Multispecies cluster expansion for the grand partition function

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.