3. The Canonical Ensemble

1. Equilibrium between a System & a Heat Reservoir

2. A System in the Canonical Ensemble

3. Physical Significance of Various Statistical Quantities in the Canonical Ensemble

4. Alternative Expressions for the Partition Function

5. The Classical Systems

6. Energy Fluctuations in the Canonical Ensemble:

Correspondence with the Microcanonical Ensemble

7. Two Theorems: the “Equipartition” & the “Virial”

8. A System of Harmonic Oscillators

9. The Statistics of Paramagnetism

10. Thermodynamics of Magnetic Systems: Negative Temperatures

Reasons for dropping the microcanonical ensemble:

1. Mathematical: Counting states of given E is difficult.

2. Physical: Experiments are seldom done at fixed E.

Canonical ensemble : System at constant T through contact with a heat reservoir.

Let r be the label of the microstates of the system.

Probablity Pr ( Er ) can be calculated in 2 ways:

1. Pr # of compatible states in reservoir.

2. Pr ~ distribution of states in energy sharing ensemble.

3.1. Equilibrium between a System & a Heat

ReservoirIsolated composite system A(0) = ( System of interest A ) + ( Heat reservoir A )

Heat reservoir : , T = const.

Let r be the label of the microstates of A.

0r rE E E r with 0

r rE E E

r rP E Probability of A in state r is 0

rE E

0

0 0lnln lnr r

E E

E E E EE

rconst E

,

ln

N VE

1rE

rP eZ

rE

r

Z e

Classical mech (Gibbs –corrected ):

3 3 ( , )3

1

!N N H q p

NZ d p d q e

N h

3.2. A System in the Canonical Ensemble

Consider an ensemble of N identical systems sharing a total energy E.

Let nr = number of systems having energy Er ( r = 0,1,2,... ).

rr

n Nr r

r

n E U E N

U EN

= average energy per system

Number of distinct configurations for a given E is

!

!r rr

r

W n W nn

N

{ nr* } = most probable distribution

Equal a priori probabilities 1r rP n W n

* maxr rW n W n

,

r

rn

W n N E

,1

r

s s rn

n n W n N E (X) means sum includes only

terms that satisfy constraint on X.

0 1 2, , ,rn n n n

Method of Most Probable Values

!

!rr

r

W nn

N ln ln ! !r

r

W n N

ln ! lnn n n n ln ln lnr r

r

W n n N N

rr

n N

To maximize lnW subjected to constraints

ln 1 0r rn E

rr

n N r rr

n E E

, are Lagrange multipliers

1* rErn e rEC e

is equivalent to minimize, without constraint ln r r rr r

W n n E N E

* rErn C e

*r

r

n N *r r

r

n E E

rE

r

C e NrE

rr

C e E E

r

r

Er

rE

r

e EU

e

EN

* r

r

Er E

r

n ee

N

Same as sec 3.1* 1

rErr

ne P

Z

N

rE

r

Z e Let

E.g.

r rr

U P E

1

kT

and set

with

Method of Mean Values

Let 0

0

!

!

rnr

rr

rr

W nn

N

Thus 1r

X X

,,

r

rn

U W n N EN r

r

r rr

n

n E U

N

E NConstraints:

,1

r

s s rn

n n W n N E

1r

s

s

,

r

rn

W n N E

ln 1s s

s s

1r Note

:

,1

r

s rn

n W n N E

r in { nr } is a dummy variable that runs from 0 to , including s.

0

!!

rnr

r rn

N

,

0

!!

r

r

nr

n r rn

N E

N

1

lnr

ss

~ means “depend on {r } ”.

Method of Steepest Descent ( Saddle Point )

,

0

, !!

r

r

nr

n r r

Un

N E

N N is difficult to evaluate due to the energy constraint.

Its asymptotic value ( N ) can be evaluate by the MSD.

Define the generating function 0

, , U

U

G z U z

NN N

,

0 0

, !!

rr

r

nEr

nU r r

zG z

n

N E

N N

r rr

n E U E N

Binomial theorem

r rn EU

r

z zN

U removes the energy constraint.

, rEr

r

G z z

N

N

0

!!

rr

r

nEr

n r r

z

n

N= N

f z N

rErr

f z zwhere

0

, , rEUr

U r

G z U z z f z

NNNN N

N U = integers = coefficient of zN U in power expansion of .

