A simple configuration-interaction calculational procedure with a finite-${\mathit{L}}^2$-basis set constructed from B splines is presented and applied to a multiphoton ionization process. An initial application of this theoretical procedure to the four-photon nonresonant ionization Mg at 532 nm has led to excellent agreement with the experimental photoelectron angular distribution measurement. The atomic structure effects due to the multielectron interactions are also examined in detail.

The basic features characterizing the dynamical evolution of the phase of a detuned-laser field under an unstable regime are physically interpreted in terms of dispersive and dynamical effects. A general method for obtaining any attractor projection containing the phase information is established, which provides evidence for the heteroclinic character of the attractor in the presence of cavity detuning for any emission regime.

We give analytical results for the fluorescence spectrum of a two-level atom interacting with fully amplitude-modulated field at resonance. The results are valid for unequal longitudinal and transverse relaxation rates and follow the experimentally observed pattern. The intensity-dependent changes in the peak widths and peak heights are presented. Novel transient effects are predicted.

In a recent paper, Ford, Lewis, and O’Connell [Phys. Rev. A 37, 4419 (1988)] considered a charged quantum particle moving in an arbitrary potential and linearly coupled to a heat bath, and they showed that the macroscopic equation describing the time development of the particle motion is in the form of a quantum generalized Langevin equation. We generalize these results to include the presence of an external magnetic field. We find that the magnetic field manifests itself in the presence of an additional term in the Langevin equation, which is the quantum generalization of the Lorentz force, but the magnetic field does not affect the memory function nor the random force appearing in the quantum Langevin equation. It follows that the noise-noise autocorrelation function, as well as the nonequal time commutator of the noise, is the same as that in the absence of a magnetic field. The case of a blackbody radiation heat bath is shown to be easily analyzed as a special case of our general formalism.

We present a method to obtain the scaling properties for multifractal shapes and distributions obtained from affine transformations. Our analysis defines two scaling exponents F(β) and F̃(β); the former relates to geometry, while the latter is associated with the distribution of gradients for networks in Laplacian fields. We determine the exact relation between F(β) and F̃(β). We argue for electrical networks that more information on scaling behavior may be obtained by only taking into account the maximal current leaving a node. Our picture suggests a new route to characterize the scaling properties of electrical networks in general.

Irreversible many-particle dynamical systems are relevant to a variety of phenomena in physics, chemistry, and biology. We present a study of an irreversible kinetic reaction model for a one-component autocatalytic reaction A+A→${\mathit{A}}_2$. In this model, if an atom adsorbing on a lattice site has any neighbors, it reacts with one of them with a probability 1-p, and the two atoms leave the lattice; otherwise the atom occupies the site. As p is varied, this model undergoes a second-order kinetic phase transition from a chemically reactive state with a partial occupation of the lattice to a completely covered state that corresponds to the ‘‘poisoning’’ phenomenon seen on catalysts. The transition is studied both analytically through various mean-field approximations and numerically in one, two, and three dimensions. Finite-size-scaling analysis of the critical behavior is used to find the static and dynamic critical exponents. These exponents are found to be consistent with the critical exponents in the Reggeon-field-theory directed-percolation universality class.

The fluctuation-spectrum theory and the generalized time-correlation function method established for the statistical characterization of univariate temporal fluctuations are extended to multivariate fluctuations by introducing the concept of the complete set of temporal fluctuations for the overall description of dynamical behaviors of the physical system. We develop the eigenfunction expansion for relevant quantities and further construct the continued-fraction expansion. The present approach enables us to study the explicit cross correlation among several time sequences as well as a detailed description of the dynamical behaviors of the system.

The formulation of phase transitions in the framework of catastrophe theory is studied, taking as a reference the tricritical two-fluid system. Adding renormalization-group ideas, a general approach to its classification is explored.

An algorithm to estimate the average local intrinsic dimension (〈${\mathit{d}}_{\mathrm{LID}}$〉) of an attractor using signal versus noise separation methods based on information-theoretic criteria is explored in this work. Using noisy sample data the 〈${\mathit{d}}_{\mathrm{LID}}$〉 is computed from an eigenanalysis of local attractor regions, indicating the local orthogonal directions along which the data are clustered. The 〈${\mathit{d}}_{\mathrm{LID}}$〉 algorithm requires the separation of signal eigenvalues, i.e., the dominant eigenvalues, from the noise eigenvalues for which thresholding mechanisms based on two information-theoretic criteria are used. Singular-value decomposition is used to calculate the eigenvalues as well as to determine the rank of the local-phase-space data matrix. The two specific information-theoretic criteria which we consider to separate signal and noise are the Akaike information criterion (AIC) [H. Akaike, IEEE Trans. Auto. Control 19, 716 (1974)] and the minimum description length (MDL) of Rissanen [Automatica 14, 465 (1978)] and Schwarz [Ann. Stat. 6, 461 (1978)]. Several test cases are presented demonstrating the use of the AIC and MDL to calculate 〈${\mathit{d}}_{\mathrm{LID}}$〉. Results are then compared to the correlation dimension as computed by the Grassberger-Procaccia method [Physica D9, 189 (1983)]. The AIC and MDL separation techniques are found to be inappropriate at higher signal-to-noise ratios (SNR), above ∼12 dB, and the MDL produces results in good overall agreement with the accepted correlation dimension in the range of SNR between 5 and 12 dB. Two of the primary benefits of the 〈${\mathit{d}}_{\mathrm{LID}}$〉 technique are that it will yield reasonable and fairly accurate estimates of the dimension of attractors and that it can probe small length scales on the attractor even for moderate SNR. Other thresholding schemes are being pursued for SNR’s above 12 dB.

