A van der Waals fluid contained in a bubble under constant surface tension and temperature is formulated as a dynamical system. The temperature and surface tension determine the nature of the equilibrium solutions. The bifurcation set is a cusp where the cusp point is a shifted critical point, thus modifying the van der Waals critical point. When the temperature is above the shifted critical temperature the model is valid for all radii; otherwise, the model is not valid for sufficiently small radii. The Hamiltonian is used to discuss the problem of using a Maxwell construction to determine a phase transition in a bubble.

A particular solution, which describes the shape-preserving evolution of an explosive localized state (ELS) in one, two, and three spatial dimensions, is obtained for the reaction-diffusion equation with a quadratic creative reaction term. This state, the ELS, is discovered to be of fundamental significance, since states which are ‘‘nearby’’ an ELS are attracted to it. A new technique of analysis is introduced, which enables one to describe the dynamical evolution of the amplitude and width of density profiles. These variables are found to be governed by two coupled nonlinear differential equations. The dynamic theory covers, in a unified form, the cases of one, two, and three spatial dimensions.

We report a nonlinear analytical solution for optimal shaping of the driving electron beam in the plasma wake-field accelerator. The upper bound of the transformer ratio is $R_u$ =[1+(2π$L_b$/${\lambda }_p$${)}^2$${]}^{1/2}$, where $L_b$ is the length of the driving electron beam, ${\lambda }_p$≊2πc/${\omega }_p$, and ${\omega }_p$ is the plasma frequency.

The occurrence of quasiperiodic orbits in typical dissipative dynamical systems is of great importance. The equations for two driven coupled Van der Pol oscillators were integrated, and four Lyapunov exponents were calculated for every orbit. The results show that for small coupling and driving parameters, most orbits are three-frequency quasiperiodic. As the coupling and driving parameters are increased, three-frequency quasiperiodicity becomes less common, being first replaced by two-frequency quasiperiodicity, and then by periodic and chaotic motions. This example reinforces past work suggesting that three-frequency quasiperiodic attractors are common in typical dynamical systems.

We use the set of all periodic points of Hénon-type mappings to develop a theory of the topological and metric properties of their attractors. The topology of a Hénon-type attractor is conveniently represented by a two-dimensional symbol plane, with the allowed and disallowed orbits cleanly separated by the ‘‘pruning front.’’ The pruning front is a function discontinuous on every binary rational number, but for maps with finite dissipation ‖b‖<1, it is well approximated by a few steps, or, in the symbolic dynamics language, by a finite grammar. Thus equipped with the complete list of allowed periodic points, we reconstruct (to resolution of order $b^n$) the physical attractor by piecing together the linearized neighborhoods of all periodic points of cycle length n. We use this representation to compute the singularity spectrum f(α). The description in terms of periodic points works very well in the ‘‘hyperbolic phase,’’ for α larger than some ${\alpha }_c$, where ${\alpha }_c$ is the value of α corresponding to the (conjectured) phase transition.

The quantum thermodynamics of the nonintegrable ${\phi }^4$ one-dimensional chain is studied by means of a classical effective potential, which includes in a fully quantum way the linear modes of the field. In contrast to the case of the sine-Gordon chain, integrable in the continuum limit, exact results are not available, so that approximate quantum calculations appear to be useful. This effective potential is determined by a variational approach developed in previous papers, which is based on the path-integral formulation of statistical mechanics. The temperature renormalization is studied, in the limit of low temperature, by means of a self-consistent saddle-point method, both for the vacuum and the one-kink sectors, and the results of the semiclassical approximation are recovered. Important results are obtained by a new low-coupling expansion for the effective potential. Its range of validity in temperature is much wider than the range of previous high-temperature expansions. The results for the nonlinear contributions to internal energy and specific heat, obtained by means of original transfer-matrix computations, are finally presented and discussed.

A fluid of hard spherocylinders is studied in the Onsager model adapted to a nonuniform system. The interfacial properties at nematic-phase–isotropic-phase coexistence are considered. It is found that the angle between the director and the normal to the interface is approximately 60° and does not depend on the length-to-width ratio L/D of the spherocylinder. The nematic-phase–isotropic-phase surface tension, however, tends linearly to zero as D/L→0. It is also argued that the anisotropic hard-core repulsion favors the perpendicular alignment at the nematic free surface. The results concerning the tilt angle are in good agreement with experimental studies for nCB (n=5,6,7,8) [(4-n-alkyl-4’-cyano)biphenyl].

The singularity spectrum f(α) and the generalized dimension D(q) of the critical orbit is numerically obtained to study the global scaling behavior of period n-tupling (n=2,3,4) in area-preserving maps. It is found that f(α) becomes quite different as n is changed, and the generalized dimension D(q) increases for all q as n increases. The global scaling behavior of conservative systems is different from that of dissipative systems. Moreover, for conservative systems, the global scaling behavior seems to depend on dimensionality.

We find that the hydrogen-bond–stretch bands of the double helix appear to be nonlinear enough to support solitary-wave energy concentration. Coupling this fact to predictions of our self-consistent theory of helix melting gives rise to speculations of a mechanism for base pair melting in RNA transcription which is consistent with the known energy needs of that process.

