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A Brief Introduction to Monte-Carlo Methods

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Basic Concepts in Computational Physics

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Abstract

The basic ideas of Monte-Carlo methods are demonstrated by means of two examples: (i) The Monte-Carlo integration as a typical example of a hit and miss technique and (ii) the Metropolis algorithm as another means to generate a sequence of random numbers from more complex pdfs. The presentation of the Monte-Carlo integration method is started with the simple task of calculating the number \(\pi \) using a sequence of uniformly distributed random numbers. This is followed by a more formal discussion of the method and of the errors involved. The mathematical background of the Metropolis algorithm will be the topic of a later chapter. The emphasis is here on the motivation of this technique as a very useful tool in the numerics of statistical physics and on the concept of detailed balance which is entirely motivated by physics.

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Notes

  1. 1.

    Please note that according to the conventions established in Appendix D capital letters denote random variables.

  2. 2.

    We give an example. Suppose \(f(x) = \exp (x)\). Then we deduce that

    $$\begin{aligned} \delta \left[ f - \exp (x) \right] = \frac{\delta (x - \ln f) }{f}, \end{aligned}$$

    and, consequently,

    $$\begin{aligned} q(f) = \frac{p( \ln f )}{f}. \end{aligned}$$
  3. 3.

    Nevertheless, there is certainly some conceptual similarity between grid-points and random numbers within this context.

  4. 4.

    The question of how one can obtain such a sequence will be discussed in Sect. 16.3.

References

  1. Chow, Y.S., Teicher, H.: Probability theory, 3rd edn. Springer Texts in Statistics. Springer, Berlin (1997)

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  2. Kienke, A.: Probability theory. Universitext. Springer, Heidelberg (2008)

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Authors and Affiliations

  1. Faculty of Physics, University of Duisburg-Essen, Lotharstrasse 1-21, 47048, Duisburg, Germany

    Benjamin A. Stickler

  2. Institute of Theoretical and Computational Physics, Graz University of Technology, Graz, Austria

    Ewald Schachinger

Authors
  1. Benjamin A. Stickler
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  2. Ewald Schachinger
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Correspondence to Benjamin A. Stickler .

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© 2014 Springer International Publishing Switzerland

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Stickler, B.A., Schachinger, E. (2014). A Brief Introduction to Monte-Carlo Methods. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-02435-6_14

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