arXiv subjects

Henry E. Kandrup

Publications and source records attributed to Henry E. Kandrup.

At least 37 records · Page 2Linked to original sources

arXiv2000

Energy relaxation in galaxies induced by an external environment and/or incoherent internal pulsations

This paper explores the phenomenon of energy relaxation for stars in a galaxy embedded in a high density environment that is subjected continually to perturbations reflecting the presence of other nearby galaxies and/or incoherent internal pulsations. The analysis is similar to earlier analyses of energy relaxation induced by binary encounters between nearby stars and between stars and giant molecular clouds in that the perturbations are idealised as a sum of near-random events which can be modeled as diffusion and dynamical friction. However, the analysis differs in one important respect: because the time scale associated with these perturbations need not be short compared with the characteristic dynamical time t_D for stars in the original galaxy, the diffusion process cannot be modeled as resulting from a sequence of instantaneous kicks, i.e., white noise. Instead, the diffusion is modeled as resulting from random kicks of finite duration, i.e., coloured noise characterised by a nonzero autocorrelation time t_c. A detailed analysis of coloured noise generated by sampling an Ornstein-Uhlenbeck process leads to a simpling scaling in terms of t_c and an effective diffusion constant D. Interpreting D and t_c following early work by Chandrasekhar (1941) (the `nearest neighbour approximation') implies that, for realistic choices of parameter values, energy relaxation associated with an external environment and/or internal pulsations could be important on times short compared with the age of the Universe.

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arXiv2000

Chaotic Mixing in Galactic Dynamics

This talk summarises what is currently understood about the phenomenon that has come to be known as {\it chaotic mixing}. The first part presents a concise statement as to what chaotic mixing actually is, and then explains why it should be important in galactic dynamics. The second discusses in detail what is currently known, describing the manifestations of chaotic mixing both for flows in time-independent Hamiltonian systems and for systems subjected to comparatively weak time-dependent perturbations. The third part discusses what one might expect if one were to allow for a strongly time-dependent potential, including possible implications for violent relaxation.

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arXiv2000

Dynamics of Cuspy Triaxial Galaxies with a Supermassive Black Hole

This talk provides a progress report on an extended collaboration which has aimed to address two basic questions, namely: Should one expect to see cuspy, triaxial galaxies in nature? And can one construct realistic cuspy, triaxial equilibrium models that are robust? Three technical results are described: (1) Unperturbed chaotic orbits in cuspy triaxial potentials can be extraordinarily sticky, much more so than orbits in many other three-dimensional potentials. (2) Even very weak perturbations can be important by drastically reducing, albeit not completely eliminating, this stickiness. (3) A simple toy model facilitates a simple understanding of why black holes and cusps can serve as an effective source of chaos. These results suggest that, when constructing models of galaxies using Schwarzschild's method or any analogue thereof, astronomers would be well advised to use orbital building blocks that have been perturbed by `noise' or other weak irregularities, since such building blocks are likely to be more nearly time-independent than orbits evolved in the absence of all perturbations.

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arXiv2000

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

(Abridged) This paper studies chaotic orbit ensembles evolved in triaxial generalisations of the Dehnen potential which have been proposed to model ellipticals with a strong density cusp that 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 degree of chaos is quantified by determining how (1) the relative number of chaotic orbits and (2) the size of the largest Lyapunov exponent depend on the steepness of the cusp and the black hole mass, and (3) the extent to which Arnold webs significantly impede phase space transport, both with and without perturbations. 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 10000 dynamical times or longer, but even very low amplitude perturbations can prove efficient in erasing many -- albeit not all -- these differences. The implications thereof 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. 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.

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arXiv1999

Chaotic mixing in noisy Hamiltonian systems

This paper summarises an investigation of the effects of low amplitude noise and periodic driving on phase space transport in 3-D Hamiltonian systems, a problem directly applicable to systems like galaxies, where such perturbations reflect internal irregularities and.or a surrounding environment. A new diagnsotic tool is exploited to quantify how, over long times, different segments of the same chaotic orbit 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 suprisingly short time scales. 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 in amplitude. For white noise, the details are unimportant: additive and multiplicative noise tend to have similar effects and the presence or absence of a 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 t here is little power at frequencies comparable to 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.

