Editing Percolation analysis
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== Connectivity rules == | == Connectivity rules == | ||
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic | The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic | ||
interaction, types of sites, ''etc''. | interaction, types of sites, ''etc''. | ||
In addition the bonding criteria can be | In addition the bonding criteria can be deterministic or probabilistic. | ||
In | In the statistical mechanics applications one can find different bonding criteria, for example: | ||
* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below). | * Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below). | ||
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: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models. | : <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models. | ||
* Energetic criteria (probabilistic): As an example, in the simulation of | * Energetic criteria (probabilistic): As an example, in the simulation of Ising models using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details). | ||
==Percolation threshold== | ==Percolation threshold== |