Statistics of Projected Motion in One Dimension of a D-Dimensional Random Walker

Abstract

We are studying the motion of a random walker in generalised d-dimensional continuum with unit step length (up to 10 dimensions) and its projected one dimensional motion numerically. The motion of a random walker in lattice or continuum is well studied in statistical physics but what will be the statistics of projected one dimensional motion of higher dimensional random walker is yet to be explored. Here in this paper, addressing this particular type of problem, we have showed that the projected motion is diffusive irrespective of any dimension, however, the diffusion rate is changing inversely with dimensions. As a consequence, it can be predicted that for the one dimensional projected motion of infinite dimensional random walk, the diffusion rate will be zero. This is an interesting result, at least pedagogically, which implies that though in infinite dimensions there is a diffusion but its one dimensional projection is motionless. At the end of the discussion we are able to make a good comparison between projected one dimensional motion of generalised d-dimensional random walk with unit step length and pure one dimensional random walk with random step length varying uniformly between −h to h where h is a 'step length renormalizing factor'.