International Journal of Theoretical Physics

, Volume 39, Issue 8, pp 2087–2105 | Cite as

Montroll–Weiss Problem, Fractional Equations, and Stable Distributions

  • Vladimir V. Uchaikin


Asymptotic solutions of the m-dimensional Montroll–Weiss'jump problem areobtained. They cover both the subdiffusive and the superdiffusive regime, obeyfractional differential equations, and are expressed in terms of stable distributions.Analytical investigation and numerical calculations of anomalous diffusiondistributions are performed and their properties are discussed.


Differential Equation Field Theory Elementary Particle Numerical Calculation Quantum Field Theory 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. V. V. Afanasiev, R. Z. Sagdeev, and G. M. Zaslavsky, (1991). Chaos 1(2), 143.Google Scholar
  2. V. Balakrishnan, (1985). Physica A 132, 569.Google Scholar
  3. E. Barkai, R. Metzler, and J. Klafter, (2000). Phys. Rev. E 61, 192.Google Scholar
  4. L. N. Bol'shev, V.M. Zolotarev, E. S. Kedrova and M.A. Rybinskaya, (1968). Theor. Probability Appl. 15, 299.Google Scholar
  5. J.-P. Bouchaud, and A. Georges, (1990). Phys. Rep. 195, 127.Google Scholar
  6. A. Compte, (1996). Phys. Rev. E 53, 4191.Google Scholar
  7. A. Compte, D. Jou, and Y. Katayama, (1997). J. Phys. A: Math. Gen. 30, 1023.Google Scholar
  8. H. E. Daniels, (1954). Ann. Math. Phys. 25, 631.Google Scholar
  9. E. F. Fama, and R. Roll. (1968). J. Amer. Statist. Assoc. 63, 817.Google Scholar
  10. W. Feller, (1971). An Introduction to Probability Theory, Vol. 2 (Wiley, New York).Google Scholar
  11. C. Fox, (1961). Trans. Am. Math. Soc. 98, 395.Google Scholar
  12. R. Hilfer, (1995). Fractals 3, 211.Google Scholar
  13. D. R. Holt and E. L. Crow, (1973). J. Res. NBS—B. Math. Sci. 77B, 143.Google Scholar
  14. M. B. Isichenko, (1992). Rev. Mod. Phys. 64, 961.Google Scholar
  15. Yu. L. Klimontovich, (1995). Statistical Theory of Opened Systems, Kluwer, Dordrecht.Google Scholar
  16. M. Kotulski, (1995). J. Stat. Phys. 81, 771.Google Scholar
  17. F. Mainardi, P. Paradisi, and R. Gorenflo, (1999). In Econophysics: An Emerging Science, J. Kertesz and I. Kondor, eds., Kluwer, Dordrecht, in press, pp. 39.Google Scholar
  18. A. M. Mathai and R. K. Saxena, (1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Lect. Not. Math. 348, (Springer Verlag, Heidelberg).Google Scholar
  19. K. S. Miller and B. Ross, (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, (Wiley, New York).Google Scholar
  20. A. S. Monin, (1955). Doklady Akad. Nauk SSSR 105, 256 (in Russian).Google Scholar
  21. E. W. Montroll and G. H. Weiss, (1965). J. Math. Phys. 6, 167.Google Scholar
  22. R. R. Nigmatullin, (1986). Phys. Stat. Sol (b) 133, 425.Google Scholar
  23. K. B. Oldham and J. Spanier, (1974). The Fractional Calculus, (Academic Press, New York).Google Scholar
  24. A. I. Saichev and G. M. Zaslawsky, (1997). Chaos 7(4), 753.Google Scholar
  25. S. G. Samko, A. A. Kilbas, and O. I. Marichev, (1993). Fractional Integrals and Derivatives—Theory and Applications, (Gordon and Breach, New York).Google Scholar
  26. W. R. Schneider, (1986). Lecture Notes in Physics, Vol. 262, Springer, Berlin, 497.Google Scholar
  27. W. R. Schneider and W. Wyss, (1989). J. Math. Phys. 30, 134.Google Scholar
  28. M. F. Shlesinger, J. Klafter, and Y. M. Wong, (1982). J. Stat. Phys. 27(3).Google Scholar
  29. H. M. Srivastava, K. C. Gupta, and S. P. Goyal, (1982). The H-Functions of One and Two Variables with Applications, (South Asian Publishers, New Delhi).Google Scholar
  30. J. K. E. Tunaley, (1974). J. Stat. Phys. 11, 397.Google Scholar
  31. V. V. Uchaikin and V. M. Zolotarev, (1999). Chance and Stability. Stable Distributions and their Applications, (VSP, Utrecht, The Netherlands).Google Scholar
  32. H. Weissman, G. H. Weiss, and S. Havlin, (1989). J. Stat. Phys. 57, 301.Google Scholar
  33. B. J. West and W. Deering, (1994). Phys. Rep. 246, 1.Google Scholar
  34. B. J. West, P. Grigolini, R. Metzler, and T. F. Nonnenmacher, (1997). Phys. Rev. E 55, 100.Google Scholar
  35. G. J. Worsdale and J. Roy, (1975). Statist. Soc. Ser. C: Appl. Statist. 24, 123.Google Scholar
  36. W. Wyss, (1986). J. Math. Phys. 27, 2782.Google Scholar
  37. G. M. Zaslavsky, (1994). Physica D 76, 110.Google Scholar
  38. V. M. Zolotarev, (1981). In Contribution to Probability (Collection Dedicated to Eugene Lukass), eds. J. Gani and V. K. Rohatgi, Academic Press, London, 283.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Vladimir V. Uchaikin
    • 1
  1. 1.Institute for Theoretical PhysicsUlyanovsk State UniversityUlyanovskRussia

Personalised recommendations