  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 initally 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 $t\gg 1$ (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 novel and 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.
