The European Physical Journal Special Topics

, Volume 226, Issue 4, pp 551–565 | Cite as

A brief history of the introduction of generalized ensembles to Markov chain Monte Carlo simulations

Review
Part of the following topical collections:
  1. Recent Advances in Phase Transitions and Critical Phenomena

Abstract

The most efficient weights for Markov chain Monte Carlo calculations of physical observables are not necessarily those of the canonical ensemble. Generalized ensembles, which do not exist in nature but can be simulated on computers, lead often to a much faster convergence. In particular, they have been used for simulations of first order phase transitions and for simulations of complex systems in which conflicting constraints lead to a rugged free energy landscape. Starting off with the Metropolis algorithm and Hastings’ extension, I present a minireview which focuses on the explosive use of generalized ensembles in the early 1990s. Illustrations are given, which range from spin models to peptides.

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References

  1. 1.
    N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953)ADSCrossRefGoogle Scholar
  2. 2.
    Z.W. Salsburg, J.D. Jacobson, W.S. Fickett, W.W. Wood, J. Chem. Phys. 30, 65 (1959)ADSCrossRefGoogle Scholar
  3. 3.
    I.R. McDonald, K. Singer, Disc. Faraday Soc. 43, 40 (1967)CrossRefGoogle Scholar
  4. 4.
    A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988)ADSCrossRefGoogle Scholar
  5. 5.
    A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 63, 1658 (1988)ADSCrossRefGoogle Scholar
  6. 6.
    R.J. Glauber, J. Math. Phys. 4, 294 (1963)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    W.K. Hastings, Biometrika 57, 97 (1970)MathSciNetCrossRefGoogle Scholar
  8. 8.
    B.A. Berg, Markov Chain Monte Carlo Simulations and their Statistical Analysis (World Scientific, 2004)Google Scholar
  9. 9.
    J.P. Valleau, D.N. Card, J. Chem. Phys. 57, 5457 (1972)ADSCrossRefGoogle Scholar
  10. 10.
    G.M. Torrie, J.P. Valleau, Chem. Phys. Lett. 28, 578 (1974)ADSCrossRefGoogle Scholar
  11. 11.
    G.N. Patey, J.P. Valleau, J. Chem. Phys. 63, 2334 (1975)ADSCrossRefGoogle Scholar
  12. 12.
    G.M. Torrie, J.P. Valleau, J. Comp. Phys. 23, 187 (1977)ADSCrossRefGoogle Scholar
  13. 13.
    J. Chandrasekar, S.F. Smith, W.K. Jorgensen, J. Am. Chem. Soc. 107, 154 (1985)CrossRefGoogle Scholar
  14. 14.
    Z. Lie, H.A. Scheraga, J. Mol. Struct. 179, 333 (1978)CrossRefGoogle Scholar
  15. 15.
    D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, 1987)Google Scholar
  16. 16.
    L. Onsager, Phys. Rev. 65, 117 (1944)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    K. Binder, Phys. Rev. A 25, 1699 (1982)ADSCrossRefGoogle Scholar
  18. 18.
    G. Bhanot, R. Salvador, S. Black, P. Carter, R. Toral, Phys. Rev. Lett. 59, 803 (1987)ADSCrossRefGoogle Scholar
  19. 19.
    G. Parisi et al., Phys. Rev. Lett. 61, 1545 (1988)ADSCrossRefGoogle Scholar
  20. 20.
    N.A. Alves, B.A. Berg, S. Sanielevici, Phys. Rev. Lett. 64, 3107 (1990)ADSCrossRefGoogle Scholar
  21. 21.
    J. Potvin, C. Rebbi, Phys. Rev. Lett. 62, 3062 (1989)ADSCrossRefGoogle Scholar
  22. 22.
    K. Kajanti, L. Kärkkäinen, K. Rummukainen, Phys. Lett. B 223, 213 (1989)ADSCrossRefGoogle Scholar
  23. 23.
    B.A. Berg, T. Neuhaus, Phys. Lett. B 267, 249 (1991)ADSCrossRefGoogle Scholar
  24. 24.
    B.A. Berg, T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992)ADSCrossRefGoogle Scholar
