Advertisement

Introduction to Monte Carlo Simulation Techniques

  • David Adams
Part of the NATO Advanced Science Institutes Series book series (volume 92)

Abstract

Starting from the assumption that matter, for our purposes, consists of interacting particles obeying classical mechanics, and using the postulates of statistical mechanics, one can model any specific material as a system of particles provided one knows what the interactions between the particles are. However, whatever interactions are chosen the integrals that it is necessary to solve are formidable. For example, the average potential energy is
$$ <U>=\int{U({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}}^{N}})}p({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}})d{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}}, $$
(1)
where the probability density for a configuration of N distinguishable particles, \( {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}}\equiv({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}_{1}},{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}_{2}},....,{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}_{N}}) \), r2,...., rN) is
$$ {{Q}_{N}}=\int{exp\left[ -U({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}}^{N}})/kT\right]d}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}}. $$
(2)
where the configurational integral QN is
$$ {{Q}_{N}}=\int{exp\left[ -U({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}}^{N}})/kT\right]}d{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r}}^{N}}. $$
(3)

Keywords

Random Walk Configuration Space Pair Potential Monte Carlo Calculation Monte CARLO Simulation Technique 
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.
    J. M. Hammersley and D. C. Handscomb, “Monte Carlo Methods”, Methuen, London (1964).MATHCrossRefGoogle Scholar
  2. 2.
    J. P. Valleau and G. M. Torrie, A Guide to Monte Carlo for Statistical Mechanics: 2. Byways, in: “Modern Theoretical Chemistry, Vol. 5A, Equilibrium Statistical Mechanics of Fluids”, B. J. Berne, ed., Plenum, New York (1977).Google Scholar
  3. 3.
    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21: 1087 (1953).ADSCrossRefGoogle Scholar
  4. 4.
    W. W. Wood, Monte Carlo Studies of Simple Liquid Models, in: “The Physics of Simple Liquids”, H. N. V. Temperley, J. S. Rowlinson, and G. S. Rushbrooke, eds., North-Holland, Amsterdam (1974).Google Scholar
  5. 5.
    J. P. Valleau and S. G. Whittington, A Guide to Monte Carlo for Statistical Mechanics: 1. Highways, in: “Modern Theoretical Chemistry, Vol. 5A, Equilibrium Statistical Mechanics of Fluids”, B. J. Berne, ed., Plenum, New York (1977).Google Scholar
  6. 6.
    C. Zannoni, Computer Simulations, in: “The Molecular Physics of Liquid Crystals”, G. R. Luckhurst and G. W. Gray, eds., Academic Press, London (1979).Google Scholar
  7. 7.
    D. M. Heyes, M. Barber and J. H. R. Clarke, J. C. S. Faraday II 73:1485 (1977); 75:1240, 1469 and 1484 (1979).Google Scholar
  8. 8.
    W. Schommers, Phys.Rev. A16: 327 (1977).ADSGoogle Scholar
  9. 9.
    M. J. L. Sangster and M. Dixon, Adv.Phys. 25: 247 (1976).ADSCrossRefGoogle Scholar
  10. 10.
