sparsity of quantum many-body systems. The energy dependence of the chaotic regime is well described by the bosonic embedded ensemble, which also reproduces the Bose–Hubbard chaotic eigenvector features, quantified by the expectation value and eigenstate-to-eigenstate fluctuations of fractal dimensions. Despite this agreement, in terms of the fractal dimension distribution, these two models depart from each other and from the Gaussian orthogonal ensemble as Hilbert space grows. These results provide further evidence of a way to discriminate among different many-body Hamiltonians in the chaotic regime.

The influence of particle size distribution on the stress distribution in granular materials

This study systematically explores the effect of the shape of the particle size distribution (PSD) on stress transmission in granular materials using three-dimensional discrete-element method simulations. Extending prior studies that have focused on bimodal mixtures of coarser and finer grains, a broad range of isotropically compressed specimens with spherical particles and linear, fractal and gap-graded particle size distributions is considered. Considering isotropic stress conditions, the nature of stress distribution was analysed by determining the mean effective particle stresses and considering the proportion of this stress transmitted by particles with different sizes. For gap-graded materials a contact-based perspective was adopted to consider the stress transmission both within and between the different size fractions. A clear correlation emerged between the cumulative distribution of particle sizes by volume and the cumulative distribution of particle sizes by mean effective stress for specimens with continuous PSDs. This correlation does not hold universally for gap-graded materials. In gap-graded materials the distribution of effective stress between the different size fractions depends upon the size ratio and the percentage of finer grains in the specimen. In contrast to specimens with continuous gradings, in gap-graded specimens the distribution of effective stress among the different size fractions exhibits a marked sensitivity to density. Basic network analysis is shown to provide a useful insight into effective stress transmission in bimodal gap-graded materials.

Immiscible/Near-Miscible relative permeability for confined fluids at high-pressure and high-temperature for a fractal reservoir

The complex interaction between fluids and solids in reservoirs includes interface slip, capillary confinement and the diffusion and mass transfer between CO2 and oil and results in intensely nonlinear flow complexity. This study proposes a relative permeability model that accommodates a fractal pore size distribution and honors these complex process interactions. The relative permeability to CO2 flooding, in the near-miscible region, is predicted through interpolation based on the Gibbs free energy (GFE). The thermodynamic phase behavior of the fluids in the nanopores is considered by applying critical shifts in the temperatures and pressures. A volume-translated Peng-Robinson equation of state is used to calculate the CO2 and n-alkane densities to high reservoir pressure. Fluid-based correlation and modified volumetric mixing rules are then used to extend the viscosity calculations to mixtures with heavy hydrocarbon components. Predictions from the proposed model better fit experimental observations relative to previous models similarly incorporating fractal theory. The nanopores are shown to increase the relative permeability of the non-wetting phase by decreasing the viscosity ratio of the two phases. Increasing key parameters that are related to the pore structure, e.g. the fractal dimension, Df, and critical pore radius, rc, increases the relative permeability of the non-wetting phase. The GFE-based interpolation contributes to the smooth and continuous change in the relative permeability parameters local to the critical point of the mixture, with the confined fluid more likely to be miscible at the same pressure than the bulk fluid. This model can be integrated with a compositional simulator to solve field-scale problems but accommodating the micro-scale physics of unconventional reservoirs.

Study on Fractal Characteristics of Mineral Particles in Undisturbed Loess and Lime-Treated Loess.

In order to explore the fractal characteristics of particle size distribution (PSD) of various minerals in loess and lime-treated loess, the Q4 undisturbed loess and lime-treated loess were studied. From the perspective of multi-scaled microstructure, the internal characteristics of loess were observed and the regularity statistics were carried out from a macroscopic view. Fractal theory was used to quantitatively study the distribution of mineral particles in undisturbed loess and lime-treated loess. It was found that the skeleton particles of undisturbed loess were obvious and the structure of soil was loose. While that of lime-treated loess decreased, the fine particles were connected with each other, and the structure of soil changed from loose to dense. The three mineral particles in the undisturbed loess and lime-treated loess did not accord with the single fractal distribution characteristics, but the total particles had fractal characteristics. The percentage content of the mineral particles in the soil varied greatly with the particle size. In addition, the non-uniform degrees of mineral particles in the two soils from large to small were carbonate minerals of lime-treated loess, carbonate minerals of undisturbed loess, quartz minerals of lime-treated loess, feldspar mineral of lime-treated loess, feldspar mineral of the undisturbed loess, and the quartz mineral of the undisturbed loess. This paper provided a basis for the future study of the different soil mechanical properties of undisturbed loess and lime-treated loess.

