Showing papers on "Phase space published in 1992"
TL;DR: In this article, a Lagrangian variational formulation of twist maps is proposed to compute the flux escaping from regions bounded by partial barriers formed from minimizing orbits, which form a scaffold in the phase space and constrain the motion of remaining orbits.
Abstract: Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamics-from simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the flux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions.
627 citations
TL;DR: In this paper, the existence and multiplicity of coherent structures in the complex Ginzburg-Landau equation was studied. But the authors focused on the competition between fronts and pulses and did not consider the non-uniformly translating front structures.
482 citations
TL;DR: One-dimensional many-body integrable systems in terms of a new set of phase space variables involving exchange operators assuming a decoupled form greatly simplifies the derivation of the conserved charges and the proof of their commutativity at the quantum level.
Abstract: We formulate one-dimensional many-body integrable systems in terms of a new set of phase space variables involving exchange operators. The Hamiltonian in these variables assumes a decoupled form. This greatly simplifies the derivation of the conserved charges and the proof of their commutativity at the quantum level.
433 citations
TL;DR: It is shown that there exist quantum states which approximate a given metric at large scales, but such states exhibit a discrete structure at the Planck scale.
Abstract: Results that illuminate the physical interpretation of states of nonperturbative quantum gravity are obtained using the recently introduced loop variables. It is shown that (i) while local operators such as the metric at a point may not be well defined, there do exist nonlocal operators, such as the area of a given two-surface, which can be regulated diffeomorphism invariantly and which are finite without renormalization; (ii) there exist quantum states which approximate a given metric at large scales, but such states exhibit a discrete structure at the Planck scale.
375 citations
TL;DR: The nonclassical properties of quantum superpositions of coherent states of light are discussed and under which conditions a superposition of two coherent states can exhibit second- and fourth-order squeezing or sub-Poissonian photon statistics are shown.
Abstract: In this paper we discuss the nonclassical properties of quantum superpositions of coherent states of light. Using general expressions for the Wigner functions of superposition states we analyze the consequences of quantum interference between coherent states. We describe in detail nonclassical properties of a superposition of two coherent states. In particular, we study the oscillatory behavior of the photon number distribution of the even and odd coherent states. We show under which conditions a superposition of two coherent states can exhibit second- and fourth-order squeezing or sub-Poissonian photon statistics. We examine the sensitivity of nonclassical effects such as oscillations in the photon number distribution or second-order squeezing to dissipation. We demonstrate that quantities such as the photon number distribution and interferences in phase space are highly sensitive to even a quite small dissipative coupling, because they depend on all moments of the field observables, and higher moments decay more rapidly than lower moments. Quantities such as quadrature squeezing, on the other hand, are more robust against dissipation because they involve only lower moments. Finally, we find a remarkable effect whereby fourth-order squeezing is generated by damping.
339 citations
TL;DR: In this article, the authors introduce reversible dynamical systems, which generalise classical mechanical systems possessing time-reversal symmetry and are found in ordinary differential equations, partial differential equations and diffeomorphisms (mappings) modelling many physical problems.
279 citations
TL;DR: In this article, the relation between partonic distributions near the phase space boundary and Wilson loop expectation values calculated along paths partially lying on the light-cone is discussed, and it is shown that the universal form of the splitting function for large x originates from the cusp anomalous dimension of Wilson loops.
Abstract: We discuss the relation between partonic distributions near the phase space boundary and Wilson loop expectation values calculated along paths partially lying on the light-cone. Due to additional light-cone singularities, multiplicative renormalizability for these expectation values is lost. Nevertheless we establish the renormalization group equation for the light like Wilson loops and show that it is equivalent to the evolution equation for the physical distributions. By performing a two-loop calculation we verify these properties and show that the universal form of the splitting function for large x originates from the cusp anomalous dimension of Wilson loops.
258 citations
TL;DR: In this article, the authors present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling, and find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts).
Abstract: We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.
236 citations
TL;DR: In this article, the authors derived the continuum equations of a dispersed phase of solid, noncolliding particles in a nonuniform turbulent gas flow from a kinetic equation for the transport of the average phase space density for particles with velocity v and position x at time t. This form for j is invariant under a random Galilean transformation.
