Solvability conditions are derived for the asymmetric Saffman-Taylor finger. In the absence of surface tension the solution contains two free parameters λ and $y_0$, where λ is the dimensionless finger width and $y_0$ is the degree of asymmetry. In the presence of surface tension, it is shown that solvability conditions are not satisfied, signaling the appearance of a cusp on the finger boundary. The mismatch angle, Δθ, due to this cusp is shown to be concave. An earlier postulate made in connection with the wire experiment is clarified here, which states that Δθ mainly depends on the tangential slope of the finger profile. The remaining portion of this paper is devoted to understanding recent experiments by G. Zocchi and co-workers and M. Rabaud and co-workers [Phys. Rev. A 35, 1894 (1987); 37, 935 (1988)], where fingers were perturbed by wires. Assuming the contact angle created by the wire to be concave, we first show within linear approximation that, for $y_0$≪1, the absolute magnitudes of the mismatch angles created by the wire and the bubble at the center are the same, establishing that all the results obtained by the bubble perturbations can be carried over to a wire experiment. We then present a theoretical prediction for δ, the distance between the tip of the asymmetric finger and the wire at the center, as a function of external parameters. In the limit of small surface tension, the results are in fair agreement with the available data.

The statistical mechanics of a fluid of hard prolate ellipsoids of revolution is addressed using density-functional methods. A variational approach is introduced for the calculation of the direct pair correlation function of the isotropic liquid, which is then used in a density-functional theory for the first-order transitions in the fluid. These calculations use, for the first time, a density functional that can properly describe the narrow real-space peaks at the crystallization transitions, and that can accommodate simultaneous or separate translational and orientational ordering. As a result, transitions to the solid, as well as to the plastic and nematic phases, are obtained. The resulting phase diagram is in good agreement with Monte Carlo results.

A weakly nonlinear theory that describes the onset of the combined Bénard-Marangoni convection is presented. All relevant transport coefficients are taken to be temperature dependent. When the boundaries are, thermally, nearly insulating, the instability is weak and the perturbed fluid-air interface is found to be proportional to the temperature field F(ζ,τ) which evolves according to $F_{\tau }$-${\pi }_1$($F_{\zeta }$${)}_{\zeta }^3$-${\pi }_2$ ($F_{\zeta }$${)}_{\zeta \zeta }^2$+${\pi }_3$$F_{\zeta \zeta \zeta \zeta }$ +${\pi }_4$$F_{\zeta \zeta }$+${\pi }_5$ ($F_{\zeta }$${)}^2$+${\pi }_6$($F^2$${)}_{\zeta \zeta }$ +${\pi }_7$($F^2$${)}_{\zeta }$+(${\beta }_1$ +${\beta }_2$)F=0 where ${\beta }_i$ are the Biot numbers, and ${\pi }_i$, i=1,...,7, are constants.

The conduction-band energy $V_0$ of excess electrons in liquid krypton is calculated as a function of number density n. Our model follows the prescription of Springett, Jortner, and Cohen [J. Chem. Phys. 48, 2720 (1968)] but avoids some of their approximations. In particular, a proper, high-precision atomic pseudopotential is used to represent the electron-krypton interaction. The theoretical values of $V_0$(n) are found to be in excellent agreement with experimental data.

The decrease of the dielectronic recombination rate coefficient by the thermal equilibration of the highly excited, bound final states produced by the recombination process with the plasma electrons is examined for the oxygen isoelectronic sequence. The potential enhancement of the effective rate of dielectronic recombination by the presence of metastable states in the ground configuration of the initial ion is analyzed for ${\mathrm{Fe}}^{18+}$. The net effect of these two processes upon the dielectronic recombination rate coefficient of ${\mathrm{Fe}}^{18+}$ is a 5–10 % enhancement from 0.8 keV to higher temperatures at plasma densities of ${10}^{14}$ electrons/${\mathrm{cm}}^3$, and a maximum 40% decrease at lower temperatures where Δn=0 radiative stabilizing transitions dominate the dielectronic recombination process.

