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Fermi-Dirac distribution

The case of fermions is a little bit similar, except that each sublevel within a degeneracy can take just ONE or ZERO particle.

Here the number of ways to place ni in gi degeneracies is:
C(ni, gi) = gi!/ni!(gi - ni)!
That is the number of combinations of ni among gi

The total number of macrostates is then: Ω = Π gi!/ni!(gi - ni)!

Using Stirling's approximation, we get:
ln(Ω) = Σ giln gi - gi - niln i + ni - (gi - ni)ln(gi - ni) + (gi - ni)
= Σ giln gi - niln ni - (gi - ni)ln(gi - ni)

Differentiating to maximaze, we get:
dln(Ω) = Σ - dniln ni - dni + dniln(gi - ni) + d ni = Σ dniln[(gi - ni)/ ni]
It follows that by adding the same above constraints:
Σ dni {ln[(gi - ni)/ ni] + α - βEi]} = 0   
Which gives: (gi - ni)/ ni = exp[- α] exp[βEi]


The Fermi-Dirac population numbers is :
ni = gi/(exp[- α] exp[βEi] + 1)

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