Multicanonical MCMC for sampling rare events: an illustrative review

  • Yukito Iba
  • Nen Saito
  • Akimasa Kitajima
Special Issue: Bayesian Inference and Stochastic Computation


Multicanonical MCMC (Multicanonical Markov Chain Monte Carlo; Multicanonical Monte Carlo) is discussed as a method of rare event sampling. Starting from a review of the generic framework of importance sampling, multicanonical MCMC is introduced, followed by applications in random matrices, random graphs, and chaotic dynamical systems. Replica exchange MCMC (also known as parallel tempering or Metropolis-coupled MCMC) is also explained as an alternative to multicanonical MCMC. In the last section, multicanonical MCMC is applied to data surrogation; a successful implementation in surrogating time series is shown. In the appendix, calculation of averages and normalizing constant in an exponential family, phase coexistence, simulated tempering, parallelization, and multivariate extensions are discussed.


Multicanonical MCMC Wang–Landau algorithm Replica exchange MCMC Rare event sampling Random matrix  Random graph Chaotic dynamical system Exact test Surrogation  



The authors would like to thank Koji Hukushima for the helpful discussions and the permission for the use of figures in Saito et al. (2010). We would also be grateful to Arnaud Doucet and the referees, for their helpful advice allowed us to improve the manuscript. This work was supported by JSPS KAKENHI Grant Numbers 22500217, 25330299, and 25240036. Saito is supported by a Grant-in-Aid for Scientific Research (No. 21120004) on Innovative Areas “Neural creativity for communication” (No. 4103), and the Platform for Dynamic Approaches to Living System from MEXT, Japan.


  1. Aazami, A., Easther, R. (2006). Cosmology from random multifield potentials. Journal of Cosmology and Astroparticle Physics, 3(03), 013.Google Scholar
  2. Agresti, A. (1992). A survey of exact inference for contingency tables. Statistical Science, 7(1), 131–153.Google Scholar
  3. Atchadé, Y. F., Liu, J.S. (2010). The Wang-Landau algorithm in general state spaces: Applications and convergence analysis. Statistica Sinica, 20(1), 209–233.Google Scholar
  4. Bachmann, M., Janke, W. (2003). Multicanonical chain-growth algorithm. Physical Review Letters, 91(20), 208105.Google Scholar
  5. Baumann, B. (1987). Noncanonical path and surface simulation. Nuclear Physics B, 285, 391–409.Google Scholar
  6. Beck, C., Schlögl, F. (1993). Thermodynamics of Chaotic Systems: An Introduction. Cambridge: Cambridge University Press.Google Scholar
  7. Belardinelli, R., Pereyra, V. (2007a). Fast algorithm to calculate density of states. Physical Review E, 75, 046701.Google Scholar
  8. Belardinelli, R., Pereyra, V. (2007b). Wang-Landau algorithm: A theoretical analysis of the saturation of the error. The Journal of Chemical Physics, 127, 184105.Google Scholar
  9. Berg, B. A. (2000). Introduction to multicanonical Monte Carlo simulations. Fields Institute Communications, 26, 1–24.Google Scholar
  10. Berg, B. A. (2004). Markov Chain Monte Carlo Simulations and Their Statistical Analysis. Singapore: World Scientific.Google Scholar
  11. Berg, B. A., Celik, T. (1992). New approach to spin-glass simulations. Physical Review Letters, 69(15), 2292–2295.Google Scholar
  12. Berg, B. A., Neuhaus, T. (1991). Multicanonical algorithms for first order phase transitions. Physics Letters B, 267(2), 249–253.Google Scholar
  13. Berg, B. A., Neuhaus, T. (1992). Multicanonical ensemble: A new approach to simulate first-order phase transitions. Physical Review Letters, 68(1), 9–12.Google Scholar
  14. Besag, J., Clifford, P. (1989). Generalized Monte Carlo significance tests. Biometrika, 76(4), 633–642.Google Scholar
  15. Binder, K., Heermann, D. (2012). Monte Carlo Simulation in Statistical Physics: An Introduction. Berlin: Springer.Google Scholar
