An analytic formalism of cumulative beam breakup in linear accelerators is developed. This formalism is applied to both low-velocity ion accelerators and high-energy electron accelerators. It includes arbitrary velocity, acceleration, focusing, initial conditions, beam-cavity resonances, finite bunch length, and arbitrary charge distribution within the bunches, and variable cavity geometry and spacing along the accelerator. For both direct-current beams and beams comprised of δ-function bunches, both the steady-state and transient displacements of the beam are calculated, and scaling laws are determined for the transient beam breakup. The steady-state transverse displacement of particles between bunches is also calculated since, if allowed to impinge on the accelerating structures, these particles could cause activation over long periods of continuous-wave operation. The formalism is then applied to high-current ion accelerators by studying the effects of finite bunch length and arbitrary charge distribution within the bunches. The role of focusing in controlling cumulative beam breakup is quantified in each of these cases. Additionally, the effects of random initial conditions and a distribution of deflecting-mode frequencies in the cavities are also quantified.

In a recent paper one of the present authors [V. V. Konotop, Phys. Rev. A 44, 1352 (1991)] put forth an analytical approach for the description of one-dimensional scattering by a Weierstrass-like layer. The method consists of splitting the fractal permittivity into two effective parts, the scales of which are long and short in comparison with a wavelength. The dependence of a splitting point on an incident wavelength is given. In the present paper we generalize such an approach, called a two-scale method (TSM) for a wide class of structures having irregularity, and provide complete analytical and numerical investigations of two particular cases: layers with Weierstrass-type and singular permittivities. A good correspondence between analytical predictions and numerical solutions of the corresponding Riccati equation, as well as qualitative agreement with outcomes published elsewhere [D. C. Jaggard and X. Sun, IEEE Trans. AP-37, 1511 (1989)], is observed. In all cases the scattering data have a strongly pronounced resonant character that is well described by the WKB approximation. Increasing the box dimension leads to growth of the internal scattering and, as a consequence, to the inapplicability of the TSM. To investigate the physical nature of this phenomena we study both the above problems in the Born approximation as well. The main feature of fractal layers consists of a noninteger-power-law dependence of scattering data on slab length. Application of the TSM allows one to find a number of harmonics (in the trigonometric-series expansion), enough to provide adequate numerical analysis of fractal layers. Validity limits of the theory developed are stated. They turn out to be approximately the same as those for the WKB approximation for the effective long-scale part of the fractal permittivity.

Two limiting expressions occur for scattering from very small spheres. One occurs when the refractive index becomes small (Rayleigh scattering), and the other occurs when the sphere becomes perfectly conducting (Thomson scattering). We explore the scatter from small spheres having real refractive indices. For such spheres, resonance conditions occur, and the resulting scattering coefficients are no longer proportional to the volume of a sphere.

The microstatistics of laser speckle has been investigated under varying incident polarization angle θ. The intensity at each point in a speckle pattern varies sinusoidally with θ and can be described completely by three parameters: amplitude, baseline, and relative phase. The probability-density distributions for these parameters were obtained by carefully measuring the intensity modulation in several thousand independent coherence areas. Samples were studied which spanned the full range of depolarization ratio ρ, recently suggested to be the main parameter needed to describe microstatistics. Excellent agreement with recent theoretical results for unity ρ was observed in the amplitude microstatistics. For different samples having the same value of ρ, differences were seen for the baseline microstatistics at low value, possibly suggesting a need to include other light-scattering parameters in the description of the microstatistics. Furthermore, the decay of the baseline microstatistics measured at high values is not in good agreement with theoretical predictions. Finally, the distributions obtained for high ρ values for speckle in transmission show excellent agreement with those in reflection.

We present results concerning kink-impurity interactions in the sine-Gordon (SG) model. In particular, we demonstrate that the SG kink may be totally reflected by an attractive impurity if its initial velocity is in some resonance windows. This effect can be predicted by a suitable collective-coordinate method, and the resonance structures can be explained with an energy exchange between the kink translational mode and the impurity mode. We also study the scattering of the kink by an excited impurity and show that such a scattering strongly depends on the amplitude and phase of the impurity mode. In particular, resonance phenomena are also observed in the scattering. In addition, we consider the interactions of the kink with an isotopic (heavy-mass) impurity. We demonstrate that if the impurity mass is not too large in comparison with the standard mass in the SG model, the kink can pass the impurity almost freely at any initial velocity. However, if the impurity mass is large enough, a higher-velocity kink will be reflected while a lower-velocity kink will pass. We explain this effect analytically and show that the impurity mode plays an important role in the scattering. In all the cases considered we find good agreements between the collective-coordinate analysis and the direct numerical simulations.

