Abstract
The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial, and temporal moments of the memory are infinite. The asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated. The resulting behavior includes localized, diffusive, wavelike, and Levy's stable laws for the appropriate scaled variable. We show that an infinite mean waiting time can lead to long time diffusive behavior, while a finite mean waiting time is not sufficient to ensure the same.
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References
E. W. Montroll and B. J. West, inFluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979).
M. F. Shlesinger and U. Landman, inApplied Stochastic Processes, G. Adomian, ed. (Academic Press, New York, 1980), and inAspects of the Kinetics and Dynamics of Surface Reactions (A.I.P. Publications, New York, 1980).
G. H. Weiss and R. J. Rubin,Adv. in Chem. Phys. (to appear).
E. W. Montroll,Proc. Natl. Acad. Sci. (USA) 78:36 (1981).
P. Levy,Calcul des Probabilities (Guthier-Villars, Paris, 1925); P. Levy,Theorie de l'Addition des Variables Aleatoires (Guthier-Villars, Paris, 1927).
W. Feller,An Introduction to Probability Theory and its Applications (Wiley, New York, 1966), Vol. II, pp. 336, 360, 548.
J. Gillis and G. H. Weiss,J. Math. Phys. 11:1308 (1970).
E. W. Montroll and H. Scher,J. Stat. Phys. 9:101 (1973).
H. Scher and E. W. Montroll,Phys. Rev. B 12:2455 (1975).
M. F. Shlesinger,J. Stat. Phys. 10:421 (1974).
H. Scher and M. Lax,Phys. Rev. B 7:4491 (1973).
J. K. E. Tunaley,J. Stat. Phys. 11:397 (1974).
E. W. Montroll and G. H. Weiss,J. Math. Phys. 6:167 (1965).
E. W. Montroll,J. Math. and Phys. 25:37 (1947).
J. Klafter and R. Silbey,Phys. Lett. 76a:143 (1980).
G. H. Weiss,J. Stat. Phys. 15:157 (1976).
L. P. Kadanoff and J. Swift,Phys. Rev. 165:310 (1968).
B. D. Hughes, M. F. Shlesinger, and E. W. Montroll,Proc. Natl. Acad. Sci. USA,78:3287 (1981); M. F. Shlesinger and B. D. Hughes,Physica 109a:597 (1981).
J. Klafter and R. Silbey,Phys. Rev. Lett. 44:55 (1980).
V. M. Kenkre and R. S. Knox,Phys. Rev. B 9:5279 (1974).
V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger,J. Stat. Phys. 9:45 (1973).
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Shlesinger, M.F., Klafter, J. & Wong, Y.M. Random walks with infinite spatial and temporal moments. J Stat Phys 27, 499–512 (1982). https://doi.org/10.1007/BF01011089
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DOI: https://doi.org/10.1007/BF01011089