Some turbulent solutions of the unaveraged Navier-Stokes equations (equations of fluid motion) are reviewed. Those equations are solved numerically in order to study the nonlinear physics of incompressible turbulent flow. Initial three-dimensional cosine velocity fluctuations and periodic boundary conditions are used in most of the work considered. The three components of the mean-square velocity fluctuations are initially equal for the conditions chosen. The resulting solutions show characteristics of turbulence, such as the linear and nonlinear excitation of small-scale fluctuations. For the stronger fluctuations, the initially nonrandom flow develops into an apparently random turbulence. Thus randomness or turbulence can arise as a consequence of the structure of the Navier-Stokes equations. The cases considered include turbulence which is statistically homogeneous or inhomogeneous and isotropic or anisotropic. A mean shear is present in some cases. A statistically steady-state turbulence is obtained by using a spatially periodic body force. Various turbulence processes, including the transfer of energy between eddy sizes and between directional components, and the production, dissipation, and spatial diffusion of turbulence, are considered. It is concluded that the physical processes occurring in turbulence can be profitably studied numerically.

The authors review the experimental evidence for the observation of single photons in hadronic reactions, summarize the phenomenology used in parametrizing the dynamics of the process, provide a survey of the current status of the experimental program, and discuss expectations from future investigations of direct-photon production at large transverse momenta.

Ad astra per aspera et per ludum

The Hubbard model is used to calculate static properties of normal-liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ at $T=0\mathrm{}$. For this, Gutzwiller's variational approach to that model is employed. The work is based on an observation by Anderson and Brinkman that the results of this method, obtained by Brinkman and Rice for the metal-insulator transition in the case of one particle per site, appear to be in qualitative agreement with the experimentally measured properties of that liquid. In this sense normal ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ can be understood to be close to a localization transition of the particles where their effective mass diverges. The incipient localization is found to determine the properties of that liquid. Hence ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is "almost localized" rather than "almost ferromagnetic," as often claimed by paramagnon theory. The author further investigates this motion. Discussing Gutzwiller's approach to the Hubbard model, he shows that it is well suited for a description of a liquid system like ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The approach and its physical implications are investigated by means of the reformulation of the solution due to Ogawa et al. It is shown explicitly that Gutzwiller's results can be placed into the concepts of Landau-Fermi-liquid theory and that within this model the Landau parameters $F_0^s$ and $F_0^a$ are related. Furthermore, the author identifies two different kinds of spin-fluctuation processes inherent to the model, one of which is shown to be responsible for the largeness of $F_0^s$. Going beyond these qualitative aspects, the author evaluates $F_0^a$ and $F_0^s$ quantitatively, finding that $F_0^a$ agrees very well with the experimentally determined values at all pressures, with $F_0^a\to -\frac{3}{4}p$ at high pressures, where $p$ is always close to unity. Hence the system is never close to a ferromagnetic transition. By means of the forward scattering sum rule for $l<\mathrm{}2\mathrm{}$ an analytic expression for $F_1^a$ is obtained. Finally, the author extends the analysis to large magnetic fields, finding that in the case of normal ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ the magnetization increases very rapidly with the magnetic field. This is due to the large zero-field effective mass. There is a line of critical values for the interaction and the magnetic field where a fully magnetized state is formed via a first-order transition. Calculating the drop in melting pressure due to the magnetic field, the author finds that it essentially removes the minimum in the melting curve. Thus the melting pressure even of fully polarized ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is larger than zero, in agreement with arguments by Castaing and Nozières.

The principles of the separation physics of the gas centrifuge were described in Part I of this review. In this second section the principles involved in spinning the rotors of these centrifuges are described. Three types of rotor can be identified, depending on the ratio of length to diameter. If the rotor is very short, length-diameter ratio less than one, it is gyroscopically stable and easy to spin. If the length-diameter ratio is in the region of 4 or 5, the rotor behaves as a rigid body and is relatively easy to accelerate to speed; however, it has a tendency at full speed to exhibit gyroscopic precessions. Finally, if the length-diameter ratio is very large, the rotor becomes easy to stabilize gyroscopically, but it is difficult to get it to speed because long rotors are very flexible and have resonant frequencies of flexure lower than the operating speed. The problems of these three types of centrifuge (the rotor dynamics, the bearings used to support the rotor, and the stress analysis of the rotating components) were investigated in the last century as part of classical mechanics because of the emergence of steam turbines during the latter part of the industrial revolution. These early principles are briefly reviewed, with particular reference to the work of De Laval, who invented the principle of self-balancing, Reynolds and Evershed, who developed hydrodynamic and magnetic bearing, respectively, and Chree, who did the most extensive early work on the stress analysis of tubes and discs. The work is described as it applies to the centrifuges developed in America and Germany during the war and in the Soviet Union after the war. The work of Beams in America is described in most detail, since he and his colleagues developed all three types of centrifuge during the Manhattan Project. The other work described is that of Groth and Beyerle, who developed subcritical machines in Germany during the war, and of Steenbeck and Zippe, who helped to develop both subcritical and supercritical centrifuges in the Soviet Union after the war. Little of this latter work has been published, but Zippe redeveloped the subcritical machine at the University of Virginia. The description of this machine concludes the present review.

