Chaotic scattering,transport, and fractals in a simple hydrodynamic flow

Advection of passive tracers in an unsteady hydrodynamic flow consisting of a background stream and a vortex is analyzed as an example of chaotic particle scattering and transport. A numerical analysis reveals a nonattracting chaotic invariant set Λ that determines the scattering and trapping of particles from the incoming flow. The set has a hyperbolic component consisting of unstable periodic and aperiodic orbits and a nonhyperbolic component represented by marginally unstable orbits in the particle-trapping regions in the neighborhoods of the boundaries of outer invariant tori. The geometry and topology of chaotic scattering are examined. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle coordinates. The hierarchy is found to have certain properties due to an infinite number of intersections of the stable manifold in Λ with a material line consisting of particles from the incoming flow. Scattering functions are singular on a Cantor set of initial conditions, and this property must manifest itself by strong fluctuations of quantities measured in experiments.     

Periodic and Quasi-Periodic Orbits¶for the Standard Map

We consider both periodic and quasi-periodic solutions for the standard map, and we study the corresponding conjugating functions, i.e. the functions conjugating the motions to trivial rotations. We compare the invariant curves with rotation numbers ω satisfying the Bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents ω _N = p _N /q _N . We prove the following results for N→ ∞: (1) for rotation numbers ω _N N we study the radius of convergence of the conjugating functions and we find lower bounds on them, which tend to a limit which is a lower bound on the corresponding quantity for ω; (2) the periodic orbits consist of points which are more and more close to the invariant curve with rotation number ω; (3) such orbits lie on analytical curves which tend uniformly to the invariant curve. Received: 14 December 2001 / Accepted: 16 March 2002?Published online: 2 October 2002     

Classical scattering theory in one dimension

We discuss various scattering properties of the classical system ofn repelling particles on the real line. In the integrable case, i.e. if the asymptotic velocities are preserved under the scattering map, the asymptotic phases behave as if the particles collided pairwise.     

Self-consistent field theory of collisions I. Scattering channels

The conventional Hartree and Hartree-Fock approaches for treating many-electron bound systems have been extended recently to positive energy scattering problems, in which both the bound and continuum orbitals are determined by the requirement of full self-consistency. Serious consequences of such a theory are that the target orbitals become energy dependent and the asymptotic boundary conditions are satisfied only approximately, in lowest order. It is important therefore to test the theory for its convergence under configuration mixing. This self-consistent field (SCF) theory for scattering has been tested here for scattering from hydrogenic target as a model where the target function is determined dynamically. Penetration of the projectile inside the bound target orbital is manifest through the SCF for the bound state. Our results show that the theory converges to the correct amplitudes and to the exact boundary conditions as more configurations are added. The use of the amputated functions and the weak asymptotic condition (WAC) upon which the SCF theory is based, is justified as the WAC converges to the correct limit. It is then applied to the positron-helium and electron-helium scattering systems where the helium function is calculated simultaneously together with the scattering function. The resulting phase shifts and the SCF target functions are compared with those obtained with the pre-determined target functions in the conventional approaches. Received 22 September 1999     

Calculation of the energy density function by the method of steepest descent

We present a unified theory of diatomic molecules which reconciles bound state spectroscopy and atomic scattering theory. The total wave-function is expanded in a complete set of atomic channel states which is entirely equivalent to an expansion in Hund's case (e) electronic-rotational states. An analysis of the coupled radial, that is vibrational, functions places strong constraints on the asymptotic properties of the molecular wave-functions. These are presented in terms of the reactance K and scattering S matrices of atomic scattering theory which offers a uniform treatment for open channels (inelastic scattering and continuum spectroscopy), closed channels (bound state spectroscopy) and mixtures of both (predissociation). The normalization of the total wavefunction is derived and related to the asymptotic boundary conditions both for continuum and bound states.     

