partition function

Pacopag

1. The problem statement, all variables and given/known data
If we have a system of N independent particles and the partition function for one particle is Z_1, then is the partition function for the N particle system Z=(Z_1)^N?


2. Relevant equations



3. The attempt at a solution
I'm pretty sure that this is true for a classical system, but I'm not sure if it's true for a quantum system. Does the Pauli exclusion principle spoil this somehow?

genneth

Even without quantum considerations, you end up with over counting if the particles are identical. See Gibb's Paradox.

Pacopag

Right. Sorry, I meant to write
[tex]Z={1\over {N!}}(Z_1)^N[/tex]
Does that take care of over counting?
What about the quantum case?

genneth

It is not true for the quantum case. The quantum case is easier handled via grand-partition function.

Pacopag

So in the quantum case, if we want to use the canonical ensemble, we have to calculate the whole partition function all in one shot?

genneth

Yep. But like I said, usually, you calculate the grand-partition function (which factorises neatly into a function of single particle states).