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Illinois PHYS 466, Lecture 12: Random Walks

By David M. Ceperley

University of Illinois at Urbana-Champaign

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Random Walks

Today we will discuss Markov chains (random walks), detailed balance and transition rules.

  • These methods were introduced by Metropolis et al. in 1953
    who applied it to a hard sphere liquid.
  • It is one of the most powerful and used algorithms


  • Equation of State Calculations by Fast Computing Machines
  • Markov chain or Random Walk
  • Properties of Random Walk
  • Random Walks Example from A&T 110-123
  • What is probability of being up on the second day?
  • Metropolis algorithm
  • Replace strong “Microscopic Reversibility” criterion
  • Rejection Method
  • The “Classic” Metropolis method
  • Picture of Metropolis Rejection
  • How to sample
  • Overview of MCMC
  • Always measure acceptance ratio. RULE: 0.1 < a.r. < 0.9
    Adjust ratio to roughly 0.5 by varying the “step size”.
  • Variance of energy (local quantity) is not as sensitive to step size.
    MC is a robust method! You don’t need to fine tune things!
  • Optimizing the moves
  • Comparison of MC and MD: Which is better?


This presentation was breezed and uploaded by Omar Sobh

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Researchers should cite this work as follows:

  • David M. Ceperley (2009), "Illinois PHYS 466, Lecture 12: Random Walks,"

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