    G. N. Patey, G. M. Torrie, and J. P. Valleau, J.Chem.Phys. 71: 96 (1979).ADSCrossRefGoogle Scholar
  11. 11.
    D. J. Adams, Chem. Phys. Lett., 62:329 (1979); in: “The Problem of Long-Range Forces in the Computer Simulation of Condensed Media”, D. Ceperely, ed., NRCC, Lawrence Berkeley Laboratory (1980).Google Scholar
  12. 12.
    I. R. McDonald and K. Singer, Disc. Faraday Soc. 43: 40 (1967).CrossRefGoogle Scholar
  13. 13.
    D. J. Adams and I. R. McDonald, Molec. Phys. 34: 287 (1977).ADSCrossRefGoogle Scholar
  14. 14.
    D. R. Squire and W. G. Hoover, J. Chem. Phys. 50: 701 (1969).ADSCrossRefGoogle Scholar
  15. 15.
    D. J. Adams and J. C. Rasaiah, Faraday Disc. 64: 22 (1978)Google Scholar
  16. 16.
    C. Pangali, M. Rao, and B. J. Berne, Chem. Phys. Lett. 55: 413 (1978).ADSCrossRefGoogle Scholar
  17. 17.
    P. J. Rossky, J. D. Doll, and H. L. Friedman, J. Chem. Phys. 69: 4628 (1978).ADSCrossRefGoogle Scholar
  18. 18.
    M. Rao, C. Pangali, and B. J. Berne, Molec. Phys. 37: 1773 (1979).ADSCrossRefGoogle Scholar
  19. 19.
    M. Rao and B. J. Berne, J. Chem. Phys. 71: 129 (1979).ADSCrossRefGoogle Scholar
  20. 20.
    I. R. McDonald, Molec. Phys. 23: 41 (1972).ADSCrossRefGoogle Scholar
  21. 21.
    D. J. Adams and I. R. McDonald, J. Phys. C 7:2761 (1974); 8: 2198 (1975).Google Scholar
  22. 22.
    D. J. Adams and I. R. McDonald, Physica 79 B: 159 (1975).Google Scholar
  23. 23.
    G. É. Norman and V. S. Filinov, High Temp. USSR 7: 216 (1969).Google Scholar
  24. 24.
    D. J. Adams, Molec. Phys. 28: 1241 (1974).ADSCrossRefGoogle Scholar
  25. 25.
    L. A. Rowley, D. Nicholson, and N. G. Parsonage, J. Comput. Phys. 17: 401 (1975).ADSCrossRefGoogle Scholar
  26. 26.
    D. J. Adams, Molec. Phys. 32, 647 (1976); 37, 211 (1979).CrossRefGoogle Scholar
  27. 27.
    J. E. Lane and T. H. Spurling, Aust. J. Chem. 29: 2103 (1976);CrossRefGoogle Scholar
  28. 31:.
    and 933 (1978).Google Scholar
  29. 28.
    L. A. Rowley, D. Nicholson, and N. G. Parsonage, Mol. Phys.Google Scholar
  30. 32:.
    and 389 (1976); J. Comput. Phys. 26:66 (1978).Google Scholar
  31. 29.
    J. E. Lane, T. H. Spurling, B. C. Freasier, J. W. Perram, and E. R. Smith, Phys. Rev. A20: 2147 (1979).ADSGoogle Scholar
  32. 30.
    W. van Megen and I. K. Snook, Molec. Phys. 39: 1043 (1980).ADSCrossRefGoogle Scholar
  33. 31.
    G. M. Torrie and J. P. Valleau, Chem. Phys. Lett. 65: 343 (1979).ADSCrossRefGoogle Scholar
  34. 32.
    J. A. Barker and D. Henderson, Rev. Mod. Phys. 48: 587 (1976).MathSciNetADSCrossRefGoogle Scholar
  35. 33.
    J. N. Cape and L. V. Woodcock, J. Chem. Phys. 72: 976 (1980).ADSCrossRefGoogle Scholar
  36. 34.
    F. F. Abraham, Phys. Rev. Lett. 44: 463 (1980).MathSciNetADSCrossRefGoogle Scholar
  37. 35.
    Stochastic Molecular Dynamics“, D. Ceperley and J. Tully, eds., NRCC, Lawrence Berkeley Laboratory (1979).Google Scholar
  38. 36.
    A. J. Stace, Molec. Phys. 38: 155 (1979).ADSCrossRefGoogle Scholar
  39. 37.
    G. E. Murch and R. J. Thorn, Phil.Mag. 35:493; 36:517 and 529 (1977).Google Scholar

Copyright information

© Plenum Press, New York 1983

Authors and Affiliations

  • David Adams
    • 1
  1. 1.Department of ChemistryUniversity of SouthamptonSouthamptonEngland

Personalised recommendations