The theoretical basis of the first-principles molecular dynamics introduced by Car and Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] is investigated. We elucidate how the classical dynamics generated by the Car-Parrinello Lagrangian approximates efficiently the quantum adiabatic evolution of a system and discuss the role played by the spectrum of the eigenvalues of the Hamiltonian of Kohn and Sham [Phys. Rev. 140, A1133 (1965)]. A detailed characterization of the statistical ensemble sampled in the numerical simulation is given. By combining theoretical arguments and numerical results we demonstrate that the motion of the electronic variables is a superposition of a direct drag due to the ions and of high-frequency normal modes. By making a connection with the averaging methods of classical mechanics, we argue that whenever it is possible to get a large separation between the time scales of these modes and the ionic frequencies, the dynamics of the ions closely approximates that resulting from the adiabatic approximation. We introduce simple n-level models, easily amenable to analytic treatment, to add clarity and study the possible mechanisms of broken adiabaticity encountered in the actual calculations.

The invariant density of discrete dynamical systems in the presence of weak Gaussian noise is studied both by means of path integrals and a WKB method. The leading order of the invariant density in the noise strength defines a generalized potential that is a Lyapunov function of the deterministic map. For this generalized potential an efficient functional recursion relation is established and applied to a variety of different maps. The typical singular behavior of the generalized potential in the neighborhood of an unstable fixed point is studied in detail. We show that noise-dependent corrections to the leading order restore the smoothness of the invariant density.

A shifted-frequency perturbation scheme is used to derive algebraic expressions for recurrence periods in the harmonic-mode energies of cubic and quartic Fermi-Pasta-Ulam (FPU) chains. Perturbation-theory solutions for the FPU period have been obtained for arbitrary particle numbers N. Although the convergence properties of the solutions for arbitrary N are not known, explicit evaluations of the solutions at specific values of N have been found to be in good agreement with numerical simulations and soliton theories. The perturbation-theory solutions also provide reliable results in the regime where continuum soliton theories do not apply. A beat mechanism for the superperiod phenomena in cubic FPU chains is also proposed, based on the perturbation-theory solutions. The superperiods predicted by this mechanism are generic in cubic chains but do not appear in quartic chains. This result is in agreement with our numerical simulations. An exceptional superperiod that does occur in quartic, chains with N=7, is accounted for by a resonance mechanism unique to this chain.

Using dynamical mean-field theory, we study the two- three-, and four-point correlations in a one-dimensional lattice gas driven by a uniform electric field. In the zero-temperature limit, the ordering is precluded when the electric field exceeds a critical value. The electric field results in a breaking of reflection symmetry in the four-point configurations.

We show how to estimate the multifractal generating functional of the structure factor of a class of substitutional sequences. These sequences are sequences of two elements, a and b, and are generated by the repetitive application of the rules a→${\mathrm{\sigma }}_1$(a,b) and b→${\mathrm{\sigma }}_2$(a,b), with the σ’s consisting of strings of a’s and b’s. We restrict ourselves to the case in which ${\mathrm{\sigma }}_1$ and ${\mathrm{\sigma }}_2$ each contain R elements. This set includes the case of the Thue-Morse sequence. Subject to a technical assumption, we present a systematic approximation scheme for the multifractal generating functional, in which the lowest-order approximation is the generating functional of an R-scale Cantor set. As a by-product of our analysis, we demonstrate the existence of a discontinuity in the multifractal spectrum of the Thue-Morse sequence, and explain the origin of that discontinuity. We present explicit results for two examples, including the Thue-Morse case, and compare the results of our approximations for the Thue-Morse system with numerical simulations.

Nonrelativistic radiation reaction theory points out the need for incorporating structure in the theory of the electron (since the assumption of a point electron leads to problems with causality). Thus the question arises as to whether other well-known results need to be modified to incorporate structure effects. Here, we show that the Larmor formula is one such result requiring modification.

We analyze chaotic motion in an experiment on a parametrically excited pendulum in terms of unstable periodic orbits. This provides a useful quantitative comparison with results of a faithful numerical simulation. Despite the presence of experimental artifacts, simulation and experiment are in good agreement. The analysis of scaling properties of both chaotic attractors along these lines remains, however, incomplete. Periodic-orbit analysis fails to account for their marginal hyperbolicity, and therefore fails to capture an important qualitative aspect of the chaotic dynamics in both experiment and simulation.