There are two sets of principles involved in the development of the gas centrifuge, the internal separation physics and the external means of spinning a rotor at very high speeds. Only the first aspect is discussed in this part of the review. First, the industrial requirement for the separation of the uranium isotopes is defined so that the separation history can be put in a modern perspective. The history of separation physics itself is then traced back to the theory of centrifugal force by Huygens and the equivalence of this force to that of gravity. The barometric equation giving the variation of atmospheric pressure with height and the law of partial pressures can then be adapted to the centrifuge to give the steady-state theory of separation. This work was completed in the last century but was not confirmed in its application to isotope separation until 1936. The detailed separation physics for non-steady-state conditions required for a production centrifuge was developed during the American wartime Manhattan Project. During this work the theory giving the maximum output of a centrifuge was developed by Dirac, and soon afterwards Cohen and Kaplan showed that the best method of operation for a production centrifuge is in a countercurrent mode of operation. This method gives a large separation factor at relatively small flow rates through the rotor. The theory of how to set up an internal countercurrent was given by Martin during an equivalent wartime German project, and refinements to the theory, showing how the countercurrent persists along a centrifuge rotor, was given by Dirac and Steenbeck, the latter during a postwar Russian project. This theory was extended by Parker, Ging, and Mayo of the University of Virginia, whose work was completed by 1962, the limit of this review.

This article gives a pedagogical introduction to relaxation methods for the numerical solution of elliptic partial differential equations, with particular emphasis on treating nonlinear problems with $\delta$-function source terms and axial symmetry, which arise in the context of effective Lagrangian approximations to the dynamics of quantized gauge fields. The authors present a detailed theoretical analysis of three models which are used as numerical examples: the classical Abelian Higgs model (illustrating charge screening), the semiclassical leading logarithm model (illustrating flux confinement within a free boundary or "bag"), and the axially symmetric Bogomol'nyi-Prasad-Sommerfield monopoles (illustrating the occurrence of topological quantum numbers in non-Abelian gauge fields). They then proceed to a self-contained introduction to the theory of relaxation methods and allied iterative numerical methods and to the practical aspects of their implementation, with attention to general issues which arise in the three examples. The authors conclude with a brief discussion of details of the numerical solution of the models, presenting sample numerical results.

A description of high-spin phenomena in atomic nuclei is presented from both the experimental and theoretical points of view. The characteristic features of collective nuclear motion, such as rotational bands, band crossings, and backbending, as well as noncollective aspects, like high-spin isomers and irregular decay patterns, are discussed in detail. Recent achievements of the cranking model including the independent quasiparticle and the shell correction methods are reviewed. Changes in the structure of nuclei excited up to the highest possible angular momenta are analyzed; in particular, angular momentum alignment effects, shape changes, possible phase transitions, and sudden rearrangements in the single-particle structure are discussed. Phenomena related to the nuclear quasicontinuum spectra are also examined.

Latent nuclear tracks are damaged zones, created along the paths of rapidly moving ions in solids. They are stored indefinitely in many insulators and can be used to initiate a phase separation process by one of several procedures that remove, collect, or transform material along the latent track. The resulting two-step technique is based on separate inscription and development stages, similar to the lithographic techniques using visible light, uv, x rays, or electrons. In contrast to these other techniques, however, it is applicable even to mechanically very stable, chemically inert, radiation-resistant dielectric materials. The most common development procedure is track etching, which acts as a chemical "amplifier" that dissolves the damaged zone of the latent track preferentially, creating etch pits or channels that can be extremely fine. Etched track diameters start around 10 nm and increase linearly with the etching time. The most outstanding feature of the technique is that one single particle is sufficient to create a developable damage. This unique combination of recording and revealing properties makes possible a single-particle structuring tool with a very wide array of applications. With the advent of high-energy, heavy-ion accelerators, with ion ranges between micrometers and centimeters, large-scale application of the technique to the imprinting of structure onto solids has become possible. At present, most frequently random track arrays are employed to induce global property changes of the solid volume or surface. In the future, scanning ion microbeam technology will make possible local generation of oriented track structures with high precision. This "ultimate" microtool should provide the versatility of a computer-controlled mechanical lathe. This review describes the processes involved in track generation. It outlines applications that are on the verge of being commercialized at present and considers future possibilities.

