We present results of the application of the Schwinger multichannel method to study the role of Feshbach resonances on the elastic and electronic excitation cross sections in e-${\mathrm{H}}_2$ collisions. Using a ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{\mathrm{g}}^{+}{}_{}{}^{}$ resonance associated with the $a^3$${\Sigma }_g^{+}$ and E${}^1$${\Sigma }_g^{+}$ states of ${\mathrm{H}}_2$ as an example, we quantitatively illustrate the dramatic influence of open channels on the decay of such Feshbach resonances in electron-molecule collisions.

An optimization of a nonlinear Mach-Zehnder interferometer to produce sub-Poissonian photon number distribution is proposed. We treat the system quantum mechanically and estimate the mirror parameters, the nonlinearity of the medium in the interferometer, and the input power to obtain minimal output uncertainty in the photon number. The power efficiency of the system is shown to be high.

We report on an interference effect arising from a two-photon entangled state produced in a potassium dihydrogen phosphate (KDP) crystal pumped by an ultraviolet argon-ion laser. Two conjugate beams of signal and idler photons were injected in a parallel configuration into a single Michelson interferometer, and detected separately by two photomultipliers, while the difference in its arm lengths was slowly scanned. The coincidence rate exhibited fringes with a visibility of nearly 50%, and a period given by half the ultraviolet (not the signal or idler) wavelength, while the singles rate exhibited no fringes.

We have analyzed the newly obtained exact enumeration series of the end-to-end distance ${\mathit{R}}_{\mathit{N}}$ and the radius of gyration ${\mathit{S}}_{\mathit{N}}$ for N-step nearest-neighbor-avoiding walks on the fcc and triangular lattices. The estimates of the scaling exponent ν for ${\mathit{R}}_{\mathit{N}}$ and ${\mathit{S}}_{\mathit{N}}$, which indicate ${\nu }_{\mathit{S}}$=${\nu }_{\mathit{R}}$, are in good agreement with the corresponding values for self-avoiding walks (SAW’s) in three and two dimensions. The respective estimates of the correction-to-scaling exponent ${\mathrm{\Delta }}_1$ for ${\mathit{R}}_{\mathit{N}}$ and ${\mathit{S}}_{\mathit{N}}$ are 0.58±0.08 and 0.35–0.55 for the fcc lattice, while those for the triangular lattice are 0.6–0.8 and 1. These results of ${\mathrm{\Delta }}_1$ are almost reconciled with those for SAW’s by considering the variance in amplitude of the correction term.

The quasiperiodic orbit families of integrable quartic polynomial potentials are always topologically compatible. Conversely, chaotic orbits are always found between quasiperiodic orbit families that cannot be smoothly deformed from one to another. A group of preferred unstable periodic orbits determines the effects of perturbations on the surfaces of section of integrable potentials. They change in two very specific ways, generating either a chaotic region about a hyperbolic point or a new family of invariant curves about one of the preferred unstable periodic orbits.

Stochastic quantization in the sense of Nelson provides an alternative interpretation of some aspects of quantum mechanics in the coordinate representation, and it was combined recently with the Ford, Kac, and Mazur (FKM) approximation [J. Math. Phys. 6, 504 (1965)] for large lattices to construct a quantum analog to the Brownian motion process. In this paper a similar approach is applied to model the effect of temperature fluctuations in a one-dimensional ordered chain of atoms with nearest-neighbor linear forces. However, we do not make use of the FKM approximation, and as a consequence the statistical properties of the involved processes are exactly determined by the lattice force field. In particular, we evaluate the covariance matrix for the fluctuations, and we examine its high- and low-temperature behavior. Because of the translation invariance of the interaction potential, the covariance matrix for the fluctuations becomes singular implying that the associated probability density has equal density along the zero eigenvector of the interaction matrix. This behavior is readily interpreted in terms of the motion of the center of mass of the system, which corresponds to a stochastically perturbed translation, while all other modes are bounded with a probability of 1. As is well known, the transformation to internal (bondlength) coordinates leads to a Hamiltonian specified by a nonsingular interaction matrix. We examine the variance of the fluctuations for the internal coordinates, and we show that in the high-temperature limit the result agrees with that of classical statistical mechanics. Both the position and bondlength of the surface atom decrease with time as is expected for a semi-infinite lattice. However, the position of the surface atom is less dependent on substrate-atom positions than is the surface bondlength on substrate bondlengths. Finally, the autocorrelation function of the surface bondlength in the case of a semi-infinite lattice limit is investigated for low- and high-temperature limits.

