In the weak noise limit ($D\to 0$), the escape rate over a one-dimensional potential barrier is $\Gamma \sim \exp (-\frac{S}{D})$. Here $S$ is calculated exactly by identifying appropriate instanton solutions in a path-integral formulation of the Langevin equation. For the "colored noise" problem, defined by the noise correlator $〈\xi \mathrm{}\left(t\right)\mathrm{}\xi \left(t^{\prime }\right)〉=\left(\frac{D}{\tau }\right)\exp \{-\frac{|t-t^{\prime }|}{\tau }\}$, the instanton satisfies a fourth-order, nonlinear, ordinary differential equation which can be readily solved numerically. Analytical results are derived for small and large $\tau$.

In this paper two representation integrals of elastic wave field, corresponding to the Rayleigh I and II integrals in the scalar wave case, are derived for the first time. The Green's functions needed for these integrals are constructed by using auxiliary functions which are singular at the "ghost-image point" of the singular point of the original Green's functions (Stoke's solutions).

We study theoretically and experimentally the effect of noise on Rydberg atoms passing through a strong coherent microwave field. We derive a master equation containing the coherent field exactly. Its solution reveals the presence of four universal dynamical regimes: (i) an initial classical regime, (ii) a subsequent coherent localized regime, (iii) a transition, induced by the noise, in which coherence and localization are destroyed, and (iv) the final regime, where equidistribution over the quasienergy states is reached.

We report results of numerical investigations of the quantum dynamics of a 1D system subject to a time-dependent perturbation with three incommensurate frequencies. These results demonstrate a transition from localized to extended states occurring at a critical value of the perturbation parameter. The dependence of the localization length and of the diffusion rate on this parameter near the critical point is analyzed and found to be in agreement with the predictions of renormalization-group theory.

The relaxation properties of a stochastic bistable system perturbed by a periodic low-frequency forcing term is investigated by means of analog simulation. The so-called stochastic resonance phenomenon is revealed under diverse experimental conditions. Its dependence on the parameters of both the periodic perturbation and the forced system is explained theoretically.

Transport in three-dimensional nonintegrable Hamiltonian flows is studied and the destruction of Kolmogorov-Arnold-Moser barriers in the presence of stochastic perturbations described. We extend the action principle to Hamiltonians with small noise, which provides a framework to determine universal scaling of characteristic times as a function of the noise.

Diffusion processes are studied which occur in three-dimensional media with traps at quenched random positions. For the first time, aspects of survival at intermediate times are derived, given that the concentration of traps is small. A first-order phase transition is found at zero concentration. The predictions are in good agreement with recent numerical data. They also have new implications for the Lifshitz band-edge singularity of the density of states of random binary mixtures, for a model of selfattracting polymers, and for the distribution of the number of distinct sites visited in a random walk.

A Monte Carlo algorithm is presented that updates large clusters of spins simultaneously in systems at and near criticality. We demonstrate its efficiency in the two-dimensional $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ $\sigma$ models for $n=1\mathrm{}$ (Ising) and $n=2\mathrm{}$ ($x-y$) at their critical temperatures, and for $n=3\mathrm{}$ (Heisenberg) with correlation lengths around 10 and 20. On lattices up to ${128}^2$ no sign of critical slowing down is visible with autocorrelation times of 1-2 steps per spin for estimators of long-range quantities.

Chaotic diffusion in periodic Hamiltonian maps is studied by the introduction of a sequence of Markov models of transport based on the partition of phase space into resonances. The transition probabilities are given by turnstile overlap areas. The master equation has a Bloch band spectrum. A general exact expression for the diffusion coefficient $D$ is derived. The behavior of $D$ is examined for the sawtooth map. We find a critical scaling law for $D$, extending a result of Cary and Meiss. The critical scaling emerges as a collective effect of many resonances, in contrast with the standard map.

Measurements have been made of the polarization properties of the two photons emitted in the decay of metastable atomic deuterium in an experiment in which three linear polarizers, instead of the normal two, are used. The quantum-mechanical predictions for these imperfect polarizers have been calculated and shown to be in good agreement with the experimental results. The results have also been used to test local realistic models which are capable of explaining all existing two-polarizer-type experiments, and have provided conclusive evidence against these theories.

