The density-functional formalism leads to several integral equations for the pair correlations in a liquid metal as a coupled electron-ion mixture, with the aid of the hypernetted-chain (HNC) approximation and the information from electrons in the jellium model: the quantal HNC equation, the integral equations of Dharma-wardana and Perrot, and those of Ichimaru and collaborators. The latter two are extended to include the effects of the electron-ion correlation, which are neglected in their earlier treatments. By solving these five integral equations numerically for the case of liquid-metallic hydrogen with fully degenerate electrons, the limitations of their approximations are investigated. Furthermore, the approximation to use the exchange-correlation effect in the jellium model in place of that for the electrons coupled with the protons is examined on the basis of the quantal HNC equation.

Unlike the case of regular networks, such as lattices, no method of general applicability exists for dealing with random walks on complex networks. We propose a method, which is based on the identification of certain types of walks on a one-dimensional segment as basic. Generating functions for complex networks or for complex types of walks can all be constructed from the generating functions corresponding to these basic walks. We also define basic walks corresponding to a network. The properties of walks in a network composed of ‘‘black boxes,’’ each containing a network by itself, can be expressed in terms of the basic walks defined for the single black box. This result can be compared to the way the Kirchhoff rules allow one to calculate properties of a network of ‘‘elements’’ in terms of the impedances of each of the elements. In the present case the combination rules are more complex than in the electrical case. Our method is demonstrated by calculating mean first-passage times on several structures: a segment, a segment with a single dangling bond, a segment with many dangling bonds, and a looplike structure. The results are analyzed and related to the question of applicability of the Einstein relation between conductance and diffusion.

Structures of composite spin operators are analyzed which appear in models of neural networks of the type which Amit et al. have recently investigated. A binary basis of size N${=2\mathrm{}}^M$ is introduced to study a problem of N quantum-mechanical spin operators. The [Z${\mathrm{}\left(2\right)\mathrm{}]}^M$ group structure of the binary basis allows for many decompositions of the [SU(2)${]}^{\mathrm{N}}$ spin algebra. These become useful in studying and solving generalized frustrated Heisenberg as well as Ising models. Using these techniques for quantum-mechanical generalized spin operators, we derive an explicit representation of the partition function of classical statistical-mechanics models, in terms of a series summation over components of collective spin variables.

High-resolution heat capacity measurements have been carried out in the smectic-A–hexatic-B–crystal-B transitions of the compound 4-propionyl-4’-(n-heptanoyloxy)azobenzene. Qualitatively, the heat-capacity anomalies of this smectic-A–hexatic-B–crystal-B transition sequence mimic those of the smectic-C–smectic-I–smectic-G transition of racemic 4-(2’-methylbutyl)phenyl-4’-(n -nonyloxy)biphenyl-4-carboxylate. The heat-capacity data near the smectic-A–hexatic-B transition are broad and asymmetric and cannot be fitted with a power-law expression.

The turbidity of a near-critical binary fluid mixture, methanol-cyclohexane, has been carefully measured above its (upper) critical consolute point. These measurements covered the reduced temperature region ${10}^{\mathrm{-}5}$${<10\mathrm{}}^{\mathrm{-}2}$ and allowed theoretical predictions for the critical exponents ν and γ to be verified. The amplitude ${\xi }_0$ of the correlation length was also determined to be 3.24±0.2 Å while ${\tau }_0$ was (6.9±0.1)×${10}^{\mathrm{-}6}$ ${\mathrm{cm}}^{\mathrm{-}1}$. Our result for ${\xi }_0$ is compared with other experimental determinations, and when combined with ${\tau }_0$ and coexistence curve data, is used to determine the ratio R=${\xi }_0$($B^2$/$k_B$$T_c$${\chi }_0$${)}^{1/}$ ${3=0.71}_5$±${0.02}_4$ which is consistent with the prediction 0.65–0.67 from two-scale-factor universality.

We report the temporal developments of the variance and distribution of fluctuations of macroscopic order in a structure-formation process using an electrohydrodynamic instability of a nematic liquid crystal at f>$f_c$, where f and $f_c$ are the applied and the critical field frequencies separating two different instabilities: a convective flow instability and a forced oscillatory instability of the director. At f>$f_c$, the dissipative structure appears essentially as the forced oscillatory instability of the director in a three-dimensional pattern in space. The temporal development of the variance is qualitatively in good agreement with the result expected from the noisy Landau equation for the early stage, which resembles the result in a transient process of laser radiation, theoretically and experimentally studied by Arecchi and Degiorgio, but not for the late stage. This comes from the spatial inhomogeneity of the macroscopic order. The distributions of the macroscopic order fluctuation have profiles similar to the Gaussian type (or lognormal type in the presence of externally applied noise) for the early stage, and have an asymmetric profile with negative skewness for the late stage. This would suggest that the growth process of the dissipative structure for the late stage is dominated by totally different kinetics from that for the initial stage.

