Given $n$ independent uniformly distributed points on $S^2$, what's the distribution of the distance between two closest points?

Consider $n$ iid uniform points on $S^1$, $Y_1, \ldots, Y_n$, in counterclockwise order. Now let $I_1 = Y_2-Y_1, \ldots, I_n = Y_1 - Y_n$ be the spacings between consecutive points. Finally order the spacing sequence into $I_{(1)} < I_{(2)} < \ldots < I_{(n)}$. They will also generate a spacing sequence, of size $n-1$, $J_1 = I_{(2)} - I_{(1)}, \ldots, J_{n-1} = I_{(n)} - I_{(n-1)}$. What's the distribution of this last sequence? In particular, what's the mean value of the smallest $J$ and largest $J$?