Abstract
The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between jumps, with a common pausing time distributionψ(t). An analytic solution in the form of an inverse Laplace transform for P(l, t), the probability of a walker being atl at timet if it started atl o att=0, is obtained in the presence of completely absorbing boundaries. Numerical results for P(l, t) are presented for characteristically different ψ(t), including one which leads to a non-Gaussian behavior for P(l, t) even for larget. Asymptotic results are obtained for the number of surviving walkers and the mean 〈l〉 showing the effect of the absorption at the boundary.
Similar content being viewed by others
References
E. W. Montrall,J. SIAM 4:241 (1956).
E. W. Montroll, inApplied Combinatorial Mathematics, E. F. Beckenbach, ed., Wiley, New York (1964), Chapter 4, p. 96.
E. W. Montroll,Proc. Symp. in Appl. Math. (Am. Math. Soc.) 16:193 (1964).
E. W. Montroll and G. H. Weiss,J. Math. Phys. 6:167 (1965).
E. W. Montroll,J. Math. Phys. 10:753 (1969).
E. W. Montroll, inEnergetics in Metallurgical Phenomenon, Gordon and Breach, New York (1967), Vol. 3, p. 123.
H. Scher and M. Lax,J. Non-Crystalline Solids 8:497 (1972).
H. Scher and M. Lax,Phys. Rev. B7:4491 (1973);Phys. Rev. B7:4502 (1973).
R. Bedeaux, K. Lakatos-Lindenberg, and K. Shuler,J. Math. Phys. 12:2116 (1970).
G. E. Roberts and H. Kaufman,Tables of Laplace Transforms, Saunders, Philadelphia, Pennsylvania (1966).
E. W. Montroll,J. Math, and Phys. 25, 37 (1946).
M. F. Shlesinger, Private communication.
B. V. Gnedendo and A. N. Kolmogorov,Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, Massachusetts (1968).
J. V. Uspensky,Introduction to Mathematical Probability, McGraw-Hill, New York (1937).
J. W. Cooley and J. W. Tukey,Math. of Comput. 19:297 (1965).
E. W. Montroll,J. Phys. Soc. Japan 26 (Suppl.):6 (1969).
K. Lakatos-Lindenberg,J. Stat. Phys. (to be published).
Author information
Authors and Affiliations
Additional information
This study was partially supported by ARPA and monitored by ONR(N00014-17-0308).
Rights and permissions
About this article
Cite this article
Montroll, E.W., Scher, H. Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries. J Stat Phys 9, 101–135 (1973). https://doi.org/10.1007/BF01016843
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01016843