Phase-Fluctuating 3D Bose-Einstein Condensates in Elongated Traps

We find that in very elongated 3D trapped Bose gases, even at temperatures far below the BEC transition temperature $T_c$, the equilibrium state will be a 3D condensate with fluctuating phase (quasicondensate). At sufficiently low temperatures the phase fluctuations are suppressed and the quasicondensate turns into a true condensate. The presence of the phase fluctuations allows for extending thermometry of Bose-condensed gases well below those established in current experiments.