Critical Percolation Density Research Articles

Topological analysis of 3D fracture networks: Graph representation and percolation threshold

This work presents a framework to analyze geometrical, topological and hydraulic properties of 3D Discrete Fracture Networks (DFN) and to estimate its properties near percolation threshold. A set of efficient algorithms has been developed to perform geometrical and topological analyses upon 3D networks of planar fractures with various shapes (mainly circular and elliptical fractures). This toolbox is capable of (i) calculating all possible intersections in the 3D network, and calculating also the resulting trace lengths, and other geometrical attributes of the DFN; (ii) extracting the percolating clusters and eliminating dead end clusters; (iii) constructing the corresponding graph of the 3D network of planar fractures. All the calculations implemented with these algorithms have been strictly validated by direct numerical simulations. The graph representation of the DFN enables the application of broad-purpose algorithms inspired by graph theory and percolation theory (among other applications). It is demonstrated that the use of our toolbox as a pre-treatment (extracting the percolation network, eliminating all dead-end fractures and clusters, and searching for shortest paths), considerably reduces the CPU time of flow/transport simulations. The gain on CPU time was of several orders of magnitude for networks containing thousands of fractures. The computational efficiency of the tools permits a broad study on the percolation in 3D fracture networks. The two main results presented in this paper concern two important issues: (i)a new procedure is developed for the numerical determination of the critical percolation density based on the marginal percolation concept; and (ii)an enhanced formula with low sensitivity to fracture shape, orientation, and size distribution, is proposed for estimating the critical percolation density.

Thermal percolation in mixtures of monodisperse spheres

The connectivity of individual species in a locally heterogeneous granular mixture strongly influences assembly-scale behavior. A behavioral transition is observed at the percolation threshold for a given constituent; that is, the mixing fraction at which the constituent has statistical connectivity between two opposing boundaries. This behavior is particularly evident in conductivity phenomena, e.g., the percolation of conductive particles (thermal, electric) or the relative degree of connectivity of the void space (hydraulic). Hard-core (first nearest-neighbor or lattice) percolation has been extensively studied experimentally, theoretically, and numerically. That hard-core percolation occurs in dense randomly packed bi-phasic mixtures of monodisperse spheres occurs at a mixing fraction of 0.15 v/v is well-accepted. Radiant conduction (e.g., heat), however, is influenced by hard-core “soft-shell” percolation, which is an nth-nearest neighbor problem and less well-studied. In the current work, we use discrete element method simulations coupled with a thermal network model that leverages a robust domain decomposition algorithm to simulate large assemblies of spheres to investigate soft-shell percolation numerically. Our results show that the thermal conductivity of a randomly packed assembly obeys a power law with respect to volume fraction of conductive particles while percolation threshold follows a power law with respect to coordination number. The ability of the pore fluid to transmit heat over a longer distance results in an increase of thermal conductivity and a decrease in thermal percolation threshold. Moreover, we observe that, contrary to previous findings, critical percolation density is not a dimensional invariant and depends on the microstructure of the assembly.

Estimating properties of kinetoplast DNA by fragmentation reactions

The mitochondrial DNA of trypanosomes, called kinetoplast DNA (kDNA) contains thousands of minicircles that are topologically linked into a single structure that resembles a medieval chainmail. This biological chainmail is characterized by two parameters: the link type between minicircles, and the number of minicircles linked to each minicircle (i.e. the minicircle valence). In previous works, a protocol was proposed to determine the mean value of the minicircle valence. In these experiments, minicircles were excised from the network and the products compared with those obtained from fragmenting idealized structures. These idealized structures assumed a negligible variance in the distribution of valences of the initial network. It is therefore unclear to what extent this theoretical analysis captures the true topology of the kDNA network when kDNA samples are extracted from unsynchronized cells or from cells with silenced kDNA replication genes. Subsequent studies proposed that there is a critical percolation density during network formation. We asked whether this density can be estimated using fragmentation reactions.The goal of this work is to develop a mathematical method that can be used to estimate the mean valence of networks when the variance of the valence is non-negligible. We first show microscopy data on Crithidia fasciculata that, in agreement with the original experimental results, show a distribution of valences with nonzero variance. Second, we use computer simulations of network fragmentation to show that the predicted and actual mean valence are different when the valence distribution has nonzero variance. We propose a more general mathematical formulation and computer simulations of kDNA fragmentation to estimate this value. Last, we show that fragmentation experiments may lead to errors in the estimation of the critical percolation density since the collapsing density depends on the initial density of the network, and on the fragmentation reaction.

