We present an algorithm which computes the standard Grassberger-Procaccia correlation dimension of a strange attractor from a finite sample of N points on the attractor. The usual algorithm involves measuring the distances between all pairs of points on the attractor, but then discarding those distances greater than some cutoff $r_0$. Our idea is to avoid computing those larger distances altogether. This is done with a spatial grid of boxes (each of size $r_0$) into which the points are organized. By computing distances between pairs of points only if those points are in the same or in adjacent boxes, we get all the distances less than $r_0$ and avoid computing many of the larger distances. The execution time for the algorithm depends on the choice of $r_0$, the smaller $r_0$, the fewer distances to calculate, and in general the shorter the run time. The minimum time scales as O(NlogN); this compares with the O($N^2$) time that is usually required. Using this algorithm, we have obtained speedup factors of up to a thousand over the usual method.

We have studied quantum-mechanical behavior of a driven Morse oscillator coupled to a bath of harmonic oscillators. The purpose is to compare the quantum behavior of such a system with the classical solutions of a driven damped Morse oscillator. We start with the quantum Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an su(2) Lie algebra. This algebra and the Markovian approximation allow us to derive the generalized master equation (for the reduced density matrix of the Morse oscillator), which contains level-dependent energy and phase relaxation terms. We have numerically integrated the differential equations of the matrix elements to obtain the time evolution of the reduced density matrix and the energy of the Morse oscillator. We show that the energy of the Morse oscillator in general varies with time and eventually reaches an asymptotic oscillatory state. The mean energy value of the asymptotic oscillatory state is studied as a function of the relaxation rates and laser frequency and amplitude. Vibrational distributions have also been found as functions of laser frequency and amplitude. The shape of the distribution changes gradually as laser amplitude increases; it first peaks at the ground state at a small field amplitude, then it peaks at some excited state at a large field amplitude, and for an amplitude in between it can have two maxima. This bimodal vibrational distribution reflects the bistability observed in the classical and semiclassical models. Thus bistability exists in quantum results, not as a hysteresis loop, but as a bimodal distribution. Finally, we show that as laser intensity increases, the time series of the oscillator energy evolves from regular to quasiperiodic behavior, and eventually a chaotic-looking series appears.

In this paper we present a new consolidated approach to the problem of determining the electromagnetic field resulting from a finite electric current source outside two electrically different media. This approach rests on properties of generalized functions and the solution of a given problem depends essentially only on the gradients of the characteristic functions of the various regions. In particular we study the scattered fields from a flat earth, from a two-layered earth, from an earth consisting of two different adjacent electrical media (the ‘‘mixed-path’’ problem), and from a rough surface. We show that each of these problems can be modeled and solved from one set of equations; in each case we carry the analysis to a stage where it agrees with that presented by the authors in specific detailed papers that treat each problem separately. As such the solutions can be shown to agree with classical approaches. The technique used here contains a minimum of sophisticated analysis and has the following properties: The source field is arbitrary, the resulting equations directly relate the unknown electric and known source field intensities without the use of intermediate Hertz potentials, and the boundary conditions evolve directly from the formulation as auxiliary equations.

Mixtures of liquid crystals of opposite diamagnetic susceptibilities exhibit a switching transition in a magnetic field between states parallel and perpendicular to the field. It is found that this transition may actually consist of two second-order transitions separated by a region in which the director is tilted, or may be first order. The behavior of the transitions in a finite geometry is investigated, and some comments on the possibilities of experimental verification are made.

A single equation is derived for the determination of the first-order correction to the frequency-dependent density, due to the perturbation of a time-varying electric field. This new expression for the first-order correction to the frequency-dependent Kohn-Sham amplitudes depends explicitly upon the velocity form of the dipole-moment operator and the square of the Kohn-Sham Hamiltonian.

The nonresonant, three-photon ionization spectrum of Kr between the ${\mathrm{Kr}}^{+}$ ${}_{}{}^2{}_{}{}^{}\mathrm{°}_{3/2}^{}{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}\mathrm{°}_{1/2}^{}{}_{}{}^{}$ fine-structure limits is reported. Members of three Rydberg series converging to the ${\mathrm{Kr}}^{+}$ ${}_{}{}^2{}_{}{}^{}\mathrm{°}_{1/2}^{}{}_{}{}^{}$ limit, corresponding to ${(}^2$P${\mathrm{°}}_{1/2}$)nd’[1/2]${\mathrm{\equiv }}_1$, ${(}^2$${\mathit{P}}_{1/2}$)nd’[3/2]${\mathrm{°}}_{,}$ and ${(}^2$P${\mathrm{°}}_{1/2}$)ng’[7/2]${\mathrm{°}}_3$, are observed. A fourth allowed series, ${(}^2$P${\mathrm{°}}_{1/2}$nd’[5/2]${\mathrm{°}}_3$, is not observed. Photoelectron angular distributions are reported for the 9s’, 7d’, and 6g’ members of the observed series. Comparison of the present results with analogous measurements in Xe indicate that transitions to J=3 final states are significantly weaker in Kr.

