Percolation theory

The percolation theory (Engl. Percolation - seepage) describes the formation of related areas ( clusters ) at random conditional occupying structures ( grids ).

The percolation theory can be used to describe phenomena such as the electrical conductivity of alloys , the spread of epidemics and forest fires or growth models. In geology / hydrology , percolation describes simple models for the spread of liquids in porous rock (see percolation (technology) ), which serve as illustrative examples of the cluster formation described below.

Subspecies are point percolation, in which grid points are occupied with a certain probability , and edge percolation, in which occupied points are connected to one another. One can imagine any randomly generated objects (e.g. droplets) being examined.

history

Historically, the percolation theory (ger .: goes percolation theory ) on Paul Flory and Walter H. Mayer floor back that the Flory-floor Mayer theory developed in the 1940s to polymerization to describe. The polymerization process comes about when molecules are lined up to form macromolecules . The combination of such macromolecules leads to a network of connections that can run through the entire system. Broadbent and Hammersley introduced the modern concept of percolation.

Modeling

Percolations are modeled on grids, with crystal grids being interpretations of mathematical grids.

Site percolation

Node and edge percolation

In general, a simple model for "node" or "space percolation" can be constructed:

The fields of a two-dimensional square grid are occupied with a certain probability. Whether a field is occupied or remains empty is independent of the occupation of all other fields. Furthermore, the grid is assumed to be so large that edge effects can be neglected (ideally infinitely large). Depending on the given distribution, groups will form on the grid; H. occupied fields in the immediate vicinity. These groups - called clusters - will be larger, the greater the probability of occupying a field. The percolation theory now deals with properties such as the size or number of these clusters.

If there is a probability that a field is occupied, larger clusters are formed as the number increases . The percolation threshold is defined as the value of at which at least one cluster reaches a size that it extends through the entire system, i.e. has an extension on the grid from the right to the left and from the upper to the lower side. They say: the cluster percolates through the system. ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p_{c}}$ ${\displaystyle p}$

Edge percolation (bond percolation)

The counterpart to this is called “edge percolation” ( bond percolation ).

Edge percolation in two dimensions with edge occupation probability p = 0.51 on a 50 × 50 section. There is a closed edge path that connects the lower and upper edges of the image.

A grid, e.g. B. the square grid mentioned above is completely occupied, and there are four connections from each field of the grid to the four neighboring fields. There is now a probability that a connection to a neighboring field is open and there is a probability that the connection is closed. This type of percolation can be compared well with the above-mentioned model in geology: the cavities in a porous rock are filled with water and connected by a network of channels; there is a probability that there is a channel between two nearest neighbors and there is a probability of none. ${\displaystyle p}$ ${\displaystyle 1-p}$ ${\displaystyle p}$ ${\displaystyle 1-p}$

A cluster is then defined as a group of grid locations that are connected by open channels. Here, too, is the percolation threshold, and for there is a cluster that percolates through the entire system, while such a cluster does not exist for. The percolation threshold is lower for edge percolation than for systems that behave according to node percolation. This applies to all types of grids. ${\displaystyle p_{c}}$ ${\displaystyle p>p_{c}}$ ${\displaystyle p<p_{c}}$

Grid type	Node percolation threshold	Edge percolation threshold
Honeycomb grid	0.6962	0.6527 ... = 1 - 2 sin (π / 18)
Square grid	0.592746	0.5
Triangular grid	0.5	0.34729 ... = 2 sin (π / 18)
Diamond lattice	0.43	0.388
simple cubic lattice	0.3116	0.2488
BCC 1.	0.246	0.1803
FCC	0.198	0.119
Hypercubic Lattice (4d)	0.197	0.1601
Hypercubic Lattice (5d)	0.141	0.1182
Hypercubic Lattice (6d)	0.107	0.0942
Hypercubic Lattice (7d)	0.089	0.0787

Directed percolation

The directed percolation (engl. Directed percolation DP) can be vividly with a coffee machine (engl. Coffee percolator or porous rocks) explained. ${\displaystyle \rightarrow }$

Based on the bond percolation, the difference between “normal” or isotropic percolation (IP) and directed percolation becomes clear.

directed percolation

When water is poured onto a porous medium, the question arises whether the medium can be penetrated; H. whether there is a channel from the top to the bottom of the medium, or whether the water is absorbed by the medium . The probability that the water will hit an open channel is given by, as with isotropic percolation . In contrast to isotropic percolation, however, there is a given preferred direction: water in porous rock as well as in the coffee machine moves in the direction that is determined by gravity . The percolation threshold is greater with directional percolation than with isotropic percolation. ${\displaystyle p}$ ${\displaystyle p_{c}}$

Applications in everyday life

Many percolation-like phase transitions occur in daily life, e.g. B. the " pudding problem" ( gel formation), the " cream stiffener problem" and the problem of "clumping". In all cases, the effect only goes towards the desired or undesired maximum when a critical value of the causal parameter is exceeded, usually according to a power law with a critical exponent , the maximum effect initially increasing very quickly when the critical value is exceeded. Chemical additives such as pudding or “cream stiffening” powder can reduce the critical value without changing the principle.

See also

• Crystal system

swell

1. ^ SR Broadbent, JM Hammersley: Percolation processes . In: Mathematical Proceedings of the Cambridge Philosophical Society . 53, No. 3, 2008, ISSN  0305-0041 , p. 629. bibcode : 1957PCPS ... 53..629B . doi : 10.1017 / S0305004100032680 .
2. ↑ Values ​​taken from uni-stuttgart.de, script (April 27, 2005): Simulation methods ( memento of the original from September 21, 2004 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 1.3 MB), pp. 41–52. 
3. ↑ ^{a b} M. F. Sykes, JW Essam: Exact critical percolation probabilities for site and bond problems in two dimensions . In: Journal of Mathematical Physics . tape 5 , no. 8 , 1964, pp. 1117–1127 , doi : 10.1063 / 1.1704215 , bibcode : 1964JMP ..... 5.1117S .
4. ↑ See, for example, the dissertation by Markus Lechtenfeld in the chemistry department of the University of Duisburg, on the subject of evaluating the rheological and optical investigations during the gelling of the gelatine / water system with the aid of percolation theory. Duisburg 2001, (online)

literature

• PJ Flory: Thermodynamics of High Polymer Solutions. In: Journal of Chemical Physics. 9, No. 8, August 1941, p. 660.
• PJ Flory: Thermodynamics of high polymer solutions. In: J. Chem. Phys. 10, 1942, pp. 51-61.
• WH Stockmayer: Theory of molecular size distribution and gel formation in branched polymers. In: J. Chem. Phys. 11, 1943, pp. 45-55.
• D. Stauffer , A. Aharony: Introduction to Percolation Theory. Taylor and Fransis, London 1994.
• D. Achlioptas et al: Explosive Percolation in Random Networks. In: Science. 2009.
• A. Bunde , HE Roman: Laws of Disorder. In: Physics in Our Time. 27, 1996, pp. 246-256.
• Vincent Beffara , Vladas Sidoravicius: Percolation. In: Encyclopedia of Mathematical Physics. Elsevier, 2006. Arxiv