Percolation analysis: Difference between revisions
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This entry focuses on the application of percolation analysis to statistical mechanics | This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref>Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref> S. Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref> | ||
=== Sites, bonds, and clusters === | === Sites, bonds, and clusters === | ||
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clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more | clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more | ||
sequences of bonds between pairs of sites. | sequences of bonds between pairs of sites. | ||
The sites of the system can belong to different types (species in the chemistry language). | The sites of the system can belong to different types (species in the chemistry language). | ||
Bonds are usually permitted only between near sites. | Bonds are usually permitted only between near sites. | ||
=== | === 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 deterministic or probabilistic. | In addition the bonding criteria can be deterministic or probabilistic. | ||
In the statistical mechanics applications one can find different bonding criteria, for example: | In the statistical mechanics applications one can find different bonding criteria, for example: | ||
Revision as of 10:58, 7 September 2009
This entry focuses on the application of percolation analysis to problems in statistical mechanics. For a general discussion see Refs. [1] [2]
Sites, bonds, and clusters
This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system. Using some connectivity rules it is possible to define bonds between pairs of sites. These bonds can be used to build up clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more sequences of bonds between pairs of sites. The sites of the system can belong to different types (species in the chemistry language). Bonds are usually permitted only between near sites.
Connectivity rules
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. In addition the bonding criteria can be deterministic or probabilistic. In the statistical mechanics applications one can find different bonding criteria, for example:
- Geometric distance: Two sites, i, j, are bonded if the distance between then satisfies: .
- Energetic criteria: Two sites i, j, has a bonding probability given by
![{\displaystyle b(r_{ij})=\max \left\{0,\exp \left[u_{ij}(r_{ij})\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23f0c0dd398044f34132fc17a4a4d2b75bc27396)
Percolation threshold
The sizes of the clusters of a given system depend on different factors: density and distribution of sites, bonding criteria (which could include the effect of temperature and energy interactions), etc.
References
- ↑ Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533
- ↑ S. Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676