Elliptic behaviour in the sawtooth standard map

This paper examines the standard map with sawtooth nonlinearity when the eigenvalues of the Jacobian lie on the unit circle. This is an area-preserving map of the torus to itself that is linear except on a line on which it is discontinuous. We discuss the closure of the set of images of the discontinuity and present numerical evidence that its Lebesgue measure is positive. Moreover, we present evidence that the measure of the closure of images of the discontinuity changes continuously with the parameter k. This means that the sawtooth standard map may exhibit coexistence of two positive measure subsets on which the dynamics is respectively regular and irregular in a certain sense. In the appendix we show that this map is equivalent to a map studied by electronics engineers as a model for the quiescent behaviour of a linear lossless digital filter with “two's complement” overflow.     

LargeN behaviour of perturbative QCD coupling

A correspondence between perturbative QCD in an axial gauge and dual topological unitarization (DTU) scheme is expressed by linking the QCD scale Λ toN, the number of flavors or colors supposed to be large through a relation which involves a typical hadronic scale.     

Diffusion in a collisional standard map

Test particle evaluation of the diffusion coefficient in the presence of magnetic field fluctuations and binary collisions is presented. Chaotic magnetic field lines originate from resonant magnetic perturbations (RMPs). To lowest order, charged particles follow magnetic field lines. Drifts and interaction (collisions) with other particles decorrelate particles from the magnetic field lines. We model the binary collision process by a constant collision frequency. The magnetic field configuration including perturbations on the integrable Hamiltonian part is such that the single particle motion can be followed by a collisional version of a Chirikov-Taylor (standard) map. Frequent collisions are allowed for. Scaling of the diffusion beyond the quasilinear and subdiffusive behaviour is investigated in dependence on the strength of the magnetic perturbations and the collision frequency. The appearance of the so called Rechester-Rosenbluth regime is verified. It is further shown that the so called Kadomtsev-Pogutse diffusion coefficient is the strong collisional limit of the Rechester-Rosenbluth formula. The theoretical estimates are supplemented by numerical simulations.     

Conformally invariant pomeron interaction in perturbative QCD with large Nc

An effective non-local quantum field theory is constructed, which describes the interaction of pomerons in high-colored QCD. The theory includes both splitting and merging triple pomeron vertexes and diagrams with pomeronic loops. The Schwinger–Dyson equations for the ‘physical’ pomeron are written. Conformal invariance allows one to reduce the theory to the old-fashioned Gribov pomeron theory with an infinite number of pomerons, one of which is supercritical.     

Elastic and vibrational excitation cross-sections in the Ps-H2 scattering: a perturbative approach

Summary A detailed account is reported of a perturbative calculation of the elastic and vibrational excitation cross-sections for the positronium-H₂ molecule scattering. The problem is studied in the context of the multichannel scattering theory for identical particles; the first-order exchange amplitude and the second-order direct amplitude are evaluated for energy from zero to the positronium excitation threshold. The lowest-order Padé approximant is also constructed and this suggests the occurrence of a strong enhancement in the vibrational cross-section and in the related energy loss in matter at the positronium excitation threshold. In comparison with a short preliminary account presented elsewhere, various improvements are introduced both in the physical approximations and in the numerical calculations.

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Riassunto Si dà un resoconto particolareggiato di un calcolo perturbativo della sezione d'urto elastica e di eccitazione vibrazionale per la collisione positronio su molecola d'idrogeno. Il problema è studiato nel contesto della teoria dello scattering a molti canali per sistemi di particelle identiche; sono calcolate l'ampiezza di scambio al primo ordine e l'ampiezza diretta al secondo ordine per energie da zero alla soglia di eccitazione del positronio. è anche valutata l'approssimante di Padé piú bassa e questo suggerisce la comparsa di una forte esaltazione della sezione d'urto vibrazionale e della perdita di energia nella materia attorno alla soglia di eccitazione del positronio. Rispetto ad un breve resoconto preliminare presentato in precedenza si sono apportati diversi miglioramenti sia nelle approssimazioni di carattere fisico che nelle valutazioni numeriche.

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Резюме Проводится пертурбационное вычисление поперечных сечений упругого рассеяния и вибрационного возбуждения в процессе рассеяния позитрония на молекуле H₂. Рассматриваемая проблема исследуется в связи с теорией многоканального рассеяния тождественных частиц. Вычисляются обменная амплитуда первого порядка и прямая амплитуда второго порядка в области энергий от нуля до порога возбуждения позитрония. Также определяется низший порядка Падэ приближения, предполагая наличие сильного увеличения в вибрационном поперечном сечении и в энергетических потерях в веществе на пороге возбуждения позитрония. По сравнению с существующими подходами, предлагаются различные улучшения в физических приблизениях и в численных расчетах.

