We consider a ball under the influence of gravity on a vibrating platform where the ball-platform collisions are completely inelastic. We present for the first time several remarkable features in the temporal behavior of the system; its phase space is divided into transmitting and absorbing regions, which are responsible for the abrupt termination of a period-doubling sequence and the onset of a locking regime, in which an appropriately defined winding number has intriguing scaling properties.

We present the theory for establishing a quantum standard of current. This is achieved by operating a scanning tunneling microscope at resonant tunneling through small metallic grains deposited on oxide layers, which exhibit Coulomb blockade. By adjusting the scanning period of the tip, a fixed number of electrons per oscillation can be made to tunnel. The current is directly related to the frequency and the electron charge.

We present a new numerical method to identify the nature of a phase transition. When combined with finite-size scaling, the method can identify unambiguously a weak first-order transition even when accessible system sizes are L/ξ<0.05 as in the five-state Potts model. At a continuous transition, exponents can be determined surprisingly accurately and the computational effort required is moderate. The method is tested on the 2D Potts and 3D Ising models, but should be quite generally applicable.

The novel phenomenon of exchange of stability among periodic orbits is studied with a view of understanding the recurrence of Kolmogorov-Arnol’d-Moser tori in Hamiltonian systems. This leads to a very complex phase diagram in the extended Frenkel-Kontorova model with cascades of first-order transitions separating nearby competing commensurate ground states. The Farey route to a given rotation number allows a very simple universal additive rule for constructing the phase diagram and understanding the self-similar structure underlying the nobel incommensurabilities.

A general concept of holonomy scattering is defined. It applies, among other things, to imply a nontrivial (generally, inelastic) scattering amplitude for the scattering of pure flux tubes off one another, when these fluxes are non-Abelian and do not commute. The amplitudes for such processes, as well as the original Aharonov-Bohm cross section, are calculated directly from an expression for the propagator with a definite winding number.

We show that the quantum spectra of simple systems whose classical counterpart is of a mixed nature, i.e., partly regular with widespread chaos, manifest the effects of classical transport through imperfect barriers. The partial barriers are characterized by the flux crossing them. We derive the relationship between this flux and quantum Hamiltonian matrix elements. This in turn predicts new statistical fluctuations properties for the spectrum and partial localization of the wave functions. The example of two coupled quartic oscillators is given in detail.

The shapes of self-avoiding continuum and lattice polygons of N monomers in a plane are studied using Monte Carlo simulations and exact enumeration. To model vesicles, a pressure increment Δp=${\mathit{p}}_{\mathrm{in}}$-${\mathit{p}}_{\mathrm{out}}$, is included. For N≫1 and Δp=0, the usual universal fractal shapes appear; but for Δp≠0, continuously variable fractal shapes are found controlled by the variable x∝Δ${\mathit{pN}}^{2\nu }$ where ν=1/${\mathit{D}}_{\mathit{F}}$=3/4. Thus, the ratio of principal radii of gyration Σ(x)=〈${\mathit{R}}_{\mathit{G},\mathrm{m}\mathrm{i}\mathrm{n}}^2$〉/〈${\mathit{R}}_{\mathit{G},\mathrm{m}\mathrm{a}\mathrm{x}}^2$〉 changes smoothly from Σ(+∞)=1, for circles, through Σ(0)≃0.39, to Σ(-∞)≃0.23, which corresponds to branched polymers.

Direct numerical measurement of the power spectrum of the number of particles on the lattice demonstrates in an example of probably broad physical relevance that 1/f behavior can arise due to self-organized criticality. Different versions of the model are studied in order to look for universality.

A method to obtain a superposition of time evolutions of a quantum system which correspond to different Hamiltonians as well as to different periods of time is derived. Its application to amplification of an effect due to the action of weak forces is considered. A quantum time-translation machine based on the same principle, utilizing the gravitational field, is also considered.

A gauged, nonlinear Schrödinger equation in two spatial dimensions is considered. This equation describes nonrelativistic matter interacting with Chern-Simons gauge fields. We find explicit static, self-dual solutions that satisfy the Liouville equation.

