PHY 341/641 Thermodynamics and Statistical Mechanics
MWF: Online at 12 PM & FTF at 2 PM
Discussion for Lecture 21:Introduction to statistical mechanics –
Microcanonical vs canonical ensembles
Reading: Chapter 6.1 and 6.5
1. Microcanonical ensemble and multiplicity factor
2. Boltzmann’s entropy in the context of a heat bath
3. Canonical ensemble
Record!!!
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Comment on last question on mid term –The curves below represent a potential function V(x)
Which of them have a stable equilibrium point?
left right
V V
x
x
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Left curve
( )V x
( )dV xdx
2
2
( )d V xd x
x
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Right curve
( )V x
( )dV xdx
2
2
( )d V xd x
x
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)
For this micro canonical ensemble, Boltzmann used the multiplicity function to calculate the
ln( ( , , )entropy --
BS k N V U= Ω
Introduction to statistical mechanics –
Up to now our discussion of statistical mechanics has been focused on the analysis of the multiplicity function Ω which, for an isolated system, describes the number of “micro” states which have the same values of the microstate variables such as N,V, U for a free electron gas. The literature refers to this as a “micro canonical ensemble”.
3/22/2021 PHY 341/641 Spring 2021 -- Lecture 21 7
( ) ( )3 /2 3
3
3/25/2
3/2
2
2
( , , ) 2! 3 / 2 !
4 3
4 5( , , ) ln3 2
N N N
B
N
N
eM
VN V U MUh N N
VN Nh
VS N V U NkN h
U
MUN
π
π
π
Ω ≈
≈
≈ +
For a mono atomic ideal gas having N particles in a volume V at an energy U --
Sackur-Tetrode expression for the entropy
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Now consider the notion of the micro canonical system in terms of probabilities. The tacit assumption is that each micro state of our system is equally likely.
1
( , , )We will label a given micro state by and its probability is
1 with 1
Boltzmann's entropy function ln c
Consider a given ensemble -- with multiplicity function
i ii
U V Ni
p p
S k
Ω
=
Ω = Ω
= =Ω
= Ω
∑
1 1
an be expressed in terms of the probabilities as1 1 ( ln ) ln ln i i
i iS k p p k k
Ω Ω
= =
= − = − = Ω Ω Ω ∑ ∑
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We will then introduce a “canonical ensemble”
.
Important equation for micro canonical ensembl
where is the m
e
ultiplic
:
ity tln
func ionS k Ω=
Ω
n
I
where is called the partition functio
mportant equation for canonical ensemble:lnF k Z
ZT= −
3/22/2021 PHY 341/641 Spring 2021 -- Lecture 21 10
Canonical ensemble:
Eb
Es
Now, we would like to extend the analysis of an isolated system to that of a system within a heat bath. The system within the heat bath will be analyzed in terms of a “canonical ensemble” --
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Canonical ensemble in terms of the internal energies of the bath (b), system (s), and total
( )( )
( )( )
( )
( ) ( )
'' ' '
'
Probability that system is energy :
ln ln
ln
( )
ln
( )
tot
tot s b s b
b tot s b tot ss
b tot s b tot ss s
t
s
s s
s b tot s
b
s
to
s
bs
U
U U U U UU
U U U UU U U U
C U U
UC U
U
UU
U
= + <<
Ω − Ω −= ≈
Ω − Ω −
= + Ω −
∂ Ω ≈ + Ω − +
Ω
∂
Ω∑ ∑
P
P
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Canonical ensemble (continued)
( ),
ln ( ) 1Note that: tot
b b
V N bU
Bk U S UU U T
∂ Ω ∂ ≈ = ∂ ∂
( )/
1ln ln
' B bs
s b tot s
U k Ts
B b
C U Uk T
C e−
≈ + Ω − +
⇒ =
P
P
( )
( ) ( )ln ln
ln ln
tot
s b tot s
bb tot s
U
C U U
UC U U
U
= + Ω −
∂ Ω ≈ + Ω − + ∂
P
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'
/
/
/
'
'1
where: "partition function"
s
B b
B b
B b
s
s
U k Ts
U k T
U k T
s
C e
eZ
Z e
−
−
−
=
=
≡ ∑
P
Canonical ensemble:
' '/ 1
' '
Calculations using the partition function:( ) where s sB
B
U k T Uk T
s sZ T e e ββ− −≡ = =∑ ∑
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Canonical ensemble continued – average energy of system:
' '/' '
' '
1 1
1 ln
s sBU k T Us s s
s sU U e U e
Z ZZ Z
Z
β
β β
− −= =
∂ ∂= − = −
∂ ∂
∑ ∑
Heat capacity for canonical ensemble:
( )
2
22 2
2 2 2 2
222
1
1 ln 1 1 1
1
B
B
s sV
s
B
s
U UC
T k T
Z Z Zk T k T Z Z
U UkT
β
β β β
∂ ∂= = −
∂ ∂
∂ ∂ ∂= = − ∂ ∂ ∂
= −
1kT
β ≡
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Comment – Note that in this formulation we are accessing the system energies Us while typically we only know the individual particle energies. For real evaluations, additional considerations will be needed.
