In an optical fiber, nonlinear optical interactions couple the sideband modes of two strong pump waves at different frequencies just as required for a quantum nondemolition measurement. Experimental measurements demonstrate that 37% of the rms phase fluctuations of one wave are caused by the quantum amplitude fluctuations of the other.

For a simple Lorentz gas, viz., a random walk on a square lattice with randomly excluded sites, we present an exact calculation of the diffusion coefficient, the static conductivity, and the long-time tail of the velocity autocorrelation function $\phi \mathrm{}\left(t\right)\mathrm{}\simeq -\frac{\beta \mathrm{}\left(c\right)\mathrm{}}{t^2}$, exact to $O\left(c^2\right)$ terms included, by making a systematic expansion in the density of impurities $c$ (fraction of excluded sites). To $O\left(c\right)\mathrm{}$ the velocity auto-correlation function is calculated exactly for all times, and shows negative correlations (cage effect).

We prove that the motion of a tagged particle in a one-dimensional fluid of hard point particles in an external potential $U$ may be asymptotically described by a diffusion process. The process is spatially homogeneous or inhomogeneous according to whether $U$ varies on a microscopic or macroscopic scale. The latter process can still be described by a simple Langevin equation, provided that one interprets it in the sense of Stratonovich.

Spherically symmetric gravitational equilibria of self-interacting scalar fields $\phi$ with interaction potential $V\left(\phi \right)=\frac{1}{4}\lambda {\left|\phi \right|}^4$ are determined. Surprisingly, the resulting configuration may differ markedly from the noninteracting case even when $\lambda \ll 1$. Contrary to generally accepted astrophysical folklore, it is found that the maximum masses of such boson stars may be comparable to the Chandrasekhar mass for fermions of mass $m_{\mathrm{fermion}}\sim {\lambda }^{-\frac{1}{4}}{m}_{\mathrm{boson}}$.

Fat fractals (sets with fractal structure, but nonzero measure) are becoming increasingly important in the study of nonlinear dynamical systems. Since fat fractals are not well characterized by their (integer) dimensions, several scaling exponents have been proposed to characterize these sets, and there has been some debate over which exponent is best. We shed light on this issue by examining these exponents and finding relationships between them. Finally, these results lead naturally to quantitative definitions of the notions of sensitivity to parameters and final-state sensitivity.

A renormalization-group analysis of a tunneling atom interacting with a degenerate electron gas is presented. We consider both electronic screening and electron-assisted tunneling. In the case where the latter is strong, it is shown that, because of the spin degeneracy, the low-temperature behavior is dominated by the nontrivial fixed point of the multichannel Kondo model.

The essential ideas of Davydov's theory of molecular solitons are used to develop a variation of the Zwanzig-Nakajima projection technique. The projection operator is time dependent and contains the feedback characteristic of the soliton dynamic. The result is a nonlinear equation of motion for a reduced density matrix. A Markoffian approximation is considered which contributes corrections to the Davydov-type evolution. The approximate equation of motion embodies the correct fluctuation-dissipation relation for this system and recovers the known exact results in the transportless limit.

In problems such as the anharmonic oscillator and the H-atom Stark effect, the energies of bound states and resonances can be formally expressed in terms of series with a radius of convergence zero. We propose a method of rational approximation to sum such series and illustrate it by application to the anharmonic oscillator. In a certain sense this method is close to the best one possible.

A Hamiltonian formulation of general relativity based on certain spinorial variables is introduced. These variables simplify the constraints of general relativity considerably and enable one to imbed the constraint surface in the phase space of Einstein's theory into that of Yang-Mills theory. The imbedding suggests new ways of attacking a number of problems in both classical and quantum gravity. Some illustrative applications are discussed.

A phenomenological quantization scheme is carried out for the collision process of identical solitons of a nonlinear Schrödinger equation. An effective potential is determined from the classical expression for the time delay. The corresponding effective $S$ matrix reproduces the exact $S$ matrix of the quantized nonlinear field as well as the ground-state binding energy for large $N$, but it is not adequate to describe the scattering process in the low-energy region.

We solve exactly the problem of diffusion in an arbitrary hierarchical space. We prove that for a given "tree silhouette" $0<s<\mathrm{}1\mathrm{}$ the dynamic critical exponent $\nu$ ranges from $\frac{s}{\mathrm{}(1-s)\mathrm{}}$, for either uniformly or randomly multifurcating trees, to $s$ for the most diverse ones, in qualitative agreement with a static measure of the tree's complexity. We conclude that uniform trees are optimal for information diffusion, that in thermally activated processes the temperature dependence of $\nu$ varies with the underlying tree structure, and that thin elongated trees are the only ones capable of producing a $\frac{1}{f}$ spectrum.

Beginning with rules for a large class of continuum analogs of cellular automata, we derive Langevin equations for the slow variables of the system by projection-operator methods. For purely dissipative rules, we obtain an exact result: The slow variable is linearly unstable due to a "fluctuation-enhancement" relation. This means that this slow variable for the cellular automata grows exponentially for early times. We discuss how this may be related to, for example, kinetic models of the growth of ordered structures.

The interaction of three-level atoms with a two-mode classical electromagnetic field is considered in the case when transitions between all levels are allowed. It is found that for exact resonance with field frequencies the dynamics is chaotic in the rotating-wave approximation, i.e., for an arbitrarily small atomic density. The possibility of experimental observation of this phenomenon for Rydberg atoms is discussed.

The free massless field in 2 + 1 dimensions is written as an "integral" over free massless fields in 1 + 1 dimensions. With the operators with fractional dimension in the Gaussian model as a spring-board, operators with fractional statistics in the former theory are constructed.

Recently, Fischbach et al. suggested that the accelerations of various materials to the Earth may be different depending on their compositions, on the basis of their reanalysis of the results of the experiment of Eötvös, Pekár, and Fekete. We find that systematic effects due to thermal gradients can account for the experimental data.

A procedure is developed for transcription of any measure for the integration over metric fields in the continuum to the Regge-calculus lattice.

For quantum bound systems whose classical versions are nonintegrable, coupled dynamical equations for both energy levels and eigenfunctions with a nonintegrability parameter $t$ taken as "time" are shown to be a completely integrable Calogero-Moser system in 1 + 1 dimensions with internal complex vector space. Lax forms and their complete algebraic solutions are given which, in place of statistical mechanics procedures, determine possible energy spectrum and wave-function patterns at an arbitrary value of $t$.

From the spectrum of dimensions of a fractal invariant measure of a dynamical system one can extract information about the dynamical process that gave rise to the measure. This is equivalent to finding the class of Hamiltonians of an Ising model with a given thermodynamics.

We present and discuss a three-parameter class of exact solutions to the localized-induction-approximation equations. These are one-soliton excitations (Bäcklund transforms) of the circular vortex motion. The corresponding generic vortex filament (of infinite or finite length) remains in the interior of the sphere (or torus) moving with a constant velocity. The above solutions generate a new class of exact solutions to the classical one-dimensional continuous Heisenberg ferromagnet model.

We determine the $\beta$ function of the $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ nonlinear $\sigma$ model in $2+\epsilon$ dimensions to four-loop order using the recently calculated four-loop-order $\zeta$ function and the critical exponent $\eta$ through order $\frac{1}{n^3}$. This $\beta$ function determines completely, according to Hikami, the four-loop-order $\beta$ function for a large class of nonlinear $\sigma$ models. As an application the conductivity exponent of the Anderson metal-insulator transition is calculated for the unitary case to order $\epsilon$. This exponent turns out to be smaller than ½.