Percolation theory and physics of compression
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Keywords:
Percolation Theory
Directed percolation
Percolation (cognitive psychology)
Exponent
Continuum percolation theory
A new method of studying percolation fronts is introduced in situations where the concentration is not constant (gradient percolation approach): it consists in an efficient way of determining subsets of the percolation cluster which are exactly at the percolation threshold. The method is applied to both lattice and continuum percolation in order to determine the percolation threshold in 3D more accurately. Two variations of this method, illumination and projection, seem to give a similar scaling behaviour at low gradients for the cubic lattice case, leading to a new estimate for the percolation threshold: pc=0.31173+or-0.00007. For continuum percolation of overlapping spheres, the new estimate for the critical volume fraction is 0.291+or-0.002, and the value of the critical exponent alpha sigma for the width of the front scaled with the concentration gradient is very close to the one for the lattice case.
Directed percolation
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Percolation (cognitive psychology)
Lattice (music)
Continuum percolation theory
Percolation Theory
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Percolation theory is a powerful framework for studying strongly disordered systems. However, almost all the interesting physics of percolation happens in a very small range in parameter space, and often it may seem difficult to justify why a physical system should happen to be near this special point, the percolation threshold. In this paper we review some aspects of percolation theory from a different viewpoint than the usual one. In this formulation, it becomes natural why the percolation threshold plays an important role in many physical systems.
Percolation (cognitive psychology)
Directed percolation
Percolation Theory
Continuum percolation theory
Physical space
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We examine the effects of introducing a wall or edge into a directed percolation process. Scaling ansatze are presented for the density and survival probability of a cluster in these geometries, and we make the connection to surface critical phenomena and field theory. The results of previous numerical work for a wall can thus be interpreted in terms of surface exponents satisfying scaling relations generalizing those for ordinary directed percolation. New exponents for edge directed percolation are also introduced. They are calculated in mean-field theory and measured numerically in 2 + 1 dimensions.
Directed percolation
Percolation (cognitive psychology)
Percolation Theory
Continuum percolation theory
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Directed percolation
Percolation (cognitive psychology)
Continuum percolation theory
Lattice (music)
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Abstract Within the framework of the percolation theory (bond problem), a new model of breaking a complex synthetic tape is proposed as a continuous-type phase transition when the state jump is zero. The percolation threshold and accompanying characteristics are calculated for the model of rupture of a synthetic reinforced tape when flowing along the first and second neighbours. The knots of the tape form a strip of a square lattice, the width of which is fixed. All nodes are intact and cannot be damaged, links (tape threads) can be intact and broken (blocked). The dependences of the percolation threshold in the bond problem and the relative deviation of the threshold from the ribbon length are calculated. It is proved that for the simplest model of one-dimensional percolation with percolation along the nearest neighbours (the problem of nodes), the percolation threshold in the thermodynamic limit is equal to unity. It is shown that, with an accuracy of 10%, the percolation threshold for a sufficiently long ribbon is equal to unity. This indicates that the system is quasi-one-dimensional. Thus, using the method of computer simulation, the percolation threshold, root-mean-square and relative threshold deviations were calculated. The critical susceptibility index was also calculated. In contrast to the usual percolation problem, in the proposed model it makes sense to consider only the region above the percolation threshold. The proposed model can be generalized to the case when nodes are also damaged (blocked), then we come to a mixed percolation model, which is supposed to be considered in the future.
Directed percolation
Square lattice
Percolation (cognitive psychology)
Continuum percolation theory
Percolation Theory
Lattice (music)
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Directed percolation
Percolation (cognitive psychology)
Exponent
Continuum percolation theory
Percolation Theory
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We investigate percolation on a randomly directed lattice, an intermediate between standard percolation and directed percolation, focusing on the isotropic case in which bonds on opposite directions occur with the same probability. We derive exact results for the percolation threshold on planar lattices, and present a conjecture for the value the percolation-threshold for in any lattice. We also identify presumably universal critical exponents, including a fractal dimension, associated with the strongly-connected components both for planar and cubic lattices. These critical exponents are different from those associated either with standard percolation or with directed percolation.
Directed percolation
Continuum percolation theory
Percolation (cognitive psychology)
Lattice (music)
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Percolation Theory
Directed percolation
Percolation (cognitive psychology)
Exponent
Continuum percolation theory
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Many modern nanostructured materials and doped polymers are morphologically too complex to be interpreted by classical percolation theory. Here, we develop the concept of a hierarchical percolating (percolation-within-percolation) system to describe such complex materials and illustrate how to generalize the conventional percolation to double-level percolation. Based on Monte Carlo simulations, we find that the double-level percolation threshold is close to, but definitely larger than, the product of the local percolation thresholds for the two enclosed single-level systems. The deviation may offer alternative insights into physics concerning infinite clusters and open up new research directions for percolation theory.
Percolation (cognitive psychology)
Continuum percolation theory
Percolation Theory
Directed percolation
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Percolation (cognitive psychology)
Directed percolation
Percolation Theory
Continuum percolation theory
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