Canonical ensemble

Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton

(Date: September 06, 2018)

Here we discuss the method of canonical ensemble, which is much simpler than that of

microcanonical ensemble. We show that both methods are equivalent. In other words, the same

results are derived using two methods.

1. Thermal equilibrium

We consider the number of accessible states of two systems in thermal contact, with constant

total energy,

21 EEE

The multiplicity ),( ENg of the combined system is

EE

EENgENgENg1

),(),(),( 122111

The largest term in ),( ENg governs the properties of the total system in thermal equilibrium.

For an extremum, it is necessary that the differential of ),( ENg be zero for an infinitesimal

change of energy.

02

2

2112

1

1

21

dEE

ggdEg

E

gdg

NN

021 dEdE

The thermal equilibrium condition is

212

2

21

1

1

11

NNE

g

gE

g

g

which may be written as

212

2

1

1 lnln

NNE

g

E

g

. (1)

We define the entropy S as

),(ln),( ENgkENS B

We write Eq.(1) in the final form

212

2

1

1

NNE

S

E

S

This is the condition for thermal equilibrium for two systems in thermal contact.

2. Temperature

The last equality leads to us to the concept of temperature. In thermal equilibrium, the

temperature of the two systems are equal.

21 TT

This rule must be equivalent to

212

2

1

1

NNE

S

E

S

NS

E

NS

NENE

NS

T

1

),(

),(

1

),(

),(1

or

NS

ET

So that T must be a function of NE

S

. If T denotes the absolute temperature in K, this function

is simply the inverse relationship,

NE

S

T

1

.

((Note))

What is the difference between the expression

NE

S

T

1

(1)

and the expression

NS

ET

. (2)

These two expressions have a slightly different meaning. In Eq.(1), S is given as a function of the

independent variables E and N as ),( NES . Hence T has the same independent variables

),( NETT . In Eq.(2), ),( NSEE . So that ),( NSTT . The definition of temperature is the

same in both cases, but it is expressed as a function of different independent variables.

REFERENCES

C. Kittel and H. Kroemer, second edition (W.H. Freeman and Company, 1980).

D.S. Lemons, Mere Thermodynamics, second edition (The John Hopkins University Press, 2009).

3. Canonical ensemble(system with constant temperature)

The theory of the micro-canonical ensemble is useful when the system depends on N, E, and

V. In principle, this method is correct. In real calculations, however, it is not so easy to calculate

the number of states W(E, E) in general case. We have an alternative method, which is much

useful for the calculation in the real systems. The formulation of the canonical ensemble is a little

different from that of the micro-canonical ensemble. Both of these ensembles lead to the same

result for the same macro system.

Canonical ensemble: (N, T, V, constant)

Suppose that the system depends on N, T, and V. A practical method of keeping the temperature

of a system constant is to immerse it in a very large material with a large heat capacity. If the

material is very large, its temperature is not changed even if some energy is given or taken by the

system in contact. Such a heat reservoir serves as a thermostat.

Fig. System one (one quantum state) and system II (thermal bath, reservoir).

We consider the case of a small system S(I) in thermal contact with a heat reservoir (II). The

system S(I) is in thermal equilibrium with a reservoir W(II). S(I) and W(II) have a common

temperature T. The system S(I) is a relatively small macroscopic system. The energy of S(I) is not

fixed. It is only the total energy of the combined system.

iIIT EEE

We assume that WII(EII) is the number of states where the thermal bath has the energy EII. If S(I)

is in the one definite state i , the probability of finding the system (I) in the state i , is

proportional to WII(EII). The thermal bath is in one of the many states with the energy ET - Ei

)()( iTIIIIIIi EEWEWp

or

ln ln[ ( )]i II T ip W E E +const

Since

iT EE

)](ln[ iTII EEW can be expanded as

iE

II

IIIITIIiTII E

dE

EWdEWEEW

T|

)(ln)(ln)](ln[

________________________________________________________________________

((Note)) Taylor expansion

( ) ln[ ( )] ln[ ( )] ln[ ( )]II II II T i II Tf x W E W E E W E x

with

i Tx E E ≪

0( ) (0) |1!

ln[ ( )]ln[ ( )] |

T

x

II IIII T E i

II

x ff x f

x

W EW E E

E

where

(0) ln[ ( )]II Tf W E

ln[ ( )]

ln[ ( )]

ln[ ( )]

IIII II

II

IIII II

i II

II II

II

EfW E

x x E

E dW E

E dE

W E

E

________________________________________________________________________

Then we obtain

]|)(ln

exp[ iE

II

IIIIi E

dE

EWdp

T (1)

Here we notice the definition of entropy and temperature for the reservoir as the micro-canonical

ensemble:

)(ln IIIIBII EWkS

and

IIII

II

TE

S 1

or

IIBII

II

BII

IIII

TkE

S

kdE

EWd 11)(ln

In thermal equilibrium, we have

TTT iII

Then Eq.(1) can be rewritten as

)exp()exp( i

B

ii E

Tk

Ep

where = 1/(kBT). This is called a Boltzmann factor. We define the partition function Z as

i

E

CieZ

)(

Quantum mechanical approach

Fig. Canonical ensemble

Here we use the quantum mechanical description to explain the canonical ensemble.

iEiH iˆ (Eigenstate in the quantum mechanics, quantum state)

The probability is given by

2

ipi

where H is the Hamiltonian of the system. The letter CZ is used because the German name is

“Zustandssumme.” (sum over states). The probability is expressed by

iE

C

i eZ

Ep

1)(

The summation in CZ is over all states i of the system. We note that

1)( i

iEp

The average energy of the system is given by

C

i

E

i

C

ZeE

ZEU i

ln1

since

i

E

i

Ci

E

i

C

C

C

C ii eEZ

eEZ

Z

Z

Z

1

)(11ln

Note that

T

ZTk

T

Z

T

ZU C

BCC

lnln1ln 2

.

since

21Tk

T

T

T

TB

In summary, the representative points of the system I are distributed with the probability density

proportional to exp(-Ei). This is called the canonical ensemble, and this distribution of

representative points is called the canonical distribution. The factor exp(-Ei) is often referred to

as the Boltzmann factor. The energy Ei is dependent on T.

((Note)) R. Baierlein Thermal Physics

iE

C

i eZ

Ep

1)(

This probability distribution, perhaps the most famous in all of thermal physics, is called the

canonical probability distribution, name introduced by J. Willard Gibbs in 1901. (The adjective

“canonical” is used in the sense of “standard.)

((Example)) Blundel-Blundell Problem 4-3

Blunde-Blundell Problem 4-3

((Solution))

We consider the two states with energy 0 and energy . Suppose that the system consists of

N particles. Each particle takes one of these two energy states.

21 NNN

The total energy of the system is

rNNE 210

Then we have

rN 2 , rNN 1

The number of ways is

!( ) ( )

( )! !

NE r r

N r r

For small amount of s )( rs

( ) [( ) ]

!

( )!( )!

! ( )! !( )

( )!( )! !

( )! !( )

( )!( )!

( 1)( 2)....( 1)( )

( 1)( 2)...( )

( )( )

s

s

E r s

N

N r s r s

N N r rE r

N r s r s N

N r rE r

N r s r s

r r r sE r

N r N r N r s

rE r

N r

or

]lnexp[)()(

r

rNEE

(1)

Note that

S

s

rN

rX

)( ,

r

rNs

rNsrsX

ln

)ln(lnln

leading to the expression

]lnexp[]lnexp[

r

rN

r

rNsX

Using the Taylor expansion, we have

TkE

dE

EdEE

B

)(ln

)(ln)(ln)(ln

or

)exp()()(Tk

EEB

(2)

From Eqs.(1) and (2), we get

r

rN

TkB

ln11

or

)1

ln(

1

ln

1

x

x

r

rN

Tky B

with

N

rx .

Using the Mathematica, we make a plot of y as a function of x.

Fig. Plot of

Tk

y B as s function of N

rx .

__________________________________________________________________________

4. Canonical ensemble: Boltzmann factor

(Kittel, Thermal Physics)

We consider the case of a small system S in thermal interaction with a heat reservoir R. We

assume that the system S is in thermal equilibrium with a reservoir R. S and R have a common

temperature T (given). The total system (R+S) is a closed system. We assume weak interaction

between R and S. So that their energies are additive. The energy of the system S is, of course, not

fixed. The total energy of the combined system (R+S) is constant E0. The energy conservation can

be written as

totalR EEE

x r N

y kBT

0.2 0.4 0.6 0.8 1.0

6

4

2

2

4

The system is in a quantum state 1, and the reservoir R has

1( )R tg E E

states accessible to it. The system is in a quantum state 2, and the reservoir R has

2( )R tg E E

states accessible to it.

R

SEt E1

g Et E1

E1

one state

R

SEt E2

g Et E2

E2

one state

Fig. The system is in quantum state 1E . The reservoir has 1( )R tg E E accessible quantum

states

The probability )(EP is proportional to the number of accessible states of the reservoir when

the state of the system is exactly specified. If we specify the state of S, the number of accessible

states of R:

1)()( RSR gg

Thus the probability )( sEP is proportional to the number of accessible states of the reservoir,

( ) ( )s R t sP E g E E .

Thus

11

2 2

( )( )

( ) ( )

R t

R t

g E EP E

P E g E E

.

