  We study the clustering of passive, non-interacting particles moving under
the influence of a fluctuating field and random noise, in one dimension. The
fluctuating field in our case is provided by a surface governed by the
Kardar-Parisi-Zhang (KPZ) equation and the sliding particles follow the local
surface slope. As the KPZ equation can be mapped to the noisy Burgers equation,
the problem translates to that of passive scalars in a Burgers fluid. We study
the case of particles moving in the same direction as the surface, equivalent
to advection in fluid language. Monte-Carlo simulations on a discrete lattice
model reveal extreme clustering of the passive particles. The resulting Strong
Clustering State is defined using the scaling properties of the two point
density-density correlation function. Our simulations show that the state is
robust against changing the ratio of update speeds of the surface and
particles. In the equilibrium limit of a stationary surface and finite noise,
one obtains the Sinai model for random walkers on a random landscape. In this
limit, we obtain analytic results which allow closed form expressions to be
found for the quantities of interest. Surprisingly, these results for the
equilibrium problem show good agreement with the results in the non-equilibrium
regime.
