Unit 4, Statistical Physics, Lecture 5
Bose-Einstein Statistics
The basic postulates of BE statistics are
1.
2.
3.
4.
5.
The associated particles are identical and indistinguishable.
Each energy state can contain any number of particles.
Total energy and total number of particles of the entire system is constant.
The particles have zero or integral spin.
The wave function of the system is symmetric under the positional exchange of any two
particles. Such particles are known as Bosons. For example, photons, phonons, all
mesons (,,) etc.
[Symmetric and Anti-symmetric wave function: Let the allowed wave function for a n-particles
system is ψ(1,2,3,…..,r,s,…n), where the integers within the argument of ψ represent the
coordinates of the n-particles relative to some fixed origin. Now, if we interchange the positions
of any two particles, say, r and s, the resulting wave function becomes ψ(1,2,3,….s,r,…..n).
The wave function ψ is said to be symmetric when ψ(1,2,3,…..,r,s,…n) = ψ(1,2,3,….s,r,…..n)
and anti-symmetric when ψ(1,2,3,…..,r,s,…n) = - ψ(1,2,3,….s,r,…..n).
In B.E. statistics all the particles are
indistinguishable. Also, the quantum states
are assumed to have equal a priori
probability. Thus gi represents the number
of quantum states with same energy Ei (gi is
degeneracy).
Each
quantum
state
corresponds to a cell in phase space. We
shall determine the number of ways in
which ni indistinguishable particles can be
distributed in gi cells.
Fig. 9. Energy levels of a system bracketed
into cells
Let the gi number of cells be numbered as 1,2,3, …., gi. Each cell contains 1 or 2 or 3 or … ni
particles at a time. Now one particle can be put in any one cell in gi ways (i.e. one particle can
be put in 1st cell or 2nd cell or gi-th cell). Two particles can be put in two ways - (i) two particles
can be put in a single cell in gi ways, i.e. we choose one cell out of gi cells in gi ways, (ii) each
of the two particles can be put in two separate cells, i.e. we choose 2 cells out of gi cells in
𝑔𝑖 𝐶 ways, i.e.
2
𝑔𝑖 (𝑔𝑖 −1)
2
ways. Thus two particles can be put in
𝑔𝑖 +
𝑔𝑖 (𝑔𝑖 −1)
2
=
𝑔𝑖 (𝑔𝑖 +1)
2
(𝑔 +1)!
𝑖
= (𝑔 −1)!2!
𝑖
ways.
Now, three particles can be put in three distinct ways – (i) three particles can be put in a single
cell. This can be done in gi ways, (ii) two particles in one cell and one particle in another cell.
1
This can be done in gi(gi – 1) ways, (iii) three particles can be put in three cells. This can be
𝑔!
𝑔𝑖 (𝑔𝑖 −1)(𝑔𝑖 −2)
𝑖
3!
done in 𝑔𝑖 𝐶 ways, i.e. 𝑔𝑖 𝐶 = 3!(𝑔 𝑖−3)! =
3
3
ways. Therefore, the total number of
ways will be
𝑔𝑖 + 𝑔𝑖 (𝑔𝑖 − 1) +
𝑔𝑖 (𝑔𝑖 −1)(𝑔𝑖 −2)
3!
=
𝑔𝑖 (𝑔𝑖 +1)(𝑔𝑖 +2)
3!
(𝑔 +2)!
𝑖
= (𝑔 −1)!3!
𝑖
Arguing in this way, the number of ways ni particles can be put in gi cells is
(𝑔𝑖 +𝑛𝑖 −1)!
(𝑔𝑖 −1)!𝑛𝑖 !
If there are
n1 particles in the energy level E1 with degeneracy g1
n2 particles in the energy level E2 with degeneracy g2
……………………………………………………….
ni particles in the energy level Ei with degeneracy gi
Hence, all these groups of particles can be distributed in, i.e. the thermodynamic probability
𝑊 = ∏𝑖
(𝑔𝑖 +𝑛𝑖 −1)!
(43)
(𝑔𝑖 −1)!𝑛𝑖 !
Now we assume, (ni + gi) >> 1, so that (𝑔𝑖 + 𝑛𝑖 − 1)! ≈ (𝑔𝑖 + 𝑛𝑖 )!
Therefore, Eqn. 43 becomes
(𝑔 +𝑛 )!
𝑖
𝑖
𝑊 = ∏𝑖 (𝑔 −1)!𝑛
!
𝑖
(44)
𝑖
Taking natural logarithm on both sides we get
𝑙𝑛𝑊 = ∑𝑖[ln (𝑔𝑖 + 𝑛𝑖 )! − 𝑙𝑛((𝑔𝑖 − 1)! − 𝑙𝑛𝑛𝑖 !]
=∑𝑖 [ (𝑔𝑖 + 𝑛𝑖 )𝑙𝑛(𝑔𝑖 + 𝑛𝑖 ) − (𝑔𝑖 + 𝑛𝑖 ) − 𝑙𝑛(𝑔𝑖 − 1)! − (𝑛𝑖 𝑙𝑛𝑛𝑖 − 𝑛𝑖 )]
[Here we have used the Sterling’s formula ln n! = n ln n – n]
= ∑𝑖 [ (𝑛𝑖 + 𝑔𝑖 )𝑙𝑛(𝑛𝑖 + 𝑔𝑖 ) − 𝑙𝑛(𝑔𝑖 − 1)! − 𝑛𝑖 𝑙𝑛𝑛𝑖 − 𝑔𝑖 ]
(45)
For most probable distribution a small variation δni in any ni does not affect the value of W.
