Dimensional Phase Space Research Articles

A localization technique for ensemble Kalman filters

AbstractEnsemble Kalman filter techniques are widely used to assimilate observations into dynamical models. The phase‐space dimension is typically much larger than the number of ensemble members, which leads to inaccurate results in the computed covariance matrices. These inaccuracies can lead, among other things, to spurious long‐range correlations, which can be eliminated by Schur‐product‐based localization techniques. In this article, we propose a new technique for implementing such localization techniques within the class of ensemble transform/square‐root Kalman filters. Our approach relies on a continuous embedding of the Kalman filter update for the ensemble members, i.e. we state an ordinary differential equation (ODE) with solutions that, over a unit time interval, are equivalent to the Kalman filter update. The ODE formulation forms a gradient system with the observations as a cost functional. Besides localization, the new ODE ensemble formulation should also find useful application in the context of nonlinear observation operators and observations that arrive continuously in time. Copyright © 2010 Royal Meteorological Society

Weak values and the Aharonov–Vaidman gauge

The Aharonov–Vaidman gauge—which additively transforms the mean value of a quantum mechanical observable into an associated weak value—is introduced. It is shown that the unusual eccentric properties of weak values are inherited from this gauge and that a weak value of an observable can be considered as its mean value measured in the Aharonov–Vaidman gauge. The total time derivative of this sum also transforms an observable's mean value equation of motion into the associated weak value equation of motion and it is shown that the weak energy of evolution which influences the evolution of a weak value is intrinsic to the rate of change of the Aharonov–Vaidman gauge. Both of these equations of motion can be expressed in terms of time varying generalized coordinates and their rates of change. These equations satisfy the Euler–Lagrange equations which—in turn—define conjugate momenta and provide for their coordinate/momentum/time Poincaré representations. The underlying mathematical forms of these two representations are identical except for three symbols which distinguish them physically and identify three simple replacement operations that are required to transform the mean value Poincaré representation into the weak value Poincaré representation. This transformational relationship between Poincaré representations defines the notion of quasi-form invariance and the replacement operations encode the peculiar physical properties induced by the Aharonov–Vaidman gauge, i.e. the complexification and increased dimension of phase space, and the absorption of the weak energy of evolution by the conjugate momenta. Simple examples are used to illustrate the theory.

Use of phase space representation to investigate points of geodynamical interest

The kinematical behavior of points on an area of geodynamical interest is analyzed in a low — dimensional Riemann phase space in contrast to classic approaches that operate in time or frequency domains or in physical space. The phase space is reconstructed from series derived from regularly repeated GPS measurements that were transformed into a unified terrestrial frame. For the reconstruction the time-delay method was used, a concept in nonlinear time series analysis as developed by Packard (Packard et al. 1980) and proved by Takens (Takens 1981). The underlying dynamical model is a Hamiltonian motion equation so the reconstructed space is extended according to holonomic conjugated Hamiltonian coordinates. The GPS measurements are selected from a small area of geodynamical interest after its investigation based on analysis of raw geoidal signals (Doufexopoulou et al. 2006). Points from a neutral zone are used also for comparison purposes.

Identification of gravitational wave signals from chaotic astrophysical systems through phase space and attractor properties

Phase space and attractor dimensions in a gravitational wave detector output can be estimated in order to identify chaotic (deterministic) signals in the presence of additive Gaussian noise. These quantities are evaluated, respectively, by means of conditional probabilities and the Grassberger-Procaccia algorithm, both methods relying on embedding in a suitable space of dimension $d$. By testing with different embedding dimensions, a deterministic---though erratic---signal can be detected by comparing the corresponding conditional probabilities via Kolmogorov-Smirnoff test and checking whether the correlation (fractal) dimension differs from $d$. Results of the two approaches are eventually compared, both for chaotic and periodic trajectories.

Extremum complexity in the monodimensional ideal gas: The piecewise uniform density distribution approximation

The extremum complexity distribution is shown to be equivalent to a piecewise uniform distribution in the accessible N -dimensional phase space of a dynamical system. This leads to piecewise exponential functions as one-particle distribution functions. It seems plausible to use these distributions in some systems out of equilibrium, thus greatly simplifying their description. In particular, an isolated ideal monodimensional gas far from equilibrium follows two non-overlapping Gaussian distribution functions. This is demonstrated by numerical simulations. Also, some previous laboratory experiments with granular systems display this kind of distribution.

