We study in more detail the properties of the generalized Beth Uhlenbeck formula obtained in a preceding article. This formula leads to a simple integral expression of the grand potential of the system, where the interaction potential appears only through the matrix elements of the second order Ursell operator
U2. Our results remain valid for significant degree of degeneracy of the gas, but not when Bose Einstein (or BCS) condensation is reached, or even too close from this transition point. We apply them to the study of the thermodynamic properties of degenerate quantum gases: equation of state, magnetic susceptibility, effects of exchange between bound states and free particles, etc. We compare our predictions to those obtained within other approaches, especially the ``pseudo potential'' approximation, where the real potential is replaced by a potential with zero range (Dirac delta function). This comparison is conveniently made in terms of a temperature dependent quantity, the ``Ursell length'', which we define in the text. This length plays a role which is analogous to the scattering length for pseudopotentials, but it is temperature dependent and may include more physical effects than just binary collision effects; for instance at very low temperatures it may change sign or increase almost exponentially, an effect which is reminiscent of a precursor of the BCS pairing transition. As an illustration, numerical results for quantum hard spheres are given.
The Beth Uhlenbeck formula gives an exact (quantum) expression of the second virial correction to the equation of state of a (slightly degenerate) dilute gas. We show how this result can be extended to arbitrary degeneracy provided that the interaction potential has a sufficiently short range. For this purpose we develop a formalism based on the use of Ursell operators, which contain no symmetrization in themselves (they correspond to an auxiliary system of distinguishable particles) and we show how they can be used for a system of identical particles. A concise expression generalizing the Beth Uhlenbeck formula is obtained, which is equally valid for bosons and fermions Higher order corrections are also introduced. The formalism is rather general and will be applied to other cases in forthcoming articles.