The lattice-covering time t is the expected time a random walk (RW) takes to visit all N lattice sites. Regular D-dimensional lattices with periodic and reflecting boundary conditions are considered. When D=1 these covering problems are equivalent to those of the first-visit type and they can be exactly solved. In contrast, when D≥2 the lattice-covering time problems are not reducible to any known lattice RW problem. The asymptotic (N→∞) behavior of t is studied using Monte Carlo methods and interesting questions regarding universality in the covering time problem are discussed.

The time evolution toward equipartition of energy is numerically investigated in nonlinear Hamiltonian systems with a large number N of degrees of freedom. The relaxation process is studied by computing the spectral entropy η; for a wide class of initial conditions, it follows a stretched exponential law η(t)∼exp[-(t/${\tau }_0$${\left)\right]}^{\xi }$, ξ<1, up to times t<${\tau }_R$, and η(t)=const for t≥${\tau }_R$. By taking advantage of this fact, a good definition of a relaxation time becomes possible. Below a critical value ${\varepsilon }_c$ (model dependent) of the energy density ɛ, the relaxation time ${\tau }_R$ is found to follow a scaling with ɛ, which is compatible with a ‘‘Nekhoroshev-like’’ law, i.e., ${\tau }_R$=${\tau }_0$exp(${\varepsilon }_0$/ɛ${)}^{\gamma }$, for both the Fermi-Pasta-Ulam (FPU) β model and the classical lattice ${\phi }^4$ model; a remarkable difference with respect to Nekhoroshev’s theorem (where the exponent γ scales as 1/$N^2$) is the N independence of numerical experiment results.

An important consequence of this fact is the existence of nonequilibrium states of arbitrary lifetimes also at large N values. On the other hand, at high-energy densities (ɛ>${\varepsilon }_c$) ${\tau }_R$ is almost independent of ɛ. The maximum Lyapunov characteristic exponent ${\lambda }_1$ is measured in both models as a function of the energy density. The striking result is a change in the scaling ${\lambda }_1$(ɛ) occurring at the same ɛ (=${\varepsilon }_c$) at which ${\tau }_R$ has its crossover. At ɛ>${\varepsilon }_c$, the scaling ${\lambda }_1$∼${\varepsilon }^{2/3}$ is found for both the FPU and the ${\phi }^4$ models; this is in agreement with a random matrix approximation of the tangent dynamics, which means that the dynamics itself mimics a random process. Below ${\varepsilon }_c$, steeper and model-dependent scalings are found. Hence, evidence for the existence of a rather sharp transition of the phase-space structure is provided, and a strong stochasticity threshold (SST) can be defined. A preliminary result, suggesting the stability of the SST in the limit of large N, is also reported.

The Harbola-Sahni exchange potential is the work needed to move an electron against the electric field of its hole charge distribution. We prove that it is not the exact exchange potential of density-functional theory, by showing that it yields the wrong second-order gradient expansion in the slowly varying limit. But we also discover that it yields the correct local-density approximation. Thus the Harbola-Sahni potential is a more physically correct version of the Slater potential, one that is better suited for molecular and solid-state applications. As a step in our derivation, we present the third-order gradient expansion of the exchange hole density, and discuss its structure. We also describe a new version of the Harbola-Sahni potential which corrects its path dependence. The exact exchange potential for an atom is given by the optimized potential model (OPM) of Talman and Shadwick. By using enhanced numerics, we confirm that the OPM potential satisfies the Levy-Perdew virial relation and exhibits correct -1/r behavior at large r. Numerical calculations also show that the intershell maxima in the exact exchange potential are needed to lower the total energy. These ‘‘bumps’’ are missing from the Harbola-Sahni and Slater potentials.

We consider qualitative and quantitative properties of ‘‘snapshot attractors’’ of random maps. By a random map we mean that the parameters that occur in the map vary randomly from iteration to iteration according to some probability distribution. By a ‘‘snapshot attractor’’ we mean the measure resulting from many iterations of a cloud of initial conditions viewed at a single instant (i.e., iteration). In this paper we investigate the multifractal properties of these snapshot attractors. In particular, we use the Lyapunov number partition function method to calculate the spectra of generalized dimensions and of scaling indices for these attractors; special attention is devoted to the numerical implementation of the method and the evaluation of statistical errors due to the finite number of sample orbits. This work was motivated by problems in the convection of particles by chaotic fluid flows.

