The continuous time random walk, still trendy: fifty-year history, state of art and outlook

Open Access
Colloquium
Part of the following topical collections:
  1. Topical issue: Continuous Time Random Walk still trendy: Fifty-year history, current state, and outlook

Abstract

In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Phys. J. B Special Issue [http://epjb.epj.org/open-calls-for-papers/123-epj-b/1090-ctrw-50-years-on] containing articles which show incredible possibilities of the CTRWs.

References

  1. 1.
    E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)ADSCrossRefGoogle Scholar
  2. 2.
    J. Klafter, G. Zumofen, J. Phys. Chem. 98, 7366 (1994)CrossRefGoogle Scholar
  3. 3.
    V. Zaburdaev, S. Denisov, J. Klafter, Rev. Mod. Phys. 87, 483 (2015)ADSCrossRefGoogle Scholar
  4. 4.
  5. 5.
    H. Scher, M. Lax, J. Non-Crystalline Solids 8-10, 497 (1972)ADSCrossRefGoogle Scholar
  6. 6.
    H. Scher, M. Lax, Phys. Rev. B 7, 4491 (1973)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    E.W. Montroll, B.J. West, in Fluctuation Phenomena, Studies in Statistical Mechanics, edited by E.W. Montroll, J.L. Lebowitz (Elsevier Amsterdam, New York, Oxford, 1979), Vol. VII, Chap. 2, p. 61Google Scholar
  8. 8.
    G.H. Weiss, R.J. Rubin, in Advances in Chemical Physics, edited by I. Prigogine, S.A. Rice (J. Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1983), Vol. LII, p. 363Google Scholar
  9. 9.
    E.W. Montroll, M.F. Shlesinger, in Nonequilibrium Phenomena II. From Stochastics to Hydrodynamics, Studies in Statistical Mechanics, edited by J.L. Lebowitz, E.W. Montroll (Elsevier, Amsterdam, New York, Oxford, 1984), Vol. XI, Chap. 1, p. 1Google Scholar
  10. 10.
    G.H. Weiss, in Fractals in Science, edited by A. Bunde, S. Havlin (Springer-Verlag, Berlin, 1995), p. 118Google Scholar
  11. 11.
    J.F. Gillis, Proc. Comb. Phil. Soc. 51, 639 (1955)ADSCrossRefGoogle Scholar
  12. 12.
    J.W. Haus, K.W. Kehr, Phys. Rep. 150, 141 (1987)CrossRefGoogle Scholar
  13. 13.
    M. Montero, J. Masoliver, Phys. Rev. E 76, 061115 (2007)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    K.W. Kehr, R. Kutner, K. Binder, Phys. Rev. B 23, 4931 (1981)ADSCrossRefGoogle Scholar
  15. 15.
    R. Kutner, J. Phys. C 18, 6323 (1985)ADSCrossRefGoogle Scholar
  16. 16.
    P. Paradisi, G. Kaniadakis, A.M. Scarfone, Chaos, Solitons, and Fractals 81, 407 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    I. Eliazer, M.F. Shlesinger, Fractional Motions, Phys. Rep. 527, 101 (2013)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Scher, E.W. Montroll, Phys. Rev. B 12, 2455 (1975)ADSCrossRefGoogle Scholar
  19. 19.
    G. Pfister, H. Scher, Adv. in Physics 27, 747 (1978)ADSCrossRefGoogle Scholar
  20. 20.
    M. Esposito, K. Lindenberg, Phys. Rev. E 77, 051119 (2008)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    J.-P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    I.M. Sokolov, J. Klafter, Chaos 15, 26103 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    G. Zumofen, J. Klafter, A. Blumen, in Disordered Effects on Relaxation Processes, Glasses, Polymers, Proteins, edited by R. Richert, A. Blumen (Springer-Verlag, Berlin, 1994), p. 251Google Scholar
  24. 24.
    R. Hilfer, in Time, Quantum and Information, edited by L. Castell, O. Ischebeck (Springer-Verlag, Berlin, 2003), p. 235Google Scholar
  25. 25.
    R. Hilfer, in Fractional Dynamics: Recent Advances, edited by J. Klafter, S. Lim, R. Metzler (World Scientific, New Yersay, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2011), p. 207Google Scholar
  26. 26.
    R. Hilfer, Mathematics 3, 626 (2015)CrossRefGoogle Scholar
  27. 27.
    R. Hilfer, Analysis 36, 49 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    W.D. Gill, J. Appl. Phys. 43, 5033 (1972)ADSCrossRefGoogle Scholar
  30. 30.
    G. Pfister, Phys. Rev. Lett. 33, 1474 (1974)ADSCrossRefGoogle Scholar
  31. 31.
