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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
- MONTE CARLO CODE
- 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?
Researchers should cite this work as follows:
David M. Ceperley (2009), "Illinois PHYS 466, Lecture 12: Random Walks," https://nanohub.org/resources/6544.