1. Introduction to (Statistical) Thermodynamics

2. Molecular SimulationsMD
Molecular dynamics: solve equations of motion
Monte Carlo: importance sampling r1 r2 rn MC r1 r2 rn

5. What is the desired distribution?
Questions
What is the desired distribution?
How can we prove that this scheme generates the desired distribution of configurations?
Why make a random selection of the particle to be displaced?
Why do we need to take the old configuration again?
How large should we take: delx?

6. Summary Thermodynamics System at constant temperature and volume
First law: conservation of energy
Second law: in a closed system entropy increase and takes its maximum value at equilibrium
System at constant temperature and volume
Helmholtz free energy decrease and takes its minimum value at equilibrium
Equilibrium:
Equal temperature, pressure, and chemical potential

7. Entropy E,V Closed system: 1st law: total energy remains constant
2nd law: entropy increases
Temperature or

8. Helmholtz free energy
Pressure:
Energy:
Chemical potential:

9. Statistical Thermodynamics
System of N molecules:
But how do we obtain the
macroscopic properties of this system
from this microscopic information?

10. Outline (2) Statistical Thermodynamics Basic Assumption
micro-canonical ensemble
relation to thermodynamics
Canonical ensemble
free energy
thermodynamic properties
Other ensembles
constant pressure
grand-canonical ensemble

11. Statistical Thermodynamics: the basics
Nature is quantum-mechanical
Consequence:
Systems have discrete quantum states.
For finite “closed” systems, the number of states is finite (but usually very large)
Hypothesis: In a closed system, every state is equally likely to be observed.
Consequence: ALL of equilibrium Statistical Mechanics and Thermodynamics

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

13. Does the basis assumption lead to something that is consistent with classical thermodynamics?
Systems 1 and 2 are weakly coupled
such that they can exchange energy.
What will be E1?
BA: each configuration is equally probable; but the number of
states that give an energy E1 is not know.

14. This can be seen as an equilibrium condition
Energy is conserved!
dE1=-dE2
This can be seen as an equilibrium condition

15. Entropy and number of configurations
Conjecture:
Almost right.
Good features:
Extensivity
Third law of thermodynamics comes for free
Bad feature:
It assumes that entropy is dimensionless but (for unfortunate, historical reasons, it is not…)

16. We have to live with the past, therefore
With kB= J/K
In thermodynamics, the absolute (Kelvin) temperature scale was defined such that
But we found (defined):

17. And this gives the “statistical” definition of temperature:
In short:
Entropy and temperature are both related to the fact that we can COUNT states.
Basic assumption:
leads to an equilibrium condition: equal temperatures
leads to a maximum of entropy
leads to the third law of thermodynamics

18. Number of configurations
How large is ?
For macroscopic systems, super-astronomically large.
For instance, for a glass of water at room temperature:
Macroscopic deviations from the second law of thermodynamics are not forbidden, but they are extremely unlikely.

19. Boltzmann distribution
Canonical ensemble
1/kBT
Consider a small system that can exchange heat with a big reservoir
Hence, the probability to find Ei:
Boltzmann distribution

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

21. Remarks (1) We have assume quantum mechanics (discrete states) but
we are interested in the classical limit
Volume of phase space (particle in a box)
Particles are indistinguishable
Integration over the momenta can be carried out for most systems:

22. Remarks (2)
Define de Broglie wave length:
Partition function:

23. Example: ideal gas Thermo recall (3) Helmholtz Free energy:
Pressure
Free energy:
Pressure:
Energy:

24. Ideal gas (2)
Chemical potential:

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

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

27. Thermodynamic properties
Probability to find a configuration:
Ensemble average:
For many properties the momenta do not matter: only the configurational part

28. MD and MC different ensembles
Other ensembles?
COURSE:
MD and MC different ensembles
In the thermodynamic limit the thermodynamic properties are
independent of the ensemble: so buy a bigger computer …
However, it is most of the times much better to think and to carefully
select an appropriate ensemble.
For this it is important to know how to simulate in the various
ensembles.
But for doing this wee need to know the Statistical Thermodynamics
of the various ensembles.

29. Example (1): vapour-liquid equilibrium mixture
Measure the composition of the coexisting vapour and liquid phases if we start with a homogeneous liquid of two different compositions:
How to mimic this with the N,V,T ensemble?
What is a better ensemble?
composition T L V
L+V

30. Example (2): swelling of clays
Deep in the earth clay layers can swell upon adsorption of water:
How to mimic this in the N,V,T ensemble?
What is a better ensemble to use?

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

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

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

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

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