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Percolation Surface Critical Behavior
Percolation surface critical behavior concerns the influence of surfaces on the critical behavior of percolation. Background Percolation is the study of connectivity in random systems, such as electrical conductivity in random conductor/insulator systems, fluid flow in porous media, gelation in polymer systems, etc. At a critical fraction of connectivity or porosity, long-range connectivity can take place, leading to long-range flow. The point where that connectivity takes place is called the percolation threshold, and considerable amount of work has been undertaken in finding those critical values for systems of various geometries, and the mathematical behavior of observables near that point. This leads to the study of critical behavior and the percolation critical exponents. These exponents allow one to describe the behavior as the threshold is approached. The behavior of the percolating network near a surface will be different from that in the main part of a system, called t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation
Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applications have since been developed that cover connectivity of many systems modeled as lattices or graphs, analogous to connectivity of lattice components in the filtration problem that modulates capacity for percolation. Background During the last decades, percolation theory, the mathematical study of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science, complex networks, epidemiology, and other fields. For example, in geology, percolation refers to filtration of water through soil and permeable rocks. The water flows to recharge the groundwater in the water table and aquifers. In places where infiltration basins or septic drain fields are planned to dispose of substantial amounts of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Threshold
The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in Randomness, random systems. Below the threshold a giant connected component (graph theory), connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified Lattice model (physics), lattice models of random systems or networks (Graph (discrete mathematics), graphs), and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability ''p'', or more generally a critical surface for a group of parameters ''p''1, ''p''2, ..., such that infinite connectivity (''percolation'') first occurs. Percolation models The most common percolation model is to take a regular lattice, like a squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Behavior
In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations among different quantities, power-law divergences of some quantities (such as the magnetic susceptibility in the ferromagnetic phase transition) described by critical exponents, universality, fractal behaviour, and ergodicity breaking. Critical phenomena take place in second order phase transitions, although not exclusively. The critical behavior is usually different from the mean-field approximation which is valid away from the phase transition, since the latter neglects correlations, which become increasingly important as the system approaches the critical point where the correlation length diverges. Many properties of the critical behavior of a system can be derived in the framework of the renormalization group. In order to explain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Critical Exponents
In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of ''universal'' critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation. Percolating systems have a parameter p\,\! which controls the occupancy of sites or bonds in the system. At a critical value p_c\,\!, the mean cluster size goes to infinity and the percolation transition takes place. As one approaches p_c\,\!, various quantities either diverge or go to a constant value by a power law in , p - p_c, \,\!, and the exponent of that power law is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Surface Transitions (corrected)
Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applications have since been developed that cover connectivity of many systems modeled as lattices or graphs, analogous to connectivity of lattice components in the filtration problem that modulates capacity for percolation. Background During the last decades, percolation theory, the mathematical study of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science, complex networks, epidemiology, and other fields. For example, in geology, percolation refers to filtration of water through soil and permeable rocks. The water flows to recharge the groundwater in the water table and aquifers. In places where infiltration basins or septic drain fields are planned to dispose of substantial amounts o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Theory
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles network theory and percolation. Introduction A representative question (and the source of the name) is as follows. Assume that some liquid is poured on top of some porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is modelled mathematically as a three-dimensional network of vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability , or closed with probability , and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Critical Exponents
In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of ''universal'' critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation. Percolating systems have a parameter p\,\! which controls the occupancy of sites or bonds in the system. At a critical value p_c\,\!, the mean cluster size goes to infinity and the percolation transition takes place. As one approaches p_c\,\!, various quantities either diverge or go to a constant value by a power law in , p - p_c, \,\!, and the exponent of that power law is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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