Fractional derivatives on cosmic scales

Abstract

Almost since the very discovery of cosmic rays, calculations of their propagation in the Galaxy have been based on the use of the local diffusion model. It is quite acceptable for modeling of the Brownian motion because displacements of the Brownian tracer are mutually independent in length and direction. However, some features of this model are incompatible with the real behavior of cosmic rays: the path of the tracer between any two points of its trajectory is infinite, the local velocity is infinite and the front from a local pulse source is absent (after a moment, the particle can be observed arbitrarily far away from its original place). In this article, we describe the current state of a new model of cosmic ray transport, being free from these imperfections. It was formulated in 2010 and is still in progress under the title NoRD (Nonlocal Relativistic Diffusion) model. Two crucial ideas underlie this approach: taking into account correlations in space-time increments and using the material derivative of a fractional order. First of them ensures the relativistic speed-limitation whereas the second one reflects the influence of interstellar medium turbulence. The numerical calculation results demonstrated in this paper do speak well for the NoRD-model as compared with the traditional one based on integer-order operators.