Skip to main content

Abstract

Some classical results from random walk theory are reviewed. Asymptotic properties are derived for random walk on a one-dimensional lattice with static disorder. Fractal time properties are illustrated on a simple example.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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. E.W. Montroll and G.H. Weiss, J. Math. Phys. 6, 167 (1965).

    Article  MathSciNet  ADS  Google Scholar 

  2. F. Spitzer, Principles of a Random Walk (Van Nostrand, Princeton, NJ, 1964).

    Book  Google Scholar 

  3. M.N. Barber and B.W. Ninham, Random and Restricted Walks, Theory and Applications (Gordon and Breach, N.Y., 1970).

    MATH  Google Scholar 

  4. N.S. Goel and N. Richter-Dyn, Stochastic Models in Biology (Academic Press, N.Y., 1974).

    Google Scholar 

  5. S. Alexander, J. Bernasconi, W.R. Schneider and R. Orbach, Rev. Mod. Phys. 53, 175 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  6. N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, (North Holland, Amsterdam, 1981).

    MATH  Google Scholar 

  7. G.H. Weiss and R.J. Rubin, Adv. Chem. Phys. 52, 363 (1983).

    Article  Google Scholar 

  8. M.E. Fisher, J. Stat. Phys. 34, 667 (1983).

    Article  ADS  Google Scholar 

  9. B.D. Hughes and R. Prager, Lect. Not. Math. 1035, 1 (1983).

    Article  MathSciNet  Google Scholar 

  10. E.W. Montroll and M.F. Shlesinger, A Wonderful World of Random Walks, in CCNY Physics Symposium: M. Lax Sixtieth Birhtday, Ed. H. Falk (City College of New York Physics Department, New York, 1983).

    Google Scholar 

  11. J.W. Haus and K.W. Kehr, Phys. Rep. 150, 264 (1987).

    Article  ADS  Google Scholar 

  12. J.V. José, in: Stochastic Processes Applied to Physics and other Related Fields, B. Gomez, S.M. Moore, A.M. Rodriguez-Vargas, A. Reuda, Eds. (World Scientific, 1983).

    Google Scholar 

  13. R. Graham and A. Schenzle, Phys. Rev. A26, 1676 (1982).

    MathSciNet  ADS  Google Scholar 

  14. G. Nicolis and J.W. Turner, Physica 89A, 326 (1977).

    ADS  Google Scholar 

  15. A.J.F. Siegert, Phys. Rev. 81, 617 (1951).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. M. Khanta and V. Balakrishnan, Phys. Rev. B29, 4679 (1984).

    ADS  Google Scholar 

  17. R.M. Mazo and C. Van den Broeck, J. Chem. Phys. 86, 454 (1986).

    Article  ADS  Google Scholar 

  18. G.H. Weiss, Adv. Chem. Phys. 13, 1 (1966);

    Article  Google Scholar 

  19. V. Seshadri, B.J. West and K. Lindenberg, J. Chem. Phys. 72, 1145 (1980).

    Article  ADS  Google Scholar 

  20. K. Lindenberg, K.E. Shuler, J. Freeman and T.J. Lie, J. Stat. Phys. 12, 217 (1975).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. C. Van den Broeck and M. Bouten, J. Stat. Phys. 45, 1031 (1986).

    Article  ADS  MATH  Google Scholar 

  22. C. Van den Broeck and V. Balakrishnan, unpublished.

    Google Scholar 

  23. W. Horsthemke and R. Lefever, Noise Induced Transitions (Springer, New York, 1983), see also

    Google Scholar 

  24. A. Rodriguez, L. Pesquèra, M. San Miguel and J.M. Sancho, J. Stat. Phys. 40, 669 (1985).

    Article  ADS  Google Scholar 

  25. M.A. Rodriguez, M. San Miguel and J.M. Sancho, Ann. Nucl. Energy 10, 263 (1983).

    Article  Google Scholar 

  26. For a review, see P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).

    Article  ADS  Google Scholar 

  27. B. Derrida, J. Stat. Phys. 31, 433 (1982) see also:

    Article  MathSciNet  ADS  Google Scholar 

  28. R. Zwanzig, J. Stat. Phys. 28, 127 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  29. B.D. Hughes and M. Sahimi, J. Stat. Phys. 29, 781 (1982);

    Article  MathSciNet  ADS  Google Scholar 

  30. B.D. Hughes, M. Sahimi and T. Davis, Physica 120A, 515 (1983).

    MathSciNet  ADS  Google Scholar 

  31. W. Feller, An Introduction to Probability Theory and its Applications (New York, Wiley, 1966).

    MATH  Google Scholar 

  32. M.F. Shlesinger and E.W. Montroll, Lect. Not. Math. 1035, 138 (1983).

    Article  MathSciNet  Google Scholar 

  33. A.E. Ingham, The distribution of prime numbers, (Cambridge, 1983).

    Google Scholar 

  34. B.B. Mandelbrot, The fractal geometry of nature (W.W. Freeman, San Francisco, 1982).

    MATH  Google Scholar 

  35. H. Scher and M. Lax, Phys. Rev. B7, 4491, (1973).

    MathSciNet  ADS  Google Scholar 

  36. H. Scher and E.W. Montroll, Phys. Rev. B12, 2455 (1975).

    ADS  Google Scholar 

  37. M. Shlesinger, J. Stat. Phys. 10, 421 (1974).

    Article  MathSciNet  ADS  Google Scholar 

  38. D. Stauffer, Phys. Rep. 54, 1 (1979).

    Article  ADS  Google Scholar 

  39. A.J. Lichtenberg and M.A. Lieberman, Appl. Math. Sci. 38, (1983).

    Google Scholar 

  40. Y. Meir and A. Aharony, Phys. Rev. A37, 596 (1988).

    MathSciNet  ADS  Google Scholar 

  41. S. Alexander and R. Orbach, J. Phys. Lett. 43, L625 (1982).

    Article  Google Scholar 

  42. R. Rammal and G. Toulouse, Phys. Rev. lett. 49, 1194 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  43. E. Domany, S. Alexander, D. Bensimon and L.P. Kadanoff, Phys. Rev. B28, 3110 (1983).

    MathSciNet  ADS  Google Scholar 

  44. R. Rammal, J. Phys. 45, 191 (1984).

    Article  MathSciNet  Google Scholar 

  45. O’Shaugnessy and I. Procaccia, Phys. Rev. Lett. 54, 455 (1985).

    Article  ADS  Google Scholar 

  46. O’Shaugnessy and I. Procaccia, Phys. Rev. A32, 3073 (1985).

    ADS  Google Scholar 

  47. A. Blumen, G. Zumofen and J. Klafter, Phys. Rev. B28, 6112 (1983).

    ADS  Google Scholar 

  48. S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987).

    Article  ADS  Google Scholar 

  49. T. Schneider, A. Politi and M.P. Sörensen, Phys. Rev. A37, 948 (1988).

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Plenum Press, New York

About this chapter

Cite this chapter

Van den Broeck, C. (1989). A Glimpse into the World of Random Walks. In: Muñoz-Cobo, J.L., Difilippo, F.C. (eds) Noise and Nonlinear Phenomena in Nuclear Systems. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-5613-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4684-5613-4_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4684-5615-8

  • Online ISBN: 978-1-4684-5613-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics