Editing Percolation analysis
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref | 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 == | ||
Line 10: | Line 10: | ||
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, | * Geometric distance: Two sites, i, j, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. | ||
* | * Energetic criteria: Two sites i, j, 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== | ||
The sizes of the clusters of a given system depend on different ''control parameters'': | 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 | 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 | 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 === | ||
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, | 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|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 | 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. | appears due to the periodic boundary conditions. | ||
== 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 | ** They are nearest neighbors 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 === | ||
Line 72: | Line 64: | ||
depend on the system size: | depend on the system size: | ||
*<math> X_{\rm per}(x_c,L) \approx | *<math> X_{\rm per}(x_c,L) \approx X_c </math> ; for large values of <math> L </math>. | ||
== 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 | 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] </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 === | ||
to be continued | |||
=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation === | === Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation === | ||
to be continued | |||
==References== | ==References== | ||
<references/> | <references/> | ||
[[Category: Confined systems]] | [[Category: Confined systems]] |