Abstract
Asymptotic distributions of the Montroll-Weiss equation for the continuous-time random walk are investigated for long times. It is shown that, for a certain subclass of the hopping waiting time distributions belonging to the domain of attraction of stable distributions, these asymptotic distributions are of stable form. This indicates that the realm of applicability of the diffusion equation is limited. The Montroll-Weiss equation is rederived to include the influence of the initial waiting interval and the role of the stable distributions in physical problems is briefly discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. W. Montroll and G. H. Weiss,J. Math. Phys. 6:167 (1965).
H. Sher and M. Lax,Phys. Rev. B 7:4491 (1973).
D. Bedeaux, K. Lakatos-Lindenberg, and K. Shuler,J. Math. Phys. 12:2116 (1971).
K. Lakatos-Lindenberg and D. Bedeaux,Physica 57:157 (1972).
V. M. Kenkre, E. W. Montroll, and M. F. Schlesinger,J. Stat. Phys. 9:45 (1973).
J. K. E. Tunaley,J. Appl. Phys. 43:4777 (1972).
W. Feller,An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed., Wiley, New York (1971).
E. W. Montroll and H. Sher,J. Stat. Phys. 9:101 (1973).
M. F. Shlesinger,J. Stat. Phys. 10:421 (1974).
J. K. E. Tunaley,J. Appl. Phys. 43:4783 (1972).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tunaley, J.K.E. Asymptotic solutions of the continuous-time random walk model of diffusion. J Stat Phys 11, 397–408 (1974). https://doi.org/10.1007/BF01026731
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01026731