6D Phase Space Research Articles

Dynamical system analysis of Einstein–Skyrme model in a Kantowski–Sachs spacetime

The present work deals with anisotropic (but zero heat flux) Skyrme fluid with a constant radial profile in locally rotational Kantowski–Sachs spacetime in the background of Einstein gravity. By suitably change of variables the field equations are transformed to an autonomous system. To examine the stability of the system critical points are determined. For hyperbolic critical points the analysis of the system is done using Hartman–Grobman theorem, while center manifold theory is used to analyze non-hyperbolic critical points. Also stability at infinity is analyzed to visualize the global evolution of the Universe. As a result the 3D-phase space is identified with the Poincaré 3-sphere embedded in R4. It is found that the parameters in the autonomous system play a crucial role for phase transition of the Universe. Finally possible bifurcation scenarios are discussed to identify the point of phase transition.

Open cluster kinematics with Gaia DR2 (Corrigendum)

Context. Open clusters are very good tracers of the evolution of the Galactic disc. Thanks to Gaia, their kinematics can be investigated with an unprecedented precision and accuracy. Aims. The distribution of open clusters in the 6D phase space is revisited with Gaia DR2. Methods. The weighted mean radial velocity of open clusters was determined, using the most probable members available from a previous astrometric investigation that also provided mean parallaxes and proper motions. Those parameters, all derived from Gaia DR2 only, were combined to provide the 6D phase space information of 861 clusters. The velocity distribution of nearby clusters was investigated, as well as the spatial and velocity distributions of the whole sample as a function of age. A high quality subsample was used to investigate some possible pairs and groups of clusters sharing the same Galactic position and velocity. Results. For the high quality sample that has 406 clusters, the median uncertainty of the weighted mean radial velocity is 0.5 km/s. The accuracy, assessed by comparison to ground-based high resolution spectroscopy, is better than 1 km/s. Open clusters nicely follow the velocity distribution of field stars in the close Solar neighbourhood previously revealed by Gaia DR2. As expected, the vertical distribution of young clusters is very flat but the novelty is the high precision to which this can be seen. The dispersion of vertical velocities of young clusters is at the level of 5 km/s. Clusters older than 1 Gyr span distances to the Galactic plane up to 1 kpc with a vertical velocity dispersion of 14 km/s, typical of the thin disc. Five pairs of clusters and one group with five members are possibly physically related. Other binary candidates previously identified turn out to be chance alignment.

Rotating Halo Traced by the K-giant Stars from LAMOST and Gaia

Abstract With the help of Gaia DR2, we are able to obtain the full 6D phase space information for stars from the Large Sky Area Multi-Object Fibre Spectroscopic Telescope DR5. With high precision of the position, velocity, and metallicity, the rotation of the local stellar halo is presented using the K-giant stars with [Fe/H] < −1 dex within 4 kpc from the Sun. By fitting the rotational velocity distribution with the three-Gaussian components, stellar halo, disk, and counterrotating hot population, we find that the local halo progradely rotates with km s−1, providing the local standard of the rest velocity of V LSR = 232 km s−1. Meanwhile, we obtain the dispersion of rotational velocity is km s−1. Although the rotational velocity strongly depends on the choice of V LSR, the trend of prograde rotation is substantial even when V LSR is set at as low as 220 km s−1. Moreover, we derive the rotation for subsamples with different metallicities and find that the rotational velocity is essentially not correlated with [Fe/H]. This may hint at a secular evolution origin of the prograde rotation. It shows that the metallicity of the progradely rotating halo is peaked within −1.9 < [Fe/H] < −1.6 without considering the selection effect. We also find a small fraction of counterrotating stars with larger dispersion and a lower metallicity. Finally, the disk component rotates with km s−1 and km s−1, which is quite consistent with the metal-weak thick disk population.

