Basic Monte Carlo (chapter 3)

1. Basic Monte Carlo(chapter 3) Algorithm Detailed Balance Other points

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

3. Does the basis assumption lead to something that is consistent with classical thermodynamics? Systems 1 and 2 are weakly coupled such that they can exchange energy. What will be E1? BA: each configuration is equally probable; but the number of states that give an energy E1 is not know.

4. Energy is conserved! dE1=-dE2 This can be seen as an equilibrium condition

5. Canonical ensemble 1/kBT Consider a small system that can exchange heat with a big reservoir Hence, the probability to find Ei: Boltzmann distribution

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

7. Monte Carlo simulation

8. Ensemble average Generate configuration using MC: with

9. Monte Carlo simulation

12. 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?

13. Detailed balance o n

14. NVT-ensemble

16. 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?

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. Mathematical Transition probability: ≠0 Probability to accept the old configuration:

20. Keeping old configuration?

21. 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?

22. Not too small, not too big!

24. Periodic boundary conditions

25. Lennard Jones potentials • The Lennard-Jones potential • The truncated Lennard-Jones potential • The truncated and shifted Lennard-Jones potential

26. Phase diagrams of Lennard Jones fluids

27. 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

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

29. 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!

30. 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:

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

32. Detailed balance! Why this bias?

33. Detailed balance o n ?

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