# Grand Canonical Partition Function for Simple System

## Homework Statement

I would like to calculate the grand canonical partition function (GCPF) for a system in which there are are $m$ lattice sites. A configuration may be specified by the numbers $(n_1, n_2, ... , n_m)$, where $n_k = 1$ if a particle occupies site $k$ and $n_k = 0$ if no particle occupies site $k$. Occupied sites have an associated energy $\epsilon$ (constant) and unoccupied sites have zero associated energy.

## Homework Equations

The general form of the GCPF in my book (Chandler) is given like this:

$$\Xi = \sum_{\nu} e^{ -\beta E_{\nu} + \beta \mu N_{\nu} }$$

where $\nu$ indicates a summation over all states. (I am confused as to what, exactly, is meant by a "state" in the context of this problem.)

## The Attempt at a Solution

For a given state $j$ the number of particles is given by $N_j = \sum_{j=1}^{m} n_i$ (summing over all sites). For the same state $j$ the energy is given by $E_j = \epsilon \sum_{i=1}^{m} n_i$.

I'm unsure of the correct direction from here. Inserting the expressions for $N_j$ and $E_j$ into $\Xi$ creates a mess of summations. Is that the only way? Is it simplify-able?

I feel that I should be able to calculate the GCPF for just one site and then extend the result to $m$ sites since the sites are independent of one another. Is this possible? If so, how?

Thank you all.

$\sum_{\nu} \rightarrow \sum_{n=0}^{m} \omega(n)$
where $\omega(n)$ is the density of states, ie. the number of microstates (n1,n2,...,nm) corresponding to the macrostate.