By definition of entropy,

1 1( ) ln[ ( )]R t B R tS E E k g E E

2 2( ) ln[ ( )]R t B R tS E E k g E E

So that the probability ratio may be written as

1

1

22

1exp[ ( )]

( ) 1exp[ ]

1( )exp[ ( )]

R t

BR

BR t

B

S E EP E k

SP E k

S E Ek

.

Here the entropy difference RS is

1 2

1

2

1 2

( ) ( )

( )

[ ( ) ]

( )

R R t R t

RR t

t

RR t

t

R

t

S S E E S E E

SS E E

E

SS E E

E

SE E

E

We know that

tE

S

T

1.

Then the entropy difference is

1 2

R

E ES

T

.

Finally we have

)exp(

)exp(

)(

)(

2

1

2

1

E

E

EP

EP

.

with

TkB

1 .

A term of )exp(Tk

E

B

is called a Boltzmann factor. It gives the ratio of the probability of finding

the system in a single quantum state 1E to the probability of finding the system in a single

quantum state 2E .

5. Pressure

The pressure P is defined as

V

Z

V

Z

Ze

V

E

Ze

ZPP CC

i C

Ei

C

E

Ci

iii

ln111)(

11

Here we define the Helmholtz free energy F as

STEF .

SdTPdV

TdSSdTPdVTdS

TdSSdTdEdF

F is a function of T and V; ),( VTFF . From the equation of dF, we have

T

V

V

FP

T

FS

((Note)) Notation A for the Helmholtz free energy

The Helmholtz free energy was developed by Hermann von Helmholtz, a German physicist, and

is usually denoted by the letter A (from the German “Arbeit” or work), or the letter F. The IUPAC

recommends the letter A as well as the use of name Helmholtz energy.

6. Helmholtz free energy and entropy

The Helmholtz free energy F is given by

lnB CF k T Z

((Proof))

We note that

CB ZT

kT

U

T

FST

T

FT

FT

T

F

Tln)(

222

,

which leads to

lnB CF k T Z

What is the expression for the entropy S in a canonical ensemble? The entropy is given by

T

FUS

where U is the average energy of the system,

CZU

ln

Then entropy S is rewritten as

i

iiB

CBi

i

CiB

CBi

i

iB

CB

C

E

i

iB

CB

i

E

i

C

CBC

ppk

ZkpZpk

ZkpEk

ZkZ

eEk

ZkeEZT

ZkZ

TS

i

i

ln

ln)lnln(

ln

ln

ln11

lnln1

or

i

iiB ppkS ln ,

where pi is that the probability of the i state and is given by

iE

C

i eZ

p

1

The logarithm of pi is described by

Cii ZEp lnln

Here we have

CB

CB

i

CiB

i

iB

ZkT

U

ZUk

ZEpkppkS

ln

)ln(

)ln(ln 11

or

CB ZTkUTS ln

or

CB ZTkSTUF ln

((Note))

We finally get a useful expression for the entropy which can be available for the information

theory.

i

iiB ppkS ln .

((Example))

Suppose that there are 50 boxes. There is one jewel in one of 50 boxes. pi is the probability of

finding one jewel in the i-th box for one trial.

(a) There is no hint where the jewel is.

p1 = p2 = ….. = p50=50

1

50

1

91.3)50

1ln(

50

1ln

s

BB

s

ssB kkppkS

(b) There is a hint that the jewel is in one of the box with even number.

p1 = p3 = ….. = p49=0

p2 = p4 = ….. = p50=25

1

evens

BB

s

ssB kkppkS 219.3)25

1ln(

25

1ln

(c) If you know that the jewel is in the 10-th box,

p10=1

ps = 0 (s ≠ 10)

0ln 1010 ppkS B

If you know more information, the information entropy becomes smaller.

7. The method (Fermi)

I found very interesting method for the derivation of the expression entropy S and the

Helmholtz free energy F in the book written by Enrico Fermi, one of the greatest scientists in the

world.

E. Fermi, Notes on Thermodynamics and Statistics (University of Chicago, 1966).

According to his book, we start with the expression

( )dU TdS PdV d TS SdT PdV

or

( )d U TS dF SdT PdV

with

F U ST or U F

ST T

Note that

1 PdS dU dV

T T

At constant V, we get

1[ ( ) ]B BdS dU k dU k d U Ud

T

Using the relation

lnU Z

we have

1

[ ( ) ]

B

B

S dUT

k dU

k d U Ud

or

(ln )

ln

ln

B B

B B

B

S k U k Z d

k U k Z

Uk Z

T

or

lnB

F US k Z

T T

The Helmholtz free energy is

1ln lnBF k T Z Z

The entropy S is

2 1[ln ln ] ( ln )B BS k Z Z k Z

This expression of S can be also derived as

2

2 1ln

V

V

B

V

B

FS

T

F

T

Fk

k Z

7. Application

7.1. Partition function ZC for ideal gas system

The partition function Z for the ideal gas can be calculated as

2/3

2

32

3

)2

(!

