We study the evolution of a small perturbation of the equilibrium of
a totally asymmetric one-dimensional interacting system. The model we take as
an example is Hammersley's process as seen from a tagged particle, which can be
viewed as a process of interacting positive-valued stick heights on the sites
of $\mathbf{Z}$. It is known that under Euler scaling (space and time scale $n$
) the empirical stick profile obeys the Burgers equation. We refine this result
in two ways. If the process starts close enough to equilibrium, then over times
$n^\nu$ for $1 \le \nu < 3$, and up to errors that vanish in hydrodynamic
scale, the dynamics merely translates the initial stick configuration. In
particular, on the hydrodynamic time scale, diffusive fluctuations are
translated rigidly. A time evolution for the perturbation is visible under a
particular family of scalings:over times $n_{\nu}, 1 < \nu < 3/2$, a
perturbation of order $n^{1-\nu}$ from equilibrium follows the inviscid Burgers
equation. The results for the stick model are derived from asymptotic results
for tagged particles in Hammersley's process.