This document introduces key concepts in statistical mechanics, including the idea that macroscopic properties are thermal averages of microscopic properties. It discusses common statistical ensembles like the microcanonical ensemble (isolated systems with constant energy) and the canonical ensemble (systems in equilibrium with a heat reservoir). The canonical partition function Z relates microscopic quantum mechanics to macroscopic thermodynamics and can be used to calculate thermodynamic variables. Properties like heat capacity can be derived from fluctuations in energy calculated from the partition function.

3. Key Concepts In Statistical Mechanics
Idea: Macroscopic properties are a
thermal average of microscopic
properties.
Replace the system with a set of systems
"identical" to the first and 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 state.
To do any calculations we have to first
Choose an Ensemble!

4. The Most Common Statistical Ensembles:
1. The Micro-Canonical Ensemble:
Isolated Systems: Constant Energy E.
Nothing happens! ⇒ Not Interesting!
4

5. The Most Common Statistical Ensembles:
1. The 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).
5

6. J. Willard Gibbs was the first to show that
An Ensemble Average is Equal to a
Thermodynamic Average:
That is, for a given property F,
The Thermodynamic Average
can be formally expressed as:
<F> ≡ ΣnFnPn
Fn ≡ Value of F in state (configuration) n
Pn ≡ Probability of the system being in state
(configuration) n.
Properties of The Canonical
& Grand Canonical Ensembles

7. Canonical Ensemble Probabilities
p
g e
Q
n
n
U
canon
N
n
=
−β
QN
canon ≡ “Canonical Partition Function”
gn ≡ Degeneracy of state n
Q g ecanon
N
n
n
Un
= ∑
−β
Note that most texts use the notation
“Z” for the partition function!

8. Grand Canonical Ensemble Probabilities:
p
g e
Q
n
n
E
grand
n
=
−β
E U Nn n n= − µ
Q g egrand n
n
En
= ∑
−β
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!

9. Partition Functions
If the volume, V, 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 PropertiesAll Thermodynamic Properties..
So, in this way,
Statistical Mechanics
provides a direct link between
Microscopic Quantum Mechanics &
Classical Macroscopic
Thermodynamics.

10. Canonical Ensemble Partition Function Z
Starting from the fundamental postulate of equal a
priori probabilities, the following are obtained:
i.ALL RESULTS of Classical Thermodynamics,
plus their statistical underpinnings;
ii.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.
The partition function for a quantum system in
equilibrium with a heat reservoir is defined
as
W
Where εi is the energy of the i’th state.
Z ≡ ∑i exp(- εi/kBT)

11. 11
Partition Function for a Quantum
System in Contact with a Heat
Reservoir:
,
F
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.

12. 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!

13. Summary for the Canonical
Ensemble Partition Function Z:
Helmholtz Free Energy
F = E – TS = – (kBT)lnZ
and
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

14. Canonical Ensemble:
Heat Capacity & Other Properties
Partition Function:
Z = Σn exp (-βEn), β = 1/(kT)

15. Canonical Ensemble:
Heat Capacity & Other Properties
Partition Function:
Z = Σn exp (-βEn), β = 1/(kT)
Mean Energy:
Ē = – ∂(ln Z)/∂β = - (1/Z)∂Z/∂β

16. Canonical Ensemble:
Heat Capacity & Other Properties
Partition Function:
Z = Σn exp (-βEn), β = 1/(kT)
Mean Energy:
Ē = – ∂(ln Z)/∂β = - (1/Z)∂Z/∂β
Mean Squared Energy:
<E2
> = ΣrprEr
2
/Σrpr= (1/Z)∂2
Z/∂β2
.

17. Canonical Ensemble:
Heat Capacity & Other Properties
Partition Function:
Z = Σn exp (-βEn), β = 1/(kT)
Mean Energy:
Ē = – ∂(ln Z)/∂β = - (1/Z)∂Z/∂β
Mean Squared Energy:
<E2
> = ΣrprEr
2
/Σrpr= (1/Z)∂2
Z/∂β2
.
nth
Moment:
<En
> = ΣrprEr
n
/Σrpr= (-1)n
(1/Z) ∂n
Z/∂βn

18. Canonical Ensemble:
Heat Capacity & Other Properties
Partition Function:
Z = Σn exp (-βEn), β = 1/(kT)
Mean Energy:
Ē = – ∂(ln Z)/∂β = - (1/Z)∂Z/∂β
Mean Squared Energy:
<E2
> = ΣrprEr
2
/Σrpr= (1/Z)∂2
Z/∂β2
.
nth
Moment:
<En
> = ΣrprEr
n
/Σrpr= (-1)n
(1/Z) ∂n
Z/∂βn
Mean Square Deviation:
<(ΔE)2
> = <E2
> - (Ē)2
= ∂2
lnZ/∂β2
= - ∂Ē/∂β .

19. Canonical Ensemble:
Constant Volume Heat Capacity
CV = ∂Ē/∂T = (∂Ē/∂β)(dβ/dT) = -
kβ2
∂Ē/∂β

20. Canonical Ensemble:
Constant Volume Heat Capacity
CV = ∂Ē/∂T = (∂Ē/∂β)(dβ/dT) = -
kβ2
∂Ē/∂β
using results for the Mean Square Deviation:
<(ΔE)2
> = <E2
> - (Ē)2
= ∂2
lnZ/∂β2
= - ∂Ē/∂β

21. Canonical Ensemble:
Constant Volume Heat Capacity
CV = ∂Ē/∂T = (∂Ē/∂β)(dβ/dT) = -
kβ2
∂Ē/∂β
using results for the Mean Square Deviation:
<(ΔE)2
> = <E2
> - (Ē)2
= ∂2
lnZ/∂β2
= - ∂Ē/∂β
CV can be re-written as:
CV = kβ2
<(ΔE)2
> = <(ΔE)2
>/kBT2