Series study of percolation moments in general dimension

High-precision anomalous dimension of three-dimensional percolation and spatial profile of the critical giant cluster

In three-dimensional percolation, we apply and test the critical geometry approach for bounded critical phenomena based on the fractional Yamabe equation. The method predicts the functional shape of the order parameter profile ϕ, which is obtained by raising the solution of the Yamabe equation to the scaling dimension Δ_{ϕ}. The latter can be fixed from outcomes of numerical simulations, from which we obtain Δ_{ϕ}=0.47846(71) and the corresponding value of the anomalous dimension η=-0.0431(14). The comparison with values of η determined by using scaling relations is discussed. A test of hyperscaling is also performed.

Connectedness percolation of fractal liquids

We apply connectedness percolation theory to fractal liquids of hard particles, and make use of a Percus-Yevick liquid state theory combined with a geometric connectivity criterion. We find that in fractal dimensions the percolation threshold interpolates continuously between integer-dimensional values, and that it decreases monotonically with increasing (fractal) dimension. The influence of hard-core interactions is significant only for dimensions below three. Finally, our theory incorrectly suggests that a percolation threshold is absent below about two dimensions, which we attribute to the breakdown of the connectedness Percus-Yevick closure.

A predictive model for fracture in human ribs based on in vitro acoustic emission data

PURPOSE

The aim of this paper is to propose a fracture model for human ribs based on acoustic emission (AE) data. The accumulation of microcracking until a macroscopic crack is produced can be monitored by AE. The macrocrack propagation causes the loss of the structural integrity of the rib.

METHODS

The AE technique was used in in vitro bending tests of human ribs. The AE data obtained were used to construct a quantitative model that allows an estimation of the failure stress from the signals detected. The model predicts the ultimate stress with an error of less than 3.5% (even at stresses 15% lower than failure stress), which makes it possible to safely anticipate the failure of the rib.

RESULTS

The percolation theory was used to model crack propagation. Moreover, a quantitative probability-based model for the expected number of AE signals has been constructed, incorporating some ideas of percolation theory. The model predicts that AE signals associated with micro-failures should exhibit a vertical asymptote when stress increases. The occurrence of this vertical asymptote was attested in our experimental observations. The total number of microfailures detected prior to the failure is N ≈ 100 and the ultimate stress is σ ∞ = 197 ± 62 MPa. A significant correlation (p < 0.0001) between σ ∞ and the predicted value is found, using only the first N = 30 micro-failures (correlation improves for N higher).

CONCLUSIONS

The measurements and the shape of the curves predicted by the model fit well. In addition, the model parameters seem to explain quantitatively and qualitatively the distribution of the AE signals as the material approaches the macroscopic fracture. Moreover, some of these parameters correlate with anthropometric variables, such as age or Body Mass Index. The proposed model could be used to predict the structural failure of ribs subjected to bending.

Anisotropic avalanches and critical depinning of three-dimensional magnetic domain walls

Simulations with more than 10^{12} spins are used to study the motion of a domain wall driven through a three-dimensional random-field Ising magnet (RFIM) by an external field H. The interface advances in a series of avalanches whose size diverges at a critical external field H_{c}. Finite-size scaling is applied to determine critical exponents and test scaling relations. Growth is intrinsically anisotropic with the height of an avalanche normal to the interface ℓ_{⊥} scaling as the width along the interface ℓ_{∥} to a power χ=0.85±0.01. The total interface roughness is consistent with self-affine scaling with a roughness exponent ζ≈χ that is much larger than values found previously for the RFIM and related models that explicitly break orientational symmetry by requiring the interface to be single-valued. Because the RFIM maintains orientational symmetry, the interface develops overhangs that may surround unfavorable regions to create uninvaded bubbles. Overhangs complicate measures of the roughness exponent but decrease in importance with increasing system size.

Ising spin glasses in dimension five

Ising spin-glass models with bimodal, Gaussian, uniform, and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin-glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two.

