Properties Of Phase Space Research Articles

The modified FitzHugh–Nagumo system as an oscillator

AbstractThe use of the modified FitzHugh–Nagumo system is extended to the limit cycle regime. Ranges of parameters for which such oscillatory behavior prevails are calculated and properties of phase space and individual pulses are obtained. Copyright © 2008 John Wiley & Sons, Ltd.

SU‐GG‐T‐320: Monte Carlo Generation of Phase Spaces for Dose Computation in TomoTherapy

Purpose: Generate phase spaces using Monte Carlo simulations for TomoTherapy® dose computation, measure accuracy of phase spaces, and present properties of phase spaces. Method and Materials: The PENELOPE 2006 Monte Carlo code system is used to generate phase spaces for the TomoTherapy® treatment device. The PENELOPE geometry package is used to model the head of the treatment device for each one of the three possible field sizes: 1 × 40 cm2, 2.5 × 40 cm2 and 5 × 40 cm2, and a phase‐space plane is positioned downstream from the collimating jaws of the device. Incident electrons are projected onto the linear accelerator's target assembly and Monte Carlo transport is performed through the head of the linear accelerator until particles cross the phase‐space plane. The incident electron energy distribution and the focal spot size are adjusted until the resulting dose profiles computed with a phase space match the TomoTherapy® open‐field gold standard profiles. Fluence maps, dose profiles and off‐axis spectra are presented. Results: A 5.5 MeV monoenergetic incident electron source produced optimal results. Computed dose profiles match the TomoTherapy® open‐field gold standard profiles within 2% / 1mm. The energy spectra, fluence maps and dose profiles exhibit the shape produced by a treatment device with no flattening filter, such as the TomoTherapy® device: minimal beam hardening along the central axis, minimal spectral shape change as a function of off‐axis angle, about 99.5% of the particles in the phase spaces are photons, and cone shape fluence maps and dose profiles in the lateral direction. Conclusion: Phase spaces for the three possible field sizes were generated. Dose computed from these phase spaces match TomoTherapy® gold standard dose profiles within 2% / 1mm and can be used for static or helical delivery dose computations.

On the dimensional properties of electron gas

A statistical model of noninteracting electrons is considered. It is assumed that free electrons can possess fractal properties in phase space and form electron gas with an arbitrary degree of degeneracy. Within the framework of the model suggested, expressions for the Fermi energy, concentration, heat capacity, and elec- trical conductivity of the electron gas have been obtained. It has been shown also that a dimensional dependence of the Lorentz number is possible. An expression for the electronic component of the surface tension has been derived.

An Extreme IMF as an Explanation for HighM/LRatios in UCDs? The CO Index as a Tracer of Bottom‐heavy IMFs

A new type of compact stellar systems, labeled ultracompact dwarf galaxies (UCDs), was discovered in the last decade. Recent studies show that their dynamical mass-to-light ratios (M/L) tend to be too high to be explained by canonical stellar populations, being on average about twice as large as those of Galactic globular clusters of comparable metallicity. If this offset is caused by dark matter in UCDs, it would imply dark matter densities as expected for the centers of cuspy dark matter halos, incompatible with cored dark matter profiles. Investigating the nature of the high M/L ratios in UCDs therefore offers important constraints on the phase space properties of dark matter particles. Here we describe an observational method to test whether a bottom-heavy IMF may cause the high M/L ratios of UCDs. We propose to use the CO index at 2.3μm—which is sensitive to the presence of low-mass stars—to test for a bottom-heavy IMF. In the case that the high M/L ratios are caused by a bottom-heavy IMF, we show that the equivalent width of the CO index should be up to 30% weaker in UCDs compared to sources in the same metallicity range that have canonical IMFs. We find that these effects are well detectable with current astronomical facilities in a reasonable amount of time (a few hours to nights). In the Fornax and Virgo Clusters, all known UCDs can be mapped within a single square degree pointing. Therefore, the best observational approach is wide-field imaging with narrowband CO filters, complemented by precise measurements of [Fe/H] from optical multi-object spectroscopy. We conclude that measuring the CO index of UCDs is a promising tool to test whether their high M/L ratios are caused by a bottom-heavy IMF.

