| 翻訳と辞書 |
| Percolation theory ： ウィキペディア英語版 |
In statistical physics and mathematics, percolation theory describes the behavior of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article percolation.==Introduction==Bond percolation and Site percolation redirect to here -->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 ''n'' × ''n'' × ''n'' vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability ''p'', or closed with probability 1 – ''p'', and they are assumed to be independent. Therefore, for a given ''p'', what is the probability that an open path exists from the top to the bottom? The behavior for large ''n'' is of primary interest. This problem, called now bond percolation, was introduced in the mathematics literature by , and has been studied intensively by mathematicians and physicists since then.In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability ''p'' or "empty" (in which case its edges are removed) with probability ''1-p''; the corresponding problem is called site percolation. The question is the same: for a given ''p'', what is the probability that a path exists between top and bottom?Of course the same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By Kolmogorov's zero-one law, for any given ''p'', the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of ''p'' (proof via coupling argument), there must be a critical ''p'' (denoted by ''p''c) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for ''n'' as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of ''p''. In some cases ''p''c may be calculated explicitly. For example, for the square lattice Z2 in two dimensions, ''p''c = 1/2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, see . A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at ''p''c = 1/(''z'' − 1) for a coordination number ''z''. For most infinite lattice graphs, ''p''c cannot be calculated exactly.==Universality==The universality principle states that the value of ''p''c is connected to the local structure of the graph, while the behavior of clusters below, at, and above ''p''c are invariant with respect to the local structure, and therefore, in some sense are more natural quantities to consider.This universality also means that for the same dimension independent of the type of the lattice or type of percolation (e.g., bond or site) the fractal dimension of the clusters at ''p''c is the same.
In statistical physics and mathematics, percolation theory describes the behavior of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article 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 ''n'' × ''n'' × ''n'' vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability ''p'', or closed with probability 1 – ''p'', and they are assumed to be independent. Therefore, for a given ''p'', what is the probability that an open path exists from the top to the bottom? The behavior for large ''n'' is of primary interest. This problem, called now bond percolation, was introduced in the mathematics literature by , and has been studied intensively by mathematicians and physicists since then.
In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability ''p'' or "empty" (in which case its edges are removed) with probability ''1-p''; the corresponding problem is called site percolation. The question is the same: for a given ''p'', what is the probability that a path exists between top and bottom?
Of course the same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By Kolmogorov's zero-one law, for any given ''p'', the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of ''p'' (proof via coupling argument), there must be a critical ''p'' (denoted by ''p''c) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for ''n'' as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of ''p''.
In some cases ''p''c may be calculated explicitly. For example, for the square lattice Z2 in two dimensions, ''p''c = 1/2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, see . A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at ''p''c = 1/(''z'' − 1) for a coordination number ''z''. For most infinite lattice graphs, ''p''c cannot be calculated exactly.
==Universality==
The universality principle states that the value of ''p''c is connected to the local structure of the graph, while the behavior of clusters below, at, and above ''p''c are invariant with respect to the local structure, and therefore, in some sense are more natural quantities to consider.
This universality also means that for the same dimension independent of the type of the lattice or type of percolation (e.g., bond or site) the fractal dimension of the clusters at ''p''c is the same.
