A review is presented of recent work related to strange attractors and chaotic motions of dynamical systems. First, simple systems capable of displaying chaotic behavior are discussed. In order of increasing dimensionality of the system, they are one-dimensional noninvertible maps, two-dimensional invertible maps, and autonomous systems of three coupled ordinary differential equations. The concept of fractional dimension of the strange attractor is stressed. Several physical examples well be reviewed, along with the possible relevance to turbulence in systems, such as fluids or plasmas, that are described by partial differential equations.

Three scenarios leading to turbulence in theory and experiment are outlined. The respective mathematical theories are explained and compared.

This article is a summary of what is know rigorously about Thomas-Fermi (TF) theory with and without the Dirac and von Weizsäcker corrections. It is also shown that TF theory agrees asymptotically, in a certain sense, with nonrelativistic quantum theory as the nuclear charge $z$ tends to infinity. The von Weizsäcker correction is shown to correct certain undesirable features of TF theory and to yield a theory in much better agreement with what is believed (but as yet unproved) to be the structure of real atoms. Many open problems in the theory are presented.

This review is devoted to the problem of critical state stability in hard superconductors and superconducting normal composites. An introduction is given to the properties of hard and composite superconductors, and to the qualitative nature of the physical processes that occur in these materials in the critical state. The dynamics of the development of instabilities of various kinds are treated in detail. Stability criteria are obtained and discussed, and theory is compared with experiment. The interaction between flux jumps and plastic strain jerks and the training phenomenon in superconductors are also covered.

The authors discuss the theory and experiments dealing with the pulsed optoacoustic effect (i.e., generation of a transient acoustic wave by absorption of an optical pulse) in condensed matter. Their primary interest lies in the measurement of small absorption coefficients (≪${10}^{-1\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$). At present an experimental capability of measuring absorption coefficients as small as ${10}^{-6\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ has been demonstrated, and further improvement is foreseen. The pulsed optoacoustic absorption measurement technique has been applied to the following linear spectroscopic studies: (1) precise measurements of the optical absorption spectra of ${\mathrm{H}}_2$O and ${\mathrm{D}}_2$O; (b) accurate determination of absorption strengths and profiles of high harmonics ($n=6, 7, \mathrm{and} 8$) of vibrational modes in transparent organic liquids (e.g., benzene); (c) quantitative absorption spectra of thin (∼ 1-10 μm) liquid films; and (d) quantitative absorption spectra of solids and finely powdered crystals. The usefulness of the pulsed optoacoustic technique to nonlinear spectroscopy has been demonstrated in the following studies: (a) quantitative two-photon absorption spectroscopy of the weak two-photon (${}_{}{}^1{}_{}{}^{}B_{2\mu }^{}{}_{}{}^{}\leftarrow {}_{}{}^1{}_{}{}^{}A_{1g}^{}{}_{}{}^{}$) transition in benzene; and (b) optoacoustic Raman-gain spectra for a variety of liquids where an ability to measure Raman gains as small as ${10}^{-5\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ has been demonstrated. In addition to reviewing the above studies the authors discuss future possible applications and compare the pulsed optoacoustic spectroscopy technique with other optoacoustic absorption measurement techniques.

An astonishing variety of systems show properties that fluctuate with approximately $\frac{1}{f}$-shaped spectral densities. In this review we deal with selected topics regarding $\frac{1}{f}$ fluctuations (or noise) in the resistance of simple condensed matter systems, especially metals. We find that considerable experimental and conceptual progress has been made, but specific physical processes mostly remain to be identified.

Various forms of the dynamo theory are presented in a graphic manner. Each of them depends on a flow pattern of presumably thermal convection in the earth's fluid core. The conducting fluid moves in magnetic fields generated by currents induced by the motion. Each of the flow patterns includes vortices, with helicity induced by the Coriolis force, that twist the magnetic fields in such a way as to regenerate an initial field. Some of them involve also differential rotation, with the parts of the core near the axis rotating more rapidly than the outer parts. Some forms of the theory are more successful than others in accounting qualitatively for the various observed aspects of the field, particularly the westward drifts and the occasional polarity reversals. The thermal energy source may be radioactivity of heat or crystallization at the inner-core surface.

