Incipient Percolation Cluster Research Articles

Superlocalization of waves and electrons and hopping conductivity on classical and superconductive fractals

We have found an unexpected paradoxical situation in the percolation transition: the superconductive behavior below and above the threshold. We have found also the two different density of states d s=4/3 and d v=1.05 and the inverse localization lengths for fractons with the scalar and vector interactions, respectively. In this concept the wave functions of electrons or waves on an incipient percolation cluster and fractal dilute structure exhibit superlocalization behavior of the form ψ( r)∝exp[− r d φ ] with values of d φ1 =1.73 and d φ2 =2.4 for the former and the latter. Applications of these results for thermally activated hopping conductivity σ( t)∝exp[−( T 0/ T) β ] between impurities on a random fractal structure give the values of β=2/5 for the scalar and β=1/2 (Mott's law) for the vector interactions, respectively. Band states are localized in classical and superlocalized in superconductive percolations.

"Generalized des Cloizeaux" exponent for self-avoiding walks on the incipient percolation cluster.

We study the asymptotic shape of self-avoiding random walks (SAW) on the backbone of the incipient percolation cluster in d-dimensional lattices analytically. It is generally accepted that the configurational averaged probability distribution function <P(B)(r,N)> for the end-to-end distance r of an N step SAW behaves as a power law for r-->0. In this work, we determine the corresponding exponent using scaling arguments, and show that our suggested "generalized des Cloizeaux" expression for the exponent is in excellent agreement with exact enumeration results in two and three dimensions.

SZNAJD SOCIAL MODEL ON SQUARE LATTICE WITH CORRELATED PERCOLATION

The Sznajd model for the spread of opinions is applied to the incipient infinite percolation cluster on the square lattice, with and without power-law correlations for the occupied sites. We trace the phase transition between the two fixed points of all spins up and all spins down.

The potential energy landscape of the ±J Ising spin glass

We calculate `inherent structures' (configurations corresponding tolocal potential energy minima) in the ±J Ising spin glass overa wide range of temperature T in two and three dimensions. Wefind that the T-dependence of the average value of the inherent-structure energy E is strikingly similar to that shown recentlyfor a glass-forming liquids. E decreases with T only weakly athigh T, but begins to decrease much more rapidly as the spin-glasstransition temperature Tsg is approached. As in the liquid, wefind that the rapid decrease of E with decreasing T occurs inthe regime of T in which the relaxation of the spinautocorrelation function becomes increasingly non-exponential. Inaddition, we show that the inherent structures of the spin glass canbe used to identify clusters of strongly correlated `frozen' spins,and that an incipient percolating cluster of these spins appears(within numerical error) at T = Tsg.

The percolation process on a tree where infinite clusters are frozen

Modify the usual percolation process on the infinite binary tree by forbidding infinite clusters to grow further. The ultimate configuration will consist of both infinite and finite clusters. We give a rigorous construction of a version of this process and show that one can do explicit calculations of various quantities, for instance the law of the time (if any) that the cluster containing a fixed edge becomes infinite. Surprisingly, the distribution of the shape of a cluster which becomes infinite at time t &gt; ½ does not depend on t; it is always distributed as the incipient infinite percolation cluster on the tree. Similarly, a typical finite cluster at each time t &gt; ½ has the distribution of a critical percolation cluster. This elaborates an observation of Stockmayer [12].

Distribution of dangling ends on the incipient percolation cluster

We study numerically and by scaling arguments the probability P(M) dM that a given dangling end of the incipient percolation cluster has a mass between M and M+d M. We find by scaling arguments that P( M) decays with a power law, P( M)∼ M −(1+ κ) , with an exponent κ= d f B / d f , where d f and d f B are the fractal dimensions of the cluster and its backbone, respectively. Our numerical results yield κ=0.83 in d=2 and κ=0.74 in d=3 in very good agreement with theory.

Critical behavior of superfluid 4He in aerogel.

