J. Phys. I France
Volume 7, Numéro 3, March 1997
Page(s) 485 - 508
DOI: 10.1051/jp1:1997171
J. Phys. I France 7 (1997) 485-508

Ursell Operators in Statistical Physics III: Thermodynamic Properties of Degenerate Gases

P. Grüter1, F. Laloë1, A.E. Meyerovich2 and W. Mullin3

1  Laboratoire Kastler Brossel de l'ENS, Laboratoire associé au CNRS, UA 18, et à l'Université Pierre et Marie Curie, 24 rue Lhomond, 75005 Paris, France
2  Department of Physics, University of Rhode Island, Kingston RI 02881, USA
3  Hasbrouck Laboratory, University of Massachusetts, Amherst Mass 01003, USA

(Received 29 August 1996, received in final form 31 October 1996, accepted le 19 November 1996)

We study in more details 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 any dilute 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 to 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, for fermions at very low temperatures, it may change sign or increase almost exponentially. As an illustration, numerical results for quantum hard spheres are given.

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