The N-k model is a dilute, k-ary spin glass in which the state of each of the N sites is affected by that site and k of its neighbors. As a function of k for large k, we explicitly compute the number of local minima of the Hamiltonian, the distribution of locally minimal energies and the first two moments of that distribution, and a number of statistical properties of ‘‘downhill’’ walks from random starting positions to local optima on these landscapes, including estimates for their length. We suggest some implications of these results for spin-glass physics and for approximating other landscapes that cannot be modeled using more conventional, quadratically coupled spin glasses.

Various aspects of stochastic resonance are investigated by analog simulation of an electric circuit. The results of the simulation are compared, in detail, with the adiabatic theory. It is found that the adiabatic theory fails in explaining essential features observed in our experiment.

Molecular-dynamics (MD) calculations with six-center Lennard-Jones potentials have been performed to model liquid benzene. Shear and bulk viscosity as well as thermal conductivity were determined for eight different states of liquid benzene. Agreement of experimental values and MD results is surprisingly good for the shear viscosity and the thermal conductivity. Even the strongly increasing shear viscosity of benzene near its triple point could be well reproduced by the MD computations. Bulk viscosity could not be compared with experiment. The calculated values are, however, expected to fall in line with measurable coefficients. In the wider neighborhood of the triple point of benzene, the time-correlation functions of the shear and the bulk viscosity begin to have an extreme form, namely, a decay to nearly zero in a short time range of only 0.3 ps followed by a positive branch of very small amplitude up to about 4 ps. This behavior of the time-correlation functions makes an accurate determination of the viscosities very difficult. MD runs of at least 0.6×${10}^6$ time steps are required for reliable computations. The molecule-number dependence of the results was found to be small, in agreement with previous findings. Systems of 32 molecules seem to be of satisfactory size.

We investigate the conformation of very stiff chains with increasing molecular weight focusing on the onset of the excluded-volume effect. A Flory argument is given for the case in which the shape of monomers has a disklike anisotropy, which causes the excluded-volume effect to set in for shorter chains. A scaling argument determines the exponent associated with the anisotropy in terms of the main exponent that controls the onset of the excluded-volume effect. We suggest a way of viewing the stiff chain as a train of mutually repelling blobs and reanalyze some data in the experiment of Murakami, Norisuye, and Fujita [Macromolecules 13, 345 (1980)]. An extensive Monte Carlo simulation of the persistent self-avoiding walk (PSAW) has been performed on cubic and diamond lattices. We find an extremely gradual crossover of the Flory exponent from the Gaussian value (${\nu }_{\mathit{F}}$=1/2) to the full self-avoiding one (${\nu }_{\mathit{F}}$≊3/5) as the chain becomes longer. Finally, we present an approximate analytic calculation of the attrition rate of an equivalent flight model for the PSAW.

In bistable systems subjected to both periodic and random forcing, the output signal-to-noise ratio may be improved by an increase in the input noise. For the square-wave periodic forcing we extend Kramers’s escape-rate theory to analyze this so-called stochastic phenomenon. Our analysis recovers the adiabatic result, and may be applied to nonadiabatic cases as well.

I describe experiments on ‘‘pulses’’ of convective traveling waves in an annular geometry, using ethanol-water mixtures with moderate negative separation ratio. In a sufficiently uniform cell at constant Rayleigh number, pulses drift in the direction of propagation of the underlying traveling waves, with no long-term change in velocity. The drift velocity increases with increasing distance ε above onset. In contrast with previous observations of motionless pulses, this result is qualitatively consistent with theories based on a subcritical Ginzburg-Landau equation. The pulse shape is also described in detail. The pulses exhibit a noticeable asymmetry, which decreases as ε is increased. I also describe experiments at high ε in which pulse destruction by convective amplification of traveling-wave fluctuations is suppressed by the existence of multiple pulses. In this case, destabilization takes place by an intrinsic mechanism: Above a certain threshold, the pulse simply expands into the rest of the system, accompanied by a large decrease in wave speed.

The first nonlinear state of traveling-wave convection in binary fluids with moderate negative separation ratio in a narrow geometry consists of ‘‘pulses’’: localized patches of convection whose spatial shape is fixed, and that drift at a velocity that depends on the local Rayleigh number. I present the results of two kinds of experiments on traveling-wave pulses. First, I study the behavior of pulses as they drift past narrow, fixed peaks in Rayleigh number. The pulses are sensitive to the Rayleigh number in a spatial domain that extends far ahead of the main body of the pulse. Second, I study collisions between pairs of counterpropagating pulses as a function of the velocity with which they approach one another. At high approach velocity, only one pulse survives the collision. At low approach velocity, a persistent double-peaked structure is formed. Under certain circumstances, this structure can be interpreted as a weakly bound state of two pulses.

