Editing Percolation analysis

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 == 

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.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications are expected to be more realistic to capture the physics of a number of real systems <ref name=lee>

== 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 either deterministic or probabilistic.
In [[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).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <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 [[Ising model]]s 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==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc. 

Let us consider some initial conditions of the control parameters, in
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 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) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition'''). 

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== 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> 
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.
* The probability of occupancy of each site is <math>  \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if: 
** They are nearest neighbours and 
** Both sites are occupied.

=== 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
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 )</math> one will have <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
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 (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0  ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1  ; {\rm for} \; \; x > x_c </math>


In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)}  </math> ; for large values of <math> L </math>.

== 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.
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=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 ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered,  an alternative definition
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:

:<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> <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>.

== 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 <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> . 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/>
[[Category: Confined systems]]