Showing papers in "Physical Review A in 1988"

Density-functional exchange-energy approximation with correct asymptotic behavior.

Abstract: Current gradient-corrected density-functional approximations for the exchange energies of atomic and molecular systems fail to reproduce the correct 1/r asymptotic behavior of the exchange-energy density. Here we report a gradient-corrected exchange-energy functional with the proper asymptotic limit. Our functional, containing only one parameter, fits the exact Hartree-Fock exchange energies of a wide variety of atomic systems with remarkable accuracy, surpassing the performance of previous functionals containing two parameters or more.

Self-organized criticality

Abstract: We show that certain extended dissipative dynamical systems naturally evolve into a critical state, with no characteristic time or length scales. The temporal ``fingerprint'' of the self-organized critical state is the presence of flicker noise or 1/f noise; its spatial signature is the emergence of scale-invariant (fractal) structure.

Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications

Abstract: The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives.

Quantum Langevin equation.

Abstract: The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i${0}^{+}$)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.

Defects in flexible membranes with crystalline order

Abstract: We study isolated dislocations and disclinations in flexible membranes with internal crystalline order, using continuum elasticity theory and zero-temperature numerical simulation. These defects are relevant, for instance, to lipid bilayers in vesicles or in the ${L}_{\ensuremath{\beta}}$ phase of lyotropic smectic liquid crystals. We first simulate defects in flat membranes, obtaining numerical results in good agreement with plane elasticity theory. Disclinations and dislocations eventually exhibit a buckling transition with increasing membrane radius. We generalize the continuum theory to include such buckled defects, and solve the disclination equations in the inextensional limit. The critical radius at which buckling starts to screen out internal elastic stresses is determined numerically. Computer simulation of buckled defects confirms predictions of the disclination energies and gives evidence for a finite dislocation energy.

Abrikosov dislocation lattice in a model of the cholesteric-to-smectic-A transition.

Abstract: The nematic--to--smectic-A transition in liquid crystals is analogous to the normal to superconducting transition in metals with the Frank director n in liquid crystals playing the role of the vector potential A in metals. The liquid-crystal analog of an external magnetic field is a field, arising, for example, from molecular chirality, leading to a nonzero \ensuremath{ abla}\ifmmode\times\else\texttimes\fi{}n in the equilibrium nematic phase. The cholesteric (twisted nematic) phase is the analog of a normal metal in an external magnetic field. In type-II superconductors in an external magnetic field, the Abrikosov flux lattice phase with partial flux penetration intervenes between the low-temperature Meissner phase and the high-temperature normal-metal phase. In this paper we study the analog in liquid crystals containing chiral molecules of the Abrikosov phase in superconductors. Using a covariant form of the de Gennes free energy, we find that in mean-field theory a state, which we call the twist-grain-boundary (TGB) state, with regularly spaced grain boundaries consisting of parallel screw dislocations, intervenes between the smectic and cholesteric phases. We calculate the liquid-crystal analogs of the upper and lower critical fields ${H}_{c2}$ and ${H}_{c1}$. The properties of the TGB phase depend on the angle 2\ensuremath{\pi}\ensuremath{\alpha} between axes of dislocations in adjacent grain boundaries. \ensuremath{\alpha} can be rational or irrational. When \ensuremath{\alpha}=p/q for mutually prime integers p and q, the TGB state has a q-fold screw axis and quasicrystalline symmetry for crystallographically forbidden q. Our calculations ignore exponentially small terms favoring lock in at rational \ensuremath{\alpha}. We calculate the x-ray scattering intensities in the cholesteric phase near the TGB phase boundary and in the TGB phase for rational and irrational \ensuremath{\alpha}. We also discuss experimental difficulties in observing the TGB state and the possible effects fluctuations not included in mean-field theory might have on its existence.

Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling.

