Partition Function in Torres-Hernandez "Photon Mass" 1984

[sayebms]

in the paper written by Jose Torres-Hernandez in 1984 titled as : "Photon mass and blackbody radiation" in the first page he writes for the partition function:

$lnZ=λ \sum_{normal modes} e^{-βε_l}$ = $\frac{-λπ}{2} \int_{ε_0}^∞ n^2 ln(1-e^{-βε}) \frac{dn}{dε}dε$

i really don't understand how he changes the summation to the integration here, and why the integration is not inside the $ln$ .In 2 days i have to prepare a summary of this paper so any help is greatly appreciated.

 

[maajdl]

First of all, I would write "Z = ..." instead of "lnZ = ..." .
See the definition of the partition function.
http://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics )

The next step would be to find out what is the exact definition of n(ε) in the paper.
I guess dn represents the somehow number of modes in the range of energies dε .
This allows an approximation of the sum by an integral.

I do not have access to the original paper.
Therefore, I can't be more precise.

 

[sayebms]

first of all id like to thank you for taking time to answer my question. I have attached the 1st page of the original paper. well here we are dealing with lnZ since, later to find the average energy we have to take partial derivative of lnZ with respect to β.