Introduction to the Theory of Lévy Flights

Anomalous Transport: Foundations and Applications

Volumn , Issue , 2008, Pages 129-162

  Chechkin, Alexei V ^a   Metzler, Ralf ^b   Klafter, Joseph ^c   Gonchar, Vsevolod Yu ^a  

^a KHARKOV INSTITUTE OF PHYSICS AND TECHNOLOGY   (Ukraine)

^b TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^c TEL AVIV UNIVERSITY   (Israel)

Author keywords

External fields; Free L vy flights; L vy flights in phase space; L vy flights theory; L vy stable distributions; Power law truncated L vy flights; Random walk processes; Semi infinite domain; Space fractional Fokker Planck equation

Indexed keywords

EID: 84871409088     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9783527622979.ch5     Document Type: Chapter

References (111)

1. B.D. Huges. Random Walks and Random Environments, vol. 1, Oxford University Press, Oxford, 1995.
2. P.P. Lévy, Théorie de l'Addition des Variables Aléatoires, 2nd edn., Gauthier-Villars, Paris 1954. (1st edn., 1937).
3. B.V. Gnedenko, A.V. Kolmogorov. Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA, 1954. (Russian edition: 1949).
4. J.-P. Bouchaud and A. Georges, Phys. Rev. A 41, 1156 (1990).
5. P.D. Ditlevsen, Geophys. Res. Lett. 26, 1441 (1999).
6. M.F. Shlesinger and J. Klafter, in: On Growth and Form, edited by H.E. Stanley and N. Ostrowsky, Martinus Nijhoff Publishers, Amsterdam, pp. 279-283 (1986).
7. G.M. Viswanathan et al., Nature 401, 911 (1999).
8. A.M. Reynolds and M.A. Frye, PLoS ONE 2(4), e354 (2007).
9. R.P.D. Atkinson, C.J. Rhodes, D.W. Macdonald, and R.M. Anderson, OIKOS 98, 134 (2002).
10. G. Ramos-Fernandez et al., Behav. Ecol. Sociobiol. 55, 223 (2003).
11. F. Bartumeus, Fractals 15, 151 (2007).
12. F. Bartumeus et al., Phys. Rev. Lett. 88, 097901 (2002).
13. M.A. Lomholt, T. Koren, R. Metzler, and J. Klafter, Proc. Natl. Acad. Sci., at press.
14. I.M. Sokolov, J. Mai, and A. Blumen, Phys. Rev. Lett. 79, 857 (1997).
15. D. Brockmann and T. Geisel, Phys. Rev. Lett. 91, 048303 (2003).
16. M.A. Lomholt, T. Ambjörnsson, and R. Metzler, Phys. Rev. Lett. 95, 260603 (2005).
17. L. Hufnagel, D. Brockmann, and T. Geisel, Proc. Natl. Acad. Sci. 101, 15124 (2004).
18. D. Brockmann, L. Hufnagel, and T. Geisel, Nature 439, 462 (2006).
19. H. Katori, S. Schlipf, and H. Walther, Phys. Rev. Lett., 79, 2221 (1997).
20. G. Zumofen and J. Klafter, Chem. Phys. Lett. 219, 303 (1994).
21. E. Barkai and R. Silbey, Chem. Phys. Lett. 310, 287 (1999).
22. E. Barkai, R. Silbey, and G. Zumofen, Phys. Rev. Lett. 84, 5339 (2000).
23. B.B. Mandelbrot, J. Bus, 36, 394 (1963).
24. R.N. Mantegna and H.E. Stanley, Nature 383, 587 (1996).
25. J.-P. Bouchaud and M. Potters, Theory of Financial Risks, Cambridge University Press, Cambridge, UK, 2000.
26. E. Lutz, Phys. Rev. Lett. 93, 190602 (2004).
27. W. Feller, An Introduction to Probability Theory and its Applications, vol. 2., Wiley, New York, 1971.
28. V.M. Zolotarev, One-Dimensional Stable Distributions, American Mathematical Society, Providence, RI, 1986 (Russian edition: 1983).
29. G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, New York, 1994, p. 619.
30. C. Fox, Trans. Am. Math. Soc. 98, No.3, 395 (1961).
