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Quantum Statistics and Bose-Einstein Condensation
Let us consider a system consisting of non-interacting molecules, denoted by a,
b, c …., whose energy levels are given by Ea, Eb, Ec, …In the classical statistical
mechanical treatment of ideal gases, the particles/molecules/atoms are assumed to be
distinguishable. Therefore, every possible occupation of energy levels by particles is
admissible. Hence the following decomposition of the Hamiltonian, total energy and the
partition function holds good.
H  H a  H b  H c  .....
E  Ea  Eb  Ec  ....
qa 
e
 Ea
k BT
a
 Eb
qb 
 e k BT
b
Q  qa qb qc .....
a
where a, b, c etc are the particle indices.
b
I
a
b
II
Owing to the distinguishability of the particles ‘a’ and ‘b’, arrangements I and II are
different and each should be counted separately.
However, when particles are indistinguishable , the arrangements I and II are one
and the same and hence should be counted only as one arrangement. Evaluating the total
partition function as
Q  qa qb qc .....
is incorrect as it has those extra terms.
……………………………………………………………………………………………....
Justification:
I. Take five balls that are distinguishable.
1
2
3
4
5
The total number of ways in which the balls can be arranged in three levels such that
2 balls in level 1
2 balls in level 2
1 ball in level 3
are
5!
 30
2!2!1!
Generically,
N!
where n1 , n2 , n3 ... are the numbers of balls occupying levels 1, 2,
n1! n 2 ! n3 !..
3.. etc respectively.
II. Supposing the balls were indistinguishable,
Then the number of ways is which the above arrangement can be done is just 1 !
Supposing an arrangement with only one ball occupying each level is desired, then
again there only one possible way to obtain it in the indistinguishable balls case. Whereas
with distinguishable balls there are N! possible ways of obtaining this arrangement.
Therefore, classical statistics vastly over-estimates the number of configurations. In
classical statistical mechanics, this is of course taken into account by the 1/N! terms
(Boltzmann factor) in front of the partition function. This method does not work at low
temperature where quantum nature of the particles become important that impose further
restrictions on the occupancy of the energy levels of the system. In a system of Bose
particles, any number of particles can occupy any given energy level. On the other hand,
particles obeying Fermi-Dirac statistics, maxium of one particle can occupy a given
energy level. That is, the occupation number of a given energy level can be 1 or 0.
………………………………………………………………………………………………
High Temperature Limit
When the temperature is high, a system has a large number of accessible energy
states such as translational, rotational, electronic…. The number of states is much more
than the number of atoms/molecules. At high temperatures, a large number of states
become accessible, especially the translational states. In such a case, the possibility of
multiple occupancy of a state will be greatly reduced. Thus the occupied energy levels
are mostly
non-degenerate. Thus, the occupation due to quantum statistics is not
important and classical Boltzmann statistics works well. This is exactly like case II in
justification. So the surplus contributions that will feature for the indistinguishable
particles in the expression
Q  qa qb qc .....
N! compared to distinguishable particles.
For an ideal gas the atoms/molecules are independent (negligible intermolecular
forces) and indistinguishable. In the expression
Q  qN
there are N! surplus contributions in the thermodynamic limit for an ideal gas. Thus the
exact partition function for an ideal gas in this limit is
qN
Q
 (1)
N!
In the thermodynamic limit, n3  1 (where n is the number density,  is the
1
 h2  2
 and n3 is a dimensionless quantity) because
de-Broglie wavelength,   
 2mkBT 
m >> 1, T >> 1 and n << 1. This is the classical regime where Maxwell-Boltzmann
statistics holds good and indistinguishability can be exactly accounted for. But as we go
on to low temperatures and high densities multiple occupancy of states increases.
n3  1 is in this limit. Under such conditions, the indistinguishability of particles
becomes much more significant. This is where the quantum effects creep in and there is
need for a revised statistics.
Quantum Statistics
All known indistinguishable particles, fall in to the class of either Fermions or
Bosons. These two statistics or distributions lay different constraints on the occupancy of
a particular state but under the conservation of two quantities, the total number of
particles and the total energy.
n
i
N
i
n 
i
i
E
i
These constraints become very significant when there is degenerate/multiple occupancy
of energy levels.
To understand the effects of these statistics on the distribution of energy levels, we first
group the energy levels with energy within a small interval together and assign a
degeneracy factor to each group according to the number of energy levels in each group.
Let us denote the energy and degeneracy of i-th group by εi and gi , respectively. The
picture below illustrates the grouping.
3
g 3 , n3
2
g 2 , n2
1
g1 , n1
Fermions obey the condition on the occupation of a state is that, a state can have 1
particle or 0 particle in it, but never two or more. So, ni is always less than or equal to gi.
This is because they obey the Pauli exclusion principle.
ni  0,1
Let ni be the number of particles occupying an energy level  i in a particular
configuration. If g i is the degeneracy of  i , (such that the degeneracy of that energy is
 the number of particles occupying that energy level), then the number of possible
arrangements amongst the degenerate levels of  i is
wF(iD) 
gi !
 (2)
ni !( g i  ni )!
(This is exactly like picking ‘n’ white balls out of ‘g’ white balls.)
Bose-Einstein Statistics
Bosons obey this statistics. There is no constraint on the occupation of a particular
state in an energy level as these particles do not obey the Pauli principle. Any number of
particles can occupy a state in an energy level.
ni  0, 1, 2, 3,……
nimax  
Let ni be the number of particles occupying an energy level  i in a particular
configuration. If g i is the degeneracy of  i , then the number of possible arrangements
amongst the degenerate levels of  i are
wB(iE) 
(ni  g i  1)!
 (3)
ni !( g i  1)!
……………………………………………………………………………………………....
Justification:
Suppose we take six indistinguishable balls (bosons, ni ) and arrange them into three
distinct boxes (degenerate levels, g i ) with no restriction on the occupation numbers in
each box.
Box 1
Box 2
Box 3
Here there are five balls and two sticks (the number of sticks will always be one lesser
than the number of degenerate levels). The total number of possible arrangements for the
balls in the boxes is
7! (ni  g i  1)!

