The Glauber dynamics of the Hopfield model is studied with use of the Mori-Zwanzig projection-operator formalism for irreversible processes and exact evolution equations for the overlaps are derived.

Two-dimensional vesicles subject to osmotic pressure imbalance may be modeled as unrestricted random rings. Assuming a pressure that couples to the square of the oriented area we are able to obtain an exactly solvable model that accommodates the deflated regime corresponding to a shriveled vesicle. Moreover, we investigated the effects of a long-range repulsive, nonlocal interaction among monomers by employing a variational approach. We find good agreement between our analytical predictions and Monte Carlo simulation results.

Comparisons are made among several three-node finite-difference schemes (FDS’s) for solving time-independent Schrödinger equations. It is shown that the Mickens FDS is, although exact in some special cases, generally two orders lower than the Numerov FDS. An alternative FDS, the combined Numerov-Mickens FDS, is introduced. It has advantages of both the Numerov and the Mickens FDS’s. Numerical comparison among these FDS’s is presented.

In this paper we discuss the Lyapunov and correlation dimensions of an attractor of a chain of unidirectionally coupled oscillators at the boundary of a chaos-hyperchaos transition. We discuss the problem of the distinction between chaos and hyperchaos based on these dimensions.

We study the probability distribution P(x,t) for the head of a directed polymer in a random medium in 1+1 dimensions. We find that, as a function of the scaling variable u=‖x‖/${\mathit{t}}^{2/3}$, the behavior of the probability distribution changes abruptly at ${\mathit{u}}_{\mathit{c}}$≊1.2. Its universal part is given by P∝exp(-α${\mathit{u}}^{\mathit{y}}$) where the exponent α is distributed for different samples according to scrP(α)=scrN exp[-f(α)${\mathit{u}}^{\mathit{y}}$]. For 0.25≲u<${\mathit{u}}_{\mathit{c}}$, y=2.1±0.1 and the multifractal measure f(α) is a convex function with a minimum at ${\mathrm{\alpha }}_{\mathrm{typ}}$, which is the exponent associated with a typical sample of the disorder. For u≫${\mathit{u}}_{\mathit{c}}$, y=3.0±0.1 and f(α) becomes trivial.

We present a simple and accurate method of estimating the translational hydrodynamic friction on rigid Brownian particles of arbitrary shape. The Brownian friction coefficient f takes the form f=6πη${\mathit{C}}_{\mathrm{\Omega }}$, where ${\mathit{C}}_{\mathrm{\Omega }}$ is mathematically equivalent to the electrostatic capacitance of the particle Ω in units where the capacity of a sphere equals its radius. This formula is particularly useful for particles consisting of a few globular subunits, for which slender body approximations are not very accurate.

We present a numerical study of the spontaneous formation of spiral patterns in Rayleigh-Bénard convection in non-Boussinesq fluids. We solve a generalized two-dimensional Swift-Hohenberg equation that includes a quadratic nonlinearity and coupling to mean flow. We show that this model predicts in quantitative detail many of the features observed experimentally in studies of Rayleigh-Bénard convection in ${\mathrm{CO}}_2$ gas. In particular, we study the appearance and stability of a rotating spiral state obtained during the transition from an ordered hexagonal state to a roll state.

We show that for rotating Rayleigh-Bénard convection over a range of dimensionless rotation rates 270<Ω<2500, bulk convection structures are Küppers-Lortz unstable near onset. We used optical shadowgraph imaging to visualize the pattern dynamics, found switching angles of approximately 60°, and observed separate spatial domains where switchings took place at different times. The characteristic switching frequencies were found to depend weakly on the rotation rate. Finite-size and boundary effects were observed to complicate the switching dynamics close to onset.

An investigation of the stochastic electron acceleration in a magnetized plasma due to an intense electron cyclotron wave is performed for the case of a stochastic regime where only a few resonances overlap and the stochastic region in phase space is bounded. The electron dynamics is described by a Hamiltonian function H(θ,I,t), and the stochastic properties of the system are investigated by means of the Poincaré-section method and the analytical estimates of the phase-correlation function. The behavior of the system is then analyzed through the Fokker-Planck-Kolmogorov (FPK) diffusion equation in action space. The theoretical FPK predictions are compared with the results of a numerical simulation of the particle motion. It is found that the system satisfies the diffusion equation at times that are short compared to the saturation time of the quasilinear diffusion. Appreciable deviations from the diffusive results are found at longer times, since the particle motion is influenced by the region of local stochasticity in phase space.

Experiments are performed that illustrate the scattering and reradiation of damped traveling waves of light energy density from objects within otherwise homogeneous turbid media. We demonstrate that simple diffractive and refractive models can be used to understand the scattering of these waves by absorptive and dispersive spheres. Furthermore, using a fluorescent dye we can reestablish the origin of the diffuse photon density waves, by converting from one optical and diffusive wavelength to another while maintaining phase coherence. The possibilities for precise medical imaging are presented.

We report control of chemical chaos observed during the electrodissolution of a rotating copper disk in a sodium acetate–acetic acid buffer. An adaptation of a simple recursive proportional-feedback control strategy, generally applicable to systems well described by one-dimensional maps, was used to stablize the chaotic response of the system on an unstable period-1 orbit. The successive minima in the measured anodic current generated a return map that was used to characterize the dynamics, and the control algorithm prescribed small changes in the anodic potential to affect the control.

