‘‘Shattering’’ transition in fragmentation

For particles undergoing fragmentation with a rate a(y)∼${\mathrm{y}}^{\mathrm{\beta }+1\mathrm{}}$, where y is the particle size, a cascading process occurs when β<1 in which smaller particles break up at increasingly rapid rates, resulting in mass being lost to a phase of ‘‘zero’’-size particles. The cascading of the breakup rate, or ‘‘shattering,’’ produces a fractal dust, similar to a Cantor dust, with dimension 0<${\mathrm{D}}_{\mathrm{f}}$<1, and is analogous to, but opposite from, the process of gelation found in aggregating systems.