We report on Monte Carlo studies of the critical behaviour of superfluid $^4$He in the presence of quenched disorder with long-range fractal correlations. According to the heuristic argument by Harris, uncorrelated disorder is irrelevant when the specific heat critical exponent $\alpha$ is negative, which is the case for the pure $^4$He. However, experiments on helium in aerogel have shown that the superfluid density critical exponent $\zeta$ changes. We hypothesize that this is a cross-over effect due to the fractal nature of aerogel. Modelling the aerogel as an incipient percolating cluster in 3D and weakening the bonds at the fractal sites, we perform XY-model simulations, which demonstrate an increase in $\zeta$ from $0.67 \pm 0.005 $ for the pure case to an apparent value of $0.722\pm 0.005$ in the presence of the fractal disorder, provided that the helium correlation length does not exceed the fractal correlation length.

Selective magnetic dilution and site percolation

The selective magnetic dilution is discussed by a new effective field theory. The phase diagram at T = 0 K is also investigated by the renormalization group, and the role of site selectivity is clarified in the study of the correlation length exponent and the fractal dimension of the incipient percolation cluster.

Monte Carlo analysis of percolation of line segments on a square lattice.

We study the percolative properties of bidimensional systems generated by a random sequential adsorption of line segments on a square lattice. As the segment length grows, the percolation threshold decreases, goes through a minimum, and then increases slowly for large segments. We explain this nonmonotonic behavior by a structural change of the percolation clusters. However, an accurate measurement of the exponent \ensuremath{\nu} and of the fractal dimension D of the incipient infinite percolating cluster for several segment lengths shows that these systems belong to the universality class of random site percolation.

Simulation of Mandelbrot's self-similarity for square site critical percolation clusters and random walks

Each site visited by a walker or occupied by an incipient infinite percolation cluster is surrounded by a neighborhood of size b. The total area within a square of size R covered by these neighborhoods then is found to obey self-similarity if 1 < < b < R < < L where L is the linear dimension of lattice or walk.

Statistical mechanics of random paths on disordered lattices

The dependence of the universality class on the statistical weight of unrestricted random paths is explicitly shown both for deterministic and statistical fractals such as the incipient infinite percolation cluster. Equally weighted paths (ideal chain) and kinetically generated paths (random walks) belong, in general, to different universality classes. For deterministic fractals exact renormalization group techniques are used. Asymptotic behaviors for the end-to-end distance ranging from power to logarithmic (localization) laws are observed for the ideal chain. In all these cases, random walks in the presence of nonperfect traps are shown to be in the same universality class of the ideal chain. Logarithmic behavior is reflected insingular renormalization group recursions. For the disordered case, numerical transfer matrix techniques are exploited on percolation clusters in two and three dimensions. The two-point correlation function scales with critical exponents not obeying standard scaling relations. The distribution of the number of chains and the number of chains returning to the starting point are found to be well approximated by a log-normal distribution. The logmoment of the number of chains is found to have an essential type of singularity consistent with the log-normal distribution. A non-self-averaging behavior is argued to occur on the basis of the results.

Irreversible phase transitions in a surface-reaction model with diffusion- and adsorption-limited reactions.