  16. Birge, J. R., Chang, C., Polson, N. G. (2012). Split sampling: Expectations, normalisation and rare events. ArXiv e-prints, 1212, 0534.Google Scholar
  17. Bononi, A., Rusch, L., Ghazisaeidi, A., Vacondio, F., Rossi, N. (2009). A fresh look at multicanonical Monte Carlo from a telecom perspective. In Global Telecommunications Conference, 2009. GLOBECOM 2009, IEEE, pp 1–8.Google Scholar
  18. Bornn, L., Jacob, P. E., Del Moral, P., Doucet, A. (2013). An adaptive interacting Wang-Landau algorithm for automatic density exploration. Journal of Computational and Graphical Statistics, 22(3), 749–773.Google Scholar
  19. Botev, Z. I., L’Ecuyer, P., Tuffin, B. (2013). Markov chain importance sampling with applications to rare event probability estimation. Statistics and Computing, 23(2), 271–285.Google Scholar
  20. Brooks, S., Gelman, A., Jones, G. L., Meng, X. L. (Eds.). (2011). Handbook of Markov Chain Monte Carlo. New York: Chapman and Hall/CRC.Google Scholar
  21. Bucklew, J. A. (2004). Introduction to Rare Event Simulation (Springer Series in Statistics). New York: Springer.Google Scholar
  22. Bunea, F., Besag, J. (2000). MCMC in \({I} \times J \times {K}\) contingency tables. Fields Institute Communications, 26, 25–36.Google Scholar
  23. Calvo, F. (2002). Sampling along reaction coordinates with the Wang-Landau method. Molecular Physics, 100(21), 3421–3427.Google Scholar
  24. Chikenji, G., Kikuchi, M. (2000). What is the role of non-native intermediates of \(\beta \)-lactoglobulin in protein folding? Proceedings of the National Academy of Sciences, 97(26), 14,273–14,277.Google Scholar
  25. Chikenji, G., Kikuchi, M., Iba, Y. (1999). Multi-self-overlap ensemble for protein folding: ground state search and thermodynamics. Physical Review Letters, 83(9), 1886–1889.Google Scholar
  26. Chopin, N., Lelièvre, T., Stoltz, G. (2012). Free energy methods for Bayesian inference: Efficient exploration of univariate Gaussian mixture posteriors. Statistics and Computing, 22(4), 897–916.Google Scholar
  27. Dean, D. S., Majumdar, S. N. (2008). Extreme value statistics of eigenvalues of Gaussian random matrices. Physical Review E, 77(4), 041108.Google Scholar
  28. de Oliveira, P. M. C., Penna, T. J. P., Herrmann, H. J. (1998). Broad histogram Monte Carlo. The European Physical Journal B-Condensed Matter and Complex Systems, 1(2), 205–208.Google Scholar
  29. Diaconis, P., Sturmfels, B. (1998). Algebraic algorithms for sampling from conditional distributions. The Annals of statistics, 26(1), 363–397.Google Scholar
  30. Donetti, L., Hurtado, P. I., Muñoz, M. A. (2005). Entangled networks, synchronization, and optimal network topology. Physical Review Letters, 95(18), 188701.Google Scholar
  31. Donetti, L., Neri, F. (2006). Muñoz MA (2006) Optimal network topologies: Expanders, cages, Ramanujan graphs, entangled networks and all that. Journal of Statistical Mechanics: Theory and Experiment, 08, P08007.Google Scholar
  32. Driscoll, T. A., Maki, K. L. (2007). Searching for rare growth factors using multicanonical Monte Carlo methods. SIAM Review, 49(4), 673–692.Google Scholar
  33. Fishman, G. S. (2012). Counting contingency tables via multistage Markov chain Monte Carlo. Journal of Computational and Graphical Statistics, 21(3), 713–738.Google Scholar
  34. Fort, G., Jourdain, B., Kuhn, E., Lelièvre, T., Stoltz, G. (2012). Convergence and efficiency of the Wang-Landau algorithm. ArXiv e-prints, 1207, 6880.Google Scholar
  35. Frenkel, D., Smit, B. (2002). Understanding Molecular Simulation, From Algorithms to Applications (Computational Science Series) (2nd ed.). San Diego: Academic Press.Google Scholar
  36. Geiger, P., Dellago, C. (2010). Identifying rare chaotic and regular trajectories in dynamical systems with Lyapunov weighted path sampling. Chemical Physics, 375(2–3), 309–315.Google Scholar
