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Ch. 6: Basic Methods & Results of Stat.
Mech. + Ch. 7: Simple Applications of Stat.
Mech. Overview + Details & Applications in Chs. 6, 7
Key Concepts In Statistical Mechanics
Basic Idea (early lectures in this course):
Macroscopic properties are thermal
averages of microscopic properties.
• 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 state. To do any
calculations we have to first
Choose an Ensemble!
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
• There are also religious meanings of “Canonical”:
1. According to or ordered by canon law
2. Canonical rites of the Catholic Church
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.
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
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
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.
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, (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.
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.
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.
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:
Z  i exp(-εi/kBT)
• εi is the energy of the i’th quantum state.
Partition Function for a Quantum
System in Contact with a Heat Reservoir:
, i/kBT)
Z  i exp(-ε
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.
14
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.
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
Note that this gives: Z = exp[-F/(kBT)]
dF = S dT – PdV, so
S = – (∂F/∂T)V, P = – (∂F/∂V)T
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.
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
Canonical Ensemble:
Heat Capacity & Other Properties
• Partition Function:
Z = nexp(-En),  = 1/(kBT)
Canonical Ensemble:
Heat Capacity & Other Properties
• Partition Function:
Z = nexp(-En),  = 1/(kBT)
• Mean Energy:
Ē = – (lnZ)/ = - (1/Z)Z/
Canonical Ensemble:
Heat Capacity & Other Properties
• Partition Function:
Z = nexp(-En),  = 1/(kBT)
• Mean Energy:
Ē = – (lnZ)/ = - (1/Z)Z/
• Mean Squared Energy:
2
2
E  = rprEr /rpr = (1/Z)2Z/2.
Canonical Ensemble:
•
•
•
•
Heat Capacity & Other Properties
Partition Function:
Z = nexp(-En),  = 1/(kBT)
Mean Energy:
Ē = – (lnZ)/ = - (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 = nexp(-En),  = 1/(kBT)
• Mean Energy:
Ē = – (lnZ)/ = - (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)V = (Ē/)(d/dT) = - k2Ē/
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/
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
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
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.
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, ...
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)]
B
/Z
<A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E)
Another 6N Dimensional
Integral!
Now, two slides from the
first class day!
A quote from Richard
Feynman:
Statistical Mechanics
(Classical or Quantum)
P(E), Z
Equations of
Motion
Calculation of
Measurable
Quantities
The Statistical/Thermal
Physics “Mountain”
P(E), Z
Equations of
Motion
Calculation of
Measurable
Quantities
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.