Clustering in a self-gravitating one-dimensional gas at zero temperature

We study a system of gravitationally interacting sticky particles. At the initial time, we have n particles, each with mass 1/n and momentum 0, independently spread on [0, 1] according to the uniform law. Due to the confining of the system, all particles merge into a single cluster after a finite time. We give the asymptotic laws of the time of the last collision and of the time of the kth collision, when n --> infinity. We prove also that clusters of size k appear at time similar to n(=1/2(k-1)). We then investigate the system at a fixed time t < 1. We show that the biggest cluster has size of order log n, whereas a typical cluster is of finite size.