Molecular-dynamics (MD) calculations are reported for three thermodynamic states of a Lennard-Jones fluid. Systems of 2048 particles and ${10}^5$ integration steps were used. The transverse current autocorrelation function, $C_t$(k,t), has been determined for wave vectors of the range 0.5<‖k‖σ<1.5. $C_t$(k,t) was fitted by hydrodynamic-type functions. The fits returned k-dependent decay times and shear viscosities which showed a systematic behavior as a function of k. Extrapolation to the hydrodynamic region at k=0 gave shear viscosity coefficients in good agreement with direct Green-Kubo results obtained in previous work. The two-exponential model fit for the memory function proposed by other authors does not provide a reasonable description of the MD results, as the fit parameters show no systematic wave-vector dependence, although the $C_t$(k,t) functions are somewhat better fitted. Similarly, the semiempirical interpolation formula for the decay time based on the viscoelastic concept proposed by Akcasu and Daniels fails to reproduce the correct k dependence for the wavelength range investigated herein.

A path-integral solution to truly nonlinear Fokker-Planck equations is derived. Such equations exhibit in the drift and diffusion coefficients a functional dependence on the distribution function. This type of implicit time dependence is shown to introduce terms into the propagator function of the exact same order in the time step τ, as does an explicit time dependence if the functional dependence is sufficiently smooth. A standard discrete lattice formulation of the path integral is then used to reproduce the appropriate, truly nonlinear Fokker-Planck equation. This discrete formulation provides a basis for an efficient numerical algorithm and is applied with excellent results to several example problems where exact solutions can be calculated.

We present an analytic solution of the problem of velocity selection in a fully nonlocal model of dendritic crystal growth. Our analysis uses a WKB technique to derive and evaluate a solvability condition for the existence of steady-state needlelike solidification fronts in the limit of small undercooling Δ. For the two-dimensional symmetric model with a capillary anisotropy of strength α, we find that the velocity is proportional to ${\Delta }^4$${\alpha }^{7/4}$. We also describe the application of our method in three dimensions.

We study the depletion of a diffusing substance [described by a scalar Laplacian field u(r)] in the vicinity of an absorbing fractal, consisting of a Gaussian random walk (RW) or a self-avoiding walk (SAW) in d dimensions. The moments 〈[u(r${\left)\right]}^n$〉 of the field u(r) at a distance r from a point on an absorber of linear size R obey 〈[u(r${\left)\right]}^n$〉∼(r/R${)}^{\lambda \left(n\right)}$, where the λ(n) are a set of independent exponents. The incident flux φ of the field u defines a fractal measure on the absorber. The scaling dimensions D(n) describing this measure obey (1-n)[D(n)-D]=nλ(1)-λ(n), with D the fractal dimension; moreover, λ(1)=D+2-d. These relations apply for any fractal absorber. For the RW and SAW, we observe that the exponents λ(n) are the same as those describing the statistics of linear chains and star polymers with selective excluded-volume interactions. This allows us to calculate λ(n) and D(n) explicitly to order ${\epsilon }^2$, where ε=4-d. We find D(n)=D-n${\epsilon }^2$/4 for the RW and D(n)=D-9n${\epsilon }^2$/64 for the SAW. We also study nonperturbatively the limit of high n. For n large, D(n)-D(∞)∼$n^{\left(3\mathrm{}\mathrm{-}d\right)/\left(d\mathrm{-}2\right)}$ for 3<d<4, whereas in d=3 dimensions, D(n)-D(∞)∼[ln(n${\left)\right]}^{\mathrm{-}1}$; here D(∞)=d-2. These high-n results apply to the RW and SAW alike. We construct for the absorbing RW part of the scaling function f(α), defined by Halsey et al. [Phys. Rev. A 33, 1141 (1986)], and find a range of α for which f(α) is negative. Identifying f(α) with the histogram of the measure expressed in logarithmic variables, we discuss the meaning of negative f(α).

The exponentially small energy-level splittings Δ$E_{2n}$≡$E_{2n+1\mathrm{}}$-$E_{2n}$ (n=0,1,. . .) for ground-state as well as excited-state doublets in a symmetric bistable potential are determined by means of a simple explicit formula. The tunneling frequencies Δ${\omega }_{2n}$≡Δ$E_{2n}$/ħ form a quasi-Weierstrass spectrum, which explains the erratic tunneling behavior of an initially squeezed wave packet. It also implies the existence of a natural fractal (Hausdorff-Besicovitch) dimension D for the chaotic tunneling trajectory, as is shown by computerized zooming towards the picture of position versus time and determining (over a fixed period of time, with sampling time intervals Δt varying over more than four orders of magnitude) the total sample path length L∼(Δt${)}^{1\mathrm{-}D}$.

