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Asymptotic solutions of continuous-time random walks

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Abstract

The continuous-time random walk of Montroll and Weiss has a complete separation of time (how long a walker will remain at a site) and space (how far a walker will jump when it leaves a site). The time part is completely described by a pausing time distributionψ(t). This paper relates the asymptotic time behavior of the probability of being at sitel at timet to the asymptotic behavior ofψ(t). Two classes of behavior are discussed in detail. The first is the familiar Gaussian diffusion packet which occurs, in general, when at least the first two moments ofψ(t) exist; the other occurs whenψ(t) falls off so slowly that all of its moments are infinite. Other types of possible behavior are mentioned. The relationship of this work to solutions of a generalized master equation and to transient photocurrents in certain amorphous semiconductors and organic materials is discussed.

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References

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

    Google Scholar 

  2. E. W. Montroll and H. Scher,J. Stat. Phys. 9:101 (1973).

    Google Scholar 

  3. D. Bedeaux, K. Latatos-Lindenberg, and K. Shuler,J. Math. Phys. 12:2116 (1970).

    Google Scholar 

  4. V. M. Kenkre, E. W. Montroll, and M. F. Schlesinger,J. Stat. Phys. 9:45 (1973).

    Google Scholar 

  5. H. Scher and M. Lax,J. Non-Crystalline Solids 8:497 (1972).

    Google Scholar 

  6. H. Scher and M. Lax,Phys. Rev. B 7:4491 (1973).

    Google Scholar 

  7. K. Lakatos-Lindenberg and D. Bedeaux,Physica 57:157 (1972).

    Google Scholar 

  8. H. Scher, inProc. 5th Int. Conf. on Amorphous and Liquid Semiconductors, Taylor and Francis, London (1973); H. Scher and E. W. Montroll (in preparation).

    Google Scholar 

  9. R. Bellman and K. L. Cooke,Difference-Differential Equations, Academic Press (1963), p. 240.

  10. H. Scher, private communication.

  11. W. Feller,An Introduction to Probability Theory and Its Applications, Wiley, New York (1966), Vol.II.

    Google Scholar 

  12. I. Ibragimov and Y. Linnik,Independent and Stationary Sequences of Random Variables, Walters-Noordhoff Publishing, Groningen (1971).

    Google Scholar 

  13. B. V. Gnedenko and A. N. Kolmogorov,Limit Distribution for Sums of Independent Random Variables, Addison-Wesley, Reading, Massachusetts (1968).

    Google Scholar 

  14. W. Feller,An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, New York (1968), Vol. I, p. 351.

    Google Scholar 

  15. J. Tunaley,J. Appl. Phys. 43:3851 (1972);J. Appl. Phys. 43:4783 (1972).

    Google Scholar 

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Authors and Affiliations

  1. Institute for Fundamental Studies, Department of Physics and Astronomy, University of Rochester, Rochester, New York

    Michael F. Shlesinger

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  1. Michael F. Shlesinger
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Additional information

This work was partially supported by NSF Grant No. 28501.

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Shlesinger, M.F. Asymptotic solutions of continuous-time random walks. J Stat Phys 10, 421–434 (1974). https://doi.org/10.1007/BF01008803

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  • DOI: https://doi.org/10.1007/BF01008803

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