Diffusive Shock Acceleration: Nonclassical Model of Cosmic Ray Transport

In this work the theory of diffusive shock acceleration is extended to the case of nonclassical particle transport with Lévy flights and Lévy traps, when the mean square displacement grows nonlinearly with time. In this approach the Green function is not a Gaussian but it exhibits power-law tails. By using the propagator appropriate for nonclassical diffusion, it is found for the first time that energy spectral index of particles accelerated at shock front is γ=[α(r+5)-6β]/[α(r-1)] , where 0<α<2 and 0<β<1 are the exponents of power-law behavior of Lévy flights and Lévy traps, respectively. We note that this result coincides with standard slope at α=2 , β=1 (normal diffusion), and also includes those obtained earlier for the subdiffusion ( α=2 , β<1 ) and superdiffusion ( α<2 , β=1 ) regimes.