For the (1+1)-dimensional Klein-Gordon equation called the ${\phi }^4$ model, there is a known asymptotic series formally representing a ‘‘breather’’ (a real-valued solution that is localized in space and periodic in time) in the limit of small amplitude and frequency just below that of spatially uniform infinitesimal oscillations. We show that even though this expansion is valid to all orders, ${\phi }^4$ theory admits no true breathers in this limit. Instead, what appear in many physical contexts are approximate breathers that slowly radiate their energy to x-±∞. We calculate this radiation rate, which lies beyond all orders in the asymptotic expansion.

In the past the universal features of the breakup of the last Kolmogorov-Arnol’d-Moser (KAM) torus has been studied by use of renormalization theory. In this case, the renormalization equations converge to a fixed point representing a map with nonsmooth noble circle. The work described in this paper strongly suggests that the breakup of any KAM torus with arbitrary winding number can be described by chaotic renormalization equations which converge on a strange attractor. The properties of this attractor are global scaling exponents characterizing the breakup of almost all KAM tori. This work unifies the quasiperiodic transitions to chaos in Hamiltonian and dissipative systems.

Destabilization of the period-doubling scenario in Hamiltonian systems due to noise is studied. A renormalization picture for stochastic area-preserving maps is introduced and a new universal characteristic number ${\gamma }_{\mathrm{H}=\mathrm{17.017.}\mathrm{}..}$ for the noise amplitude is found. This factor governs scaling properties of the escape time from stable regions, which is demonstrated numerically as well.

We present the two basic mechanisms leading to spatial complexity in one-dimensional patterns. They are shown to be the counterparts of the horseshoe formation mechanisms studied by Melnikov and Shilnikov in dynamical systems theory.

Using accurate calculations of the critical-to-classical crossover performed within the ${\mathrm{\phi }}^4$ theory, we show that it is possible to observe nonuniversal power laws extending over almost three decades of temperature as seems to be the case near the cloud point of micellar solutions. The possibility of a nonmonotonic crossover is quantitatively examined and confronted with measurements made near the critical point of xenon. We emphasize the difficulties of determining the mechanism responsible for the nonmonotonicity.

Supercomputer simulations of the elementary A+B→0, diffusion-limited reaction were performed under steady-source conditions, on a cubic and a fractal lattice. While both reaction orders have the classical value (X=2) a dramatic segregation (‘‘morphogenesis’’) appears for the Sierpiński gasket ‘‘lattice.’’ This is relevant to chemical reactions on heterogeneous catalysts, to annealing of radiation damage and electron-hole recombination in inhomogeneous media, to charge polarization in clouds, and to possible matter-antimatter distributions in a steady-state universe.

The quantum energy eigenvalues near the separatrix energy are found by combination of previous analysis of the quantum states near a barrier top with recent results for the behavior of the classical action near a separatrix. It is found that the minimum energy separation vanishes only logarithmically with Planck’s constant. As a result, quantum aspects of the states of a Penning trap should be observable.

Kadanoff transformations are discussed in terms of a metric in the space of Gibbs measures. Direct Monte Carlo computation of this metric is illustrated by trial calculations for small Ising systems. Applications to gauge-field quantization are suggested.

The connected-moments expansion (CMX) for the ground-state energy is derived from the Horn-Weinstein theorem. This new technique possesses several striking advantages over traditional approaches to calculation of the ground-state energy. Like perturbational theories, the CMX series is conceptually simple but is easier to derive and program and is usually convergent even for very ‘‘crude’’ trial kets. Numerical examples reveal potential applicability of CMX for calculation of the correlation energy when the trial ket has the form of multideterminant wave function.

A new approach to Monte Carlo simulations is presented, giving a highly efficient method of simulation for large systems near criticality. The algorithm violates dynamic universality at second-order phase transitions, producing unusually small values of the dynamical critical exponent.

