1. Chapter 6: Basic Methods & Results of Statistical Mechanics + Chapter 7: Simple Applications of Statistical Mechanics Overview + Details & Applications

2. Statistical Mechanics “Canonical Ensemble”
Chapter 6: Basic Methods & Results of Statistical Mechanics + Chapter 7: Simple Applications of Statistical Mechanics Overview + Details & Applications
Subtitle:
Statistical
Mechanics
in the
“Canonical
Ensemble”

3. Key Concepts In Statistical Mechanics
The Basic Idea (early lectures in this course): Macroscopic properties are thermal averages of microscopic properties. 3

4. Key Concepts In Statistical Mechanics
The Basic Idea (early lectures in this course): Macroscopic properties are thermal averages of microscopic properties.
First, 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. Key Concepts In Statistical Mechanics
The Basic Idea (early lectures in this course): Macroscopic properties are thermal averages of microscopic properties.
First, 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 calculations we have to first
Choose an Ensemble! 5

6. First,
What in the world
does the term
“Canonical”
mean? 6

7. Some Science Definitions of “Canonical”:7

8. Some Science Definitions of “Canonical”: 1
Some Science Definitions of “Canonical”: 1. Accepted as being Accurate & Authoritative 8

9. Some Science Definitions of “Canonical”: 1
Some Science Definitions of “Canonical”: 1. Accepted as being Accurate & Authoritative 2. According to Rules or Scientific Laws 9

10. Some Science Definitions of “Canonical”: 1
Some Science Definitions of “Canonical”: 1. Accepted as being Accurate & Authoritative 2. According to Rules or Scientific Laws 3. Of or Relating to a General Rule or Standard Formula 10

11. 1. According to or Ordered by Canon Law 2. The Canonical Rites of the
There are also religious meanings of “Canonical”:
1. According to or Ordered by
Canon Law
2. The Canonical Rites of the
Catholic Church
3. The Canonical Gospels of
the New Testament. 11

12. 3 Common Statistical Ensembles:12

13. 3 Common Statistical Ensembles:
Micro-Canonical Ensemble:
Isolated Systems: Constant Energy E.
Nothing happens!  Not Interesting! 13

14. 3 Common Statistical Ensembles:
Micro-Canonical Ensemble:
Isolated Systems: Constant Energy E.
Nothing happens!  Not Interesting!
The Canonical Ensemble:
Systems with a fixed number N of molecules
in equilibrium with a Heat Reservoir (Heat Bath). 14

15. 3 Common Statistical Ensembles:
Micro-Canonical Ensemble:
Isolated Systems: Constant Energy E.
Nothing happens!  Not Interesting!
The Canonical Ensemble:
Systems with a fixed number N of molecules
in equilibrium with a Heat Reservoir (Heat Bath).
The Grand Canonical Ensemble:
Systems in equilibrium with a Heat Bath
which is also a Source of Molecules.
Their chemical potential is fixed. 15

16. Example: If Systems of Interest are Gases

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

18. Microcanonical Ensemble Canonical Ensemble
Example: 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

19. Microcanonical Ensemble Canonical Ensemble Grand Canonical Ensemble
Example: 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

20. All Thermodynamic Properties Can Be Calculated With Any Ensemble
So, it is up to the problem solver which ensemble is used. So, we should choose the most convenient one for a particular problem
Note that, Historically, the first systems people applied Statistical Mechanics to were gases.
It is much more general than that, but when introducing the theory, many books, including Reif’s, still focus on gases at first & some of the first examples discussed are applications to gases. 20

21. All Thermodynamic Properties Can Be Calculated With Any Ensemble
1. For Gases: PVT properties usually use
The “Canonical Ensemble” 21

22. All Thermodynamic Properties Can Be Calculated With Any Ensemble
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 22

23. 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: 23

24. 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 24

25. 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 F in state (configuration) n
Pn  Probability of the system being in
state (configuration) n. 25

26. Canonical Ensemble Probabilities
QNcanon  “Canonical Partition Function”
Often called simply the “Partition Function”
gn  Degeneracy of state n
Note: Un  En  n = Quantum Energy of state n
Also, most texts use the notation
“Z” for the partition function! 26

27. Note that most texts use the notation
Grand Canonical Ensemble Probabilities:
Qgrand  “Grand Canonical Partition Function”
Often called simply the
“Grand Partition Function”
gn  Degeneracy of state n, μ  “Chemical Potential”
Note that most texts use the notation
“ZG” for the Grand Partition Function! 27

28. The Partition Function Z
Partition Functions
We’ll show that if the volume V, (or other external parameter) the temperature T, & the energy levels En (from some microscopic model) of a system are known, in principle
The Partition Function Z
can be calculated. 28

29. The Partition Function Z
Partition Functions
We’ll show that if the volume V, (or other external parameter) the temperature T, & the energy levels En (from some microscopic model) of a system are known, in principle
The Partition Function Z
can be calculated.
We’ll also show that if the partition function Z is known, it can be used to
Calculate all Thermodynamic Properties of the System!!. 29