We have done numerical simulations of a lattice model for irreversible diffusion-dominated reactions of the type A+B→C in one to four dimensions with (I) C inert; (II) A=B, C inert; (III) A=B=C; and (IV) C=B with B immobile. Where possible, the results are compared with theoretical predictions of the densities of the various species at long times. Except for case (IV), the simulation results are consistent with these predictions, aside from size effects, which are not easily overcome in all instances.

Critical exponents of the continuum-percolation system of two-dimensional distributions of disks in the penetrable concentric shell model are determined by Monte Carlo simulations and by real-space Monte Carlo renormalization-group techniques. In this model, each disk of diameter σ is composed of a mutually impenetrable core of diameter λσ (0≤λ≤1) encompassed by a perfectly penetrable shell of thickness (1-λ)σ/2. Pairs of particles are connected when the interparticle distance is less than σ. We find that the susceptibility exponent γ is given by 2.50±0.03 for an impenetrability parameter λ=0.8 and the correlation-length exponent ν to be between 1.30 and 1.35 for various values of λ. Both results consistently suggest that continuum percolation in the penetrable concentric shell model for nonzero hard-core radii belongs to the same universality class as that of ordinary lattice percolation and of randomly centered disks, as far as the geometrical critical exponents are concerned. We also present the critical reduced number densities and critical area fractions for selected values of λ.

Self-avoiding walks (SAW’s) and manifolds (SAM’s) in random environments are studied using a combination of Lifshitz arguments and field-theoretic methods. The number of N-step SAW’s starting at the origin, Z, is shown to be a broadly distributed quantity whose typical value, ${\mathit{Z}}_{\mathrm{typ}}$, behaves as ${\mathit{Z}}_{\mathrm{typ}}$∼〈Z〉exp(-${\mathit{cN}}^{\mathrm{\alpha }}$) below four dimensions. Here α=2-dν and 〈Z〉 is the average number of SAW’s at the origin. On the other hand, the integer moments of Z are exponentially larger than the average, i.e., 〈${\mathit{Z}}^{\mathit{k}}$〉∼〈Z${\mathrm{〉}}^{\mathit{k}}$exp[${\mathit{ck}}^{1/\mathrm{\alpha }}$(k-1)N] for the range 1<k<${\mathit{k}}_{\mathit{c}}$. Similar results hold for SAM’s. Within the field theory for SAW’s the results for 1<k<${\mathit{k}}_{\mathit{c}}$ arise from a fluctuation-driven first-order phase transition in the k-replicated theory. Above ${\mathit{k}}_{\mathit{c}}$, Griffiths singularities control the moments of Z.

We calculate noninteger moments 〈${\mathit{t}}^{\mathit{q}}$〉 of first passage time to trapping, at both ends of an interval (0,L), for some diffusion and dichotomous processes. We find the critical behavior of 〈${\mathit{t}}^{\mathit{q}}$〉, as a function of q, for free processes. We also show that the addition of a potential can destroy criticality.

It has long been known that the time course of a bimolecular reaction occurring in a condensed host depends on the behavior of the nonequilibrium pair-correlation function for reactant pairs. The classical analysis of such reactions has led to a kind of standard rule: The association rate constant for a diffusion-controlled reaction is 4πDR and this rate constant produces the fastest possible kinetics. This result is only (approximately) true for the case of an irreversible reaction, however. Here, we reexamine this old problem, looking closely at the reversible case. We report a result that challenges the standard wisdom: When the reaction is highly reversible the relaxation of the related kinetics to equilibrium can be much faster than the model in which 4πDR is the association rate constant. We suggest that our work provides a natural resolution to a well-known, long-standing controversy in the study of electrically active impurities in silicon grown by the Czochralski method.

The quantum-mechanical version of the Kramers turnover problem is considered. The multidimensional character of the problem is taken into account via transformation to normal modes. This eliminates the coupling to the bath near the barrier top allowing the use of a simple harmonic transmission coefficient for the barrier dynamics. The well dynamics is described by a continuum form of a master equation for the energy in the unstable normal mode. Within first-order perturbation theory, the equations of motion for the stable normal modes have the form of a forced oscillator. The transition probability kernel is found using the known solution for the quantum forced oscillator problem. An expression for the quantum escape rate is derived. It encompasses all previously known limiting results in the thermally activated tunneling regime. The depopulation factor, which accounts for the nonequilibrium energy distribution is evaluated. The quantum transition probability kernel is broader than the classical and is skewed towards lower energies. Interplay between these two effects, together with a positive tunneling contribution, determines the relative magnitude of the quantum rate compared to the classical one. The theory is valid for arbitrary dissipation. Its use is illustrated for the case of a cubic potential with Ohmic (Markovian) dissipation.