For a change δni in ni, δln Wmax = 0. Therefore,
𝛿𝑙𝑛𝑊𝑚𝑎𝑥 = ∑[𝛿( (𝑛𝑖 + 𝑔𝑖 )𝑙𝑛(𝑛𝑖 + 𝑔𝑖 )) − 𝛿𝑙𝑛(𝑔𝑖 − 1)! − 𝛿(𝑛𝑖 𝑙𝑛𝑛𝑖 ) − 𝛿𝑔𝑖 ] = 0
𝑖
or
∑𝑖[𝛿𝑛𝑖 𝑙𝑛(𝑛𝑖 + 𝑔𝑖 ) + (𝑛𝑖 + 𝑔𝑖 ) (𝑛
1
𝑖 +𝑔𝑖 )
1
𝛿𝑛𝑖 − 𝛿𝑛𝑖 𝑙𝑛𝑛𝑖 − 𝑛𝑖 𝑛 𝛿𝑛𝑖 ] = 0
𝑖
[Since δgi = 0]
or
∑𝑖[𝑙𝑛(𝑛𝑖 + 𝑔𝑖 ) − 𝑙𝑛𝑛𝑖 ]𝛿𝑛𝑖 = 0
(46)
2
We incorporate the conservation of particles as
𝛿 ∑𝑖 𝑛𝑖 = 𝛿𝑁 = 0 (N being the total number of particles), i.e.
∑𝑖 𝛿𝑛𝑖 = 0
(i)
and the conservation of energy expressed as
∑𝑖 𝐸𝑖 𝛿𝑛𝑖 = 0
(ii)
[Note: ∑𝑖 𝑛𝑖 𝐸𝑖 = 𝐸 (𝑡𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦) or 𝛿 ∑𝑖 𝑛𝑖 𝐸𝑖 = 𝛿𝐸 = 0, we change ni not in Ei]
Multiplying (i) by – α and (ii) by – β and adding to Eqn. 46, we get
∑𝑖[𝑙𝑛(𝑛𝑖 + 𝑔𝑖 ) − 𝑙𝑛𝑛𝑖 − 𝛼 − 𝛽𝐸𝑖 ]𝛿𝑛𝑖 = 0
Since the δni’s are independent, the quantity in bracket of the above equation must vanish for
each i. Hence
𝑙𝑛(𝑛𝑖 + 𝑔𝑖 ) − 𝑙𝑛𝑛𝑖 − 𝛼 − 𝛽𝐸𝑖 = 0
or
or
𝑙𝑛
𝑛𝑖 +𝑔𝑖
𝑛𝑖
𝑛𝑖 +𝑔𝑖
𝑛𝑖
= 𝛼 + 𝛽𝐸𝑖
= 𝑒 𝛼 𝑒 𝛽𝐸𝑖
𝑔𝑖
or
1+
or
𝒏𝒊 =
𝑛𝑖
= 𝑒 𝛼 𝑒 𝛽𝐸𝑖
𝒈𝒊
𝜷𝑬
𝜶
𝒆 𝒆 𝒊 −𝟏
(47)
This is the general form of Bose-Einstein (BE) distribution law.
From Eqn. 31, we have β = 1/kT. Therefore, Eq. 47 can be rewritten as
𝒏𝒊 =
or
𝒈𝒊
𝑬𝒊
𝒆𝜶 𝒆𝒌𝑻 −𝟏
𝒇(𝑬𝒊 ) =
𝒏𝒊
𝒈𝒊
=
𝟏
𝑬𝒊
𝒆𝜶 𝒆𝒌𝑻 −𝟏
f(Ei) is known as Bose-Einstein distribution function.
3
(48)
Comparison of MB, FD and BE-statistics
Features
Particle
Particle
Spin
Wave
function
No. of
particles
per energy
state
MB
BE
FD
The particle of the
system in equilibrium
are indistunguishable
and Pauli's exclusion
principle is not obeyed.
The particle of the system in
equilibrium are indistunguishable
and Pauli's exclusion principle is
obeyed.
0,1,2, …
1/2, 3/2, 5/2, ….
Symmetric under
interchange of the
coordinates of any two
bosons.
Antisymmetric under interchange
of the coordinates of any two
fermions.
No upper limit.
No upper limit as Pauli's
exclusion principle is not
obeyed.
Maximum one fermion per
quantum state is allowed as Pauli's
exclusion principle is obeyed.
Common gases at
normal temperature.
Applies to photons,
phonons, particles with
integral or zero spin, like
π-mesons.
Applies to electron gas in metals,
particles having half-integer spin
like protons, neutrinos etc.
The particle of the
system in equilibrium
are distunguishable and
Pauli's exclusion
principle doesn't apply.
Spinless.
-
Distribution
function
Applies to
N.B.
Both FD and BE-statistics reduces to MB-statistics when gi >> Ni, i.e. when the particle number
is quite small, and when the temperature T is high. The reduction of quantum statistics to the
MB statistics at sufficiently low concentration or sufficiently high temperature is known as the
classical limit of quantum statistics. A gas in the classical limit is called nondegenerate,
whereas for concentrations and temperatures where FD and BE distribution function is valid,
the gas is called degenerate.
4
Fig. 10. Plot of MB, FD and BE distribution functions as a function of (ε-μ)/kT. Each
system is at the same temperature and has the same number of particles.
5