Phase-space Domain Research Articles

Dynamics of an equation with a large coefficient of delay control

Under condition (2), we analyze the behavior of all solutions to Eq. (3) in an arbitrary bounded (as K → ∞) domain of the phase space C [–T ,0 ] , i.e., of the space of

Modification of particle distributions by MHD instabilities I

The modification of particle distributions by low amplitude magnetohydrodynamic modes is an important topic for magnetically confined plasmas. Low amplitude modes are known to be capable of producing significant modification of injected neutral beam profiles, and the same can be expected in burning plasmas for the alpha particle distributions. Flattening of a distribution due to phase mixing in an island or due to portions of phase space becoming stochastic is a process extremely rapid on the time scale of an experiment but still very long compared to the time scale of guiding center simulations. Thus it is very valuable to be able to locate significant resonances and to predict the final particle distribution produced by a given spectrum of magnetohydrodynamic modes. In this paper we introduce a new method of determining domains of phase space in which good surfaces do not exist and use this method for quickly determining the final state of the particle distribution without carrying out the full time evolution leading to it.

Dynamics of actively regulated gene networks

A popular way of modelling gene regulatory networks is to design the models as if the genes were acting directly on each other. Genes are activated or inhibited by transcription factors which are direct gene products. The action of a transcription factor on a gene is modelled as a binary on–off dose-response function or a sigmoid around a certain threshold concentration. A transcription factor may act on several genes with equal or different thresholds. The combined effect of several transcription factors on a gene is generally assumed to obey Boolean-like composition rules. A mathematical challenge related to such network models is to analyse the behaviour in the narrow domains in phase space where at least one variable is close to one of its thresholds, called switching domains. In the present paper we analyse the motion in switching domains for models with steep sigmoidal response functions under the assumption that each transcription factor only regulates one gene at each of its thresholds. This biologically reasonable assumption allows us to establish rules that determine the passage through any switching domain and the sequence of domains through which the system passes. The rules can be effectively implemented in a sound automated analyser for such networks. The assumption simplifies certain mathematical and computational issues, but unfortunately may be at variance with one of the necessary conditions for the singular perturbation theory which our derivations are based on. We give sufficient conditions ensuring that, despite this, the difficulties raised by our assumption can be evaded and singular perturbation approximation safely applied.

Characterization of Healthy and Pathological Voice Through Measures Based on Nonlinear Dynamics

In this paper, we propose to quantify the quality of the recorded voice through objective nonlinear measures. Quantification of speech signal quality has been traditionally carried out with linear techniques since the classical model of voice production is a linear approximation. Nevertheless, nonlinear behaviors in the voice production process have been shown. This paper studies the usefulness of six nonlinear chaotic measures based on nonlinear dynamics theory in the discrimination between two levels of voice quality: healthy and pathological. The studied measures are first- and second-order Renyi entropies, the correlation entropy and the correlation dimension. These measures were obtained from the speech signal in the phase-space domain. The values of the first minimum of mutual information function and Shannon entropy were also studied. Two databases were used to assess the usefulness of the measures: a multiquality database composed of four levels of voice quality (healthy voice and three levels of pathological voice); and a commercial database (MEEI Voice Disorders) composed of two levels of voice quality (healthy and pathological voices). A classifier based on standard neural networks was implemented in order to evaluate the measures proposed. Global success rates of 82.47% (multiquality database) and 99.69% (commercial database) were obtained.

Unbiased computation of transition times by pathway recombination

In many systems, the time scales of the microscopic dynamics and macroscopic dynamics of interest are separated by many orders of magnitude. Examples abound, for instance, nucleation, protein folding, and chemical reactions. For these systems, direct simulation of phase space trajectories does not efficiently determine most physical quantities of interest. The past decade has seen the advent of methods circumventing brute force simulation. For most dynamical quantities, these methods all share the drawback of systematical errors. We present a novel method for generating ensembles of phase space trajectories. By sampling small pieces of these trajectories in different phase space domains and piecing them together in a smart way using equilibrium properties, we obtain physical quantities such as transition times. This method does not have any systematical error and is very efficient; the computational effort to calculate the first passage time across a free energy barrier does not increase with the height of the barrier. The strength of the method is shown in the Ising model. Accurate measurements of nucleation times span almost ten orders of magnitude and reveal corrections to classical nucleation theory.

