1. Basic Methods of Stat Mech: Overview of Common Ensembles

2. Topic Coverage for the Short Term
Over the next few classes, we’ll summarize the basic methods of Statistical Mechanics.
We’ll also talk about some applications by treating some common examples.
We will end up covering most of the material in Chapters 2 and 3 of the book by Pathria. However, the topics covered may be in a different order than in that book.
Further, many formal math details will be skipped. Instead, we will try to discuss the physics rather than the math. 2

3. Energy Functions in Classical Thermo:
1. Internal Energy: E  E(S,V)
2. Enthalpy: H = H(S,p)  E + pV
3. Helmholtz Free Energy: F = F (T,V)  E – TS
4. Gibbs Free Energy: G = G(T,p)  E – TS + pV
1. dE = TdS – pdV
2. dH = TdS + Vdp
3. dF = - SdT – pdV
4. dG = - SdT + Vdp
Combined 1st
& 2nd Laws 3

4. Key Concepts In Statistical Mechanics
Basic Idea (Chapters 2 & 3 of Text) Macroscopic properties are thermal
averages of microscopic properties.
To treat the system of interest with probability theory, we replace the system with a set of a large number of systems "identical" to the first & average over all of the systems. We call the set of systems
“The Statistical Ensemble” 4

5. “The Statistical Ensemble”
To use probability theory, replace the system with a set of a large number of systems "identical" to the first & average over all of the systems. We call the set of systems
“The Statistical Ensemble”
Identical Systems means that they are all in the same thermodynamic macrostate. To do any calculations we have to first
Choose an Ensemble! 5

6. Science Definitions of “Canonical”
Accepted as being accurate & authoritative.
According to recognized rules or scientific laws.
Of or relating to a general rule or standard formula.
There are also religious meanings of “Canonical”
According to or ordered by canon law
Canonical rites of the Roman Church 6

7. 3 Common Statistical Ensembles:
1. Micro-Canonical Ensemble:
Isolated Systems: Constant Energy E.
Nothing happens!  Not Interesting! 7

8. 3 Common Statistical Ensembles:
1. Micro-Canonical Ensemble:
Isolated Systems: Constant Energy E.
Nothing happens!  Not Interesting!
2. The Canonical Ensemble:
Systems with a fixed number N of molecules
in equilibrium with a Heat Reservoir (Heat Bath). 8

9. 3 Common Statistical Ensembles:
1. Micro-Canonical Ensemble:
Isolated Systems: Constant Energy E.
Nothing happens!  Not Interesting!
2. The Canonical Ensemble:
Systems with a fixed number N of molecules
in equilibrium with a Heat Reservoir (Heat Bath).
3. The Grand Canonical Ensemble:
Systems in equilibrium with a Heat Bath
which is also a Source of Molecules.
Their chemical potential is fixed. 9

10. Microcanonical Ensemble
Examples: If Systems of Interest are Gases
Microcanonical Ensemble
E, V, N fixed, S = kB lnΩ(E,V,N)
Ω(E,V,N)  # Accessible States

11. Microcanonical Ensemble Canonical Ensemble
Examples: If Systems of Interest are Gases
Microcanonical Ensemble
E, V, N fixed, S = kB lnΩ(E,V,N)
Ω(E,V,N)  # Accessible States
Canonical Ensemble
T, V, N fixed, F = -kBT lnZ(T,V,N)
Z(T,V,N)  Partition Function

12. Microcanonical Ensemble Canonical Ensemble Grand Canonical Ensemble
Examples: If Systems of Interest are Gases
Microcanonical Ensemble
E, V, N fixed, S = kB lnΩ(E,V,N)
Ω(E,V,N)  # Accessible States
Canonical Ensemble
T, V, N fixed, F = -kBT lnZ(T,V,N)
Z(T,V,N)  Partition Function
Grand Canonical Ensemble
T, V,  fixed,  = -kBTln (T,V,)
(T,V,)  Grand Partition Function

13. All Thermodynamic Properties Can Be Calculated With Any Ensemble
Choose the most convenient one for a particular problem.
1. For Gases: PVT properties usually use
The Canonical Ensemble
2. Systems which Exchange Particles:
Such as Vapor-Liquid Equilibrium use
The Grand Canonical
Ensemble 13