Numerical study and test of the APS linac transverse deflecting cavity

The transverse deflecting cavity can be used to transform particle distributions in the 6D phase space, which makes it a promising component in phase space beam diagnostics and beam manipulations. In the Advanced Photon Source (APS) Linac, a LOLA-type traveling wave deflecting cavity was installed for the diagnostics of beam characteristics, such as the bunch length, bunch temporal profile, time-dependent energy spread and slice (time-correlated) transverse emittance. In this paper, this deflecting cavity is modeled and analyzed with the numerical method. The effects of the center cell and coupler cell dimensions on the performance of the whole structure are studied, which shows the coupler cell radius has a dominant effect over the coupler cell length and slot width on the global reflection coefficient and field flatness. Important RF parameters, such as S11, field flatness and phase advance etc., of the LOLA-II cavity in tuning are calculated and discussed. After optimization, the field flatness of the cavity is 2.1%, the phase advance is 119.66° degrees with a standard deviation smaller than 0.5° and the bandwidth is 15.738 MHz when VSWR < 1.1 (i.e. the S11 <−26 dB). Bead-pull measurements and RF conditioning have been also performed and discussed. Finally, the initial cavity commissioning with electron beam of 375 MeV is reported, and compared with simulation.

Proper motions and dynamics of the Milky Way globular cluster system fromGaiaDR2

We use Gaia Data Release 2 to determine the mean proper motions for 150 Milky Way globular clusters (almost the entire known population), with a typical uncertainty of 0.05 mas/yr limited mainly by systematic errors. Combining them with distance and line-of-sight velocity measurements from the literature, we analyze the distribution of globular clusters in the 6d phase space, using both position/velocity and action/angle coordinates. The population of clusters in the central 10 kpc has a mean rotational velocity reaching 50-80 km/s, and a nearly isotropic velocity dispersion 100-120 km/s, while in the outer galaxy the cluster orbits are strongly radially anisotropic. We confirm a concentration of clusters at high radial action in the outer region of the Galaxy. Finally, we explore a range of equilibrium distribution function-based models for the entire globular cluster system, and the information they provide about the potential of the Milky Way. The dynamics of clusters is best described by models with the circular velocity between 10 and 50 kpc staying in the range 210-240 km/s.

Gaia-DR2 extended kinematical maps

Context. The Gaia Collaboration has used Gaia-DR2 sources with six-dimensional (6D) phase space information to derive kinematical maps within 5 kpc of the Sun, which is a reachable range for stars with relative error in distance lower than 20%. Aims. Here we aim to extend the range of distances by a factor of two to three, thus adding the range of Galactocentric distances between 13 kpc and 20 kpc to the previous maps, with their corresponding error and root mean square values. Methods. We make use of the whole sample of stars of Gaia-DR2 including radial velocity measurements, which consists in more than seven million sources, and we apply a statistical deconvolution of the parallax errors based on the Lucy’s inversion method of the Fredholm integral equations of the first kind, without assuming any prior. Results. The new extended maps provide lots of new and corroborated information about the disk kinematics: significant departures of circularity in the mean orbits with radial Galactocentric velocities between −20 and +20 km s−1 and vertical velocities between −10 and +10 km s−1; variations of the azimuthal velocity with position; asymmetries between the northern and the southern Galactic hemispheres, especially towards the anticenter that includes a larger azimuthal velocity in the south; and others. Conclusions. These extended kinematical maps can be used to investigate the different dynamical models of our Galaxy, and we will present our own analyses in the forthcoming second part of this paper. At present, it is evident that the Milky Way is far from a simple stationary configuration in rotational equilibrium, but is characterized by streaming motions in all velocity components with conspicuous velocity gradients.

The anisosphere model: a novel differential phase space representation for Foucault pendulums and 2D oscillators

It is customary to describe the behaviour and stability of oscillators with the help of phase space representation. However, two-dimensional (2D) oscillators like the Foucault pendulum call for a 4D phase space that is not simple to visualize. Applying celestial body perturbation theory to the Foucault pendulum in his doctor dissertation, Nobel laureate Kamerlingh Onnes showed that the essential features of a Foucault pendulum are its inherent circular and linear anisotropies. A spherical differential 2D sub-space can be defined, where the group of the points of a spherical surface with respect to the operation rotation about a diametral axis is isomorphic with the group of sequential states of oscillation of a 2D pendulum with respect to the operation translation in time. Any Foucault pendulum is then characterized by two elliptical eigenstates which are represented by the poles of that rotation axis on the so-called anisosphere. Such poles play the role of attractor/repellor when “dichroic” damping is present. Moreover, they move drastically within a meridian plane when nonlinear restoring torque giving rise to Airy precession occurs. The concept of anisosphere constitutes a very powerful tool for analysing and optimizing actual Foucault pendulum implementations. That feature is illustrated by a numerical model.

