We show that a network of quenched random grain boundaries, terminating in polymerized patches containing excess disclinations, destabilizes the flat phase of sparsely polymerized membranes, in contrast to membranes completely polymerized in the liquid state. We argue that many of these grain boundaries will buckle, even in the absence of excess disclinicity, leading to a glassy wrinkled phase consistent with recent experiments on partially polymerized vesicles by Mutz, Bensimon, and Brienne [Phys. Rev. Lett. 67, 923 (1991)].

The definition of a Lyapunov exponent can be extended to include an imaginary part. This extension requires the definition of a coordinate frame on the tangent space of the differential equation and an extension of the concept of a limit. The definition of extended Lyapunov exponents is based on the eigenvalues of the fundamental matrix. It is shown that the extended exponent agrees completely with the constant-coefficient case. It is shown that the eigenvectors and eigenvalues obey differential equations and can be propagated numerically without constructing the fundamental matrix itself. Bifurcation of eigenvalues and eigenvectors can also be followed numerically without recourse to the fundamental matrix. Two example applications of the method to the calculation of extended Lyapunov exponents are given. In the Lorenz problem, the real parts of the extended Lyapunov exponents agree quite well with previous results. Fourier-transform methods are used to show that the power spectrum of relative motion is discrete, with fundamental frequency quite close to the calculated imaginary part of the extended Lyapunov exponent. In the simple pendulum, the extended Lyapunov exponents are usually purely imaginary and are the relative oscillation frequencies of adjacent trajectories.

The mean field in a globally coupled system of chaotic logistic maps does not obey the standard rules of statistics, even for systems of very large sizes. This indicates the existence of intrinsic instabilities in its evolution. Here these instabilities are related to the very nonsmooth behavior of mean values in a single logistic map, as a function of its parameter. Problems of this kind do not affect a similar system of coupled tent maps, where good statistical behavior has been found. We also explore the transition between these two regimes.

A diffusion equation is derived for the distribution of energies of an ensemble of systems governed by an ergodic adiabatic Hamiltonian. We compare the predictions of this equation with previously derived results and find some disagreement. We obtain explicit expressions for the degree to which the ergodic adiabatic invariant is conserved. We demonstrate that our equation gives the expected results for two special cases. Finally, we discuss applications and tests of our theory, as well as directions for future study.

We analyze the behavior of a damped quantum harmonic oscillator immersed in a heat bath with a nonlinear coupling. For this purpose, we construct a generalized master equation (GME) for the reduced density of the oscillator. We specialize the results for a quadratic coupling in the presence of a bosonic reservoir and an Ohmic dissipation model, analyzing the evolution equation for the mean value of the oscillator coordinate. In the asymptotic case we obtain a gain-loss master equation whose transition rates are polynomials in the number of quanta. No explicit solutions other than the equilibrium one can be written, since the standard methods do not apply. The equilibrium solution is investigated and the eigenvalues of the corresponding spectral problem are numerically computed to study the effect of the nonlinearity on the evolution of the system. By means of the Wigner transformation of the GME, we extract a third-order partial derivative equation that represents the semiclassical evolution of the damped oscillator. In the classical limit this is a nonlinear Fokker-Planck equation.

Video polarization microscopy, optical diffraction, and digital image analysis have been employed to investigate the transformation of a well-ordered lamellar (‘‘smectic’’) domain phase of ferrimagnetic garnet films into a globally disordered, ‘‘labyrinthine’’ pattern. Surprisingly, in view of the presence of nonlocal interactions, the ensuing pattern evolution in response to temperature-induced strain is characterized, in its first stages, by a (local) transverse elastic response which manifests itself in the form of ‘‘smectic’’ instabilities and generates undulation and chevron (‘‘zigzag’’) patterns. At a characteristic limit of accumulated strain, the nucleation of topological defects in the form of disclination dipoles initiates the second stage of the evolution. The subsequent continuous, topologically constrained ‘‘unbinding’’ of these dipoles represents the essential mechanism mediating the loss of global orientational order in the pattern. The application of a set of algorithms for line-pattern analysis to track the motion of individual topological charges permits the quantitative description of this process. The emerging labyrinthine patterns represent the result of a constrained optimization and, while globally disordered, have in fact been shown to exhibit a robust and well-defined local structure in the form of ‘‘cybotactic’’ clusters.

