Self similarity and correlations in percolation

The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform

This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT.

Percolation perspective on sites not visited by a random walk in two dimensions

We consider the percolation problem of sites on an L×L square lattice with periodic boundary conditions which were unvisited by a random walk of N=uL^{2} steps, i.e., are vacant. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with u. The clusters of vacant sites are not fractal but have fractal boundaries of dimension 4/3. The lattice size L is the only large length scale in this problem. The typical mass (number of sites s) in the largest cluster is proportional to L^{2}, and the mean mass of the remaining (smaller) clusters is also proportional to L^{2}. The normalized (per site) density n_{s} of clusters of size (mass) s is proportional to s^{-τ}, while the volume fraction P_{k} occupied by the kth largest cluster scales as k^{-q}. We put forward a heuristic argument that τ=2 and q=1. However, the numerically measured values are τ≈1.83 and q≈1.20. We suggest that these are effective exponents that drift towards their asymptotic values with increasing L as slowly as 1/lnL approaches zero.

Emergence of logarithmic-periodic oscillations in contact process with topological disorder

We present a model of contact process on Domany-Kinzel cellular automata with a geometrical disorder. In the 1D model, each site is connected to two nearest neighbors which are either on the left or the right. The system is always attracted to an absorbing state with algebraic decay of average density with a continuously varying complex exponent. The log-periodic oscillations are imposed over and above the usual power law and are clearly evident as p→1. This effect is purely due to an underlying topology because all sites have the same infection probability p and there is no disorder in the infection rate. An extension of this model to two and three dimensions leads to similar results. This may be a common feature in systems where quenched disorder leads to effective fragmentation of the lattice.

Critical p=1/2 in percolation on semi-infinite strips

We study site percolation on lattices confined to a semi-infinite strip. For triangular and square lattices we find that the probability that a cluster touches the three sides of such a system at the percolation threshold has a continuous limit of 1/2 and argue that this limit is universal for planar systems. This value is also expected to hold for finite systems for any self-matching lattice. We attribute this result to the asymptotic symmetry of the separation lines between alternating spanning clusters of occupied and unoccupied sites formed on the original and matching lattice, respectively.

On the question of fractal packing structure in metallic glasses

This work addresses the long-standing debate over fractal models of packing structure in metallic glasses (MGs). Through detailed fractal and percolation analyses of MG structures, derived from simulations spanning a range of compositions and quenching rates, we conclude that there is no fractal atomic-level structure associated with the packing of all atoms or solute-centered clusters. The results are in contradiction with conclusions derived from previous studies based on analyses of shifts in radial distribution function and structure factor peaks associated with volume changes induced by pressure and compositional variations. The interpretation of such shifts is shown to be challenged by the heterogeneous nature of MG structure and deformation at the atomic scale. Moreover, our analysis in the present work illustrates clearly the percolation theory applied to MGs, for example, the percolation threshold and characteristics of percolation clusters formed by subsets of atoms, which can have important consequences for structure-property relationships in these amorphous materials.

Appropriate Domain Size for Groundwater Flow Modeling with a Discrete Fracture Network Model

When a discrete fracture network (DFN) is constructed from statistical conceptualization, uncertainty in simulating the hydraulic characteristics of a fracture network can arise due to the domain size. In this study, the appropriate domain size, where less significant uncertainty in the stochastic DFN model is expected, was suggested for the Korea Atomic Energy Research Institute Underground Research Tunnel (KURT) site. The stochastic DFN model for the site was established, and the appropriate domain size was determined with the density of the percolating cluster and the percolation probability using the stochastically generated DFNs for various domain sizes. The applicability of the appropriate domain size to our study site was evaluated by comparing the statistical properties of stochastically generated fractures of varying domain sizes and estimating the uncertainty in the equivalent permeability of the generated DFNs. Our results show that the uncertainty of the stochastic DFN model is acceptable when the modeling domain is larger than the determined appropriate domain size, and the appropriate domain size concept is applicable to our study site.

