Nonequilibrium indentation simulations for two-dimensional crystals composed of up to 1036 800=720×1440 atoms are described. The forces used include smoothly truncated Lennard-Jones force laws, both with and without added embedded-atom contributions typical of copper or nickel. Both low and intermediate temperatures are considered over a wide range of indentor speeds. Typical microhardness yield strengths for these two-dimensional materials, force divided by projected area, exceed 10% of the shear modulus. For the most part these simulations were carried out on the 64-transputer SPRINT computer at Livermore.

The dynamics of the classical XYZ model with uniform interaction is investigated by the recursion method and, in part, by exact analysis. The time evolution is anharmonic for arbitrary N (number of spins); only the cases N=2 and ∞ are completely integrable. For the special (uniaxially symmetric) equivalent-neighbor XXZ model, the nonlinearities in the equations of motion disappear in the limit N→∞, and the spin autocorrelation functions are determined exactly for infinite temperature: The function 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$(t)${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$(t)${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$〉 decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=∞ dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.

We study the dynamics of a system with a nonconserved complex order parameter, following a deep quench. In contrast to the case when the order parameter has a discrete symmetry, we observe an effective value of the dynamical exponent at early times of φ=0.375±0.005. In this regime, finite-size scaling of the scattering function is observed. At later times, we observe a crossover to φ=0.5±0.02. Our results indicate that any renormalization of the kinetic coefficient must be small.

The evolution of an imperfectly mixed system—mimicked in terms of a distribution of lamellae—is studied. Two reactants A and B, initially placed in alternate striations, diffuse and undergo a reaction A+B→2P with intrinsic rate r=${\mathit{k}}_{\mathit{r}}$(${\mathit{c}}_{\mathit{A}}$${\mathit{c}}_{\mathit{B}}$${)}^{\mathrm{\alpha }}$. Simulations, scaling analysis, and space-averaged (fractal) kinetics are used to study the evolution of the system for different values of α and ${\mathit{k}}_{\mathit{r}}$. For α=1 and short times, a model based on the dynamics of reaction for a single lamella with infinite neighbors predicts the overall rate of reaction. For α<2.5, diffusion takes control of the dynamics for moderate to large times, and the kinetic parameters become irrelevant. Under these conditions, critical self-organization determines the behavior of the system, and the spatial structure evolves into a self-similar form that is independent of both ${\mathit{k}}_{\mathit{r}}$ and initial conditions. En route to scaling, the system undergoes two independent transitions: (i) from intrinsic chemical kinetics control to diffusion control, and (ii) from a system with several characteristic lengths to a system with only one characteristic length; these transitions might occur in any order, depending on controlling parameters. A combination of both short- and long-time regimes gives an efficient prediction for the average concentration of reactants for all times.

The standard map has a divided phase space in which two-dimensional regions of stochasticity are isolated by one-dimensional Kolmogorov-Arnold-Moser (KAM) curves that form a barrier to diffusion in action. When two standard maps are coupled together, the two-dimensional KAM surfaces no longer divide the four-dimensional phase space, and particles diffuse slowly along stochastic layers by the process of Arnold diffusion. We compare an analytic calculation of the rate of localized Arnold diffusion with numerically determined rates in regions having rotational and librational KAM curves for a single map, in the weak-coupling limit for which the three-resonance model holds. We then determine the rate of global Arnold diffusion across many cells of the 2π-periodic mapping. The global diffusion rate depends both on the local diffusion rate and on the relative volume occupied by the various stochastically accessible regions in the four-dimensional phase space.

Experimental realization of a Hele-Shaw cell with periodic boundary conditions (PBC’s) is achieved by building an azimuthal cell with two coaxial cylinders separated by a small gap. The development of viscous-fingering patterns formed by a critical binary liquid mixture at very low viscosity contrast has been observed and recorded from the onset of instability to very late stages. Comparison with the experimental results measured in cells which have sidewalls shows that PBC’s yield few differences of results. At the early stage there is no sidewall disturbance so the Fourier transforms show less noise in the low-wave-number modes. Because the annular cell is larger than the earlier cells, it was possible to follow the flow to a very late nonlinear stage where, instead of showing steadily lengthening and broadening fingers, the necks of the longer fingers, crowded by fattening bulbs of the tips of the less-long fingers, constrict until their width is no longer negligible in comparison to the cell gap, at which point the pattern breaks up into a rich variety of bubbles.

