Many natural phenomena are governed by nonlinear recursive relations of the type $x_{t+1\mathrm{}}$=f($x_t$), where f does depend on t. We focus our interest on the particularly simple case $x_{t+1\mathrm{}}$=$r_t$$x_t$(1-$x_t$), where $r_t$ adopts either periodically or at random the values A and B. Graphical representations of the Lyapunov exponent on the AB plane show unexpected features, like self-similarity and early chaos (i.e., chaos for very low parameter values). In relation with the latter we discuss a novel mechanism to induce chaotic behavior. The meaning of the Lyapunov exponent for random processes is examined.

A detailed framework for a new formalism of phase-space path integral is presented, the outline of the theory of which has been reported elsewhere [K. Takatsuka, Phys. Rev. Lett. 61, 503 (1988)]. This path integral is described in terms of a phase-space distribution function, which was proposed earlier by the present author under the name of the dynamical characteristic function (DCF). The DCF is characterized by two independent wave functions and with two phase spaces. In the present paper, we extend the DCF by utilizing the Feynman kernel in place of the two independent wave functions. This is called the identity DCF. The dynamics of the DCF for general wave packets is reduced to that of the identity DCF. Some characteristics of the identity DCF are presented. For example, the quantum-mechanical time-evolution operator is represented in terms of the identity DCF and of a complete set of quantum q (coordinate) and p (momentum) operators. A semiclassical theory for the identity DCF is also developed, its primary aim being the study of heavy-particle dynamics such as molecular collisions.

It is shown that the propagation of the identity DCF is represented by classical trajectories, each of which is associated with an amplitude factor and with the action integral as the quantum phase factor. This amplitude factor is free from singularity, unlike the primitive semiclassical expression of the Feynman kernel, which is well known to diverge at caustics. The physical significance of the identity DCF is analyzed in the context of chaotic dynamics. In particular, it is shown that its amplitude factor is an explicit function of the eigenvalues of a Jacobian matrix [∂(${\mathit{q}}_{\mathit{f}}$,${\mathit{p}}_{\mathit{f}}$)/∂(${\mathit{q}}_{\mathit{i}}$,${\mathit{p}}_{\mathit{i}}$)], where (${\mathit{q}}_{\mathit{i}}$,${\mathit{p}}_{)}$ and ($q_f$,$p_f$) are the initial and final points, respectively, of a classical trajectory. These eigenvalues are well known in the study of the classical chaos of Hamilton systems and have been used to construct a measure of the linear (and local) stability of an orbit, the Liapounov characteristic exponents, and so on. Further, if the identity DCF is applied to a periodic orbit, the amplitude factor becomes equivalent to the square root of the determinant of a matrix, which is the so-called monodromy matrix minus the unit matrix. This matrix is one of the standard quantities used in bifurcation theory to investigate the stability of fixed points. In fact, the amplitude factor is closely related to the so-called residue, which was proposed by Greene [J. Math. Phys. 20, 1183 (1979)] in order to predict the occurrence of global chaos in classical systems. It is emphasized that all these classical quantities arise naturally in our representation of quantum mechanics.

A new kinetic real-space renormalization-group (KRG) approach to diffusion-limited aggregation (DLA) is presented. It is based on the combined length- and time-scale invariance of the clusters. The probabilities relating different local configurations under rescaling are determined from the growth process itself. Recursion relations between the masses of the cluster and of its interface on consecutive iterations are obtained. The common fractal dimension ($D_f$) of the cluster and of its interface is calculated. The approach is applied to DLA on square, cubic, and two regular fractal lattices. We find $D_f$=1.727 (2.494) for the square two-dimensional (cubic three-dimensional) lattices, in very good agreement with the most extensive numerical results. Corrections to scaling exponents are found from the smaller eigenvalues. In addition, the multifractal spectrum D(q) of the local-growth probability’s moments and the f-α curve are derived.

By linearizing the Langevin equation and the fluctuation-dissipation theorem using ensemble-averaged quantities, new approaches for calculating friction in nonlinear ultrafast dynamical processes are presented. Numerical testing of these methods confirms their accuracy. The harmonic approximation commonly used in barrier-crossing dynamics is also discussed.

The stability problem posed in statistical mechanics by self-gravitating systems is discussed for the simpler case of systems with a purely attractive nonsingular pair potential of short range. Molecular-dynamics simulations of such systems are reported and are found to agree remarkably well with a modified version of the Hertel-Thirring cell model. A first-order phase transition is observed between a homogeneous phase at high energies and a collapsing phase with a single, very compact cluster in a diluted homogeneous background at low energies. The latter is not an extensive or thermodynamic phase. According to the cell model, really large systems are most likely to be found in a critical state hesitating between these two phases.

By means of Monte Carlo simulations, diffusion-limited recombination reactions, followed by desorption and the desorption of single particles from fractal and multifractal substrata, are studied. From all these results, we conjecture that for both processes (a) the dissociative adsorption and associative desorption of diatomic molecules and (b) the nondissociative adsorption and desorption of single particles, the relationship between the particle coverage Θ and the applied pressure p at the steady state (i.e., the adsorption isotherm) may be written, in the limit Θ→0 as ${\mathrm{\Theta }}^{1/2+1/2\mathrm{f}}$(1-Θ${)}^{\mathrm{-}1}$∝${\mathrm{p}}^{1/2}$ for (a) and Θ(-lnΘ${)}^{1\mathrm{-}1/\mathrm{f}}$(1-Θ${)}^{\mathrm{-}1}$∝${\mathit{p}}^{1/2}$ for (b), where f is the site-visitation random-walk exponent of the substratum. Classical isotherms are recovered for homogeneous surfaces with f=1.

We formulate a continuous-time random-walk model for the motion of conducting ions in β-alumina which accounts for correlation effects as well as free and jump diffusion mechanisms in a unified way. The calculated frequency-dependent conductivity and incoherent intermediate scattering function agree with experimental results, suggesting that the model provides a good description of the underlying processes, including sublattice disorder effects.

A revision of the Douglas-Kroll transformation showed that the sign of the second-order term in the resulting transformed Dirac Hamiltonian has to be changed. This is in accordance with its use in practical applications. A brief review of the theory is given and a slight simplification of the second-order term is presented.

Energy levels are investigated within a square well containing δ-function barriers on a Cantor set. Particular attention is drawn to the dependence on the strength λ of the δ functions. It turns out that multiple avoided level crossings occur for λ>0 connecting the limiting cases at λ=0 and λ=∞, which are amenable analytically. The intermediate range is investigated numerically by two different methods: matrix diagonalization and implementation of a boundary condition for the wave function.

Measurements of superelastic scattering of electrons by laser-excited ${}_{}{}^{138}{}_{}{}^{}\mathrm{Ba}$(. . .6s6p ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$) atoms were carried out. An asymmetry observed by us and previously by Register et al. [Phys. Rev. A 28, 151 (1983)] has been explained using a model of scattering from a target with finite dimensions. This model employed coherence parameters which we calculated in the distorted-wave approximation. The results indicated that the interpretation of coherence experiments in terms of scattering from a pointlike target can lead to serious errors in the deduction of coherence parameters at low scattering angles.

Our normal understanding of the dissipative relaxation of a quantum system interacting with a reservoir or heat bath leads us to expect that the final state of the quantum system is a mixed state characteristic of one subsystem in equilibrium with the larger reservoir. We show that the broadband squeezed vacuum has such strong correlations in its quantum noise that, although it represents a true reservoir, the decay of subsystems, such as pairs of atoms or oscillators interacting with a squeezed reservoir, relaxes to a pure state that reflects these strong correlations.

We report the observation of an instability in a bistable optical system with a homogeneously broadened two-level medium involving the onset of the cavity side mode that is two modes away from the source frequency and three modes away from the molecular resonance frequency. These measurements extend the range of a recent report [Phys. Rev. A 39, 703 (1989)] of observations of the multimode instability in optical bistability predicted by Lugiato and Bonifacio [Lett. Nuovo Cimento 21, 505 (1978); 21, 510 (1978)], with the key difference being that the experimental observations in the previous report were limited to instabilities of the side mode adjacent to the source frequency.