])2

exp([!

NB

N

N

B

N

N

CN

h

Tmk

N

V

Tmk

pdp

hN

VZ

where !N is a factor coming from indistinguishable particles (Gibbs paradox). Note that

NCCN ZN

Z 1!

1 .

1CZ is the one-particle partition function.

2/3

2

3

2

3

222

3

222

31

)2

(

])2

exp([

])2

exp()2

exp()2

exp([

)2

exp([

h

TmkV

Tmk

pdp

h

V

Tmk

pdp

Tmk

pdp

Tmk

pdp

h

V

Tmk

pppdpdpdp

h

VZ

B

B

xx

B

zz

B

y

y

B

xx

B

zyx

zyxC

In other words, many-particle problem reduces to one particle problem. Using this expression of

CNZ , the Helmholtz free energy F can be calculated as

]1)2

ln(2

3)ln([ln

2

h

Tmk

N

VTNkZTkF B

BCNB

.

The internal energy E is

TNkZ

U BCN

2

3ln

.

The entropy S is

]2

5)

2ln(

2

3)[ln(

2

ℏTmk

N

VNk

T

FES B

B.

The pressure P is

V

TNk

V

FP B

T

)( . (Boyle’s law).

or

UPV3

2

7.2. Partition function ZC for photon gas system

We consider the photon gas system with the energy dispersion cp . The N-photon partition

function is given by

NCCN ZN

Z 1!

1 .

1CZ is the one-photon partition function and is given by

3

33

0

2

31

)(

8

)(

!24

4

ch

V

ch

V

epdph

VZ cp

C

the Helmholtz free energy F can be calculated as

]1)

8ln()ln(3[ln

!ln

ln

33

1

chTk

N

VTNk

N

ZTk

ZTkF

BB

N

CB

CNB

.

where

]1)8

ln()ln(3[lnln33

chTk

N

VNZ BCN

.

The internal energy E is

ln3CN

B

ZU Nk T

.

The pressure P is

V

TNk

V

FP B

T

)( . (Boyle’s law).

or

UPV3

1 .

7.3 Maxwell’s distribution function

The Maxwell distribution function can be derived as follows.

dvvTk

mvAdvvfdvvvndn

B

22

2 4)2

exp()(4)()( vv

The normalization condition:

1)(4)()( 2 dvvfdvvvndn vv

The constant A is calculated as

2/3

2

Tk

mA

B.

Then we have

)2

exp(2

4)(2

2

2/3

Tk

mvv

Tk

mvf

BB

.

Since M = m NA and R = NA kB, we have

)2

exp(42

)(2

2

2/3

RT

Mvv

RT

Mvf

which agrees with the expression of f(v) in Chapter 19.

((Mathematica))

8. Comparison of the expression of S in the canonical ensemble with the original

definition of S in the microcanonical ensemble

The partition function Z can be written as

dEeEeZ E

i

E

Ci

)(

The partition function ( )CZ is the Laplace transform of the density of states, )( . The density

of states )( is related to the partition function ( )CZ , through an inverse Laplace transform,

*

*

1( ) ( )

2

i

E

C

i

E e Z di

(Bromwich-Wagner integral)

where * 0 .

Here we define the function () by

( ) ( ) EE E e .

The function Ee decreases with increasing E while ( )E increases with increasing E.

f1 = IntegrateBExpB−m v2

2 kB TF 4 π v2, 8v, 0, ∞<,

GenerateConditions −> FalseF

2 2 π3ê2

J m

kB TN3ê2

eq1 = A f1 � 1;

Solve@eq1, AD

::A →

J m

kB TN3ê2

2 2 π3ê2>>

We assume that ( )E has a local maximum at *E E

* *

2* * * 2

2

ln ( ) 1 ln ( )ln ( ) ln ( ) ( ) | ( ) | ...

2!E E E E

E EE E E E E E

E E

We choose *E and E such that

*

ln ( )|E E

E

E

=0,

*

2

22

ln ( ) 1|

2E E

E

E

E

Thus we have

* * 2

2

1ln ( ) ln ( ) ( )

2 E

E E E E

() can be approximated by a Gaussian function

])(2

)(exp[)()(

2

2**

E

EEEE

where

* * *( ) ( ) exp( )E E E

e E E

E E

E

Fig. () vs . () has a Gaussian distribution with the width E around *EE

Since

ln ( ) ln ( )d E d E

dE dE

we have

* *

ln ( ) ln ( )| | 0E E E E

d E d E

dE dE

or

*

ln ( )|E E

d E

dE

Here we define the number of states ),( EEW by

* * *( , ) ( ) 2 ( )EW E E E E E

with

*2 EE .

