  A simple model of ballistic aggregation and fragmentation is proposed. The
model is characterized by two energy thresholds, Eagg and Efrag, which
demarcate different types of impacts: If the kinetic energy of the relative
motion of a colliding pair is smaller than Eagg or larger than Efrag, particles
respectively merge or break; otherwise they rebound. We assume that particles
are formed from monomers which cannot split any further and that in a
collision-induced fragmentation the larger particle splits into two fragments.
We start from the Boltzmann equation for the mass-velocity distribution
function and derive Smoluchowski-like equations for concentrations of particles
of different mass. We analyze these equations analytically, solve them
numerically and perform Monte Carlo simulations. When aggregation and
fragmentation energy thresholds do not depend on the masses of the colliding
particles, the model becomes analytically tractable. In this case we show the
emergence of the two types of behavior: the regime of unlimited cluster growth
arises when fragmentation is (relatively) weak and the relaxation towards a
steady state occurs when fragmentation prevails. In a model with mass-dependent
Eagg and Efrag the evolution with a cross-over from one of the regimes to
another has been detected.