(Invited) Long Range Coherent Transport in Proteins via Electron Pairing

Recently, Lindsay et al. showed in a series of papers (summarized in Lindsay, S. Life 2020, 10, 72.) that single proteins coupled to metal electrodes have unexpectedly good conductance properties. It became clear that much of the resistance drop happens at the metallic contacts, and the internal conductivity of the protein itself can be very close to the quantum limit 2e^2/h for short proteins. The localization length can be as large as ten nanometers for longer ones. Cahen et al. showed that the conductance is temperature independent in the range of 30-300 Kelvins in a series of measurements. All this indicates that proteins support fast and at least partially coherent electron transport at room temperature. According to electronic structure calculations, proteins have significant HOMO-LUMO gaps surrounded by localized orbitals, a typical arrangement for insulators ruling out efficient single-electron quantum transport even in the presence of photon mediation. However, due to the repetitive nature of the basic peptide backbone of the proteins, there are states deeper below the HOMO energy, which are intermediary between extended states typical in metallic conductance bands and the localized ones. These so-called critical states (Vattay G. et al. 2015 J. Phys.: Conf. Ser. 626 012023) have fractal density distributions and show large electron density correlations between other orbitals in the same energy region. It has been shown that in the presence of electron-electron attraction, these states are especially suited for pair formation of electrons, and an insulator-superconductor transition can occur even at high temperatures. Here we argue that polarizable sidechains of amino acids can create an attractive force between electrons propagating along the backbone. We show that the typical size and polarization of amino acids make the proper magnitude of pairing energy to maintain superconducting patches even at room temperature. We calculate and display the geometry of the superconducting gap over the protein structure and propose this new mechanism as a possible explanation for the observed coherent transport.

Some Notes on Granular Mixtures with Finite, Discrete Fractal Distribution

Why fractal distribution is so frequent? It is true that fractal dimension is always less than 3? Why fractal dimension of 2.5 to 2.9 seems to be steady-state or stable? Why the fractal distributions are the limit distributions of the degradation path? Is there an ultimate distribution? It is shown that the finite fractal grain size distributions occurring in the nature are identical to the optimal grading curves of the grading entropy theory and, the fractal dimension n varies between –¥ and ¥. It is shown that the fractal dimensions 2.2–2.9 may be situated in the transitional stability zone, verifying the internal stability criterion of the grading entropy theory. Micro computed tomography (μCT) images and DEM (distinct element method) studies are presented to show the link between stable microstructure and internal stability. On the other hand, it is shown that the optimal grading curves are mean position grading curves that can be used to represent all possible grading curves.

Cancer as a Model System for Testing Metabolic Scaling Theory

Biological allometries, such as the scaling of metabolism to mass, are hypothesized to result from natural selection to maximize how vascular networks fill space yet minimize internal transport distances and resistance to blood flow. Metabolic scaling theory argues two guiding principles—conservation of fluid flow and space-filling fractal distributions—describe a diversity of biological networks and predict how the geometry of these networks influences organismal metabolism. Yet, mostly absent from past efforts are studies that directly, and independently, measure metabolic rate from respiration and vascular architecture for the same organ, organism, or tissue. Lack of these measures may lead to inconsistent results and conclusions about metabolism, growth, and allometric scaling. We present simultaneous and consistent measurements of metabolic scaling exponents from clinical images of lung cancer, serving as a first-of-its-kind test of metabolic scaling theory, and identifying potential quantitative imaging biomarkers indicative of tumor growth. We analyze data for 535 clinical PET-CT scans of patients with non-small cell lung carcinoma to establish the presence of metabolic scaling between tumor metabolism and tumor volume. Furthermore, we use computer vision and mathematical modeling to examine predictions of metabolic scaling based on the branching geometry of the tumor-supplying blood vessel networks in a subset of 56 patients diagnosed with stage II-IV lung cancer. Examination of the scaling of maximum standard uptake value with metabolic tumor volume, and metabolic tumor volume with gross tumor volume, yield metabolic scaling exponents of 0.64 (0.20) and 0.70 (0.17), respectively. We compare these to the value of 0.85 (0.06) derived from the geometric scaling of the tumor-supplying vasculature. These results: (1) inform energetic models of growth and development for tumor forecasting; (2) identify imaging biomarkers in vascular geometry related to blood volume and flow; and (3) highlight unique opportunities to develop and test the metabolic scaling theory of ecology in tumors transitioning from avascular to vascular geometries.

