Phase Space Transport Research Articles

From long-range interaction to collective behaviour and from Hamiltonian chaos to stochastic models

Assemblies of charged particles are the prototype of systems with long range interactions. Their behaviour involves effective degrees of freedom of two types. Collective components emerge, commonly classified as modes or waves, which are (often) nonlinearly coupled, or resonant, with a small subset of the initial assembly of particles. The reduction from many bodies to the effective model for this self-consistent dynamics may be performed in the Lagrangian setting, and the resulting dynamics is Hamiltonian. For regular interactions (e.g. regularized Coulomb) the many-body limit leads to the Vlasov equation, but its validity for long times is questionable. The self-consistent dynamics generates chaotic behaviour, leading to some transport in phase space. However, this transport is generally not a diffusion process, and when it is diffusion-like its parameters may have unexpected values. In the strongly chaotic limit (with many overlapping resonances), however, the transport does tend to a diffusion process, with parameters determined by a straightforward prescription—though the physical elementary process underlying short-term transport is not the same process which underlies long-time transport, due to the interplay of chaos and granularity (finite N) effects.

Polynomial dispersion of trajectories in sticky dynamics

Hamiltonian chaotic dynamics is, in general, not ergodic and the boundaries of the ergodic or quasiergodic area (stochastic sea, stochastic layers, stochastic webs, etc.) are sticky, i.e., trajectories can spend an arbitrarily long time in the vicinity of the boundaries with a nonexponentially small probability. Segments of trajectories imposed by the stickiness are called flights. The flights have polynomial dispersion that can lead to non-Gaussian statistics of displacements and to anomalous transport in phase space. In particular, the presence of flights influences the distribution of Poincaré recurrences. We use the distribution function of (l,t;epsilon, epsilon0) -separation of trajectories that at time instant t and trajectory length l are separated for the first time by epsilon<<1, being initially at a distance epsilon0 <<epsilon. The connection of this function, called the complexity function [Afraimovich and Zaslavsky, Chaos 13, 519 (2003)], with the distribution of Poincaré recurrences is established for three cases: (i) for the case of superdiffusion in standard and web maps for which the stickiness is defined by the boundaries of hierarchical sets of islands; (ii) for the case of the Sinai billiard with infinite horizon, where the stickiness is defined by zero-measure slits in the phase space; (iii) for the square billiard with a slit (bar-in-square billiard) where the Lyapunov exponent is zero and the stickiness is defined by the vicinity of the trajectory to the closest periodic trajectories obtained from the Diophantine approximation. Finally, the powerwise asymptotics of the Poincaré recurrences can be connected, in some cases, to the anomalous transport exponent.

Escape from planetary neighbourhoods

In this paper we use recently developed phase-space transport theory coupled with a so-called classical spectral theorem to develop a dynamically exact and computationally efficient procedure for studying escape from a planetary neighbourhood. The ‘planetary neighbourhood’ is a bounded region of phase space where entrance and escape are only possible by entering or exiting narrow ‘bottlenecks’ created by the influence of a saddle point. The method therefore immediately applies to, for example, the circular restricted three-body problem and Hill's lunar problem (which we use to illustrate the results), but it also applies to more complex, and higher-dimensional, systems possessing the relevant phase-space structure. It is shown how one can efficiently compute the mean passage time through the planetary neighbourhood, the phase-space flux in, and out, of the planetary neighbourhood, the phase-space volume of initial conditions corresponding to trajectories that escape from the planetary neighbourhood, and the fraction of initial conditions in the planetary neighbourhood corresponding to bound trajectories. These quantities are computed for Hill's problem. We study the dependence of the proportions of these quantities on energy and dimensionality (two-dimensional planar and three-dimensional spatial Hill's problem). The methods and quantities presented are of central interest for many celestial and stellar dynamical applications such as, for example, the capture and escape of moons near giant planets, the formation of binaries in the Kuiper belt and the escape of stars from star clusters orbiting about a galaxy.

Diffusion in the asteroid belt

In the beginning we review briefly the evolution of the ideas on the motion of the bodies in our solar system, from Newton's clockwork Universe to the presently accepted ubiquity of chaotic transport in the asteroid belt. Then we discuss the result of chaotic motion, which is transport in phase space, and we introduce the concept of diffusion of an asteroid in action space. We proceed by reviewing recent work on numerical as well as analytical study of asteroids following chaotic trajectories and we summarize the main results. We present several applications of the theoretical modelling of asteroid motion as diffusion in action space, to problems of specific interest.

A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill's problem

A computational procedure that allows the detection of a new type of high-dimensional chaotic saddle in Hamiltonian systems with three degrees of freedom is presented. The chaotic saddle is associated with a so-called normally hyperbolic invariant manifold (NHIM). The procedure allows to compute appropriate homoclinic orbits to the NHIM from which we can infer the existence a chaotic saddle. NHIMs control the phase space transport across an equilibrium point of saddle-centre-...-centre stability type, which is a fundamental mechanism for chemical reactions, capture and escape, scattering, and, more generally, ``transformation'' in many different areas of physics. Consequently, the presented methods and results are of broad interest. The procedure is illustrated for the spatial Hill's problem which is a well known model in celestial mechanics and which gained much interest e.g. in the study of the formation of binaries in the Kuiper belt.

Chaos and chaotic phase mixing in cuspy triaxial potentials

This paper investigates chaos and chaotic phase mixing in triaxial Dehnen potentials which have been proposed to describe realistic ellipticals. Earlier work is extended by exploring the effects of (1) variable axis ratios, (2) `graininess' associated with stars and bound substructures, idealised as friction and white noise, and (3) large-scale organised motions presumed to induce near-random forces idealised as coloured noise with finite autocorrelation time. Three important conclusions are: (1) not all the chaos can be attributed to the cusp; (2) significant chaos can persist even for axisymmetric systems; and (3) introducing a supermassive black hole can increase both the relative number of chaotic orbits and the size of the largest Lyapunov exponent. Sans perturbations, distribution functions associated with initially localised chaotic ensembles evolve exponentially towards a nearly time-independent form at a rate L that correlates with the finite time Lyapunov exponents associated with the evolving orbits. Perturbations accelerate phase space transport by increasing the rate of phase mixing in a given phase space region and by facilitating diffusion along the Arnold web that connects different phase space regions, thus facilitating an approach towards a true equilibrium. The details of the perturbation appear unimportant. All that matters are the amplitude and the autocorrelation time, upon which there is a weak logarithmic dependence. Even comparatively weak perturbations can increase L by a factor of three or more, a fact that has potentially significant implications for violent relaxation.

Intramolecular vibrational energy redistribution in DCO (X̃2A′): Classical-quantum correspondence, dynamical assignments of highly excited states, and phase space transport

Intermolecular dynamics of highly excited DCO (X^2A') is studied from a classical-quantum perspective using the effective spectroscopic Hamiltonian proposed recently by Trollch and Temps (Z. Phy. Chem. 215, 207 (2001)). This work focuses on the polyads P = 3 and P = 4 corresponding to excitation energies E_v ~ 5100 cm^-1 and 7000 cm^-1 respectively. The majority of states belonging to these polyads are dynamically assigned, despite extensive stochasticity in the classical phase space, using the recently proposed technique of level velocities. A wavelet based time-frequency analysis is used to reveal the nature of phase space transport and the relevant dynamical bottlenecks. The local frequency analysis clearly illustrates the existence of mode-specific IVR dynamics i.e., differing nature of the IVR dynamics ensuing from CO stretch and the DCO bend bright states. In addition the role of the weak Fermi resonance involving the CO stretch and DCO bend modes is investigated. A key feature of the present work is that the techniques utilized for the analysis i.e., parametric variations and local frequency analysis are not limited by the dimensionality of the system. This study, thus, explores the potential for understanding IVR in large molecules from both time domain and frequency domain perspectives.

