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

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Presentation transcript:

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

2 Molecular Simulations u Molecular dynamics: solve equations of motion u Monte Carlo: importance sampling r1r1 MD MC r2r2 rnrn r1r1 r2r2 rnrn

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

4 Monte Carlo simulation

5 Ensemble average Generate configuration using MC: with

6 Monte Carlo simulation

7

8

9 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?

10 Detailed balance o n

11 NVT-ensemble

12

13 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?

14

15 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?

16 Mathematical Transition probability: Probability to accept the old configuration:

17 Keeping old configuration?

18 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?

19 Not too small, not too big!

20 Non-Boltzmann sampling We perform a simulation at T=T 2 and we determine A at T=T 1 T 1 is arbitrary! We only need a single simulation! Why are we not using this?

21 T1T1 T2T2 T5T5 T3T3 T4T4 E P(E) Overlap becomes very small

22 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!

23 Parallel Monte Carlo Algorithm (WRONG): 1.Generate k trial configurations in parallel 2.Select out of these the one with the lowest energy 3.Accept and reject using normal Monte Carlo rule:

24 Conventional acceptance rule Conventional acceptance rules leads to a bias

25 Why this bias? Detailed balance!

26 Detailed balance o n ?

27

28

29 Modified acceptance rule Modified acceptance rule remove the bias exactly!