LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Fractal Dimension ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Average Distance ◽  
Local Maximum ◽  
Fractal Dimensions ◽  
Dimension Formula ◽  
Phonon Spectra ◽  
Wide Range ◽  
Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

Application of Fractal Theory in Aggregate Gradation Research

2012 ◽  
Vol 204-208 ◽  
pp. 1923-1928
Author(s):  
Bo Tan ◽  
Rui Hua Yang ◽  
Yan Ting Lai
Keyword(s):  
Fractal Dimension ◽  
Fractal Theory ◽  
Asphalt Mixture ◽  
Fractal Dimensions ◽  
Aggregate Gradation ◽  
Structure Characteristics ◽  
Aggregate Distribution ◽  
Dimension Formula ◽  
Mass Fractal ◽  
Fractal Characteristics

The paper presents the fractal dimension formula of distribution of asphalt mixture aggregate diameter by the deducing mass fractal characteristics function. Taking AC-20 and SMA-20 as examples, selected 6 groups of representative grading curves within the grading envelope proposed by the present specification, and calculated their fractal dimensions. The asphalt mixture gradation has fractal dimension D (D∈(1,3)), and the fractal of continuous gradation is single while the fractal of gap-gradation shows multi-fractal with 4.75 as the dividing point. Fractal dimension of aggregate gradation of asphalt mixture reflect the structure characteristics of aggregate distribution, that is, finer is aggregate, bigger is the fractal dimension.

scholarly journals Seismo Electric Transfer Function Fractal Dimension for Characterizing Shajara Reservoirs Of The Permo-Carboniferous Shajara Formation, Saudi Arabia

2018 ◽  
Vol 1 (1) ◽  
Keyword(s):  
Fractal Dimension ◽  
Transfer Function ◽  
Capillary Pressure ◽  
Fractal Dimensions ◽  
Pressure Profile ◽  
Wide Range ◽  
Phase Saturation ◽  
Second Procedure ◽  
Bottom To Top ◽  
The Relationship

The quality of a reservoir can be described in details by the application of seismo electric transfer function fractal dimension. The objective of this research is to calculate fractal dimension from the relationship among seismo electric transfer fuction, maximum seismo electric transfer function and wetting phase saturation and to confirm it by the fractal dimension derived from the relationship among capillary pressure and wetting phase saturation. In this research, porosity was measured on real collected sandstone samples and permeability was calculated theoretically from capillary pressure profile measured by mercury intrusion techniques. Two equations for calculating the fractal dimensions have been employed. The first one describes the functional relationship between wetting phase saturation, seismo electric transfer function, maximum seismo electric transfer function and fractal dimension. The second equation implies to the wetting phase saturation as a function of capillary pressure and the fractal dimension. Two procedures for obtaining the fractal dimension have been developed. The first procedure was done by plotting the logarithm of the ratio between seismo electric transfer function and maximum seismo electric transfer function versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was completed by plotting the logarithm of capillary pressure versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. On the basis of the obtained results of the constructed stratigraphic column and the acquired values of the fractal dimension, the sandstones of the Shajara reservoirs of the Shajara Formation were divided here into three units. The gained units from bottom to top are: Lower Shajara Seismo Electric Transfer Function Fractal Dimension Unit, Middle Shajara Seismo Electric Tranfser Function Fractal dimension Unit, and Upper Shajara Seismo Electric Transfer Function Fractal Dimension Unit. The results show similarity between seismo electric transfer tunction fractal dimension and capillary pressure fractal dimension. It was also noted that samples with wide range of pore radius were characterized by high values of fractal dimension due to an increase in their connectivity and seismo electric transfer function. In our case , and as conclusions the higher the fractal dimension, the higher the permeability, the better the shajara reservoir characteristics.

Synchronizability analysis of three kinds of dynamical weighted fractal networks

2021 ◽  
pp. 2150243
Author(s):  
Jian-Hui Li ◽  
Jin-Long Liu ◽  
Zu-Guo Yu ◽  
Bao-Gen Li ◽  
Da-Wen Huang
Keyword(s):  
Fractal Dimension ◽  
Correlation Dimension ◽  
Average Distance ◽  
Laplacian Matrix ◽  
Scaling Factor ◽  
Type I ◽  
Type Ii ◽  
Maximum Eigenvalue ◽  
Dimension Formula ◽  
Text Information

In this paper, we study the synchronizability of three kinds of dynamical weighted fractal networks (WFNs). These WFNs are weighted Cantor-dust networks, weighted Sierpinski networks and weighted Koch networks. We calculated some features of these WFNs, including average distance ([Formula: see text]), fractal dimension ([Formula: see text]), information dimension ([Formula: see text]), correlation dimension ([Formula: see text]). We analyze two representative types of synchronizable dynamical networks (the type-I and the type-II). There are two indexes ([Formula: see text] and [Formula: see text]) that can be used to characterize the synchronizability of the two types of dynamical network. Here, [Formula: see text] and [Formula: see text] are the minimum nonzero eigenvalue and the maximum eigenvalue of the Laplacian matrix of the network, respectively. We find that the larger scaling factor [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] implies stronger synchronizability for the type-I dynamical WFNs.

