We study diffusion in a linear chain where random fields are located at each site and can accept the values ±E with equal probability. While the average density distribution of a random walker scales and is described by a single exponent, it requires an infinite hierarchy of exponents α to characterize the fluctuations. Their density distribution f(α) is a single-hump function and depends continuously on the magnitude of the field E.

We show that the two-photon nonresonant contribution to the lifetime of excited 3s and 3d states of hydrogen, as given by a formula recently derived by Cresser et al. [Phys. Rev. A 33, 1677 (1986)], can be evaluated in an economic way using the analytic expressions of the amplitudes of 3s-1s and 3d-1s two-photon transitions. Our numerical evaluation confirms the numbers given by Cresser et al.

In gels, a critical point at which swelling transition becomes continuous, is characterized by vanishing of the bulk modulus $K$. At this point, however, an instability is triggered by a single mode representing homogeneous expansion while fluctuations with finite wave numbers are still suppressed by the finite shear modulus $\mu$. Critical divergence of the fluctuation spectrum is expected not at this critical point but only in an unstable region. This unusual feature has been overlooked because of small values of $\mu$ in previous experiments.

An associative memory network with local variables assuming one of $q$ equidistant positions on the unit circle ($q$-state phasors) is introduced, and its recall behavior is solved exactly for any $q$ when the interactions are sparse and asymmetric. Such models can describe natural or artificial networks of (neuro-)biological, chemical, or electronic limit-cycle oscillators with $q$-fold instead of circular symmetry, or similar optical computing devices using a phase-encoded data representation.

We elucidate the non-Abelian gauge structure associated with nuclear quadrupole resonance. The Abelian part of this structure has been experimentally observed. The phases to be observed in various non-Abelian experiments are computed.

We study isolated dislocations and disclinations in flexible membranes with internal crystalline order, using continuum elasticity theory and zero-temperature numerical simulation. These defects are relevant, for instance, to lipid bilayers in vesicles or in the $L_{\beta }$ phase of lyotropic smectic liquid crystals. We first simulate defects in flat membranes, obtaining numerical results in good agreement with plane elasticity theory. Disclinations and dislocations eventually exhibit a buckling transition with increasing membrane radius. We generalize the continuum theory to include such buckled defects, and solve the disclination equations in the inextensional limit. The critical radius at which buckling starts to screen out internal elastic stresses is determined numerically. Computer simulation of buckled defects confirms predictions of the disclination energies and gives evidence for a finite dislocation energy.

Computer simulations of diffusion-limited aggregation (DLA) show that surface tension, as a short-range interaction, does not change the fractal structure for large aggregates. Clusters grown with surface tension develop three regions. (1) At the center, the cluster is compact and regular; the effective fractal dimension $D_f^{\mathrm{eff}}$ is close to 2. (2) Next to the center is a transition region where $D_f^{\mathrm{eff}}$ decreases quickly from 2 to about 1.70±0.05. (3) Past the transition region, the cluster grows as a DLA fractal and $D_f^{\mathrm{eff}}$ extrapolates to 1.70±0.05. An increased surface tension is found to produce clusters with rounder shapes, thicker fingers, and larger compact centers. The thickness of the fingers and the size of the compact center are determined theoretically, and these estimations compare well with results obtained from the numerical simulations.

The nearest-neighbor spacing distributions P(S) for the variationally determined eigenvalues of two separable model Hamiltonians (two-dimensional quartic oscillators) are studied as functions of the dimension n of the variational basis. For large basis sizes the spacing histograms for a given energy range can be well fitted by a Poisson distribution, reflecting the regularity of the spectrum of a completely separable Hamiltonian. The decrease of n leads to histograms of qualitatively different spacing (for the same energy range as before), which can be well fitted by Brody distributions $P_q$(S). The Brody parameter q controls the transition between a Poisson (q=0) and a Wigner distribution (q=1). For large n, q=0 holds. When n falls below a critical value, q=1 results, i.e., $P_q$(S) turns over into a Wigner distribution. The latter is assumed to characterize a completely irregular spectrum in the classically chaotic energy range of a nonseparable Hamiltonian. The transition between a regular and an irregular spectrum is shown to be induced only by a change of numerical accuracy. We conjecture that similar behavior should also be observed in nonseparable systems.

Copper electrodeposits from an acid solution were grown, supported by laboratory filter paper. The paper provides good sample preservation and prevents convective solvent motion. Structural data from electron microscope and image analysis are reported. On decreasing the applied voltage, a transition from dense radial growth to open growth was observed. Values of the fractal dimension measured by image analysis for a series of growths were found in the range 2.0–1.7. The growths were often inhomogeneous and anisotropic, both on the microscopic and macroscopic scales. A number of growths were not fully characterized on the macroscopic scale by a single value to the fractal dimension. Wide variation in the microstructure of the growths was observed. On the micrometer scale, irregular microcrystalline, dendritic microcrystalline, and granular deposits were found. Copper surface densities were measured by x-ray microanalysis; fractal interpretations of observed scaling in these data were found to be unsatisfactory.

The density-wave theory of Ramakrishnan and Yussouff is used to study phase transitions between liquid, liquid-crystalline, and crystalline phases. The different phases considered are liquid, nematic, smectic, discotic, bcc plastic crystal, orientationally ordered bcc, and a new incommensurate bcc crystal with orientational order. The direct correlation function, required as an input for the theory, is expressed approximately in terms of five generalized Fourier coefficients. The theory is then used to obtain sections through the phase diagram in the five-dimensional space of these coefficients. Simple approximations for the direct correlation function of hard ellipsoids of revolution are used to compare these phase diagrams with those obtained from experiments and numerical simulations. Molecular-field theories of smectic and discotic ordering are reexamined, and, given the potentials they use, it is shown that an orientationally ordered bcc crystal has a lower free energy than either the smectic or the discotic phase. The conditions required to stabilize smectic and discotic phases are examined.

The Löwdin alpha functions, which are the functions associated with the spherical-harmonic expansion of a displaced Slater-type orbital, are expressed using C matrices to represent the polynomials in terms of the displacement distance a and the radial distance r. These polynomials are multiplied by the sum and difference of exponentials. The expansion of the exponentials leads to the use of E and F matrices. By keeping only the r variable identifiable, further simplifications of the alpha functions are possible, which makes for easy programming of all multicenter integrals. Also, no singularities appear in these developments. Everything is demonstrated by using 1s orbitals as prototypes.

The Thomas-Fermi equation for the electronic structure of the planar uniform-background model is solved with use of relativistic kinematics.

Stochastic cooling in storage rings was a great success: the present paper gives a complete theory for the application of the same general concepts to Penning traps. The structural difference between a storage ring and a Penning trap requires a complete revision of the starting points on which to base the theory. Not surprisingly, however, the final results exhibit many similarities. One of the main differences is that the noise effects are best treated, in a Penning trap, by introducing a limiting energy ${\varepsilon }_{\lim }$, proportional to the noise amplitude, that represents the minimum energy attainable with stochastic cooling. The theory shows that stochastic cooling in a Penning trap can be very useful when one wants rapidly to cool the energy of the trapped particles down to limiting energies greater than kT, T being the noise temperature associated with the detection circuits.

The matrix element between arbitrary Slater determinants of an operator of the form V’GV, where V and V’ are symmetric operators and G is the total resolvent operator, is considered. This element is obtained as the stationary value of a functional and the resulting variational equations have the form of driven Hartree-Fock equations. We show that these mean-field equations allow one to define a dynamical, canonical representation of the orbitals in the initial and final determinants. This thus defines new shell models, which are significant for the special transition under investigation.

We consider the response of an isolated two-level atom to strong-field excitation in the special case where the atom decays via coupling to a frequency-dependent photon-mode density. It is found that atomic squeezing substantially in excess of that found in the case of free-space atoms arises. The enhanced squeezing is related to a polarization of the atom-field dressed states that is also found to occur.

Elastic $e^{+}$-H scattering has been investigated by a perturbative approach at medium energies. The effect of the positronium-formation channel on the elastic one has been taken into account explicitly. In the perturbative series, the first term is the solution of the coupled static equation. The conventional perturbative series including the rearrangement channel has also been employed. The effect of the capture channel on the elastic one is found to enhance the cross section near the forward direction. The present results are compared with existing elaborate theoretical predictions.

The triple-differential cross sections for the electron-impact ionization of helium in coplanar asymmetric geometry have been calculated by use of a second Born approximation which employs an improved choice for the target continuum-state wave function. The results are compared with the absolute experimental data of K. Jung et al. [J. Phys. B 18, 2955 (1985)] at an incident energy of 600 eV and are found to be in very good agreement with them. The corresponding results for the positron-impact case are also presented.

We will determine the effect of dissipation and nonunit quantum efficiency on the oscillations in the photon-number distribution for squeezed states. These oscillations are evidence of interference in phase space and thus our results show the effect of dissipation on such interference effects. Our method, based on the Q function, enables a direct comparison of the quantum and classical phase-space descriptions.

In this Comment certain questions of higher-order and intrinsic higher-order squeezing from an anharmonic oscillator recently discussed by Gerry and Rodrigues [Phys. Rev. A 35, 4440 (1987)] are clarified.

We report theoretical results for autoionizing states of systems with a closed-shell core and that lie above several ionization thresholds. The energy positions and widths are given for the (${1s}^{2}3l{3l}^{\prime }$) ${}_{}{}^1{}_{}{}^{}S$ and ${}_{}{}^{1,3}{}_{}{}^{}P$ resonant states of the ${\mathrm{C}}^{2+}$, ${\mathrm{N}}^{3+}$, ${\mathrm{O}}^{4+}$, and ${\mathrm{Ne}}^{6+}$ ions, which are of great experimental interest at the present moment. Our results show that a generalization of the method of F. Martín et al. [Europhys. Lett. 4, 799 (1987); J. Chem. Phys. 87, 6635 (1987)] provides a very useful and simple procedure to obtain energy positions and resonance widths, as well as the contributions of the different continua to the width.