Percolation Description Research Articles

Logarithmic correlation functions in 2D critical percolation

It is believed that the large-scale geometric properties of two-dimensional critical percolation are described by a logarithmic conformal field theory, but it has been challenging to exhibit concrete examples of logarithmic singularities and to find an explanation and a physical interpretation, in terms of lattice observables, for their appearance. We show that certain percolation correlation functions receive independent contributions from a large number of similar connectivity events happening at different scales. Combined with scale invariance, this leads to logarithmic divergences. We study several logarithmic correlation functions for critical percolation in the bulk and in the presence of a boundary, including the four-point function of the density (spin) field. Our analysis confirms previous findings, provides new explicit calculations and explains, in terms of lattice observables, the physical mechanism that leads to the logarithmic singularities we discover. Although we adopt conformal field theory (CFT) terminology to present our results, the core of our analysis relies on probabilistic arguments and recent rigorous results on the scaling limit of critical percolation and does not assume a priori the existence of a percolation CFT. As a consequence, our results provide strong support for the validity of a CFT description of critical percolation and a step in the direction of a mathematically rigorous formulation of a logarithmic CFT of two-dimensional critical percolation.

Conformal model for gravitational waves and dark matter: a status update

We present an updated analysis of the first-order phase transition associated with symmetry breaking in the early Universe in a classically scale-invariant model extended with a new SU(2) gauge group. Including recent developments in understanding supercooled phase transitions, we compute all of its characteristics and significantly constrain the parameter space. We then predict gravitational wave spectra generated during this phase transition and by computing the signal-to-noise ratio we conclude that this model is well-testable (and falsifiable) with LISA. We also provide predictions for the relic dark matter abundance. It is consistent with observations in a rather narrow part of the parameter space. We strongly constrain the so-called supercool dark matter scenario based on an improved description of percolation and reheating after the phase transition as well as the inclusion of the running of couplings. Finally, we devote attention to the renormalisation-scale dependence of the results. Even though our main results are obtained with the use of renormalisation-group improved effective potential, we also perform a fixed-scale analysis which proves that the dependence on the scale is not only qualitative but also quantitative.

Percolation description of charge transport in amorphous oxide semiconductors

The charge transport mechanism in amorphous oxide semiconductors (AOS) is a matter of controversial debates. Most theoretical studies so far neglected the percolation nature of the phenomenon. In this article, a recipe for theoretical description of charge transport in AOSs is formulated using the percolation arguments. Comparison with the previous theoretical studies shows a superiority of the percolation approach. The results of the percolation theory are compared to experimental data obtained in various InGaZnO materials revealing parameters of the disorder potential in such AOS.

Percolation description of charge transport in the random barrier model applied to amorphous oxide semiconductors

Charge transport in amorphous oxide semiconductors is often described as the band transport affected by disorder in the form of random potential barriers (RB). Theoretical studies in the framework of this approach neglected so far the percolation nature of the phenomenon. In this article, a recipe for theoretical description of charge transport in the RB model is formulated using percolation arguments. Comparison with the results published so far evidences the superiority of the percolation approach.

A Percolation Model for Piezoresistivity in Conductor–Polymer Composites

AbstractInsulating polymer composites with conductive filler particles are attractive for sensor applications due to their large piezoresistive response. Composite samples composed of a polymer matrix filled with particles of doped semiconductor that gives a piezoresistive response that is 105 times larger than that of bulk semiconductor sensors are prepared here. The piezoresistance of such composite materials is typically described by using a tunneling mechanism. However, it is found that a percolation description not only fits prior data better but provides a much simpler physical mechanism for the more flexible and soft polymer composite prepared and tested in this study. A simple model for the resistance as a function of applied pressure is derived using percolation theory with a conductivity exponent, s. The model is shown to fit experimental piezoresistive trends with the resistance measured both perpendicular and parallel to the pressure direction.

