Alessandro Attanasi (603657), Andrea Cavagna (278274), Lorenzo Del Castello (603658) et al. · Figshare · 2016
<p>Percolation threshold as a function of the nearest-neighbour distance in natural swarms. The linear fit (black line) gives, Inset: Fraction of midges belonging to the largest cluster as a function of the clustering threshold In correspondence of the percolation threshold there is the formation of a giant cluster. We define as the point where (red dashed line). Because of the sharp nature of the percolation transition, the value of does not depend greatly on the threshold used.</p>
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Jiřı́ Vohlı́dal, Carlos F. O. Graeff, Roger C. Hiorns et al. · IUPAC Standards Online · 2023
These recommendations are specifically for polymers and polymer systems showing a significant response to an electromagnetic field or one of its components (electric field or magnetic field), i.e., for electromagnetic-field-responsive polymer materials. The structures, processes, phenomena and quantities relating to this interdisciplinary field of materials science and technology are herein defined. Definitions are unambiguously explained and harmonized for wide acceptance by the chemistry, physics, polymer and materials science communities. A survey of typical electromagnetic-field-responsive polymers is included.
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2024
Citation: 'percolation threshold' in the IUPAC Compendium of Chemical Terminology, 5th ed.; International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.08848 • License: The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International for individual terms. Requests for commercial usage of the compendium should be directed to IUPAC.
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Serge W. Wiltshire (4926124) · Figshare · 2018
<p>Lines plot average values for the 100 runs at each of 150 <i>N</i><sub><i>p</i></sub> levels, with corresponding color fields indicating 95% CI. Top left plot shows infection duration. Top right shows mean proportion infected (cumulative). Bottom left shows the fraction of runs resulting in a systemic network-level infection lasting the full duration of the model run (4135 model days). Bottom right shows the fraction of runs in which 95% or more of the agents became infected.</p>
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Vipin Srivastava, Enzo Granato · AIP conference proceedings · 1992
Stronger then exponential localization, or ‘‘superlocalization,’’ on the infinite cluster in a percolating system is expected to arise due to the fractal nature of the percolating cluster. This assertion is substantiated here through the calculation of the fractal dimension of the percolating cluster using results on inverse‐participation‐ratio versus composition calculations.
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North-Holland series in applied mathematics and mechanics · 1996
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I. Avramov · Journal of Physics Condensed Matter · 2009
The mathematical limits for the percolation of rigid and floppy clusters throughout continuous networks are considered. Results are compared to the previous prediction of Thorpe based on the assumption that the threshold condition is the equality of degrees of freedom and the number of constraints. It is demonstrated that, in two-dimensional systems, the thresholds values predicted here and in the Thorpe model are relatively close. Our investigation demonstrates that, in 3D, there is a range of strong bond concentrations in which both floppy and rigid clusters percolate simultaneously. Depending on the particular property, either the rigid or the floppy cluster plays the key role. This explains the existence of solid electrolytes. They are solid because of the percolating rigid skeleton and at the same time have high conductivity controlled by the floppy cluster that also percolates.
Cited by 11
Srivastava, Vipin, Granato, Enzo · Biblioteca Digital da Memória Científica do INPE (National Institute for Space Research)
Stronger than exponential localization, or "superlocalization", on the infinite cluster in a percolating system is expected to arise due to the fractal nature of the percolating cluster. This assertion is substantiated here through the calculation of the fractal dimension of the percolating cluster using results on inverseparticipation-ratio versus composition calculations.
