We present a new method for optimizing the analysis of data from multiple Monte Carlo computer simulations over wide ranges of parameter values. Explicit error estimates allow objective planning of the lengths of runs and the parameter values to be simulated. The method is applicable to simulations in lattice gauge theories, chemistry, and biology, as well as statistical mechanics.

A hyperfine transition in the ground state of ${}_{}{}^9{}_{}{}^{}\mathrm{Be}_{}^{+}{}_{}{}^{}$ was used to test a nonlinear generalization of quantum mechanics recently formulated by Weinberg. We searched for a dependence of the frequency of a coherent superposition of two hyperfine states on the populations of the states. We are able to set a limit of 4×${10}^{\mathrm{-}27}$ on the fraction of binding energy per nucleon of the ${}_{}{}^9{}_{}{}^{}\mathrm{Be}_{}^{+}{}_{}{}^{}$ nucleus that could be due to nonlinear corrections to quantum mechanics.

The onset of chaotic behavior in a class of classical scattering problems is shown to occur in two possible ways. One is abrupt and is related to a change in the topology of the energy surface. The other arises as a result of a complex sequence of saddle-node and period doubling bifurcations. The abrupt bifurcation represents a new generic route to chaos and yields a characteristic scaling of the frac- tal dimension associated with the scattering function as [ln($E_c$-E${)}^{\mathrm{-}1}$${]}^{\mathrm{-}1}$, for particle energies E near the critical value $E_c$ at which the scattering becomes chaotic.

Critical behavior associated with intermittent temporal bursting accompanying the sudden widening of a chaotic attractor was observed and investigated experimentally in a gravitationally buckled, parametrically driven, magnetoelastic ribbon. As the driving frequency, f, was decreased through the critical value, $f_c$, we observed that the mean time between bursts scaled as ‖$f_c$-f${\Vert }^{\mathrm{-}\gamma }$. .AE

We present a procedure of practical use for representing functions of several variables as superpositions of functions of only one variable. We show how the procedure works when applied, for example, to the location of global minima. Our numerical examples are restricted here, for simplicity, to functions of two variables. The straightforward extension to functions of more variables will be discussed elsewhere.

Exact one-soliton and two-soliton solutions of generalized Calogero-Moser (gCM) equations are presented. We explain how these solutions describe the origin of the successive avoided crossings observed in the energy-level structure of bounded systems when a parameter is varied. A new statistical description, based on the grand-canonical ensemble for the gCM system, is developed for the study of the parametric properties of irregular spectra. The relationship with random matrix theory is discussed.

We show that under certain circumstances a simple quantum harmonic oscillator driven by a quantum current evolves to unique pure states even if started as a mixed state. In various limits, these states exhibit nonclassical properties such as sub-Poissonian statistics, or more interestingly resemble macroscopic superpositions.

A numerical technique for the characterization of the chaotic regime of dissipative maps through unstable periodic orbits is presented. It is shown that although the maps are dissipative their trajectories can be derived from a Hamiltonian, which allows us to calculate unstable periodic orbits of arbitrary length finding all points to any desired accuracy. Applying the method to the Hénon map we found that in a previously unexplored region of parameter space the topological entropy exhibits plateaus on which it is constant while the dynamics is characterized by a strange repeller.

We demonstrate the utility of the periodic-orbit description of chaotic motion by computing from a few periodic orbits highly accurate estimates of a large number of quantum resonances for the classically chaotic 3-disk scattering problem. The symmetry decompositions of the eigenspectra are the same for the classical and the quantum problem, and good agreement between the periodic-orbit estimates and the exact quantum poles is observed.

We prove the rigorous lower bound $z_{\mathrm{sw}\ge \mathrm{\alpha }/\nu }$ for the dynamic critical exponent of the Swendsen-Wang algorithm. For two-dimensional q-state Potts models with q=2,3,4, this implies $z_{\mathrm{sw}\ge 0}$,(2/5,1. We present numerical data indicating that $z_{\mathrm{sw}=0.55\mathrm{}\pm 0.03}$, 0.89±0.05 for q=3,4 (95% confidence limits, statistical errors only). The discrepancy for q=4 appears to be caused by multiplicative logarithmic corrections.

A line-charge–neutron composite is proposed as a new model for fractional spin. Considering time-varying line-charge configurations, the kinetic angular momentum is a constant of motion while the canonical angular momentum is not. This result is contrary to the case with magnetic flux tubes. The fractional angular momentum (spin) is, therefore, determined without ambiguity.

We establish three results on the 3D Schrödinger equation with a potential having no spherical symmetry: the existence of a Wiener-Hopf factorization of the scattering operator, a method to recover the potential from the scattering data, and the existence of the 3D Jost operator.

The reversible one-species diffusion-limited reaction X+X⇄X is solved exactly in one dimension. We find that the asymptotic approach to equilibrium exhibits a sharp (second-order-type) transition depending on the initial density ${\rho }_0$. The exponential relaxation time becomes infinitely long as ${\rho }_0$→0. .AE

We propose definite experiments to provide realizations of a number of important models in the field of quantum chaos. These models share the feature that they are driven impulsively in time, i.e., kicked. In quantum language, a kick drives the system from one energy level to a neighboring one. The most fundamental question is whether localization occurs, i.e., whether eventually the system is driven far from its initial level or not. The most straightforward realizations are in optical fibers where the language of modes and propagation constants replaces quantum terminology.

The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0≤γ≤π/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.

We consider the problem of multiplication in a diffusive random medium. For time-independent one-dimensional white-noise multiplication ‘‘potential’’ we find asymptotic growth ∼exp(${\mathrm{t}}^3$), while for a ‘‘potential’’ with finite-range correlation we find asymptotic growh ∼exp(${\mathrm{t}}^2$).

We study the phenomena of self-organized criticality originally proposed by Bak, Tang, and Wiesenfeld. A continuous-energy model is introduced. Using numerical simulations, we find that energy is homogeneously and isotropically distributed in space, and that it is concentrated around discrete values. We propose a scaling theory to estimate the various exponents. The activation cluster size distribution is found to be D(s)∼1/$s^{\tau }$, τ=2-2/d; and the dispersion relation t∼$r^z$, z=(d+2)/3.

We find a representation for the finite Fourier series of a vector whereby any constant constraint on its magnitude is completely solved and automatically satisfied. The representation is a product of rotations, one set for each harmonic, such that the independent degrees of freedom are identified as rotational angles and an infinite series is replaced by an infinite product. This can be applied to the study of relativistic strings, and other applications and generalizations are anticipated.

A globally coupled map lattice is investigated. A simple coding of many attractors with clustering is shown. Through the coding, the attractors are organized so that their change exhibits the period-doubling bifurcation. By a simple input on a site, we can switch among attractors and tune the strength of chaos. A threshold on the cluster size is found beyond which a peculiar ‘‘posi-nega’’ switch occurs.

Statistical mechanics is used to describe the observed information processing complexity of nonlinear dynamical systems. We introduce a measure of complexity distinct from and dual to the information theoretic entropies and dimensions. A technique is presented that directly reconstructs minimal equations of motion from the recursive structure of measurement sequences. Application to the period-doubling cascade demonstrates a form of superuniversality that refers only to the entropy and complexity of a data stream.