Introduction to Organic Semiconductors Using Accessible Undergraduate Chemistry Concepts

Organic semiconductors (OSCs) are making great progress as active components in alternative energy and flexible electronics technologies that are of interest to many undergraduates. However, in materials governed by the confluence of multiple length scales (molecular (Angstrom), intermolecular (nm), domain (100s nm)), providing a pedagogically accessible pathway to incorporating OSCs into undergraduate education can be difficult. Here, we provide a multiscale description of OSCs that relies only on concepts covered in typical undergraduate chemistry and chemical engineering curricula: a tight-binding description of molecular orbitals using Huckel theory, the miscibility of intermolecular domains using thermodynamic principles based on the Ising spin model, the incorporation of thermal disorder of both the electronic and molecular states using the Boltzmann distribution, and a simple description of charge percolation using low-level graph theory. We illustrate these topics on a small organic molecule, 1,3,...

Random geometric graph description of connectedness percolation in rod systems.

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function). We employ the observation that contributions from closed loops of connected rods are negligible in the limit of large aspect ratios to obtain percolation thresholds that are fully equivalent to those calculated within the second-virial approximation of the connectedness Ornstein-Zernike equation. Our formulation can account for effects due to interactions between the rods, and many-body features can be partially addressed by suitable choices for the edge probabilities.

Effective medium approximations for anisotropic composites with arbitrary component orientation

A Maxwell Garnett approximation (MGA) and a symmetric effective medium approximation (SEMA) are derived for anisotropic composites of host-inclusion and symmetric-grains morphologies, respectively, with ellipsoidal grains of arbitrary intrinsic, shape and orientation anisotropies. The effect of anisotropy on the effective dielectric tensor is illustrated in both cases. The MGA shows negative and non-monotonic off-diagonal elements for geometries where the host and inclusions are not mutually aligned. The SEMA leads to an anisotropy-dependent nonlinear behaviour of the conductivity as a function of volume fraction above a percolation threshold of conductor-insulator composites, in contrast to the well-known linear behaviour of the isotropic effective medium model. The percolation threshold obtained for composites of aligned ellipsoids is isotropic and independent of the ellipsoids aspect ratio. Thus, the common identification of the percolation threshold with the depolarization factors of the grains is unjustified and a description of anisotropic percolation requires explicit anisotropic geometric characteristics.

Busemann functions and the speed of a second class particle in the rarefaction fan

In this paper we will show how the results found in [Probab. Theory Related Fields 154 (2012) 89–125], about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process or the Hammersley interacting particle process. The method will be to use the well-known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well-known connection between second class particles and competition interfaces.

Percolation Description of the Global Topography of Earth and the Moon

Remarkable global correlations exist between geometrical features of terrestrial surfaces on Earth, current mean sea level, and its geological internal processes whose origins have remained an essential goal in the earth sciences. Theoretical modeling of the ubiquitous self-similar fractal patterns observed on Earth and their underlying rules is indeed of great importance. Here I present a percolation description of the global topography of Earth in which the present mean sea level is automatically singled out as a critical level in the model. This finding elucidates the origins of the appearance of scale invariant patterns on Earth. The criticality is shown to be accompanied by a continental aggregation, unraveling an important correlation between the water and long-range topographic evolutions. To have a comparison point in hand, I apply such an analysis to the lunar topography which reveals various characteristic features of the Moon.

No coincidence of center percolation and deconfinement in SU(4) lattice gauge theory

We study the behavior of center sectors in pure SU(4) lattice gauge theory at finite temperature. The center sectors are defined as spatial clusters of neighboring sites with values of their local Polyakov loops near the same center elements. We study the connectedness and percolation properties of the center clusters across the deconfinement transition. We show that for SU(4) gauge theory deconfinement cannot be described as a percolation transition of center clusters, a finding which is different from pure SU(2) or pure SU(3) Yang–Mills theory, where the percolation description even allows for a continuum limit.

Logarithmic observables in critical percolation

Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q → 1 limit of the Q-state Potts model, and analyzing the underlying SQ symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q → 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates two propagating clusters. In d = 2 dimensions this agrees with general LCFT results, and in particular the universal prefactor of the logarithm can be computed exactly. We confirm its numerical value by carrying out extensive Monte Carlo simulations.

