The average distance of the n-th neighbour in a uniform distribution of random points

Details The average distance of the n-th neighbour in a uniform distribution of random points The average distance of the n-th neighbour in a uniform distribution of random points

2001-04-09

arxiv.org/abs/cond-mat/0104139v2

Physics

Condensed Matter

Statistical Mechanics

6 pages, no figures

Scientific paper

We first review the derivation of the exact expression for the average distance $$ of the n-th neighbour of a reference point among a set of N random points distributed uniformly in a unit volume of a D-dimensional geometric space. Next we propose a `mean-field\rq theory of $$ and compare it with the exact result. The result of the `mean-field\rq theory is found to agree with the exact expression only in the limit $D \to \infty$ and $n \to \infty$. Thus the `mean-field\rq approximation is useless in this context.

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