Dimensionality and size scaling of coordinated Ca(2+) dynamics in MIN6 β-cell clusters

Pancreatic islets of Langerhans regulate blood glucose homeostasis by the secretion of the hormone insulin. Like many neuroendocrine cells, the coupling between insulin-secreting β-cells in the islet is critical for the dynamics of hormone secretion. We have examined how this coupling architecture regulates the electrical dynamics that underlie insulin secretion by utilizing a microwell-based aggregation method to generate clusters of a β-cell line with defined sizes and dimensions. We measured the dynamics of free-calcium activity ([Ca(2+)]i) and insulin secretion and compared these measurements with a percolating network model. We observed that the coupling dimension was critical for regulating [Ca(2+)]i dynamics and insulin secretion. Three-dimensional coupling led to size-invariant suppression of [Ca(2+)]i at low glucose and robust synchronized [Ca(2+)]i oscillations at elevated glucose, whereas two-dimensional coupling showed poor suppression and less robust synchronization, with significant size-dependence. The dimension- and size-scaling of [Ca(2+)]i at high and low glucose could be accurately described with the percolating network model, using similar network connectivity. As such this could explain the fundamentally different behavior and size-scaling observed under each coupling dimension. This study highlights the dependence of proper β-cell function on the coupling architecture that will be important for developing therapeutic treatments for diabetes such as islet transplantation techniques. Furthermore, this will be vital to gain a better understanding of the general features by which cellular interactions regulate coupled multicellular systems.

Elastic contact mechanics: percolation of the contact area and fluid squeeze-out

The dynamics of fluid flow at the interface between elastic solids with rough surfaces depends sensitively on the area of real contact, in particular close to the percolation threshold, where an irregular network of narrow flow channels prevails. In this paper, numerical simulation and experimental results for the contact between elastic solids with isotropic and anisotropic surface roughness are compared with the predictions of a theory based on the Persson contact mechanics theory and the Bruggeman effective medium theory. The theory predictions are in good agreement with the experimental and numerical simulation results and the (small) deviation can be understood as a finite-size effect. The fluid squeeze-out at the interface between elastic solids with randomly rough surfaces is studied. We present results for such high contact pressures that the area of real contact percolates, giving rise to sealed-off domains with pressurized fluid at the interface. The theoretical predictions are compared to experimental data for a simple model system (a rubber block squeezed against a flat glass plate), and for prefilled syringes, where the rubber plunger stopper is lubricated by a high-viscosity silicon oil to ensure functionality of the delivery device. For the latter system we compare the breakloose (or static) friction, as a function of the time of stationary contact, to the theory prediction.

Application of percolation theory to microtomography of structured media: percolation threshold, critical exponents, and upscaling

Percolation theory provides a tool for linking microstructure and macroscopic material properties. In this paper, percolation theory is applied to the analysis of microtomographic images for the purpose of deriving scaling laws for upscaling of properties. We have tested the acquisition of quantities such as percolation threshold, crossover length, fractal dimension, and critical exponent of correlation length from microtomography. By inflating or deflating the target phase and percolation analysis, we can get a critical model and an estimation of the percolation threshold. The crossover length is determined from the critical model by numerical simulation. The fractal dimension can be obtained either from the critical model or from the relative size distribution of clusters. Local probabilities of percolation are used to extract the critical exponent of the correlation length. For near-isotropic samples such as sandstone and bread, the approach works very well. For strongly anisotropic samples, such as highly deformed rock (mylonite) and a tree branch, the percolation threshold and fractal dimension can be assessed with accuracy. However, the uncertainty of the correlation length makes it difficult to accurately extract its critical exponents. Therefore, this aspect of percolation theory cannot be reliably used for upscaling properties of strongly anisotropic media. Other methods of upscaling have to be used for such media.

Quantitative muscle ultrasound in neuromuscular disorders using the parameters 'intensity', 'entropy', and 'fractal dimension'

BACKGROUND AND PURPOSE

Ultrasound is a useful non-invasive instrument in visualizing physiological and pathological morphology in skeletal muscle. Here, we evaluate the possibility that quantitative muscle ultrasound using the parameters 'intensity', 'entropy', and 'fractal dimension' is a feasible method to distinguish between dystrophic myopathies (DM), inflammatory myopathies (IM), and motor neuron disorders.

METHODS

Seven patients with IM, 12 patients with DM, nine patients with motor neuron diseases, and 24 healthy subjects underwent an identical ultrasound examination protocol, applied on gastrocnemius and tibialis anterior muscle. Analysis parameters were applied on grey scale images as well as on gradient images.