A model is proposed and studied for the heterogeneously catalyzed monomer-monomer reaction of the type A+B\ensuremath{\rightarrow}AB. In two dimensions, the active sites (AS's) of the catalyst belong to a (fractal) incipient percolation cluster, while the remaining surface, i.e., the inactive sites (IS's), is reaction passive. B adsorption is only allowed on AS's while IS's are embedded on a (fully) A-covered state. A monomers can diffuse from IS to AS and reaction takes place when A and B monomers are adsorbed on a pair of nearest-neighbor AS's. The model has a single parameter, namely, the adsorption probability of B monomers (${\mathit{p}}_{\mathit{B}}$). Starting from an empty configuration of AS's, two points ${\mathit{p}}_{\mathit{B}1}$ and ${\mathit{p}}_{\mathit{B}2}$ are found such that for ${\mathit{p}}_{\mathit{B}}$\ensuremath{\le}${\mathit{p}}_{\mathit{B}1}$ (${\mathit{p}}_{\mathit{B}}$\ensuremath{\ge}${\mathit{p}}_{\mathit{B}2}$) the fractal cluster of AS's becomes poisoned by A (B) monomers. For ${\mathit{p}}_{\mathit{B}1}$${\mathit{p}}_{\mathit{B}}$${\mathit{p}}_{\mathit{B}2}$ the reaction reaches a stationary regime with AB production. So, both ${\mathit{p}}_{\mathit{B}1}$\ensuremath{\simeq}0.2075\ifmmode\pm\else\textpm\fi{}0.0025 and ${\mathit{p}}_{\mathit{B}2}$\ensuremath{\simeq}0.3600\ifmmode\pm\else\textpm\fi{}0.0025 are critical points at which second-order irreversible phase transitions take place. The dynamic critical behavior of the reaction is also studied and the obtained critical exponents strongly suggest that the model does not belong to the Reggeon-field-theory universality class, in contrast to rather well-established conjectures.

Vibrational excitations in percolation: Localization and multifractality.

We discuss localized excitations on the incipient infinite percolation cluster. Assuming a simple exponential decay of the amplitudes ${\mathrm{\ensuremath{\psi}}}_{\mathit{i}}$ in terms of the chemical (minimal) path, we show theoretically that the \ensuremath{\psi}'s are characterized by a logarithmically broad distribution, and display multifractal features as a function of the Euclidean distance. The moments of ${\mathrm{\ensuremath{\psi}}}_{\mathit{i}}$ exhibit novel crossover phenomena. Our numerical simulations of fractons exhibit a nontrivial distribution of localization lengths, even when the chemical distance is fixed. These results are explained via a generalization of the theory.

Monte Carlo simulation of transient currents induced by a random walk on percolation clusters

Abstract The transient current induced by the random walk of particles in site percolation clusters in a 30 × 30 × 30 simple-cubic lattice is calculated with the Monte Carlo technique. The calculated transient current could be fitted by a pair of hyperbolic decay functions and can describe the dispersive transient currents observed in time-of-flight experiment on amorphous semiconductors. We have found that the dispersion parameter α1 calculated for incipient percolation clusters is given by δ/[dbar], where [dbar] and δ are the fractal and fracton dimensions, respectively.

Monte Carlo simulation of dispersive transient transport in percolation clusters

Abstract Transient currents induced by the random walk of particles in site percolation clusters have been investigated by the Monte Carlo technique. The features of the calculated transient currents could be fitted by a pair of hyperbolic decay functions and can describe dispersive transient currents observed in time-of-flight experiments on amorphous solids. The dispersion parameter αi calculated for the incipient percolation cluster was found to be given by T/d where d and d are fractal and fracton dimensions, respectively. We indicate the similarity between the dispersion parameters calculated and those obtained experimentally for amorphous solids and further show that the temperature evolution of the dispersion parameters of a-Si: H can be explained by a random walk in the percolation clusters.

The compaction behaviour of particulate materials. An elucidation based on percolation theory

An approach based on percolation theory and principles of mechanics has been used to elucidate the relationship between the normalized solid fraction P of compacts and compressive pressure σo during compaction. The series of linear regions comprising the P=f(ln σ) relationship has been shown to be a manifestation of the different stages, namely, powder → flexible compact → rigid compact → continuum solid body, which the particulate system goes through during compaction. The transitions from one state to another are signified by percolation thresholds which occur in crossover regions. Near the percolation thresholds, there is a deviation in the P=f(ln σ,) relationship. Thus, different relationships exist between P and ln σ, near the percolation thresholds and away from the percolation thresholds. All the powders exhibited a particulate bond percolation threshold P cs, where a continuous interparticle network of bonds spanning the system is formed. Other percolation thresholds were identified as the rigidity threshold P R and the percolation threshold of the pores P ca. The rigidity threshold P R is the point where the flexible incipient percolating cluster of the particles becomes rigid and was exhibited by soft materials. The pore percolation threshold P ca is the point where the pores cross over from a percolating continuous interconnected structure to disconnected isolated structures. In other words, it signifies the transition from a particulate system to a continuum. Under the conditions of cold compression used in this study, only highly ductile materials exhibited P ca. A brittle—ductile transition P T, where the deformation behaviour gradually changes from a predominantly brittle to a predominantly plastic one, was also identified. The exhibition of P R, P T and P ca in the P=f(in σ) profile is therefore dependent on the mechanical properties of the materials. Thus, the approach offers a way of classifying materials into soft or hard and ductile or brittle.

