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

Abstract

Context. The acceleration and transport of energetic particles in astrophysical plasmas can be described by the so-called Parker equation, which is a kinetic equation comprising diffusion terms both in coordinate space and in momentum space. In the past years, it has been found that energetic particle transport in space can be anomalous, for instance, superdiffusive rather than normal diffusive. This requires a revision of the basic transport equation for such circumstances. Aims. Here, we extend the Parker equation to the case of anomalous diffusion by means of fractional derivatives that generalize the usual second-order spatial diffusion operator. Methods. We introduce the left and right Caputo fractional derivatives in space. These derivatives are one of the tools used to describe anomalous transport. We consider the case of steady-state solutions upstream and downstream of a planar shock. Results. We obtain an estimate of the particle acceleration time at shocks in the case of superdiffusion. An analytical solution of the steady-state fractional Parker equation is given by the Mittag-Leffler functions, which correspond to a power-law profile for the energetic particle intensity far upstream of the shock, in agreement with the results obtained from a probabilistic approach to superdiffusion. These functions also correspond to a stretched exponential close upstream of the shock. Conclusions. These results can help to model more precisely the measured fluxes of energetic particles that are accelerated at both interplanetary shocks and supernova remnant shocks.

On the Fractional Diffusion-Advection Equation for Fluids and Plasmas

“…We recall that several definitions of fractional derivatives are available [36,40,50], and that this variety reflects the enhanced possibility offered by fractional derivatives to adapt the non-local flux operator, corresponding to the fractional Fick's law, to the diverse physical systems under consideration. In particular, in space plasmas the density of high-energy particles upstream of a shock wave is found to exhibit a sharp decrease close to the shock, and a slow power-law decay far upstream [12,13,33,43,47,51]. Such energetic particle density profiles are very similar to the properties of the Mittag-Leffler functions shown above.…”

“…The fact that the Mittag-Leffler functions are solutions of the above equation is shown in, for example, [50], and also by [43] for the case when the x-axis is pointing in the same direction as the flow velocity V. In addition, it can be easily verified by recalling the effect of fractional derivatives on power functions f (x) = x γ for γ > −1, γ = 0, and…”

“…The problem of transport in the presence of both anomalous diffusion and advection is of great interest for many applications. This includes the case of energetic particle transport in front of a shock wave, considered by [39,43], where the plasma turbulence is generated by backstreaming particles upstream of the shock [44,45]; the case of plasma transport in the presence of turbulence and of an external electric field, as in the Earth's magnetotail; and the case of plasma and impurity transport in the presence of turbulence and an imposed velocity field, as in recent plasma medical applications [46]. Fractional diffusion-advection equations have already been considered in the literature (e.g., [4]), but mostly considering the time fractional derivative appropriate for the subdiffusive case.…”

Several Results of Fractional Differential and Integral Equations in Distribution

“…Up to now, fractional operators have been applied in various areas, such as anomalous diffusion, long-range interactions, long-memory processes and materials, waves in liquids, and physics. Recently, Zimbardo et al [1] generalized the Parker equation, which describes the acceleration and transport of energetic particles in astrophysical plasmas, to the case of anomalous by Caputo fractional derivatives. As far as we know, fractional calculus is one of best tools to construct certain electro-chemical problems and characterizes long-term behaviors, allometric scaling laws, nonlinear operations of distributions [2] and so on.…”

Powerlaw spectra from stochastic acceleration

“…More generally speaking, the Fokker-Planck formalism can be used to describe the transport in momentum space. By contrast, heavy-tailed distributions, who do not possess a finite mean, characterize Lévy flights and are more properly described by fractional transport equations, see for instance Zimbardo & Perri (2013), Zimbardo et al (2017) and Isliker et al (2017a).…”