Analyzing intramolecular vibrational energy redistribution via the overlap intensity-level velocity correlator

Numerous experimental and theoretical studies have established that intramolecular vibrational energy redistribution in isolated molecules has a hierarchical tier structure. The tier structure implies strong correlations between the energy level motions of a quantum system and its intensity-weighted spectrum. A measure, which explicitly accounts for this correlation, was first introduced by one of us as a sensitive probe of phase space localization. It correlates eigenlevel velocities with the overlap intensities between the eigenstates and some localized state of interest. In this work we investigate the usefulness of the correlator in understanding the classical-quantum correspondence of effective spectroscopic Hamiltonians. Application to a model two dimensional effective spectroscopic Hamiltonian shows that the correlation measure can provide information about the terms in the molecular Hamiltonian which play an important role in an energy range of interest and the character of the dynamics. Moreover, the correlation function is capable of highlighting relevant phase space structures including the local resonance features associated with a specific bright state. In addition to being ideally suited for multidimensional systems with a large density of states, the measure can also be used to gain insights into phase space transport and localization. It is argued that the overlap intensity-level velocity correlation function provides a novel way of studying vibrational energy redistribution in isolated molecules. The correlation function is ideally suited to analyzing the parametric spectra of molecules in external fields.

Particle Transport through Scattering Regions with Clear Layers and Inclusions

This paper introduces generalized diffusion models for the transport of particles in scattering media with nonscattering inclusions. Classical diffusion is known as a good approximation of transport only in scattering media. Based on asymptotic expansions and the coupling of transport and diffusion models, generalized diffusion equations with nonlocal interface conditions are proposed which offer a computationally cheap, yet accurate, alternative to solving the full phase-space transport equations. The paper shows which computational model should be used depending on the size and shape of the nonscattering inclusions in the simplified setting of two space dimensions. An important application is the treatment of clear layers in near-infrared (NIR) spectroscopy, an imaging technique based on the propagation of NIR photons in human tissues.

Chaos and the continuum limit in the gravitational N-body problem. II. Nonintegrable potentials.

This paper continues a numerical investigation of the statistical properties of "frozen-N orbits," i.e., orbits evolved in frozen, time-independent N-body realizations of smooth density distributions rho corresponding to both integrable and nonintegrable potentials, allowing for 10(2.5)</=N</=10(5.5). The focus is on distinguishing between, and quantifying, the effects of graininess on initial conditions corresponding, in the continuum limit, to regular and chaotic orbits. Ordinary Lyapunov exponents chi do not provide a useful diagnostic for distinguishing between regular and chaotic behavior. Frozen-N orbits corresponding in the continuum limit to both regular and chaotic characteristics have large positive chi even though, for large N, the "regular" frozen-N orbits closely resemble regular characteristics in the smooth potential. Alternatively, viewed macroscopically, both regular and "chaotic" frozen-N orbits diverge as a power law in time from smooth orbits with the same initial condition. However, convergence towards the continuum limit is much slower for chaotic orbits. For regular orbits, the time scale associated with this divergence t(G) approximately N(1/2)t(D), with t(D) a characteristic dynamical, or crossing, time; for chaotic orbits t(G) approximately (ln N)t(D). For N>10(3)-10(4), clear distinctions exist between the phase mixing of initially localized ensembles, which, in the continuum limit, exhibit regular versus chaotic behavior. Regular ensembles evolved in a frozen-N density distribution diverge as a power law in time, albeit more rapidly than ensembles evolved in the smooth distribution. Chaotic ensembles diverge in a fashion that is roughly exponential, albeit at a larger rate than that associated with the exponential divergence of the same ensemble evolved in smooth rho. For both regular and chaotic ensembles, finite-N effects are well mimicked, both qualitatively and quantitatively, by energy-conserving white noise with amplitude eta proportional, variant 1/N. This suggests strongly that earlier investigations of the effects of low amplitude noise on phase space transport in smooth potentials are directly relevant to real physical systems.

Phase-space transport of stochastic chaos in population dynamics of virus spread.

A general way to classify stochastic chaos is presented and applied to population dynamics models. A stochastic dynamical theory is used to develop an algorithmic tool to measure the transport across basin boundaries and predict the most probable regions of transport created by noise. The results of this tool are illustrated on a model of virus spread in a large population, where transport regions reveal how noise completes the necessary manifold intersections for the creation of emerging stochastic chaos.

