RSS feeds
ÐóññêèéEnglish
Issue 12, 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

Îðèãèíàë: Ó÷àéêèí Â Â «Àâòîìîäåëüíàÿ àíîìàëüíàÿ äèôôóçèÿ è óñòîé÷èâûå çàêîíû» ÓÔÍ 173 847–876 (2003); DOI: 10.3367/UFNr.0173.200308c.0847

References (76) Cited by (156) ↓ Similar articles (20)

  1. Ivanov V V, Dlugach Ja M Journal Of Quantitative Spectroscopy And Radiative Transfer 323 108999 (2024)
  2. Sheremetyeva O V Mathematics Of Planet Earth Vol. Hereditary Models of Dynamic Processes in GeospheresHereditarian Properties of Foreshock Distributions in the Framework of a Fractional Model of the Deformation Process12 Chapter 4 (2024) p. 69
  3. Petukhov B V Kristallografiâ 69 876 (2024)
  4. Xin X, khan Ibrar et al 14 (3) (2024)
  5. Iqbal M A, Akbar M A, Islam M A Results In Physics 56 107251 (2024)
  6. Tarasov V E Physica A: Statistical Mechanics And Its Applications 609 128366 (2023)
  7. Tarasov V E Fractal Fract 7 137 (2023)
  8. Petukhov B V J. Surf. Investig. 17 971 (2023)
  9. Petukhov B V Poverhnostʹ. Rentgenovskie, Sinhrotronnye I Nejtronnye Issledovaniâ (9) 30 (2023)
  10. Zhmakin A I Non-Fourier Heat Conduction Chapter 9 (2023) p. 225
  11. Uchaikin V V, Erlykin A D, Sibatov R T Uspekhi Fizicheskikh Nauk 193 233 (2023)
  12. [Uchaikin V V, Erlykin A D, Sibatov R T Phys. Usp. 66 221 (2023)]
  13. Singh R K, Burov S Phys. Rev. E 108 (5) (2023)
  14. Barna I F, Mátyás L Mathematics 10 3281 (2022)
  15. Tarasov V E Mathematics 10 3848 (2022)
  16. Saenko V V, Kovalnogov V N et al Mathematics 10 511 (2022)
  17. Sheremetyeva O, Shevtsov B Fractal Fract 6 372 (2022)
  18. Areshi M, Khan A et al MATH 7 6936 (2022)
  19. Uchaikin V V J Math Sci 260 335 (2022)
  20. Bikulov A Kh, Zubarev A P P-Adic Num Ultrametr Anal Appl 14 265 (2022)
  21. Gharari F, Arias-Calluari K et al Phys. Rev. E 104 (5) (2021)
  22. Bianucci M Chaos, Solitons & Fractals 148 110961 (2021)
  23. Golmankhaneh A Kh, Sibatov R T Mathematics 9 613 (2021)
  24. Saenko V V, Kovalnogov V N et al Mathematics 9 3219 (2021)
  25. Svetukhin V Mathematics 9 740 (2021)
  26. Ivanov V V Springer Series in Light Scattering Springer Series In Light Scattering Chapter 1 (2021) p. 1
  27. Kukushkin A B, Kulichenko A A et al Symmetry 13 394 (2021)
  28. Strelkova T A, Lytyuga A P, Kalmykov A S Examining Optoelectronics in Machine Vision and Applications in Industry 4.0 Advances In Computational Intelligence And Robotics chapter 5 (2021) p. 134
  29. Shapovalov A V, Brons R Russ Phys J 64 704 (2021)
  30. Aleksandrov D A, Voronov A A et al Phys. Part. Nuclei Lett. 18 574 (2021)
  31. dos Santos M A F, Colombo E H, Anteneodo C Chaos, Solitons & Fractals 152 111422 (2021)
  32. Kostadinova E G, Banka R et al 28 (7) (2021)
  33. Sibatov R T, Sun HongGuang Fractal Fract 4 42 (2020)
  34. Likhomanova P, Kalashnikov I Phys. Rev. E 102 (2) (2020)
  35. Sibatov R T Mathematics 8 1991 (2020)
  36. Kitsyuk E P, Sibatov R T, Svetukhin V V Energies 13 213 (2020)