Anti-nuclei and nuclei production in Pb+Pb collisions at CERN SPS energies

We present new results on production of 3He and t obtained by the NA49 experiment in 20A, 30A, 40A and 80A GeV central Pb+Pb collisions at the CERN SPS. Transverse mass spectra and rapidity distributions for clusters measured over a large phase space domain are discussed. We observe a weak dependence of the mid-rapidity t/3He ratio on collision energy at SPS. The energy dependence of the total yield for 3He is remarkably reproduced by a statistical hadron gas model. A gradual decrease of the coalescence parameters B3 for 3He with s√SNN is observed. In addition, a measurement of the yield of antideuterons in Pb+Pb reactions at SPS's top energy of s√SNN = 17.2 GeV is presented.

Chaos, Order, and Periodic Orbits in 3 : 1 Resonant Planetary Dynamics

This study addresses the long-term evolution of possible two-planet extrasolar systems that are initially trapped in 3:1 mean motion resonance. A planar general three-body problem model is used, and its resonant dynamics are examined by computing periodic orbits in a rotating frame, Poincare maps, and maps of dynamical stability. We computed the families of symmetric resonant periodic orbits that obey bifurcations, giving rise to families of asymmetric periodic orbits. The linear stability of such orbits has also been computed, and their relation to the long-term stability of their nearby phase-space domain has been studied. The maps of dynamical stability reveal a complicated structure in the phase space, where chaos and order coexist and alternate as the initial eccentricities or the phases of the planets change. The regular orbits are classified into various types according to the librating or rotating evolution of the resonant angles. Apsidal symmetric librations are common in the domain of resonant motion, but asymmetric ones are associated exclusively with the existence of asymmetric periodic orbits. Such a stable asymmetric configuration seems to correspond to the companions b and c of the 55 Cnc extrasolar system, which are trapped in the 3:1 mean motion resonance according to the study of McArthur et al. However, a recent study by Fischer et al. shows the existence of a new planet (the companion f) in the system, and that the planets b and c are not in mean motion resonance.

ON CONVERGENCE TO EQUILIBRIUM IN STRONGLY COUPLED BOGOLIUBOV'S OSCILLATOR MODEL

We examine classical Bogoliubov's model of a particle coupled to a heat bath which consists of infinitely many stochastic oscillators. Bogoliubov's result1 suggests that, in the stochastic limit, the model exhibits convergence to thermodynamical equilibrium. It has recently been shown that the system does attain the equilibrium if the coupling constant is small enough.12 We show that in the case of the large coupling constant, the distribution function ρS (q, p, t) → 0 pointwise as t → ∞. This implies that if there is convergence to equilibrium, then the limit measure has no finite momenta. Besides, the probability to find the particle in any finite domain of phase space tends to zero. This is also true for domains in the coordinate space and in the momentum space.