Tuning of LANSCE 805-MHz high-energy linear accelerator with reduced beam losses

Suppression of beam losses is essential for successful operation of high-intensity linac. Historically, the values of the field amplitudes and phases of the side-coupled, 805-MHz LANSCE linac modules are maintained using a well-known delta-t tuning procedure. Transverse matching of the beam with accelerator is performed through adjustments of beam ellipses in 4D phase space with accelerator lattice using matching quadrupoles. Control of the beam-energy ramp along the length of a proton linear accelerator is required to improve tune of accelerator and decrease beam losses. Time-of-flight measurements of the H− beam energy are now being used to confirm and improve the overall control of the energy ramp along the linac. The time-of-flight method utilizes absolute measurements of beam energy using direct signals from beam at an oscilloscope, as well as the difference in RF phases measured as the beam passes installed delta-t pickup loops. A newly developed BPPM data acquisition system is used. Beam energy measurement along accelerator together with phase scans, klystrons output power control, and delta-t method, allow tuning of accelerator much more close to original design and reduce losses generated by linac. Details of the upgraded tuning procedure and results of tuning and operation are presented.

Harnessing olfactory bulb oscillations to perform fully brain-based sleep-scoring and real-time monitoring of anaesthesia depth.

Real-time tracking of vigilance states related to both sleep or anaesthesia has been a goal for over a century. However, sleep scoring cannot currently be performed with brain signals alone, despite the deep neuromodulatory transformations that accompany sleep state changes. Therefore, at heart, the operational distinction between sleep and wake is that of immobility and movement, despite numerous situations in which this one-to-one mapping fails. Here we demonstrate, using local field potential (LFP) recordings in freely moving mice, that gamma (50–70 Hz) power in the olfactory bulb (OB) allows for clear classification of sleep and wake, thus providing a brain-based criterion to distinguish these two vigilance states without relying on motor activity. Coupled with hippocampal theta activity, it allows the elaboration of a sleep scoring algorithm that relies on brain activity alone. This method reaches over 90% homology with classical methods based on muscular activity (electromyography [EMG]) and video tracking. Moreover, contrary to EMG, OB gamma power allows correct discrimination between sleep and immobility in ambiguous situations such as fear-related freezing. We use the instantaneous power of hippocampal theta oscillation and OB gamma oscillation to construct a 2D phase space that is highly robust throughout time, across individual mice and mouse strains, and under classical drug treatment. Dynamic analysis of trajectories within this space yields a novel characterisation of sleep/wake transitions: whereas waking up is a fast and direct transition that can be modelled by a ballistic trajectory, falling asleep is best described as a stochastic and gradual state change. Finally, we demonstrate that OB oscillations also allow us to track other vigilance states. Non-REM (NREM) and rapid eye movement (REM) sleep can be distinguished with high accuracy based on beta (10–15 Hz) power. More importantly, we show that depth of anaesthesia can be tracked in real time using OB gamma power. Indeed, the gamma power predicts and anticipates the motor response to stimulation both in the steady state under constant anaesthetic and dynamically during the recovery period. Altogether, this methodology opens the avenue for multi-timescale characterisation of brain states and provides an unprecedented window onto levels of vigilance.

Open cluster kinematics with Gaia DR2

Context. Open clusters are very good tracers of the evolution of the Galactic disc. Thanks to Gaia, their kinematics can be investigated with an unprecedented precision and accuracy. Aims. The distribution of open clusters in the 6D phase space is revisited with Gaia DR2. Methods. The weighted mean radial velocity of open clusters was determined, using the most probable members available from a previous astrometric investigation that also provided mean parallaxes and proper motions. Those parameters, all derived from Gaia DR2 only, were combined to provide the 6D phase-space information of 861 clusters. The velocity distribution of nearby clusters was investigated, as well as the spatial and velocity distributions of the whole sample as a function of age. A high-quality subsample was used to investigate some possible pairs and groups of clusters sharing the same Galactic position and velocity. Results. For the high-quality sample of 406 clusters, the median uncertainty of the weighted mean radial velocity is 0.5 km s−1. The accuracy, assessed by comparison to ground-based high-resolution spectroscopy, is better than 1 km s−1. Open clusters nicely follow the velocity distribution of field stars in the close solar neighbourhood as previously revealed by Gaia DR2. As expected, the vertical distribution of young clusters is very flat, but the novelty is the high precision to which this can be seen. The dispersion of vertical velocities of young clusters is at the level of 5 km s−1. Clusters older than 1 Gyr span distances to the Galactic plane of up to 1 kpc with a vertical velocity dispersion of 14 km s−1, typical of the thin disc. Five pairs of clusters and one group with five members might be physically related. Other binary candidates that have been identified previously are found to be chance alignments.