In a series of two papers, we present a comprehensive approach to the pattern formed via viscous ‘‘fingering’’ in a Hele Shaw cell, the Saffman-Taylor finger pattern. We explain how the ‘‘finger’’ width is selected and why this selection cannot be computed via asymptotic analysis in the (large) capillary number. Also, we derive the spectrum of small oscillations around the steady-state shape and show that the selected shape is stable. In this paper, we set up the mathematical and numerical formalism, and demonstrate stability for the λ=(1/2) solution. We also consider the effects of noise on the stability analysis.

We generalize the stability results of the preceding paper to arbitrary λ. Coupled with a new approach to the selection of ‘‘finger’’ width, this stability computation allows us to predict the dependence of allowed widths on capillary number. At the same time, we show that of all possible finger shapes at fixed capillary number, only one [denoted ${\lambda }^{\mathrm{*}}$(γ)] is stable. Finally, we show how finite noise can shift ${\lambda }^{\mathrm{*}}$(γ), causing observed widths and stability to diverge from the noiseless calculation at large velocity. We conclude with a discussion of the paradigm of microscopic solvability for diffusive controlled pattern formation, into which the above analysis neatly falls.

A modified manifestly self-consistent Langer-Zittartz method is applied to the nonperturbative treatment of the electrostatic excluded-volume effects in the polyelectrolyte chain. The method is tested for the case of the ordinary excluded-volume problem for which it produces the standard Flory result, (〈$R^2$〉${)}^{1/2}$∝$N^{3/5}$. When it is applied to the polyelectrolyte chain, for the totally unscreened case, it also gives the Flory result (〈$R^2$〉${)}^{1/2}$∝N, which is confirmed by recent Monte Carlo simulations performed by Baumgaaumlrtner. Complications associated with effects of partial screening are considered as well. Explanations for the failure of the conventional renormalization-group methods for the polyelectrolyte chain are presented on the basis of comparisons between the polyelectrolyte problem and mathematically similar problems in solid-state physics.

A generalized mode-coupling theory that takes into account molecular-scale effects is presented. Thermodynamic as well as dissipative nonlinearities are included. The theory is used to calculate molecular-scale renormalizations or mode-coupling contributions to the zero-wave-number shear viscosity and to the time-dependent stress-tensor autocorrelation function. For a range of times the theory leads to a stress-tensor autocorrelation function that appears to decay as $t^{\mathrm{-}3/2}$. The results are in reasonable agreement with computer simulations.

The mode-coupling theory recently used to calculate the intermediate-time behavior of the stress-tensor autocorrelation in dense simple liquids is generalized to treat tagged-particle fluctuations. The velocity autocorrelation function for hard-sphere fluids is calculated in the same time regime. The results agree well with computer simulations. The calculation establishes a connection between the intermediate-time behavior of these correlation functions.

An algorithm related to diffusion-limited aggregation is used to generate patterns of interface motion in the Hele Shaw geometry. With increasing surface tension, the objects generated change from fractal structures to smooth boundary ‘‘fingers.’’ The fingers’ profiles agree with experiments and numerical solutions. The fingers’ width as the function of the surface tension is consistent with the results obtained through other simulational methods. As the surface tension is decreased, apparently unstable patterns of flow develop. For very small values of the surface tension, a kind of dendritic growth is produced.

A perturbation procedure is employed to derive a formula for the field-line displacement in a magnetic configuration consisting of a dominant (unperturbed) field with closed field lines and a small additional (perturbation) field that causes the perturbed field lines to fail to close on themselves. An expression for the spatial displacement of the perturbed field line per circuit, obtained in a first-order perturbation approximation in the Clebsch representation, is given in terms of an action integral representing a longitudinal adiabatic invariant. Numerical results are also presented to compare the first-order approximation with the exact relation.