This is the case if all Er , except the ground state E0 = 0, are integer multiples

of a basic unit.

1

1,

2 UC

f zU d z

i z

N

NN

C : |z| < R

analytic for |z| < R

Let 1

g z

U

f ze

z

N

NN

( For { r ~ 1 }, sharp min at z = x0 )

1ln lng z f z U z

N

,U N G

f z

Mathematica

1ln lng z f z U z

N

1 1fg U

zf

N

0

0

00

1 10

f xU

xf x

N

2

22

1 1f fg U

zf f

N

N >>1

00

0

f xU x U

f x

2

00 2

0 00

f x U Ug x

x xf x

Fo z real, has sharp min at x0

20 0 0

1

2g x i y g x g x y

0 0g x 0 0g x

For z complex : max along ( i y )-axis

x0 is a saddle point of .

g

g

20 0 0

1

2g x i y g x g x y

1

g z

U

f ze

z

N

NN

0 20 01

0

1exp

2U

f x i yg x g x y

x i y

N

N N N

0 2

010

1exp

2U

f xg x y

x

N

N N

1

1,

2 UC

f zU d z

i z

N

NN

MSD: On C, integrand has sharp max near x0 .

0

0

1

0

1

2 UNear x

f x i yi d y

i x i y

N

N

0 2

010

1 1exp

2 2U

f xg x y d y

x

N

N N

Gaussian dies quickly

0

10 0

1 2

2 U

f x

x g x

N

N N

0

10 0

1 2,

2 U

f xU

x g x

N

NNN

20

0 00

1ln ln ln 2

2U

f xg x x

x N N

0

0

lnU

f x

x

N

N

rErr

f z z

0 0ln ln lnf x U x N

0

0

s

r

E

E

r

x

x

N

0

0

s

r

Es

Er

r

x

x

N

1

lnr

s ss

n

10 00 0s rE E

r rrs s

f x xx E x

00

0

sE

s

U xx f

x

0

0

r

r

Er r

rE

rr

E xU U

x

0 0 0

0 0

lnsE

s ss s s

x U x U x

x xf

N

0sE

s

x

fN

0x e

0

0

s

r

E

s E

r

xn

x

N

0

0

r

r

Er r

rE

rr

E xU

x

0 0ln ln lnrEr

r

x U x

N

C.f.* 1

rErr

ne P

Z

N

With { r = 1 } :

r

r

Er

rE

r

E eU

e

ln ln rE

r

e U N

rr

r

nE N

1s

s

r

EEs

E

r

n ee

e Z

Nso that

ln Z U N

rE

r

Z e

0 1r

Z f x

rE

(r) r is a dummy variable

sEs rs

sr r sr

nn eP P

Z

N N

r rr

E P

Fluctuations

0

!!

r

r r

nr

rn n r r

W nn

N

2

1

1

r

s s ss s

n

1r

ss

s

n

22

s s sn n n 22 2s s s sn n n n

22s sn n

ln ss s s

s s s s

22

2

1 ss s

s s s

22

1

ln

r

s s s ss s

n n

2

sn

where

0lnsE

ss

s

x

f

N

0rE

r

Z x

0 0s sE E

s

x xn

Z f N N

00

0

sE

s s

f U xx f

x

0lnsE

ss s s

s s s

x

f

N

10 0 0 0 0

020

s s s

s

E E EEs s s

ss s

x E x x x U xx f

xf f f

N

2

0

0

s ss s s s s

s

n xn n E U

x

N

0

0

r

r

Er r

rE

rr

E xU U

x

0rE

r rrs s

fE x U

12 0 00 0 0

0

s srE EEs r r

r s s

x U xE x E x U x f

x

1r

12 0 00 0 0

0

s srE EEs r r

r s s

x U xE x E x U x f

x

20

0

0

1rE

r rsr

ss

E xnx

UU U Ex f

N

2 2 2

2

s s ss

r

n n E Un

E U

N N

2

0

01r

s ss s s s s

s

n xn n E U

x

N

0

2 20 1

1

r

ss

s r

nU Ex

x E U

N

2

1

ln

r

s s ss s

n

2

ss

r

nU E

E U

N

2 2

2

1 1 1 ss

s sr

E Un

n n E U

N NRelative fluctuation

0sand hence n

N

* if non-zeror r rn n n

rU E

3.3. Physical Significance of Various Statistical Quantities in the Canonical Ensemble

Canonical distribution :rE

rr

n eP

Z

N

1

rErr

r

U E E eZ

rE

r

Z e

ln Z

Helmholtz free energy A ( T, V, N ) :