Computational Analysis of Biophilic Scale Distributions of Façades in Kaunas City Centre

The results of numerous studies which are performed on the concepts of Biophilic architecture demonstrate that it can influence emotional tension and health of the observers. Moreover Biophilic research exhibits that not only natural plants induce biophilic response, but also artificial, human creations with certain fractal dimensions or distributions of scales can have an impact. In that regard, the aim of this research is to describe the relation between measurable Biophilic properties of façades and the emotional tension inducing health problems measured with the count of medical emergency arrivals in the vicinity of the façades. To achieve the aim several tasks were completed, such as the development of a methodology of façade analysis, and application of it in an experiment to test the validity. The engineered features found by this research are based on statistical analysis of distributions of line lengths and distances between lines in a drawing of a façade. To test the methodology, a linear regression model with six features was trained and it achieved a 37 % confidence, measured with R² adjusted, predicting the number of medical emergency arrivals. Simplicity of the model allowed to make additional insights into the specificity of façade properties, and their importance to Biophilia, which establishes the scientific novelty and the significance of this research.

Numerical Simulation of Granular Column Collapse with Fractal Particle Size Distribution Using Discrete Element Method

This study conducts numerical simulations of the granular column collapse with Fractal Particle Size Distributions (FPSDs) via the Discrete Element Method (DEM) and investigated kinetic behaviours of dry granular flows. The aim of this paper is to explore the effects of the fractal dimension of FPSD on the kinetics of dry granular flows. When the fractal dimension of the flows consisting of granular materials increases, the horizontal particle translational velocities become greater and the mobility improves, whereas the particle rotational velocities decrease. Meanwhile, the change in the potential energy increases, and the particle kinetic energy in the rotational form reduces; thus, the particle kinetic energy in the translational form increases. The reducing particle rotational movement may be related to the reducing particle shearing behaviours because only the contact shearing can affect particle rotational motion. In conclusion, a larger fractal dimension of FPSD of a dry granular flow leads to a longer spreading distance and a smaller rotational velocity.

Are hierarchically formed embedded star clusters surviving gas expulsion depending on their initial conditions?

ABSTRACT We investigate the dissolution process of young embedded star clusters with different primordial mass segregation levels using fractal distributions by means of N-body simulations. We combine several star clusters in virial and subvirial global states with Plummer and uniform density profiles to mimic the gas. The star clusters have masses of Mstars = 500 M⊙ that follow an initial mass function where the stars have maximum distance from the centre of r = 1.5 pc. The clusters are placed in clouds that at the same radius have masses of Mcloud = 2000 M⊙, resulting in star formation efficiency of 0.2. We remove the background potential instantaneously at a very early phase, mimicking the most destructive scenario of gas expulsion. The evolution of the fraction of bound stellar mass is followed for a total of 16 Myr for simulations with stellar evolution and without. We compare our results with previous works using equal-mass particles where an analytical physical model was used to estimate the bound mass fraction after gas expulsion. We find that independent of the initial condition, the fraction of bound stellar mass can be well predicted just right after the gas expulsion but tends to be lower at later stages, as these systems evolve due to the stronger two-body interactions resulting from the inclusion of a realistic initial mass function. This discrepancy is independent of the primordial mass segregation level.

