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]].

{{orphan | date=9 March 2009}}

The '''Wolff algorithm''' is an [[algorithm]] for [[Monte Carlo simulation]] of the [[Ising model]] in which an equal-spin cluster is formed around one spin. That cluster is then flipped. The Wolff algorithm is an improvement over the [[Swendsen-Wang algorithm]] because it tends to form bigger clusters.

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 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 }}

*{{citation | doi=10.11421103/S0129183195000150PhysRevLett.62.361 | title=Collective Monte Carlo Updating for Spin Systems | year=19951989 | author=BaeWolff, S.Ulli | journal=InternationalPhysical JournalReview ofLetters Modern| Physicsvolume=62 C| pages=361–364 | volumepmid=610040213 | pagesissue=1974 | bibcode=1989PhRvL..62..361W}}

*{{citation | doi=10.11031142/PhysRevLett.69.3382S0129183195000150 | title=MonteParallel CarloWolff simulations:cluster Hiddenalgorithms errors| fromyear=1995 ‘‘good’’| randomauthor1=Bae, number generatorsS. | yearauthor2=1992Ko, S.H. | authorauthor3=FerrenbergCoddington, Alan MP.D. | journal=PhysicalInternational ReviewJournal Lettersof Modern Physics C | volume=696 | issue=2 | pages=3382197 |bibcode = 1995IJMPC...6..197B | citeseerx=10.1.1.138.1448 }}

*{{citation | doi=10.1103/PhysRevLett.69.3382 | title=Monte Carlo simulations: Hidden errors from ''good'' random number generators | year=1992 | author1=Ferrenberg, Alan M. | author2=Landau, D.P. | author3=Wong, Y. Joanna | journal=Physical Review Letters | volume=69 | pages=3382–3384 | pmid=10046804 | issue=23 | bibcode=1992PhRvL..69.3382F}}

==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

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[[Category:Monte Carlo methods]]

[[Category:Statistical mechanics]]

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