Download presentation

Presentation is loading. Please wait.

Published byDillon Sliman Modified over 2 years ago

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

Similar presentations

OK

1 CE 530 Molecular Simulation Lecture 6 David A. Kofke Department of Chemical Engineering SUNY Buffalo

1 CE 530 Molecular Simulation Lecture 6 David A. Kofke Department of Chemical Engineering SUNY Buffalo

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Pptm to ppt online converter Ppt on non renewable energy sources Free download ppt on revolt of 1857 Ppt on normalization Ppt on diode characteristics conclusion Ppt on obesity diet camp Ppt on surface water quality Ppt on waves tides and ocean currents affect Ppt on google cloud storage Ppt on chromosomes and genes activities