Statistics of Projected Motion in one dimension of a d-dimensional Random Walker

TL;DR

This paper investigates the one-dimensional projected motion of a d-dimensional random walker, revealing that it remains diffusive with a dimension-dependent diffusion rate that approaches zero as dimension increases.

Contribution

It is the first to analyze the statistics of projected one-dimensional motion of higher-dimensional random walkers, showing diffusive behavior regardless of dimension.

Findings

Projected motion is diffusive in all dimensions studied.

Diffusion rate decreases inversely with dimension.

At infinite dimension, the projected motion becomes motionless.

Abstract

We are studying the motion of a random walker in generalized 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 dimension. As a consequence, we can say that at infinite dimension the diffusion rate becomes zero. This is an interesting result, at least pedagogically, which implies that though in infinite dimension 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 generalized 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.