    M.E. Scharfe, Phys. Rev. B 2, 5025 (1974)ADSCrossRefGoogle Scholar
  32. 32.
    G. Pfister, Phys. Rev. Lett. 36, 271 (1976)ADSCrossRefGoogle Scholar
  33. 33.
    A.J. Campbell, D.D.C. Bradley, D.G. Lidzey, J. Appl. Phys. 82, 6326 (1997)ADSCrossRefGoogle Scholar
  34. 34.
    J.K.E. Tunaley, Phys. Rev. Lett. 33, 1037 (1974)ADSCrossRefGoogle Scholar
  35. 35.
    J.K.E. Tunaley, J. Stat. Phys. 11, 397 (1974)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    J.K.E. Tunaley, J. Stat. Phys. 12, 1 (1975)ADSCrossRefGoogle Scholar
  37. 37.
    J.K.E. Tunaley, J. Stat. Phys. 14, 461 (1976)ADSCrossRefGoogle Scholar
  38. 38.
    M.M. Meerscheart, P. Straka, The Annals of Probability 42, 1699 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    V.S. Barbu, N. Limnios, Lecture Notes in Statistical Physics (Springer-Verlag, Berlin, Heidelberg, 2008), Vol. 192Google Scholar
  40. 40.
    J. Klafter, G. Zumofen, Physica A 196, 102 (1993)ADSCrossRefGoogle Scholar
  41. 41.
    S. Denisov, V. Zaburdaev, P. Hänggi, Phys. Rev. Lett. 109, 069903(E) (2012)ADSCrossRefGoogle Scholar
  42. 42.
    N. Leibovich, E. Barkai, Phys. Rev. Lett. 115, 080602 (2015)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    N. Leibovich, A. Dechant, E. Lutz, E. Barkai, Phys. Rev. E 94, 052130 (2016)ADSCrossRefGoogle Scholar
  44. 44.
    R. Hilfer, Chaos, Solitons & Fractals 5, 1475 (1995)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    R. Hilfer, Fractals 3, 549 (1995)MathSciNetCrossRefGoogle Scholar
  46. 46.
    R. Hilfer, Physica A 221, 89 (1995)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    J.-P. Bouchaud, J. Phys. I (France) 2, 1705 (1992)CrossRefGoogle Scholar
  48. 48.
    G. Bel, E. Barkai, Phys. Rev. Lett. 94, 240602 (2005)ADSCrossRefGoogle Scholar
  49. 49.
    G. Margolin, E. Barkai, Phys. Rev. Lett. 94, 080601 (2005)ADSCrossRefGoogle Scholar
  50. 50.
    Y. He, S. Burov, R. Metzler, E. Barkai, Phys. Rev. Lett. 101, 058101 (2008)ADSCrossRefGoogle Scholar
  51. 51.
    A. Rebenshtok, E. Barkai, Phys. Rev. Lett. 99, 210601 (2007)ADSCrossRefGoogle Scholar
  52. 52.
    A. Rebenshtok, E. Barkai, J. Stat. Phys. 133, 565 (2008)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    A. Lubelski, I.M. Sokolov, J. Klafter, Phys. Rev. Lett. 100, 250602 (2008)ADSCrossRefGoogle Scholar
  54. 54.
    J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, R. Metzler, Phys. Rev. Lett. 106, 048103 (2011)ADSCrossRefGoogle Scholar
  55. 55.
    A.V. Weigel, B. Simon, M.M. Tamkun, D. Krapf, Proc. Natl. Sci. U.S.A. 108, 6438 (2011)ADSCrossRefGoogle Scholar
  56. 56.
    F. Thiel, I.M. Sokolov, Phys. Rev. E 89, 012136 (2014)ADSCrossRefGoogle Scholar
  57. 57.
    X. Brokmann, J.-P. Hermier, G. Messin, P. Desbiolles, J.-P. Bouchaud, M. Dahan, Phys. Rev. Lett. 90, 120601 (2003)ADSCrossRefGoogle Scholar
  58. 58.
    E. Barkai, Diffusion Fundamentals 20, 4 (2013)Google Scholar
  59. 59.
    R. Metzler, J.-H. Jeon, A.G. Cherstvy, E. Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014)CrossRefGoogle Scholar
  60. 60.
    R. Metzler, Int. J. Mod. Phys. Conf. Ser. 36, 1560007 (2015)CrossRefGoogle Scholar
  61. 61.
    R. Kutner, Chem. Phys. 284, 481 (2002)ADSCrossRefGoogle Scholar
  62. 62.
    M. Kozłowska, R. Kutner, Physica A 357, 282 (2005)ADSCrossRefGoogle Scholar
  63. 63.
