On fractional differential models for cosmic ray diffusion

V.V. Uchaikin and R.T. Sibatov1


We consider a model of anomalous cosmic ray diffusion with a finite velocity of free particle motion. Inclusion of the finite velocity substantially modifies the anomalous diffusion equation and its solutions. The propagator in a one-dimensional model is presented in an analytic form while in the three-dimensional case numerical results have been obtained. The propagators in models with restricted and unrestricted anomalous diffusion are compared.


  1. K. Iosida. Functional Analysis (Mir, Moscow, 1967).
  2. C. Godreche and J. M. Luck, J. Stat. Phys. 104, 489 (2001).
  3. V. V. Uchaikin and R. T. Sibatov. Subrecoil laser cooling dynamics: fractional derivative approach. Statistical Mechanics: Theory and Experiment, P04001, 1-16 (2009).
  4. V. V. Uchaikin and R. T. Sibatov, A statistical model of scintillating fluorescence. Zh. Eksp. Teor. Fiz. 136 4 (10), 627-638 (2009).
  5. R. T. Sibatov and V. V. Uchaikin, Statistics of photon counts at scintillating fluorescence of quantum dots. Optika i Spektroskipiya 108 (5), 804-811 (2010).
  6. I. M. Sokolov and R. Metzler, Towards deterministic equations for Levy walks: The fractional material derivative. Phys. Rev. E 67, 010101 (R) (2003).
  7. V. V. Uchaikin and R. T. Sibatov, A one-dimensional fractal walk with a finite velocity of free motion. Pis'ma v Zh. Tekhn. Fiz. 30 (8), 27-33 (2004).
  8. V. Yu. Zaburdaev and K. V. Chukbar. Accelerated superdiffusion and a finite velocity of free motion. Zh. Eksp. Teor. Fiz. 121 (2), 299-307 (2002).
  9. A. A. Lagutin, Yu. A. Nikulin, and V. V. Uchaikin, A break in the cosmic ray spectrum as a consequence of the fractal magnetic field of the Galaxy. Preprint AGU-2000/4, Barnaul, 2000.
  10. A. A. Lagutin and A. G. Tyumentsev, The spectrum, mass composition and anisotropy of cosmic rays in a fractal galaxy. Izvestiya Altaiskogo Gosuniversiteta No. 5, 4 (2004).
  11. A. A. Lagutin and V. V. Uchaikin, Nucl. Instr. Meth. B201, 212 (2003).
  12. L. I. Dorman, Experimental and Theoretical Foundations of Cosmic Ray Astrophysics (Nauka, Moscow, 1975).
  13. V. V. Lagutin and V. V. Uchaikin, in: Proc. 27th Int. Cosmic Ray Conf., Hamburg, Vol 5 (2001), p. 1896.
  14. A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications. Chaos 7(4), 753 (1997).
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