The effect of high momentum transfer on scattering from oscillators and crystals

We examine two aspects of scattering at high energy and momentum transfer; asymptotic forms of scattering laws and comparison of approximate and exact results for a particular system, a harmonic lattice.By treating model systems of harmonically bound particles, an asymptotic analysis shows that, under conditions of high momentum transfer the particle looks as though it is unbound, in the sense that the weighting (envelope) of the various inelastic channels is that of a free particle, although the discreteness of the allowed energy transfers reflects the fact that the particles are, in reality, bound. The discreteness of the energy transfers requires an extension of the usual impulse approximation, which had been realised by Wick for a total cross section calculation, and this is treated in an appendix.An exact numerical technique for handling multiphonon effects directly, that is, without expansions, for the case of scattering from a lattice is used as a check of the Sjölander approximation. The latter treats elastic and one-phonon processes exactly, the remainder in a Gaussian-like approximation. As expected this scheme works well at low and high momentum transfer. At intermediate values the scattering law turns out to have a lot of structure which isnot reproduced. Simple interpretations would then give split peaks and such effects which in reality are simply manifestations of multiphonon processes.     

Chaotic scattering on a double well: Periodic orbits,symbolic dynamics,and scaling

We investigate classical scattering of particles by a double-well potential. Irregularity in the scattering functions, such as scattering angle and escape time, appears when the collision energy is lowered below a threshold value. This threshold is closely related to the appearance of periodic orbits with energies above the potential maxima. We study the scattering as a function of the energy and impact parameter. In this initial parameter space the scattering functions consist of regular regions interlaced with chaotic rivers. A symbolic dynamics has been developed to organize these structures and used to reveal their scaling properties.     

Asymptotics of Rydberg States for the Hydrogen Atom

The asymptotics of Rydberg states, i.e., highly excited bound states of the hydrogen atom Hamiltonian, and various expectations involving these states are investigated. We show that suitable linear combinations of these states, appropriately rescaled and regarded as functions either in momentum space or configuration space, are highly concentrated on classical momentum space or configuration space Kepler orbits respectively, for large quantum numbers. Expectations of momentum space or configuration space functions with respect to these states are related to time-averages of these functions over Kepler orbits. Received: 8 November 1996 / Accepted: 13 January 1997     

Boundary perturbation methods for high-frequency acoustic scattering: shallow periodic gratings

Despite significant recent advances in numerical methodologies for simulating rough-surface acoustic scattering, their applicability has been constrained by the limitations of state-of-the-art computational resources. This has been particularly true in high-frequency applications where the sheer size of the full-wave simulations render them impractical, and engineering processes must therefore rely on asymptotic models [e.g., Kirchhoff approximation (KA)]. However, the demands for high precision can make the latter inappropriate, thus efficient, error-controllable methodologies must be devised. This paper presents a computational strategy that combines the virtues of rigorous solvers (error control) with those of high-frequency asymptotic models (frequency-independent computational costs). These methods are based on high-order "boundary perturbations," which display high precision and unparalleled efficiency. This is accomplished by incorporating asymptotic phase information to effect a significant decrease in computational effort, simultaneously retaining the full-wave nature of the approach. The developments of this contribution are constrained to configurations that preclude multiple scattering; it is further explained how the schemes can be made applicable to general scattering scenarios, though implementation details are left for future work. Even for single-scattering configurations, the approach presented here gives significant gains in accuracy when compared to asymptotic theories (e.g., KA) with modest additional computational cost.     

High-Speed Scattering of Charged and Uncharged Particles in General Relativity

After a brief consideration of the high-speed scattering of two point charges we thoroughly discuss high-speed scattering for a charged particle by a fixed mass and of two uncharged particles of comparable masses. We use perturbation technique over Minkowski spacetime in the de Donder gauge and solve the field equations and the resulting equations of motion (which take the reaction of the particles' quasistatic self-field into account) by iteration. The obtained energy-momentum conservation laws allow the computation of second-order corrections for the scattering angle and the cross section. The asymptotic structure of the far-field indicates synchrotron radiation (electromagnetic and gravitational, respectively) which causes an energy loss whose reaction on the motion is briefly considered in the low-velocity limit including bound motion. (For neutral particles this is a third-order effect).     