Efficiency improvements for ion chamber calculations in high energy photon beams

This article presents the implementation of several variance reduction techniques that dramatically improve the simulation efficiency of ion chamber dose and perturbation factor calculations. The cavity user code for the EGSnrc Monte Carlo code system is extended by photon cross-section enhancement (XCSE), an intermediate phase-space storage (IPSS) technique, and a correlated sampling (CS) scheme. XCSE increases the density of photon interaction sites inside and in the vicinity of the chamber and results-in combination with a Russian Roulette game for electrons that cannot reach the cavity volume-in an increased efficiency of up to a factor of 350 for calculating dose in a Farmer type chamber placed at 10 cm depth in a water phantom. In combination with the IPSS and CS techniques, the efficiency for the calculation of the central electrode perturbation factor Pcel can be increased by up to three orders of magnitude for a single chamber location and by nearly four orders of magnitude when considering the Pcel variation with depth or with distance from the central axis in a large field photon beam. The intermediate storage of the phase-space properties of particles entering a volume that contains many possible chamber locations leads to efficiency improvements by a factor larger than 500 when computing a profile of chamber doses in the field of a linear accelerator photon beam. All techniques are combined in a new EGSnrc user code egs_chamber. Optimum settings for the variance reduction parameters are investigated and are reported for a Farmer type ion chamber. A few example calculations illustrating the capabilities of the egs_chamber code are presented.

Time-of-arrival probabilities and quantum measurements. II. Application to tunneling times

We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.

Distinguishing quasiperiodic dynamics from chaos in short-time series

We propose a procedure to distinguish quasiperiodic from chaotic orbits in short-time series, which is based on the recurrence properties in phase space. The histogram of the return times in a recurrence plot is introduced to disclose the recurrence property consisting of only three peaks imposed by Slater's theorem. Noise effects on the statistics are studied. Our approach is demonstrated to be efficient in recognizing regular and chaotic trajectories of a Hamiltonian system with mixed phase space.

Connecting Blackbody Radiation, Relativity, and Discrete Charge in Classical Electrodynamics

It is suggested that an understanding of blackbody radiation within classical physics requires the presence of classical electromagnetic zero-point radiation, the restriction to relativistic (Coulomb) scattering systems, and the use of discrete charge. The contrasting scaling properties of nonrelativistic classical mechanics and classical electrodynamics are noted, and it is emphasized that the solutions of classical electrodynamics found in nature involve constants which connect together the scales of length, time, and energy. Indeed, there are analogies between the electrostatic forces for groups of particles of discrete charge and the van der Waals forces in equilibrium thermal radiation. The differing Lorentz- or Galilean-transformation properties of the zero-point radiation spectrum and the Rayleigh-Jeans spectrum are noted in connection with their scaling properties. Also, the thermal effects of acceleration within classical electromagnetism are related to the existence of thermal equilibrium within a gravitational field. The unique scaling and phase-space properties of a discrete charge in the Coulomb potential suggest the possibility of an equilibrium between the zero-point radiation spectrum and matter which is universal (independent of the particle mass), and an equilibrium between a universal thermal radiation spectrum and matter where the matter phase space depends only upon the ratio mc^2/kT. The observations and qualitative suggestions made here run counter to the ideas of currently accepted quantum physics.

Nonclassicality and decoherence of photon-subtracted squeezed states

We discuss nonclassical properties of single-photon subtracted squeezed vacuum states in terms of the sub-Poissonian statistics and the negativity of the Wigner function. We derive a compact expression for the Wigner function from which we find the region of phase space where Wigner function is negative. We find an upper bound on the squeezing parameter for the state to exhibit sub-Poissonian statistics. We then study the effect of decoherence on the single-photon subtracted squeezed states. We present results for two different models of decoherence, viz. amplitude decay model and the phase diffusion model. In each case we give analytical results for the time evolution of the state. We discuss the loss of nonclassicality as a result of decoherence. We show through the study of their phase-space properties how these states decay to vacuum due to the decay of photons. We show that phase damping leads to very slow decoherence, slower than the photon-number decay.

Attosecond-time-scale multielectron collisions in the Coulomb four-body problem: Traces in classical probability densities

In the triple ionization of the Li ground state by single photon absorption the three electrons escape to the continuum mainly through two collision sequences with individual collisions separated by time intervals on the attosecond scale. We investigate the traces of these two collision sequences in the classical probability densities. We show that each collision sequence has characteristic phase space properties which distinguish it from the other. Classical probability densities are the closest analog to quantum mechanical densities allowing our results to be directly compared to quantum mechanical results.

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

The cubic nonlinear Schrödinger (NLS) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The “nonlinear” spectrum of the associated Lax pair reveals topological properties of the NLS phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete.

Nilpotent classical mechanics:s-geometry

We introduce specific type of hyperbolic spaces. It is not a general linear covariant object, but of use in constructing nilpotent systems. In the present work necessary definitions and relevant properties of configuration and phase spaces are indicated. As a working example we use a D=2 isotropic harmonic oscillator.

