We have measured the fractal dimensions and the Kolmogorov entropies of periodic and chaotic attractors for a ${\mathrm{CO}}_2$ laser system with modulated losses. In particular, we find an increase in dimension near the accumulation point of the periodic cascade according to the Feigenbaum scaling law, besides the expected usual increase of the attractor dimension in the chaotic region. Numerical solutions of the theoretical model yield dimensions in quantitative agreement with the experiments, thus demonstrating a close match of experiment and theory for this physical system.

The apparent universality of the fractal dimension of the set of quasiperiodic windings at the onset of chaos in a wide class of circle maps is described by construction of a universal one-parameter family of maps which lies along the unstable manifold of the renormalization group. The manifold generates a universal ‘‘devil’s staircase’’ whose dimension agrees with direct numerical calculations. Applications to experiments are discussed.

We predict the manner in which a physical system, of size ${\mathrm{L}}^{\mathrm{d}\mathrm{-}\mathrm{d}\mathcal{’}}$×${\mathrm{\infty }}^{\mathrm{d}\mathcal{’}}$, subject to periodic boundary conditions, approaches bulk critical behavior as L→∞. While for T>${\mathrm{T}}_{\mathrm{c}}$(∞) the approach is exponential, for T<${\mathrm{T}}_{\mathrm{c}}$(∞) it is generally governed by power laws whose indices are determined by the bulk exponents for the corresponding d- and d’-dimensional systems. Specific predictions on the spherical model of ferromagnetism and the relativistic Bose gas with pair production are verified by analytical results.

Two qualitatively different types of dynamical behavior can be so tightly interwoven that it becomes impossible predict when a small change in parameters will cause a change in qualitative properties. For quadratic mappings of the interval, for example, the chaotic to parameter values form a Cantor set of positive measure, broken up by periodic intervals. This set can be described by a global scaling law, which makes it possible to form a good estimate of the fraction of chaotic parameter values. Sensitive dependence on parameters occurs when the scaling exponent β<1. β is conjectured to display universal behavior.

Field contributions and the Gauss theorem for retarded functions lead to a new formula for gravitational radiation energy loss. It clarifies the applicability of the quadrupole formula for systems with ${\mathrm{T}}^{\mathrm{ik}}$-dominated motion. Applied to the axially symmetric two-body problem with equation of state ε=ε(P) for short free-fall times, the new formula yields the bulk-motion quadrupole-formula result. This contrasts our earlier larger flux with unconventional equation of state and that of critics claiming a quadrupole formula flux from internal motions. An order of magnitude estimate yields potential corrections of quadrupole formula size after ∼${10}^2$ yr for the binary pulsar.

We construct the evolution operator U of the quantum Chirikov map explicitly. When 2πħ=M/N is a rational number, corresponding to the quantum resonance condition of Casati et al., we can reduce the operator U as a direct sum of independent N×N unitary matrices. We then obtain numerically the eigenstates of the same system. We describe these eigenstates in the coherent-state representation and find that they follow closely the Kolmogorov-Arnol’d-Moser curves and other classical orbits. We discuss the long-time behavior of initially localized quantum states.

Effects that could distinguish the Lorentz ether theory from Einstein’s special relativity, and their measurability, are analyzed.

A general approximation scheme for solving the quantum Liouville equation has been developed and is presented here specifically within the Hartree approximation. This method uses the connection between the density matrix and the Wigner phase-space representations. By use of the Wigner representation the off-diagonal elements of the density matrix are expanded in powers of deviations from the mean momentum value. The time evolution of these deviations is governed by a closed self-consistent set of equations.

Monte Carlo Fourier path-integral techniques are explored. It is shown that fluctuation renormalization techniques provide an effective means for treating the effects of high-order Fourier contributions. The resulting formalism is rapidly convergent, is computationally convenient, and has potentially useful variational aspects.

Near the onset of a period-doubling bifurcation, any dynamical system can be used to amplify perturbations near half the fundamental frequency: The closer the bifurcation point, the greater the amplification. An analytic expression for the frequency response curve is derived explicitly for the driven Duffing oscillator. Results of analog simulations are presented to check the main features of the theory. We propose that the superconducting Josephson parametric amplifier is an example of this amplification process.

The stroboscopic map of symmetric self-oscillators driven by pulses of alternating sign is constructed, for the regime of strong relaxation, by means of the so-called phase-transition curve. An ordering for the symmetries of the phase-locked orbits is found which is conjectured to be universal. Both results are illustrated by an exactly solvable example of such a system. When the approximation fails, unusual period doubling from symmetric orbits and strange attractors of two-dimensional structure are encountered.

We suggest that ionic conductors containing a dispersed insulating phase are suitable materials for investigating static and kinetic aspects of percolation. We develop a lattice model for the two-phase mixture with special emphasis on the role of an enhanced interface conductivity. By using two-dimensional Monte Carlo simulations we show that our model correctly describes the distinct conduction properties of those materials, and in addition displays the critical behavior of both random-superconducting and random-resistor networks at two different critical concentrations.

It is pointed out that a continuous random medium can cause an average advance of the arrival time of a pulse. This advance will occur for unsaturated and partially saturated propagation, but not in full saturation (which corresponds to the discrete scatterer case). The effect, which is associated with Fermat’s principle of least time, can be observed by measuring the difference between intensity-weighted and unweighted average arrival times.

It is shown that it is impossible to construct, in any reaction in atomic, nuclear, or particle physics, a null experiment that would unambiguously test the validity of time-reversal invariance independently of dynamical assumptions.

Operationalistic definition of the dimension of space-time leads to the possibility of its experimental determination. Several reasons may be given for the fractional dimension of space-time to be slightly smaller than four, yielding a finite quantum electrodynamics. Comparison between the best experimental values for the electron $g-2\mathrm{}$ factor and theoretical prediction gives the value 4- (5.3±2.5)×${10}^{-7\mathrm{}}$ for the dimension of space-time.

Models of self-avoiding random surfaces with chemical potentials for topological quantities in three and four dimensions are investigated by Monte Carlo methods. We explore the phase diagrams and find first- and second-order transition lines and triple points.

Measurements of the position $x$ of a free mass $m$ are thought to be governed by the standard quantum limit (SQL): In two successive measurements of $x$ spaced a time $\tau$ apart, the result of the second measurement cannot be predicted with uncertainty smaller than ${\left(\frac{\hslash \tau }{m}\right)}^{\frac{1}{2}}$. Yuen has suggested that there might be ways to beat the SQL. Here I give an improved formulation of the SQL, and I argue for, but do not prove, its validity.

The electrostatic field distribution due to the contact potential difference in a bimetallic wire introduces a quantum phase difference which can be detected in Fresnel and Fraunhofer diffraction experiments by means of an electron microscope.

A new Monte Carlo method is introduced which generates configurations according to any desired probability distribution. Unlike previous techniques, which require the relative probability of any two configurations to be computed exactly, this method allows the prescence of large but unbiased noise in this computation. The method has important applications in including the effects of dynamical fermions in Monte Carlo calculations, amongst other problems.

It is shown that the Migdal approximate-renormalization recursion formulas always bring the initial state into the high-temperature region for the U(1)-invariant models at the critical dimensions. This means that the Migdal recursion formulas fail to exhibit the Kosterlitz-Thouless transitions. This is also the case for the recursion formulas of Kadanoff type. The method developed here is extended to non-Abelian systems by some additional tricks.