On the Fractional Version of Cosmic Ray Acceleration Nonlocality

The commonly accepted CR transport theory is based on the diffusion equation, which in turn follows from the Boltzmann kinetic equation. The latter was derived for multiple collisions of particles with local formations under the assumption of their statistical independence, more precisely, of the Poisson nature of the ISM ensemble. This property is responsible for the appearance of local operators in diffusion-type equations. However, the interstellar cloud structure, coupled with its turbulent nature, is an example of a strongly correlated medium where CR transport does not obey the ordinary diffusion theory. To describe it, the authors use the generalized (non-local) transport equation for describing the distributed acceleration of CRs as multiple scattering in the momentum phase space, taking into account the long-range correlations of magnetic field in ISM. The latter is realized by means of solving fractional order integro-differential equation for momentum distribution by Monte Carlo method. Numerical calculations have confirmed analytical results obtained earlier and demonstrate steepening of the energy spectrum due to the increease in medium correlations.