Semiclassical cross section correlations

We calculate within a semiclassical approximation the autocorrelation function of cross sections. The starting point is the semiclassical expression for the diagonal matrix elements of an operator. For general operators with a smooth classical limit the autocorrelation function of such matrix elements has two contributions with relative weights determined by classical dynamics. We show how the random matrix result can be obtained if the operator approaches a projector onto a single initial state. The expressions are verified in calculations for the kicked rotor.Comment: 6 pages, 2 figure

Symmetry Decomposition of Chaotic Dynamics

Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations in turn reduce the labor and improve the convergence of cycle expansions for classical and quantum spectra associated with the flow. In this paper the general formalism is developed, with the $N$-disk pinball model used as a concrete example and a series of physically interesting cases worked out in detail.Comment: CYCLER Paper 93mar01

Small Disks and Semiclassical Resonances

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wave lengths smaller than the order of the disk radius we recover the usual semi-classical approximation; however, for wave lengths larger than the order of the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks.Comment: Final published version - some changes in the discussion and the labels on one figure are correcte

Non-sequential triple ionization in strong fields

We consider the final stage of triple ionization of atoms in a strong linearly polarized laser field. We propose that for intensities below the saturation value for triple ionization the process is dominated by the simultaneous escape of three electrons from a highly excited intermediate complex. We identify within a classical model two pathways to triple ionization, one with a triangular configuration of electrons and one with a more linear one. Both are saddles in phase space. A stability analysis indicates that the triangular configuration has the larger cross sections and should be the dominant one. Trajectory simulations within the dominant symmetry subspace reproduce the experimentally observed distribution of ion momenta parallel to the polarization axis.Comment: 9 pages, 8 figures, accepted for publication in Phys. Rev.

Second moment of the Husimi distribution as a measure of complexity of quantum states

We propose the second moment of the Husimi distribution as a measure of complexity of quantum states. The inverse of this quantity represents the effective volume in phase space occupied by the Husimi distribution, and has a good correspondence with chaoticity of classical system. Its properties are similar to the classical entropy proposed by Wehrl, but it is much easier to calculate numerically. We calculate this quantity in the quartic oscillator model, and show that it works well as a measure of chaoticity of quantum states.Comment: 25 pages, 10 figures. to appear in PR

How does flow in a pipe become turbulent?

The transition to turbulence in pipe flow does not follow the scenario familiar from Rayleigh-Benard or Taylor-Couette flow since the laminar profile is stable against infinitesimal perturbations for all Reynolds numbers. Moreover, even when the flow speed is high enough and the perturbation sufficiently strong such that turbulent flow is established, it can return to the laminar state without any indication of the imminent decay. In this parameter range, the lifetimes of perturbations show a sensitive dependence on initial conditions and an exponential distribution. The turbulence seems to be supported by three-dimensional travelling waves which appear transiently in the flow field. The boundary between laminar and turbulent dynamics is formed by the stable manifold of an invariant chaotic state. We will also discuss the relation between observations in short, periodically continued domains, and the dynamics in fully extended puffs.Comment: for the proceedings of statphys 2

The spectral form factor is not self-averaging

The spectral form factor, k(t), is the Fourier transform of the two level correlation function C(x), which is the averaged probability for finding two energy levels spaced x mean level spacings apart. The average is over a piece of the spectrum of width W in the neighborhood of energy E0. An additional ensemble average is traditionally carried out, as in random matrix theory. Recently a theoretical calculation of k(t) for a single system, with an energy average only, found interesting nonuniversal semiclassical effects at times t approximately unity in units of {Planck's constant) /(mean level spacing). This is of great interest if k(t) is self-averaging, i.e, if the properties of a typical member of the ensemble are the same as the ensemble average properties. We here argue that this is not always the case, and that for many important systems an ensemble average is essential to see detailed properties of k(t). In other systems, notably the Riemann zeta function, it is likely possible to see the properties by an analysis of the spectrum.Comment: 4 pages, RevTex, no figures, submitted to Phys. Rev. Lett., permanent e-mail address, [email protected]

Classical, semiclassical, and quantum investigations of the 4-sphere scattering system

A genuinely three-dimensional system, viz. the hyperbolic 4-sphere scattering system, is investigated with classical, semiclassical, and quantum mechanical methods at various center-to-center separations of the spheres. The efficiency and scaling properties of the computations are discussed by comparisons to the two-dimensional 3-disk system. While in systems with few degrees of freedom modern quantum calculations are, in general, numerically more efficient than semiclassical methods, this situation can be reversed with increasing dimension of the problem. For the 4-sphere system with large separations between the spheres, we demonstrate the superiority of semiclassical versus quantum calculations, i.e., semiclassical resonances can easily be obtained even in energy regions which are unattainable with the currently available quantum techniques. The 4-sphere system with touching spheres is a challenging problem for both quantum and semiclassical techniques. Here, semiclassical resonances are obtained via harmonic inversion of a cross-correlated periodic orbit signal.Comment: 12 pages, 5 figures, submitted to Phys. Rev.

Pathways to double ionization of atoms in strong fields

We discuss the final stages of double ionization of atoms in a strong linearly polarized laser field within a classical model. We propose that all trajectories leading to non-sequential double ionization pass close to a saddle in phase space which we identify and characterize. The saddle lies in a two degree of freedom subspace of symmetrically escaping electrons. The distribution of longitudinal momenta of ions as calculated within the subspace shows the double hump structure observed in experiments. Including a symmetric bending mode of the electrons allows us to reproduce the transverse ion momenta. We discuss also a path to sequential ionization and show that it does not lead to the observed momentum distributions.Comment: 10 pages, 10 figures; fig.6 and 7 exchanged in the final version accepted for publication in Phys. Rev.

An experimental evaluation of software redundancy as a strategy for improving reliability

The strategy of using multiple versions of independently developed software as a means to tolerate residual software design faults is suggested by the success of hardware redundancy for tolerating hardware failures. Although, as generally accepted, the independence of hardware failures resulting from physical wearout can lead to substantial increases in reliability for redundant hardware structures, a similar conclusion is not immediate for software. The degree to which design faults are manifested as independent failures determines the effectiveness of redundancy as a method for improving software reliability. Interest in multi-version software centers on whether it provides an adequate measure of increased reliability to warrant its use in critical applications. The effectiveness of multi-version software is studied by comparing estimates of the failure probabilities of these systems with the failure probabilities of single versions. The estimates are obtained under a model of dependent failures and compared with estimates obtained when failures are assumed to be independent. The experimental results are based on twenty versions of an aerospace application developed and certified by sixty programmers from four universities. Descriptions of the application, development and certification processes, and operational evaluation are given together with an analysis of the twenty versions