Bethe ansatz cluster expansion method for a one-dimensional δ-function Bose gas

A one-dimensional repulsive δ-function Bose system is studied at finite temperature. By only using the Bethe ansatz (BA) equation, n-particle partition functions are exactly calculated. Such method is named direct method or BA cluster expansion method. From the expression for the n-particle partition function, the n-particle cluster integral is derived. The results completely agree with those of the thermal Bethe ansatz (TBA). This directly proves the validity of the TBA. The theory of partitions and graphs is used to make the presentation simple and transparent.