The neutron interferometer is a unique instrument that allows one to construct a neutron wave packet of macroscopic size, divide it into two components separated by centimeters, and then coherently recombine them. A number of experiments clearly showing the difference between quantum and classical theory have been performed with it, which are suitable for presentation in elementary quantum courses. This article presents a simple mathematical model of the interferometer, which can be used to illustrate clearly many of the surprising features of quantum systems. For example, one can describe an experiment to determine which component beam the neutron takes (an analog of the two-slit electron experiment). One can then trace in detail the loss of coherence of the wave function, rather than merely invoke the usual "handwaving" uncertainty arguments. The author discusses the effect of gravity on the neutron beam [the classic COW (Colella, Overhauser, and Werner) experiment], including a simple analysis in an accelerated reference frame, and its relation to the equivalence principle, the red shift, and the twin paradox. Also described are the effect of rotation of the neutron by 360° to change its phase, the effect on the wave function of measuring the absence of the particle from a beam ("Dicke's paradox"), and a realizable version of Wheeler's "delayed-choice" experiments, as well as their relation to the problem of "Schrödinger's cat." The treatment is suitable for bright undergraduates and first-year graduate students.

Correlations between the orientations of two particles with spin ½ are formulated in terms of a complete set of operators appropriate to display the spectrum of oscillations under the influence of the spin interactions. The operators are classified for this purpose according to space and permutation symmetries. The treatment is intended as a model for any pair of two-level systems. Its extension to multilevel systems and the eventual emergence of dissipative behavior are outlined. The correlation of two-photon polarizations in the Einstein-Rosen-Podolsky paradox is described as an illustration.

The author reviews in a pedagogical fashion some of the recent developments in lattice quantum chromodynamics. This review emphasizes explicit examples and illustrations rather than general proofs and analyses. It begins with a discussion of the heavy-quark potential in continuum quantum chromodynamics. Asymptotic freedom and renormalization-group improved perturbation theory are discussed. A simple dielectric model of confinement is considered as an intuitive guide to the vacuum of non-Abelian gauge theories. Next, the Euclidean form of lattice gauge theory is introduced, and an assortment of calculational methods are reviewed. These include high-temperature expansions, duality, Monte Carlo computer simulations, and weak coupling expansions. A $\Lambda$- parameter calculation for asymptotically free-spin models is presented. The Hamiltonian formulation of lattice gauge theory is presented and is illustrated in the context of flux tube dynamics. Roughening transitions, Casimir forces, and the restoration of rotational symmetry are discussed. Mechanisms of confinement in lattice theories are illustrated in the two-dimensional electrodynamics of the planar model and the U(1) gauge theory in four dimensions. Generalized actions for SU(2) gauge theories and the relevance of monopoles and strings to crossover phenomena are considered. A brief discussion of the continuity of fields and topological charge in asymptotically free lattice models is presented. The final major topic of this review concerns lattice fermions. The species doubling problem and its relation to chiral symmetry are illustrated. Staggered Euclidean fermion methods are discussed in detail, with an emphasis on species counting, remnants of chiral symmetry, Block spin variables, and the axial anomaly. Numerical methods for including fermions in computer simulations are considered. Jacobi and Gauss-Siedel inversion methods to obtain the fermion propagator in a background gauge field are reviewed. Langevin, pseudofermion Monte Carlo methods, and random walk methods are discussed, as are proposals for including the fermion determinant in computer simulations. Finally, spontaneous chiral symmetry breaking in lattice models is discussed. The review ends with brief remarks about present research directions.