EVOLUTION OF FRACTAL DIMENSIONS AND GAS TRANSPORT MODELS DURING THE GAS RECOVERY PROCESS FROM A FRACTURED SHALE RESERVOIR

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950129 ◽  
Author(s):  
BOWEN HU ◽  
J. G. WANG ◽  
ZHONGQIAN LI ◽  
HUIMIN WANG
Keyword(s):  
Fractal Dimension ◽  
Surface Diffusion ◽  
Gas Transport ◽  
Knudsen Diffusion ◽  
Fractal Dimensions ◽  
Gas Extraction ◽  
Multilayer Adsorption ◽  
Permeability Model ◽  
Dimension Formula ◽  
Shale Reservoir

Previous studies ignore the evolutions of pore microstructure parameters (pore diameter fractal dimension [Formula: see text] and tortuosity fractal dimension [Formula: see text]) but these evolutions may significantly impact the gas transport during gas extraction. In order to investigate these evolutions of fractal dimension properties during gas extraction, following four aspects are studied. Firstly, surface diffusion in adsorbed multilayer is modeled for fractal shale matrix. Our new matrix permeability model considers the slip flow, Knudsen diffusion and surface diffusion. This model is verified by experimental data. Secondly, a new fracture permeability model is proposed based on fractal theory and the coupling of viscous flow and Knudsen diffusion. Thirdly, the multilayer adsorption and these permeability models are introduced into the equations of gas flow and reservoir deformation. Finally, sensitivity analysis is performed to determine the key factors on fractal dimension evolution. The results show that the multilayer adsorption can accurately describe the adsorption properties of real shale reservoir. Shale reservoir deformation and gas desorption govern the evolutions of fractal dimensions. The multilayer adsorption and adsorbed gas porosity [Formula: see text] play an important role in the evolutions of fractal dimensions during gas extraction. The monolayer saturated adsorption volume [Formula: see text] is the most sensitive parameter affecting the evolution of fractal dimensions. Therefore, the effects of gas adsorption on the evolution of fractal dimensions cannot be neglected in shale reservoirs.

LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS II: MULTIFRACTALITY OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1453-1474 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Lattice Dynamics ◽  
Energy Spectra ◽  
Legendre Transformation ◽  
Model Parameters ◽  
Local Scaling ◽  
Multifractal Formalism ◽  
Phonon Spectra ◽  
Wide Range ◽  
Multifractal Spectra ◽  
Density Of Phonon States

The curdling of the phonon eigenvalues (PEV) on energy spectra of the binary generalized Fibonacci and non-Fibonaccian chains of atoms are numerically studied. A multifractal formalism based upon a new numerically efficient Legendre transformation from (q, τ) to (α, f) variables is proposed. The multifractal spectra of the normalized integrated density of phonon states (NIDOPS) for aperiodic chains of atoms are calculated in a wide range of model parameters. It is found out that the interval (α min , α max ) of magnitudes of the exponent α, determining the local scaling of the NIDOPS, shows a considerable shift to smaller values. This tendency is most pronounced for the NIDOPS of the so-called copper-mean, nickel-mean, structural circle and Rudin-Shapiro chain, where 0<α min <0.1. It is verified numerically that this effect is a manifestation of a strong curdling of PEV which take place in optical regions of the phonon spectra.

scholarly journals Flow Simulation of Artificially Induced Microfractures Using Digital Rock and Lattice Boltzmann Methods

Energies ◽  
2018 ◽  
Vol 11 (8) ◽  
pp. 2145 ◽  
Author(s):  
Yongfei Yang ◽  
Zhihui Liu ◽  
Jun Yao ◽  
Lei Zhang ◽  
Jingsheng Ma ◽  
...  
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Lattice Boltzmann ◽  
Fractal Theory ◽  
Fractal Dimensions ◽  
Geometrical Parameters ◽  
Fracture Aperture ◽  
Berea Sandstone ◽  
Wide Range ◽  
Induced Fractures

Microfractures have great significance in the study of reservoir development because they are an effective reserving space and main contributor to permeability in a large amount of reservoirs. Usually, microfractures are divided into natural microfractures and induced microfractures. Artificially induced rough microfractures are our research objects, the existence of which will affect the fluid-flow system (expand the production radius of production wells), and act as a flow path for the leakage of fluids injected to the wells, and even facilitate depletion in tight reservoirs. Therefore, the characteristic of the flow in artificially induced fractures is of great significance. The Lattice Boltzmann Method (LBM) was used to calculate the equivalent permeability of artificially induced three-dimensional (3D) fractures. The 3D box fractal dimensions and porosity of artificially induced fractures in Berea sandstone were calculated based on the fractal theory and image-segmentation method, respectively. The geometrical parameters (surface roughness, minimum fracture aperture, and mean fracture aperture), were also calculated on the base of digital cores of fractures. According to the results, the permeability lies between 0.071–3.759 (dimensionless LB units) in artificially induced fractures. The wide range of permeability indicates that artificially induced fractures have complex structures and connectivity. It was also found that 3D fractal dimensions of artificially induced fractures in Berea sandstone are between 2.247 and 2.367, which shows that the artificially induced fractures have the characteristics of self-similarity. Finally, the following relations were studied: (a) exponentially increasing permeability with increasing 3D box fractal dimension, (b) linearly increasing permeability with increasing square of mean fracture aperture, (c) indistinct relationship between permeability and surface roughness, and (d) linearly increasing 3D box fractal dimension with increasing porosity.