Spectroscopic measurements were made of lithiumlike Al emission lines from laser-produced plasmas. Flat targets, consisting of thin layers of Al coated on a Mylar backing, were irradiated with up to eight beams of the OMEGA laser system employing a line-focus configuration. The Al xi 4f-3d line at 154.7 Å exhibited a gain coefficient of 4.1±1.2 ${\mathrm{cm}}^{\mathrm{-}1}$ from comparisons of line intensities from 3- and 6-mm-long plasmas. Similarly, the 4d-3p line at 150.7 Å and the 5f-3d line at 105.7 Å had gain coefficients of 4.5±1.3 and 3.5±0.8 ${\mathrm{cm}}^{\mathrm{-}1}$, respectively. The linewidth of the 5f-3d line was measured to be 0.35 Å, which is much larger than the expected Doppler and instrumental width and can be attributed to Stark broadening. This Stark broadening corresponds to an electron density of 3×${10}^{19}$ ${\mathrm{cm}}^{\mathrm{-}3}$.

The first-order excluded-volume correction to the dynamical viscosity of dilute polymer solutions is studied within the path-integral formulation of the Kirkwood diffusion equation. The renormalization of the dynamical viscosity [η(Ω)] is explicitly performed in two limits: (a) zero frequency Ω=0, and (b) Ω≠0, large contour length L (or finite L and Ω→∞). The effect of screening the hydrodynamic interaction by increasing the excluded-volume strength is predicted. The renormalization at the Ω-dependent renormalization point enabled us to derive the scaling law for the dynamical viscosity [η(Ω)]≊${\Omega }^{\mathrm{-}(d\nu \mathrm{-}1)/d\nu }$ within the renormalization group.

Three principles are presented as the basis for group-theoretical statistical mechanics (GTSM), incorporating Neumann’s (or Curie’s) principle as the first. The second extends Neumann’s principle to molecule fixed axes, (x,y,z), as defined in molecular point-group theory, and the third deals with the symmetry of applied fields and their effect on molecular ensembles in the field-on steady state. The power of GTSM is illustrated with reference to the set of field-induced n-time correlation functions in the laboratory frame (X,Y,Z) and frame (x,y,z).

The formation of spatial patterns characterized by nontrivial fractal dimension often results from processes where the outermost parts of the pattern ‘‘screen’’ inner sites from growth. The threshold-screening model uses a simplified measure of screening to define active sites on a growing pattern. As the strength of screening is increased, the pattern geometry changes from compact to one dimensional, passing through an intermediate critical state of nontrivial fractal dimension. Precise knowledge about the active-growth sites makes possible the use of information theory to compute the rate of entropy production for the different growing patterns. We find that the specific heat computed from the information-theoretic part of the entropy is singular at the critical point.

Electron correlations in ${\mathrm{H}}^{\mathrm{-}}$ are represented with a model interaction potential, 1/($r_1$+$r_2$). Hyperspherical-coordinate wave functions describing strong electron correlations near the origin are matched onto dipole field quantum-defect functions far from the nucleus. The results of this calculation complement and in some respects differ from those of Altick [Phys. Rev. A 38, 33 (1988)].

A new method of constructing approximate Kohn-Sham exchange-correlation potentials, $v_{\mathrm{xc}}$, from the orbital-dependent potentials ${\mathit{v}}_{\mathrm{xc}}^{\left(\mathit{i}\right)}$=δ${\mathit{E}}_{\mathrm{xc}}$[{${\mathit{n}}_{\mathit{i}}$}]/δ${\mathit{n}}_{\mathit{i}}$ is proposed in which $v_{\mathrm{xc}}$ is obtained from integrating the weighted average of electric (WAE) field produced by the $v_{\mathrm{xc}{}^{\left(i\right)}}$. Application of the WAE method to atoms with closed subshells employing exchange-only local-spin-density self-interaction correction and gradient-expansion-approximation self-interaction correction energy functionals yields $v_x$ and n(r) which closely approximate the OPM potentials and Hartree-Fock electron densities, respectively.

Barium clusters are produced by the inert-gas condensation technique. The observed size distribution shows large peaks at n=13, 19, 23, 26, 29, 32, as well for singly charged clusters as for doubly and triply charged clusters. These results can be interpreted by icosahedral close-packing structures, similar to those observed in noble-gas clusters.