Physica A 168 (1990) 637-645 North-Holland Ll~VY WALK APPROACH TO ANOMALOUS DIFFUSION J. KLAFTER a, A. BLUMEN b, G. ZUMOFEN c and M.F. SHLESINGER d aSchool of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel bphysics Institute and B1MF, University of Bayreuth, D-8580, Bayreuth, Fed. Rep. Germany CLaboratorium fiir Physikalische Chemie, ETH-Zentrum, CH-8092 Ziirich, Switzerland dphysics Division, Office of Naval Research, Arlington, VA 22217, USA The transport properties of L6vy walks are discussed in the framework of continuous time random walks (CTRW) with coupled memories. This type of walks may lead to anomalous diffusion where the mean squared displacement (r2(t))~ t~ with ct ~ 1. We focus on the enhanced diffusion limit, ct > 1, in one dimension and present our results on (r2(t)), the mean number of distinct sites visited S(t) and P(r, t), the probability of being at position r at time t. 1. Introduction Recently there has been growing interest in systems that display anomalous diffusion, a fact which is manifested through the dependence of the mean squared displacement <r2(t)> on time. Whereas for simple diffusion (r2(t)) - t holds, anomalous diffusion is characterized by [1-13] (rZ(t)> ~ t ~ with a ~ 1. Cases with a < 1 (known as the dispersive transport regime) have been the topic of many theoretical and experimental studies [11-15] and have been related to the presence of geometrical (fractal) [12] or of temporal disorder [13, 14]. For t~ < 1 it is established that the probability distributions P(r, t) of being at position r at time t are generally non-Gaussian [16, 17]. In a review article [14] we have summarized different scale-invaraint models- continuous-time random walks (CTRW), fractals and ultrametric spaces- which give rise to ot < 1. Less explored has been the enchanced transport regime, which corresponds to a > 1. Although a number of experimental and theoretical examples are already well-documented, a basic theoretical framework for an overall descrip- tion of enhanced transport is still missing. Examples for enhanced diffusion are 0378-4371/90/$03.50 © 1990-Elsevier Science Publishers B.V. (North-Holland)