1. Monte Carlo in different ensembles Chapter 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. Ensemble average
Generate configuration using MC:
with

4. Detailed balance o n

5. NVT-ensemble

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

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

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

10. The perfect simulation ensemble: partition function

11. 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:

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

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

14. Detailed balance o n

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

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

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

21. NPT simulations

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

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

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

25. The Murfect simulation ensemble: partition function

26. 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:

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

28. Detailed balance o n

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

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

33. Application: equation of state of Lennard-Jones

34. Application: adsorption in zeolites

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

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

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

39. Phase diagram: alcohol

40. 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.

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

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