Recent from talks
Percolation theory
Knowledge base stats:
Talk channels stats:
Members stats:
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 (cognitive psychology).
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 (meaning a path, each of whose links is an "open" bond) 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 Broadbent & Hammersley (1957), 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? Similarly, one can ask, given a connected graph at what fraction 1 – p of failures the graph will become disconnected (no large component).
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 pc) 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.
The Flory–Stockmayer theory was the first theory investigating percolation processes.
The history of the percolation model as we know it has its root in the coal industry. Since the industrial revolution, the economical importance of this source of energy fostered many scientific studies to understand its composition and optimize its use. During the 1930s and 1940s, the qualitative analysis by organic chemistry left more and more room to more quantitative studies.
In this context, the British Coal Utilisation Research Association (BCURA) was created in 1938. It was a research association funded by the coal mines owners. In 1942, Rosalind Franklin, who then recently graduated in chemistry from the university of Cambridge, joined the BCURA. She started research on the density and porosity of coal. During the Second World War, coal was an important strategic resource. It was used as a source of energy, but also was the main constituent of gas masks.
Coal is a porous medium. To measure its 'real' density, one was to sink it in a liquid or a gas whose molecules are small enough to fill its microscopic pores. While trying to measure the density of coal using several gases (helium, methanol, hexane, benzene), and as she found different values depending on the gas used, Rosalind Franklin showed that the pores of coal are made of microstructures of various lengths that act as a microscopic sieve to discriminate the gases. She also discovered that the size of these structures depends on the temperature of carbonation during the coal production. With this research, she obtained a PhD degree and left the BCURA in 1946.
Hub AI
Percolation theory AI simulator
(@Percolation theory_simulator)
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 (cognitive psychology).
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 (meaning a path, each of whose links is an "open" bond) 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 Broadbent & Hammersley (1957), 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? Similarly, one can ask, given a connected graph at what fraction 1 – p of failures the graph will become disconnected (no large component).
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 pc) 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.
The Flory–Stockmayer theory was the first theory investigating percolation processes.
The history of the percolation model as we know it has its root in the coal industry. Since the industrial revolution, the economical importance of this source of energy fostered many scientific studies to understand its composition and optimize its use. During the 1930s and 1940s, the qualitative analysis by organic chemistry left more and more room to more quantitative studies.
In this context, the British Coal Utilisation Research Association (BCURA) was created in 1938. It was a research association funded by the coal mines owners. In 1942, Rosalind Franklin, who then recently graduated in chemistry from the university of Cambridge, joined the BCURA. She started research on the density and porosity of coal. During the Second World War, coal was an important strategic resource. It was used as a source of energy, but also was the main constituent of gas masks.
Coal is a porous medium. To measure its 'real' density, one was to sink it in a liquid or a gas whose molecules are small enough to fill its microscopic pores. While trying to measure the density of coal using several gases (helium, methanol, hexane, benzene), and as she found different values depending on the gas used, Rosalind Franklin showed that the pores of coal are made of microstructures of various lengths that act as a microscopic sieve to discriminate the gases. She also discovered that the size of these structures depends on the temperature of carbonation during the coal production. With this research, she obtained a PhD degree and left the BCURA in 1946.
Recent media