Bose-Einstein condensation in a two-dimensional system at constant pressure

Using techniques developed by Greenspoon and Pathria, a rigorous, asymptotic analysis of the onset of Bose-Einstein condensation in a finite two-dimensional system at constant pressure is carried out. Some of the results obtained by Imry, Bergman, and Gunther, who first considered this problem, are upheld while others get modified. In particular, a macroscopic occupation of the single-particle ground state at finite temperatures is indeed possible. As the critical region, $T\simeq T_c$, is approached from above, the volume of the system becomes subextensive, $V_c=O\left(\frac{N}{\ln N}\right)$, which is both necessary and sufficient for the onset of Bose-Einstein condensation in the system. For instance, if the system is now cooled even at constant volume $V(=V_c)$, the condensate fraction gradually builds up and becomes nonnegligible as $T$ approaches $T_0\simeq \frac{\left(T_c\right)}{2}$; below $T_0$, it grows steadily as $(1-\frac{T}{T_0})$. On the other hand, if we continued to cool the system at constant $P$ its volume, over a thermodynamically negligible range of temperatures, would reduce to values $O\left(N^{\frac{1}{2}}\right)$ which, in turn, would be accompanied by an abrupt accumulation of practically all the particles of the system into the single-particle ground state ${\epsilon }_0$. The specific heat $C_P$, after having passed through a maximum, would also finally reduce to subextensive values. Phase transitions of this kind may be of relevance to the physical behavior of thin helium films and helium submonolayers.