Introduction to Statistical Thermodynamics (Recall)

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

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

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

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

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

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

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

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

15 Detailed balance o n

16 NVT-ensemble

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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 Detailed balance o n

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

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

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

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26 NPT simulations

27 Grand-canonical ensemble What are the equilibrium conditions?

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

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 Detailed balance o n

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

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

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35 Application: equation of state of Lennard-Jones

36 Application: adsorption in zeolites

37 Exotic ensembles What to do with a biological membrane?

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

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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 Phase diagram: alcohol

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

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