Derivation of the Grand Canonical Partition Function for Fermions

Question

Regarding the derivation on this page: http://lampx.tugraz.at/~hadley/ss2/fermigas/thermo/thermo.php

I'm stuck with the summation over macrostates {$q$} being the same as the sum over microstates {$n_i$}. I have [equation 3 in the above page] $$\mathcal{Z}=\sum_q \prod_i exp\big(-\beta n_{q,i}(\varepsilon_i-\mu)\big) $$ but I'm stuck on how this summation evaluates to [equation 5] $$\mathcal{Z}=\prod_i \bigg(1+exp(-\beta(\varepsilon_i-\mu))\bigg)$$

The mediary step they provide is replacing the macrostate sum with the microstate sum: $$\sum_q \rightarrow \sum_{n_1=0}^1\sum_{n_2=0}^1...\sum_{n_{q_{max}}=0}^1$$ so they have [equation 4] $$\mathcal{Z}=\sum_{n_1=0}^1 \sum_{n_2=0}^1...\sum_{n_{q_{max}}=0}^1 \prod_i exp\big(-\beta n_{q,i}(\varepsilon_i-\mu)\big)$$ but I haven't been able to manipulate or expand this to produce [equation 5] without having additional factors. I'm not sure if $n_{q,i}$ has a new meaning in the microstate sum (e.g. $n_{q,i}$ being different from $[n_{j}]_i$), if I'm mistaking the way $n_{q,i}$ is defined (I'm looking at it as though $q$ and $i$ are indices like that of a rank-2 tensor), if I'm making a mistake with evaluating the sums, or something I'm not thinking of, but I'm stuck. Could anyone explain how to get from [equation 3] to [equation 5]?

Answer 1

In this context, $q=(n_1,n_2,\ldots)$ is a list of the occupation numbers of each energy level. When we sum over $q$, we are summing over every possible such list where each $n_i\in\{0,1\}$.

As a toy example, let's assume that our system has 2 energy levels with energies $\epsilon_1$ and $\epsilon_2$. We would then have

$$\sum_q \prod_i \exp\big[-\beta n_{q,i}(\epsilon_i-\mu)\big]= \sum_q \exp\big[-\beta n_{q,1}(\epsilon_1-\mu)\big] \exp\big[-\beta n_{q,2}(\epsilon_2-\mu)\big]$$ Now we need to sum over the four possibilities $q\in\{(0,0),(1,0),(0,1),(1,1)\}$. A quick bit of algebra should convince you that this is equal to

$$\left(\sum_{n_{q,1}=0}^1 \exp\big[-\beta n_{q,1}(\epsilon_1-\mu)\big]\right)\left(\sum_{n_{q,2}=0}^1 \exp\big[-\beta n_{q,2}(\epsilon_2-\mu)\big]\right)$$ $$= \prod_i \left(1+\exp\big[-\beta(\epsilon_i- \mu)\big]\right)$$

I'm stuck with the summation over macrostates $\{q\}$ being the same as the sum over microstates $\{n_i\}$.

I would also like to point out that the terminology used by your source is non-standard in a way that invites enormous confusion. Each $q$ is a microstate of the system, not a macrostate. It is a precise specification of the occupancy of each energy level, and consists of a list $q=(n_1,n_2,\ldots)$. The individual energy levels are neither macrostates nor microstates of the system.