Stream sediment geochemical exploration for gold in Central Eastern Desert, Egypt: Application of the concentration-number fractal model, factor analysis, and geochemical mineralization probability index

Geochemical exploration by stream sediment sampling using bulk leach extractable gold (BLEG) technique and applying concentration-number (C-N) fractal model, factor analysis (FA), and geochemical mineralization probability index (GMPI) resulted in the recognition of new Au occurrences around the Sukari gold mine in the central Eastern Desert of Egypt. The geochemical data of 128 stream sediment samples collected from the study area was used for delineating the geochemical anomalies and characterizing the dispersion trains of ore and associated elements (Au, Ag, As, Sb, Cu, Pb, Zn, Mo). Statistical analysis of the geochemical data applying the C-N fractal modeling enabled us to identify significant anomaly and background populations of the investigated elements and to construct reliable geochemical anomaly maps. Factor analysis using centered log-ratios (CLR), to address the problem of closed compositional data, revealed significant element associations for mineralization (Au, As, Mo, Zn, Ba), country rock compositions (Rb, Li, Be, Sn, Bi for granite, and Co, Cr, Ni for mafic rocks), and element mobility (e.g. Sb, Zr, and Ag). Weak and moderate Au anomalies that cannot be detected by factor score maps can be delineated clearly by using the C-N fractal method and GMPI distribution map. Our study revealed that Ag, As, and Sb are the main pathfinder elements for gold mineralization in arid to semiarid regions exemplified by the Sukari gold district. Silver can be used as a “direct” pathfinder, whereas As and Sb are “indirect” pathfinders for Au in such regions. The spatial distribution of Au and Ag anomalies indicate that gold mineralization in the Sukari district is structurally controlled. However, the spatial distribution of Cu, Pb, Zn, and Mo is controlled by mineralogical and lithological factors and is not related to any significant base metal deposits.

Threshold pressure gradient for helium seepage in coal and its application to equivalent seepage channel characterization

Seepage is the main gas migration form in coal seam. The experiments in this study show that there exist threshold pressure gradient and low-velocity nonlinear seepage phenomenon when non-adsorbable helium flows in coal briquette. To further explain that from microscopic view, we establish a capillary seepage model and use a new experimental method to characterize the equivalent seepage channel size distribution (ESCSD) of coal under different effective stress conditions (3–48 MPa). The experimental results prove that the ESCSD of coal obeys fractal distribution. The fractal dimensions in this study are in the range of 2.30–3.36, and decrease significantly with the increase of effective stress. Under the condition of effective stress <48 MPa, the seepage channels with size >1 μm in the experimental coal sample always control most seepage discharge (>90%). Based on the theoretically simplified seepage model and the measured threshold pressure gradient for seepage, the relationship between the critical seepage channel size (dcr) of helium in coal and the pressure gradient (∂P/∂L) is obtained, that is, dcr=α∂P/∂L and α is in the range of 0.0024–0.0033 Pa. Note that the critical condition here represents the transition not from flow to no flow but from seepage to diffusion.

Effect of fractal particle size distribution on the mobility of dry granular flows

This paper performs a numerical investigation on the kinetic behaviors of dry granular flows generated by the collapse of granular columns with different fractal particle size distributions (FPSDs) using the discrete element method. The FPSD has a remarkable impact on the regime and mobility of a granular flow. As the fractal dimension of a granular assembly increases, the translational velocities of the particles increase in the horizontal direction, whereas the rotational velocities decrease. The number of particles involved in the horizontal spreading increases due to the decreasing intensity of the contact shearing particle behavior. These increased translational velocities in the horizontal direction effectively facilitate particle spreading. A boundary layer is formed at the front bottom of the granular body in which the clockwise rotational velocities of particles are significantly increased, thereby benefiting particle spreading. The whole front of the granular flow obtains an immense spreading velocity. Moreover, the translational and rotational characteristics are dependent on the particle size. Smaller particles tend to translate and rotate faster than larger particles because of the greater particle contact forces of the former. As the fractal dimension increases, medium- and large-size particles obtain less kinetic energy than do small-size particles. Therefore, small-size particles play a crucial role in the mobility of granular flows in a granular assembly with a high fractal dimension.

