Phase Space Boundary Research Articles

Urey prize lecture: Chaotic dynamics in the solar system

Newton's equations have chaotic solutions as well as regular solutions. There are several physical situations in the solar system where chaotic solutions of Newton's equations play an important role. There are examples of both chaotic rotation and chaotic orbital evolution. Hyperion is currently tumbling chaotically. Many of the other irregularly shaped satellites in the solar system have had chaotic rotations in the past. This episode of chaotic tumbling could have had a significant effect on the orbital histories of these satellites. Chaotic orbital evolution seems to be an essential ingredient in the explanation of the Kirkwood gaps in the distribution of asteroids. The phase space boundary of the chaotic zone at the 3 1 mean-motion commensurability with Jupiter is in excellent agreement with the boundary of the observed population of asteroids. Chaotic trajectories at the 3 1 commensurability have the correct properties to provide a dynamical route for the transport of meteoritic material from the asteroid belt to Earth. There is a large chaotic zone at the 2 1 commensurability, where there is a Kirkwood gap, but the phase space near the Hilda group of asteroids at the 3 2 commensurability is dominated by quasiperiodic behavior. Chaotic trajectories in the 2 1 chaotic zone reach very high eccentricities by a path that temporarily takes them to high inclinations. The long-term evolution of Pluto is suspiciously complicated, but objective criteria have not yet indicated that the motion is chaotic.

Transition to Stochasticity for a Periodically Perturbed Area Preserving System

The motion of a hard sphere between walls, under the influence of a periodic square wave force, is studied analytically to determine the existence of KAM curves. It is shown that their existence depends on the fact that any segment of them can not be transformed by the dynamics into a "folded" structure. It is shown that a special family of KAM curves are the most stable KAM curves in their respective neighborhoods of the phase space. One of these curves constitutes the boundary, such that there exists no KAM curve with smaller momentum for the same value of the position coordinate. Analytic estimates of this boundary in phase space is obtained, and are shown to closely agree with the numerical results. We discuss the fact that the present criterion for the existence of the KAM curves can be applied to a number of other systems.

Determination of large QCD corrections near the boundary of phase space

Here we present a rule which determines the coefficients of the leading and next to leading logarithms which appear in the Wilson coefficient function near the boundary of phase space. The validity of this rule is checked up to two-loop order in perturbation theory for two model field theories. In order to include the next to leading logarithms we propose a new resummation technique of the large corrections.

Angular distributions of Drell Yan pairs in the leading twist approximation

We compute the angular decay distribution of lepton pairs, integrated over the transverse momentum within the soft gluons resummation procedure. The deviations from the naive Drell-Yan model are found very small. Therefore, the phase space boundary region should provide a clear signal for higher twist contributions.

A numerical code for the phase-space boundary integration of water Bag plasmas

A numerical code for the phase-space boundary integration of water Bag plasmas

A threshold effect for spacecraft charging

The existence of a threshold energy dependence is shown for the charging of geostationary spacecraft in eclipse. Applied Technology Satellite 6 and P78‐2 (SCATHA) data show that plasma sheet fluxes must extend above 10 keV for these satellites to charge in eclipse. The existence of a threshold energy is significant because magnetospheric convection boundaries produce relatively sharp phase space boundaries, with electron fluxes falling rapidly above the energy associated with the local Alfven layer. The threshold effect is due to the shape of the normal secondary yield curve, in particular the high energy crossover, where the secondary yield drops below 1. A large portion of the ambient electron flux must exceed this energy for a negative current to exist.

Production of gluon jets via theZ0resonance

We study the production of two gluons and one photon in e/sup +/e/sup -/ collisions, in particular the process in which the intermediate resonance is the neutral weak vector boson Z/sup 0/. We compare the cross section of this process with the cross section of the lower-order process in which two quarks and one photon are produced. We find that gg..gamma.. production is unimportant in comparison with qq..gamma.. production except near phase-space boundaries on the Z/sup 0/ resonance.

Singularities in quantum-chromodynamic perturbation theory forx∼1

The QCD perturbation series has been observed to converge poorly near phase-space boundaries; for example, as $x\ensuremath{\rightarrow}1$ in structure functions. Methods have been proposed for improving the convergence by including kinematical corrections. We compare these methods, and apply them to the calculation of the photon structure function.

On the use of electron data to infer the structure of parallel electric fields

We investigate the use of electron data to probe the structure of parallel electrostatic fields. Low‐energy electrons can be trapped between their magnetic mirror points below an observer and the upward directed electric field above. The shape of the trapping region in phase space gives information about the field structure. We use equivalent potential diagrams and plots of phase space boundaries in (µ, E) space both to explain our results and to demonstrate the utility of these formalisms for studying particle motion in time‐independent electric fields. A comparison of theoretical boundaries in (ν∥, ν⊥) space with S3‐3 data shows that it is difficult to determine the shape of the low‐energy trapping boundary with present particle experiments. We discuss the instrumentation required to measure the low‐energy electron population with sufficient detail to resolve these boundaries.