Abstract: The continuum equations of a dispersed phase of solid, noncolliding particles in a nonuniform turbulent gas flow are derived from a kinetic equation for the transport of the average phase space density 〈W(v,x,t)〉 for particles with velocity v and position x at time t. The crucial feature of this equation is the form given for the phase space diffusion current j representing the net acceleration of a particle from interactions with turbulent eddies. This is based on Kraichnan’s Lagrangian history direct interaction approximation which gives j=−[(∂/∂v)⋅μ+(∂/∂x)⋅λ+γ]〈W(v,x,t)〉, where μ, λ, and γ are dispersion tensors dependent upon displacements in the velocity and position of a particle about v,x in times of order of the time scale of the fluctuating aerodynamic driving force. Most important these tensors are affected by spatial variations in the mean carrier flow velocity and external force as well as inhomogeneities in the carrier phase turbulence; γ is zero for homogeneous turbulence. This form for j is invariant under a random Galilean transformation. The dispersed phase momentum and energy equations deduced from the kinetic equation contain local gradient forms for the net fluctuating interphase force (per unit volume) and its rate of working. The former contains in general an asymmetric stress component (which adds to the particle Reynolds stress) as well as a body force dependent upon inhomogeneities in the turbulence. Conditions are examined under which the mass and momentum equations reduce to a convection–diffusion equation. The convection velocity in this case is the local carrier flow velocity plus a drift velocity proportional to local gradients of the turbulence (zero if the particles follow the flow). The diffusion coefficient is linearly related to the pressure/mean density of the dispersed phase via the particle response time. It is shown to be influenced by the local mean shearing of the carrier flow. Conditions are derived when this contribution can be ignored, the diffusion coefficient reducing to the form for homogeneous turbulence.
182 citations
TL;DR: Two methods are presented to calculate the optimal embedding parameters for Takens's delay-time coordinates, including a procedure that yields a global measure of phase-space utilization for (quasi) periodic and strange attractors and leads to a maximum separation of trajectories within the phase space.
Abstract: Fractal dimensions and Lyapunov exponents can be estimated faster and more accurately and efficiently with the knowledge of the optimal embedding dimension and delay time. Two methods are presented to calculate the optimal embedding parameters for Takens's delay-time coordinates. The first, called the fill factor, is a procedure that yields a global measure of phase-space utilization for (quasi) periodic and strange attractors and leads to a maximum separation of trajectories within the phase space. The second, which we call local deformation, is complementary to the fill factor
172 citations
TL;DR: The Van Vleck formula as discussed by the authors is an approximate, semiclassical expression for the quantum propagator, which is the starting point for the Gutzwiller trace formula, and through this, a variety of other expansions representing eigenvalues, wave functions, and matrix elements in terms of classical periodic orbits.
Abstract: The Van Vleck formula is an approximate, semiclassical expression for the quantum propagator. It is the starting point for the derivation of the Gutzwiller trace formula, and through this, a variety of other expansions representing eigenvalues, wave functions, and matrix elements in terms of classical periodic orbits. These are currently among the best and most promising theoretical tools for understanding the asymptotic behavior of quantum systems whose classical analogs are chaotic. Nevertheless, there are currently several questions remaining about the meaning and validity of the Van Vleck formula, such as those involving its behavior for long times. This article surveys an important aspect of the Van Vleck formula, namely, the relationship between it and phase space geometry, as revealed by Maslov's theory of wave asymptotics. The geometrical constructions involved are developed with a minimum of mathematical formalism.
TL;DR: In this article, the quantum concept of a Wigner distribution function is used to define a representation of an ultrashort light pulse in terms of a real function in the time-frequency phase space.
Abstract: The quantum concept of a Wigner distribution function is borrowed in order to define a representation of an ultrashort light pulse in terms of a real function in the time-frequency phase space. This conceptual framework is used to describe the action of several linear devices and the nonlinear propagation of ultrashort pulses. An experimental determination of chronocyclic representations based on spectrally resolved cross correlations is proposed. This is generalized to an analysis of phase and amplitude measurements of ultrashort pulses. >
TL;DR: The first path-integral calculations of the properties of a strongly correlated continuum fermion system are described, and restricting the walks with the noninteracting density matrix gives good results for liquid 3 He above 1 K.