 21
5!2! ni !( g i  1)!
We are justified in doing ( g i  1)! because only the boxes are distinct, the sticks are not.
The sticks are indistinguishable.
………………………………………………………………………………………………
The total number of arrangements possible for a particular configuration for both
F-D as well as B-E are (this is over all the energy levels of a particular configuration)
W ni    w (i ) (ni )  (4)
i
The total number of arrangements possible over all configurations is
( N ,V , E )  W (ni )  (5)
ni 
where ( N , V , E ) is the partition function in microcanonical ensemble.
Thus the entropy is
S  k B ln 
 
 k B ln W ni
Maximizing the entropy and finding out the maximum term by Lagrange’s method of
undetermined multipliers.
 ln W ni     ni     i ni   0  (7)

i

i
W ni    w (i ) (ni )
i

ln W ni   ln w (i ) (ni )

  ln w (i ) (ni )
i
 ln W ni     ln w(i ) (ni )  (8)
i
For F-D statistics,
w(i ) 
gi !
ni !( g i  ni )!
ln w (i )  ln g i ! ln ni ! ln( g i  ni )!
 g i ln g i  g i  ni ln ni  ni  ( g i  ni ) ln( g i  ni )  g i  ni
 g i ln g i  ni ln ni  ( g i  ni ) ln( g i  ni )
 gi
 g i ln 
 g i  ni

 g  ni
  ni  i

 ni
 g  ni
  g i ln  i
 gi




 g  ni
  ni  i

 ni



 n 
g

  g i ln 1  i   ni ln  i  1  (9)
 gi 
 ni

For B-E statistics,
w(i ) 
(ni  g i  1)!
ni !( g i  1)!
ln w (i )  ln( ni  g i  1)! ln ni ! ln( g i  1)!
 (ni  gi  1) ln( ni  gi  1)  ni  gi  1  ni ln ni  ni  ( gi  1) ln( gi  1)  gi  1
 (ni  g i  1) ln( ni  g i  1)  ni ln ni  ( g i  1) ln( g i  1)
ln w(i )  ni ln
(ni  g i  1)
(n  g i  1)
 ( g i  1) ln i
ni
( g i  1)
 g 
 n 
 ni ln 1  i   g i ln 1  i   (10)
ni 

 gi 
General form of ln w(i ) for F-D and B-E statistics is
g
 g 
n
ln w(i )  ni ln  i  a   i ln 1  a i
gi
 ni
 a 

  (11)

a  1 for BE
where
 ln w (i )
a 1
for FD
a0
for MB


g

 ln  i  a   ni
  ni




  gi 
 a

2
ni  g i 
gi



ni
 gi
 a
  a 
1 a
gi

 ni



 

 
 g
 gi  1   1
i

 ln   a  


 
  ni
 ni  g i  a   1  a ni

gi
 ni
 




 g

1
i
 ln   a   

  ni
  1  a ni

gi


 
 
