Critical Fractal Dimension Research Articles

The Pathologically Evolving Aggregation-State of Cells in Cancerous Tissues as Interpreted by Fractal and Multi-Fractal Dispersion Theory in Saturated Porous Formations.

A recent author's fractal fluid-dynamic dispersion theory in porous media has focused on the derivation of the associated nonergodic (or effective) macrodispersion coefficients by a 3-D stochastic Lagrangian approach. As shown by the present study, the Fickian (i.e., the asymptotic constant) component of a properly normalized version of these coefficients exhibits a clearly detectable minimum in correspondence with the same fractal dimension (d ≅ 1.7) that seems to characterize the diffusion-limited aggregation state of cells in advanced stages of cancerous lesion progression. That circumstance suggests that such a critical fractal dimension, which is also reminiscent of the colloidal state of solutions (and may therefore identify the microscale architecture of both living and non-living two-phase systems in state transition conditions) may actually represent a sort of universal nature imprint. Additionally, it suggests that the closed-form analytical solution that was provided for the effective macrodispersion coefficients in fractal porous media may be a reliable candidate as a physically-based descriptor of blood perfusion dynamics in healthy as well as cancerous tissues. In order to evaluate the biological meaningfulness of this specific fluid-dynamic parameter, a preliminary validation is performed by comparison with the results of imaging-based clinical surveys. Moreover, a multifractal extension of the theory is proposed and discussed in view of a perspective interpretative diagnostic utilization.

Collective evolution of surface microcrack for compacted graphite iron under thermal fatigue with variable amplitude

With the growing demand of good performance and high reliability of the heated components, the cracking failure caused by the complex thermal loading of variable amplitude has become a crucial problem. The collective evolution of surface microcrack for compacted graphite iron under thermal fatigue with variable amplitude is studied in this paper, which is induced by pulsed laser. The thermal microcrack is analyzed with statistic method and fractal method systematically. The result shows that, the secondary microcrack is the primary crack pattern, and the number of main microcrack is the least. As the test goes on, the fractal dimension increases following the Hill’s function. Furthermore, the effect of maximum temperature Tmax and superimposed number NHCF on the crack evolution is investigated. Tmax in the heating stage mainly affects the number of main microcrack. With the increase of plastic strain amplitude, the fractal dimension increases exponentially, and gradually tends to be the critical fractal dimension D0 of 1.395. The superimposed number NHCF in the high-cycle stage mainly affects the number of secondary microcrack. The fractal dimension increases exponentially with the increase of NHCF, and tends to be the critical fractal dimension D0 of 1.404. The analysis of the collective behavior of surface microcrack is helpful for evaluating the damage degree and predicting the lifetime, which can be applied to other materials working under thermal loading of variable amplitude.

A modified scale method based on fractal theory for rockfill materials

To study the theoretical scale effect of rockfill materials, fractal theory was included in the grading design. Next, the existing scale methods were discussed relative to fractal theory and normalised to obtain a unified formula. A modified scale method was suggested based on fractal theory and a critical fractal dimension Dc concept was proposed. The application of the suggested scale method was verified regarding the minimum porosity, which indicated that the suggested scale method was rational. The following improvements were made relative to existing scale methods: confusion in the selecting scale method was eliminated, and the fine grain contents were determined theoretically to reflect the influence of the maximum diameter dmax during grading. The results of this study provide a solid foundation for studying the scale effects of rockfill materials with fractal theory.

Proposed connection between critical exponents and fractal dimensions in the Ising model

Scaling properties near the critical point indicates the existence of self-similarity behavior for the critical phenomena. Although the system considered here is not a truly dynamic one, we propose a specific set of relations between fractal dimensions and critical exponents in the Ising model of statistical mechanics. In particular, we put forward, corresponding to six critical exponents for the Ising model, six fractal dimensions. Assuming the latter proposals, we can then derive relationships between such fractal dimensions.

Fractal phenomenon in sound absorbing behavior of down fiber assembly

X-ray computed tomography (CT) can quantify variations in microstructure of materials. In this paper, we first demonstrated that by using a micro-CT method, we can characterize the internal porous structures of different fiber assemblies. Furthermore, the fractal dimension of the porous areas can be calculated based on the acquired two-dimensional cross-sectional images of the assemblies. Then after measuring the sound absorption behaviors of the assemblies, correlations between the fractal dimension, the pore structural characteristics, and the acoustic absorption property are analyzed. Results show that for down fiber assemblies, the fractal dimension increases with fiber mass, and decreases with the porosity. Also the maximum absorption coefficient αm reaches a maximum value at a critical fractal dimension, while critical frequency fr continuously decreases with the fractal dimension all the time.

