We study an interacting one-dimensional quantum lattice gas, based on the Heisenberg-Ising ring. The particles are given a charge, and the ring is threaded by a magnetic flux. We then calculate exactly the energy of the state, which begins as the ground state with zero magnetic flux, when the magnetic flux is adiabatically increased. We find the result that the period of the ground state is two flux quanta, which can be interpreted as charge carriers with half-integer charge.

A Bell inequality is derived for a state of n spin-1/2 particles which superposes two macroscopically distinct states. Quantum mechanics violates this inequality by an amount that grows exponentially with n.

For dynamical systems of large spatial extension giving rise to transport phenomena, like the Lorentz gas, we establish a relationship between the transport coefficient and the difference between the positive Lyapunov exponent and the Kolmogorov-Sinai entropy per unit time, characterizing the fractal and chaotic repeller of trapped trajectories. Consequences for nonequilibrium statistical mechanics are discussed.

For an arbitrary quantum evolution, it is shown that the integral of the uncertainty of energy with respect to time is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space. It is the distance along this curve measured by the Fubini-Study metric. This gives a new time-energy uncertainty principle. New geometric meanings to time as measured by a clock and the transition probability during a quantum measurement are also obtained.

We analyze a large dynamical system of limit-cycle oscillators with mean-field coupling and randomly distributed natural frequencies. Depending on the choice of coupling strength and the spread of natural frequencies, the system exhibits frequency locking, amplitude death, and incoherence, as well as novel unsteady behavior characterized by periodic, quasiperiodic, or chaotic evolution of the system’s order parameter. The phase boundaries between several of these states are obtained analytically.

We examine the question of accurately determining Lyapunov exponents for a time series. We find that it is advantageous to use local mappings with higher-order Taylor series, rather than linear maps as done earlier. We demonstrate this procedure for the Ikeda map and the Lorenz system. We present methods for identifying spurious exponents by analyzing data-set singularities and by determining the Lyapunov direction vectors. The behavior of spurious exponents in the presence of noise is also investigated, and found to be different from that of the true exponents.

For a chaotic system, a noisy trajectory diverges rapidly from the true trajectory with the same initial condition. To understand in what sense the noisy trajectory reflects the true dynamics of the actual system, we developed a rigorous procedure to show that some true trajectories remain close to the noisy one for long times. The procedure involves a combination of containment, which establishes the existence of an uncountable number of true trajectories close to the noisy one, and refinement, which produces a less noisy trajectory. Our procedure is applied to noisy chaotic trajectories of the standard map and the driven pendulum.

The title statement is numerically shown for a globally coupled chaotic system. With an increasing number of elements, N, the distribution of the mean field approaches a Gaussian distribution, but the decrease of its mean-square deviation with N stops for large N. This violation of the law of large numbers is found to be caused by the emergence of a subtle coherence among elements, as is measured by the mutual information. With the inclusion of noise, the law of large numbers is restored. The mean-square deviation decreases in proportion to ${\mathit{N}}^{\mathrm{-}\mathrm{\beta }}$ with an exponent β<1 depending on the noise strength.

It is shown that critical behavior of systems with self-organized criticality can be explained as a Goldstone-mode phenomenon. The nonlinear interaction of the Goldstone mode causes the nontrivial critical exponents. Two more models are introduced with self-organized criticality features.

Variational upper bounds are derived for the escape rate of a particle trapped in a metastable well and interacting wth a dissipative medium. The theory leads to a reduced two-degrees-of-freedom Hamiltonian involving the unstable normal mode and a newly defined collective bath mode. Explicit treatment of strong nonlinearities or low-barrier systems present no special problem.

We consider the 1D Schrödinger equation with a potential proportional to energy. When the spatial part of the potential is twice continuously differentiable, is less than 1 everywhere, and satisfies a certain integrability condition, we compute the scattering matrix. For the same class of potentials, under the further assumption that the potential is non-negative, we obtain the potential from one of the reflection coefficients.

Scattering connected with symmetry is studied. Explicit expressions for the S operator are provided for the scattering related to any symmetric space of the noncompact type.

One-dimensional reductions of the self-dual Yang-Mills equations yield various classical systems depending on the choice of the Lie algebra associated with the self-dual fields. Included are the Euler-Arnold equations for rigid bodies in n dimensions, the Kovalevskaya top, and a generalization of the Nahm equation which is related to a classical third-order differential equation possessing a movable natural boundary in the complex plane.

We present a novel cluster algorithm for Monte Carlo simulations of the fully frustrated Ising model on the square lattice. The new method does not suffer from problems of metastability, and is extremely efficient even at T=0. Our algorithm is a special case of a more general Monte Carlo simulation scheme. The general scheme unifies many cluster algorithms that were developed recently in order to accelerate Monte Carlo simulations.

A new approach to clustering based on statistical physics is presented. The problem is formulated as fuzzy clustering and the association probability distribution is obtained by maximizing the entropy at a given average variance. The corresponding Lagrange multiplier is related to the ‘‘temperature’’ and motivates a deterministic annealing process where the free energy is minimized at each temperature. Critical temperatures are derived for phase transitions when existing clusters split. It is a hierarchical clustering estimating the most probable cluster parameters at various average variances.

We study a deterministic version of the cellular automaton first shown to display self-organized criticality. Detailed analysis shows that there exist many coexisting periodic attractors, with a period that is independent of initial condition. This leads us to picture the critical state as the union of many such coexisting, neutrally stable orbits. Dhar’s recently developed formalism [Phys. Rev. Lett. 64, 1613 (1990)] can be used to explain many of the observed regularities.

The coherent scattering rate from n particles bound in a periodic lattice of N potential wells is considered. The quantum theory predicts a total coherent scattering rate proportional to ${\mathit{n}}^2$/N, whereas the conventional semiclassical approach predicts a rate proportional to n. These effects allow an experimental test of certain nonlocal assumptions inherent in quantum statistical mechanics and may be useful in analyzing the properties of crystalline materials.

A numerical study of a set of random matrices which interpolate Poisson and Gaussian orthogonal ensembles is reported. The result indicates that the transition from Poisson to Wigner distributions for the nearest-level spacing does not depend on the details of the random-matrix parametrization and is essentially governed by a single parameter. Brody’s one-parameter interpolation formula is found to describe the transition rather well.

Evidence is presented for the existence of a strange nonchaotic attractor in a two-frequency quasiperiodically driven, buckled, magnetoelastic ribbon experiment. Scaling behavior in the Fourier amplitude spectrum is observed in agreement with predicted scaling behavior for strange nonchaotic attractors. Dimension measurements also support the existence of a strange nonchaotic attractor.

We propose here a new method to characterize the loss of memory with time in a chaotic system from a time series. This is done by introducing time-dependent generalized exponents. The asymptotic behavior can distinguish between chaotic systems which lose memory of the initial conditions completely, those which partially retain the memory, and those (borderline of chaos) which fully retain the memory. We give illustrative examples of the logistic and Hénon maps.