On the self-poisoning of small particles upon island formation of the reactants in a model for a heterogeneously catalyzed reaction

Assuming the Langmuir–Hinshelwood mechanism, the reaction A+(1/2) B2■AB is studied on both small homogeneous and disordered surfaces by means of computer simulations. The occurrence of a new self-poisoning regime, where for a determined initial condition the substrata could be completely covered for each of the reactants with a certain probability, is observed and discussed. Large fluctuations in the size of adsorbed islands formed by A and B particles would cause the observed behavior. A crossover from the new self-poisoning regime to a steady state with AB production occurs at L≂30±5 and pA≂0.36±0.02 for incipient percolation clusters and at L≂12±2 and pA≂0.51±0.02 on the square lattice, where L and pA are the lattice size and the mole fraction of the specie A in the gas phase, respectively. It is shown that surface diffusion of A particle does not influence the location of the self-poisoning regime.

Diffusion on two-dimensional percolation clusters with multifractal jump probabilities

By means of Monte Carlo simulations we studied the properties of diffusion limited recombination reactions (DLRR's) and random walks on two dimensional incipient percolation clusters with multifractal jump probabilities. We claim that, for these kind of geometric and energetic heterogeneous substrata, the long time behavior of the particle density in a DLRR is determined by a random walk exponent. It is also suggested that the exploration of a random walk is compact. It is considered a general case of intersection ind euclidean dimension of a random fractal of dimension DF and a multifractal distribution of probabilities of dimensionsD q (q real), where the two dimensional incipient percolation clusters with multifractal jump probabilities are particular examples. We argue that the object formed by this intersection is a multifractal of dimensionsD' q =D q +D F -d, for a finite interval ofq.

Diffusion-limited aggregation near the percolation threshold

Diffusion-limited aggregation (DLA) and dielectric breakdown (DB) models have been used to simulate growth controlled by a Laplacian field on a square lattice network. A fraction ƒ (near the percolation threshold ƒ c ) of the bonds had a high conductivity (equal to 1), while the others had a low conductivity, equal to R . We used 10 -5 < R < 1 for DLA and R = 10 -8 for DB. We find crossover from growth on an incipient percolation cluster, with fractal dimensionality D ≅ 1.3, for small length scales, to that on a uniform substrate ( D ≅ 1.7), for a large length scales. The crossover length behaves as L R ≈ - a , with the crossover exponent a ≅ 0.25. The results were using a scaling theory.

Multifractal structure of the incipient infinite percolating cluster.

By analyzing the voltage distribution of a random resistor network, we show that the backbone of the percolating cluster can be partitioned into an infinity of subsets, each one characterized by a fixed value of x\ensuremath{\equiv}lnV/ln${V}_{\mathrm{max}}$, where V is the voltage across each bond and ${V}_{\mathrm{max}}$ is its maximum value. Each subset is characterized by a distinct value of the fractal dimension \ensuremath{\varphi}(x), and as a consequence an infinite set of order parameters is required to describe the backbone structure. A new scaling approach and a real-space renormalization-group treatment are presented to treat the novel aspects of this problem. The mechanism for multifractality based on an underlying multiplicative process is illustrated on a hierarchical model.