We examine the transition to Gaussian orthogonal ensemble (GOE) behavior in the statistical properties of the eigenspectrum and transition amplitudes of a class of model Hamiltonians, which simulates the crossing region between a bright potential surface with a low density of states and a dense set of states on a dark potential surface. The Hamiltonian contains the highly delocalized coupling of multiple bright states with the regular, Poisson-distributed, bath of dark states expected for a classically integrable system. Coupling between rovibrational states on electronic surfaces with different equilibrium molecular conformations like the ${\mathrm{NO}}_2^2$${\mathrm{A}}_2$${\mathrm{-}}^2$${\mathrm{B}}_2$ or acetylene-vinylidene couplings, may be of this type. Statistical properties of the eigenspectrum and of the delocalized bright state intensity are shown to follow those of the Gaussian orthogonal ensemble for increasingly longer ranges and in higher-order correlations, as the number of simultaneously interacting bright states increases, but are shown to be limited by intermediate behavior at long range, which is stable to increases in the coupling strength and depends on the number of bright states. Molecular systems that follow this model would show statistics dependent on experimental limitations on resolution and range, such as a spacing distribution function with level repulsion at short range and a nearly exponential tail, and values of ${\Delta }_3$ that have GOE behavior at short range, but linear at long range. Delocalization of the bright states over the sampled energy range is required for statistics stable to changes in coupling strength and is shown to be measured by the short-time behavior of the spectral Fourier transform.

The simplest example of a reacting polymer system is studied, that of a single chain with reacting end groups. The long-time cyclization rate is calculated systematically using renormalization-group methods. Several types of single-chain dynamics are treated, with and without hydrodynamical and excluded-volume interactions. The hydrodynamical cases (‘‘non-free-draining’’) describe cyclization in dilute polymer solutions. For free-draining polymers we find ‘‘diffusion-controlled’’ behavior for long chains: The ratio of the reaction rate to the longest relaxation rate of the unperturbed polymer is a universal constant independent of the chemistry of the reacting ends. For non-free-draining models the analogous ratio vanishes to order ε (ε=4-d). In good solvents (hydrodynamics plus excluded volume) we predict that a ‘‘mass action’’ law holds asymptotically: The molecular weight dependence of the reaction rate is that of the probability that the chain ends are together in equilibrium. In all cases, the asymptotic behavior, either diffusion controlled or mass action, is determined by the class of dynamics and is independent of the reactivity of the end groups.

A linear stability analysis of the Saffman-Taylor instability in a radial Hele-Shaw cell is presented. It is shown that the limit of validity of the linear stability analysis can be estimated without having to solve the problem to higher orders. The formation of fingers initiated by the presence of white noise is also examined. We find that, depending on the amplitude of the noise, the number of fingers that is observed as the limit of the linear regime is reached may differ significantly from the number of fingers predicted by the fastest growing mode.

The potential of Fokker-Planck equations with a noninvertible diffusion matrix is calculated by systematically employing a polynomial-expansion approach. About Hopf bifurcation and degenerate Hopf bifurcation points the concrete form of the potential is specified. The analysis is extended to general Fokker-Planck equations, which include Fokker-Planck equations with both invertible and noninvertible diffusion matrices.

The solution of the Ornstein-Zernike equation in the mean-spherical approximation (MSA) is used to study the spinodal region of the gas-liquid transition, between the mathematical limits of the van der Waals (vdW) fluid and the adhesive-hard-sphere fluid. We study a liquid of spherically symmetric molecules interacting via a Yukawa potential (hard core plus screened Coulomb potential) of constant finite integrated strength (which leads to a constant temperature scale). By changing the inverse range parameter in this potential, one interpolates smoothly between the adhesive-hard-sphere and vdW limits. We compare our potential with other potentials with less convenient scaling properties used in earlier studies. We confirm that mean-spherical critical-exponent values apply for adhesive hard spheres, whereas classical critical exponents are found for the van der Waals fluid.

