Dimensional Phase Space Research Articles

The phase-space density of fermionic dark matter haloes

We have performed a series of numerical experiments to investigate how the primordial thermal velocities of fermionic dark matter particles affect the physical and phase space density profiles of the dark matter haloes into which they collect. The initial particle velocities induce central cores in both profiles, which can be understood in the framework of phase space density theory. We find that the maximum coarse-grained phase space density of the simulated haloes (computed in 6 dimensional phase space using the EnBid code) is very close to the theoretical fine-grained upper bound, while the pseudo phase space density, Q ~ {\rho}/{\sigma}^3, overestimates the maximum phase space density by up to an order of magnitude. The density in the inner regions of the simulated haloes is well described by a 'pseudo-isothermal' profile with a core. We have developed a simple model based on this profile which, given the observed surface brightness profile of a galaxy and its central velocity dispersion, accurately predicts its central phase space density. Applying this model to the dwarf spheroidal satellites of the Milky Way yields values close to 0.5 keV for the mass of a hypothetical thermal warm dark matter particle, assuming the satellite haloes have cores produced by warm dark matter free streaming. Such a small value is in conflict with the lower limit of 1.2 keV set by observations of the Lyman-{\alpha} forest. Thus, if the Milky Way dwarf spheroidal satellites have cores, these are likely due to baryonic processes associated with the forming galaxy, perhaps of the kind proposed by Navarro, Eke and Frenk and seen in recent simulations of galaxy formation in the cold dark matter model.

Simulating rare events using a weighted ensemble-based string method.

We introduce an extension to the weighted ensemble (WE) path sampling method to restrict sampling to a one-dimensional path through a high dimensional phase space. Our method, which is based on the finite-temperature string method, permits efficient sampling of both equilibrium and non-equilibrium systems. Sampling obtained from the WE method guides the adaptive refinement of a Voronoi tessellation of order parameter space, whose generating points, upon convergence, coincide with the principle reaction pathway. We demonstrate the application of this method to several simple, two-dimensional models of driven Brownian motion and to the conformational change of the nitrogen regulatory protein C receiver domain using an elastic network model. The simplicity of the two-dimensional models allows us to directly compare the efficiency of the WE method to conventional brute force simulations and other path sampling algorithms, while the example of protein conformational change demonstrates how the method can be used to efficiently study transitions in the space of many collective variables.

Gating of maxi channels observed from pseudo-phase portraits

Phase space has been used to visualize and analyze the dynamic behavior of stochastic and chaotic systems. We applied this concept to maxi channels recorded from excised inside-out patches of in situ interstitial cells of Cajal. Pseudo-phase portraits of channel current were fairly homogeneous from patch to patch. They showed three main peaks, α, β, and γ, in increasing conductance. These represented single or near aggregated states. The α-peak was the closed state. The β-peak was small, consisting of a single conductance state, or in some cases two (a doublet). The β-peak state(s) had a long lifetime and displayed a characteristic behavior of frequent short transitions to γ but not to α. It was always preceded by a short series of α/γ-transitions. The γ-peak was the largest and consisted of a large number of conductance states with fast state transitions, sometimes to the extent of causing a diffusive-type behavior. Phase portraits allowed us to construct a provisional gating scheme for the maxi channel and suggest that further analysis of recordings in higher dimensional phase space and with related techniques may be promising.

Non-Gaussian Test Models for Prediction and State Estimation with Model Errors

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems: the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for the real time prediction and the state estimation.

