  We study the dynamics of a system of hard-core particles sliding downwards on
a one dimensional fluctuating interface, which in a special case can be mapped
to the problem of a passive scalar advected by a Burgers fluid. Driven by the
surface fluctuations, the particles show a tendency to cluster, but the
hard-core interaction prevents collapse. We use numerical simulations to
measure the auto-correlation function in steady state and in the aging regime,
and space-time correlation functions in steady state. We have also calculated
these quantities analytically in a related surface model. The steady state
auto-correlation is a scaling function of t/L^z, where L is the system size and
z the dynamic exponent. Starting from a finite intercept, the scaling function
decays with a cusp, in the small argument limit. The finite value of the
intercept indicates the existence of long range order in the system. The
space-time correlation, which is a function of r/L and t/L^z, is non-monotonic
in t for fixed r. The aging auto-correlation is a scaling function of t_1 and
t_2 where t_1 is the waiting time and t_2 the time difference. This scaling
function decays as a power law for t_2 \gg t_1; for t_1 \gg t_2, it decays with
a cusp as in steady state. To reconcile the occurrence of strong fluctuations
in the steady state with the fact of an ordered state, we measured the
distribution function of the length of the largest cluster. This shows that
fluctuations never destroy ordering, but rather the system meanders from one
ordered configuration to another on a relatively rapid time scale.
