 Solution

For a homogeneous ideal gas in a two-dimensional box of area $A$, with dispersion $ϵ=p^2/(2m)$ and one internal state, the density of states is $g(ϵ)=Am/(2\pi {\hslash }^2)$. The excited-particle density at temperature $T>0$ isAs $\mu \to 0^{-}$, this diverges. Hence at every fixed positive temperature the excited states can accommodate any finite areal density with $\mu <0$; they never saturate at a finite critical density. There is no finite-temperature Bose-Einstein condensation in the homogeneous two-dimensional ideal gas thermodynamic limit.

The low-energy logarithmic divergence is the reason, not merely the small number of spatial dimensions. A finite box has a low-energy cutoff and a crossover; a trap changes the density of states and can change the conclusion. Condensation at zero temperature and an interacting gas's Berezinskii–Kosterlitz–Thouless transition are separate issues. The proof assumes the uniform quadratic-dispersion model, rather than every conceivable two-dimensional trapping potential.