Asymptotically optimal quantum circuits for d-level systems

Scalability of a quantum computation requires that the information be processed on multiple subsystems. However, it is unclear how the complexity of a quantum algorithm, quantified by the number of entangling gates, depends on the subsystem size. We examine the quantum circuit complexity for exactly universal computation on many d-level systems (qudits). Both a lower bound and a constructive upper bound on the number of two-qudit gates result, proving a sharp asymptotic of theta(d(2n)) gates. This closes the complexity question for all d-level systems (d finite). The optimal asymptotic applies to systems with locality constraints, e.g., nearest neighbor interactions.     

Analytic continuation of the effective-range expansion as a method for determining the features of bound states: Application to the 6Li nucleus

The scattering function (effective-range function) for the two-channel elastic scattering of charged particles is used to analyze dα scattering at low energies. In order to construct this function, use is made of various sets of phase shifts and mixing parameter, both those that were obtained by solving Faddeev equations in the three-body (n, p, α) model and those that were deduced from available phaseshift analyses. By means of an analytic continuation of the scattering function to the point of the pole corresponding to the bound state of the ⁶Li nucleus, the values of the vertex constants and asymptotic normalization coefficients are found for the process ⁶Li → α + d. Possible means for refining these results are discussed.     

Structure and Occurrence of Three-Body Halos in Two Dimensions

 The quantum-mechanical three-body problem is reformulated in two dimensions by use of hyperspherical coordinates and an adiabatic expansion of the Faddeev equations. The effective radial potentials are calculated and their large-distance asymptotic behavior is derived analytically for short-range two-body interactions. Energies and wave functions are computed numerically for various potentials. An infinite series of Efimov states does not exist in two dimensions. Borromean systems, i.e. bound three-body systems without bound binary subsystems, can only appear when a short-range repulsive barrier at finite distance is present in the two-body interaction. The corresponding Borromean state is never spatially extended. For a system of three weakly interacting identical bosons we find two bound states with both binding energies proportional to the two-body binding energy. In the limit of small binding these states are spatially located at the very large distances characterized by the scattering length. Their properties are universal and independent of the details of the potential. We compare throughout with the corresponding properties in three dimensions. Received September 25, 1998; accepted for publication January 30, 1999     

Charged particle motion in an electromagnetic field on Kerr background geometry

In this paper we study the trajectories of charged particles in an electromagnetic field superimposed on the Kerr background. The electromagnetic fields considered are of two types: (i) a dipole magnetic field with an associated quadrupole electric field, (ii) a uniform magnetic field. The contribution of the background geometry to the electromagnetic field is taken through the solutions of Petterson and Wald respectively. The effective potential is studied in detail for ther-motion of the particles in the equatorial plane and the orbits are obtained. The most interesting aspect of the study is the illustration of the effect of inertial frame dragging due to the rotation of the central star. This appears through the existence of nongyrating bound orbits at and inside the ergo surface. The presence of the magnetic field seems to increase the range of stable orbits, as was found in a previous study involving the Schwarzschild background.     

Charged particle trajectories in a magnetic field on a curved space-time

In this paper we have studied the motion of charged particles in a dipole magnetic field on the Schwarzscbild background geometry. A detailed analysis has been made in the equatorial plane through the study of the effective potential curves. In the case of positive canonical angular momentum the effective potential has two maxima and two minima giving rise to a well-defined potential well rear the event horizon. This feature of the effective potential categorises the particle orbits into four classes, depending on their energies. (i) Particles, coming from infinity with energy less than the absolute maximum ofV _eff, would scatter away after being turned away by the magnetic field. (ii) Whereas those with energies higher than this would go into the central star seeing no barrier. (iii) Particles initially located within the potential well are naturally trapped, and they execute Larmor motion in bound gyrating orbits. (iv) and those with initial positions corresponding to the extrema ofV _eff follow circular orbits which are stable for non-relativistic particles and unstable for relativistic ones. We have also considered the case of negative canonical angular momentum and found that no trapping in bound orbits occur for this case. In the case when particles are not confined to the equatorial plane we have found that the particles execute oscillatory motion between two mirror points if the magnetic field is sufficiently high, but would continuously fall towards the event horizon otherwise. An erratum to this article is available at .     

