Percolation theory

The percolation theory (English percolation - seepage ) describes the formation of contiguous areas ( clusters ) at A random occupation of structures ( grids ). Examples are the Punktperkolation or Kantenperkolation.

General

When Punktperkolation grid points are occupied with a certain probability, in the Kantenperkolation occupied points to be interconnected. But you can be arbitrary objects randomly generated (such as droplets) present, which are then examined.

With the percolation phenomena such as electrical conductivity of alloys, spreads of epidemics and forest fires or growth models are described.

Historically, the percolation theory goes (English: percolation theory ) on Paul Flory and Walter H. Stockmayer back, they developed during the Second World War to describe polymerization. The polymerization process is concluded by the juxtaposition of molecules thereby forming macromolecules. The composite of these macromolecules leads to a network of connections that can be drawn through the whole system.

In geology / hydrology percolation describes simple models for the spreading of liquids in porous rock (see percolation (technology) ), which serve as illustrative examples of clustering described below.

Modeling

Percolations be modeled on grids, with the crystal lattice (see crystal structure ) interpretations of mathematical gratings.

Knotenperkolation (site percolation )

General can be a simple model for the " node " or " Platzperkolation " construct:

The fields of a two-dimensional square lattice are occupied with a certain probability. Whether a field is filled or left empty, is independent of the occupation of all the other fields; occupied spaces " ignore " so to speak, the neighboring fields. Further, the mesh assumed to be so large that the edge effects can be neglected (ideally infinite ). Depending on the given distribution will form groups on the grid, that is occupied spaces in the immediate vicinity. These groups - called clusters - will be greater, the greater is the probability of occupation of a field. The percolation theory now deals with properties such as size or number of clusters.

If the probability is that a square is occupied, formed from the rise of larger clusters. The so-called percolation threshold is defined as the value of which at least one cluster to a size that it extends through the entire system, that is an extension to the grid from the right to left and from the upper to the lower side has. They say: The cluster percolates through the system.

Kantenperkolation ( bond percolation )

The counterpart to this is " Kantenperkolation " (English: bond percolation ) called.

A grating, e.g., the above square lattice is fully occupied and there are four of each square of the grid connections to the four neighboring fields. Now a probability is open to an adjacent field, and with a probability closed compound is a compound. This type of percolation can be compared well with the above model in geology. The voids in a porous rock are filled with water and connected by a network of channels. With a probability of a channel between two nearest neighbors, and with a probability of exists no.

A cluster is then defined as the set of grid locations, which are connected by open channels. Again, the percolation threshold again, and there is a cluster, which percolates through the entire system, while such does not exist in the cluster. The percolation threshold is lower in the Kantenperkolation than in systems which behave as Knotenperkolation. This applies to all types of gratings.

Directed percolation

The directed percolation ( directed percolation DP) can be explained very clearly with a coffee maker (English coffee percolator ) or with the already mentioned porous rock.

On the basis of bond percolation is the difference between " normal", or isotropic percolation (IP) and the directed percolation clear.

When water is poured into a porous medium, the question arises as to whether the media can be penetrated, that is, 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 hits an open channel is given as a percolation through isotropic. In contrast to the isotropic percolation exists a given preferred direction. Water in porous rocks as well as in the coffee maker is moving in the direction that is determined by the gravity. The percolation threshold is higher than in the directed percolation for the isotropic percolation.