Gender and age classification using a new Poincare section-based feature set of ECG

In the medicine, the implication of individual differences has frequently been emphasized. Gender- and age-related differences can be mentioned as the most important individual parameters. On the other hand, electrocardiogram (ECG) signals are the subject of these differences. However, limited information is available regarding these individual dissimilarities in ECG dynamics. This study was aimed to evaluate gender and age differences by means of novel Poincare section indices. Our focus was to detect and classify dynamical behaviors of the ECG trajectories using three binary classification strategies: (1) gender-, (2) age-, (3) gender- and age-based classification. After constructing the 2D phase space of ECG, linear Poincare sections in distinct angles were developed and some geometric indices were extracted. The effect of delayed phase space on ECG measures was also inspected. We tested our algorithm on 79 healthy subjects. Using support vector machine, the maximum correct rate of 93.33% was achieved for the gender- and age-based classification strategies. Considering the information of both age and gender, the highest rate was 94.66%. The best results were achieved with delays of 5 and 6. In conclusion, our results showed that basin geometry of the ECG phase states is affected by individual differences.

Plausible Home Stars of the Interstellar Object ‘Oumuamua Found in Gaia DR2

Abstract The first detected interstellar object, ‘Oumuamua, which passed within 0.25 au of the Sun on 2017 September 9, was presumably ejected from a stellar system. We use its newly determined non-Keplerian trajectory together with the reconstructed Galactic orbits of 7 million stars from Gaia DR2 to identify past close encounters. Such an “encounter” could reveal the home system from which ‘Oumuamua was ejected. The closest encounter, at 0.60 pc (0.53–0.67 pc, 90% confidence interval), was with the M2.5 dwarf HIP 3757 at a relative velocity of 24.7 , 1 Myr ago. A more distant encounter (1.6 pc) but with a lower encounter (ejection) velocity of 10.7 was with the G5 dwarf HD 292249, 3.8 Myr ago. Two more stars have encounter distances and velocities intermediate to these. The encounter parameters are similar across six different non-gravitational trajectories for ‘Oumuamua. The ejection of ‘Oumuamua by scattering from a giant planet in one of the systems is plausible, but requires a rather unlikely configuration to achieve the high velocities found. A binary star system is more likely to produce the observed velocities. None of the four home candidates have published exoplanets or are known to be binaries. Given that the 7 million stars in Gaia DR2 with 6D phase space information is just a small fraction of all stars for which we can eventually reconstruct orbits, it is a priori unlikely that our current search would find ‘Oumuamua’s home star system. As ‘Oumuamua is expected to pass within 1 pc of about 20 stars and brown dwarfs every Myr, the plausibility of a home system also depends on an appropriate (low) encounter velocity.

Single-shot reconstruction of core 4D phase space of high-brightness electron beams using metal grids

This paper details the development of a single-shot diagnostic technique for the 4D average and core phase space densities of low-charge, high-brightness electron beams, based on the analysis of shadow point-projection images of metal grids. This technique is similar to the standard pepper pot method, although it allows much greater transmission of the beam and therefore is more suitable for low-charge electron beams. Transverse coupling terms are included in the analysis, allowing the complete 4D transverse beam matrix to be reconstructed. The 4D beam phase space information is extremely important for the characterization of nonround beams. An analysis of the resolution limits and experimental benchmarking of the technique with pepper-pot emittance measurements are presented.

