  We consider the one-dimensional asymmetric simple exclusion process (ASEP) in
which particles jump to the right at rate $p\in(1/2,1]$ and to the left at rate
$1-p$, interacting by exclusion. In the initial state there is a finite region
such that to the left of this region all sites are occupied and to the right of
it all sites are empty. Under this initial state, the hydrodynamical limit of
the process converges to the rarefaction fan of the associated Burgers
equation. In particular suppose that the initial state has first-class
particles to the left of the origin, second-class particles at sites 0 and 1,
and holes to the right of site 1. We show that the probability that the two
second-class particles eventually collide is $(1+p)/3p$, where a_collision_
occurs when one of the particles attempts to jump over the other. This also
corresponds to the probability that two ASEP processes, started from
appropriate initial states and coupled using the so-called "basic coupling",
eventually reach the same state. We give various other results about the
behaviour of second-class particles in the ASEP. In the totally asymmetric case
($p=1$) we explain a further representation in terms of a multi-type particle
system, and also use the collision result to derive the probability of
coexistence of both clusters in a two-type version of the corner growth model.
