Local Phase Space Research Articles

Bi-Stable Spiral Waves: Stepped Spiral Waves

A new type of bi-stable spiral waves called “stepped spiral waves”, is investigated in this study in an oscillatory medium exhibiting period-doubling bifurcations. Prior to the period-doubling bifurcation of this system, the stepped spiral waves are produced by an unwanted phase trajectory event; the loss of symmetry takes the form of synchronization defect lines, where the trajectory in the local oscillation phase space changes into two different ways. The formation principle of this type of bi-stable spiral wave and the internal structure and geometry of these synchronization defects are studied, and several potential categories of stepped spiral waves are discussed.

Dark matter in 3D

We discuss the relevance of directional detection experiments in the post-discovery era and propose a method to extract the local dark matter phase space distribution from directional data. The first feature of this method is a parameterization of the dark matter distribution function in terms of integrals of motion, which can be analytically extended to infer properties of the global distribution if certain equilibrium conditions hold. The second feature of our method is a decomposition of the distribution function in moments of a model independent basis, with minimal reliance on the ansatz for its functional form. We illustrate our method using the Via Lactea II N-body simulation as well as an analytical model for the dark matter halo. We conclude that O(1000) events are necessary to measure deviations from the Standard Halo Model and constrain or measure the presence of anisotropies.

Proposal for a macroscopic test of local realism with phase-space measurements

We propose a new test of local realism based on correlation measurements of continuum valued functions of positions and momenta, known as modular variables. The Wigner representation of these observables are bounded in phase space, and therefore, the associated inequality holds for any state described by a non-negative Wigner function. This agrees with Bell's remark that positive Wigner functions, serving as a valid probability distribution over local (hidden) phase space coordinates, do not reveal non-locality. We construct a class of entangled states resulting in a violation of the inequality, and thus truly demonstrate non-locality in phase space. The states can be realized through grating techniques in space-like separated interferometric setups. The non-locality is verified from the spatial correlation data, collected from the screens.

Revealing the flux: Using processed Husimi maps to visualize dynamics of bound systems and mesoscopic transport

We elaborate upon the ``processed Husimi map'' representation for visualizing quantum wave functions using coherent states as a measurement of the local phase space to produce a vector field related to the probability flux. Adapted from the Husimi projection, the processed Husimi map is mathematically related to the flux operator under certain limits but offers a robust and flexible alternative since it can operate away from these limits and in systems that exhibit zero flux. The processed Husimi map is further capable of revealing the full classical dynamics underlying a quantum wave function since it reverse engineers the wave function to yield the underlying classical ray structure. We demonstrate the capabilities of processed Husimi maps on bound systems with and without electromagnetic fields, as well as on open systems on and off resonance, to examine the relationship between closed system eigenstates and mesoscopic transport.

Is the effect of the Sun's gravitational potential on dark matter particles observable?

We consider the effect of the Sun's gravitational potential on the local phase space distribution of dark matter particles, focusing on its implication for the annual modulation signal in direct detection experiments. We perform a fit to the modulation signal observed in DAMA/LIBRA and show that the allowed region shrinks if Solar gravitational focusing (GF) is included compared to the one without GF. Furthermore, we consider a possible signal in a generic future direct detection experiment, irrespective of the DAMA/LIBRA signal. Even for scattering cross sections close to the current bound and a large exposure of a xenon target with 270 ton yr it will be hard to establish the presence of GF from data. In the region of dark matter masses below 40 GeV an annual modulation signal can be established for our assumed experimental setup, however GF is negligible for low masses. In the high mass region, where GF is more important, the significance of annual modulation itself is very low. We obtain similar results for lighter targets such as Ge and Ar. We comment also on inelastic scattering, noting that GF becomes somewhat more important for exothermic scattering compared to the elastic case.

Dynamically optimized Wang-Landau sampling with adaptive trial moves and modification factors

The density of states of continuous models is known to span many orders of magnitudes at different energies due to the small volume of phase space near the ground state. Consequently, the traditional Wang-Landau sampling which uses the same trial move for all energies faces difficulties sampling the low-entropic states. We developed an adaptive variant of the Wang-Landau algorithm that very effectively samples the density of states of continuous models across the entire energy range. By extending the acceptance ratio method of Bouzida, Kumar, and Swendsen such that the step size of the trial move and acceptance rate are adapted in an energy-dependent fashion, the random walker efficiently adapts its sampling according to the local phase space structure. The Wang-Landau modification factor is also made energy dependent in accordance with the step size, enhancing the accumulation of the density of states. Numerical simulations show that our proposed method performs much better than the traditional Wang-Landau sampling.

Semiclassical deconstruction of quantum states in graphene

We build upon previous work that used coherent states as a measurement of the local phase space and extended the flux operator by adapting the Husimi projection to produce a vector field called the Husimi map. In this article, we extend its definition from continuous systems to lattices. This requires making several adjustments to incorporate effects such as group velocity and multiple bands. Several phenomena which uniquely occur in lattice systems, like group-velocity warping and internal Bragg diffraction, are explained and demonstrated using Husimi maps. We also show that scattering points between bands and valleys can be identified in the divergence of the Husimi map.

Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

We give a detailed study of the infinite‐energy solutions of the Cahn–Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well‐posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties. Copyright © 2013 John Wiley & Sons, Ltd.

Dissipation and entropy production in deterministic heat conduction of quasi-one-dimensional systems

We explore the consequences of a deterministic microscopic thermostat-reservoir contact mechanism. With different temperature reservoirs at each end of a two-dimensional system, a heat current is produced and the system has an anomalous thermal conductivity. The microscopic form for the local heat flux vector is derived and both the kinetic and potential contributions are calculated. The total heat flux vector is shown to satisfy the continuity equation. The properties of this nonequilibrium steady state are studied as functions of system size and temperature gradient, identifying key scaling relations for the local fluid properties and separating bulk and boundary effects. The local entropy density calculated from the local equilibrium distribution is shown to be a very good approximation to the entropy density calculated directly from the velocity distribution even for systems that are far from equilibrium. The dissipation and kinetic entropy production and flux are compared quantitatively and the differing mechanisms discussed within the Bhatnagar-Gross-Krook approximation. For equal-temperature reservoirs the entropy production near the reservoir walls is shown to be proportional to the local phase space contraction calculated from the tangent space dynamics. However, for unequal temperatures, the connection between local entropy production and local phase space contraction is more complicated.

The destruction of tori in volume-preserving maps

Invariant tori are prominent features of symplectic and volume-preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an n-dimensional volume-preserving map, such tori are prevalent when the map is nearly “integrable,” in the sense of having one action and n − 1 angle variables. As the map is perturbed, numerical studies show that the originally connected image of the frequency map acquires gaps due to resonances and domains of nonconvergence due to chaos. We present examples of a three-dimensional, generalized standard map for which there is a critical perturbation size, ε c , above which there are no tori. Numerical investigations to find the “last invariant torus” reveal some similarities to the behavior found by Greene near a critical invariant circle for area preserving maps: the crossing time through the newly destroyed torus appears to have a power law singularity at ε c , and the local phase space near the critical torus contains many high-order resonances.

Migration scheme for imaging offset VSP data within local phase space

The imaging of offset VSP data in local phase space can improve the image of the subsurface structure near the well. In this paper, we present a migration scheme for imaging VSP data in a local phase space, which uses the Gabor-Daubechies tight framebased extrapolator (G-D extrapolator) and its high-frequency asymptotic expansion to extrapolate wavefields and also delineates an improved correlation imaging condition in the local angle domain. The results for migrating synthetic and real VSP data demonstrate that the application of the high-frequency G-D extrapolator asymptotic expansion can effectively decrease computational complexity. The local angle domain correlation imaging condition can be used to weaken migration artifacts without increasing computation.

CHAOTIC DIFFUSION OF RESONANT KUIPER BELT OBJECTS

We carried out extensive numerical orbit integrations to probe the long-term chaotic dynamics of the two strongest mean motion resonances of Neptune in the Kuiper belt, the 3:2 (Plutinos) and 2:1 (Twotinos). Our primary results include a computation of the relative volumes of phase space characterized by large- and small-resonance libration amplitudes, and maps of resonance stability measured by mean chaotic diffusion rate. We find that Neptune's 2:1 resonance has weaker overall long-term stability than the 3:2 -- only ~15% of Twotinos are projected to survive for 4 Gyr, compared to ~27% of Plutinos, based on an extrapolation from our 1-Gyr integrations. We find that Pluto has only a modest effect, causing a ~4% decrease in the Plutino population that survives to 4 Gyr. Given current observational estimates, and assuming an initial distribution of particles proportional to the local phase space volume in the resonance, we conclude that the primordial populations of Plutinos and Twotinos formerly made up more than half the population of the classical and resonant Kuiper Belt. We also conclude that Twotinos were originally nearly as numerous as Plutinos; this is consistent with predictions from early models of smooth giant planet migration and resonance sweeping of the Kuiper Belt, and provides a useful constraint for more detailed models.

Pulse propagation and classification in time/frequency and position/wave number phase space.

In dispersive pulse propagation, different frequencies travel at different velocities, and, hence, the pulse is nonstationary in that it changes over time and distance. Joint phase space representations, such as the Wigner distribution and the spectrogram, are often employed to study nonstationary signals, and, hence, it is natural to consider their application to dispersive propagation. We consider position/wave number and time/frequency representations of a pulse propagating with dispersion and damping. We give the Wigner distribution of the pulse at a particular time/position in terms of the Wigner distribution of the initial pulse. Approximations of this result are derived, which are accurate and remarkably simple to apply, as compared to the stationary phase approximation. The approximations provide insights as well, in that they show how each point in phase space propagates at the group velocity, and lead to new features for classification, based on local phase space moments, that are invariant to t...