Abstract: We present a computationally efficient scheme of modeling the phase-ordering dynamics of thermodynamically unstable phases. The scheme utilizes space-time discrete dynamical systems, viz., cell dynamical systems (CDS). Our proposal is tantamount to proposing new Ansa$iuml---tze for the kinetic-level description of the dynamics. Our present exposition consists of two parts: part I (this paper) deals mainly with methodology and part II [S. Puri and Y. Oono, Phys. Rev. A (to be published)] gives detailed demonstrations. In this paper we provide a detailed exposition of model construction, structural stability of constructed models (i.e., insensitivity to details), stability of the scheme, etc. We also consider the relationship between the CDS modeling and the conventional description in terms of partial differential equations. This leads to a new discretization scheme for semilinear parabolic equations and suggests the necessity of a branch of applied mathematics which could be called ``qualitative numerical analysis.''

Singular-Value Decomposition and the Grassberger-Procaccia Algorithm

Abstract: A singular-value decomposition leads to a set of statistically independent variables which are used in the Grassberger-Procaccia algorithm to calculate the correlation dimension of an attractor from a scalar time series. This combination alleviates some of the difficulties associated with each technique when used alone, and can significantly reduce the computational cost of estimating correlation dimensions from a time series.

Finite basis sets for the Dirac equation constructed from B splines

Abstract: A procedure is given for constructing basis sets for the radial Dirac equation from B splines. The resulting basis sets, which include negative-energy states in a natural way, permit the accurate evaluation of the multiple sums over intermediate states occurring in relativistic many-body calculations. Illustrations are given for the Coulomb-field Dirac equation and tests of the resulting basis sets are described. As an application, relativistic corrections to the second-order correlation energy in helium are calculated. Another application is given to determine the spectrum of thallium (where finite--nuclear-size effects are important) in a model potential. Construction of B-spline basis sets for the Dirac-Hartree-Fock equations is described and the resulting basis sets are applied to study the cesium spectrum.

Capillary Waves on the Surface of Simple Liquids Measured by X-Ray Reflectivity

Abstract: The properties of the liquid-vapor interface for three simple liquids (water, carbon tetrachloride, and methanol) have been measured using x-ray reflectivity. The measured surface roughness is interpreted using a model that combines the effects of thermally induced capillary waves and the dimensions of the constituent molecules.

Field equation for interface propagation in an unsteady homogeneous flow field.

Abstract: The nonlinear scalar field equation governing the propagation of an unsteadily convected interface is used to derive a convenient expression for the average volume flux through such an interface in a homogeneous flow field. For a particular choice of the initial scalar field, the average volume flux through any such interface is expressed as a volume-averaged functional of the evolving scalar field, facilitating analysis based on renormalized perturbation theory and numerical simulation. It is noted that this process belongs to a different universality class from the propagation model of M. Kardar, G. Parisi, and Y.-C. Zhang (Phys. Rev. Lett. 56, 889 (1986)).

Fast, accurate algorithm for numerical simulation of exponentially correlated colored noise.

Abstract: Traditionally, stochastic differential equations used in the physical sciences have involved Gaussian white noise. ' In recent times, however, white noise has been replaced by colored noise in a variety of contexts. Laser noise problems and first passage time problems have been shown to necessitate the use of colored noise instead of white noise. Even the mathematical foundations for the theory of stochastic differential equations call for colored noise if the Stratonovich perspective is adopted, as it is when physical arguments are invoked. ' In each of these contexts, many speci6c problems require numerical simulation as a component of a complete analysis. This is usually a consequence of nonlinearity and the resulting intractability in purely analytic terms. Consequently, numerical-simulation algorithms have been developed, originally for white noise, and recently for colored noise as well. The simplest type of colored noise to generate is exponentially correlated colored noise. Such noise introduces only one more parameter, the correlation time for the exponential correlation, and it is easily generated by a linear damping equation driven by white noise. Our new algorithm is for this kind of colored noise. In Sec. II we review the white-noise algorithm and the differential version of the exponentially correlated, colored-noise algorithm. In Sec. III we present the integral version of the colored-noise algorithm and demonstrate its superior properties.

Numerical simulations of multiphoton ionization and above-threshold electron spectra.

