Dimensional Crossover for the Bose – Einstein Condensation of an Ideal Bose Gas∗

We study an ideal Bose gas of N atoms contained in a box formed by two identical planar and parallel surfaces S , enclosed by a mantle of height a perpendicular to them. Calling r0 the mean atomic distance, we assume S ≫ r 0 , while a may be comparable to r0. In the bidimensional limit (a/r0 ≡ γ << 1) we find a macroscopic number of atoms in the condensate at temperatures T ∼ O(1/ log(N)); therefore, condensation cannot be described in terms of intensive quantities; in addition, it occurs at temperatures not too low in comparison to the tridimensional case. When condensation is present we also find a macroscopic occupation of the low–lying excited states. In addition, the condensation phenomenon is sensitive to the shape of S. The former two effects are significant for a nanoscopic system. The tridimensional limit is slowly attained for increasing γ, roughly at γ ∼ 10 − 10.