Tribological characterization of hard coating film for elliptical surface peak

The hard coating film on optical devices are the major methods to control wear and friction for industrial applications. This paper used fractal characteristics of surface topography such as elliptic ratio of surface roughness, coating hardness, and surface energy to build microcontact model of rough surface, in order to research contact behaviors between elliptical surfaces. Results of analysis showed that increased elliptic ratio, soft coating and adhesive force caused increase of contact area. When fractal roughness parameter and external force decreased, the true contact area also increased. When adhesive force remained unchanged, the increase of hardness value coefficient value resulted in the increase of true contact area. True contact area also increased as the fractal dimension increased from 1.1, and reached its maximum when fractal dimension reached 1.7, but would decreased when fractal dimension continued to increase. Such critical fractal dimension was affected by external force, fractal roughness coefficient and material characteristics.

Analysis of wear characteristics of natural rubber nanocomposites

Tribological characteristics of natural rubber nanocomposites for wear resistant applications have been studied by sliding against a steel blade, in a specially designed abrader. Testing parameters have been optimized for minimum wear based on Taguchi orthogonal design with four important parameters, viz., nanofiller loading, applied normal load, speed and time of run. Amongst these, nanofiller loading has the most significant influence on wear characteristics. In spite of the high normal pressure acting at the line of contact, certain well established power law relations are found to obey in principle, as wear increases with normal load and frictional work, F w. Analysis of the micrographs of the abraded surface and the corresponding debris reveals that the specific wear rate of both the nanocomposites (sepiolite and carbon nanofiber filled) is found to increase beyond a critical fractal dimension, i.e., with increasing structural complexity of the debris formed. The rate of wear decreases steeply with nanofiber loading as compared to sepiolite. The changes in temperature build up and dynamic coefficients of friction are found to be concomitant. The wear mechanism is found to be fatigue at low frictional work, followed by frictional wear at high F w.

Application of Percolation Theory to Rock Damage and Fracture

Rock is a kind of complex and high-disordered geological material, its damage and fracture process usually shows obvious criticality. In this paper, percolation theory is applied to analyze and describe this critical property. First, we discuss the critical fracture probability of rock through percolation and renormalization analysis, and present the equivalence between fracture probability and damage variable. Based on scaling law and the relationship between critical exponents, a critical fractal dimension is obtained. Furthermore, according to the analysis of relationship between damage and fractal dimension, we suggest a damage-fractal formula, ω=ω0+ (D-D0)/Dc. This formula can not only be used to describe the damage evolution through the variation of fractal dimension, but also to define initial damage in rock. Finally, the theoretical conclusions are validated by a series of model experiments, and the experimental results agree with that of theoretical.

Breakdown of universality in directed spiral percolation

Directed spiral percolation (DSP), percolation under both directional and rotational constraints, is studied on the triangular lattice in two dimensions (2D). The results are compared with that of the 2D square lattice. Clusters generated in this model are generally rarefied and have chiral dangling ends on both the square and triangular lattices. It is found that the clusters are more compact and less anisotropic on the triangular lattice than on the square lattice. The elongation of the clusters is in a different direction than the imposed directional constraint on both the lattices. The values of some of the critical exponents and fractal dimension are found considerably different on the two lattices. The DSP model then exhibits a breakdown of universality in 2D between the square and triangular lattices. The values of the critical exponents obtained for the triangular lattice are not only different from that of the square lattice but also different form other percolation models.

A shell-model approach to fractal-induced turbulence

We present a shell-model of fractal induced turbulence which predicts that structure function scaling exponents decrease in absolute value as the fractal dimension of the turbulence-inducing fractal object increases. This qualitative prediction is in agreement with laboratory measurements. Finer details of the fractal induced turbulence statistics and dynamics depend on the fractal force's phases, i.e. on the detailed construction of the fractal stirrer. In a case of deterministic forcing phases, a critical fractal dimension exists below which the average rate of inter-scale energy transferhTni is a decreasing function of the wavenumber kn and the structure function scaling exponents take close to Kolmogorov values. Above this critical fractal dimension, hTni is an increasing function of kn and the structure function scaling exponents deviate signicantly from Kolmogorov values. PACS. 47.27.Ak Fundamentals { 47.27.Gs Isotropic turbulence; homogeneous turbulence

Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media

Considerable effort has been directed towards the application of percolation theory and fractal modeling to porous media. We combine these areas of research to investigate percolation in prefractal porous media. We estimated percolation thresholds in the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. The percolation thresholds for these simulations were found to increase beyond the 0.5927l... porosity expected in Bernoulli (uncorrelated) percolation networks. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media and can be used to bound modeling efforts. More fundamental applications are also possible. Only a limited range of parameters has been explored empirically but extrapolations allow the critical fractal dimension to be estimated for a large combination of b and i values. Extrapolation to infinite iterations suggests that there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 – the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks.

Uncertainty in fractal dimension estimated from power spectra and variograms

The reliability of using fractal dimension (D) as a quantitative parameter to describe geological variables is dependent mainly on the accuracy of estimated D values from observed data. Two widely used methods for the estimation of fractal dimensions are based on fitting a fractal model to experimental variograms or power-spectra on a log-log plot. The purpose of this paper is to study the uncertainty in the fractal dimension estimated by these two methods. The results indicate that both spectrum and variogram methods result in biased estimates of the D value. Fractal dimension calculated by these two methods for the same data will be different unless the bias is properly corrected. The spectral method results in overestimated D values. The variogram method has a critical fractal dimension, below which overestimation occurs and above which underestimation occurs. On the bases of 36,000 simulated realizations we propose empirical formulae to correct for biases in the spectral and variogram estimated fractal dimension. Pitfalls in estimating fractal dimension from data contaminated by white noise or data having several fractal components have been identified and illustrated by simulated examples.

Characterization of the electronic states in weakly disordered one-dimensional systems

The problem of characterization of electronic states in disordered systems is readdressed. For this task, we present our ensemble-averaged calculations of both density of states and dc-conductivity for linear chains containing up to 3500 sites and modeling the Anderson Hamiltonian with on-site energies randomly chosen from a box distribution of width W. The analysis of the spatial behaviour of eigenfunctions when increasing the disorder shows an increase of the curdling of the wavefunction amplitude, which reflects a stronger localization. The dc-conductivity results show that, for low disorder W/V < 2, the states at the midband seem to have localization length λ at least as large as the system sizes considered here. For larger disorder, the system exhibits "almost an Anderson transition" between localized and quasi-extended states, which are localized with very large λ. Our dc-conductivity results suggest a critical fractal dimension d*c = 0.70 ± 0.09 to discriminate between the exponentially and the power-law localized states. The relatively large error bar Δd*c may reflect the nature of a continuous transition between the two regimes.

Forest spread and phase transitions at forest-prairie ecotones in Kansas, U.S.A.

The spread of gallery forest habitat into upland areas is of substantial interest to resource managers because such spread has many implications for the management of grassland and forest habitats. This study used a dynamic percolation model to examine the potential rates of spread or invasion of forest in eastern Kansas. Aerial photos taken 16 years apart at the Fort Riley training base were used to calibrate a spatially explicit contagion model of forest spread to interpolate and extrapolate the forest spread processes. Results fit the actual pattern of spread well, as measured by both visual inspection and a multiscale fractal measure of pattern. Comparisons to a long-term fire-exclusion experiment in Geary County, Kansas, and to the Konza Prairie also provided validation. Both the simulation and the 100-year Geary County series showed an interesting pattern of forest spread. Spread was slow and steady until about 20% forest cover was reached, at which point the rate increased. We conclude that this self-accelerating response is due to spatial patterns created by the spreading forest that tend to accelerate the growth process after a critical point is reached. On the basis of theoretical study and experimental simulation of the percolation phase transitions, we suggest that fractal dimensions in a transient ecotone of binary mixtures (e.g., trees and grasses) should range between 1.56 and 1.8958, and the critical fractal dimension during ecotonal phase transitions should be 1.7951. This critical point of about 18.5% forest cover that we predicted was close to the observed result and might represent a phase transition at the forest-prairie ecotone.