Abstract: The first path-integral calculations of the properties of a strongly correlated continuum fermion system are described. The paths are restricted to the region of phase space with a positive trial density matrix, thereby avoiding the fermion sign problem. This restriction is exact if the nodes of the trial density matrix are correctly placed, but otherwise gives a physically reasonable approximation generalizing the ``fixed-node'' approximation used at zero temperature. Computations show that restricting the walks with the noninteracting density matrix gives good results for liquid $^{3}\mathrm{He}$ above 1 K. Using imaginary-time-independent nodes or not allowing atomic exchange results in substantially poorer agreement with experimental energies.
TL;DR: In this article, the authors consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers, and show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement is Gaussian, but the normalization factor is (t logt)1/2 and nott 1/2 as in the classical case.
Abstract: We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the timet goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t logt)1/2 and nott1/2 as in the classical case. We find the covariance matrix of the limit distribution.
TL;DR: In this paper, a new and improved version of second order analytical proper elements has been obtained by taking into account the secular perturbations of the asteroids, by the four major planets, and also part of the effect of the inner planets.
TL;DR: In this paper, the interplay of various quantum effects on barrier crossing for a one-dimensional system with dissipation is discussed based on a numerical study using a hierarchy of kinetic equations introduced by Tanimura and Kubo.
Abstract: We discuss the interplay of various quantum effects on barrier crossing for a one‐dimensional system with dissipation. This is based on a numerical study using a hierarchy of kinetic equations introduced by Tanimura and Kubo. The numerical work uses a grid in phase space for the Wigner distribution and deals with both the classical limit and the tunneling regimes.
TL;DR: In this article, a mathematical and numerical framework has been worked out to represent the density operator in phase space and to propagate it in time under dissipative conditions, based on the Fourier pseudospectral method which allows a description both in configuration as well as in momentum space.
Abstract: A mathematical and numerical framework has been worked out to represent the density operator in phase space and to propagate it in time under dissipative conditions. The representation of the density operator is based on the Fourier pseudospectral method which allows a description both in configuration as well as in momentum space. A new propagation scheme which treats the complex eigenvalue structure of the dissipative Liouville superoperator has been developed. The framework has been designed to incorporate modern computer architecture such as parallelism and vectorization. Comparing the results to closed-form solutions exponentially fast convergence characteristics in phase space as well as in the time propagation is demonstrated. As an example of its usefulness, the new method has been successfully applied to dissipation under the constraint of selection rules. More specifically, a harmonic oscillator which relaxes to equilibrium under the constraint of second-order coupling to the bath was studied.
TL;DR: In this article, numerical results of three-dimensional (3-D) resistive magnetohydrodynamic (MHD) plasma simulations are presented, where a system of coupled nonlinear differential equations is evolved in time over a significant fraction of the macroscopic resistive diffusion time scale.
Abstract: In this paper numerical results of three‐dimensional (3‐D) resistive magnetohydrodynamic (MHD) plasma simulations are presented. A system of coupled nonlinear differential equations is evolved in time over a significant fraction of the macroscopic resistive diffusion time scale. The dynamical evolution resembles the main features of the famous Lorenz system. In fact, sensitivity of MHD equations on initial distribution of spectral energy and stochastic oscillations in phase space have been found. At least two dynamic attractors of the motion have been identified. Moreover, in analogy with Lorenz’s system, the stochastic motion can be damped by an enhanced dissipation and the fixed point can be recovered. In this paper more specific topics are also considered, which are relevant to the reversed field pinch (RFP), such as the role of different modes in the ‘‘dynamo’’ mechanism for plasma sustainment and the associated transport due to stochastic diffusion.
TL;DR: In this paper, a sharp interpretation of Liapunov stability of relative equilibria for Hamiltonian systems is given in terms of concepts on the unreduced space under mild assumptions.