Abstract: We study above-threshold ionization (ATI) of a one-dimensional model atom in a time-varying external laser field. The time-dependent Schr\"odinger equation is integrated in space and time using the Crank-Nicholson method, and the photoelectron energy spectrum is then computed by projecting the wave function onto the energy eigenstates of the time-independent zero-field Hamiltonian. We demonstrate an intensity-dependent ponderomotive shift of the ionization threshold, find a free-electron scaling of the number of ATI peaks with intensity and frequency of the field, and contrast the numerical simulations with two simple Keldysh-type models. For certain field parameters we encounter turn-on transients in the form of ``energy-nonconserving'' substructure within each ATI peak. Effects on the results of the physical parameters such as length and shape of the laser pulse on one hand, and of the iteration parameters on the other, are discussed in detail.

Density-functional theory for ensembles of fractionally occupied states. I. Basic formalism.

Abstract: A density-functional theory for ensembles of unequally weighted states is formulated on the basis of the generalized Rayleigh-Ritz principle of the preceding paper. From this formalism, two alternative approaches to the computation of excitation energies are derived, one equivalent to the equiensemble method proposed by Theophilou [J. Phys. C 12, 5419 (1979)], the other grounded on an expression relating the excitation energies to the Kohn-Sham single-particle eigenvalues.

Effect of closed classical orbits on quantum spectra: Ionization of atoms in a magnetic field. I. Physical picture and calculations

Abstract: This is the first of two papers that develop the theory of oscillatory spectra. When an atom is placed in a magnetic field, and the absorption spectrum into states close to the ionization threshold is measured at finite resolution, so that individual energy levels are not resolved, it is found that the absorption as a function of energy is a superposition of sinusoidal oscillations. These papers present a quantitative theory of this phenomenon. In this first paper, we describe the physical ideas underlying the theory in the simplest possible way, and we present our first calculations based upon the theory. In the second paper, the theory is developed in full detail, proofs of all of the assertions are given, and we describe the algorithm that was used to make the calculations.

Rayleigh-Ritz variational principle for ensembles of fractionally occupied states

Abstract: The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues ${E}_{1}$\ensuremath{\le}${E}_{2}$\ensuremath{\le}...\ensuremath{\le}${E}_{M}$ of a Hamiltonian H, and given M real numbers ${w}_{1}$\ensuremath{\ge}${w}_{2}$\ensuremath{\ge}...\ensuremath{\ge}${w}_{M}$g0, an upper bound for the weighted sum ${w}_{1}$${E}_{1}$ +${w}_{2}$${E}_{2}$+...+${w}_{M}$${E}_{M}$ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (${w}_{1}$=${w}_{2}$=...=${w}_{M}$). Applications of the generalized principle are discussed.

Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection

Abstract: In this paper the propagation of fronts into an unstable state are studied. Such fronts can occur e.g., in the form of domain walls in liquid crystals, or when the dynamics of a system which is suddenly quenched into an unstable state is dominated by domain walls moving in from the boundary. It was emphasized recently by Dee et al. that for sufficiently localized initial conditions the velocity of such fronts often approaches the velocity corresponding to the marginal stability point, the point at which the stability of a front profile moving with a constant speed changes. I show here when and why this happens, and advocate the marginal stability approach as a simple way to calculate the front velocity explicitly in the relevant cases. I sketch the physics underlying this dynamical mechanism with analogies and, building on recent work by Shraiman and Bensimon, show how an equation for the local ``wave number'' that may be viewed as a generalization of the Burgers equation, drives the front velocity to the marginal stability value. This happens provided the steady-state solutions lose stability because the group velocity for perturbations becomes larger than the envelope velocity of the front.For a given equation, our approach allows one to check explicitly that the marginal stability fixed point is attractive, and this is done for the amplitude equation and the Swift-Hohenberg equation. I also analyze an extension of the Fisher-Kolmogorov equation, obtained by adding a stabilizing fourth-order derivative -\ensuremath{\gamma} ${\ensuremath{\partial}}^{4}$\ensuremath{\varphi}/\ensuremath{\partial}${x}^{4}$ to it. I predict that for \ensuremath{\gamma}(1/12 the fronts in this equation are of the same type as those occurring in the Fisher-Kolmogorov equation, i.e., localized initial conditions develop into a uniformly translating front solution of the form \ensuremath{\varphi}(x-vt) that propagates with the marginal stability velocity. For \ensuremath{\gamma}g(1/12, localized initial conditions may develop into fronts propagating at the marginal stability velocity, but such front solutions cannot be uniformly translating. Differences between the propagation of uniformly translating fronts \ensuremath{\varphi}(x-vt) and envelope fronts are pointed out, and a number of open problems, some of which could be studied numerically, are also discussed.

