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I have a question about the FD and BE grand canonical potential. We derived the expression for both cases, but probably the expressions must be wrong.

$$J_{FD}=kT\sum_i\ln\left(1+ e^{-\beta(\epsilon_i - \mu)}\right)$$ for Fermi-Dirac.

$$J_{BE}= -kT\sum_i\ln\left(1 - e^{-\beta(\epsilon_i - \mu)}\right)$$ for Bose-Einstein.

But shouldn't it be the other way around, meaning plus sign below and negative sign above?

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  • $\begingroup$ Why do you think that? $\endgroup$
    – J. Murray
    Jun 4, 2021 at 17:41
  • $\begingroup$ because when you derive the grand canonical partition functions, for the bose einstein you have an expression on power -1 which means when you try to find the potential witht he formula -ktlY, you get a 2nd minus, which will result in a plus $\endgroup$
    – imbAF
    Jun 4, 2021 at 17:54
  • $\begingroup$ Are you referring to the minus sign before the $kT$, or the minus sign in $1 \pm e^{-\beta(\epsilon-\mu)}$? My initial assumption was that you meant the latter. $\endgroup$
    – J. Murray
    Jun 4, 2021 at 19:20

1 Answer 1

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The Grand Canonical Partition function look like $$\ln \mathcal{Z}=\pm\sum_i\ln(1\pm e^{\beta(\mu-E_i)})$$ where the $\pm$ sign means $+$ for fermions and $-$ for bosons. $$\Phi_G=-k_BT\ln\mathcal{Z}=\mp k_BT\sum_i \ln(1\pm e^{\beta(\mu-E_i)})$$

So indeed your expression should have flipped sign.

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  • $\begingroup$ Exactly! But then it comes this 2nd problem in my script. Later we try to derive the grand canonical partition function for the case of continues energy/momentum values. If i make the change, then i should get a minus by then end for the FD case, while the professor, has no minus. What should i do??? $\endgroup$
    – imbAF
    Jun 4, 2021 at 19:01
  • $\begingroup$ how can that be? I have a plus there, not a minus. That's who we derived it $\endgroup$
    – imbAF
    Jun 4, 2021 at 19:05
  • $\begingroup$ Ah, that's alright. $\endgroup$ Jun 4, 2021 at 19:07
  • $\begingroup$ $Y_{FD} = \Pi_i (1+ e^{-\beta (\epsilon_i - \mu)})$ for the fermi dirac grand canonical partition function. $Y_{BE} = \Pi_i (1- e^{-\beta (\epsilon_i - \mu)})^{-1}$ Bose einstein. If you use this expressions, you will get the minus for FD and plus for BE $\endgroup$
    – imbAF
    Jun 4, 2021 at 19:08
  • $\begingroup$ @imbAF Can you show your calculation? The one you are talking of in your first comment $\endgroup$ Jun 4, 2021 at 19:09

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