An exact enumeration approach is developed for the directed polymer problem. The probability distribution of the number of directed self-avoiding walks that can reach a certain level t is obtained exactly up to t=10. This enables us to calculate some properties of directed polymers that are not attainable by Monte Carlo simulations. Specifically, we find that the fluctuation of the logarithm of the number of directed self-avoiding walks that can reach level t, when averaged over the configurations that can reach level t, scales as ${\mathit{t}}^{1/2}$ well below the directed percolation threshold ${\mathit{p}}_{\mathit{c}\mathit{D}}$, contrary to the behavior ${\mathit{t}}^{1/3}$, which is known to be valid when the bond probability p is above ${\mathit{p}}_{\mathit{c}\mathit{D}}$. When p is close to 1, these fluctuations scale as ${\mathit{t}}^{1/5}$ for a very long time before the true asymptotic behavior ${\mathit{t}}^{1/3}$ is recovered. The method can also be used to obtain the behavior of averages of moments of the number of directed self-avoiding walks that can reach level t. Below ${\mathit{p}}_{\mathit{c}\mathit{D}}$ these quantities are dominated by rare configurations and cannot be obtained by Monte Carlo simulations.

The correspondence limit is illustrated for the periodically kicked pendulum. The classical dynamics of this system can be represented by a discrete map. A corresponding quantum map is derived and then rendered in the Ehrenfest formulation for expectation values. The Ehrenfest representation is studied for a minimum uncertainty Gaussian wave packet. It is shown that as ħ gets very small, followed by a decrease in the variance of the wave packet, the quantum map shadows the classical map with an error approaching zero for a length of time approaching infinity. The Gaussian form of the wave packet is preserved by the time evolution provided ${\mathrm{\omega }}_0$T≪1. This constraint implies that the classical map is predominantly not chaotic with very small regions of very weak chaos. As ${\mathrm{\omega }}_0$T approaches 1, where the classical map becomes strongly chaotic, the propagation in time of the Gaussian wave packet completely breaks down. The possible significance of this breakdown is discussed.

Chaos is characterized by sensitive dependence on initial conditions. Trajectories determined by coupled, ordinary differential equations show sensitive dependence when their associated Liapunov exponent is positive. The Liapunov exponent is positive if the Jacobi matrix associated with the coupled differential equations has an eigenvalue with a positive real part, on the average, as the Jacobi matrix evolves along the trajectory. For macrovariable equations, there are also fluctuation equations which follow the macrovariable trajectories. The covariance matrix for these fluctuations evolves according to an equation in which the Jacobi matrix for the deterministic motion plays the dominant role. For a chaotic trajectory, the covariance matrix grows exponentially. This means that for macrovariable equations that imply chaos, the construction of the macrovariable equations out of an underlying master equation is no longer valid. The macrovariable equations cease to be physical, and the physical description must be done entirely at the master equation level of description where the fluctuations, which are very large scale, can be properly treated. In parallel with this analysis, the correspondence limit connecting the time evolution of the Wigner distribution with Liouville’s equation breaks down when the classical motion is strongly chaotic. This implies that strongly chaotic classical dynamics must be treated quantum mechanically in order to properly treat the quantum fluctuations which have grown macroscopically large. Experimental confirmation of these ideas is discussed.

We consider the Brownian motion of a particle in a force field exposed to external periodic modulations. Discussed are ergodic properties, the long-time behavior of correlation functions, and spectral densities for general (nonlinear) force fields. For a parabolic potential we present analytical results for the quasispectrum (quasieigenvalues and quasieigenfunctions) at all values of the damping constant as well as spectral densities and cycle-averaged probability distributions. The theory is also applied to a Brownian oscillator with memory damping, yielding additional resonances between the external field and internal degrees of freedom.

A two-level system subject to quasiperiodically modulated kicking is investigated under the aspects of ergodic theory. For weak kicking strength κ the correlation functions exhibit quasiperiodic behavior, whose character changes with increasing κ as recurrences become infrequent. A quantum instability with a transition to mixing behavior that was suggested for similar models does not arise. Analytic results are obtained for the case where two of three characteristic frequencies are incommensurate.

