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Biased diffusion in a piecewise linear random potential

  • Statistical and Nonlinear Physics
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Abstract

We study the biased diffusion of particles moving in one direction under the action of a constant force in the presence of a piecewise linear random potential. Using the overdamped equation of motion, we represent the first and second moments of the particle position as inverse Laplace transforms. By applying to these transforms the ordinary and the modified Tauberian theorem, we determine the short- and long-time behavior of the mean-square displacement of particles. Our results show that while at short times the biased diffusion is always ballistic, at long times it can be either normal or anomalous. We formulate the conditions for normal and anomalous behavior and derive the laws of biased diffusion in both these cases.

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References

  1. J.P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  2. S. Scheidl, Z. Phys. B 97, 345 (1995)

    Article  ADS  Google Scholar 

  3. P. Le Doussal, V.M. Vinokur, Physica C 254, 63 (1995)

    Article  ADS  Google Scholar 

  4. P.E. Parris, M. Kuś, D.H. Dunlap, V.M. Kenkre, Phys. Rev. E 56, 5295 (1997)

    Article  ADS  Google Scholar 

  5. D.A. Gorokhov, G. Blatter, Phys. Rev. B 58, 213 (1998)

    Article  ADS  Google Scholar 

  6. S.I. Denisov, W. Horsthemke, Phys. Rev. E 62, 3311 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  7. A.V. Lopatin, V.M. Vinokur, Phys. Rev. Lett. 86, 1817 (2001)

    Article  ADS  Google Scholar 

  8. C. Monthus, Lett. Math. Phys. 78, 207 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. P. Reimann, R. Eichhorn, Phys. Rev. Lett. 101, 180601 (2008)

    Article  ADS  Google Scholar 

  10. M.N. Popescu, C.M. Arizmendi, A.L. Salas-Brito, F. Family, Phys. Rev. Lett. 85, 3321 (2000)

    Article  ADS  Google Scholar 

  11. L. Gao, X. Luo, S. Zhu, B. Hu, Phys. Rev. E 67, 062104 (2003)

    Article  ADS  Google Scholar 

  12. D.G. Zarlenga, H.A. Larrondo, C.M. Arizmendi, F. Family, Phys. Rev. E 75, 051101 (2007)

    Article  ADS  Google Scholar 

  13. S.I. Denisov, T.V. Lyutyy, E.S. Denisova, P. Hänggi, H. Kantz, Phys. Rev. E 79, 051102 (2009)

    Article  ADS  Google Scholar 

  14. H. Kunz, R. Livi, A. Sütő, Phys. Rev. E 67, 011102 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  15. S.I. Denisov, M. Kostur, E.S. Denisova, P. Hänggi, Phys. Rev. E 75, 061123 (2007)

    Article  ADS  Google Scholar 

  16. S.I. Denisov, M. Kostur, E.S. Denisova, P. Hänggi, Phys. Rev. E 76, 031101 (2007)

    Article  ADS  Google Scholar 

  17. S.I. Denisov, H. Kantz, Phys. Rev. E 81, 021117 (2010)

    Article  ADS  Google Scholar 

  18. E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)

    Article  MathSciNet  ADS  Google Scholar 

  19. B.D. Hughes, Random Walks and Random Environments (Clarendon Press, Oxford, 1995), Vol. 2

  20. D. ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000)

  21. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. G. Zaslavsky, Phys. Rep. 371, 461 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Anomalous Transport: Foundations and Applications, edited by R. Klages, G. Radons, I.M. Sokolov (Wiley-VCH, Berlin, 2008)

  24. M.F. Shlesinger, B. West, J. Klafter, Phys. Rev. Lett. 58, 1100 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  25. J. Masoliver, K. Lindenberg, G.H. Weiss, Physica A 157, 891 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  26. G. Zumofen, J. Klafter, Phys. Rev. E 47, 851 (1993)

    Article  ADS  Google Scholar 

  27. E. Barkai, Chem. Phys. 284, 13 (2002)

    Article  ADS  Google Scholar 

  28. I.M. Sokolov, R. Metzler, Phys. Rev. E 67, 010101(R) (2003)

    Article  ADS  Google Scholar 

  29. V. Zaburdaev, M. Schmiedeberg, H. Stark, Phys. Rev. E 78, 011119 (2008)

    Article  ADS  Google Scholar 

  30. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1971), Vol. 2

  31. B.V. Gnedenko, V.Yu. Korolev, Random Summation: Limit Theorems and Applications (CRC Press, Boca Raton, FL, 1996)

  32. A. Erdélyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1

  33. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972)

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

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187, Dresden, Germany

    S. I. Denisov & H. Kantz

  2. Sumy State University, 2 Rimsky-Korsakov Street, 40007, Sumy, Ukraine

    S. I. Denisov & E. S. Denisova

Authors
  1. S. I. Denisov
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  2. E. S. Denisova
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  3. H. Kantz
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Correspondence to S. I. Denisov.

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Denisov, S., Denisova, E. & Kantz, H. Biased diffusion in a piecewise linear random potential. Eur. Phys. J. B 76, 1–11 (2010). https://doi.org/10.1140/epjb/e2010-00185-3

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  • DOI: https://doi.org/10.1140/epjb/e2010-00185-3

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