Transport properties of the continuous-time random walk with a long-tailed waiting-time density |
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Authors: | Haim Weissman George H Weiss Shlomo Havlin |
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Institution: | (1) Department of Physics, Bar-Ilan University, 52100 Ramat Gan, Israel;(2) National Institutes of Health, 20892 Bethesda, Maryland |
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Abstract: | We derive asymptotic properties of the propagatorp(r,t) of a continuous-time random walk (CTRW) in which the waiting time density has the asymptotic form (t) T
/t
+1 whent T and 0< <1. Several cases are considered; the main ones are those that assume that the variance of the displacement in a single step of the walk is finite. Under this assumption we consider both random walks with and without a bias. The principal results of our analysis is that one needs two forms to characterizep(r,t), depending on whetherr is large or small, and that the small-r expansion cannot be characterized by a scaling form, although it is possible to find such a form for larger. Several results can be demonstrated that contrast with the case in which t =
0
![tau](/content/g349q21571555r74/xxlarge964.gif) ( )d is finite. One is that the asymptotic behavior ofp(0,t) is dominated by the waiting time at the origin rather than by the dimension. The second difference is that in the presence of a fieldp(r,t) no longer remains symmetric around a moving peak. Rather, it is shown that the peak of this probability always occurs atr=0, and the effect of the field is to break the symmetry that occurs when t![rang](/content/g349q21571555r74/xxlarge9002.gif) . Finally, we calculate similar properties, although in not such great detail, for the case in which the single-step jump probabilities themselves have an infinite mean. |
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Keywords: | Random walks disordered media transport properties |
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