Recent determinations of the volt and the quantized Hall resistance at the Australian National Measurement Laboratory have yielded values having sub-ppm uncertainties for a number of important fundamental constants. The direct experimental results are 2e/h=483.597 90(15) THz ${\mathrm{V}}^{\mathrm{-}1}$ and h/$e^2$=25 812.809 9(20) Ω, both in SI units. Derived values are given for the electron charge, Planck’s constant, the magnetic flux quantum, and the inverse fine-structure constant.

The nuclear magnetic resonance frequencies of ${}_{}{}^{201}{}_{}{}^{}\mathrm{Hg}$ (I=(3/2)) and ${}_{}{}^{199}{}_{}{}^{}\mathrm{Hg}$ (I=(1/2)) have been compared by use of optically pumped atomic light absorption oscillators to see if the relative frequencies depend on the orientation of the quantization axis in space. The null result δ${\nu }_2$<5×${10}^{\mathrm{-}7}$ Hz improves by over 3 orders of magnitude recent limits on any quadrupolar interactions violating spatial isotropy, and can be interpreted as the most precise test yet of local Lorentz invariance. The experiment also improves the limits on anisotropic dipole interactions.

For a large class of $d$-dimensional disordered systems, we prove that if an appropriately defined finite-size scaling correlation length diverges at a nontrivial value of the disorder with an exponent $\nu$, then $\nu$ must satisfy the bound $\nu \ge \frac{2}{d}$. Given the assumption that such a correlation length can be defined, the result applies to, e.g., percolation, disordered magnets, and Anderson localization, both with and without interactions. For localization, this puts stringent constraints on scaling theories and interpretation of experiments.

The maximum-entropy formalism is used to characterize the fluctuations in transition strengths for a bound quantum-mechanical system. In the chaotic limit only one, ever present, sum rule is required as a constraint. The resulting distribution is that of Porter and Thomas, which can also be derived from random-matrix theory. For nonchaotic systems the distribution of transition strengths has a lower entropy. A possible additional constraint, operative during the onset of chaos, is proposed. The distribution of maximal entropy subject to both constraints accords with computed intensities in a system of two degrees of freedom.

Classical Kolmogorov-Arnol'd-Moser tori and cantori are found to act as barriers in the quantum dynamics of a kicked rotator. In their vicinity the asymptotic distribution decays exponentially. The penetration depth of a Kolmogorov-Arnol'd-Moser torus scales as ${\hslash }^{0.66}$ and the penetration probability as ${\hslash }^{2.5}$. Cantori can inhibit the diffusive growth of mean square displacements and thus act as barriers more drastically than in classical systems.

Our formalism for describing quantum measurements is generalized to the multitime case and used to derive rigorous time-energy uncertainty relations. For a free particle, we find that $T\delta H\ge \frac{1}{2}$, where $\delta H$ is the dispersion in energy and $T$ is the measurement duration as given by an external clock of arbitrarily high accuracy. Moreover, any system used as a clock obeys $\delta t\delta H\ge \frac{1}{2}$, where $\delta t$ is the dispersion in the values of time as measured by the system.

We apply a formalism for the description of multitime measurements to the derivation of the quantum limit on the precision with which a pair of successive position measurements can be performed on a free mass under conditions that would be optimal for detecting very weak forces. The result depends on the position resolution of the measuring device as well as on the time interval between the two measurements, and it spans a range of values which include a presently controversial result known as the standard quantum limit; any controversy is thereby resolved.

A simple fluid in uniform isothermal shear flow is studied for conditions used in recent nonequilibrium computer simulations. The short-time dynamics for the local conserved densities is found to be unstable at large shear rates for wave vectors near the peak of the structure factor. The critical shear rates obtained are similar to those for an order-disorder transition observed in the computer simulations.

The semiclassical approximation serves as an ideal basis for Monte Carlo evaluation of Euclidean path integrals. When the semiclassical approximation is good, the averaging converges rapidly. Comparison with conventional techniques shows a great advantage in computational efficiency. A simple one-dimensional example is presented, along with a discussion of how the technique is extended to several degrees of freedom.

A new Monte Carlo method is presented for simulations of systems with quenched random interactions. The approach greatly reduces the long correlation times characteristic of standard methods, allowing the investigation of lower temperatures with less computer time than previously necessary.