TITLE:
Statistics of Projected Motion in One Dimension of a D-Dimensional Random Walker
AUTHORS:
Jayeeta Chattopadhyay, Muktish Acharyya
KEYWORDS:
Higher Dimensions, Random Walk, Projected Walk, Renormalizing Factor, Diffusive
JOURNAL NAME:
Applied Mathematics,
Vol.9 No.6,
June
13,
2018
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”.