A recursive method for enumerating clusters on a hypercubic lattice in d spatial dimensions is presented from which the weak embedding constants are determined as polynomials in d. A tabulation for all clusters having no free ends is available for $n_b$≤15, where $n_b$ is the number of bonds. As illustrated here and elsewhere, this tabulation can be used to generate many series expansions. A novel method of checking the enumeration with an algebraic calculation is presented.

The effects of nonequilibrium kinetic undercooling on the velocity selection in the two-dimensional nonlocal model of dendritic growth are studied with use of recently developed analytical and numerical methods. We study and compare the effect of anisotropy on the selected velocity in detail for both terms.

We describe the first nondestructive technique for measuring the fractal dimension of a large three-dimensional object (size of the order of 1 cm), based on the balance between capillarity and buoyancy effects between two immiscible fluids. The wetting fluid covers the object like a cocoon with an interface ruled by the radius of curvature which is tuned by changing the difference of density between the two fluids. The volume of the wetting fluid measured versus the radius of curvature leads to the fractal dimension of the external surface. The experimental setup is described and the first results are analyzed and compared to the results of simulations on known two-dimensional fractal clusters, in particular, diffusion-limited-aggregation structures.

Melting of the Wigner crystal is studied in a new expansion scheme which is much more powerful than the Wigner-Kirkwood expansion of the quantum correction. The theory does not include the symmetry effect. The free energy including the quantum correction is evaluated with full use of the recent Monte Carlo results obtained for the classical one-component plasma. The melting entropy as well as the melting curve is obtained as a result of various extrapolations of properties for the liquid phase, in the approximation up to the sixth ‘‘reduced moment’’ of our expansion.

In this work we analyze the relation between the stochastic representation of nonrelativistic quantum mechanics and the Newtonian stochastic mechanics. In order to understand how the nodes of a wave function are accounted for in stochastic quantum mechanics, we solve the Fokker-Planck equation associated with the quantum-mechanical density ${\rho }_0$=$x^2$ and current $j_0$=0, and find that the stochastically perturbed particle does not cross the nodal surface x=0 of the density ${\rho }_0$. Further we consider the classical motion of a particle subject to a stochastic perturbation, in the absence of conventional external forces and friction, and moreover we consider the classical stochastic motion of a harmonically bound particle in the presence of a friction force proportional and opposite to the velocity of the particle, to show that although the quantum-mechanical equations for the probability density and current can be put in a form resembling Newton’s second law, the actual significance of those equations is not related to Newtonian stochastic mechanics. From the fact that, in the stochastic representation of quantum mechanics, the particles have continuous trajectories, we infer that the interactions would possess, in this theory, a global character. We finally show that the property of reversibility in stochastic quantum mechanics depends on the existence of suitably chosen compensating drifts in the mean forward velocity or acceleration of the particle under consideration.

We analyze the dynamics of a transient pattern formation in the Fréedericksz transition corresponding to a twist geometry. We present a calculation of the time-dependent structure factor based on a dynamical model which incorporates consistently the coupling of the director field with the velocity flow and also the effect of fluctuations. The appearance and development of a characteristic periodicity is described in terms of the time dependence of the maximum of the structure factor. We find a well-defined time for the appearance of the pattern and a subsequent stage of pattern development in which the characteristic periodicity tends to an asymptotic value.

We present an experimental study and a numerical simulation of the effect of time-independent, localized perturbations applied to the interface in the Saffman-Taylor fingering problem. When the perturbation is applied at a specific spot near the tip of the finger, the selection of the steady-state shape is drastically changed. In particular, one can obtain fingers with a width well below λ=1/2. A perturbation applied far away from the tip has no effect. We observe the same behavior in the simulation and in the experiment.

High-resolution volumetric measurements have been made on pure 8¯S5 and 8OCB in the vicinity of the smectic-A–to–nematic (A-N) transition. The molar-volume variations show in both compounds a continuous second-order A-N transition. For 8¯S5 the A-N transition is well described by a critical three-dimensional XY exponent α=α’=-0.026, but for 8OCB by the exponent α=α’=0.19±0.07. For 8OCB the anisotropic hyperscaling relation 2-α=${\nu }_{?}$+2${\nu }_{\perp }$ is satisfied to within experimental error. The thermal-expansion coefficient ${\alpha }_p$ has been determined as a function of temperature.

Continuum diffusion of a third species in a two-phase system comprised of stationary globules, randomly distributed in a uniform background medium, is considered. The diffusing species can dissolve in either phase, with different solubilities and diffusion constants. A density expansion of the average diffusion coefficient is presented. In the strong- (but not-perfect-) trap limit, both the first- and second-order globule-density contributions to the space Fourier transforms of the average Green’s and velocity autocorrelation functions decay as (t${)}^{\mathrm{-}5/2}$ for long times t. In addition, various intermediate time scales are examined. Finally, repeated-ring approximations are formulated and solved. In general, the repeated-ring approximations break down for sufficiently strong traps.

The Coulomb interaction of an electron with each electron or nucleus in its environment is Fourier analyzed into components with wave vectors q. These components are classified as long (short) range when q is smaller (larger) than the electron’s wave number p. The contrasting characteristics of each class are discussed. Electron reflection at a material’s surface measures the effect of short-range components only.

An uncertainty-type lower bound [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)] to the information-entropy sum in complementary spaces has recently been reformulated by Gadre et al. [Phys. Rev. A 32, 2602 (1985)] in terms of the respective one-particle probability densities. This bound has been exploited to derive rigorous upper as well as lower bounds to the information entropies and their sum in terms of the corresponding second moments of their distributions. Thus the present work establishes a direct connection, as suggested by Sears, Parr, and Dinur [Israel J. Chem. 19, 165 (1980)], between the quantum-mechanical kinetic energy and information entropy in position space. It has also been demonstrated that given at least one arbitrary moment-type constraint in each space, it is possible to derive an upper bound to the information entropy sum in complementary spaces.