    J.K.E. Tunaley, J. Stat. Phys. 15, 149 (1976)ADSCrossRefGoogle Scholar
  64. 64.
    M. Nelkin, A.K. Harrison, Phys. Rev. B 26, 6696 (1982)ADSCrossRefGoogle Scholar
  65. 65.
    I. Goychuk, P. Hänggi, Phys. Rev. Lett. 99, 200601 (2007)ADSCrossRefGoogle Scholar
  66. 66.
    E. Barkai, Y.-C. Cheng, J. Chem. Phys. 118, 6167 (2003)ADSCrossRefGoogle Scholar
  67. 67.
    V.M. Kenkre, E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 9, 45 (1973)ADSCrossRefGoogle Scholar
  68. 68.
    M.F. Shlesinger, J. Stat. Phys. 10, 421 (1974)ADSCrossRefGoogle Scholar
  69. 69.
    V.M. Kenkre, R.S. Knox, Phys. Rev. B 9, 5279 (1974)ADSCrossRefGoogle Scholar
  70. 70.
    V.M. Kenkre, T.S. Rahman, Phys. Lett. A 50, 170 (1974)ADSCrossRefGoogle Scholar
  71. 71.
    V.M. Kenkre, Phys. Rev. B 12, 2150 (1975)ADSCrossRefGoogle Scholar
  72. 72.
    U. Landmann, E.W. Montroll, M.F. Shlesinger, Proc. Natl. Acad. Sci. USA 74, 430 (1977)ADSCrossRefGoogle Scholar
  73. 73.
    U. Landmann, M.F. Shlesinger, Phys. Rev. B 19, 6207 (1979)ADSCrossRefGoogle Scholar
  74. 74.
    U. Landmann, M.F. Shlesinger, Phys. Rev. B 19, 6220 (1979)ADSCrossRefGoogle Scholar
  75. 75.
    D.T. Gillespie, Phys. Lett. A 64, 22 (1977)ADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    M.O. Caceres, H.S. Wio, Z. Phys. B 54, 175 (1984)ADSCrossRefGoogle Scholar
  77. 77.
    J. Masoliver, G. Weiss, Physica A 183, 537 (1990)ADSCrossRefGoogle Scholar
  78. 78.
    J. Klafter, R. Silbey, Phys. Rev. Lett. 44, 55 (1980)ADSCrossRefGoogle Scholar
  79. 79.
    R. Zwanzig, in Lectures in Theoretical Physics, edited by W.E. Brittin, B.W. Downs, J. Downs (Interscience, New York, 1961), Vol. 3, p. 106Google Scholar
  80. 80.
    F. Haake, Springer Tracts in Modern Physics 66, 98 (1973)CrossRefGoogle Scholar
  81. 81.
    G. Schehr, P. Le Doussal, J. Stat. Mech.: Theo. Exp. 2010, P01009 (2010)Google Scholar
  82. 82.
    J. Machta, Phys. Rev. B 24, 5260 (1981)ADSCrossRefGoogle Scholar
  83. 83.
    J. Machta, J. Stat. Phys. 30, 305 (1983)ADSMathSciNetCrossRefGoogle Scholar
  84. 84.
    W. Kuhn, Kolloid-Z. 76, 258 (1936) (in German)CrossRefGoogle Scholar
  85. 85.
    W. Kuhn, Kolloid-Z. 87, 3 (1939) (in German)CrossRefGoogle Scholar
  86. 86.
    J. Bardeen, C. Herring, in Imperfections in Nearly Perfect Crystals, edited by W. Shockley (Wiley, New York, 1952), p. 261Google Scholar
  87. 87.
    J.R. Manning, Diffusion Kinetics for Atoms in Crystals (Van Nostrand Comp. Inc., Princeton, 1968)Google Scholar
  88. 88.
    R. Hilfer, R. Orbach, Chem. Phys. 128, 275 (1988)ADSCrossRefGoogle Scholar
  89. 89.
    J.W. Haus, K.W. Kehr, Solid State Commun. 26, 753 (1978)ADSCrossRefGoogle Scholar
  90. 90.
    J.W. Haus, K.W. Kehr, J. Phys. Chem. Solids 40, 1019 (1979)ADSCrossRefGoogle Scholar
  91. 91.
    J. Masoliver, K. Lindeberg, G.H. Weiss, Physica A 157, 891 (1989)ADSMathSciNetCrossRefGoogle Scholar
  92. 92.
    V. Halpern, Physica A 223, 329 (1996)ADSMathSciNetCrossRefGoogle Scholar
  93. 93.
    J. Perelló, J. Masoliver, A. Kasprzak, R. Kutner, Phys. Rev. E 78, 036108 (2008)ADSCrossRefGoogle Scholar
  94. 94.
