Shattering Transitions in Collision-Induced Fragmentation
P.L. Krapivsky and Ben-Naim
We investigate the kinetics of nonlinear collision-induced
fragmentation. We obtain the fragment mass distribution
analytically by utilizing its travelling wave behavior. The system
undergoes a shattering transition in which a finite fraction of the
mass is lost to infinitesimal fragments (dust). The nature of the
shattering transition depends on the fragmentation process. When
the larger of the two colliding fragments splits, the transition is
discontinuous and the entire mass is transformed into dust at the
transition point. When the smaller fragment splits, the transition
is continuous with the dust gaining mass steadily on the account of
the fragments. At the transition point, the fragment mass
distribution diverges algebraically for small masses, $c(m)\sim
m^{-\alpha}$, with $\alpha=1.20191\ldots$
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