Swendsen–Wang algorithm: Difference between revisions

Revision as of 11:58, 23 April 2017

Revision as of 06:26, 14 July 2017

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Let's consider a typical ferromagnetic Ising model with only nearest-neighbour interaction.

* Starting from a given configuration of spins, we associate to each couplepair of nearest neighbours on sites <math>n,m</math> a random variable <math>b_{n,m}\in \lbrace 0,1\rbrace</math> which is interpreted in the following way: if <math>b_{n,m}=0</math> there is no link between the sites <math>n</math> and <math>m</math>; if <math>b_{n,m}=1</math>, <math>\sigma_n</math> and <math>\sigma_m</math> are connected. These values are assigned according to the following probability distribution:

<math>P\left[b_{n,m}=0|\sigma_n\neq\sigma_m\right]=1</math>;

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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 couplepair 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: