A class of two-dimensional maps with sensitive dependence on initial conditions introduced by Kaplan and Yorke is studied, including the influence of external noise. Exact solutions of the stochastic equations of motion are obtained and used to calculate moments, correlation functions, and probability densities. The relation of the steady-state probability density and the attractor of the maps is considered. The models are also used to test the applicability of other methods of solution from the theory of stochastic processes.

It is found that the modulational instability of a finite-amplitude Buchsbaum mode can give rise to enhanced vortex (convection cells) and magnetostatic modes in a D-T fusion plasma. The relevance of our investigation to the anomalous particle and heat losses in a Tokamak device is briefly pointed out.

Wiener-Hopf factorization technique as suggested by Baxter has been used to solve the Ornstein-Zernike equation for a system of hard spheres in the framework of a generalized mean spherical model. The method reported here yields unambiguous and accurate values of the radial distribution function $g\left(r\right)\mathrm{}$, direct correlation function $C\left(r\right)\mathrm{}$, and the structure function $S\left(k\right)\mathrm{}$. These thermodynamic quantities have been calculated from low to fairly high densities and, with a normalized $g\left(R\right)\mathrm{}$, they have been compared with the molecular-dynamics values of Alder and Hecht.

We consider a model for the relativistic Bose gases with certain forms of the energy spectra in $d$ spatial dimensions. First we discuss how some of the effects of the interactions can be brought into the energy spectra for independent particles. The thermodynamical properties arising from the energy spectra are calculated with particular attention to the spatial dependence of Bose-Einstein condensation. Then we explicitly interpret some special cases of the energy spectra through Fourier transformation to the usual coordinate-dependent potentials.

Giving a statistical-mechanical formulation of structure-function dynamics, we present a formulation of a scaling law for the first-order phase transition. The basic idea proposed, which previously appeared in a specific form, is that any function of the form $F_k(t_1,t_2,\dots )$, where $k$ is the wave number and $t_1$,$t_2$,$\dots$ are times, is scaled as $F_k(t_1,t_2,\dots )={\left[R\right(t_1\left)\right]}^{x_1}{\left[R\right(t_2\left)\right]}^{x_2}\cdots \tilde{F}\left(kR\right(t_1),kR(t_2),\dots ),$ where $R\left(t\right)\mathrm{}$ is a relevant scale length, such as a linear dimension of an average cluster size, which is found to behave as $R\left(t\right)\mathrm{}\propto t^{\frac{1}{z}}$ with $z$ being a constant. The autocorrelation function of the density fluctuation is found to obey the scaling law for the conserved system: $J_k(t,t^{\prime })={\mathrm{}\left[R\right(t\left)\right]}^{\theta }{\left[R\right(t^{\prime }\left)\right]}^{d-\theta }\tilde{J}\mathrm{}\left(kR\right(t),kR(t^{\prime }\left)\right), \theta \le \frac{d}{2}$ where $d$ is the dimensionality. For $\theta =0\mathrm{}$ this scaling law can be naturally derived on the basis of the dynamic-scaling assumption and the conservation law. However, for large $t$ the possibility of an anomalous scaling law ($\theta \ne 0$) is found. On the computer simulation for the three-dimensional spin-exchange kinetic Ising model we examine such a scaling law for the autocorrelation function. A remarkable difference in the temporal behaviors of the autocorrelation function is found. That observation strongly suggests the existence of the spinodal-like line.

Assuming that the nucleation occurs in a small region with the volume of the order ${\mathrm{}\left(\mathrm{Dt}\right)\mathrm{}}^{\frac{3}{2}}$, where $D$ is the diffusion constant and $t$ is the time passing after a quench, we find that the droplet-formation barrier undergoes a transition from an infinitely large value to a finite value indicating the existence of the onset of the nucleation reaction. This finite value of the droplet-formation barrier is smaller in deeply supersaturated phases than the value required for the stabilization of the supersaturated phase, i.e., $\sim 60k_{B}T-70\mathrm{}{k}_{B}T$. Thus the cloudiness should occur just after the droplet-formation barrier has undergone the transition. For a deeply supersaturated phase the locus of the cloud point is described by $y_1\propto {\epsilon }^{-\phi }{t}_1^{-\frac{3}{8}}$, where $y_1\propto \frac{\delta \rho }{\Delta \rho }$ and $T_c(1-\epsilon )$ are a reduced density (with $\Delta \rho$ being the coexistence curve) measured from the coexistence state, and the temperature of supersaturated phase with $T_c$ being the critical temperature, respectively. $t_1$ is the observation time after a quench. The exponent $\phi$ is found to be 0.70 for the liquid and 0.47 for the kinetic Ising-spin system. At low temperatures or for large $t_1$, the cloud point is given by the traditional one. The comparisons of the present result with experimental data are presented.

The propagation properties of laser-driven shock waves in pure and layered polyethylene and aluminum slab targets are studied for a set of laser intensities and pulse widths. The laser-plasma simulations were carried out by means of our one-dimensional Lagrangian hydrodynamic code. It is shown that the various parts of a laser-driven compression wave undergo different thermodynamic trajectories: The shock front portion is on the Hugoniot curve whereas the rear part is closer to an adiabat. It is found that the shock front is accelerated into the cold material till $t\approx 0.8\tau$ (where $\tau$ is the laser pulse width) and only later is a constant velocity propagation attained. The scaling laws obtained for the pressure and temperature of the compression wave in pure targets are in good agreement with those published in other works. In layered targets, high compression and pressure were found to occur at the interface of C${\mathrm{H}}_2$ on Al targets due to impedance mismatch but were not found when the layers were reversed. The persistence time of the high pressure on the interface in the C${\mathrm{H}}_2$ on Al case is long enough relative to the characteristic times of the plasma to have an appreciable influence on the shock-wave propagation into the aluminum layer. This high pressure and compression on the interface can be optimized by adjusting the C${\mathrm{H}}_2$ layer thickness.

An ideal relativistic Bose-gas model with pair production is solved exactly in $d$ dimensions. The critical behavior is shown to be the same as the usual nonrelativistic Bose gas. Scaling functions (including nonuniversal parameters) for the equation of state and the correlation functions are obtained in a closed form. The scale-factor universality is verified by the formation of the expected universal ratios. Various limiting cases of the model depending on the relative magnitudes of the three basic length scales, i.e., Compton wave-length, thermal wavelength, and interparticle distance are discussed. In these limiting cases, previously known results are recovered, wherever these are available.

A new kinetic theory for a particle moving in a random array of static spherical scatterers is derived. By decoupling a higher-order dynamical correlation function, a kinetic equation is obtained comprising known approximations like ring, repeated-ring theory, and their self-consistent versions and, in particular, the equivalence of a recent mode-coupling theory and self-consistent ring theory is established. Vertex corrections, as represented by a new class of collision sequences, are shown to be essential for the Lorentz gas. New results for the Burnett coefficient are also presented.

Using a formalism adapted to slow spatio-temporal evolutions of phase perturbations to a perfect roll pattern, we show that "zigzags" which relax in a diffusive way for the case with "no-slip" (or "rigid") boundaries are replaced by Busse's oscillatory mode for the case with "stress-free" boundaries. The diffusion coefficient $D_{\perp }$ for zigzags, which is determined analytically using a perturbation expansion, has the form $D_{\perp }=\frac{{\xi }_0^2}{{\tau }_0}\left[\frac{q-q_c}{q_c}+\frac{N\left(\mathcal{P}\right)}{R_2\left(\mathcal{P}\right)}\frac{\mathcal{R}-{\mathcal{R}}^c}{{\mathcal{R}}^c}\right],$ where $\mathcal{R} [={\mathcal{R}}^c+R_2(\mathcal{P}){\epsilon }^2+\cdots ]$ is the Rayleigh number, $\mathcal{P}$ is the Prandtl number, ${\xi }_0$ is the coherence length, and ${\tau }_0$ is the relaxation time; $\frac{{\xi }_0^2}{{\tau }_0}=\frac{5.685}{(1.95+{\mathcal{P}}^{-1\mathrm{}})}$, $N\left(\mathcal{P}\right)=0.166+23.04\mathrm{}{\mathcal{P}}^{-1\mathrm{}}+6.196\mathrm{}{\mathcal{P}}^{-2\mathrm{}}$, $R_2\left(\mathcal{P}\right)=10.76-0.073\mathrm{}{\mathcal{P}}^{-1\mathrm{}}+0.128\mathrm{}{\mathcal{P}}^{-2\mathrm{}}$, and ${\mathcal{R}}^c=1707.8\mathrm{}$ at the critical wave vector $q_c=3.116\mathrm{}$ for no-slip boundaries. The curvature is seen to induce a large-scale flow which tends to convect the roll pattern. This flow is forbidden for concentric rolls in cylindrical geometry. This leads to a new wavelength-selection criterion which no longer corresponds to $D_{\perp }=0\mathrm{}$. The corrected criterion is calculated for both no-slip and stress-free boundaries and is seen to behave in a similar way at the limit $\mathcal{P}\to 0$ in the two cases. Our findings are in quantitative agreement with earlier numerical results and also with more recent approaches using the amplitude-equation formalism.

