A new class of exact scalar Green’s functions is presented. These functions solve fluid dynamics problems analogous to lens design problems in optics. In particular, given a point signal at a specified location, the Green’s functions discussed here describe waves that propagate asymptotically to specified image points. The geometry of the target determines the structure of the background medium, which may be interpreted to be an acoustic or optical fluid, or a curved space-time. There is no restriction to spherical or axial symmetry. Included in the class are the known propagators for massless fields in Rindler space and in a space of constant negative curvature. In the latter case, the Green’s function derived here is the analog of the Green’s function that describes free-particle nonrelativistic quantum mechanics in a space of constant negative curvature. In general, the waveforms of this class are multiple bubbles that separate from a spherical initial pulse and converge monotonically on point singularities or zero-equipotential surfaces of a pair of underlying electrostatic potentials.

The average degree of ionization for a dense plasma is calculated with the aid of a newly established atomic model based on the Thomas-Fermi (TF) method. This new model is characterized by the following features: (i) the bound electrons and free electrons are treated separately, (ii) a physically reasonable definition of the bound electrons is given, (iii) the system is described as a strongly coupled plasma of free electrons and TF ions, (iv) the source density in the Poisson equation is determined by the electron-ion and ion-ion correlation functions, and (v) the degree of ionization is calculated through the minimization of the total free energy. Results from the first and second approximations are presented, corresponding to models of increasing sophistication for the ion-ion correlation function. A comparison of these results with the results from earlier TF calculations and with the results from the Saha equation is also provided.

By an asymptotic procedure, the complete water wave equation is shown to have a reflected wave after overtaking collisions of two solitary waves [C.-H. Su and Q. S. Zou, Phys. Rev. A 35, 4738 (1987)]. This is a result of a nonlocal behavior of the solution. Using the asymptotic procedure to a model equation of water waves, which is shown to be solvable by the inverse scattering method [D.J. Kaup, Progr. Theor. Phys. 54, 396 (1975)], we show that the solution to the model equation does not have such a nonlocal behavior. Hence the asymptotic method is consistent with the inverse scattering method. It provides a complementary way to check complete integrability for some systems in addition to the inverse scattering method and the Painlevé analysis, etc.

The use of f(α) spectra to characterize multifractals has gained increasing attraction during recent years. There are basically two procedures to determine f(α) from a given point distribution, both offering a number of approaches with different details in the methods. Three of these approaches are applied to the specific example of the galaxy distribution as compiled in the Zwicky et al. Catalogue of Galaxies and Clusters of Galaxies (California Institute of Technology, Pasadena, 1962-1968), Vols. 1–6. As points of particular importance with respect to the finiteness of the sample, the influence of boundary effects, the behavior of rms errors in α and f(α), and differing notions of locality in the different approaches are presented and discussed.

In this paper we present a combined analytical and numerical study of transport properties of Lévy walks. Here, within the framework of continuous-time random walks (CTRW’s) with coupled memories, we focus on the probability $P_0$(t) of being at the initial site at time t and on S(t), the mean number of distinct sites visited in time t. We use the connection between $P_0$(t) and S(t), which are related via their Laplace transform, and we reanalyze our previous findings for 〈$r^2$(t)〉, the mean-squared displacement. Furthermore, S(t) shows, as a function of the memory parameters, a very interesting, nonuniversal, nonmonotonic behavior, which we corroborate by numerical simulations in one dimension. We compare the findings with those for decoupled CTRW’s on regular lattices and on fractals.

We study the transition between the twisted and untwisted states of a cholesteric liquid crystal bounded between two plates and constrained to have homeotropic anchoring at the plate surfaces. Previous theories predict that the twisted-untwisted transition should be second order, while experiments show that it is first order. In this paper we use a variational technique to show that either a first- or a second-order transition is possible, depending on the values of the Frank elastic constants $K_{11}$, $K_{22}$, and $K_{33}$. The previous experiments were conducted on materials whose elastic constants are such that a first-order transition is predicted.

