Skip to main content

Basic Monte Carlo Models: Equilibrium and Kinetics

  • Chapter
Handbook of Materials Modeling

Abstract

Monte Carlo (MC) is a very general computational technique that can be used to carry out sampling of distributions. Random numbers are employed in the sampling, and often in other parts of the code. One definition of MC based on common usage in the literature is, any calculation that involves significant applications of random numbers. Historical accounts place the naming of this method in March 1947, when Metropolis suggested it for his method of evaluating the equilibrium properties of atomic systems, and this is the application that we will discuss in this section [1]. An important sampling technique is the one named after Metropolis, which we will describe below.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 709.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. Metropolis, “The beginning of the Monte Carlo method,” Los Alamos Sci., Special Issue, 125, 1987.

    Google Scholar 

  2. E.J. Janse van Rensburg and G.M. Torrie “Estimation of multidimensional integrals: is Monte Carlo the best method?” J. Phys. A: Math. Gen., 26, 943–953, 1993.

    Article  MATH  ADS  Google Scholar 

  3. A.R. Kansal and S. Torquato, “Prediction of trapping rates in mixtures of partially absorbing spheres,” J. Chem. Phys., 116, 10589, 2002.

    Article  ADS  Google Scholar 

  4. H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods, Part 2, Chaps 10–12, 14, 15, Addison-Wesley, Reading, 1988.

    ADS  Google Scholar 

  5. D.W. Hermann, Computer Simulation Methods, 2nd edn., Chap 4, Springer-Verlag, Berlin 1990.

    Google Scholar 

  6. K. Binder and D.W. Hermann, Monte Carlo Simulation in Statistical Physics, An Introduction, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  7. E.E. Lewis and W.F. Miller, Computational Methods of Neutron Transport, Chap 7, American Nuclear Society, La Grange Park, IL, 1993.

    Google Scholar 

  8. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys., 21, 1087, 1953.

    Article  ADS  Google Scholar 

  9. M.H. Kalos and P.A. Whitlock, Monte Carlo Methods, Wiley, New York, 1986.

    Book  MATH  Google Scholar 

  10. S. Kirkpatrick, C.D. Gelatt, and M.P. Vecchi, “Optimization by simulated annealing,” Science, 220, 671, 1983.

    Article  MathSciNet  ADS  Google Scholar 

  11. G.M. Torrie and J.P. Valleau, “Non-physical sampling distributions in Monte Carlo free energy estimation — umbrella sampling,” J. Comput. Phys., 23, 187, 1977.

    Article  ADS  Google Scholar 

  12. M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, 1987.

    MATH  Google Scholar 

  13. M. Rao, C. Pangali, and B.J. Berne, “On the force bias Monte Carlo simulation of water: methodology, optimization and comparison with molecular dynamics,” Mol. Phys., 37, 1773, 1979.

    Article  ADS  Google Scholar 

  14. J. Dalla Torre, C.-C. Fu, F. Willaime, and J.-L. Bocquet, Simulations multi-echelles des experiences de recuit de resistivite isochrone dans le Fer-ultra pur irradie aux electrons: premiers resultants, CEA Annuel Rapport, p. 94, 2003.

    Google Scholar 

  15. D. T. Gillespie, “General method for numerically simulating stochastic time evolu-tion of coupled chemical-reactions,” Comp. Phys., 22, 403–434, 1976.

    Article  MathSciNet  ADS  Google Scholar 

  16. A.B. Bortz, M.H. Kalos, and J. L. Lebowitz, J. Comput. Phys., 17, 10, 1975.

    Article  ADS  Google Scholar 

  17. G. H. Gilmer, “Growth on imperfect crystal faces,” J. Cryst, Growth, 36, 15, 1976.

    Article  ADS  Google Scholar 

  18. R.H. Swendsen and J.S. Wang, “Replica Monte Carlo simulation of spin-glasses,” Phys. Rev. Lett., 57, 2607, 1986.

    Article  MathSciNet  ADS  Google Scholar 

  19. M.T. Robinson, “The binary collision approximation: background and introduction, Rod. Eff. Defects Sol., 130–131, 3, 1994.

    Google Scholar 

  20. M.E. Law, G.H. Gilmer, and M. Jaraiz, “Simulation of defects and diffusion phenomena in silicon,” MRS Bull., 25, 45, 2000.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer

About this chapter

Cite this chapter

Gilmer, G., Yip, S. (2005). Basic Monte Carlo Models: Equilibrium and Kinetics. In: Yip, S. (eds) Handbook of Materials Modeling. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-3286-8_31

Download citation

Publish with us

Policies and ethics