Bound Dirac states,different Lorentz-type couplings of central potentials and the non-relativistic limit

The analysis of bound radial Dirac states is shown to simplify for problems with an equal mixture of Lorentz vector and Lorentz scalar potentials, thus satisfying a so-called spin symmetry of the energy spectrum. Typical relativistic restrictions on potentials that are singular at the origin then disappear. Such potentials may even be strongly singular at the origin like the well known Lennard-Jones potentials modelling many atom-atom interactions, and they reduce to non-relativistic potentials of identical form. Bound state energies for potentials with equal vector- and scalar couplings are compared with those of a pure vector coupling of the same radial (attractive screened and unscreened Coulomb) shapes, and with non-relativistic results.     

Relation between the projection operator formalism and the Faddeev theory

In an earlier paper it was shown by means of the Faddeev equations that the reduction of the three-body problem to formal two-body equations with energy-dependent complex potentials — a procedure which was already well known for separable potentials — can be generalized rigorously to arbitrary potentials. In the present paper we show that the projection operator approach of Feshbach appears as a special, and in a certain sense singular case of our general treatment. Optical potentials for rearrangement scattering are given explicitly and questions of how to calculate them are discussed, showing that the Faddeev approach is better suited in these respects.     

Test of a Fokker-Planck-Kramers equation treatment for short-time dynamics of diffusion-controlled reactions by molecular dynamics simulation in Lennard-Jones fluids

We test the extended Harris (EXH) theory for the dynamics of diffusion-controlled reactions using the molecular dynamics simulations in Lennard-Jones fluids. The EXH theory is based on the Fokker-Planck-Kramers equation in which the inertia effect on the molecular migration is taken into account. The time dependence of the theoretical survival probabilities of the reactant agrees with the simulation results if the potential of mean force and the initial equilibrium distribution are appropriately taken into account. The position dependence of the diffusion coefficient is not necessary in explaining the simulation results. On the other hand, the simulation results cannot be reproduced if the two-body or the uniform potentials are employed in the theoretical calculations.     

Derivation of three-body collision-induced light-scattering spectrum moments for argon,krypton, xenon

The expressions for the zeroth and second moments of the three-body component of the collision-induced light-scattering spectrum are derived within the pair-potential and pair-polarizability approximations.

Computer calculation of the three-dimensional integrals which appear in the moment expressions is performed for Lennard-Jones and Barker potentials of argon, krypton and xenon within the dipole-induced-dipole approximation for the pair-polarizability.

The comparison of the results which are obtained with the two different potentials shows the very important role which is played by the choice of the potential in the determination of the values of the moments. An example of the applicability of the moment analysis of the three-body spectrum is also given for argon and the results compared with those of two-body.     

Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions

We construct infinite-dimensional Wiener processes with interactions by constructing specific quasi-regular Dirichlet forms. Our assumptions are very mild; accordingly, our results can be applied to singular interactions such as hard core potentials, Lennard-Jones type potentials, and Dyson's model. We construct nonequilibrium dynamics.Dedicated to Professor Masatoshi Fukushima on his 60th birthday     

The Bethe-Goldstone equation with singular interactions: A fully off-shell solution for the boundary condition model

The mutual interaction of a pair of fermions imbedded in a many-body system of identical particles when they are excited out of the filled Fermi sea, is studied via the T-matrix or transition amplitude specified by the Bethe-Goldstone (BG) equation. The role of the bare two-body interaction is emphasised, and in particular the consequences are elucidated of whether the potential is “well-behaved” (nonsingular) or not. The properties of the BG T-matrix, including generalized orthonormality and completeness relations, are derived both for nonsingular potentials and for singular potentials containing an infinite hard core. General analytic properties are exploited to derive relations that express the fully off-shell BG T-matrix purely in terms of the half-shell amplitude (and the properties of any possible bound states in the medium). The general formalism is illustrated by deriving exact analytic expressions for the fully off-shell BG T-matrices for a pair of particles with equal and opposite momenta interacting via either of two singular model interactions; namely, the pure hard-core interaction and the boundary condition model. Results for both models are expressed in terms of the solution to a simple one-dimensional Fredholm integral equation. The analytic properties of the solutions are discussed and exploited to prove both their uniqueness and that they satisfy the various general relations derived. To our knowledge, these results represent the first exact nontrivial solution to the fully off-shell BG equation for any local potential, or singular limiting case thereof.     

Harnack Inequalities and Discrete—Continuous Error Estimates for a Chain of Atoms with Two—Body Interactions

We consider deformations in ℝ³ of an infinite linear chain of atoms where each atom interacts with all others through a two-body potential. We compute the effect of an external force applied to the chain. At equilibrium, the positions of the particles satisfy an Euler–Lagrange equation. For large classes of potentials, we prove that every solution is well approximated by the solution of a continuous model when applied forces and displacements of the atoms are small. We establish an error estimate between the discrete and the continuous solution based on a Harnack lemma of independent interest. Finally we apply our results to some Lennard-Jones potentials.     

Application of cellular automata to N-body systems

A two-dimensional cellular automaton model is introduced to deal with the dynamics of a finite system of particles whose interactions are simulated by two-body step potentials. The method is illustrated for a potential approximating the standard Lennard-Jones potential, representative for the problem of heavy ion collisions in nuclear physics. From the cellular automaton dynamics thermodynamic equilibrium state variables are introduced in the usual way. The numerical experiments indicate the occurrence of a phase transition. Macroscopically the transition is marked by a singularity in the equation of state; microscopically it manifests itself by the formation of clusters of particles of all sizes, obeying a mass distribution in the form of a power law of exponent 1.35.     