A general exact procedure for separation of the energy of the overall rotation from the "internal" (vibrational) energy in any (arbitrarily floppy) $N$-body system is formulated. It is based on following the detailed dynamics of the system and it retains all the dynamical couplings allowing for the flow of energy between the rotation and vibrations. Still, it leads to a complete instantaneous separation of the energies in that the Coriolis term in the Hamiltonian is zero at all times. The utility of the new scheme is illustrated.

We use an electronic four-level model, optically prepared in a coherent superposition of energy eigenstates, to discuss the dynamics of a measurement process leading to the preparation of pure energy eigenstates. The coupling strength to the environment (test laser) is found to control the speed at which information can be gained. An explicit detection strategy is given which can be evaluated as regards its average error probability. The model helps to clarify the conditions under which axiomatic measurement theory (i.e., the collapse of the wave function) applies; it also gives some clues on the recently revived discussion on "quantum jumps."

Combining the methods of scattering theory and supersymmetric quantum mechanics we obtain relations between the $S$ matrix and its supersymmetric partner. These relations involve only asymptotic quantities and do not require knowledge of the dynamical details. For example, for coupled channels with no threshold differences the relations involve the asymptotic normalization constant of the bound state removed by supersymmetry.

The possibility is investigated that impurity spins in SQUID tunnel junctions might have a strong dissipative effect on macroscopic flux tunneling. A microscopic theory is given which shows that for temperatures lower than the Kondo temperature $T_K$ of the spins, there is a strong effect if $T_K\simeq {\omega }_J$, the Josephson plasma frequency. The connection with quantum measurement theory is discussed.

We argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible, but is also a theoretically optimal measurement of the invariant properties of a dynamical system.

We compare the classical and quantum mechanics of a system whose classical dynamics are dominated by chaos. We observe a quantum "scars" of short periodic orbits. By examining the states on a quantum surface of section, we find that the stable and unstable manifolds of the periodic orbits have a clear effect on the eigenstates, in addition to the orbits themselves. A local density operator, constructed from a band of energy eigenstates, is strongly scarred by short periodic orbits. We also demonstrate a relation between the quantum eigenvalue spectrum and the actions of the periodic orbits.

We present a new method for using the data from Monte Carlo simulations that can increase the efficiency by 2 or more orders of magnitude. A single Monte Carlo simulation is sufficient to obtain complete thermodynamic information over the entire scaling region near a phase transition. The accuracy of the method is demonstrated by comparison with exact results for the d=2 Ising model. New results for d=2 eight-state Potts model are also presented. The method is generally applicable to statistical models and lattice-gauge theories.

The physical basis for the description of phenomena by a rotating observer is investigated. A simple, yet tentative, extension of the hypothesis of locality is used to determine the interference phase shift induced by the rotation of a neutron interferometer. The result consists of the Sagnac term, which is due to the coupling of the orbital angular momentum of the neutron with the rotation of the frame, and a new term which arises from a similar coupling because of the neutron spin. The latter effect is generally smaller than the Sagnac phase shift by the ratio of de Broglie wavelength of the neutron to the dimension of the interferometer. The possibility of detecting the new effect is briefly discussed.

We define a Lyapunov exponent for a class of quantum dissipative systems which in the classical limit can undergo a cascade of period-doubling bifurcations into chaos. We do so by computing the average of a functional over a semiclassical trajectory for a dynamical system whose Poincaré section corresponds to the Hénon map. In the strongly dissipative limit we establish a scaling law which determines the way in which chaos can set in for finite values of Planck's constant.

We consider in two dimensions $L (L\ge 2)$ independent Brownian paths of common lengths $S$, all starting at the origin, and the probability $P_L$ that their trajectories do not intersect. For $S$ large, $P_L\sim S^{{-\zeta }_L}$ where ${\zeta }_L$ is universal. In 2D the ${\zeta }_L\text{'}\mathrm{s}$ are identified as Kac conformal dimensions with $c=0\mathrm{}$ central charge ${\zeta }_L=h_{0,L}=\frac{({4L}^2-1)}{24, L\ge 2}$. This is generalized to $L$ walks in a half-plane, with a common origin on the boundary.