Gaia DR2 in 6D: searching for the fastest stars in the Galaxy

ABSTRACT We search for the fastest stars in the subset of stars with radial velocity measurements of the second data release (DR2) of the European Space Agency mission Gaia. Starting from the observed positions, parallaxes, proper motions, and radial velocities, we construct the distance and total velocity distribution of more than 7 million stars in our Milky Way, deriving the full 6D phase space information in Galactocentric coordinates. These information are shared in a catalogue, publicly available at http://home.strw.leidenuniv.nl/~marchetti/research.html. To search for unbound stars, we then focus on stars with a probability greater than $50 $ per cent of being unbound from the Milky Way. This cut results in a clean sample of 125 sources with reliable astrometric parameters and radial velocities. Of these, 20 stars have probabilities greater than 80 per cent of being unbound from the Galaxy. On this latter subsample, we perform orbit integration to characterize the stars’ orbital parameter distributions. As expected given the relatively small sample size of bright stars, we find no hypervelocity star candidates, stars that are moving on orbits consistent with coming from the Galactic Centre. Instead, we find seven hyperrunaway star candidates, coming from the Galactic disc. Surprisingly, the remaining 13 unbound stars cannot be traced back to the Galaxy, including two of the fastest stars (around 700 km s−1). If conformed, these may constitute the tip of the iceberg of a large extragalactic population or the extreme velocity tail of stellar streams.

Magnetoactive acoustic metamaterials

Acoustic metamaterials with negative constitutive parameters (modulus and/or mass density) have shown great potential in diverse applications ranging from sonic cloaking, abnormal refraction and superlensing, to noise canceling. In conventional acoustic metamaterials, the negative constitutive parameters are engineered via tailored structures with fixed geometries; therefore, the relationships between constitutive parameters and acoustic frequencies are typically fixed to form a 2D phase space once the structures are fabricated. Here, by means of a model system of magnetoactive lattice structures, stimuli-responsive acoustic metamaterials are demonstrated to be able to extend the 2D phase space to 3D through rapidly and repeatedly switching signs of constitutive parameters with remote magnetic fields. It is shown for the first time that effective modulus can be reversibly switched between positive and negative within controlled frequency regimes through lattice buckling modulated by theoretically predicted magnetic fields. The magnetically triggered negative-modulus and cavity-induced negative density are integrated to achieve flexible switching between single-negative and double-negative. This strategy opens promising avenues for remote, rapid, and reversible modulation of acoustic transportation, refraction, imaging, and focusing in subwavelength regimes.

Efficient Parallel Algorithm for Calculating Electric Currents in a Toroidal Plasma

In the six-dimensional (6D) phase space, a new statement of the problem of calculating the electric current due to the pressure gradient in a toroidal plasma is considered. A semi-Lagrangian approach is applied. A new efficient parallel algorithm is developed for the numerical solution of a 6D kinetic equation with the Coulomb collision operator. By the DiFF-PK code developed by the authors, a 5D problem is solved under the conditions of ITER-scale facilities. The bootstrap current of electrons is calculated. A good agreement with the previously known limit cases is demonstrated. The method proposed can be applied to the high-precision computation of a large class of problems whose solution within existing approaches is complicated. For these purposes, the mini-supercomputer at the Research Institute for Systems Analysis, Russian Academy of Sciences, or pentaflop class supercomputers can be used. The problems include, for example, the calculation of the bootstrap current of alpha particles–a product of thermonuclear fusion, the calculation of the electric current produced by neutrals injected into a plasma, and simulation of radial electric fields and instabilities of plasma.