Computing the Maslov index of solitary waves, Part 1: Hamiltonian systems on a four-dimensional phase space

When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We develop a new robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithm reported here is developed in the exterior algebra representation, which leads to a fast algorithm with some novel properties. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg–de Vries equation, and the long-wave–short-wave resonance equations are presented. Part 1 considers the case of a four-dimensional phase space, and Part 2 considers the case of a 2 n -dimensional phase space with n > 2 .

Canonical structure of topologically massive gravity with a cosmological constant

We study the canonical structure of three-dimensional topologically massive gravity with a cosmological constant, using the full power of Dirac's method for constrained Hamiltonian systems. It is found that the dimension of the physical phase space is two per spacetime point, which corresponds to a single Lagrangian degree of freedom. The analysis of the AdS asymptotic region reveals a remarkable relation to 3D gravity with torsion: in the limit of vanishing torsion, the conserved charges and asymptotic symmetries of the two theories become identical.

Quasiprobability distribution functions for finite-dimensional discrete phase spaces: Spin-tunneling effects in a toy model

We show how quasiprobability distribution functions defined over ${N}^{2}$-dimensional discrete phase spaces can be used to treat physical systems described by a finite space of states which exhibit spin-tunneling effects. This particular approach is then applied to the Lipkin-Meshkov-Glick model in order to obtain the time evolution of the discrete Husimi function, and as a by-product the energy gap for a symmetric combination of ground and first excited states. Moreover, we also show how an angle-based potential approach can be efficiently employed to explain qualitatively certain features of the energy gap in terms of a spin tunneling. Entropy functionals are also discussed in this context. Such results reinforce not only the formalism per se but also the possibility of some future potential applications in other branches of physics.

Elucidation of Cardiovascular System Pathology Using Pulse Wave Phase Space Analysis

Human body pathology and normal states are docu� mented by the World Health Organization. Strict mathe� matical equations and language substitute descriptive and qualitative terms. The goal of this work was to provide express diagno� sis based on twodimensional analysis of cardiovascular system pathology using pulse wave phase space analysis in sphygmography. N. A. Amosov studied contractile myocardium function in phase space with coordinates of amplitude and amplitude change (1). The diagnostic value of this study is due to the fact that damage to the cardiovascular system modifies the signal and its deriva� tive with respect to time. The goal of this work was also to formulate integral numerical coefficient providing identification of cardio� vascular system pathology. One phase space dimension is signal itself (pulse wave pressure P), whereas another phase space dimension is the derivative of this parameter with respect to time dP/dt. The shape of loop of pressure derivative with respect to time dP/dt and its dependence on pressure P revealed specific features of blood flow in the aorta, e.g., specific features of blood flow induced by artery wall myorelaxants (2, 4, 5). It was suggested in (3) that the number of phase variables required to describe homeostasis was several thousands. This requires large computation time and modern computer hardware. Analysis of twodimensional information array requires modern computer hardware and software. The patients in this work were tested sphygmograph� ically for cardiovascular system pathology. A MEDISON 8000EX scanner was used for primary atherosclerosis diagnosis. Carotid artery wall state was assessed in 2�D ultrasonic mode at the site of ultrasonic probe applica� tion. The atherosclerosis diagnosis criteria were: intima� media thickness >1 mm and/or atherosclerosis plaques. The patients with atherosclerosis diagnosis were tested sphygmographically. The results of the tests were processed using pulse wave phase space analysis. An FML�12T�9.2A1�50 piezoelectric element (resonance frequency, 1500 Hz) was used as the sphygmographic sen� sor. The sensor signal was modulated with respect to amplitude and applied to a linear input of the sound card of the computer. The signal frequency was 500 Hz and sampling frequency was 10,000 Hz. The preamplifier input resistance was 100 MΩ. The preamplifier output signal amplitude was proportional to piezoelectric ele� ment current integral. The piezoelectric sensor was installed at the patient's carotid artery. The sensor posi� tioning accuracy was 1 cm. The shape of the sphygmo�

Dynamics of Symplectic Subvolumes

In this paper we analyze the evolution of $2k$-dimensional submanifolds in the $2n$-dimensional Hamiltonian phase space, with $k<n$. We show that parasymplectic subvolumes (to be defined in the paper), which can be thought of as even-dimensional subvolumes that are initially parallel to k of the symplectic planes, have a minimal obtainable $2k$-volume. We define volume expansion factors along the flow, and also the projection factors one obtains by projecting the volumes onto various planes. This structure can be computed by looking at various subdeterminants or Gram determinants of the state transition matrix. We also relate these various subdeterminants to classical integral invariants, giving a practical analytical method for their computation. We then indicate how this can be used to quantify probability expansions in asteroid or space debris tracking problems, with large uncertainty distributions along one of the symplectic planes. We also show how local subvolume expansion leads to the local collaps...