Phase-space transport in cuspy triaxial potentials: can they be used to construct self-consistent equilibria?

This paper focuses on the statistical properties of chaotic orbit ensembles evolved in triaxial generalizations of the Dehnen potential which have been proposed recently to model realistic ellipticals that have a strong density cusp and manifest significant deviations from axisymmetry. Allowance is made for a possible supermassive black hole, as well as low-amplitude friction, noise, and periodic driving which can mimic irregularities associated with discreteness effects and/or an external environment. The chaos exhibited by these potentials is quantified by determining (1) how the relative number of chaotic orbits depends on the steepness of the cusp, as probed by γ, the power-law exponent with which density diverges, and MBH, the black hole mass, (2) how the size of the largest Lyapunov exponent varies with γ and MBH, and (3) the extent to which Arnold webs significantly impede phase-space transport, both with and without perturbations. The most important conclusions dynamically are (1) that, in the absence of irregularities, chaotic orbits tend to be extremely ‘sticky’, so that different pieces of the same chaotic orbit can behave very differently for times ∼10000tD or more, but (2) that even very low-amplitude perturbations can prove efficient in erasing many — albeit not all — of these differences. The implications of these facts are discussed both for the structure and evolution of real galaxies and for the possibility of constructing approximate near-equilibrium models using Schwarzschild's method. For example, when trying to use Schwarzschild's method to construct model galaxies containing significant numbers of chaotic orbits, it seems advantageous to build libraries with chaotic orbits evolved in the presence of low-amplitude friction and noise, since such noisy orbits are more likely to represent reasonable approximations to time-independent building blocks. Much of the observed qualitative behaviour can be reproduced with a toy potential given as the sum of an anisotropic harmonic oscillator and a spherical Plummer potential, which suggests that the results may be generic.

Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model

We study a model derived by Fei et al. [Phys. Rev. A 45 (1992) 6019] of a kink solution to the sine-Gordon equation interacting with an impurity mode. The model is a two degree of freedom Hamiltonian system. We investigate this model using the tools of dynamical systems, and show that it exhibits a variety of interesting behaviors including transverse heteroclinic orbits to degenerate equilibria at infinity, chaotic dynamics, and an extremely complex and delicate structure describing the interaction of the kink with the defect. We interpret this in terms of phase space transport theory.

Phase space transport in a map of asteroid motion

We have studied the chaotic transport in a nonlinear model directly applicable to asteroid motion. An exponential and an algebraic diffusion law are observed in different regions of the phase space. We have also investigated the effects of small perturbations and found they can not only accelerate but also decelerate the transport.

Noise-induced phase space transport in two-dimensional Hamiltonian systems.

First passage time experiments were used to explore the effects of low amplitude noise as a source of accelerated phase space diffusion in two-dimensional Hamiltonian systems, and these effects were then compared with the effects of periodic driving. The objective was to quantify and understand the manner in which "sticky" chaotic orbits that, in the absence of perturbations, are confined near regular islands for very long times, can become "unstuck" much more quickly when subjected to even very weak perturbations. For both noise and periodic driving, the typical escape time scales logarithmically with the amplitude of the perturbation. For white noise, the details seem unimportant: Additive and multiplicative noise typically have very similar effects, and the presence or absence of a friction related to the noise by a fluctuation-dissipation theorem is also largely irrelevant. Allowing for colored noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that there is little power at frequencies comparable to the natural frequencies of the unperturbed orbit. Similarly, periodic driving is relatively inefficient when the driving frequency is not comparable to these natural frequencies. This suggests that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. The logarithmic dependence of the escape time on amplitude reflects the fact that the time required for perturbed and unperturbed orbits to diverge a given distance scales logarithmically in the amplitude of the perturbation.

