  We study the dynamics of density fluctuations in the steady state of a
non-equilibrium system, the Zero-Range Process on a ring lattice. Measuring the
time series of the total number of particles in a \emph{segment} of the
lattice, we find remarkable structures in the associated power spectra, namely,
two distinct components of damped-oscillations. The essential origin of both
components is shown in a simple pedagogical model. Using a more sophisticated
theory, with an effective drift-diffusion equation governing the stochastic
evolution of the local particle density, we provide reasonably good fits to the
simulation results. The effects of altering various parameters are explored in
detail. Avenues for improving this theory and deeper understanding of the role
of particle interactions are indicated.