14. A  nAnPn The Thermodynamic Average
Properties of The Canonical & Grand Canonical Ensembles
J. Willard Gibbs was the first to show that
An Ensemble Average is Equal to a Thermodynamic Average:
That is, for a given property A,
The Thermodynamic Average
can be formally expressed as:
A  nAnPn
An  Value of A in state (configuration) n
Pn  Probability of the system being in
state (configuration) n. 14

15. Canonical Ensemble Probabilities
QNcanon  “Canonical Partition Function”
or “Partition Function”
gn  Degeneracy of state n
Note that most texts use the notation
“Z” for the partition function! 15

16. Note that most texts use the notation
Grand Canonical Ensemble Probabilities:
Qgrand  “Grand Canonical Partition Function” or
“Grand Partition Function”
gn  Degeneracy of state n, μ  “Chemical Potential”
Note that most texts use the notation
“ZG” for the Grand Partition Function! 16

17. The Partition Function Z
Partition Functions
If the volume, V, (or other external parameter) the temperature T, & the energy levels En, of a system are known, in principle
The Partition Function Z
can be calculated.
If the partition function Z is known, it can be used to Calculate
All Thermodynamic Properties. 17

18. All Thermodynamic Properties.
Partition Functions
If the partition function Z is known, it can be used to Calculate
All Thermodynamic Properties.
So, in this way,
Statistical Mechanics is a direct link between Microscopic Quantum Mechanics & Classical Macroscopic Thermodynamics. 18

19. Canonical Ensemble Partition Function Z
Starting from the postulate of equal a priori probabilities, the following are obtained:
ALL RESULTS of Classical Thermo,
plus their statistical underpinnings.
A MEANS OF CALCULATING the
thermodynamic variables (E, H, F, G, S)
from a single statistical parameter, the partition function Z (or Q), which may be obtained from energy-levels of a system. 19

20. Canonical Partition Function Z
The Partition Function Z is
A MEANS OF CALCULATING
all thermodynamic variables (E, H, F, G, S)
Z is obtained from the energy-levels of a quantum system. In a quantum system in equilibrium with a heat reservoir, Z is defined as:
εn  Energy of the n’th Quantum State.
Z  n exp(-εn/kBT) 20

21. εn  energy of the n’th quantum state.
Partition Function for a Quantum System in Contact with a Heat Reservoir: ,
εn  energy of the n’th quantum state.
The connection to the macroscopic entropy function S is through the microscopic parameter Ω, which, as we already know, is the number of microstates in a given macrostate.
The connection between them, as discussed in the text & many other places, is
Z  n exp(-εn/kBT)
S = kBln Ω. 21 21

22. Relationship of Z to Macroscopic Parameters
Summary for the Canonical Ensemble Partition Function Z:
(Derivations are in the book!)
Internal Energy: Ē  E = - ∂(lnZ)/∂β
<(ΔE)2> = [∂2(lnZ)/∂β2]
β = 1/(kBT), kB = Boltzmann’s constantt. 22

23. Relationship of Z to Macroscopic Parameters
Summary for the Canonical Ensemble Partition Function Z:
(Derivations are in the book!)
Internal Energy: Ē  E = - ∂(lnZ)/∂β
<(ΔE)2> = [∂2(lnZ)/∂β2]
β = 1/(kBT), kB = Boltzmann’s constantt.
Entropy: S = kBβĒ + kBlnZ
An important, frequently used result! 23

24. Summary for the Canonical Ensemble Partition Function Z:
Helmholtz Free Energy:
F = E – TS = – (kBT)lnZ
Note that this gives: Z = exp[-F/(kBT)]
dF = S dT – PdV, so
S = – (∂F/∂T)V, P = – (∂F/∂V)T 24

25. Summary for the Canonical Ensemble Partition Function Z:
Helmholtz Free Energy:
F = E – TS = – (kBT)lnZ
Note that this gives: Z = exp[-F/(kBT)]
dF = S dT – PdV, so
S = – (∂F/∂T)V, P = – (∂F/∂V)T
Gibbs Free Energy:
G = F + PV = PV – kBT lnZ. 25

