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

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• Philip Koeck
In summary, the conversation discusses the possibility of finding an expression similar to the Sackur-Tetrode equation for the statistical entropy of fermions or bosons, particularly in the context of electron or photon gases. The speakers mention sources such as a PDF from ETH Zürich and lecture notes from David L. Feder on statistical mechanics. They also mention a potential mistake in Feder's notes and reference a book by J. E. Mayer and M. G. Mayer for comparison.
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?

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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Philip Koeck

## 1. What is the expression for entropy of fermions or bosons?

The expression for entropy of fermions or bosons is given by S = kBlnΩ, where kB is the Boltzmann constant and Ω is the number of microstates or possible arrangements of the particles.

## 2. How is the entropy of fermions or bosons different from classical particles?

The entropy of fermions or bosons is different from classical particles because they obey different statistical distributions. Fermions follow the Fermi-Dirac distribution, which results in a maximum entropy at zero temperature. Bosons, on the other hand, follow the Bose-Einstein distribution, which results in a non-zero entropy at zero temperature.

## 3. Does the expression for entropy of fermions or bosons apply to all systems?

No, the expression for entropy of fermions or bosons only applies to systems in thermal equilibrium, where the particles are indistinguishable and obey either the Fermi-Dirac or Bose-Einstein distribution.

## 4. Can the expression for entropy of fermions or bosons be derived from first principles?

Yes, the expression for entropy of fermions or bosons can be derived from statistical mechanics and quantum mechanics principles, such as the Pauli exclusion principle and the indistinguishability of particles.

## 5. How does the entropy of fermions or bosons relate to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system always increases or remains constant. In the case of fermions and bosons, the entropy is related to the number of available microstates, which increases with increasing temperature. Therefore, as the temperature increases, the entropy of fermions and bosons also increases, in accordance with the second law of thermodynamics.

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