A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems

This thesis concerns the numerical simulation of creeping rays and their contribution to high frequency scattering problems. Creeping rays are a type of diffracted rays which are generated at the shadow line of the scatterer and propagate along geodesic paths on the scatterer surface. On a perfectly conducting convex body, they attenuate along their propagation path by tangentially shedding diffracted rays and losing energy. On a concave scatterer, they propagate on the surface and importantly, in the absence of dissipation, experience no attenuation. The study of creeping rays is important in many high frequency problems, such as design of sophisticated and conformal antennas, antenna coupling problems, radar cross section (RCS) computations and control of scattering properties of metallic structures coated with dielectric materials. First, assuming the scatterer surface can be represented by a single parameterization, we propose a new Eulerian formulation for the ray propagation problem by deriving a set of escape partial differential equations in a three-dimensional phase space. The equations are solved on a fixed computational grid using a version of fast marching algorithm. The solution to the equations contain information about all possible creeping rays. This information includes the phase and amplitude of the ray field, which are extracted by a fast post-processing. The advantage of this formulation over the standard Eulerian formulation is that we can compute multivalued solutions corresponding to crossing rays. Moreover, we are able to control the accuracy everywhere on the scatterer surface and suppress the problems with the traditional Lagrangian formulation. To compute all possible creeping rays corresponding to all shadow lines, the algorithm is of computational order O(N3 log N), with N3 being the total number of grid points in the computational phase space domain. This is expensive for computing the wave field for only one shadow line, but if the solutions are sought for many shadow lines (for many illumination angles), the phase space method is more efficient than the standard methods such as ray tracing and methods based on the eikonal equation. Next, we present a modification of the single-patch phase space method to a multiple-patch scheme in order to handle realistic problems containing scatterers with complicated geometries. In such problems, the surface is split into multiple patches where each patch has a well-defined parameterization. The escape equations are solved in each patch, individually. The creeping rays on the scatterer are then computed by connecting all individual solutions through a fast post-processing. We consider an application to mono-static radar cross section problems where creeping rays from all illumination angles must be computed. The numerical results of the fast phase space method are presented.

Quantum baker maps with controlled-not coupling

The characteristic stretching and squeezing of chaotic motion is linearized within the finite number of phase space domains which subdivide a classical baker map. Tensor products of such maps are also chaotic, but a more interesting generalized baker map arises if the stacking orders for the factor maps are allowed to interact. These maps are readily quantized, in such a way that the stacking interaction is entirely attributed to primary qubits in each map, if each subsystem has power-of-two Hilbert space dimension. We here study the particular example of two baker maps that interact via a controlled-not interaction. Numerical evidence indicates that the control subspace becomes an ideal Markovian environment for the target map in the limit of large Hilbert space dimension.

Models for the determination of kinetic phase diagrams and kinetic phase separation domains

Several theoretical models for the determination of kinetic phase diagrams for solid solution growth from the liquid phase are presented and compared to each other. These models include a Monte Carlo simulation model, used as a reference model, a previously defined analytical model, based on linear non-equilibrium thermodynamics, and a new model, rooted in the kinetics at kink sites. All models have in common that the composition of the growing solid phase tends to the liquid phase composition for increasing undercooling, enhancing mixing even for systems with a strong tendency to phase separation. However, depending on the system parameters considerable quantitative differences can occur between the results from the model based on non-equilibrium thermodynamics and the MC model. Instead, the new model follows very well the trends of the MC simulations, both for well-mixing systems and for phase separating systems. For phase separating systems the analytical models predict kinetic phase separation domains, zones in the kinetic phase diagram yielding steady state growth of more than one solid phase with different compositions. According to MC simulations such domains in phase space correspond to domain formation in real space. Also in this case the new model is consistent with the MC results.

Direct Lyapunov exponent analysis enables parametric study of transient signalling governing cell behaviour

Computational models aid in the quantitative understanding of cell signalling networks. One important goal is to ascertain how multiple network components work together to govern cellular responses, that is, to determine cell 'signal-response' relationships. Several methods exist to study steady-state signals in the context of differential equation-based models. However, many biological networks influence cell behaviour through time-varying signals operating during a transient activated state that ultimately returns to a basal steady-state. A computational approach adapted from dynamical systems analysis to discern how diverse transient signals relate to alternative cell fates is described. Direct finite-time Lyapunov exponents (DLEs) are employed to identify phase-space domains of high sensitivity to initial conditions. These domains delineate regions exhibiting qualitatively different transient activities that would be indistinguishable using steady-state analysis but which correspond to different outcomes. These methods are applied to a physicochemical model of molecular interactions among caspase-3, caspase-8 and X-linked inhibitor of apoptosis--proteins whose transient activation determines cell death against survival fates. DLE analysis enabled identification of a separatrix that quantitatively characterises network behaviour by defining initial conditions leading to apoptotic cell death. It is anticipated that DLE analysis will facilitate theoretical investigation of phenotypic outcomes in larger models of signalling networks.