The use of variational principles as a calculational tool is reviewed, with special emphasis on methods for constructing such principles. In particular, it is shown that for a very wide class of problems it is possible to construct a variational principle (VP) for just about any given quantity $Q$ of interest, by routine procedures which do not require the exercise of ingenuity; the resultant VP will yield an estimate of $Q$ correct to second order whenever the quantities appearing in the VP are known to first order. The only significant requirement for application of the routine procedures is that the entities which enter into the definition of $Q$ be uniquely specified by a given set of equations; the equations may involve difference or differential or integral operators, they may be homogeneous or inhomogeneous, linear or nonlinear, self-adjoint or not, and they may or may not represent time-reversible systems. No numerical calculations are presented, but procedures for the construction of VP's are illustrated for numerous quantities $Q$ of physical interest, particularly those $Q$ arising in quantum-mechanical scattering and transition probability calculations. For pedagogical purposes VP's are also derived for several problems in classical and (simple) mathematical physics which the authors hope will prove instructive and perhaps even amusing. The quantum-mechanical quantities $Q$ whose VP's are examined include various matrix elements and the quantum-mechanical eigenfunctions themselves. Topics examined include some points which have not always been appreciated in the literature, such as the necessity for properly specifying the phase when complex eigenfunctions are involved, and the importance of avoiding, wherever possible, formulations requiring the inversion of singular operators. The basic element of the technique is the recognition that the defining equations of a system can be incorporated into the VP as constraints through the use of (generalized) Lagrange multipliers; these can be constants, scalar, vector, or tensor functions of one or more variables, operators, etc. In the typical problem, these Lagrange multipliers $\mathcal{L}$ serve as a new set of adjoint functions or entities, and the construction of the VP simultaneously provides well-defined equations for the $\mathcal{L}$. Moreover, these $\mathcal{L}$ often have ready physical significance themselves; for example, they often may be regarded as generalized Green's functions. The construction of the VP also readily yields so-called variational identities for the quantity $Q$ of interest; these identities give explicit (if formal) expressions for the error in the variational estimate of $Q$. In some cases this error can be shown to have a definite sign, so that the VP actually is an extremum principle, that is, that it yields an upper or lower variational bound for $Q$; however, our routine procedures for constructing VP's do not routinely yield extremum principles.

The research field "flow acoustics" is defined, and problems of current interest are discussed. The physical interpretation of fluid-mechanical sound sources for an acoustic medium, both at rest and in motion, is addressed. In the first case, it has been possible to relate the sound pressure field for low Mach numbers to sound sources that depend linearly on velocity; progress has been achieved by means of the causality correlation method with the application of laser-Doppler velocity measurements. In the theory of the acoustic medium in motion, it has been found that vortices downstream from a nozzle discharge can function as sound sinks; for a unidirectional mean flow, conditions for the sound sources are developed on the basis of the causality principle and the boundedness of the flow quantities. Experimental and theoretical results for flowfield oscillations in a high-velocity duct flow with sudden duct enlargement are discussed. Finally, some points concerning the influence of shear flow on sound propagation are described.

This review presents in a comprehensive and tutorial form the basic principles of the Monte Carlo method, as applied to the solution of transport problems in semiconductors. Sufficient details of a typical Monte Carlo simulation have been given to allow the interested reader to create his own Monte Carlo program, and the method has been briefly compared with alternative theoretical techniques. Applications have been limited to the case of covalent semiconductors. Particular attention has been paid to the evaluation of the integrated scattering probabilities, for which final expressions are given in a form suitable for their direct use. A collection of results obtained with Monte Carlo simulations is presented, with the aim of showing the power of the method in obtaining physical insights into the processes under investigation. Special technical aspects of the method and updated microscopic models have been treated in some appendixes.

Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the values of its nearest neighbors. With simple initial configurations, the cellular automata either tend to homogeneous states, or generate self-similar patterns with fractal dimensions ≃ 1.59 or ≃ 1.69. With "random" initial configurations, the irreversible character of the cellular automaton evolution leads to several self-organization phenomena. Statistical properties of the structures generated are found to lie in two universality classes, independent of the details of the initial state or the cellular automaton rules. More complicated cellular automata are briefly considered, and connections with dynamical systems theory and the formal theory of computation are discussed.

Part I [Rev. Mod. Phys. 54, 1183 (1982)] explored stellar evolution leading up to supernovae and observations and models of the events themselves. Part II addresses the aftermath: supernova remnants, products, and by-products, including nucleosynthesis, and the future of supernova research. Some of the important questions are: (1) How close is the association among supernova events, pulsar production, and remnant production? (2) Where does most of the energy from neutron star formation go? and (3) How do supernovae interact with the rest of the universe, for instance in heating and stirring the interstellar medium, accelerating cosmic rays, and triggering or inhibiting star formation?