A model for describing the influence of the nearest-neighbor interaction on the thermodynamic properties of plasmas is presented. This model is a generalization of the Thomas-Fermi model to cylindrical symmetry. The model was applied to the calculation of the electric potential of a pair of identical ions. An iron plasma at an ion density of ${10}^{23}$ ${\mathrm{cm}}^{\mathrm{-}3}$ at a range of temperatures of practical interest was used to illustrate the predictions of the model. The ionization-potential reduction, the ion-pair free energy, and the average distance to the nearest neighbor were calculated. Our results indicate that the inclusion of the nearest-neighbor interaction predicts a plasma which is slightly more compressible than predicted by the spherically symmetric Thomas-Fermi model.

The transitions from 3$d^{10}$nf (n=7,8) to 3$d^{10}$4d, from 3$d^{10}$nd (n=6,7) and 3$d^{10}$7s to 3$d^{10}$4p, and from 3$d^{10}$6p to 3$d^{10}$4s are identified in the spectrum of laser-irradiated gold ${\mathrm{Au}}^{50+}$. The ionization limit derived is in good agreement with ab initio evaluations by means of the relativistic parametric potential method. The small deviations between experimental and theoretical ionization limits are consistent throughout the copper sequence and accurate predictions are given for unobserved elements. Similar methods are used for deriving ionization limits of nickel-like ions from 3$d^{10}$ J=0-3${\mathrm{d}}_{5/2}^9$${\mathrm{nf}}_{7/2}$ J=1 known transitions and from ab initio estimates.

Invariant measures generated by the backward iteration of one-dimensional maps are studied. The method yields the generalized dimensions as eigenvalues. For a particular backward iteration, earlier results concerning the natural measure are recovered. As an application, the case of the one-dimensional random-field Ising model is discussed.

We study a system consisting of two oscillators coupled to each other through a Hamiltonian interaction. The ‘‘oscillator of interest’’ and the ‘‘irrelevant oscillator’’ are also coupled, via rigorous Hamiltonian interactions, to two independent thermal baths at the temperatures $T_1$ and $T_2$, respectively. Since $T_1$<$T_2$, these two oscillators are also termed cold and hot oscillators, respectively. In a preceding short communication [L. Fronzoni and P. Grigolini, Phys. Lett. 106A, 289 (1984)] it was noted that when $T_1$=0, the vanishing effective temperature of the cold oscillator does not change upon increase of $T_2$ until a certain critical value $T_c$ is reached. The threshold $T_2$=$T_c$ is characterized by a critical slowing down. This theory is extended so as to take into account the influence of the finite time scale of the hot oscillator and the effect of the higher-order perturbation terms, both neglected in the preceding short report. This allows us to make predictions also on the statistical behavior of the hot oscillator, i.e., on how its dynamics depends on that of the cold oscillator. It is shown that in the white-noise limit our system is indistinguishable from the van der Pol oscillator. The quantitative determination of the threshold now relies on a more reliable criterion, based on the evaluation of the norm of the energy distribution. The freezing state of the cold oscillator corresponds to an infinite value of this norm and the regime where the heating up of the cold oscillator is made possible means a finite value of it. The analytical expression for the threshold transition from the former to the latter regime is accordingly determined. The role of a weak and spurious additive noise causing the breakdown of a neat threshold is discussed. To get rid of these disturbances, unavoidably affecting analog simulation, an experiment of digital simulation is carried out. The agreement with the theoretical predictions is proven to be excellent.

The ionic channels associated with proteins embedded in cell membranes often show a fluctuating behavior between open and closed states. A feature commonly observed is the long-time tail in the distribution f(t) of closed-state durations. A generalization of a defect-diffusion model recently proposed to explain this behavior is solved analytically for a one-dimensional geometry. The analysis involves the solution of a target annihilation problem with a finite annihilation rate for the target hit by a walker. While at short times f(t) may decay exponentially, at long times the solution is dominated either by a power law or a stretched exponential, depending on the initial defect configuration. The predictions of the model are shown to be in agreement with experimental data.

We use the linearized Boltzmann equation to discuss the time dependence of atomic collision cascades. The atoms are presumed to interact via a power-law potential, the power being denoted (-2/a). We wish to ascertain whether the equation, and/or certain special solutions, show striking changes in behavior at certain special values of a. We find a value of a which distinguishes energy-conserving cascades from anomalous cascades, and another which signals the failure of the linear approximation. Scaling, and also the emergence of similarity solutions, are discussed.