Cited by 0
Jennifer Chayes, L. Chayes, Charles M. Newman · The Annals of Probability · 1987
Using the invasion percolation process, we prove the following for Bernoulli percolation on $\mathbb{Z}^d (d > 2)$: (1) exponential decay of the truncated connectivity, $\tau'_{xy} \equiv P(x$ and $y$ belong to the same finite cluster$) \leq \exp(-m\|x - y\|)$; (2) infinite differentiability of $P_\infty(p)$, the infinite cluster density, and of $\chi'(p)$, the expected size of finite clusters, as functions of $p$, the density of occupied bonds; and (3) upper bounds on the cluster size distribution tail, $P_n \equiv P$(the cluster of the origin contains exactly $n$ bonds) $\leq \exp(-\lbrack c/\log n\rbrack n^{(d-1)/d})$. Such results (without the $\log n$ denominator in (3)) were previously known for $d = 2$ and $p > p_c$, the usual percolation threshold, or for $d > 2$ and $p$ close to 1. We establish these results for all $d > 2$ when $p$ is above a limit of "slab thresholds," conjectured to coincide with $p_c$.
Cited by 60Open access
Eva Kroener, Mutez Ali Ahmed, Andrea Carminati · Physical Review E · 2015
The rhizosphere is the layer of soil around the roots where complex and dynamic interactions between plants and soil affect the capacity of plants to take up water. The physical properties of the rhizosphere are affected by mucilage, a gel exuded by roots. Mucilage can absorb large volumes of water, but it becomes hydrophobic after drying. We use a percolation model to describe the rewetting of dry rhizosphere. We find that at a critical mucilage concentration the rhizosphere becomes impermeable. The critical mucilage concentration depends on the radius of the soil particle size. Capillary rise experiments with neutron radiography prove that for concentrations below the critical mucilage concentration water could easily cross the rhizosphere, while above the critical concentration water could no longer percolate through it. Our studies, together with former observations of water dynamics in the rhizosphere, suggest that the rhizosphere is near the percolation threshold, where small variations in mucilage concentration sensitively alter the soil hydraulic conductivity. Is mucilage exudation a plant mechanism to efficiently control the rhizosphere conductivity and the access to water?
Cited by 26Open access
А. В. Гончаренко, E. F. Venger · Physical Review E · 2004
Using a simple phenomenological approach, we calculate the percolation threshold for Bruggeman composites having microgeometry of two kinds. Both kinds of composites consist of spheroids whose shape follows the Beta distribution. At the same time, the first one is a mixture of spheroids equally oriented along their revolution axis. In this case the percolation threshold is shown to be the same as for an assembly of equally oriented identical spheroids whose shape corresponds to the most probable shape of the distribution. For such composites the percolation threshold can vary between 0 and 1. The second one is a random mixture of the spheroids. In this case the percolation threshold is expressed in terms of the Gauss hypergeometric function; it is shown to vary between 0 and 1/3. The derived analytical results are supplemented with numerical calculations carried out for different values of the Beta distribution parameters.
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Jiřı́ Škvor, Ivó Nezbeda · Physical Review E · 2009
Extensive Monte Carlo simulations on three qualitatively different model supercritical fluids (square-well fluid, Lennard-Jonesium, and primitive water) have been performed to examine percolation threshold parameters for continuum (correlated) models and their relation to general results valid for random lattice models; random-site percolation simple-cubic lattice has therefore been considered as well. Two different bond criteria, the configurational and self-bound ones, defining a cluster have been used. In addition to the percolation threshold occupation probability pc and the percolation threshold fluid density rhoc, the correlation length exponent nu and the wrapping probability at the percolation threshold Rw,c have also been evaluated. It is found that parameters nu and Rw,c exhibit not only strong temperature dependence but also, unlike the case of lattice systems, dependence on the nature of the system considered and the employed definition of the cluster.
Cited by 34
Alexa W. Harter, Greg Mohler, D.C. Maybury et al. · Digest of INTERMAG 2003. International Magnetics Conference (Cat. No.03CH37401) · 2004
In this paper, the threshold is defined as the volume/surface fraction at which the DC magnetic susceptibility increases dramatically and is studied using micromagnetics and experimental measurements.