An analysis is presented of the role of topological defects in the evolution of disordered stripe-domain phases of ferrimagnetic garnet films. Proliferation by continued nucleation and topologically constrained unbinding of disclination dipoles are identified as the essential mechanisms mediating the strain-induced disordering process. Two distinct classes of metastable, disordered states are distinguished on the basis of the relative predominance of defect-pair proliferation and unbinding, and the fundamentally different corresponding pattern morphologies which accommodate the emerging disclination network. Striking patterns containing linear chains of interdigitated disclination dipoles are representative of the first class, while labyrinthine patterns, exhibiting a robust and well-defined local structure in the form of ‘‘cybotactic’’ clusters, are typical of the second class. Results are discussed with reference to defect-mediated melting and the structure of glasses. Comparison is also made with pertinent phenomena in related systems such as ferrofluids and amphiphilic films.

The frequency- and wave-vector-dependent dielectric function of a waterlike fluid is expressed in terms of the self-correlation part of the dipole-dipole correlation function and a static, wave-vector-dependent local-field factor. The self-correlation function is calculated in a number-conserving relaxation-time approximation which correctly interpolates between diffusive behavior at low frequency and long wavelength and free-particle behavior at high frequency and short wavelength. The local-field factor is obtained by matching the static limit of the dielectric function to the one obtained by the reference-hypernetted-chain approximation. The resulting behavior of the dielectric function is compared with that obtained in previous theoretical calculations and with available experimental data for water. The refractive index shows a ‘‘free-rotation’’ peak, and the longitudinal dielectric function exhibits a collective resonance.

We have derived, from a general quantum-mechanical statistical approach, the density expansion of the neutron-scattering dynamic structure factor both for coherent and incoherent scattering. The theory is based on the differentiation of appropriate functionals and leads to a series in which the first term is the perfect-gas contribution, the second the contribution of pairs, the third the contribution of triplets. The term due to pairs has been analyzed in detail, and quantum-mechanical rigorous expressions have been given that can be calculated once the pair-interaction potential is known and wave functions of the pairs are derived. Also the exact expressions of the first four moments of the pair contribution are given.

A multistate cellular automaton model for computational simulation of reaction-diffusion processes at the microscopic level is presented. Diffusion is introduced through a standard stochastic scheme. On the other hand, the reaction dynamics, which applies to generic chemical reactions, consists of a deterministic algorithm driven by a discrete mapping. By means of a proper average on the states of the cellular automaton, the microscopic dynamics is fully identified with the corresponding macroscopic reaction-diffusion laws. As an illustration of the capability of the model for describing the behavior of reaction-diffusion systems, some results on pattern formation for various chemical systems are presented.

Taylor-Couette flow subject to a nonaxisymmetric external Coriolis force is studied experimentally. The Coriolis force is induced by placing the Taylor-Couette apparatus on a turntable such that the flow is nonaxisymmetrically perturbed. This symmetry-breaking perturbation affects the entire sequence of bifurcations found in the unperturbed system. We identify the dynamical regimes encountered as a function of two control parameters: a Reynolds number and a dimensionless turntable rotation rate. The bifurcation map for this system is presented, and we show that there is nonhysteretic reemergent order as a function of either control parameter. We also find a possible codimension-2 bifurcation point beyond which occurs a nonhysteretic direct bifurcation to turbulence from the time-independent base flow.

By performing molecular-dynamics simulations of a fluid under the influence of a sinusoidal transverse force, we point out that heat flow can be induced in the absence of a temperature gradient. This heat flux is observed to be proportional to the gradient of the square of the strain-rate tensor. We show that this shear-gradient-induced heat flux is of the same order in the pressure gradient as the heat flow produced by viscous heating.

The dynamics of defects against the background of a regular system of rolls is investigated within the Swift-Hohenberg model with periodically-space-modulated pumping. For thermal convection, this corresponds to periodic nonuniform heating. Amplitude equations obtained for a parametric-resonance case generalize the equations derived earlier and describe both longitudinal and transverse defects. It is shown that a pair of longitudinal defects either attract one another until they annihilate or they diverge to infinity. Transverse defects are able to form bound static states. We investigate the influence of small nongradient effects on the dynamics of single longitudinal and transverse defects as well as on their interaction. We also discuss the effect of ‘‘elastic’’ scattering of transverse defects on each other in the presence of small dispersion.

A passive-scalar field is considered to evolve under the influence of a turbulent fluid governed by the Navier-Stokes equation. Turbulent-transport coefficients are calculated by small-scale elimination using a renormalization-group method. Turbulent processes couple both the viscosity and the diffusivity. In the absence of any correlation between the passive-scalar fluctuations and any component of the fluid velocity, the renormalized diffusivity is essentially the same as if the fluid velocity were frozen, although the renormalized equation does contain higher-order nonlinear terms involving viscosity. This arises due to the nonlinear interaction of the velocity with itself. In the presence of a finite correlation, the turbulent diffusivity becomes coupled with both the velocity field and the viscosity. There is then a dependence of the turbulent decay of the passive scalar on the turbulent Prandtl number.

