  Bose-Einstein condensation (BEC) of a noninteracting Bose gas of N particles
in a two-dimensional 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. The
conventionally-defined transition temperature TE for an infinite 3D system is
shown to correspond in a 2D system with finite N to a crossover temperature
between a slow and rapid increase in the fractional boson occupation N0/N of
the ground state with decreasing T. We further show that TE ~ 1/log(N) at fixed
area per boson, so in the thermodynamic limit there is no significant BEC in 2D
at finite T. Thus, paradoxically, BEC only occurs in 2D at finite N with no
phase transition associated with it. 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
volume (area) CV and at constant pressure Cp, isothermal compressibility
kappa_T and thermal expansion coefficient alpha_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_T and alpha_p
at large N, T and A, but fails for smaller values of these three parameters for
which BEC becomes significant, whereas the CE formalism gives accurate results
for all thermodynamic properties of finite systems even at low T and/or A where
BEC occurs.
