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 == | ||
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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>[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)]</ref>). | <ref>[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)]</ref>). | ||
The algorithm for an Ising system with [[periodic boundary conditions]] can be implemented as follows: | The algorithm for an Ising system with [[boundary conditions |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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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 | |||
<ref>[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)]</ref>. | |||
== References == | == References == |