TL;DR: In this paper, the Hamiltonian dynamics of coupled map systems is used to find the clustering motion of particles in the phase space of KAM tori and islands, and Lyapunov analysis distinguishes global instability from local fluctuations.
Abstract: Clustered motion of particles is found in Hamiltonian dynamics of symplectic coupled map systems. Particles assemble and move with strong correlation. The motion is chaotic but is distinguishable from random chaotic motion. Lyapunov analysis distinguishes global instability from local fluctuations. Clustered motions have finite lifetime. They have fractal geometric structure in the phase space, as the orbits are trapped to ruins of KAM tori and islands.
TL;DR: In this paper, the connection between maximal complexity and power law noise or correlations can be derived from a simple variational principle, and it is shown that for a 1D signal, one can find 1/f noise, in accordance with Zhang.
Abstract: We elaborate in some detail on a new phase space approach to complexity, due to Y.-C. Zhang. We show in particular that the connection between maximal complexity and power law noise or correlations can be derived from a simple variational principle. For a 1D signal we find 1/f noise, in accordance with Zhang.
TL;DR: In this article, the authors apply the method of coadjoint orbits of W∞-algebra to the problem of non-relativistic fermions in one dimension.
Abstract: We apply the method of coadjoint orbits of W∞-algebra to the problem of non-relativistic fermions in one dimension. This leads to a geometric formulation of the quantum theory in terms of the quantum phase space distribution of the Fermi fluid. The action has an infinite series of expansion in the string coupling, which to leading order reduces to the previously discussed geometric action for the classical Fermi fluid based on the group w∞ of area-preserving diffeomorphisms. We briefly discuss the strong coupling limit of the string theory which, unlike the weak coupling regime, does not seem to admit a two-dimensional space-time picture. Our methods are equally applicable to interacting fermions in one dimension.
TL;DR: Using the time-dependent variational principle with a group theoretical coherent state defining the wave functions for electrons and nuclei, a system of coupled, first-order, nonlinear differential equations is obtained for a general molecular system.
Abstract: Using the time‐dependent variational principle with a group theoretical coherent state defining the wave functions for electrons and nuclei, a system of coupled, first‐order, nonlinear differential equations is obtained for a general molecular system. The equations form a classical Hamiltonian system within a generalized phase space that allows a systematic time‐dependent study of molecular processes. The approach is general and provides a computational framework for a variety of properties such as transition and excitation probabilities in atomic and molecular collisions, and molecular spectra such as vibrational spectra with anharmonicities. The basic approximation corresponding to the choice of a single determinantal wave function for the electrons and classical nuclei is analyzed. Illustrative applications to the p+H collision process and to vibrations of the H2O molecule exhibit good agreement with experiment and with other theoretical work.
TL;DR: In this paper, the authors considered a general class of piecewise expanding single-component transformations and treated the coupled system as a perturbation of the uncoupled one, and obtained existence and stability results for T-invariant measures with absolutely continuous finite-dimensional marginals.
Abstract: Let L denote a finite or infinite one-dimensional lattice. To each lattice site is attached a copy of a dynamical system with phase space [0, 1] and dynamics described by a transformation τ: [0, 1] → [0, 1], which is the same on each component. Denote the direct product of these identical systems by T: X → X where X = [0, 1]L. From this product system we obtain a coupled map lattice (CML) Se: X → X, if we introduce some interaction between the components, e.g. by averaging between nearest neighbours. The strength of the coupling depends upon some parameter e.For a broad class of piecewise expanding single-component-transformations τ we study such systems via their transfer operators and treat the coupled system as a perturbation of the uncoupled one. This yields existence and stability results for T-invariant measures with absolutely continuous finite-dimensional marginals.
TL;DR: In this article, a comparison of the cosmological models derived from Lagrangian densities with powers higher than two in the Ricci scalar of curvature and the Starobinsky model is carried out.
