canonical ensemble masatsugu sei suzuki department of
TRANSCRIPT
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