We have carried out an extensive simulation program to study the behavior of sine-Gordon breathers initially at rest in the presence of perturbations that are periodic in space and constant in time. We report here a number of different observed phenomena and the range of the relevant parameters (the ratio of the breather width to the perturbation wavelength and the perturbation magnitude) for which each one of them occurs. We also propose some qualitative explanations valid for certain regimes.

The correspondence between Eigen’s model [Naturwissenschaften 58, 465 (1971)] for molecular quasispecies and the equilibrium properties of a lattice system proposed by Leuthäusser [J. Chem. Phys. 84, 1884 (1986); J. Stat. Phys. 48, 343 (1987)] is used to characterize the error thresholds for the existence of quasispecies as phase transitions. For simple replication landscapes the error threshold is related to a first-order phase transition smoothed by the complete wetting of the time surface. Replication landscapes based on the Hopfield Hamiltonian for neural networks allow for the tuning of the landscape complexity and reveal the existence of two error thresholds, bracketing a region of spin-glass quasispecies between the simple quasispecies and the fully disordered mixture of sequences.

Statistical mechanics is applied to estimate the maximal capacity per weight (${\mathrm{\alpha }}_{\mathit{c}}$) of a network consisting of N binary units and NC binary weights, where C is the average connectivity. For some range of η, which measures the average correlation between ${\mathit{J}}_{\mathit{i}\mathit{j}}$ and ${\mathit{J}}_{\mathit{j}\mathit{i}}$, the replica-symmetric solution gives the exact ${\mathrm{\alpha }}_{\mathit{c}}$, where in the symmetric limit the capacity per bit is greater than 1. However, for symmetric systems with deeper weight depths, the capacity per bit is less than 1, and no perfect information engine exists.

Learning from examples in feedforward neural networks is studied within a statistical-mechanical framework. Training is assumed to be stochastic, leading to a Gibbs distribution of networks characterized by a temperature parameter T. Learning of realizable rules as well as of unrealizable rules is considered. In the latter case, the target rule cannot be perfectly realized by a network of the given architecture. Two useful approximate theories of learning from examples are studied: the high-temperature limit and the annealed approximation. Exact treatment of the quenched disorder generated by the random sampling of the examples leads to the use of the replica theory. Of primary interest is the generalization curve, namely, the average generalization error ${\mathrm{\epsilon }}_{\mathit{g}}$ versus the number of examples P used for training. The theory implies that, for a reduction in ${\mathrm{\epsilon }}_{\mathit{g}}$ that remains finite in the large-N limit, P should generally scale as αN, where N is the number of independently adjustable weights in the network. We show that for smooth networks, i.e., those with continuously varying weights and smooth transfer functions, the generalization curve asymptotically obeys an inverse power law. In contrast, for nonsmooth networks other behaviors can appear, depending on the nature of the nonlinearities as well as the realizability of the rule. In particular, a discontinuous learning transition from a state of poor to a state of perfect generalization can occur in nonsmooth networks learning realizable rules.

We illustrate both gradual and continuous learning with a detailed analytical and numerical study of several single-layer perceptron models. Comparing with the exact replica theory of perceptron learning, we find that for realizable rules the high-temperature and annealed theories provide very good approximations to the generalization performance. Assuming this to hold for multilayer networks as well, we propose a classification of possible asymptotic forms of learning curves in general realizable models. For unrealizable rules we find that the above approximations fail in general to predict correctly the shapes of the generalization curves. Another indication of the important role of quenched disorder for unrealizable rules is that the generalization error is not necessarily a monotonically increasing function of temperature. Also, unrealizable rules can possess genuine spin-glass phases indicative of degenerate minima separated by high barriers.

