We study interacting particle (spin) systems on a lattice under the combined influence of Glauber (spin flip) and simple exchange (Kawasaki) dynamics. We prove that when the conserving exchanges occur on a microscopically fast scale the macroscopic density (magnetization) evolves according to an autonomous nonlinear diffusion-reaction equation. Microscopic fluctuations about the deterministic macroscopic evolution are found explicitly. They grow, with time, to become infinite, when the deterministic solution is unstable.

The Krylov-Bogoliubov method of averaging is applied to the time-dependent quantum anharmonic oscillator. A regular perturbation expansion contains secular terms. The averaging approximation does not, and as a result has a validity over larger time intervals. A new variant of the usual averaging transformation is used and rigorous error bounds are derived. Rigorous averaging methods have been applied extensively to ordinary differential equations but our work appears to be the first generalization to partial differential equations.

Higher-order corrections to the Korteweg-de Vries equation are examined by Hamiltonian methods. Starting from the underlying Hamiltonian systems (e.g., the two-fluid equations in the case of ion-acoustic waves), one finds that the corrected equations have the same Poisson bracket as the Korteweg-de Vries equation at every order. One also finds that the underlying equations become nonlocal at sufficiently high order.

We use the $d=2\mathrm{}$ Ising model as a concrete example to show that truncation of the derivative matrix $T$ to a subspace $S$ of short-range interactions needs some justification, since each row of $T$ contains arbitrarily large elements in the long-range sector. We point out that despite this a small parameter justifying truncation may exist, we show how to find it, and we perturbatively correct the errors due to truncation, all without leaving $S$. The bulk of our analysis carries over to other Monte Carlo renormalization-group studies, particularly the $d=3\mathrm{}$ Ising model.

We extend the region of applicability of phase-space techniques, for the study of quantum systems, by developing an expansion for the Wigner distribution function and correlation functions, in powers of an interaction potential. These results are not restricted to the near-classical regime (expansion in powers of ħ), in contrast to the Wigner-Kirkwood expansion. The relation to other perturbation methods (Lindstedt-Poincaré, Green’s function) is explored.

The steady-state particle density n¯ and the relaxation time τ of homogeneous density fluctuations are calculated for one-dimensional systems in which particles move diffusively and annihilate irreversibly, and steady sources of either single particles (model I) or pairs of neighboring particles (model II) are also present. For small particle-production rates h, we find n¯∼${\mathrm{h}}^{1/\mathrm{\delta }}$ and τ∼${\mathrm{h}}^{\mathrm{-}\mathrm{\Delta }}$ with δ=3, Δ=(2/3) for model I and δ=2, Δ=1 for model II. If we interpret particles as solitons, model II is used to account for some aspects of the experimental data on the photoinduced absorption of trans-(CH${)}_{\mathrm{x}}$.

Successive-subharmonic-oscillation cascade up to the fifth order followed by chaotic emission (parametric instability) is experimentally observed for the first time in an AlGaAs laser diode coupled to an external cavity. This phenomenon is attributed to the nonlinear interaction between the lasing and amplified-spontaneous-emission modes through the refractive-index dependence on carrier density in the active layer. Observed instabilities are successfully reproduced by the numerical simulations based on the generalized van der Pol equations including the asymmetric intermode interaction.

Stochastic quantization is used to obtain upper and lower bounds on the vacuum expectation value of ${\mathrm{\phi }}^{\mathrm{n}}$ in a cutoff λ${\mathrm{\phi }}^{\mathrm{n}}$ scalar theory in d dimensions. These determine the behavior of 〈${\mathrm{\phi }}^{\mathrm{n}}$〉 as the cutoff is removed. The results are given by formulas which depend on whether the values of n and d define a perturbatively superrenormalizable, strictly renormalizable, or nonrenormalizable theory.

We present a new result which shows the transitions from chaos to order and again to chaos as the coupling parameter between two nonlinearly coupled oscillators of a Hamiltonian system is varied continuously from $-\infty \mathrm{to} +\infty$. By exploiting the symmetry of the system, we show that there is no general correspondence between the classical chaotic motion and the Gaussian-orthogonal-ensemble distributions of the energy-level fluctuations of the corresponding quantum system.