Chaotic mixing in noisy Hamiltonian systems

This paper summarizes an investigation of the effects of low-amplitude noise and periodic driving on phase-space transport in three-dimensional Hamiltonian systems, a problem directly applicable to systems like galaxies, where such perturbations reflect internal irregularities and/or a surrounding environment. A new diagnostic tool is exploited to quantify the extent to which, over long times, different segments of the same chaotic orbit evolved in the absence of such perturbations can exhibit very different amounts of chaos. First-passage-time experiments are used to study how small perturbations of an individual orbit can dramatically accelerate phase-space transport, allowing ‘sticky’ chaotic orbits trapped near regular islands to become unstuck on surprisingly short time-scales. The effects of small perturbations are also studied in the context of orbit ensembles with the aim of understanding how such irregularities can increase the efficacy of chaotic mixing. For both noise and periodic driving, the effect of the perturbation scales roughly logarithmically in amplitude. For white noise, the details are unimportant: additive and multiplicative noise tend to have similar effects and the presence or absence of friction related to the noise by a fluctuation—dissipation theorem is largely irrelevant. Allowing for coloured noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that there is little power at frequencies comparable with the natural frequencies of the unperturbed orbit. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. Potential implications for galaxies are discussed.

Phase space transport in noisy Hamiltonian systems.

This paper analyzes the effect of low amplitude friction and noise in accelerating phase space transport in time-independent Hamiltonian systems that exhibit global stochasticity. Numerical experiments reveal that even very weak non-Hamiltonian perturbations can dramatically increase the rate at which an ensemble of orbits penetrates obstructions like cantori or Arnold webs, thus accelerating the approach toward an invariant measure, i.e., a microcanonical population of the accessible phase space region. An investigation of first passage times through cantori leads to three conclusions, namely: (i) that, at least for white noise, the detailed form of the perturbation is unimportant, (ii) that the presence or absence of friction is largely irrelevant, and (iii) that, overall, the amplitude of the response to weak noise scales logarithmically in the amplitude of the noise.

Orbital complexity, short-time Lyapunov exponents, and phase space transport in time-independent Hamiltonian systems.

This paper compares two alternative characterizations of chaotic orbit segments, one based on the complexity of their Fourier spectra, as probed by the number of frequencies n(k) required to capture a fixed fraction k of the total power, and the other based on the computed values of short-time Lyapunov exponents chi. An analysis of orbit ensembles evolved in several different two- and three-dimensional potentials reveals that there is a strong, roughly linear correlation between these alternative characterizations, and that computed distributions of complexities, N[n(k)] and short-time chi, N[chi] often assume similar shapes. This corroborates the intuition that chaotic segments which are especially unstable should have Fourier spectra with particularly broad-band power. It follows that orbital complexities can be used as probes of phase space transport and other related phenomena in the same manner as can short-time Lyapunov exponents.

Nonlinear dynamics in periodic phase space

Regular and chaotic dynamics of a system with periodic phase space perturbed by an alternating external field is considered. It is relevant for the electronic motion in two dimensions in the presence of a uniform magnetic field and a perpendicular alternating electric field. The phase space is divided into cells embedded in a chaotic mesh. Bifurcations of resonances within the cells are studied. Transport takes place in the chaotic mesh. It is analyzed in the framework of the separatrix map. Accelerator modes are found for some values of parameters and their bifurcations are investigated. Their effects on transport in phase space are discussed.

Multiterm solution of the reactive space–time-dependent Boltzmann equation

The effect of spatial gradients, whether associated with surfaces or otherwise, on charged particle phase-space distribution functions and transport properties is a long-standing problem in the kinetic theory of gases. This article first discusses the necessary ingredients for an accurate kinetic theory analysis of the space–time behavior of electrons in gases and compares the different approaches currently in use. We then focus upon the electrons in a radio-frequency discharge away from the walls, where gradients are weak, and point out that the fundamental kinetic theory of even the boundary-free problem still warrants more careful attention than has been given to it in much of the contemporary literature. We highlighted the importance of careful analysis by presenting results for the first comprehensive “multiterm” solution of Boltzmann’s equation furnishing the complete set of hydrodynamic transport coefficients for electrons in an ac field undergoing ionization and attachment.