  37. Geyer, C. J. (1991). Markov chain Monte Carlo maximum likelihood. In E. Keramidas (Ed.), Computing science and statistics: Proceedings of 23rd Symposium on the Interface (pp. 156–163). Fairfax Station: Interface Foundation.Google Scholar
  38. Geyer, C. J., Thompson, E. A. (1995). Annealing Markov chain Monte Carlo with applications to ancestral inference. Journal of the American Statistical Association, 90(431), 909–920.Google Scholar
  39. Gilks, W. R., Richardson, S., Spiegelhalter, D. J. (Eds.). (1996). Markov Chain Monte Carlo in Practice. London: Chapman and Hall.Google Scholar
  40. Grün, S., Rotter, S. (Eds.). (2010). Analysis of Parallel Spike Trains (Springer Series in Computational Neuroscience). New York: Springer.Google Scholar
  41. Hartmann, A. K. (2002). Sampling rare events: Statistics of local sequence alignments. Physical Review E, 65(5), 056102.Google Scholar
  42. Higo, J., Nakajima, N., Shirai, H., Kidera, A., Nakamura, H. (1997). Two-component multicanonical Monte Carlo method for effective conformation sampling. Journal of computational chemistry, 18(16), 2086–2092.Google Scholar
  43. Higo, J., Ikebe, J., Kamiya, N., Nakamura, H. (2012). Enhanced and effective conformational sampling of protein molecular systems for their free energy landscapes. Biophysical Reviews, 4, 27–44.Google Scholar
  44. Hirata, Y., Katori, Y., Shimokawa, H., Suzuki, H., Blenkinsop, T. A., Lang, E. J., et al. (2008). Testing a neural coding hypothesis using surrogate data. Journal of Neuroscience Methods, 172(2), 312–322.Google Scholar
  45. Holzlöhner, R., Menyuk, C. R. (2003). Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems. Optics Letters, 28(20), 1894–1896.Google Scholar
  46. Holzlöhner, R., Mahadevan, A., Menyuk, C. R., Morris, J. M., Zweck, J. (2005). Evaluation of the very low BER of FEC codes using dual adaptive importance sampling. IEEE Communications Letters, 9(2), 163–165.Google Scholar
  47. Hukushima, K. (2002). Extended ensemble Monte Carlo approach to hardly relaxing problems. Computer Physics Communications, 147(1–2), 77–82.Google Scholar
  48. Hukushima, K., Iba, Y. (2008). A Monte Carlo algorithm for sampling rare events: application to a search for the Griffiths singularity. Journal of Physics: Conference Series, 95, 012005.Google Scholar
  49. Hukushima, K., Nemoto, K. (1996). Exchange Monte Carlo method and application to spin glass simulations. Journal of the Physical Society of Japan, 65(6), 1604–1608.Google Scholar
  50. Iba, Y. (2001). Extended ensemble Monte Carlo. International Journal of Modern Physics C, 12(05), 623–656.Google Scholar
  51. Iba, Y., Hukushima, K. (2008). Testing error correcting codes by multicanonical sampling of rare events. Journal of the Physical Society of Japan, 77(10), 103801.Google Scholar
  52. Iba, Y., Takahashi, H. (2005). Exploration of multi-dimensional density of states by multicanonical Monte Carlo algorithm. Progress of Theoretical Physics Supplements, 157, 345–348.Google Scholar
  53. Iba, Y., Chikenji, G., Kikuchi, M. (1998). Simulation of lattice polymers with multi-self-overlap ensemble. Journal of the Physical Society of Japan, 67, 3327–3330.Google Scholar
  54. Jacob, P. E., Ryder, R. J. (2011). The Wang-Landau algorithm reaches the flat histogram criterion in finite time. ArXiv e-prints, 1110, 4025.Google Scholar
  55. Jacobson, M. T., Matthews, P. (1996). Generating uniformly distributed random Latin squares. Journal of Combinatorial Designs, 4(6), 405–437.Google Scholar
  56. Janke, W. (1998). Multicanonical Monte Carlo simulations. Physica A: Statistical Mechanics and its Applications, 254(1–2), 164–178.Google Scholar
  57. Jerrum, M., Sinclair, A. (1996). The Markov chain Monte Carlo method: An approach to approximate counting and integration. Approximation algorithms for NP-hard problems, pp. 482–520.Google Scholar
