Advertisement

On fractional diffusion and its relation with continuous time random walks

  • R. Hilfer
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 519)

Abstract

Time evolutions whose infinitesimal generator is a fractional time derivative arise generally in the long time limit. Such fractional time evolutions are considered here for random walks. An exact relationship is established between the fractional master equation and a separable continuous time random walk of the Montroll-Weiss type. The waiting time density can be expressed using a generalized Mittag-Leffler function. The first moment of the waiting density does not exist.

Keywords

Continuous Time Master Equation Random Environment Infinitesimal Generator Fractional Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barber, M., Ninham, B. (1970): Random and Restricted Walks. New York: Gordon and Breach Science Publ.zbMATHGoogle Scholar
  2. Compte, A. (1996): Stochastic foundations of fractional dynamics, Phys. Rev. E, 55, 4191CrossRefADSGoogle Scholar
  3. Erdelyi, A., et al. (1955): Higher Transcendental Functions, vol. III. New York: Mc Graw Hill Book Co.zbMATHGoogle Scholar
  4. Feller, W. (1971): An Introduction to Probability Theory and Its Applications, vol. II. New York: WileyzbMATHGoogle Scholar
  5. Fox, C. (1961): The G and H functions as symmetrical Fourier kernels, Trans. Am. Math. Soc. 98, 395zbMATHCrossRefGoogle Scholar
  6. Haus, J., Kehr, K. (1987): Diffusion in regular and disordered lattices, Phys. Rep. 150, 263CrossRefADSGoogle Scholar
  7. Hilfer, R. (1993): Classification theory for anequilibrium phase transitions, Phys. Rev. E 48, 2466CrossRefADSMathSciNetGoogle Scholar
  8. Hilfer, R. (1995a): Fractional dynamics, irreversibility and ergodicity breaking, Chaos, Solitons & Fractals 5, 1475zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. Hilfer, R. (1995b): Foundations of fractional dynamics, Fractals 3, 549zbMATHCrossRefMathSciNetGoogle Scholar
  10. Hilfer, R. (1995c): An extension of the dynamical foundation for the statistical equilibrium concept, Physica A 221, 89CrossRefADSMathSciNetGoogle Scholar
  11. Hilfer, R. (1995d): Exact solutions for a class of fractal time random walks, Fractals 3(1), 211zbMATHCrossRefMathSciNetGoogle Scholar
  12. Hilfer, R. (1998): Applications of Fractional Calculus in Physics. Singapore: World Scientific Publ. Co., in VorbereitungGoogle Scholar
  13. Hilfer, R., Anton, L. (1995): Fractional master equations and fractal time random walks, Phys. Rev. E, Rapid Commun., 51, 848ADSGoogle Scholar
  14. Hughes, B. (1995): Random Walks and Random Environments, vol. 1. Oxford: Clarendon PresszbMATHGoogle Scholar
  15. Hughes, B. (1996): Random Walks and Random Environments, vol. 2. Oxford: Clarendon PresszbMATHGoogle Scholar
  16. Klafter, J., Blumen, A., Shlesinger, M. (1987): Stochastic pathway to anomalous diffusion, Phys. Rev. A 35, 3081CrossRefADSMathSciNetGoogle Scholar
  17. Metzler, J.K.R., Sokolov, I., PreprintGoogle Scholar
  18. Montroll, E., West, B. (1979): On an enriched collection of stochastic processes, in Fluctuation Phenomena (E. Montroll and J. Lebowitz, eds.), (Amsterdam), p. 61, North Holland Publ. Co.Google Scholar
  19. Shlesinger, M. (1974): Asymptotic solutions of continuous time random walks, J. Stat. Phys. 10, 421CrossRefMathSciNetADSGoogle Scholar
  20. Shlesinger, M., Klafter, J., Wong, Y. (1982): Random walks with infinite spatial and temporal moments, J. Stat. Phys. 27, 499zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. Tunaley, J. (1974): Asymptotic solutions of the continuous time random walk model of diffusion, J. Stat. Phys. 11, 397CrossRefMathSciNetADSGoogle Scholar
  22. Tunaley, J. (1975): Some properties of the asymptotic solutions of the Montroll-Weiss equation, J. Stat. Phys. 12, 1CrossRefADSGoogle Scholar
  23. Weiss, G., Rubin, R. (1983): Random walks: Theory and selected applications, Adv. Chem. Phys. 52, 363CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • R. Hilfer
    • 1
    • 2
  1. 1.Universität StuttgartICA-1Stuttgart
  2. 2.Institut für PhysikUniversität MainzMainzGermany

Personalised recommendations