THE ROCKSTAR PHASE-SPACE TEMPORAL HALO FINDER AND THE VELOCITY OFFSETS OF CLUSTER CORES

We present a new algorithm for identifying dark matter halos, substructure, and tidal features. The approach is based on adaptive hierarchical refinement of friends-of-friends groups in six phase-space dimensions and one time dimension, which allows for robust (grid-independent, shape-independent, and noise-resilient) tracking of substructure; as such, it is named Rockstar (Robust Overdensity Calculation using K-Space Topologically Adaptive Refinement). Our method is massively parallel (up to 10^5 CPUs) and runs on the largest current simulations (>10^10 particles) with high efficiency (10 CPU hours and 60 gigabytes of memory required per billion particles analyzed). A previous paper (Knebe et al 2011) has shown Rockstar to have class-leading recovery of halo properties; we expand on these comparisons with more tests and higher-resolution simulations. We show a significant improvement in substructure recovery as compared to several other halo finders and discuss the theoretical and practical limits of simulations in this regard. Finally, we present results which demonstrate conclusively that dark matter halo cores are not at rest relative to the halo bulk or satellite average velocities and have coherent velocity offsets across a wide range of halo masses and redshifts. For massive clusters, these offsets can be up to 350 km/s at z=0 and even higher at high redshifts. Our implementation is publicly available at http://code.google.com/p/rockstar .

Rarefied gas flow through a cylindrical tube due to a small pressure difference

Flow of a rarefied gas through a cylindrical tube connecting two reservoirs maintained at a small pressure difference is considered using the axisymmetric version of the linearised BGK kinetic model equation subject to Maxwell diffuse–specular boundary conditions. This is a problem of five dimensions in phase space, solved in a fully deterministic manner using a parallelised discrete velocity algorithm. Results include flow rates as well as distributions of density and velocity perturbations, from the free molecular up to the slip regime and for length-over-radius (L/R) ratios ranging from zero (orifice flow) up to 20. The dependency of the results on gas rarefaction, wall accommodation and tube length is analysed and discussed. It is found that the Knudsen minimum appears only at L/R=20. Furthermore, in the case of L/R=0 it is confirmed that the results are practically independent of the accommodation coefficient. Comparing the present linear results with corresponding non-linear ones, it is seen that linearised analysis can capture the correct behaviour of the flow field not only for infinitesimally small but also for small but finite pressure differences and that its range of applicability is wider than expected. Also, the error introduced by the assumption of fully developed flow for channels of moderate length is estimated through a comparison with the present corresponding results.

Time Series Prediction with Neural Networks for the Athens Stock Exchange Indicator

1. Introduction The common characteristic of all stock exchange indicators have the term of uncertainty, which is related with their short and long-term future state. This feature is undesirable for the investor but it is also unavoidable whenever the stock exchange indicator is selected as an investment tool. Among the best possible alterations that the researcher desires to make is to reduce this uncertainty. Stock exchange prediction or forecasting is one of the interesting issues in this process as it is also referred in other works (Helstrom T. & Holmstrom K., 1998, Tsibouris G. & Zeidenberg M., 1996, White H., 1993). Time series forecasting, or time series prediction, takes an existing series of data [x.sub.t-n],...,[x.sub.t-2], [x.sub.t-1], [x.sub.t] and forecasts the [x.sub.t+1], [x.sub.t+2],..., data values. The goal is to observe or model the existing data series to enable future unknown data values to be forecasted accurately. Examples of data series include financial data series (stocks, indices, rates, etc.), physically observed data series (sunspots, weather, etc.), and mathematical data series (Fibonacci sequence, integrals of differential equations, etc.). A neural network (Tsibouris G. & Zeidenberg M., 1996) is a computational model that is loosely based on the neuron cell structure of the biological nervous system. Given a training set of data, the neural network can learn the data with a learning algorithm; in this research the most common algorithm, back propagation, is used. Through back propagation, the neural network forms a mapping between inputs and desired outputs from the training set by altering weighted connections within the network. The origin of neural networks dates back to the 1940s. McCulloch and Pitts (1943) and Hebb (1949) researched networks of simple computing devices that could model neurological activity and learning within these networks, respectively. Later, the work of Rosenblatt (1962) focused on computational ability in perceptrons, or single-layer feed-forward networks. Proofs showing that perceptrons, trained with the Perceptron Rule on linearly separable pattern class data, could correctly separate the classes generated excitement among researchers and practitioners. This excitement waned with the discouraging analysis of perceptrons presented by Minsky and Papert (1969). The analysis pointed out that perceptrons could not learn the class of linearly inseparable functions. It also stated that the limitations could be resolved if networks contained more than one layer, however no effective training algorithm for multi-layer networks was available. Rumelhart, Hinton, and Williams (1986) revived interest in neural networks by introducing the generalized delta rule for learning by back propagation, which is today the most commonly used training algorithm for multi-layer networks. 2. Athens Stock Exchange Indicator Time Series The Athens Stock Exchange Indicator is presented as a signal x=x(t) as it shown in Figure 1. It covers data from the period 1998 to 2005. The sampling rate is [DELTA]t=1 day. [FIGURE 1 OMITTED] 3. Neural Network Constructions In order to predict the time series for the Athens Stock Exchange Indicator we construct a back propagation network (Wan A.E., 1990, Widrow B., and Lehr M., 1990, Rumelhart D., McClelland J. et al., 1986) that consists of 1 input layer 1 middle or hidden layer and 1 output layer. The input layer has 3 processing elements (PE) neurons; the middle layer has 2x3+1=7 neurons according to Kolmogorov theorem (Kolmogorov N. A., 1950, Blackwell D., 1947). We choose the input layer to have 3 inputs because of minimum embedding dimension 3, and the attractor is embedded at a three -dimensional phase space (Hanias P.M., Curtis G.P., and Thalassinos E.J., 2006). Beginning with the first value of the time series, the first set of inputs data is x1, x2, x3 and our outputs are x4, x5, x6, x7, x8, x9, x10, x11, x12. …

