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Basic Monte Carlo (chapter 3)

Basic Monte Carlo (chapter 3). Algorithm Detailed Balance Other points non-Boltzmann sampling. Molecular Simulations. MD. Molecular dynamics: solve equations of motion Monte Carlo: importance sampling. r 1. r 2. r n. MC. r 1. r 2. r n. Statistical Thermodynamics.

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Basic Monte Carlo (chapter 3)

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  1. Basic Monte Carlo(chapter 3) Algorithm Detailed Balance Other points non-Boltzmann sampling

  2. Molecular Simulations MD • Molecular dynamics: solve equations of motion • Monte Carlo: importance sampling r1 r2 rn MC r1 r2 rn

  3. Statistical Thermodynamics Partition function Ensemble average Probability to find a particular configuration Free energy

  4. Monte Carlo simulation

  5. Ensemble average Generate configuration using MC: with

  6. Monte Carlo simulation

  7. Questions • How can we prove that this scheme generates the desired distribution of configurations? • Why make a random selection of the particle to be displaced? • Why do we need to take the old configuration again? • How large should we take: delx?

  8. Detailed balance o n

  9. NVT-ensemble

  10. Questions • How can we prove that this scheme generates the desired distribution of configurations? • Why make a random selection of the particle to be displaced? • Why do we need to take the old configuration again? • How large should we take: delx?

  11. Questions • How can we prove that this scheme generates the desired distribution of configurations? • Why make a random selection of the particle to be displaced? • Why do we need to take the old configuration again? • How large should we take: delx?

  12. Mathematical Transition probability: Probability to accept the old configuration:

  13. Keeping old configuration?

  14. Questions • How can we prove that this scheme generates the desired distribution of configurations? • Why make a random selection of the particle to be displaced? • Why do we need to take the old configuration again? • How large should we take: delx?

  15. Not too small, not too big!

  16. Non-Boltzmann sampling Why are we not using this? T1 is arbitrary! We only need a single simulation! We perform a simulation at T=T2 and we determine A at T=T1

  17. P(E) E T1 T2 T3 T4 T5 Overlap becomes very small

  18. How to do parallel Monte Carlo • Is it possible to do Monte Carlo in parallel • Monte Carlo is sequential! • We first have to know the fait of the current move before we can continue!

  19. Parallel Monte Carlo Algorithm (WRONG): • Generate k trial configurations in parallel • Select out of these the one with the lowest energy • Accept and reject using normal Monte Carlo rule:

  20. Conventional acceptance rule Conventional acceptance rules leads to a bias

  21. Detailed balance! Why this bias?

  22. Detailed balance o n ?

  23. Modified acceptance rule Modified acceptance rule remove the bias exactly!

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