Fragmentation of a sheet by propagating, branching and merging cracks

We consider a model of the fragmentation of a sheet by cracks that move with a velocity in a preferred direction but which undergo random transverse displacements as they move. There is a non-zero probability of crack-splitting and the split cracks move independently. If two cracks meet, they merge, and move as a single crack. In the steady state, there is non-zero density of cracks and the sheet left behind by the moving cracks is broken into a large number of fragments of different sizes. The evolution operator for this model reduces to the Hamiltonian of a quantum XY spin chain, which is exactly integrable. This allows us to determine the steady state and to also determine the distribution of the sizes of the fragments. (paper)