Tracing the dark matter sheet in phase space

The primordial velocity dispersion of dark matter is small compared to the velocities attained during structure formation. The initial density distribution is close to uniform and it occupies an initial sheet in phase space that is single valued in velocity space. Because of gravitational forces this three dimensional manifold evolves in phase space without ever tearing, conserving phase-space volume and preserving the connectivity of nearby points. N-body simulations already follow the motion of this sheet in phase space. This fact can be used to extract full fine-grained phase-space-structure information from existing cosmological N-body simulations. Particles are considered as the vertices of an unstructured three dimensional mesh, moving in six dimensional phase-space. On this mesh, mass density and momentum are uniquely defined. We show how to obtain the space density of the fluid, detect caustics, and count the number of streams as well as their individual contributions to any point in configuration-space. We calculate the bulk velocity, local velocity dispersions, and densities from the sheet - all without averaging over control volumes. This gives a wealth of new information about dark matter fluid flow which had previously been thought of as inaccessible to N-body simulations. We outline how this mapping may be used to create new accurate collisionless fluid simulation codes that may be able to overcome the sparse sampling and unphysical two-body effects that plague current N-body techniques.

Non-linear density–velocity divergence relation from phase space dynamics

We obtain the non-linear relation between cosmological density and velocity perturbations by examining their joint dynamics in a two dimensional density-velocity divergence phase space. We restrict to spatially flat cosmologies consisting of pressureless matter and non-clustering dark energy characterized by a constant equation of state $w$. Using the spherical top-hat model, we derive the coupled equations that govern the joint evolution of the perturbations and examine the flow generated by this system. In general, the initial density and velocity are independent, but requiring that the perturbations vanish at the big bang time sets a relation between the two. This relation, which we call the `Zel'dovich curve', acts like an attracting solution for the phase space dynamics and is the desired non-linear extension of the density-velocity divergence relation. We obtain a fitting formula for the curve, which is a generalization of the formulae by Bernardeau and Bilicki & Chodorowski, and find that as in the linear regime, the explicit dependence on the dark energy parameters stays weak even in the non-linear regime. Although the result itself is somewhat expected, the new feature of this work is the interpretation of the relation in the phase space picture and the generality of the method. Finally, as an observational implication, we examine the evolution of galaxy cluster profiles using the spherical infall model for different values of $w$. We demonstrate that using only the density or the velocity information to constrain $w$ is subject to degeneracies in other parameters such as $\sigma_8$ but plotting observations onto the joint density-velocity phase space can help lift this degeneracy.

