Skip to main content
SpringerLink
Log in

Mittag-Leffler Pattern in Anomalous Diffusion

  • Chapter
Advances in the Theory and Applications of Non-integer Order Systems

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 257))

  • 1242 Accesses

Abstract

Various systems described by the bi-fractional Fokker-Planck-Smoluchowski equation display some very general and universal properties. These universal characteristics originate in the underlying competition between long jumps (fractional space derivative) and long waiting times (fractional time derivative). Using a few selected model examples the universal features of anomalous diffusion will be demonstrated.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Montroll, E.W., Weiss, G.H.: Random walks on lattices II. J. Math. Phys. 6, 167 (1965)

    Article  MathSciNet  Google Scholar 

  2. Scher, H., Montroll, E.W.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12(6), 2455 (1975)

    Article  Google Scholar 

  3. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339(1), 1 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Klages, R., Radons, G., Sokolov, I.M.: Anomalous transport: Foundations and applications. Wiley-VCH, Weinheim (2008)

    Book  Google Scholar 

  5. Fogedby, H.C.: Langevin equations for continuous time Lévy flights. Phys Rev. E 50(2), 1657 (1994)

    Article  MathSciNet  Google Scholar 

  6. Helmstetter, A., Sornette, D.: Diffusion of epicenters of earthquake aftershocks, Omori’s law, and generalized continuous-time random walk models. Phys. Rev. E 66(6), 061104 (2002)

    Article  Google Scholar 

  7. Dybiec, B., Gudowska-Nowak, E.: Discriminating between normal and anomalous random walks. Phys. Rev. E 80(4), 061122 (2009)

    Article  Google Scholar 

  8. Metzler, R., Barkai, E., Klafter, J.: Deriving fractional Fokker-Planck equations from a generalized Master equation. Europhys. Lett. 46(4), 431 (1999)

    Article  MathSciNet  Google Scholar 

  9. Sokolov, I.M.: Solutions of a class of non-Markovian Fokker-Planck equations. Phys. Rev. E 66(4), 041101

    Google Scholar 

  10. Magdziarz, M., Weron, A.: Competition between subdiffusion and Lévy flights: A Monte Carlo approach. Phys. Rev. E 75(5), 056702 (2007)

    Article  Google Scholar 

  11. Magdziarz, M., Weron, A., Klafter, J.: Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: The case of a time-dependent force. Phys. Rev. Lett. 101(21), 210601 (2008)

    Article  Google Scholar 

  12. Magdziarz, M., Weron, A., Weron, K.: Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation. Phys. Rev. E 75(1), 016708 (2007)

    Article  MathSciNet  Google Scholar 

  13. Dybiec, B., Gudowska-Nowak, E., Hänggi, P.: Lévy-Brownian motion on finite intervals: Mean first passage time analysis. Phys. Rev. E 73(4), 046104 (2006)

    Article  Google Scholar 

  14. Dybiec, B.: Anomalous diffusion: Temporal non-Markovianity and weak ergodicity breaking. J. Stat. Mech., P08025 (2009)

    Google Scholar 

  15. Janicki, A., Weron, A.: Simulation and chaotic behavior of α-stable stochastic processes. Marcel Dekker, New York (1994)

    Google Scholar 

  16. Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: Solutions and applications. Chaos 7, 753 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Barkai, E.: CTRW pathways to the fractional diffusion equation. Chem. Phys. 284(1-2), 13 (2002)

    Article  Google Scholar 

  18. Metzler, R., Klafter, J.: The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37(31), R161 (2004)

    Article  MathSciNet  Google Scholar 

  19. Dybiec, B., Chechkin, A.V., Sokolov, I.M.: Stationary states in a single-well potential under Lévy noises. J. Stat. Mech., P07008 (2010)

    Google Scholar 

  20. Dybiec, B.: Approaching stationarity: Competition between long jumps and long waiting times. J. Stat. Mech., P03019 (2010)

    Google Scholar 

  21. Dybiec, B., Sokolov, I.M., Chechkin, A.V.: Relaxation to stationary states for anomalous diffusion. Comm. Nonlinear. Sci. Numer. Simulat. 16, 4549 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Scalas, E., Gorenflo, R., Mainardi, F.: Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. Phys. Rev. E 69(1), 011107 (2004)

    Article  MathSciNet  Google Scholar 

  23. Scalas, E.: The application of continuous-time random walks in finance and economics. Physica A 362, 225 (2006)

    Article  Google Scholar 

  24. Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  25. Glöckle, W.G., Nonnenmacher, T.F.: Fractional relaxation and the time-temperature superposition principle. Rheol. Acta 33(4), 337 (1994)

    Article  Google Scholar 

  26. Zoia, A., Rosso, A., Kardar, M.: Fractional Laplacian in bounded domains. Phys. Rev. E 76(2), 021116 (2007)

    Article  MathSciNet  Google Scholar 

  27. Dybiec, B.: Anomalous diffusion on finite intervals. J. Stat. Mech., P01011 (2010)

    Google Scholar 

  28. Dybiec, B.: Universal character of escape kinetics from finite intervals. Acta Phys. Pol. B 41(5), 1127 (2010)

    Google Scholar 

  29. Dybiec, B.: Escape from the potential well: Competition between long jumps and long waiting times. J. Chem. Phys. 133(43), 244114 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. Reymonta 4, 30-059, Kraków, Poland

    Bartłomiej Dybiec

Authors
  1. Bartłomiej Dybiec
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. , Dept. of Automatics & Biomedical Eng., AGH University of Science and Technology, Al. Mickiewicza 20, Cracow, 30-059, Poland

    Wojciech Mitkowski

  2. , Systems Research Institute, Polish Academy of Sciences, Ul. Newelska 6, Warsaw, 01-447, Poland

    Janusz Kacprzyk

  3. , Dept. of Automatics & Biomedical Eng., AGH University of Science and Technology, Al. Mickiewicza 30, Cracow, 30-059, Poland

    Jerzy Baranowski

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Dybiec, B. (2013). Mittag-Leffler Pattern in Anomalous Diffusion. In: Mitkowski, W., Kacprzyk, J., Baranowski, J. (eds) Advances in the Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol 257. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00933-9_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-00933-9_12

  • Publisher Name: Springer, Heidelberg

  • Print ISBN: 978-3-319-00932-2

  • Online ISBN: 978-3-319-00933-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation