In an extended model, a rigorous proof is given for sound-wave amplifications in an imploding plasma shell. It is shown that, in the absence of a massless free surface, the boundary conditions give the exact eigenvalues which determine the asymptotic solution to the problem.

In this paper the propagation of fronts into an unstable state are studied. Such fronts can occur e.g., in the form of domain walls in liquid crystals, or when the dynamics of a system which is suddenly quenched into an unstable state is dominated by domain walls moving in from the boundary. It was emphasized recently by Dee et al. that for sufficiently localized initial conditions the velocity of such fronts often approaches the velocity corresponding to the marginal stability point, the point at which the stability of a front profile moving with a constant speed changes. I show here when and why this happens, and advocate the marginal stability approach as a simple way to calculate the front velocity explicitly in the relevant cases. I sketch the physics underlying this dynamical mechanism with analogies and, building on recent work by Shraiman and Bensimon, show how an equation for the local ‘‘wave number’’ that may be viewed as a generalization of the Burgers equation, drives the front velocity to the marginal stability value. This happens provided the steady-state solutions lose stability because the group velocity for perturbations becomes larger than the envelope velocity of the front.

For a given equation, our approach allows one to check explicitly that the marginal stability fixed point is attractive, and this is done for the amplitude equation and the Swift-Hohenberg equation. I also analyze an extension of the Fisher-Kolmogorov equation, obtained by adding a stabilizing fourth-order derivative -γ ${\partial }^4$φ/∂$x^4$ to it. I predict that for γ<(1/12 the fronts in this equation are of the same type as those occurring in the Fisher-Kolmogorov equation, i.e., localized initial conditions develop into a uniformly translating front solution of the form φ(x-vt) that propagates with the marginal stability velocity. For γ>(1/12, localized initial conditions may develop into fronts propagating at the marginal stability velocity, but such front solutions cannot be uniformly translating. Differences between the propagation of uniformly translating fronts φ(x-vt) and envelope fronts are pointed out, and a number of open problems, some of which could be studied numerically, are also discussed.

Recent analytical results for the shape of viscous fingers in a Hele-Shaw cell have established that the steady-state equations with nonzero surface tension are overdetermined and hence do not admit finger solutions with arbitrary finger widths. We develop a simple counting argument for the asymptotic region (the tails of the finger), based on the symmetry of the system for zero surface tension, which indicates in a simple way how this overcompleteness arises. Similarities and differences with counting arguments for breather solutions in soliton theory and for dendritic needle solutions are also briefly discussed.

We consider the propagation of kinks in an elastic chain in a bistable or multistable potential under the action of a driving force. Each element of the chain is subject to a damping force proportional to its velocity. We show that both the propagation velocity of the kinks as a function of the driving field, and the kink width as a function of propagation velocity, are determined by characteristic functions which depend only on the form of the potential. These functions can be found by considering a single particle moving in the upside-down potential of the chain. The general properties of these functions are studied and illustrated by several examples. The stability of these driven kinks is discussed. Interestingly we find in addition to the expected discrete localized eigenmodes a two-dimensional continuum of oscillatory modes with a localized envelope.

In microcanonical molecular dynamics the conservation of total momentum implies that the trajectories generate an ensemble with constant total momentum. We present the exact statistical mechanics of this microcanonical molecular dynamics ensemble. Using the adiabatic invariant phase volume to define the entropy, exact formulas, containing thermodynamic response functions, are derived for the microcanonical molecular dynamics ensemble and a numerical comparison of these formulas is made with fluctuation formulas which are valid in the thermodynamic limit and often used in molecular dynamics. No significant difference is found between the results calculated using the different formulas for a system of 432 particles; however, for simulations with small numbers of particles or small-cluster studies, the differences could be important.

The microscopic mechanics discovered by Nosé, of which Gauss’s isokinetic mechanics is a special case, makes it possible to simulate macroscopic irreversible nonequilibrium flows with purely reversible equations of motion. The Gauss-Nosé and Nosé-Hoover equations of motion explicitly include time-reversible momentum and energy reservoirs. Computer simulations of nonequilibrium steady-state systems described by Gauss-Nosé mechanics invariably evolve in such a way as to increase entropy. The corresponding phase-space distribution functions, which include reservoir degrees of freedom, collapse onto stable strange attractors. Hypothetical time-reversed motions, which would violate the second law of thermodynamics, cannot be observed for two reasons: First, such reversed motions would occupy zero volume in the phase space; second, they would be dynamically unstable. Thus, Nosé’s reversible mechanics is fully consistent with irreversible thermodynamics, in the way forecast by Prigogine. That is, the consistency follows from the formulation of new microscopic equations of motion.

A new superposition method is presented for evaluating the matrix representation of the generators in the unitary-group approach. This superposition method is based on the Weyl graphical method for the calculation of matrix elements; the latter is an extension of Harter’s jawbone formula for the evaluation of the matrix elements of $E_i$,i-1 to the more general case $E_i$,j and can also deal with both fermion and boson.

