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This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato">Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref> | This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore 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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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 | 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>. | ||
: <math> | |||
* Energetic criteria | * Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math> | ||
==Percolation threshold== | ==Percolation threshold== | ||
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== Percolation and finite-size scaling analysis == | == Percolation and finite-size scaling analysis == | ||
=== Example: Site-percolation on a square lattice === | === Example: Site-percolation on a square lattice === | ||
Let us consider a standard example of percolation theory, <ref name=deng> [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref> | Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref> | ||
a two-dimensional [[building up a square lattice|square lattice]] in which: | a two-dimensional [[building up a square lattice|square lattice]] in which: | ||
* Each site of the lattice can be occupied (by one ''particle'') or empty. | * Each site of the lattice can be occupied (by one ''particle'') or empty. | ||
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== Computation of the percolation threshold == | == Computation of the percolation threshold == | ||
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here. | A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here. | ||
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name=deng/> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref> | These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref> | ||
<ref name= | <ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details). | ||
=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes === | === Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes === | ||
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:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math> | :<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math> | ||
where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"/> <ref name=Torquato/>for details). Therefore, by fitting the results of <math> x_c(L) </math> it is | where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is | ||
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>. | possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>. | ||
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In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref> | In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref> | ||
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories", Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> | <ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details). | ||
. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details). | |||
==References== | ==References== | ||
<references/> | <references/> | ||
[[Category: Confined systems]] | [[Category: Confined systems]] |