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 Nin Yang. In one dimensional case correlation functions also were evaluated. The In one dimension Bose gas is equivalent to quantum non-linear Schroedinger equation.
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