Information-theory techniques are employed in order to deal with the dynamics of nonlinear quantal Hamiltonians. Some interesting new results are obtained that tend to dispel some widely shared notions concerning the Hamiltonian approach to quantal dissipation effects.

A one-dimensional kinetic Ising model with dynamics characterized by a combination of spins flips at temperature T and spin exchanges at T=∞ is studied. The two-spin correlations in the steady state are calculated exactly and the decay times describing the relaxation of both the magnetization and the two-spin correlations are also given. We find that neither the steady-state nor the dynamic quantities show any sign of a phase transition that could exist in this one-dimensional, nonequilibrium system. Two remarkable features of the solution are that (i) the correlation length in the steady state with random spin exchanges is larger than the correlation length in the corresponding equilibrium state without spin exchanges, and (ii) a fluctuation-dissipation theorem is satisfied in the nonequilibrium steady state.

Liapunov exponents have been determined from a time series of acoustic turbulence. It is found that a low-dimensional chaotic attractor with one positive largest exponent rules the motions in phase space in the broadband noise region. The method for calculating the spectrum of Liapunov exponents is analyzed in detail for its applicability in experimental situations. It is shown, in particular, that spurious exponents arising from the embedding procedure do not spoil the calculation and that the calculation of the Liapunov dimension is possible.

A connected web of stochasticity can be generated by a mapping derived from a linear oscillator perturbed by a periodic δ function. Such a stochastic web is useful for investigating global diffusion through a phase space in which the local diffusion within the web is nonuniform. An analytic expression for the global diffusion rate has been obtained using (1) the basic phase-space concept that the ergodic region is uniformly populated in the asymptotic limit, (2) a local calculation of the thickness of the stochastic web, and (3) the average local period for traversing a single mesh of the web. The results are compared with numerical computations of the diffusion rate and are found to be in good agreement. Although the linearity of the kicked oscillator leads to a connected grid in phase space, the diffusion rate, unlike Arnol’d diffusion, is related to that in two-dimensional phase space, with the diffusion coefficient $D_{\mathrm{web}\mathrm{\equiv }{L}_{\mathrm{rms}{}^2/\mathrm{T}}}$ scaling as $K_{\alpha }^3$, where $K_{\alpha }$ is a perturbation parameter. Discrepancies are discussed, and the effect of extrinsic stochasticity is briefly considered.

In Landau–de Gennes theory, the free energy f of liquid crystals is expanded into powers of a symmetric, traceless tensor order parameter $Q_{\alpha \beta }$ and its derivatives $Q_{\alpha \beta }$,γ. The expansion is subject to the condition that f is a scalar, i.e., invariant under all rotations of the group SO(3). Using the method of integrity basis, we have established the most general SO(3)-invariant free-energy density up to all powers in $Q_{\alpha \beta }$ and up to second order in $Q_{\alpha \beta }$,γ. It turns out that this free-energy density is composed of 39 invariants, which are multiplied by arbitrary polynomials in Tr$Q^2$ and Tr$Q^3$. On the other hand, these 39 invariants can be expressed as polynomials of 33 so-called irreducible invariants. Interestingly, among the irreducible invariants there are only three chiral terms (i.e., linear in $Q_{\alpha \beta }$,γ). They locally give rise to three independent helix modes in chiral, biaxial liquid crystals. This conclusion generalizes results of Trebin [J. Phys. (Paris) 42, 1573 (1981)] and Govers and Vertogen [Phys. Rev. A 31, 1957 (1985); 34, 2520 (1986)] and contradicts a statement of Pleiner and Brand [Phys. Rev. A 24, 2777 (1981); 34, 2528 (1986)], according to which only one twist term is supposed to exist.

Some possibilities, leading to a smaller number of the irreducible invariants, are also discussed in detail. These special forms of the elastic free-energy expansion are obtained by imposing additional symmetry restrictions on the field Q. One arrives, for example, at a system of ‘‘soft biaxial nematic’’ phases by requiring Tr$Q^2$=const, at a system of biaxial phases with Tr$Q^3$=const, or at a system with fixed moduli of the order parameter (‘‘hard biaxial’’ phases) by requiring both. In all cases the number of irreducible invariants is reduced significantly. For example, hard biaxial phases can be described using only 12 irreducible invariants among which three are chiral. Furthermore, three irreducible invariants can be eliminated if surface terms vanish. In all limiting cases relations between irreducible invariants are given explicitly.