In this paper a two-dimensional lattice model with two coupled degrees of freedom per site is defined. The degrees of freedom are (i) a standard Ising spin σ=±1 and (ii) a continuous variable τ which represents small molecular displacement perpendicular to the lattice plane. The model is an example of a class of models which may be used to describe complex systems such as lipid bilayers below the lipid-chain melting temperature. Using methods applied to deformable lattice models the two-variable Hamiltonian is separated into two independent single-variable terms. This greatly simplifies the statistical-mechanical effort required to determine the model properties. In the present case the spin portion of the separated Hamiltonian becomes an anisotropic Ising model with four-spin interactions. The properties of this model are determined by mean-field and Monte Carlo techniques, and a phase diagram based upon the results is constructed.

The shape of a polymer molecule in the presence of randomly distributed stationary impurities is investigated using the path-integral formulation. The polymer molecule is assumed to be described by the Edwards minimal model. Calculation of the mean-square end-to-end displacement 〈$R^2$〉 shows that at a critical density of the scatterers the polymer molecule makes a transition from a swollen state (chain with excluded-volume interactions) to a disordered, random coiled state (a Gaussian chain). At this density there is a crossover in the value of the exponent ν (in three dimensions), characterizing the scaling of 〈$R^2$〉, with the length of the polymer from about 0.6 (corresponding to the excluded-volume regime) to 0.5 (corresponding to the coiled state). For densities below the critical density, the exponent ν is well described by the Flory value.

The rate of energy transfer between the radial and axial degrees of freedom of protons with measured temperature and number stored in a radio-frequency quadrupole ion trap is quantified. The results are discussed in terms of the theory of charged-particle collisional relaxation to thermal equilibrium.

We have measured charge-changing cross sections with ${\mathrm{He}}^{+}$ and ${\mathrm{He}}^{2+}$ ions impinging upon target gases He, Ne, Ar, and Kr. The energy range of the helium ions was 0.75–4 MeV and the cross sections were obtained either by the initial-growth method or by the attenuation method. The measured values were compared with the corresponding cross sections obtained with equal velocity ${\mathrm{H}}^{+}$ and H projectiles. Deviations from the simple scaling rules usually applied to the capture process are discussed.

Using the Monte Carlo method the behavior of a system of true hard cylinders has been studied. Values of the length-to-breadth ratio L/D and packing fraction η have been chosen similar to those of real nematic liquid crystals. Results include radial distribution function g(r), structure factor S(k), and orientational order parameter M. These results lead to the conclusion that the hard cylinder model may be a useful reference for real mesomorphic phases.

For the first time reentrances below smectic $A_1$ are obtained with the molecular "spin-gas" model of liquid crystals, in agreement with recent experiments. The model further predicts sextuple (and octuple) reentrance, $\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}-\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{c}-A_d-\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}-\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{c}-A_d-\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}-\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{c}-A_1-\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}-\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{c}-A_d(-\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}-\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{c}-A_1)$, as temperature is lowered. Two distinct reentrance mechanisms are found: microscopic frustration of dipoles and competing short-range orders.

A simple expression is derived for all multipole nonadiabatic corrections to the long-range effective potential at threshold energy due to the exchange of two Coulomb photons in electronatom scattering. Multipole and nonadiabatic corrections are treated on equal footing. Transverse photon exchanges and hence retardation effects are ignored. The same results hold for positronatom scattering.

Extensive and previously unknown fluorescence and excitation spectra of the HgZn excimer have been observed and correlated with a potential-energy (PE) diagram. The spectra were excited in a Hg-Zn mixture contained in a quartz vapor cell by successive pulses from two dye lasers, using the pump-and-probe methods, and the PE curves were derived from Hg and Zn atomic parameters, previously known ${\mathrm{Hg}}_2$ and ${\mathrm{Zn}}_2$ curves, and from molecular-orbital considerations. There is good agreement between the measured and calculated transition energies.

The frame transformation theory for diatomic molecules is generalized to the case of triatomic molecules with $D_{3h}$ symmetry. Analytic expressions are formulated to explain the difference and similarity of the $l$-uncoupling effect in ${\mathrm{H}}_3$ with those in ${\mathrm{H}}_2$. An anomalous photoemission of $p$-Rydberg series in ${\mathrm{H}}_3$ is predicted.

A high-resolution temperature-concentration diagram and detailed differential scanning calorimetry data near an expected $\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{c}-A_1-\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}-\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{c}-A_d$ bicritical point in a binary liquid-crystal system are presented. These results show that the bicritical point has split into a critical end point and a tricritical point, a situation which has not been predicted by any of the theories so far.

A procedure is given for constructing basis sets for the radial Dirac equation from B splines. The resulting basis sets, which include negative-energy states in a natural way, permit the accurate evaluation of the multiple sums over intermediate states occurring in relativistic many-body calculations. Illustrations are given for the Coulomb-field Dirac equation and tests of the resulting basis sets are described. As an application, relativistic corrections to the second-order correlation energy in helium are calculated. Another application is given to determine the spectrum of thallium (where finite–nuclear-size effects are important) in a model potential. Construction of B-spline basis sets for the Dirac-Hartree-Fock equations is described and the resulting basis sets are applied to study the cesium spectrum.

Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density ρ(r) can be used to estimate the potential field v(r) that will give rise to a given ρ(r), or the ρ(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizsäcker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizsäcker type can never predict multiple maxima in ρ if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.