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# Percolation threshold

About: Percolation threshold is a research topic. Over the lifetime, 9299 publications have been published within this topic receiving 294605 citations.

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01 Jan 1985

TL;DR: In this paper, a scaling solution for the Bethe lattice is proposed for cluster numbers and a scaling assumption for cluster number scaling assumptions for cluster radius and fractal dimension is proposed.

Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques

9,830 citations

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01 Jan 1992

TL;DR: In this article, a scaling solution for the Bethe lattice is proposed for cluster numbers and a scaling assumption for cluster number scaling assumptions for cluster radius and fractal dimension is proposed.

Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques

7,349 citations

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IBM

^{1}TL;DR: In this article, an extension of percolation theory to treat transport is described, and a general expression for the conductance of such networks is derived, which relates to the spin-stiffness coefficient of dilute ferromagnet.

Abstract: Extensions of percolation theory to treat transport are described. Resistor networks, from which resistors are removed at random, provide the natural generalization of the lattice models for which percolation thresholds and percolation probabilities have previously been considered. The normalized conductance, $G$, of such networks proves to be a sharply defined quantity with a characteristic concentration dependence near threshold which appears sensitive only to dimensionality. Numerical results are presented for several families of $3D$ and $2D$ network models. Except close to threshold, the models based on bond percolation are accurately described by a simple effective medium theory, which can also treat continuous media or situations less drastic than the percolation models, for example, materials in which local conductivity has a continuous distribution of values. The present calculations provide the first quantitative test of this theory. A "Green's function" derivation of the effective medium theory, which makes contact with similar treatments of disordered alloys, is presented. Finally, a general expression for the conductance of a percolation model is obtained which relates $G$ to the spin-stiffness coefficient, $D$, of an appropriately defined model dilute ferromagnet. We use this relationship to argue that the "percolation channels" through which the current flows above threshold must be regarded as three dimensional.

4,342 citations

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TL;DR: In this paper, a comprehensive survey of electrical percolation of carbon nanotubes (CNT) in polymer composites is presented, together with an attempt of systematization.

Abstract: We review experimental and theoretical work on electrical percolation of carbon nanotubes (CNT) in polymer composites. We give a comprehensive survey of published data together with an attempt of systematization. Parameters like CNT type, synthesis method, treatment and dimensionality as well as polymer type and dispersion method are evaluated with respect to their impact on percolation threshold, scaling law exponent and maximum conductivity of the composite. Validity as well as limitations of commonly used statistical percolation theories are discussed, in particular with respect to the recently reported existence of a lower kinetic (allowing for re-aggregation) and a higher statistical percolation threshold.

1,815 citations

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TL;DR: In this article, the scaling theory of phase transition has been used to explain percolation through the cluster properties; it can also be used as an introduction to critical phenomena at other phase transitions for readers not familiar with scaling theory.

Abstract: For beginners : This review tries to explain percolation through the cluster properties; it can also be used as an introduction to critical phenomena at other phase transitions for readers not familiar with scaling theory. In percolation each site of a periodic lattice is randomly occupied with probability p or empty with probability 1−p. An s-cluster is a group of s occupied sites connected by nearest-neighbor distances; the number of empty nearest neighbors of cluster sites is the perimeter t. For p above pc also one infinite cluster percolates through the lattice. How do the properties of s-clusters depend on s, and how do they feel the influence of the phase transition at p = pc? The answers to these questions are given by various methods (in particular computer simulations) and are interpreted by the so-called scaling theory of phase transitions. The results presented here suggest a qualitative difference of cluster structures above and below pc: Above p c some cluster properties suggest the existence of a cluster surface varying as s 2 3 in three dimensions, but below pc these “surface” contributions are proportional to s. We suggest therefore that very large clusters above pc (but not at and below pc) behave like large clusters of Swiss cheese: Inspite of many internal holes the overall cluster shape is roughly spherical, similar to raindrops. For experts : Scaling theory suggests for large clusters near the percolation threshold pc that the average cluster numbers n s vary as s −τ ƒ(z) , with z ≡ (p − pc)sσ. Analogously the average cluster perimeter is ts = s · (1 − p)/p + sσ · ψ1(z), the average cluster radius Rs varies as sσv · R1(z), and the density profile Ds(r), which depends also on the distance r from the cluster center, varies as s −1 δ · D 1 (rs −σv , z) . These assumptions relate the seven critical exponents α,β,γ,δ,v,σ,τ in d dimensions through the well-known five scaling laws 2 − α = γ + 2β = βδ + β = dv = β + 1 σ = (τ − 1)/σ , leaving only two exponents as independent variables to be fitted by “experiment” and not predicted by scaling theory. For the lattice “animals”, i.e. the number gst of geometrically different cluster configurations, a modified scaling assumption is derived: g st s s t 1 /(s + t) s + 1 ∝ s −τ− 1 2 · ƒ(z) , with z ∝ (ac − t/s)sσ and ac = (1 − pc)/pc. All these expressions are variants of the general scaling idea for second-order phase transitions that a function g(x,y) of two critical variables takes the homogeneous form xcG(x/yb) near the critical point, with two free exponents b and c and a scaling function G of a single variable. These assumptions, which may be regarded as generalizations of the Fisher droplet model, are tested “experimentally” by Monte Carlo simulation, series expansion, renormalization group technique, and exact inequalities. In particular, detailed Monte Carlo evidence of Hoshen et al. and Leath and Reich is presented for the scaling of cluster numbers in two and three dimensions. If the cluster size s goes to infinity at fixed concentration p, not necessarily close to pc, three additional exponents ξ, θ, ϱ are defined by: cluster numbers ∝ s−θ exp(−const · sξ) and cluster radii ∝ sϱ. These exponents are different on both sides of the phase transition; for example ξ(p pc) = 1 − 1/d was found from inequalities, series and Monte Carlo data. The behavior of θ and of ϱ(p This article does not cover experimental applications, correlation functions and “classical” (mean field, Bethe lattice, effective medium) theories. For the reader to whom this abstract is too short and the whole article is too long we recommend sections 1 and 3.

1,763 citations