The 3$d_{3/2}$ photoelectron peak from atomic cesium and the shake-up satellites up to ∼30-eV-higher binding energy have been studied with Al Kα x rays (1487 eV). At ∼22 eV from the 3$d_{3/2}$ peak satellites corresponding to excitation from the 5p levels have been observed with an intensity 5% that of 3$d_{3/2}$. Accordingly, the shoulder observed previously by Mathews et al. on the high-binding-energy side of the 6s→7s shake-up associated with 3$d_{3/2}$ ionization has now been assigned to shake-up from the 5p levels accompanying 3$d_{5/2}$ ionization. This conclusion has necessitated a lowering of the estimated 6s→7s relative intensity from ∼18% to ∼14%. Comparison with sudden-approximation intensity calculation shows that they overestimate the relative intensities for ns→(n+1)s shake-up in Na, K, and Cs. These intensities decrease in going down the group, as shown by both theory and experiment.

In a previous Comment on recent papers by Hong and Mandel, I was able to show that the ‘‘Jackiw state,’’ which is infinitesimally different from the coherent state, displays fourth-order squeezing (as defined by Hong and Mandel’s criterion) of both quadrature field components; the magnitude of such squeezing is, however, infinitesimal. In this paper an explicit perturbation calculation is given which results in a state with finite fourth-order squeezing of both quadrature components.

A line at 1462.65 Å in the emission spectrum of Alcator-C plasmas seeded with krypton has been identified as the 3$s^2$3$p^2$ ${}^3$$P_0$–3$s^2$3$p^2$ ${}^3$$P_1$ magnetic dipole transition in Si-like krypton.

We discuss the use of rotating-cylinder viscometers to determine absolute shear viscosities of classical fluids and of helium II in the context of past and current knowledge of the stability and flow of these fluids between concentric cylinders. We identify a problem in measuring the absolute viscosity when the inner cylinder is rotating and the outer cylinder is at rest. We conclude by discussing the design of viscometers for absolute viscosity measurements in helium I and helium II.

The pair distribution function and thermodynamic quantities of the classical one-component plasma obtained by some approximation schemes are compared with the values given by numerical experiments in the supercooled liquid state. It is shown that the behavior of the pair distribution function is not reproduced with sufficient accuracy even by those approximations whose results are in excellent agreement with experiments in the liquid domain.

Fluctuation expressions are developed for the derived properties of steady states. We give expressions for the compressibility and the specific heats at constant volume and pressure. In contrast to the compressibility, the specific-heat fluctuation formulas are not simple generalizations of the corresponding equilibrium expressions.

The step size of random walks is shown to exhibit an important role in controlling the structure of clusters produced by the diffusion-limited aggregation (DLA) processes. In this study, the structural compactness is found to increase significantly as the random-walk size is increased. The step-size effects are precisely quantified by the radius of gyration evaluated during the cluster growth process. It is also observed that by using a very long random walk the DLA model approaches the Vold-Sutherland ballistic aggregation model.

The debate over the validity of the principle of material frame indifference of modern continuum mechanics, as recently discussed by Ryskin [Phys. Rev. A 32, 1239 (1985)], is examined from a more rigorous theoretical standpoint based on statistical mechanics. It is shown that constitutive equations can only represent special solutions of the microscopic dynamics and, as such, can have a larger invariance group than the Galilean group. While there are reasons to question the general validity of material frame indifference, these doubts arise exclusively from the presence of unbalanced molecular Coriolis forces whose effect becomes vanishingly small in the classical continuum limit. Hence it is argued that statistical mechanics is in excellent approximate agreement with material frame indifference for the description of physical phenomena which lie strongly within the classical continuum limit.

In my recent Brief Report [Phys. Rev. A 32, 1239 (1985)] it was pointed out that the widespread confusion over the nature of material frame-indifference (MFI) is due to the attempts to justify this useful approximation by an appeal to the correct, but irrelevant, idea of material behavior being independent of the motion of an observer. The Comment by C. G. Speziale does not address this point, but concentrates instead on discussing how good an approximation the MFI is. This discussion is criticized below.

The second-order chain rule of functional calculus is used to obtain a second-order Ornstein-Zernike equation. This identity leads directly to an exact expression for the three-particle direct correlation function that has previously been derived from graph theory.

A quantum Monte Carlo technique to compute ground-state energies of He, ${\mathrm{H}}^{-}$, and ${\mathrm{Ps}}^{-}$ is described. Coordinates appropriate to a molecular description of atoms are used, and results with good accuracy are obtained upon input of essentially just reduced masses and the nuclear charge. The method minimizes the need to construct sophisticated variational or basis functions.

Energy distributions of secondary electrons emitted backward from Au, Ag, Cu, Al, and Be targets were measured in the energy range from 100 to 1000 eV for impacts of ${\mathrm{He}}^{+}$ and ${\mathrm{He}}^{2+}$ ions with velocities corresponding to 1.1 MeV/amu. Energies, $E_p$, at which electron loss peaks for ${\mathrm{He}}^{+}$ impact take maximum values are found to show pronounced target dependence. The order of $E_p$ for these targets is Au>Cu>Ag>Al>Be This dependence of $E_p$ is explained in terms of energy loss of loss electrons in their transportation to the surface.