dA dU Td S SdT

A U T S

SdT PdV d N

,V N

AS

T

,T N

AP

V

,V T

A

N

U A T S ,V N

AA T

T

2

,V N

AT

T T

,V N

A

lnU Z

,V N

A

lnA kT Z

= Partition function ( Zustandssumme / sum over states ) ,NZ Q V T

; , ,rEr

r

Z e Z T E V N A , & hence lnZ , must be extensive.

,

V

N V

UC

T

,N V

AA T

T T

2

2

,N V

AT

T

G A PV

Gibbs free energy G ( T, P, N ) :

,T N

AA V

V

N,T V

AN

N

Prob 3.5

P

lnA kT Z

,T N

AP

V

rE

r

Z e

1rEr

r N

Ee

Z V

1rE

r Nr

PdV e dEZ

r r Nr

P dE 1rE

rP eZ

r rrr

U E P E , rr rN P N

r

dU P dE PdV

,N S

UP

V

c.f.

F

Er is indep of T

( Fixed { Pr } = Fixed S )

S1

rErP e

Z ln lnr rP Z E

ln lnr rP Z E A U

lnA kT Z

T SS

k

ln rS k P

lnr rr

S k P P

T = 0, non-degenerate ground state 0r rP 0S ( 3rd law )

1rP

1

1ln

r

S k

lnk ( microcanonical

)

Disorder Unpredictability S Information theory (Shannon)

3.4. Alternative Expressions for the Partition

Function ,, , ,rE N V

Nr

Z N V T e Q V T Non-degenerate systems:

Degenerate systems: rEr

r

Z g e gr = degeneracy of Er

r rr

X X P 1rE

r rr

g X eZ

rE

rr

g eP

Z

Thermodynamic limit ( N , V ) continuum approx. :

EZ d E g E e 1 EP E g E eZ

X dE X E P E 1 Ed E X E g E eZ

0

EZ d E g E e Z( > 0 ) = Laplace transform of g(E)

Inverse transform:

1

2

i E

ig E e Z d

i

If g diverges, then > 0 is realsuch that all poles of Z are to the left of

1

2i Ee Z i d

3.5. The Classical Systems

Quantum Classical states = d

1rE

r r rr r

X X P X eZ

3 3

3 3

, ,

,

N N

N N

d q d p X q p q pX

d q d p q p

H

H

d X e

d e

where 3 3N Nd d q d p

Gibbs’ prescription: 3

1

!H

NZ d e

N h

,H H q p

,NQ V T3! N

dd

N h

Ideal Gas2

1 2

Ni

i

Hm

p

2

3 33

1 1

1, , exp

! 2

N Ni

i iNi i

Z T V N d q d pN h m

p

( In Cartesian coordinates, sum has 3N terms )

23

3exp

! 2

NN

N

Vd p

N h m

p

2 23 2

0

exp 4 exp2 2

pd p d p p

m m

p 3/22 m k T

2

10

1 1 1

2 2n x

n

nd x x e

where

3 1 1

2 2 2 2

3

1,

!

N

N

VZ Q T V

N

1

1,

!N

Q T VN

3h

3

1,

!

N

N

VZ Q T V

N

lnA kT Z 3ln ln

VkT N N N NkT

3

ln 1N

A NkTV

3

,

ln 1T V

A N NkTkT

N V N

3

lnN

kTV

,T N

A NkTP

V V

3

,

3ln 1

2N V

A N NkTS Nk

T V T

3

5ln

2

VNk

N

lnU Z

lnA kT Z ,N V

A

,N V

AA

,N V

AU A T

T

A T S

3

2U NkT

3

ln 1N

A NkTV

3

5ln

2

VS Nk

N

Non-interacting (free) particles : 1

1, , , ,

!N

NZ N T V Q T V Q T VN

g EE

3 /2

3

2

3 / 2 !