    A. Kasprzak, R. Kutner, J. Perelló, J. Masoliver, Eur. Phys. J. B 76, 513 (2010)ADSMathSciNetCrossRefGoogle Scholar
  95. 95.
    P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, A. Vulpiani, Physica D 134, 75 (1999)ADSCrossRefGoogle Scholar
  96. 96.
    R. Artuso, G. Cristadoro, Phys. Rev. Lett. 90, 244101 (2003)ADSCrossRefGoogle Scholar
  97. 97.
    D.P. Sanders, H. Larralde, Phys. Rev. E 73, 026205 (2006)ADSMathSciNetCrossRefGoogle Scholar
  98. 98.
    P. de Anna, T. Le Borgne, M. Dentz, A.M. Tartakovsky, D. Bolster, P. Davy, Phys. Rev. Lett. 110, 184502 (2013)ADSCrossRefGoogle Scholar
  99. 99.
    A. Rebenshtok, S. Denisov, P. Hänggi, E. Barkai, Phys. Rev. Lett. 112, 110601 (2014)ADSCrossRefGoogle Scholar
  100. 100.
    A. Rebenshtok, S. Denisov, P. Hänggi, E. Barkai, Phys. Rev. E 90, 062135 (2014)ADSCrossRefGoogle Scholar
  101. 101.
    L. Seuront, H.E. Stanley, Proc. Natl. Acad. Sci. U.S.A. 111, 2206 (2014)ADSCrossRefGoogle Scholar
  102. 102.
    M. Magdziarz, R. Metzler, W. Szczotka, P. Żebrowski, Phys. Rev. E 85, 051103 (2012)ADSCrossRefGoogle Scholar
  103. 103.
    M. Magdziarz, A. Weron, K. Burnecki, J. Klafter, Phys. Rev. Lett. 103, 180602 (2009)ADSCrossRefGoogle Scholar
  104. 104.
    G.H. Weiss, S. Havlin, J. Stat. Phys. 63, 127 (1990)Google Scholar
  105. 105.
    H. Scher et al., Geophys. Rev. Lett. 29, 1061 (2002)ADSCrossRefGoogle Scholar
  106. 106.
    B. Berkowitz, A. Cortis, M. Dentz, H. Scher, Rev. Geophys. 44, RG2003 (2006)ADSCrossRefGoogle Scholar
  107. 107.
    A. Klemm, R. Metzler, R. Kimmich, Phys. Rev. E 65, 021112 (2002)ADSCrossRefGoogle Scholar
  108. 108.
    S.S. Rogers, C.V.D. Walle, T.A. Waigh, Langmuir 24, 13549 (2008)CrossRefGoogle Scholar
  109. 109.
    I. Golding, E.C. Cox, Phys. Rev. Lett. 96, 098102 (2006)ADSCrossRefGoogle Scholar
  110. 110.
    S.C. Weber, A.J. Spakowitz, J.A. Theriot, Phys. Rev. Lett. 104, 238102 (2010)ADSCrossRefGoogle Scholar
  111. 111.
    J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sorensen, L. Oddershede, R. Metzler, Phys. Rev. Lett. 106, 048103 (2011)ADSCrossRefGoogle Scholar
  112. 112.
    I. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Shav-Tal, E. Barkai, Y. Garini, Phys. Rev. Lett. 103, 018102 (2009)ADSCrossRefGoogle Scholar
  113. 113.
    A.J. Bray, S.N. Majumdar, G. Schehr, Adv. Phys. 62, 225 (2013)ADSCrossRefGoogle Scholar
  114. 114.
    S. Eule, R. Friedrich, J. Stat. Mech.: Theory and Experiment 2014, P12005 (2014)CrossRefGoogle Scholar
  115. 115.
    D. ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, Cambridge, 2000)Google Scholar
  116. 116.
    E.J. Moore, J. Phys. C 7, 339 (1974)ADSCrossRefGoogle Scholar
  117. 117.
    V. Uchaikin, R. Sibatov, Fractional Kinetics in Solids. Anomalous Transport in Semiconductors, Dielectric and Nanosystems (World Scientific New Yersay, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2013)Google Scholar
  118. 118.
    R. Hilfer, L. Anton, Phys. Rev. E 51, R848 (1995)ADSCrossRefGoogle Scholar
  119. 119.
    R. Hilfer, Fractals, 3, 211 (1995)MathSciNetGoogle Scholar
  120. 120.
    R. Hilfer, in Anomalous Transport: Foundations and Applications, edited by R. Klages, G. Radons, I. Sokolov (Wiley-VCH, Weinheim, 2008), p. 17Google Scholar
  121. 121.
