No bose-Einstein condensate for a 2-d gas, how to prove it?

[fluidistic]

Homework Statement

Consider a 2-dimensional Bose-Einstein ideal gas.
1)Calculate the grand partition function of that system.
2)Calculate the mean number of particles per unit area in function of T and z, the fugacity.
3)Show that there's no Bose-Einstein condensate for this system.

Homework Equations

Several...

The Attempt at a Solution

1)Done.
2)Done. Yielded $\frac{\langle N \rangle}{A}=\frac{gmkT}{2\pi \hbar ^2} \ln \left ( \frac{1}{1-z} \right )$.
3)I have no idea on how to answer this question.
I guess I would have to calculate the mean number of particles per unit area that are in the ground state (<e>=0) and show that it doesn't "blow up" for any temperature...
Or maybe I could do it simply by using my result obtained in part 2)?

I also took the limit of the expression that I got in part 2), for when T goes to 0. And I reached <N>/A tends to 0. (With a software because I do not know how to tackle that limit which has an undetermined form as 0 times positive infinity).

Thank you for any help!

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[fluidistic]

I am thinking deeply about it, I still don't understand why there isn't a B-E condensate for a 2-dimensional boson gas.
Is it because the ground state would be macroscopically populated at T=0K, while in the 3-dimensional case this would be true for T>0K? Is that the reason?
I really don't understand what's going on... please help me.