An exact general solution is presented for a multiply infinite class of irreversible filling or exchange processes on a rectangular lattice. There are many known applications in physics, chemistry, and biology. The dynamics of this class of models follows from the generating function for the number of permutations of n elements with specified cycles. As examples, general solutions are given for random sequential adsorption on a 2×∞ strip with nearest-neighbor exclusion and for a conserved-order dynamics model recently formulated by Privman [Phys. Rev. Lett. 69, 3686 (1992)].

A head-on collision between two solitary waves from opposite directions in a Rayleigh-Bénard convecting fluid is investigated by use of the Poincaré-Lighthill-Kuo method. The fluid system is bounded below by an isothermic plane and above a free deformable surface on which a heat flux is fixed. The results show that, near the transition of the long-wavelength oscillatory instability, the solitary waves emerging from the collision can preserve their original identities to the second order. The phase shifts due to the collision are calculated explicitly.

We performed static and dynamic light scattering measurements for the binary liquid mixture, carbon disulfide–nitromethane, imbedded in a 1000-Å porous glass. The system exhibits strong spatial but weak temporal fluctuations, giving rise to interesting nonergodic behavior. A reasonably sharp transition was detected at a temperature ∼1 K below the bulk ${\mathit{T}}_{\mathit{C}}$, with the system showing critical slowing-down on both sides of the transition. Despite the large pore size (in comparison with the correlation length), the intensity-intensity autocorrelation function is nonexponential with a characteristic relaxation time that does not diverge even at low temperatures, T≪${\mathit{T}}_{\mathit{C}}$. Our observations cannot be explained by the random-field model.

Solid nematic liquids, formed by cross-linking polymer liquid crystals into elastomers, are shown to display novel and complex elasticity. The internal (nematic) direction suffers a barrier to its rotation and this couples to standard elasticity. By considering imposed strains we illuminate this elasticity and also demonstrate an entirely new nematic phase transition. At a critical strain there is a discontinuity of director rotation.

A recently observed stable regime in the form of periodically colliding counterpropagating wave packets (pulses) in an annular convection channel at very small positive overcriticalities is described analytically in terms of coupled Ginzburg-Landau equations. First, the existence of this regime is demonstrated in the framework of the simplest system including only the group-velocity difference, weak gain, and nonlinear dissipative coupling between two modes. In this approximation, the shape of the counterpropagating waves remains indefinite. It is demonstrated that additional dispersive terms, regarded as a small perturbation, provide shaping of the wave packets and also give rise to the deviation of the phase velocity from that for purely linear waves.

The problem of nonlinear pulse propagation in optical fibers near the zero dispersion point is investigated. It is shown, both analytically and numerically, that stable double-humped solitary-wave solutions potentially can be used for transmission. The results clarify previous problems with the existence of single-humped localized solitary waves, i.e., fundamental solitons, which are forbidden in the strict mathematical sense.

We introduce a method based on the hybrid Monte Carlo (HMC) algorithm [S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Phys. Lett. B 195, 216 (1987)] to sample the canonical distribution in systems with a conserved order parameter. We show that the standard HMC method and the one introduced here are special cases of a general class of HMC methods. As an application we compute the scaling function of the interface excess energy for the ${\mathrm{\phi }}^4$ model.

We derive a Fokker-Planck equation for the joint probability density of the displacement and the velocity of a free particle subjected to an exponentially correlated Gaussian force. This equation is solved analytically in the limits t≪τ, t≫τ and for τ=0 (white noise), where τ is the correlation time. The parameters (moments) which determine the joint density are calculated including terms up to order ${\mathit{t}}^2$/${\mathrm{\tau }}^2$ for t≪τ, and up to order τ/t for t≫τ. For t≪τ the marginal distribution of displacements is exactly Gaussian, to the considered order. A Gaussian distribution derived approximately for t≫τ is suggested to be exact, on the basis of independent, exact calculations of low-order moments. For Gaussian white noise, the joint density is obtained exactly and yields a Gaussian distribution of displacements, with the familiar superdiffusive form for the mean-square deviation. The marginal distribution of velocity obeys an exact diffusion equation with a variable diffusion coefficient, for arbitrary τ.

