Title: The second critical density and anisotropic generalised condensation
Author(s):

	M. Beau (Université de la Méditerranée and Centre de Physique Théorique - UMR 6207, Luminy - Case 907, 13288 Marseille, Cedex 09, France) ,
	V.A. Zagrebnov (Université de la Méditerranée and Centre de Physique Théorique - UMR 6207, Luminy - Case 907, 13288 Marseille, Cedex 09, France)

In this letter we discuss the relevance of the 3D Perfect Bose gas (PBG) condensation in extremely elongated vessels for the study of anisotropic condensate coherence and the "quasi-condensate". To this end we analyze the case of exponentially anisotropic (van den Berg) boxes, when there are two critical densities ρ_c<ρ_m for a generalised Bose-Einstein Condensation (BEC). Here ρ_c is the standard critical density for the PBG. We consider three examples of anisotropic geometry: slabs, squared beams and "cigars" to demonstrate that the "quasi-condensate" which exists in domain ρ_c<ρ<ρ_m is in fact the van den Berg-Lewis-Pulé generalised condensation (vdBLP-GC) of the type III with no macroscopic occupation of any mode. We show that for the slab geometry the second critical density ρ_m is a threshold between quasi-two-dimensional (quasi-2D) condensate and the three dimensional (3D) regime when there is a coexistence of the "quasi-condensate" with the standard one-mode BEC. On the other hand, in the case of squared beams and "cigars" geometries, critical density ρ_m separates quasi-1D and 3D regimes. We calculate the value of the difference between ρ_c, ρ_m (and between corresponding critical temperatures T_m, T_c) to show that the observed space anisotropy of the condensate coherence can be described by a critical exponent γ(T) related to the anisotropic ODLRO. We compare our calculations with physical results for extremely elongated traps that manifest "quasi-condensate".

Key words: anisotropic generalized Bose-condensation, van den Berg-Lewis-Pule condensation of type III, effect of exponential anisotropy, the second critical point for the BEC
PACS: 05.30.Jp, 03.75.Hh, 67.40.-w

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