Is there an Expression for Entropy of Fermions or Bosons?

[Philip Koeck]

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]

There are expressions for the entropy (or heat capacity) of ideal Fermi or Bose gases derived from quantum statistical physics (see, for example: [PDF]Statistical Physics - ETH Zürich).

 

[Philip Koeck]

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

n_i = g_i e^-α e^{-β u_i}

ln W ≈ ∑n_i (1 - ln (n_i / g_i)) = 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?

 

[Lord Jestocost]

Maybe, section 4.1.3 of David L. Feder's lecture notes on statistical mechanics might be of help: [PDF]Physics 451 - Statistical Mechanics II - Course Notes

 

[Philip Koeck]

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

 

[Philip Koeck]

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?

 

[Philip Koeck]

And here's the file I forgot to upload.

 

[Lord Jestocost]

Compare your expression for lnW to equation (5.9) in “Statistical Mechanics” by J. E. Mayer and M. G. Mayer (https://archive.org/details/statisticalmecha029112mbp). It seems to me that David L. Feder made a sign error in his course notes.