Abstract
Markov-chain Monte-Carlo is revisited in this chapter with the emphasis on importance sampling. This method allows to reduce the variance of expectation values measured during a computer simulation. It is shown that Markov-chain Monte-Carlo techniques correspond indeed to importance sampling as long as detailed balance is obeyed. The Metropolis algorithm with its symmetric acceptance probability is one realization of Markov-chains with detailed balance. Another realization can be established by the Metropolis–Hastings algorithm which uses an asymmetric acceptance probability. It also obeys detailed balance and results in an improved variance over the symmetric Metropolis algorithm. In an example the Potts model is studied in a computer simulation based on the Metropolis algorithm. The results of this simulation prove that the particular phase transition properties of this model can be reproduced faithfully. More advanced sampling techniques are discussed in passing.
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Notes
- 1.
Please note that it is common in the literature to refer even to Eq. (18.12) as a Metropolis–Hastings algorithm, despite the fact that here \(P_p(S' \rightarrow S) = P_p(S \rightarrow S')\).
- 2.
We calculate the magnetization in a particular spin configuration \(Q\) via
$$\begin{aligned} \fancyscript{M}_Q(\fancyscript{C}) = \left( \sum _i \delta _{\sigma _i,Q} \right) _\fancyscript{C}. \end{aligned}$$(18.18)
References
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Stickler, B.A., Schachinger, E. (2014). Markov-Chain Monte Carlo and the Potts Model. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-02435-6_18
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DOI: https://doi.org/10.1007/978-3-319-02435-6_18
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