Morphology and statistical statics of simple microclusters

In this paper we discuss the statics of small assemblies of soft, rare-gas type atoms (N=4 to 13) interacting under simple two-body central forces such as the Lennard-Jones and Morse type. Our main concern is to characterize the problem of isomer multiplicity in small packed structures and to devise practical algorithms for the discovery of a representative majority, if not all, stable soft-packing structures for clusters of atoms in the above size-range.

We illustrate one such possible ‘Aufbau algorithm’ by demonstrating the existence of no less than 988 distinct, stable minimum configurations for 13 Lennard-Jones atoms and of correspondingly smaller numbers in the range N=7 to 12. The minimal structures obtained may be classified broadly into ‘crystallographic’ and ‘non-crystallographic’ types, the latter predominating among those of greatest binding energy.

A surprising result is that, when the softer (α=3) Morse potential is used, the great majority of the Lennard-Jones minima are not supported and only a much smaller class of distinct structures survive. Moreover, broadly speaking, crystallographic configurations are favoured by the softer potential, in confirmation of the intuitive view that relatively hard potentials dispose to amorphous structure.

These results, representing the probable structures of free condensation nuclei near 0 K, provide an interesting statistical morphology for small nuclei, as well as being a necessary first step in the construction of a statistical thermodynamics valid at finite temperatures.     

Theory of quasimagnetic impurity in a metal: An exactly soluble model of interacting electrons

We study an impurity atom, on which two-body forces are important, dissolved in a metal, where they are negligible. With the aid of the well-known boson excitation spectrum of the electronic Fermi sea, we predict the low-energy effects of one- and two-body potentials on the impurity, in the nonmagnetic regime. We obtain for the first time exact expressions for the cutoff independent contributions to the specific heat and paramagnetic susceptibility, the spectral amplitudes or one-electron density of states on the impurity, and the scattering cross-section. The entire spectrum of manybody eigenstates is explicitly obtained. The onset of a local magnetic moment appears as a sudden breakdown of the model Hamiltonian, and occurs when the two-body potential exceeds a critical value U_c which is O(E_F) in magnitude. A study of the renormalization of the interaction parameters terminates the paper.     

HNC variational calculations of boson matter

A simple and reliable numerical technique is given for determining the two-body distribution function which minimizes the HNC energy of boson matter. Numerical results are presented for the neutron matter homework problem and the ⁴He Lennard-Jones potential. The resulting distribution function is found to have proper asymptotic behaviour and yields reasonable binding energies.     

Differential equations in momentum space for the three-body problem in the case of pointlike pair interactions

Corrects schemes for solving equations of three-body dynamics for systems governed by a zerorange two-body interaction are considered. Correlations between spectral features of a three-boson system are obtained. The results are compared with the results obtained by calculating the spectra and scattering lengths in the system of three helium atoms with realistic two-body interaction potentials.     

The convergence of the one-dimensional ground states to an infinite lattice

For one-dimensional systems interacting via a two-body potential, the sequence of ground states is proved to converge to an infinite lattice, for a large open class of interactions, containing in particular the Lennard-Jones potential.     

Computation of some thermodynamics, transport, structural properties, and new equation of state for fluid methane using two-body and three-body intermolecular potentials from molecular dynamics simulation

Molecular dynamics simulation has been performed to obtain pressure, radial distribution function, and self-diffusion coefficient of fluid methane using one site OPLS (optimized potentials for liquid simulations), five sites OPLS-SITE, and two-body HFD (Hartree-Fock dispersion)-like potentials. To take higher-body forces into account, three-body potential of Hauschild and Prausnitz (1993) has been used with the two-body HFD-like potential. The significance of this work is that the three-body potential of Hauschild and Prausnitz extended as a function of density and temperature and used with the HFD-like potential to improve the prediction of the results of pressure of fluid methane without requiring an expensive three-body calculation. The molecular dynamics simulation of methane has been also used to determine a new equation of state. The results are in a good agreement with experimental and theoretical values.     

Generalized Flinn operators with applications to multi-component alloys

The relationship between the occupation operator, spin operator and Flinn operator formalisms for a substitutional binary alloy is presented. These three methods are generalized to multicomponent alloys with irreducible static n-body potentials. In this manner the concept of generalized Flinn operators is introduced. The configurational Hamiltonians and generalized Flinn operators are then explicitly presented for ternary and quaternary alloys with static two-body potentials. The merits of the generalized Flinn operator approach are discussed.     

Semi-Relativistic Two-Body States of Spinless Particles with a Scalar-Type Interaction Potential

A semi-relativistic quantum approximation for mutual scalar interaction potentials is outlined and discussed.Equations are consistent with two-body Dirac equations for bound states of zero total angular momentum. Two-body effects near the non-relativistic limit for a linear scalar potential is studied in some detail.     

Oscillator model for the relativistic fermion-boson system

The solvable quantum mechanical model for the relativistic two-body system composed of spin-1/2 and spin-0 particles is constructed. The model includes the oscillator-type interaction through a combination of Lorentz-vector and Lorentz-tensor potentials. The analytical expressions for the wave functions and the order of the energy levels are discussed.     

Quantum Mechanical Second Virial Coefficient for Cutoff Potentials

The high- and low temperature approximations of the second virial coefficient for cutoff potentials are discussed and compared with numerical results.