# Is there an Expression for Entropy of Fermions or Bosons?

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Is there an expression similar to the Sackur-Tetrode equation that describes the statistical entropy of fermions or bosons, maybe for the electron gas in a metal or the photon gas in a cavity?

Lord Jestocost
Gold Member
There are expressions for the entropy (or heat capacity) of ideal Fermi or Bose gases derived from quantum statistical physics (see, for example:

I was thinking more along these lines: For the BoItzmann distribution I would write:

ni = gi e e-β ui

ln W ≈ ∑ni (1 - ln (ni / gi)) = N + αN + βU

α = - μ / kT ; β = 1 / kT

S = k ln W = k N - μ N / T + U / T

Inserting expressions for μ and U for an ideal gas gives the Sackur-Tetrode equation.

Is something similar possible for FD or BE?

Last edited:
Thanks for your help. I'll look at both texts during summer.

Now I've looked at Feder's lecture notes. I've appended a text of my own where in the expression for ln W in sections 1 and 3 I get the same as equation 4.6 in Feder, apart from a minus in front of one of the terms. Don't know where the mistake is.
Anyway: I'd like to know if it's possible to simplify this expression as sketched in the text. My problem is that I'm left with 1 term I can't simplify at the end of section 3. Do you know if this is done anywhere?

And here's the file I forgot to upload.

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• Fermions.pdf
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Lord Jestocost
Gold Member
Philip Koeck