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Multichannel approach to clustering matter

Vyacheslav I. Yukalov, Elizaveta P. Yukalova

Published Jan 1, 1997• 20 citations
OpenAlex

Abstract

An approach is developed, combining the ideas of quantum statistical mechanics and multichannel theory of scattering, for treating statistical systems whose constituents can possess different bound states realized as compact clusters. The main principles for constructing multichannel cluster Hamiltonians are formulated: principle of statistical correctness, principle of cluster coexistence, and principle of potential scaling. The importance of the principle of statistical correctness is emphasized by showing that when it does not hold the behaviour of thermodynamic functions becomes essentially distorted. And moreover, unphysical instabilities can appear. The ideas are carefully illustrated by a statistical model of hot nuclear matter.

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An approach is developed, combining the ideas of quantum statistical mechanics and multichannel theory of scattering, for treating statistical systems whose constituents can possess different bound states realized as compact clusters. The main principles for constructing multichannel cluster Hamiltonians are formulated: principle of statistical correctness, principle of cluster coexistence, and principle of potential scaling. The importance of the principle of statistical correctness is emphasized by showing that when it does not hold the behaviour of thermodynamic functions becomes essentially distorted. And moreover, unphysical instabilities can appear. The ideas are carefully illustrated by a statistical model of hot nuclear matter.
In this paper we review the coupled cluster formulation of quantum many-body theory.Apart from a brief and heuristic, but essentially complete, discussion of 8 the by now rather standard tr-eatments of both ground and excited states within the 2 coupled cluster method (CeM), our main aim is to.give a rather careful critique of the various choices that have to be made in order to implement .themethod.These lS--•divide essentially into two classes.On the one hand we have the inputs to the method, 11 which amount largely to.achoice.ofstarting (or zero-order or model or uncorrelated) wayefunctions; while on the other hand we have the choice of decomposition of the ~equations into a natural hierarchy, and which amounts in practice to a particular approximation or truncati on scheme.We shall show expl tc+t ly .thatwhil e these two 13 ---choices are in principle independent of each other, in practice they are intimately 14
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.
Bethe's treatment of three-body clusters is extended to the entire linked-cluster perturbation series. The result is a completely rearranged expansion in terms of "compact clusters." The spatial correlations within these terms are all of quite short range. As a direct consequence, each of these new compact-cluster terms is proportional to ${\ensuremath{\kappa}}^{{h}^{\ensuremath{'}}\ensuremath{-}1}$, where ${h}^{\ensuremath{'}}$ is the number of momenta inside the Fermi sea which can be summed independently after allowing for momentum conservation. The "small parameter" of the expansion is $\ensuremath{\kappa}=\ensuremath{\rho}\ensuremath{\int}{|\ensuremath{\zeta}|}^{2}d\ensuremath{\tau}$, where $\ensuremath{\rho}$ is the ordinary nuclear density and $\ensuremath{\zeta}=\ensuremath{\varphi}\ensuremath{-}{\ensuremath{\psi}}_{\mathrm{BG}}$ is the "wound" in the correlated two-body (bethe-Goldstone) wave function. In nuclear matter this $\ensuremath{\kappa}$ is of order 10%. This result requires that the single-particle potentials be defined in terms of a certain subset of all the self-energy insertions. The allowed insertions are those which can be evaluated entirely on the energy shell, by means of a "generalized time-ordering" factorization. The totality of these insertions is said to constitute an "on-energy-shell mass operator" Mon{M}^{\mathrm{on}}Mon. The resulting occupied-state potentials are quantitatively very similar to those of most previous versions of nuclear-matter theory. However, the present intermediate-state potentials are small enough (being only of order +1 MeV) to be safely ignored in practical calculations. This simplification is offset by the need for a separate calculation of the three-body clusters. These single-particle potentials are "physically meaningful" in the sense that they lead to an optimum treatment of the short-range correlations. The bulk properties of nuclear systems are determined almost entirely by these short-range correlations. It is therefore proposed that similar potentials be used in a many-body theory of finite nuclei. True occupation numbers occur in a simple and natural way throughout the entire expansion. The formal and practical consequences of this feature are carefully examined. It is argued that this is significant for a theory of finite nuclei. Comparisons are made with several different "renormalized" formulations of quantum statistical mechanics, and also with the Green's-function theory of nuclear matter. The present formulation tends to artificially suppress the long-range correlations (e.g., pairing correlations) between particles near the Fermi surface. This defect can be eliminated by using a degenerate analog of the Goldstone expansion.
The field theoretical formulation of quantum mechanics is used to consider the nonrelativistic multichannel scat~i~ng theory. With the help of appropriately constructed time dependent creation operators, Hilbert vectors are formed whose limits in time can be defined as multiehanne) scattering states in the usual sense. The existence of these states is proved under certain assumptions for the potential, by showing the convergence of the above mentioned operators. The commutation relations for the limits of these operators are given.