Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
We present a field theoretic method for the calculation of the second and third virial coefficients
b2 and
b3 of two-species fermions interacting via a contact interaction. The method is mostly analytic. We find a closed expression for
b3 in terms of the two- and three-body
T matrices. We recover numerically, at unitarity, and also in the whole Bose-Einstein-condensate--BCS crossover, previous numerical results for the third virial coefficient
b3.
The quantum theory of the third virial coefficient
C is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of
CBE are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for
He4 and
He3, by means of a comparison of the corresponding expansion for the second virial coefficient
B with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to
B. Then the leading term of
C is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for
C is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
We derive three-particle higher-order sum rules in two dimensions in analogy with similar rules existing in three dimensions. We employ these rules to study the Planck-Larkin structure of the third cluster integral in the presence of three-particle processes like rearrangement scattering and breakup. In particular, in three dimensions we also show the cancellation between bound-state and continuum contributions at the Efimov point. As a byproduct we obtain a Wigner-Kirkwood-type expansion for the third virial coefficient in two and three dimensions.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.
Expansion formulas for density matrices are derived with the use of the calculus of ordered exponentials. Using the formulas the expressions for virial coefficients are explicitly obtained. The expression of the third virial coefficient is calculated to obtain its expansion formula in terms of
ℏ2β/M. The expansion formulas for density matrices can be used to derive a new formula of multiple scattering, which involves Luttinger and Kohn's formula as a special case. The expansion formula for the normalized density matrix is also given. Expansion formulas which are to the applied to irreversible processes and relaxation phenomena are also given.