Chaotic particle transport in time-dependent Rayleigh-Bénard convection

Abstract: The transport of passive impurities in nearly two-dimensional, time-periodic Rayleigh-B\'enard convection is studied experimentally and numerically. The transport may be described as a one-dimensional diffusive process with a local effective diffusion constant ${D}^{\mathrm{*}}$(x) that is found to depend linearly on the local amplitude of the roll oscillation. The transport is independent of the molecular diffusion coefficient and is enhanced by 1--3 orders of magnitude over that for steady convective flows. The local amplitude of oscillation shows strong spatial variations, causing ${D}^{\mathrm{*}}$(x) to be highly nonuniform. Computer simulations of a simplified model show that the basic mechanism of transport is chaotic advection in the vicinity of oscillating roll boundaries. Numerical estimates of ${D}^{\mathrm{*}}$ are found to agree semiquantitatively with the experimental results. Chaotic advection is shown to provide a well-defined transition from the slow, diffusion-limited transport of time-independent cellular flows to the rapid transport of turbulent flows.

Unstable periodic orbits and the dimensions of multifractal chaotic attractors

Abstract: The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily fine-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and Chaotic repellers are considered.

Universality classes for deterministic surface growth.

Abstract: A scaling theory for the generalized deterministic Kardar-Parisi-Zhang (1986) equation with beta greater than 1, is developed to study the growth of a surface through deterministic local rules. A one-dimensional surface model corresponding to beta = 1 is presented and solved exactly. The model can be studied as a limiting case of ballistic deposition, or as the deterministic limit of the Eden (1961) model. The scaling exponents, the correlation functions, and the skewness of the surface are determined. The results are compared with those of Burgers' (1974) equation for the case of beta = 2.

Density-functional theory for ensembles of fractionally occupied states. II. Application to the He atom.

Abstract: The two density-functional methods of calculating excitation energies proposed in the preceding paper, combined with the recently formulated quasi-local-density approximation for the equiensemble exchange-correlation energy functional [W. Kohn, Phys. Rev. A 34, 737 (1986)], are applied to the He atom. Although the splittings between nearly degenerate levels with different angular momenta are badly overestimated, in both approaches the averages over angular momentum and spin of the experimental excitation energies measured from the ionization threshold are reproduced within a few percent. The computed self-consistent ensemble-averaged densities and the Kohn-Sham potentials associated with them are discussed.

Intrinsic bending force in anisotropic membranes made of chiral molecules.

Abstract: A linear term linked to molecular chirality occurs in the bending energy of membranes with C/sub 2/ or D/sub 2/ symmetry. We find it to alternate in sign, depending on the direction of a cylindrical curvature, and discuss consequences for anisotropic solid membranes, tilted fluid bilayers, and ferroelectric smectic liquid crystals.

Nonadiabatic coupled-rearrangement-channel approach to muonic molecules.

Abstract: A new variational approach to muonic molecules is proposed. The total three-body wave function is expanded in terms of basis functions spanned over the three rearrangement channels in the Jacobian coordinate system. Energy levels of the dt\ensuremath{\mu} molecule are calculated with a high accuracy and a short computation time. For the weakly bound state with J=v=1, which is a key to the muon-catalyzed d-t fusion, the calculated energy ${\ensuremath{\varepsilon}}_{11}$ is better than the literature data. With the use of the most up-to-date, 1986 CODATA--recommended [E. R. Cohen and B. N. Taylor, CODATA Bull. 63 (1986)] values of physical constants, we obtained ${\ensuremath{\varepsilon}}_{11}$=-0.660 264 eV with 2662 basis functions and ${\ensuremath{\varepsilon}}_{11}$(\ensuremath{\infty})=-0.66030\ifmmode\pm\else\textpm\fi{}000 02 eV by extrapolation.

