Fermi-Bose Correspondence and Bose-Einstein Condensation in The Two-Dimensional Ideal Gas

TL;DR

This paper explores the relationship between 2D Fermi and Bose gases, deriving May's Theorem for their internal energies in the thermodynamic limit and examining finite-size effects leading to pseudo-Bose-Einstein condensation.

Contribution

It derives May's Theorem for 2D gases and analyzes finite-size effects, revealing the emergence of pseudo-Bose-Einstein condensation in finite 2D Bose gases.

Findings

Fermi and Bose gases have the same heat capacity in the thermodynamic limit.

Finite 2D Bose gases exhibit pseudo-Bose-Einstein condensation at low temperatures.

Finite-size effects cause deviations from May's Theorem in 2D gases.

Abstract

The ideal uniform two-dimensional (2D) Fermi and Bose gases are considered both in the thermodynamic limit and the finite case. We derive May's Theorem, viz. the correspondence between the internal energies of the Fermi and Bose gases in the thermodynamic limit. This results in both gases having the same heat capacity. However, as we shall show, the thermodynamic limit is never truly reached in two dimensions and so it is essential to consider finite-size effects. We show in an elementary manner that for the finite 2D Bose gas, a pseudo-Bose-Einstein condensate forms at low temperatures, incompatible with May's Theorem. The two gases now have different heat capacities, dependent on the system size and tending to the same expression in the thermodynamic limit.