Problem understanding Fermi - Dirac distribution function

Question

I am studying Statistical Mechanics by R K Pathria. There the author tries to calculate the partition function of in canonical ensemble as:

$$Q_N(V,T) = \sum_{[n_\epsilon]} (e^{-\beta\sum_{\epsilon}n_\epsilon \epsilon})$$

(In above the expression the summation runs over the occupation sets complying to total particle number restriction and also restriction on $n_\epsilon$)

This is easily evaluated for Boltzmann-Maxwell distribution, but due to restricted values of $n_\epsilon$ that can be taken in case FD distribution ($0$ and $1$), he argues that it is difficult to evaluate. So he jumps to grand canonical partition function and there he doesn't care about the values that can be taken by $n_\epsilon$.

Why is that in case of Grand Canonical Partition Function, there are no such restrictions, after all we are still dealing with Fermi-Dirac statistics?

Answer 1

You are confusing the two restrictions. The restriction the author is referring to which makes the derivation difficult using the canonical ensemble is the restriction on total particle number $N_0$ in the canonical ensemble, not the restriction on the individual occupancies $n_{\epsilon}$ to be $0$ or $1$.

$$ \sum_{\epsilon} n_{\epsilon} = N_0 $$

In the grand canonical ensemble it is still true that each $n_{\epsilon}$ can only take on the values $0$ or $1$, but there is no constraint on the sum of all of the $n_{\epsilon}$. The author certainly uses the fact that each $n_{\epsilon}$ can only be $0$ or $1$ in the derivation of the Fermi-Dirac distribution using the grand canonical ensemble.

The reason we are licensed to use the grand canonical ensemble instead of the canonical ensemble is that in the thermodynamic limit the statistics of a system with a fixed number of particles, $N_0$, are the same as the statistics of a system which can exchange particles with a reservoir so long as the average number of particles, $\langle N \rangle$ is equal to the same number of particles as the isolated system: $\langle N \rangle = N_0$.