Wolff algorithm: Difference between revisions

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The '''Wolff algorithm''',<ref>{{Cite journal|last=Wolff|first=Ulli|date=1989-01-23|title=Collective Monte Carlo Updating for Spin Systems|url=https://link.aps.org/doi/10.1103/PhysRevLett.62.361|journal=Physical Review Letters|volume=62|issue=4|pages=361–364|doi=10.1103/PhysRevLett.62.361|pmid=10040213|bibcode=1989PhRvL..62..361W |url-access=subscription}}</ref> named after [[Ulli Wolff]], is an [[algorithm]] for [[Monte Carlo simulation]] of the [[Ising model]] and [[Potts model]] in which the unit to be flipped is not a single spin (as in the [[Glauber dynamics|heat bath]] or [[Metropolis–Hastings algorithm|Metropolis algorithms]]) but a cluster of them. This cluster is defined as the set of connected spins sharing the same spin states, based on the [[Random cluster model|Fortuin-Kasteleyn representation]].

{{Merge|Monte Carlo method| discuss=Talk:Monte Carlo method|date=June 2012}}

The Wolff algorithm is similar to the [[Swendsen–Wang algorithm]], but different in that the former only flips one randomly chosen cluster with probability 1, while the latter flip every cluster independently with probability 1/2. It is shown numerically that flipping only one cluster decreases the [[autocorrelation]] time of the spin statistics.

The '''Wolff algorithm''', named after [[Ulli Wolff]], is an [[algorithm]] for [[Monte Carlo simulation]] of the [[Ising model]] in which the unit to be flipped is not a single spin, as in the heat bath or Metropolis algorithms, but a cluster of them. This cluster is defined as the set of neighbouring spins sharing the same value of the spin. The Wolff algorithm is an improvement over the [[Swendsen–Wang algorithm]] because it has a larger probability of flipping bigger clusters.

The advantage of Wolff algorithm over other algorithms for magnetic spin simulations like single spin flip is that it allows non-local moves on the energy. One important consequence of this is that in some situations (e.g. ferromagnetic Ising model or fully frustrated Ising model), the scaling of the Multicanonic simulation is <math>N^2</math>, better than <math>N^{2+z}</math>, where z is the exponent associated with the critical slowing down phenomena.

==References==

{{Reflist}}

*{{citation | doi=10.1103/PhysRevLett.62.361 | title=Collective Monte Carlo Updating for Spin Systems | year=1989 | author=Wolff, Ulli | journal=Physical Review Letters | volume=62 | pages=361 | pmid=10040213 | issue=4 | bibcode=1989PhRvL..62..361W}}

*{{citation | doi=10.11421103/S0129183195000150PhysRevLett.62.361 | title=ParallelCollective WolffMonte Carlo Updating for clusterSpin algorithmsSystems | year=19951989 | author1author=BaeWolff, S.Ulli | author2journal=Ko,Physical S.H.Review Letters | author3volume=Coddington, P.D.62 | journalpages=International Journal of Modern Physics C361–364 | volumepmid=610040213 | issue=24 | pages=197 |bibcode = 1995IJMPC1989PhRvL..62.6..197B 361W}}

*{{citation | doi=10.11031142/PhysRevLett.69.3382S0129183195000150 | title=MonteParallel CarloWolff simulations:cluster Hidden errors from ‘‘good’’ random number generatorsalgorithms | year=19921995 | author1=FerrenbergBae, Alan MS. | author2=LandauKo, DS.PH. | author3=WongCoddington, YP.D. Joanna | journal=PhysicalInternational ReviewJournal Lettersof Modern Physics C | volume=696 | pagesissue=33822 | pmidpages=10046804197 |bibcode issue=23 1995IJMPC...6..197B | bibcodeciteseerx=1992PhRvL10.1.691.3382F138.1448 }}

*{{citation | doi=10.1103/PhysRevLett.6269.3613382 | title=Collective Monte Carlo Updatingsimulations: forHidden Spinerrors Systemsfrom ''good'' random number generators | year=19891992 | authorauthor1=WolffFerrenberg, UlliAlan M. | author2=Landau, D.P. | author3=Wong, Y. Joanna | journal=Physical Review Letters | volume=6269 | pages=3613382–3384 | pmid=1004021310046804 | issue=423 | bibcode=1989PhRvL.1992PhRvL.62.69.361W3382F}}

==External links==

*[http://www.netlib.org/utk/lsi/pcwLSI/text/node292.html ''Cluster Algorithms''] at [[Netlib]]

*Implementation in Julia: https://github.com/cossio/SquareIsingModel.jl

[[Category:Monte Carlo methods]]

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