Survey on the role of accelerator modes for anomalous diffusion: The case of the standard map

Abstract

We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis on understanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and their immediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is still much affected ("dragged") by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic approach rests upon the following applications: the GALI method to detect the regular and chaotic regions and thus to describe in detail the structure of the phase space, the description of the momentum distribution in terms of the Lévy stable distributions, and the numerical calculation of the diffusion exponent and of the corresponding diffusion constant. We use this approach to analyze in detail and systematically the standard map at all values of the kick parameter K, up to K = 70. All complex features of the anomalous diffusion are well understood in terms of the role of the accelerator modes, mainly of period 1 at large K ≥ 2π, but also of higher periods (2,3,4,...) at smaller values of K ≤ 2π.

Diffusion in randomly perturbed dissipative dynamics

“…We use f 0 = 4.0, which results in multiple attractors when ν = 0 [1]. At this parameter value and ν = 0 the standard map displays superdiffusion, due to the existence of accelerator modes [24]. If we evolve our system under the presence of random noise beyond a certain amplitude ξ ≥ ξ 0 the attracting sets lose their stability, as discussed in the section Escape.…”

Diffusion phenomena in a mixed phase space

“…around a KAM island may lead to very complicated behavior. The stability structures influence directly the transport properties of chaotic orbits 4 , often generating so-called anomalous diffusion 5,6 .…”

Normal and Anomalous Diffusion in Soft Lorentz Gases

“…We now focus on the pronounced irregular fine structure of D(w) in the soft system, which is in sharp contrast to the diffusion coefficient of the hard disk model. Irregular diffusion coefficients have been re- ported for parameter-dependent deterministic diffusion in much simpler chaotic dynamical systems, such as onedimensional maps [50][51][52][53], the standard map [54,55] and particle billiards [49,[56][57][58]. To our knowledge this is the first time that a diffusive fine structure has been unambiguously revealed in quite a realistic soft Hamiltonian system.…”

“…Analogous results hold for varying the energy E as a parameter [68], which experimentally corresponds to changing the temperature of the system. Note that in superdiffusive parameter regions ergodicity is broken, hence for single particle experiments there will be a dependence on initial conditions [54,55]. In real systems with thermal noise we expect these superdiffusive regions, ergodicity breaking and irregularities on fine scales to disappear, however, larger irregularities should persist under noise [56,71].…”