NN mEV

h N

( from sec 1.4 )

3 /2 3 /2 1

3

2

3

1

! / 2 1 !

NN Nm EVg E

h NN

( Gibbs factor added by hand )

EZ d E g E e

3 /2

3 /2 1

30

2

3 / 2 1 !

1

!

NNN EmV

dE E eh NN

3 /2 1 3 /2

0

3 / 2 1 !N E Nd E E e N

3 /2

3

1

!

2NN

V mZ

hN

3

1

!

N

N

V

1-particle DOS : 3/2 1/2

3

2

/ 2

mVa

h

1

0

Q d a e

3/2

3

2V m

h

1

1

!, N

NZ Q T VN

Q 3 /2

3

1

!

2NN

V

N

m

h

( same as before )

1

2

i E

ig E e Z d

i

3 /2 3 /2

3

12

2

1

!

NiN E N

i

Vm d

hNe

i

11 0

Res 01!

20 0

s x ns xs i nn ss i

e xxe

d s s ni s

x

3 /2 3 /2 1

2

20

3 / 2 1 !

0

!

0

N NNV m EE

g E hN N

E

contour closes on the left

contour closes on the right

( same as before )

Prob 3.15

3

V

3.6. Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble

rU E E 1rE

rr

E eZ

rE

r

Z e

2

22

,

1 1r rE E

r rr rN V

UE e E e

Z Z

22E E 2E

2

,N V

UE

2

,N V

UkT

T

,r rE E N V

2VkT C

Relative root-mean-square fluctuation in E : 2

2V

E kT C

E U

1

N

Almost all systems in a canonical ensemble have energy U .

( Just like the microcanonical ensemble )

P(E)

EP E g E e

max P at E* satisfies : 0E EP ge g e

E E

*

0E E

gg

E

*

ln

E E

g

E

or

lnS k g *E E

Sk

E

1

T

c.f.,

1

N V

S

U T

*E U E

( Every system in ensemble has same N & V )

i.e., Most probable E = mean E

*g E E U

2

2

2

1ln ln ln

2E U E

E U

g E e g U e g E e E UE

lnS k g U

2

2

2

1ln

2E

E U

SU g E e E U

k E

1 S

k U

2 2

2 2

1ln E

E U

Sg E e

E k U

ln lnE

E U

g E e g U UE U

, ,S S U N V ,

1

N V

S S

U U T

2

2 2

, ,

1

N V N V

S T

U T U

2

1

VT C

2

1

Vk T C

2

2

1ln

2E

V

g E e U T S E Uk T C

Everything, except E, are kept const.

2

2

1ln

2E

V

g E e U T S E Uk T C

EP E g E e 2

2exp

2U T S

V

E Ue

k T C

P(E) is a Gaussian with mean U and dispersion (rms) 2 2VE E k T C

P(E/U ) is a Gaussian with mean 1 and dispersion (rms)

2 2

1 Vk T CE

U U

1~O

N

P(E) (E U ) as N

Ideal Gas

3 /2 3 /2 1

2

2

! 3 / 2 1 !

N NNV m Eg E

N h N

EP E g E e 2

2exp

2U T S

V

E Ue

k T C

*

ln

E E

g

E

13 / 2 1

*N

E

3 / 2 1*

NE

0

E dE P E E

0

0

E

E

d E g E e E

d E g E e

3 /2

0

3 /2 1

0

E N

NE

d E e E

d E e E

3 / 2N

*E E U for N >> 1

N = 10, = 1

3 /2 1

3 /2 1

N

N

g E E

g U U

3 / 2 1*

NE

3 / 2N

U

2VkT C 2

1 3

2Nk

k

3 / 2

U

N

Mathematica

Z

2

2exp

2U T SE

V

E Ug E e e

k T C

0

, ENZ Q V T dE g E e 2

20exp

2U T S

V

E Ue d E

k T C

22U T SVe k T C

lnA kT Z 21ln 2

2 VU T S kT k T C lnU T S O N

O(N)

U T S

2

0

22U T S xV x

e k T C d x e 0 22 V

Ux O N

k T C

2xd x e