The effect of concentration-dependent diffusivity on diffusion-limited aggregation (DLA) is investigated by a computational method. The growth process is governed by a nonlinear diffusion equation with a concentration-dependent diffusivity. The aggregate grows proportionally to the mass flux on the surface of the aggregate. The development of distinct morphologies is found in nonlinear diffusive pattern-forming systems. When the dependence of diffusivity on the concentration increases, the aggregate crosses over from the DLA fractal to dense aggregation. The crossover phenomenon also occurs with an increase of the concentration difference between the aggregate and the outer boundary. A new dimensionless parameter is found to govern the crossover phenomenon. The morphology diagram is described in terms of the coefficient of the nonlinear part of the diffusivity and the concentration difference between the boundaries. Similarity with pattern formation in electrochemical deposition is discussed.

The Lyapunov exponents describe the time-averaged rates of expansion and contraction of a Lagrangian hypersphere made up of comoving phase-space points. The principal axes of such a hypersphere grow, or shrink, exponentially fast with time. The corresponding set of phase-space growth and decay rates is called the ‘‘Lyapunov spectrum.’’ Lyapunov spectra are determined here for a variety of two- and three-dimensional fluids and solids, both at equilibrium and in nonequilibrium steady states. The nonequilibrium states are all boundary-driven shear flows, in which a single boundary degree of freedom is maintained at a constant temperature, using a Nosé-Hoover thermostat. Even far-from-equilibrium Lyapunov spectra deviate logarithmically from equilibrium ones. Our nonequilibrium spectra, corresponding to planar-Couette-flow Reynolds numbers ranging from 13 to 84, resemble some recent approximate model calculations based on Navier-Stokes hydrodynamics. We calculate the Kaplan-Yorke fractal dimensionality for the nonequilibrium phase-space flows associated with our strange attractors. The far-from-equilibrium dimensionality may exceed the number of additional phase-space dimensions required to describe the time dependence of the shear-flow boundary.

The application of Padé and inhomogeneous differential approximants to the low-temperature series for the surface tension Σ(T ) of the simple cubic Ising model provides accurate values for temperatures T≲$T_R$, where $T_R$ is the roughening temperature. The roughening transition is marked by significant structure in the variation of Σ(T ) above $T_R$. Approximants with surface-tension amplitudes, $A_{\Sigma }$$a^2$/$k_B$$T_c$, between 1.4 and 2.0 agree well with Monte Carlo data for T<$T_c$.

A version of the two-dimensional site-diluted spin-(1/2 Ising model is proposed as a microscopic interaction model which governs solidification and growth processes controlled by vacancy diffusion. The Ising Hamiltonian describes a solid-fluid phase transition and it permits a thermodynamic temperature to be defined. The dynamics of the model are taken to involve (i) solid-fluid conversion and (ii) diffusion of vacancies in the fluid phase. By means of Monte Carlo computer-simulation techniques the solidification and growth processes following rapid thermal quenches below the transition temperature are studied as functions of temperature, time, and concentration. At zero temperature and high dilution, the growing solid is found to have a fractal morphology and the effective fractal exponent D varies with concentration and ratio of time scales of the two dynamical processes. The mechanism responsible for forming the fractal solid is shown to be a buildup of a locally high vacancy concentration in the active growth zone. The growth-probability measure of the fractals is analyzed in terms of multifractality by calculating the f(α) spectrum. It is shown that the basic ideas of relating probability measures of static fractal objects to the growth-probability distribution during formation of the fractal apply to the present model. The f(α) spectrum is found to be in the universality class of diffusion-limited aggregation. At finite temperatures, the fractal solid domains become metastable and a crossover to compact equilibrium solidification is observed as a function of both temperature and time. At low temperatures in the metastable fractal growth regime, the time dependence of the particle content of the domains is found to obey the scaling law, N(t)∼$t^{D/\left(2\mathrm{}d\mathrm{-}D\mathrm{-}1\right)}$. At higher temperatures where the growth is stable and leads to compact domains, the time dependence of N(t) can be described by a simple hyperbolic function. The various results of the theoretical model study are related to fractal and compact growth patterns observed in experimental studies of impure lipid monolayer films at air-water interfaces.