Thirty years of turnstiles and transport

Abstract

To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the action of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions. The problem of transport in dynamical systems is to quantify the motion of collections of trajectories between regions of phase space with physical significance, for example, to determine chemical reaction rates, mixing rates in a fluid, or particle confinement times in an accelerator or fusion plasma device. For Hamiltonian systems with a phase space containing regular (elliptic islands and tori) and irregular (chaotic) components, transport is impeded by partial barriers bounding resonance zones or by remnants of invariant tori. In the former case the barriers are formed from the broken separatrices of an unstable periodic orbit, and in the latter, they are formed from similar stable and unstable manifolds of a remnant torus-a cantorus. Thirty years ago, MacKay, Meiss, and Percival discovered that cantori form robust partial barriers for area-preserving maps: in any region of phase space the most resistant barriers are the cantori. In this review, I discuss this work and survey subsequent developments. While much has been done, there are still many open questions.

Transition from normal to ballistic diffusion in a one-dimensional impact system

“…A natural observable allows the study of the statistical properties of the transport, in particular ρ(n), the probability (given a suitable distribution of initial conditions) that an orbit does not escape through a hole until a time n. Here the hole is defined as a predefined subset of the phase space. The most important aspect of this analysis is that the escape rate is very sensitive to the system dynamics [55,56]. For strongly chaotic systems the decay is typically exponential [57][58][59], while for systems that present a mixed phase space the decay can be slower, presenting a mix of exponential with a power law [9], or even stretched exponential decay [60].…”

“…The survival probability, described in terms of an escape formalism [54][55][56], is then obtained by the integration of this escape frequency histogram, as…”

Recurrence-time statistics in non-Hamiltonian volume-preserving maps and flows

“…Is there an universal decay exponent of recurrences? Similar questions were proposed recently [31], where the author summarizes both, the state of the art in the theory of transport for conservative dynamical systems, based on the last thirty years of investigations, and tries to point out some open problems that could be addressed next.…”

Pivotal issues on relativistic electrons in ITER

“…Even a narrow annulus of magnetic surfaces will prevent the escape of energetic electrons, which have a small gyroradius compared to system size in ITER. Boozer and Punjabi [12] have used the Hamiltonian mechanics concept of turnstiles [28] to show that relativistic electrons confined in a large region of stochastic magnetic field lines by a narrow annulus of magnetic surfaces are prone to fast, extremely concentrated depositions on the surrounding walls.…”