Abstract: The phase space portrait of the cosmological models deduced from fourth-order gravity theories is discussed with the analytical and numerical methods of our previous paper. A comparison is carried out between models inferred from Lagrangian densities containing powers higher than two in the Ricci scalar of curvature, and the Starobinsky model. Some peculiar structures, such as attractors and singular points, emerging neatly from both theories, have a close physical affinity, in addition to the mathematical one. Trajectories of interest in both scenarios are those undergoing an inflationary expansion and then reaching a Friedmannian asymptotic stable phase. These features are moreover discussed through a potential U(R) in RN-models. Three kinds of potential regions are recognized. They are the allowed regions (a-regions), in which trajectories can reach the Friedmannian phase after possibly undergoing an inflationary period, the disconnected regions (d-regions), in which trajectories, although physical, never reach the Friedmannian stage and the forbidden regions (f-regions), in which there are no physical solutions. A general survey of the global phase space for Starobinsky and RN-models is given via Poincare projections of suitable variables. a-, d-, and f-regions are represented.
TL;DR: In this article, the finiteness of the spin phase space is shown to strongly influence the systematic behaviour of periodic orbits, and the consequences of the chaotic dynamics and of the superradiant phase transition are discussed.
TL;DR: In this paper, it was shown that Hilbert-space quantum mechanics and many other statistical theories can be represented on some phase space, in the sense that states can be identified with probability measures and observables can be described by functions.
Abstract: It is shown that Hilbert‐space quantum mechanics and many other statistical theories can be represented on some phase space, in the sense that states can be identified with probability measures and observables can be described by functions In the general context of statistical dualities, informationally complete observables are introduced and a theorem on their existence is proven The correspondence between these observables and the injective affine mappings from the states into the probability measures on phase space, ie, the phase‐space representations, is pointed out In particular, a description of all observables by functions is presented, such that all expectation values can, in arbitrarily good physical approximation, be calculated as integrals Moreover, some new aspects of the particular case of those phase‐space representations of quantum mechanics that are related to certain joint position‐momentum observables are discussed
TL;DR: In this paper, the quantum dynamics of a particle in a double-well potential under a monochromatic external driving force is computed by using a minimum-uncertainty Gaussian wave packet as an initial state.
Abstract: The quantum dynamics of a particle in a double-well potential under a monochromatic external driving force is computed by using a minimum-uncertainty Gaussian wave packet as an initial state. Irregular or regular time development of dynamical variables is obtained, depending on whether the initial wave packet is located in the classical chaotic sea or the regular region in phase space, respectively. Tunnelings between nonresonant islands and between resonant islands are obtained. Tunneling times between nonresonant islands decrease smoothly, in general, with the increase of the amplitude of the driving force, but tunneling times between resonant islands are more erratic.
TL;DR: In this article, the behavior of the quantum baker's transformation was investigated for time scales where the classical mechanics generated phase space structures on a scale smaller than Planck's constant. But it was not shown that a semiclassical theory can accurately reproduce many features of quantum evolution of a wave packet in this strongly mixing time regime.
TL;DR: In this paper, the Karhunen-Loeve (K-L) decomposition is used to analyze complex spatio-temporal structures in PDE simulations in terms of concepts from dynamical systems theory.
Abstract: We propose the Karhunen-Loeve (K-L) decomposition as a tool to analyze complex spatio-temporal structures in PDE simulations in terms of concepts from dynamical systems theory. Taking the Kuramoto-Sivashinsky equation as a model problem we discuss the K-L decomposition for 4 different values of its bifurcation parameter α. We distinguish two modes of using the K-L decomposition: As an analytic and synthetic tool respectively. Using the analytic mode we find unstable fixed points and stable and unstable manifolds in a parameter regime with structurally stable homoclinic orbits (α=17.75). Choosing the data for a K-L analysis carefully by restricting them to certain burst events, we can analyze a more complicated intermittent regime at α=68. We establish that the spatially localized oscillations around a so called “strange” fixed point which are considered as fore-runners of spatially concentrated zones of turbulence are in fact created by a very specific limit cycle (α=83.75) which, for α=87, bifurcates into a modulated traveling wave. Using the K-L decomposition synthetically by determining an optimal Galerkin system, we present evidence that the K-L decomposition systematically destroys dissipation and leads to blow up solutions.