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Useful theories for growth of surfaces under random deposition of materialhave been developed by several authors. The simplest theory is that introducedby Edwards and Wilkinson (EW), which is linear and soluble. Its non lineargeneralization by Kardar, Parisi and Zhang (KPZ), resulted in many subsequentstudies. Yet both theories EW and KPZ contain an unphysical feature. Whendeposition of material is stopped both theories predict that as time tends toinfinity, the surface becomes flat. In fact, of course, the final surface isnot flat, but simply has no gradients larger than the gradient related to theangle of repose. We modify the EW and KPZ to accommodate this feature and studythe consequences for the simpler system which is a modification of the EWequation. In spite of the fact that the equation describing the evolution ofthe surface is not linear, we find that the steady state in the presence ofnoise is not very different in the long wave length limit from that of thelinear EW. The situation is quite different from that of EW when depositionstops. Initially there is still some rearrangement of the surface but thatstops as everywhere on the surface the gradient is less than that related tothe angle of repose. The most interesting feature observed after depositionstops is the emergence of history-dependent steady state distributions

Persisting roughness when deposition stops