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.
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′ → S) = P p (S → S′).
- 2.
We calculate the magnetization in a particular spin Q via
$$\displaystyle{ \mathcal{M}_{Q}(\mathcal{C}) = \left (\sum _{i}\delta _{\sigma _{i},Q}\right )_{\mathcal{C}}\;. }$$(18.18) - 3.
This are regions in which all spins point in the same direction, the so-called Weiss domains [15].
References
Norris, J.R.: Markov Chains. Cambridge Series in Statistical and Probabilistic. Cambridge University Press, Cambridge (1998)
Kendall, W.S., Liang, F., Wang, J.S.: Markov Chain Monte Carlo: Innovations and Applications. Lecture Notes Series, vol. 7. Institute for Mathematical Sciences, National University of Singapore. World Scientific, Singapore (2005)
Modica, G., Poggiolini, L.: A First Course in Probability and Markov Chains. Wiley, New York (2012)
Graham, C.: Markov Chains: Analytic and Monte Carlo Computations. Wiley, New York (2014)
Potts, R.B.: Some generalized order-disorder transformations. Math. Proc. Camb. Philos. Soc. 48, 106–109 (1952). doi:10.1017/S0305004100027419
Doucet, A., de Freitas, N., Gordon, N. (eds.): Sequential Monte Carlo Methods in Practice. Information Science and Statistics. Springer, Berlin/Heidelberg (2001)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C++, 2nd edn. Cambridge University Press, Cambridge (2002)
Kalos, M.H., Whitlock, P.A.: Monte Carlo Methods, 2nd edn. Wiley, New York (2008)
von der Linden, W., Dose, V., von Toussaint, U.: Bayesian Probability Theory. Cambridge University Press, Cambridge (2014)
Berg, B.A.: Markov Chain Monte Carlo Simulations and Their Statistical Analysis. World Scientific, Singapore (2004)
German, S.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984). doi:10.1109/TPAMI.1984.4767596
Neal, R.M.: Slice sampling. Ann. Stat. 31, 705–767 (2003). doi:10.1214/aos/1056562461
Kardar, M., Berker, A.N.: Commensurate-incommensurate phase diagrams for overlayers from a helical potts model. Phys. Rev. Lett. 48, 1552–1555 (1982). doi:10.1103/PhysRevLett.48.1552
Wu, F.Y.: The potts model. Rev. Mod. Phys. 54, 235–268 (1982). doi:10.1103/RevModPhys.54.235
White, R.M.: Quantum Theory of Magnetism, 3rd edn. Springer Series in Solid-State Sciences. Springer, Berlin/Heidelberg (2007)
Swendsen, R.H., Wang, J.S.: Nonuniversal critical dynamics in monte carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987). doi:10.1103/PhysRevLett.58.86
Wolff, U.: Collective monte carlo updating for spin systems. Phys. Rev. Lett. 62, 361–364 (1989). doi:10.1103/PhysRevLett.62.361
Evertz, H.G.: The loop algorithm. Adv. Phys. 52, 1–66 (2003). doi:10.1080/0001873021000049195
Newman, M.E.J., Barkema, G.T.: Monte Carlo Methods in Statistical Physics. Clarendon Press, Oxford (1999)
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Stickler, B.A., Schachinger, E. (2016). Markov-Chain Monte Carlo and the Potts Model. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27265-8_18
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