PHY 341/641 Thermodynamics and Statistical Mechanics MWF:...

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!!!

3/22/2021 PHY 341/641 Spring 2021 -- Lecture 21 1


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


Left curve

( )V x

( )dV xdx

2

2

( )d V xd x

x


Right curve

( )V x

( )dV xdx

2

2

( )d V xd x

x


)

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 /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


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 ∑ ∑


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= −


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” --


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


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


'

/

/

/

'

'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− −≡ = =∑ ∑


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 ≡


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.


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= −


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 ∂


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 ),,( −=


( ) ( )

( )

( )VTVT

NTNT

NVNVNV

NZkT

NF

VZkTP

VF

TZkTZk

TZkTS

TF

,,

,,

,,,

ln

ln

lnlnln

∂∂

−==

∂∂

∂∂

−=−=

∂∂

∂∂

−−=

∂∂−

=−=

∂∂

µ

Generalization --

NdF SdT P ddV µ= − − +


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


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− −∆ −∆ − ∆ −∆ − ∆= = + + + = + +∑


/ / / 2 /

/ 2 /

1

1 2

sU kT kT kT kT

skT kT

Z e e e e

e e

− −∆ −∆ − ∆

−∆ − ∆

= = + + +

= + +

∑

kT/∆

T =∆/k


kT/∆

F(T)

U(T)

Model results for Helmholtz free energy and internal energy

PHY 341/641 Thermodynamics and Statistical Mechanics MWF:...

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