Modeling of nonlinear effects due to head-on beam-beam interactions

The beam-beam interaction is one of the most severe limitations on the performance of circular colliders, as it is an unavoidable strong nonlinear effect. As one aspires for greater luminosity in future colliders, one will simultaneously achieve stronger beam-beam interactions. We study the limitations caused by strong incoherent head-on beam-beam interactions, using a new code (cabin) that calculates on a graphics processing unit (GPU), allowing for a detailed description of the long-term particle trajectories in 6D phase space. The evolution of the beam emittance and beam intensity has been monitored to study the impact quantitatively, while frequency map analysis has been performed to understand the impact qualitatively. Results from cabin have shown good quantitative agreement with dedicated experiments in the Large Hadron Collider (LHC). For large beam-beam tune shifts, alternatives to the LHC tunes have been found to improve the beam quality. Schemes devised to cancel beam-beam driven resonances, by use of specific intermediate phase advances between the interaction points, work very well with zero crossing angle, and the accuracy required is achievable. Due to lack of symmetry, these schemes have an almost negligible impact with a significant crossing angle. The hourglass effect has been found to reduce the detrimental effects caused by the chromaticity and vice versa. The optimal level of the hourglass effect has been achieved when ${\ensuremath{\beta}}^{*}=1.5{\ensuremath{\sigma}}_{s}$. The ultimate parameters of the Future Circular Hadron Collider (FCC-hh) seem within reach, in absence of residual odd resonances.

Poincaré's section analysis for PPG-based automatic emotion recognition

Given the importance and abundant use of emotional recognition in the human–computer interfaces and multimedia applications, several methods for designing such systems have been reported in the literature. However, the performance of the photoplethysmography (PPG) signal in these devices has not been sufficiently studied. In addition, details of the phase space geometry of this signal have not been investigated in the emotional states up to now. In connection with this issue, Poincaré's sections can quantify the geometric patterns of the trajectory in the high-dimensional phase space. Therefore, this work attempted to propose an automatic emotion recognizer which can characterize the PPG signals during the exposure of emotional music-video. Our focus was to detect and classify dynamical behaviors of the PPG trajectories in three emotional classes of love, hate, and fun using Poincaré's section measures. To this effect, the 2D phase space of PPG was firstly reconstructed. Then, forming the Poincaré's sections in different angles, some geometric indices were extracted. Finally, using support vector machine, the PPG measures were classified into emotional states. Our results showed that basin geometry of the PPG phase states was significantly different in different emotional states. The maximum accuracy rates of 96.67% and 91.11% were achieved for a binary and multi-class classification scheme, respectively. The proposed framework was fast and operated using the single-sensor signal. In conclusion, the dynamical Poincaré's section indices of PPG signals during three emotional states paved the way for designing an online emotion recognition system.

Discrimination of multi-class EEG signal in phase space of variability for epileptic seizure detection using error correcting output code (ECOC)

A new approach for detection of epileptic seizure based on multiclass decision tree classier combining with error correcting output codes (ECOC) using ensemble learning is presented in this paper. The dynamical properties in electroencephalogram (EEG) recorded data of electrical character of the brain during seizure is different from a normal behavior of the brain. Based on the electrical characteristics of the brain, the classification of epileptic seizure in EEG signal is performed in two steps: extraction of feature by first-order difference (FOD) whose variability has been used in 95% confidence ellipse area computation in 2D phase space reconstruction (PSR) and Interquartile range (IQR) of Euclidean distance of FOD plot in 3D PSR; and for classification, bagging algorithm of ensemble learning with decision tree as a base classifier and ECOC is used. This shows a great discriminating ability among datasets belonging to five groups (denoted A–E) at different state of the brain. The proposed method shows 100% classification accuracy.

High-order discretization of a gyrokinetic Vlasov model in edge plasma geometry

We describe a new spatial discretization of a continuum gyrokinetic Vlasov model in axisymmetric tokamak edge plasma geometries. The geometries are represented using a multiblock decomposition in which logically distinct blocks are smoothly mapped from rectangular computational domains and are aligned with magnetic flux surfaces to accommodate strong anisotropy induced by the magnetic field. We employ a fourth-order, finite-volume discretization in mapped coordinates to mitigate the computational expense associated with discretization on 4D phase space grids. Applied to a conservative formulation of the gyrokinetic system, a finite-volume approach expresses local conservation discretely in a natural manner involving the calculation of normal fluxes at cell faces. In the approach presented here, the normal fluxes are computed in terms of face-averaged velocity normals in such a way that (i) the divergence-free property of the gyrokinetic velocity is preserved discretely to machine precision, (ii) the configuration space normal velocities are independent of mapping metrics, and (iii) the configuration space normal velocities are computed from exact pointwise evaluation of magnetic field data except for one term. The algorithms described in this paper form the foundation of a continuum gyrokinetic edge code named COGENT, which is used here to perform a convergence study verifying the accuracy of the spatial discretization.