Algorithm for the Detection of Changes of the Correlation Structure in Multivariate Time Series

In this research, an improved algorithm for the detection of changes of the correlation structure in multivariate time series is proposed. The starting point of the technique is a covariance matrix whose entries are the largest entries of a cross-covariance matrix which is composed of all pairs of the time series reconstructed to an M -dimensional phase space. Principal component analysis is performed on this maximized cross-covariance matrix, and the overall degree of synchronization among multiple-channel signals is defined, by synchronization index, as the Shannon entropy of the eigenvalue spectrum. Throughout the experiment, the effectiveness of the proposed algorithm is validated with simulated data – a network of time series generated by autoregressive models and a network of coupled chaotic Roessler oscillators. DOI: http://dx.doi.org/10.5755/j01.eee.18.8.2625

On the Hamiltonian form of 3D massive gravity

We present a "Chern-Simons-like" action for the "general massive gravity" model propagating two spin-2 modes with independent masses in three spacetime dimensions (3D), and we use it to find a simple Hamiltonian form of this model. The number of local degrees of freedom, determined by the dimension of the physical phase space, agrees with a linearized analysis except in some limits, in particular that yielding "new topologically massive gravity", which therefore suffers from a linearization instability.

A new approach for embedding dimension determination based on empirical mode decomposition

PurposeEmbedding dimension determination in phase space reconstruction is difficult. The purpose of this paper is to present a new approach for embedding dimension determination based on empirical mode, showing that embedding dimensions for phase space reconstruction could be easily determined according to the number of intrinsic mode functions decomposed by empirical mode decomposition.Design/methodology/approachThrough the relation analysis of intrinsic mode functions and embedding dimensions, the approach for embedding dimension determination by the number of intrinsic mode functions is presented. First, a time series is decomposed into several intrinsic mode functions. Second, correlation analysis between intrinsic mode functions and original signals is investigated, and then false intrinsic mode functions could be eliminated by the analysis of correlation coefficient thresholds, which makes the embedding dimension precise. Finally, the method presented is applied to the Lorenz system, Chen's system, and the Duffing equation. Simulation results prove this method is feasible.FindingsA new approach for embedding dimension determination based on empirical mode decomposition is presented. Compared with G‐P algorithms, this new method is effective and decreases computational complexity.Research limitations/implicationsThis method provides an effective qualitative criterion to the selection of embedding dimensions in phase space reconstruction.Practical implicationsThis method could be used to determine embedding dimensions of phase space reconstruction and degree‐of‐freedom of nonlinear dynamical systems.Originality/valueThe paper proposes a new method of embedding dimension determination in phase space reconstruction.

Development of Chemistry Coordinate Mapping Approach for Turbulent Partially Premixed Combustion

Direct numerical simulation (DNS) coupling with chemistry coordinate mapping (CCM) is presented to simulate flame propagation and auto-ignition in a partially premixed syngas/air mixture. In the CCM approach, the physical domain is mapped into a low dimensional phase space with a few thermodynamic variables as the independent variables. The integration of the chemical reaction rates and heat release rate are done on the grid in the phase space. Previously we showed that for premixed mixtures, two variables temperature and specific element mass fraction of H atom, can be sufficient to construct the phase space for a satisfactory mapping. However, for partially premixed combustion mode, a third phase space variable is required to map the physical cell into the phase space. It is shown that scalar dissipation rate of the element mass ratio of H atom can be used as the third dimension of the phase space. An investigation is carried out on the behavior of CCM and the choice over the element on which the local element mass ratio should be based. Mapping error in the CCM is investigated. It is shown that if the element mass ratio is based on the element involved in the most diffusive molecules, the error of the mapping can approach zero when the grid in the phase space is refined. To validate the CCM approach the results of DNS coupled with CCM (DNS-CCM) are compared with full DNS that integrates the chemical reaction rates and heat release rate directly in physical space. Good agreement between the results from DNS and DNS-CCM is obtained while the computational time is reduced at least by 70 %.

