Editing Cluster algorithms
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'''Cluster algorithms''' are mainly used in the simulation of [[Ising Models|Ising-like models]] | '''Cluster algorithms''' are mainly used in the simulation of [[Ising Models|Ising-like models]]. The essential feature is the use of collective motions of particles (spins) in a single [[Monte Carlo]] step. | ||
An interesting property of some of these | An interesting property of some of these application is the fact that the [[percolation analysis]] of the clusters can | ||
be used to study [[phase transitions]]. | be used to study [[phase transitions]]. | ||
== Swendsen-Wang algorithm == | == Swendsen-Wang algorithm == | ||
As an introductory example to the Swendsen-Wang algorithm we shall discuss the technique | As an introductory example to the Swendsen-Wang algorithm we shall discuss the technique (Ref 1) in the simulation of the | ||
in the simulation of the | |||
[[Ising Models |Ising model]]. In one [[Monte Carlo]] step of the algorithm the following recipe is used: | [[Ising Models |Ising model]]. In one [[Monte Carlo]] step of the algorithm the following recipe is used: | ||
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The procedure to create a given bond is the same as in the Swendsen-Wang algorithm. However in Wolff's method | The procedure to create a given bond is the same as in the Swendsen-Wang algorithm. However in Wolff's method | ||
the whole set of interacting pairs is not tested to generate (possible) bonds. Instead, a single cluster | the whole set of interacting pairs is not tested to generate (possible) bonds. Instead, a single cluster | ||
is built. See | is built. See Ref 2 for details. | ||
for details. | |||
#The initial cluster contains one site, which is selected at random. | #The initial cluster contains one site, which is selected at random. | ||
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== Invaded Cluster Algorithm == | == Invaded Cluster Algorithm == | ||
The purpose of this algorithm is to locate [[critical points]] (i.e. the critical temperature). So, in this case | The purpose of this algorithm is to locate [[critical points]] (i.e. the critical temperature). So, in this case | ||
the probability of bonding neighbouring sites with equal spins is not set ''a priori'' (See | the probability of bonding neighbouring sites with equal spins is not set ''a priori'' (See Ref 3). | ||
The algorithm for an Ising system with [[boundary conditions |periodic boundary conditions]] can be implemented as follows: | |||
The algorithm for an Ising system with [[periodic boundary conditions]] can be implemented as follows: | |||
Given a certain configuration of the system: | Given a certain configuration of the system: | ||
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== Probability-Changing Cluster Algorithm == | == Probability-Changing Cluster Algorithm == | ||
This method was proposed by Tomita and Okabe | This method was proposed by Tomita and Okabe (See Ref 4). This procedure is orientated towards computing [[critical points]]. | ||
It applies when the symmetry of the interactions imply that the critical | It applies when the symmetry of the interactions imply that the critical | ||
temperature is that in which the clusters, built using a Swendsen-Wang type algorithm, reach | temperature is that in which the clusters, built using a Swendsen-Wang type algorithm, reach | ||
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In addition, extensions have been proposed in the literature | In addition, extensions have been proposed in the literature | ||
to build up very efficient cluster algorithms to simulate more complex lattice systems (for example the [[XY model]], | to build up very efficient cluster algorithms to simulate more complex lattice systems (for example the [[XY model]], | ||
[[Heisenberg model]], [[Lebwohl-Lasher model]] | [[Heisenberg model]], [[Lebwohl-Lasher model]], etc). | ||
== Application to continuous (atomistic) models == | == Application to continuous (atomistic) models == | ||
It is sometimes possible (and very convenient) to include cluster algorithms in the simulation of | It is sometimes possible (and very convenient) to include cluster algorithms in the simulation of | ||
models with continuous translational degrees of freedom. In most cases the cluster algorithm has | models with continuous translational degrees of freedom. In most cases the cluster algorithm has | ||
to be complemented with other sampling moves to ensure [[Ergodic hypothesis |ergodicity]]. Examples: | to be complemented with other sampling moves to ensure [[Ergodic hypothesis |ergodicity]]. Examples: | ||
* Spin fluids | * Spin fluids | ||
* Binary | * Binary mixtures having interaction symmetry | ||
* Continuous versions of the [[XY model]], [[Heisenberg model]], [[Lebwohl-Lasher model]], etc. | * Continuous versions of the [[XY model]], [[Heisenberg model]], [[Lebwohl-Lasher model]], etc. | ||
In these cases, the usual approach is to combine one-particle moves (e.g. particle translations), | In these cases, the usual approach is to combine one-particle moves (e.g. particle translations), | ||
with cluster procedures. In the cluster steps, multiparticle modification of -composition, orientations, etc.- | with cluster procedures. In the cluster steps, multiparticle modification of -composition, orientations, etc.- | ||
is carried out. | is carried out. | ||
== | == Other (not so smart) applications of cluster algorithms == | ||
Monte Carlo simulation of atomistic systems with multiparticle moves, for example see: | |||
* | *[http://dx.doi.org/10.1063/1.2759924 N. G. Almarza and E. Lomba "Cluster algorithm to perform parallel Monte Carlo simulation of atomistic systems", Journal of Chemical Physics '''127''' 084116 (2007)] | ||
== References == | == References == | ||
#[http://dx.doi.org/10.1103/PhysRevLett.58.86 Robert H. Swendsen and Jian-Sheng Wang, "Nonuniversal critical dynamics in Monte Carlo simulations", Physical Review Letters '''58''' pp. 86 - 88 (1987) ] | |||
#[http://dx.doi.org/10.1103/PhysRevLett.62.361 Ulli Wolff, "Collective Monte Carlo Updating for Spin Systems" , Physical Review Letters '''62''' pp. 361 - 364 (1989) ] | |||
#[http://dx.doi.org/10.1103/PhysRevLett.75.2792 J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, and L. V. Chayes, "Invaded Cluster Algorithm for Equilibrium Critical Points" , Physical Review Letters '''75''' pp. 2792 - 2795 (1995)] | |||
#[http://dx.doi.org/10.1103/PhysRevLett.86.572 Yusuke Tomita and Yutaka Okabe, "Probability-Changing Cluster Algorithm for Potts Models", Physical Review Letters '''86''' pp. 572 - 575 (2001)] | |||
[[category: computer simulation techniques]] | [[category: computer simulation techniques]] | ||
[[category: Monte Carlo]] | [[category: Monte Carlo]] |