Derive the probability distribution \(w(r)\) for the distance to the closest neighboring particle using the pair correlation function \(g(r)\) and the number density \(n\). Show that in three dimensions

\[w(r)=4 \pi n r^{2} g(r) \exp \left(-\int_{0}^{r} 4 \pi n s^{2} g(s) d s\right)\]

and the average closest-neighbor distance for an ideal gas is

\[r_{1}=\int_{0}^{\infty} r w(r) d r=\Gamma\left(\frac{4}{3}\right)\left(\frac{4 \pi n}{3}\right)^{-1 / 3}\]