Then we have

*|

),(lnEEdE

EEWd (1)

since

*ln ( , ) ln ( ) lnW E E E E

** |

)(ln|

),(lnEEEE E

E

E

EEW

with fixed E . We note that

)exp()(2

2)(

])(2

)(exp[)(

)(

**

*

0

2

2**

EE

E

dEEE

E

dEEZ

E

E

E

C

where

* 2

2

0

1 ( )exp[ ] 1

2( )2 EE

E EdE

(Gaussian distribution)

Then we have

* *ln ln[ 2 ( )]C EZ E E

We note that

lnB CF k T Z ,

CC

C

ZZ

ZEE

ln1*

The entropy S is calculated as

),(ln

)](2ln[

)](2ln[

lnln1

*

**

*

EEWk

Ek

T

EEk

T

E

ZkZ

T

T

F

T

ES

B

EB

EB

CBC

(2)

Using Eqs.(1) and (2), we get

Tk

E

SB

1

or

TE

EEW 1),(ln

In other words, the thermodynamic properties derived from the canonical ensemble is equivalent

to those from the microcanonical ensemble. Since the calculations for the microcanonical

ensemble is much more complicated compared to those for the canonical ensemble, it is suggest

that one may choose the method of canonical ensemble if it is allowed.

((Note-1)) Expression for )(E

)(])(2

)(exp[)()(

2

2***

EeEE

EeE E

E

E

or

])(2

)()(exp[)()(

2

2***

E

EEEEEE

This function takes a maximum at

* 2

EE E .

9. Boltzmann-Planck’s method

Finally we show the standard method of the derivation, which characterizes well the theory of

canonical ensembles.

Fig. Canonical ensembles with the states (E1, E2, …). ...21 EEEtot = constant. M1

ensembles for the energy E1, ,M2 ensembles for the energy E2, and so on. In general, Mi

ensembles in the energy level Ej.

We consider the way of distributing M total ensembles among states with energies Ej. Let Mj

be the number of ensembles in the energy level Ej; M1 ensembles for the energy E1, the M2

ensembles for the energy E2, and so on. The number of ways of distributing M ensembles is given

by

!...!

!

21 MM

MW

where

MMj

j

and the average energy E is given by

j

j

j

j

jjM

MEEEPE )(

We note that the probability of finding the system in the state j is simply given by

M

MEP

j

j )( .

The entropy S is proportional to lnW,

j

jMMW )!ln(!lnln

Using the Stirling's formula

j

jj

j

jj

MMMM

MMMMW

lnln

)1(ln)1(lnln

in the limit of large M and Mj. Then we have

j

jj

j

jj

j

jj

j

jj

EPEP

MEPEPM

EMPEMPM

M

MMM

MWM

)](ln[)(

]ln)(ln[)(ln

)](ln[)(1

ln

ln1

lnln1

which is subject to the conditions

1)( j

jEP , EEPEj

jj )(

Treating P(Ej) as continuous variables, we have the variational equation

0)}()1()({ln

)()()(ln)([

j

jjj

j

jjjjj

EPEEP

EPEEPEPEP

which gives P(Ej) for the maximum W. Here and are Lagrange’s indeterminate multipliers.

Thus we obtain

0)1()(ln jj EEP

or

( ) exp[ ]j jP E C E

or

)exp()(

1)( jj E

ZEP

where

j

jEZ )exp()(

and

= 1/kBT.

With the above )( jEP , the entropy S is expressed by

j

jjB

B

EPEPMk

WkS

)](ln[)(

ln

for the total system composed of M ensembles. Therefore, the entropy of each ensemble is

j

jjB EPEPkS )](ln[)(

10. Density of states for quantum box (ideal gas)

(a) Energy levels in 1D system

We consider a free electron gas in 1D system. The Schrödinger equation is given by

)()(

2)(

2)(

2

222

xdx

xd

mx

m

pxH kk

kkk

ℏ, (1)

where

dx

d

ipℏ

,

and k is the energy of the particle in the orbital.

The orbital is defined as a solution of the wave equation for a system of only one

electron:one-electron problem.

Using a periodic boundary condition: )()( xLx kk , we have

ikx

k ex ~)( , (2)

with

22

22 2

22

nLm

km

k

ℏℏ,

1ikLe or nL

k2

,

where n = 0, ±1, ±2,…, and L is the size of the system.

(b) Energy levels in 3D system

We consider the Schrödinger equation of an electron confined to a cube of edge L.

kkkkk

p 2

22

22 mmH

ℏ. (3)

It is convenient to introduce wavefunctions that satisfy periodic boundary conditions. Boundary

condition (Born-von Karman boundary conditions).

),,(),,( zyxzyLx kk ,

),,(),,( zyxzLyx kk ,

),,(),,( zyxLzyx kk .