RESULTS

Statistical evaluation revealed no significant differences in the evaluated parameters for differentiation of the distinct disease groups. Compared with healthy controls however we found statistically significant differences between almost of all the investigated parameters, even in disease cases with clinically unaffected distal musculature.

CONCLUSION

The parameters are able to distinguish between healthy and affected musculature but not between distinct disease entities. Studies are needed to establish whether or not the parameters are helpful to monitor muscle involvement and disease progression in neuromuscular diseases.

Percolation in hierarchical scale-free nets

We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small world (the diameter grows either algebraically or logarithmically with the net size), assortative or disassortative (a measure of the tendency of like-degree nodes to be connected to one another), or possess various degrees of clustering. The percolation phase transition can be analyzed exactly in all these cases, due to the self-similar structure of the hierarchical nets. We find different types of criticality, illustrating the crucial effect of other structural properties aside from the scale-free degree distribution of the nets.

Surfaces of percolation systems in lattice problems

The internal and external surface area of a percolation cluster along with a full surface area of whole percolation system are investigated both analytically and numerically. Numerical simulation is performed by a Monte Carlo method for site and bond problems on square and simple cubic lattices. It is shown that both the external and full surface areas of a percolation cluster as well as the full surface area of the whole percolation system have maxima for a certain share of occupied sites (for the site problem) or permeable bonds (for a bond problem). On the basis of a probabilistic approach, analytical expressions are obtained which relate the surface area of percolation cluster to its density. The last value has been studied in more details at present that allows to analyze the behavior of the above-mentioned surface for various lattices. Two particular technological processes are discussed where the surface area of a percolation cluster plays an important part: generation of electric current in a fuel cell and self-propagating high-temperature synthesis in heterogeneous condensed systems.

Diffusion on random-site percolation clusters: theory and NMR microscopy experiments with model objects

Quasi-two-dimensional random-site percolation model objects were fabricated based on computer-generated templates. Samples consisting of two compartments, a reservoir of H2O gel attached to a percolation model object, which was initially filled with D2O, were examined with nuclear magnetic resonance microscopy for rendering proton spin density maps. The propagating proton/deuteron interdiffusion profiles were recorded and evaluated with respect to anomalous diffusion parameters. The deviation of the concentration profiles from those expected for unobstructed diffusion directly reflects the anomaly of the propagator for diffusion on a percolation cluster. The fractal dimension of the random walk d(w) evaluated from the diffusion measurements on the one hand and the fractal dimension d(f) deduced from the spin density map of the percolation object on the other permits one to experimentally compare dynamical and static exponents. Approximate calculations of the propagator are given on the basis of the fractional diffusion equation. Furthermore, the ordinary diffusion equation was solved numerically for the corresponding initial and boundary conditions for comparison. The anomalous diffusion constant was evaluated and is compared to the Brownian case. Some ad hoc correction of the propagator is shown to pay tribute to the finiteness of the system. In this way, anomalous solutions of the fractional diffusion equation could experimentally be verified.

Flow through percolation clusters: NMR velocity mapping and numerical simulation study

Three- and (quasi-)two-dimensional percolation objects have been fabricated based on Monte Carlo generated templates. The object size was up to 12 cm (300 lattice sites) in each dimension. Random site, semicontinuous swiss-cheese, and semicontinuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by nuclear magnetic resonance (NMR) imaging and, after exerting a pressure gradient, by NMR velocity mapping. The spatial resolutions of the fabrication process and the NMR experiments were 400 microm and better than 300 microm, respectively. The experimental velocity resolution was 60 microm/s. The fractal dimension, the correlation length, and the percolation probability can be evaluated both from the computer generated templates and the corresponding NMR spin density maps. Based on velocity maps, the percolation backbones were determined. The fractal dimension of the backbones turned out to be smaller than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law, the exponent of which was not yet considered in the theoretical literature. The experimental results favorably compare to computer simulations based on the finite-element method (FEM) or the finite-volume method (FVM). This demonstrates that NMR microimaging as well as FEM/FVM simulations reliably reflect transport features in percolation clusters.