It is shown on a specific example that fractal boundary conditions drastically alter the properties of wave excitations in space. The low-frequency part of the vibration spectrum of a finite-range fractal drum is computed using an analogy between the Helmoltz equation and the diffusion equation. The irregularity of the frontier is found to influence strongly the density of states at low frequency. The fractal perimeter generates a specific screening effect. Very near the frontier, the decrease of the wave form is related directly to the behavior of the harmonic measure. The possibility of localization of the vibrations is qualitatively discussed and we show that localized modes may exist at low frequencies if the geometrical structures possess narrow paths. Possible application of these results to the interpretation of thermal properties of binary glasses is briefly discussed.

We carry out a detailed numerical study of the Cahn-Hilliard equation in two dimensions for phase separation in very large systems and for different values of the area fraction φ. We present results for the scaling function obtained from the pair-correlation function, the structure factor, and the droplet distribution function. We find that dynamical scaling is satisfied at late times for all of the above functions and for different area fractions. We study how the shape of these scaling functions changes with the area fractions and compare these results with available theoretical predictions. We have also analyzed the growth law for the characteristic domain size for various area fractions. Our analysis of the time dependence of various measures for the characteristic length supports a modified Lifshitz-Slyozov law in which the asymptotic-growth-law exponent is 1/3 for all area fractions.

The unbinding of three strings in a plane which interact via short-ranged attractive potentials is studied using both transfer-matrix methods and Monte Carlo techniques. In the symmetric case, for which the two outer strings have the same line tension, the critical behavior depends on the relative flexibility of the inner string in a nonuniversal fashion. In the asymmetric case, the strings unbind in two successsive unbinding transitions at two different temperatures. Renormalization-group arguments predict that these results are also applicable to the adhesion and unbinding of three membranes.

A bivariate nonlinear model perturbed by external white noises is investigated stochastically. Attention is concentrated on the transient properties before the nonequilibrium phase is achieved. Effects of both additive and multiplicative noises are found to weaken stability and to slow down transient processes. The critical exponent describing this slowing-down phenomenon near a noise-induced instability is estimated for various types of noises. Results derived with two versions of stochastic calculus are compared systematically.

We examine a model of rupture propagation on a one-dimensional earthquake fault, focusing both on the role of small length scales in controlling the dynamics and on the mechanisms by which propagating modes are selected. In order to obtain a well-defined continuum theory, we add a small amount of viscous dissipation to a previously studied model in which rupture propagation was controlled by a finite-difference grid spacing. We find that the dynamically selected rupture mode is one in which nearly free slip occurs in a pulse whose speed and shape are determined by a novel selection mechanism.

The time delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension. The minimum necessary global embedding dimension ${\mathit{d}}_{\mathit{E}}$ may still be larger than the actual dimension of the underlying dynamics ${\mathit{d}}_{\mathit{L}}$. The embedding theorem only guarantees that the attractor of the system is fully unfolded using ${\mathit{d}}_{\mathit{E}}$ greater than 2${\mathit{d}}_{\mathit{A}}$, with ${\mathit{d}}_{\mathit{A}}$ the fractal attractor dimension. Using the idea of local false nearest neighbors, we discuss methods for determining the integer-valued ${\mathit{d}}_{\mathit{L}}$.

We present an algorithm for carrying out time-dependent canonical Monte Carlo simulations on a lattice with single occupancy of each site and arbitrary site-site interactions. The method incorporates velocities at each site whose components are proportional to the probability of exchange with each of the neighboring sites. Kinetic and potential energy are exchanged through the choice of trial final velocities for attempted exchanges which are accepted or rejected using the Metropolis algorithm. Caculations are carried out for the two-dimensional Ising model for which the exact partition function is known. Our method reproduces the results of standard Monte Carlo simulations with comparable accuracy, and in addition allows the calculation of dynamical properties previously inaccessible with traditional methods. Results are presented for self-diffusion, interfacial tension, and shear viscosity.