  37. Kostadinova E G, Padgett J L et al Phys. Rev. Research 2 (4) (2020)
  38. Bol’shov L A, Kondratenko P S, Matveev L V Uspekhi Fizicheskikh Nauk 189 691 (2019) [Bolshov L A, Kondratenko P S, Matveev L V Phys.-Usp. 62 649 (2019)]
  39. Sibatov R T, Sun HongGuang Fractal Fract 3 47 (2019)
  40. Shapovalov A V, Trifonov A Yu Russ Phys J 62 835 (2019)
  41. Sibatov R T, Svetukhin V V et al Electronics 8 650 (2019)
  42. Uchaikin V V, Uchaikin V V Itogi Nauki Tekhniki. Seriya «Sovremennaya Matematika Ee Prilozheniya. Tematicheskie Obzory» 167 62 (2019)
  43. Kostadinova E G, Liaw C D et al Phys. Rev. B 99 (2) (2019)
  44. Shapovalov A V, Obukhov V V Russ Phys J 61 1093 (2018)
  45. Levin D M, Kolmakov Yu N DDF 383 74 (2018)
  46. Feng J, Xu W et al Physica A: Statistical Mechanics And Its Applications 512 367 (2018)
  47. Kiselev B V Geomagn. Aeron. 57 326 (2017)
  48. Shevtsov B, Sheremetyeva O et al E3S Web Conf. 20 02013 (2017)
  49. Arkashov N S, Seleznev V A Theor Math Phys 193 1508 (2017)
  50. Naeem I, Khan M D Communications In Nonlinear Science And Numerical Simulation 42 560 (2017)
  51. Arkashov N S, Seleznev V A Sib Math J 58 752 (2017)
  52. Pandey R K, Mishra H K Adv Comput Math 43 365 (2017)
  53. Okorokov V A Advances In High Energy Physics 2017 1 (2017)
  54. Sorokin M V, Dubinko V I, Borodin V A Phys. Rev. E 95 (1) (2017)
  55. Rudoi Yu G, Kotel’nikova O A Theor Math Phys 189 1818 (2016)
  56. Lukashchuk S Yu Bulletin Of The SUSU. MMP 9 105 (2016)
  57. Korichi Z, Korichi Z i dr Teoreticheskaya Matematicheskaya Fizika 186 433 (2016) [Korichi Z, Meftah M T Theor Math Phys 186 374 (2016)]
  58. Kukushkin A B, Sdvizhenskii P A J. Phys. A: Math. Theor. 49 255002 (2016)
  59. Rudoi Yu G, Rudoi Yu G i dr Teoreticheskaya Matematicheskaya Fizika 189 477 (2016)
  60. Geneston E, Tuladhar R et al Phys. Rev. E 94 (1) (2016)
  61. Arkashov N S, Seleznev V A Theory Probab. Appl. 60 173 (2016)
  62. Kharcheva A A, Dubkov A A et al J. Stat. Mech. 2016 054039 (2016)
  63. Lukashchuk S Yu, Lukashchuk S Yu Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya «Fiziko-matematicheskie Nauki» 20 603 (2016)
  64. Guarcello C, Valenti D et al J. Stat. Mech. 2016 054012 (2016)
  65. El-Wakil S A, Abulwafa E M Nonlinear Dyn 80 167 (2015)
  66. Svetukhin V V, Sibatov R T J. Exp. Theor. Phys. 120 678 (2015)
  67. Sibatov R T, Svetukhin V V Chaos, Solitons & Fractals 81 519 (2015)
  68. Prozorov A A, Trifonov A Yu, Shapovalov A V Russ Phys J 58 399 (2015)
  69. Xu F, Gao Y, Zhang W Advances In Mathematical Physics 2015 1 (2015)
  70. Zaburdaev V, Denisov S, Klafter J Rev. Mod. Phys. 87 483 (2015)
  71. Sibatov R T, Svetukhin V V Theor Math Phys 183 846 (2015)
  72. Arkashov N S, Seleznev V A Teoriya Veroyatnostei Ee Primeneniya 60 209 (2015)
  73. Galiyarova N M Bull. Russ. Acad. Sci. Phys. 78 971 (2014)
  74. Wolfendale A, Erlykin A Astroparticle Physics 53 115 (2014)
  75. Mészáros C, Kirschner I, Bálint Á Continuum Mech. Thermodyn. 26 447 (2014)
