Henry Kandrup's Ideas About Relaxation of Stellar Systems

Henry Kandrup wrote prolifically on the problem of relaxation of stellar systems. His picture of relaxation was significantly more refined than the standard description in terms of phase mixing and violent relaxation. In this article, I summarize Henry's work in this and related areas.Comment: 11 pages. To appear in "Nonlinear Dynamics in Astronomy and Physics, A Workshop Dedicated to the Memory of Professor Henry E. Kandrup", ed. J. R. Buchler, S. T. Gottesman and M. E. Maho

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.Comment: 17 pages, 17 figures -- revised and extended manuscript to appear in Monthly Notices of the Royal Astronomical Societ

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.Comment: 16 pages LaTeX, including 5 figures, no macros require

Semi-Analytic Estimates of Lyapunov Exponents in Lower-Dimensional Systems

Recent work has shown that statistical arguments, seemingly well-justified in higher dimensions, can also be used to derive reasonable, albeit less accurate, estimates of the largest Lyapunov exponent ${\chi}$ in lower-dimensional Hamiltonian systems. This letter explores the detailed assumptions incorporated into these arguments. The predicted values of ${\chi}$ are insensitive to most of these details, which can in any event be relaxed straightforwardly, but {\em can} depend sensitively on the nongeneric form of the auto-correlation function characterising the time-dependence of an orbit. This dependence on dynamics implies a fundamental limitation to the application of thermodynamic arguments to such lower-dimensional systems.Comment: 6 pages, 3 PostScript figure

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.Comment: 16 pages plus 9 figures, LaTeX, no macros required to appear in Physical Review

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.Comment: 13 pages, 3 Postscript figures, latex, no macors. Annals of the New York Academy of Sciences, in pres

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.Comment: 14 pages, LaTeX, no macro

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.Comment: 18 pages, including 19 figures; Accepted for publication by MNRAS; higher quality figures available from http://www.astro.ufl.edu/~siopis/papers

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.Comment: 19 pages + 11 Postscript figures, latex, no macros. Monthly Notices of the Royal Astronomical Society, in pres