This paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations. Such a procedure is called ‘‘targeting.’’ Previous work on targeting for chaotic attractors has been in the context of one- and two-dimensional maps. Here it is shown that targeting can also be done in higher-dimensional cases. The method is demonstrated with a mechanical system described by a four-dimensional mapping whose attractor has two positive Lyapunov exponents and a Lyapunov dimension of 2.8. The target is reached by making very small successive changes in a single control parameter. In one typical case, 35 iterates on average are required to reach a target region of diameter ${10}^{\mathrm{-}4}$, as compared to roughly ${10}^{11}$ iterates without the use of the targeting procedure.

We consider the fully developed chaos in a class of driven one-degree-of-freedom nonlinear systems. In analogy to Kubo relations in statistical mechanics, we have quantitatively related the maximal positive Lyapunov exponent, which is characteristic of divergence of trajectories, to the spectral density of fluctuations of the appropriate dynamical variable. A numerical experiment is carried out to confirm the qualitative validity of the theoretical prediction. A generalization of the relationship for N-dimensional Hamiltonian system has been given.

For a variety of ‘‘random configurations’’ of equal disks, the probability distributions of the Voronoi-cell areas are obtained from a computer simulation and a comparison is made in terms of the relation between the regularity of the cell areas and the mean bulk fraction of the area occupied by the disks.

The relationship between the stochastic and deterministic approaches to the description of the dynamic behavior for a nonlinear reacting system is studied. A self-propagating exothermal reaction in a heterogeneous medium has been chosen as an example. A stochastic model for this phenomenon is developed. The system is composed of cells characterized by temperature and degree of conversion, their transformation probability being dependent on temperature. Computer simulation showed that stochasticity reveals itself in the generation of disturbances that are absent in the deterministic model. For a well-developed steady-state regime, the distributions of the mean temperature and degree of conversion are close to those given by the deterministic model. Under the conditions of planar-wave-front instability, the stochastic model possesses a mechanism for spontaneous transfer to a stable regime from arbitrary initial conditions (temperature distribution, etc.) due to origination, propagation, and disintegration of disturbances. Such behavior agrees with experimental data and is not predicted by the deterministic model.

The Boltzmann equation for reacting charged-particle swarms in neutral gases in the presence of both electric and magnetic fields is decomposed into a hierarchy of kinetic equations by expanding the velocity dependence of the phase-space distribution function in terms of spherical harmonics. No limit is set on the number of spherical harmonics and no approximation is made concerning the mass of the charged particles related to that of the neutrals species. The space-time dependence is treated by making the hydrodynamic assumption which is taken to second order in density gradients. Spherical tensors are used throughout. The resulting heirarchy of equations has universal validity and is amenable to a range of numerical solutions. The structure of these equations is discussed and the inadequacies of a Legendre-polynomial expansion are pointed out. The special configurations of the magnetic field parallel and perpendicular to the electric field are discussed in detail.

Algorithms for molecular dynamics (MD) at constant temperature and pressure are investigated. The ability to remain in a regular orbit in an intermittent chaotic regime is used as a criterion for long-time stability. A simple time-centered algorithm (leap frog) is found to be the most stable of the commonly used algorithms in MD. A model of N one-dimensional dimers with a double-well intermolecular potential, for which the distribution functions at constant temperature T and pressure P can be calculated, is used to investigate MD-NPT dynamics. A time-centered NPT algorithm is found to sample correctly and to be very robust with respect to volume scaling.

Anomalous viscoelastic, diffusion, and heat-transfer phenomena observed in spatially inhomogeneous simple and complex fluids are analyzed in this paper in the setting of weakly nonlocal hydrodynamics. Governing equations of this generalized hydrodynamics involve higher-order derivatives with respect to the position coordinate. The governing equations are obtained on the basis of the following consideration. The time evolution in both local and weakly nonlocal hydrodynamics is generated by a thermodynamic potential in a state space equipped with a Poisson structure and a dissipative potential. The Poisson structure is an expression of kinematics in the chosen state space. In weakly nonlocal hydrodynamics the Poisson structure remains the same as in local hydrodynamics but the potentials are generalized. The potentials are allowed to depend on higher-order derivatives of the hydrodynamic fields that are chosen as state variables. This way of introducing the governing equations guarantees that the equations are intrinsically compatible and that their solutions agree with certain fundamental macroscopic observations (e.g., the observations constituting the basis of equilibrium thermodynamics).

