Expanded out, the grand potential is:
All thermodynamic properties can be computed from this potential. The following table lists various thermodynamic quantities calculated in the limit of low temperature and high temperature, and in the limit of infinite particle number. An equal sign (=) indicates an exact result, while an approximation symbol indicates that only the first few terms of a series in  is shown.
It is seen that all quantities approach the values for a classical ideal gas in
the limit of large temperature. The above values can be used to calculate other
thermodynamic quantities. For example, the relationship between internal energy and
the product of pressure and volume is the same as that for a classical ideal gas over
all temperatures:
A similar situation holds for the specific heat at constant volume
The entropy is given by:
Note that in the limit of high temperature, we have
which, for α=3/2 is simply a restatement of the Sackur–Tetrode equation. In one dimension bosons with delta interaction behave as fermions, they obey Pauli exclusion principle. In one dimension Bose gas with delta interaction can be solved exactly by Bethe ansatz. The bulk free energy and thermodynamic potentials were calculated by Chen-Ning Yang. In one dimensional case correlation functions also were evaluated. In one dimension Bose gas is equivalent to quantum non-linear Schrödinger equation.