We report critical densities and temperatures for the fluids, in both limits, the latter being finite and nonzero using our temperature scale. The phase diagram of these fluids is investigated by using the inverse range parameter in the Yukawa potential as the third dimension, along which the $T^{\mathrm{*}}$-ρ phase diagram is projected. The pair correlation functions have been calculated within the MSA. As noticed previously, the region within the (mean-field) spinodal decomposes into a space of real solutions and a space of complex solutions. The space of real solutions within the spinodal vanishes for the van der Waals fluid, but for adhesive hard spheres, real solutions are found in a considerable region of the spinodal. The one-dimensional manifolds that correspond (a) to the loci of critical points (that is, maxima of the spinodal manifold) and (b) to the maxima of the manifold (that is, the envelope surface of the space of complex solutions) coalesce into a single line for the van der Waals fluid but remain distinct for adhesive hard spheres. Such behavior can be taken as one criterion for identifying a model fluid as having either van der Waals or adhesive-hard-sphere properties. Most importantly, the coalescence of these lines in phase space is associated with the onset (or disappearance) of classical values for the critical exponents δ and γ. This connection between the evolution of critical exponent values and classical liquid pair correlation functions has not, to our knowledge, been reported previously.

We present a lattice model for the nematic–smectic-A phase transition in liquid crystals. It reduces to the de Gennes model in the continuum limit and, in contrast to earlier lattice attempts, is capable of describing the phase transition in both the second- and first-order regimes, thus comprising also the experimentally observed tricritical point. The model is investigated using mean-field techniques and Monte Carlo simulations.

A ‘‘window’’ concept, used previously in a phase-space representation, is extended to an energy representation. The new definition of trace is used to evaluate the properties of the density matrix and the entropy for electromagnetic fields. It is shown that the ‘‘ensemble’’ entropy behaves more closely to the ‘‘thermodynamic’’ entropy than the ‘‘physical’’ one. The source of negative entropy values is clearly shown as being due to higher contributions of virtual states.

Molecular-dynamics simulations (MD) have been carried out on a soft-sphere model for binary alloys quenched into glassy states at different quench rates. The main purpose of the present work is to investigate slow relaxation phenomena of the quenched glassy states close to the glass transition by calculating both the static and dynamical structural changes upon aging (annealing) of them. For this purpose the MD simulations were performed for a time interval as long as an order of 2000τ, where τ is a microscopic time scale, equivalent to an order of the Einstein period, which is the characteristic period of oscillation of atoms. It is shown that the static properties exhibit neither significant changes during annealing of samples, nor changes for different samples. On the other hand, the dynamic properties show remarkable aging effects as well as sample-dependent behavior, meaning that the quenched glassy state cannot attain to an equilibrium state for the time scale of our simulations, due to dynamical slowing-down phenomena. The quenched sample is, however, shown to tend to a seemingly quasiequilibrium state after a sufficient annealing. Detailed discussions are made on both aging effects and sample-dependent behaviors of quenched glassy states, by paying particular attention to the behavior of the single-atom motion, mean-square displacement, and self-diffusion constant of the quenched glassy state. A non-Gaussian parameter is also calculated, which we propose as a good candidate to represent an order parameter of the glass transition.

The electromagnetic field generated in a nonabsorbing anisotropic multilayer by a plane incident wave with given wave vector is considered. By taking into account the fact that the field within a given layer is a superposition of four proper waves, namely, four waves that propagate without changing their polarization state, the field is associated with a four-dimensional complex vector space. In this space a new algebra is defined such that the norm of a vector gives the energy flux density of the corresponding field, and the scalar product of two proper waves is zero. This allows one (1) to simply derive the amplitudes of the four proper waves for any layer, (2) to explicitly write the reflection and transmission coefficients of a structure whose layers are uniaxial media with arbitrary direction of the optical axis, and (3) to deduce the propagation equations for the four waves in the limiting case of a plane stratified medium with continuous variation of the dielectric tensor. Similar equations are found in the literature only for some very particular cases and are generally used to obtain approximate solutions for the wave equation. Here the general case is considered. The given equations contain as specific cases most of the already known approximations and give a unifying method for their discussion and generalization. In particular, they contain the geometrical-optics approximation (GOA) for uniaxial stratified media with arbitrary directions of the optical axis and of the incidence angle, as well as a generalization of the GOA, where the coupling between the ordinary and extraordinary waves is taken into account. The theory developed here has a wide range of applications in many fields of physics, as for instance the propagation of electromagnetic waves in magnetoactive plasmas, the optical properties of liquid crystals, and more generally the optics of anisotropic media.