Intercomparison of Inversion Algorithms for Particle‐Sizing Using Mie Scattering

The applicability of different inversion algorithms to retrieve a size distribution of particles in air from light scattering is examined. The investigation is focused on an optical measurement setup with an elliptical mirror as the main optical element. In order to evaluate the capabilities of the individual inversion methods, light scattering by spherical particles is simulated in the size ranges of 0.1 – 10 μm and 0.05 – 1 μm. The distribution of the particle diameters is modeled with three different parametric functions, i.e., RRSB, logarithmic‐normal and a more specific distribution from an ultrasonic nebulizer. Different kinds of noise, e.g., additive and/or multiplicative, are applied in different levels to the simulated scattering measurement to include real physical measurement conditions. The convergence properties of the scattering simulation are investigated with respect to the number of size classes, and thus, information concerning the size resolution required to simulate a measurement for a given particle size distribution is obtained. Further parameters of interest are the minimum angular resolution of the measurements, the number of size classes of the retrieved particle size distribution and the measured polarization of the scattered light.     

大气气溶胶粒子数密度谱和折射率虚部的测量

介绍一种综合利用光学粒子计数器和能见度仪测量大气气溶胶折射率虚部的新方法。首先，使用光学粒子计数器测量出大气气溶胶粒子的数密度谱(待订正)，用能见度仪同步测量出水平能见度。然后，根据球形粒子的米氏(Mie)散射理论，通过分析气溶胶粒子的折射率虚部、分档半径、粒子数密度谱、消光系数和能见度之间的关系，对分档半径进行订正，得到折射率虚部和能见度之间的对应关系。结合同步测量的能见度，反演出大气气溶胶粒子的折射率虚部。最后，利用折射率虚部对光学粒子计数器数据进行订正，得到大气气溶胶粒子的数密度谱。     

Ultraviolet renormalons in abelian gauge theories

We analyze the large-order behaviour in perturbation theory of classes of diagrams with an arbitrary number of chains (i.e. photon lines, dressed by vacuum polarization insertions). We derive explicit formulae for the leading and subleading divergence as n, the order in perturbation theory, tends to infinity, and a complete result for the vacuum polarization at the next-to-leading order in an expansion in l / N f, where N f is the number of fermion species. In general, diagrams with more chains yield stronger divergence. We define an analogue of the familiar diagrammatic R-operation, which extracts ultraviolet renormalon counterterms as insertions of higher-dimension operators. We then use renormalization group equations to sum the leading (in n/ N f ) k corrections to all orders in l INf and find the asymptotic behaviour in n up to a constant that must be calculated explicitly order by order in 1/N_f.     

Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light

We show that the phase sensitivity Deltatheta of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit Deltatheta approximately 1/N(T), where N(T) is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, Deltatheta approximately 1/square root[|alpha|(2)e(2r)+sinh(2)r], can be saturated for arbitrary values of the squeezing parameter r and the amplitude of the coherent mode alpha by using a Bayesian phase inference protocol.     

Classical and quantum dynamics of BPS monopoles

The dynamics of two non-relativistic BPS monopoles is described using the Atiyah-Hitchin metric on the space of collective coordinates of the monopoles. M. Classical orbits correspond to geodesics on M. Quantum states are obtained by supposing the hamiltonian is proportional to the covariant laplacian on M. We analyse in detail the asymptotic equations describing monopoles whose separation is large compared with the individual monopole radii. These give a non-trivial generalization of the Coulomb problem and are exactly soluble both in the classical and quantum regime. A variety of scattering processes and bound states are described.