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Marcelo A. Moret, Monia Cristina Gomes de Araújo Santana, Eliane Maria de Souza Nogueira et al. · Physica A Statistical Mechanics and its Applications · 2005
Abstract The major factor that drives a protein chain toward collapse and folding is the hydrophobic effect. Amino acids with apolar side chains join forming a solvent-shielded hydrophobic core. This process pack the structure in the native one. Here we investigate the average packing density of 5526 protein chains deposited in the Brookhaven Protein Data Bank. This analysis is carried out from the scaling analysis of the mass-size exponent and it shows that the exponent is δ = 2.47 . This fractal dimension of the protein chain is close to the one obtained in the randomly packed spheres near their percolation threshold. The present findings supply a measure of the protein compactness, that tends to a constant protein chain packing. The average packing density tends to ρ = 0.86 a . u . /A3.
Cited by 22Open access
Antonio Coniglio · Journal of Physics A Mathematical and General · 1982
Derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc. In particular, the author proves the relations p(dpij/dp)=( lambda ij) where p is the bond density, pij is the pair connectedness function and ( lambda ij) is the average number of cutting bonds between i and j. From this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi , diverges as mod p-pc mod -1. The remaining (multiply connected) bonds in the percolating backbone, which lump together in 'blobs', diverge with a dimensionality-dependent exponent. He also shows that in the cell renormalisation group of Reynolds et al. (1978, 1980) the 'thermal' eigenvalue is simply related to the average number of cutting bonds in the spanning cluster. He discusses a percolation model in which the 'blobs' can be controlled by varying a parameter, and study the influence on the critical exponents.
Cited by 335
Wolf B. Dapp, Martin H. Müser · Scientific Reports · 2016
Percolation is a concept widely used in many fields of research and refers to the propagation of substances through porous media (e.g., coffee filtering), or the behaviour of complex networks (e.g., spreading of diseases). Percolation theory asserts that most percolative processes are universal, that is, the emergent powerlaws only depend on the general, statistical features of the macroscopic system, but not on specific details of the random realisation. In contrast, our computer simulations of the leakage through a seal--applying common assumptions of elasticity, contact mechanics, and fluid dynamics--show that the critical behaviour (how the flow ceases near the sealing point) solely depends on the microscopic details of the last constriction. It appears fundamentally impossible to accurately predict from statistical properties of the surfaces alone how strongly we have to tighten a water tap to make it stop dripping and also how it starts dripping once we loosen it again.
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Stéphane Boucheron, I. Benjamini, Gábor Lugosi et al. · HAL (Le Centre pour la Communication Scientifique Directe) · 2012
International audience
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Sergio Alonso (270169), Rodrigo Weber dos Santos (3359138), Markus Bär (422625) · Figshare · 2016
<p>Dependence of the percolation threshold in a slab of tissue for different thickness. For each thickness <i>N</i> we generate 100 realizations for each value of <i>ϕ</i> and check the probability of finding a pathway from one side of the system to the other. After a fitting process, we evaluate the fraction <i>ϕ</i><sub><i>c</i></sub> which correspond to a probability of 50%. The size of the system is 4 <i>cm</i> × 4 <i>cm</i> × <i>N</i> <i>layers</i>, where each layer corresponds to 0.01 <i>cm</i>.</p>
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Clinton DeW. Van Siclen · arXiv (Cornell University) · 2016
Connections are found between the two-component percolation problem and the conductor/insulator percolation problem. These produce relations between critical exponents, and suggest formulae connecting the conductivity exponents in different dimensions. Values for the critical exponents are obtained from calculations on the incipient infinite cluster in two and three dimensions.
Cited by 1Open access
Junmo Lee, Jaewook Nam · Physical review. E · 2021
In this study, we investigate the percolation threshold of curved linear objects, describing them as quadratic Bézier curves. Using Monte Carlo simulations, we calculate the critical number densities of the curves with different curviness. We also obtain the excluded area of the curves. When an excluded area is given, we can find the critical number density of the curves with arbitrary curviness. Apparent conductivity exponents are computed for the curves, and these values are found to be analogous to that of sticks in the percolative region for a junction resistance dominant system. These results can be used to analyze the optoelectrical performance of metal nanowire films because the high-aspect-ratio metal nanowires can be easily curved during coating.
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