1. Fermi statistics and Bose Statistics
The occupation numbers, or number of particles in each one-particle state n are restricted by a general principle of quantum mechanics. The wave function of a system of identical particles must be either symmetric (Bosons) or antisymmetric (Fermions) in the permutation of the particle quantum numbers (including spin). It can be shown that there can be only 2 cases:
for Fermions (Fermi-statistics) n = 0 or 1
for Bosons (Bose-statistics) n= 0,1,2,
The differences between the two are determined by the nature of the particle. Particles which follow Fermi-statistics are called Fermi-particles (Fermions) and those which follow Bose-statistics are called Bose- particles (Bosons).
Electrons, positrons, protons & neutrons are Fermions.
Photons are Bosons.
Fermions have spin = (½)*(odd integer)
Bosons have spin = integer

2. Fermions & Bosons

3. Wolfgang Pauli

4. Fermi-Dirac Distribution
Enrico Fermi
P.A.M. Dirac
(5.46)
Now, we’ll discuss in detail the physics of this distribution.

5. Fermi-Dirac Distribution
Consider a system of identical independent non-interacting particles sharing a common volume & obeying anti-symmetric statistics: That is, spin = (½)n, n = odd integer. Therefore, according to the Pauli principle, the total wave function is anti-symmetric on interchange of any two particles.
Because the particles are non-interacting, it is convenient to discuss the system in terms of the energy states i of one particle in a volume V. Specify the system by specifying the number of particles ni , occupying the eigenstate of energy i . i denotes a single state, not the set of degenerate states which may have the same energy.
The Pauli principle allows only the values ni =1,0. This is, of course, the elementary statement of the Pauli principle:
A given single-particle state may not be occupied by more than one identical particle.

6. (5.29)
Partition Function
subject to Note that the  in the exponent runs over all one-particle states of the system; {ni} represents n allowed set of values of the ni ; and runs over all such sets. Each ni is 0 or 1.
Example: Consider a system with two states with energies 1 & 2.
The exponential then reads
(5.30)
the partition function is then:
(5.31)
But, we must include the fact that n1+n2 = N.
If we take N=1, we have
(5.32)

7. For a system with many states and many particles it is difficult analytically to take care of the condition ni=N. It is more convenient to work with grand canonical ensemble. We have for the grand partition function
(5.33)
so that
(5.34)
A simple consideration shows that we may reverse the order of the  and  in (5.34). We note that the significance of the  changes entirely, from {ni}=0,1. Every term, which occurs, for one order will occur for the other order
(5.35)
where
(5.36)

8. Now from the definition of the grand partition function
(5.37)
we have
(5.38)
where
(5.39)
For ni restricted to 0,1, we have
(5.40)
Now
(5.41)

9. with it appears reasonable to set
(5.42)
The same result can be provided by direct use of averaging in the grand canonical ensemble
(5.43)
This may be simplified using the form (5.36):
(5.44)

10. Fermi-Dirac distribution function.or
(5.45)
in agreement with (5.42).
This is the Fermi-Dirac distribution law. It is often written in terms of f(), where f is the probability that a state of energy is occupied:
(5.46)
It is implicit in the derivation that  is the chemical potential. Often  is called the Fermi level, or, for free electron gas, the Fermi energy EF.

11. Classical limit
For sufficiently large  we will have (-)/kT>>1, and in this limit
(5.47)
This is just the Boltzmann distribution. The high-energy tail of the Fermi-Dirac distribution is similar to the Boltzmann distribution. The condition for the approximate validity of the Boltzmann distribution for all energies  0 is that
(5.48)

12. Bose-Einstein Distribution
S.N. Bose
Albert Einstein
(5.51)
Now, we’ll discuss in detail the physics of this distribution.

13. Bose-Einstein Distribution
Particles of integral spin (bosons) must have symmetric wave functions. There is no limit on the number of particles in a state, but states of the whole system differing only by the interchange of two particles are identical and must not be counted as distinct. For bosons we can use the results (5.38) and (5.39), but with ni = 0,1,2,3,...., so that
(5.38)
(5.49)
where
(5.50)
Thus
(5.51) or
This is the Bose-Einstein distribution

14. We can confirm (5. 50) by a direct calculation on nj
We can confirm (5.50) by a direct calculation on nj. Using the previous result
we have
(5.52) or
(5.53)
in agreement with (5.50).