STOCHASTIC BRIGHTNESS VARIATIONS IN THE CENTRAL STAR OF PLANETARY NEBULA NGC 6826

One year of continuous Kepler Space Telescope data for the central object of planetary nebula NGC?6826 (KIC 12071221) were analyzed. KIC 12071221 has long-cadence light curves with stochastic brightness variations and broadband power spectra. The exception is a one-month long period during which variability was detected with a period of 14.8?hr. In search of the origin of this periodicity and in an attempt to better understand the complex NGC?6826 system, long-cadence KIC 12071221 data were analyzed using nonlinear time series methods. Nonlinear projective noise reduction from a phase-space dimension of six lowered the tail of the power spectra at the higher frequencies by factors of 38-133, an unexpected result for light curves with no obvious dominant frequencies. Even more notable is the identification of a structured distribution of trajectories, or coherence, in the reconstructed phase space for vectors spanning an interval of 1.5?hr. Organization weakens as the delay increases and is lost for phase-space vectors spanning times longer than 10?hr. This may be indicative of a binary object which shares a common envelope and/or pulsations but for a complete understanding one or both may have to be combined with wind structure. Thus, an estimate is made of the timescale of the response to what we propose is stochastic driving. This estimate of the timescale of the organized response is made by monitoring the loss of structure in the distribution of trajectories in phase space, a novel application of nonlinear methodology that may be used on data sampled with any sampling time.

Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis

The standard approach to study symmetric Hopf bifurcation phenomenon is based on the usage of the equivariant singularity theory developed by M. Golubitsky et al. In this paper, we present the equivariant degree theory based method which is complementary to the equivariant singularity approach. Our method allows systematic study of symmetric Hopf bifurcation problems in non-smooth/non-generic equivariant settings. The exposition is focused on a network of eight identical van der Pol oscillators with hysteresis memory, which are coupled in a cube-like configuration leading to S4-equivariance. The hysteresis memory is the source of non-smoothness and of the presence of an infinite dimensional phase space without local linear structure. Symmetric properties and multiplicity of bifurcating branches of periodic solutions are discussed in the context showing a direct link between the physical properties and the equivariant topology underlying this problem.

An Indian monsoon intraseasonal oscillations (MISO) index for real time monitoring and forecast verification

The wet/dry spells of the Indian summer monsoon (ISM) rainfall are governed by northward propagating boreal summer monsoon intraseasonal oscillations (MISO). Unlike for the Madden Julian Oscillation (e.g. RMM indices, Wheeler and Hendon in Mon Weather Rev 132:1917–1932, 2004), a low dimensional real-time monitoring and forecast verification metric for the MISO is not currently available. Here, for the first time, we present a real time monitoring index developed for identifying the amplitude and phase of the MISO over the ISM domain. The index is constructed by applying extended empirical orthogonal function (EEOF) analysis on daily unfiltered rainfall anomalies averaged over the longitudinal domain 60.5°E–95.5°E. The gravest two modes of the EEOFs together explain about 23 % of the total variance, similar to the variance explained by MISO in observation. The pair of first two principal components (PCs) of the EEOFs is named as MISO1 and MISO2 indices which together represent the evolution of the MISOs in a low dimensional phase space. Power spectral analysis reveals that the MISO indices neatly isolate the MISO signal from the higher frequency noise. It is found that the current amplitude and phase of the MISO can be estimated by preserving a memory of at least 15 days. Composite pictures of the spatio-temporal evolution of the MISOs over the ISM domain are brought out using the MISO indices. It is further demonstrated that the MISO indices can be used in the quantification of skill of extended range forecasts of MISOs. Since the MISO index does not rely on any sort of time filtering, it has great potential for real time monitoring of the MISO and may be useful in developing some prediction scheme.