Precise calculation of a bond percolation transition and survival rates of nodes in a complex network

Through precise numerical analysis, we reveal a new type of universal loopless percolation transition in randomly removed complex networks. As an example of a real-world network, we apply our analysis to a business relation network consisting of approximately 3,000,000 links among 300,000 firms and observe the transition with critical exponents close to the mean-field values taking into account the finite size effect. We focus on the largest cluster at the critical point, and introduce survival probability as a new measure characterizing the robustness of each node. We also discuss the relation between survival probability and k-shell decomposition.

Surface and bulk criticality in midpoint percolation

The concept of midpoint percolation has recently been applied to characterize the double percolation transitions in negatively curved structures. Regular d-dimensional hypercubic lattices are investigated in the present work using the same concept. Specifically, the site-percolation transitions at the critical thresholds are investigated for dimensions up to d=10 by means of the Leath algorithm. It is shown that the explicit inclusion of the boundaries provides a straightforward way to obtain critical indices, both for the bulk and surface parts. At and above the critical dimension d=6, it is found that the percolation cluster contains only a finite number of surface points in the infinite-size limit. This is in accordance with the expectation from studies of lattices with negative curvature. It is also found that the number of surface points, reached by the percolation cluster in the infinite limit, approaches 2d for large dimensions d. We also note that the size dependence in proliferation of percolating clusters for d>or=7 can be obtained by solely counting surface points of the midpoint cluster.

Ferromagnetic phase transition for the spanning-forest model (q-->0 limit of the Potts model) in three or more dimensions

We present Monte Carlo simulations of the spanning-forest model (q-->0 limit of the ferromagnetic Potts model) in spatial dimensions d=3, 4, 5. We show that, in contrast to the two-dimensional case, the model has a ferromagnetic second-order phase transition at a finite positive value w(c). We present numerical estimates of w(c) and of the thermal and magnetic critical exponents. We conjecture that the upper critical dimension is 6.

High-temperature series expansions for the q-state Potts model on a hypercubic lattice and critical properties of percolation

We present results for the high-temperature series expansions of the susceptibility and free energy of the q-state Potts model on a D-dimensional hypercubic lattice ZD for arbitrary values of q. The series are up to order 20 for dimension D<or=3, order 19 for D<or=5, and up to order 17 for arbitrary D. Using the q-->1 limit of these series, we estimate the percolation threshold pc and critical exponent gamma for bond percolation in different dimensions. We also extend the 1/D expansion of the critical coupling for arbitrary values of q up to order D-9.

Dynamic scaling in entangled mean-field gelation polymers

We present a simple reaction kinetics model to describe the polymer synthesis used by Lusignan et al. [Phys. Rev. E 60, 5657 (1999)] to produce randomly branched polymers in the vulcanization class. Numerical solution of the rate equations gives probabilities for different connections in the final product, which we use to generate a numerical ensemble of representative molecules. All structural quantities probed in the experiments are in quantitative agreement with our results for the entire range of molecular weights considered. However, with detailed topological information available in our calculations, our estimate of the "rheologically relevant" linear segment length is smaller than that estimated from the experimental results. We use a numerical method based on a tube model of polymer melts to calculate the rheological properties of such molecules. Results are in good agreement with experiment, except that in the case of the largest molecular weight samples our estimate of the zero-shear viscosity is significantly lower than the experimental findings. Using acid concentration as an indicator for closeness to the gelation transition, we show that the high-molecular-weight polymers considered are at the limit of mean-field behavior--which possibly is the reason for this disagreement. For a truly mean-field gelation class of model polymers, we numerically calculate the rheological properties for a range of segment lengths. Our calculations show that the tube theory with dynamical dilation predicts that, very close to the gelation limit, the contribution to viscosity for this class of polymers is dominated by the contribution from constraint-release Rouse motion and the final viscosity exponent approaches a Rouse-like value.

