Coarsening of Sand Ripples in Mass Transfer Models with Extinction

Abstract

Coarsening of sand ripples is studied in a one-dimensional stochastic model, where neighboring ripples exchange mass with algebraic rates, $Γ(m) \sim m^γ$, and ripples of zero mass are removed from the system. For $γ< 0$ ripples vanish through rare fluctuations and the average ripples mass grows as $\avem(t) \sim -γ^{-1} \ln (t)$. Temporal correlations decay as $t^{-1/2}$ or $t^{-2/3}$ depending on the symmetry of the mass transfer, and asymptotically the system is characterized by a product measure. The stationary ripple mass distribution is obtained exactly. For $γ> 0$ ripple evolution is linearly unstable, and the noise in the dynamics is irrelevant. For $γ= 1$ the problem is solved on the mean field level, but the mean-field theory does not adequately describe the full behavior of the coarsening. In particular, it fails to account for the numerically observed universality with respect to the initial ripple size distribution. The results are not restricted to sand ripple evolution since the model can be mapped to zero range processes, urn models, exclusion processes, and cluster-cluster aggregation.