Two recently reported treatments [J. M. Porrà et al., Phys. Rev. A 44, 4866 (1991) and I. L’Heureux and R. Kapral, J. Chem. Phys. 88, 7468 (1988)] of the problem of bistability driven by dichotomous colored noise with a small correlation time are brought into agreement with each other and with the exact numerical results of L’Heureux and Kapral [J. Chem. Phys. 90, 2453 (1989)].

A scaling description is given for the long-wavelength behavior of avalanche models. Two examples are analyzed in detail: the ‘‘two-state’’ model and the ‘‘local-limited’’ model. Scaling estimates for the long-wavelength behavior of coherence length, relaxation time, and diffusion coefficient are derived for both models. The results agree well with the estimates obtained from previous simulations.

Numerical Monte Carlo studies and analytical considerations are reported, indicating that in certain models of multilayer irreversible surface adsorption the density may actually increase away from the substrate. This unexpected conclusion is contrary to simple intuitive considerations in the formation of amorphous deposits. The behavior of the density is found to obey a universal power law. Some time-dependent properties are also examined.

Using the results of a recent Monte Carlo simulation and analytical studies on self-avoiding walks (SAW’s) on critical percolation clusters (CPC’s), the various Flory-type formulas for SAW’s on fractals and disordered media are examined. The probability density formulas for a random walker on the fractals that are needed to derive the various Flory approximants are also discussed. We also try to resolve some controversial problems in the recent studies on SAW’s on CPC’s from a most plausible Flory approximant.

In this paper, we study the problem of the covering of the bonds of a finite lattice by a random walk that visits all the lattice sites. One-, two-, three-, and four-dimensional regular periodic lattices are considered. While in one-dimension the problem is trivial, our numerical results show very interesting features in higher dimensions, concerning the set of nonvisited bonds when the site covering has been completed. The number of such bonds as a function of the lattice size follows a square-logarithmic law in two dimensions and a power law in three and four dimensions, suggesting the presence of a fractal geometry.

The relations between the homoclinic structure and spatial coherent pattern in Zakharov equations (ZE’s) are discussed. Our results present Kolmogorov-Arnold-Moser curves and homoclinic crossing for ZE’s, which exhibit the property of a near-integrable system, and Hamiltonian chaos in the ZE’s is revealed.

We present a first-principles density-functional study of the free surface of a Bose Lennard-Jones (BLJ) fluid at zero temperature. Following the Weeks-Chandler-Andersen perturbation scheme for classical LJ fluids, we treat the repulsive part of the BLJ by a Bose hard-sphere (BHS) reference system, where the equivalent BHS diameter is taken from the approach of Kalos, Levesque, and Verlet [Phys. Rev. A 9, 2178 (1974)]. When the de Boer–Michels parameters of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are chosen for the BLJ fluid, the calculated surface properties are in good agreement with those from experiment and Monte Carlo simulation.

The kinetic evolution of the mass distribution of cluster-particle aggregates formed from random-walking monomers is examined. It is shown that in Euclidean dimensions greater than or equal to 3, the variances in the configurationally averaged radii of gyrations (with respect to which the mass scaling law is defined) tend to zero at large times for all random-walk aggregates and tend to a nonzero constant for ballistic-limited aggregates. In Euclidean dimensions less than or equal to 2, the variances diverge with the mean radius of gyration. These results follow crucially from the well-known causality bounds for the fractal dimension of random aggregates.

The kinetics of Brownian particle annihilation on randomly distributed correlated traps is considered. It is proved that trap attraction (repulsion) leads to a slowdown (acceleration) of the process, as compared to the case of noncorrelated traps. The importance of trap correlation length is demonstrated.

The control of the very complex behavior of a plasma with laser interaction by smoothing with induced spatial incoherence or other methods was related to improving the lateral uniformity of the irradiation. While this is important, it is shown from numerical hydrodynamic studies that the very strong temporal pulsation (stuttering) will mostly be suppressed by these smoothing methods too.

The results of the analytical and numerical investigations of the stopping power of a fast ion in an electron plasma under the action of the potential perturbation induced by the transit of a foregoing similar particle are presented. The analysis is based on the assumptions that the velocity of the two ions is the same and is fixed and that their motion occurs on the same straight line. The energy deposition profile in a dense plasma is computed by retaining two-ion correlation effects and is compared with the results of the individual-projectile approximation.