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.
Third- and higher-order quantum virial coefficients require the solution of the corresponding quantum many-body problem. Nevertheless, in an earlier paper [Phys. Rev. Lett. 108, 260402 (2012)], we proposed that the higher-order cluster integrals of a dilute unitary fermionic gas may be approximated in terms of the two-body cluster together with an appropriate suppression factor. Although not exact, this ansatz gave a fair agreement up to fugacity $z\ensuremath{\approx}6$ with the experimentally obtained equation of state. The objective of the present Brief Report is to give some physical arguments in favor of this ansatz.
This paper studies the problem of a quantum gas of particles whose interparticle potential is attractive enough to support stable bound clusters. The approach to this problem is to employ the time-dependent two-Hilbert-space scattering theory---first developed by Faddeev to treat the three-body problem. From this theory the asymptotic completeness theorem is taken and utilized to give a new definition of the canonical partition function suitable for describing states that are a mixture of free particles and bound clusters. A multispecies cluster expansion is obtained for the grand partition function. The theory of few-particle time delay is employed to determine the values of these cluster integrals. From these solutions are obtained the multidensity virial equation of state, the internal energy, and a density expansion of the chemical mass-action law that gives the effect of interactions between particles and clusters. All of these solutions exhibit the fact that the only aspect of the collision process that has any effect on the macroscopic behavior of the system is the time delay. From another perspective this problem and the solutions obtained represent a specific nontrivial realization of the
S-matrix description of statistical mechanics put forth by Dashen.
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.
Extending some previous work of Boll\'e [Phys. Rev. A 36, 3259 (1987)], we construct expressions for the n-particle cluster coefficients that remedy difficulties associated with the presence of both an infinite number of bound states and strongly repulsive potentials. Applying the existence theorems for cluster coefficients implicit in the work of Ginibre [J. Math. Phys. 6, 238 (1965); 6, 252 (1965); 6, 1432 (1965)], we demonstrate the validity of these expressions. We apply the formalism to obtain a well-behaved expression for
b3 at the Efimov point. As a by-product we are able to directly observe the divergence in
b3 due to the continuum part of the three-body resolvent together with the counterterms that render it convergent.
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.