We have studied a time-independent nonlinear Schrödinger equation of the tight-binding form on a one-dimensional lattice. The real and complex wave functions as solutions to the equation are considered separately for different physical problems. For each case, an area-preserving map for the discrete nonlinear Schrödinger equation is introduced and analyzed. The bounded solutions can be organized in hierarchies composed of periodic, quasiperiodic, as well as chaotic orbits on the phase plane of the nonlinear map. A ‘‘stability-zone’’ diagram, where the bounded orbits exist, is displayed in the parameter space, serving as ‘‘phase diagram’’ of the nonlinear Schrödinger equation under appropriate boundary conditions. Studies of the stability zone yield useful information for the physical problems considered. The periodic orbits and their stabilities can be obtained by a convergent perturbation method. Finally, we remark on several physical problems where these results might be applicable. In particular, we discuss the stabilities of the large polaron solution in the Holstein model.

A Monte Carlo simulation method is used to study a dilute gas of hard spheres under uniform shear flow in the hydrodynamic regime over a wide range of shear rates. The results for the components of the pressure tensor and also for the distribution function are presented and compared with those obtained by using the Bhatnagar-Gross-Krook model kinetic equation. The agreement is fairly good for the pressure tensor and also for the velocity distribution function in the small-velocity region, while discrepancies appear for large velocities. The possibility of using a maximum-entropy method to construct an approximate distribution function is discussed. It is shown that the inclusion of the pressure tensor leads to a distribution function significantly different from the one obtained in the simulation.

We have performed small-angle Rayleigh-scattering measurements in toluene subject to a large stationary temperature gradient ∇T. The experiments demonstrate the presence of long-range entropy and viscous fluctuations enhancing and modifying the Rayleigh spectrum. The experiments confirm that the viscous-heat-mode contributions vary with (∇T${)}^2$/$q^4$, where q is the wave number of the fluctuations. The observed amplitudes of both the entropy and viscous fluctuations are in excellent agreement with the theoretical predictions for the fluctuations in nonequilibrium fluids.

A stationary, nonlinear collisional-radiative model for high-temperature atomic oxygen is presented. Populations of electrons, ions, and excited atoms and intensities of spectral, continuum, and dielectronic recombination lines are calculated in a wide range of conditions (11 000 K≤$T_e$≤15000 K, ${10}^{10}$≤$N_t$≤${10}^{18}$ ${\mathrm{cm}}^{\mathrm{-}3}$). Transport of radiation is included by coupling the rate equations for production of the electrons, ions, and excited atoms with the concept of the escape factors that are not constant but dependent upon plasma conditions. The calculated total continuum emission is in good agreement with existing measurements.

A system that consists of a monolayer spread on a solution containing a monovalent and a divalent ion is investigated. The solution of the Poisson-Boltzmann-Stern equation for this system indicates that the metal ions segregating to the surface can be found in two distinct states. Divalent ions are chemically condensed on the monolayer, while monovalent ions are electrically attracted to it. We derive simple expressions for the charge left on the surfactant monolayer and the amount of metal ions condensed on the monolayer. These formulas reproduce very accurately (to within pro milles) the values obtained using the nonlinear Grahame equation and eliminate the need to solve that equation. That permits a simple identification of the state of the surfactant monolayer and we propose a universal condensation chart that characterizes the state of the surfactant. We further derive a chemical equilibrium equation for the surface components that has considerable range of validity. This equation requires a knowledge of the bulk concentrations only, and thus allows in many cases the identification of the state of the monolayer, avoiding the need to solve the full nonlinear Poisson-Boltzmann equation. All existing experimental results on Langmuir systems are in good agreement with the one-dimensional Poisson-Boltzmann-Stern model with no adjustable parameters. Several of these fits are presented in this work and are also mapped on the condensation chart. Our calculations point to some characteristic differences between the monovalent and the divalent ions that explain why it is possible to build Langmuir-Blodgett multilayers from divalent compensated surfactants but not from monovalent ones.

An analysis is conducted on the coupling between hydrodynamic behaviors and a solid-liquid interface that forms due to solidification of a portion of a layer of fluid located between almost completely insulating plates. Beginning with a long-wavelength expansion, an evolution equation is derived that includes the effects of the field variables in the liquid phase, which describes the temporal and spatial behavior of the interface. An expression is obtained that shows that increases in the thickness of the solid layer increase the critical value of the Rayleigh number and that deformations in the interface can either decrease or increase this critical value. The interface undergoes a transition from the planar state to a given pattern at the critical value of the Rayleigh number. Analysis of the evolution equation leads to the prediction that the interfacial pattern consists of squares at very small values for the thickness of the solid layer. A critical value of the thickness is found, and for a thickness greater than this value squares and rolls are predicted to coexist as stable interfacial patterns.

The effect of the logarithmic terms on the ground-state energy level and wave function of a dtμ system is investigated. No significant contribution of the logarithmic terms on either the energy level or wave function is found. At the same time, we find the lowest upper bound of the ground-state energy ever obtained by the variational method using the Hylleraas-type trial function and that the corresponding wave function satisfies the cusp condition as $r_{\mathrm{dt}}$→0 automatically to a reasonable accuracy for r<3 (muonic a.u.), where r is the distance between the fused dt nuclear compound and the muon.