The Matsuoka-Clementi-Yoshimine (MCY) configuration interaction potential for rigid water-water interactions has been extended to include the intramolecular vibrations. The extended potential (MCYL), using no empirical parameters other than the atomic masses, electron charge, and Planck constant, is used in a molecular-dynamics simulation study of the static and dynamic properties of liquid water. Among the properties studied are internal energy, heat capacity, pressure, radial distribution functions, dielectric constant, static structure factor, velocity autocorrelation functions, self-diffusion coefficients, dipole autocorrelation function, and density and current fluctuations. Comparison with experiments is made whenever possible. Most of these properties are found to improve slightly relative to the MCY model. The simulated high-frequency sound mode seems to support the results and interpretation of a recent coherent inelastic neutron scattering experiment.

The statistical dynamics of the response of the logistic map, $x_{n+1\mathrm{}}$=${\mathrm{rx}}_n$(1-$x_n$), towards additively and multiplicatively coupled fluctuating forces is studied analytically and numerically in quantitative detail in the range of control parameter r where the unforced system shows the first transcritical and the first pitchfork bifurcation.

Selection of steady needle crystals in the full nonlocal symmetric model of dendritic growth is considered. The diffusion equation and associated kinematic and thermodynamic boundary conditions are recast into a nonlinear integral equation which is solved numerically. For the range of Péclet numbers and capillarity lengths considered it is found that a smooth solution exists only if anisotropy is included in the capillarity term of the Gibbs-Thomson condition. The behavior of the selected velocity and tip radius as a function of undercooling is also examined.

Temperature and velocity correlation functions in a fluid subjected to conditions creating both a temperature and a velocity gradient are computed up to second order in the gradients. Temperature and velocity fluctuations are coupled due to convection and viscous heating. When the viscosity goes to infinity one gets the temperature correlation function for a solid under a temperature gradient, which contains a long-ranged contribution, quadratic in the temperature gradient. The velocity correlation function also exhibits long-range behavior. In a particular case its equilibrium term is diagonal whereas the nonequilibrium correction contains nondiagonal terms.

The theory for isotropic-nematic transition at constant pressure described in an earlier paper is extended to include isotropic-plastic transition. The transition is located from the structural information about the liquid using the first-principles order-parameter theory of freezing. This theory makes the role of the structure of the medium explicit and the role of the intermolecular interaction implicit. For the plastic phase, order parameters are the coefficients of a Fourier expansion of the spatially varying single-particle density ρ(r,Ω) in terms of the reciprocal lattice of the plastic. For the nematic phase, order parameters are the coefficients of a spherical harmonic expansion of an orientational singlet distribution. The theory predicts that the equilibrium positional freezing (plastic) on fcc lattice takes place for the value of c¯${^}_0$,$0^1$ [=1-1/S(‖${\mathrm{G}}_m$‖), where S(‖${\mathrm{G}}_m$‖) is the first peak in the structure factor of the center of mass] ≃0.67 [or S(‖${\mathrm{G}}_m$‖)≃3.07]. The equilibrium orientational freezing (nematic) takes place when the orientational correlation c¯${^}_2$,$2^0$≃4.45. For a simple model of hard ellipsoids of revolution parametrized by length-to-width ratio $X_0$, we find that the plastic phase stabilizes first for 0.57≲$X_0$≲1.75 and the nematic phase for $X_0$<0.57 and $X_0$>1.75. These values are in reasonable agreement with the computer-simulation results. We also find, in agreement with computer simulation, a remarkable symmetry between the systems with inverse length-to-width ratios.

An orientational order mechanism, which may lead to a symmetry change from a higher-temperature hexagonal columnar phase to lower-symmetry phases with two-dimensional rectangular symmetry, is considered. A tensor density function is invoked to describe the symmetry change through the phase transition. For a transition corresponding to a symmetry breaking in the plane perpendicular to the column axes, the molecular orientation in the plane at the low-temperature phase is determined. Transitions in which there is a symmetry breaking out of the plane are briefly discussed.

The $D_2$-$D_1$ phase transition in columnar liquid crystals of the HAT series [e.g., HAT11 (triphenelene hexa-n-dodecanoate)] is discussed within the framework of Landau theory. The order parameters which describe the transition are abstracted from a tensor density function, and are associated with two irreducible representations of the symmetry group of the high-temperature $D_2$ phase. A mechanism for a first-order transition is then suggested in accordance with both theoretical considerations and the experimental result for the $D_2$-$D_1$ transition. Two possible arrangements of the herringbone structure of the $D_1$ phase are obtained, each of which gives six orientational states in the low-temperature $D_1$ phase.