    R. Hilfer, in Applications of Fractional Calculus in Physics, edited by R. Hilfer (World Scientific, New Yersay, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2000), p. 87Google Scholar
  122. 122.
    R. Hilfer, in Anomalous Diffusion. From Basics to Applications, Lecture Notes in Physics, edited by R. Kutner, A. Pȩkalski, K. Sznajd-Weron (Springer-Verlag, Berlin, 1999), p. 77Google Scholar
  123. 123.
    R. Hilfer, J. Phys. Chem. B 104, 3914 (2000)CrossRefGoogle Scholar
  124. 124.
    R. Hilfer, Physica A 329, 35 (2003)ADSMathSciNetCrossRefGoogle Scholar
  125. 125.
    R. Hilfer, Fractional Calculus and Applied Analysis 18, 333 (2015)MathSciNetCrossRefGoogle Scholar
  126. 126.
    F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Physica A 287, 468 (2000)ADSCrossRefGoogle Scholar
  127. 127.
    E. Scalas, R. Gorenflo, F. Mainardi, Physica A 284, 376 (2000)ADSMathSciNetCrossRefGoogle Scholar
  128. 128.
    R. Gorenflo, F. Mainardi, in Fractals and Fractional Calculus in Continuuum Mechanics, edited by A. Carpinteri, F. Mainardi (Springer-Verlag, Wien, 1997), p. 223Google Scholar
  129. 129.
    F. Mainardi, Chaos, Solitons, Fractals 7, 1461 (1996)ADSMathSciNetCrossRefGoogle Scholar
  130. 130.
    F. Mainardi, in Fractals and Fractional Calculus in Continuuum Mechanics, edited by A. Carpinteri, F. Mainardi (Springer-Verlag, Wien, 1997), p. 291Google Scholar
  131. 131.
    R. Metzler, J. Klafter, J. Phys. A 37, R161 (2004)ADSCrossRefGoogle Scholar
  132. 132.
    A.I. Saichev, G.M. Zaslavsky, Chaos 7, 753 (1997)ADSMathSciNetCrossRefGoogle Scholar
  133. 133.
    E. Barkai, Chem. Phys. 284, 13 (2002)ADSCrossRefGoogle Scholar
  134. 134.
    A. Compte, Phys. Rev. E 53, 4191 (1996)ADSCrossRefGoogle Scholar
  135. 135.
    H.C. Fogedby, Phys. Rev. Lett. 73, 2517 (1994)ADSCrossRefGoogle Scholar
  136. 136.
    J. Honkonen, Phys. Rev. E 53, 327 (1996)ADSMathSciNetCrossRefGoogle Scholar
  137. 137.
    S. Jespersen, R. Metzler, H.C. Fogedby, Phys. Rev. E 59, 2736 (1999)ADSCrossRefGoogle Scholar
  138. 138.
    R. Metzler, J. Klafter, I. Sokolov, Phys. Rev. E 58, 1621 (1998)ADSCrossRefGoogle Scholar
  139. 139.
    A. Chechkin, I.M. Sokolv, J. Klafter, in Fractional Dynamics. Recent Advances, edited by J. Klafter, S.C. Lim, R. Metzler (World Scientific, New Yersay, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2012), p. 107Google Scholar
  140. 140.
    M. Shlesinger, J. Klafter, G. Zumofen, Fractals 3, 491 (1995)CrossRefGoogle Scholar
  141. 141.
    J.B. Keller, PNAS 101, 1120 (2004)ADSCrossRefGoogle Scholar
  142. 142.
    M. Abdel Aziz, S. Gavin, Phys. Rev. C 70, 034905 (2004)ADSCrossRefGoogle Scholar
  143. 143.
    J. Dunkel, P. Talkner, P. Hänggi, Phys. Rev. E 75, 043001 (2007)ADSGoogle Scholar
  144. 144.
    J. Masoliver, G.H. Weiss, Eur. J. Phys. 17, 190 (1996)CrossRefGoogle Scholar
  145. 145.
    G. H. Weiss, Physica A 311, 381 (2002)ADSMathSciNetCrossRefGoogle Scholar
  146. 146.
    E. Orshinger, X. Zhao, Chinese Ann. Math. B, 1 (2003)Google Scholar
  147. 147.
    E. Orshinger, L. Beghin, Probab. Theory Relat. Fields 128, 141 (2004)CrossRefGoogle Scholar
  148. 148.
    M. D’Ovidio, E. Orshinger, B. Toaldo, Stochastic Analysis and Applications 32, 1009 (2014)MathSciNetCrossRefGoogle Scholar
  149. 149.
    A. Compte, R. Metzler, J. Phys. A 30, 7277 (1997)ADSMathSciNetCrossRefGoogle Scholar
  150. 150.
