Thermodynamics of the Noninteracting Bose Gas in a Two-Dimensional Box

Bose-Einstein condensation (BEC) of a noninteracting Bose gas of $N$ particles in a two-dimensional (2D) box with Dirichlet boundary conditions is studied. Confirming previous work, we find that BEC occurs at finite $N$ at low temperatures $T$ without the occurrence of a phase transition. We further show that the crossover temperature between weak and strong increases in BEC upon cooling is $T_{\rm E} \sim 1/\log(N)$ at fixed area per boson, so in the thermodynamic limit there is no significant BEC in 2D at finite $T$. Calculations of thermodynamic properties versus $T$ and area $A$ are presented, including Helmholtz free energy, entropy $S$, pressure $p$, ratio of $p$ to the energy density $U/A$, heat capacity at constant area $C_{\rm V}$ and at constant pressure $C_{\rm p}$, isothermal compressibility $\kappa_{\rm T}$ and thermal expansion coefficient $\alpha_{\rm p}$, obtained using both the grand canonical ensemble (GCE) and canonical ensemble (CE) formalisms. The GCE formalism gives acceptable predictions for $S$, $p$, $p/(U/A)$, $\kappa_{\rm T}$ and $\alpha_{\rm p}$ at large $N$, $T$ and $A$, but fails when $N$ is small or BEC is significant, whereas the CE formalism gives accurate results even at low $T$ and/or $A$ where BEC occurs.