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Oct 1993

Volume 3, Issue 4, pp. 417-782

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Chaotic scattering: An introduction

Edward Ott and Tamás Tél

Chaos 3, 417 (1993); http://dx.doi.org/10.1063/1.165949 (10 pages) | Cited 83 times

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In recent years chaotic behavior in scattering problems has been found to be important in a host of physical situations. Concurrently, a fundamental understanding of the dynamics in these situations has been developed, and such issues as symbolic dynamics, fractal dimension, entropy, and bifurcations have been studied. The quantum manifestations of classical chaotic scattering is also an extremely active field, with new analytical techniques being developed and with experiments being carried out. This issue of Chaos provides an up‐to‐date survey of the range of work in this important field of study.
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11.80.-m Relativistic scattering theory
05.45.-a Nonlinear dynamics and chaos

What is the role of chaotic scattering in irreversible processes?

Pierre Gaspard

Chaos 3, 427 (1993); http://dx.doi.org/10.1063/1.165950 (16 pages) | Cited 33 times

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We study kinetic properties of simple mechanical models of deterministic diffusion like the scattering of a point particle in a billiard of Lorentz type and the multibaker area‐preserving map. We show how dynamical chaos and, in particular, chaotic scattering are related to the transport property of diffusion. Moreover, we show that the Liouvillian dynamics of the multibaker map can be decomposed into the eigenmodes of diffusive relaxation associated with the Ruelle resonances of the Perron–Frobenius operator.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.45.-a Nonlinear dynamics and chaos
05.60.-k Transport processes

Phase space structure and chaotic scattering in near‐integrable systems

B.‐P. Koch and B. Bruhn

Chaos 3, 443 (1993); http://dx.doi.org/10.1063/1.165951 (15 pages) | Cited 9 times

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We investigate the bifurcation phenomena and the change in phase space structure connected with the transition from regular to chaotic scattering in classical systems with unbounded dynamics. The regular systems discussed in this paper are integrable ones in the sense of Liouville, possessing a degenerated unstable periodic orbit at infinity. By means of a McGehee transformation the degeneracy can be removed and the usual Melnikov method is applied to predict homoclinic crossings of stable and unstable manifolds for the perturbed system. The chosen examples are the perturbed radial Kepler problem and two kinetically coupled Morse oscillators with different potential parameters which model the stretching dynamics in ABC molecules. The calculated subharmonic and homoclinic Melnikov functions can be used to prove the existence of chaotic scattering and of elliptic and hyperbolic periodic orbits, to calculate the width of the main stochastic layer and of the resonances, and to predict the range of initial conditions where singularities in the scattering function are found. In the second example the value of the perturbation parameter at which channel transitions set in is calculated. The theoretical results are supplemented by numerical experiments.
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05.45.-a Nonlinear dynamics and chaos
45.05.+x General theory of classical mechanics of discrete systems

Truncated horseshoes and formal languages in chaotic scattering

G. Troll

Chaos 3, 459 (1993); http://dx.doi.org/10.1063/1.165952 (15 pages) | Cited 7 times

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In this paper we study parameter families of truncated horseshoes as models of multiscattering systems which show a transition to chaos without losing hyperbolicity, so that the topological features of the transition are completely describable by a parametrized family of symbolic dynamics. At a fixed parameter value the corresponding horseshoe represents the set of orbits trapped in the scattering region. The bifurcations are a pure boundary effect and no other bifurcations such as saddle center bifurcations occur in this transition scenario. Truncated horseshoes actually arise in concrete potential scattering under suitable conditions. It is shown that a simple scattering model introduced earlier can realize this scenario in a certain parameter range (the ‘‘truncated sawshoe’’). For this purpose, we solve the inverse scattering problem of finding the central potential associated to the sawshoe model. Furthermore, we review classification schemes for the transition to chaos of truncated horseshoes originating from symbolic dynamics and formal language theory and apply them to the truncated double horseshoe and the truncated sawshoe.
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11.80.-m Relativistic scattering theory
05.45.-a Nonlinear dynamics and chaos