A numerical code for the phase-space boundary integration of water bag plasmas

A numerical code for the phase-space boundary integration of water bag plasmas

Trilocal structures. I. Secular equation

The principles used with the bilocal photon are applied to the analysis of trilocal structures. From the trilocal wave equation, and the phase-space boundary condition (that the structure float on a Fermi sea filling the vacuum), a secular equation is obtained, an eighth-order polynomial in the energy of the structure. Requiring that the structure move as a particle, with w2=k2+m 2, provides explicit expressions for two auxiliary parameters, h and #, to be used later in an expansion of the wave function.

Hadrons from jets

A dynamical ansatz on the nature of the interaction that binds the constituents into physical hadrons makes it possible to compute perturbatively the spectrum and the quantum numbers of hadrons near the phase space boundary of a QCD jet. Nonperturbative effects are important for less extreme values of the momentum of the detected hadrons. They can be incorporated in the model using the algorithm of the jet calculus.

Quark fragmentation model

We propose a simple model for quark fragmentation in the spirit of the fragmentation model, and use it to predict the behaviour of high-energy inclusive spectra near the phase-space boundary.

Nonlinear oscillations in warm plasmas with initial velocity perturbations

The Vlasov equation is solved by a phase-space boundary integration method in order to investigate the nonlinear frequency of an electron plasma mode. Unlike the familiar discrete particle simulation codes in which a limited number of particles may be considered, the present model represents a more realistic plasma case in which a very large number (practically unlimited) of plasma particles evolving in their self-consistent collective field is investigated. The unperturbed collisionless one-dimensional plasma system consists of warm electrons having a water-bag distribution to simulate equilibrium, and of static ions. The initial perturbation is introduced by changing the boundaries of the electron plasma such that υu = υ0 (1 + α.sin kx) and υl = − υ0(1 − α sin kx), υu and υl representing the upper and the lower boundaries, respectively. This corresponds to a standing wave perturbation. The results obtained for different wavelength perturbations λ ≡ 2π/k, and for several perturbation amplitudes α, are presented and discussed.

The fragmentation of pions near the phase-space boundary

Recent experimental measurements have shown the two inclusive spectra ϱ1(π−→π0)=ϱ1(π−→π+) to be remarkably similar. The fragmentation model can provide a natural answer to this equality.

The effects of electron-electron scattering and thermal vibrations on the motion of swift electrons through single crystals

Abstract A computer simulation of the transmission of 20 MeV electrons through a (001) MgO single crystal was performed. The calculation model represents a combination of the usual Monte Carlo simulation of projectile transmission through a heterogeneous absorber and the binary collision model of trajectory tracing. The dependence on the penetration depth of electrons moving within the limits of separate Seitz unit cells was obtained. The influence of electron-electron scattering for such projectiles was studied in comparison with that of thermal lattice vibrations. The boundaries of the occupied phase space of transverse energies and angular momenta for the swift electrons motion along the atomic strings were determined.

Measurement of inclusive two-particle rapidity correlations at the ISR

We present data on two-particle distributions in the approximate rapidity variable η = −1n tan θ/2 for secondaries produced in proton-proton collisions at the CERN Intersecting Storage Rings (ISR). The data at centre-of-mass energies of 23 and 62 GeV show the existence of energy-independent positive, short-range correlations between secondaries within the central region. Deviations from short-range correlation exist when one (or both) of the particles is near the boundary of the available phase space. In part, such deviations are interpreted as the effect of a diffractive-like component in multiparticle production.

Cuts in the Angular Momentum Plane and Inclusive-Type Sum Rules

Explicit forms for the two-Reggeon cut in particle-particle scattering are derived iron the sum rule for inclusive-type discontinuities in the kinematic region near the phase-space boundary. When the dual resonance model is used as a representation for the six-point amplitude, the detailed results are in agreement with those of Abarbanel.

Pion exchange and missing-mass spectra near the phase-space boundary

Pion exchange and missing-mass spectra near the phase-space boundary

Triple-Regge Scaling and the Behavior of Form Factors for Resonance Couplings

The restrictions imposed by Feynman's scaling relation for single-particle spectra near the boundary of phase space are examined in relation to the production of a resonance of large mass Regge exchange in the sense of a Van Hove model. We obtain certain conditions on the form factor's coupling resonances of different spins. These are discussed in relation to the various form factors that have been used lately.