Phase space of solids under deformation

This paper analyzes sequences of various strain-induced defects. The defect is considered as an internal stress source, and hence the deformation is described by a phase curve in the 2D stress-strain space. Notice that the defect as a local discontinuity has eigenenergy and this, if taken into account, increases the phase space dimension because of the parameters to be introduced to characterize defect structures formed under deformation. Analysis of the potential energy of a deformed system makes it possible to elucidate the origin of various defects and to introduce the notion of strain levels in the phase space. Thus, we can distinguish strain levels for which the nature and the number of parameters of the system remain unchanged under deformation and associate the transition of the deformed system from one strain level to another with qualitative and quantitative changes. In this context, the deformation from elasticity to fracture is described by the phase space curve and the phase space dimension is determined, in addition to stresses and strains, by the parameters of defect structures formed under deformation. A standard stress-strain curve is the projection of the phase space curve on the stress-strain plane.

Equilibrium and dynamic methods when comparing an English text and its Esperanto translation

A comparison of two English texts written by Lewis Carroll, one (Alice in Wonderland), also translated into Esperanto, the other (Through the Looking Glass) are discussed in order to observe whether natural and artificial languages significantly differ from each other. One dimensional time series like signals are constructed using only word frequencies (FTS) or word lengths (LTS). The data is studied through (i) a Zipf method for sorting out correlations in the FTS and (ii) a Grassberger–Procaccia (GP) technique based method for finding correlations in LTS. The methods correspond to an equilibrium and a dynamic approach respectively to human texts features. There are quantitative statistical differences between the original English text and its Esperanto translation, but the qualitative differences are very minutes. However different power laws are observed with characteristic exponents for the ranking properties, and the phase space attractor dimensionality. The Zipf exponent can take values much less than unity (∼0.50 or 0.30) depending on how a sentence is defined. This variety in exponents can be conjectured to be an intrinsic measure of the book style or purpose, rather than the language or author vocabulary richness, since a similar exponent is obtained whatever the text. Moreover the attractor dimension r is a simple function of the so called phase space dimension n , i.e., r = n λ , with λ = 0.79 . Such an exponent could also be conjectured to be a measure of the author style versatility, — here well preserved in the translation.

Completing Irving and Kirkwood’s Molecular Theory of Transport Processes: Nonequilibrium Entropy Conservation

Following Irving and Kirkwood’s (J. Chem. Phys. 1950, 18, 817) classical approach to the statistical mechanics of transport processes and the conservation equations for mass, momentum, and energy, we introduce a particular dynamical variable for entropy and derive the general nonequilibrium entropy conservation equation. This particular formalism is shown to encompass both Boltzmann’s and Gibbs’ entropy definitions as special cases. Entropy generation is shown to follow from phase-space dimensionality loss and truncations or approximations in higher-order space, the latter of which is consistent with the thesis of Jaynes (Am. J. Phys. 1965, 33, 391). The general approach to entropy conservation given here not only completes Irving and Kirkwood’s treatment of the transport equations but also allows for a consistent analysis of all transport equations for any given system. Following standard perturbation expansion methods about local equilibrium states, we derive the closed form of the entropy conservation equation for isolated systems, which is shown to be in agreement with well-known phenomenological results and the principles of irreversible thermodynamics. In addition, the generalized nonequilibrium entropy developed here is fully consistent with its equilibrium counterpart. As an example, our formalism allows the analysis of entropy changes in dense gases and liquids through the introduction of a nonequilibrium Green’s entropy. This study provides a firm molecular basis of entropy conservation by consistent methods across the transport equations, allowing ready extensions to complex systems. Such foundations are of contemporary importance in designing energy-efficient or minimum entropy generating engineering systems.

Energy dependence of non-linear dynamical features in e+e− collisions**Supported by NSFC (10375025) and Science Foundation of Education Department of Hubei Province (EJK0316)

A study of the dynamical fluctuation properties at various c.m. energies in e+e− collisions is performed using the Monte Carlo method. The results suggest that, after the normalized factorial moments of 3-dimensional phase space are analyzed using an isotropical phase space partition, the NFM describing nonlinear dynamical properties show a power-law scaling, i.e., the dynamical fluctuations in higher dimensional phase space are isotropic. For c.m. energies √s ⩽ 80 GeV, the scaling exponents ϕq increase rapidly with the c.m. energy and for c.m. energies √s > 80 GeV, the ϕq gradually saturate.

