PHYSICAL REVIEW A VOLUME 43, NUMBER 10 15 MAY 1991 Diffusion of walkers with persistent velocities Mariela Araujo Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215 Shlomo Havlin Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215 and Physical Sciences Laboratory, Division of Computer Research and Technology, National Institutes of Health, Bethesda, Maryland 20892 George H. Weiss Physical Sciences Laboratory, Division of Computer Research and Technology, National Institutes of Health, Bethesda, Maryland 20892 H. Eugene Stanley Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215 (Received 2 January 1991) We describe some properties for a phenomenological model of superdiffusion based on a generali- zation of the persistent random walk in one dimension to continuous time. The time spent moving to either increasing or decreasing x is characterized by a fractal-time pausing time density, g(t) — T /t +', with 1 &ct &2. For this system it is shown that asymptotically p(0, t) — 1/t'~ . The form of the profile is shown to be Gaussian near the peak and to fall off'like tx " ' near the tails, and the survival probability is asymptotically proportional to exp( — Bt/L ). These results are confirmed by numerical calculations based on the method of exact enumeration. I. INTRODUCTION Considerable recent interest has been focused on the attempt to understand superdiftusive physical systems. The definition of a superdiffusive system is that the dis- placement of a random walker in the dimension scales 2/d with time as a power law, (r ) -t, where the fractal dimension d„of the walk is less than, rather than greater than, 2. A number of physical systems exhibit such be- havior. ' One example of such a system, originally studied by Matheron and de Marsily was suggested as a model for ground-water transport in stratified media characterized by varying pressure in the direction of the strata (so that the average velocity in each stratum is a random variable). A second example is provided by coherent wave propagation through disordered multiple- scattering media. Properties of the continuous-time random walk (hence- forth CTRW) on a translationally invariant lattice have been used by a number of investigators as a simple way to mimic those of transport in a disordered medium since the pioneering work of Scher and Lax. ' Most models that are based on the CTRW (Refs. 11 and 12) for trans- port in random media use an assumption that the pausing-time density P(t) has a fractal-time behavior, that is g(t)- T /t'+ for t /T )) 1, where 0 & o; & 1, and T is a parameter hav- ing the dimensions of time. Such a pausing-time density has no finite integer moments greater than 0, and in consequence, is known to change the probability distribu- tion for the displacement of the random walker at time t. '3 In this paper we explore a number of properties of the continuous-time generalization' of the persistent random walk' which incorporates superdiftusion with a continu- ously variable exponent in contrast to the model in Ref. 7 in which it is shown that d = —,. The pausing-time densi- ty in this model is characterized by a pausing-time densi- ty having a finite first, but infinite second moment. The model for which an expression for (x ) was given in Ref. 14 consists of a persistent random walk in continuous time for which the pausing-time density has the property given in Eq. (1) with 1 & a & 2, and in which the displace- ment x is related to the time in a single sojourn (that is, moving either right or left) of duration t by f (x, t)=o(x+vt), where U is a constant velocity. The further assumption made is that the initial step by the random walker is equally likely to be in the positive or negative direction. Such a model is obviously symmetric, and was shown' to have the property that the mean-squared displacement has the asymptotic behavior (x')-t' For the purpose of studying further properties of the model we simulated a lattice version of the persistent ran- dom walk in one dimension in which time is discrete. It 43 5207 1991 The American Physical Society