By means of Monte Carlo simulations, we examine the behavior of random packings of hard ellipses formed by pouring into a two-dimensional container. The particles pack so that their semimajor axes are preferentially aligned with the horizontal, and their orientational alignment increases with increasing aspect ratio. This self-organizational effect is accompanied by a corresponding reduction in the translational order of the centers of masses of the particles. For sufficiently elongated particles no translational order is present: The packings are amorphous, but still exhibit long-range orientational order. We call this state a ‘‘nematic glass.’’ We find that the orientational order also increases as the rate of deposition is reduced. The orientational order decreases if the packings are shaken. The behavior of the packings can be explained heuristically as being the result of competition between two local-potential-energy-minimization processes. The relevance of our results to materials science and petroleum engineering is discussed.

We describe a method for noise reduction in chaotic systems that is based on projection of the set of points comprising an embedded noisy orbit in ${\mathit{openR}}^{\mathit{d}}$ toward a finite patchwork of best-fit local approximations to an m-dimensional surface M’⊂${\mathit{openR}}^{\mathit{d}}$, m≤d. We generate the orbits by the delay coordinate construction of Ruelle and Takens [N. H. Packard et al., Phys. Rev. Lett. 45, 712 (1980); F. Takens, in Dynamical Systems and Turbulence, Warwick, 1980, edited by D. A. Rand and L.-S. Young (Springer, Berlin, 1981)] from time series v(t), which in an experimental situation we would assume to have come, together with additional high-dimensional background noise, from an underlying dynamical system ${\mathit{f}}^{\mathit{t}}$: M→M existing on some low m-dimensional manifold M. The surface M’ in ${\mathit{openR}}^{\mathit{d}}$ is the assumed embedded image of M. We give results of systematic studies of linear (tangent plane) projection schemes. We describe in detail the basic algorithm for implementing these schemes. We apply the algorithm iteratively to known map and flow time series to which white noise has been added. In controlled studies, we measure the signal-to-noise ratio improvements, iterating ${\mathit{n}}_{\mathit{M}}$ times until a stable maximum ${\mathrm{\delta }}_{\mathit{M}}$ is achieved. We present extensive results for ${\mathrm{\delta }}_{\mathit{M}}$ and ${\mathit{n}}_{\mathit{M}}$ for a wide range of values of embedding trial dimension d, projection dimension k, number of nearest-neighbor points for local approximation ν, embedding delay Δ, sampling interval ΔT, initial noise amplitude scrN, and trajectory length N.

We give results for very low and very high noise amplitudes 0%≤scrN≤100%. We develop an empirical method for estimating the initial noise level for a given experimental time series, and for the optimal choice of algorithm parameters to achieve peak reduction. We present interesting results of application of the noise-reduction algorithm to a chaotic time series produced from a periodically driven magnetoelastic ribbon experiment on the control of chaos. Two noteworthy elements of the noise-reduction method we describe result in certain stabilizing and efficiency features. The first is our use of a physical replacement time series, which is a unique scalar time series with the property that its corresponding delay coordinate construction data state vector time series in ${\mathit{openR}}^{\mathit{d}}$ is optimally close to the noise-reduced replacement vector time series generated by the projection. The second is the introduction of a ‘‘measure-ordered’’ cover, which produces notable improvement in reliability, control, and computational efficiency of the whole algorithm.

Selection formulas for the wavelength and frequency of uniformly rotating spiral waves are derived for the general class of three-variable models of single-diffusive excitable-oscillatory media and applied to make quantitative predictions that could be experimentally tested for the model proposed by Sales, Turner, and Maple [Surf. Sci. 114, 381 (1982)] to describe the oscillatory oxidation of CO over polycrystalline metal catalysts.

We describe a simple and intuitive model that relates the catastrophe theory unfolding of a double-well potential with the unfolding of a bimodal map characterizing the dynamics of an autonomous chaotic van der Pol–Duffing oscillator. The construction of the model was facilitated by developing an analogy with the stochastic dynamics of a particle in a double-well potential. Experimental evidence supporting the model is given.