A renormalization-group recursion formula in a two-dimensional O(N)-invariant model is described by an integral kernel ${\mathit{K}}_{\mathit{N}}$, a probability density exhibiting perturbative asymptotic freedom for all N>1. The ${\mathit{K}}_{\mathit{N}}$’s are qualitatively the same for all N>1 and thus there may be Kosterlitz-Thouless–type transitions in some non-Abelian systems. In fact, we establish that the correlation functions of the O(4) σ model are bounded from below by ones of a random-coupling XY model in which the strong-coupling regions percolate over ${\mathit{Z}}^2$ as the original coupling constant becomes large.

The relativistic generalization of the quantum-mechanical virial theorem is derived and used to clarify the connection between the nonrelativistic and (semi)relativistic treatment of bound states.

We derive an exact expression for the low-temperature correlation function in the eight-vertex-equivalent Ising model. We provide a phenomenological interpretation of the qualitative nature of the transfer-matrix spectrum in terms of bubble excitations and speculate about the behavior in nonzero magnetic field.

We extend a nonlinear generalization of quantum mechanics, recently formulated by Weinberg, to systems of composite spin, such as atoms. We have used hydrogen (H) masers to set a limit of 3.7×${10}^{20}$ eV (8.9 μ Hz) on the magnitude of a nonlinear correction to the quantum mechanics of atomic spins. This result is of comparable magnitude to the limit recently set in a single-valued (nuclear) spin system. In the absence of nonlinear effects, the experiment provides a new and stringent test of H-maser clock performance and the applicability of standard maser theory.

The replica-symmetry-breaking (RSB) solution of the infinite-range Ising spin glass in the presence of a transverse field is obtained. The quenched free energy and the phase boundary of the glass transition temperature versus the transverse field are calculated at first-step RSB without using the static approximation. We demonstrate that replica symmetry (RS) has to be broken in the spin-glass phase by comparing the free energies of the RSB and RS solutions. No evidence is found to support an intermediate spin-glass phase with replica symmetry.

Chaos in area-preserving maps is organized on the basis of the unstable periodic orbits (UPO’s) and the partition of phase space into resonances. Each UPO is of a well defined type, which specifies the sequence of resonances visited and the number of rotations performed in a resonance. Chaotic orbits are grouped into sets of well defined type, topological and metric characterizations. The effect of regular motion on global diffusion may be characterized by an infinity of diffusion coefficients, associated with ensembles of chaotic orbits that are trapped in extended sets of resonances.

A new type of resonance phenomenon in a class of bistable oscillators driven by a periodic dichotomous noise process is described. The phenomenon manifests itself as troughs or peaks in the stationary density and has its origin in the appearance of periodic orbits in the noisy dynamics. The stationary density exhibits many self-similar scaling features as the system parameters are tuned, and these features are discussed. Since the noise process is easy to implement and the phenomenon is quite general, it should be possible to observe such resonances in experiments.

It is shown that there is an exact equivalence between the Aharonov-Bohm effect for spin-1/2 particles and the Aharonov-Casher effect. The demonstration of this precise relationship between the two is seen furthermore to be independent of whether relativistic or nonrelativistic kinematics are used. The only remaining substantive distinction between the two effects may well be the fact that the scattering cross section for polarized beams has a considerably greater sturcture in the Aharonov-Casher case despite the mathematical equivalence of the scattering amplitudes for the two effects.

We show how to characterize a strange attractor by a set of integers. These are extracted from the chaotic time-series data by first reconstructing the low-period orbits and then determining the template, or knot holder, which supports all periodic orbits embedded in the strange attractor, and the strange attractor itself. The template is identified by a set of integers which therefore characterize the strange attractor. This identification is explicitly demonstrated for the Pirogon using a relatively small data set (5000 points).

We show that motion with as few as three degrees of freedom (for instance, a particle moving in a three-dimensional potential) can be equivalent to a Turing machine, and so be capable of universal computation. Such systems possess a type of unpredictability qualitatively stronger than that which has been previously discussed in the study of low-dimensional chaos: Even if the initial conditions are known exactly, virtually any question about their long-term dynamics is undecidable.