Number of spanning clusters at the high-dimensional percolation thresholds

A scaling theory is used to derive the dependence of the average number k of spanning clusters at threshold on the lattice size L. This number should become independent of L for dimensions d<6 and vary as ln L at d=6 . The predictions for d>6 depend on the boundary conditions, and the results there may vary between L(d-6) and L0. While simulations in six dimensions are consistent with this prediction [after including corrections of order ln(ln L)], in five dimensions the average number of spanning clusters still increases as ln L even up to L=201 . However, the histogram P(k) of the spanning cluster multiplicity does scale as a function of kX(L), with X(L) =1+const/L, indicating that for sufficiently large L the average k will approach a finite value: a fit of the five-dimensional multiplicity data with a constant plus a simple linear correction to scaling reproduces the data very well. Numerical simulations for d>6 and for d=4 are also presented.

Mass-fractal clustering and power-law decay of cluster size in 1-propanol aqueous solution

Mesoscale structure of 1-propanol aqueous solutions with propanol mole fraction xp ranging from 0.1 to 0.33 has been studied by means of small angle neutron scattering (SANS) and large-scale reverse Monte Carlo (RMC) technique. Analysis of the SANS intensities in terms of a fractal model shows that the fractal dimension df of mesoscale structure of the solution is about 1.8-1.9 for water-rich solution and about 1.5 for propanol-rich solution. Percolation analysis on the RMC results reveals that the water molecules and the propanol molecules cluster, respectively, as a mass fractal, the dimension dM of which is about 2.3-2.5 for both clusters for water-rich solution. Furthermore, the distribution of the cluster size is expressed by a simple power law with an exponent tau of about 1.35-1.5 for the propanol clusters and 1.05-1.2 for the water clusters. These results imply that the current solution is characterized by polydisperse mass fractals. In fact, a theoretical relation for polydisperse system of mass fractals, df = dM(2-tau), holds well in the current solution. The characteristic change in df from 1.8-1.9 to 1.5 described above is attributed to the crossover between the water-rich regime and the propanol-rich regime. Most of the water molecules and the propanol molecules are located on the interface between clusters, and the water molecules form thin layers of about 10 A thick irrespective of 1-propanol content studied.

Critical percolation in high dimensions

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4-13 dimensions. For d<6 they are preliminary, for d>or =6 they are between 20 and 10(4) times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing that allowed us to simulate clusters of millions of sites on computers with less than 500 Mbytes memory; (ii) a histogram method that allowed us to obtain information for several p values from a single simulation; and (iii) a variance reduction technique that is especially efficient at high dimensions where it reduces error bars by a factor of up to approximately 30 and more. Based on these data we propose a scaling law for finite cluster size corrections.

Cluster diffusion at the gelation point

We use molecular dynamics simulations to study a model of the gelation transition with a dynamic bond forming procedure. After establishing evidence for three-dimensional (3D) percolation as the static universality class, we turn our attention to the dynamics of clusters at the gelation point, focusing in particular on the behavior of the diffusion constant D(s) as a function of cluster size s. We find a very clear power law behavior D(s) approximately s(-k) with a nontrivial exponent k approximately 0.69.

Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions

We develop a method of constructing percolation clusters that allows us to build very large clusters using very little computer memory by limiting the maximum number of sites for which we maintain state information to a number of the order of the number of sites in the largest chemical shell of the cluster being created. The memory required to grow a cluster of mass s is of the order of s(straight theta) bytes where straight theta ranges from 0.4 for two-dimensional (2D) lattices to 0.5 for six (or higher)-dimensional lattices. We use this method to estimate d(min), the exponent relating the minimum path l to the Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site and bond percolation, we find d(min)=1.607+/-0.005 (4D) and d(min)=1.812+/-0.006 (5D). In order to determine d(min) to high precision, and without bias, it was necessary to first find precise values for the percolation threshold, p(c): p(c)=0.196889+/-0.000003 (4D) and p(c)=0.14081+/-0.00001 (5D) for site and p(c)=0.160130+/-0.000003 (4D) and p(c)=0.118174+/-0.000004 (5D) for bond percolation. We also calculate the Fisher exponent tau determined in the course of calculating the values of p(c): tau=2.313+/-0.003 (4D) and tau=2.412+/-0.004 (5D).

Are proteins well-packed?