It now appears that the general nature of the deviations from uniformity in the spectrum of a complicated nucleus is essentially the same in all regions of the spectrum and over the entire Periodic Table. This behavior, moreover, is describable in terms of standard Hamiltonian ensembles which could be generated on the basis of simple information-theory concepts, and which give also a good account of fluctuation phenomena of other kinds and, apparently, in other many-body systems besides nuclei. The main departures from simple behavior are ascribable to the moderation of the level repulsion by effects due to symmetries and collectivities, for the description of which more complicated ensembles are called for. One purpose of this review is to give a self-contained account of the theory, using methods—sometimes approximate—which are consonant with the usual theory of stochastic processes. Another purpose is to give a proper foundation for the use of ensemble theory, to make clear the origin of the simplicities in the observable fluctuations, and to derive other general fluctuation results. In comparing theory and experiment, the authors give an analysis of much of the nuclear-energy-level data, as well as an extended discussion of observable effects in nuclear transitions and reactions and in the low-temperature thermodynamics of aggregates of small metallic particles.

This review deals with quantitative descriptions of electronic transitions in atom-atom and ion-atom collisions. In one type of description, the nuclear motion is treated classically or semiclassically, and a wave function for the electrons satisfies a time-dependent Schrödinger equation. Expansion of this wave function in a suitable basis leads to time-dependent coupled equations. The role played by electron-translation factors in this expansion is noted, and their effects upon transition amplitudes are discussed. In a fully quantum-mechanical framework there is a wave function describing the motion of electrons and nuclei. Expansion of this wave function in a basis which spans the space of electron variables leads to quantum-mechanical close-coupled equations. In the conventional formulation, known as perturbed-stationary-states theory, certain difficulties arise because scattering boundary conditions cannot be exactly satisfied within a finite basis. These difficulties are examined, and a theory is developed which surmounts them. This theory is based upon an intersecting-curved-wave picture. The use of rotating or space-fixed electronic basis sets is discussed. Various bases are classified by Hund's cases (a)-(e). For rotating basis sets, the angular motion of the nuclei is best described using symmetric-top eigenfunctions, and an example of partial-wave analysis in such functions is developed. Definitions of adiabatic and diabatic representations are given, and rules for choosing a good representation are presented. Finally, representations and excitation mechanisms for specific systems are reviewed. Processes discussed include spin-flip transitions, rotational coupling transitions, inner-shell excitations, covalent-ionic transitions, resonant and near-resonant charge exchange, fine-structure transitions, and collisional autoionization and electron detachment.

The properties of exactly integrable two-dimensional quantum systems are reviewed and discussed. The nature of exact integrability as a physical phenomenon and various aspects of the mathematical formalism are explored by discussing several examples, including detailed treatments of the nonlinear Schrödinger (delta-function gas) model, the massive Thirring model, and the six-vertex (ice) model. The diagonalization of a Hamiltonian by Bethe's Ansatz is illustrated for the nonlinear Schrödínger model, and the integral equation method of Lieb for obtaining the spectrum of the many-body system from periodic boundary conditions is reviewed. Similar methods are applied to the massive Thirring model, where the fermion-antifermion and bound-state spectrum are obtained explicitly by the integral equation method. After a brief review of the classical inverse scattering method, the quantum inverse method for the nonlinear Schrödinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique. In the quantum inverse method, an auxiliary linear problem is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length. The particular operators for which the string is infinitely long (free boundary conditions) or forms a closed loop around a cylinder (periodic boundary conditions) correspond to the quantized scattering data and have a special significance. One of them creates the Bethe eigenstates, while the other is the generating function for an infinite number of conservation laws. The analogous operators on a lattice are constructed for the symmetric six-vertex model, where the object which corresponds to a solution of the auxiliary linear problem is a string of vertices contracted over horizontal links (arrows). The relationship between the quantum inverse method and the transfer matrix formalism is exhibited. The inverse Gel'fand-Levitan transform which expresses the local field operator as a functional of the quantized scattering data is formulated for the nonlinear Schrödinger equation, and some interesting properties of this transformation are noted, including its reduction to a Jordan-Wigner transformation in the limit of infinitely repulsive coupling.