A version of the no-feedback-control method developed by Hübler and Lüscher [Helv. Phys. Acta 62, 544 (1989)] is applied to two-dimensional open-channel flow past a plate. The control is applied in the vortex-shedding regime with the aim of reducing the effective Reynolds number, thereby suppressing the vortex shedding. It is shown that the method works well not only when the flow is globally forced but also when the forcing is restricted to the boundary-layer region.

The process of pattern formation in the two-dimensional Swift-Hohenberg equation is examined through numerical and analytic methods. Dynamic scaling relationships are developed for the collective ordering of convective rolls in the limit of infinite aspect ratio. The stationary solutions are shown to be strongly influenced by the strength of noise. Stationary states for small and large noise strengths appear to be quasiordered and disordered, respectively. The dynamics of ordering from an initially inhomogeneous state is very slow in the former case and fast in the latter. Both numerical and analytic calculations indicate that the slow dynamics can be characterized by a simple scaling relationship, with a characteristic dynamic exponent of ¼ in the intermediate-time regime.

Interferometric measurements of the chirality-induced biaxiality in the vicinity of the uniaxial-to-biaxial cholesteric transition of the lyotropic system potassium laurate, decylammonium chloride, ${\mathrm{H}}_2$O, and brucine sulfate are reported. The mean-field approach to describe the cholesteric-to-cholesteric transition allows an estimate of the ratio between coefficients of the Landau expansion. The values of the bare correlation length, ${\xi }_0$∼1000 Å, calculated are larger than in lyotropic systems with alcohol. The chirality-induced biaxiality measurements in the limit of zero elastic field (${\mathit{q}}^2$=0) and the birefringence measurements in the nematic phase confirm critical properties of the uniaxial-to-biaxial nematic transition. The critical exponents for the order parameter (β) and the susceptibility (γ) are in good agreement with the values calculated for the XY model.

Nucleation and growth in a mixture of isobutyric acid and water are studied by small-angle light scattering under a shear flow. The study is restricted to off-critical quenches, just below the coexistence curve,where nucleation is essentially heterogeneous. When no shear is applied, the distribution of the droplet radii is peaked around its mean value, which behaves as R(t)=${\mathit{R}}_0$${\mathit{t}}^{\mathit{a}}$, with a=0.6±0.1. A continuous shear flow is shown to accelerate the growth. An advection-diffusion process is proposed, which allows the data to be expressed in a scaled form, over a wide range of shear rates. The effects of shear-induced coalescence and gravity-induced flows are also investigated.

Binuclear copper (II) carboxylates, ${\mathrm{Cu}}_2$(${\mathrm{C}}_{\mathit{n}}$${\mathrm{H}}_{2\mathit{n}+1\mathrm{}}$${\mathrm{O}}_2$${)}_4$, crystallize at room temperature in layered systems in which planes of polar cores are separated by a double layer of alkyl chains. These compounds are mesomorphic in nature above ca. 100 °C. Pseudopolymeric chains of regularly stacked binuclear cores are located at the nodes of a two-dimensional hexagonal lattice and are surrounded by disordered aliphatic chains. The transition from the crystal to the columnar mesophase is characterized by a change in the repeat distance of the binuclear cores along the pseudopolymeric axis. In the crystalline phase, these cores are all oriented in the same direction with a repeat distance of 5.2 Å; in the columnar mesophase, the polar cores are perpendicular to the columnar axis and superposed in a fourfold helicoidal fashion, at least on a local scale, with a repeat distance of 4.7 Å. We present here the effect of pressure on these anisotropic systems in a direction parallel to the columnar axis, and in the plane of the two-dimensional lattice. In a first part, we report the pressure-volume-temperature (P-V-T) relationship of these compounds (n=12, 18, and 24) in the temperature range from 30 to 200 °C, and in the pressure range from 1 to 2000 bars. Isothermal compressibility and isobaric expansion are determined in the crystalline and mesomorphic phases. In the mesophase, pressure-volume isotherms can be described by the Tait equation, as in most liquids or molten polymers. In a second part, we discuss the x-ray-diffraction experiments performed under pressure. In the mesophase, the area of the two-dimensional lattice decreases with increasing pressure and, at sufficiently high pressure, the columnar mesophase transforms into a crystalline lamellar phase. By combining P-V-T and x-ray results, we deduce an increase of the stacking period of the binuclear cores as a function of increasing pressure.

A discretization method is presented in order to describe structural properties of Yukawa-type fluids consisting of particles with a continuous size and charge distribution. The accuracy of the method is tested by comparing the results from the Rogers-Young closure scheme of the Ornstein-Zernike equation for the correlation functions with the corresponding Monte Carlo data. The relevance of this method for the interpretation of light- and neutron-scattering data of colloidal dispersions is also briefly discussed.