1. Chapter 6: Basic Methods & Results of Statistical Mechanics

2. 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! 2

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

4. 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). 4 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).
3. The Grand Canonical Ensemble:
Systems in equilibrium with a Heat Bath
which is also a Source of Molecules.
Their chemical potential is fixed. 5

6. All Thermodynamic Properties Can Be Calculated With Any Ensemble
Choose the most convenient one for a particular problem.
For Gases: PVT properties
use
The Canonical Ensemble
For Systems which Exchange Particles:
Such as Vapor-Liquid Equilibrium
The Grand Canonical Ensemble 6

7. F  nFnPn 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 F,
The Thermodynamic Average
can be formally expressed as:
F  nFnPn
Fn  Value of F in state (configuration) n
Pn  Probability of the system being in state
(configuration) n. 7

8. Canonical Ensemble Probabilities
QNcanon  “Canonical Partition Function”
gn  Degeneracy of state n
Note that most texts use the notation
“Z” for the partition function! 8

9. 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! 9

10. The Partition Function Z Statistical Mechanics
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 Properties.
So, in this way,
Statistical Mechanics
provides a direct link between
Microscopic Quantum Mechanics &
Classical Macroscopic Thermodynamics. 10

11. 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.
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

12. εi = Energy of the i’th state.
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 12

13. 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

14. 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

15. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = n exp (-En),  = 1/(kT) 15

16. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = n exp (-En),  = 1/(kT)
Mean Energy:
Ē = – (ln Z)/ = - (1/Z)Z/ 16

17. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = n exp (-En),  = 1/(kT)
Mean Energy:
Ē = – (ln Z)/ = - (1/Z)Z/
Mean Squared Energy:
E2 = rprEr2/rpr = (1/Z)2Z/2. 17

18. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = n exp (-En),  = 1/(kT)
Mean Energy:
Ē = – (ln Z)/ = - (1/Z)Z/
Mean Squared Energy:
E2 = rprEr2/rpr = (1/Z)2Z/2.
nth Moment:
En = rprErn/rpr = (-1)n(1/Z) nZ/n 18

19. Canonical Ensemble: Heat Capacity & Other Properties
Partition Function:
Z = n exp (-En),  = 1/(kT)
Mean Energy:
Ē = – (ln Z)/ = - (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/ . 19

20. Canonical Ensemble: Constant Volume Heat Capacity
CV = Ē/T = (Ē/)(d/dT) = - k2Ē/ 20

21. Canonical Ensemble: Constant Volume Heat Capacity
CV = Ē/T = (Ē/)(d/dT) = - k2Ē/
using results for the Mean Square Deviation:
(ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/ 21

22. Canonical Ensemble: Constant Volume Heat Capacity
CV = Ē/T = (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 22

23. Canonical Ensemble: Constant Volume Heat Capacity
CV = Ē/T = (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 23

24. Canonical Ensemble: Constant Volume Heat Capacity
CV = Ē/T = (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 and (ii) Ē/T ≥ 0. 24

25. 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 such as
energy, number of particles, volume, ... 25

26. 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) 26

27. P(E) ≡ e[-E/(kBT)]/Z 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!