Energy and centrality dependent study of deconfinement phase transition in a color string percolation approach at RHIC energies

We take the experimental data for transverse momentum spectra of identified charged hadrons in different centrality classes for nucleus-nucleus (A+A) collisions at various Relativistic Heavy-Ion Collider (RHIC) energies measured by the STAR collaboration. We analyse these data in the framework of color string percolation model (CSPM) in order to extract various percolation parameters at different centralities at RHIC energies to study the effect of collision geometry and collision energy. We use these parameters to study the centrality dependent behaviour of initial temperature of the percolation cluster, energy density, average transverse momentum, shear viscosity to entropy density ratio ($\eta/s$) and trace anomaly for different energies at RHIC from $\sqrt{s_{\rm NN}}$ = 19.6 to 200 GeV. These observables are found to strongly depend on centrality at various collision energies. The critical percolation density, which is related to the deconfinement phase transition is achieved in the most central nucleus-nucleus collisions, while it fails in the peripheral collisions. A universal scaling is observed in color suppression factor and initial temperatures when studied as a function of charged particle pseudorapidity distribution scaled by nuclear overlap area at RHIC energies. The minimum of $\eta/s$ is observed in the most central collisions at $\sqrt{s_{\rm NN}}$ = 130 and 200 GeV.

Study of metal insulator transition: percolation-type in GaAs/AlGaAs 2D

In this work the existence of a possible percolation transition in a dilute two dimensional hole gas in GaAs/AlGaAs within a parallel magnetic field at a fixed temperature T = 260mk is studied. Following the evolution of the electrical conductivity as a function of magnetic field at high carrier densities, we have showed the creation of a densitity inhomogeneity beyond the critical percolation leading to metallic side. These results were proved in the present article by studying the exponent characterizing the type of transition and the variation of the critical percolation densities, which are considered good indicators to observe the percolation transition. In this investigation, we have reanalyzed the data obtained by Kumar et al published in the reference [M. Kumar et al, Solid State Communications Vol. 135. pp. 57–61 (2005)].

The effect of volume exclusion on the formation of DNA minicircle networks: implications to kinetoplast DNA

Kinetoplast DNA (kDNA) is the mitochondrial of DNA of disease causing organisms such as Trypanosoma Brucei (T. Brucei) and Trypanosoma Cruzi (T. Cruzi). In most organisms, KDNA is made of thousands of small circular DNA molecules that are highly condensed and topologically linked forming a gigantic planar network. In our previous work we have developed mathematical and computational models to test the confinement hypothesis, that is that the formation of kDNA minicircle networks is a product of the high DNA condensation achieved in the mitochondrion of these organisms. In these studies we studied three parameters that characterize the growth of the network topology upon confinement: the critical percolation density, the mean saturation density and the mean valence (i.e. the number of mini circles topologically linked to any chosen minicircle). Experimental results on insect-infecting organisms showed that the mean valence is equal to three, forming a structure similar to those found in medieval chain-mails. These same studies hypothesized that this value of the mean valence was driven by the DNA excluded volume. Here we extend our previous work on kDNA by characterizing the effects of DNA excluded volume on the three descriptive parameters. Using computer simulations of polymer swelling we found that (1) in agreement with previous studies the linking probability of two minicircles does not decrease linearly with the distance between the two minicircles, (2) the mean valence grows linearly with the density of minicircles and decreases with the thickness of the excluded volume, (3) the critical percolation and mean saturation densities grow linearly with the thickness of the excluded volume. Our results therefore suggest that the swelling of the DNA molecule, due to electrostatic interactions, has relatively mild implications on the overall topology of the network. Our results also validate our topological descriptors since they appear to reflect the changes in the physical properties of the polymeric chains and at the same time remain faithful to their description of kDNA.