31. W.R. Schneider, Stable distributions: Fox function representation and generalization. in: S. Albeverio, G. Casati, D. Merlini (eds.), Stochastic Processes in Classical and Quantum Systems, Lecture Notes in Physics, Springer, Berlin, 1986, pp. 497-511.
32. A.M. Mathai, R.K. Saxena, The HFunctions with Applications in Statistics and Other Disciplines, Wiley Eastern, New Delhi, 1978, p. 235.
33. H.M. Srivastava, K.G. Gupta, S.P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publisher, New Delhi, 1982, p. 186.
34. H. Pollard, Bull. Am. Math. Soc. 52, 908 (1946).
35. H. Bergström, Ark. Math. 2, 375 (1952).
36. Y.V. Linnik, Dokl. Akad. Nauk SSSR 94, 619 (1954).
37. A.V. Skorokhod, Dokl. Akad. Nauk SSSR 98, 731 (1954).
38. E.W. Montroll and G.H. Weiss, J. Math. Phys. 10, 753 (1969).
39. E.W. Montroll and H.J. Scher, J. Stat. Phys. 9, 101 (1973).
40. J. Klafter, A. Blumen, and M.F. Shlesinger, Phys. Rev. A 35, 3081 (1987).
41. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
42. R. Metzler and J. Klafter, J. Phys. A 37, R161 (2004).
43. A. Erdélyi, Bateman Manuscript Project Higher Transcendental Functions, Vol. III, McGraw-Hill, New York, 1955.
44. R. Metzler, E. Barkai, and J. Klafter, Phys. Rev. Lett. 82, 3563 (1999).
45. R. Metzler and T.F. Nonnenmacher, Chem. Phys. 284, 67 (2002).
46. M.F. Shlesinger, G.M. Zaslavsky, and J. Klafter, Nature 363, 31 (1993).
47. I.I. Gikhman and A.V. Skorohod, Theory of Stochastic Processes II, Springer, Berlin, 1975.
48. H.C. Fogedby, Phys. Rev. Lett. 73, 2517 (1994);
49. H.C. Fogedby, Phys. Rev. E 58, 1690 (1998).
50. R. Metzler, E. Barkai, and J. Klafter, Europhys. Lett. 46, 431 (1999).
51. S. Jespersen, R. Metzler, and H.C. Fogedby, Phys. Rev. E 59, 2736 (1999).
52. A. Dubkov and B. Spagnolo, Fluctuat. Noise Lett. 5, L267-L274 (2005).
53. A.I. Saichev and G.M. Zaslavsky, Chaos 7, 753 (1997).
54. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, New York, 1993.
55. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1998.
56. R. Metzler, Euro. Phys. J. B19, 249 (2001).
57. G.H. Weiss. Aspects and Applications of the Random Walk, North-Holland, Amsterdam, 1994.
58. I.M. Sokolov and R. Metzler, J. Phys. A 37, L609 (2004).
59. I.M. Sokolov, J. Klafter, and A. Blumen, Phys. Rev. E 64, 021107 (2001).
60. D. Brockmann and I.M. Sokolov, Chem. Phys. 284, 409 (2002).
61. Ya. Gambin, G. Massiera, L. Ramos, Ch. Ligoure, and W. Urbach, Phys. Rev. Lett. 94, 110602 (2005).
62. T. Koren, M.A. Lomholt, A.V. Chechkin, J. Klafter, and R. Metzler, Phys. Rev. Lett. 99, 160602 (2007).
63. E. Sparre Andersen, Math. Scand. 1, 263 (1953);
64. E. Sparre Andersen, Math. Scand. 2, 195 (1954).
65. S. Redner. A Guide to First-Passage Processes, Cambridge University Press, Cambridge, UK, 2001.
66. U. Frisch and H. Frisch, in: Lévy Flights and Related Topics in Physics, edited by M.F. Shlesinger, G.M. Zaslavsky, and U. Frisch, Lecture notes in physics, vol. 450, Springer, Berlin, 1995.
67. G. Zumofen and J. Klafter, Phys. Rev. E 51, 2805 (1995).
68. I. Eliazar and J. Klafter, Physica A 336, 219 (2004).
69. F. Mainardi, in: Fractals and Fractional Calculus in Continuum Mechanics, edited by A. Carpinteri and F. Mainardi, CISM Courses and Lectures, vol. 378, Springer, Wien, 1997, p. 291.
70. T. Koren, A.V. Chechkin, and J. Klafter, Physica A 379, 10 (2007).