26. Summary for the Canonical Ensemble Partition Function Z:
Helmholtz Free Energy:
F = E – TS = – (kBT)lnZ
Note that this gives: Z = exp[-F/(kBT)]
dF = S dT – PdV, so
S = – (∂F/∂T)V, P = – (∂F/∂V)T
Gibbs Free Energy:
G = F + PV = PV – kBT lnZ.
Enthalpy:
H = E + PV = PV – ∂(lnZ)/∂β 26

27. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = nexp(-En),  = 1/(kBT) 27

28. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = nexp(-En),  = 1/(kBT)
Mean Energy:
Ē = – (lnZ)/ = - (1/Z)Z/ 28

29. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = nexp(-En),  = 1/(kBT)
Mean Energy:
Ē = – (lnZ)/ = - (1/Z)Z/
Mean Squared Energy:
E2 = npnEn2/npn = (1/Z)2Z/2. 29

30. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = nexp(-En),  = 1/(kBT)
Mean Energy:
Ē = – (lnZ)/ = - (1/Z)Z/
Mean Squared Energy:
E2 = npnEn2/npn = (1/Z)2Z/2.
nth Moment:
En = rprErn/rpr = (-1)n(1/Z) nZ/n 30

31. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = nexp(-En),  = 1/(kBT)
Mean Energy:
Ē = – (lnZ)/ = - (1/Z)Z/
Mean Squared Energy:
E2 = rprEr2/rpr = (1/Z)2Z/2.
nth Moment:
En = rprErn/rpr = (-1)n(1/Z) nZ/n
Mean Square Deviation:
(ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/ . 31

32. Canonical Ensemble: Constant Volume Heat Capacity
CV = (Ē/T)V = (Ē/)(d/dT) = - k2Ē/ 32

33. Canonical Ensemble: Constant Volume Heat Capacity
CV = (Ē/T)V = (Ē/)(d/dT) = - k2Ē/
using results for the Mean Square Deviation:
(ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/ 33

34. Canonical Ensemble: Constant Volume Heat Capacity
CV = (Ē/T)V = (Ē/)(d/dT) = - k2Ē/
using results for the Mean Square Deviation:
(ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/
CV can thus be re-written as:
CV = k2(ΔE)2 = (ΔE)2/kBT2 34

35. Canonical Ensemble: Constant Volume Heat Capacity
CV = (Ē/T)V = (Ē/)(d/dT) = - k2Ē/
using results for the Mean Square Deviation:
(ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/
CV can be re-written as:
CV = k2(ΔE)2 = (ΔE)2/kBT2
so that:
(ΔE)2 = kBT2CV 35

36. Canonical Ensemble: Constant Volume Heat Capacity
CV = (Ē/T)V = (Ē/)(d/dT) = - k2Ē/
using results for the
Mean Square Deviation:
(ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/
CV can be re-written as:
CV = k2(ΔE)2 = (ΔE)2/kBT2
so that:
(ΔE)2 = kBT2CV
Note that, since (ΔE)2 ≥ 0
(i) CV ≥ 0 & (ii) Ē/T ≥ 0. 36

37. Ensembles in Classical Statistical Mechanics
As we’ve seen, classical phase space for a system with f degrees of freedom is f generalized coordinates & f generalized momenta (qi,pi).
The classical mechanics problem is done in the Hamiltonian formulation with a Hamiltonian energy function H(q,p).
There may also be a few constants of motion:
energy, particle number, volume, ... 37

38. The Partition Function
The Canonical Distribution in
Classical Statistical Mechanics
The Partition Function
has the form:
Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT)
A 6N Dimensional Integral!
This assumes that we have already solved the classical mechanics problem for each particle in the system so that we know the total energy E for the N particles as a function of all positions ri & momenta pi.
E  E(r1,r2,r3,…rN,p1,p2,p3,…pN) 38

39. P(E) ≡ e[-E/(kBT)] /Z Another 6N Dimensional Integral!
CLASSICAL Statistical Mechanics:
Let A ≡ any measurable, macroscopic quantity. The thermodynamic average of A ≡ <A>. This is what is measured. Use probability theory to calculate <A> :
P(E) ≡ e[-E/(kBT)] /Z
<A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E)
Another 6N Dimensional Integral!