The loop structure generated by the percolation hulls in two dimensions is investigated both for random percolation and for the percolation properties of interacting, diffusing particles using the gradient diffusion method. Scaling forms for the loop distribution function are proposed and verified numerically. The results show that while bonding nearby particles on the original hull reduces its fractal dimension from ${\mathrm{D}}_{\mathrm{H}}$=74 to ${\mathrm{D}}_{\mathrm{H}\prime }$=43, no further change in ${\mathrm{D}}_{\mathrm{H}\prime }$ is observed when additional bonds are placed on the already reduced hull. A similar behavior holds for strongly interacting particles during phase separation on length scales larger than the characteristic droplet size.

We have studied numerically the properties of the logistic map with a single sectional discontinuity at x=${\mathit{x}}_{\mathit{d}}$. We give the main features of these maps over a wide range of ${\mathit{x}}_{\mathit{d}}$, including accumulation points, inverse cascades, bifurcation diagrams, basins of attraction, and a new superimposition rule. We find that the main characteristics of the logistic map with a discontinuity at the origin [T. T. Chia and B. L. Tan, Phys. Rev. A 45, 8441 (1992)], such as the occurrence of inverse cascades, and the validity of rule I, rule II (which are rules for determining whether higher-level cascades exist), and the summation rule, are still retained in these new discontinuous maps, implying that these properties and rules are universal in discontinuous maps. However, there are important differences as well, such as the number of inverse cascades and the types of routes to chaos which may include a period-doubling route, with period doublings occurring at the same values of ${\mathit{a}}_{\mathrm{PD}}$ for different values of ${\mathit{x}}_{\mathit{d}}$, and an ‘‘alternating’’ route. Further, we find that in the chaotic regions of these maps, either the modified summation rule holds or there exists the following period-doubling sequence: 2->4→8→16→32→..., which may also exist in the periodic regions, depending on the values of ${\mathit{x}}_{\mathit{d}}$.

The scaling properties of three nontrivial one-dimensional avalanche models are analyzed. The first two of them are the local limited model with one open, one closed, and with periodic boundary conditions, respectively. A theory for the scaling properties of these models based on the existence of two fundamental length scales, which diverge in the thermodynamic limit, is developed. The third model studied is a trapless version of the nonperiodic local limited model. We find that it is scale invariant. Our theoretical predictions are compared with extensive computer simulations in all three cases.

Recent research on the dynamics of certain fluid-dynamical instabilities shows that when there is a slow invariant manifold subject to fast time-scale instability the dynamics are extremely sensitive to noise. The behavior of such systems can be described in terms of a one-dimensional map, and previous work has shown how the effect of noise can be modeled by a simple adjustment to the map. Here we undertake an in-depth investigation of a particular set of equations, using the methods of stochastic integration. We confirm the prediction of the earlier studies that the noise becomes important when μ‖lnε‖=O(1), where μ is the small time-scale ratio and ε is the noise level. In addition, we present detailed information about the statistics of the solution when the noise is a dominant effect; the analytical results show excellent agreement with numerical simulations.

Galton boards, found in museum exhibits devoted to science and technology, are often used to demonstrate visually the ubiquity of so-called ‘‘laws of probability’’ via an experimental realization of normal distributions. A detailed theoretical study of Galton-board phase-space dynamics and statistical behavior is presented. The study is based on a simple inelastic-collision model employing a particle fall- ing through a spatially periodic lattice of rigid, convex scatterers. We show that such systems exhibit indeterminate behavior through the presence of strange attractors or strange repellers in phase space; nevertheless, we also show that these systems exhibit regular and predictable behavior under specific circumstances. Phase-space strange attractors, periodic attractors, and strange repellers are present in numerical simulations, confirming results anticipated from geometric analysis. The system’s geometry (dictated by lattice geometry and density as well as the direction of gravity) is observed to play a dominant role in stability, phase-flow topology, and statistical observations. Smale horseshoes appear to exist in the low-lattice-density limit and may exist in other regimes. These horseshoes are generated by homoclinic orbits whose existence is dictated by system characteristics. The horseshoes lead directly to deterministic chaos in the system. Strong evidence exists for ergodicity in all attractors. Phase-space complexities are manifested at all observed levels, particularly statistical ones. Consequently, statistical observations are critically dependent upon system details. Under well-defined circumstances, these observations display behavior which does not constitute a realization of the ‘‘laws of probability.’’