We study the influence of system-bath correlations in the initial state on the dynamics of biased two-state systems. The formal solution for the equilibrium correlation function in the form of a power series in the number of transitions between the system states is determined. For Ohmic damping and a special value of the coupling strength (K=1/2), the exact solution for all times and temperatures is presented. We find that at low temperatures the long-time tails of the decay into equilibrium are strongly affected by the initial preparation. Correspondingly, the low-frequency behavior of the neutron-scattering function is qualitatively changed by the presence or absence of correlations in the initial state.

In the context of a phase transition, the critical singularity is characterized by a set of numbers, the critical exponents. It is widely believed that for transitions in systems with pair interactions on Bravais lattices (as opposed to hierarchical lattices), the critical exponents are determined solely by parameters d and n, representing the dimensionality of space and of the order parameter, respectively, and by σ, the interaction range. In this paper we consider a certain generalization of the Lifshitz singularity in the n-vector model in which a new parameter λ, describing a type of anisotropy of the interactions, is introduced. This is described by a model Hamiltonian in k space in which the leading term is of the form [${\mathrm{\Sigma }}_{\mathit{i}=1\mathrm{}}^{\mathit{d}}$(${\mathit{k}}_{\mathit{i}}^2$${)}^{\mathrm{\sigma }/(2\mathrm{}\lambda )}$]λ. It should be noted that, as for the usual Lifshitz case, what would otherwise be the dominant ${\mathit{k}}^2$ term is supposed to have already been cancelled. For this model the parameters d, n, and σ are not sufficient to determine the value of the critical exponents to O(1/n), although they are sufficient for the first order in the ε expansion.

Calculations exact to O(1/n) show that, for a given set d, n, and σ, the exponents vary continuously with the parameter λ and include as a special case (for λ=1) the critical exponents for an anisotropic Lifshitz point. The critical exponents for this Lifshitz point differ markedly in value from those for the usually studied isotropic Lifshitz point, which is also a special case (for λ=σ/2) of our model. For σ=2, however, the two cases corresponding to λ=1 and σ/2 become degenerate. We provide another explicit example of the dependence of the critical exponents on λ by performing, in addition to the 1/n expansion, an expansion in σ-2. This helps to shed some light on the approach to the degeneracy at σ=2. The analysis of the Feynman integrals arising in the O(1/n) calculation provides interesting examples of the applications of hypergeometric functions; these integrals have themselves led to new results (published separately) in the theory of hypergeometric functions.

It is well known that the Belousov-Zhabotinskii reaction can exhibit planar traveling waves in the excitable-kinetics regime. Using the previously formulated piecewise-linear Oregonator model, we study the stability of these waves to general disturbances. We find possible linear instabilities, either to transverse structure or to a longitudinal oscillation; these set in below a critical velocity, and which one occurs first depends on the ratio of the diffusivities of the two chemical species. We also consider the weakly nonlinear bifurcation analysis for the steady-state bifurcation, showing that the bifurcation is supercritical over a wide parameter range. The implications of these results for experimental systems are briefly discussed.

The period-M (M=1 and 2) scaling pattern of period doubling in a symmetric four-dimensional volume-preserving map is studied. The period-doubling sequence repeats itself asymptotically from one bifurcation to the next in a period-1 bifurcation route, and to every other one in a period-2 bifurcation route. The parameter-scaling factors γ1 and γ2 in a bifurcation route depend on the bifurcation path. They are some combination of δ1 and δ2 (divergence rates from the period-M map of the renormalization transformation) and δ1′ and δ2′ (convergence rates to the period-M map). Therefore each bifurcation route is characterized by these four scaling factors. The values of δ1, δ2, δ1′, and δ2′ are obtained by a numerical calculation and a renormalization analysis. We find that there are three kinds of period-1 scaling patterns and one kind of period-2 scaling pattern. δ1 and δ1′ in any bifurcation route are the same as those in area-preserving maps; however, δ2 and δ2′ depend on the bifurcation route.

A simplified model of the evolution of molecular sequences is proposed. An ensemble of strings is considered that consists of two letters and undergoes random point mutations and natural selections. A set of evolution equations is deduced. From the solution it is found that the first-order (second-order) informational parameters (redundancies) ${\mathit{D}}_1$ decrease (${\mathit{D}}_2$ increase) in the course of evolution. Furthermore, the statistical correlations of the letters (bases) in the sequences are investigated in detail and the short-distance correlation is demonstrated. These results give a preliminary explanation of some physical aspects in the evolution of nucleic acid sequences.