Abstract
Numerical studies of statistical systems aim at sampling the Boltzmann-Gibbs distribution defined over the system configuration space. In the large-volume limit, the number of configurations becomes large and the distribution very narrow, so that independent-sampling methods do not work and importance sampling is needed. In this case, the dynamic Monte Carlo (MC) method, which only samples the relevant “equilibrium” configurations, is the appropriate tool.
However, in the presence of ergodicity breaking in the thermodynamic limit (for instance, in systems showing phase coexistence) standard MC simulations are not able to sample efficiently the Boltzmann-Gibbs distribution. Similar problems may arise when sampling rare configurations. We discuss here MC methods that are used to overcome these problems and, more generally, to determine thermodynamic/statistical properties that are controlled by rare configurations, which are indeed the subject of the theory of large deviations.
We first discuss the problem of data reweighting, then we introduce a family of methods that rely on non-Boltzmann-Gibbs probability distributions, umbrella sampling, simulated tempering, and multicanonical methods. Finally, we discuss parallel tempering which is a general multipurpose method for the study of multimodal distributions, both for homogeneous and disordered systems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A precise definition of D β is not necessary for our purposes. For example, we can consider for D β the smallest set of configurations such that \(\sum _{x\in D_{\beta }}\pi _{\beta }(x) > 1-\varepsilon\).
- 2.
It is interesting to observe that, for R = 2, the multiple histogram method is equivalent to Bennett’s acceptance ratio method [14] which was developed for liquid systems.
- 3.
In the case the measures are correlated with an autocorrelation time τ i , then an effective \(\tilde{n}_{i} = n_{i}/(2\tau _{i} + 1)\) should be used in all following formulae.
- 4.
We remind the reader of a few basic facts. If A i are different estimates of the same quantity, i.e., they all satisfy \(\langle A_{i}\rangle = a\), any weighted average \(A_{\mathrm{wt}} =\sum w_{i}A_{i}\), \(\sum _{i}w_{i} = 1\), is correct in the sense that \(\langle A_{\mathrm{wt}}\rangle = a\). Usually, one takes w i = kσ i −2 (k is the normalization factor) because this gives the optimal estimator, that is the one with the least error. Here, however, robustness and not optimality is the main issue.
- 5.
If the inverse temperatures β i are ordered, one could determine Z i ∕Z i−1 by using the reweighting method and then \(\hat{Z}_{i} = (Z_{i}/Z_{i-1})(Z_{i-1}/Z_{i-2})\ldots Z_{2}/Z_{1}\).
- 6.
There are instances of second-order transitions which show bimodal distributions in finite volume [23, 24]: however, in these cases the two peaks get closer and the gap decreases as the volume increases. ST should work efficiently in these instances. Note, however, that the algorithm may not work in some disordered systems, even if the transition is of second order. One example is the random field Ising model.
- 7.
In principle the swapping can be attempted among any pair of replicas, but only for nearby replicas the swap has a reasonable probability of being accepted.
- 8.
If the PT method is applied to a system undergoing a first-order transition, the swapping procedure would be highly inefficient, because HT replicas would hardly swap with LT replicas. The two sets of replicas would remain practically non-interacting.
- 9.
The condition of unimodality is not required in the proofs of the theorems. However, the theorems have physically interesting consequences only if a unimodal decomposition is possible.
