Advertisement

Introduction to Cluster Monte Carlo Algorithms

  • E. Luijten
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 703)

Abstract

This chapter provides an introduction to cluster Monte Carlo algorithms for classical statistical-mechanical systems. A brief review of the conventional Metropolis algorithm is given, followed by a detailed discussion of the lattice cluster algorithm developed by Swendsen and Wang and the single-cluster variant introduced by Wolff. For continuum systems, the geometric cluster algorithm of Dress and Krauth is described. It is shown how their geometric approach can be generalized to incorporate particle interactions beyond hardcore repulsions, thus forging a connection between the lattice and continuum approaches. Several illustrative examples are discussed.

Keywords

Ising Model Detailed Balance Break Bond Pair Energy American Physical Society 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N.Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller (1953) Equation of state calculations by fast computing machines. J. Chem. Phys. 21, pp. 1087–1092CrossRefADSGoogle Scholar
  2. 2.
    D. Frenkel and B. Smit (2002) Understanding Molecular Simulation. San Diego: Academic, 2nd ed.Google Scholar
  3. 3.
    R. H. Swendsen and J.-S. Wang (1987) Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58(2), pp. 86–88CrossRefADSGoogle Scholar
  4. 4.
    P. W. Kasteleyn and C. M. Fortuin (1969) Phase transitions in lattice systems with random local properties. J. Phys. Soc. Jpn. Suppl. 26s, pp. 11–14ADSGoogle Scholar
  5. 5.
    C. M. Fortuin and P. W. Kasteleyn (1972) On the random-cluster model. I. Introduction and relation to other models. Physica 57, pp. 536–564CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    S.-K. Ma (1976) Modern Theory of Critical Phenomena. Redwood City, Calif.: Addison-WesleyCrossRefGoogle Scholar
  7. 7.
    U. Wolff (1989) Collective Monte Carlo updating for spin systems. Phys. Rev. Lett. 62(4), pp. 361–364CrossRefADSGoogle Scholar
  8. 8.
    R. J. Baxter, S. B. Kelland, and F. Y.Wu (1976) Equivalence of the Potts model or Whitney polynomial with an ice-type model. J. Phys. A 9, pp. 397–406CrossRefADSGoogle Scholar
  9. 9.
    E. Luijten and H. W. J. Blöte (1995) Monte Carlo method for spin models with long-range interactions. Int. J. Mod. Phys. C 6, pp. 359–370CrossRefADSGoogle Scholar
  10. 10.
    E. Luijten, H. W. J. Blöte, and K. Binder (1996) Crossover scaling in two dimensions. Phys. Rev. E 54, pp. 4626–4636CrossRefADSGoogle Scholar
  11. 11.
    E. Luijten and H. W. J. Blöte (1997) Classical critical behavior of spin models with long-range interactions. Phys. Rev. B 56, pp. 8945–8958CrossRefADSGoogle Scholar
  12. 12.
    E. Luijten and H. Meßingfeld (2001) Criticality in one dimension with inverse square-law potentials. Phys. Rev. Lett. 86, pp. 5305–5308CrossRefADSGoogle Scholar
  13. 13.
    A. Hucht (2002) On the symmetry of universal finite-size scaling functions in anisotropic systems. J. Phys. A 35, pp. L481–L487zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    C. Dress and W. Krauth (1995) Cluster algorithm for hard spheres and related systems. J. Phys. A 28, pp. L597–L601CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    J. R. Heringa and H. W. J. Blöte (1996) The simple-cubic lattice gas with nearest-neighbour exclusion: Ising universality. Physica A 232, pp. 369–374CrossRefADSGoogle Scholar
  16. 16.
    J. Liu and E. Luijten (2005) Generalized geometric cluster algorithm for fluid simulation. Phys. Rev. E 71, p. 066701CrossRefADSGoogle Scholar
  17. 17.
    A. Buhot and W. Krauth (1998) Numerical solution of hard-core mixtures. Phys. Rev. Lett. 80, pp. 3787–3790CrossRefADSGoogle Scholar
  18. 18.
    L. Santen and W. Krauth (2000) Absence of thermodynamic phase transition in a model glass former. Nature 405, pp. 550–551CrossRefADSGoogle Scholar
  19. 19.
    J. Liu and E. Luijten (2004) Rejection-free geometric cluster algorithm for complex fluids. Phys. Rev. Lett. 92(3), p. 035504CrossRefADSGoogle Scholar
  20. 20.
    J. G. Malherbe and S. Amokrane (1999) Asymmetric mixture of hard particles with Yukawa attraction between unlike ones: a cluster algorithm simulation study. Mol. Phys. 97, pp. 677–683CrossRefADSGoogle Scholar
  21. 21.
    J. R. Heringa and H. W. J. Blöte (1998) Geometric cluster Monte Carlo simulation. Phys. Rev. E 57(5), pp. 4976–4978CrossRefADSGoogle Scholar
  22. 22.
    J. R. Heringa and H. W. J. Blöte (1998) Geometric symmetries and cluster simulations. Physica A 254, pp. 156–163CrossRefGoogle Scholar
  23. 23.
    A. Coniglio and W. Klein (1980) Clusters and Ising critical droplets: a renormalisation group approach. J. Phys. A 13, pp. 2775–2780CrossRefADSGoogle Scholar
  24. 24.
    M. E. Fisher (1967) The theory of condensation and the critical point. Physics 3, pp. 255–283Google Scholar
  25. 25.
    M. P. Allen and D. J. Tildesley (1987) Computer Simulation of Liquids. Oxford: ClarendonzbMATHGoogle Scholar
  26. 26.
    J. Liu and E. Luijten (2004) Stabilization of colloidal suspensions by means of highly charged nanoparticles. Phys. Rev. Lett. 93, p. 247802CrossRefADSGoogle Scholar
  27. 27.
    S. Asakura and F. Oosawa (1954) On interaction between two bodies immersed in a solution of macromolecules. J. Chem. Phys. 22, pp. 1255–1256ADSGoogle Scholar
  28. 28.
    K. Binder and E. Luijten (2001) Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models. Phys. Rep. 344, pp. 179–253zbMATHCrossRefADSGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • E. Luijten
    • 1
  1. 1.Department of Materials Science and Engineering, Frederick Seitz Materials Research LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaU.S.A.

Personalised recommendations