Evolution of slow magnetoacoustic perturbations is used to demonstrate a new mechanism of plane Couette flow shear energy transformation. The initial uniform magnetic field is considered to be directed along the unperturbed flow velocity. We find that the amplification of waves is nonordinary—it occurs in a limited time interval and is anomalously strong. The unusual behavior is manifested also by the fact that the characteristic length scale of perturbation diminishes down to the dissipative scales not only due to nonlinear cascade processes, but due to a linear drift in the k space as well.

We consider the effect of viscosity on Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities by deriving a moment equation for fluids with arbitrary density and viscosity profiles, including surface tension. We apply our result to the classical case of two semi-infinite fluids with densities ${\mathrm{\rho }}_1$ and ${\mathrm{\rho }}_2$ and viscosities ${\mathrm{\mu }}_1$ and ${\mathrm{\mu }}_2$. Treating a shock as an instantaneous acceleration we find that perturbations at the interface undergo damped oscillations when viscosity and surface tension are both present. For pure viscosity the amplitude η(t) evolves according to η(t)/η(0)=1+(ΔvA/2kν)(1-${\mathit{e}}^{\mathrm{-}2\mathit{k}2}$νt) where Δv is the jump velocity imparted by the shock, A=(${\mathrm{\rho }}_2$-${\mathrm{\rho }}_1$)/(${\mathrm{\rho }}_2$+${\mathrm{\rho }}_1$), ν=(${\mathrm{\mu }}_1$+${\mathrm{\mu }}_2$)/(${\mathrm{\rho }}_1$+${\mathrm{\rho }}_2$), k=2π/λ is the wave number of the perturbation, and t is time. We also consider the turbulent energy in accelerating fluids and calculate the reduction in ${\mathit{E}}_{\mathrm{turbulent}}$ as a function of ν, and propose experiments to measure the effect of viscosity on RT and RM instabilities.

The problem of Brownian motion in shear flows is analyzed using a path-integral approach and a piecewise-linearized velocity profile. In this way, the complete two-dimensional conditional probability distribution is found and the unimodal-bimodal transition in the longitudinal distribution is consistently described. A remarkable aspect is the appearance of a transient trimodal regime.

In a previous publication [Z. Y. Chen, A. Abbaci, S. Tang, and J.V. Sengers, Phys. Rev. A 42, 4470 (1990)] a renormalized Landau expansion was constructed for the thermodynamic free energy of one-component fluids that incorporates the crossover from singular thermodynamic behavior at the critical point to regular behavior far away from the critical point. In the present paper the approach is extended to obtain a crossover free energy for binary fluid mixtures in the region around the vapor-liquid critical line. The thermodynamic equations thus obtained are compared with experimental equation-of-state and specific-heat data for mixtures of carbon dioxide and ethane.

We present results of computer simulations of the motion of shock fronts in a variety of one-dimensional stochastic lattice models with parallel and serial dynamics, infinite and finite temperatures, and ferromagnetic and antiferromagnetic particle interactions. We find that fluctuations in the shock location, about an average determined by evolving an ensemble of systems with the same initial conditions, generically grow in time like ${\mathit{t}}^{1/3}$. We discuss the robustness of the ${\mathit{t}}^{1/3}$ growth and determine the density dependence of the coefficient in a simple case. We compare this with models where the dynamics are specially tuned so that the growth is reduced to ${\mathit{t}}^{1/4}$, and with the situation where the ensemble members have different initial conditions, in which case the growth is like ${\mathit{t}}^{1/2}$.

The two-point correlation function of an ensemble of interacting closed self-avoiding surfaces on a cubic lattice is analyzed in the disordered phase, which corresponds to the paramagnetic region in a related spin formulation. Mean-field theory and Monte Carlo simulations predict the existence of a disorder line which corresponds to a transition from an exponential decay to an oscillatory damped behavior of the two-point correlation function. The relevance of the results for the description of amphiphilic systems in a microemulsion phase is discussed. The scattering function is also calculated for a bicontinuous phase coexisting with the paramagnetic phase.

The energy ${\mathit{V}}_0$ of the bottom of the conduction band (relative to vacuum) of quasifree positrons in rare-gas fluids is calculated as a function of fluid density. The calculations are performed within the framework of the Wigner-Seitz approximation [Phys. Rev. 43, 804 (1933)] for nonpolar fluids, using a semiempirical analytical potential to model the positron–rare-gas-atom interactions. For all the rare gases studied, ${\mathit{V}}_0$ is negative and decreases almost linearly with increasing density. Extended to the solid-phase density range, our ${\mathit{V}}_0$ calculations are in good agreement with available experimental data for rare-gas crystals.

