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

TL;DR

This paper investigates the thermodynamics of a noninteracting Bose gas in a 2D box, revealing that Bose-Einstein condensation occurs only at finite particle numbers without a true phase transition, and analyzes ensemble differences.

Contribution

It provides a detailed analysis of BEC in 2D finite systems, showing the absence of a phase transition in the thermodynamic limit and comparing grand canonical and canonical ensemble predictions.

Findings

BEC occurs at finite N without a phase transition.

Transition temperature scales as 1/log(N).

Canonical ensemble accurately predicts thermodynamics at all sizes.

Abstract

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.