How to Write a Canonical Ensemble for a System Using the Einstein Model

[romeo6]

Hi there,

I have a system with the following energy using the einstein model:

$$E_\nu=\sum_{i=1}^{2N} h\omega n_i+\sum_{j=1}^{N} h\omega n_j$$
I need to set up a canonical ensemble for this.

How would I write the partition function please?

 

[Dr Transport]

Isn't the partition function defined as

$$\int \frac{d^{3N}p d^{3N}q}{N! h^{3N}} e^{-\beta H(p,q)}$$

for $$N$$ particles. You just need to substitute for the energy that you have and perform the integration (hint integrate over all the freuqncies $$\omega$$ and you should only have to do one and simplify).

 

[dextercioby]

First, it't $\hbar$ instead of "h" in E_{\nu}. Second, this looks like an application of quantum statistics, and so the classical partition function won't be of any use.

Daniel.

 

[romeo6]

Thanks, I think my problem is that I don't know how to set this up as thesummations have different upper values.

I want to write something like:

$$Q=\sum_{i=1}^{2N} e^{-\beta \hbar n_i}+\sum_{j=1}^N e^{-\beta \hbar n_j}$$

where Q is the partition function.

Does this look anything like the partition function?