Local phase space control and interplay of classical and quantum effects in dissociation of a driven Morse oscillator

This work explores the possibility of controlling the dissociation of a monochromatically driven one-dimensional Morse oscillator by recreating barriers, in the form of invariant tori with irrational winding ratios, at specific locations in the phase space. The control algorithm proposed by Huang et al. [Phys. Rev. A 74, 053408 (2006)] is used to obtain an analytic expression for the control field. We show that the control term, approximated as an additional weaker field, is efficient in recreating the desired tori and suppresses the classical as well as the quantum dissociation. However, in the case when the field frequency is tuned close to a two-photon resonance the local barriers are not effective in suppressing the dissociation. We establish that in the on-resonant case quantum dissociation primarily occurs via resonance-assisted tunneling and controlling the quantum dynamics requires a local perturbation of the specific nonlinear resonance in the underlying phase space.

CONTROLLING TURBULENCE VIA TARGET WAVES GENERATED BY LOCAL PHASE SPACE COMPRESSION

The control of spatiotemporal chaos by developing target waves via local phase space compression in dynamical systems described by partial differential equation is reported. By considering spatially one-dimensional (1D) and two-dimensional (2D) complex Ginzburg–Landau equations (CGLE), we find that by using local system perturbation and confining variable in a finite range within a controlled area, periodic signals are generated and spread into an uncontrolled area in 1D spatiotemporal systems with weak turbulence. In 2D systems, target waves can be developed to suppress spatiotemporal chaos by using this compression technique in the convective instability area. The controllable parameters are investigated by performing numerical simulations.

Chaos in models of the solar neighbourhood

Aims. We quantify the amount of chaos that exists in the local phase space. Methods. A sample of orbits from four different models of the solar neighbourhood phase space are analysed by a new chaos identification (and quantification) technique. While three of the models we used bear the signature of the perturbation due to both the Galactic bar and the spiral pattern, the last of the models is a bar-only one. We explore the models by comparing the corresponding values of chaos strength that are induced at the various energy levels. Results. We find that all the viable models that have been demonstrated to successfully reproduce the local phase space structure -models that include the bar as well as the spiral - bear strong chaoticity, though the model that implies the highest degree of chaos is the one in which overlap of the major resonances of the bar and the spiral occurs. The bar-only model is found to display regularity. We advance chaos as primarily responsible for the splitting of the Hyades-Pleiades mode (the larger mode) of the local velocity distribution.

Propagation-invariant classification of sonar signals

Underwater sound waves can undergo frequency-dependent effects, such as dispersion and damping, that cause the wave to change as it propagates. In active sonar, these changes can be detrimental to automatic classification because the observed backscatter depends on the propagation environment and how far the wave has traveled. We aim to develop signal processing techniques to enable propagation-invariant classification. Toward that end, we have developed a local phase space view of propagation with dispersion and damping that is physically illuminating and suggests new features for classification. In this talk, we will present moment-like features extracted from the backscattered wave that are invariant to dispersion and damping, per mode. Environmental knowledge is not needed, beyond knowing the general form of the damping. Two forms of damping are considered. Classification results on simulations of the backscatter from different steel shells propagating in a Pekeris waveguide with damping and random variations will be presented. A dispersion-invariant matched-filter receiver approach will also be presented. [Supported by ONR Grant No. N00014-06-1-0009.]

SPATIALLY NONDECAYING SOLUTIONS OF THE 2D NAVIER-STOKES EQUATION IN A STRIP

AbstractThe weighted energy theory for Navier-Stokes equations in 2D strips is developed. Based on this theory, the existence of a solution in the uniformly local phase space (without any spatial decaying assumptions), its uniqueness and the existence of a global attractor are verified. In particular, this phase space contains the 2D Poiseuille flows.

Reducing colored noise for chaotic time series in the local phase space

A two step method is proposed to reduce colored noise for chaotic data in the local phase space. With the observation that the energy of colored noise is mainly distributed in a low dimensional subspace, a noise dominated subspace is first estimated by the energy distribution of colored noise. At step 1, for the reference phase point, the components projected into the noise dominated subspace are deleted and the enhanced data are reconstructed with the remaining components. The residual error of the output of step 1 tends to distribute on each direction uniformly. So at step 2, the local projection method is further applied to the output of step 1, treating the residual error as white noise. Experiments show that our method performs well in eliminating colored noise for chaotic data.

Enhancement of Chinese speech based on nonlinear dynamics

Based on recently observed nonlinear dynamic features of human speech, the local projection (LP) method, originally developed for noisy chaotic time series, is generalized and adapted to the enhancement of Chinese speech. The analysis of minimum embedding dimensions estimated by the false nearest neighbor algorithm shows that all the basic phonemes and syllables in Chinese can be faithfully embedded in some low-dimensional phase space. Over-embedding is applied to reconstruct the dynamics of continuous speech in some extended phase space of higher dimension, thus solving the problem of nonstationarity in continuous speech. A generalization of the LP method, named the local subspace method, is presented for speech enhancement in the phase space. It is demonstrated that, the local subspace method is essentially an extension of the well-known linear subspace technique in the local phase space, and the LP method is the least square case of this generalization. Noise reduction is then carried out in the local phase space. Results show that the LP method, with 2 or 3 iterations, achieves better performances than the local subspace method. For both isolated and continuous speech with additive white noise, experiments show the superiority of the LP method over two other popular algorithms.