    R. Metzler, A. Compte, Physica A 268, 454 (1999)ADSCrossRefGoogle Scholar
  151. 151.
    R. Metzler, T. F. Nonnenmacher, Phys. Rev. E 57, 6409 (1998)ADSMathSciNetCrossRefGoogle Scholar
  152. 152.
    J. Masoliver, Phys. Rev. E 93, 052107 (2016)ADSCrossRefGoogle Scholar
  153. 153.
    J. Masoliver, K. Lindenberg, G.H. Weiss, Physica A 182, 891 (1989)ADSCrossRefGoogle Scholar
  154. 154.
    E. Barkai, R. Metzler, J. Klafter, Phys. Rev. E 61, 132 (2000)ADSMathSciNetCrossRefGoogle Scholar
  155. 155.
    E. Heinsalu, M. Patriarca, I. Goychuk, P. Hänggi, Phys. Rev. E 79, 041137 (2009)ADSCrossRefGoogle Scholar
  156. 156.
    E. Heinsalu, M. Patriarca, I. Goychuk, P. Hänggi, Phys. Rev. Lett. 99, 120602 (2007)ADSCrossRefGoogle Scholar
  157. 157.
    T. Srokowski, Phys. Rev. E 79, 040104(R) (2009)ADSMathSciNetCrossRefGoogle Scholar
  158. 158.
    I. Goychuk, P. Hänggi, in Fractional Dynamics. Recent Advances, edited by J. Klafter, S.C. Lim, R. Metzler (World Scientific, New Yersay, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2012), p. 307Google Scholar
  159. 159.
    E. Lutz, in Fractional Dynamics. Recent Advances, edited by J. Klafter, S.C. Lim, R. Metzler (World Scientific, New Yersay, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2012), p. 285Google Scholar
  160. 160.
    S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943)ADSCrossRefGoogle Scholar
  161. 161.
    R. Metzler, J. Klafter, Phys. Rev. E 61, 6308 (2000)ADSCrossRefGoogle Scholar
  162. 162.
    R. Metzler, J. Klafter, J. Phys. Chem. B 104, 3858 (2000)CrossRefGoogle Scholar
  163. 163.
    R. Metzler, Phys. Rev. E 62, 6233 (2000)ADSMathSciNetCrossRefGoogle Scholar
  164. 164.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981)Google Scholar
  165. 165.
    R. Metzler, E. Barkai, J. Klafter, Phys. Rev. Lett. 82, 3563 (1999)ADSCrossRefGoogle Scholar
  166. 166.
    R. Metzler, E. Barkai, J. Klafter, Europhys. Lett. 46, 431 (1999)ADSMathSciNetCrossRefGoogle Scholar
  167. 167.
    E. Lutz, Phys. Rev. Lett. 86, 2208 (2001)ADSCrossRefGoogle Scholar
  168. 168.
    F.E. Peseckis, Phys. Rev. A 36, 892 (1987)ADSCrossRefGoogle Scholar
  169. 169.
    D. Kusnezov, A. Bulgac, G.D. Dang, Phys. Rev. Lett. 82, 1136 (1999)ADSCrossRefGoogle Scholar
  170. 170.
    E. Lutz, Phys. Rev. A 67, 051402 (2003)ADSCrossRefGoogle Scholar
  171. 171.
    V. Seshadri, B.J. West, Proc. Natl. Acad. Sci. USA 79, 4501 (1982)ADSCrossRefGoogle Scholar
  172. 172.
    R. Kutner, Physica A 264, 84 (1999)ADSCrossRefGoogle Scholar
  173. 173.
    V.Yu. Zaburdaev, J. Stat. Phys. 123, 871 (2006)ADSMathSciNetCrossRefGoogle Scholar
  174. 174.
    M.F. Shlesinger, J. Klafter, Y. Wong, J. Stat. Phys. 27, 499 (1982)ADSCrossRefGoogle Scholar
  175. 175.
    E. Barkai, Chem. Phys. 284, 13 (2002)ADSCrossRefGoogle Scholar
  176. 176.
    R. Kutner, M. Regulski, Physica A 264, 107 (1999)ADSCrossRefGoogle Scholar
  177. 177.
    H. Katori, S. Schlipf, H. Walther, Phys. Rev. Lett. 79, 2221 (1997)ADSCrossRefGoogle Scholar
  178. 178.
    L. Turgeman, S. Carmi, E. Barkai, Phys. Rev. Lett. 103, 190201 (2009)ADSMathSciNetCrossRefGoogle Scholar
  179. 179.
    S. Carmi, E. Barkai, in Fractional Dynamics. Recent Advances, edited by J. Klafter, S.C. Lim, R. Metzler (World Scientific, New Yersay, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2012), p. 185Google Scholar
  180. 180.