     

Stationary condition in a perturbative approach for mass varying neutrinos

A perturbative approach for arbitrary choices of the equation of state of the universe is introduced in order to treat scenarios for mass varying neutrinos (MaVaNs) coupled to the dark sector. The generalized criterion for the applicability of such an approach is expressed through a constraint on the coefficient of the linear perturbation on the dark sector scalar field. This coefficient depends on the ratio between the variation of the neutrino energy and the scalar field potential. Upon certain conditions, the usual stationary condition found in the context of MaVaN models together with the perturbative contribution can be employed to predict the dynamical evolution of the neutrino mass. Our results clearly indicate that the positiveness of the squared speed of sound of the coupled fluid and the model stability are not conditioned by the stationary condition.     

Fractal properties of critical invariant curves

We examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers, through both the thermodynamic formalism and numerical computation. The maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed. Our results indicate that the dimension is universal for a given type of singularity and rotation number, and that among all rotation numbers, the golden mean produces the largest dimension.     

Critical behaviour in a quark plasma

We study the effect which critical behaviour in a strongly interacting plasma has on the spectrum of photons in equilibrium with such a system.     

Unreasonable postulate in the perturbative approach to quantum gravity

The conventional perturbative approach to quantum gravity is based on the expansion in powers of k, wherek denotes the Einstein gravitational constant. The introduction of a square root is due to the unreasonable postulate that thek0 limit of the gravitational field is ac-number. It is more natural that it is aq-number, which can be determined explicitly by the theory, and then the expansion becomes that in powers ofk but not of k. Thus the nonrenormalizability of Einstein gravity should be completely reconsidered in the light of the new expansion.     

Large-mass and high-temperature behaviour in perturbative quantum field theory

Bounds for large-mass behaviour in renormalized perturbation expansions at zero temperature, which were previously obtained by Manoukian and Caswell-Kennedy in momentum space, are rederived in the parametric representation. A very simple unified proof of the BPHZ theorem and the decoupling theorem is also given. A new technique for asymptotic analysis, based on a generalized Kontorovich-Lebedev integral transform, is introduced. This method is applied to find the leading high-temperature behaviour of perturbative field theories in the imaginary-time formalism. We prove that diagrams containing nonstatic modes, which at high temperature behave like particles with a large mass, are suppressed relative to purely static diagrams. This rigorously proves a limited form of dimensional reduction at infinite temperature.     

Breakup of an invariant curve in a dissipative standard mapping

A critical value for the existence of an invariant curve in a dissipative standard mapping is calculated by a new method in relation to Arnold's tongue. It is shown that the golden mean invariant curve is not the last curve for the strongly dissipative parameter region.     

A new perturbative approach for the anharmonic oscillator

A new perturbative approach is presented and applied to the anharmonic oscillator. Estimating the large-order behaviour by means of the Lipatov method, the new perturbation expansion turns out to be convergent for the partition function at non-zero temperature. Although the new perturbation expansion is not convergent for the ground-state energy, it nevertheless provides accurate approximations, even in situations which cannot be treated by standard perturbation techniques, like the strong coupling regime.     

On the invariant regularization of the standard model

We show that the gauge invariant regularization of the Standard Model proposed by Frolov and Slavnov describes a nonlocal theory with quite simple Lagrangian.     

Weak coupling perturbative calculations of the Wilson loop for the standard action

Wilson loops of size m × n with m, n≤6 are evaluated to second order in weak coupling perturbation theory. The results are compared with earlier estimates by Hattori and Kawai.     

A new perturbative approach to the classical anharmonic oscillator

The periodic motion of the classical anharmonic oscillator characterized by the potentialV(x)=1/2x ²+λ/2k x ^2k is considered. The period is first determined to all orders inλ in a perturbative series. Making use of this, the solution of the nonlinear equation of motion is then expressed in the form of a Fourier series. The Fourier coefficients are obtained by solving simple algebraic relations. Secular terms are inherently absent in this perturbative scheme. Explicit solution is presented for generalk up to the second order, from which the Duffing and the sextic oscillator results follow as special cases.     

Visual explorations of dynamics: The standard map

The Macintosh application StdMap allows easy exploration of many of the phenomena of area-preserving mappings. This tutorial explains some of these phenomena and presents a number of simple experiments centered on the use of this program. Dedicated to the memory of John Greene. His ideas are as timeless as his generosity is legendary.     

Critical behaviour of the ising model in a transverse field

A theory for the critical behaviour of the Ising model in a transverse field is presented. The existence of a central component in the spectral density of the longitudinal spin relaxation function is shown to result self-consistently from decay into itself and energy-fluctuation diffusion mode. Estimates of the various quantities are made forT nearT _c , the damping is seen to be temperature dependent. Our theory is appropriate for the caseΓ ～J whereΓ andJ are the magnitudes of the transverse field and the exchange interaction between the spins respectively. Its relevance to neutron scattering data on Van Vleck paramagnets is discussed.     

Near threshold anomalous transport in the standard map

Anomalous transport is investigated near threshold in the standard map. Very long time flights, and a large anomaly in the transport, are shown to be associated with a new form of multi-island structures causing orbit sticking. The phase space structure of these traps, and the exponents of the characteristic long time tails associated with them are determined. In general these structures are very complex, but some cases, consisting of layers of islands, allow simple modeling. (c) 1998 American Institute of Physics.