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Physical Review A

Multispecies cluster expansion for the grand partition function

T. A. Osborn

Published Jul 1, 1977• 19 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.16.334

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Physical Review A•Mar 1, 1989•1 citation
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 SSS-matrix description of statistical mechanics put forth by Dashen.
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.
The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.
A new cluster development for the logarithm of the grand partition function of a system of interacting particles is derived. The leading term in this expansion is the pressure exerted by an ideal Bose or Fermi gas at the same temperature and absolute activity ZZZ as the actual system. Succeeding terms involve quantum cluster integrals which themselves depend upon ZZZ, unlike their classical analogs. The definition of these cluster integrals follows in a natural fashion using techniques illustrated by construction (in closed form) of the successive ZZZ derivatives of the Bose and Fermi ideal gas grand partition functions. It is not possible (except in the classical limit) to eliminate ZZZ explicitly between the pressure and density series, so that the equation of state must remain in parametric form.
Macro properties of cold atomic gases are driven by few-body correlations, even if the gas has thousands of particles. Quantum systems composed of two and three particles with attractive zero\=/range pairwise interactions are considered for general masses and interaction strengths in two and three dimensions (2D and 3D). The Faddeev decomposition is used to derive the equations for the bound state, which is the starting point for the investigation of universal properties of few\=/body systems, i.e. those that all potentials with the same physics at low energy are able to describe in a model\=/independent form. In 2D, the number of bound states in a three\=/body system increases without bound as the mass of one particle becomes much lighter than the other two. The analytic form of an effective potential between the heavy particles explains the mass\=/dependence on the number of bound energy levels. An exact analytic expression for the large\=/momentum asymptotic behaviour of the spectator function in the Faddeev equation is presented. The spectator function and its asymptotic form define the two- and three\=/body contact parameters. The two\=/body parameter is found to be independent of the quantum state in some specific 2D systems. The 2D and 3D momentum distributions have a distinct sub\=/leading form whereas the 3D term depends on the mass of the particles. A model that interpolates between 2D and 3D is proposed and a sharp transition in the energy spectrum of three-body systems is found.
We discuss the relationship between the classical Lagrange theorem in mathematics and the quantum statistical mechanics and thermodynamics of an ideal gas of multispecies quasiparticles with mutual fractional exclusion statistics. First, we show that the thermodynamic potential and the density of the system are analytically expressed in terms of the language of generalized cluster expansions, where the cluster coefficients are determined from Wu's functional relations for describing the distribution functions of mutual fractional exclusion statistics. Second, we generalize the classical Lagrange theorem for inverting the one complex variable functions to that for the multicomplex variable functions. Third, we explicitly obtain all the exact cluster coefficients by applying the generalized Lagrange theorem.
In this paper we present a realization of the concept, first put forth by V. Efimov [Few Body Dynamics, edited by A. N. Mitra et al. (North-Holland, Amsterdam, 1976)], of enhancing the Efimov effect by placing the three particles in a medium whose parameters vary. We do this in the context of equilibrium statistical mechanics by incorporating three-body potentials and subsequently writing their contribution to the cluster series in terms of effective, density-dependent, two-body potentials. We then perform approximate partial summations of the cluster series expressing the sums in terms of cluster coefficients involving the effective potentials. From the high-temperature expansion of the corresponding equation of state for a model system, we find that, at the Efimov point, behavior resembling that of a first-order phase transition appears.