A detailed analysis of existing neutral-current data has been performed in order (a) to determine as fully as possible the structure of the hadronic and leptonic neutral currents without recourse to a specific weak-interaction model; (b) to search for the effects of small deviations from the Weinberg-Salam (WS-GIM) model; and (c) to determine the value of ${{sin}^2\theta }_w$ as accurately as possible. The authors attempt to incorporate the best possible theoretical expressions in the treatment of each of the reactions. For deep-inelastic scattering, for example, the effects of quantum chromodynamics, including the contributions of the $s$ and $c$ quarks, have been included. The sensitivity of the results both to systematic uncertainties in the data and to theoretical uncertainties in the treatment of deep-inelastic scattering, semi-inclusive pion production, $\nu$ elastic scattering from protons, and the asymmetry in polarized $\mathrm{eD}$ scattering have been considered; the systematic errors are generally found to be smaller than the statistical uncertainties. In the model-independent analyses the authors find that the hadronic neutral-current parameters are uniquely determined to lie within a small domain consistent with the WS-GIM model. The leptonic couplings are determined to within a twofold ambiguity; one solution, the axial-vector-dominant, is consistent with the WS-GIM model. If factorization is assumed then the axial-dominant solution is uniquely determined and null atomic parity violation experiments are inconsistent with other neutral-current experiments. Within generalized SU(2)×U(1) models we find the following limits on mixing between right-handed singlets and doublets: ${sin}^2{\alpha }_u\le 0.103$, ${sin}^2{\alpha }_d\le 0.348$, and ${sin}^2{\alpha }_e\le 0.064$. Assuming these mixing angles to be zero, a fit to the most accurate data (deep-inelastic and the polarized $\mathrm{eD}$ asymmetry) yields $\rho =0.992\mathrm{}\pm 0.017(\pm 0.011)$ and ${{sin}^2\theta }_w=0.224\mathrm{}\pm 0.015(\pm 0.012)$, where $\rho =\frac{M_W^2}{M_Z^2}{{cos}^2\theta }_w$ and the numbers in parentheses are the theoretical uncertainties. The value of $\rho$ is remarkably close to 1.0 and strongly suggests that the Higgs mesons occur only as doublets and singlets. If one makes this assumption, then the limit on $\rho$ implies $m_L\le 500$ GeV, where $m_L$ is the mass of any heavy lepton with a massless partner. In addition, for $\rho =1.0\mathrm{}$, the authors determine ${{sin}^2\theta }_w=0.229\mathrm{}\pm 0.009(\pm 0.005)$. Fits which also include the semi-inclusive, elastic, and leptonic data yield very similar results. A two-parameter fit gives $\rho =1.002\mathrm{}\pm 0.015(\pm 0.011)$ and ${{sin}^2\theta }_w=0.234\mathrm{}\pm 0.013(\pm 0.009)$, while a one-parameter fit to ${{sin}^2\theta }_w$ gives ${{sin}^2\theta }_w=0.233\mathrm{}\pm 0.009(\pm 0.005)$. Finally, the authors have found no evidence for a violation of factorization or for the existence of additional $Z$ bosons. Fits to two explicit two-boson models yield the lower limits $\frac{M_{Z_2}}{M_{Z_1}}>1.61\mathrm{} \mathrm{and} 3.44$ for the mass of the second $Z$ boson. The desirability of a complete analysis of radiative and higher-order weak corrections, which have not been included in the authors' theoretical uncertainties, is emphasized.