Unstable flow patterns are produced when a viscous fluid (o) is displaced by an inviscid one (s). These patterns, viscous fingers, become fractallike in media such as porous rock that promote extensive fluid-mobility fluctuations. The porous rock is also effective in dispersing the concentrations of the fluids. The rate of (geometric) dispersion competes with the growth rate of the fluctuations and establishes a lower limit to the length scale of the fractal pattern. We have solved the equations describing the coupled dynamics of unstable miscible flow and dispersion by a generalization of the previously reported ‘‘probabilistic’’ method. The inclusion of dispersion provides both a well-posed mathematical problem and a means for quantitative comparison with experiment. Numerical calculations show, for large viscosity ratio (M==${\mu }_o$/${\mu }_s$), that scaling of the pattern area with the amount of physical dispersion is the same as scaling with system size (in the absence of dispersion). In the case of small dispersion we confirm the validity of the discovery of a crossover from fractal scaling to stable flow based only on the finite viscosity ratio (${\mu }_o$>${\mu }_s$≠0). We have designated this global-scale stabilization as viscous relaxation. In this limit of vanishing dispersion, we show that the individual finger width scales with the numerical grid size, but the areal density of the fingers increase with decreasing M. We have developed a simple analytic expression for the areal displacement density as a function of M. Only for M→∞ does this density vanish (corresponding to fractal scaling). Comparisons are made with two-dimensional physical model flow experiments. Using independently determined dispersion coefficients we account quantitatively for flood displacement over a wide range of viscosity ratio M.

The ellipticity of the vapor-liquid interface above mixtures of methylcyclohexane (${\mathrm{C}}_7$${\mathrm{H}}_{14}$) plus perfluoromethylcyclohexane (${\mathrm{C}}_7$${\mathrm{F}}_{14}$) has been measured near the consolute point $T_c$=318.6 K. The data are consistent with a model of the interface that combines a short-ranged density-versus-height profile in the vapor phase with a much longer-ranged composition-versus-height profile in the liquid. The one-parameter model profile fits the data well for reduced temperatures t=(T-$T_c$)/$T_c$>7×${10}^{\mathrm{-}4}$. The model predicts a maximum ellipticity at t=7×${10}^{\mathrm{-}4}$, which occurs at t=2×${10}^{\mathrm{-}4}$ in the experiment. The value of the free parameter produced by fitting the model to the data is consistent with results from two other simple mixtures and a mixture of a polymer and solvent. This experiment combines precision ellipsometry of the vapor-liquid interface with in situ measurements of refractive indices of the liquid phases, and it precisely locates the consolute point.

The radial Hele-Shaw experiment with constant-flux boundary conditions is considered. Our results suggest that substantial differences may exist between the final morphologies associated with constant-flux and constant-pressure boundary conditions. In addition, we show that arguments based on an extended linear analysis to motivate the dense-branching morphology in the constant-pressure case do not lead to a compact structure for the constant-flux case.

A high-Reynolds-number turbulent flow subsumes several intermittent fields; some examples are the rates of dissipation of turbulent energy and scalar variance, square of turbulent vorticity and rate of strain, etc. These intermittent fields display different degrees of correlation among them. Motivated by the need for characterizing such coexisting distributions of intermittent fields in fully developed turbulence, the multifractal formalism—which we have already found useful in describing such intermittent distributions singly—is extended to more than one variable. The formalism is first illustrated by studying joint log-normal as well as joint binomial distributions. It is then applied to simultaneous measurements in several classical turbulent flows of the joint distribution of a component of the dissipation of kinetic energy, the dissipation rate of passive scalar variance, as well as the square of a component of turbulent vorticity. This allows simple but realistic models of simultaneous cascades of more than one variable to be developed.

The collapse transition of a self-avoiding walk (SAW) on a two-dimensional lattice with directed bonds, the Manhattan lattice, is shown to occur at temperature $T_{\theta }$=2ɛ/ln2, where ɛ is the attractive energy between nearest-neighbor pairs of monomers. The exact tricritical exponents are ${\nu }_t$=(4/7 and ${\gamma }_t$=(6/7. The latter result differs from the value for undirected two-dimensional lattices ${\gamma }_t$=(8/7 because self-trapping configurations do not occur on the Manhattan lattice. The exact tricritical temperature and exponents for a constrained self-avoiding trail (SAT) on the square lattice are obtained by mapping the problem onto the self-avoiding walk on the Manhattan lattice. The mapping also shows that these SAT and SAW collapse transitions are in the same universality class. Finally, it is argued that the kinetic self-avoiding trail must be compact on the square lattice.