A comprehensive analysis of elastic light scattering in cholesteric liquid-crystal systems, based upon the Mueller matrix approach, is given. Emphasis is placed upon analyzing the polarization state of the scattered light for arbitrarily polarized incident radiation. The theory is applied to the ordered cholesteric blue phases BPI and BPII, which exhibit long-range orientational ordering. Both the polycrystalline and single-crystal cases are considered and the role played by optical activity is studied. It is shown how light-scattering studies yield information on the structural properties of the ordered phases. The quasielastic light scattering due to excitations in the disordered phase is also analyzed in the harmonic approximation. It is shown that this scattering cannot explain the strong anomolous scattering observed in the amorphous BPIII phase.

It has been demonstrated that the Weizsäcker inhomogeneity term may be replaced by a gradient-free form for atomic systems. This forms a realization of a previous conjecture made by the authors [Phys. Rev. A 25 668 (1982)]. In the variational context, numerical computations have been performed with such a representation with the first-row atoms employed at test cases. The resulting energies are seen to match very well with their Hartree-Fock counterparts.

With use of the classical concepts for electron capture and ionization, a general scaling relationship for the electron-loss cross section in collisions of atoms with highly charged ions is derived. The validity of the obtained formula is determined by the combined validity regions of the classical approximations for electron capture and ionization and covers a broad range of parameters in which most of present-day experiments are carried out. An analogous scaling formula for the ${\mathrm{H}}^{*}\mathrm{}\left(n\right)\mathrm{}$ + ${\mathrm{H}}^{+}$ electron-loss collisions is also derived.

Differential and integrated cross sections summed over all angular momentum sublevels are calculated for the process ${\mathrm{H}}^{+}+\mathrm{H}\mathrm{}(n=1\mathrm{})\mathrm{}\to {\mathrm{H}}^{+}+\mathrm{H}\mathrm{}(n=4\mathrm{})\mathrm{}$ in the incident-proton energy range 15-400 keV, employing the Glauber approximation. The integrated cross sections are in very good agreement in curve shape with the available experimental measurement. Results are also presented for the integrated Glauber cross sections for $1s-4d$ and $1s-4f$ excitations of hydrogen by proton impact and are compared with available theoretical results.

The energy dependence of the total charge-transfer cross section for the collision of helium ions with cesium atoms at epithermal energies (0.1 to 1 eV) has been measured. It was observed that the cross section increases with $E^{-\frac{1}{2}}$. This can be attributed to orbiting collisions. The charge-transfer cross section is expressed as the product of the classical orbiting cross section and a quantum-mechanical charge-transfer probability which is shown to be energy independent.

In this work we use the first term of a multiple-scattering expansion for the calculation of the distribution of electrons emitted in ${\mathrm{H}}^{+}$-He scattering, in a neighborhood of the cusp that corresponds to the electron capture to continuum. The theoretical values that determine the peak shape are compared with recent experimental values at intermediate energies, and the influence of the projectile—target-core interaction in the asymmetry of the peak is discussed.

A Coulomb-Born-Oppenheimer approach to the calculation of cross sections for electronimpact excitation of the doubly excited $2p^2{}_{}{}^3{}_{}{}^{}P$ state in heliumlike ions is presented. It is shown that closed-form expressions can be obtained for both differential and total cross sections. The analysis is applied to calculate the total cross section for the process $e^{-}+{\mathrm{Li}}^{+}\left(1\mathrm{}{s}^2{}_{}{}^1{}_{}{}^{}S\right)\to e^{-}+{\mathrm{Li}}^{+}\left(2\mathrm{}{p}^2{}_{}{}^3{}_{}{}^{}\mathrm{P}\right)$.

A coherent two-potential approach (in which multiple scattering is allowed) is used to calculate the integrated and momentum-transfer cross sections, and differential cross sections, for electron-${\mathrm{O}}_2$ elastic scattering in the energy range 40-500 eV. Above 100 eV, the present theory is in excellent agreement with experiment. However, below 100 eV, there are discrepancies at intermediate angles in the differential cross sections.

The absorption cross section of ${\mathrm{H}}_2$ has been measured in the region between 77.5 and 82.4 nm and the effects of electric fields up to 22 kV/cm on the cross sections have been investigated. The apparent cross section is found to be increased in the neighborhood of optically allowed transitions to $\mathrm{np}\sigma {{}_{}{}^1{}_{}{}^{}\Sigma _u^{}{}_{}{}^{}}^{+}$ and $\mathrm{np}\pi {}_{}{}^1{}_{}{}^{}\Pi _u^{}{}_{}{}^{}$ states with $n=4-7\mathrm{}$. This increase in absorption is attributed to field mixing with nearby optically forbidden states.

The electrostatic stability of the free-electron laser is studied for a configuration in which a relativistic electron beam propagates through combined helical wiggler and axial guide fields. Instability is found for certain specific parameter regimes which, in the beam frame, is shown to be purely growing and to require the presence of both the wiggler and axial guide fields. The electrostatic stability is also studied for a confuguration which consists of a linearly polarized wiggler and an axial guide field, for which analogous results are found.