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Analyzing the relationship of the partition function with the notion of entropy
( )
l
A
l
ccording to our previous statements, let us suppose that
for ln
n ln
1ln ln
ln
n
s
s s s
U
s
U U U
s
s s s
ss
Z
U ZZ Z Z
Z ZT
Z
F Z
eS k
e e eS k k
S k U k U k
ST U kT
kT
β
β β β
β
β
−
− − −
= − =
=
− −
+
− = −
⇒ = +
⇒
−
=
=
−
=
∑
∑ ∑
P P P
- Helmholtz free energy
Note that the Helmholtz free energy is given byF U TS= −
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Recap:
( )
, ,
, ,
, ,
Microcanonical ensemble: Internal energy ( , , ) ln ( , , )( , , )
1V N V N
U N U N
E V U V
B
B
B
B
US U V N k U V NU S V N dU TdS PdV dN
kST U U
kP ST V V
kSN N
µ
µ
= Ω
⇒ = − +
∂ ∂Ω = = ∂ Ω ∂
∂ ∂Ω = = ∂ Ω ∂
∂ ∂Ω = = ∂ Ω ∂
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Canonical ensemble
' '/
' '
Partition function; keep constant ( , ) ( , )
( , ) ln ( , )
s sBU k T U
s s
NZ e e Z T V Z V
F T V U TS k Z T V
β β− −≡ ≡ = ≡
= − = −
∑ ∑
( )NVTZkTNVTF ,,ln ),,( −=
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( ) ( )
( )
( )VTVT
NTNT
NVNVNV
NZkT
NF
VZkTP
VF
TZkTZk
TZkTS
TF
,,
,,
,,,
ln
ln
lnlnln
∂∂
−==
∂∂
∂∂
−=−=
∂∂
∂∂
−−=
∂∂−
=−=
∂∂
µ
Generalization --
NdF SdT P ddV µ= − − +
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Example: Canonical distribution for free particles
( )
( )2/
2
2
2
0
2!
1
2
1
length ofbox in dimensionsin moving mass of particles free for on distributi canonical Classical
2
dNdN
dN/dN
dN
pmdN
Lr
dNdN
hmkTL
N
mkTN!h
L
epdrd N!h
Z(T,V,N)
L d mN
ii
i
=
=
∑= ∫∫
−
≤≤
π
π
β
( )
3
3 /2
2
3 /2
232
For 3,
2( , , )!
Compare with microcanonical ensemble:
2( , , )! 1
NN
NN
N
d L V
V mkTZ T V N N h
V mUU V NN h
π
π
= ≡
=
= Γ + Ω
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Example of a canonical ensemble consisting of 2 particles
Consider a system consisting of 2 distinguishable particles. Each particle can be in one of two microstates with single-particle energies 0 and ∆. The system is in equilibrium with a heat bath at temperature T.
s ε1 ε2 Us
1 0 0 02 0 ∆ ∆
3 ∆ 0 ∆
4 ∆ ∆ 2∆
/ / / 2 / / 2 /1 1 2sU kT kT kT kT kT kT
sZ e e e e e e− −∆ −∆ − ∆ −∆ − ∆= = + + + = + +∑
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/ / / 2 /
/ 2 /
1
1 2
sU kT kT kT kT
skT kT
Z e e e e
e e
− −∆ −∆ − ∆
−∆ − ∆
= = + + +
= + +
∑
kT/∆
T =∆/k
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kT/∆
F(T)
U(T)
Model results for Helmholtz free energy and internal energy
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