 1
 
ni
 1 a
gi
 
  gi

 a ni  (13)

  ni
 ln w (i )  ln 
From eqns (7) and (8),
 ln W ni     ni     i ni   0

i
i

 ln W ni     ln w(i ) (ni )
i
Thus,
 g


 ln  i  a      i ni  0
i

  ni


g

ln  i  a      i  0
 ni

ni 
gi
e   i  a
 (14)


n
 i




n
 i




n
 i


n i is the distribution that maximizes the entropy. Hence
 
S
 ln W ni
kB
 
ln W ni   ln w (i ) (ni )
i
From eqn (11)

g
 g 
n  
S
  ni ln  i  a   i ln 1  a i 
kB
g i 
i 

 ni
 a 


gi
  ni ln 
   i
i 
 gi e

 gi 
g e    i
  ln 1  a i
 a 
gi


g
S


  ni (   i )  i ln( 1  ae    i )  (15)
kB
a

i 
n

i
N
n 

i i
;
i
E
i
S
1
 N  E   g i ln( 1  ae    i )
kB
a i
 

k BT

;
1
k BT
Thus,
S
N
E
1


  g i ln( 1  ae    i )
k B k BT k BT a i
From thermodynamics,
S
N
E
G  ( E  TS ) PV




 (16)
k B k BT k BT
k BT
k BT
Equation of state for a quantum ideal gas is
PV 
k BT
a
g
i
ln( 1  ae    i )  (17)
i
In the high temperature limit,
,   0
Hence Taylor expansion can be done, ln( 1  x)  x




PV 1
  g i ae    i
k BT a i
  g i e    i  (18)
i
For an ideal gas with Maxwell-Boltzmann distribution, the equation of state is
PV
  ni  N
k BT
i
(Ideal gas law)
But from eqn (14)
ni 
gi
e
   i
Thus,
PV
  g i e    i  (19)
k BT
i
(Compare eqn (18) and (19)). Irrespective of whether it is a Bose gas or a Fermi gas, in
the thermodynamic limit it goes over to Maxwell-Boltzmann distribution.
Ideal Bose gas
From eqn (17), the equation of state of a Bose gas is
PV
  g i ln( 1  e    i )
k BT
i

e

e
kBT
 z , where z is the fugacity of the gas.
The equation of state can be rewritten as
PV
  g i ln( 1  ze   i )  (20)
k BT
i
N   ni  
i
i

i
gi
e
   i
1
g
 (21)
z e 1
i
1  i
The summations in eqn (20) and (21) are sum over (energy) levels. Supposing we do a
sum over states, then it is necessary that we give a statistical weight to each state. Since
there is no restriction on the number of particles occupying a particular state, each state
can be given a statistical weight of 1.
The equation of state of the Bose gas as a sum over all states is
PV
  ln( 1  ze   k )  (22)
k BT
k
The total number of particles is
N 
k
1
1  k
z e
1
 (23) , where the index k is for the states.
Eqns (22) and (23) are oft found expressions for a Bose ideal gas in texts.
The summation can be replaced by an integral if the density of states in the
neighbourhood of each energy level is included.
PV
   dG ( ) ln( 1  ze   )
k BT
N   dG ( )
1
1 
z e
1
Density of states around each energy level is
1
3
 2V 
G ( )d   3 2m  2  2 d
 h 
The equation of state thus becomes

1
3
PV
 2V 
  3 2m 2  d ln( 1  ze   )  2
k BT
 h 
0


By incorporating the density of states, no weight is given to   0 state (zero momentum
state) as the kinetic energy of the particles goes to zero in this energy level. In quantum
mechanical treatment this is incorrect as the particles essentially occupy the lower energy
states in the low temperature and high density limit where quantum effects become
dominant. Hence it is necessary that we take this term out of the summation (eqn (20))
before carrying out the integration. Now,


1 
3 
PV
 2V 
  3 2m 2  d ln( 1  ze   )  2   ln( 1  z )
k BT
 h 
 0






1 
3 
P
1
 2 
2
  3 2m   d ln( 1  ze   )  2   ln( 1  z )
k BT
h 
 V
 0



Similarly for the total number of particles
1

2
3
N  2 

1 z 

  3 2m 2  d 1 

V h 
z e 1 V 1 z 
 0

The total number of particles present in the system can be classified into
N  Ne  N0
1