A new Langevin equation for dye-laser fluctuations below and near threshold is used to obtain the intensity distribution function in closed form. The effect of strongly colored multiplicative pump noise is incorporated by means of an ansatz of Hanggi et al. [Phys. Rev. A 32, 695 (1985)] for an effective diffusion coefficient at steady states. Excellent agreement with the measurements of Lett, Short, and Mandel [Phys. Rev. Lett. 52, 341 (1984)] is obtained.

It is shown that the excess free energy due to hard-core repulsions in terms of the geometrical asymmetry of chiral molecules is negligibly small and that the corresponding contribution in terms of the shape anisotropy of rodlike molecules stabilizes the smectic-A phase with respect to the smectic-C phase.

We study period doubling in a symmetric four-dimensional volume-preserving quadratic map, i.e., two symmetrically coupled two-dimensional area-preserving Hénon maps. We must vary two parameters and thus obtain two Feigenbaum constants, ${\delta }_1$ and ${\delta }_2$. It is a very important point that for each region of stability (belonging to some period-q orbit) in this parameter plane we find two regions of stability for the period-2q orbit, four regions for the period-4q orbit, and so on. Hence we have an infinite number of stability regions and infinities of bifurcation ‘‘paths’’ through these regions. Almost all self-similar bifurcation paths fall into one of three possible ‘‘universality classes,’’ i.e., each class is characterized by its own two Feigenbaum constants, ${\delta }_1$ and ${\delta }_2$. We find ${\delta }_2$=+4.000. . .,-2.000. . .,-4.404. . ., respectively, for the three classes. These ${\delta }_2$ values are also recovered here from some approximate (numerical) renormalization scheme. The ${\delta }_1$ is, in all cases, the same as in two-dimensional area-preserving maps, ${\delta }_1$=8.721. . . . The ${\delta }_2$=-15.1. . ., reported in an earlier paper [J. M. Mao, I. Satija, and B. Hu, Phys. Rev. A 32, 1927 (1985)], applies to only two exceptional paths.

A simulation of a one-dimensional coagulating system of droplets with mass-dependent diffusion coefficients is presented. The total droplet density is shown to decay as $t^{\mathrm{-}1/3}$ and the density of droplets of mass l decays as exp(-${\mathrm{ct}}^{1/3}$/l) in the long-time limit. Simple kinetic arguments are derived which predict these results. The scaling behavior and the similarity to phase-separation dynamics is investigated. The scattering function is calculated. It develops a well-defined peak which grows in intensity and shifts to lower wave vectors as a function of time. The scattering function and the droplet-size distribution are both shown to assume scaled forms in the late stages.

In the present paper, extending the Weinhold theory, we obtain some differential-geometry properties of the ideal gas and the ideal paramagnetic gas. In particular, we show that the scalar curvature of the ideal paramagnetic gas can be a function of temperature.

We report x-ray reflectivity measurements on the free surface of 4-n-heptylphenyl-4’-(4’’-nitrobenzoyloxy)benzoate (${\mathrm{DB}7\mathrm{NO}}_2$) at the nematic to smectic-A phase transition, $T_{\mathrm{NA}}$=99.9 °C. The free surface in the nematic phase exhibits smecticlike ordering at two q vectors, one which is commensurate with the smectic-A monolayer q vector $q_2$. The other q vector is incommensurate corresponding to ordering at ∼0.59$q_2$. The commensurate peak constructively interferes with the air-liquid interface while the incommensurate peak destructively interferes. These results are compared with bulk-phase x-ray scattering measurements.

Dynamics near a singular Hopf bifurcation under conditions of nonlinear self-focusing is dominated by intermediate-scale solitons with amplitudes and speeds evolving on a slower time scale. The long-scale evolution can result in selection of either standing or propagating solitons with either stationary or oscillating amplitude and speed.

We devise and implement a technique to study the evolution of the interface for a one-sided model of directional solidification on a lattice with biased random walkers. Results include the appearance of a marginally stable wavelength at the onset of nonlinearity. Also, as the velocity of the interface is increased, there is a morphological transition from flat to cellular, and then to cellular with grooves. Anisotropy in dynamics is not included and the transition from cells to dendrites has not been seen.

We propose to look at first-return maps into a specified region of phase space as a basis for a unified renormalization scheme for dynamical systems. The choice of the region for first return is dictated by the symbolic dynamics (e.g., kneading sequence) of the relevant trajectories. The renormalization group can be formulated on the symbolic level, but once translated to maps it yields the said renormalization scheme. We show how the well-studied examples of the onset of chaos via period doubling and quasiperiodicity fit into this approach, and argue that these problems get in fact unified. The unification leads also to a generalization that allows us to study the onset of chaos in maps that belong to larger spaces of functions than those usually considered. In these maps we discover a host of new scenarios for the onset of chaos. These scenarios are physically relevant since the maps considered are reductions of simple flows. We present a theoretical analysis of some of these new scenarios, and report universal results. Finally we show that all the available renormalization groups can be found using symbolic manipulations only.