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

Vincent Daniels, Michel Vallières, and Jian‐Min Yuan

Chaos 3, 475 (1993); http://dx.doi.org/10.1063/1.165953 (11 pages) | Cited 10 times

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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.
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11.80.-m Relativistic scattering theory
05.45.-a Nonlinear dynamics and chaos
45.05.+x General theory of classical mechanics of discrete systems

Scaling and decay in periodically driven scattering systems

Arne Beeker and Peter Eckelt

Chaos 3, 487 (1993); http://dx.doi.org/10.1063/1.165954 (8 pages) | Cited 6 times

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We investigate irregular scattering in a periodically driven Hamiltonian system of one degree of freedom. The potential is asymptotically attracting, so there exist parabolically escaping scattering orbits, i.e. orbits with asymptotic energy Eout=0. The scattering functions (i.e. the asymptotic out‐variables as functions of an asymptotic in‐variable) show a characteristic algebraic scaling in the vicinity of these orbits. This behavior is explained by asymptotic properties of the interaction. As a consequence, the number N(Δt) of temporarily bound particles decays algebraically with the delay time Δt, although no KAM scenario can be found in phase space. On the other hand, we find the number Nn of temporarily bound particles to decay exponentially with the number n of zeros of x(t).
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05.45.-a Nonlinear dynamics and chaos

Conditions for the abrupt bifurcation to chaotic scattering

Tamás Tél, Celso Grebogi, and Edward Ott

Chaos 3, 495 (1993); http://dx.doi.org/10.1063/1.165955 (9 pages) | Cited 8 times

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One of the generic ways in which chaotic scattering can come about as a system parameter is varied is the so‐called ‘‘abrupt bifurcation’’ in which the scattering is nonchaotic on one side of the bifurcation and is chaotic and hyperbolic on the other side. Previous work demonstrating the abrupt bifurcation [S. Bleher et al., Phys. Rev. Lett. 63, 919 (1989); Physica D 46, 87 (1990)] was primarily for the case where the scattering potential had maxima (‘‘hilltops’’) which had locally circular isopotential contours. Here we extend these considerations to the more general case of locally elliptically shaped isopotential contours at the hilltops. It turns out that the conditions for the abrupt bifurcation change drastically as soon as even a small amount of noncircularity is included (i.e., the circular case is singular). The illustrative case of scattering from three isolated potential hills is dealt with in detail. One interesting result is a simple geometrical sufficient condition for an abrupt bifurcation in the case of large enough ellipticity of the hill with lowest potential at its hilltop.
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11.80.-m Relativistic scattering theory
05.45.-a Nonlinear dynamics and chaos

The simplest case of chaotic wave scattering

Arkady S. Pikovsky

Chaos 3, 505 (1993); http://dx.doi.org/10.1063/1.165995 (2 pages) | Cited 3 times

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We consider scattering of nondispersive linear waves on a discrete nonlinear element. The problem reduces to the dynamics of a forced damped nonlinear oscillator. Chaotic motions of the oscillator produce chaotic reflected and transmitted waves.
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05.45.-a Nonlinear dynamics and chaos
41.20.Jb Electromagnetic wave propagation; radiowave propagation
11.80.-m Relativistic scattering theory

Chaotic scattering in the gravitational three‐body problem

Patricia T. Boyd and Stephen L. W. McMillan

Chaos 3, 507 (1993); http://dx.doi.org/10.1063/1.165956 (17 pages) | Cited 20 times

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We summarize some results of an ongoing study of the chaotic scattering interaction between a bound pair of stars (a binary) and an incoming field star. The stars are modeled as point masses and their equations of motion are numerically integrated for a large number of initial conditions. The global features of the resulting initial‐value space maps are presented, and their evolution as a function of system parameters is discussed. We find that the maps contain regular regions separated by rivers of chaotic behavior. The probability of escape within the chaotic regions is discussed, and a straightforward explanation of the scaling present in these regions is reviewed. We investigate a statistical quantity of interest, namely the cross section for temporarily bound interactions, as a function of the third star’s incoming velocity and mass. Finally, a new way of considering long‐lived trajectories is presented, allowing long data sets to be qualitatively analyzed at a glance.
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95.10.Fh Chaotic dynamics
05.45.-a Nonlinear dynamics and chaos
45.05.+x General theory of classical mechanics of discrete systems