Phase space and black-hole entropy of higher genus horizons in loop quantum gravity

In the context of loop quantum gravity, we construct the phase space of isolated horizons with genus greater than 0. Within the loop quantum gravity framework, these horizons are described by genus g surfaces with N punctures and the dimension of the corresponding phase space is calculated including the genus cycles as degrees of freedom. From this, the black-hole entropy can be calculated by counting the microstates which correspond to a black hole of fixed area. We find that the leading term agrees with the A/4 law and that the sub-leading contribution is modified by the genus cycles.

Determination of the phase projection of a dynamic system from strain field observations

The paper analyzes the possibility of extracting a geophysical field component from a time series considered as a phase projection of a dynamic system. Based on model calculations, the method of averaging of phase trajectories is shown to be applicable for the recovery of time variations of a dynamic system and more reliable determination of its quantitative characteristics if the variations are superimposed by a high-dimension process (a purely random process) and a low-dimension process (linear interpolation). The analysis of the extraction of dynamic system variations from strain records obtained at various points of seismically active regions of China provided the following estimates of dynamic parameters: the phase space dimension is m = 5–7, and the correlation dimension varies, depending on the observation point, within the range d 2 = 2.6–3.5.

Model for high‐frequency Strombolian tremor inferred by wavefield decomposition and reconstruction of asymptotic dynamics

We study the volcanic tremor time series recorded by a broadband three‐component seismic network installed at Stromboli volcano during 1997. By using decomposition methods in both frequency and time domains, we prove that Strombolian tremor can be described as a linear combination of nonlinear signals in time domain. These “components” are similar to those obtained for explosion quakes, with the only difference being the amplitude enhancement. We characterize each of these nonlinear signals both in terms of their wavefield properties as well as dynamic systems. Moreover, we take into account the complex processes of magma flow and turbulent degassing, looking at time and amplitude modulation of tremor on a suitable scale. The distribution of tremor amplitudes is Gaussian while the intertimes between the maxima in a suitable scale are described by a Poisson clustered process. Starting from these analyses, a first approximate model for volcanic tremor field can be deduced. The recorded signals, i.e., the elastic vibrations at a point, can be described by a nonlinear equation which gives limit cycles (different observed “nonlinear modes”). This equation is governed by a time‐dependent threshold which represents the variability of bubble flux. We take into account some inelasticity in the medium perturbing the elastic potential with a Gaussian function on a suitable scale. It acts as a radiance function modulating the frequency of the limit cycle. This proposed model is able to reproduce waveform, Fourier spectrum, and phase space dimension of one of the extracted nonlinear wave packets.

Coarse Grained Variables and Deterministic Chaos in an Excitable System

Temporal coarse graining was applied to the dynamical variables of a semiconductor laser with optical feedback. The chaotic low frequency fluctuations obtained in numerical and experimental data are shown to have properties of a self-excitable deterministic system. External exciting noise is replaced by the ultrafast chaotic oscillations of the system. A low dimensional coarse-grained phase space is defined and time constants are introduced and measured for the exponential drop and recovery of the randomly excited equally shaped spikes.

METHODS OF ELLIPSOIDAL ESTIMATION FOR LINEAR CONTROL SYSTEMS

We present explicit formulas for ellipsoids bounding reachable sets for linear control dynamic systems with geometric bounds on control. We study both locally and globally optimal ellipsoidal estimates with regard to different optimality criteria. In particular, we solve some essentially nonlinear boundary problems related to the search for globally optimal ellipsoids with regard to the volume criterion. It is shown that by using the explicit formulas one can efficiently pass to limits in several asymptotic problems, including passing to the limit when the phase space dimension goes to infinity.

Quantum mechanics as an approximation of statistical mechanics for classical fields

We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the term of the second order. To escape technical difficulties related to the infinite dimension of phase space for quantum mechanics, we start with a detailed presentation of our approach for the finite-dimensional quantum mechanics. We also separate real and complex cases, because the reproduction of the complex structure of quantum mechanics is a special problem which is not related to approximation of classical averages. In our approach quantum mechanics is an approximative theory. It predicts statistical averages only with some precision. In principle, there might be found deviations of averages calculated within the quantum formalism from experimental averages (which are supposed to be equal to classical averages given by our model).