Enlarging the depth of focus by filtering in the phase-space domain

The ambiguity function was employed as a merit function to design an optical system with a high depth of focus. The ambiguity function with the desired enlarged-depth-of-focus characteristics was obtained by using a properly designed joint filter to modify the ambiguity function of the original pupil in the phase-space domain. From the viewpoint of the filter theory, we roughly propose that the constraints of the spatial filters that are used to enlarge the focal depth must be satisfied. These constraints coincide with those that appeared in the previous literature on this topic. Following our design procedure, several sets of apodizers were synthesized, and their performances in the defocused imagery were compared with each other and with other previous designs.

Planar periodic orbits in exterior resonances with Neptune

In the framework of the planar restricted three-body problem we study a considerable number of resonances associated to the basic dynamical features of Kuiper belt and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrounded by regular librations in phase space and in such domains the capture of trans-Neptunian object is possible. All the periodic orbits found are symmetric and there is an indication of the existence of asymmetric ones only in a few cases. In the present work first, second and third order resonances are under consideration. In the planar circular case we found that most of the periodic orbits are stable. The families of periodic orbits are temporarily interrupted by collisions but they continue up to relatively large values of the Jacobi constant and highly eccentric regular motion exists for all cases. In the elliptic problem and for a particular eccentricity value of the primary bodies, the periodic orbits are isolated. The corresponding families, where they belong to, bifurcate from specific periodic orbits of the circular problem and seem to continue up to the rectilinear problem. Both stable and unstable orbits are obtained for each case. In the elliptic problem, the unstable orbits found are associated with narrow chaotic domains in phase space. The evolution of the orbits, which are located in such chaotic domains, seems to be practically regular and bounded for long time intervals.

Phase-space analysis of acoustics fields and its application to waveguide

A description of two-dimensional acoustic fields by means of a joint "space-wave number" representation is discussed. A function defined in the phase-space domain (x,y,k(x),k(y)) is associated with a signal which is a function of spatial coordinates (x,y). This paper presents two methods to realize it. The first is to associate with each point (x,y) of the wave field a two-dimensional wave number spectrum (k(x),k(y)), called local spectrum. The second is to process by other coordinates the wave field along an arbitrary direction, introduced in quantum mechanics for the study of classical billiards, and provided by the Birkhoff variables (s,cos phi). Phase-space diagrams are given by quadratic phase-space distributions. Simulations are presented for wave fields in a 2D planar waveguide for a pedagogical point of view with Gaussian beam or point-source excitation, and nonuniform waveguides as a sudden area expansion chamber and an open billiard with a single incoming mode at the entrance of each of them. In these problems, local spectrum and Birkhoff analysis are used in order to identify the structures hidden in the field. The result is the contribution of different wave vectors which contribute to the field value at the analysis point or at a certain section of the boundary, and show complicated structure of the acoustic field like whispering gallery or diffracted waves.

Spin‐dependent Cyclotron Decay Rates in Strong Magnetic Fields

Cyclotron decay and absorption rates have been well studied in the literature, focusing primarily on spectral, angular, and polarization dependence. Astrophysical applications usually do not require retention of information on the electron spin state, and these are normally averaged in obtaining the requisite rates. In magnetic fields, higher order quantum processes such as Compton scattering become resonant at the cyclotron frequency and its harmonics, with the resonances being formally divergent. Such divergences are usually eliminated by accounting for the finite lifetimes of excited Landau states. This practice requires the use of spin-dependent cyclotron rates in order to obtain accurate determinations of process rates very near cyclotronic resonances, the phase-space domain most relevant for certain applications to pulsar models. This paper develops previous results in the literature to obtain compact analytic expressions for cyclotron decay rates/widths in terms of a series of Legendre functions of the second kind; these expressions can be used expediently in astrophysical models. The rates are derived using two popular eigenstate formalisms, namely, that due to Sokolov & Ternov and that due to Johnson & Lippmann. These constitute two sets of eigenfunctions of the Dirac equation that diagonalize different operators and accordingly yield different spin-dependent cyclotron rates. This paper illustrates the attractive Lorentz transformation characteristics of the Sokolov & Ternov formulation, which is another reason why it is preferable when electron spin information must be explicitly retained.