Results of a bifurcation analysis are given for the model of a Van der Pol–Duffing autonomous electronic oscillator. The oscillator is described by three ordinary differential equations and consists of a RC oscillator resistively coupled to an LC oscillator. The steady-state problem is described by the unfolding of the quartic potential F=1/4${\mathit{X}}^4$-1/2α${\mathit{X}}^2$+μX, giving rise to the elementary cusp catastrophe. We show how the bifurcation diagram evolves with μ and recover a ‘‘cross-shaped diagram’’ reminiscent of the one obtained by Boissonade and De Kepper for the Belousov-Zhabotinskii chemical system [J. Phys. Chem. 84, 501 (1980)]. We also show that nonzero values of μ result in coexisting attractors with different dynamics. Specifically, we show a limit cycle attractor in one potential well coexisting with a chaotic attractor in the other well.

We present a computational method to determine if an observed time series possesses structure statistically distinguishable from high-dimensional linearly correlated noise, possibly with a nonwhite spectrum. This method should be useful in identifying deterministic chaos in natural signals with broadband power spectra, and is capable of distinguishing between chaos and a random process that has the same power spectrum. The method compares nonlinear predictability of the given data to an ensemble of random control data sets. A nonparametric statistic is explored that permits a hypothesis testing approach. The algorithm can detect underlying deterministic chaos in a time series contaminated by additive random noise with identical power spectrum at signal to noise ratios as low as 3 dB. With less noise, this method can also be used to get good estimates of the parameters (the embedding dimension and the time delay) needed to perform the standard phase-space reconstruction of a chaotic time series.

The conformation and scaling properties of two-dimensional randomly triangulated self-avoiding surfaces embedded in three-dimensional space are studied using Monte Carlo methods. Results for the fractal dimension ${\mathit{d}}_{\mathit{f}}$ and the spectral dimension ${\mathit{d}}_{\mathit{s}}$, as well as the ratios of the eigenvalues of the moment of inertia tensor (which characterize the mean shapes of the surfaces), are presented. It is shown that these surfaces belong to the self-avoiding branched polymer universality class.

Three simple models of steady-state crack propagation are examined in a search for clues about the nature of dynamic velocity-selection mechanisms in these systems. The first is a one-dimensional model with stick-slip friction which includes, as a special case, a model of an earthquake fault. The conclusion here is that velocity-weakening slipping friction generally causes the crack to accelerate to the limiting wave speed independent of loading strength. The second model is also one dimensional, but the dissipation mechanism is the analog of a Kelvin viscosity. In this case, steady-state solutions at large applied stresses exhibit oscillating crack-opening displacements and propagate at speeds comparable to or higher than the nominal wave speed. The third model is the two-dimensional analog of the second. Its qualitative behavior turns out to be essentially the same as in the one-dimensional version.

We report on the spatial distribution of particles in the reaction A+B→0. For the spatial dimension d≤4, this process exhibits anomalously slow kinetics which stems from the formation of a mosaic of continuously growing domains which contain only one of the two species. We investigate the temporal evolution of the distribution of domain sizes, as well as the distribution of interparticle distances between closest-neighbor particles, both between the same and opposite species. Our results are considerably richer than might at first be expected. The average distance between closest-neighbor AB pairs scales differently than the corresponding distance between same-species pairs. The full distribution of AA separations is found to reflect the competing influences of these two length scales. Many of our observations can be accounted for in terms of simple scaling arguments. Rather surprisingly, many of our results are drastically altered if one of the species is immobile. The spatial distribution of the immobile reactant exhibits a self-similar character, leading to complex behavior for the moments of the interparticle distance distribution.

