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Physics LibreTexts

8.3: Fermi-Dirac Distribution

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The counting of distinct arrangements for fermions is even simpler than for the Bose-Einstein case, since each state can have an occupation number of either zero or 1. Thus consider g states with n particles to be distributed among them. There are n states which are singly occupied and these can be chosen in g!(n!(gn)!) ways. The total number of distinct arrangements is thus given by

W({nα})=αgαnα!(gαnα)

The function to be maximized to identify the equilibrium distribution is therefore given by

SkβU+βµN=nαlognα(gαnα)log(gαnα)β(ϵαμ)nα+constant

The extremization condition reads

log[(gαnα)nα]=β(ϵαμ)

with the solution

nα=gαeβ(ϵαμ)+1

So, for fermions in equilibrium, we can take the occupation number to be given by

n=1eβ(ϵμ)+1

with the degeneracy factor arising from summation over states of the same energy. This is the Fermi-Dirac distribution. The normalization conditions are again,

d3xd3p(2πħ)31eβ(ϵμ)+1=Nd3xd3p(2πħ)3ϵeβ(ϵμ)+1=U

As in the case of the Bose-Einstein distribution, we can write down the partition function for free fermions as

logZ=log(1+eβ(ϵμ))Z=11+eβ(ϵμ)

Notice that, here too, the partition function for each state is nenβ(ϵμ) ; it is just that, in the present case, n can only be zero or 1.


This page titled 8.3: Fermi-Dirac Distribution is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by V. Parameswaran Nair via source content that was edited to the style and standards of the LibreTexts platform.

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