Infiltration processes and flow rates in developed karst vadose zone using tracers in cave drips

AbstractThe purpose of this research was to identify groundwater recharge mechanisms and measure flow velocities through the epikarst and the vadose zone in a typical site of Mediterranean carbonate karst in natural rain conditions. By avoiding artificial flushing more realistic travel times could be measured, and the effective resultant vadose flow velocity could be inferred. Additionally, detailed monitoring enabled better description and quantification of infiltration and percolation processes.Nine different cave drips were monitored and sampled for three hydrological years in fractured karst lithology on Mount Carmel, Israel. At three drips, discharge was measured continuously by tipping buckets. Rainfall and soil water content were recorded above the cave. An artificial tracer experiment was conducted using uranine placed in a joint and at the soil–rock interface at soil pockets, both 27 m above the cave.Four hydrological types of drips: post‐storm, seasonal, perennial, and overflow, were identified; each demonstrating different characteristics in terms of discharge and chemistry.The maximum effective flow velocities (uranine dye arrival times) were 41–76 cm/h; tracer mass flux was computed from the uranine breakthrough curves and drip hydrographs. The dominant flow velocities, derived from the peak of tracer mass flux, were 0·35–0·41 cm/h from post‐storm drips, and 22–24 cm/h from the other drips.‘Pulse‐through’ (intra‐event piston flow) was indicated by the time lags between the increase in drip rate and the time of drop in salinity and appearance of the tracer in most of the measured drips. Chloride concentrations were used to distinguish between ‘old/matrix water’ and ‘preferential‐flow water’ in the perennial drips, demonstrating a two‐component mixing model. In 2005–2006 ‘preferential‐flow water’ reached up to 20–25% from annual discharge of these drips. This study promoted a method to compute dominant effective flow rates in the vadose zone in natural rain conditions. Copyright © 2010 John Wiley & Sons, Ltd.

Transport properties of low-angle grain boundaries and granular coated conductors

Grain boundaries in oxide superconductors are of central importance both for engineeringapplications and for issues in fundamental science. Although they have been closely studiedfor more than a decade, there is still by no means a consensus on the properties ofindividual boundaries, or their collective behaviour when current percolates in a granularmaterial. A quantitative description of current and flux percolation is essential forthe practical specification of conductors and for charting routes to conductoroptimization. There are three components to the problem: (i) the properties of individualgrains and grain boundaries, (ii) microstructure, texture and grain morphology, and(iii) the macroscopic geometrical constraints such as conductor size and shape. Onthe basis of new results on the angular variation in an applied magnetic fieldof the grain boundary critical current, it will be argued that grain morphologyas well as texture is important for the critical current of a coated conductor.Progress in characterizing and modelling current and flux percolation will besurveyed and particular gaps in understanding and areas of neglect discussed.

Impact of wettability alteration on two‐phase flow characteristics of sandstones: A quasi‐static description

We describe a two‐phase pore network simulator of drainage and imbibition which integrates a realistic representation of pore connectivity and morphology, a quasi‐static description of fluid displacement mechanisms, and a sound representation of the wetting properties of a sedimentary rock and of their alteration. The simulator works with 3‐D disordered networks of cylindrical ducts with triangular, square, and circular cross sections obtained directly from the analysis of microfocused computed tomography (CT) images of rock samples. All pore‐level displacement mechanisms (piston type, snap off, and cooperative pore body filling) are considered with arbitrary receding and advancing contact angles. Bond invasion percolation description is used in primary drainage, while bond site invasion percolation with ordinary percolation on a dual network and compact cluster growth is used in secondary imbibition. In this paper, we resolve how to calculate the relative permeability of nonaqueous phase liquid (NAPL) in the quasi‐static approximation of imbibition and illustrate spatial distribution of the clusters of trapped NAPL using our generalization to disordered networks of the Hoshen‐Kopelman cluster‐labeling algorithm. To understand the impact of wettability alteration on the capillary pressure and relative permeability functions, we perform a series of drainage and imbibition simulations by changing the range of advancing contact angles. Our study indicates that in imbibition, transport properties of a permeable solid are quite sensitive to the hysteresis between the receding and advancing contact angle. This sensitivity reflects competition among the different displacement mechanisms, which shapes the relative permeabilities, capillary pressures, and the distribution of the trapped NAPL.

From the zero-field metal-insulator transition in two dimensions to the quantum Hall transition: A percolation-effective-medium theory

Effective-medium theory is applied to the percolation description of the metal-insulator transition in two dimensions with emphasis on the continuous connection between the zero-magnetic-field transition and the quantum Hall transition. In this model the system consists of puddles connected via saddle points, and there is loss of quantum coherence inside the puddles. The effective conductance of the network is calculated using appropriate integration over the distribution of conductances, leading to a determination of the magnetic field dependence of the critical density. Excellent quantitative agreement is obtained with the experimental data, which allows an estimate of the puddle physical parameters.

