Patterns In Phase Space Research Articles

Phase-space patterns of quantum transport on ordered and disordered networks

In this paper, we consider the quantum-mechanical phase space patterns on ordered and disordered networks. For ordered networks in which each node is connected to its $2m$ nearest neighbors ($m$ on either side), the phase space quasiprobability of Wigner function shows various patterns. In the long time limit, on even-numbered networks, we find an asymmetric quasiprobability between the node and its opposite node. This asymmetry depends on the network parameters and specific phase space positions. For disordered networks in which each edge is rewired with probability $p>0$, the phase space displays regional localization on the initial node.

Localization of coherent exciton transport in phase space

We study numerically the dynamics of excitons on discrete rings in the presence of static disorder. Based on continuous-time quantum walks we compute the time evolution of the Wigner function (WF) both for pure diagonal (site) disorder, as well as for diagonal and off-diagonal (site and transfer) disorder. In both cases, large disorder leads to localization and destroys the characteristic phase space patterns of the WF found in the absence of disorder.

Dynamics of chaotic driving: Rotation in the restricted three-body problem

We investigate the rotation of a small nonspherical body in the planar restricted three-body problem along periodic, quasi-periodic, and chaotic orbits of the small body's center of mass. The rotation dynamics is chaotic in all three cases, but a systematic overview of it via stroboscopic mappings is possible only in the periodic case. We propose to explore the structured phase space patterns by following an ensemble of trajectories, a droplet, in the phase space. The temporal evolution of the pattern can be characterized by a time-dependent fractal dimension. It is shown to converge exponentially to a time-independent value for long times. In the presence of dissipation, the droplet typically converges to a so-called snapshot chaotic attractor whose shape might change chaotically in time, but whose asymptotic fractal dimension is constant.

Novel regimes of electron dynamics in superlattices

A superlattice (SL) is an artificial crystal in which alternating nanometre-thick layers of two or more different semiconductor materials provide a periodic potential for conduction electrons. Strong magnetic and electric fields applied to this type of structure provide a means of exploring novel regimes of electron dynamics. The applied fields lower the dimensionality of the electronic states and lead to qualitative changes in the electronic conduction. This discovery is of fundamental interest and highly relevant to the properties of other low-dimensional conductors, such as nanowires and quantum dot SLs, which are presently attracting the attention of the physics and device communities. In addition, a rare type of chaotic electron dynamics, called non-Kolmogorov-Arnold-Moser (KAM) chaotic motion, which has been theoretically studied for several decades, is observed experimentally in SLs. The onset of chaos at discrete values of the applied electric and magnetic fields is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns in phase space. Therefore, non-KAM chaos could provide a new mechanism for controlling the electrical conductivity of the electronic devices with extreme sensitivity.

Symmetry-breaking analysis for the general Helmholtz–Duffing oscillator

Abstract The symmetry breaking phenomenon for a general Helmholtz–Duffing oscillator as a function of a symmetric parameter in the nonlinear force is investigated. Different values of this parameter convert the general oscillator into either the Helmholtz or the Duffing oscillator. Due to the variation of the symmetric parameter, the phase space patterns of the unperturbed Helmholtz–Duffing oscillator will cause a huge difference between the left-hand homoclinic orbit and the right-hand one. In particular, the area of the left-hand homoclinic orbits is a strictly monotonously decreasing function, while the area of the right-hand homoclinic orbit varies only in a very small range. There exist distinct local supercritical and subcritical saddle-node bifurcations at two different centers. The left-hand and the right-hand existing regions of the harmonic solutions of the Helmholtz–Duffing oscillator created by the left-hand and the right-hand saddle-node bifurcation curves will lead to different transition in the amplitude–frequency plane. There exists also a critical frequency which has the effect that the left-hand homoclinic bifurcation value is equal to the right-hand homoclinic bifurcation value. And, if the amplitude coefficient of the Helmholtz–Duffing oscillator is used as the control parameter, and it is larger than the same left-hand and right-hand homoclinic bifurcation, then the global stability of the system will be destroyed at a lowest cost. Besides this critical frequency, the left-hand and the right-hand homoclinic bifurcations are not only unequal, but also their effects for the system’s stability are different. Among them, the effect resulting from the small homoclinic bifurcation for the system’s stability is local and negligible, while the effect from the large homoclinic bifurcation is global but this is accomplished at a quite larger cost.

Continuous-time quantum walks in phase space

We formulate continuous time quantum walks (CTQW) in a discrete quantum mechanical phase space. We define and calculate the Wigner function (WF) and its marginal distributions for CTQWs on circles of arbitrary length $N$. The WF of the CTQW shows characteristic features in phase space. Revivals of the probability distributions found for continuous and for discrete quantum carpets do manifest themselves as characteristic patterns in phase space.

