  We investigate the level population statistics and degree of coherence
encoded in the single-particle density matrix of harmonically trapped
low-dimensional [quasi-one-dimensional (quasi-1D) or quasi-two-dimensional
(quasi-2D)] Bose gases with repulsive contact interactions. Using a variational
analysis, we derive fragmentation of the condensate in the weakly confining
directions into two (quasi-1D) respectively three (quasi-2D) mutually
incoherent macroscopic pieces, upon increasing a dimensionless interaction
measure beyond a critical value. Fragmented condensate many-body states in
low-dimensional systems therefore occur well before the thermodynamic limit of
infinite extension is reached, in which phase fluctuations of the matter wave
field create an infinite number of nonmacroscopic fragments.