Flow, diffusion, and thermal convection in percolation clusters: NMR experiments and numerical FEM/FVM simulations

Percolation objects were fabricated based on computer-generated, two- or three-dimensional templates. Random-site, semi-continuous swiss cheese, and semi-continuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by NMR imaging and, in the presence of a pressure gradient, NMR velocity mapping. The fractal dimension, the correlation length, and the percolation probability were evaluated both from the computer-generated templates and the corresponding NMR spin density maps. Based on velocity maps, the percolation backbones were determined. The fractal dimension of the backbones turned out to be smaller than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law the exponent of which was not yet considered in the theoretical literature. The experimental results favorably compare to computer simulations based on the finite-element method (FEM) or the finite-volume method (FVM). Percolation theory suggests a relationship between the anomalous diffusion exponent and the fractal dimension of the cluster, i.e., between a dynamic and a structural parameter. We examined interdiffusion between two compartments initially filled with H2O and D2O, respectively, by proton imaging. The results confirm the theoretical expectation. As a third transport mechanism, thermal convection in percolation clusters of different porosities was studied with the aid of NMR velocity mapping. The velocity distribution is related to the convection roll size distribution. Corresponding histograms consist of a power law part representing localized rolls, and a high-velocity cut-off for cluster-spanning rolls. The maximum velocity as a function of the porosity clearly visualizes the percolation transition.

Dynamic contrast-enhanced MRI and fractal characteristics of percolation clusters in two-dimensional tumor blood perfusion

Dynamic contrast-enhanced magnetic resonance imaging (DE-MRI) of the tumor blood pool is used to study tumor tissue perfusion. The results are then analyzed using percolation models. Percolation cluster geometry is depicted using the wash-in component of MRI contrast signal intensity. Fractal characteristics are determined for each two-dimensional cluster. The invasion percolation model is used to describe the evolution of the tumor perfusion front. Although tumor perfusion can be depicted rigorously only in three dimensions, two-dimensional cases are used to validate the methodology. It is concluded that the blood perfusion in a two-dimensional tumor vessel network has a fractal structure and that the evolution of the perfusion front can be characterized using invasion percolation. For all the cases studied, the front starts to grow from the periphery of the tumor (where the feeding vessel was assumed to lie) and continues to grow toward the center of the tumor, accounting for the well-documented perfused periphery and necrotic core of the tumor tissue.

Criticality of D=2 and D=3 Ising models: cluster structure versus populations

The energy and the specific heat of two- and three-dimensional Ising systems are analyzed in terms of cluster properties. The energy and the specific heat are decomposed into two components, which are defined by quantities pertaining to cluster populations and cluster structure expressed in terms of average cluster perimeters. It is shown that the structural component of the energy as well as of the specific heat represents the dominant contribution. Indications are presented that the critical exponent of structural and populational components of specific heat matches the exponent of the entire specific heat.

Characterization of human head vasculature by percolation parameters

A data reduction procedure, originally proposed for characterization of fractals and random percolation clusters, has been used to evaluate the vascular system of the human head. The motivation behind this study arose from the wish to study empirically transport properties of vascular systems and to find a suitable formalism for their description. MR angiographic data acquired by a standard 3D inflow method were used. The evaluated parameters refer to the backbone fractal dimensionality and the correlation length. The fractal dimensionality of the backbone was found to be 1.71 for the human head vasculature. This value fits the theoretical range of random percolation networks. It is concluded that concepts of percolation theory might have some value for characterizing the structure and transport properties of the vascular system.

Six-dimensional spin density/velocity NMR microscopy of percolation clusters

Using computer-simulated random-site percolation networks as templates, three-dimensional percolation cluster objects were fabricated. The pore space was filled with water and experimentally investigated with the aid of NMR microimaging. A pulse sequence for six-dimensional spin density/velocity NMR imaging was employed for the combined record of the three-dimensional spin-density distribution and the three-dimensional velocity vector field of water percolating through the pore space. An evaluation procedure for the NMR image data was established that reliably renders the characteristic parameters (fractal dimensionality, fractal dimensionality of the backbone, correlation length).

Stochastic webs and continuum percolation in quasiperiodic media

We report the results of an analytical and numerical study of the contour line and surface geometry in two models of continuum percolation with quasiperiodic properties. Both the fractal dimension of long isolines and the scaling coefficient nu are determined analytically for the two-dimensional percolation problem. The scaling characteristics of the isosurfaces of the three-dimensional potential function with an icosahedral symmetry are obtained using computer graphic representation.