Chaotic scattering and acceleration of particles by waves

A. A. Chernikov and G. Schmidt

Chaos 3, 525 (1993); http://dx.doi.org/10.1063/1.165957 (4 pages) | Cited 11 times

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Relativistic charged particles in a uniform magnetic field and traveling waves of large amplitude can be accelerated limitlessly. Since much of phase space is chaotic the process can be viewed as chaotic scattering on waves. For multiple waves an explicit map arises with unusual properties. It contains infinite island chains around nonperiodic orbits and it is also an example of transient chaos in a Hamiltonian system.
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05.45.-a Nonlinear dynamics and chaos
41.20.Jb Electromagnetic wave propagation; radiowave propagation
11.80.-m Relativistic scattering theory
03.30.+p Special relativity

Transient chaos in room acoustics

Fabrice Mortessagne, Olivier Legrand, and Didier Sornette

Chaos 3, 529 (1993); http://dx.doi.org/10.1063/1.165958 (13 pages) | Cited 16 times

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The decay of sound in an auditorium due to absorption is central to the theory and practice of room acoustics. Within geometrical acoustics, this problem involves the partial trapping of chaotic ray trajectories in billiards, hence transient chaos. We first present a theoretical and numerical analysis of the decay of rays in 2‐D chaotic billiards (2‐D room acoustic models) and show that the existence of fluctuations (in the mean‐free path and in the rate of phase space exploration) leads to modifications from the standard statistical theory. An ergodic wave theory of room acoustics based on a wave formulation is then discussed and tested by direct numerical calculations of the eigenmodes in 2‐D billiards. Finally, we present a semiclassical calculation of the acoustic Green’s functions in the time domain, based on a summation over rays, viewed as generalized wave impulses, and successfully compare its predictions with a direct numerical integration of the wave equation. This formalism provides a framework to link the geometrical ray description to the ergodic wave theory.
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43.55.Br Room acoustics: theory and experiment; reverberation, normal modes, diffusion, transient and steady-state response
43.20.Dk Ray acoustics
05.45.-a Nonlinear dynamics and chaos

Chaotic capture of vortices by a moving body. I. The single point vortex case

James B. Kadtke and Evgeny A. Novikov

Chaos 3, 543 (1993); http://dx.doi.org/10.1063/1.165959 (11 pages) | Cited 15 times

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The study of the dynamical properties of vortex systems is an important and topical research area, and is becoming of ever increasing usefulness to a variety of physical applications. In this paper, we present a study of a model of a rotational singularity which obeys a logarithmic potential interacting with a bluff body in a uniform inviscid laminar flow, e.g., a line vortex interacting with a cylinder in three dimensions or a point vortex with a circular boundary in two dimensions. We show that this system is Hamiltonian and simple enough to be solved analytically for the stagnation points and separatrices of the flow, and a bifurcation diagram for the relevant parameters and classification of the various types of motion is given. We also show that, by introducing a periodic perturbation to the body, chaotic motion of the vortex can be readily generated, and we present analytic criteria for the generation of chaos using the Poincaré–Melnikov–Arnold method. This leads to an important dynamical effect for the model, i.e., that the possibility exists for the vortex to be chaotically captured around the body for periods of time which are extremely sensitive to initial conditions. The basic mechanism for this capture is due to the chaotic dynamics and is similar to that of other chaotic scattering phenomena. We show numerically that cases exist where the vortex can be captured around an elliptic point external to (and possibly far from) the body, and the existence of other very complicated motions are also demonstrated. Finally, generalizations of the problem of the vortex–body interaction are indicated, and some possible applications are postulated such as the interaction of line vortices with aircraft wings.
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47.15.ki Inviscid flows with vorticity
47.52.+j Chaos in fluid dynamics
47.32.C- Vortex dynamics
05.45.-a Nonlinear dynamics and chaos

Application of scattering chaos to particle transport in a hydrodynamical flow

C. Jung, T. Tél, and E. Ziemniak

Chaos 3, 555 (1993); http://dx.doi.org/10.1063/1.165960 (14 pages) | Cited 67 times

