I have a uniform random variable $X \sim \mathcal{U}(a,b)$ and I take a sample of $N$ i.i.d. realization from it: $\{X_1, X_2, ..., X_N\}$. I'm interested in the statistics of the interval between every two successive realizations after they are sorted out in increasing order. More precisely, if the sorted realizations are $\{X'_1, X'_2, ..., X'_N\}$ (with $X'_i < X'_{i+1}$), I want to know what is the distribution of $X_{i+1} - X_i$ with respect to the one of $X.$
UPDATE: The question is updated for clarification. Thanks for the constructive comments.
For future reference, the sorted sample values are called the order statistics of the sample.
Consider $n+1$ points independently uniformly randomly chosen on a circle. Uniformly randomly choose one of them to break the circle into an interval, and the other $n$ as points on that interval. The distribution of these $n$ points is that of $n$ points independently uniformly randomly chosen on the interval. By the symmetry of the circle, all distances between the points are identically distributed.
The probability density of each distance $l$ is proportional to the volume that it leaves for the remaining $(n+1)-2=n-1$ points, which is $(L-l)^{n-1}$ (where $L=b-a$ is the length of the interval / the perimeter of the circle). Then normalization yields the density $\frac nL\left(1-\frac lL\right)^{n-1}$.