3
 2
 2V 
N   3 2m  2  d 1 

z e 1
h 
0


3  z 
 2 
 3 2m  2 
  (24)
1 
z
h
Ne
N0
where N e refers to the number of particles in the excited states and N 0 is the number of
particles in the   0 state.
When the temperature is very high (classical limit), then z << 1 as
 (chemical potential) is a –ve quantity. Under such conditions, the term
ze

k BT
and
z
is well
1 z
behaved and does not give a significant contribution to N. In other words N 0 << N and
N e is large. Hence most particles are in the excited states and not in the   0 state. But
as the temperature approaches 0 K, z  1 and the term
z
diverges as z = 1 at
1 z
T = 0 K. The contribution from singularity terms becomes more significant under these
conditions. Here N 0 becomes much more significant compared to N e . Thus the   0
level gets densely populated. This is what is called Bose-Einstein condensation.
B-E Integral
In the following expressions,


1 
3 
P
1
 2 
  3 2m 2  d ln( 1  ze   )  2   ln( 1  z )  (25)
k BT
h 
 0

 V


1

3
N  2 
 2
1 z 
2
  (26)
  3 2m  d 1 

V h 
z e 1 V 1 z 
 0

N0
z
 N 0 (from eqn (24)) and thus, z 
.
1 z
N0 1
The last terms in eqn (24) and (25) are significant only at low temperatures. The last
terms in eqn (24) is

1
1
ln( 1  z )  ln( N 0  1)  (27)
V
V
Volume V, generally scales as N. At low temperature most particles are in the   0
state. Hence N 0  N . The upper bound on eqn (27) is N 1 ln N which is negligible for
all z and hence the last term can be dropped. The eqns (25) and (26) can now be rewritten
as

1
3
P
 2 
  3 2m 2  d ln( 1  ze   )  2  (28)
k BT
h 
0


1

3
N  N 0  2 
 2
2
  3 2m   d 1 
 (29)
V
z e 1
h 
0
Substituting   x

1
3
P
 2 
  3 2mk BT  2  dx ln( 1  ze x )  2  (30)
k BT
h 
0


1

3
N  N 0  2 
x 2
2
  3 2mk B T   dx 1 x
 (31)
V
z e 1
h 
0
Integrals like those in eqns (30) and (31) are very common in Bose-Einstein systems.
……………………………………………………………………………………………....
Aside
They essentially of the form

x 1dx
G ( z )   1 x
 (32)
0 z e 1

Lim G ( z )   ze  x x 1dx  z( )
z 0
0
0  z  1,   0 ; z  1,  1
It is useful to introduce another function g (z ) such that

g ( z ) 
1
1
x 1dx
G ( z ) 
 (31)
( )
( ) 0 z 1e x  1
For small vales of z, the integrand in eqn (33) can be in powers of z as


1
 1
g ( z ) 
x
( ze  x ) l dx


( ) 0
l 1

zl
z2 z3
    z      ....
2
3
l 1 l
For z << 1, the function g (z ) , for all  , behaves like z itself. For z  1,  1 g (z )
approaches the Riemann zeta function  ( )

g (1)  
l 1
1
1
1
 1      ...      (34)

l
2
3
………………………………………………………………………………………………
Rewriting eqns (30) and (31) in terms of g (z ) ,

1
3
P
1
 2 
  3 2mk BT  2  dx ln( 1  ze x )  2  3 g 5 ( z )  (32)
2
k BT

h 
0


(Integration of eqn (30) by parts will give eqn (32))
1

3
N  N 0  2 
x 2
1
  3 2mk B T  2  dx 1 x
 3 g 3 ( z )  (33)
2
V
z e 1 
h 
0
1
 h2  2

where   
 2mkBT 
Eqn (32) gives

1
3
PV
V
 2V 
  3 2mk BT  2  dx ln( 1  ze x )  2  3 g 5 ( z )  (34)
2
k BT

 h 
0


What are the conditions necessary for Bose-Einstein condensation?
Eqns (33) and (24) give us that
(2mk B T )
N e  3 g 3 ( z)  V
2

h3
1
3
2
g 3 ( z )  (35)
2
At high temperatures, z << 1 and N e  N i.e. most particles reside in the excited energy
levels. g 3 ( z ) can be written as a summation in terms of z
2
z2
g 3 ( z)  z 
2
3
2

2
z3
3
3
 ....
2
The upper bound on the value of g 3 ( z ) is when z = 1
2
g 3 (1)  1 
2
1
2
3

2
1
3
3
 ...      2.612
2
The upper bound on the value on N e is
(2mk B T )
Ne  V
h3
3
2
 3
2
    (36)
Thus at high temperatures with z of interest,
g 3 ( z)   ( z)  (37)
2
From eqns (35) and (36)
(2mk B T )
Ne  V
h3
3
2
 3
2
    (37)
So long as the total number of particles is lesser that the limiting value of N e , almost all
particles occupy the excited states, N e  N . But if the total number of particles exceeds
the upper bound of this upper bound of N e , then the excited states receive only as much
as it can hold (as dictated by the upper bound on N e ). The rest of the particles will be
pushed to   0 ground state.   0 energy level has no limitations on the number of
particles that it can hold under all conditions. Number of particles occupying the ground
state now becomes
 (2mk T ) 3 2  3 
B
N 0  N  V
    (38)
3
h
 2 

