Advertisement

Random Numbers and Monte Carlo Methods

  • Philipp O. J. Scherer
Part of the Graduate Texts in Physics book series (GTP)

Abstract

Many-body problems often involve the calculation of integrals of very high dimension which cannot be treated by standard methods. For the calculation of thermodynamic averages Monte Carlo methods are very useful which sample the integration volume at randomly chosen points. After summarizing some basic statistics, we discuss algorithms for the generation of pseudo-random numbers with given probability distribution which are essential for all Monte Carlo methods. We show how the efficiency of Monte Carlo integration can be improved by sampling preferentially the important configurations. Finally the famous Metropolis algorithm is applied to classical many-particle systems. Computer experiments visualize the central limit theorem and apply the Metropolis method to the traveling salesman problem.

Keywords

Partition Function Cumulative Distribution Function Travel Salesman Problem Pseudo Random Number Monte Carlo Integration 
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.

References

  1. 31.
    E. Bonomi, J.-L. Lutton, SIAM Rev. 26, 551 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 34.
    G.E.P. Box, M.E. Muller, Ann. Math. Stat. 29, 610 (1958) zbMATHCrossRefGoogle Scholar
  3. 49.
    R.E. Caflisch, Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7, 1–49 (1998) MathSciNetCrossRefGoogle Scholar
  4. 85.
    G.S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications (Springer, New York, 1995). ISBN 038794527X Google Scholar
  5. 168.
    G. Marsaglia, A. Zaman, Ann. Appl. Probab. 1, 462 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 174.
    N. Metropolis, S. Ulam, J. Am. Stat. Assoc. 44, 335 (1949) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 175.
    N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953) ADSCrossRefGoogle Scholar
  8. 202.
    H. Pollard, Q. J. Pure Appl. Math. 49, 1 (1920) Google Scholar
  9. 215.
    J. Rice, Mathematical Statistics and Data Analysis, 2nd edn. (Duxbury Press, N. Scituate, 1995). ISBN 0-534-20934-3 zbMATHGoogle Scholar
  10. 218.
    R.D. Richtmyer, Principles of Modern Mathematical Physics I (Springer, New York, 1978) CrossRefGoogle Scholar
  11. 220.
    C.P. Robert, G. Casella, Monte Carlo Statistical Methods, 2nd edn. (Springer, New York, 2004). ISBN 0387212396 zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Philipp O. J. Scherer
    • 1
  1. 1.Physikdepartment T38Technische Universität MünchenGarchingGermany

Personalised recommendations