71. A.V Chechkin, R. Metzler, V.Yu. Gonchar, J. Klafter, and L.V. Tanatarov, J. Phys. A: Math. Gen. 36, L537 (2003).
72. A.V. Chechkin, J. Klafter, V.Yu. Gonchar, R. Metzler, and L.V. Tanatarov, Phys. Rev E 67, 010102(R) (2003).
73. A.V. Chechkin, V.Yu. Gonchar, J. Klafter, R. Metzler, L. Tanatarov, Chem. Phys. 284, 233 (2002).
74. A.V. Chechkin, V.Yu. Gonchar, J. Klafter, R. Metzler, and L.V. Tanatarov, J. Stat. Phys. 115, 1505 (2004).
75. I. Eliazar and J. Klafter, J. Stat. Phys. 111, 739 (2003).
76. P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).
77. A.V. Chechkin, V.Yu. Gonchar, J. Klafter, and R. Metzler, Europhys. Lett. 75, 348 (2005).
78. B. Dybiec, E. Gudowska-Nowack, and P. Hänggi, Phys. Rev. E 75, 021109 (2007).
79. A.V. Chechkin, O.Yu. Sliusarenko, R. Metzler, and J. Klafter, Phys. Rev. E 75, 041101 (2007).
80. P. Imkeller and I. Pavlyukevich, J. Phys. A 39, L237 (2006);
81. P. Imkeller and I. Pavlyukevich, Stochastic Proc. Appl. 116, 611 (2006).
82. P.D. Ditlevsen, Phys. Rev. E 60, 172 (1999).
83. H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1989.
84. Yu.L. Klimontovich, Statistical Physics, Nauka, Moscow, 1982;
85. Harwood Academic, New York, 1986.
86. B.J. West and V. Seshadri, Physica A 113, 203 (1982).
87. F.E. Peseckis, Phys. Rev. A 36, 892 (1987).
88. A.V. Chechkin, and V. Yu. Gonchar, Journ. Exp. Theor. Phys. 91, 635 (2000).
89. A.V. Chechkin, V. Yu. Gonchar, and M. Szydlowski, Phys. Plasmas 9, 78 (2002).
90. B. Kursunoglu, Ann. Phys. (N.Y.) 17, 259 (1962).
91. B. Kursunoglu, Phys. Rev. 132, 21 (1963).
92. J.A. Krommes, Phys. Rep. 360, 1 (2002).
93. A.V. Chechkin, V.Yu. Gonchar, J. Klafter, and R. Metzler, Phys. Rev. E 72, 010101(R) (2005).
94. N.N. Bogoliubov and Y.A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Hindustan Publication, Delhi, 1961 (distributed by Gordon& Breach, New York).
95. H. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962.
96. A.A. Andronov, C.E. Chaikin, and S. Lefschetz, Theory of Oscillations, University Press, Princeton, NJ, 1949.
97. Yu.L. Klimontovich, Turbulent Motion and the Structure of Chaos: A New approach to the Statistical Theory of Open Systems, Kluwer, Dordrecht, 1992.
98. M.F. Shlesinger, B.J. West, and J. Klafter, Phys. Rev. Lett. 58, 1100 (1987).
99. R.N. Mantegna and H.E. Stanley, Phys. Rev. Lett. 73, 2946 (1994).
100. I. Koponen, Phys. Rev. E 52 1197 (1995).
101. R.N. Mantegna and H.E. Stanley, J. Stat. Phys. 89, 469 (1997).
102. H.E. Stanley, Physica A 318 (2003) 279.
103. B. Dubrulle, J.-Ph. Laval, Eur. Phys. J. B 4, 143 (1998).
104. R. Jha, P.K. Kaw, D.R. Kulkarni, and J.C. Parikh, ADITYA Team, Phys. Plasmas 10, 699 (2003).
105. I.M. Sokolov, A.V. Chechkin, and J. Klafter, Physica A 336, 245 (2004).
106. A.V. Chechkin, R. Gorenflo, and I.M. Sokolov, Phys. Rev. E 66, 046129 (2002).
107. M.F. Shlesinger, Phys. Rev. Lett. 74, 4959 (1995).
108. P.K. Clark, Econometrica 41, 135 (1973).
109. A.V. Chechkin, V.Yu. Gonchar, J. Klafter, and R. Metzler, in: Fractals, Diffusion, and Relaxation in Disordered Complex Systems, edited by W.T. Coffee and Yu.P. Kalmykov, A Special Volume of Advances in Chemical Physics 133B, 439 (2006).
110. J. Klafter, M.F. Shlesinger, and G. Zumofen, Phys. Today 49, No.2, 33 (1996).
111. I.M. Sokolov and J. Klafter, A. Blumen, Phys. Today 55, November, 48 (2002).