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The dynamics of a passive particle in a hydrodynamical flow behind a cylinder is investigated. The velocity field has been determined both by a numerical simulation of the Navier–Stokes flow and by an analytically defined model flow. To analyze the Lagrangian dynamics, we apply methods coming from chaotic scattering: periodic orbits, time delay function, decay statistics. The asymptotic delay time statistics are dominated by the influence of the boundary conditions on the wall and exhibit algebraic decay. The short time behavior is exponential and represents hyperbolic effects.
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47.52.+j Chaos in fluid dynamics
05.45.-a Nonlinear dynamics and chaos

Chaotic electronic scattering with He+

Jian‐Min Yuan and Yan Gu

Chaos 3, 569 (1993); http://dx.doi.org/10.1063/1.165961 (12 pages) | Cited 10 times

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We investigate classical electronic collisions with a He+ ion. Scattering functions, such as the scattering angle, collisional time, or energy of the outgoing electron, all exhibit an interesting hierarchial self‐similar structure, which can be interpreted in terms of the indefinite number of electronic returns to the vicinity of the nucleus, encounters between electrons, and Keplerian excursions of electrons during the collisional processes. Based on this mechanism a binary coding is introduced to organize the dynamics of this three‐body system and to provide an understanding of the self‐similarity among generations of scale magnification, which yields escape rates that vary with the sectional cut into the parameter space. The self‐similarity displayed within a single generation, on the other hand, can be simply tied to the periods of the two independent electronic excursions. The physical interpretation and the symbolic dynamics introduced here are generally useful for three‐body collisional systems, including atomic, molecular, or stellar collisions.
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34.80.-i Electron and positron scattering
05.45.-a Nonlinear dynamics and chaos
11.80.-m Relativistic scattering theory

A scattering approach to the quantization of billiards— The inside–outside duality

Barbara Dietz and Uzy Smilansky

Chaos 3, 581 (1993); http://dx.doi.org/10.1063/1.165962 (9 pages) | Cited 35 times

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We present some recent results on the semiclassical quantization of billiards using an approach which is based on the strong link between the billiard interior and exterior problems. That is, the spectrum of the interior problem is extracted from the scattering matrix of the exterior problem. Once this is put on a rigorous basis, the semiclassical approximation is used to derive the semiclassical ζ function and the spectral density. The duality between the inside and outside problems prevails also in the classical description and offers new insight into this quantization procedure. The relation between the present approach and the more standard quantization methods is also discussed and illustrated with some numerical results.
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03.65.Sq Semiclassical theories and applications
11.80.-m Relativistic scattering theory
05.45.-a Nonlinear dynamics and chaos
46.05.+b General theory of continuum mechanics of solids

Poincaré surface of section and quantum scattering

Martin C. Gutzwiller

Chaos 3, 591 (1993); http://dx.doi.org/10.1063/1.165963 (9 pages) | Cited 7 times

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A very simple model for the quantum‐mechanical scattering of a particle is studied with a dual goal: The chaotic nature of the corresponding classical problem should be quite obvious, and the method of solution should use an approach that is closely related to the surface of section in classical mechanics. Moreover, the mathematical operations should be elementary so that the errors in a semiclassical approximation or in any computational work have a chance of being controllable. Finally, the mode of presentation is such as to be understandable for a newcomer to the field of chaos. The model is a variation of the Sinai billiard where the circular hard wall inside a box (parallelogram) is replaced by a trombone‐shaped surface for the particle to enter and exit the box. The rim (circular boundary between trombone and box) is the surface of section, with the total current at fixed energy in either direction providing the measure for the wave functions. The Poincaré map then becomes the product of two unitary transformations, where the first is diagonal in angular momentum, while the second is diagonal in angle.
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03.65.Nk Scattering theory
03.65.Sq Semiclassical theories and applications
05.45.-a Nonlinear dynamics and chaos

ℏ expansion for the periodic orbit quantization of chaotic systems

D. Alonso and P. Gaspard

Chaos 3, 601 (1993); http://dx.doi.org/10.1063/1.165964 (12 pages) | Cited 24 times