Radiation and lifetimes of atoms inside dielectric particles.

Abstract: The transition rates of atoms inside spherical dielectric particles are computed and studied as functions of the transition frequency and of the physical properties of the host particle. The rates are found to range from about 0.2 to more than 1500 times the free-space value, depending on the location of the atom and other relevant physical parameters. The radiation from distributions of atoms inside liquid drops is also studied. Analytic and numerical results are given for the case of a uniform distribution of excited atoms. Large enhancements of the power output (some 100 times the value for the same distribution in bulk material) are found to occur under resonant conditions.

Relaxation of spins due to field inhomogeneities in gaseous samples at low magnetic fields and low pressures.

Abstract: We have developed a theory for the effect of magnetic-field inhomogeneities on the spin relaxation of gases in cells with negligible relaxation at the walls. There is a characteristic pressure ${p}^{\mathrm{*}}$ at which the time ${\ensuremath{\tau}}_{d}$ required for an atom to diffuse across the cell is equal to the time ${\ensuremath{\tau}}_{l}$ required for the spin to precess by one radian in the mean magnetic field. For ``high pressures,'' p\ensuremath{\gg}${p}^{\mathrm{*}}$, the longitudinal spin-relaxation time ${T}_{1}$ is inversely proportional to the pressure. This is the classic pressure dependence discussed in the literature. The new results reported in this paper are that at ``low pressures,'' p\ensuremath{\ll}${p}^{\mathrm{*}}$, the pressure dependence changes and the longitudinal relaxation time becomes directly proportional to the pressure; that is, motional narrowing occurs. We show that the transverse relaxation time ${T}_{2}$ will ordinarily be proportional to the pressure at both low and high pressures, but with different coefficients. There is also a small, pressure-dependent shift of the Larmor frequency associated with the field inhomogeneity.

Topological and metric properties of Hénon-type strange attractors.

Abstract: We use the set of all periodic points of H\'enon-type mappings to develop a theory of the topological and metric properties of their attractors. The topology of a H\'enon-type attractor is conveniently represented by a two-dimensional symbol plane, with the allowed and disallowed orbits cleanly separated by the ``pruning front.'' The pruning front is a function discontinuous on every binary rational number, but for maps with finite dissipation \ensuremath{\Vert}b\ensuremath{\Vert}l1, it is well approximated by a few steps, or, in the symbolic dynamics language, by a finite grammar. Thus equipped with the complete list of allowed periodic points, we reconstruct (to resolution of order ${b}^{n}$) the physical attractor by piecing together the linearized neighborhoods of all periodic points of cycle length n. We use this representation to compute the singularity spectrum f(\ensuremath{\alpha}). The description in terms of periodic points works very well in the ``hyperbolic phase,'' for \ensuremath{\alpha} larger than some ${\ensuremath{\alpha}}_{c}$, where ${\ensuremath{\alpha}}_{c}$ is the value of \ensuremath{\alpha} corresponding to the (conjectured) phase transition.

Electron collision excitations in complex spectra of ionized heavy atoms.

Abstract: A theory for calculating a many-transition spectrum of electron-ion collisional excitations in the distorted-wave approximation (DWA) is presented. First, it is shown that the collision strength including exchange can be factorized into (i) a radial part, involving one-electron wave functions only, and the summation over partial waves of the continuum electron; and (ii) an angular part, involving the coupling between bound electrons in the target states only, specific to each transition. Factorized representations of the collision strengths are derived in various coupling schemes. Second, the computationally involved radial part is shown to be a smooth function of transition energies over a very wide range, allowing easy interpolation. These two results enable one to obtain a complete collisional-excitation array with a drastic reduction of the number of time-consuming radial calculations compared with standard methods. This allows the solution of problems which were heretofore considered impractical. As an illustration, the whole array of excitation rate coefficients for Ni-like Gd xxxvii including the lowest 107 levels (5671 transitions) was calculated in the DWA, and used in a steady-state collisional-radiative model. Resulting population inversions are presented versus plasma density.