Physica 1lOA (1982) 535-549 North-Holland Publishing Co. RANDOM WALK ON A RANDOM WALK K.W. KEHR and R. KUTNERt lnstitut fiir Festkorperforschung der Kernforschungsanlage Jiilich, D-5170 Jiilich, Federal Repu- blic of Germany Received 8 June 1981 The authors investigate the random walk of a particle on a one-dimensional chain which has been constructed by a random-walk procedure. Exact expressions are given for the mean-square displacement and the fourth moment after n steps. The probability density after n steps is derived in the saddle-point approximation, for large n. These quantities have also been studied by numerical simulation. The extension to continuous time has been made where the particle jumps according to a Poisson process. The exact solution for the self-correlation function has been obtained in the Fourier and Laplace domain. The resulting frequency-dependent diffusion coefficient and incoherent dynamical structure factor have been discussed. The model of random walk on a random walk is applied to self-diffusion in the concentrated one-dimensional lattice gas where the correct asymptotic behavior is found. 1. Introduction The random walk of a particle on a linear chain is a basic problem of statistics, and the result for the mean-square displacement after n steps (x2), = n is common knowledge. Consider the random walk of a particle on a linear chain which does not extend uniformly in one direction, but which has been constructed by a random-walk description itself, see fig. 1. A particle performing random walk on this folded chain will have a reduced mean- square displacement after n steps. It is intuitively clear that the mean-square displacement is proportional to n I” after sufficiently many steps. Namely after n steps the mean-square displacement (v’) of the particle in the auxiliary variable v is proportional to n, when the first moment of elementary transition probability is zero, and the mean-square displacement (x2) is proportional to V. Combination of both laws suggests (x2) 0~ n”‘. Physical situations which can be described by random walk of a particle along random paths are easily visualized. For instance a particle in an amorphous substance might find an irregular path along which it can hop rather easily. A tagged particle on a linear chain which is occupied by a finite t Fellow of the Heinrich-Hertz-Foundation. Permanent address: Institute of Experimental Physics, Warsaw University, Hoza 69, 00-681 Warsaw, Poland. 0378-4371/82/0000-0000/$02.75 @ 1982 North-Holland