  We consider the one-dimensional totally asymmetric zero-range process
starting from a step decreasing profile leading in the hydrodynamic limit to
the rarefaction fan of the associate hydrodynamic equation. Under that initial
condition, we show that the weighted sum of joint probabilities for second
class particles sharing the same site, is convergent and we compute its limit.
We derive the Law of Large Numbers for the position of a second class particle
initially at the origin under the initial state in which all positive sites are
empty and all negative sites are occupied and also for a slight perturbation of
the invariant state.