Two separate lines of investigation have recently converged to produce a highly detailed picture of the behavior of helium atoms physisorbed on graphite basal plane surfaces. Atomic beam scattering experiments on single crystals have yielded accurate values for the binding energies of several states for both ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, as well as matrix elements of the largest Fourier component of the periodic part of the interaction potential. From these data, a complete three-dimensional description of the potential has been constructed, and the energy band structure of a helium atom moving in this potential calculated. At the same time, accurate thermodynamic measurements were made on submonolayer helium films adsorbed on Grafoil. The binding energy and low-coverage specific heat deduced from these measurements are in excellent agreement with those calculated from the band structures.

The authors investigate the asymptotic (low-frequency or long-time) behavior of random one-dimensional systems described by a master equation of the form $\frac{C_{n}d{P}_n}{\mathrm{dt}}=W_{n,n-1\mathrm{}}(P_{n-1\mathrm{}}-P_n)+W_{n,n+1\mathrm{}}(P_{n+1\mathrm{}}-P_n),$ where either the $C_n$ or the $W_{n,n+1\mathrm{}}(=W_{n+1,n})$ are independent positive random variables. This problem can be mapped onto a variety of physical problems, including one-dimensional particle or excitation diffusion with random spatial transfer rates or random trap depths, low-temperature properties of the random one-dimensional Heisenberg ferromagnet, one-dimensional tight-binding fermion systems with correlated diagonal and off-diagonal disorder, and electrical lines of random conductances or capacitances. Replacing $\frac{d{P}_n}{\mathrm{dt}}$ by $\frac{d^2{P}_n}{d{t}^2}$, the above equations also describe a harmonic chain with random force constants or random masses. Both the random $W$ and the random $C$ problem are shown to reduce to the same mathematical problem, and the authors review the derivation and consequences of its exact asymptotic solution for some general classes of probability densities. In particular, they are able to determine the exact $\omega \to 0$ asymptotic behavior of the single-site Green function, and thus the low-energy density of states and (for diffusion-type problems) the long-time behavior of the autocorrelation function. For diffusion-type problems, the investigators further introduce a scaling hypothesis (based on the assumption of the existence of a single characteristic length) for the time dependence of the excitation amplitude at sites other than the initially excited site. This allows the calculation of the low-frequency diffusion constant, and with the help of the Einstein relation, the low-frequency conductivity. The authors apply their results to several physical systems and, in particular, are able to account quantitatively for the complex low-frequency conductivity of the one-dimensional superionic conductor hollandite. The results may also be relevant to quasi-one-dimensional electronic systems for times (or frequencies) such that diffusion is restricted to a single dimension. The expansion of an earlier scaling approach gives additional insight into the physical meaning of the asymptotic solutions. The researchers further discuss effective-medium-type approximations, which, apart from a numerical prefactor, lead to the correct asymptotic dependences. Comparison is made with the results of alternative approaches to this type of problem, specifically with those of the continuous time random walk. The authors exhibit the remarkable satisfaction of the hyperscaling relations for the specific heat and correlation function critical exponents for the random Heisenberg ferromagnet, even though the exponents themselves are not universal. Finally, aspects of the problem which remain unsolved, or which are only partly resolved, are discussed. These include the derivation of asymptotic expansions beyond the leading term, the derivation of rigorous asymptotic expressions for the conductivity, the calculation of fluctuations in the site occupancy probabilities for the diffusion problem, and the treatment of the random one-dimensional antiferromagnet and random one-dimensional $\mathrm{xy}$ model. The investigators close with a short description of an application of this work to electronic conduction in highly anisotropic materials.