Stochastic Carrier Dynamics in Semiconductor Superlattices

is an integer, the electron orbits change fromlocalised Bloch-like trajectories to unbounded stochastic orbits, which difiuserapidly through intricate web patterns in phase space. To quantify how thesewebs afiect electron transport, we make drift-difiusion calculations of thecurrent{voltage curves including the efiects of space-charge build up. Whenthe magnetic fleld is tilted, our simulations reveal a large resonant peak,which originates from stochastic delocalisation of the electron orbits. Weshow that the corresponding quantised eigenstates change discontinuouslyfrom a highly localised character when the system is ofi resonance to a fullydelocalised form when the resonance condition is satisfled.PACS numbers: 73.20.At, 73.21.Cd

Chaotic electron diffusion through stochastic webs enhances current flow in superlattices.

Understanding how complex systems respond to change is of fundamental importance in the natural sciences. There is particular interest in systems whose classical newtonian motion becomes chaotic as an applied perturbation grows. The transition to chaos usually occurs by the gradual destruction of stable orbits in parameter space, in accordance with the Kolmogorov-Arnold-Moser (KAM) theorem--a cornerstone of nonlinear dynamics that explains, for example, gaps in the asteroid belt. By contrast, 'non-KAM' chaos switches on and off abruptly at critical values of the perturbation frequency. This type of dynamics has wide-ranging implications in the theory of plasma physics, tokamak fusion, turbulence, ion traps, and quasicrystals. Here we realize non-KAM chaos experimentally by exploiting the quantum properties of electrons in the periodic potential of a semiconductor superlattice with an applied voltage and magnetic field. The onset of chaos at discrete voltages is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns in phase space. Non-KAM chaos therefore provides a mechanism for controlling the electrical conductivity of a condensed matter device: its extreme sensitivity could find applications in quantum electronics and photonics.

Hyperbolic scar patterns in phase space

We develop a semiclassical approximation for the spectral Wigner andHusimi functions in the neighbourhood of a classically unstable periodicorbit of chaotic 2-D maps. The prediction of hyperbolicfringes for the Wigner function, asymptotic to the stable and unstablemanifolds, is verified computationally for a (linear) cat map, after thetheory is adapted to a discrete phase space appropriate to a quantizedtorus. The characteristic fringe patterns can be distinguished even forquasi-energies where the fixed point is not Bohr-quantized. Thecorresponding Husimi function dampens these fringes with a Gaussianenvelope centred on the periodic point. Even though the hyperbolicstructure is then barely perceptible, more periodic points stand out dueto the weakened interference.

Motion of three point vortices in a periodic parallelogram

The motion of three interacting point vortices with zero net circulation in a periodic parallelogram defines an integrable dynamical system. A method for solving this system is presented. The relative motion of two of the vortices can be ‘mapped’ onto a problem of advection of a passive particle in ‘phase space’ by a certain set of stationary point vortices, which also has zero net circulation. The advection problem in phase space can be written in Hamiltonian form, and particle trajectories are given by level curves of the Hamiltonian. The motion of individual vortices in the original three-vortex problem then requires one additional quadrature. A complicated structure of the solution space emerges with a large number of qualitatively different regimes of motion. Bifurcations of the streamline pattern in phase space, which occur as the impulse of the original vortex system is changed, are traced. Representative cases are analysed in detail, and a general procedure is indicated for all cases. Although the problem is integrable, the trajectories of the vortices can be surprisingly complicated. The results are compared qualitatively to vortex paths found in large-scale numerical simulations of two-dimensional turbulence.

Patterns of solid–fluid phase equilibria: new possibilities?

Thermodynamic phase space patterns of solid–liquid–vapor (slg) behavior in binary systems are reviewed. The van der Waals equation of state is used in combination with a common mathematical artifice for the solid–phase fugacity function to map out the slg loci for a model binary homologous series of solvent+solute mixtures as a function of the solute characterization. The computational results suggest the possible existence of a richer thermodynamic phase space topography than previously envisioned.

Local Toeplitz operators based on wavelets: phase space patterns for rough wavelets

We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-

Elementary excitations and nonlinear dynamics of a magnetic domain wall.

Two-dimensional magnons localized in the polarized 180\ifmmode^\circ\else\textdegree\fi{} Bloch domain wall and its nonlinear dynamics at high ac drive field in an yttrium iron garnet single-crystal plate are investigated. The interaction between translational and flexural modes of the wall vibration is studied in detail. The excitation spectrum of the moving wall is shown to depend on its velocity. An unexpected asymmetric dependence of flexural mode eigenfrequencies and resonance linewidths on the wall velocity is found. The flexural waves themselves are found to influence the amplitude of the translational wall vibrations. Two kinds of instability were found at increasing drive field. The first is the transition from the periodic linear vibrations at low field and low wall mobility to weakly nonharmonic oscillations at high mobility. They are shown to be due to the excitation of the standing flexural modes of the wall. And the second one is the transition to the region of chaotic wall motion and low mobility at the highest field studied. The real-time signals, spectral data, and phase-space patterns corresponding to different regions were obtained. The phase portrait at the highest field studied is found to be of the chaotic attractor type.

Circulation Patterns in Phase Space: A Multinormal Distribution?