    J. Klafter, G. Zumofen, Phys. Rev. E 49, 4873 (1994)ADSCrossRefGoogle Scholar
  181. 181.
    V. Zaburdaev, K. Chukbar, J. Exp. Ttheor. Phys. 94, 252 (2002)ADSCrossRefGoogle Scholar
  182. 182.
    J. Klafter, I. Sokolov, First Steps in Random Walks: From Tools to Applications (Oxford University Press, Oxford, 2011)Google Scholar
  183. 183.
    J. Klafter, A. Blumen, M.F. Shlesinger, Phys. Rev. A 35, 3081 (1987)ADSMathSciNetCrossRefGoogle Scholar
  184. 184.
    G. Zumofen, J. Klafter, Phys. Rev. E 47, 851 (1993)ADSCrossRefGoogle Scholar
  185. 185.
    R. Kutner, F. Świtała, Quant. Fin. 3, 201 (2003)CrossRefGoogle Scholar
  186. 186.
    E. Barkai, J. Klafter, in Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas, edited by S. Benkadda, G.M. Zaslavsky (Springer-Verlag, Berlin, Heidelberg, 1998), p. 373Google Scholar
  187. 187.
    V.Yu. Zaburdaev, M. Schmiedeberg, H. Stark, Phys. Rev. E 78, 011119 (2008)ADSCrossRefGoogle Scholar
  188. 188.
    V.Yu. Zaburdaev, S. Denisov, P. Hänggi, Phys. Rev. Lett. 106, 180601 (2011)ADSCrossRefGoogle Scholar
  189. 189.
    V.Yu. Zaburdaev, S. Denisov, P. Hänggi, Phys. Rev. Lett. 110, 170604 (2013)ADSCrossRefGoogle Scholar
  190. 190.
    P. Tong, W. Goldberg, Phys. Lett. A 127, 147 (1988)ADSCrossRefGoogle Scholar
  191. 191.
    I.A. Min, I. Mezić, A. Leonard, Phys. Fluids 8, 1169 (1996)ADSMathSciNetCrossRefGoogle Scholar
  192. 192.
    K.V. Chukbar, Plasma Phys. Rep. 25, 77 (1999)ADSGoogle Scholar
  193. 193.
    E. Ben-Naim, B. Machta, J. Machta, Phys. Rev. E 72, 021302 (2005)ADSCrossRefGoogle Scholar
  194. 194.
    E. Trizac, A. Barrat, M.H. Ernst, Phys. Rev. E 76, 031305 (2007)ADSCrossRefGoogle Scholar
  195. 195.
    A. Hasegawa, K. Mima, M. Duang-van, Phys. Rev. Lett. 54, 2608 (1985)ADSCrossRefGoogle Scholar
  196. 196.
    Z. Meng, R.M. Thorne, D. Summers, J. Plasma Phys. 47, 445 (1992)ADSCrossRefGoogle Scholar
  197. 197.
    C. Tsallis, J. Stat. Phys. 52, 479 (1988)ADSCrossRefGoogle Scholar
  198. 198.
    C. Tsallis, Braz. J. Phys. 29, 1 (1999)ADSCrossRefGoogle Scholar
  199. 199.
    H. Takagi, M.J. Sato, T. Yanagida, M. Ueda, PLoS One 3, e2648 (2008)ADSCrossRefGoogle Scholar
  200. 200.
    H. Aref et al., arXiv:1403.2953 (2014)
  201. 201.
    T.H. Solomon, E.R. Weeks, H.L. Swinney, Phys. Rev. Lett. 71, 3975 (1993)ADSCrossRefGoogle Scholar
  202. 202.
    R.R. Weeks, T.H. Solomon, J.S. Urbach, H.L. Swinney, in Lévy Flights and Related Topics in Physics, edited by M.F. Shlesinger, G.M. Zaslavsky, U. Frisch, Lectures Notes in Physics (Springer-Verlag, Berlin, Heidelberg, 1995), Vol. 450, p. 4Google Scholar
  203. 203.
    M.F. Shlesinger, G.M. Zaslavsky, J. Klafter, Nature (London) 363, 31 (1993)ADSCrossRefGoogle Scholar
  204. 204.
    A. Davis, A. Marshak, in Fractal Frontiers, edited by M.M. Novak, T.G. Dewey (World Scientific, New Yersay, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 1997), p. 63Google Scholar
  205. 205.
    M. Nirmal, B.O. Dabbousi, M.G. Bawendi, J.J. Macklin, J.K. Trautman, T.D. Harris, L.E. Brus, Nature 383, 802 (1996)ADSCrossRefGoogle Scholar
  206. 206.