Time and Phase-Space Stability Analysis of the Jupiter-Sun System

A comparison of dynamic stability determination for a celestial body is made between the Newtonian time domain and the phase-space domain. A new method of using the line of syzygies for determination of the stability of a third body moving in the region of two primary bodies is reviewed for accuracy via more proven or investigated phase-space methods such as Poincare phase space surface of section methods. Phase-space surface of section methods appear to verify that the use of the line of syzygies as a stability criteria is valid, shedding new information of the stable motion of the third body, especially in the time domain. resulting data and not requiring a Hamiltonian constant. However, the disadvantage of time-domain analysis surfaces in the difficult attempt to investigate the system for an infinite time. These two different approaches are investigated and compared in this paper to study the long-term stability of a dynamic system. The dynamic system investigated is the circular restricted problem of three bodies as will be explained. Szebehely, 9 Tuckness, 10 and McKenzie and Szebehely 11 have had success in studying the dynamic stability of the three-body system using a Newtonian time-domain boundary envelope approach. The boundary envelope approach is a method of bounding the kinetic en- ergy of the third body as it departs (or passes through) the L4 point. This study compares this Newtonian time-domain boundary enve- lope measure, or interpretation, of stability to the classical Poincare surface of section interpretation found using phase-space investiga- tive techniques. Both methods are used to investigate the motion of a third body for stability in the same Jupiter-sun system. This study compares the phase-space-domain results with the time-domain re- sults applied to the stability of a third body in motion near one of the equilibrium points, the L 4 Lagrange point. In particular, it is shown that the time-domain boundary envelope approach used as a stability measure in Refs. 9-11 offers very similar and yet totally in- dependent results compared to the traditional phase-space methods. Furthermore, the boundary envelope approach presents the stability criteria in a time domain Cartesian coordinate system, which can be easier to interpret for real-time application and analysis. It is also shown that the time-domain studies can be used to better understand the results of the phase-space studies that can sometimes be difficult to analyze.

Charge fluctuations in π+p and K+p collisions at 250 GeV/c

We report on measurements of event-by-event charge fluctuations in $\pi^{+}\rp$ and $\rK^{+}\rp$ collisions at 250 GeV/$c$. The dependence of these fluctuations on the size of the rapidity windows are presented for the first time in the full phase space domain. The corrections for the influence of global charge conservation and leading-particle stopping are tested by the data. The discrepancy due to incomplete correction given by STAR and PHENIX are estimated. The dependence of the fluctuations on the position of the rapidity bin and on the multiplicity at different rapidity windows are also presented.

Study of anomalous couplings of gauge bosons in the reaction $$e^ - \gamma \to W^ - \nu \to \ell \bar \nu \nu$$

The possibility of observing an anomaly in triple interactions of a W boson and a photon in the process eγ → Wν is considered in the leptonic mode of W-boson decay. It is shown that, in some cases, the use of a bounded phase-space domain instead of the total phase space improves upper limits on the anomalous couplings. The estimates obtained in the above mode under conditions of the TESLA project are poorer than analogous current estimates for all modes of the e+e− channel.

Pattern formation of stationary transcellular ionic currents in Fucus.

Stationary and nonstationary spatiotemporal pattern formations emerging from the cellular electric activity are a common feature of biological cells and tissues. The nonstationary ones are well explained in the framework of the cable model. Inversely, the formation of the widespread self-organized stationary patterns of transcellular ionic currents remains elusive, despite their importance in cell polarization, apical growth, and morphogenesis. For example, the nature of the breaking symmetry in the Fucus zygote, a model organism for the experimental investigation of embryonic pattern formation, is still an open question. Using an electrodiffusive model, we report here an unexpected property of the cellular electric activity: a phase-space domain that gives rise to stationary patterns of transcellular ionic currents at finite wavelength. The cable model cannot predict this instability. In agreement with experiments, the characteristic time is an ionic diffusive one (<2 min). The critical radius is of the same order of magnitude as the cell radius (30 microm). The generic salient features are a global positive differential conductance, a negative differential conductance for one ion, and a difference between the diffusive coefficients. Although different, this mechanism is reminiscent of Turing instability.