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

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We first review the derivation of the exact expression for the average distance <rₙ> 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 theory of <rₙ> and compare it with the exact result. The result of the `mean-field theory is found to agree with the exact expression only in the limit D → ∞ and n → ∞. Thus the `mean-field approximation is useless in this context.