RSS feeds
РусскийEnglish
Issue 11, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2003

 / 

August

←  →

Reviews of topical problems


Self-similar anomalous diffusion and Levy-stable laws


Ul'yanovsk State University, ul. L. Tolstogo 42, Ulyanovsk, 432700, Russian Federation

Stochastic principles for constructing the process of anomalous diffusion are considered, and corresponding models of random processes are reviewed. The self-similarity and the independent-increments principles are used to extend the notion of diffusion process to the class of Levy-stable processes. Replacing the independent-increments principle with the renewal principle allows us to take the next step in generalizing the notion of diffusion, which results in fractional-order partial space-time differential equations of diffusion. Fundamental solutions to these equations are represented in terms of stable laws, and their relationship to the fractality and memory of the medium is discussed. A new class of distributions, called fractional stable distributions, is introduced.

Fulltext pdf (634 KB)
Fulltext is also available at DOI: 10.1070/PU2003v046n08ABEH001324
PACS: 02.50.−r, 05.40.Fb, 05.40.Jc (all)
DOI: 10.1070/PU2003v046n08ABEH001324
URL: https://ufn.ru/en/articles/2003/8/c/
000187205200003
Citation: Uchaikin V V "Self-similar anomalous diffusion and Levy-stable laws" Phys. Usp. 46 821–849 (2003)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Учайкин В В «Автомодельная аномальная диффузия и устойчивые законы» УФН 173 847–876 (2003); DOI: 10.3367/UFNr.0173.200308c.0847

References (76) ↓ Cited by (155) Similar articles (20)