High-dimensional asymptotics for percolation of Gaussian free field level sets

We consider the Gaussian free field on $Z^d$, $d\geq3$, and prove that the critical density for percolation of its level sets behaves like $1/d^{1+o(1)}$ as $d$ tends to infinity. Our proof gives the principal asymptotic behavior of the corresponding critical level $h_*(d)$. Moreover, it shows that a related parameter $h_{**}(d)$ introduced by Rodriguez and Sznitman in [23] is in fact asymptotically equivalent to $h_*(d)$.

Percolation phenomena in disordered topological networks

Topological polymer networks consist of circular polymers that are topologically linked. Topological networks made of small circular DNA or protein molecules are of great interest in biology and nanotechnology because they are found in living organisms and can be constructed in-vitro. The physical factors that determine the topology of a network as well as the pathways that are followed for its formation remain poorly understood. In our previous work we proposed a novel biophysical/computational approach to model the formation of planar DNA minicircle networks in trypanosomatid parasites. This model suggests that minicircle networks in trypanosomatid parasites emerged from topologically free minicircles upon space confinement through a percolation pathway. Our model however is somewhat idealized because it assumes that the centers of the minicircles in the network are positioned following a regular planar lattice. Here we propose an extension of the model by allowing the centers of the minicircles to be randomly displaced from the vertices of the lattice. We numerically show that networks form following a percolation pathway upon increasing minicircle density. Our model suggests that the critical percolation density increases as Dperc = 0.8357 − 1.4297 exp(0.6439x) with x is the maximum displacement of the centers of the minicircles. Our results therefore show that the plane distribution of minicircles does not dramatically affect the percolation of minicircles and therefore supports they hypothesis that DNA minicircle networks in trypanosomes evolved through a percolation pathway.

The Effect of Angle Restriction on the Topological Characteristics of Minicircle Networks

Networks of topologically linked minicircle polymers are found in diverse natural systems and are a subject of intense research in nanotechonology. In a recent report the authors introduced a new theoretical model to study the effects of polymer density on the formation and on the topological properties of minicircle networks. Three key topological characteristics were identified in the formation and characterization of a network: the critical percolation density, the average saturation density and the mean valence of the network. In this work we report how these characteristics change when an orientation bias is imposed on the minicircles forming the network. We observe that such restrictions have significant effects on the key topological characteristics of the network. In particular while the effects of restriction of the tilting angle can be predicted we find that those of the azimuthal angle can have somewhat unexpected results.

The effects of density on the topological structure of the mitochondrial DNA from trypanosomes

Trypanosomatida parasites, such as trypanosoma and lishmania, are the cause of deadly diseases in many third world countries. A distinctive feature of these organisms is the three dimensional organization of their mitochondrial DNA into maxi and minicircles. In some of these organisms minicircles are confined into a small disk volume and are topologically linked, forming a gigantic linked network. The origins of such a network as well as of its topological properties are mostly unknown. In this paper we quantify the effects of the confinement on the topology of such a minicircle network. We introduce a simple mathematical model in which a collection of randomly oriented minicircles are spread over a rectangular grid. We present analytical and computational results showing that a finite positive critical percolation density exists, that the probability of formation of a highly linked network increases exponentially fast when minicircles are confined, and that the mean minicircle valence (the number of minicircles that a particular minicircle is linked to) increases linearly with density. When these results are interpreted in the context of the mitochondrial DNA of the trypanosome they suggest that confinement plays a key role on the formation of the linked network. This hypothesis is supported by the agreement of our simulations with experimental results that show that the valence grows linearly with density. Our model predicts the existence of a percolation density and that the distribution of minicircle valences is more heterogeneous than initially thought.

