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, by addressing this particular type of problem, it shows 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 diffusion, 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”.
Highlights
The basis of random walk theory can be described by the irregular motion of individual pollen particle, famously studied by botanist Brown (1828), known as Brownian motion
If we consider random walker moving in a two dimensional continuum, the walker has infinitely possible directions to choose with equal probability and equal step length
In this paper we have discussed random walk problem extensively in generalised d-dimensional continuum by numerical simulation where we have observed how the statistical properties of random walker are changing with dimensions
Summary
The basis of random walk theory can be described by the irregular motion of individual pollen particle, famously studied by botanist Brown (1828), known as Brownian motion. 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.
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