A proof is given that self-sustaining ionic-wave propagations—heuristically inferred by the author in a previous paper [Phys. Rev. A 32, 3618 (1985)] concerning a new holographic theory of animal memory—are possible in animal tissues at normal physiological conditions. The proof is obtained by standard electrochemical methods purely on the basis of well-known properties of cell membranes and of molecular devices found in them. It is shown that interstitial pockets filled with extracellular fluid, when viewed as functional units dispersed all over the cell tissue, promote ion currents proportionally to linear combinations of macroscopic sodium and potassium concentration gradients. Oscillations of sodium and potassium macroscopic concentrations prove to be possible thanks to the voltage-driven amplification function for ion fluxes exerted by ${\mathrm{Na}}^{+}$-${\mathrm{K}}^{+}$-ATPase (adenosinetriphosphatase) pumps and to the feedback control role exerted by the (${\mathrm{Na}}^{+}$ antiports ${\mathrm{Ca}}^{2+}$)→(${\mathrm{Ca}}^{2+}$ activates ${\mathrm{K}}^{+}$) channel system. All steps in deriving the wave equations are coarse-graining invariant; this ensures the correctness of the macroscopic view in treating the problem. Theoretical wave patterns and their general features are in excellent agreement with EEG (electroencephalogram) patterns detected on brain cortices and on scalps. Epileptic foci artificially generated by injection of ${\mathrm{Na}}^{+}$ ions into glial tissue and inhibition of EEG by ${\mathrm{K}}^{+}$ superfusion of brain cortex, are correctly accounted for by the theory.

Using the Löwdin α-function method with a computer-generated C-matrix associated with a displaced Slater-type orbital, an exact closed formula as well as a Taylor series with exact coefficients is produced for the overlap integral over two 7gγ orbitals. It is shown how ten-decimal-digit accuracy is achieved throughout the entire range of parameter values. A noteworthy development is the great simplification of this overlap formula made possible by factoring out (1-t${)}^{20}$ by means of computer algebra.

The radial Schrödinger equation is solved for the screened Coulomb potential with use of a power-series expansion for the wave function. For given n and l, the energy eigenvalue can be obtained to arbitrary order in the screening parameter; the coefficients of expansion for the corresponding radial wave function are deduced from a recursion relation between the various expansion coefficients which is in general dependent on the energy eigenvalue.

The nuclear-motion fine-structure isotope shifts in the ground states of C ii and N ii are computed by use of many-body perturbation theory. There are three different kinds of nuclear motion corrections to the atomic fine structure: normal mass effects on the spin-orbit and spin-spin interactions, electron-nucleus relativistic interactions, and higher-order cross terms between the specific-mass-shift operator and spin-dependent interactions. All three types of corrections have been considered and computed to third order in the many-body perturbation expansion. The computed results are in excellent agreement with recent accurate experimental values.

We report the electric field ionization of the n=8–15 levels of Na using field pulses of rise time ∼1 μs. The passage from low to high field is adiabatic, and the fields required for ‖$m_l$‖=0, 1, and 2 states are 3.12(9)×${10}^8$$n_s^{\mathrm{-}4}$ V/cm, 3.37(11)×${10}^8$$n_s^{\mathrm{-}4}$ V/cm, and 3.68(21)×${10}^8$$n_s^{\mathrm{-}4}$ V/cm. Here $n_s$ is the effective quantum number in the ionizing field. These values are in agreement with those determined previously for higher-lying states.

Recoil ions ejected at 70° from 20-keV proton collisions with ${\mathrm{O}}_2$, CO, ${\mathrm{CO}}_2$, and Ne were analyzed by an electrostatic analyzer. For the molecular targets, only singly ionized ions of oxygen or carbon were detected while in the spectrum of neon ions, ${\mathrm{Ne}}^{+}$ and ${\mathrm{Ne}}^{2+}$ were both observed. The absence of ${\mathrm{O}}_2$${\mathrm{}}^{+}$, ${\mathrm{CO}}^{+}$, and ${\mathrm{CO}}_2$${\mathrm{}}^{+}$ is due to the fact that the recoil ions detected in this experiment result only from very close collisions with one of the atoms in the molecule resulting in dissociation. A possible explanation for the absence of ${\mathrm{O}}^{2+}$ and ${\mathrm{C}}^{2+}$ peaks based on the molecular nature of the targets is proposed.