  1. Zaidel’ A N Oshibki Izmerenii Fizicheskikh Velichin (Errors in Measurements of Physical Quantities, Leningrad: Nauka, 1974)
  2. Einstein A, von Smoluchowski M, in Brounovskoe Dvizhenie (Brownian Movement, Ed. B I Davydov, Moscow – Leningrad: ONTI, 1936)
  3. Einstein A Ann. Phys. (Leipzig) 17 549 (1905)
  4. Langevin P CR Acad. Sci. 146 530 (1908)
  5. von Smoluchowski M Phys. Z. 13 1069 (1912)
  6. Bouchaud J P, Georges A Phys. Rev. A 41 1156 (1990)
  7. Isichenko M B Rev. Mod. Phys. 64 961 (1992)
  8. Shlesinger M F, Zaslavsky G M, Klafter J Nature 363 31 (1993)
  9. West B J, Deering W Phys. Rep. 246 1 (1994)
  10. Klafter J, Shlesinger M F, Zumofen G Phys. Today 49 (2) 33 (1996)
  11. Pekalski A, Sznajd-Weron K (Eds) Anomalous Diffusion: From Basics to Applications (Lecture Notes in Physics, Vol. 519, Berlin: Springer, 1999)
  12. Metzler R, Klafter J Phys. Rep. 339 1 (2000)
  13. Gikhman I I, Skorokhod A V Teoriya Sluchainykh Protsessov (Theory of Random Processes) Vol. 1 (Moscow: Nauka, 1971) [Translated into English: The Theory of Stochastic Processes Vol. 1 (Berlin: Springer-Verlag, 1974)]
  14. Prokhorov Yu V (Ed.) Veroyatnost’ i Matematicheskaya Statistika: Entsiklopediya (Encyclopedia of Probabilities and Mathematical Statistics, Moscow: Bol’shaya Rossiiskaya Entsiklopediya, 1999)
  15. Lévy P Processus Stochastiques et Mouvement Brownien 2nd ed. (Paris: Gauthier-Villars, 1965) [Translated into Russian (Moscow: Nauka, 1972)]
  16. Prokhorov Yu V, Rozanov Yu A Teoriya Veroyatnostei (Probability Theory, Moscow: Nauka, 1973) [Translated into English (Berlin: Springer-Verlag, 1969)]
  17. Feller W An Introduction to Probability Theory and Its Applications 2nd ed. (New York: Wiley, 1971) [Translated into Russian (Moscow: Mir, 1984)]
  18. Keiser J Statistical Thermodynamics of Nonequilibrium Processes (New York: Springer-Verlag, 1987) [Translated into Russian (Moscow: Mir, 1990)]
  19. Wentzel’ A D Kurs Teorii Sluchainykh Protsessov (Textbook of the Theory of Random Processes, Moscow: Nauka, 1975) [Translated into English: A Course in the Theory of Stochastic Processes (New York: McGraw-Hill Intern., 1981)]
  20. Faddeev L D (Ed.) Matematicheskaya Fizika: Entsiklopediya (An Encyclopedia of Mathematical Physics, Moscow: Bol’shaya Rossiiskaya Entsiklopediya, 1998)
  21. Zolotarev V M Odnomernye Ustoichivye Raspredeleniya (Stable One-Dimensional Distributions, Moscow: Nauka, 1983)
  22. Samorodnitsky G, Taqqu M S Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (New York: Chapman & Hall, 1994)
  23. Zolotarev V M, Uchaikin V V, Saenko V V Zh. Eksp. Teor. Fiz. 115 1411 (1999) [JETP 88 780 (1999)]
  24. Uchaikin V V Int. J. Theor. Phys. 38 2377 (1999)
  25. Samko S G, Kilbas A A, Marichev O I Integraly i Proizvodnye Drobnogo Poryadka i Nekotorye Ikh Prilozheniya (Fractional-Order Integrals and Derivatives and Some Their Applications, Minsk: Nauka i Tekhnika, 1987) [Translated into English: Fractional Integrals and Derivatives: Theory and Applications (Philadelphia, Pa.: Gordon and Breach Sci. Publ., 1993)]
  26. Van Kampen N G Stochastic Processes in Physics and Chemistry (Amsterdam: North-Holland, 1981) [Translated into Russian (Moscow: Vysshaya Shkola, 1990)]
  27. Klyatskin V I Stokhasticheskie Uravneniya Glazami Fizika (Physicist’s View of Stochastic Equations, Moscow: Fizmatlit, 2001)
  28. Chechkin A et al. Chem. Phys. 284 233 (2002)
  29. Kotulski M J. Stat. Phys. 81 777 (1995)
  30. Kolokoltsov V, Korolev V, Uchaikin V V J. Math. Sci. 105 2569 (2001)
  31. Uchaikin V V Int. J. Theor. Phys. 39 2087 (2000)
  32. Uchaikin V V Physica A 305 205 (2002)
  33. Uchaikin V V "Fractional stable distributions 2", Preprint No. 12/02 (Nottingham: Nottingham Trent Univ., 2002)
  34. Uchaikin V V Chem. Phys. 284 507 (2002)
  35. Montroll E W, Weiss G H J. Math. Phys. 6 167 (1965)
  36. Montroll E W, Schlesinger M F, in Nonequilibrium Phenomena II: From Stochastic to Hydrodynamics (Studies in Statistical Mechanics, Vol. 11, Eds J L Lebowitz, E W Montroll, Amsterdam: North-Holland, 1984) p. 1
  37. Cox D R, Smith V L Renewal Theory (London: Methuen, 1962) [Translated into Russian (Moscow: Sov. Radio, 1967)]
