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

Abstract

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 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. PACS number(s): 05.60.+w, 51.10.+y, 05.40.+j I. MOTIVATIONIt is well known that a complex phase space containing large measures of both regular and chaotic orbits is often partitioned by such partial obstructions as cantori [1] or Arnold webs [2] which, although not serving as absolute barriers, can significantly impede the motion of a chaotic orbit through a connected phase space region. Indeed, the fact that, in two-dimensional Hamiltonian systems, chaotic orbits can be "stuck" near regular islands for very long times was discovered empirically [3] long before the existence of cantori was proven [4].It has also been long known that low amplitude stochastic perturbations can accelerate Hamiltonian phase space transport by enabling orbits to traverse these partial barriers. This was, e.g., explored by Lieberman and Lichtenberg [5], who investigated how motion described by the simplified Ulam version of the Fermi acceleration map [6] is impacted by random perturbations, allowing for the modified equations [7] where the "noise" ∆ψ corresponds to a random phase shift uniformly sampling an interval [−ϕ, +ϕ].That stochastic perturbations can have such effects on Hamiltonian systems is important in understanding the limitations of simple models of real systems. In the absence of all "perturbations" and any other irregularities, the chaotic phase space associated with so...

Collective Modes and Colored Noise as Beam-Halo Amplifiers

“…The special case of white noise corresponds to the limit t c $IWHU JHQHUDWLQJ D FRORUHG-noise signal using, e.g., the technique developed in Ref. [6], the next step is to compute |A| 〈|δ |〉 which then constitutes the measure of noise strength. In a beam each particle will have its own distinct initial conditions and thus experience a manifestation of the noise differing from that seen by each of the other particles.…”

Phase Mixing in Unperturbed and Perturbed Hamiltonian Systems

“…This seems reasonable when considering noise as a source to 'fuzz' out trajectories within a single, easily accessible phase space region; and an analysis of diffusion through cantori and/or along an Arnold web [15] suggests that, at least for these sorts of entropy barriers, this is also the case. By continually wiggling the orbits, noise helps them 'hunt' for holes in cantori.…”

“…Motivated by the visual impression that, at least for chaotic ensembles, the orbits usually approach a nearly time-independent equilibrium state, the last q snapshots of a sequence of p snapshots can be used to derive a near-invariant distribution 15) and convergence towards f niv can then be probed by computing discretised L p norms…”

“…These effects are easily understood qualitatively, given the expectation that noise impacts individual orbits via a resonant coupling [15]. Most of the power in an orbit, either regular or chaotic, is concentrated at frequencies ω ∼ 2π/t cr , with t cr the natural orbital time scale, and an efficient coupling requires noise with substantial power at comparable frequencies.…”

“…And similarly, it has been recognised that even very low amplitude periodic driving can have important implications for phase space transport in systems like plasmas [14]. Recent work on continuous Hamiltonian systems [15,16] has served to quantify this effect, demonstrating that weak noise and/or periodic driving can significantly impact phase space transport on time scales much shorter than the relaxation time t R on which they significantly change the values of quantities like energy, which would be conserved absolutely in the absence of perturbations. All this would suggest that low amplitude perturbations can significantly impact the phase mixing of chaotic orbits, a possibility that has direct physical implications for the destabilisation of various structures in a galaxy [17] or the defocusing of an accelerator beam [18].…”