  58. Kastner, C. A., Braumann, A., Man, P. L. W., Mosbach, S., Brownbridge, G. P. E., Akroyd, J., et al. (2013). Bayesian parameter estimation for a jet-milling model using Metropolis-Hastings and Wang-Landau sampling. Chemical Engineering Science, 89, 244–257.Google Scholar
  59. Kimura, K., Taki, K. (1991). Time-homogeneous parallel annealing algorithm. In Proceedings of the 13th IMACS World Congress on Computation and Applied Mathematics (IMACS’91), Vol. 2, pp. 827–828.Google Scholar
  60. Kirkpatrick, S., Gelatt, C. D., Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220(4598), 671–680.Google Scholar
  61. Kitajima, A., Iba, Y. (2011). Multicanonical sampling of rare trajectories in chaotic dynamical systems. Computer Physics Communications, 182(1), 251–253.Google Scholar
  62. Körner, M., Katzgraber, H. G., Hartmann, A. K. (2006). Probing tails of energy distributions using importance-sampling in the disorder with a guiding function. Journal of Statistical Mechanics: Theory and Experiment, 04, P04005.Google Scholar
  63. Kumar, S. (2013). Random matrix ensembles: Wang-Landau algorithm for spectral densities. Europhysics Letters, 101(2), 20002.Google Scholar
  64. Kwon, J., Lee, K.M. (2008). Tracking of abrupt motion using Wang-Landau Monte Carlo estimation. In Proceedings of the 10th European Conference on Computer Vision: Part I, ECCV ’08, Heidelberg: Springer, pp. 387–400.Google Scholar
  65. Laffargue, T., Lam, K. D. N. T., Kurchan, J., Tailleur, J. (2013). Large deviations of Lyapunov exponents. Journal of Physics A: Mathematical and Theoretical, 46(25), 254002.Google Scholar
  66. Landau, D.P., Binder, K. (2009). A Guide to Monte Carlo Simulations in Statistical Physics (3rd ed.). Cambridge University Press.Google Scholar
  67. Landau, D. P., Tsai, S. H., Exler, M. (2004). A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling. American Journal of Physics, 72(10), 1294–1302.Google Scholar
  68. Lee, H. K., Okabe, Y., Landau, D. P. (2006). Convergence and refinement of the Wang-Landau algorithm. Computer Physics Communications, 175(1), 36–40.Google Scholar
  69. Lee, J. (1993). New Monte Carlo algorithm: Entropic sampling. Physical Review Letters, 71(2), 211–214.Google Scholar
  70. Liang, F. (2005). A generalized Wang-Landau algorithm for Monte Carlo computation. Journal of the American Statistical Association, 100(472), 1311–1327.Google Scholar
  71. Liang, F., Liu, C., Carroll, R. J. (2007). Stochastic approximation in Monte Carlo computation. Journal of the American Statistical Association, 102(477), 305–320.Google Scholar
  72. Liang, F., Liu, C., Carroll, R. J. (2010). Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples (Wiley Series in Computational Statistics). West Sussex: Wiley.Google Scholar
  73. Lyubartsev, A. P., Martsinovski, A. A., Shevkunov, S. V., Vorontsov-Velyaminov, P. N. (1992). New approach to Monte Carlo calculation of the free energy: Method of expanded ensembles. The Journal of Chemical Physics, 96(3), 1776–1783.Google Scholar
  74. Majumdar, S. N., Vergassola, M. (2009). Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices. Physical Review Letters, 102(6), 060601.Google Scholar
  75. Marinari, E., Parisi, G. (1992). Simulated tempering: A new Monte Carlo scheme. Europhysics Letters, 19(6), 451–458.Google Scholar
  76. Matsuda, Y., Nishimori, H., Hukushima, K. (2008). The distribution of Lee-Yang zeros and Griffiths singularities in the \(\pm \) J model of spin glasses. Journal of Physics A: Mathematical and Theoretical, 41(32), 324012.Google Scholar
  77. May, R. M. (1972). Will a large complex system be stable? Nature, 238, 413–414.Google Scholar
  78. Mezei, M. (1987). Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias. Journal of Computational Physics, 68(1), 237–248.Google Scholar
