What is the difference between randomness and chaos ? Although one can define randomness and one can define chaos, one cannot easily assess the difference in a practical situation. Here we compare the results of these two antipodal approaches on a specific example. Specifically, we study how well the logistic map in its chaotic regime can be used as quasirandom number generator by calculating pertinent properties of a well-known random process: invasion percolation. Only if λ>${\lambda }_1^{\mathrm{*}}$ (the first reverse bifurcation point) is a smooth extrapolation in system size possible, and percolation exponents are retrieved. If λ≠1, a sequential filling of the lattice with the random numbers generates a measurable anisotropy in the growth sequence of the clusters, due to short-range correlations.

We have developed a renormalization transformation, based on the existence of higher-order nonlinear resonances in the double-resonance model, that gives good predictions for the extension of the wave function in that system due to nonlinear resonance overlap. The double-resonance model describes the qualitative behavior, in local regions of the Hilbert space, of many quantum systems with two degrees of freedom whose dynamics is described by a nonlinear Hamiltonian but a linear Schrödinger equation.

We investigate the dynamics of a spin-1/2 particle in a constant magnetic field driven by two transverse magnetic noise fields. These noise fields are exponentially correlated in time. By introducing suitable marginal averages and suitable expansions of these averages it is shown that the problem reduces to the solution of two independent tridiagonally coupled vector recurrence relations with vectors of dimension 3. One of these relations describes the decay of the spin component parallel to the constant field, the other the decay of the spin component transverse to the constant field. The equations are solved by a direct time integration and by determining the eigenvalues. It is shown that the Green’s functions are determined by ordinary continued fractions. These continued fractions agree with those obtained by Shibata and Sato and are thus rederived by a completely different method. The long-time dependence is governed by the eigenvalue with the lowest real part and by its weight. The eigenvalues are found by determining the poles of the Green’s function, the weights of the eigenvalues by evaluating the Green’s function near the poles. The relaxation times ${\mathit{T}}_1$ and ${\mathit{T}}_2$ as well as a frequency shift of the perpendicular component are thus obtained. For certain parameters a single decay constant is not sufficient.

The dynamical properties of two measures to probe ergodic behavior in Hamiltonian systems with a large number of degrees of freedom are investigated. The measures, namely, the energy metric d(t) and the fluctuation metric Ω(t) are both based on the time-averaged energies of the individual particles comprising the system. The energy metric d(t) is obtained by following the dynamics of the system using two independent configurations, whereas Ω(t) is expressed in terms of the properties of a single trajectory. Both measures obey a dynamical scaling law for long but finite times. The scaling law for d(t), which was previously established numerically, and for Ω(t) is derived for systems that are effectively ergodic. The scaling function suggests that the configuration space is explored by a ‘‘diffusive’’ process in the space of the variables used to construct d(t) and Ω(t). Furthermore, a single parameter, namely ${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\Omega }}$, characterizes the time scales needed for effective ergodicity to be obtained. These ideas are used to study the behavior of supercooled and glassy states of soft-sphere binary alloys using microcanonical molecular dynamics. The scaling forms for d(t) and Ω(t) are explicitly demonstrated. The temperature dependence of the numerically computed diffusion constants (${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\Omega }}$) reveals that the nature of phase-space exploration changes both near the fluid-solid transition and near the liquid-to-glass transition. The changes in ${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\Omega }}$ with temperature are well described by a Vogel-Fulcher equation close to the glass transition temperature. In addition, the distribution of the time-averaged individual particle energies P(e;t), moments of which are related to Ω(t), is shown to be broad in the glassy state. We argue that the time dependence of P(e;t) can be used to probe local structural relaxations in glasses.

The inverse profile equation for the zero-thickness membrane interface model is derived from the Schrödinger equation with an arbitrary external time-dependent potential.

Exact enumeration data have been analyzed by the partial-differential-approximation method for the adsorption (special) transition in self-avoiding walks attached to penetrable and impenetrable surfaces. Estimates of the crossover exponent cphi are consistent with 0.28±0.05 (D=2) and 0.40±0.01 (D=3) for the penetrable surface, and 0.51±0.04 (D=2) and 0.54±0.07 (D=3) for the impenetrable surface.

We study superlinear diffusion in a layered medium containing random velocity fields, where the mean-squared displacement grows as 〈${\mathit{x}}^2$(t)〉∼${\mathit{t}}^{\mathrm{\alpha }}$ with α>1 [S. Redner, Physica D 38, 287 (1989)]. For a two-dimensional system with preassigned random velocities in the longitudinal x direction and with diffusional motion in the transversal direction, we determine exactly the asymptotic behavior of the first three nontrivial moments ${\mathit{M}}_{2\mathit{m}}$=〈${\mathit{x}}^{2\mathit{m}}$(t)〉/〈${\mathit{x}}^2$(t)${\mathrm{〉}}^{\mathit{m}}$ of the displacement. Furthermore, we succeed in relating the diffusional problem to the one-dimensional trapping problem. We then are in a position to analyze the scaling form of the propagator P(x,t)∼${\mathit{t}}^{\mathrm{-}3/4}$f(${\mathit{x}}^{3/4}$), where the function f(z) obeys a complicated stretched exponential behavior. We also generalize the problem to transverse motion on fractals and ultrametric spaces that leads to α values that interpolate between 1 and 2. We support our theoretical analytical results by simulation calculations.

