Lévy flights on a comb and the plasma staircase

Abstract

We formulate the problem of confined Lévy flight on a comb. The comb represents a sawtoothlike potential field V(x), with the asymmetric teeth favoring net transport in a preferred direction. The shape effect is modeled as a power-law dependence V(x)∝|Δx|^{n} within the sawtooth period, followed by an abrupt drop-off to zero, after which the initial power-law dependence is reset. It is found that the Lévy flights will be confined in the sense of generalized central limit theorem if (i) the spacing between the teeth is sufficiently broad, and (ii) n>4-μ, where μ is the fractal dimension of the flights. In particular, for the Cauchy flights (μ=1), n>3. The study is motivated by recent observations of localization-delocalization of transport avalanches in banded flows in the Tore Supra tokamak and is intended to devise a theory basis to explain the observed phenomenology.

Self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity

“…From a scientific perspective, the nonlinear dynamics of the plasma staircase occupies an interesting niche where microscale and mesoscale nonlinearities can appear on an equal footing [3,4]. From a practical perspective, the periodic dynamical patterning due to the plasma staircase offers a unique environment to control the avalanche activity by fine tuning the shape and the radial positions of the barriers [4,7]. These practical aspects are dictated by the understanding that the avalanche transport may have a deteriorating effect on the confinement properties of thermal plasma and charged fusion products [8,9], while significant losses could be detrimental.…”

Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes

“…In magnetised plasma experiments, a predator–prey system exists with avalanches (strong driver of transport) and zonal flows (sheared flows that decorrelate turbulent eddies reducing transport). It has been suggested that an Fractional Fokker–Planck Equation on a comb-like potential background can be applied where meso-scale transport events (avalanching) occurs in between regions of strong zonal flow activity (see Milovanov and Rasmussen [ 55 ]). This method is straightforward for applications in this setting; by assuming the used potential in between the zonal flow regions, it is suggested that the potential should be of degree 4 (or higher), as has been used here.…”

Subdiffusive Lévy flights in quantum nonlinear Schrödinger lattices with algebraic power nonlinearity

“…A derivation using transition probabilities in reciprocal space has been obtained in Refs. [50,51]. As is well known, the Riesz operator in Eq.…”

“…which corresponds to the elusive gray swans in the vocabulary of Ref. [51]. The gray swans [51] are a specific type of large-amplitude events [55] that occur at the border of localization-delocalization in complex systems.…”

“…[51]. The gray swans [51] are a specific type of large-amplitude events [55] that occur at the border of localization-delocalization in complex systems. Tuning µ to its lower limit at µ = 1 (i.e., s = 1/2), one also obtains χ(∆n) ∝ ∆n −2 .…”