We consider the classical Chirikov map as an eigenvalue problem and analyze the implications of its translational symmetry in the momentum coordinate p. The map is then formulated on a discrete lattice of size N×N, and solutions are obtained for the eigenstates of the classical evolution operator, and for the spectrum of eigenvalues as a function of the lattice momentum. A comparison is made with the analogous system defined on a discrete finite quantum lattice. We obtain the semiclassical limit on this lattice and show how it corresponds to a modified classical evolution operator, where δ functions are relaxed into Gaussians and some phases are introduced. These modifications reduce the number of participating eigenstates from ${\mathit{N}}^2$ to N. The possibility is raised that an arbitrary area-preserving map could be quantized in a similar way.

The subtraction technique is a powerful tool for studying nonequilibrium phenomena in many-body systems at short times. Its major limitation lies in the long-time noise due to the Lyapunov divergence of initially close trajectories in phase space. We present a method to compute transport coefficients even when long-time tails are present, with a subtraction technique modified by adding a kind of frictional force field to monitor the phase-space divergence. The price to be paid is to perform an extrapolation of the results to zero field.

We apply the recently developed ladder approximation for three- and four-particle correlation functions to thermodynamic perturbation theory. We calculate the first correction in the perturbative expansion of the pair correlation function and the second correction in the expansion of the free energy of a fluid of particles interacting via a hard-sphere reference potential plus a perturbing term. Both calculated quantities depend on the three- and four-particle correlation functions of the reference system. The results for the free energy are in good overall agreement with computer-simulation data over a wide range of densities, up to the hard-sphere fluid-solid phase transition. The results for the pair correlation function are in good agreement with computer-simulation data except near contact at high densities.

A very simple model is designed to understand dynamical states and bifurcations ranging up to chaos in thermal lens oscillations and associated hot-wire experiments. The present system is governed by partial-derivative equations in the case of bulk liquid and also at a free surface which is a boundary of the liquid. It is reduced to a set of three nonlinear ordinary differential equations with two control parameters to be studied within the framework of the theory of nonlinear dynamical systems. The model may evolve to chaos in three domains of the parameter plane. Metric and dynamical properties (generalized dimensions and entropies, and associated singularity spectra) of a (presumably) strange chaotic attractor produced by the model are determined. The overall good and sometimes striking agreement between the model and previously reported experimental results shows that we have reached a fair understanding of the experimental phenomena, much better than what has been hitherto possible.

We have obtained results for the activation energy in bistable overdamped systems driven by colored noise. Our results cover the region of small diffusion coefficients and small to large correlation times. A scaling relation between the escape rate from one attractor to the other and the diffusion coefficient allows us to obtain results for the activation energy in the limit of vanishingly small diffusion coefficients. Numerical results for both the activation energy and the mean first-passage time are compared with theoretical predictions.

We consider the problem of placing self- and mutually avoiding chains, each formed by M consecutive monomers, on the Cayley and Husimi trees. As particular cases, we could mention the dimer (M=2) and the polymer (M→∞) problems. Defining an activity x for a monomer, we obtain the density ρ(x) of sites occupied by chains in the central region of the trees, as well as the entropy as a function of ρ. On the Bethe lattice (central region of the Cayley tree), we were able to obtain closed expressions for general M. On the Husimi lattice (central region of the Husimi tree), only for the case of dimers were closed expressions found, and the other cases were treated numerically.