The wavefunctions are of the form of a traveling plane wave.

rk

k r ie)( , (4)

with

kx = (2/L) nx, (nx = 0, ±1, ±2, ±3,…..),

ky = (2/L) ny, (ny = 0, ±1, ±2, ±3,…..),

kz = (2/L) nz, (nz = 0, ±1, ±2, ±3,…..).

The components of the wavevector k are the quantum numbers, along with the quantum number

ms of the spin direction. The energy eigenvalue is

22

2222

2)(

2)( kk

mkkk

mzyx

ℏℏ . (5)

Here

)()()( rkrrp k kkki

ℏℏ

. (6)

So that the plane wave function )(rk is an eigen-function of p with the eigenvalue kℏ . The

ground state of a system of N electrons, the occupied orbitals are represented as a point inside a

sphere in k-space.

(c) Density of states

Because we assume that the electrons are non-interacting, we can build up the N-electron

ground state by placing electrons into the allowed one-electron levels we have just found. The one-

electron levels are specified by the wave-vectors k and by the projection of the electron’s spin

along an arbitrary axis, which can take either of the two values ±ħ/2. Therefore associated with

each allowed wave vector k are two levels:

,k , ,k .

Fig. Density of states in the 3D k-space. There is one state per (2/L)3.

There is one state per volume of k-space (2/L)3. We consider the number of one-electron

levels in the energy range from to +d; D()d

dkkL

dD 2

3

3

42

)(

, (13)

where D() is called a density of states.

11. Application of canonical ensemble for ideal gas

(a) Partition function for the system with one atom; 1CZ

The partition function ZC1 is given by

2/3

2

22

2

3

22

3

22

1

8

)2

exp(4)2(

)2

exp()2(

)2

exp(

CV

km

dkkV

md

V

mZ

k

C

ℏ

ℏ

ℏ

kk

k

where 3LV ,

mC

2

2ℏ

, TkB

1

((Mathematica))

Then the partition function 1CZ can be rewritten as

VnTmk

V

Tmk

V

Tmk

VZ Q

B

BB

C

2/3

22/32

2/32

2

122

28

ℏℏℏ

where Qn is a quantum concentration and is defined by

Clear "Global` " ;

f1V

2 34 k

2Exp C1 k

2;

Integrate f1, k, 0,

Simplify , C1 0 &

V

8 C13 2 3 2

2/3

22

ℏTmk

n BQ

.

nQ is the concentration associated with one atom in a cube of side equal to the thermal average de

Broglie wavelength.

Fig. Definition of quantum concentration. The de Broglie wavelength is on the order of

interatomic distance.

ℏ2

h

vmp ,

vm

ℏ

2 ,

where v is the average thermal velocity of atoms. Using the equipartition law, we get the

relation

Tkvm B2

3

2

1 2 , or

m

Tkv B3 ,

Then we have

3/13/1

2 11447.1

2

3

2

3

2

3

22

QQBBB nnTmkTmk

m

Tkm

vm

ℏℏℏℏ

where

2/3

22

ℏTmk

n BQ

It follows that

3

1

Qn .

((Definition))

1Qn

n → classical regime

An ideal gas is defined as a gas of non-interacting atoms in the classical regime.

((Example)) 4He gas at P = 1 atm and T = 300 K, the concentration n is evaluated as

1910446.2 Tk

P

V

Nn

B

/cm3.

The quantum concentration nQ is calculated as

24108122.7 Qn /cm3

which means that Qnn in the classical regime. Note that the mass of 4He is given by

24106422.64 um g.

where u is the atomic unit mass.

((Mathematica))

(b) Partition function of the system with N atoms

Suppose that the gas contains N atoms in a volume V. The partition function ZN, which takes

into account of indistinguishability of the atoms (divided by the factor N!), is given by

!

1

N

ZZ

N

N .

Using VnZ Q1 , we get

]ln1)[ln(

!ln)ln(ln

NVnN

NVnNZ

Q

QN

where we use the Stirling’s formula

)1(lnln! NNNNNN ,

Clear "Global` " ;

rule1 kB 1.3806504 1016,

NA 6.02214179 1023,

1.054571628 1027,

amu 1.660538782 1024,

atm 1.01325 106;

T1 300; P1 1 atm . rule1;

m1 4 amu . rule1

6.64216 1024

nQm1 kB T1

2 2

3 2

. rule1

7.81219 1024

n1P1

kB T1. rule1

2.44631 1019

in the limit of large N. The Helmholtz free energy is given by

]1)2

ln(2

3ln

2

3[ln

)ln(

]1)[ln(

]ln1)[ln(

ln

2

ℏB

B

B

Q

B

Q

B

QB

NB

mkT

N

VTNk

TNkn

nTNk

N

VnTNk

NVnTNk

ZTkF

since

12

ln2

3ln

2

3ln]

2ln[)ln(

2

2/3

2

ℏℏ

BBQ mkT

N

V

N

VTmk

N

Vn

The entropy S is obtained as

)ln2

3(ln

]2

5)

