Swendsen–Wang algorithm: Difference between revisions

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The '''Swendsen–Wang algorithm''' is the first non-local or cluster [[algorithm]] for [[Monte Carlo simulation]] for large systems near criticality. It has been introduced by [[Robert Swendsen]] and [[Jian-Sheng Wang]] in 1987 at [[Carnegie Mellon University|Carnegie Mellon]].

[[Robert Swendsen]] and Jian-Sheng Wang in 1987.

The original algorithm was designed for the Ising and Potts models, and it was later generalized to other systems as well, such as the XY model by [[Wolff algorithm]] and particles of fluids. AThe key ingredient iswas the [[random cluster model]], a representation of the Ising or [[Potts model|Potts]] model through percolation models of connecting bonds, due to Fortuin and Kasteleyn. It has been generalized by Barbu and Zhu (2005) to arbitrary sampling probabilities by viewing it as a [[Metropolis–Hastings algorithm]] and computing the acceptance probability of the proposed Monte Carlo move.

It has been generalized by Barbu and Zhu (2005) to arbitrary sampling probabilities by viewing it as a [[Metropolis–Hastings algorithm]] and computing the acceptance probability of the proposed Monte Carlo move.

== Motivation ==

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where <math>J_{nm}>0</math> is the ferromagnetic interaction intensity.

This probability distribution has been derived in the following way: the Hamiltonian of the Ising model is

<math>H[\sigma]=\sum\limits_{<i,j>}-J_{i,j}\sigma_i\sigma_j</math>,

and the [[Partition function (statistical mechanics)|partition function]] is <math>Z=\sum\limits_{\lbrace\sigma\rbrace}e^{-\beta H[\sigma]}</math>.

Consider the interaction between a pair of selected sites <math>n</math> and <math>m</math> and eliminate it from the total Hamiltonian, defining

<math>H_{nm}[\sigma]=\sum\limits_{<i,j>\neq<n,m>}-J_{i,j}\sigma_i\sigma_j.</math>

Define also the restricted sums:

<math>Z_{n,m}^{same}=\sum\limits_{\lbrace\sigma\rbrace}e^{-\beta H_{nm}[\sigma]}\delta_{\sigma_n,\sigma_m}</math>;

<math>Z_{n,m}^{diff}=\sum\limits_{\lbrace\sigma\rbrace}e^{-\beta H_{nm}[\sigma]}\left(1-\delta_{\sigma_n,\sigma_m}\right).</math>

<math>Z=e^{\beta J_{nm}}Z_{n,m}^{same}+e^{-\beta J_{nm}}Z_{n,m}^{diff}.</math>

Introduce the quantity

<math>Z_{nm}^{ind}=Z_{n,m}^{same}+Z_{n,m}^{diff}</math>;

the partition function can be rewritten as

<math>Z=\left(e^{\beta J_{nm}}-e^{-\beta J_{nm}}\right)Z_{n,m}^{same}+e^{-\beta J_{nm}}Z_{n,m}^{ind}.</math>