This paper illustrates the possibility, in simple loaded string models, of trapping most of the system energy in a single degree of freedom for very long times, demonstrating in particular that the robustness of the trapping is enhanced by increasing the connectance of the system, that is, the extent to which many degrees of freedom are coupled directly by the interaction Hamiltonian and/or the strength of the couplings.

Both galaxies and charged particle beams can exhibit collisionless evolution on surprisingly short time scales. This can be attributed to the dynamics of chaotic orbits. The chaos is often triggered by resonance caused by time dependence in the bulk potential, which acts almost identically for attractive gravitational forces and repulsive electrostatic forces. The similarity suggests that many physical processes at work in galaxies, although inaccessible to direct controlled experiments, can be tested indirectly via controlled experiments with charged particle beams, such as those envisioned for the University of Maryland electron ring currently nearing completion.

This paper summarises a numerical investigation of phase mixing in time-independent Hamiltonian systems that admit a coexistence of regular and chaotic phase space regions, allowing also for low amplitude perturbations idealised as periodic driving, friction, and/or white and coloured noise. The evolution of initially localised ensembles of orbits was probed through lower order moments and coarse-grained distribution functions. In the absence of time-dependent perturbations, regular ensembles disperse initially as a power law in time and only exhibit a coarse-grained approach towards an invariant equilibrium over comparatively long times. Chaotic ensembles generally diverge exponentially fast on a time scale related to a typical finite time Lyapunov exponent, but can exhibit complex behaviour if they are impacted by the effects of cantori or the Arnold web. Viewed over somewhat longer times, chaotic ensembles typical converge exponentially towards an invariant or near-invariant equilibrium. This, however, need not correspond to a true equilibrium, which may only be approached over very long time scales. Time-dependent perturbations can dramatically increase the efficiency of phase mixing, both by accelerating the approach towards a near-equilibrium and by facilitating diffusion through cantori or along the Arnold web so as to accelerate the approach towards a true equilibrium. The efficacy of such perturbations typically scales logarithmically in amplitude, but is comparatively insensitive to most other details, a conclusion which reinforces the interpretation that the perturbations act via a resonant coupling.

Both galaxies and charged particle beams can exhibit collective relaxation on surprisingly short time scales. This can be attributed to the effects of chaos, often triggered by resonances caused by time-dependences in the bulk potential, which act almost identically for attractive gravitational and repulsive electrostatic forces. These similarities suggest that many physical processes at work in galaxies, albeit not subject to direct controlled experiments, can be tested indirectly using facilities such as the University of Maryland Electron Ring (UMER) currently nearing completion.

Orbits in a three-dimensional potential subjected to periodic driving, V(x^i,t)=[1+m_0 sin(omega t) V_0(x^i), divide naturally into two types, regular and chaotic, between which transitions are seemingly impossible. The chaotic orbits divide in turn into two types, apparently separated by entropy barriers, namely `sticky' orbit segments, which are `locked' to the driving frequency and exhibit little systematic energy diffusion, and `wildly' chaotic segments, which are not so locked and can exhibit significant energy diffusion. Attention focuses on how the relative abundance of these different orbit types and the transition rate between sticky and wildly chaotic orbits depends on amplitude m_0, and on the extent to which these quantities can be altered by weak friction and/or noise and by pseudo-random variations in the driving frequency, idealized as an Ornstein-Uhlenbeck process with (in general) nonzero autocorrelation time t_c. When, in the absence of perturbations, there exist large measures of both regular and chaotic orbits, the primary effect of weak noise is to increase the relative measure of chaotic orbits. Alternatively, when almost all the orbits are already chaotic, noise serves primarily to accelerate transitions from sticky to wildly chaotic behavior. The presence or absence of friction is unimportant and the details of the noise seem largely immaterial. In particular, there is only a weak, roughly logarithmic dependence on amplitude, and additive and multiplicative noise typically have virtually identical effects. By contrast, allowing for high frequency `noisy' variations in omega tends to weaken effects of the driving, decreasing the relative measure of chaotic orbits and suppressing large scale energy diffusion.

Phase mixing of chaotic orbits exponentially distributes the orbits through their accessible phase space. This phenomenon, commonly called "chaotic mixing", stands in marked contrast to phase mixing of regular orbits which proceeds as a power law in time. It is inherently irreversible; hence, its associated e-folding time scale sets a condition on any process envisioned for emittance compensation. We numerically investigate phase mixing in the presence of space charge, distinguish between the evolution of regular and chaotic orbits, and discuss how phase mixing potentially influences macroscopic properties of high-brightness beams.

Cuspy triaxial potentials admit a large number of chaotic orbits, which moreover exhibit extreme "stickiness" that makes the process of chaotic mixing surprisingly inefficient. Environmental effects, modeled as noise and/or periodic driving, help accelerate phase space transport but probably not as much as in simpler potentials. This could mean that cuspy triaxial ellipticals cannot exist as time-independent systems.

This review summarises several different lines of argument suggesting that one should not expect cuspy nonaxisymmetric galaxies to exist as robust, long-lived collisionless equilibria, i.e., that such objects should not be idealised as time-independent solutions to the collisionless Boltzmann equation.

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.

This paper uses the assumptions of ergodicity and a microcanonical distribution to compute estimates of the largest Lyapunov exponents in lower-dimensional Hamiltonian systems. That the resulting estimates are in reasonable agreement with the actual values computed numerically corroborates the intuition that chaos in such systems can be understood as arising generically from a parametric instability and that this instability may be modeled by a stochastic-oscillator equation [cf. Casetti, Clementi, and Pettini, Phys. Rev. E 54, 5969 (1996)], linearized perturbations of a chaotic orbit satisfying a harmonic-oscillator equation with a randomly varying frequency.

This paper focuses on the statistical properties of chaotic orbit ensembles evolved in triaxial generalisations 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.