2ln(

2

3ln

2

3[ln

0

2

TN

VNk

mkT

N

VNk

T

FS

B

BB

V

ℏ

where

)2

ln(2

3

2

520ℏ

Bmk

Note that S can be rewritten as

B

Q

B Nkn

nNkS

2

5ln (Sackur-Tetrode equation)

((Sackur-Tetrode equation))

The Sackur–Tetrode equation is named for Hugo Martin Tetrode (1895–1931) and Otto

Sackur (1880–1914), who developed it independently as a solution of Boltzmann's gas statistics

and entropy equations, at about the same time in 1912.

https://en.wikipedia.org/wiki/Sackur%E2%80%93Tetrode_equation

In the classical region ( 1Qn

n or 1

n

nQ), we have

0ln n

nQ

The internal energy E is given by

TNk

TNkn

nTNkTNk

n

nTNk

STFE

B

B

Q

BB

Q

B

2

3

2

5ln)ln(

Note that E depends only on T for the ideal gas (Joule’s law). The factor 3/2 arises from the

exponent of T in Qn because the gas is in 3D. If Qn were in 1D or 2D, the factor would be 1/2 or

2, respectively.

(c) Pressure P

The pressure P is defined by

V

TNk

V

FP B

T

leading to the Boyle’s law. Then PV is

3

2ETNkPV B (Bernoulli’s equation)

(d) Heat capacity

The heat capacity at fixed volume V is given by

B

V

V NkT

STC

2

3

.

When ANN , we have

RCV2

3 .

Cp is the heat capacity at constant P. Since

PdVTdSdE

or

PdVdETdS

then we get

ppp

pT

VP

T

E

T

STC

RkNP

kNP

T

VP BA

BA

p

We note that

TkNE BA2

3 .

E is independent of P and V, and depends only on T. (Joule’s law)

VBA

Vp

CkNT

E

T

E

2

3

Thus we have

RRRRCC VP2

5

2

3 .

((Mayer’s relation))

RCC VP for ideal gas with 1 mole.

The ratio is defined by

3

5

V

P

C

C .

(e) Isentropic process (constant entropy)

The entropy S is given by

]ln)[ln()ln2

3(ln 0

2/3

0 NVTNkTN

VNkS BB

The isentropic process is described by

2/3VT =const, or 3/2TV =const,

Using the Boyle’s law ( RTPV ), we get

3/2VR

PV=const,, or 3/5PV =const

Since = 5/3, we get the relation

PV = constant

12. The expression of entropy: )(ln EWkS B

The entropy is related to the number of states. It is in particular, closely related to Wln . In

order to find such a relation, we start with the partition function

dEfN

EEWdE

EEWdE

EEdEW

EZC

]exp[

])(exp[ln

)exp()](exp[ln

)exp()(

)exp(

where )(EW is the number of states with the energy E. The function )(Ef is defined by

)](ln[1)(ln

)( EWTkENN

EWEEf B

In the large limit of N, )(Ef is expanded using the Taylor expansion, as

...)(|)(

)()( ***

EEE

EfEfEf

EE

where

0])(ln

1[1)(

E

EWTk

NE

EfB

at *EE

or

*|)(ln1

EEBE

EWk

T

At *EE ,

*exp[ ( )]CZ N f E

For simplicity, we use E instead of *E . The Helmholtz free energy F is dsefined by

ln [ ( )] ( )B C BF k T Z k T N f E Nf E

or

STEEWTkEF B )(ln

leading to the expression of the entropy S as

)(ln EWkS B ,

and

E

S

T

1

.

13. The expression of entropy:

ppkS B ln

We consider the probability given by

Ee

Zp

1

,

where

EeZ ,

EZp lnln ,

The energy E is given by

Z

Z

Z

eEZ

EpU

E

ln

1

1

The entropy is a logarithmic measure of the number of states with significant probability of

being occupied. The Helmholtz energy F is defined by

ZTkSTUF B ln .

The entropy S is obtained as

T

UZk

T

FUS B

ln

We note that

S

T

FE

ZT

Tk

T

E

ZEk

Zpkppk

B

B

BB

ln

)ln(

)ln(ln

Thus it follows that the entropy S is given by

ppkS B ln .