We have computed collision rates between passive marker particles subject to the blinking vortex flow, a chaotic flow. The case of low number densities is considered, which is relevant for the later stages of the aggregation processes. The theory pertaining to this case is first reviewed. The computational results indicate that the time-averaged collision frequency achieves an asymptotic value (〈f${\mathrm{〉}}_{\mathrm{\infty }}$) for all number densities that is nearly independent of the initial positions of the particles. The dependence of 〈f${\mathrm{〉}}_{\mathrm{\infty }}$ on the system parameters was found to be similar to the predictions of the gradient theory of Levich $[—Physico-Chemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1962), Sec. 40]; the magnitude of the collision rate was significantly smaller. The deviation, attributed to the structure in the flow, decreases rapidly with increasing number density. The results imply validity of the gradient theory for low number densities for chaotic flows.

The characterization of singular point sets in phase diagrams is surveyed in the light of catastrophe theory. The critical end point and the phase diagram near it, containing the unstable critical point, are particularly examined. Some diagrams of its sections are given, which extend beyond the standard analysis of the butterfly catastrophe. The interplaying phenomena between these points are also briefly discussed.

We have shown that, depending on some critical values of the phase and amplitude, an external periodic resonant driving field can inhibit classical deterministic stochasticity in a bistable Hamiltonian system. Our analysis is based on a coupled oscillator model describing a harmonic oscillator quadratically coupled to a nonlinear oscillator with a double-well potential.

A free-energy model for the inhomogeneous hard-sphere fluid mixture was derived recently [Phys. Rev. Lett. 63, 980 (1989)], which is based on the fundamental geometric measures of the particles. It yields explicit analytic predictions for the direct correlations of the uniform fluid in three dimensions, which are in overall good agreement with extensive Monte Carlo simulation results for the pair and triplet functions. In the present paper, general features of the model are further explicated, with special emphasis on those features enabling a straightforward application to the inhomogeneous hard-disk fluid mixture. In particular, an analytic theory for the structure factors of the hard-disk mixture is derived. The present results should be useful as reference for the statistical-thermodynamic analysis of two-dimensional systems.

We compute the viscosity η of a fluid consisting of a large number of particles, N=108 and 864, as a function of shear rate γ from its maximum and minimum Lyapunov exponents. The calculation is based on an extension of Smale’s pairing rule of Lyapunov exponents for Hamiltonian systems to non-Hamiltonian systems in contact with a heat bath. The numerical values of these maximal Lyapunov exponents as a function of γ are determined using nonequilibrium molecular dynamics (NEMD) computer simulations. The η(γ) computed this way agree with those obtained directly from NEMD within the experimental error of 2% for the triple-point N=108 system. A ${\gamma }^{1/2}$ dependence of η(γ) for large γ is found up to a Péclet number of 5.

A report about experimental studies of the dynamics and interaction of topological defects in the roll structure of electrohydrodynamic convection in nematic liquid crystals is given. It is found that the motion of defects of opposite topological charges towards annihilation has two stages. At large distances they move with a constant velocity that depends mainly linearly on the wave-number mismatch. At a later stage, when the defects come closer, they are accelerated due to attraction. In order to compare these results quantitatively with available theories that are based on the Ginzburg-Landau equation, all coefficients of this equation are measured.

We consider the simplest dynamical model of the Ginzburg-Landau (GL) type with a trivial state that is stable with respect to infinitesimal disturbances but may be triggered into a traveling-wave (TW) state by a finite disturbance. Treating the dispersion coefficients in the GL model as small parameters, we construct a kink solution interpolating between a TW and a trivial state. We find the equilibrium velocity of the kink and demonstrate that it uniquely selects a wave number of the TW. Next we find analytically a stable kink-antikink bound state (a ‘‘soliton’’). In particular, the size of the soliton is found in an explicit form. We also discuss possible implementations of the soliton in particular physical systems.

Small-angle neutron scattering (SANS) is used to study the scattering length density correlations in a dry porous silica glass (Vycor) and in the glass filled with anionic and cationic surfactant molecules dispersed in water. SANS intensity distributions from empty pores of the glass can be satisfactorily analyzed using a standard late stage spinodal decomposition model in which the porous structure is described by a superposition of random density waves. This model, however, agrees with experiment only if one allows for a suitable distribution of wave numbers for the density waves. Micellar formation is observed in Vycor filled with surfactant solutions. We discuss the interpretation of the intermicellar correlations, measured by SANS, within the pores of the Vycor glass.