We test the predictions of the solvability theory in the Saffman-Taylor problem by a random-walk simulation. By a suitable modification of the surface tension in accordance with the solvability theory, we observe a dynamically stable cusp to develop on the finger profile leading to a substantial reduction in a finger width. Identifying this cusp with that originating from the solvability theory, we find an excellent agreement between the simulation results and the solvability theory.

We investigate both dynamic and time-averaged chaos in a series of problems ranging from a simple pendulum to many-body chains and strings of particles in a gravitational field. Chain and spring systems typically display time-averaged Lyapunov spectra resembling those of one- and two-dimensional statistical-mechanical systems, but with details depending strongly on the nature of the links and the choice of canonical coordinates.

Following our earlier work on quantum kinetic equations and the use of the Boltzmann-Lorentz model to describe the collisional broadening of spectra, we develop a new approach to that problem in which the center-of-mass motion of the emitter as well as the interaction during the collision between the emitter and the buffer gas are treated quantum mechanically. Possible further extensions of our model are discussed.

An exact analytic method of calculating relaxation times of nonlinear systems driven by colored noise, applicable to a range of systems described by separable stochastic differential equations, is presented. Analog experiments and digital simulations performed on a quadratic model are shown to yield results in satisfactory agreement with the theoretical predictions.

Critical phenomena in optical bistability (OB) are studied in detail. The theoretical analyses, mainly based on the steady-state solution of the ‘‘thermodynamics’’ Fokker-Planck equation [Fa Ou and Zixiong Qin, Opt. Commun. 65, 455 (1988)] for absorptive OB, show that the critical phenomena in OB also can be brought into the framework of Landau’s theory for the second-order phase transition, and the critical exponents obey the scaling laws.

The structure of the resonances in two-dimensional (2D) Hamiltonian systems is analyzed in the frequency space. We define the frequency-frequency curves (FFC’s), discuss some of their mathematical characteristics, and with that basis in mind, we classify 2D Hamiltonian systems from the point of view of the shape of their FFC’s. We discuss this classification through a number of prototypical examples. The analytical conditions allow us to predict relevant features of the resonances’ structure of integrable systems. We also consider under which conditions the FFC picture persists (in a coarse-grained sense) for near-integrable systems. This allows us to qualitatively understand and quantitatively predict the energy of the transition to global stochasticity by using Chirikov’s criterion for the overlapping of resonances. These ideas are applied to Sinai’s billiard with two different perturbations.

We address the issue of statistical error in the estimation of the correlation dimension of a (possibly fractal) set from a finite sample of N points on the set. Although both numerical and analytical investigations show that this error generically scales with the number of points as O(1/ √N ) for N→∞, it is also shown that this is not always the case. The exceptions exhibit anomalously precise O(1/N) scaling, and can be characterized by their extreme uniformity. The existence of these exceptions points out that the central limit theorem is not directly applicable. The exceptions are not generic, but it is common for a fractal to exhibit 1/N scaling that crosses over to 1/ √N scaling only after N becomes very large. The coefficients of both the 1/ √N term and the 1/N term are derived, and the results are applied to practical problems in dimension estimation. In particular, the practice of computing dimension with only a few reference points is discussed. The choice of optimal scaling region and the overall scaling of error (systematic and statistical) with N are also considered in light of these results.

Random walks are used to obtain the diffusion constant for continuum percolation models of composite materials in two and three dimensions. An Einstein relation is then used to find the conductivity. The same calculation gives the dielectric constant for the composite. First-passage-time methods and special boundary conditions are used for systems where both materials have finite conductivity, where one component is a superconductor, and where one component does not conduct. The percolation models consist of randomly placed overlapping spheres in three dimensions or disks in two dimensions. Our results are consistent with known results where applicable and are far better than effective medium theories. Estimates for anomalous diffusion exponents at percolation were also found.

We study the Hebb rule for learning several Boolean functions (random and linearly separable functions) defined on the hypercube of dimension N. Learning and generalization rates are derived in the N→∞ limit versus α=P/N, where P is the number of learned patterns. In the linearly separable case, the generalization rate grows monotonically from 1/2 to 1, whereas the learning rate first decreases from 1 to a minimum value, and then increases again towards 1. This result is interpreted as an interference phenomenon, like in the learning for associative memories implemented with the same rule. Comparisons are then made with the case of random Boolean functions, associative memories, and their clipped version. The behavior of the Hebb rule is decomposed in two distinct contributions, referred to as the ‘‘rote’’ and the ‘‘conceptual’’ learnings. Illustrative numerical simulations are given.