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We report the results of a periodic orbit quantization of classically chaotic billiards beyond Gutzwiller approximation in terms of asymptotic series in powers of the Planck constant (or in powers of the inverse of the wave number κ in billiards). We derive explicit formulas for the κ−1 approximation of our semiclassical expansion. We illustrate our theory with the classically chaotic scattering of a wave on three disks. The accuracy on the real parts of the scattering resonances is improved by one order of magnitude.
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03.65.Sq Semiclassical theories and applications
05.45.-a Nonlinear dynamics and chaos
41.20.Jb Electromagnetic wave propagation; radiowave propagation

Correlations in quantum time delay

Bruno Eckhardt

Chaos 3, 613 (1993); http://dx.doi.org/10.1063/1.165925 (5 pages) | Cited 24 times

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The semiclassical periodic orbit content of the form factor K(λ ), the Fourier transform of the autocorrelation function, for the quantum time delay is analyzed. In analogy to the case of bounded systems, three regimes can be identified. For small λ isolated periodic orbits can be identified. For intermediate λ, there is a λ exp(−Γλ) regime, where Γ is the classical escape rate. For large λ, this changes into an exp( − γqmλ) law, where now γqm is related to an inverse lifetime of the resonances. The transition between the latter two regimes is determined by the density of resonances. The theoretical analysis is supported by numerical data for the three disk scattering system.
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03.65.Sq Semiclassical theories and applications
05.45.-a Nonlinear dynamics and chaos

A Fredholm determinant for semiclassical quantization

Predrag Cvitanović, Per E. Rosenqvist, Gábor Vattay, and Hans Henrik Rugh

Chaos 3, 619 (1993); http://dx.doi.org/10.1063/1.165992 (18 pages) | Cited 24 times

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We investigate a new type of approximation to quantum determinants, the ‘‘quantum Fredholm determinant,’’ and test numerically the conjecture that for Axiom A hyperbolic flows such determinants have a larger domain of analyticity and better convergence than the Gutzwiller–Voros zeta functions derived from the Gutzwiller trace formula. The conjecture is supported by numerical investigations of the 3‐disk repeller, a normal‐form model of a flow, and a model 2‐D map.
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03.65.Sq Semiclassical theories and applications
05.45.-a Nonlinear dynamics and chaos

Quantum irregular scattering induced by tunneling

András Csordás and Petr Šeba

Chaos 3, 637 (1993); http://dx.doi.org/10.1063/1.165926 (6 pages) | Cited 2 times

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Irregular behavior in a simple two‐dimensional scattering model is investigated in the quantum domain. The scattering potential is composed from Dirac deltas on a stadium shaped curve. The unusual feature of the model is that the irregular patterns disappear in the classical limit because the main mechanism leading to resonances in the cross section data is the quantum tunneling. Calculations for the standard characteristics such as nearest‐neighbor distribution of eigenphases of the S‐matrix, the distribution of the S‐matrix elements and the correlation function of the total cross section are performed. Deviations from the usual predictions for irregular scattering have been found in certain regions, which can be traced back to the fact that the model does not have such a characteristic time like the classical escape rate, which survives the classical limit.
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03.65.Nk Scattering theory
11.80.-m Relativistic scattering theory
05.45.-a Nonlinear dynamics and chaos

Conductance fluctuations and quantum chaotic scattering in semiconductor microstructures

C. M. Marcus, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard

Chaos 3, 643 (1993); http://dx.doi.org/10.1063/1.165927 (11 pages) | Cited 60 times

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We review recent experiments on aperiodic conductance fluctuations in ballistic GaAs/AlGaAs microstructures in the shape of a stadium billiard and a circle with point‐contact leads, measured at millikelvin temperatures. Much of the observed behavior can be analyzed within a semiclassical approach to quantum chaotic scattering. After a brief review of the Landauer–Büttiker formulation of coherent transport, a variety of novel experimental phenomena and comparisons to semiclassical theory are presented. In particular, we discuss quantum‐enhanced backscattering, the power spectrum of conductance fluctuations, crossover to the high‐magnetic‐field and tunneling regimes, and an application allowing the rate of phase‐randomizing scattering to be measured in chaotic ballistic microstructures.
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72.20.Dp General theory, scattering mechanisms
05.45.-a Nonlinear dynamics and chaos
03.65.Sq Semiclassical theories and applications
03.65.Nk Scattering theory