There are a number of apparently disparate problems in multifractal scaling whose solutions have remained unclear, ranging from rather pathological cases where the standard Legendre transformations do not produce effective measures for the Hölder exponent and Hausdorff-Besicovitch dimension to the problem of describing the scaling of point-point correlation functions of moments of multifractal measures. We prove that an equivalent statement of multifractal scaling is the invariance of the generating functions of the scaling transformation. We show that the invariance of the generating functions is what allows the moment integrals to scale with simple power laws. We show that this definition can be successfully extended to cover the scaling of point-point correlation functions of moments of multifractal measures. Previous attempts to solve this problem have lead to non-scale-invariant behavior, presented as an inconsistency by Cates and Deutsch [Phys. Rev. A 35, 4907 (1987)]. We propose that the invariance of generating functions under their own transformations is the central defining characteristic of scale invariance in multifractal scaling.

This contribution presents four results. First, calculations indicate that when examined by the Grassberger-Procaccia algorithm alone, filtered noise can mimic low-dimensional chaotic attractors. Given the ubiquity of signal filtering in experimental investigations, this is potentially important. Second, a criterion is derived which provides an estimate of the minimum data accuracy needed to resolve the dimension of an attractor. Third, it is shown that a criterion derived by Eckmann and Ruelle [Physica D 56, 185 (1992)] to estimate the minimum number of data points required in a Grassberger-Procaccia calculation can be used to provide a further check on these dimension estimates. Fourth, it is shown that surrogate data techniques recently published by Theiler and his colleagues [in Nonlinear Modeling and Forecasting, edited by M. Casdagli and S. Eubanks (Addison Wesley, Reading, MA, 1992)] can successfully distinguish between linearly correlated noise and nonlinear structure. These results, and most particularly the first, indicate that Grassberger-Procaccia results must be interpreted with far greater circumspection than has previously been the case, and that the algorithm should be used in combination with additional procedures such as calculations with surrogate data. When filtered signals are examined by this algorithm alone, a finite noninteger value of ${\mathit{D}}_2$ is consistent with low-dimensional chaotic behavior, but it is certainly not a definitive diagnostic of chaos.

A correlation technique is developed to characterize the distribution of fractal structures. This method generalizes the concept of lacunarity to a scaling function and is naturally suited to distinguish between fractal structures with similar fractal dimensions for already small systems. We therefore propose it as a convenient tool to quantify the vaguely defined concept of morphology. Our lacunarity function is sensitive to logarithmic periodicities and we conjecture that its asymptotic behavior yields information on corrections to scaling in simpler measurements. The formalism we use can be generalized to analyze multifractal structures.

We use lattices with randomly distributed site-barrier energies to study diffusion properties as a function of disorder and temperature. We study the case of ‘‘dynamic’’ disorder whereby the random environment is renewed at each successive jump of the hopping particle, and also the case of ‘‘static’’ disorder with frozen randomly distributed barriers. The transition characteristics are governed by Boltzmann statistics. We employ standard Monte Carlo techniques to monitor properties such as the mean-square displacement. The trends of the motion are shown to include local trapping at early times, allowing for the search of a crossover time to the conventional diffusive regime, 〈${\mathit{R}}^2$〉∼t, as a function of temperature. We find that the crossover time versus temperature dependence is of the Arrhenius type determined by an effective activation energy barrier for percolation in the case of static disorder. For a uniform distribution of the barriers, this activation barrier is shown to coincide with the threshold concentration for bond percolation, as simple arguments suggest. We also demonstrate that an increase in the degree of dynamic disorder leads to an increase in the particle mobility. Some relationships of the present model to several experimental systems are discussed.

The correlation dimension is calculated for data from a partial-differential-equation model of a magnetic-domain wall—the Bloch wall in a magnetic thin film. The data were extracted from different locations along the height of the wall and two types of embedding were used: a time-delay method from data taken at a single location in the wall (reconstructed phase space) and data taken from ten locations in the wall simultaneously (physical phase space). Three chaotic attractors of the Bloch wall were studied. The time-delay method for some attractors gives a dependence of the correlation dimension on the location at which the data were extracted. This dependence has a specific symmetry with respect to the film in which the Bloch wall resides and a surface (or boundary) effect is seen—the correlation dimension is different at the surfaces of the film. Several scaling regions are found in some cases. The physical phase-space embedding yields a correlation dimension which is an average of the spatial dependence obtained in the time-delay method, but usually this embedding gives a somewhat lower value of the correlation dimension.

Low-temperature expansions are presented for the surface tension and second moment of the density gradient in the three-dimensional axial next-nearest-neighbor Ising model. Padé approximants for the second moment of the density gradient show a definite line of singularities in the space of coupling parameters. The surface tension is shown to be positive along this line. We conclude that our low-temperature expansion is internally consistent to seventh order, that the singularity structure we found is an exact property of the system, and that it is the roughening transition.

Nonlinear-response effects in stochastic resonance are investigated. An analytic solution is obtained in the framework of the adiabatic approximation. It is found that the input signal strength plays an important role in affecting the stochastic resonance behavior. Our analytic predictions fit well both numerical calculations and experimental results.

Centrifuge experimentation (10–1000 m/${\mathrm{s}}^2$) on sand avalanches has shown that (i) grain cohesion and elasticity are negligible; (ii) the avalanche size δθ and the maximum angle of repose ${\mathrm{\theta }}_{\mathit{m}}$ depend on the initial pile density d; and (iii) the pile surface evolves spontaneously towards an asymptotic state after a few avalanches, this state being characterized by a mean surface angle equal to the macroscopic friction angle φ and a mean avalanche size equal to 1.7°. This demonstrates that the density d is a controlling parameter of the avalanche process. These results are compatible with a unique internal friction angle φ with φ=34°; φ corresponds to the value measured with other soil-mechanics devices; however, φ fluctuates within 2° from pile to pile or test to test. These data are also consistent with a dilatancy effect which depends on density, but the existence of at least one other parameter controlling the avalanche size has been also proved.