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

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Presentation transcript:

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

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

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 E 1 ? BA: each configuration is equally probable; but the number of states that give an energy E 1 is not know.

4 Energy is conserved! dE 1 =-dE 2 This can be seen as an equilibrium condition

5 Canonical ensemble Consider a small system that can exchange heat with a big reservoir 1/k B T Hence, the probability to find E i : Boltzmann distribution

6 Thermodynamics What is the average energy of the system? Compare: Hence: Thermo recall (2) First law of thermodynamics Helmholtz Free energy:

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

8 Monte Carlo simulation

9 Ensemble average Generate configuration using MC: with

10 Monte Carlo simulation

11

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 Detailed balance o n

15 NVT-ensemble

16

17 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

19 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?

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

21 Keeping old configuration?

22 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?

23 Not too small, not too big!

24

25 Periodic boundary conditions

26 Lennard Jones potentials The truncated and shifted Lennard-Jones potential The truncated Lennard-Jones potential The Lennard-Jones potential

27 Phase diagrams of Lennard Jones fluids

28 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?

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

30 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!

31 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:

32 Conventional acceptance rule Conventional acceptance rules leads to a bias

33 Why this bias? Detailed balance!

34 Detailed balance o n ?

35

36

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