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

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Document pages: 16 pages

Abstract: We are studying the motion of a random walker in generalised d-dimensionalcontinuum with unit step length (up to 10 dimensions) and its projected onedimensional motion numerically. The motion of a random walker in lattice orcontinuum is well studied in statistical physics but what will be the statistics ofprojected one dimensional motion of higher dimensional random walker isyet to be explored. Here in this paper, by addressing this particular type ofproblem, it shows that the projected motion is diffusive irrespective of anydimension; however, the diffusion rate is changing inversely with dimensions.As a consequence, it can be predicted that for the one dimensional projectedmotion of infinite dimensional random walk, the diffusion rate will be zero.This is an interesting result, at least pedagogically, which implies that thoughin infinite dimensions there is diffusion, its one dimensional projection is motionless.At the end of the discussion we are able to make a good comparisonbetween projected one dimensional motion of generalised d-dimensionalrandom walk with unit step length and pure one dimensional random walkwith random step length varying uniformly between -h to h where h is a “steplength renormalizing factor”.

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