The spatial-localization properties of general lattice distributions (e.g., charge distribution of one-particle eigenstates) are studied. We have introduced a localization quantity, the structural entropy characteristic of the decay form (shape) of the distribution. We have found that there exists a nontrivial relation between the structural entropy and the well-known delocalization index (participation ratio) where the latter parameter measures the spatial extension of the distribution. This relation is universal in the sense that it is independent of the geometrical arrangement of the atomic network, of the lattice constant, and of the size of the system, and is fully determined by the decay form of the eigenstates and the dimension d of the lattice. Based on this relation we have developed a classification scheme for the characterization of the shape and extension of the one-particle states, which makes possible the identification of various decay forms (Gaussian, exponential, power law, etc.) in numerical calculations even on finite systems. The implementation of our method in already existing program systems is particularly easy. Besides presenting the theoretical background we demonstrate its applicability on some numerical calculations.

The importance of periodic orbits, both elliptic and hyperbolic, in analyzing driven quantum systems is now well established. We present a detailed analysis of the quantum mechanics in the vicinity of these orbits for both a piecewise-linear and the original standard maps. We begin by constructing effective Hamiltonian operators valid locally near nondegenerate periodic orbits. Our general formula allows for the calculation of the quasienergy spectrum near elliptic oribits as well as of the strength of scarring on hyperbolic orbits. Preliminary results on the dependence of this measure of scarring on the instability of the classical orbit and on ħ are also presented.

Rapid temperature quenches have long been known to produce metastable thermodynamic phases. We analyze a Landau-Ginzburg model of front propagation and show that metastable phases can arise via a transition that splits the front separating the disordered from the stable phase into one front between the disordered and the metastable phases and another between the metastable and the stable phases. For systems described by a single nonconserved order parameter, the splitting transition is continuous with no hysteresis as the control parameter is varied. For two order parameters, the transition can be continuous, hysteretic, or require a finite-amplitude perturbation. We also discuss briefly applications to pattern-forming systems, where the pattern formed behind a propagating front may change discontinuously as the front velocity is increased.

We study a (generalized) globally coupled system whose elements are two-dimensional chaotic maps, and find clear evidence of nonstatistical behavior: the mean-square deviation (MSD) of both components of the mean field saturate with respect to an increase in the number of coupled elements, N, after a critical value of N is reached, and their distributions are clearly non-Gaussian. We also find that the power spectrum of both components of the mean field display well-defined peaks, indicating a subtle coherence among different elements, even in the ‘‘turbulent’’ phase. This system is a higher-dimensional example of coupled maps, and its study confirms that the phenomena observed in a wide class of coupled one-dimensional maps (and also in an example of coupled complex maps) are present here as well. This gives more evidence to support that such nonstatistical behavior is probably generic in globally coupled systems. We also investigate the influence of parametric fluctuations on the MSD and power spectra, and find that noise restores the statistical behavior, after a critical value of the number of coupled elements is reached.

We study analytically and numerically modulational instabilities in discrete nonlinear chains, taking the discrete Klein-Gordon model as an example. We show that discreteness can drastically change the conditions for modulational instability; e.g., at small wave numbers a nonlinear carrier wave is unstable to all possible modulations of its amplitude if the wave amplitude exceeds a certain threshold value. Numerical simulations show the validity of the analytical approach for the initial stage of the time evolution, provided that the harmonics generated by the nonlinear terms are considered. The long-term evolution exhibits chaoticlike states.

