URSELL OPERATORS IN STATISTICAL PHYSICS .3. THERMODYNAMIC PROPERTIES OF DEGENERATE GASES

We study in more details the properties of the generalized Beth Uhlenb eck formula obtained in a preceding article. This formula leads to a s imple integral expression of the grand potential of any dilute system, where the interaction potential appears only through the matrix eleme nts of the second order Ursell operator U-2. Our results remain valid for significant degree of degeneracy of the gas, but not when Bose Ein stein (or BCS) condensation is reached, or even too close to this tran sition point. We apply them to the study of the thermodynamic properti es of degenerate quantum gases: equation of state, magnetic susceptibi lity, 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 po tential is replaced by a potential with zero range (Dirac delta functi on). This comparison is conveniently made in terms of a temperature de pendent quantity, the ''Ursell length'', which we define in the text. This length plays a role which is analogous to the scattering length f or pseudopotentials, but it is temperature dependent and may include m ore physical effects than just binary collision effects; for instance, for fermions at very low temperatures, it may change sign or increase almost exponentially. As an illustration, numerical results for quant um hard spheres are given.