Towards deterministic equations for Lévy walks: The fractional material derivative

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 67, Issue 1 1, 2003, Pages 101011-101014

  Sokolov, Igor M ^a   Metzler, Ralf ^b  

^a HUMBOLDT UNIVERSITY   (Germany)

^b NORDITA   (Denmark)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; BROWNIAN MOVEMENT; DIFFUSION; EQUATIONS OF MOTION; FOURIER TRANSFORMS; LAPLACE TRANSFORMS; PHASE SPACE METHODS; PROBABILITY DENSITY FUNCTION; SPATIAL VARIABLES MEASUREMENT;

ANOMALOUS DIFFUSION; CONTINOUS TIME RANDOM WALK (CTRW);

MARKOV PROCESSES;

EID: 0038732260     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (29)

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 et al., Phys. Rev. A 35, 3081 (1987).
4. D. Bedeaux et al., J. Math. Phys. 12, 2116 (1971); J. Klafter and R. Silbey, Phys. Rev. Lett. 44, 55 (1980); 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.
5. D. Bedeaux et al., J. Math. Phys. 12, 2116 (1971); J. Klafter and R. Silbey, Phys. Rev. Lett. 44, 55 (1980); 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.
6. D. Bedeaux et al., J. Math. Phys. 12, 2116 (1971); J. Klafter and R. Silbey, Phys. Rev. Lett. 44, 55 (1980); 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 et al., Phys. Rev. Lett. 82, 3563 (1999); Europhys. Lett. 46, 431 (1999).
9. R. Metzler et al., Phys. Rev. Lett. 82, 3563 (1999); Europhys. Lett. 46, 431 (1999).
10. R. Metzler, Phys. Rev. E 62, 6233 (2000); R. Metzler and J. Klafter, J. Phys. Chem. B 104, 3851 (2000).
11. R. Metzler, Phys. Rev. E 62, 6233 (2000); R. Metzler and J. Klafter, J. Phys. Chem. B 104, 3851 (2000).
12. I.M. Sokolov et al., Phys. Rev. E 64, 021107 (2001).
13. E. Barkai, Phys. Rev. E 63, 046118 (2001).
14. c with c>2, the mean squared displacement of a Lévy flight is finite, A. Chechkin et al., Chem. Phys. 284, 233 (2002).
15. E. Barkai and V.N. Fleurov, Phys. Rev. E 56, 6355 (1997); P. Levitz, Europhys. Lett. 39, 593 (1997).
16. E. Barkai and V.N. Fleurov, Phys. Rev. E 56, 6355 (1997); P. Levitz, Europhys. Lett. 39, 593 (1997).
17. I.M. Sokolov et al., Europhys. Lett. 47, 152 (1999); I.M. Sokolov et al., Phys. Rev. E 61, 2717 (2000).
18. I.M. Sokolov et al., Europhys. Lett. 47, 152 (1999); I.M. Sokolov et al., Phys. Rev. E 61, 2717 (2000).
19. E. Barkai and R. Silbey, J. Phys. Chem. B 104, 3866 (2000); E. Barkai, Chem. Phys. 284, 13 (2002)
20. E. Barkai and R. Silbey, J. Phys. Chem. B 104, 3866 (2000); 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. α-1.
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. t μf(x(t))dt,t); however, there is no direct Fourier-Laplace transform of this operator.
29. M.M. Meerschaert and H.-P. Schefler (unpublished).