14. Thermal average of energy fluctuation

EU

2

2

222 )(11

EEeE

ZeE

ZEE

][1

])([11

]1

[1

1

22

2

2

22

2

2

EETk

eEd

dZeEZ

ZTk

eEZd

d

Tk

Ed

d

Tk

Ed

d

dT

dE

dT

d

B

EE

B

E

B

B

where

ZUEZeEd

dZ E

Then we have

][1 22

2EE

TkU

dT

d

B

Since VCU

dT

d , we get the relation

][)(

1 22

2EE

Tkk

C

BB

V

15. Example: 4He atom as ideal gas

We consider the 4He atom with mass

um 4 = 6.64216 x 10-24 g

The number density n at at P = 1 atm and T = 300 K, is

n = 2.44631 x 1019/cm3

The number of atoms in the volume V = 103 cm3 is

N = nV = 2.44631 x 1022

The internal energy

987.1512

3 TNkU B

J

The entropy S

]2

5)

2ln(

2

3ln

2

3[ln

2

ℏB

B

mkT

N

VNkS = 5.125 J/K.

((Mathematica))

Clear "Global` " ;

rule1 kB 1.3806504 1016,

NA 6.02214179 1023,

1.054571628 1027,

amu 1.660538782 1024,

atm 1.01325 106, bar 10

6, J 10

7;

n1 2.44631 1019; V1 10

3; T1 300;

N1 n1 V1

2.44631 1022

m1 4 amu . rule1

6.64216 1024

P1kB N1

V1T1 . rule1

1.01325 106

16. Link

Entropy (Wikipedia)

http://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)

_______________________________________________________________________

REFERENCES

M. Toda, R. Kubo, and N. Saito, Statistical Physics I: Equilibrium Statistical (Springer, 1983).

L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon Press, 1980).

R.P. Feynman, Statistical Mechanics A Set of Lectures (The Benjamin/Cummings, 1982).

E. Fermi, Thermodynamics (Dover, 1956).

S.J. Blundell and K.M. Blundel, Concepts in Thermal Physics (Oxford, 2006).

C. Kittel and H. Kroemer, Thermal Physics (W.H. Freeman and Company (1980).

C. Kittel, Elementary Statistical Physics (Wiley & Sons, 1958).

P. Atkins, The Laws of Thermodynamics: A Very Short Introduction (Oxford, 2010).

D.V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley, 2000).

R. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).

F. Bloch, Fundamentals of Statistical Mechanics (World Scientific, 2000).

P1 bar . rule1

1.01325

E13

2kB N1 T1 . rule1

1.51987 109

E1 J . rule1

151.987

S1 kB N13

2Log T1 Log

V1

N1

3

2Log

m1 kB

2 2

5

2. rule1

5.12503 107

S1 J . rule1

5.12503

D. ter Haar, Elements of Statistical Mechanics (Butterworth Heinemann, 1995).

F. Schwabl, Statistical Mechanics, 2nd edition (Springer, 2006).

J.W. Halley, Statistical Mechanics (Cambridge, 2007).

H. Kawamura, Statistical Physics (Maruzen, 1997) [in Japanese]

D. Yoshioka, Statistical Physics, An Introduction (Springer, 2007).

______________________________________________________________________________

APPENDIX

Partition Zustandssumme

In the classical mechanics, the partition function Z is given by

])(

exp[1

),,(3 Tk

Hd

hTVNZ

B

NN

N (phase integral)

where )( NH is the Hamiltonian in the phase space.

Laplace transformation

0

),,(),,(),,( dEeEVNWeEVNWeTVNZ E

E

E

i

E i

Thus the partition function is the Laplace transform of the density of states ),,( EVNW

Separation of Z;

Suppose the energy of the system is a sum of subsystems which are independent to each

other

)3()2()1(

iiii EEEE

The partition function is obtained as the product of each Z

)3()2()1( ZZZZ

where

i

EieZ)1(

)1( ,

i

EieZ)2(

)2( ,

i

EieZ)3(

)3(

Calculation of mean values in a canonical ensemble

i

E

CieZ

i

E

iC ieE

Z

,

i

E

i

i

E

iC ii eEeE

Z

22

2

2

)(

CC

Ci

E

i

C

ZZ

ZeE

ZUE i

ln11,

2

222 11

C

Ci

E

i

C

Z

ZeE

ZE i

2

2

2

22

2

22

22

ln

11

)(

C

C

C

C

Z

Z

Z

Z

Z

EE

EEE

or

CTkUT

TkUE BB

222

where C is the heat capacity.

Similarly we have

n

C

nnn Z

E

ln)1(

Since 2E can never be negative, it follows that 0

E

(or equivalently, that

0

ET

).

NCTkE B 22, NE

NN

N

E

E 1

In the limit of N (thermodynamic limit), the energy fluctuation becomes zero.

((Adiabatic approximation))

______________________________________________________________________

Derivation of entropy

i

iiB ppkS ln

iE

C

i eZ

p

1, 1

i

ip

The entropy S is defined by

CB

CBB

p

iCiiB

p

iiB

ZkET

ZkEk

pZEpk

ppkS

ln1

ln

ln)ln(

ln

where

CC

Ci

E

i

C

ZZ

ZeE

ZU i

ln11

We define the Helmholtz free energy

CB ZTkSTUF ln