A new lattice model for nematic and/or smectic liquid crystals with rigid (inflexible) molecular cores and semiflexible pendent tails is presented and an approximate expression for its canonical ensemble partition function is derived. The model system consists of N rodlike molecules on a face-centered-cubic (fcc) lattice of M sites; each molecule has a rigid central core of r segments and two semiflexible tails of f/2 segments each and occupies x=r+f contiguous sites on the lattice. Molecular cores are restricted to three mutually orthogonal orientations while each tail bond can have five possible orientations with respect to the preceding bond, corresponding (roughly) to five of the nine possible rotational-isomeric states of a pair of successive C-C bonds on a saturated molecular tail. In addition to hard-core exclusions, there are interactions between pairs of segments on different molecules separated by the first-, second-, and third-nearest-neighbor distances on the fcc lattice, and a positive energy is required to make each bend in a molecular tail. The canonical ensemble partition function is derived using the Bragg-Williams approximation to treat intermolecular interactions (other than hard-core exclusions) and an extension of the DiMarzio counting scheme to evaluate the number of ways to pack the molecules on the lattice. This model and statistical-mechanical approach are improvements over earlier lattice-model treatments of nematic and smectic liquid-crystal-forming substances in the following major respects: (1) The treatment of smectic positional order is much less approximate than the earlier treatment of Dowell. (2) The representation of the rotational-isomeric states of a molecular tail is considerably more realistic than was possible with previous lattice models. (3) Positional, orientational, and conformational order are all coupled. (4) The conformational statistics of the model molecular tails are treated exactly within the maximum-term approximation.

The fcc lattice model presented in the preceding paper [Phys. Rev. A 40, 6021 (1989)] is used to study smectic-A and nematic mesophases formed by molecules with a central rigid core and two symmetric semiflexible pendent tails. The method used to solve the complex nonlinear equations for the density and the orientational, positional, and conformational order parameters of the model system at a particular temperature and pressure is described, as is the procedure used to find smectic-nematic (Sm-N), nematic-isotropic (N-I), and smectic-isotropic (Sm-I) phase-transition temperatures. Calculated mesophase diagrams, i.e., plots of transition temperatures versus tail length f/2, are presented for 12 model homologous series with various values of the tail-bending energy ξ and the segmental interaction energies.

For small values of ξ, the model systems exhibit no stable nematic phase and the Sm-I transition temperature $T_{\mathrm{Sm}\mathrm{-}\mathit{I}}$ decreases with increasing tail length f/2. For moderate values of ξ, behavior qualitatively very similar to that exhibited by many homologous series of real mesogens is obtained, i.e., $T_{N\mathrm{-}I}$ decreases and $T_{\mathrm{Sm}\mathrm{-}\mathit{N}}$ increases with increasing f/2 until the two curves meet to form a falling $T_{\mathrm{Sm}\mathrm{-}\mathit{I}}$ curve. For very large values of ξ, the $T_{\mathrm{Sm}\mathrm{-}\mathit{N}}$ curve is nearly flat and the $T_{N\mathrm{-}I}$ curve rises sharply as f/2 increases. Over a narrow range of moderately large ξ values, reentrant behavior is exhibited by the homologs with longer tails. A detailed analysis of this model behavior leads to a coherent physical explanation of the behavior of the Sm-N, N-I, and Sm-I transition temperatures as a function of tail length for many real homologous series. Our results indicate that tail flexibility plays a major role in determining the nature and stability of smectic-A, nematic, and isotropic liquid phases of rodlike mesogens with alkyl or alkoxy tails. In particular, it is strongly suggested that conformational entropic stabilization of the smectic relative to the nematic mesophase is at least as important as energetic stabilization of the smectic phase by strong attractions between molecular cores.