On uncertainty bounds and growth estimates for fractional fourier transforms

Gelfand–Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand–Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space.

Complexity of reachability problems for finite discrete dynamical systems

Sequential Dynamical Systems (SDSs) are a special type of finite discrete dynamical systems that can be used to model simulation systems. We focus on the computational complexity of testing several phase space properties of SDSs. Our main result is a sharp delineation between classes of SDSs whose behavior is easy to predict and those whose behavior is hard to predict. Specifically, we show the following. 1. Several state reachability problems for SDSs are PSPACE-complete, even when restricted to SDSs whose underlying graphs are of bounded bandwidth (and hence of bounded pathwidth and treewidth), and the function associated with each node is symmetric. Moreover, this result holds even when the underlying graph is d-regular for some constant d and all the nodes compute the same symmetric Boolean function. An immediate corollary of this result is a PSPACE-hard lower bound on the complexity of reachability problems for regular generalized 1D-Cellular Automata and undirected systolic networks with Boolean totalistic local transition functions. 2. In contrast, the above reachability problems are solvable in polynomial time for SDSs when the Boolean function associated with each node is symmetric and monotone. The PSPACE-completeness results follow as corollaries of simulation results which show for several classes of SDSs, how one class of SDSs can be efficiently simulated by another (more restricted) class of SDSs. We also prove several structural properties concerning the phase space of an SDS. SDSs are closely related to Cellular Automata (CA), concurrent transition systems, discrete Hopfield networks and systolic networks. This observation in conjunction with our lower bounds for SDSs, yields new PSPACE-hard lower bounds on the complexity of state reachability problems for these models, extending some of the earlier results in [K. Culik II, J. Karhumäki, On totalistic systolic networks, Inform. Process. Lett. 26 (5) (1988) 231–236; P. Floréen, E. Goles, G. Weisbuch, Transient length in sequential iterations of threshold functions, Discrete Appl. Math. 6 (1983) 95–98; P. Floréen, P. Orponen, Complexity issues in discrete Hopfield networks, Research Report No. A-1994-4, Department of Computer Science, University of Helsinki, 1994. Also appears in: I. Parberry (Ed.), Comp. and Learning Complexity of Neural Networks: Advanced Topics, 1999; D. Harel, O. Kupferman, M.Y. Vardi, On the complexity of verifying concurrent transition systems, Inform. and Comput. 173 (2002) 143–161; S.K. Shukla, H.B. Hunt III, D.J. Rosenkrantz, R.E. Stearns, On the complexity of relational problems for finite state processes, in: International Colloquium on Automata Programming and Languages, ICALP, 1996, pp. 466–477; A. Rabinovich, Complexity of equivalence problems for concurrent systems of finite agents, Inform. and Comput. 127 (2) (1997) 164–185].

ON ASYNCHRONOUS CELLULAR AUTOMATA

We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v1,…,vn} and where each vertex vi has a state xvi in a finite field K, (b) a sequence of functions (Fvi,Y)i, and (c) a word w = (w1,…,wk), where each wi is a vertex in Y. The function Fvi,Y updates the state of vertex vi as a function of the state of vi and its Y-neighbors and maps all other vertex states identically. The SDS is the composed map [Formula: see text]. In the particular case of ACA, the graph is the circle graph on n vertices (Y = Circ n), and all the maps Fvi are induced by a common Boolean function. Our main result is the identification of all w-independent ACA, that is, all ACA with periodic points that are independent of the word (update schedule) w. In general, for each w-independent SDS, there is a finite group whose structure contains information about for example SDS with specific phase space properties. We classify and enumerate the set of periodic points for all w-independent ACA, and we also compute their associated groups in the case of Y = Circ 4. Finally, we analyze invertible ACA and offer an interpretation of S35 as the group of an SDS over the three-dimensional cube with local functions induced by nor3 + nand3.

Operator product expansions as a consequence of phase space properties

The paper presents a model-independent, nonperturbative proof of operator product expansions in quantum field theory. As an input, a recently proposed phase space condition is used that allows a precise description of point field structures. Based on the product expansions, we also define and analyze normal products (in the sense of Zimmermann).