  76. Valenti D, Guarcello C, Spagnolo B Phys. Rev. B 89 (21) (2014)
  77. Grafov B M Russ J Electrochem 50 92 (2014)
  78. Milovanov A V, Iomin A Phys. Rev. E 89 (6) (2014)
  79. Lagutin A, Tyumentsev A J. Phys.: Conf. Ser. 409 012043 (2013)
  80. Bakunin O G Uspekhi Fizicheskikh Nauk 183 257 (2013) [Bakunin O G Phys.-Usp. 56 243 (2013)]
  81. Uchaikin V V Uspekhi Fizicheskikh Nauk 183 1175 (2013) [Uchaikin V V Phys.-Usp. 56 1074 (2013)]
  82. Uchaikin V V Fractional Derivatives for Physicists and Engineers Nonlinear Physical Science Chapter 3 (2013) p. 107
  83. Uchaikin V V J. Exp. Theor. Phys. 116 897 (2013)
  84. Uchaikin V V Fractional Derivatives for Physicists and Engineers Nonlinear Physical Science Chapter 5 (2013) p. 257
  85. Arkashov N S, Seleznev V A Sib Math J 54 968 (2013)
  86. Erlykin A D, Wolfendale A W Astroparticle Physics 42 70 (2013)
  87. (Eleventh International Conference on Correlation Optics) Vol. Eleventh International Conference on Correlation OpticsInvestigation of Cauchy and nonlocal problems of diffusion equationOleg V.AngelskyV. V.GorodetskyYa. M.Drin9066 (2013) p. 90660T
  88. Butkovskii A G, Postnov S S, Postnova E A Autom Remote Control 74 543 (2013)
  89. D’Ovidio M Stochastic Processes And Their Applications 122 3513 (2012)
  90. Shevtsov B M, Sagitova R N J. Volcanolog. Seismol. 6 116 (2012)
  91. Abrashina-Zhadaeva N, Romanova N 17 447 (2012)
  92. Stanislavsky A, Weron K J. Stat. Mech. 2012 P07020 (2012)
  93. Maksyuta N V, Koshcheev V P, Panina T A J. Synch. Investig. 6 1019 (2012)
  94. Balankin A S, Elizarraraz B E Phys. Rev. E 85 (5) (2012)
  95. Balankin A S, Mena B et al Phys. Rev. E 86 (5) (2012)
  96. Zagainov V A, Timashev S F et al Russ. J. Phys. Chem. B 6 89 (2012)
  97. Grafov B M Russ J Electrochem 48 142 (2012)
  98. Khamzin A A, Nigmatullin R R et al J. Phys.: Conf. Ser. 324 012005 (2011)
  99. Sibatov R T Jetp Lett. 93 503 (2011)
  100. Parovik R I, Shevtsov B M Math Models Comput Simul 2 180 (2010)
  101. Sibatov R T, Uchaikin V V Opt. Spectrosc. 108 761 (2010)
  102. Augello G, Valenti D, Spagnolo B Eur. Phys. J. B 78 225 (2010)
  103. Astaf’ev M G, Grafov B M Russ J Electrochem 46 247 (2010)
  104. Tarasov V E Nonlinear Physical Science Vol. Fractional DynamicsFractional Statistical Mechanics0 Chapter 15 (2010) p. 335
  105. La Cognata A, Valenti D et al Eur. Phys. J. B 77 273 (2010)
  106. Dybiec B J. Stat. Mech. 2010 P03019 (2010)
  107. Dybiec B, Gudowska-Nowak E 20 (4) (2010)
  108. Tarasov V E Nonlinear Physical Science Vol. Fractional DynamicsFractional Dynamics of Media with Long-Range Interaction0 Chapter 8 (2010) p. 153
  109. La Cognata A, Valenti D et al Phys. Rev. E 82 (1) (2010)
  110. Timashev S F, Polyakov Yu S et al Phys. Rev. E 81 (4) (2010)
  111. Povstenko Yu Nonlinear Dyn 59 593 (2010)
  112. Penson K A, Górska K Phys. Rev. Lett. 105 (21) (2010)
  113. Timashev S F, Polyakov Yu S et al Laser Phys. 19 836 (2009)