A general approximate expression for the nonrelativistic ground-state energies of Coulomb systems with charges $Z_a$=$Z_b$=±1 and $Z_c$=∓1 and arbitrary masses $m_a$, $m_b$, and $m_c$ is suggested. The results obtained with use of this expression are in excellent agreement with precise variational and adiabatic calculations of three-particle electron, muon, and excitonic molecular systems and can be used for the prediction of ground-state energies of yet-untested systems.

Using the quantum theoretical version of the Thomas double-scattering model given by Shakeshaft and Spruch, we have obtained the collision cross sections for the reaction ${\mathrm{H}}^{+}$+HCl→${\mathrm{H}}_2$${\mathrm{}}^{+}$+Cl.

The partial cross sections for production of slow-recoil ${\mathrm{Ne}}^{\mathrm{i}+}$ ions in 1.05 MeV/amu ${\mathrm{Ne}}^{\mathrm{q}+}$ (q=2,6,8–10) and ${\mathrm{Ar}}^{\mathrm{q}+}$ (q=4, 10, 12, 14) ion impact have been determined. The ratios of the cross sections ${\sigma }_i$ of recoil ${\mathrm{Ne}}^{\mathrm{i}+}$ in ${\mathrm{Ne}}^{10+}$ ion impact to those in ${\mathrm{Ar}}^{10+}$ ion impact are found to increase rapidly for i>7, whereas those for i≤7 are unity, indicating that the production recoil ions are dependent on only the projectile charge and that the main contribution comes from direct ionization processes. This suggests that a significant part of the high-charge-state recoil ${\mathrm{Ne}}^{\mathrm{i}+}$ ions (i>7) in ${\mathrm{Ne}}^{10+}$-ion impact is produced mainly by transfer processes of K-shell electrons between the projectiles and the target atoms.

Well-resolved state-to-state data are reported for the vibrational predissociation of specific rotational levels of ${\mathrm{HeCl}}_2$(B) with eight vibrational quanta in the Cl-Cl stretching mode. The state resolution is sufficient to clearly reveal the symmetry constraints on the dynamics as well as a node in the product-rotational-state distribution. ‘‘Parity-selected’’ excitation spectra are obtained which reduce spectral congestion and aid in the assignment of individual rovibronic transitions.

We report the first complex-rotation calculation which incorporates spin-dependent perturbations. By using the intermediate $\mathrm{LSJ}$ coupling scheme, the lifetimes due to the spin-induced autoionization of the lithiumlike metastable $\mathrm{}\left(1s2s2p\right)\mathrm{}{}_{}{}^4{}_{}{}^{}P_{\frac{5}{2}}^{}{}_{}{}^{}$ state are determined along the isoelectronic series for $Z=2-10\mathrm{}$ and $Z=16-18\mathrm{}$. The results are compared with experimental measurements and other theoretical results.

The contribution of transfer ionization (TI) to total electron capture has been measured for ${\mathrm{O}}^{q+}$ ions ($q=5, 6, 7, \mathrm{and} 8$) colliding with helium at energies from 0.5 to 1.5 MeV/u. These measurements, along with other published results, suggest a maximum TI contribution to total capture of $\sim 0.15q^{0.5}$ at $\frac{E(\mathrm{in}\mathrm{keV}/\mathrm{u})}{q^{0.5}}\simeq 100$ The results demonstrate that the failure to account for transfer ionization in total single-charge-transfer cross sections may lead to large discrepancies between experiment and theory.

A homodyne detector measures a field-amplitude component of the incoming signal. If the incoming signal is in an $n$-photon eigenstate the homodyne detector's output probability distribution exhibits $n$ fringes. Here the degree to which the visibility of these fringes is degraded by a homodyne detector with less than unit quantum efficiency is evaluated. An experiment employing conjugate pairs of photons generated via a parametric down conversion or four-wave mixing is proposed by which these fringes could be observed.

Nonlinear neural networks are studied near saturation, when the number $q$ of stored patterns is proportional to the system size $N$, i.e., $q=\alpha N$. The statistical mechanics is obtained for arbitrary nonlinearity. For a wide class of models, including the original Hopfield model and clipped synapses, it is shown that there exists a critical ${\alpha }_c$ above which the system looses its memory completely. Furthermore, ${\alpha }_c$ never exceeds ${\alpha }_c^{\mathrm{hopfield}}$ and is determined by a universal expression. A moderate dilution of the bonds may improve the memory function.

In nonequilibrium growth such as diffusion-limited aggregation (DLA), the growth-site probability distribution characterizes these growth processes. By solving the Laplace equation numerically, we calculate the growth probability $P_g\mathrm{}\left(x\right)\mathrm{}$ at the perimeter site $x$ of clusters for the DLA and its generalized version called the $\eta$ model, and obtain the generalized dimension $D\left(q\right)\mathrm{}$ and the $f-\alpha$ spectrum proposed by Halsey et al. [Phys. Rev. A 33, 1141 (1986)]. It is found that $D\left(q\right)\mathrm{}$ depends strongly on $q$ and that the $f-\alpha$ spectrum is continuous. Our results suggest that these growth processes cannot be described by a simple scaling theory with a few scaling exponents. This is in clear contrast to the Botet-Jullien model [Phys. Rev. Lett. 55, 1943 (1985)] which yields equilibrium patterns whose $D\left(q\right)\mathrm{}$ is constant. It is also found that the information dimension $D\left(1\right)\mathrm{}$ which represents the properties of the unscreened surface is in good agreement with our recent theory.