In continuation of a previous paper on the static properties of the Widom-Rowlinson (WR) model mixture, we determine the thermal transport coefficients of that model by molecular-dynamics (MD) calculations and kinetic theory. The MD computations indicate that good accuracy can be obtained for the diffusion coefficient, the bulk and the shear viscosities, as well as for the thermal conductivity in a large region of densities. In contrast, the determination of the Soret coefficient poses problems and is planned to be treated in a separate publication. Comparison of the MD-generated transport coefficients with the kinetic theoretical results show remarkably good agreement over a range of densities that even includes the phase separation region. One main reason for this good agreement is certainly the accurate information for the thermodynamic properties of the WR model used in the kinetic theory expressions.

We consider the rf superconducting quantum interference device above its homoclinic threshold. The effects of weak additive (i.e., Langevin) noise on the dynamics of the system are analyzed from the standpoint of the effects on the chaotic attractors and the maximal Liapunov exponents that characterize the system in this regime. It is seen that noise has a ‘‘smoothing’’ effect on chaotic attractors. On the other hand, the injection of noise can lead to a change in sign of the Liapunov exponent that characterizes a periodic point in its absence, leading to ‘‘noise-induced chaos.’’ We also consider the cases of additive fluctuations that manifest themselves as a fluctuating dc driving term, and multiplicative fluctuations (at initial times) in the nonlinearity parameter. In these cases, we study the motion of the system, in the mean, by averaging over numerous realizations of the fluctuating driving term. Depending on the strength of the fluctuations, one obtains mixtures of periodic and chaotic motion in the multiplicative-noise case.

The maximum-entropy formalism provides an exponential ansatz, ρ¯, for the phase-space distribution that can be used in the information-theoretic entropy functional to calculate nonequilibrium thermodynamic potentials. ρ¯, like the Grad function, gives a finite number of moments exactly. If ρ¯ is used to derive a Gibbs equation, the thermodynamic pressure is not in general one-third the trace of the momentum flux, as commonly assumed phenomenologically. One can modify the exponential ansatz by adding terms. Coefficients in these terms satisfy physical conditions such as requirements that thermodynamic pressure and entropy flux have their classical forms. These conditions alter the thermodynamic forces. However, in terms of the new forces, one can modify the projection operator of Grabert to derive a set of nonlinear extended thermodynamic kinetic equations exhibiting Onsager symmetry. Depending on conditions imposed, one has statistical bases for several equivalent forms of nonlinear extended thermodynamics.

We show that a solid-state device, the long Josephson junction, behaves very much like other turbulent systems exhibiting regimes of weak and hard turbulence. Because of the discrete nature of our simulations, we conclude that while a large number of active degrees of freedom are necessary for the existence of turbulence, their number can indeed be small. We show that the transition to the turbulent state is intimately related to the spatiotemporal symmetry breaking and to the existence and breakdown of spontaneous pattern formation in the system.

Analytic expressions for the critical percolation density in the continuum are derived using the direct-connectedness expansion method. Demonstrating a systematic application of this method for systems of permeable cubes or spheres, it is concluded that the present expansion is quite practical for the derivation of critical parameters of continuum systems. Correspondingly we also utilize the method to find the percolation threshold of the so-far unstudied continuum systems of oriented hypercubes. The results obtained are in excellent agreement with Monte Carlo simulations.

A numerical technique for the calculation of unstable periodic orbits of chaotic maps in the complex plane is presented. Applying this technique to the complex Hénon map we show that it can find all the $2^{\mathit{p}}$ real and complex periodic orbits of any given order p. The real periodic orbits coincide with those obtained by a similar algorithm for the ordinary Hénon map that we have proposed earlier, and thus verify its completeness. The method provides a new definition of symbolic dynamics for the Hénon map, which holds for both the real and complex periodic orbits. It provides a computational framework that applies to all limits of this map on a common footing, unlike the conventional techniques in which one needs different algorithms for calculations involving strange attractors, strange repellers, and various types of Julia sets.

The quantized standard map (kicked rotor) is coupled to macroscopic systems acting as measuring devices for the probability distribution of the action variable or some of its moments. As a result of such measurements, which are continuous in time but have a limited time resolution, localization of the action variable is destroyed in sufficiently long time scales and replaced by diffusion. The diffusion constant, in general, differs from that of the classical chaotic diffusion and depends on the measurement performed.

