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

We first review the derivation of the exact expression for the average distance $<r_n>$ 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 $<r_n>$ 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.