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## Homework Statement

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

## Homework Equations

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

[tex] \Xi = \sum_{\nu} e^{ -\beta E_{\nu} + \beta \mu N_{\nu} } [/tex]

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

I'm unsure of the correct direction from here. Inserting the expressions for [itex] N_j [/itex] and [itex] E_j [/itex] into [itex] \Xi [/itex] 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 [itex]m[/itex] sites since the sites are independent of one another. Is this possible? If so, how?

Thank you all.