Statistics of the total number of collisions and the ordering time in a freely expanding hard-point gas

We consider a Jepsen gas of N hard-point particles undergoing free expansion on a line, starting from random initial positions of the particles having random initial velocities. The particles undergo binary elastic collisions upon contact and move freely in-between collisions. After a certain ordering time To, the system reaches a 'fan' state where all the velocities are completely ordered from left to right in an increasing fashion and there is no further collision. We compute analytically the distributions of (i) the total number of collisions and (ii) the ordering time To. We show that several features of these distributions are universal