1. Introduction to Statistical Thermodynamics (Recall)

2. 2 Basic assumption Each individual microstate is equally probable …, but there are not many microstates that give these extreme results If the number of particles is large (>10) these functions are sharply peaked

3. 3 Consistent with classical thermodynamics? Systems 1 and 2 are weakly coupled such that they can exchange energy. What will be E 1 ?

4. 4 Summary: micro-canonical ensemble (N,V,E) Partition function: Probability to find a particular configuration Free energy

5. 5 Ensembles Micro-canonical ensemble: E,V,N Canonical ensemble: T,V,N Constant pressure ensemble: T,P,N Grand-canonical ensemble: T,V,μ

6. 6 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

7. 7 Thermodynamics What is the average energy of the system? Compare: Hence: Thermo recall (2) First law of thermodynamics Helmholtz Free energy:

8. 8 Summary: Canonical ensemble (N,V,T) Partition function: Probability to find a particular configuration Free energy

9. 9 Constant pressure simulations: N,P,T ensemble Consider a small system that can exchange volume and energy with a big reservoir 1/k B T Hence, the probability to find E i,V i : p/k B T Thermo recall (4) First law of thermodynamics Hence and

10. 10 N,P,T ensemble (2) In the classical limit, the partition function becomes The probability to find a particular configuration:

11. 11 Grand-canonical simulations: μ,V,T ensemble Consider a small system that can exchange particles and energy with a big reservoir 1/k B T Hence, the probability to find E i,N i : -μ/k B T Thermo recall (5) First law of thermodynamics Hence and

12. 12 μ,V,T ensemble (2) In the classical limit, the partition function becomes The probability to find a particular configuration:

13. Monte Carlo in different ensembles Chapter 5 NVT ensemble NPT ensemble Grand-canonical ensemble Exotic ensembles

14. 14 Statistical Thermodynamics Partition function Ensemble average Free energy Probability to find a particular configuration

15. 15 Detailed balance o n

16. 16 NVT-ensemble

17. 17

18. 18 NPT Ensemble Partition function: Probability to find a particular configuration: Sample a particular configuration: Change of volume Change of reduced coordinates Acceptance rules ?? Detailed balance

19. 19 Detailed balance o n

20. 20 NPT-ensemble Suppose we change the position of a randomly selected particle

21. 21 NPT-ensemble Suppose we change the volume of the system

22. 22 Algorithm: NPT Randomly change the position of a particle Randomly change the volume

23. 23

24. 24

25. 25

26. 26 NPT simulations

27. 27 Grand-canonical ensemble What are the equilibrium conditions?

28. 28 Grand-canonical ensemble We impose: –Temperature –Chemical potential –Volume –But NOT pressure

29. 29 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 ?? Detailed balance

30. 30 Detailed balance o n

31. 31  VT-ensemble Suppose we change the position of a randomly selected particle

32. 32  VT-ensemble Suppose we change the number of particles of the system

33. 33

34. 34

35. 35 Application: equation of state of Lennard-Jones

36. 36 Application: adsorption in zeolites

37. 37 Exotic ensembles What to do with a biological membrane?

38. 38 Model membrane: Lipid bilayer hydrophilic head group two hydrophobic tails water

39. 39

40. 40 Questions What is the surface tension of this system? What is the surface tension of a biological membrane? What to do about this?

41. 41 Phase diagram: alcohol

42. 42 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.

43. 43 Constant surface tension simulation A A’ LL’ A L = A’ L’ = V

44. 44  (A o ) = -0.3 +/- 0.6  (A o ) = 2.5 +/- 0.3  (A o ) = 2.9 +/- 0.3 Tensionless state:  = 0