  38. Saichev A I, Zaslavsky G M Chaos 7 753 (1997)
  39. Uchaikin V V, Zolotarev V M Chance and Stability. Stable Distributions and Their Applications (Utrecht: VSP, 1999)
  40. Zaslavsky G M, in Applications of Fractional Calculus in Physics (Ed. R Hilfer, Singapore: World Scientific, 2000) p. 203
  41. Gorenflo R, Mainardi F Arch. Mech. 50 377 (1998)
  42. Metzler R, Nonnenmacher T F Chem. Phys. 284 67 (2002)
  43. Krein S G (Ed.) Funktsional’nyi Analiz (Functional Analysis, Moscow: Nauka, 1972) [Translated into English (Groningen: Wolters-Noordhoff, 1972)]
  44. Uchaikin V V Zh. Eksp. Teor. Fiz. 115 2113 (1999) [JETP 88 1155 (1999)]
  45. Landa P S Nelineinye Kolebaniya i Volny (Nonlinear Oscillations and Waves, Moscow: Nauka – Fizmatlit, 1997)
  46. Ivanova V S et al. Sinergetika i Fraktaly v Materialovedenii (Synergetics and Fractals in Material Science, Moscow: Nauka, 1994)
  47. Nigmatullin R R Phys. Status Solidi B 123 739 (1984)
  48. Chukbar K V Zh. Eksp. Teor. Fiz. 109 1335 (1996) [JETP 82 719 (1996)]
  49. Rutman R S Teor. Mat. Fiz. 100 476 (1994) [Theor. Math. Phys. 100 1154 (1994)]
  50. Rutman R S Teor. Mat. Fiz. 105 393 (1995) [Theor. Math. Phys. 105 1509 (1995)]
  51. Volterra V Leçons sur la théorie mathématique de la lutte pour la vie (Paris: Gauthier-Villars et cie, 1931) [Translated into Russian (Moscow: Nauka, 1976)]
  52. Volterra V Theory of Functionals and of Integral and Integro-Differential Equations (New York: Dover Publ., 1959) [Translated into Russian (Moscow: Nauka – Fizmatlit, 1982)]
  53. Schroeder M R Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (New York: W.H. Freeman, 1991) [Translated into Russian (Moscow – Izhevsk: RKhD, 2001)]
  54. Maslennikov M V Aksiomatichaskaya Model’ Yavlenii Perenosa Chastits (Axiomatic Model of Particle-Transport Phenomena, Moscow: Nauka, 1989)
  55. Case K M, Zweifel P F Linear Transport Theory (Reading, Mass.: Addison-Wesley Publ. Co., 1967) [Translated into Russian (Moscow: Mir, 1972)]
  56. Uchaikin V V, Gusarov G G J. Math. Phys. 38 2453 (1997)
  57. Baryshev Yu V et al. Vistas Astron. 38 419 (1994)
  58. Barkai E, Fleurov V, Klafter J Phys. Rev. E 61 1164 (2000)
  59. Uchaikin V V, in Emergent Nature: Patterns, Growth and Scaling in the Sciences (Ed. M M Novak, Singapore: World Scientific, 2002) p. 411
  60. Uchaikin V V, in Paradigms of Complexity: Fractals and Structures in the Sciences (Ed. M M Novak, Singapore: World Scientific, 2000) p. 41
  61. Lifshits I M, Gredeskul S A, Pastur L A Vvedenie v Teoriyu Neuporyadochennykh System (Introduction to the Theory of Disordered Systems, Moscow: Nauka, 1982) [Translated into English (New York: Wiley, 1988)]
  62. Gantmakher V F, Feigel’man M V Usp. Fiz. Nauk 168 113 (1998) [Phys. Usp. 41 105 (1998)]
  63. Grinstein G, Mazenko G (Eds) Directions in Condensed Matter Physics: Memorial Volume in Honor of Prof. S.K. Ma (World Scientific Ser. on Directions in Condensed Matter Physics, Vol. 1, Singapore: World Scientific, 1986)
  64. Godoy S, García-Colín L S Physica A 258 414 (1998)
  65. Uchaikin V V Izv. Vyssh. Uchebn. Zaved. Fiz. 43 (5) 23 (2000)
  66. Uchaikin V V, Yarovikova I V Izv. Vyssh. Uchebn. Zaved. Fiz. 43 (2) 88 (2000) [Russ. Phys. J. 43 161 (2000)]
  67. Uchaikin V V, Yarovikova I V, Saenko V V Izv. Vyssh. Uchebn. Zaved. Fiz. 43 (10) 11 (2000) [Russ. Phys. J. 43 871 (2000)]
  68. García-Pelayo R Physica A 216 299 (1995)
  69. Wilks S S Mathematical Statistics (New York: Wiley, 1962) [Translated into Russian (Moscow: Nauka, 1967)]
  70. Monin A S Izv. Akad. Nauk SSSR Ser. Geofiz. (3) 234 (1955)
  71. Yarovikova I V Uchenye Zapiski Ul’yanovskogo Gos. Univ. Ser. Fiz. (2(9)) 12 (2000)
  72. Uchaikin V V, Saenko V V Zh. Tekh. Fiz. 71 (2) 8 (2001) [Tech. Phys. 46 139 (2001)]
  73. Uchaikin V V, Saenko V V J. Phys. Studies 4 371 (2000)
  74. Uchaikin V V Physica A 255 65 (1998)
  75. Zaburdaev V Yu, Chukbar K V Zh. Eksp. Teor. Fiz. 121 299 (2002) [JETP 94 252 (2002)]
  76. Uchaikin V V Obozrenie Prikl. Promyshl. Mat. 9 477 (2002)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions