Partition Function for Phonons

[Master J]

In looking at phonons, and their energy, I came across the Partition function. THis was needed to calculate the internal energy of the solid.

But howcome the Partition function is used, and not the GRAND Partition function? The number of phonons is not conserved, I know that, but isn't N, the number, fixed in the Partition function? Surely one should used the GPF to take into acount the varying number??


I guess I have misunderstood something here. Any help appreciated!

 

[kanato]

Well, within the thermodynamic limit, both partition functions give equivalent results, so which one you use is a matter of convenience. Varying particle numbers can be taken into account in the canonical ensemble, it just is not as convenient. Since particle number for phonons is never conserved, the chemical potential is always zero, which makes the grand partition function for phonons trivially related to the partition function for a collection of harmonic oscillators.