Kinetics of ballistic annihilation and branching

We consider a one-dimensional model consisting of an assembly of two-veloci ty particles moving freely between collisions. When two particles meet, the y instantaneously annihilate each other and disappear from the system. More over, each moving particle can spontaneously generate an offspring having t he same velocity as its mother with probability 1-q. This model is solved a nalytically in the mean-field approximation and studied by numerical simula tions. It is found that for q=1/2 the system exhibits a dynamical phase tra nsition. For q<1/2, the slow dynamics of the system is governed by the coar sening of clusters of particles having the same velocities, while for q>1/2 the system relaxes rapidly towards its stationary state characterized by a distribution of small cluster sizes. [S1063-651X(99)00501-2].