Irreversibility Lines in Halogen- and Hydrogen-Doped YBa2Cu3O7−δ Compounds

We measure the real and imaginary parts of the ac magnetic permeability of YBa2Cu3O7−δA x (A = Cl, F, H) ceramics and powders as functions of temperature and ac magnetic field amplitude H0, applying uncommonly low and widely ranged excitation fields (1 mOe ≤ Ho ≤ 200 Oe). We determine the temperature dependence of two loss peaks, Hp(T) ∼ (Tc − T)2.7 at low fields in ceramics and Hm(T) ∼ Ts0 − T at higher fields in both powders and ceramics. The extrapolated field Hp(0) and characteristic temperature Tc depend on x and δ, whereas Hm(0) and Ts0 depend mainly on δ. The real part of the permeability indicates a two-step flux-penetration process with two threshold fields for flux penetration. The latter govern the temperature dependence of maxima in the imaginary part as well as the peaks' widths. We propose a field-dependent percolation description of this process, derived from a previous temperature-dependent percolation model. It involves two penetration depths, the classic Ginzburg–Landau one and a low-field abnormally large penetration depth due to chemical disorder. Correspondingly, two types of vortices describe high- and low-field dissipation peaks.

Methods for Examining Anisotropies in Early Combustion Part II: Theoretical Developments

ABSTRACT This study reports some preliminary steps in the development of techniques for simplifying the problem of analysing spatially propagating reactions in anisotropic multi-phase systems. The target situation is combustion in a spark ignition petrol engine; nevertheless the method described has a much broader range of application. Both present and future experimental techniques and C.A.D. methods concerned with spatially resolved combustion data are and will be faced with enormous complexity of detail. This is essentially due to the geometry and dimensionality of the problem of the propagation of in-cylinder combustion. However, many features of interest concern the fundamental structural properties of the burning air/fuel system, such as how far combustion has progressed, how a given burning region is linked to others, how robust are these connections with respect to flow field disturbance. These are the type of issues the formalism is best suited to address. The approach is mainly analytical and the combustion description is treated via a set of mathematically tractable special cases of increasing complexity. From these a general computational approach is proposed for real situations. The background of this work is the recent introduction of a percolation description of combustion.

Discrete and integral hydrodynamic models of percolation networks

The hydrodynamic description of percolation is used to introduce permeability network models via discretization and leads to the study of random walks on lattices by decimation transformations. Decimation is exact in one dimension and provides the basis for understanding an approximate renormalization that has been reported for two and three dimensional lattices. We discuss the question of existence of a block permeability picture by analysing the question whether decimation (“renormalization”) can be used to obtain a hierarchy of coarse-grained descriptions of an arbitrary permeability distribution. We show that both the block permeability and multiplicative multifractal pictures are valid in one dimension but that the method of proof breaks down in higher dimensions. We conclude that there is presently no justification for assuming a multifractal permeability distribution or even a hierarchy of nonmultifractal block permeabilities for two and three dimensional permeability networks. The difference between real rocks and reservoirs and artificial ones is emphasized. To construct an artificial reservoir that agrees with observed nonuniversal scaling, percolation models with variable topology are introduced based upon the integrated form of hydrodynamics. The models considered are multifractal whenever the distribution of capillary tube exponents is binomial. In connection with this modelling, we argue that constraints derived from the experimental study of small rock samples can be used to test whether a given distribution of permeability applies to an entire geological layer within a reservoir. Finally, the method of symbolic dynamics is adapted to the generation of pseudorandom permeability distributions on lattices.

Ultrasonic investigation of granular superconducting films

The dc resistance and surface-acoustic-wave (SAW) attenuation has been measured in granular Pb/${\mathrm{PbO}}_{\mathit{x}}$ and In/${\mathrm{InO}}_{\mathit{x}}$ films as a function of temperature. The behavior of resistance and attenuation data has been found to be in agreement with the predictions of a recently developed model that is based on a percolation description of granular superconducting films. In this paper we will present the resistance and SAW attenuation data and compare them to the predictions of the percolation model.