  25. 25.
    C. Borgs, S. Kappler, Phys. Lett. A 171, 37 (1992)ADSCrossRefGoogle Scholar
  26. 26.
    W. Janke, B.A. Berg, M. Katoot, Nucl. Phys. B 382, 649 (1992)ADSCrossRefGoogle Scholar
  27. 27.
    C. Borgs, W. Janke, J. Phys. I France 2, 2011 (1992)CrossRefGoogle Scholar
  28. 28.
    E. Buffenoir, S. Wallon, Saclay preprint SPHT/92-077Google Scholar
  29. 29.
    A. Billoire, T. Neuhaus, B.A. Berg, Nucl. Phys. B 413, 795 (1994)ADSCrossRefGoogle Scholar
  30. 30.
    B.A. Berg, U. Hansmann, T. Neuhaus, Phys. Rev. B 47, 497 (1993)ADSCrossRefGoogle Scholar
  31. 31.
    B.A. Berg, U. Hansmann, T. Neuhaus, Z. Phys. B 90, 229 (1993)ADSCrossRefGoogle Scholar
  32. 32.
    W. Janke, S. Kappler, Phys. Rev. Lett. 74, 212 (1995)ADSCrossRefGoogle Scholar
  33. 33.
    B.A. Berg, W. Janke, Phys. Rev. Lett. 98, 040602 (2007)ADSCrossRefGoogle Scholar
  34. 34.
    B.A. Berg, T. Celik, Phys. Rev. Lett. 69, 2292 (1992)ADSCrossRefGoogle Scholar
  35. 35.
    B.A. Berg, T. Celik, U. Hansmann, Phys. Rev. B 50, 16444 (1994)ADSCrossRefGoogle Scholar
  36. 36.
    B.A. Berg, A. Billoire, W. Janke, Phys. Rev. B 61, 12143 (2000)ADSCrossRefGoogle Scholar
  37. 37.
    U. Hansmann, Y. Okamoto, J. Comp. Chem. 14, 1333 (1993)CrossRefGoogle Scholar
  38. 38.
    U. Hansmann, Y. Okamoto, J. Chem. Phys. 110, 1267 (1999)ADSCrossRefGoogle Scholar
  39. 39.
    U. Hansmann, Y. Okamoto, J. Chem. Phys. 111, 1339 (1999)ADSCrossRefGoogle Scholar
  40. 40.
    U. Hansmann, Y. Okamoto, Ann. Rev. Comp. Phys., edited by D. Stauffer (World Scientific, 1999) vol VI, pp. 129–157Google Scholar
  41. 41.
    A.P. Lyubartsev, A.A. Martsinovski, S.V. Shevkanov, P.N. Vorontsov-Velyaminov, J. Chem. Phys. 96, 1776 (1992)ADSCrossRefGoogle Scholar
  42. 42.
    E. Marinari, G. Parisi, Europhys. Lett. 19, 451 (1992)ADSCrossRefGoogle Scholar
  43. 43.
    G.J. Geyer, in Computing Science and Statistics, Proceedings of the 23rd Symposium on the Interface, edited by E.M. Keramidas, S.M. Kaufman (Interface Foundation, Fairfax, VA, 1991) pp. 156–163Google Scholar
  44. 44.
    K. Hukusima, K. Nemoto, J. Phys. Soc. Japan 65, 1604 (1996)ADSCrossRefGoogle Scholar
  45. 45.
    R.H. Swendsen, J.-S. Wang, Phys. Rev. Lett. 57, 2607 (1986)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    U.H. Hansmann, Chem. Phys. Lett. 281, 140 (1997)ADSCrossRefGoogle Scholar
  47. 47.
    E. Bittner, A. Nuß baumer, W. Janke, Phys. Rev. Lett. 101, 130603 (2008)ADSCrossRefGoogle Scholar
  48. 48.
    Y. Sugita, Y. Okamoto, Chem. Phys. Lett. 314, 141 (1999)ADSCrossRefGoogle Scholar
  49. 49.
    T. Neuhaus, J.S. Hager, J. Stat. Phys. 113, 47 (2003)CrossRefGoogle Scholar
  50. 50.
    B.A. Berg, J. Stat. Phys. 82, 323 (1996)ADSCrossRefGoogle Scholar
  51. 51.
    F. Wang, D.P. Landau, Phys. Rev. Lett. 86, 2050 (2001)ADSCrossRefGoogle Scholar
  52. 52.
    E. Bittner, W. Janke, in preparationGoogle Scholar
  53. 53.
    A. Bazavov, B.A. Berg, Comp. Phys. Comm. 180, 2339 (2009)ADSCrossRefGoogle Scholar
  54. 54.
    T. Vogel, Y.W. Li, T. Wüst, D.P. Landau, Phys. Rev. Lett. 110, 210603 (2013)ADSCrossRefGoogle Scholar
  55. 55.
    J. Zierenberg, M. Marenz, W. Janke, Comp. Phys. Comm. 184, 1155 (2013)ADSCrossRefGoogle Scholar
  56. 56.
    T. Vogel, D. Perez, Phys. Rev. Lett. 115, 190602 (2015)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.Department of PhysicsFlorida State University, TallahasseeFLUSA

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