We investigate the validity of the time-dependent Hartree-Fock (TDHF) approximation for a finite fermion model system. With the help of coherent states we derive equations of motion for the one-particle density, which are of classical Hamiltonian form and can be interpreted in terms of generalized coordinates and momenta. The corresponding trajectories in phase space show transitions between regular and chaotic regimes depending on the parameters of the model. A chaotic behavior indicates a failure of the TDHF approximation because the exact solutions are always regular due to the linearity of the Schrödinger equation.

We present a theoretical calculation of the Frank elastic constants of a nematic liquid crystal made of hard spherocylinders. We use the existing relations for these elastic constants together with the direct correlation function which we obtain consistently from a free-energy density-functional approximation recently derived by us. We compare our results with previous theoretical calculations, computer simulations, and experimental results.

It is pointed out that the rms fluctuation of the local energy encountered in variational Monte Carlo calculations, together with the expected energy, can be used to produce a nontrivial lower bound on the ground-state energies of atoms and molecules. This lower bound has the desirable properties of general accessibility (via Monte Carlo calculations) and linear sensitivity to deviations of the trial wave function from the exact wave function. These qualities of accessibility and sensitivity are demonstrated with numerical data obtained from recent Monte Carlo calculations on the helium atom and the hydride ion.

A very simple variational parameter is introduced in the Hamiltonian of the quantum anharmonic oscillator. A suitable choice of this parameter not only removes the difficulties encountered by various authors in connection with the application of the Hill determinant method, but also makes the eigenvalues rapidly convergent for a small size of the determinant.

The exchange-correlation potential of the Kohn-Sham density-functional theory has recently been interpreted as the work required to move an electron against the electric field of its Fermi-Coulomb hole charge distribution. In this paper we present self-consistent results for ground-state total energies and highest occupied eigenvalues of closed subshell atoms as obtained by this formalism in the exchange-only approximation. The total energies, which are an upper bound, lie within 50 ppm of Hartree-Fock theory for atoms heavier than Be. The highest occupied eigenvalues, as a consequence of this interpretation, approximate well the experimental ionization potentials. In addition, the self-consistently calculated exchange potentials are very close to those of Talman and co-workers [J. D. Talman and W. F. Shadwick, Phys. Rev. A 14, 36 (1976); K. Aashamar, T. M. Luke, and J. D. Talman, At. Data Nucl. Data Tables 22, 443 (1978)].

Partial cross sections of single-electron capture on the n=3 levels in the ${\mathrm{N}}^{5+}$+He collision have been calculated by means of a semiclassical method using ab initio potential-energy curves and radial and rotational coupling matrix elements.

Calculations of the photoionization cross sections of inner shells of Mg, ${\mathrm{Mg}}^{2+}$, and ${\mathrm{Mg}}^{8+}$ along with Ar, ${\mathrm{Ar}}^{6+}$, and ${\mathrm{Ar}}^{8+}$ have been performed at the relativistic-random-phase approximation level. The results confirm earlier predictions made using the simple Hartree-Slater model that removal of outer-shell electrons does not affect the photoionization cross sections of inner-shell electrons, except for a shift of threshold.

We compute the generating function for the photon-number distribution for single-mode fields described by a Gaussian Wigner distribution considered previously by Agarwal and Adam [Phys. Rev. A 38, 750 (1988)] and obtain simpler formulas for the photon-number distribution than those given by them. We also give analytical results for the factorial cumulants of the photon-number distribution and consider some limiting cases.

We have studied the effects of perpendicular electric and magnetic fields on a liquid-crystal sample aligned between glass plates. One field is applied to stabilize the initial alignment, and the other is applied in order to drive the transition; the field and initial alignment directions define six possible geometries. We find that, in each case in which an electric field is present parallel to the glass plates, the Freedericksz transition from a uniform to an elastically deformed state can be first order irrespective of whether the electric field drives or stabilizes the transition. Using material parameters characteristic of the liquid-crystal 4-cyano-4’-n-pentylbiphenyl (5CB), we show that two of the transitions should be first order for this material in all values of stabilizing fields and that two should exhibit a tricritical point as the stabilizing field is increased.