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All derivations I have seen for the Fermi-Dirac statistics presuppose the grand canonical ensemble. However, all applications of it, e.g. ideal quantum gases, electrons in a metal and semiconductors, are assuming a system where $N$ is fixed.

Why can the Fermi-Dirac Statistics even be used, and what is the meaning of $\mu$ in those cases, since the chemical potential

$$ \mu = \left(\frac{\partial U}{\partial N}\right)_{S,V} $$

is defined as the energy needed in order to increase the particle number in system by one? Why the derivation doesn't consider a canonical ensemble instead?

I understand that, in the thermodynamic limit, canonical and grand canonical descriptions are equivalent for $N$ and $\langle N \rangle$, but I don't see the meaning of $\mu$ for a system with constant $N$.

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When dealing with fermions it is analytically easier to handle Fermi–Dirac statistics in the grand canonical ensemble. If not you have the challenge of handling combinatorics, examples of the derivation here: Wikipedia Fermi-Dirac statistics (look how tedious the calculation is for other ensembles).

The chemical potential has different meanings in different ensembles. For the microcanonical ensemble, $\mu/T$ is a Lagrange multiplier (related to the constraint of having a fixed number of particles $N$) and represents the change of entropy $S$ when a particle is added (at constant energy $E$ and volume $V$). It is given by

$$\frac{\mu}{T}=-\left(\frac{\partial S}{\partial N}\right)_{E,V}$$

where $T$ is temperature.

In the canonical ensemble $\mu$ represents a change of free energy $F$ when a particle is added (at constant $T$ and $V$), and is given by $$\mu=\left(\frac{\partial F}{\partial N}\right)_{T,V}$$

Note that in the canonical and microcanonical ensembles $N$ is usually an integer, so the derivatives are differences and are exactly valid in the thermodynamic limit.

In fermionic systems, when dealing with micro- or canonical systems, in practice what people do is to calculate everything in the grand canonical ensemble and then check for what value of $\mu$ you get the desired $N$, however this is usually done numerically.

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  • $\begingroup$ $\mu$ is related to change of entropy associated with change of $N$, but does it does not represent this change, because there is a factor of $-T$. $\endgroup$ Jan 16 at 17:29
  • $\begingroup$ @JánLalinský you are right corrected. $\endgroup$
    – Mauricio
    Jan 16 at 17:36

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