The condition necessary for the onset of Bose-Einstein condensation can hence be
summarized as
I
Number of particles in the system should be greater than the limiting value on N e
N  VT
3
2
(2mk B )
h3
3
2
3
2
    (39)
II If we hold N and V constant and vary only T, we find that T has to be lesser than a
particular value Tc such that N is greater than the limiting value of N e .




2
h
 N 
T  Tc 


2mk B   3  
V  
  2  
2
3
 (40)
The following plot shows how the complementary fractions
Ne
N
and 0 vary with
N
N
temperature. For T  Tc , the particles are essentially in the excited states and negligible in
the ground state. Below Tc , there is significant population   0 energy level which drops
with increase in temperature. The drop in N 0 when approached from T = 0 follows the
equation
T
N0
 1  
N
 Tc



3
2

3 Tc  T
 (41)
2 Tc
This eqn (41) is obtained from eqns (38) and (40).
Thermodynamic properties of an ideal Bose gas
Pressure dependence on temperature
From eqn (34), it can be seen that,
P(T ) 
k BT
3
g 5 ( z )  (42)
2
For unit fugacity,
P (T ) 
3 has a T
3
2
k BT  5 
 
3  2 
dependence. Thus P(T ) , in all has T
5
2
dependence.
At the critical temperature the value of pressure is,
 2m 
P(Tc )   2 
 h 
3
2
kTc  2   5   (43)
5
2
From eqn (40), eqn (43) can be rewritten as
5

N

2 N
P (Tc ) 
 k B Tc   0.5134 k B Tc   (44)
3 V

V

  
2
 
 
According to the ideal gas law,
P
N
k BT
V
Hence the pressure exerted by an ideal Bose gas is almost half of that applied by an
equivalent Boltzmannian (classical ideal gas) at the critical temperature.
The pressure exerted by the ideal Bose gas above the critical temperature, T  Tc is
P(Tc ) 
g 5 ( z) N


2
 k BT 
g 3 ( z)  V

2
Internal energy of an ideal Bose gas
Statistical thermodynamics gives us
PV
 ln 
k BT
where  is the grand canonical partition function.
The internal energy of a system can be evaluated using the grand canonical partition
function (using eqn (34)).
   PV 
 

 d  1 
  k BT 2Vg 5 ( z )  3 
U  
ln    k BT 2  
2
 dT   
 
 z ,V
 T  k BT  z ,V
U
3
V
k B T 3 g 5 ( z )  (35)
2
2

For an ideal gas
P
2 U 
 
3V 
Specific heat of a Bose ideal
Specific heat is given by
Cv 
U 3
d
 k BVg 5 ( z )
2
T 2
dT
T 
 3   (36)
 
An implicit assumption is that
g 3 ( z) 
2
N
h3
 (37)
V (2mk T ) 3 2
B
For unit fugacity,
Cv
3V  5  d  T

 

Nk B 2 N  2  dT  3
3
 15 (2mk B T ) 2 V  5 
    (38)

Nh 3
 4
2
It has a T
3
2
dependence on temperature. This is how the specific heat behaves in the low
temperature limit.
At the critical temperature T  Tc (from (37) and (38)),
5
C v (Tc ) 15  2 

 1.925
Nk B
4 3
 
2
 
For T  Tc ( z  0) , it is the classical limit of an ideal gas. Assuming the gas to be
monoatomic, the specific heat is
CV 
3
2
A cusp can be seen in the plot. This is a consequence of a phase transition that occurs at
T  Tc which gives rise to a sudden change in the density.
An experimental evidence for Bose –Einstein condensation came from the (CV , T )
relationship of He4. Phase transition of He4 was found to occur at 2.19 K. When the mass,
m  6.65 * 10 24 and volume, V  27.6 cm3/mole were substituted in eqn (40) to find
theoretically find out the transition temperature it was found to be 3.13. The two numbers
were not drastically different and the similarity between the ( C V vs. T) plot of an ideal
Bose gas and that of He4 was enough evidence to prove that the phase transition in He4 is
actually Bose-Einstein condensation.
Struck by the shape of this graph, the phase transition was given the name   transition
by Keesom and the transition point was called   transition point.