PHY 341/641 Thermodynamics and Statistical Mechanics

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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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3/22/2021 PHY 341/641 Spring 2021 -- Lecture 21 2

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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” --

3/22/2021 PHY 341/641 Spring 2021 -- Lecture 21 11

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

3/22/2021 PHY 341/641 Spring 2021 -- Lecture 21 13

'

/

/

/

'

'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