  79. Mitsutake, A., Sugita, Y., Okamoto, Y. (2001). Generalized-ensemble algorithms for molecular simulations of biopolymers. Biopolymers (Peptide Science), 60(2), 96–123.Google Scholar
  80. Monthus, C., Garel, T. (2006). Probing the tails of the ground-state energy distribution for the directed polymer in a random medium of dimension d = 1, 2, 3 via a Monte Carlo procedure in the disorder. Physical Review E, 74(5), 051109.Google Scholar
  81. Newman, M. E. J., Barkema, G. T. (1999). Monte Carlo Methods in Statistical Physics. New York: Clarendon Press.Google Scholar
  82. Ott, E. (2002). Chaos in Dynamical Systems. Chambridge: Cambridge University Press.Google Scholar
  83. Pinn, K., Wieczerkowski, C. (1998). Number of magic squares from parallel tempering Monte Carlo. International Journal of Modern Physics C, 09(04), 541–546.Google Scholar
  84. Prellberg, T., Krawczyk, J. (2004). Flat histogram version of the pruned and enriched Rosenbluth method. Physical Review Letters, 92(12), 120602.Google Scholar
  85. Robert, C. P., Casella, G. (2004). Monte Carlo Statistical Methods (2nd ed.). New York: Springer.Google Scholar
  86. Rubino, G., Tuffin, B. (Eds.). (2009). Rare Event Simulation using Monte Carlo Methods. West Sussex: Wiley.Google Scholar
  87. Rubinstein, R. Y., Kroese, D. P. (2008). Simulation and the Monte Carlo Method (Wiley Series in Probability and Statistics) (2nd ed.). Hoboken: Wiley-Interscience.Google Scholar
  88. Saito, N., Iba, Y. (2011). Probability of graphs with large spectral gap by multicanonical Monte Carlo. Computer Physics Communications, 182(1), 223–225.Google Scholar
  89. Saito, N., Iba, Y., Hukushima, K. (2010). Multicanonical sampling of rare events in random matrices. Physical Review E, 82(3), 031142.Google Scholar
  90. Sasa, S., Hayashi, K. (2006). Computation of the Kolmogorov-Sinai entropy using statistical mechanics: Application of an exchange Monte Carlo method. Europhysics Letters, 74(1), 156–162.Google Scholar
  91. Schreiber, T. (1998). Constrained randomization of time series data. Physical Review Letters, 80(10), 2105–2108.Google Scholar
  92. Schreiber, T., Schmitz, A. (2000). Surrogate time series. Physica D: Nonlinear Phenomena, 142(3–4), 346–382.Google Scholar
  93. Schulz, B. J., Binder, K., Müller, M., Landau, D. P. (2003). Avoiding boundary effects in Wang-Landau sampling. Physical Review E, 67(6), 067102.Google Scholar
  94. Shell, M. S., Debenedetti, P. G., Panagiotopoulos, A. Z. (2002). Generalization of the Wang-Landau method for off-lattice simulations. Physical Review E, 66(5), 056703.Google Scholar
  95. Shirai, N. C., Kikuchi, M. (2013). Multicanonical simulation of the Domb-Joyce model and the Gō model: new enumeration methods for self-avoiding walks. Journal of Physics: Conference Series, 454(1), 012039.Google Scholar
  96. Shteto, I., Linares, J., Varret, F. (1997). Monte Carlo entropic sampling for the study of metastable states and relaxation paths. Physical Review E, 56(5), 5128–5137.Google Scholar
  97. Sweet, D., Nusse, H. E., Yorke, J. A. (2001). Stagger-and-step method: Detecting and computing chaotic saddles in higher dimensions. Physical Review Letters, 86(11), 2261–2264.Google Scholar
  98. Tailleur, J., Kurchan, J. (2007). Probing rare physical trajectories with Lyapunov weighted dynamics. Nature Physics, 3(3), 203–207.Google Scholar
  99. Takemura, A., Aoki, S. (2004). Some characterizations of minimal Markov basis for sampling from discrete conditional distributions. Annals of the Institute of Statistical Mathematics, 56(1), 1–17.Google Scholar
  100. Torrie, G. M., Valleau, J. P. (1974). Monte Carlo free energy estimates using non-Boltzmann sampling: Application to the sub-critical Lennard-Jones fluid. Chemical Physics Letters, 28(4), 578–581.Google Scholar
  101. Tracy, C. A., Widom, H. (1994). Level-spacing distributions and the Airy kernel. Communications in Mathematical Physics, 159(1), 151–174.Google Scholar
