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 new, completely rigorous way of proving the relation between the second virial coefficient and the
S matrix. The method involves the use of the ${U}_{2,\ensuremath{\infty}}$ function introduced by Lee and Yang and proceeds in the most straightforward manner. Although difficulties are encountered for virial coefficients of higher order, there is hope that this type of approach might shed some light on the question of the connection between the virial series and the scattering matrix.
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 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.
A general method of calculation is described for quantum statistical mechanics. It is based on a simplification of the Laplace transform of the density matrix which follows from a theorem due to Hugenholtz. The basic result is that an element of the density matrix can be written as a sum over graphs, with the contribution of each graph factored into contributions from connected or linked graphs. Applied to the grand partition function, the exponential formula of Bloch and DeDominicis is obtained in a simple way. A similar formula is then derived for the canonical ensemble for the case of a nondegenerate gas. In this way the familiar result of Uhlenbeck and Beth is obtained for the second virial coefficient. Techniques are also introduced for evaluating ensemble averages of operators. In this connection, some care must be exercised in the case of diagonal operators. Finally, these methods are used to calculate the pair-correlation function for a system of fermions interacting through short-range forces.
Cluster coefficients for a quantum gas can be related by means of a Laplace transform to the resolvent of an interacting system. Techniques developed by Faddeev are applied in order to express resolvents in terms of quantities which satisfy coupled integral equations. The resulting theory for the cluster coefficients is free of convergence difficulties encountered in series expansions of those coefficients in terms of a binary-collision kernel or two-body scattering matrix. Present computational difficulties necessitate an approximate solution of the Faddeev equations. The assumption of a separable two-body scattering matrix makes possible such a solution and a subsequent calculation of cluster coefficients.