  114. Artyomov V G, Kapralov P O et al Bull. Lebedev Phys. Inst. 36 209 (2009)
  115. Dybiec B, Gudowska-Nowak E Phys. Rev. E 80 (6) (2009)
  116. Timashev S F Russ. J. Phys. Chem. B 3 320 (2009)
  117. Nigmatullin R R Phys. Wave Phen. 17 100 (2009)
  118. Uchaikin V V, Sibatov R T J. Exp. Theor. Phys. 109 537 (2009)
  119. Bryukhanov V V, Samusev I G et al Russ Phys J 52 119 (2009)
  120. Uchaikin V V, Sibatov R T J. Stat. Mech. 2009 P04001 (2009)
  121. Dubkov A A, La Cognata A, Spagnolo B J. Stat. Mech. 2009 P01002 (2009)
  122. Lagutin A A, Tyumentsev A G, Volkov N V Bull. Russ. Acad. Sci. Phys. 73 561 (2009)
  123. Datsko B Y, Gafiychuk V V Phys. Scr. T136 014027 (2009)
  124. Milovanov A V Phys. Rev. E 79 (4) (2009)
  125. Sibatov R T, Uchaikin V V Uspekhi Fizicheskikh Nauk 179 1079 (2009)
  126. Kolesnichenko A V Sol Syst Res 42 391 (2008)
  127. Zaburdaev V Yu, Popov P V et al J. Exp. Theor. Phys. 106 999 (2008)
  128. Potapov A A, Gil’mutdinov A Kh, Ushakov P A J. Commun. Technol. Electron. 53 977 (2008)
  129. Shkilev V P J. Exp. Theor. Phys. 107 892 (2008)
  130. Lagutin A A, Tyumentsev A G, Yushkov A V Nuclear Physics B - Proceedings Supplements 175-176 555 (2008)
  131. DUBKOV A A, SPAGNOLO B, UCHAIKIN V V Int. J. Bifurcation Chaos 18 2649 (2008)
  132. Bryukhanov V V, Samusev I G et al Russ. J. Phys. Chem. 82 258 (2008)
  133. Zaburdaev V Yu J Stat Phys 133 159 (2008)
  134. UCHAIKIN V V, CAHOY D O, SIBATOV R T Int. J. Bifurcation Chaos 18 2717 (2008)
  135. Nigmatullin R R Phys. Wave Phen. 16 (2) (2008)
  136. Sibatov R T, Uchaikin V V Semiconductors 41 335 (2007)
  137. Lagutin A A, Tyumentsev A G, Yushkov A V Bull. Russ. Acad. Sci. Phys. 71 590 (2007)
  138. Nigmatullin R R, Arbuzov A A et al Journal Of Non-Crystalline Solids 353 4143 (2007)
  139. Nigmatullin R R, Arbuzov A A et al Physica B: Condensed Matter 388 418 (2007)
  140. Tarasov V E, Zaslavsky G M Physica A: Statistical Mechanics And Its Applications 383 291 (2007)
  141. Petukhov B V Phys. Solid State 49 251 (2007)
  142. Uchaikin V V, Sibatov R T Jetp Lett. 86 512 (2007)
  143. Mészáros C, Bálint Á et al Drying Technology 25 1297 (2007)
  144. Zaburdaev V Yu J Stat Phys 123 871 (2006)
  145. Tarasov V E 16 (3) (2006)
  146. Tarasov V E 13 (5) (2006)
  147. Tarasov V E J. Phys. A: Math. Gen. 39 14895 (2006)
  148. Tarasov V E, Zaslavsky G M Communications In Nonlinear Science And Numerical Simulation 11 885 (2006)
  149. Tarasov V E, Zaslavsky G M J. Phys. A: Math. Gen. 39 9797 (2006)
  150. Tarasov V E, Zaslavsky G M 16 (2) (2006)
  151. Shpyrko S, Ryazanov V V Eur. Phys. J. B 54 345 (2006)
  152. Tarasov V E J. Phys. A: Math. Gen. 39 8395 (2006)
  153. Tarasov V E J. Phys. A: Math. Gen. 39 8409 (2006)
  154. Tarasov V E 47 (9) (2006)
  155. Dykhne A M, Dranikov I L et al Phys. Rev. E 72 (6) (2005)
  156. Chukbar K, Zaburdaev V Phys. Rev. E 71 (6) (2005)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions