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Chapter 6: Basic Methods & Results
of Statistical Mechanics
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!
The Most Common Statistical Ensembles:
1. The Micro-Canonical Ensemble:
Isolated Systems: Constant Energy E.
Nothing happens!  Not Interesting!
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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).
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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.
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
use
The Grand Canonical Ensemble
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.
Canonical Ensemble Probabilities
U n
g ne
pn  N
Qcanon
N
Qcanon
  g ne
U n
n
QNcanon  “Canonical Partition Function”
gn  Degeneracy of state n
Note that most texts use the notation
“Z” for the partition function!
Grand Canonical Ensemble Probabilities:
E n
g ne
pn 
Q grand
Qgrand   g ne
E n
n
E n  Un  N n
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!
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.
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
Z  i exp(- εi/kBT)
Where εi is the energy of the i’th state.
Partition Function for a Quantum
System in Contact with a Heat Reservoir:
,
Z  i exp(- εi/kBT)
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
S = kBln Ω.
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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!
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
Canonical Ensemble:
Heat Capacity & Other Properties
Partition Function:
Z = n exp (-En),  = 1/(kT)
Canonical Ensemble:
Heat Capacity & Other Properties
Partition Function:
Z = n exp (-En),  = 1/(kT)
Mean Energy:
Ē = – (ln Z)/ = - (1/Z)Z/
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:
2
2
E  = rprEr /rpr = (1/Z)2Z/2.
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:
2
2
E  = rprEr /rpr = (1/Z)2Z/2.
nth Moment:
n
n
n
n
n
E  = rprEr /rpr = (-1) (1/Z)  Z/
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:
2
2
E  = rprEr /rpr = (1/Z)2Z/2.
nth Moment:
n
n
n
n
n
E  = rprEr /rpr = (-1) (1/Z)  Z/
Mean Square Deviation:
(ΔE)2 = E2 - (Ē)2 = 2lnZ/2 = - Ē/ .
Canonical Ensemble:
Constant Volume Heat Capacity
CV = Ē/T = (Ē/)(d/dT) = - k2Ē/
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/
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
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
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.
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, ...
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)
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/(k T)]/Z
B
<A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E)
Another 6N Dimensional Integral!