  We analyze the equilibrium properties of a weakly interacting, trapped
quasi-one-dimensional Bose gas at finite temperatures and compare different
theoretical approaches. We focus in particular on two stochastic theories: a
number-conserving Bogoliubov (ncB) approach and a stochastic Gross-Pitaevskii
equation (sGPe) that have been extensively used in numerical simulations.
Equilibrium properties like density profiles, correlation functions, and the
condensate statistics are compared to predictions based upon a number of
alternative theories. We find that due to thermal phase fluctuations, and the
corresponding condensate depletion, the ncB approach loses its validity at
relatively low temperatures. This can be attributed to the change in the
Bogoliubov spectrum, as the condensate gets thermally depleted, and to large
fluctuations beyond perturbation theory. Although the two stochastic theories
are built on different thermodynamic ensembles (ncB: canonical, sGPe:
grand-canonical), they yield the correct condensate statistics in a large BEC
(strong enough particle interactions). For smaller systems, the sGPe results
are prone to anomalously large number fluctuations, well-known for the
grand-canonical, ideal Bose gas. Based on the comparison of the above theories
to the modified Popov approach, we propose a simple procedure for approximately
extracting the Penrose-Onsager condensate from first- and second-order
correlation functions that is computationally convenient. This also clarifies
the link between condensate and quasi-condensate in the Popov theory of
low-dimensional systems.