A crucial issue in both mammalian and machine vision is how depth is perceived. In this paper we explore the relationships between the optimization tasks involved in stereoscopic vision and statistical mechanics. The ability of different algorithms in processing visual information to find stereoscopic disparity is examined. An algorithm to perform this task is introduced by expressing the optimization constraints in terms of a generalized spin-glass Hamiltonian. Our main results concern the derivation of the exact free-energy surface of a global version of the model in the spirit of the Sherrington-Kirkpatrick approach to spin glasses.

The addition of external noise to a dynamical system described by an iterated map causes the orbit to escape from the attractor. The escape time τ has the behavior τ≊${\tau }_0$exp($E_0$/Γ), where Γ is the noise temperature, $E_0$ is the minimum escape energy, and ${\tau }_0$ is the inverse of the attempt rate. We will describe an analytical method for calculating the mean escape time based on the principle of minimum escape energy. Analytical solutions for $E_0$ are presented for values of the mapping control parameter a close to tangent bifurcations and interior crises. The minimum escape energy displays a power-law dependence on the control parameter near tangent bifurcations ($E_0$∼‖a-$a_t$${\Vert }^{3/2}$) and near interior crises ($E_0$∼‖a-$a_c$${\Vert }^2$). Numerical solutions are given for control-parameter values throughout the range of the attractor. The results agree with the results of Monte Carlo simulations of the logistic map and with independent work on the noise stability of rf-driven Josephson junctions.

It is shown that Monte Carlo methods can be fruitfully applied to the computation of quantum processes in the real-time (Minkowski) regime. The method is first explained in the context of a simple quantum-mechanical model, then applied to the problem of the quantized electromagnetic field interacting with a large number of discrete two-state systems (the problem of superradiance). Numerical results previously obtained by operator methods are reproduced in this framework and the calculations extended to much larger systems.

We study the propagation of classical waves through a network of waveguides with a randomly varying index of refraction in one, two, and three dimensions. We find evidence for localization of these waves by disorder, although the localization length appears to be very sensitive to the shape of the probability distribution of the random variable. In a medium with a strong dielectric contrast, we find always good evidence for the existence of localized states in some range of frequencies. In three dimensions they are separated in frequency by large regions of extended states. On the other hand, when the dielectric contrast is weak, localized states, even in two dimensions, have a large localization length.

It is shown that the specific configuration of a chiral molecule gives rise to a spontaneous polarization in the smectic-C phase due to the molecular interaction in terms of the electrostatic quadrupole moments that produces the tilt of the director in the smectic-C phase.

High-resolution heat-capacity measurements are reported for the vicinity of the smectic-$A_d$–smectic-$A_2$ critical point in mixtures of undecyloxyphenyl-cyanobenzyloxy benzoate (11OPCBOB) and nonyloxybiphenyl cyanobenzoate (9OBCB). This type of critical point, which has similarities to the familiar liquid-gas (Ising) critical point, is predicted to belong to a new universality class. The variation of $C_{\mathrm{px}}$ in two near-critical mixtures with 11OPCBOB mole fractions X=0.655 and 0.641 can be characterized by the critical exponent γ/Δ=0.6±0.2. Previously reported smectic layer thicknesses d(T) for a mixture with X=0.642 vary like $t^{1\mathrm{-}(\gamma /\Delta )}$ with γ/Δ=0.62. Extensive rounding of the $C_{\mathrm{px}}$(T) peak near $T_c$, which complicates the critical exponent determination, appears to be an intrinsic effect that may be associated with the unusual x-ray intensity variation near $T_c$.

We describe a general procedure to locate periodic saddle orbits in a chaotic attractor reconstructed from experimental data. The method is applied to data from a Belousov-Zhabotinskii chemical reaction. The eigenvalues associated with the saddle orbits are used to estimate the Lyapunov exponents. An analysis of the next amplitude map determines the allowable periodic orbits and yields an estimate of the topological entropy.

