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Journal de Physique I

Ursell operators in statistical physics I: Generalizing the Beth Uhlenbeck formula

P. Grüter, Franck Laloë

Published Feb 1, 1995• 6 citations
Publisher

Abstract

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.

DOI
10.1051/jp1:1995120

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We generalize our previous work on the compatibilty of kinetic equations with second virial corrections to the inclusion of spin and particle indistinguishability; the system is supposed to be sufficiently dilute for higher order virial density corrections (interactions and statistics) to be negligible. We show that the general idea of the “free Winger transform” can be extended to this situation; the function which appears in the kinetic equation becomes here a matrix which acts in the space of spin states of the particles. Assuming that the collisions are described by a hamiltonian which does not act on the spins (a very good approximation for nuclear spins), we write explicitly a kinetic equation which is valid for this case. The right hand side of the equation is an 18 dimension integral, as for spinless distinguishable particles, but here it contains an additional term due to statistics, which introduces commutators and anticommutators. We discuss the local conservation laws in this formalism and find, as expected, a total number of 8 conserved quantities for spin 1/2 particles (including three components of the magnetization). When the gas is at equilibrium, we obtain a pressure dependence which is in agreement with known calculations on spin polarized gases. We finally study the gradient expansion of the collision integral, and show that the zero-order (local) part is identical with the 4 terms (including identical spin rotation terms) obtained previously by Lhuillier et al. The first order (non-local) part contains many terms, wich we compare with those obtained by Silin in a context more closely related to the Landau theory.

Journal de Physique I•Feb 1, 1991•5 citations
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.
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 U2U_{2}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.
We study the properties of quantum gases which are dilute in terms of the interactions but degenerate; this may be the case if the thermal wavelength of the particles is much larger than the interatomic potential range. We assume that the potential contains a short range repulsive part, which creates strong correlations between particles that cannot be treated by ordinary perturbation theory, even if the region of configuration space where they are dominant is relatively small. Our method combines two basic ingredients: Ursell operators, which generalize the Ursell functions that are usually used to generate quantum cluster expansions, but are not in themselves symmetrized (the operators can be related to an auxiliary system of distinguishable particles); in a distinct step, an exact treatment of particle indistinguishability is given through the inclusion of exchange cycles of arbitrary length. For non-condensed boson systems, as well as for fermion systems, the method provides a compact expression of the partition function that generalizes the well-known Beth–Uhlenbeck formula to arbitrary degeneracy. For the study of the Bose–Einstein condensation, our treatment retains more information on the short range correlations than, for instance, the Hartree–Fock method. It also introduces a natural and explicit distinction between particles and quasiparticles; the latter may condense into a single one-particle state, which obeys a non-linear equation somewhat similar to the Gross–Pitaevskii equation.
We apply the formalism of Ursell operators, introduced in a previous article, to the calculation of reduced density operators (one and two-body density operators) in a dilute quantum gas; the calculation is not a fugacity expansion and is therefore not limited to low degrees of degeneracy. We obtain quantum corrections for the one-particle density operator as a function of the second Ursell operator. For the two-body density operator, we examine how statistics and interactions combine their effects on the correlations between particles; in particular we discuss in detail how hard cores potentials affect the short range correlations, a non-perturbative effect.
We generalize our previous work on the compatibilty of kinetic equations with second virial corrections to the inclusion of spin and particle indistinguishability; the system is supposed to be sufficiently dilute for higher order virial density corrections (interactions and statistics) to be negligible. We show that the general idea of the “free Winger transform” can be extended to this situation; the function which appears in the kinetic equation becomes here a matrix which acts in the space of spin states of the particles. Assuming that the collisions are described by a hamiltonian which does not act on the spins (a very good approximation for nuclear spins), we write explicitly a kinetic equation which is valid for this case. The right hand side of the equation is an 18 dimension integral, as for spinless distinguishable particles, but here it contains an additional term due to statistics, which introduces commutators and anticommutators. We discuss the local conservation laws in this formalism and find, as expected, a total number of 8 conserved quantities for spin 1/2 particles (including three components of the magnetization). When the gas is at equilibrium, we obtain a pressure dependence which is in agreement with known calculations on spin polarized gases. We finally study the gradient expansion of the collision integral, and show that the zero-order (local) part is identical with the 4 terms (including identical spin rotation terms) obtained previously by Lhuillier et al. The first order (non-local) part contains many terms, wich we compare with those obtained by Silin in a context more closely related to the Landau theory.