  102. Tracy, C. A., Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Communications in Mathematical Physics, 177(3), 727–754.Google Scholar
  103. Vogel, T., Li, Y. W., Wüst, T., Landau, D. P. (2013). Generic, hierarchical framework for massively parallel Wang-Landau sampling. Physical Review Letters, 110, 210603.Google Scholar
  104. Vorontsov-Velyaminov, P. N., Broukhno, A. V., Kuznetsova, T. V., Lyubartsev, A. (1996). Free energy calculations by expanded ensemble method for lattice and continuous polymers. The Journal of Physical Chemistry, 100(4), 1153–1158.Google Scholar
  105. Vorontsov-Velyaminov, P. N., Volkov, N. A., Yurchenko, A. A. (2004). Entropic sampling of simple polymer models within Wang-Landau algorithm. Journal of Physics A: Mathematical and General, 37(5), 1573–1588.Google Scholar
  106. Wang, F., Landau, D. P. (2001a). Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram. Physical Review E, 64(5), 056101.Google Scholar
  107. Wang, F., Landau, D. P. (2001b). Efficient, multiple-range random walk algorithm to calculate the density of states. Physical Review Letters, 86(10), 2050–2053.Google Scholar
  108. Wang, J. S., Swendsen, R. H. (2002). Transition matrix Monte Carlo method. Journal of Statistical Physics, 106(1–2), 245–285.Google Scholar
  109. Wolfsheimer, S., Hartmann, A. K. (2010). Minimum-free-energy distribution of RNA secondary structures: Entropic and thermodynamic properties of rare events. Physical Review E, 82(2), 021902.Google Scholar
  110. Wolfsheimer, S., Herms, I., Rahmann, S., Hartmann, A. K. (2011). Accurate statistics for local sequence alignment with position-dependent scoring by rare-event sampling. BMC Bioinformatics, 12(1), 47.Google Scholar
  111. Wüst, T., Landau, D. P. (2012). Optimized Wang-Landau sampling of lattice polymers: Ground state search and folding thermodynamics of HP model proteins. The Journal of Chemical Physics, 137(6), 064903.Google Scholar
  112. Yan, Q., Faller, R., de Pablo, J. J. (2002). Density-of-states Monte Carlo method for simulation of fluids. The Journal of Chemical Physics, 116(20), 8745–8749.Google Scholar
  113. Yanagita, T., Iba, Y. (2009) Exploration of order in chaos using the replica exchange Monte Carlo method. Journal of Statistical Mechanics: Theory and Experiment, 02, P02043.Google Scholar
  114. Yevick, D. (2002). Multicanonical communication system modeling-Application to PMD statistics. IEEE Photonics Technology Letters, 14(11), 1512–1514.Google Scholar
  115. Yu, K., Liang, F., Ciampa, J., Chatterjee, N. (2011). Efficient p-value evaluation for resampling-based tests. Biostatistics, 12(3), 582–593.Google Scholar
  116. Zhan, L. (2008). A parallel implementation of the Wang-Landau algorithm. Computer Physics Communications, 179(5), 339–344.Google Scholar
  117. Zhang, C., Ma, J. (2007). Simulation via direct computation of partition functions. Physical Review E, 76(3), 036708.Google Scholar
  118. Zhang, C., Ma, J. (2009). Counting solutions for the N-queens and Latin-square problems by Monte Carlo simulations. Physical Review E, 79(1), 016703.Google Scholar
  119. Zhou, C., Su, J. (2008). Optimal modification factor and convergence of the Wang-Landau algorithm. Physical Review E, 78(4), 046705.Google Scholar
  120. Zhou, C., Schulthess, T. C., Torbrügge, S., Landau, D. P. (2006). Wang-Landau algorithm for continuous models and joint density of states. Physical Review Letters, 96(12), 120201.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2014

Authors and Affiliations

  1. 1.The Institute of Statistical Mathematics and SOKENDAITachikawa, TokyoJapan
  2. 2.Research Center for Complex Systems BiologyThe University of TokyoMeguro-ku, TokyoJapan
  3. 3.Digital Information Services Division, Digital Information DepartmentNational Diet LibraryChiyoda-ku, TokyoJapan

Personalised recommendations