抄文引用元・出典: フリー百科事典『 bond percolation, was introduced in the mathematics literature by , and has been studied intensively by mathematicians and physicists since then.In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability ''p'' or "empty" (in which case its edges are removed) with probability ''1-p''; the corresponding problem is called site percolation. The question is the same: for a given ''p'', what is the probability that a path exists between top and bottom?Of course the same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By Kolmogorov's zero-one law, for any given ''p'', the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of ''p'' (proof via coupling argument), there must be a critical ''p'' (denoted by ''p''c) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for ''n'' as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of ''p''. In some cases ''p''c may be calculated explicitly. For example, for the square lattice Z2 in two dimensions, ''p''c = 1/2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, see . A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at ''p''c = 1/(''z'' − 1) for a coordination number ''z''. For most infinite lattice graphs, ''p''c cannot be calculated exactly.==Universality==The universality principle states that the value of ''p''c is connected to the local structure of the graph, while the behavior of clusters below, at, and above ''p''c are invariant with respect to the local structure, and therefore, in some sense are more natural quantities to consider.This universality also means that for the same dimension independent of the type of the lattice or type of percolation (e.g., bond or site) the fractal dimension of the clusters at ''p''c is the same.">ウィキペディア（Wikipedia）』
■bond percolation, was introduced in the mathematics literature by , and has been studied intensively by mathematicians and physicists since then.In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability ''p'' or "empty" (in which case its edges are removed) with probability ''1-p''; the corresponding problem is called site percolation. The question is the same: for a given ''p'', what is the probability that a path exists between top and bottom?Of course the same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By Kolmogorov's zero-one law, for any given ''p'', the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of ''p'' (proof via coupling argument), there must be a critical ''p'' (denoted by ''p''c) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for ''n'' as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of ''p''. In some cases ''p''c may be calculated explicitly. For example, for the square lattice Z2 in two dimensions, ''p''c = 1/2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, see . A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at ''p''c = 1/(''z'' − 1) for a coordination number ''z''. For most infinite lattice graphs, ''p''c cannot be calculated exactly.==Universality==The universality principle states that the value of ''p''c is connected to the local structure of the graph, while the behavior of clusters below, at, and above ''p''c are invariant with respect to the local structure, and therefore, in some sense are more natural quantities to consider.This universality also means that for the same dimension independent of the type of the lattice or type of percolation (e.g., bond or site) the fractal dimension of the clusters at ''p''c is the same.">ウィキペディアで「In statistical physics and mathematics, percolation theory describes the behavior of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article percolation.==Introduction==Bond percolation and Site percolation redirect to here -->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 ''n'' × ''n'' × ''n'' vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability ''p'', or closed with probability 1 – ''p'', and they are assumed to be independent. Therefore, for a given ''p'', what is the probability that an open path exists from the top to the bottom? The behavior for large ''n'' is of primary interest. This problem, called now bond percolation, was introduced in the mathematics literature by , and has been studied intensively by mathematicians and physicists since then.In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability ''p'' or "empty" (in which case its edges are removed) with probability ''1-p''; the corresponding problem is called site percolation. The question is the same: for a given ''p'', what is the probability that a path exists between top and bottom?Of course the same questions can be asked for any lattice dimension. As is quite typical, it is actually easier to examine infinite networks than just large ones. In this case the corresponding question is: does an infinite open cluster exist? That is, is there a path of connected points of infinite length "through" the network? By Kolmogorov's zero-one law, for any given ''p'', the probability that an infinite cluster exists is either zero or one. Since this probability is an increasing function of ''p'' (proof via coupling argument), there must be a critical ''p'' (denoted by ''p''c) below which the probability is always 0 and above which the probability is always 1. In practice, this criticality is very easy to observe. Even for ''n'' as small as 100, the probability of an open path from the top to the bottom increases sharply from very close to zero to very close to one in a short span of values of ''p''. In some cases ''p''c may be calculated explicitly. For example, for the square lattice Z2 in two dimensions, ''p''c = 1/2 for bond percolation, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, see . A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at ''p''c = 1/(''z'' − 1) for a coordination number ''z''. For most infinite lattice graphs, ''p''c cannot be calculated exactly.==Universality==The universality principle states that the value of ''p''c is connected to the local structure of the graph, while the behavior of clusters below, at, and above ''p''c are invariant with respect to the local structure, and therefore, in some sense are more natural quantities to consider.This universality also means that for the same dimension independent of the type of the lattice or type of percolation (e.g., bond or site) the fractal dimension of the clusters at ''p''c is the same.」の詳細全文を読む
スポンサード リンク
| 翻訳と辞書 : 翻訳のためのインターネットリソース |
Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.
|
|