Below we derive in standard textbook fashion the Fermi-Dirac and Bose-Einstein distributions from the perspective of the grand canonical ensemble, which in a way is the natural ensemble to consider.
Consider a system of identical quantum particles. Let be the number of particles occupying the single-particle states . Let be the energy for the state . The occupation number is limited to for fermions due to the Pauli exclusion principle. For bosons the occupation numbers can be arbitrarily large.
The total energy is thus given by
Our goal is to calculate the mean occupation number for fermions and bosons when they are in thermodynamic equilibrium, at temperature and with chemical potential .
1. The grand canonical partition function
For fixed chemical potential , temperature parameter and volume , the grand canonical partition function is given by
In the situation previously described, we can write it as
In other words, can be factored as product over a sum over occupation numbers for the -th single-particle state. The latter sum is easy to calculate for bosons and even easier for fermions.
The mean occupation numbers can then be calculated as a weighted average for the grand canonical ensemble:
Substituting for from (4) we get
Unless the derivative vanishes, so that
2. Fermi-Dirac statistics
For fermions cannot be greater than 1. So
Finally we get
3. Bose-Einstein Statistics
For bosons we have
which is a geometric series.
Recall that for we can write
from which we get
Substituting, we get
Finally we get
The Fermi-Dirac and Bose-Einstein distributions can be written as
with the plus sign for Fermi-Dirac statistics and the minus sign for Bose-Einstein statistics.