Monte Carlo in different ensembles Chapter 5

1. Monte Carlo in different ensemblesChapter 5 NVT ensemble NPT ensemble Grand-canonical ensemble Exotic ensembles

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

3. Detailed balance o n

4. NVT-ensemble

6. We control the temperature, pressure, and number of particles. NPT ensemble

7. Scaled coordinates Partition function Scaled coordinates The energy depends on the real coordinates This gives for the partition function

8. The perfect simulation ensemble Here they are an ideal gas Here they interact What is the statistical thermodynamics of this ensemble?

9. The perfect simulation ensemble: partition function

10. To get the Partition Function of this system, we have to integrate over all possible volumes: Now let us take the following limits: As the particles are an ideal gas in the big reservoir we have:

11. We have To make the partition function dimension less This gives:

12. Detailed balance NPT Ensemble Partition function: Probability to find a particular configuration: • Sample a particular configuration: • Change of volume • Change of reduced coordinates Acceptance rules ??

13. Detailed balance o n

14. NPT-ensemble Suppose we change the position of a randomly selected particle

15. NPT-ensemble Suppose we change the volume of the system

16. Algorithm: NPT • Randomly change the position of a particle • Randomly change the volume

20. NPT simulations

21. Grand-canonical ensemble What are the equilibrium conditions?

22. We impose: Temperature Chemical potential Volume But NOT pressure Grand-canonical ensemble

23. The Murfect ensemble Here they are an ideal gas Here they interact What is the statistical thermodynamics of this ensemble?

24. The Murfect simulation ensemble: partition function

25. To get the Partition Function of this system, we have to sum over all possible number of particles Now let us take the following limits: As the particles are an ideal gas in the big reservoir we have:

26. Detailed balance MuVT Ensemble Partition function: Probability to find a particular configuration: • Sample a particular configuration: • Change of the number of particles • Change of reduced coordinates Acceptance rules ??

27. Detailed balance o n

28. mVT-ensemble Suppose we change the position of a randomly selected particle

29. mVT-ensemble Suppose we change the number of particles of the system

32. Application: equation of state of Lennard-Jones

33. Application: adsorption in zeolites

34. Exotic ensembles What to do with a biological membrane?

35. Model membrane: Lipid bilayer hydrophilic head group two hydrophobic tails water water

37. Questions • What is the surface tension of this system? • What is the surface tension of a biological membrane? • What to do about this?

38. Phase diagram: alcohol

39. Simulations at imposed surface tension • Simulation to a constant surface tension • Simulation box: allow the area of the bilayer to change in such a way that the volume is constant.

40. A A’ L L’ A L = A’ L’ = V Constant surface tension simulation

41. Tensionless state:g = 0 g(Ao) = -0.3 +/- 0.6 g(Ao) = 2.5 +/- 0.3 g(Ao) = 2.9 +/- 0.3