30. All Thermodynamic Properties.
Partition Functions
If the partition function Z is known, it can be used to Calculate
All Thermodynamic Properties. 30

31. 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. 31

32. Canonical Ensemble Partition Function Z
Starting from the fundamental postulate of equal a priori probabilities, the following are obtained: 32

33. Canonical Ensemble Partition Function Z
Starting from the fundamental postulate of equal a priori probabilities, the following are obtained:
ALL RESULTS of Classical Thermodynamics, plus their statistical underpinnings; 33

34. Canonical Ensemble Partition Function Z
Starting from the fundamental postulate of equal a priori probabilities, the following are obtained:
ALL RESULTS of Classical Thermodynamics, 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 the energy-levels of a quantum system. 34

35. 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. For a quantum system in equilibrium with a heat reservoir, Z is defined as:
εi is the energy of the i’th quantum state.
Z  i exp(-εi/kBT) 35

36. εi = Energy of the i’th state.
The Partition Function for a Quantum System in Contact with a Heat Reservoir: ,
εi = Energy of the i’th 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 previous chapters, is
Z  i exp(-εi/kBT)
S = kBln Ω. 36 36

37. Relationship of Z to Macroscopic Parameters37

38. Relationship of Z to Macroscopic Parameters
Summary:
Canonical Ensemble Partition Function Z:
(Derivation for Ē on the board.
Other derivations are in the book!)
Internal Energy: Ē  E = - ∂(lnZ)/∂β 38

39. Relationship of Z to Macroscopic Parameters
Summary:
Canonical Ensemble Partition Function Z:
(Derivation for Ē on the board.
Other derivations are in the book!)
Internal Energy: Ē  E = - ∂(lnZ)/∂β
rms Energy Fluctuations:
<(ΔE)2> = [∂2(lnZ)/∂β2]
β = 1/(kBT), kB = Boltzmann’s constantt. 39

40. Relationship of Z to Macroscopic Parameters
Summary:
Canonical Ensemble Partition Function Z:
(Derivation for Ē on the board.
Other derivations are in the book!)
Internal Energy: Ē  E = - ∂(lnZ)/∂β
rms Energy Fluctuations:
<(ΔE)2> = [∂2(lnZ)/∂β2]
β = 1/(kBT), kB = Boltzmann’s constantt.
Entropy: S = kB[βĒ + lnZ]
An important, frequently used result! 40

41. Canonical Ensemble Partition Function Z:
(Derivations are in the book!)
Helmholtz Free Energy:
F = E – TS = – (kBT)lnZ 41

42. F = E – TS = – (kBT)lnZ Canonical Ensemble Partition Function Z:
(Derivations are in the book!)
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 42

43. F = E – TS = – (kBT)lnZ G = F + PV = PV – kBT lnZ.
Canonical Ensemble Partition Function Z:
(Derivations are in the book!)
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. 43

44. F = E – TS = – (kBT)lnZ G = F + PV = PV – kBT lnZ.
Canonical Ensemble Partition Function Z:
(Derivations are in the book!)
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 44

45. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = nexp(-En),  = 1/(kBT) 45

46. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = nexp(-En),  = 1/(kBT)
Mean Energy:
Ē = – (lnZ)/ = - (1/Z)Z/ 46

47. 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. 47

48. 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 48

49. 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/ . 49

50. Canonical Ensemble: Constant Volume Heat Capacity
CV = (Ē/T)V = (Ē/)(d/dT) = - k2Ē/ 50

51. 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/ 51

52. 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 52

53. 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 53

54. 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. 54

55. Ensembles in Classical Statistical Mechanics
We’ve seen that, for a classical description, it is
convenient to use a classical phase space for a system with f degrees of freedom. That is, use 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, ... 55

56. The Partition Function
The Canonical Distribution in
Classical Statistical Mechanics
The Partition Function
has the form:
Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT) 56

57. 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! 57

58. 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) 58

59. CLASSICAL Statistical Mechanics:
Let A ≡ any measurable, macroscopic quantity. The thermodynamic average of A ≡ <A>. This is what is measured.

60. 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

61. 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!

62. Now, two slides from the first class day!
A quote from Richard Feynman:

63. P(E), Z The Statistical/Thermal Physics “Mountain”
Statistical Mechanics
(Classical or Quantum)
P(E), Z
Calculation of
Measurable
Quantities
Equations of
Motion
The Statistical/Thermal Physics “Mountain”

64. Statistical/Thermal Physics “Mountain”
P(E), Z
Calculation of
Measurable
Quantities
Equations of
Motion
Statistical/Thermal Physics “Mountain”
The entire subject is either the “climb” UP to the summit (calculation of P(E), Z) or the slide DOWN (use of P(E), Z to calculate measurable properties).
On the way UP: Thermal Equilibrium & Temperature are defined from statistics. On the way DOWN, all of Thermodynamics can be derived, beginning with microscopic theory.
We are now near the peak of this mountain!