Estimating the largest Lyapunov exponent and noise level from chaotic time series

A novel method for estimating simultaneously the largest Lyapunov exponent (LLE) and noise level (NL) from a noisy chaotic time series is presented in this paper. We research the influence of noise on the average distance of different pairs of points in an embedding phase space and provide a rescaled formula for calculating the LLE when the time series is contaminated with noise. Our algorithm is proposed based on this formula and the invariant of the LLE in different dimensional embedding phase spaces. With numerical simulation, we find that the proposed method provides a reasonable estimate of the LLE and NL when the NL is less than 10% of the signal content. The comparison with Kantz algorithm shows that our method gives more accurate results of the LLE for the noisy time series. Furthermore, our method is not sensitive to the distribution of the noise.

Noncommutative Dirac oscillator in an external magnetic field

We show that (2+1)-dimensional noncommutative Dirac oscillator in an external magnetic field is mapped onto the same but with reduced angular frequency in absence of magnetic field. We construct the relativistic Landau levels by solving corresponding Dirac equation in (2+1)-dimensional noncommutative phase space. All the Landau levels become independent of noncommutative parameter for a critical value of the magnetic field. Several other interesting features along with the relevance of such models in the study of atomic transitions in a radiation field have been discussed.

Photonic Nonlinear Transient Computing with Multiple-Delay Wavelength Dynamics

We report on the experimental demonstration of a hybrid optoelectronic neuromorphic computer based on a complex nonlinear wavelength dynamics including multiple delayed feedbacks with randomly defined weights. This neuromorphic approach is based on a new paradigm of a brain-inspired computational unit, intrinsically differing from Turing machines. This recent paradigm consists in expanding the input information to be processed into a higher dimensional phase space, through the nonlinear transient response of a complex dynamics excited by the input information. The computed output is then extracted via a linear separation of the transient trajectory in the complex phase space. The hyperplane separation is derived from a learning phase consisting of the resolution of a regression problem. The processing capability originates from the nonlinear transient, resulting in nonlinear transient computing. The computational performance is successfully evaluated on a standard benchmark test, namely, a spoken digit recognition task.

A unified gas kinetic scheme with moving mesh and velocity space adaptation

There is great difficulty for direct Boltzmann solvers to simulate high Knudsen number flow due to the severe steep slope and high concentration of the gas distribution function in a local particle velocity space. Local mesh adaptation becomes necessary in order to make the Boltzmann solver to be a practical tool in aerospace applications. The present research improves the unified gas-kinetic scheme (UGKS) in the following two aspects. First, the UGKS is extended in a physical space with moving mesh. This technique is important to study a freely flying object in a rarefied environment. Second, the adaptive quadtree method in the particle velocity space is implemented in the UGKS. Due to the new improvements in the discretization of a gas distribution function in the six dimensional phase space, the adaptive unified gas kinetic scheme (AUGKS) is able to deal with a wide range of flow problems under extreme flying conditions, such as the whole unsteady flying process of an object from a highly rarefied to a continuum flow regime. After validating the scheme, the capability of AUGKS is demonstrated in the following two challenge test cases. The first case is about the free movement of an ellipse flying at initial Mach number 5 in a rarefied flow at different Knudsen numbers. The force on the ellipse and the unsteady trajectory of the ellipse movement are fully captured. The gas distribution function around the ellipse is analyzed. The second case is about the study of unsteady flight of a nozzle under a bursting process of the compressed gas expanding into a rarefied environment. Due to the strong expansion wave and the huge density difference between interior and exterior regions around the nozzle, the particle distribution function changes dramatically in the particle velocity space. The use of an adaptive velocity space in the AUGKS becomes necessary to simulate such a flow and to control the computational cost to a tolerable level. The second test is a challenge problem for any existing rarefied flow solver.