The average packing density inside proteins is as high as in crystalline solids. Does this mean proteins are well-packed? We go beyond average densities, and look at the full distribution functions of free volumes inside proteins. Using a new and rigorous Delaunay triangulation method for parsing space into empty and filled regions, we introduce formal definitions of interior and surface packing densities. Although proteins look like organic crystals by the criterion of average density, they look more like liquids and glasses by the criterion of their free volume distributions. The distributions are broad, and the scalings of volume-to-surface, volume-to-cluster-radius, and numbers of void versus volume show that the interiors of proteins are more like randomly packed spheres near their percolation threshold than like jigsaw puzzles. We find that larger proteins are packed more loosely than smaller proteins. And we find that the enthalpies of folding (per amino acid) are independent of the packing density of a protein, indicating that van der Waals interactions are not a dominant component of the folding forces.

Random walks on fractals and stretched exponential relaxation

Stretched exponential relaxation [exp-(t/tau)(beta(K))] is observed in a large variety of systems but has not been explained so far. Studying random walks on percolation clusters in curved spaces whose dimensions range from 2 to 7, we show that the relaxation is accurately a stretched exponential and is directly connected to the fractal nature of these clusters. Thus we find that in each dimension the decay exponent beta(K) is related to well-known exponents of the percolation theory in the corresponding flat space. We suggest that the stretched exponential behavior observed in many complex systems (polymers, colloids, glasses...) is due to the fractal character of their configuration space.

Viscoelasticity of randomly branched polymers in the vulcanization class

We report viscosity, recoverable compliance, and molar mass distribution for a series of randomly branched polyester samples with long linear chain sections between branch points. Molecular structure characterization determines tau=2.47+/-0.05 for the exponent controlling the molar mass distribution, so this system belongs to the vulcanization (mean-field) universality class. Consequently, branched polymers of similar size strongly overlap and form interchain entanglements. The viscosity diverges at the gel point with an exponent s=6.1+/-0.3, that is significantly larger than the value of 1.33 predicted by the branched polymer Rouse model (bead-spring model without entanglements). The recoverable compliance diverges at the percolation threshold with an exponent t=3.2+/-0.2. This effect is consistent with the idea that each branched polymer of size equal to the correlation length stores k(B)T of elastic energy. Near the gel point, the complex shear modulus is a power law in frequency with an exponent u=0.33+/-0.05. The measured rheological exponents confirm that the dynamic scaling law u=t/(s+t) holds for the vulcanization class. Since s is larger and u is smaller than the Rouse values observed in systems that belong to the critical percolation universality class, we conclude that entanglements profoundly increase the longest relaxation time. Examination of the literature data reveals clear trends for the exponents s and u as functions of the chain length between branch points. These dependencies, qualitatively explained by hierarchical relaxation models, imply that the dynamic scaling observed in systems that belong to the vulcanization class is nonuniversal.

Transport on an annealed disordered lattice

We study the diffusion on an annealed disordered lattice with a local dynamical reorganization of bonds. We show that the typical rearrangement time depends on the renewal rate like tr approximately tau(alpha) with alpha not equal 1. This implies that the crossover time to normal diffusion in a slow rearrangement regime shows a critical behavior at the percolation threshold. Additional scaling relations for the dependence of the diffusion coefficient on the renewal rate are obtained. The derivation of scaling exponents confirms the crucial role of singly connected bonds in transport properties. These results are checked by numerical simulations in two and three dimensions.

Entropic rigidity of randomly diluted two- and three-dimensional networks

In recent work, we presented evidence that site-diluted triangular central-force networks, at finite temperatures, have a nonzero shear modulus for all concentrations of particles above the geometric percolation concentration p(c). This is in contrast to the zero-temperature case where the (energetic) shear modulus vanishes at a concentration of particles p(r)>p(c). In the present paper we report on analogous simulations of bond-diluted triangular lattices, site-diluted square lattices, and site-diluted simple-cubic lattices. We again find that these systems are rigid for all p>p(c) and that near p(c) the shear modulus mu approximately (p-p(c))(f), where the exponent f approximately 1.3 for two-dimensional lattices and f approximately 2 for the simple-cubic case. These results support the conjecture of de Gennes that the diluted central-force network is in the same universality class as the random resistor network. We present approximate renormalization group calculations that also lead to this conclusion.

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.