References
N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953)
M.N. Rosenbluth, A.W. Rosenbluth, J. Chem. Phys. 22, 881 (1954)
B.J. Alder, S.P. Frankel, V.A. Lewinson, J. Chem. Phys. 23, 417 (1955)
B.J. Alder, T.E. Wainwright, J. Chem. Phys. 27, 1208 (1957)
S.R.S. Varadhan, Ann. Prob. 36, 397 (2008)
A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988); Erratum ibid. 63, 1658 (1989)
R.H. Swendsen, A.M. Ferrenberg, in Computer Studies in Condensed Matter Physics II, ed. by D.P. Landau, K.K. Mon, H.B. Schüttler (Springer, Berlin, 1990), pp. 179–183
A.M. Ferrenberg, D.P. Landau, R.H. Swendsen, Phys. Rev. E 51, 5092 (1995)
B. Efron, Ann. Stat. 7, 1 (1979)
B. Efron, Biometrika 68, 589 (1981)
L. Onsager, Phys. Rev. 65, 117 (1944)
B.M. McCoy, T.T. Wu, The Two-Dimensional Ising Model (Harvard University Press, Cambridge, 1973)
A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989)
C.H. Bennett, J. Comp. Phys. 22, 245 (1976)
G.M. Torrie, J.P. Valleau, J. Comp. Phys. 23, 187 (1977)
A.P. Lyubartsev, A.A. Martsinovski, S.V. Shevkunov, P.N. Vorontsov-Velyaminov, J. Chem. Phys. 96, 1776 (1991)
E. Marinari, G. Parisi, Europhys. Lett. 19, 451 (1992)
N. Madras, M. Piccioni, Ann. Appl. Prob. 9, 1202 (1999)
R.H. Swendsen, J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987)
R.G. Edwards, A.D. Sokal, Phys. Rev. D 38, 2009 (1988)
W.K. Hastings, Biometrika 57, 97 (1970)
N. Bhatnagar, D. Randall, in Proceeding of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans (ACM, New York, 2004), pp. 478–487
M. Fukugita, H. Mino, M. Okawa, A. Ukawa, J. Phys. A 23, L561 (1990)
J.F. McCarthy, Phys. Rev. B 41, 9530 (1990)
B. Berg, T. Neuhaus, Phys. Lett. B 267, 249 (1991)
B. Berg, T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992)
C.J. Geyer, in Computing Science and Statistics, Proceedings of the 23rd Symposium on the Interface, ed. by E.M. Keramidas, S.M. Kaufman (American Statistical Association, New York, 1991), pp. 156–163
M.C. Tesi, E.J. Janse van Rensburg, E. Orlandini, S.G. Whittington, J. Stat. Phys. 82, 155 (1996)
K. Hukushima, K. Nemoto, J. Phys. Soc. Jpn. 65, 1604 (1996)
U.H.E. Hansmann, Chem. Phys. Lett. 281, 140 (1997)
D.J. Earl, M.W. Deem, Phys. Chem. Chem. Phys. 7, 3910 (2005)
N. Madras, Z. Zheng, Random Struct. Alg. 22, 66 (2003)
Z. Zheng, Stoch. Proc. Appl. 104, 131 (2003)
D.B. Woodard, S.C. Schmidler, M. Huber, Ann. Appl. Prob. 19, 617 (2009)
N. Madras, D. Randall, Ann. Appl. Prob. 12, 581 (2002)
F. Martinelli, E. Olivieri, R. Schonmann, Comm. Math. Phys. 165, 33 (1994)
D.A. Kofke, J. Chem. Phys. 117, 6911 (2002)
A. Kone, D.A. Kofke, J. Chem. Phys. 122, 206101 (2005)
C. Predescu, M. Predescu, C.V. Ciobanu, J. Chem. Phys. 120, 4119 (2004)
D. Sabo, M. Meuwly, D.L. Freeman, J.D. Doll, J. Chem. Phys. 128, 174109 (2008)
H.G. Katzgraber, S. Trebst, D.A. Huse, M. Troyer, J. Stat. Mech. P03018 (2006)
E. Bittner, A. Nußbaumer, W. Janke, Phys. Rev. Lett. 101, 130603 (2008)
R. Alvarez Baños et al. (Janus collaboration), J. Stat. Mech. P06026 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pelissetto, A., Ricci-Tersenghi, F. (2014). Large Deviations in Monte Carlo Methods. In: Vulpiani, A., Cecconi, F., Cencini, M., Puglisi, A., Vergni, D. (eds) Large Deviations in Physics. Lecture Notes in Physics, vol 885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54251-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-54251-0_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54250-3
Online ISBN: 978-3-642-54251-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)