The multifractal character of the electronic states in disordered two-dimensional systems

The nature of electronic states in disordered two-dimensional (2D) systems is investigated. With this aim, we present our calculations of both density of states and d.c. conductivity for square lattices modelling the Anderson Hamiltonian with on-site energies randomly chosen from a box distribution of width W. For weak disorder (W), the eigenfunctions calculated by means of the Lanczos diagonalization algorithm display spatial fluctuations reflecting their (multi)fractal behaviour. For increasing disorder the observed increase of the curdling of the wavefunction reflects its stronger localization. However, as a function of energy, the eigenstates at energy mod E mod /V approximately=1.5 are found to be the least localized over the band. Our d.c. conductivity results suggest a critical fractal dimension dc*=1.48+or-0.17 to discriminate between me exponentially and the power-law-localized states. Consequences of the localization for transport properties are also discussed.

A singular zeta function at the onset of chaos

A Selberg type zeta function with a parameter of fractal dimensions is derived from the assumption of the scaling property of invariant measures. The analyticity of this zeta function breaks down at the critical fractal dimensions for the transition to chaos through the period doubling bifurcation. We also find the convergence property of the relevant critical parameter and we conjecture that the zeros of this zeta function asymptotically accumulate on the unit circle. This suggests a phase transition as in “Lee-Yang” systems.

Potts model on infinitely ramified Sierpinski-gasket-type fractals and algebraic order at antiferromagnetic phases.

We propose families of infinitely ramified fractals, which we call the m-sheet Sierpinski gasket with side b [(mSG${)}_{\mathit{b}}$], on which the q-state Potts model can be exactly solvable through a real-space renormalization-group (RSRG) technique for which there are phase transitions at finite temperatures for m&gt;1. We also propose, within a cell-to-cell RSRG scheme, a criterion for a suitable choice of cells in the study of antiferromagnetic (AF) classical spin models defined on (or approximated by) multirooted hierarchical lattices, and apply it for the AF Potts model on some (mSG${)}_{\mathit{b}}$ fractals. Concerning the Ising model on the (mSG${)}_{2}$ family, we obtain the exact para (P)--ferromagnetic (F) critical temperature as a function of m and verify that, for m=1 and 2, there is no AF order (not even at zero temperature). We calculated the exact P-F and possibly exact P-AF critical frontiers and the corresponding correlation-length critical exponents for q=2-, 3-, and 4-state Potts model on the (mSG${)}_{4}$. The AF q=2 and 4 cases have highly degenerate ground states and each one presents, above a certain critical fractal dimension ${\mathit{D}}_{\mathit{f}}^{\mathit{c}}$(q,m), an unusual low-temperature phase whose attractor occurs at a non-null temperature. For q=2 and ${\mathit{D}}_{\mathit{f}}$(2,m)\ensuremath{\ge}5.1, we prove that the correlations have a power-law decay with distance along this entire phase.

Irreversible phase transitions in the dimer-monomer surface reaction process on fractal media

The dimer-monomer surface reaction process of the type 1 2 A 2+B→AB is studied on different media with fractal dimension ( D F) 1 ⩽ D F ⩽ 2. It is found that the first order irreversible phase transition (IPT) observed at D F = 2 becomes of second order for D F < 2. Also the finite width stationary reaction window, which lies between both IPTs characteristic of the process, becomes narrowed when decreasing D F. It is suggested that, for some specific kind of fractals, there is a critical fractal dimension ( D ∗ F, such that for D F ⩽ D ∗ F ≌ 1.6 ± 0.05 the reaction window becomes of zero width.

Fractal eigenstates in disordered systems

The wave functions of the non-interacting electrons in disordered systems described by a tight-binding model with site-diagonal disorder are investigated by means of the inverse participation ratio. The wave functions are shown to be fractal objects. In three-dimensional samples, a critical fractal dimension can be defined for the mobility edge in the band centre, which yields the mobility edge trajectory in the whole energy range in good agreement with previous calculations based on the investigation of the exponentially decaying transmission coefficient.

Fractal Aggregation Near the Threshold11Projects supported by the National Natural Science Foundation of China.

In this paper, we present two fractal aggregation models, line pattern seed model (model 1) and point pattern seed model (model 2), which are particle-cluster models. Using the current models, we investigate the critical transition in fractal aggregation processes in two dimensions, and suggest a method for finding the critical transition point. The computer simulation results show that the critical concentration is for model 1 and for model 2, critical fractal dimension. for model 1 and for model 2, which are in good agreement with those of DLA model and experimental data. The results also show that the critical transition point in two dimensions seems to be independent of the size of lattices and the initial seed patterns. The results seem to belong to the same universality class.