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1 - Introduction

Published online by Cambridge University Press:  24 November 2021

David Landau
Affiliation:
University of Georgia
Kurt Binder
Affiliation:
Johannes Gutenberg Universität Mainz, Germany
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Summary

In a Monte Carlo simulation we attempt to follow the ‘time dependence’ of a model for which change, or growth, does not proceed in some rigorously predefined fashion (e.g. according to Newton’s equations of motion) but rather in a stochastic manner which depends on a sequence of random numbers which is generated during the simulation. With a second, different sequence of random numbers the simulation will not give identical results but will yield values which agree with those obtained from the first sequence to within some ‘statistical error’. A very large number of different problems fall into this category: in percolation an empty lattice is gradually filled with particles by placing a particle on the lattice randomly with each ‘tick of the clock’. Lots of questions may then be asked about the resulting ‘clusters’ which are formed of neighboring occupied sites. Particular attention has been paid to the determination of the ‘percolation threshold’, i.e. the critical concentration of occupied sites for which an ‘infinite percolating cluster’ first appears. A percolating cluster is one which reaches from one boundary of a (macroscopic) system to the opposite one. The properties of such objects are of interest in the context of diverse physical problems such as conductivity of random mixtures, flow through porous rocks, behavior of dilute magnets, etc. Another example is diffusion limited aggregation (DLA), where a particle executes a random walk in space, taking one step at each time interval, until it encounters a ‘seed’ mass and sticks to it. The growth of this mass may then be studied as many random walkers are turned loose. The ‘fractal’ properties of the resulting object are of real interest, and while there is no accepted analytical theory of DLA to date, computer simulation is the method of choice. In fact, the phenomenon of DLA was first discovered by Monte Carlo simulation.

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Publisher: Cambridge University Press
Print publication year: 2021

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References

Hartmann, A. (2009), Practical Guide to Computer Simulations (World Scientific, Singapore).CrossRefGoogle Scholar
Histoire de l’Académie royale des sciences ANNÉE M.DCCXXXIII. (Académie royale des sciences, France, 1735), https://books.google.de/books?id=GOAEAAAAQAAJ.Google Scholar
Kadau, K., Germann, T. C., and Lomdahl, P. S. (2006), Int. J. Mod. Phys. C 17, 1755.CrossRefGoogle Scholar
Metropolis, N. and Ulam, S. (1949), J. Amer. Stat. Assoc. 44, 335.CrossRefGoogle Scholar

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  • Introduction
  • David Landau, University of Georgia, Kurt Binder, Johannes Gutenberg Universität Mainz, Germany
  • Book: A Guide to Monte Carlo Simulations in Statistical Physics
  • Online publication: 24 November 2021
  • Chapter DOI: https://doi.org/10.1017/9781108780346.002
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  • Introduction
  • David Landau, University of Georgia, Kurt Binder, Johannes Gutenberg Universität Mainz, Germany
  • Book: A Guide to Monte Carlo Simulations in Statistical Physics
  • Online publication: 24 November 2021
  • Chapter DOI: https://doi.org/10.1017/9781108780346.002
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Introduction
  • David Landau, University of Georgia, Kurt Binder, Johannes Gutenberg Universität Mainz, Germany
  • Book: A Guide to Monte Carlo Simulations in Statistical Physics
  • Online publication: 24 November 2021
  • Chapter DOI: https://doi.org/10.1017/9781108780346.002
Available formats
×