Fracture strength in randomly perforated plates

We have measured the forces that need to be applied quasistatically to randomly perforated rectangular aluminum plates to tear them apart. Concomitantly, for identical geometrical structures, we have carried out numerical simulations of the tearing process. The two results agree quite reasonably, and especially so for perforated area densities (P) that are rather less than the critical percolation densities. A given value of P can be achieved through various different realizations (patterns of perforations), each with a somewhat different fracture (or failure) strength F. Our results predict empirically an “expected F” [F¯(P)] and an expected standard deviation [root variance, S(P)] as functions of P. This enables the risk analysis of failures in plates under tensile stress, which is needed in, e.g., the estimation of the flying safety of a spacecraft subject to multiple perforations.

Percolation of color sources and critical temperature

We argue that clustering of color sources, leading to the percolation transition, may be the way to achieve deconfinement in heavy ion collisions. The critical density for percolation is related to the effective critical temperature of the thermal bath associated to the presence of strong color fields inside the percolating cluster. We find that the temperature is rapidity, centrality and energy dependent. We emphasize the similarities of percolation of strings with color glass condensate.

Continuum Percolation with Unreliable and Spread-Out Connections

We derive percolation results in the continuum plane that lead to what appears to be a general tendency of many stochastic network models. Namely, when the selection mechanism according to which nodes are connected to each other, is sufficiently spread out, then a lower density of nodes, or on average fewer connections per node, are sufficient to obtain an unbounded connected component. We look at two different transformations that spread-out connections and decrease the critical percolation density while preserving the average node degree. Our results indicate that real networks can exploit the presence of spread-out and unreliable connections to achieve connectivity more easily, provided they can maintain the average number of functioning connections per node.

Clustering and continuum percolation of hard spheres near a hard wall: Monte Carlo simulation and connectedness theory

The effect of a hard wall on the clustering and continuum percolation of a hard spheres fluid is studied using Monte Carlo simulations and connectedness theory. We calculate an averaged pair-connectedness function ρ†(r;z) which is the probability density of finding two particles in the same cluster and separate by a distance r under the assumption that one of them is fixed at a distance z from the wall. We also obtain the mean size S for the cluster containing the fixed sphere and the critical percolation density ρc at which it becomes macroscopically large. Monte Carlo results allow us to conclude that, for given number density and connectedness distance, the wall causes the decrease of S and the increase of ρc in comparison with those found for the bulk in the absence of the wall. Both effects diminish with increasing z. The simulation data also show that, in the presence of the wall, the clusters are eccentric with cylindrical symmetry, slightly flattened in the region of contact with the wall. The theoretical calculations involve the solution for ρ†(r;z) of an integral equation. It is derived from the one proposed some time ago by Giaquinta and Parrinello to obtain the average of the ordinary pair correlation function in the presence of the hard wall [J. Chem. Phys. 78, 1946 (1983)]. Integrating the pair-connectedness function over r we have S whose divergence determines the theoretical critical density. The results so obtained are in satisfactory agreement with Monte Carlo data.

Permeability changes in granite with crack growth during immersion in hot water

Permeability tests and observation of induced, as well as inherent, cracks have been carried out to quantitatively explain permeability changes in granite taking place during a weathering test in terms of micro-structure by cracks. To this end, permeability tensors were estimated from observations of crack systems using the Replica Method and of the crack apertures with a scanning electron microscope. The following conclusions were reached: the drastic increase in permeability around an immersion of 1000 d in 90°C water could not be explained by the existence of a percolation threshold because all the specimens were percolative in the sense that the traces of crack tensor were above the critical density for percolation. On the other hand, SEM observation revealed that, with immersion, crack apertures widened and their distribution spread. Permeability estimated using the third (not the first) moment of observed apertures showed the increase in measured permeability well. When evaluating the connectivity parameter λ to give a quantitative accordance between measured and estimated permeability, it had a constant value as low as 0.17, irrespective of the immersion period, in spite of the higher density of micro-cracks than the percolation threshold. It can be said that, in considering the natural three-dimensional crack system, the third moment of physical apertures plays a more important role in predicting the permeability and λ is greatly affected by the spread of crack apertures.