We present detailed results on the form factors of two-dimensional systems undergoing phase-ordering processes, using both deterministic and stochastic cell dynamical systems. We show the robustness of the asymptotic form factors against quench depth, noise amplitude, etc. The effect of noise is essentially to delay the number of steps needed to reach the asymptotic behavior. In the case with a nonconserved order parameter, we demonstrate that the form factor obtained by T. Ohta, D. Jasnow, and K. Kawasaki [Phys. Rev. Lett. 49, 1223 (1982)] is asymptotically very accurate. We also present preliminary results for off-critical quenches.

The global organization of the periodic orbits in the periodically driven Duffing oscillator is described in terms of the relative rotation rates of pairs of orbits. We compare this organization with that determined by the second return of a horseshoe map and find a complete agreement, up to the global torsion, for all orbit pairs and all the regions of the bifurcation diagram studied. In addition, we characterize topologically different flows in the Duffing family by their global torsion.

The behavior of a two-dimensional orientation field has been studied directly in space and time. In a freely suspended thin film of tilted smectic liquid crystal the local projection onto the film plane of the molecular orientation averaged through the film forms a two-dimensional (2D) vector field C^(x,y) with local orientation Φ(x,y). This orientation field was imaged in real space by depolarized reflection microscopy. By probing the light at two distinct (x,y) locations in the image and employing cross-correlation intensity-fluctuation spectroscopy, we have measured the space-time behavior of ${\sigma }^2$(ρ,τ)≡〈‖Φ(0,0) -Φ(ρ,τ)${\Vert }^2$〉. At large τ, σ increases logarithmically with τ, consistent with the expected diffusional dynamics and thermally excited fluctuation spectrum of Φ. This logarithmic (Landau-Peierls) divergence confirms that this 2D orientational system is at its lower marginal dimensionality. In addition, by fitting the behavior of ${\sigma }^2$(ρ,τ), we extract the 2D orientational diffusion constants for splay and bend in both the smectic-C and smectic-I phases.

A simple definition of Berry phases based on the Pancharatnam phase convention is stated in the language of ordinary quantum mechanics. It is shown to be equivalent to previous definitions for a cycle of transformations built continuously from infinitesimal transformations. It also applies to discrete transformations and partial cycles. The phase for a partial cycle can be the same as for the complete cycle. This suggests possibilities for simpler experiments. In an example considered here, each part of a split beam of light is put through two half-wave plates. The light is not brought back to the original polarization state, but interference produced by the Berry phase can be observed because the different parts of the beam are brought to the same polarization state.

Thirty-five configurations, built up from 15 orbitals obtained in multiconfigurational Hartree-Fock calculations, have been used in configuration-interaction calculations (involving 53 states with J=0) for the two-electron systems He i, Liii, and Beiii. The Hamiltonian operator consisted of the electronic, SL-nonsplitting, and fine-structure terms, and the results have been corrected by consideration of the appropriate nuclear mass effects. The accuracy of the resulting total energies is 99.986%, 99.994%, and 99.998%, respectively.

Eigenenergies are calculated for the potential V(r)=-(a/r)[1+(1+br)$e^{\mathrm{-}2\mathrm{br}}$] by the shifted 1/N expansion method and the results are compared with those obtained by Sever and Tezcan for the same potential by the 1/N method. The shifted 1/N expansion results show a large improvement over the 1/N expansion results.

We carry out a simple calculation of the rate at which atoms confined within an atom trap are kicked out of the trap by elastic collisions with background gas molecules. The calculated rates are in good agreement with the observed rates and are remarkably insensitive to the trap depth.

The ion production that results when K(nd) Rydberg atoms are incident at grazing incidence on a metal surface is investigated. The data show that such ion production forms the basis of a simple technique with which to detect Rydberg atoms, even at relatively low values of n, and suggest that Rydberg atoms may provide a novel means to investigate local electric fields present near a surface.

Electron-impact excitation of sodium from the ground state to the $4^2$S state is investigated for impact energies 5eV≤E≤100 eV. Configuration-interaction-type wave functions are employed in a close-coupling (CC) approximation to determine the significance of the 3p and 3d intermediate states on the 3s→4s excitation cross section by comparing six-, four-, three-, and two-state CC results (6CC, 4CC, 3CC, and 2CC). Surprisingly, the 2CC and 6CC calculated cross sections are in excellent agreement with each other throughout the energy range investigated, and with measurement at 54.4 eV. At other energies the agreement with experimental integral cross sections is less close. Calculations which couple only the 3p or the 3p and 3d intermediate states such as the present 3CC and 4CC approximations may overestimate significantly the Na(4s) cross sections. The second-Born-approximation analysis leads to incorrect results even at 100 eV.

Absolute photon absorption spectra at the K edges of Ti and Ni and at the $L_3$ edges of Pt and Au, recorded using synchrotron radiation, are reported. The average photon cross section at $L_3$ edges of Pt and Au are found to be in general agreement with the theoretical data of Scofield. A possible explanation of why such agreement is not found for Ti and Ni at K edges is presented.

Differential cross sections have been calculated for electron transfer and elastic scattering of 20- to 125-keV α particles on H(1s) atoms. The transition amplitudes in the eikonal expression for the scattering amplitude are those recently determined in a large triple-center, coupled-state calculation with up to 34 states on the centers, including a third center at the saddle point of the electronic potential. The differential cross sections are compared at 20 and 70 keV with recently reported results using a coupled–molecular-state approach. There is very good agreement at the lower energy, but significant disagreement at the higher energy, particularly in the smaller, more sensitive elastic differential cross section.