Universal extremal statistics in a freely expanding Jepsen gas

We study the extremal dynamics emerging in an out-of-equilibrium one-dimensional Jepsen gas of (N+1) hard-point particles. The particles undergo binary elastic collisions, but move ballistically in between collisions. The gas is initially uniformly distributed in a box [-L,0] with the "leader" (or the rightmost particle) at X=0 , and a random positive velocity, independently drawn from a distribution phi(V) , is assigned to each particle. The gas expands freely at subsequent times. We compute analytically the distribution of the leader's velocity at time t , and also the mean and the variance of the number of collisions that are undergone by the leader up to time t . We show that in the thermodynamic limit and at fixed time t>>1 (the so-called "growing regime"), when interactions are strongly manifest, the velocity distribution exhibits universal scaling behavior of only three possible varieties, depending on the tail of phi(V) . The associated scaling functions are entirely different from the usual extreme-value distributions of uncorrelated random variables. In this growing regime the mean and the variance of the number of collisions of the leader up to time t increase logarithmically with t , with universal prefactors that are computed exactly. The implications of our results in the context of biological evolution modeling are pointed out.