scholarly journals Critical temperatures of the Ising model on Sierpiñski fractal lattices

2020 ◽  
Vol 244 ◽  
pp. 01013
Author(s):  
Michel Perreau
Keyword(s):  
Fractal Dimension ◽  
Critical Temperature ◽  
Ising Model ◽  
Partition Function ◽  
Fractal Dimensions ◽  
Simple Expression ◽  
Square Lattice ◽  
Critical Temperatures ◽  
Wide Range ◽  
Sierpinski Carpets

We report our latest results of the spectra and critical temperatures of the partition function of the Ising model on deterministic Sierpiñski carpets in a wide range of fractal dimensions. Several examples of spectra are given. When the fractal dimension increases (and correlatively the lacunarity decreases), the spectra aggregates more and more tightly along the spectrum of the regular square lattice. The single real root vc, comprised between 0 and 1, of the partition function, which corresponds to the critical temperature Tc through the formula vc = tanh(1/Tc), reliably fits a power law of exponent k where k is the segmentation step of the fractal structure. This simple expression allows to extrapolate the critical temperature for k → ∞. The plot of the logarithm of this extrapolated critical temperature versus the fractal dimension appears to be reliably linear in a wide range of fractal dimensions, except for highly lacunary structures of fractal dimensions close from 1 (the dimension of a quasilinear lattice) where the critical temperature goes to 0 and its logarithm to −∞.

Temporal Changes in Habitat Selection and Nest Spacing in a Colony of Ross' and Lesser Snow Geese

The Auk ◽  
1983 ◽  
Vol 100 (2) ◽  
pp. 335-343 ◽  
Author(s):  
M. Robert McLandress
Keyword(s):  
Nearest Neighbor ◽  
Average Distance ◽  
Habitat Preferences ◽  
Nearest Neighbors ◽  
Nest Density ◽  
Snow Goose ◽  
Snow Geese ◽  
Lesser Snow Geese ◽  
Average Size ◽  
Island Habitats

Abstract I studied the nesting colony of Ross' Geese (Chen rossii) and Lesser Snow Geese (C. caerulescens caerulescens) at Karrak Lake in the central Arctic of Canada in the summer of 1976. Related studies indicated that this colony had grown from 18,000 birds in 1966-1968 to 54,500 birds in 1976. In 1976, geese nested on islands that were used in the late 1960's and on an island and mainland sites that were previously unoccupied. Average nest density in 1976 was three-fold greater than in the late 1960's. Consequently, the average distance to nearest neighbors of Ross' Geese in 1976 was half the average distance determined 10 yr earlier. The mean clutch size of Ross' Geese was greater in island habitats where nest densities were high than in less populated island or mainland habitats. The average size of Snow Goose clutches did not differ significantly among island habitats but was larger at island than at mainland sites. Large clutches were most likely attributable to older and/or earlier nesting females. Habitat preferences apparently differed between species. Small clutches presumably indicated that young geese nested in areas where nest densities were low. The establishment of mainland nesting at Karrak Lake probably began with young Snow Geese using peripheral areas of the colony. Young Ross' Geese nested in sparsely populated habitats on islands to a greater extent than did Snow Geese. Ross' Geese also nested on the mainland but in lower densities than Ross' Geese nesting in similar island habitats. Successful nests with the larger clutches had closer conspecific neighbors than did successful nests with smaller clutches. The species composition of nearest neighbors changed significantly with distance from Snow Goose nests but not Ross' Goose nests. Nesting success was not affected by the species of nearest neighbor, however. Because they have complementary antipredator adaptations, Ross' and Snow geese may benefit by nesting together.

FRACTAL DIMENSION, PRIMES, AND THE PERSISTENCE OF MEMORY

2003 ◽  
Vol 06 (02) ◽  
pp. 241-249
Author(s):  
JOSEPH L. PE
Keyword(s):  
Number Theory ◽  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Distinguishing Characteristic ◽  
Local Behavior ◽  
Remarkable Similarity ◽  
Recursive Procedures

Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales — a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimension as a distinguishing characteristic of the graph. Also, it notes a remarkable similarity between two apparently unrelated sequences: the persistence of a number, and the memory of a prime. This similarity is quantified using fractal dimension.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

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