Towards deterministic equations for Lévy walks: The fractional material derivative

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 67, Issue 1, 2003, Pages 4-

  Sokolov, Igor M ^a   Metzler, Ralf ^b  

^a HUMBOLDT UNIVERSITY   (Germany)

^b NORDITA   (Denmark)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84885101996     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.67.010101     Document Type: Article

References (28)

1. J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990).
2. B.D. Hughes, Random Walks and Random Environments (Oxford University Press, Oxford, 1995).
3. J. Klafter, Phys. Rev. A 35, 3081 (1987).
4. D. Bedeaux, J. Math. Phys. 12, 2116 (1971);
5. J. Klafter and R. Silbey, Phys. Rev. Lett. 44, 55 (1980);
6. R. Metzler, Eur. Phys. J. B 19, 249 (2001). Often, the memory kernels in generalized master equations of the CTRW type correspond to fractional differintegral operators.
7. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
8. R. Metzler, Phys. Rev. Lett. 82, 3563 (1999);
9. R. MetzlerEurophys. Lett. 46, 431 (1999).
10. R. Metzler, Phys. Rev. E 62, 6233 (2000);
11. R. Metzler and J. Klafter, J. Phys. Chem. B 104, 3851 (2000).
12. I.M. Sokolov, Phys. Rev. E 64, 021107 (2001).
13. E. Barkai, Phys. Rev. E 63, 046118 (2001).
14. However, in an external force field (Formula presented) with (Formula presented) the mean squared displacement of a Lévy flight is finite, A. Chechkin, Chem. Phys. 284, 233 (2002).
15. E. Barkai and V.N. Fleurov, Phys. Rev. E 56, 6355 (1997);
16. P. Levitz, Europhys. Lett. 39, 593 (1997).
17. I.M. Sokolov, Europhys. Lett. 47, 152 (1999);
18. I.M. Sokolov, Phys. Rev. E 61, 2717 (2000).
19. E. Barkai and R. Silbey, J. Phys. Chem. B 104, 3866 (2000);
20. E. Barkai, Chem. Phys. 284, 13 (2002)
21. R. Metzler and I.M. Sokolov, Europhys. Lett. 58, 482 (2002).
22. A.I. Shushin, Phys. Rev. E 64, 051108 (2001)
23. This is the Laplace transform of a pdf, since (Formula presented) and (Formula presented) is a completely monotonic function: it has a form (Formula presented) where (Formula presented) is a completely monotonic function (it is positive, and the signs of its derivatives alternate), and (Formula presented) is a function with a completely monotonic derivative (Formula presented)
24. I.M. Sokolov, Phys. Rev. E 63, 011104 (2001).
25. K.B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New York, 1974).
26. G. Cattaneo, Atti Semin. Mat. Fis. Univ. Modena 3, 83 (1948).
27. G. Zumofen and J. Klafter, Phys. Rev. E 47, 851 (1993).
28. M.M. Meerschaert and H.-P. Schefler (unpublished).