Order and chaos in semiconductor microstructures

W. A. Lin, J. B. Delos, and R. V. Jensen

Chaos 3, 655 (1993); http://dx.doi.org/10.1063/1.165994 (10 pages) | Cited 44 times

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The semiclassical theory of ballistic electron transport in semiconductor microstructures provides a description of the quantum conductance fluctuations in terms of the classical distributions for the lengths and directed areas of the scattering trajectories. Because the classical dynamics differs for integrable (circular) and chaotic (stadium) scattering domains, experimental measurements of the conductance of these microstructures provide a unique probe of the quantum properties of classically regular and chaotic systems. To advance these theoretical and experimental studies we compare geometrical formulas for the classical distributions of lengths and areas with numerical simulations for microstructures examined in recent experiments, we assess the effects of lead size and placement, and we provide a critical analysis of the role of scattering ‘‘noise’’ on the classical and semiclassical predictions. Finally, we present a detailed comparison of the semiclassical theory with recent experimental measurements of the conductance fluctuations in circular‐ and stadium‐shaped microstructures.
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72.20.Dp General theory, scattering mechanisms
05.45.-a Nonlinear dynamics and chaos
03.65.Sq Semiclassical theories and applications

Quantum‐chaotic scattering effects in semiconductor microstructures

Harold U. Baranger, Rodolfo A. Jalabert, and A. Douglas Stone

Chaos 3, 665 (1993); http://dx.doi.org/10.1063/1.165928 (18 pages) | Cited 141 times

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We show that classical chaotic scattering has experimentally measurable consequences for the quantum conductance of semiconductor microstructures. These include the existence of conductance fluctuations—a sensitivity of the conductance to either Fermi energy or magnetic field—and weak‐localization—a change in the average conductance upon applying a magnetic field. We develop a semiclassical theory and present numerical results for these two effects in which we model the microstructures by billiards attached to leads. We find that the difference between chaotic and regular classical scattering produces a qualitative difference in the fluctuation spectrum and weak‐localization lineshape of chaotic and nonchaotic structures. While the semiclassical theory within the diagonal approximation accounts well for the weak‐localization lineshape and for the spectrum of the fluctuations, we uncover a surprising failure of the semiclassical diagonal‐approximation theory in describing the magnitude of these quantum transport effects.
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72.20.Dp General theory, scattering mechanisms
05.45.-a Nonlinear dynamics and chaos
11.80.-m Relativistic scattering theory
03.65.Nk Scattering theory

Quantum chaotic scattering with CsI molecules

R. Blümel

Chaos 3, 683 (1993); http://dx.doi.org/10.1063/1.165929 (8 pages) | Cited 6 times

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A molecular beam of CsI molecules passed through the electric field created by a pair of charged wires is proposed as a laboratory experiment on quantum chaotic scattering. It is shown that the proposed experiment is realistic.
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33.90.+h Other topics in molecular properties and interactions with photons (restricted to new topics in section 33)
11.80.-m Relativistic scattering theory
05.45.-a Nonlinear dynamics and chaos
03.65.Nk Scattering theory

Chaotic scattering in heavy‐ion reactions

M. Baldo, E. G. Lanza, and A. Rapisarda

Chaos 3, 691 (1993); http://dx.doi.org/10.1063/1.165930 (16 pages) | Cited 12 times

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We discuss the relevance of chaotic scattering in heavy‐ion reactions at energies around the Coulomb barrier. A model in two and three dimensions which takes into account rotational degrees of freedom is discussed both classically and quantum mechanically. The typical chaotic features found in this description of heavy‐ion collisions are connected with the anomalous behavior of several experimental data.  
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25.70.-z Low and intermediate energy heavy-ion reactions
11.80.-m Relativistic scattering theory
05.45.-a Nonlinear dynamics and chaos
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