The definition, properties, and applications of the single-particle (electron) density $\rho \left(\mathrm{r}\right)$ are discussed in this review. Since the discovery of Hohenberg-Kohn theorem, which gave a theoretical justification for considering $\rho \left(\mathrm{r}\right)$, rather than the wave function, for studying both nondegenerate and degenerate ground states of many-electron systems, $\rho \left(\mathrm{r}\right)$ has been acquiring increasing attention. The quantum subspace concept of Bader et al. has further highlighted $\rho \left(\mathrm{r}\right)$ since a rigorous decomposition of the three-dimensional (3D) space of a molecule into quantum subspaces or virial fragments is possible, the boundaries of such subspaces being defined solely in terms of $\rho \left(\mathrm{r}\right)$. Further, $\rho \left(\mathrm{r}\right)$ is a very useful tool for studying various chemical phenomena. The successes and drawbacks of earlier models, such as Thomas-Fermi-Dirac, incorporating $\rho \left(\mathrm{r}\right)$ are examined. The applications of $\rho \left(\mathrm{r}\right)$ to a host of properties—such as chemical binding, molecular geometry, chemical reactivity, transferability, and correlation energy—are reviewed. There has been a recent trend in attempting to bypass the Schrödinger equation and directly consider single-particle densities and reduced density matrices, since most information of physical and chemical interest are encoded in these quantities. This approach, although beset with problems such as $N$-representability, and although unsuccessful at present, is likely to yield fresh concepts as well as shed new light on earlier ideas. Since charge density in 3D space is a fundamental quantum-mechanical observable, directly obtainable from experiment, and since its use in conjunction with density-functional theory and quantum fluid dynamics would provide broadly similar approaches in nuclear physics, atomic-molecular physics, and solid-state physics, it is not unduly optimistic to say that $\rho \left(\mathrm{r}\right)$ may be the unifying link between the microscopic world and our perception of it.

This paper is a review of the present status of the problem of the properties of the dielectric function of matter. It is shown that the dielectric function $\epsilon (\mathrm{q}, \omega )$ obeys the Kramers-Kronig relations only when momentum q is identically equal to zero. For an arbitrary momentum $\mathrm{q}\ne 0$ the dielectric function does not obey the Kramers-Kronig relations, and the static dielectric function at $\mathrm{q}\ne 0$ may have a negative sign [$\epsilon (\mathrm{q}, 0)<0$]. It is shown that the appearance of a negative sign for the static dielectric function does not contradict the requirements of system stability. Special attention is paid to the possible existence of a negative sign for the static dielectric function in real physical systems. It is shown that the inequality $\epsilon (\mathrm{q}, 0)<0$ at $\mathrm{q}\ne 0$ is valid for a rather wide class of condensed media (simple metals, nonideal plasma, etc.). The interrelation between the existence of a negative sign for the dielectric function and the problem of high-temperature superconductivity is briefly discussed.

The current understanding of the behavior of quantum chromodynamics at finite temperature is presented. Perturbative methods are used to explore the high-temperature dynamics. At sufficiently high temperatures the plasma of thermal excitations screens all color electric fields and quarks are unconfined. It is believed that the high-temperature theory develops a dynamical mass gap. However in perturbation theory the infrared behavior of magnetic fluctuations is so singular that beyond some order the perturbative expansion breaks down. The topological classification of finite-energy, periodic fields is presented and the classical solutions which minimize the action in each topological sector are examined. These include periodic instantons and magnetic monopoles. At sufficiently high temperature only fields with integral topological charge can contribute to the functional integral. Electric screening completely suppresses the contribution of fields with nonintegral topological charge. Consequently the $\theta$ dependence of the free energy at high temperature is dominated by the contribution of instantons. The complete temperature dependence of the instanton density is explicitly computed and large-scale instantons are found to be suppressed. Therefore the effects of instantons may be reliably calculated at sufficiently high temperature. The behavior of the theory in the vicinity of the transition from the high-temperature quark phase to the low-temperature hadronic phase cannot be accurately computed. However, at least in the absence of light quarks, semiclassical techniques and lattice methods may be combined to yield a simple picture of the dynamics valid for both high and low temperature, and to estimate the transition temperature.