  We analyze the cross-over of a homogeneous one-dimensional Bose gas from the
ideal gas into the dense quasi-condensate phase. We review a number of
mean-field theories, perturbative or self-consistent, and provide accurate
evaluations of equation of state, density fluctuations, and correlation
functions. A smooth crossover is reproduced by classical-field simulations
based on the stochastic Gross-Pitaevskii equation, and the Yang-Yang solution
to the one-dimensional Bose gas.