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arXiv1999

Diffusion and scaling in escapes from two-degree-of-freedom Hamiltonian systems

This paper summarises an investigation of the statistical properties of orbits escaping from three different two-degree-of-freedom Hamiltonian systems which exhibit global stochasticity. Each H=H_{0}+eH', with H_{0} integrable and eH' a nonintegrable correction, not necessarily small. For e below a critical e_{0} escapes are impossible energetically. For somewhat higher values, escape is allowed energetically but many orbits never escape and the escape probability P for a generic orbit ensemble decays exponentially. At or near a critical e_{1}>e_{0}, there is an abrupt qualitative change in behaviour. Above e_{1}, P typically exhibits (1) a rapid evolution towards a nonzero P_{0}(e) followed by (2) a much slower subsequent decay towards zero which, in at least one case, is well fit by a power law P=const x t^{m}, with m=0.35-0.40. In all three cases, P_{0} and the time T required to converge towards P_{0} scales in e-e_{1}, i.e., P_{0}=const x (e-e_{1})^{a} and T=const x (e-e_{1})^{b}, and T also scales in the size r of the region sampled for initial conditions, i.e., T=const x r^{-d}. To within statistical uncertainties, the best fit values of the critical exponents are the same for all three potentials, namely: a=0.5, b=0.4, and d=0.1, and satisfy a-b-d=0. The transitional behaviour observed near e_{1} is attributed to the breakdown of some especially significant KAM tori or cantori. The power law behaviour at late times is interpreted as reflecting intrinsic diffusion of chaotic orbits through cantori surrounding islands of regular orbits.

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arXiv1999

Chaos in Hamiltonians with a Cosmological Time Dependence

This paper summarises a numerical investigation of how the usual manifestations of chaos and regularity for flows in time-independent Hamiltonians can be alterred by a systematic time-dependence of the form arising naturally in an expanding Universe. If the time-dependence is not too strong, the observed behaviour can be understood in an adiabatic approximation. One still infers sharp distinctions between regular and chaotic behaviour, even though ``regular'' does not mean ``periodic'' and ``chaotic'' will not in general imply strictly exponential sensitivity towards small changes in initial conditions. However, these distinctions are no longer absolute, as it is possible for a single orbit to change from regular to chaotic and/or visa versa. If the time-dependence becomes too strong, the distinction between regular and chaotic can disappear so that no sensitive dependence on initial conditions is manifest.

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arXiv1999

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 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 strongly 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.

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arXiv1998

Chaos in cosmological Hamiltonians

This paper summarises a numerical investigation which aimed to identify and characterise regular and chaotic behaviour in time-dependent Hamiltonians H(r,p,t) = p^2/2 + U(r,t), with U=R(t)V(r) or U=V[R(t)r], where V(r) is a polynomial in x, y, and/or z, and R = const * t^p is a time-dependent scale factor. When p is not too negative, one can distinguish between regular and chaotic behaviour by determining whether an orbit segment exhibits a sensitive dependence on initial conditions. However, chaotic segments in these potentials differ from chaotic segments in time-independent potentials in that a small initial perturbation will usually exhibit a sub- or super-exponential growth in time. Although not periodic, regular segments typically exhibit simpler shapes, topologies, and Fourier spectra than do chaotic segments. This distinction between regular and chaotic behaviour is not absolute since a single orbit segment can seemingly change from regular to chaotic and visa versa. All these observed phenomena can be understood in terms of a simple theoretical model.

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arXiv1998

Collisionless Relaxation of Stellar Systems

The objective of the work summarised here has been to exploit and extend ideas from plasma physics and accelerator dynamics to formulate a unified description of collisionless relaxation that views violent relaxation, Landau damping, and phase mixing as (manifestations of) a single phenomenon. This approach embraces the fact that the collisionless Boltzmann equation (CBE), the basic object of the theory, is an infinite-dimensional Hamiltonian system, with the distribution function f playing the role of the fundamental dynamical variable, and that, interpreted appropriately, an evolution described by the {\it CBE} is no different fundamentally from an evolution described by any other Hamiltonian system. Equilibrium solutions correspond to extremal points of the Hamiltonian subject to the constraints associated with Liouville's Theorem. Stable equilibria correspond to energy minima. The evolution of a system out of equilibrium involves (in general nonlinear) phase space oscillations which may -- or may not -- interfere destructively so as to damp away.

