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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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which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and | which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and | ||
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase | the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase | ||
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s | the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition'''). | ||
=== Percolation and boundary conditions === | === Percolation and boundary conditions === | ||
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== Percolation and finite-size scaling analysis == | == Percolation and finite-size scaling analysis == | ||
=== Example | |||
Let us consider a standard example of percolation theory, | === Example === | ||
a two-dimensional [[building up a square lattice|square lattice]] in which: | Let us consider a standard example of percolation theory, 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, and | ||
* The probability of occupancy of each site is <math> | * The probability of occupancy of each site is <math> x </math>. | ||
* Two sites are considered to be bonded if and only if: | * Two sites are considered to be bonded if and only if: | ||
** They are nearest neighbours and | ** They are nearest neighbours and | ||
** Both sites are occupied. | ** Both sites are occupied. | ||
=== Fraction of percolating realizations === | === Fraction of percolating realizations === | ||
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using | On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using | ||
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute | periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute | ||
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 | the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x </math>, (<math> x \rightarrow 0 </math>) we will get <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different | ||
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more | values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more | ||
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math> | abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math> | ||
at which the transition would take place for an infinite system ( | at which the transition would take place for an infinite system (so to say, in the thermodynamic limit). | ||
=== Finite-size scaling === | === Finite-size scaling === | ||
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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> | 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> | ||
=== 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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for <math> x_c\left(L \right) </math> can be taken, for instance: | for <math> x_c\left(L \right) </math> can be taken, for instance: | ||
<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>. | |||
The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as: | The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as: | ||
<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 | 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>. | ||
== Percolation threshold and critical thermodynamic transitions == | == Percolation threshold and critical thermodynamic transitions == | ||
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 | 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. 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]] |