Consider an ideal Bose gas confined to a region of area \(A\) in two dimensions. Express the number of particles in the excited states, \(N_{e}\), and the number of particles in the ground state, \(N_{0}\), in terms of \(z, T\), and \(A\), and show that the system does not exhibit Bose-Einstein condensation unless \(T ightarrow 0 \mathrm{~K}\).

Refine your argument to show that, if the area \(A\) and the total number of particles \(N\) are held fixed and we require both \(N_{e}\) and \(N_{0}\) to be of order \(N\), then we do achieve condensation when

\[
T \sim \frac{h^{2}}{m k l^{2}} \frac{1}{\ln N}
\]

where \(l[\sim \sqrt{ }(A / N)]\) is the mean interparticle distance in the system. Of course, if both \(A\) and \(N ightarrow \infty\), keeping \(l\) fixed, then the desired \(T\) does go to zero.