FRACTAL EVIDENCE OF REACTIVE TRANSPORT IN POROUS MEDIA UNDER ACID GAS INJECTION

In the geologic and soil science literature, the fractal model of porous systems abound. Consequently, any process, be it physicochemical or geomechanical that impacts pore structure will equally impact the fractal nature of the system. Several studies have reported the effect of physicochemical processes such as water–rock interations, on the fractal dimension of the porous system using different approaches involving sophisticated instrumentation. Others have also reported the effect of geomechanical processes such as compaction on fractal dimension using similar approaches. To the best of our knowledge, none or little has been reported regarding the use of multiphase flow hydraulic models for revealing the effect of water–rock interactions on the fractal dimension and pore size distribution of porous systems under acid gas injection. Consequently, to fill the knowledeg gap, data on two-phase flow of acid gas and brine in porous media have been used to show that the fractal dimension and pore size distribution of the porous system change in response to water–rock interactions. The bases of the study are hydraulic models integrating the pore size distribution index parameter that have intimate associations with fractal dimension. Most importantly, the results of this study show that data on two-phase flow in porous media involoving acid gas and brine can be used to show changes in the fractal nature of the porous system, consistent with measurements made using sophisticated methods for core analysis.

The dependence of the hierarchical distribution of star clusters on galactic environment

ABSTRACT We use the angular two-point correlation function (TPCF) to investigate the hierarchical distribution of young star clusters in 12 local (3–18 Mpc) star-forming galaxies using star cluster catalogs obtained with the Hubble Space Telescope (HST) as part of the Treasury Program Legacy ExtraGalactic UV Survey. The sample spans a range of different morphological types, allowing us to infer how the physical properties of the galaxy affect the spatial distribution of the clusters. We also prepare a range of physically motivated toy models to compare with and interpret the observed features in the TPCFs. We find that, conforming to earlier studies, young clusters ($T \lesssim 10\, \mathrm{Myr}$) have power-law TPCFs that are characteristic of fractal distributions with a fractal dimension D2, and this scale-free nature extends out to a maximum scale lcorr beyond which the distribution becomes Poissonian. However, lcorr, and D2 vary significantly across the sample, and are correlated with a number of host galaxy physical properties, suggesting that there are physical differences in the underlying star cluster distributions. We also find that hierarchical structuring weakens with age, evidenced by flatter TPCFs for older clusters ($T \gtrsim 10\, \mathrm{Myr}$), that eventually converges to the residual correlation expected from a completely random large-scale radial distribution of clusters in the galaxy in $\sim 100 \, \mathrm{Myr}$. Our study demonstrates that the hierarchical distribution of star clusters evolves with age, and is strongly dependent on the properties of the host galaxy environment.

Intermediate Structural Hierarchy in Biological Networks Modulates the Fractal Dimension and Force Distribution of Percolating Clusters.

Globular protein hydrogels are an emerging class of materials with the potential for rational design, and a generalized understanding of how their network properties emerge from the structure and dynamics of the building block is a key challenge. Here we computationally investigate the effect of intermediate (polymeric) nanoscale structure on the formation of protein hydrogels. We show that changes in both the cross-link topology and flexibility of the polymeric building block lead to changes in the force transmission around the system and provide insight into the dynamic network formation processes. The preassembled intermediate structure provides a novel structural coordinate for the hierarchical modulation of macroscopic network properties, as well as furthering our understanding of the general dynamics of network formation.

On generalized Lemaitre–Tolman–Bondi metric: Fractal matter at the end of matter–antimatter recombination

Many recent researches have investigated the deviations from the Friedmannian cosmological model, as well as their consequences on unexplained cosmological phenomena, such as dark matter and the acceleration of the Universe. On one hand, a first-order perturbative study of matter inhomogeneity returned a partial explanation of dark matter and dark energy, as relativistic effects due to the retarded potentials of far objects. On the other hand, the fractal cosmology, now approximated by a Lemaitre–Tolman–Bondi (LTB) metric, results in distortions of the luminosity distances of SNe Ia, explaining the acceleration as apparent. In this work, we extend the LTB metric to ancient times. The origin of the fractal distribution of matter is explained as the matter remnant after the matter–antimatter recombination epoch. We show that the evolution of such a inhomogeneity necessarily requires a dynamical generalization of LTB, and we propose a particular solution.

Input files for DEM simulation of dry granular flows generated by the collapse of granular columns with different fractal particle size distributions

Input files for DEM simulation of dry granular flows generated by the collapse of granular columns with different fractal particle size distributions