A freely rotating linear chain, formed by N (N≥2) atoms with N-1 ‘‘bonds’’ of fixed lengths, is studied in three spatial dimensions. The classical (c) theory of that constrained system is formulated in terms of the classical transverse momentum -${\mathbf{a}}_{\mathit{j},}$c and angular momentum ${\mathrm{l}}_{\mathit{j},}$c associated to the jth ‘‘bond’’ (j=1,...,N-1). The classical Poisson brackets of the Cartesian components of -${\mathbf{a}}_{\mathit{j},}$c and ${\mathrm{l}}_{\mathit{j},}$c are shown to close an algebra. The quantization of the chain in spherical polar coordinates is carried out. The resulting ‘‘curved-space’’ quantization yields modified angular momenta ${\mathrm{l}}_{\mathit{j}}$. Quantum-mechanical transverse momenta (${\mathbf{e}}_{\mathit{j}}$) are constructed. The commutators of the Cartesian components of ${\mathbf{e}}_{\mathit{j}}$ and ${\mathrm{l}}_{\mathit{j}}$ satisfy a closed Lie algebra, formally similar to the classical one for Poisson brackets. Using ${\mathbf{e}}_{\mathit{j}}$’s and ${\mathrm{l}}_{\mathit{j}}$’s, the quantum theory is shown to be consistent by itself and, via the correspondence principle, with the classical one. Several properties of ${\mathbf{e}}_{\mathit{j}}$ and the modified ${\mathrm{l}}_{\mathit{j}}$ are given: some sets of eigenfunctions (modified spherical harmonics, etc.) and uncertainty relations. As an example, the case of N=3 atoms in two spatial dimensions is worked out. The peculiar properties of the chain regarding distinguishability at the quantum level play an important role in justifying the absence of a ‘‘Boltzmann counting’’ factor [(N-1)!${]}^{\mathrm{-}1}$ in its classical statistical distribution. The physical limitations and the methodological virtues of the model at the classical and quantum levels, and its relationship to previous works by different authors, are discussed.

The long-wavelength properties of the Kuramoto-Sivashinsky equation are studied in 2+1 dimensions using numerical and analytic techniques. It is shown that this equation is not in the universality class of the Kardar-Parisi-Zhang model. Its roughening exponents are (up to logarithmic corrections) like those of the free-field theory, with dimension 2 being the marginal dimension for roughening. Assuming that the solution has logarithmic corrections, we derive a scaling relation for the exponents of the logarithmic terms. This solution is consistent order by order with the Dyson-Wyld diagrams. We explain why previous renormalization-group treatments failed.

We present an algorithm for calculating the thermodynamic properties of a system of nonrelativistic bosons on a three-dimensional spatial lattice. The method, which maps the three-dimensional quantum system onto a four-dimensional classical system, uses Monte Carlo sampling of configurations in either the canonical or the grand canonical ensemble. Our procedure is applicable to any system of lattice bosons with arbitrary short-range interactions. We test the algorithm by computing the temperature dependence of the energy, the heat capacity, and the condensate fraction of the free Bose gas.

We analyze the irreversible behavior of a quantum harmonic oscillator immersed in an arbitrary thermal reservoir. For the fully coupled (FC) model, i.e., the coupling between subsystems linear in the coordinate of the oscillator, we construct a generalized master equation for the reduced density operator of the oscillator. We apply the rotating-wave approximation (RWA) and investigate its connection with the FC master equation. By means of the Wigner distribution, we obtain the corresponding Fokker-Planck equations in the semiclassical representation. We find that, when the RWA is not imposed, the correct classical limit is obtained (Kramer’s equation). Then, we establish the conditions for the equivalence between the FC Fokker-Planck equation and that obtained in the RWA. Quantum maps for a kicked oscillator immersed in a heat bath are obtained in both cases. These maps are studied in the classical and semiclassical limits, and it is shown that they coincide for low kicking frequency compared to the damping rate.

Electric analog circuits are used to simulate stochastic resonance (SR) systems. It is found that the signal-to-noise ratio (SNR) obtained by the SR device can exceed the upper limit of the SNR obtained by the optimal linear filter. By applying two SR devices successively, the SNR obtained by the SR devices may be about eight times larger than that of the optimal linear filter.

A bistable system is simulated by an electric circuit. The system is driven by an informational aperiodic signal and a noise force. Under the stochastic resonance condition, the portion of information received by the bistable system can be greatly enhanced.