The intermittent property is investigated for numerically simulated incompressible three-dimensional fluid turbulence at Taylor microscale Reynolds number ${\mathit{R}}_{\lambda }$∼100, 120, and 180. Cluster statistics applied to energy dissipation and velocity (spatial-) derivative fields characterize several intermittent properties, such as the degrees of concentration and the connectivity of regions of large magnitude. Some power-law behavior with a universal exponent appears in the statistics of the energy-dissipation field. Cluster statistics for the derivative field confirm the well-known experimental result that a higher-order derivative field is more intermittent. Moreover, the longitudinal- and the lateral-derivative velocity fields show some differences in the statistics: The former field has smaller concentration, and its cluster pattern is more spotty than the latter field.

We report a one-parameter family of solutions for the problem of the motion of an interface between a viscous and a nonviscous two-dimensional fluid. The solutions have the interface moving uniformly while the viscous fluid has a nontrivial potential flow. In an alternative interpretation, the existence of gravitational forces in the plane allow the interface to be at rest while the fluid is in motion. This family of solutions is a generalization of a solution reported previously [L. P. Kadanoff, Phys. Rev. Lett. 65, 2986 (1990)]. The solutions presented here are found to be related to the traveling-wave solutions of the ‘‘Harry Dym equation,’’ which is a completely integrable nonlinear evolution equation.

The formulation of generalized transport laws leading to causal hyperbolic hydrodynamical equations gives rise to a critical Mach number ${\mathit{M}}_{\mathit{c}}$ at which a singularity develops in the shock wave layer. This happens in kinetic theory of gases (Grad’s 13 moments method) and extended irreversible thermodynamics as well. In this paper, it is shown that if nonlinear and nonlocal effects are taken into account, ${\mathit{M}}_{\mathit{c}}$ significantly increases. Specifically, nonlocal effects raise ${\mathit{M}}_{\mathit{c}}$ from 2.01 up to 3.03. Further consideration of some nonlinear terms raise ${\mathit{M}}_{\mathit{c}}$ from the latter value up to 4.67.

We have studied the dynamical and equilibrium scaling behaviors of a coarse-grained model for microphase separation of block copolymers. Our numerical study has been carried out in three dimensions. We compute the power-law exponent θ characterizing the equilibrium microdomain size ${\mathit{R}}_{\mathrm{eq}}$ as a function of the molecular weight of the copolymers N as ${\mathit{R}}_{\mathrm{eq}}$∼${\mathit{N}}^{\mathrm{\theta }}$, and conclude that we are able to study the weak-segregation regime, but not the true strong-segregation regime of the model. We have also tested scaling hypotheses connecting the microdomain size and the equilibrium structure factor of the block copolymers.

We use a nonlinear least-squares-fitting program to find an analytic fit to the one-component plasma fluid structure factor S(q,Γ) over the ranges 0≤q≤21.75 and 1≤Γ≤225. The S(q,Γ) function is generated as a table from the solution of the modified hypernetted chain integral equation. The table is fitted with polynomials in q and Γ, using a total of 175 coefficients. The overall fit is good, suggesting that the fitting method might be applied successfully to other distribution functions.

We study steady symmetric lamellar eutectic growth in directional solidification by an extensive analysis of the pertinent boundary integral equation. We find a discrete set of solutions that differ in their average undercoolings. As the wavelength λ increases, the branches coalesce by pairs to form fold singularities above which axisymmetric solutions cease to exist. Front shapes are computed in a wide range of wavelengths, and a systematic comparison with an improved Jackson and Hunt theory [Trans. Metall. Soc. AIME 236, 1129 (1966)] is made. The last one turns out to be accurate in general, for the lowest branch, but does not provide any hint at other branches. In the experimentally relevant parameter range, the front equation reduces to a similarity equation containing two dimensionless parameters σ==${\mathit{d}}_0$l/${\lambda }^2$ and χ==l/${\mathit{l}}_{\mathit{T}}$, where ${\mathit{d}}_0$,l,${\mathit{l}}_{\mathit{T}}$ are the capillary, diffusion, and thermal lengths. We explicitly demonstrate the similarity properties of the pattern. The selected wavelength scales as λ≃ √${\mathit{d}}_0$l f(l/${\mathit{l}}_{\mathit{T}}$). At the minimum undercooling, λ varies with the growth velocity V as ${\mathit{V}}^{\mathrm{-}\mathrm{\beta }}$, where β is an increasing function of V that saturates to about 1/2 at large V. This feature is in agreement with experiments [G. Lesoult, Ann. Chim. Fr. 5, 154 (1980)]. The general scaling of the wavelength allows the exact prediction of a new scaling exponent that is independent of the nature of the selection criterion. This prediction inspires another experimental test of the theory.