On the critical density for continuum percolation of spheres of variable radii

We propose an approximate formula to determine the critical percolation density for continuum percolation of non-interacting spheres when the radii of spheres are also random variables. We conjecture that this gives a strict lower bound to the critical density for all distributions of radii.

An exponential approximation for continuum percolation in dipolar hard-sphere fluids

We consider a generalized mean spherical approximation for the pair-connectedness function of a dipolar hard-sphere fluid. Based on its analytical solution, we propose an exponential approach to the continuum percolation of dipolar fluids. The mean cluster size and the critical percolation density so obtained agree well with previously reported Monte Carlo simulations. The Kirkwood factor calculated among connected dipoles, a magnitude that can be taken as a measure of the dipolar ordering inside the cluster, also compares well with the simulation results.

Nonuniversal diffusivity exponent for the soft-percolation process in two dimensions.

Soft percolation is a dynamic percolation process in the spatially disordered system, where the jump rate between two sites is assumed to have power-law dependence on the distance between the sites with a cutoff. We present a numerical study of the soft percolation model in two dimensions. We first derive an exact expression for the diffusion constant of a system with periodic boundary conditions. We obtain the diffusion constant as a function of the density of sites for various samples generated by a Monte Carlo technique and find, using the finite size scaling, that for 01, \ensuremath{\mu} is almost equal to the known value ${\mathrm{\ensuremath{\mu}}}_{\mathit{u}\mathit{n}}$\ensuremath{\approxeq}1.30 for the ordinary percolation process and for \ensuremath{\alpha}>1, \ensuremath{\mu} increases linearly in \ensuremath{\alpha}, where \ensuremath{\alpha} denotes the exponent of the power-law dependence. This result indicates that the contrauniversality holds, that is, the critical percolation density is independent of \ensuremath{\alpha} and the diffusivity exponent \ensuremath{\mu} depends on \ensuremath{\alpha}.

Finite-size effects in line-percolating systems

Continuous percolation of finite lines randomly distributed in two dimensions is studied numerically. Two finite-size effects in the distributions of critical percolation densities as a function of the system size are clearly distinguished for the first time in this system. The first one is a rounding of the distribution due to the finite size of the system. The second effect is a shift of the percolation threshold average. The rounding is well described by standard finite-size scaling whereas the shift is not and stems from a perimeter over surface correction. The numerical values of the critical exponents governing the rounding and the probability P ∞ for a line to belong to the percolating cluster are found to be in good agreement with theoretical predictions.

Classical and quantum continuum percolation with hard core interactions

We study the classical and quantum percolation of spheres in a three-dimensional continuum. Each sphere has an impenetrable hard core of diameter σ, and two spheres are considered to be directly connected if the distance between their centers is less than d. We calculate the critical percolation density as a function of σ/d. In the classical problem this is the density ρc at which an infinite cluster of connected spheres first forms. In the quantum problem, we study a tight-binding model where the hopping matrix element between two spheres is nonzero only if they are directly connected. In this case the critical density ρq is the density at which the eigenstates of the Hamiltonian first become extended. Our method uses Monte Carlo simulation and finite-size scaling techniques, and for the quantum problem, the concept of quantum connectivity. We find that both ρc and ρq exhibit nonmonotonic behavior as a function of σ/d. We also find that for all values of σ/d, ρq>ρc, although the ratio of the thresholds decreases with increasing σ/d. We argue that a better understanding of this ratio is obtained by considering the average coordination number. We speculate about the nature of both classical and quantum percolation as σ/d approaches 1.