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Fermi-Bose Correspondence and Bose-Einstein Condensation in The Two-Dimensional Ideal Gas
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.Comment: 18 pages, 3 figures in EPS format, to be published in Journal of Low
Temperature Physic
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