The complex Ginzburg-Landau equation with weak noise, the normal form of the amplitude equation for the order parameter in a spatially distributed system undergoing a continuous Hopf bifurcation, is solved in certain limits for its time-independent probability distribution, which governs the steady state in one spatial dimension. The method used consists of solving the Hamilton-Jacobi equation of the nonequilibrium potential associated with the steady-state distribution. The solution is obtained in the limit of weak spatial diffusion of the order parameter. The nonequilibrium potential serves as a Lyapunov functional for the order-parameter field. We use our result to discuss the Newell-Kuramoto instability and the Eckhaus-Benjamin-Feir instability in one spatial dimension, and to calculate potential barriers of the saddles separating plane-wave attractors. The latter ones provide us with a global measure of stability for these attractors.

Simple two- and three-dimensional models for droplet drift and coalescence have been explored. In these models droplets of equal size are injected randomly along the z=0 edge or z=0 surface of two-dimensional (x-z) or three-dimensional (x-y-z) substrates. The droplets move in the z direction with a velocity V(r) given by V(r)∼${\mathit{r}}^{\mathrm{\eta }}$, where r is the droplet radius. If two droplets contact, they are coalesced with conservation of their mass and center of mass, and the combined droplet continues to move with a velocity given by V(R)∼${\mathit{R}}^{\mathrm{\eta }}$, where R is the radius of the combined droplet. Computer simulations indicated that the asymptotic values of mean droplet size S(z), number density of droplets N(z), and mass density M(z) at a distance z from the injection line or surface are given by S(z)∼${\mathit{z}}^{\mathrm{\omega }}$, N(z)∼${\mathit{z}}^{\mathrm{-}\mathrm{\epsilon }}$, and M(z)∼${\mathit{z}}^{\mathrm{-}\gamma }$, where the exponents z, ε, and γ have values of ω=D/(D+η+1-d), ε=(D+η)/(D+η+1-d), and γ=η/(D+η+1-d), respectively, where D is the droplet dimensionality and d is the dimensionality of the substrate. A simple scaling model indicates that these results are exact, that the mean droplet size S(t) for the entire system grows as ${\mathit{t}}^{\mathrm{\zeta }}$, and that the mass-weighted mean distance Z(t) grows as ${\mathit{t}}^{\mathrm{\theta }}$, where ζ=D/(D-d+1) and θ=1+η/(D-d+1).

Fluid flows in a two-dimensional channel with atomically rough walls were studied by molecular-dynamics simulations. Both sinusoidally and randomly roughened walls with various roughness amplitudes A were used. It was found that the no-slip condition arises when the molecular mean free path l is comparable with A. Therefore the ratio l/A should be employed to validate the no-slip condition, rather than the traditionally used ratio of the mean free path to the channel or tube diameter (Knudsen number).

Steady and oscillatory convection in a binary fluid mixture heated from below is considered. Stress-free and fixed temperature and concentration boundary conditions are used at the top and bottom, with periodic boundary conditions in the horizontal. A minimal Galerkin truncation is constructed such that the local branching behavior near both the steady state and Hopf bifurcations is correctly determined. To do this, modes generated through fourth order in perturbation theory must be retained. The properties of the steady states and standing, traveling, and modulated traveling waves in the resulting system are systematically analyzed. Broad agreement with the results of an analysis of a codimension-two bifurcation with O(2) symmetry is found.

We propose a variational principle applicable to the heat-transport equations derived some years ago by Guyer and Krumhansl [Phys. Rev. 133, 1411 (1964); 148, 766 (1966); 148, 778 (1966)] to describe heat-wave propagation in dielectric crystals at low temperature. The variational principle appears as a generalization of Prigogine’s minimum-entropy production criterion. The analysis is based on extended irreversible thermodynamics, which is used to derive the Guyer-Krumhansl equations from macroscopic arguments.

In a three-phase equilibrium system, the middle phase either wets or does not wet the interface between the upper and lower phases. In this work, we perform a systematic experimental search for wetting transitions between wetting and nonwetting regimes in three-component water–n-alkane–${\mathrm{C}}_{\mathit{i}}$${\mathrm{E}}_{\mathit{j}}$ systems, where ${\mathrm{C}}_{\mathit{i}}$${\mathrm{E}}_{\mathit{j}}$ denotes the surfactant polyoxyethylene alcohol ${\mathrm{C}}_{\mathit{i}}$${\mathrm{H}}_{2\mathit{i}+1\mathrm{}}$(${\mathrm{OC}}_2$${\mathrm{H}}_4$${)}_{\mathit{j}}$OH. It is found that the water–n-hexadecane–${\mathrm{C}}_6$${\mathrm{E}}_2$ and water–n-octadecane–${\mathrm{C}}_6$${\mathrm{E}}_2$ systems exhibit a wetting transition lying at least 10° C below the upper critical temperature, and one system, water–n-tetradecane–${\mathrm{C}}_6$${\mathrm{E}}_2$, exhibits a wetting transition lying 4.3° C below the upper critical temperature. Measurements of the contact angles as a function of temperature indicate that the wetting transition is of first order.