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arXiv1998

Phase mixing in time-independent Hamiltonian systems

Everything you ever wanted to know about what has come to be known as ``chaotic mixing:'' This paper describes the evolution of localised ensembles of initial conditions in 2- and 3-D time-independent potentials which admit both regular and chaotic orbits. The coarse-grained approach towards an invariant, or near-invariant, distribution was probed by tracking (1) phase space moments through order 4 and (2) binned reduced distributions f(Z_a,Z_b,t) for a,b=x,y,z,p_x,p_y,p_z, computed at fixed time intervals. For ``unconfined'' chaotic orbits in 2-D systems not stuck near islands by cantori, the moments evolve exponentially: Quantities like the dispersion in p_x, which start small and eventually asymptote towards a larger value, initially grow exponentially in time at a rate comparable to the largest short time Lyapunov exponent. Quantities like | |, that can start large but eventually asymptote towards zero, decrease exponentially. With respect to a discrete L^p norm, reduced distributions f(t) generated from successive decay exponentially towards a near-invariant f_{niv}, although a plot of Df(t)=||f(t)-f_{niv}|| can exhibit considerable structure. Regular ensembles behave very differently, both moments and Df evolving in a fashion better represented by a power law time dependence. ``Confined'' chaotic orbits, initially stuck near regular islands because of cantori, exhibit an intermediate behaviour. The behaviour of ensembles evolved in 3-D potentials is qualitatively similar, except that, in this case, it is relatively likely to find one direction in configuration space which is ``less chaotic'' than the other two, so that quantities like L_{ab} depend more sensitively on which phase space variables one tracks.

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arXiv1998

Phase Space Transport in Noisy Hamiltonian Systems

This paper analyses 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 towards an invariant measure, i.e., a near-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.

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arXiv1997

Invariant distributions and collisionless equilibria

This paper discusses the possibility of constructing time-independent solutions to the collisionless Boltzmann equation which depend on quantities other than global isolating integrals such as energy and angular momentum. The key point is that, at least in principle, a self-consistent equilibrium can be constructed from any set of time-independent phase space building blocks which, when combined, generate the mass distribution associated with an assumed time-independent potential. This approach provides a way to justify Schwarzschild's (1979) method for the numerical construction of self-consistent equilibria with arbitrary time-independent potentials, generalising thereby an approach developed by Vandervoort (1984) for integrable potentials. As a simple illustration, Schwarzschild's method is reformulated to allow for a straightforward computation of equilibria which depend only on one or two global integrals and no other quantities, as is reasonable, e.g., for modeling axisymmetric configurations characterised by a nonintegrable potential.

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arXiv1997

Violent Relaxation, Phase Mixing, and Gravitational Landau Damping

This paper proposes a geometric interpretation of flows generated by the collisionless Boltzmann equation (CBE), focusing on the coarse-grained approach towards equilibrium. The CBE is a noncanonical Hamiltonian system with the distribution function f the fundamental dynamical variable, the mean field energy H[f] playing the role of the Hamiltonian and the natural arena of physics being the infinite-dimensional phase space of distribution functions. Every time-independent equilibrium f_0 is an energy extremal with respect to all perturbations that preserve the constraints associated with Liouville's Theorem, local energy minima corresponding to linearly stable equilibria. If an initial f(t=0) is sufficiently close to some linearly stable lower energy f_0, its evolution involves linear phase space oscillations about f_0 which, in many cases, would be expected to exhibit linear Landau damping. If f(t=0) is far from any stable extremal, the flow will be more complicated but, in general, one would anticipate that the evolution involves nonlinear oscillations about some lower energy f_0. In this picture, the coarse-grained approach towards equilibrium usually termed violent relaxation is interpreted as nonlinear Landau damping. The evolution of a generic initial f(t=0) involves a coherent initial excitation, not necessarily small, being converted into incoherent motion associated with nonlinear oscillations about some equilibrium f_0 which, in general, will exhibit destructive interference.