An analytical study of the dynamics of a piecewise cubic map depending on two parameters is carried out in this paper. For this type of map, it is shown that a stable fixed point coexists with a chaotic attractor. The way in which the deterministic dynamics of the map undergoes chaos is not representable by means of the standardly proposed routes to chaos. It appears that in this case the chaos is initiated by the appearance of an unstable solution born out of a tangent bifurcation. The general geometric approach presented here in obtaining the analytical results does not make use of the Schwarzian curvature of the map. Moreover, it is shown that only the first few iterates essentially describe the global dynamics of the map. The methodology presented here could be used to good advantage for understanding the dynamics of other types of maps as well. Computational results are provided by the cell-mapping method to expose and confirm the analytical results given in this paper.

The principle of synapse distribution on dendrites in neural networks is proposed. The principle consists of two parts: minimum information transfer and self-similarity in the synapse distribution. The first principle assumes that energy needed for transferring information through the neural network is minimized, while the second principle states that the synapse distribution is invariant under scale transformation. It is shown that the macroscopic distribution of synaptic terminals based on the proposed principle is consistent with the experimental data.

We extend a recent instanton calculation of the escape rate Γ over a one-dimensional potential barrier in the weak-noise limit (D→0), from the case where the noise is Gaussian and exponentially correlated to a more general process. Specifically, the system we initially consider consists of a Langevin equation ẋ=-V’(x)+ξ, where ξ is a Gaussian noise with zero mean and correlator 〈ξ(t)ξ(t’)〉=(D/τ)C(‖t-t’‖/τ), τ being the noise correlation time. Using a path-integral formulation of this process, we find that, in the weak-noise limit, Γ∼exp(-S/D) and calculate S order ${\tau }^6$ for an arbitrary double-well potential. The extension to non-Gaussian processes is briefly discussed.

The dynamics of a highly excited Morse oscillator in an intense laser field are examined using Floquet theory. Two classes of quasienergy states are found, broad states, which extend over a wide range of the Morse basis, and narrow states, which are localized to a small number of Morse states. A comparison to a quantum simulation of the dissociation dynamics via the fast-Fourier-transform grid method reveals that the projection of the initial oscillator state onto the broad states leads to dissociation, whereas the projection onto narrow states implies localization of the oscillator. The Floquet analysis explains a number of observations from the simulation including the slowdown of the dissociation rate and concomitant localization of the oscillator, the threshold dependence of dissociation on field intensity, and the variation in dissociation probability with field frequency and initial oscillator state.

The analytic properties of nonequilibrium potentials are studied in a class of two-variable models undergoing a bifurcation of codimension higher than 1. Several methods for the construction of nonequilibrium potentials are given. Cases are exhibited explicitly where a polynomial expansion does not exist due to logarithmic terms, even though the potential remains smooth. It is concluded (i) that the nonexistence of polynomial expansions near bifurcation points of higher order, recently reported by several authors, does not imply the nonexistence of a smooth potential, and (ii) that even in cases where the Hamilton-Jacobi equation has a particular solution in the form of a power series, that particular solution may still fail to represent the nonequilibrium potential by failing to satisfy the necessary boundary condition at the attractor.

The relativistic generalization of the standard map is introduced, based on the problem of particle acceleration in the electric field of an electrostatic wave packet. It is shown that the diffusion is, in general, less effective than that for nonrelativistic particles. In the ultrarelativistic case chaotic motion appears in the form of intermittency.

The different conditions that could be imposed on a Markovian master equation for a nonlinear oscillator weakly coupled to a thermal reservoir are formulated. They concern preservation of trace and positivity of a density matrix, return to a proper equilibrium state, and the detailed balance condition. It is shown that only one of the known master equations satisfies all of these conditions. Then the validity of the Markovian approximation is reanalyzed using certain non-Markovian weak-coupling approximations, and the existence of different stages of evolution associated with different time scales of the Hamiltonian dynamics is predicted. The consequences of these facts for the description of a damped nonlinear oscillator are discussed.