On certain morphisms of sequential dynamical systems

We study a class of discrete dynamical systems that consist of the following data: (a) a finite (labeled) graph Y with vertex set { 1 , … , n } , where each vertex has a binary state, (b) a vertex labeled multi-set of functions ( F i , Y : F 2 n → F 2 n ) i and (c) a permutation π ∈ S n . The function F i , Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions F i , Y are applied. By composing the functions F i , Y in the order given by π we obtain the sequential dynamical system (SDS) [ F Y , π ] = ∏ i = 1 n F π ( i ) , Y : F 2 n ⟶ F 2 n . Let G [ F Y , π ] be the graph with vertex set F 2 n and edge set { ( x , [ F Y , π ] ( x ) ) ∣ x ∈ F 2 n } . An SDS-morphism between [ F Y , π ] and [ F Z , σ ] is a triple ( ϕ , η , Φ ) , where ϕ : Y ⟶ Z is a graph-morphism, η : S | Z | ⟶ S | Y | is a map such that η ( σ ) = π and Φ is a digraph-morphism Φ : G [ F Z , σ ] ⟶ G [ F Y , π ] . Our main result is that locally bijective graph-morphisms (coverings) between dependency graphs of SDS naturally induce SDS-morphisms. We show how these SDS-morphisms allow for a new proof for the upper bound on the number of inequivalent SDS obtained by only varying their underlying permutations. Here, two SDS are called inequivalent if they are inequivalent as dynamical systems. Furthermore, we apply our result in order to obtain phase space properties of SDS.

Phase space properties and the short distance structure in quantum field theory

The paper investigates relations between the phase space structure of a quantum field theory (“nuclearity”) and the concept of pointlike localized fields. Given a net of local observable algebras, a phase space condition is introduced that allows a very detailed description of the theory’s field content. An appendix discusses noninteracting models as examples.

Nonlinear multivariable analysis of SOI and local precipitation and temperature

Abstract. Global climate variability affects important local hydro-meteorological variables like precipitation and temperature. The Southern Oscillation (SO) is an easily quantifiable major driving force that gives impact on regional and local climate. The relationships between SO and local climate variation are, however, characterized by strongly nonlinear processes. Due to this, teleconnections between global-scale hydro-meteorological variables and local climate are not well understood. In this paper, we suggest to study these processes in terms of nonlinear dynamics. Consequently, the nonlinear dynamic relationship between the Southern Oscillation Index (SOI), precipitation, and temperature in Fukuoka, Japan, is investigated using a nonlinear multivariable approach. This approach is based on the joint variation of these variables in the phase space. The joint phase-space variation of SOI, precipitation, and temperature is studied with the primary objective to obtain a better understanding of the dynamical evolution of local hydro-meteorological variables affected by global atmospheric-oceanic phenomena. The results from the analyses display rather clear low-order phase space trajectories when treating the time series individually. However, when plotting phase space trajectories for several time series jointly, complicated higher-order nonlinear relationships emerge between the variables. Consequently, simple data-driven prediction techniques utilizing phase-space characteristics of individual time series may prove successful. On the other hand, since either the time series are too short and/or the phase-space properties are too complex when analysing several variables jointly, it may be difficult to use multivariable statistical prediction techniques for the present investigated variables. In any case, it is essential to further pursue studies regarding links between the SOI and observed local climatic and other geophysical variables even if these links are not fully understood in physical terms.

Existence and Stability of Localized Oscillations in 1-Dimensional Lattices With Soft-Spring and Hard-Spring Potentials

In this paper, we use the method of homoclinic orbits to study the existence and stability of discrete breathers, i.e., spatially localized and time-periodic oscillations of a class of one-dimensional (1D) nonlinear lattices. The localization can be at one or several sites and the 1D lattices we investigate here have linear interaction between nearest neighbors and a quartic on-site potential Vu=12Ku2±14u4, where the (+) sign corresponds to “hard spring” and (−) to “soft spring” interactions. These localized oscillations—when they are stable under small perturbations—are very important for physical systems because they seriously affect the energy transport properties of the lattice. Discrete breathers have recently been created and observed in many experiments, as, e.g., in the Josephson junction arrays, optical waveguides, and low-dimensional surfaces. After showing how to construct them, we use Floquet theory to analyze their linear (local) stability, along certain curves in parameter space (α,ω), where α is the coupling constant and ω the frequency of the breather. We then apply the Smaller Alignment Index method (SALI) to investigate more globally their stability properties in phase space. Comparing our results for the ± cases of Vu, we find that the regions of existence and stability of breathers of the “hard spring” lattice are considerably larger than those of the “soft spring” system. This is mainly due to the fact that the conditions for resonances between breathers and linear modes are much less restrictive in the former than the latter case. Furthermore, the bifurcation properties are quite different in the two cases: For example, the phenomenon of complex instability, observed only for the “soft spring” system, destabilizes breathers without giving rise to new ones, while the system with “hard springs” exhibits curves in parameter space along which the number of monodromy matrix eigenvalues on the unit circle is constant and hence breather solutions preserve their stability character.