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arXiv1997

Geometric Interpretation of Chaos in Two-Dimensional Hamiltonian Systems

Time-independent Hamiltonian flows are viewed as geodesic flows in a curved manifold, so that the onset of chaos hinges on properties of the curvature two-form entering into the Jacobi equation. Attention focuses on ensembles of orbit segments evolved in 2-D potentials, examining how various orbital properties correlate with the mean value and dispersion, and k, of the trace K of the curvature. Unlike most analyses, which have attributed chaos to negative curvature, this work exploits the fact that geodesics can be chaotic even if K is everywhere positive, chaos arising as a parameteric instability triggered by regular variations in K along the orbit. For ensembles of fixed energy, with both regular and chaotic segments, simple patterns connect the values of and k for different segments, both with each other and with the short time Lyapunov exponent X. Often, but not always, there is a near one-to- one correlation between and k, a plot of these quantities approximating a simple curve. X varies smoothly along this curve, chaotic segments located furthest from the regular regions tending systematically to have the largest X's. For regular orbits, and k also vary smoothly with ``distance'' from the chaotic phase space regions, as probed, e.g., by the location of the initial condition on a surface of section. Many of these observed properties can be understood qualitatively in terms of a one-dimensional Mathieu equation.

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arXiv1997

Collisionless Relaxation in Galactic Dynamics and the Evolution of Long Range Order

This talk provides a critical assessment of collisionless galactic dynamics, focusing on the interpretation and limitations of the collisionless Boltzmann equation and the physical mechanisms associated with collisionless relaxation. Numerical and theoretical arguments are presented to motivate the idea that the evolution of a system far from equilibrium should be interpreted as involving nonlinear gravitational Landau damping, which implies a greater overall coherence and remembrance of initial conditions than is implicit in the conventional theory of violent relaxation.

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arXiv1994

Chaos, Regularity, and Noise in Self-Gravitating Systems

This paper summarises a number of new, potentially significant, results, obtained recently by the author and his collaborators, which impact on various issues related to the gravitational N-body problem, both Newtonianly and in the context of general relativity. Topics addressed include: (1) direct N-body simulations and their interpretation, with reference to the observed exponential instability towards small changes in initial conditions and the phenomenon of ``nonviolent relaxation;'' (2) the Hamiltonian structure of the collisionless Boltzmann equation of general relativity, i.e., the Vlasov-Einstein system; (3) ``transient ensemble dynamics,'' i.e.,,the short time statistical characterisation of collections of orbits in nonintegrable mean field potentials; and (4) the structural stability of the smooth potential approximation typically used in galactic dynamics.

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arXiv1994

Short times characterisations of stochasticity in nonintegrable galactic potentials

This paper proposes an alternative characterisation of the degree of stochasticity exhibited by orbits in a fixed galactic potential. This differs from earlier work involving Liapounov exponents by focusing on the statistical properties of ensembles of trajectories, rather than individual orbits, and by restricting attention to the properties of these ensembles over time scales shorter than the age of the Universe, t_H. For many potentials, generic ensembles of initial conditions corresponding to stochastic orbits exhibit a rapid evolution towards an invariant measure $Γ$, the natural unit to consider if one is interested in self-consistent equilibria. The basic idea is to compute short time Liapounov exponents $χ(Dt)$ over time intervals $Dt$ for orbits in an ensemble that samples $Γ$, and to analyse the overall distribution of these $χ$'s. One finds that time averages and ensemble averages coincide, so that the form of the distribution of short time $χ(Dt)$'s for such an ensemble is actually encoded in the calculation of $χ(t)$ for a single orbit over long times $t>>t_{H}$. The distribution of short time $χ$'s is analysed as a function of the energy E of orbits in the ensemble and the length of the short time sampling interval ${\D}t$. For relatively high energies, the distribution is essentially Gaussian, the dispersion decreasing with time as $t^{-p}$, with an exponent 0<p<1/2 that depends on the energy E. (to appear in Astronomy and Astrophysics)

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