We report the first observation of quantized energy levels for a macroscopic variable, namely the phase difference across a current-biased Josephson junction in its zero-voltage state. The position of these energy levels is in quantitative agreement with a quantum mechanical calculation based on parameters of the junction that are measured in the classical regime.

Data are presented on the temperature-dependent escape rates of a macroscopic system, a SQUID containing a highly damped Josephson junction, from its metastable state. These data provide the first quantitative confirmation of recent predictions of the exponential effects of temperature and damping on quantum tunneling in macroscopic systems in contrast to previously reported discrepancies. Predicted quantum corrections to thermally activated escape at high temperatures are not observed.

By the writing of the translational Langevin equation in a rotating frame of reference (1,2,3)′ and the rotational equation in the usual moving frame (1,2,3), it is shown that the statistical correlation between the angular and linear center-of-mass motion of a diffusing asymmetric top can be described naturally and in detail. Computer simulation is used to illustrate the symmetry class and time dependence of many new types of autocorrelation and cross-correlation functions suggested by the structure of these equations.

A system of $N$ identical point vortices on a plane, e.g., a superfluid-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ thin film, is studied with use of elementary quantum mechanics. We find the surprising result that the system obeys fractional or $\theta$ statistics which depends on $N$, where $\theta \mathrm{}\left(N\right)=\frac{\pi }{N}$ or $\frac{\pi }{N}+\pi$. This result arises from the angular momentum contained in the zero-point motion of the vortices.

We show that several schemes for compactification of the extra dimensions in Kaluza-Klein theories are unstable to a quantum gravitational process of barrier penetration: The Universe can tunnel from a state with static extra dimensions to a de Sitter expansion of all dimensions. We estimate the tunneling rate and find that the present state of the Universe is probably long-lived (in good agreement with observation).

A fractal-looking basin boundary for forced periodic motions of a particle in a two-well potential is observed in numerical simulation. The fractal structure seems to be correlated with the appearance of homoclinic orbits in the Poincaré map as calculated by Holmes using the method of Melnikov. Below this critical forcing amplitude the basin boundary appears to be smooth and nonfractal. This example raises questions about predictability in nonchaotic dynamics of nonlinear systems.

This Letter presents a model for tunneling-induced disorder in solids which encompasses both hindered rotors (such as ${\mathrm{H}}_2$ and C${\mathrm{H}}_4$) and hydrogen-bonded ferroelectrics (such as K${\mathrm{H}}_2$P${\mathrm{O}}_4$). The model indicates that while the interaction between tunneling systems decreases tunneling, coupling to phonons actually increases tunneling.

A recent theory of two-dimensional percolation predicts critical singularities identical to those appropriate for the associated dilute Ising model. This implies the absence of a separate universality class for two-dimensional percolation processes. Novel numerical and series-expansion studies are presented in support of this unexpected result.

We show that for a pure gauge theory in a general curved space-time, the effective value of the gauge coupling constant can become small as a result of high curvature.

A simple computational technique related to Feynman path integrals and linear control systems is derived for the dynamical properties of $N$-level model quantum mechanical systems in arbitrary classical fields: periodic, pulses, or sequences of pulses. Since the Hamiltonian is not required to be Hermitian, the effects of natural lifetimes and other relaxation parameters can be determined. The time-dependent Hamiltonian is approximated by a sequence of time-independent Hamiltonians ${\mathrm{H}}^{\mathrm{}\left(n\right)\mathrm{}}$.

Commensurate-incommensurate transitions in uniaxial two-dimensional systems are examined in the presence of impurities that interact with domain walls. Exact Bethe-Ansatz calculations, tested by numerical studies, indicate that randomness is relevant and leads to new critical behavior—a discontinuity in the specific heat and divergence of domain size with an exponent of 1. The results are interpreted in terms of the behavior of a single interface. Global phase diagrams are presented and experimental implications are discussed.