In this paper daily wintertime extratropical Northern Hemisphere (NH) circulation analogs are studied. First the analog forecasts are compared to various common benchmark methods such as random or persistence forecasts and the climate mean as a forecast. In line with earlier work, it is concluded that beyond a few days lead time the analogs offer no advantage over any of these benchmark methods. The same is true for derivative analogs (where only the time derivative of the analogs is used and added to the base case), although they perform considerably better than the traditional analogs on the first time step(s). Even though the circulation analogs have no extended-range forecast capability, they nevertheless offer a convenient way of studying the gross structure of the phase space of circulation patterns. A thorough study of the root-mean-square distances (rmsd) between the best circulation analogs, considered as an indicator for the relative frequency in the phase space, has been performed. It was shown that in a phase-average sense, when the density characteristics are considered only as a function of distance from the climate mean, the distribution of circulation patterns is statistically indistinguishable from a multinormal distribution. This simple but previously unobserved fact has a series of consequences, some of which are presented here. First, since the density of the distribution of circulation patterns is increasing with decreasing distance from the climate mean, the best analog to a particular circulation pattern is more likely to be closer to the climate mean than the base case. A second observation is that the persistence of the flow increases with decreasing distance from the climate mean. A double stratification of the analogs according to their initial difference and the persistence of the flow showed no enhanced predictability in persistent cases; the forecast error depends only on the initial error of the analogs. This is an indication that the higher numerical forecast skill in persistent cases (reported in earlier studies) may be related to the fact that those cases are relatively close to the climate mean, where even random forecasts have smaller rms error. A third point is that analog predictability does not depend on the initial flow's distance from the climate mean either. The phase-average multinormality of the wintertime extratropical NH circulation phase space discussed in this study does not rule out the possibility of a fine structure with several local maxima (multiple equilibria) embedded in the overall gross structure of approximate normality. Indeed, a refined methodology revealing the existence of such “dense” areas will be reported in a later paper.

Canonical ensembles from chaos

We present methods to compute thermodynamic properties of classical systems which involve extending the phase space by two degrees of freedom. These two additional degrees of freedom are used to replicate the coupling of the original system to the infinite degrees of freedom of a heat bath. In the extended phase space, the trajectories are ergodic. This feature enables one to replace phase space averages by time averages, which are extremely simple to compute. We examine phase space patterns, thermal distributions, correlations, ergodicity, Lyapunov exponents, mixing, and rate of convergence in an analysis of several simple systems.

Simulation of intense particle beams with regularly distributed Gaussian subbeams

Techniques for the simulation of intense particle beams are investigated with respect to the required number of simulation particles. It is shown that for nonchaotic systems it is advantageous if the particles initially are not distributed in a statistical manner but rather arranged in a regular pattern in phase space. This reduces the number of required simulation particles drastically. In the case of such an initially regular arrangement of particles the algorithm which assigns the charges of the particles to the computation mesh becomes of prime importance. The performances of different commonly used algorithms are investigated. The Gaussian assignment algorithm proved far superior to other more commonly used techniques, allowing simulations even at the theoretical limit of 1 particle per cell. Examples for very accurate simulations of beam dynamics with very few particles using an initially regular mesh of particles and Gaussian assignment are given.

Phase space electroencephalography (EEG): a new mode of intraoperative EEG analysis.

Intraoperative monitoring of electroencephalography (EEG) data can help assess brain integrity and/or depth of anesthesia. We demonstrate a computer generated technique which provides a visually robust display of EEG data plotted as 'phase space trajectories' and a mathematically derived parameter ('dimensionality') which may correlate with depth of anesthesia. Application of nonlinear mathematical analysis, used to describe complex dynamical systems, can characterize 'phase space' EEG patterns by identifying attractors (geometrical patterns in phase space corresponding to specific ordered EEG data subjects) and by quantifying the degree of order and chaos (calculation of dimensionality). Dimensionality calculations describe the degree of complexity in a signal and may generate a clinically useful univariate EEG descriptor of anesthetic depth. In this paper we describe and demonstrate phase space trajectories generated for sine waves, mixtures of sine waves, and white noise (random chaotic events). We also present EEG phase space trajectories and dimensionality calculations from a patient undergoing surgery and general anesthesia in 3 recognizable states: awake, anesthetized, and burst suppression. Phase space trajectories of the three states are visually distinguishable, and dimensionality calculations indicate that EEG progresses from 'chaos' (awake) to progressively more 'ordered' attractors (anesthetized and burst suppression).

Some remarks about the extraction system of the AEG compact cyclotron

Extraction calculations for the AEG compact cyclotron are described, and the phase space pattern for precessional extraction of protons is discussed.

Experimental determination of the nonlinear interaction in a one dimensional beam-plasma system

Recent theories have shown that the instability driven by a cold, weak electron beam in a one-dimensional plasma is stabilized when the unstable wave grows to an amplitude sufficient to trap the electron beam. Quantitative measurements of the instability confirm every prediction of the theory: the maximum wave amplitude, monochromaticity of the unstable mode, harmonic content of the mode, and wavelength of the slow oscillation in wave amplitude. The electron beam goes through the same patterns in phase space and time-averaged f(v) as predicted. Both scaling and absolute magnitudes agree with theory.