We study an electrorheological fluid consisting of spherical dielectric particles in a liquid of low dielectric constant. At a fixed temperature, there are two critical electric fields ${\mathit{E}}_{\mathit{c}2}$<${\mathit{E}}_{\mathit{c}1}$. When the applied field E<${\mathit{E}}_{\mathit{c}2}$, the system is a fluid with no long-range order. When ${\mathit{E}}_{\mathit{c}2}$<E<${\mathit{E}}_{\mathit{c}1}$, the particles form chains between two electrodes, but the distribution of these chains is random. This state is similar to an induced nematic liquid crystal. As E exceeds ${\mathit{E}}_{\mathit{c}1}$, the system is a solid whose ideal structure is a body-centered tetragonal lattice with ${\mathbf{a}}_1$= √6 ax^, ${\mathbf{a}}_2$= √6 ay^, and ${\mathbf{a}}_3$=2az, where a is the radius of dielectric spheres and z^ is the field direction.

Thermodynamic perturbation theory and density-functional approximations are systematically combined to produce a model of Lennard-Jones solids and solid-liquid coexistence. The perturbation theory is based on expanding the free energy about that of the fcc hard-sphere solid, which is described by an accurate nonperturbative density-functional theory. Approximations made throughout the development are systematically checked against results of Monte Carlo simulations. The Gaussian approximation used for describing the solid density is shown to be a good approximation for the stable solid; however, anisotropies in the structure that are not captured by the Gaussian approximation become pronounced at densities corresponding to metastable solids. The free energies of both the solid and the liquid Lennard-Jones phases and the phase diagram predicted by the density-functional thermodynamic perturbation model are in good agreement with Monte Carlo simulations for temperatures in the range 0.75≤kT/ɛ≤100.

We describe a light-scattering technique capable of probing the dynamics of thickness fluctuations in lipid bilayers. The technique, which we call reflectance fluctuation spectroscopy (RFS), is keenly sensitive to light scattered from the squeeze modes of motion in a thin liquid film, and insensitive to light scattered from the bend modes. A laser beam is focused to a small spot on the film, and the power in the specularly reflected beam is recorded in real time. Thickness fluctuations associated with the squeeze modes of motion give rise to fluctuations in the power of the specularly reflected light. The frequency spectrum of the fluctuations in detected power (RFS spectrum) can be analyzed to yield values for the film viscosity and thickness compressibility. We present two independent theoretical derivations of the RFS spectrum and show that scattering from the bend mode can be neglected. The theoretical expression for the RFS spectrum is compared with experimental spectra obtained from glycerylmono-oleate–decane bilayers. The fit of the theory to the data is excellent and the values deduced for the film viscosity and thickness compressibility are reasonable.

We have quantitatively determined the nature of the surface order of the outermost layer of monolayer and multilayer Langmuir-Blodgett films of cadmium arachidate, in air and under water, using atomic-force microscopy. Molecular-resolution images of 1200-${\mathrm{nm}}^2$ areas show that the alkyl chains of 2–5-layer films have a noncentered rectangular lattice with lattice constants of 0.482±0.004 and 0.748±0.006 nm, in good agreement with diffraction measurements of bulk aliphatic systems. We show unambiguously that the unit cell consists of two molecules. In contrast with the highly ordered multilayers, monolayer films have a disordered surface. Images taken under water of the head-group surface show that the bilayer surface is disordered, while thicker films show order of the alkyl chains. The presence of an underlying head-group–head-group interface with its associated cadmium ion bridging is shown to be the critical feature needed to obtain a stable and ordered surface layer.

Based on the same bending energy of the spontaneous curvature model, three shape equations for axisymmetric vesicles are derived from different variational methods. They are degenerate for the spherical vesicle, while for the cylindrical vesicle, two of them are the same. They all have a special toroidal solution, Clifford tori, but the constraints on the Lagrange multipliers ΔP and λ and the spontaneous curvature ${\mathit{c}}_0$ are different. We consider the physical mode of variation and introduce an arbitrary parameter for the axisymmetric action; we get the shape equation in terms of this parameter from it. When this parameter is identified as the parameter ρ, it reduces to the same equation that is from the general shape equation.

Results obtained on electrical conductivity, dielectric relaxation (static permittivity and characteristic frequency), and dynamic viscosity are presented for microemulsions with very different components (water-based or waterless, ternary or quaternary). However, they do have in common the fact that they all correspond to the model of a distribution of spheres, subjected to Brownian motion, dispersed in a continuous medium. For all these systems the associated dynamic percolation model is qualitatively verified. Moreover when the conditions of application of the asymptotic laws of the theory are satisfied, the scale exponents are the same (approximately equal to 2 above the threshold and - 1.2 below it) for all the systems, in agreement with the theoretical predictions. This reflects the fact that all these systems belong to the same class of universality.