At a critical temperature (very close to absolute zero), ideal 3D bose gasses will condense into a Bose-Einstein condensate, where many atoms occupy the same quantum state. Fluctuations in the density of the gas are suppressed and there is phase coherence across the system (the condensate may be described by a complex-valued wavefunction). Such a system is exhibits the famous quantum-mechanical particle-wave duality, but on a (nearly) macroscopic scale.

This paper (arxiv) shows the phase diagram (below) of a 1D bose gas is somewhat more complicated.

In an infinite-sized 1D system, a Bose gas cannot condense into such a Bose-Einstein condensate, since phase fluctations destroy the coherence across the system. In this case, the density fluctations are still supressed, and the system is called a quasi-condensate. In a finite system, at sufficiently low temperature and high atom number, the phase fluctuations may be of such large wavelength that they are bigger than the size of the system itself, and the gas behaves exactly as a true Bose-Einstein condensate.

This is not the entire picture; if a gas is sufficiently dilute, or the repulsive interactions are strong, the atoms can be thought of as localised individual particles. In 1D, the particles cannot pass each other, as in higher dimensional systems. As a consequence of this, the atoms gain some properties of Fermi atoms (such Fermionised bosons cannot occupy the same position state - however, unlike Fermions, they can still occupy the same momentum state). This system is called a Tonks gas.

Moreover, when the inter-particle collisions in a Tonks gas are elastic, the system becomes integrable (it has "regular motion"), and the atoms cannot obtain a thermal equilibrium since the energy cannot divide itself equally between the particles. This system acts like a quantum Newton's cradle (this Newton's cradle gif animation is taken from Wikipedia).

The moral of this post is that there is a lot to consider when modelling a 1D Bose gas.

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