Grand canonical partition function: factorization

Question

The grand canonical ensemble partition function is defined to be $$ \mathcal{Z} := \sum_{\forall |n\rangle} e^{\beta \mu N_{|n\rangle} - \beta E_{|n\rangle}} $$ being $|n\rangle$ a notation for each microstate (not necessarily quantum, just a microstate), $N_{|n\rangle}$ the number of particles of that microstate and $E_{|n\rangle}$ the energy corresponding to this microstate. This is the formal definition.

Now I have from my lecture notes that when we are dealing with non-interacting and indistinguishable particles (e.g. an ideal gas) the partition function can be written as $$ \mathcal{Z} = \prod_{\forall \epsilon_i} \sum_{\forall \text{ allowed } n}(z e^{-\beta \epsilon _i})^n $$ now being $\epsilon _i$ each "monopartiuclar state" and "$\forall \text{ allowed } n$" is $\{0,1\}$ for fermions and $\{0,1,2,\dots\}$ for bosons. The "monoparticular states" are the energy levels that each particle can be in.

The question: How do we go from the definition to the second formula?

---

My approach (wrong, or at least incomplete)

If particles do not interact between one another then the energy of the microstate $|n\rangle$ can be written as a summation $$ E_{|n\rangle} = \epsilon_1 + \epsilon_2 + \dots + \epsilon_{|n\rangle} = \sum_{i=1}^{N_{|n\rangle}} \epsilon_i $$ where $\epsilon_i$ is the energy of each particle. Thus the partition function writes $$ \mathcal{Z} = \sum_{\forall |n\rangle} e^{\beta \mu N_{|n\rangle}} \prod_{i=1}^{N_{|n\rangle}} e^{-\beta \epsilon _i}$$ where I have already expanded the exponential of a summation as a product of exponentials.

Now, if particles are identical then the allowed values for $\epsilon _i$ are for all the particles the same, say $\epsilon _i \in \{\varepsilon_0, \varepsilon_1, \dots\}$. Using this notation, $\epsilon_8 = \varepsilon_3$ reads as "particle number 8 is in the third energy level". This allows to arrange the microstates of the system as follows (sorry for changing the language in the pic):

Now I don't know how to go on... Any help is appreciated.

Answer 1

You are misunderstanding what is meant by a product over monopartiuclar states. This is not a product over the states of the $N$ particles in the system, it is a product over all possible single particles states. The sum is then over the occupation number of those states, i.e. the number of particles actually in that state (which may be $0$). The advantage of this occupation number approach to over keeping track of individual particles is that it automatically takes account of particle indistinguishably.

The total number of particles in a state $\gamma$ is clearly simply the sum of the number of particles in each single particle state $$ N_\gamma = \sum_i n_i $$ The total energy is the energy of each single particle state, times the number of particles in that state $$ E_\gamma = \sum_i \epsilon_i n_i $$

Now the formula follows from a simple manipulation \begin{align} \mathcal{Z} &= \sum_{\gamma} e^{-\beta(E_\gamma - \mu N_\gamma)}\\ &= \sum_{\gamma} e^{-\beta\sum_i n_i(\epsilon_i-\mu)}\\ &= \sum_{n_0}\sum_{n_1}\ldots\; \left(\prod_i e^{\beta n_i(\mu - \epsilon_i)}\right)\\ &= \prod_i \sum_{n_i} \left(ze^{-\beta\epsilon_i}\right)^{n_i} \end{align}