Interpreting data on solar cosmic ray fluxes via the fractional derivative method

Abstract

Solar cosmic ray propagation through the interplanetary magnetic field is considered as a random process of particles traveling along magnetic lines at a finite velocity of free motion and with a free path dis tributed according to an inverse power law. The propagator is presented as a sum of direct (nonscattered) flux (singular part of solution) and multiple scattered flux (regular part). In the long time asymptotic, the regular part is described by an equation with a fractional order derivative. Using analytical expressions for the prop agator, we numerically calculate fluxes of energetic particles accelerated by shock waves generated by solar flares. The presented model is in better agreement with Ulysses and Voyager 2 data than the Perri-Zimbardo model and may therefore be recommended for use in interpreting the results of further experiments.

Fractional Parker equation for the transport of cosmic rays: steady-state solutions

“…Thus, the transport properties strongly depend on the turbulence features and the adopted numerical representation. The implications of superdiffusion for particle acceleration and transport at shocks have recently been considered by Perri & Zimbardo (2012b); Zimbardo & Perri (2013); Lazarian & Yan (2014), and these works have attracted considerable theoretical interest (Uchaikin et al 2015;Rocca et al 2015Rocca et al , 2016Saenko 2016). Nondiffusive processes can be described by several tools, including Lévy walks, Lévy flights, and fractional derivatives (e.g., Perrone et al 2013).…”

Numerical Solution to Anomalous Diffusion Equations for Levy Walks

“…This solution describes the spatial distribution of particles with random walks of a particle at a constant velocity, with an arbitrary distribution of paths. This solution is not new and was obtained earlier (see, for example, [40,42,[48][49][50][51][52]). In this paper, we will consider the asymptotic solution to this equation in the case when the distribution of paths has asymptotics of the form:…”