    G. Margolin, V. Protasenko, M. Kuno, E. Barkai, J. Phys. Chem. B 110, 19053 (2006)CrossRefGoogle Scholar
  207. 207.
    S. Sadegh, E. Barkai, D. Krapf, New J. Phys. 16, 113054 (2014)ADSCrossRefGoogle Scholar
  208. 208.
    E. Barkai, Y. Garini, R. Metzler, Physics Today 65, 29 (2012)CrossRefGoogle Scholar
  209. 209.
    F.D. Stefani, J.P. Hoogenboom, E. Barkai, Physics Today 62, 34 (2009)CrossRefGoogle Scholar
  210. 210.
    A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988)ADSCrossRefGoogle Scholar
  211. 211.
    F. Bardou, J.-P. Bouchaud, O. Emile, A. Aspect, C. Cohen-Tannoudji, Phys. Rev. Lett. 72, 203 (1994)ADSCrossRefGoogle Scholar
  212. 212.
    J. Reichel, F. Bardou, M. Ben Dahan, E. Peik, S. Rand, C. Salomon, C. Cohen-Tannoudji, Phys. Rev. Lett. 75, 4575 (1995)ADSCrossRefGoogle Scholar
  213. 213.
    E. Barkai, E. Aghion, D.A. Kessler, Phys. Rev. X 4, 021036 (2014)Google Scholar
  214. 214.
    D.A. Kessler, E. Barkai, Phys. Rev. Lett. 108, 230602 (2012)ADSCrossRefGoogle Scholar
  215. 215.
    M.F. Shlesinger, J. Klafter, in On Growth and Form, Proceedings of the Cargese Summer School, edited by H.E. Stanley, N. Ostrowski (Nijhoff Publ., 1985), p. 279Google Scholar
  216. 216.
    G.M. Viswanathan, V. Afanasyev, S.V. Buldyrev, E.J. Murphy, E.P. Prince, H.E. Stanley, Nature 381, 413 (1996)ADSCrossRefGoogle Scholar
  217. 217.
    G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. da Luz, E.P. Rapso, H.E. Stanley, Nature 401, 911 (1999)ADSCrossRefGoogle Scholar
  218. 218.
    O. Bénichou, C. Loverdo, M. Moreau, R. Voituriez, Rev. Mod. Phys. 83, 81 (2011)ADSCrossRefGoogle Scholar
  219. 219.
    G.M. Viswanathan, M. da Luz, E. Raposo, H. Stanley, The Physics of Foraging: An Introduction to Random Searches and Bilogical Encounters (Cambridge University Press, Cambridge, 2011)Google Scholar
  220. 220.
    V. Méndez, D. Campos, F. Bartumeus, Stochastic Foundations in Movement Ecology (Springer-Verlag, Berlin, Heidelberg, 2014)Google Scholar
  221. 221.
    E. Scalas, in The Complex Networks of Economic Interactions, Lecture Notes in Economics and Mathematical Systems (Springer-Verlag, Berlin, Heidelberg, 2006), Vol. 567, p. 3Google Scholar
  222. 222.
    E. Scalas, R. Gorenflo, F. Mainardi, Physica A 287, 486 (2000)Google Scholar
  223. 223.
    J. Masoliver, M. Montero, G.H. Weiss, Phys. Rev. E 67, 021112 (2003)ADSCrossRefGoogle Scholar
  224. 224.
    J. Masoliver, M. Montero, J. Perelló, G.H. Weiss, J. Econ. Behavior and Organization 61, 577 (2006)CrossRefGoogle Scholar
  225. 225.
    Kyungsik Kim, Seong-Min Yoon, Soo Yong Kim, Dong-In Lee, Enrico Scalas, J. Korean Phys. Soc. 50, 182 (2007)CrossRefGoogle Scholar
  226. 226.
    T. Gubiec, R. Kutner, Phys. Rev. E 82, 046119 (2010)ADSMathSciNetCrossRefGoogle Scholar
  227. 227.
    M. Denys, T. Gubiec, R. Kutner, M. Jagielski, H.E. Stanley, Phys. Rev. E 94, 042305 (2016)ADSCrossRefGoogle Scholar
  228. 228.
    E. Barkai, V. Fleurov, Phys. Rev. E 58, 1296 (1998)ADSCrossRefGoogle Scholar
  229. 229.
    A. Dechant, E. Lutz, D.A. Kessler, E. Barkai, Phys. Rev. X 4, 011022 (2014)Google Scholar

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Authors and Affiliations

  1. 1.Faculty of Physics, University of WarsawWarsawPoland
  2. 2.Departament de Fisica Matèria Condensada and Institute of Complex Systems, Universitat de BarcelonaBarcelonaSpain

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