Simple dynamical models for hierarchical bunching

Simple one-dimensional models for hierarchical bunching are proposed. A uniform state with equal spacing is linearly unstable and bunching clusters are created. The bunching clusters are further merged into even larger clusters. The coarsening process towards the larger clusters obeys a power law for the long-range forces. The exponent of the power law depends on the long-range forces. A continuum version of the lattice model with linear repulsive force is studied more in detail. The model has a form of a kind of spinodal decomposition. The coarsening dynamics is similar to a one-dimensional version of the Ostwald ripening.