The temperature dependence and dispersion relation of phason and amplitudon relaxation times were measured in Sm-A and Sm-$C^{\mathrm{*}}$ phases of the ferroelectric liquid crystal S-(+)-[4-(2’-methylbutyl) phenyl 4’-n-octylbiphenyl-4-carboxylate] (CE8) by photon correlation spectroscopy. In the Sm-A phase both modes are degenerate, their inverse relaxation time is strongly temperature dependent and has a parabolic dispersion relation with a minimum at a finite wave vector. In the Sm-$C^{\mathrm{*}}$ phase, the amplitudon relaxation time is temperature dependent, while the phason relaxation time is temperature independent and shows parabolic dispersion. The ratios of two orientational elastic constants and the viscosity were obtained at different temperatures. In addition, the temperature dependence of the helical pitch has been measured in the Sm-$C^{\mathrm{*}}$ phase of CE8 in homeotropic and bookshelf-geometry alignments.

Dynamics of atoms near the glass transition of simple classical liquids is studied on the basis of the mesoscopic stochastic-trapping diffusion model recently developed by Odagaki [J. Phys. A 20, 6455 (1987); Phys. Rev. B. 38, 9044 (1988)]. The jump rate of an atom (tracer) is assumed to have a distribution following a power-law function with exponent ρ, where ρ is a phenomenological parameter. A sharp transition is predicted at ρ=0, that is, the self-diffusion vanishes when ρ<0 and takes a nonzero finite value when ρ>0. This transition is identified as the glass transition. With use of the coherent-medium approximation, the mean-square displacement is shown to exhibit a power-law dependence on time with exponent less than unity, and hence the incoherent scattering function for small wave vectors shows stretched exponential decay when ρ<0. The non-Gaussian parameter at time t=∞ is shown to be nonzero in the glassy state (ρ<0) and vanishes in the fluid state (ρ>0), indicating that this quantity may be used as an order parameter of the glass transition. The mean-square displacement and the non-Gaussian parameter are obtained in the intermediate time scale as well from the frequency-dependent diffusion constant. The apparent diffusion constant determined by the derivative of the mean-square displacement at an imtermediate time shows a smooth transition instead of the sharp one, which coincides with observations in molecular dynamics studies. The incoherent scattering function in the intermediate time scale agrees qualitatively with experiments and the exponent of its stretched exponential decay deviates from unity before the glass transition takes place, in agreement with observations made via computer experiments.

We report the results of a measurement of the Lamb shift in the n=2 state of muonium. Muonium atoms in the metastable 2S state were produced by the beam-foil method and the formation rate was determined by observation of the Ly-α photons emitted when the metastable atoms were quenched in a static electric field. The Lamb shift (${\mathrm{scrS}}_{\mathrm{Mu}}$) was measured through observation of the microwave-induced 2 ${}_{}{}^2{}_{}{}^{}\mathrm{S}_{1/2}^{}{}_{}{}^{}$–2 ${}_{}{}^2{}_{}{}^{}P_{1/2}^{}{}_{}{}^{}$ transition with the result ${\mathrm{scrS}}_{\mathrm{Mu}{=1042\mathrm{}}_{\mathrm{-}23}^{+21\mathrm{}}}$ MHz. This result is in agreement with the current theoretical value.

The results obtained for the conductivity, dielectric relaxation, and viscosity of ternary water-AOT-oil microemulsions are presented and discussed in the context of percolation theory. [Here, AOT is an abbreviation for sodium bis(2-ethylhexyl) sulfosuccinate.] Several parameters are considered: the volume fraction φ(water + AOT), the molar ratio n=[water]/[AOT], the temperature, the salt content of the aqueous solution, and the nature of the oil. Certain results obtained by replacing the water with glycerol (waterless microemulsion) are reported. Eight different experimental paths were systematically followed. At constant temperature and salt content we have the paths n=const, φ=const, $P_w$=const (water content by weight), $P_o$=const (oil content by weight), $P_s$=const [surfactant (AOT) content by weight], $P_S$/($P_S$+$P_o$)=const (i.e., $P_S$/$P_o$=const). We also considered variations of temperature and salt content for constant values of all the other parameters. The experimental curves have various different shapes for conductivity, dielectric permittivity, relaxation frequency, or viscosity. Depending on the path followed we observed either monotonic variations or, in contrast, curves presenting maxima and minima. Discussion of the results in connection with percolation theory and the concept of ‘‘percolation threshold lines’’ [M. Moha-Ouchane, J. Peyrelasse, and C. Boned, Phys. Rev. A 35, 3027 (1987)] is satisfactory and the theoretical curves are comparable in all respects with the experimental curves. Finally, in the Discussion, certain analogies of behavior (and therefore of interpretation) are emphasized between the microemulsions studied here and certain quaternary systems.